i
i
6TH INTERNATIONAL EURASIAN CONFERENCE
ON MATHEMATICAL SCIENCES
AND APPLICATIONS
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FOREWORD
Following five very successful previous conferences (IECMSA-2012, Prishtine,
Kosovo, IECMSA-2013, Sarajevo, Bosnia and Herzegovina, IECMSA-2014, Vienna, Austria, and
IECMSA-2015, Athens, Greece, IECMSA-2016, Belgrade, Serbia) now the “6th International
Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)’’ has been held
in Budapest, Hungary on August 15-18, 2017 .
This conference is supported by Sakarya University, Kocaeli University, Namik Kemal
University, Amasya University, Bilecik Seyh Edebali University, Baku State University, Institute
of Applied Mathematics, Turkic World Mathematical Society, and International Balkan
University.
The major goal of the series of IECMSA is to facilitate bridge building across the disparate
fields within Mathematical Sciences and their Applications. Since 2012, these conferences have been
held regularly every year and featured many distinguished speakers from the across the globe, with
talks focusing on algebra, analysis, applied mathematics, geometry, mathematics education,
statistics, topology, physics, economics or computer sciences.
The scientific program of IECMSA-2017 features invited talks, followed by contributed oral
and poster presentations in parallel sessions. Scientific topics cover the latest developments in all
subfields of mathematics. The abstracts of all presentations have been substituted in this book.
Also, the electronic version of the abstracts of all presentations can be found in the Conference
Abstracts Book at www.iecmsa.org
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I wish to thank all members of scientific committee and sponsors for their continued support
to the IECMSA. And finally, I would like to sincerely thank all participants of IECMSA-2017 for
contributing to this great meeting in many different ways. I believe and hope that each of them will
get the maximum benefit from the conference.
Welcome to Budapest!
Prof. Dr. Murat TOSUN
Chairman
On behalf of the Organizing Committee
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HONORARY COMMITTEE
Prof. Dr. Muzaffer ELMAS (Rector of Sakarya University)
Prof. Dr. Sadettin HÜLAGÜ (Rector of Kocaeli University)
Prof. Dr. İbrahim TAŞ (Rector of Bilecik Seyh Edebali University)
Prof. Dr. Osman ŞİMŞEK (Rector of Namık Kemal University)
Prof. Dr. Metin ORBAY (Rector of Amasya University)
Prof. Dr. İsmail KOCAYUSUFOĞLU (Rector of International Balkan University)
Prof. Dr. H. Hilmi HACISALİHOĞLU (Honorary President of TWMS
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SCIENTIFIC COMMITTEE
Prof. Dr. Abdel Salhi University of Essex
Prof. Dr. Bayram Sahin Ege University
Prof. Dr. Cengizhan Murathan Uludag University
Prof. Dr. Ding-Xuan Zhou Zhejiang University
Prof. Dr. Emine Mısırlı Ege University
Prof. Dr. Etibar Penakhov Baku State University
Prof. Dr. Fikret A. Aliev Baku State University
Prof. Dr. Gennady A. Leonov Saint-Petersburg State University
Prof. Dr. Gerald Beer California State University
Prof. Dr. Hans Peter Kunzi University of Cape Town
Prof. Dr. Hellmuth Stachel Vienna Technical University
Prof. Dr. İsmail Ekincioglu Dumlupınar University
Prof. Dr. Ioan Rasa Technical University of Cluj-Napoca
Prof. Dr. Josef Slapal Brno University of Technology
Prof. Dr. Maria Isabel Garrido Complutense University of Madrid
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SCIENTIFIC COMMITTEE
Prof. Dr. Mehmet Ali Güngör Sakarya University
Prof. Dr. Mo Sal Moslehian Ferdowsi University of Mashhad
Prof. Dr. Musatafa Çalışkan Gazi University
Prof. Dr. Mohammad Mursaleen Aligarh Muslim University
Prof. Dr. Naime Ekici Çukurova University
Prof. Dr. Nuri Kuruoğlu İstanbul Gelişim University
Prof. Dr. Nesip Aktan Necmettin Erbakan University
Prof. Dr. Neville Robbins San Francisco State University
Prof. Dr. Sadık Keleş İnönü University
Prof. Dr. Siamak Yassemi University of Tehran
Prof. Dr. Sujatha Ramdorai University of British Columbia
Prof. Dr. Sidney A. Morris Federation University of Australia
Prof. Dr. Varga Kalantarov Koç University
Prof. Dr. Vladimir Touraev Indiana University
Prof. Dr. Wolfgang Sproessig Technische Uni. Bergakademie Freiber
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ORGANIZING COMMITTEE
GENERAL COORDINATOR:
Prof. Dr. Murat Tosun Sakarya University
VICE-GENERAL COORDINATOR:
Prof. Dr. Ljubisa Kočinac Nis University
VICE-GENERAL COORDINATOR:
Prof. Dr. István Juhász Alfred Renyi Institute of Mathematics
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CHAPTERS
INVITED TALKS………………………………...……………………………………………………………... 2
ALGEBRA………………………………………………………………………………… 13
ANALYSIS…………………………………………………………………………….. 34
APPLIED MATHEMATICS…………………………………………………. 93
DISCRETE MATHEMATICS…………………………………………………... 186
GEOMETRY…………………………………………………………………………… 189
MATHEMATICS EDUCATION………………………………………… 230
STATISTICS……………………………………………………………………………… 265
TOPOLOGY…………………………………………………………………………….. 272
THE OTHER AREAS…………………………………………………………. 285
POSTERS……………………………………………………………………………………….. 301
PARTICIPIANTS……………………………………………………………………….. 256
ORGANIZING COMMITTEE
Prof. Dr. Arunabha Chanda Jadavpur University
Prof. Dr. Cristina Flaut Ovidius University
Prof. Dr. Cihan Özgür Balikesir University
Prof. Dr. Debasis Giri Haldia Institute of Technology
Prof. Dr. Kadri Arslan Uludağ University
Prof. Dr. Kailash C. Madan Ahlia University
Prof. Dr. Kazım İlarslan Kırıkkale University
Prof. Dr. Laura Ventura Universita Degli Studi Di Padova
Prof. Dr. Levent Kula Ahi Evran University
Prof. Dr. Mahmut Ergüt Namık Kemal University
Prof. Dr. Mao-Ting Chien Soochow University
Prof. Dr. Moiz ud Din Khan COMSATS Institute of Information Technology
Prof. Dr. Muhammad Saeed Khan COMSATS Institute of Information Technology
Prof. Dr. Nejat Ekmekçi Ankara University
Prof. Dr. Nirmal C. Sacheti Sultan Qaboos University
Prof. Dr. Pallath Chandran Sultan Qaboos University
Prof. Dr. Pranesh Kumar University of Northern British Columbia
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ORGANIZING COMMITTEE
Prof. Dr. Ram N. Mohapatra University of Central Florida
Prof. Dr. Slavica Ivelić Bradanović University of Split
Prof. Dr. Vichian Laohakosol Kasetsart University
Prof. Dr. Victor Jimenez Lopez Universidad De Murcia
Prof. Dr. Victor Martinez-Luaces Universidad De La República
Prof. Dr. Yue Kuen Kwok Hong Kong University
Assoc. Prof. Dr. Emrah Evren Kara Duzce University
Assoc. Prof. Dr. İsmet Altıntaş Sakarya University
Assoc. Prof. Dr. Soley Ersoy Sakarya University
Assoc. Prof. Dr. Şenol Dost Hacettepe University
Assist. Prof. Dr. Ayse Zeynep Azak Sakarya University
Assist. Prof. Dr. Mahmut Akyiğit Sakarya University
Assist. Prof. Dr. Murat Sarduvan Sakarya University
Assist. Prof. Dr. Hidayet Hüda Kösal Sakarya University
Assist. Prof. Dr. Tülay Erişir Erzincan University
Dr. Handan Akyiğit Sakarya Universitesi
Lecturer Furkan Aydın Kahramanmaraş Sütçü imam University
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CONTENTS
Invited Talks
Resolvability of Topological Spaces 2
István JUHÁSZ ........................................................................................................................... 1
Boundedness properties of function spaces 2
Ljubiša D. R. KOČINAC and L’ubica HOLÁ ............................................................................ 2
Finite Differences Method of the Riemann Type Problem for 2D Conservative Laws in a Class
of Discontinuous Functions 3
Mahir RASULOV ........................................................................................................................ 3
On Some Applications of Measures of Noncompactness 4
Mohammad MURSALEEN ......................................................................................................... 4
Mathematical Literacy and a New Classification Proposal for Mathematical Literacy
Problems 6
Murat ALTUN ............................................................................................................................ 6
Fixed point property for some Alexandroff topological spaces with T0-separation axiom and
its applications 8
Sang-Eon HAN ........................................................................................................................... 8
The Role of Convergence Theory in Mathematics 9
Szymon DOLECKI ..................................................................................................................... 9
From Dual Bodies to the Kneser-Poulsen Conjecture 11
Karoly BEZDEK ...................................................................................................................... 11
Algebra
Some Results for Matrix Representation of Fibonacci Octonions 13
Adnan KARATAŞ and Serpil HALICI ...................................................................................... 13
On Special Elements of Split Octonions 14
Adnan KARATAŞ and Serpil HALICI ...................................................................................... 14
Numerical solutions of mKdV equation via modified cubic B-spline based Differential
Quadrature Method 16
Ali BAŞHAN ............................................................................................................................. 16
New Bounds for the Levels and Sublevels of Algebras Obtained by the Cayley-Dickson
Process 17
Cristina FLAUT........................................................................................................................ 17
Some Results on Generalized Derivations and (,) Lie Ideals 19
Evrim GÜVEN .......................................................................................................................... 19
A Note on the Sequence of Balancing Numbers 20
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Gül Karadeniz GÖZERİ ........................................................................................................... 20
Graph Operations of Randic Index 21
Gülistan Kaya GÖK and Şerife BÜYÜKKÖSE ........................................................................ 21
On Codes Written by Matrices Lexicographically Ordered 22
Mustafa ÖZKAN and Figen ÖKE ............................................................................................. 22
About Quasi-Cyclic Codes over the Field 23
Mustafa ÖZKAN and Figen ÖKE ............................................................................................. 23
On Certain Type of Band Matrices with the Generalized Fibonacci Numbers 24
Neşe ÖMÜR and Cemile Duygu ÇOLAK ................................................................................. 24
On Generalization of Quaternions 25
Serpil HALICI and Adnan KARATAŞ ...................................................................................... 25
Generalized Quaternions and Dual Fibonacci Octonions 27
Serpil HALICI and Adnan KARATAŞ ...................................................................................... 27
Formulæ for Two Weighted Binomial Identities with the Falling Factorials 29
Sibel Koparal, Neşe Ömür and Emrah Kılıç ............................................................................ 29
S3–graded iso(1,2) 30
Yasemen UÇAN , Reşat KÖŞKER and Ramazan TEKERCİOĞLU……………………………... 30
On Multiplicative Generalized Derivations in 3-Prime Near-Rings 32
Zeliha BEDİR and Öznur GÖLBAŞI 2..................................................................................... 32
Analysis
f -Lacunary statistical convergence of order for double sequences 34
Abdulkadir KARAKAŞ, Hıfsı ALTINOK and Yavuz ALTIN ..................................................... 34
Some Special Values of Vertices of Trees on the Suborbital Graphs 35
Ali Hikmet DEĞER and Ümmügülsün AKBABA ..................................................................... 35
On Some Special Values of Fibonacci and Lucas Sequence with the Suborbital Graphs 36
Ali Hikmet DEĞER and Ümmügülsün AKBABA ..................................................................... 36
A Note on the Ambarzumyan's Theorem 37
Alp Arslan KIRAÇ .................................................................................................................... 37
Polars in Locally Convex Cones 37
A. RANJBARI ........................................................................................................................... 37
A Sufficient Condition for Complex-Valued Summable Functions to be Absolutely Continuous
a.e. 40
Alp Arslan KIRAÇ .................................................................................................................... 40
On Quasi Subordinations for Analytic and Bi-Univalent Function Class 42
Arzu AKGÜL ............................................................................................................................ 42
Quadratic Time-Frequency Representations on Mixed Lorentz Type Modulation Spaces 44
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Ayşe SANDIKÇI ....................................................................................................................... 44
Bilinear Multipliers on nM P,Q 45
Ayşe SANDIKÇI ....................................................................................................................... 45
Reguler Equivalents of a Measure Space 46
Bahaettin CENGİZ and Banu GÜNTÜRK ............................................................................... 46
Hilbert Matrix on Spaces of Holomorphic Functions 47
Boban KARAPETROVIĆ .......................................................................................................... 47
Some Relation on the Poly-Genocchi Numbers and Polynomials with a q-Parameter 48
Burak KURT ............................................................................................................................. 48
The Fixed Point of the Sum of Nonlinear Operators in WC-Banach Algebras Relative to the
Weak Topology 49
Cesim TEMELand Musa ÇAKIR .............................................................................................. 49
Investigating the Center of L-weakly and M-weakly Compact Operators 50
E. BAYRAM, A.W. WICKSTEAD ............................................................................................. 50
The Generating Function of ( , )p q -Bernstein Polynomials and Their Properties Based on
( , )p q -Calculus 51
Erkan AGYUZ and Mehmet ACIKGOZ ................................................................................... 51
Characterizations of Matrix and Compact Operators on the Space 𝑬𝒓𝜽𝒑 52
Fadime GÖKÇE and Mehmet Ali SARIGÖL ........................................................................... 52
Matrix and Compact Operators on the Absolute Fibonacci Spaces 54
Fadime GÖKÇE and Mehmet Ali SARIGÖL ........................................................................... 54
The Theory of 𝒏-Scales 56
Furkan Semih DÜNDAR .......................................................................................................... 56
Some Specific Properties of Concave Functions Defined by the Generalized Srivastava-Attiya
Operator 57
Hasan BAYRAM and Şahsene ALTINKAYA ............................................................................ 57
A Note On Weighted Composition Operators on Besov-Type Spaces 59
Hamid VAEZI .......................................................................................................................... 59
Generalized Lupaş Operators 59
Hatice Gül İNCE İLARSLAN, Ali ARAL and Gülen BAŞCANBAZ-TUNCA ........................... 59
Statistical Convergence of order (β,γ) for Sequences of Fuzzy Numbers 60
Hıfsı ALTINOK and Mikail ET ................................................................................................. 60
Statistical Convergence of order β for Double Sequences of Fuzzy Numbers Defined by a
Modulus Function 61
Hıfsı ALTINOK, Yavuz ALTIN and Mahmut IŞIK.................................................................... 61
Δnm-lacunary Statistical Convergence of Order α 62
Hıfsı ALTINOK , Mikail ET and Mahmut IŞIK ....................................................................... 62
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Generalized Statistical Convergence of order β for Sequences of Fuzzy Numbers 63
Hıfsı ALTINOK, Abdulkadir KARAKAŞ and Yavuz ALTIN ..................................................... 63
A New Application of Quasi Power Increasing Sequences 64
Hikmet SEYHAN ÖZARSLAN .................................................................................................. 64
Almost Increasing Sequences and Their New Applications 65
Hikmet SEYHAN ÖZARSLAN and Bağdagül KARTAL ........................................................... 65
Wavelet-Type Transforms Generated by Some Semigroups and Their Applications 66
İlham A. ALIEV ........................................................................................................................ 66
Higher Order Fractional Boundary Value Problems with Integral Boundary Conditions 67
İsmail YASLAN ......................................................................................................................... 67
Boundedness of Products of Weighted Composition Operators and Differentiation Operators
between Weighted Bergman Spaces and Weighted Zygmund Spaces of Analytic Functions 68
Jasbir S. MANHAS ................................................................................................................... 68
The Vitali Convergence Theorem for Nonlinear Integrals 69
Jun KAWABE ........................................................................................................................... 69
Berezin Number Inequalities and Engliš Algebras 70
M. GÜRDAL, M.B. HUBAN and M.T. GARAYEV ................................................................... 70
Hadamard Type Inequalities for m-Convex and (α,m)-Convex Functions via Fractional
Integrals 71
Merve Avcı ARDIÇ, Alper EKİNCİ, Ahmet Ocak AKDEMİR and M. Emin ÖZDEMİR ......... 71
On Wijsman Asymptotically Deferred Statistical Equivalence of Order α for Set Sequences 73
Mikail ET, Hıfsı ALTINOK and Rifat ÇOLAK ........................................................................ 73
On Infinite Bernoulli Matrices 74
Murat KİRİŞCİ ......................................................................................................................... 74
Curious Bounds for the Gamma Function 75
Necdet BATIR ........................................................................................................................... 75
Some Fixed-Circle Theorems and Discontinuity at Fixed Circle 77
Nihal Yılmaz ÖZGÜR and Nihal TAŞ ...................................................................................... 77
Bergman Projections on the Weighted Harmonic Bloch Spaces on the Ball, Atomic
Decompositions and Gleason’s Problem 78
Ömer Faruk DOĞAN ............................................................................................................... 78
Majorization Problem on a Subclass of Analytic Functions 80
Öznur ÖZKAN KILIÇ, Osman ALTINTAŞ ............................................................................... 80
A Note on the Extended Type Riemann-Liouville Fractional Derivative Operator 81
Praveen AGARWAL, İ. Onur KIYMAZ2, Shilpi JAIN3 and Ayşegül ÇETİNKAYA4 ................. 81
On the Harmonic Averages of Sequences of Fuzzy Numbers 83
Rahmet SAVAŞ and Sefa Anıl SEZER ...................................................................................... 83
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Malliavin Calculus of Bismut Type for an Operator of Order Four on a Lie Group 84
Rémi LÉANDRE ....................................................................................................................... 84
On Convergence Properties of Sequences of k-Positive Operators in the Subspace of Analytic
Functions 85
Seda KARATEKE and Tülin COŞKUN .................................................................................... 85
Some Operator Inequalities Related to Means and Entropies 88
Shigeru FURUICHI .................................................................................................................. 88
On a Subclass of Harmonic Univalent Functions Involving a Linear Operator 88
Sibel Yalçın TOKGÖZ and Şahsene ALTINKAYA ................................................................... 88
On a Coefficient Problem for a Subclass of Bi-Univalent Functions Defined by Using a q-
Derivative Operator 90
Sibel Yalçın TOKGÖZ and Şahsene ALTINKAYA ................................................................... 90
On Asymptotically f-Statistical Equivalent Set Sequences 92
Sukran KONCA and Mehmet KUCUKASLAN ......................................................................... 92
Reverse Inequalities for the Berezin Numbers of Operators 93
U. YAMANCI, M. GÜRDAL and M. T. GARAYEV .................................................................. 93
On the (p,q)-Analogue of Poly-Bernoulli Polynomials 94
Veli KURT ................................................................................................................................ 94
Applied Mathematics
An Affine Scaling Method Using a Class of Differential Barrier Functions 96
Abdessamad BARBARA ............................................................................................................ 96
Hyperbolic Tangent Solution to the Conformable Time Fractional Zakharov-Kuznetsov
Equation in 3D Space 97
Alper KORKMAZ and Ozlem Ersoy HEPSON ........................................................................ 97
Asymptotic Properties of the Lasota Equation in Various Functional Spaces 98
Anna POSKROBKO and Antoni Leon DAWIDOWICZ ........................................................... 98
Iterative Solutions to the Systems of Linear Differential Equations 99
Arzu GÜLEROĞLU .................................................................................................................. 99
A Mathematical Model on Coral-Population Dynamics 100
Arzu ÜNAL ............................................................................................................................. 100
The Class of 𝑳 ∩ 𝑫 and Its Application to Renewal Reward Process 101
Asli Bektaş KAMIŞLIK, Tülay KESEMEN, Tahir KHANIYEV3 ............................................ 101
Mathematical Modeling for Real Life Problems 103
Aysegul SAGLAM ARSLAN .................................................................................................... 103
Higher Order Trigonometric B-Spline Algorithms to the Solution of Coupled Burgers’
Equation 104
Aysun Tok ONARCAN and Ozlem ERSOY HEPSON ............................................................ 104
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Numerical Solution of Nonlinear Parabolic Type Equation in the Class of Generalized
Functions with Degeneration 105
Bahaddin SINSOYSAL, Hakan BALand Kubra YILMAZ ....................................................... 105
A Finite Differences Scheme for Solving the System of Differential Equations of Non-
Isentropic Flow in a Class of Discontinuous Functions 106
Bahaddin SINSOYSAL, Hasan CARFI and Nazife KOC ....................................................... 106
A Finite Difference Method for Solving Generalized FitzHugh-Nagumo Equation 107
Bilge İNAN ............................................................................................................................. 107
Numerical Solutions of Boundary Value Problems for Nonlinear Delay Differential Equations 108
Bülent YILMAZ and Volkan YAMAN2 .................................................................................... 108
The Analytical Solutions of the Space-Time Fractional Modified Kawahara Equation 109
Dilek Varol BAYRAM, Sevil ÇULHA, Ayşegül DAŞCIOĞLU ............................................... 109
An Approximation Method for the Fractional Linear Fredholm Integrodifferential Equations
by Laguerre Polynomials 110
Dilek VAROL BAYRAM and Ayşegül DAŞCIOĞLU .............................................................. 110
Convergence of Backward Semi-Lagrangian Scheme with BDF2 for Two-Dimensional
Burgers’ Equation 111
Dojin KIM, Soyoon BAK, Philsu KIM, ................................................................................... 111
Compressed Sensing with Cyclic-S Hadamard Matrix for TeraHertz Imaging Applications 113
Esra Şengün ERMEYDAN and İlyas ÇANKAYA ................................................................... 113
Symbolic Regression: An Application to the Development of Models for the Surface Tension 115
Eva T. López SANJUÁN and María Isabel Parra ARÉVALO ............................................... 115
On the Stability Analysis of Sturm-Liouville Operator with Jump Conditions 119
Etibar S. PANAKHOV and Ahu ERCAN ........................................................................ 119116
Inverse Nodal Problem for p-Laplacian Diffusion Equation with Polynomially Dependent
Spectral Parameter 119
Etibar S. PANAKHOV , Emrah YILMAZ and Tuba GULSEN ....................................... 119117
On the Stress Distribution in the Elastic Body with a Locally Curved Triple Walled Carbon
Nanotube 119
Fatma COBAN and Reşat KOSKER ...................................................................................... 119
Construction of Exact Solutions to Partial Differential Equations via CRE Method 120
Filiz TAŞCAN and Arzu AKBULUT ....................................................................................... 120
Numerical Solution of an Inverse Problem for the Stokes Equations 121
Gullazata DAIRBAYEVA ........................................................................................................ 121
Existence of Solution for the Problem with a Concentrated Source in a Subdiffusive Medium 122
H. Terence LIU ....................................................................................................................... 122
Numerical Solution of the Conformable Fractional Diffusion Equation 123
Handan Çerdik YASLAN ........................................................................................................ 123
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Oscillation Criteria Fractional Difference Equations with Mixed Nonlinearities ................ 124
Hande BÜYÜKÇAVUŞOĞLU, Mustafa Kemal YILDIZ and Tuğba YALÇIN UZUN............ 124
Free Vibration of a Tapered Euler-Bernoulli Beam with 3D Tip Mass by Multi-Step
Differential Transformation Method 125
Hilal Doğanay KATI and Hakan GÖKDAĞ .......................................................................... 125
Image Segmentation with Fan-shaped Gradients 127
Hosook KIM and Hyoungseok KIM ....................................................................................... 127
The Computational and Experimental Studies on (E)-4-Methyl-N'-(1-(Pyridin-2-Yl)
Ethylidene) Benzenesulfonohydrazide 128
Hümeyra BATI, Erbil Murat AYDIN, Murat ÇINARLI, Nezihe ÇALIŞKAN ......................... 128
An Application of Integration Interpolation Method to Cauchy Type Singular Integral
Equation System 129
Hüseyin OĞUZ and Elçin YUSUFOĞLU .............................................................................. 129
A New Analysis for Fractional Order Ebola Model with Non-Local and Non-Singular Kernel 130
Ilknur KOCA .......................................................................................................................... 130
Linearization of the State Equation of a Nonlinearly Elastic Material of a Circular Composite
Shaft for Estimation of Natural Frequencies of Torsional Vibrations 131
I.A. TARASYUK and A.S. KRAVCHUK ................................................................................. 131
Parameter Dependent Generalized Sylvester and T-Sylvester Equations 133
Ivana KUZMANOVIĆ and Ninoslav TRUHAR ...................................................................... 133
Generalization of the Linear Momentum Operator in Quantum Mechanics: A Position-
Dependent Effective Mass Approach 134
J. J. PEÑA, J. MORALES, J. GARCÍA-RAVELO, Leonardo SALINAS ................................ 134
On Trace Inequalities for Generalized Quasi-Metric Adjusted Skew Informations 135
Kenjiro YANAGI ..................................................................................................................... 135
The Solutions of Special Second Order Ordinary Differential Equations Using Symmetries 137
Kısmet KASAPOĞLU ............................................................................................................. 137
Schrödinger Equation with Variable Effective Mass: Linear and Staggered Mass
Distributions 138
Leonardo SALINAS, H. LUNA-GARCÍA, J. GARCÍA-RAVELO, J. J. PEÑA ........................ 138
Weak Solutions to Interdiffusion Models with Vegard Rule 139
Lucjan SAPA, Bogusław BOŻEK and Marek DANIELEWSKI .............................................. 139
Self-Adjoint Extensions of Differential Operators with Potentials-Point Interactions 141
Manaf Dzh. MANAFOV ......................................................................................................... 141
Detecting a Hyperbolic Quadratic Eigenvalue Problem Using a Subspace Algorithm 142
Marija MILOLOŽA PANDUR ............................................................................................... 142
A Generalization of Midpoint Type Inequality of m-Logarithmically Convex Functions 143
Mehmet Eyüp KİRİŞ ............................................................................................................... 143
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Some Approximation Results for Lupaş-Kantorovich-Schurer Operators Based on (p, q )-
Calculus 146
Melek SOFYALIOĞLU and Kadir KANAT ........................................................................... 146
Conformable Fractional Klein-Gordon and Cahn-Allen Equations 147
Meryem ODABASI ................................................................................................................. 147
On the Finite Element Approximation in the Maximum Norm of Elliptic Quasi-Variational
Inequalities with Nonlinear Source Terms 148
Messaoud BOULBRACHENE ................................................................................................ 148
Exponentially Cusped Prismatic Shells in the 0N Approximation of I. Vekua’s Hierarchical
Models 149
Miranda GABELAIA .............................................................................................................. 149
Bounds for Blow up Time in Nonlinear Pseudo-Parabolic Equations 150
Müge MEYVACI ..................................................................................................................... 150
Oscillation Criteria for Higher Order Fractional Difference Equations with Nonlinearities 152
Mustafa Kemal YILDIZ , Ebru YILMAZ and Umut Mutlu ÖZKAN ....................................... 152
Lyapunov Type Inequalities for nth-order Three point BVP Differential Equations with Mixed
Nonlinearities 154
Naceri MOSTEPHA and Abdullah OZBEKLER .................................................................... 154
3D FEM Analysis of Buckling of a Piezoelectric Rectangular Thick Plate 155
Nazmiye YAHNIOGLU and Fatih AYLIKCI .......................................................................... 155
Identities for the double L22-integral transform 156
Neşe DERNEK, Fatih AYLIKÇI ............................................................................................. 156
Simplified Hirota Method for a Nonlinear Partial Differential Equation 158
Ömer ÜNSAL, Ahmet BEKIR, Murat KOPARAN .................................................................. 158
Extended Transformed Rational Function Method for Some (3+1) Dimensional Nonlinear
Partial Differential Equations 160
Ömer ÜNSAL, Ahmet BEKIR, Murat KOPARAN .................................................................. 160
Some Results on Cooperative Grey Games 162
Osman PALANCI, Rana S.M. BAKRI, Marwah Imad Najm NAJM, Sirma Zeynep
ALPARSLAN GOK ................................................................................................................. 162
A Cubic Subdomain Galerkin Method over the Geometrically Graded Mesh to the Singularly
Perturbed Problem 163
Ozlem Ersoy HEPSON and Idris DAG .................................................................................. 163
Optimal Control of Distributed Systems in Problems of Quartz Optical Fiber Production 164
Pervadchuk V.P., Vladimirova D.B., Gordeeva I.V. .............................................................. 164
On the Kinetics of the Dimer-Dimer Reactions over Supported Catalysts 166
Pranas KATAUSKIS ............................................................................................................... 166
The Importance of Algebraic Geometry and Cusp Singularities for Phase Transition
Dynamics of Neural Populations in Cerebral Cortex 167
xviii
R Murat DEMİRER , Oya DEMİRER .................................................................................... 167
Stress Distribution in the Infinite Elastic Body with Two Neighboring Periodical Curved
Carbon Nanotube Located on Different and Parallel Planes 168
Reşat KÖŞKER and İsmail GÜLTEN ..................................................................................... 168
Discrete Fractional Solutions of a Legendre Equation 170
Resat YILMAZER ................................................................................................................... 170
Discrete Fractional Solutions of a Associated Laguerre Equation 171
Resat YILMAZER and Erdal BAS .......................................................................................... 171
Asymptotic Stability of an Evolutionary Nonlinear Boltzmann-Type Equation 172
Roksana BRODNICKA and Henryk GACKI .......................................................................... 172
Hybrid Interval Algorithms for the Simultaneous Inclusion of Polynomial Complex Zeros: An
Experimental Comparison 174
Roseleine Neves MACHADO and Luiz Guerreiro LOPES .................................................... 174
An Investigation on the Free Vibration of a Sphere with Inhomogeneous Initial Stresses 175
S. D. AKBAROV, , N. YAHNIOGLU and H. H. GULIYEV ..................................................... 175
GDP and Efficiency of Russian Economy 176
Sergey M. BORODACHEV .................................................................................................... 176
Stabilization of Discrete Systems by High Order Compensators 177
Şerife YILMAZ İRTEM, Taner BÜYÜKKÖROĞLU, Vakif DZHAFAROV ............................ 177
Numerical Investigation of a Steady Flow of an Incompressible Pseudoplastic Fluid in a Lid
Driven Cavity 179
Serpil ŞAHİN and Hüseyin DEMİR ....................................................................................... 179
Numerical Comparision of Newtonian and Dilatant Fluids in an Enclosed Cavity Region 180
Serpil ŞAHİN and Hüseyin DEMİR ....................................................................................... 180
New Hybrid Conjugate Gradient Method as a Convex Combination of HS and FR Conjugate
Gradient Methods 181
Snezana S. DJORDJEVIC ...................................................................................................... 181
A New Formula for Higher Order Derivatives of Type xfdx
dx
n
k
and Its Applications 182
Telhat ÖZDOĞAN, Melek ERASLAN, Metin ORBAY............................................................ 182
Asymptotic Behaviour for Functional Three-Dimensional Navier-Stokes-Voigt Equations 183
T. CARABALLO and Antonio M. Márquez DURÁN .............................................................. 183
Mixing Problems: Associated Matrices and Stability of Linear ODE Systems 184
Victor MARTINEZ-LUACES .................................................................................................. 184
Backward Semi-Lagrangian Scheme for Guiding Center Problems 186
Xiangfan PIAO, Dojin KIM and Philsu KIM ......................................................................... 186
A Comparison of Multi-step and Multi-stage Method 187
Yonghyeon JEON and Sunyoung BU, Philsu KIM ................................................................. 187
xix
Green's Functions For Problems Simulating Potential Fields in Thin-Walled Assemblies of
Irregular Configuration 188
Yu. A. MELNIKOV and V. N. BORODIN............................................................................... 188
The Hurwitz’s Formula and Boundary Value Problems for Second-Order Equations 190
Yu. E. ANIKONOV, İsmet GÖLGELEYEN and Mustafa YILDIZ .......................................... 190
Discrete Mathematics
Secret Sharing Schemes Based on Extension Fields 192
Selda ÇALKAVUR .................................................................................................................. 192
Geometry
Pseudo Spherical Indicatrix Elastic Curves 195
Ahmet YÜCESAN Gözde ÖZKAN TÜKEL and Tunahan TURHAN ...................................... 195
Problems of g- lifts 196
Arif SALIMOV and Rabia CAKAN ......................................................................................... 196
Position Vectors of Curves with Respect to Darboux Frame in the Galilen 3-Space 197
Buket CEYLAN DİRİŞEN and Tevfik ŞAHİN ......................................................................... 197
Polynomial Helices in 𝑬𝒏 198
Bülent ALTUNKAYA and Levent KULA ................................................................................ 198
On Normal Magnetic Curves in 3-Dimensional Heisenberg Group H3 199
Cihan ÖZGÜR ........................................................................................................................ 199
On Chen Invariants for Submanifolds of Riemannian Product Manifolds 200
Erol KILIÇ, Mehmet GÜLBAHAR and Sadık KELEŞ ........................................................... 200
Special Curves in Finsler Space 201
Fatma ATEŞ, Zehra ÖZDEMİR, F. Nejat EKMEKCİ and Mustafa ÇALIŞKAN ................... 201
A Differential Calculus on Rh(1|2) 202
Fatma BULUT ........................................................................................................................ 202
Application Example of the Processing of the Cubes in the Geometry Lesson in Elementary
Schools 203
Gönül TÜRKAN DEMİR, Keziban ORBAY and Emine ALTUNAY ŞAM .............................. 203
A study on Elastic Strips 205
Gözde ÖZKAN TÜKELAhmet YÜCESAN .............................................................................. 205
Involutes of Order k of a Space-like Curve in 4
1IE 206
Günay ÖZTÜRK, İlim KİŞİ2, Sezgin BÜYÜKKÜTÜK and Kadri ARSLAN .......................... 206
Similarity Geometry of Null Cartan Curves 207
Hakan ŞİMŞEK ...................................................................................................................... 207
xx
Semisimilarity and Consemisimilarity for matrices over a Commutative Quaternion 209
Hidayet Hüda KÖSAL , Mahmut AKYİĞİT and Murat TOSUN ............................................ 209
On an Almost Contact Metric Manifold with a Type of Semi-symmetric Non-metric
Connection 210
Hulya Bagdatli YILMAZ ......................................................................................................... 210
On Invariants of a System of Vectors in Minkowski Spacetime 212
İdris ÖREN ............................................................................................................................. 212
Determining by Control Points of Null Bézier Curves with Degree 2 and 3 Curves in 3-
Dimensional Minkowski Spacetime 214
İdris ÖREN ............................................................................................................................. 214
The Elliptic Matrices Associated with Eliptic Biquaternions and De-Moivre’s Formula for
These Matrices 215
Murat Tosun and Kahraman Esen Özen ................................................................................ 215
New Results for General Helix in Euclidean 3-Space 216
Kazım İLARSLAN ................................................................................................................... 216
On Cardan Position for the Lorentzian Plane Motion of a Rigid Body 217
Kemal EREN, Soley ERSOY and Mahmut ERGÜT ................................................................ 217
Some Characterizations for Screen Isotropic Leaves on Lightlike Hypersurfaces 219
Mehmet GÜLBAHAR, Sadık KELEŞ and Erol KILIÇ ........................................................... 219
Special Smarandache Curves with Respect to Darboux Frame in the Galilean 3-Space 220
Merve OKUR and Tevfik ŞAHİN ............................................................................................ 220
Elliptical Motion on a Given Ellipsoid 221
Mustafa ÖZDEMİR ................................................................................................................ 221
Ruled Surfaces in Contact Geometry 223
Nural YÜKSEL, Murat Kemal KARACAN, Hasibe İKİZ and Çağlar Zeki ODABAŞI .......... 223
A Note on Spherical Orthotomic of Lorentzian Spherical Spacelike Curves 224
Önder Gökmen YILDIZ .......................................................................................................... 224
Local Diffeomorphic Image of Spherical Antiorthotomic of Lorentzian Spherical Timelike
Curves 225
Önder Gökmen YILDIZ .......................................................................................................... 225
The Kinetic Energy Formula for the Closed Planar Homothetic Motions in Complex Plane 226
Önder ŞENER, Ayhan TUTAR and Serdar SOYLU ............................................................... 226
New Developments in Lightlike Submanifolds Theory 227
Sadık KELEŞ, Erol KILIÇ and Mehmet GÜLBAHAR ........................................................... 227
Holditch Type Theorem for Kinetic Energy of Projective Curve Under the 1-Parameter
Spatial Motion 228
Serdar SOYLU , Ayhan TUTAR and Önder ŞENER .............................................................. 228
xxi
On the Casual Characteristics of Higher Order Bertand Mate, Involute Curve, Mannheim of
a Non-Null Curve in IL³ 230
Şeyda KILIÇOĞLU and Süleyman ŞENYURT ....................................................................... 230
On the Second Order Involute of a Space-like Curve with Time-like Binormal in IL³ 232
Şeyda KILIÇOĞLU and Süleyman ŞENYURT ....................................................................... 232
On Biharmonic Hypersurfaces in Semi-Euclidean Spaces 233
Sibel SEVİNÇ, Gülşah AYDIN ŞEKERCİ and A. Ceylan ÇÖKEN ........................................ 233
Timelike and Spacelike Split Quaternion Ruled Surfaces in Minkowski 3-space 234
Kıvanç KARAKAŞ , Mesut ALTINOK and Levent KULA ...................................................... 234
Mathematics Education
Parameterized Examination in Econometrics 236
Anna MALINOVA, Vesselin KYURKCHIEV and Georgi SPASOV ....................................... 236
The Effect of Differentiated Instruction on Mathematical Attitudes of Students 238
Ayten Pinar BAL and Onur EKİNCİ ...................................................................................... 238
Investigation of the Test Anxieties of 8th Grade Students 239
Bülent ALTUNKAYA, Kamile ŞANLI KULA and Cahit AYTEKİN ........................................ 239
Examining Pre-service Teachers’ Perceptions and Evaluations about Student-Invented
Strategies 241
Büşra KARTAL and Yasemin KIYMAZ .................................................................................. 241
Investigation of Parent Expectations of Middle School Students from Mathematics Education* 242
Cahit AYTEKİN, Serdal BALTACI, Bülent ALTUNKAYA3, ................................................... 242
The Impact of the Probability Explorer Software on Preparing Probability Teaching Plans of
Elementary Mathematics Teacher Candidates* 244
Cahit AYTEKİN, Serdal BALTACI, Yasemin KIYMAZ3, Avni Yıldız4 .................................... 244
Proof Evaluation: Mathematicians and Prospective Mathematics Teachers’ Perspectives 246
Esra Selcen YAKICI-TOPBAŞ, Elçin EMRE-AKDOĞAN, Fatma Nur AKTAŞ, Gönül
YAZGAN-SAĞ and Ziya ARGÜN ........................................................................................... 246
New Method of Grading Exams based on Computational Statistics 248
Evgeny GERSHIKOV ............................................................................................................. 248
Difference Equations in Economics 250
Jelena STANOJEVIC and Katarina KUKIC .......................................................................... 250
Factors Affecting Scoring Open-Ended Questions 251
Kübra AKKAYA and Selahattin ARSLAN .............................................................................. 251
Development Process of In-Service Training Intended for Teachers to Perform Teaching of
Mathematics with Computer Algebra Systems 252
Mehmet Alper ARDIÇ and Tevfik İŞLEYEN .......................................................................... 252
xxii
Examination of the Activities about Enciphering Done with Fifth Grade Students and Their
Views about the Activities 253
Muhammet ŞAHAL and Ahmet Şükrü ÖZDEMİR ................................................................ 253
The Concept of Zero: A Comparison of Students with Learning Disability vs Non-Learning
Disability 254
N. Dilşad GÜVEN and Esra Selcen YAKICI TOPBAŞ and Ziya ARGÜN ............................. 254
Preservice Mathematics Teachers’ Mathematizing Skills in The Process of Solving a Real Life
Problem 256
Nuray Çalışkan DEDEOĞLU ................................................................................................ 256
Learning Implementations about Cooperative Learning Method: A Case Study in Türkiye 258
Perihan DINC ARTUT and Ayten Pınar BAL ........................................................................ 258
Investigation of Number Sense Strategies Used By the 8th Grade Students in Turkey 260
Perihan DINC ARTUT and Zübeyde ER ................................................................................ 260
Examining Teacher Candidates’ Thinking Skills 262
Perihan DİNÇ ARTUTand Ayten Pınar BAL ......................................................................... 262
Difficulties of High School Students in Complex Numbers 264
Selahattin ARSLAN ................................................................................................................ 264
Investigation of Mathematical Viewpoints of Primary School Teacher Program Students 265
Tuğba BARAN KAYA and Ahmet IŞIK ................................................................................... 265
A Framework Suggestion for the Analyze of the Solving Process of a Geometrical
Construction Problem 266
Tuğçe KOZAKLI ÜLGER, Işıl BOZKURT and Murat ALTUN ............................................. 266
Active Learning in Flipped College Algebra Class 268
Violeta VASILEVSKA ............................................................................................................. 268
Statistics
New Method for Obtaining of Weighted Distributions 271
Božidar V. POPOVİĆ ............................................................................................................. 271
Transmuted Exponential Power Distribution and its Distributional Properties 273
Buğra SARAÇOĞLU .............................................................................................................. 273
A New Bivariate Archimedean Copula Based on a Transformed Generator Function 274
Çiğdem TOPÇU GÜLÖKSÜZ, Nuri ÇELİK .......................................................................... 274
The Comparison of Population Means under Unequal Variances 275
Neriman AKDAM, M. Fedai KAYA2 and Buğra SARAÇOĞLU3 ........................................... 275
A New Lifetime Distribution 276
Nuri CELİK and Cigdem TOPCU GULOKSUZ .................................................................... 276
Topology
xxiii
A Topological Study of Harmony and Counterpoint in Music Using Quotient Orbifolds 278
Aditya SIVAKUMAR and Dmitri TYMOCZKO ...................................................................... 278
New Functors from Fuzzy Normed Spaces Category 279
Deniz Pınar SUNAOĞLU and Erdal GÜNER ...................................................................... 279
Soft Metric Spaces 280
Ebru Aydoğdu, Abdülkadir Aygünoğlu and Halis Aygün ...................................................... 280
A New Property between Compactness and Completeness in Generalized Metric Spaces 281
Emrah Evren KARA and Merve İLKHAN .............................................................................. 281
Extensions of T_0-quasi-metrics 282
Filiz YILDIZ and Hans-Peter KÜNZİ .................................................................................... 282
Some Categorical Approaches to the Relations between Inverse Systems of Bitopological and
Ditopological Spaces 283
Filiz YILDIZ ........................................................................................................................... 283
Normal Product Adjacency for Simplicial Homology Groups of Digital Images 284
Gülseli BURAK ...................................................................................................................... 284
Bracket Polynomials of Torus Links as Fibonacci Polynomials 285
Kemal TAŞKÖPRÜ, İsmet ALTINTAŞ and Merve BEYAZTAŞ.............................................. 285
Controllability of Affine Control Systems on Solvable Lie Groups 286
Memet KULE .......................................................................................................................... 286
Some Effective Results Related to the New Statistical Cauchy Sequence 288
Merve İLKHAN and Emrah Evren KARA .............................................................................. 288
Products of Hopfian Manifolds and Their Shape Fibrators' Properties 289
Violeta VASILEVSKA ............................................................................................................. 289
Soft AB-Sets and Soft αAB-Sets in Soft Topological Spaces 291
Zehra Güzel ERGÜL and Naime TOZLU .............................................................................. 291
The Other Areas
Approximation by Certain Linear Positive Operators of Three Variables 294
Afşin Kürşat GAZANFER ....................................................................................................... 294
Extension of Silver-Meal Algorithm with Variable Demand and Delivery Time to Cope with
Bullwhip Effect in Multi-Echelon Supply Chains 295
Halil İbrahim CEBECİ and Doğan ÜNAL ............................................................................. 295
Consumer and Producer Surpluses of the Quadratic Demand and Supply Functions by Using
Trapezoidal Fuzzy Numbers and Signed Distance 297
İsmail ÖZCAN and Salih AYTAR ........................................................................................... 297
On Creating Tessellations with Using Transformational Geometry 298
Kübra ÖZLÜ DEĞER and Ali Hikmet DEĞER ..................................................................... 298
xxiv
Relationship between Electricity Consumption and Economic Growth in some Developing
Countries: MS-VECM Analysis 299
Melike BİLDİRİCİ , Fazıl KAYIKÇI ...................................................................................... 299
Test of validity of Baltic Dry index with MS-ARMA and SET-ARMA models 300
Melike BİLDİRİCİ , Fazıl KAYIKÇI and Işıl Şahin ONAT .................................................... 300
Chaotic Examination of Turkish Financial Market 301
Melike E. BİLDİRİCİ , Fulya ÖZAKSOY and Bahri SONÜSTÜN ........................................ 301
Chaotic Structure of Oil Price 302
Melike E. BİLDİRİCİ , Fulya ÖZAKSOY .............................................................................. 302
Optimization of MANET Routing Tables with Plant Propagation Algorithm 303
Mevlüt ERSOY and Tuncay YİĞİT ......................................................................................... 303
The Integrability and the Zero-Hopf Bifurcation of the Three-Dimensional Lotka-Volterra
Systems 305
Rizgar H. SALIH .................................................................................................................... 305
Posters
Estimates of Initial Coefficients of a New Analytic and Bi-Univalent Function Class Defined
by Integral Operator Involving Polylogarithm Function 307
Arzu AKGÜL .......................................................................................................................... 307
Finding Initial Coefficients for a Class of Bi-Univalent Functions Given by q-Derivative
Operators 310
Arzu AKGÜL .......................................................................................................................... 310
Dual-Hyperbolic Fibonacci and Lucas Numbers 313
Arzu CİHAN, Ayşe Zeynep AZAK and Mehmet Ali GÜNGÖR .............................................. 313
Toeplitz Operators with Symbols in Some Function Spaces 315
Ayşe SANDIKÇI ..................................................................................................................... 315
A Further Generalization of Gamma, Beta and Hypergeometric Functions 316
Ayşegül ÇETINKAYA, İ. Onur KIYMAZ, Praveen AGARWAL and Shilpi JAIN ................... 316
Teachers’ Perceptions About Written Examination Preparation Process 318
Ayten Pınar BAL and Fatma SADIK ...................................................................................... 318
The Analysis of Mistakes and Solution Strategies Applıed in Algebraic Verbal Problems 319
Ayten Pınar BAL and Ahmet KARACAOĞLU ....................................................................... 319
Examination of Burnout and Life Satisfaction of Nurses 320
Emrah GÜRLEK and Kamile ŞANLI KULA and Mehmet YETİŞ and Aysu YETİŞ ............... 320
Ruled Surface Pair Generated by Darboux Vectors of a Curve and Its Natural Lift in 𝑰𝑹𝟏𝟑 322
Evren ERGÜN , Mustafa ÇALIŞKAN and Keziban ORBAY .................................................. 322
Caratheodory’s Theorem in B-1-convexity 323
Gabil ADILOV and Ilknur YESILCE ..................................................................................... 323
xxv
The Simulation of the Order Parameter Probability Distribution for the Four Dimensional
Ising Model on the Creutz Cellular Automaton with the Linear Dimensions L=24, 26 and 28 324
Ganimet MÜLAZIMOĞLU KIZILIRMAK ............................................................................. 324
Some Numerical Experiments for a Sharper Version of Hölder Inequality 325
Gültekin TINAZTEPE ............................................................................................................. 325
Conformable variational iteration method for solving the time-fractional Fornberg-Whitham
equation 326
Handan Çerdik YASLAN ........................................................................................................ 326
A Further Generalization of Fractional Operators 328
İ. Onur KIYMAZ, Ayşegül ÇETİNKAYA1, Praveen AGARWAL and Shilpi JAIN3 ................ 328
Higher-Order Multi-Point Fractional Boundary Value Problems 330
İsmail YASLAN ....................................................................................................................... 330
Elliptic Biquaternion Algebra 331
Kahraman Esen ÖZEN and Murat TOSUN ........................................................................... 331
Beliefs about Mathematical Problem Solving of University Students 332
Kamile ŞANLI KULA and Ezgi ÇAĞATAY İN ....................................................................... 332
Investigation of Structural Phase Transformation in Iridium Dioxide (IrO2) under High
Pressure 333
Köksal KIZILIRMAK, Ganimet M. KIZILIRMAK and Hülya ÖZTÜRK................................ 333
Helixes on Clifford Surfaces in a Hyperbolic Space of Positive Curvature 335
Lyudmila ROMAKINA ............................................................................................................ 335
A New Generalization of Whittaker Function and its Properties 336
M. Baki YAĞBASAN, Ayşegül ÇETİNKAYA1 and İ. Onur KIYMAZ1 .................................... 336
Classification of Homothetical Hypersurfaces and Its Applications to Production Functions in
Economics 338
Mahmut ERGUT and Muhittin Evren AYDIN ...................................................................... 338
Generation of Pseudo-Random Numbers from Given Probabilistic Distribution with the Use
of Chaotic Maps 339
Marcin LAWNIK .................................................................................................................... 339
A New Approach on Slant Curves in three Dimensional Lie Groups 340
Osman Zeki OKUYUCU, Caner DEĞIRMEN and İsmail GÖK ............................................ 340
Analysis of Antimicrobial Resistance among Clinical Isolates from Denizli 341
Selma KIRAÇ, Dilek KESKIN and Muradiye YARAR ............................................................ 341
An Approach to Volterra and Fredholm Integral Equations by Homotopy Perturbation
Method 343
Serpil ŞAHİN .......................................................................................................................... 343
Composition Formulae Associated Fractional Integral Operator with the Multi-Index Mittag-
Leffler Functions 344
Shilpi JAIN, Praveen AGARWAL, Ayşegül ÇETINKAYA and İ. Onur KIYMAZ ................... 344
xxvi
On a General Subclass of Univalent Functions Based on the q-Derivative Operator 346
Sibel Yalçın TOKGÖZ and Şahsene ALTINKAYA ................................................................. 346
Estimates of Coefficients for a Subclass of Bi-Univalent Functions by Making Use of Faber
Polynomial Expansions 348
Sibel Yalçın TOKGÖZ and Şahsene ALTINKAYA ................................................................. 348
On a Subclass of Univalent Functions Defined by Chebyshev Polynomials 350
Sibel Yalçın TOKGÖZ and Şahsene ALTINKAYA ................................................................. 350
Coefficient Estimates for Analytic Bi-Bazilevi cFunctions of Order and Type 351
Sibel Yalçın TOKGÖZ and Şahsene ALTINKAYA ................................................................. 351
Singularities of the Darboux Ruled Surface of a Space Curve in the Pseudo-Galilean Space 353
Tevfik ŞAHİN .......................................................................................................................... 353
On the Hyperbolic Spinors and Split Quaternions 354
Tülay ERİŞİR and Mehmet Ali GÜNGÖR.............................................................................. 354
An Introduction to Fibonaci, Lucas and Generalized Fibonacci Commutative Quaternions 355
Ömer TETIK and Mahmut AKYIĞIT ..................................................................................... 355
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
0
INVITED
TALKS
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
1
Resolvability of Topological Spaces
István Juhász1
Abstract. A topological space X is called λ-resolvable, where λ is a (finite or
infinite) cardinal, if X contains λ many pairwise disjoint dense subsets. X is maximally
resolvable if it is X -resolvable, where
min : ,X G G open G
The expectation is that “nice” spaces should be maximally resolvable, as verified e.g.
by the well-known facts that both metric and linearly ordered spaces, as well as compact
Hausdorff spaces, are maximally resolvable. There is, however, a countable regular (hence
“nice”) space with no isolated points that is not even 2-resolvable.
In this talk we present resolvability results about spaces that are more general than
the above. On one hand, we consider the class of monotonically normal spaces that includes
both metric and linearly ordered spaces, on the other we consider spaces with properties that
are more general than compactness, namely Lindel¨ofness, countable compactness, and
pseudocompactness.
1 Alfred Renyi Institute of Mathematics, Hungary, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
2
Boundedness Properties of Function Spaces
Ljubiša D. R. KOČINAC1 and L’ubica HOLÁ2
Abstract. Let X be a Tychonoff space, (Y,d) a metric space, and C(X,Y) be the set of
continuous mappings from X to Y. There is a variety of natural uniformities and the
corresponding topologies on this set: the topology of pointwise convergence, the compact-
open topology, the topology of uniform convergence, the m-topology, the graph topology, and
others. We investigate several boundedness properties of these uniform spaces related to the
classical covering properties by Menger, Hurewicz, Rothberger and Gerlits-Nagy.
When Y is the real line R with the standard metric, then we write C(X) instead of
C(X,R). The set C(X) equipped with each of the mentioned topologies is a topological group.
In this case we consider boundedness properties of these Hausdorff topological groups with
the operation of pointwise addition.
Keywords. M-bounded, H-bounded, R-bounded, GN-bounded.
AMS 2010. 46E10, 54C35, 54D20, 54E15, 54H11.
References
[1] Babinkostova, L., Kočinac, Lj.D.R., Scheepers, M., Combinatorics of open covers (XI):
Menger- and Rothberger-bounded groups, Topology Appl., 154, 1269--1280, 2007.
[2] Di Maio, G., Holá, L'., Holy, D., McCoy, R.A., Topologies on the space of continuous
functions, Topology Appl., 86, 105-122, 1998.
[3] Holá, L'., Zsilinszky, L., Completeness properties of the graph topology, Topology Proc.,
46, 1-14, 2015.
[4] Kočinac, Lj.D.R., Selection principles in uniform spaces, Note Mat., 22, 127-139,
2003/2004.
1University of Niš, Niš, Serbia, [email protected] 2Slovak Academy of Sciences, Bratislava, Slovakia, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
3
Finite Differences Method of the Riemann Type Problem for 2D Conservative Laws in a
Class of Discontinuous Functions
Mahir RASULOV1
Abstract. In this study a new method for finding the numerical solution of the Cauchy
problem with a discontinuous initial profile for the nonlinear 2D dimensional scalar
conservation laws is suggested. Firstly, some properties of the weak solution of the linearized
equation are investigated. Taking these properties into consideration, we introduce an
auxiliary problem since it has some advantages over the main problem, and it is equivalent to
the main problem in a definite sense. The proposed auxiliary problems allows us to develop
effective finite differences schemes for finding the solution with higher accuracy, such that
the obtained solution expresses all of the physical properties of the problem under
investigation. Using of the solution of the introduced auxiliary problem the method for
obtaining the location of shock appearing in the solution of main problem and it is evolution
is investigated. Moreover, using the auxiliary problem we can write the higher order
numerical scheme with respect to the time variable.
Keywords. Scalar conservation laws, auxiliary problem, exact and numerical solution
in a class of discontinuous functions
AMS 2010. 53A40, 20M15.
1 Beykent University,, Istanbul, Turkey, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
4
On Some Applications of Measures of Noncompactness
Mohammad MURSALEEN1
Abstract. In this talk, we present a brief survey of theory and applications of measures
of noncompactness. The classical measures of noncompactness are discussed and their
properties are compared. The approaches for constructing measure of noncompactness in a
general metric or linear space are described, along with the classical results for existence of
fixed point for condensing operators. Also several generalization of classical results are
mentioned and their applications in various problems of analysis such as linear equation,
differential equations, integral equations and common solutions of equations are discussed.
The most effective way in the characterization of compact operators between the
Banach spaces is applying the Hausdorff measure of noncompactness. In this chapter, we
present some identities or estimates for the operator norms and the Hausdorff measures of
noncompactness of certain operators given by infinite matrices that map an arbitrary BK-
space into the sequence space 0 , ,c c l and 1.l Many linear compact operators may be
represented as matrix operators in sequence spaces or integral operators in function spaces
[1].
We apply the technique of measures of noncompactness to the theory of infinite
systems of differential equations in some Banach sequence space 0 , , pc c l (1 p ).
Infinite systems of ordinary differential equations describe numerous world real problems
which can be encountered in the theory of branching processes, the theory of neural nets, the
theory of dissociation of polymers and so on. Let us also mention that several problems
investigated in mechanics lead to infinite systems of differential equations. Moreover, infinite
systems of differential equations can be also used in solving some problems for parabolic dif-
differential equations investigated via semidiscretization. We adopt the technique of measures
of noncompactness to the theory of infinite systems of differential equations. Particularly, we
are going to present a few existence results for infinite systems of differential equations
formulated with the help of convenient and handy conditions. We study of the solvability of
the infinite systems of differential equations in some classical Banach sequence spaces.
References
1 Aligarh Muslim University, India, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
5
[1] Banas, J. and Mursaleen, M., Sequence Spaces and Measures of Noncompactness with
Applications to Differential and Integral Equations, Springer, 2014.
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
6
Mathematical Literacy and a New Classification Proposal for Mathematical Literacy
Problems
Murat ALTUN1
Abstract. The improvement of mathematical literacy level remains one of the
fundamental problems of primary education. Mathematical literacy is defined as “an
individual’s capacity to identify and understand the role that mathematics plays in the world,
make well-founded judgements and use and engage with mathematics in ways that meet the
needs of that individual’s life as a constructive, concerned and reflective citizen. In a short
term, mathematical literacy is defined as “an individual’s capacity to formulate, employ and
interpret mathematics in a variety of contexts”. Assessments for identifying mathematical
literacy levels are conducted by allowing students to use their mathematical competencies
through addressing them as contextual, conceptual, and operational problems. Countries need
to know students’ difficulties at solving mathematical literacy problems to be able to take the
necessary precautions.
Four different types of mathematical literacy classification have been employed so far:
Subject areas, capabilities, competency clusters and process skills. However, there are some
problems in assessments, so, there is a need to make a new classification in order to inhibit the
uncertainty in assessments. In this study, a new classification has been identified and
suggested.
In this study, middle school students (8th grade) were asked mathematical literacy
questions. The data obtained from the student responses were subjected to factor analysis. A
six-factor structure was obtained at the end of the analysis. The obtained factors were found to
have adequate variance in explaining the students’ mathematical literacy. These factors were
named as making algorithmic operations, mastering rich mathematical content, mathematical
inference, developing mathematical proposals and interpreting a developed proposal,
understanding the mathematical equivalence of real world situations, and understanding the
counterpart of mathematical language in life. The data analysis revealed that the students
failed in the factors of mathematical inference, developing mathematical proposals and
interpreting a developed proposal, and understanding the mathematical equivalence of real
world situations. The results of this study can help and guide future researches for analyzing
1 Uludag University, Bursa, Turkey, : [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
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mathematical literacy achievement, developing a mathematical literacy scale, developing
mathematics curricula and organizing its instruction.
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
8
Fixed point property for some Alexandroff topological spaces with T0-separation axiom
and its applications
Sang-Eon Han1
Abstract. In this talk we study the fixed point property (FPP for short) and the almost
fixed point property (AFPP for brevity) for non-Hausdorff topological spaces (or digital
spaces). Besides, we refer to some applicatoions of this approach. More precisely, we focus
on the following approaches.
(1) FPP for digital images motivated by the Rosenfeld model [1]
(2) AFPP for digital images motivated by the Rosenfeld model [1]
(3) FPP for Khalimsky topological spaces [3], [5]
(4) FPP for Marcus-Wyse topological spaces [2], [4]
(5) FPP for Alexandroff spaces with T0-separation axiom
(6) Order theoretical approach
Keywords. Fixed point property, Khalimsky topological space, Alexandroff space.
AMS 2010. 54A10, 68U10.
References
[1] Han, S.-E., Banach fixed point theorem from the viewpoint of digital topology, Journal of
Nonlinear Sciences and Applications 9(3) 895-905, 2016.
[2] Han, S.-E., Almost fixed point property for digital spaces associated with Marcus-Wyse
topological spaces, Journal of Nonlinear Sciences and Applications 10, 34-47, 2017.
[3] Han, S.-E., Fixed point property for digital spaces, Journal of Nonlinear Sciences and
Applications 10, 2510-2523, 2017.
[4] Han, S.-E., The FPP of an M-retract and its applications, Topology and its Applications,
Online First press, 2017.
[5] Han, S.-E., The fixed point property of the smallest open neighborhood of the Khalimsky
topological plane, Filomat, Online First press, 2017.
1 Chonbuk National University, Jeonju, Korea, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
9
The Role of Convergence Theory in Mathematics
Szymon Dolecki 1
Abstract. A convergence on a set X is a relation between filers F on X and
elements x of X
limx F
such that F G implies lim lim ,F G and lim 2 :Xx F x F for each .x X
Every topology can be seen as a convergence given by
lim ,x F N x F
where N x is the filter of all neighborhoods of x with respect to .
Actually, the study of topologies is equivalent to the study of certain families of filters; on one
hand, each topology is determined by its neighborhoods, on the other, each filter H on X
gives rise either to a topology on ,X or to a topology on X where X so that H
is the neighborhood filter either of some elements of X or of .
The emergence of convergence theory was triggered by a realization that some natural and
important phenomena correspond to non-topological convergences, for instance, the
restrictions of topologies to sequential filters, some convergences occurring in measure
theory, the natural convergence on the space of continuous maps, and the dual convergences
related to the Arens-Mackey duality in functional analysis.
In [1] Choquet discovers that certain hyperspaces are non-topological, but are always
pseudotopological. The introduction of pseudotopologies2, an exponential category including
the (non-exponential) category of topologies, may be considered as an actual starting point of
convergence theory. Pseudotopologies are a completion of topologies in a similar way as
complex numbers of real numbers.
These new perspectives enable one to view many topological properties and classes of spaces
and maps as solutions of functorial inequalities in the category of convergences. This is the
case of sundry variants of quotient maps; they turn out to be the quotients with respect to
certain subcategories of convergences.
1 Mathematical Institute of Burgundy, Dijon, France, [email protected] 2 With continuous maps as morphisms.
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
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Variants of compactness appear as a single property with respect to various subcategories of
convergences.
Compactness and completeness turn out to be degrees of a single unifying quality.
One can see ([2]) that the convergence-theoretic framework is simpler, more natural, far-
reaching and powerful than that of topologies.
References
[1] Choquet G., Convergences, Ann. Univ. Grenoble, 23, 55-112, 1947-48.
[2] Dolecki, S., and Mynard, F., Convergence foundations of topology, Word Scientific, 2016.
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
11
From Dual Bodies to the Kneser-Poulsen Conjecture
Karoly Bezdek 1
Abstract. The Kneser–Poulsen Conjecture (1955) states that if the centers of a family
of N congruent balls in Euclidean d-space are contracted, then the volume of the intersection
does not decrease. The first half of my talk will give a summary of the status of this long-
standing conjecture in geometry. The second half will discuss the following latest
development. A uniform contraction is a contraction where all the pairwise distances in the
first set of centers are larger than all the pairwise distances in the second set of centers. I will
present a proof of the Kneser- Poulsen conjecture for uniform contractions whenever N is
sufficiently large (depending only on d) in Euclidean, spherical as well as hyperbolic d-space
for all 1d . The method of proof is centered around a Blaschke-Santalo type inequality for
dual bodies.
1 University of Calgary, Canada, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
12
ALGEBRA
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
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Some Results for Matrix Representation of Fibonacci Octonions
Adnan KARATAŞ1 and Serpil HALICI2
Abstract. In this study, we investigate non-standard matrix representation of octonion
algebra. This representation allows us to calculate octonion multiplication without knowing
the octonion multiplication table. Also, we use mentioned matrix representation for Fibonacci
octonions and we obtain two different Cassini identities related with the matrix representation.
Keywords. Matrix representation, Fibonacci numbers, octonions.
AMS 2010. 11B39, 17A20
References
[1] Smith, Jonathan D. H. An introduction to quasigroups and their representations, CRC
Press, 2006.
[2] Lounesto, P., Clifford algebras and spinors, Clifford Algebras and Their Applications in
Mathematical Physics. Springer Netherlands, 25-37, 1986.
[3] Koshy, T., Fibonacci and lucas numbers with applications, A Wiley-Interscience
Publication, 2001.
[4] Keçilioğlu, O., and Akkus, I., The fibonacci octonions, Advances in Applied Clifford
Algebras 25.1, 151-158, 2015.
[5] Halici, S., and Karataş, A., Some matrix representations of fibonacci quaternions and
octonions, Advances in Applied Clifford Algebras: 1-10.
[6] Günaydin, M., and Gürsey F., Quark structure and octonions, Journal of Mathematical
Physics 14.11, 1651-1667, 1973.
[7] Gürsey, F., and Chia-Hsiung T., On the role of division, Jordan and related algebras in
particle physics, World Scientific, 1996.
[8] Baez, J., The octonions, Bull. of the American Mathematical Society 39.2, 145-205, 2002.
1 Pamukkale University, Denizli, TURKEY, [email protected] 2 Pamukkale University, Denizli, TURKEY, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
14
On Special Elements of Split Octonions
Adnan KARATAŞ1 and Serpil HALICI2
Abstract. In this study, we focus on split octonion algebras over finite fields and
investigate non trivial idempotent and nilpotent elements. We present formulas which gives
idempotent and nilpotent elements according to chosen finite fields. Also, we obtain some
fundamental properties including them. Furthermore, we show that for some subalgebras of
octonions our formulas compatible with earlier studies in this subject.
Keywords. Idempotent, nilpotent, octonions.
AMS 2010. 16N40, 17A20.
References
[1] Aristidou, M.; Demetre, A., idempotent elements in quaternion rings over Zp,
International Journal of Algebra, 6.5-8, 249-254, 2012.
[2] Aristidou, M.; Demetre, A. A note on nilpotent elements in quaternion rings over Zp,
International Journal of Algebra, 6.14, 663-666, 2012.
[3] Baez, J. The Octonions, Bulletin of the American Math. Soc., 39.2, 145-205, 2002.
[4] Schafer, R. D., An introduction to nonassociative algebras, Vol. 22, Courier Corporation,
1966.
[5] Ward, J. P., Quaternions and cayley numbers, Algebra and Applications. Vol. 403,
Springer Science and Business Media, 2012.
[6] Miguel, C. J.; Serodio, R., On the structure of quaternion rings over Zp, International
Journal of Algebra, 5.27, 1313-1325, 2011.
[7] Conway, J. H.; Smith, D. A., On quaternions and octonions: their geometry, arithmetic,
and symmetry, Bulletin of the American Mathematical Society, 42, 2, 229-243, 2003.
1 Pamukkale University, Denizli, TURKEY, [email protected] 2 Pamukkale University, Denizli, TURKEY, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
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[8] Okubo, S., Introduction to octonion and other non-associative algebras in physics, Vol.2,
Cambridge University Press, 1995.
[9] Smith, J. Dh., An Introduction to quasigroups and their representations, CRC Press, 2006.
[10] Dickson, L. E., On quaternions and their generalization and the history of the eight
square theorem, Annals of Mathematics, 155-171,1919..
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
16
Numerical solutions of mKdV equation via modified cubic B-spline based Differential
Quadrature Method
Ali BAŞHAN1
Abstract. In this study, an effective approach is used to obtain numerical solutions of
the third order non-linear modified Korteweg-de Vries (mKdV) equation by using differential
quadrature method based on modified cubic B-spline functions. Single soliton, interaction of
double solitons, evolution of solitons, and Train of the solitons test problems have been
studied. To check the accuracy of the method, error norms L2, and L∞ and the three lowest
invariants I1, I2 and I3 are calculated, and compared with some earlier works. Stability analysis
is also investigated.
Keywords. Differential quadrature method, mKdV equation, Strong stability-
preserving Runge-Kutta method.
AMS 2010. 65D07, 65M99, 65L06, 65L07.
References
[1] Ablowitz, M. J. and Clarkson, P. A., Solitons, nonlinear evolution equations and inverse
Scattering, Cambridge University Press, Cambridge, 1991.
[2] Ablowitz, M. J. and Segur, H., Solitons and inverse acattering transform, SIAM,
Philadelphia, 1981.
[3] Başhan, A., Karakoç, S. B. G., Geyikli, T., Approximation of the KdVB equation by the
quintic B-spline differential quadrature method, Kuwait Journal of Science, 42(2),67-92,
2015.
[4] Başhan, A., Uçar, Y., Yağmurlu, N. M., Esen, A. Numerical solution of the complex
modified korteweg-de vries equation by DQM, Journal of Physics: Conference Series 766
(2016) 012028 doi:10.1088/1742-6596/766/1/012028.
1 Bulent Ecevit University, Zonguldak, TURKEY, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
17
New Bounds for the Levels and Sublevels of Algebras Obtained by the Cayley-Dickson
Process
Cristina FLAUT1
Abstract. In this talk, we present some new bounds for the level and sublevel of
algebras obtained by the Cayley-Dickson process when their level and sublevel are greater
than dimension of the algebras.
References
[1] Brown, R. B., On generalized Cayley-Dickson algebras, Pacific J. of Math., 20,3, 415-
422, 1967.
[2] Dai, Z. D., Lam, T. Y., Peng, C. K., Levels in algebra and topology, Bull. Amer. Math.
Soc., 3, 845-848, 1980.
[3] Elman, R., Lam, T. Y., Pfister forms and K-theory of fields, Journal of Algebra 23, 181–
213, 1972.
[4] Flaut, C., Isotropy of some quadratic forms and its applications on levels and sublevels of
algebras, J. Math. Sci. Adv. Appl., 12, 2, 97-117, 2011.
[5] Flaut, C., Levels and sublevels of algebras obtained by the Cayley–Dickson process, Ann.
Mat. Pura Appl., 192, 6, 1099-1114, 2013.
[6] Hoffman, D. W., Isotropy of quadratic forms over the function field of a quadric, Math. Z,
220, 3, 461-476, 1995.
[7] Hoffman, D. W., Levels of quaternion algebras, Archiv der Mathematik, 90, 5, 401-411,
2008.
[8] Karpenko, N.A., Merkurjev, A.S., Essential dimension of quadratics, Inventiones
Mathematicae, 153, 361-372, 2003.
1 Ovidius University, Constanta, ROMANIA, [email protected] ; [email protected],
http://cristinaflaut.wikispaces.com/
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
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[9] Knebusch, M., Generic splitting of quadratic forms I, Proc. London Math. Soc. 33, 65-93,
1976.
[9] Laghribi A., Mammone P., On the level of a quaternion algebra, Comm. Algebra, 29, 4,
1821-1828, 2001.
[10] Lewis, D. W., Levels and sublevels of division algebras, Proc. Roy. Irish Acad. Sect. A,
87, 1, 103-106, 1987.
[11] Lewis, D. W., Levels of quaternion algebras, Rocky Mountain J, Math. 19, 787-792,
1989.
[12] O’ Shea, J., New values for the levels and sublevels of composition algebras, preprint.
[13] O’ Shea, J., Levels and sublevels of composition algebras, Indag. Mathem., 18, 1, 147-
159, 2007.
[14] O’ Shea, J., Bounds on the levels of composition algebras, Mathematical Proceedings of
the Royal Irish Academy 110A, 1, 21-30, 2010.
[15] O’ Shea, J., Sums of squares in certain quaternion and octonion algebras, C.R. Acad.
Sci. Paris S´er. I Math, 349, 239-242, 2011.
[16] Pfister, A., Zur Darstellung von-I als Summe von quadraten in einem Korper, J. London
Math. Soc. 40, 159-165, 1965.
[17] Pumpl¨un, S., Sums of squares in octonion algebras, Proc. Amer. Math. Soc., 133, 3143-
3152, 2005.
[18] Schafer, R. D., An Introduction to Nonassociative Algebras, Academic Press, New-York,
1966.
[19] Schafer, R. D., On the algebras formed by the Cayley-Dickson process, Amer. J. Math.,
76, 435-446, 1954.
[20] Scharlau,W., Quadratic and Hermitian Forms, Springer Verlag, 1985.
[21] Tignol, J.-P., Vast, N., Representation de -1 comme somme de carr´e dans certain
alg`ebres de quaternions, C.R. Acad. Sci. Paris S´er. Math. 305, 13, 583-586, 1987.
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
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Some Results on Generalized Derivations and (,) Lie Ideals
Evrim GÜVEN1
Abstract. Let R be a prime ring with characteristic not 2 and , , , , , µ,
automorphisms of R. Let h:RR be a nonzero left-generalized (, )-derivation, a,bR and
V≠0 a left (, )-Lie ideal of R. The main object in this article is to study the situations.
(1) h(I)C,µ(V), (2) ah(I)C,µ(V), (or h(I)aC,µ(V)), (3) h(V)=0, (4) h(V)a=0 or
ah(V)=0.
Keywords. Prime rings, generalized derivation, Lie Ideal.
AMS 2010. 16N60, 16W25.
References
[1] Aydın, N., Kaya, K., Some generalizations in prime rings with (𝜎,𝜏)-derivation, Doga-Tr.
J. of Math. 16, 1992.
[2] Bresar, M. On the distance of the compositions of two derivations to the generalized
derivations, Glasgow Math. J. 33, 80-93, 1991.
[3] Chang, J. C., On the identity h(x)=af(x)+g(x)b, Taiwanese J. Math, 7(1), 103-113, 2003.
[4] Güven, E., Kaya, K. and Soytürk, M., Some results on (𝜎,𝜏)-lie ideals, Math. J. Okayama
Univ. 49, 2007.
[5] Güven, E. One sided (𝜎,𝜏)-lie Ideals and generalized derivations in prime rings,5th
International Eurasian Conference on Mathematical Sciences and Applications (IECMSA),
Belgrade SERBIA, 2016.
[6] Kaya, K and Aydin, N, Some results on generalized lie Ideals, A Scienti.c Journal Issued
by Jordan University for Women, Vol. 3, No. 1, 1999.
[7] Kaya, K., (𝜎, 𝜏)-Lie ideals in prime rings. An Univ Timisoara Ser Stiinte Math. 30, 1992.
[8] Mayne, J. H., Centralizing mappings of prime rings, Canad. Math. Bull, 27, 122-126,
1984.
[9] Park, K.-H. and Jung, Y.-S., Some result concerning (,)-derivations on prime rings, J.
Korea Soc. Math. Educ. Ser. B: Pure Appl. Math.,Volume 10, Number 4, 2003.
1 Kocaeli University, Kocaeli, TURKEY, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
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A Note on the Sequence of Balancing Numbers
Gül Karadeniz GÖZERİ1
Abstract. In this work, we consider balancing numbers which are the square roots of
the square triangular numbers. The definition of the sequence of balancing numbers, Binet
formula for this sequence and usefull techniques are presented. The purpose of this work is to
provide some new results on the sequence of balancing numbers.
Keywords. Balancing numbers, triangular numbers, recurrence relation.
AMS 2010. 11B37, 11B39, 11B83.
References
[1] Behera A., Panda G. K., On the square roots of triangular numbers, The Fibonacci
Quarterly, 37, 2, 98-105, 1999.
[2] Castillo R. C., A survey on triangular number, factorial and some associated numbers,
Indian Journal of Science and Technology, 9, 41, 1-7, 2016.
[3] Catarino P., Campos H., Vasco P., On some identities for balancing and cobalancing
numbers, Annales Mathematicae et Informaticae, 45, 11-24, 2015.
[4] Garge A. S., Shirali S.A., Triangular numbers, Resonance, 17,7, 672-81, 2012.
[5] Hoggatt V. E., Bicknell M., Triangular numbers, Fibonacci Quarterly, 12, 221-30, 1974.
[6] Olajos P., Properties of balancing, cobalancing and generalized balancing numbers,
Annales Mathematicae et Informaticae, 37, 125-138, 2010.
[7] Panda G.K., Some fascinating properties of balancing numbers, Proceedings of the
Eleventh International Conference on Fibonacci Numbers and their Applications, Congr.
Numer. 194 , 185-189, 2009.
1Istanbul University, Istanbul, TURKEY, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
21
Graph Operations of Randic Index
Gülistan Kaya GÖK1 and Şerife BÜYÜKKÖSE2
Abstract. The Randic index 𝑅(𝐺) of a graph 𝐺 whose vertex 𝑖 and 𝑗 has degree 𝑑𝑖 and
𝑑𝑗 respectively is defined by sum of the 1
√𝑑𝑖𝑑𝑗 if the vertices i and j are adjacent. We introduce
the weighted Randic index, we establish basic mathematical properties of Randic index and
we investigate its behavior under some graph operations in this paper.
Keywords. Randic index, Weighted graph, Graph operations.
AMS 2010. 05C22, 05C50.
References
[1] Alizadeh, Y., Iranmanesh, A., Doslic, T., Additively weighted harary index of some
composite graphs, Discrete Mathematics, 313, 26-34, 2013.
[2] Gutman, I., Furtula, B.(Eds.), Recent results in the theory of randic index, Univ.,
Kragujevac, 2008.
[3] Gutman, I., Furtula, B., Bozkurt, S., B., On the randic energy, Linear Algebra and its
Applications, 442, 50-57, 2014.
[4] Liu, H., Lu, M., Tian, F., On the randic index, Journal of Mathematical Chemistry, 38.3,
2005.
[5] Randic, M., On history of the randic index and emerging hostility toward chemical graph
theory, MATCH Commun. Math. Comput. Chem., 59, 5-124, 2008.
[6] Vasudev, C., Graph theory with applications, (1), New Delhi/ Indian: New Age
International Publishers, 4-5, 21, 56-57, 2006.
1 Hakkari University, Hakkari, TURKEY, [email protected] 2 Gazi University, Ankara, TURKEY, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
22
On Codes Written by Matrices Lexicographically Ordered
Mustafa ÖZKAN1 and Figen ÖKE2
Abstract. In this study, the paper obtained by authors M. Özkan and F. Öke [1] is
extended the ring 23
[ ]u
u . Certain matrices lexicographically ordered are written using
the elements of 23
[ ]u
u . The relations between the codes generate by these matrices and
Hadamard codes are described.
Key Words. Lexicographically ordered, Hadamard Codes , Codes over Ring.
AMS 2010.94B05 , 94B15.
References
[1] Özkan M. and Öke F., A relation between Hadamard codes and some special codes over
2 2u , App Mathematics and Inf. Sci. vol. 10 no 2, pp 701-704,2016.
[2] Özkan M. and Öke F. Codes defined via especial matrices over the ring and Hadamard
codes, Mathematical Sciences and Applications E-Notes, Volume 5, No :1, 93-98,2017.
[3] Krotov, D. S., 4-linear perfect codes, Diskretn. Anal. Issled. Oper. Ser.1.Vol. 7, 4,78–
90, 2000.
[4] Özkan M. and Öke F. Construction of Hadamard codes with rings, International
Workshop on Mathematical Methods in Engineering 2017 , Ankara-TURKEY 27- 29.04.2017
[5] Vermani, L. R., Elements of Algebraic Coding Theory, Chapman Hall , India., 1996.
[6] Udomkavanich P., Jitman, S., On the gray image of ( mu1 )-cyclic codes
...k k k
m
p p pu u , Int. J. Contemp. Math. Sciences, Vol.26, 4, 1265-1272, 2009.
1 Trakya University Edirne, TURKEY, [email protected] 2 Trakya University, Edirne, TURKEY, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
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About Quasi-Cyclic Codes over the Field
Mustafa ÖZKAN1 and Figen ÖKE2
Abstract. In this paper quasi-cyclic codes over the field p are studied and they are classified
according to their lengths for each prime number p , using the theories obtained before us for 2p
and 3p .
Key Words. Gray Map, Quasi-cyclic Codes , Finite Field.
AMS 2010. 94B15 , 94B60.
References
[1] Özkan M. and Öke F., Some Special Codes Over 2
3 3 3 3v u u , Mathematical Sciences
and Applications E-Notes. vol. 4 no 1, pp 40-44, 2016.
[2] Özkan M. and Öke F., Gray images of (1 + v)-constacyclic codes over a particular ring, Palestine
Journal of Mathematics, Vol. 6, pp : 241-245.,2017
[3] F.J. MacWilliams, N.J.A. Sloane, The Theory of Error Correcting Codes, (North- Holland
Publishing Company, 1977)
[4] S.Karadeniz, B.Yıldız, On (1+ v) -constacyclic codes 2 2 2 2u v uv , Journal of the
Franklin Institude , 348, 2625-2632, 2011
[5] S. Roman, Coding and Information Theory, (Springer Verlag, 1992)
1 Trakya University, Edirne, TURKEY, [email protected] 2 Trakya University, Edirne, TURKEY, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
24
On Certain Type of Band Matrices with the Generalized Fibonacci Numbers
Neşe ÖMÜR1 and Cemile Duygu ÇOLAK2
Abstract. In this study, considering the technique used in [2], we show their arbitrary
powers for band matrices, whose entries are expressed in the terms of generalized Fibonacci
numbers tnU and tnV , where 0n , 0t .
Keywords. The generalized Fibonacci numbers, band matrices.
AMS 2010. 11B39, 15A15, 15A36.
References
[1] Kılıç, E., Stanica, P., Factorizations and representations of second order linear
recurrences with indices in arithmetic progressions, Bulletin of the Mexican Mathematical
Society, 15, 1, 23-36, 2009.
[2] Hirasaka, M., Komatsu, T., On computation of arbitrary integer Powers for certain type of
band matrices with Fibonacci numbers, Journal of Combinatorics and Number Theory, 7, 1,
65-78, 2015.
[3] Rimas, J., On computing of arbitrary positive integer Powers for one type of symmetric
tridiagonal matrices of even order-I, Applied Mathematics and Computation, 168, 783-787,
2005.
1 Kocaeli University, Kocaeli, TURKEY, [email protected] 2 Kocaeli University, Kocaeli, TURKEY, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
25
On Generalization of Quaternions
Serpil HALICI1 and Adnan KARATAŞ 2
Abstract. In this study, generalized Horadam quaternions are defined. These
quaternions generalize the well-known quaternions such as Fibonacci, Lucas, Pell, Jacobsthal,
Pell-Lucas, Jacobsthal-Lucas quaternions. For these new quaternions, we obtain the Binet
formulas that generalize the Binet formulas of all special quaternions.
Keywords: Quaternions, Horadam Sequences, Generalized Fibonacci Sequence.
AMS: 11B37, 11B39, 11B83, 16G37.
References
[1] Catarino, P., A note on h (x)− fibonacci quaternion polynomials, Chaos, Solitons&
Fractals 77, 1-5, 2015.
[2] Çimen, C. Bolat and Ipek A., On Pell quaternions and Pell-Lucas quaternions, Advances
in Applied Clifford Algebras 26.1, 39-51, 2016.
[3] Flaut, C., and Savin, D., Quaternion algebras and generalized fibonacci–lucas
quaternions, Advances in Applied Clifford Algebras 25.4, 853-862, 2015.
[4] Halici, S., On fibonacci quaternions, Advances in Applied Clifford Algebras 22.2, 321-
327, 2012.
[5] Haukkanen, Pentti., A note on Horadam's sequence, Fib. Quart.. 40.4, 358-361, 2002.
[6] Horadam, A. F., Complex fibonacci numbers and fibonacci quaternions, The American
Mathematical Monthly 70.3, 289-291, 1963.
[7] Horadam, A. F., A generalized fibonacci sequence, The American Math. Mont. 68.5, 455-
459, 1961.
[8] Horadam, A. F., Complex fibonacci numbers and fibonacci quaternions, The American
Mathematical Monthly 70.3, 289-291, 1963.
1 Pamukkale University, Denizli, TURKEY, [email protected] 2 Pamukkale University, Denizli, TURKEY, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
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[9] Horadam, A. F., Special properties of the sequence Wn (a, b; p,q), Fib. Quart.5(5), 424-
434, 1967.
[10] Horadam, A. F., Quaternion recurrence relation, Ulam Quarterly 2.2, 23-33, 1993.
[11] Horzum, T. and E. G. Kocer., On some properties of Horadam polynomials, Int. Math.
Forum. Vol. 4. No. 25. 2009.
[12] Larcombe, P., O. Bagdasar, and E. Fennessey., Horadam sequences: a survey, Bulletin
of the ICA 67, 49-72, 2013.
[13] Lounesto, P., Clifford algebras and spinors, vol. 286. Cambridge university press,
Cambridge, 2001.
[14] Polatli, E., and Seyhun K., On quaternions with generalized fibonacci and lucas number
components, Advances in Difference Equations 1, 2015
[15] Swamy, M. N. S., On generalized fibonacci quaternions, Fibonacci Quart.11(5), 547-
549, 1973.
[16] Szynal-Liana, Anetta, and Iwona Włoch., The pell quaternions and the pell octonions,
Advances in Applied Clifford Algebras 26.1, 435-440, 2016.
[17] Szynal-Liana, A., and Iwona W., A note on jacobsthal quaternions, Advances in Applied
Clifford Algebras 26, 441-447, 2016.
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
27
Generalized Quaternions and Dual Fibonacci Octonions
Serpil HALICI1 and Adnan KARATAŞ2
Abstract. In this study, we investigate the dual Fibonacci quaternions and then we
define the dual Fibonacci octonions. We give some algebraic properties of these octonions.
Moreover, we give the Binet formulas and the generating functions of them.
Key words: Recurrence Relations, Quaternions, Octonions .
2000 Mathematics Subject Classification. 11B37, 11B39, 20G20.
References
[1] Horadam, A. F., Complex fibonacci numbers and fibonacci quaternions, Amer.Math.
Monthly, 70, 289,291, 1963.
[2] Horadam, A. F., Quaternion recurrence relations, Ulam Quat. 2, 23-33, 1993.
[3] Iakin, A. L., Generalized Quaternions of higher order, The Fib. Quarterly,15, 343-346,
1977.
[4] Iakin, A. L., Generalized quaternions with quaternion components, The Fib.Quarterly, 15,
350-352, 1977.
[5] Iyer, M. R., A note on fibonacci quaternions, The Fib. Quarterly, 3,225-229, 1969.
[6] Swamy, M. N. S., On generalized fibonacci quaternions, The Fib. Quarterly,5, 547-550,
1973.
[7] Koshy, T., Fibonacci and lucas numbers with applications, A Wiley-Interscience
publication, U.S.A, 2001.
[8] Sangwine, S. J., Ell, T. A. and Bihan, N. L., Fundamental represantations and algebraic
properties of biquaternions or complexified quaternions, Adv. Appl. Clifford Algebras, 21,
607-636, 2011.
1 Pamukkale University, Denizli, TURKEY, [email protected] 2 Pamukkale University, Denizli, TURKEY, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
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[9] Ward, J. P., Quaternions and cayley numbers: algebra and applications, volume 403 of
Mathematics and Its Applications. Kluwer, Dordrecht, 1997.
[10] Kuipers, J. B., Quaternions and rotation sequences, Princeton University, Press,
Princeton, New Jersey, 1999.
[11] Kantor, I. L. and Solodnikov, A. S., Hypercomplex numbers, an elementary introduction
to algebras, Springer-Verlag, New York, 1989.
[12] Halıcı, S., On complex fibonacci quaternions, Advances in Applied Clifford Algebras,
105-112 pp., 2013.
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
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Formulæ for Two Weighted Binomial Identities with the Falling Factorials
Sibel Koparal1, Neşe Ömür2 and Emrah Kılıç3
Abstract. In this study, we will give closed formulæ for weighted and alternating weighted
binomial sums with the generalized Fibonacci and Lucas numbers including both falling
factorials and powers of indices. Furthermore we will derive closed formulæ for weighted
binomial sums including odd powers of the generalized Fibonacci and Lucas numbers.
Keywords. Binary linear recurrences, binomial sums, closed formula, operator.
AMS 2010. 05A19, 11B37, 11B39.
References
[1] Khan, M. and Kwong, H., Some binomial identities associated with the generalized
natural number sequence, The Fibonacci Quarterly, 49, 1, 57-65, 2011.
[2] Kılıç, E., Ulutaş, Y.T. and Ömür, N., Formulas for weighted binomial sums with the
powers of terms of binary recurrences, Miskolc Math. Notes, 13, 1, 53-65, 2012.
[3] Kılıç, E., Ömür, N. and Ulutaş, Y.T., Alternating sums of the powers of Fibonacci and
Lucas numbers, Miskolc Math. Notes, 12, 1, 87-103, 2011.
1 Kocaeli University, Kocaeli, TURKEY, [email protected] 2 Kocaeli University, Kocaeli, TURKEY, [email protected] 3 TOBB University, Ankara, TURKEY,[email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
30
S3–graded iso(1,2)
Yasemen UÇAN1 , Reşat KÖŞKER2 and Ramazan TEKERCİOĞLU3
Abstract. Lie algebras and groups have an important role in physics, engineering and
applied mathematics. For example, a great majority of symmetries are expressed by Poincare
symmetries [1]. Inspired by this, we intend to construct Fractional supersymmetric (1,2)iso
algebra in this study. There are other approaches for Fractional Supersymmetry in the
literature [2, 3, 4]. However, in our study, we obtained 3S –graded (1,2)iso algebra using the
method in [4], which is consistent with the co-algebra structure. Firstly, we have introduced
3
1,2(1,2) (1,2)ISO SO R group, which is the semi-direct product of (1,2)SO and 3
1,2R
groups in three dimensional space. Here, (1,2)SO group consists of matrices preserving the
quadratic form 2 2 2 2
1 2 3x x x R in Pseudo Euclid space and 3
1,2R translations group in 3IR
space [5]. Then, to arrive 3S - graded (1,2)iso algebra one adds new elements Q to
generators jX of the corresponding (1,2)U iso algebra where , j
jQ Q Q b X . We
denoted this algebra with 3 (1,2)NU iso .
Keywords. Poincaré symmetries, Fractional Supersymmetry, Semi-direct product.
References
[1] Goze, M., Raush deTraunbenberg, M. and Tanasa, A J. Math. Phys., 48, 093507-1-
093507-24, 2007.
[2] Ahn, C., Bernard, D. and Leclair, A. Nucl. Phys. B, 346, 409-439, 1990.
[3] Ahmedov, H. and Dayi, O. F. Mod. Phys. Lett. A, 15, 1801-18111, 2000.
[4] Ahmedov, H., Yildiz, A., and Ucan, Y., J. Phys. A:Math.Gen., 34, 6413-6423, 2001.
1 Yildiz Technical University, İstanbul, TURKEY, [email protected] 2 Yildiz Technical University, İstanbul, TURKEY, [email protected] 3 Yildiz Technical University, İstanbul, TURKEY, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
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[5] Vilenkin, N. Ya. And Klimyk, A. U., Representations of Lie Groups and Special
Functions, Kluwer Academic, Dordrecht, Netherlands, 1991.
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
32
On Multiplicative Generalized Derivations in 3-Prime Near-Rings
Zeliha BEDİR 1 and Öznur GÖLBAŞI 2
Abstract. An additively written group equipped with a binary operation
, such that and for all
is called a left near-ring. Any ring is a near-ring, hence a near-ring is a generalization of a
ring. A near-ring is said to be prime if implies or . An additive
map 𝑑: 𝑁 → 𝑁 is called a derivation if 𝑑(𝑥𝑦) = 𝑥𝑑(𝑦) + 𝑑(𝑥)𝑦 holds for all 𝑥, 𝑦 ∈ 𝑁. In the
present study, we deal with the generalizations of the notion of derivation on 3-prime near-
ring 𝑁.
Keywords. Near-ring, derivation, multiplicative generalized derivation
AMS 2010. 16Y30, 16W25.
References
[1] Bell, H. and Mason, G., On derivations in near rings, near rings and near Öelds,
NorthHolland Mathematical Studies, 137, 31-35, 1987.
[2] Bell, H. E., On derivations in near-rings II, Kluwer Academic Pub. Math. Appl., Dordr.,
426, 191-197, 1997.
[4] Daif, M. N. , Bell, H. E., Remarks on derivations on semiprime rings, Int. J. Math. Math.
Sci., 15(1), 205-206, 1992.
[5] Daif, M. N. and Tammam El-Sayiad, M. S., Multiplicative generalized derivations which
are additive, East-west J. Math., 9 (1), 31-37, 2007.
[6] Bedir, Z., Gölbaşı, Ö, Notes on prime near rings with multiplicative derivation,
Cumhuriyet University Faculty of Science, Science Journal (CSJ), Vol. 38, No. 2,355-363,
2017.
[8] Kamal, A. M. and Al-Shaalan, K. H., Existence of derivations on near-rings, Math.
Slovaca, 63, no:3, 431-438, 2013.
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
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ANALYSIS
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
34
f -Lacunary statistical convergence of order for double sequences
Abdulkadir KARAKAŞ1, Hıfsı ALTINOK2 and Yavuz ALTIN3
Abstract. New concepts of f -statistical convergence for double sequences of order
and strongly f -Cesàro summability for double sequences of order are introduced for
sequences of (complex or real) numbers. Futhermore we give the relationship between the
spaces 2
,0( , )w f , 2 ( , )w f and
2
, ( , )w f . Furthermore we expressed that the properties of
the strong f -Cesàro summability of order which is related to strong f -Cesàro
summability of order . Also some relations between f -statistical convergence of order
and strongly f -Cesàro summability of order are given.
Keywords. Double sequences, Statistical convergence, Cesàro summability.
AMS 2010. 40A05, 40C05, 46A45.
References
[1] Aizpuru, A., Listán-García, M. C.; Rambla-Barreno, F. Density by moduli and statistical
convergence, Quaest. Math., 37, 4, 525-530, 2014.
[2] Aizpuru, A., Listán-García, M. C.; Rambla-Barreno, F. Double density by moduli and
statistical convergence, Bull. Belg. Math. Soc. Simon Stevin., 19, 4, 663-673, 2012.
[3] Altin, Y., Altinok, H.; Çolak, R., Statistical convergence of order for difference
sequences,. Quaest. Math. 38, 4, 505-514, 2015.
[4] Bhardwaj, Vinod K., Dhawan, Shweta, f-statistical convergence of order and strong
Cesàro summability of order with respect to a modulus, J. Inequal. Appl., 332 pp.14,
2015.
[5] Bhardwaj, Vinod K., Dhawan, Shweta. Density by moduli and lacunary statistical
convergence. Abstr. Appl. Anal., Art. ID 9365037, 11 pp, 2016.
1 Siirt University, Siirt, TURKEY, [email protected] 2 Fırat University, Elazığ, TURKEY, [email protected] 3 Fırat University, Elazığ, TURKEY, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
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Some Special Values of Vertices of Trees on the Suborbital Graphs
Ali Hikmet DEĞER1 and Ümmügülsün AKBABA2
Abstract. In this research some special vertices of paths of minimal length on
suborbital graphs are investigated. Using edge conditions for these types of graphs,
relationship with the values of vertices on graphs and some elements of special sequence
values are also described.
Keywords. Suborbital graphs, Modular group, Continued fractions.
AMS 2010. 20H10, 20H05, 05C05, 05C20.
References
[1] Jones, G. A., Singerman, D., Wicks, K., The modular group and generalized farey
graphs, London Math. Soc. Lect. Note Ser., Cambridge Univ. Press, Vol. 160, pp. 316-338,
1991.
[2] Sims, C.C., Graphs and finite permutation groups, Math. Zeitsch., Vol. 95, pp. 76-86,
1967.
[3] Akbas, M., On suborbital graphs for the modular group, Bull. London Math. Soc. 33,
647-652, 2001.
[4] Değer, A. H., Beşenk, M., Güler, B. Ö., On Suborbital graphs and related continued
fractions, Applied Mathematics and Computation, 218, 3, 746-750, 2011.
[5] Değer, A. H., Vertices of paths of minimal lengths on suborbital graphs, Filomat, 31, 3,
913-923, 2017.
[6] Sarma, R., Kushwaha, S., Krishnan, R., Continued fractions arising from 𝐹1,2, Journal of
Number Theory, Vol. 154, pp. 179-200, 2015.
1 Karadeniz Technical University, Trabzon, TURKEY, [email protected] 2 Karadeniz Technical University, Trabzon, TURKEY, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
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On Some Special Values of Fibonacci and Lucas Sequence with the Suborbital Graphs
Ali Hikmet DEĞER1 and Ümmügülsün AKBABA2
Abstract. Using the action of the modular group and a special equivalence relation on
the extended rational set, Jones, Singerman and Wicks [1] defined the suborbital graphs.
Modular group acts transitively and imprimitively on this set. So suborbital graphs have
vertices from the infty block from equivalence relation and stabilizer of this block is the
congruence subgroup of the modular group.
In the present study vertices of these suborbital graphs and relationships with
Fibonacci and Lucas sequences are investigated.
Keywords. Suborbital graphs, Modular group, Fibonacci sequence, Lucas sequence.
AMS 2010. 20H10, 20H05, 05C05, 05C20.
References
[1] Jones, G. A., Singerman, D., Wicks, K., The modular group and generalized farey
graphs, London Math. Soc. Lect. Note Ser., Cambridge Univ. Press, Vol. 160, pp. 316-338,
1991.
[2] Akbas, M., On suborbital graphs for the modular group, Bull. London Math. Soc. 33,
647-652, 2001.
[3] Değer, A. H., Beşenk, M., Güler, B. Ö., On suborbital graphs and related continued
fractions, Applied Mathematics and Computation, 218, 3, 746-750, 2011.
[4] Değer, A. H., Vertices of paths of minimal lengths on suborbital graphs, Filomat, 31, 3,
913-923, 2017.
[5] Değer A.H., Imprimitive action of the normalizer of Gamma_0(N) and suborbital graphs,
AIP Conf. Proc., vol.1676, pp.1-4, 2015.
[6] Beşenk M., Değer A.H., Güler B.Ö., An application on suborbital graphs, AIP Conf.
Proc., vol.1470, pp.187-190, 2012.
1 Karadeniz Technical University, Trabzon, TURKEY, [email protected] 2 Karadeniz Technical University, Trabzon, TURKEY, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
37
A Note on the Ambarzumyan's Theorem
Alp Arslan KIRAÇ1
Abstract. The classical Ambarzumyan’s theorem was proved under a more general
inverse assertion. Then, Yurko proved that it is enough to specify only the first eigenvalue
rather than the whole spectrum. We weaken slightly the Yurko’s conditions on the first
eigenvalue.
Keywords. Inverse spectral problems, Hill operator, Ambarzumyan theorem.
AMS 2010. 34A55, 34B30, 34L05, 47E05, 34B09.
References
[1] Ambarzumian, V., Über eine Frage der Eigenwerttheorie. Zeitschrift für Physik, 53, 690–
695, 1929.
[2] Borg, G., Eine umkehrung der Sturm-Liouvilleschen eigenwertaufgabe bestimmung der
differentialgleichung durch die eigenwerte. Acta Math. 78, 1–96, 1946.
[3] Cheng, Y. H., Wang, T.E., Wu, C.J., A note on eigenvalue asymptotics for Hill’s equation.
Appl. Math. Lett. 23(9), 1013–1015, 2010.
[4] Chern, H. H., Lawb, C.K., Wang, H.J., Corrigendum to Extension of Ambarzumyan’s
theorem to general boundary conditions. J. Math. Anal. Appl. 309, 764–768, 2005.
[5] Chern, H. H., Shen, C.L., On the n-dimensional Ambarzumyan’s theorem. Inverse
Problems 13, 1, 15–18, 1997.
[6] Freiling, G., Yurko, V.A., Inverse Sturm-Liouville Problems and Their Applications.
NOVA Science Publishers, New York, 2001.
[7] Hochstadt, H., Lieberman, B., An inverse sturm-liouville problem with mixed given data.
SIAM J. Appl. Math. 34, 676–680, 1978.
[8] Levitan, B. M., Gasymov, M.G., Determination of a differential equation by two of its
spectra. Usp. Mat. Nauk 19, 3–63, 1964.
[9] Kıraç, A. A., On the Ambarzumyan’s theorem for the quasi-periodic problem.
Anal.Math.Phys. 6, 297–300, 2016.
1 Pamukkale University, Denizli, TURKEY, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
38
[10] Veliev, O. A., Kıraç, A. A., On the nonself-adjoint differential operators with the
quasiperiodic boundary conditions. International Mathematical Forum 2, 35, 1703–1715,
2007.
[11] Yang, C. F., Huang, Z.Y., Yang, X. P., Ambarzumyan’s theorems for vectorial sturm-
liouville systems with coupled boundary conditions. Taiwanese J. Math. 14, 4, 1429–1437,
2010.
[12] Yurko, V. A., On Ambarzumyan-type theorems, Applied Mathematics Letters 26, 506–
509, 2013.
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
39
Polars in Locally Convex Cones
A. Ranjbari1 and S. Jafarizad2
Abstract. The theory of locally convex cones uses an order theoretical structure on a
cone. We consider three types of polarity from [4] and [7] and we investigate some properties
of them. Also, we prove some relations between these polars.
Keywords. Locally convex cones, polar, duality.
AMS 2010. 46A03, 46A20, 46A40
References
[1] Ayaseh, D., Ranjbari, A., Bornological Convergence in Locally Convex Cones, Mediterr.
J. Math., 13 (2016) 1921-1931.
[2] Ayaseh, D., Ranjbari, A., Bornological locally convex cones, Matematiche (Catania) 69 (2)
(2014) 267-284.
[3] Ayaseh, D., Ranjbari, A., Locally convex quotient lattice cones, Math. Nachr. 287 (10)
(2014) 1083-1092.
[4] Plotkin, G.D., A Domain-Theoretic Banach-Alaoglu Theorem, Math. Struct. Compute.
Sci. 16 (2) (2006) 299-313.
[5] Ranjbari, A., Saiflu, H., Projective and inductive limits in locally convex cones, J. Math.
Anal. Appl. 332 (2) (2007)1097 -1108.
[6] Ranjbari, A., Strict inductive limits in locally convex cones, Positivity 15 (3) (2011) 465-
471.
[7] Roth, W., Operator-valued measures and integrals for cone-valued functions, Vol. 1964
of Lecture Notes in Mathematics, Springer-Verlag, Berlin, 2009.
1 University of Tabriz, Tabriz, Iran, [email protected] (Speaker) 2 University of Tabriz, Tabriz, Iran, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
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A Sufficient Condition for Complex-Valued Summable Functions to be Absolutely
Continuous a.e.
Alp Arslan KIRAÇ1
Abstract. In this study, we consider Gasymov potantial. We prove that if a complex-
valued summable potential has the Fourier coefficients q(n)= 0 for n ≤ 0, then the potential
of Hill’s equation is absolutely continuous, a.e.
Keywords. Schrödinger operator, smoothness of potential, gap lengths.
AMS 2010. 47E05 , 58F19
References
[1] ] Hochstadt, H., Estimates on the stability intervals for the Hill’s equation, Proc. Amer.
Math. Soc. 14, 930–932, 1963.
[2] Marchenko, V. A., Sturm–Liouville operators and applications. Oper. Theory Adv. Appl.,
Vol. 22, Birkhauser, 1986.
[3] Magnus, W., Winkler, S., Hill’s Equation, Interscience, 1969.
[4] Marchenko, V.A., Ostrovskii, I. V., Characterization of the spectrum of Hill’s operator,
Mat. Sb. 97, 1975, 540–606; English transl.: Math. USSR-Sb. 26 (175).
[5] J. Pöschel, E. Trubowitz, Inverse Spectral Theory, Academic Press, 1987.
[6] Trubowitz, E., The inverse problem for periodic potentials, CPAM 30, 321–342, 1977.
[7] Kappeler, T. and Mityagin, B., Estimates for periodic and Dirichlet eigenvalues of the
Schrödinger operator, SIAM J. Math. Anal. 33, 113–152, 2001.
[8] Djakov, P., Mityagin, B., Smoothness of Schrödinger operator potential in the case of
Gevrey type asymptotics of the gaps. J. Funct. Anal. 195, 89–128, 2002.
[9] Gasymov, M. G., Spectral analysis of a class of second order nonselfadjoint differential
operators, Funksional. Anal. i Prilozhen. 14, 14–19, 1980.
1 Pamukkale University, Denizli, TURKEY, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
41
[10] Sansuc, J. J. and Tkachenko, V., Spectral properties of non-selfadjoint Hill’s operators
with smooth potentials. In: A. Boutet de Monvel and V. Marchenko (eds.), Algebraic and
Geometric Methods in Mathematical Physics. 371–385, Kluwer, 1996.
[11] Tkachenko, V. A., Spectral analysis of nonselfadjoint Hill operatör, Doklady AN SSSR,
322, 2, 248-252, 1992.
[12] Tkachenko, V., Characterization of Hill operators with analytic potentials. Integral
Equations Operator Theory 41, 360–380, 2001.
[13] Djakov, P., Mityagin, B., Spectral triangles of Schrödinger operators with complex
potentials. Selecta Math. (N.S.) 9, 495–528, 2003.
[14] Pöschel, J., Hill’s potentials in weighted Sobolev spaces and their spectral gaps, Math.
Ann. 349:433–458, 2001.
[15] Kıraç, A. A., Inverse problems associated with the Hill operator. Electron. J. Diff. Equ.
41, 1–12, 2016.
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
42
On Quasi Subordinations for Analytic and Bi-Univalent Function Class
Arzu AKGÜL1
Abstract: In this study, certain subclasses of analytic bi-univalent functions are
defined and established bounds for the coefficients for this subclass. Also several related
classes are considered and connections to earlier known results are made.
Keywords: Bi-univalent functions, bi-starlike functions, coefficient estimates.
Mathematics Subject Classification: 2010: 30C45, 30C50.
References
[1] Ali, R. M., Lee, S. K., Ravichandran, V. and Supramaniam, S., Coefficient estimates for
bi-univalent Ma-Minda starlike and convex functions, Appl. Math. Lett., 25, 344-351,2012.
[2] Brannan, D.A., Taha, T.S., On some classes of bi-univalent functions, in: S.M. Mazhar, A.
Hamoui, N.S. Faour (Eds.), Math. Anal. and Appl., Kuwait; February 18.21, 1985, in: KFAS
Proceedings Series, vol. 3, Pergamon Press, Elsevier Science Limited, Oxford, 1988, pp.
53.60. see also Studia Univ. Babe¸s-Bolyai Math. 31 (2) (1986) 70.77.
[3] Brannan, D. A. and Clunie, J. G., Aspects of comtemporary complex analysis,
(Proceedings of the NATO Advanced Study Instute Held at University of Durham:July 1-20,
1979). New York: Academic Press, 1980.
[4] Frasin, B. A. and Aouf, M. K., New subclasses of bi-univalent functions, Applied
Mathematics Letters, 24, 1569-1573, 2011.
[5] Deniz, E., Certain subclasses of bi-univalent functions satisfying subordinate conditions,
Journal of Classical analysis, 2, 1, 40-60, 2012.
[6] Jahangiri, J. M. and Hamidi, S. G., Coefficient estimates for certain classes of bi-univalent
functions, Int. J. Math. Math. Sci., ArticleID 190560, 4, 2013.
[7] Lee, S. Y., Quasi-subordinate functions and coefficient conjectures, J. Korean Math. Soc.,
12, no. 1, 43.50, 1975.
1 Kocaeli University, Kocaeli, TURKEY, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
43
[8] Kumar, S. S., Kumar, V. and Ravichandran, V., Estimates for the initial coefficients of bi-
univalent functions, Tamsui Oxf. J. Inf. and Math. Sci. 29, 4, 487-504, 2013.
[9] Levin, M., On a coefficient problem for bi-univalent functions, Proceeding of the
American Mathematical Society, 18, 63-68, 1967.
[10] Ma, W. C. and Minda, D., A unified treatment of some special classes of univalent
functions, in Proceedings of the Conference on Complex Analysis (Tianjin, 1992), 157.169,
Conf. Proc. Lecture Notes Anal. I Int. Press, Cambridge, MA.
[11] Netanyahu, E., The minimal distance of the image boundary from the orijin and the
second coefficient of a univalent function in j, Archive for Rational Mechanics and Analysis,
32, 100-112, 1969.
[12] Patil, A. B. and Naik, U. H., Estimates on initial coe¢ cients of certain subclass es of bi-
univalent functions associated with quas-subordination, Global Journal of Mathematical
Analysis, 5, 2, 6-10, 2017.
[13] Ch. Pommerenke, Univalent functions, Vandenhoeck and Rupercht, Göttingen, 1975.
[14] Ren, F. Y., Owa, S. and Fukui, S., Some inequalities on quasisubordinate functions, Bull.
Aust. Math. Soc., 43, 2, 317-324, 1991.
[15] Robertso, M. S., Quasi-subordination and coefficient conjecture, Bull. Amer. Math.
Soc., 76, 1.9, 1970.
[16] Srivastava, H. M., Mishra, A. K., and Gochhayat, P., Certain subclasses of analytic and
bi-univalent functions, Applied Mathematics Letters, 23, 10, 1188-1192, 2010.
[17] Srivastava, H. M., Joshi, S. B., Joshi, S. S., Pawar, H., Coefficient estimates for certain
subclasses of meromorphically bi-univalent functions, Palest. J. Math., 5 (Special Issue: 1),
250-258, 2016.
[18] Xu, Q.-H.,Gui, Y.-C. and Srivastava, H. M., Coincident estimates for a certain subclass
of analytic and biunivalent functions, Appl. Math. Lett. 25, 990.994, 2012.
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
44
Quadratic Time-Frequency Representations on Mixed Lorentz Type Modulation Spaces
Ayşe SANDIKÇI1
Abstract. In this work we investigate the continuity properties of a class of time
frequency representations which is based on the short-time Fourier transform on the Lorentz
mixed normed modulation space dM P,Q which is the set of all tempered distributions
df S such that the short-time Fourier transform gV f of f is in the Lorentz mixed norm
space 2dL P,Q , where dg S is a non-zero window function, 1 21 P p ,p
and 1 21 Q q ,q . Some key references are given below.
Keywords. Lorentz mixed normed space, Lorentz mixed normed modulation space,
time frequency representations.
AMS 2010. 42B10, 47B38.
References
[1] Boggiatto, P., De Donno, G. and Oliaro, A., Time-frequency representations of Wigner
type and pseudodiffererential operators, Trans. Amer. Math. Soc., 362, 4955-4981, 2010.
[2] Cordero, E. and Gröchenig, K., Time-frequency analysis of localization operators, J.
Funct. Anal., 205, 1, 107-131, 2003.
[3] Fernandez, D. L., Lorentz spaces, with mixed norms, J. Funct. Anal., 25, 128-146, 1, 1977.
[4] Gröchenig, K., Foundation of time-frequency analysis. Birkhäuser, Boston, ISBN 0-8176-
4022-3, 2001.
[5] Hunt, R.A., On L(p,q) spaces, Extrait de L'Enseignement Mathematique, T.XII, fasc.4,
249-276, 1966.
[6] Sandıkçı, A., On Lorentz mixed normed modulation spaces, J. Pseudo-Differ. Oper. Appl.,
3, 263-281, 2012.
[7] Sandıkçı, A., Boundedness of localization operators on Lorentz mixed normed modulation
spaces, Journal of Inequalities and Applications, 2014, 430, 2014.
1 Ondokuz Mayis University, Samsun, TURKEY, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
45
Bilinear Multipliers on nM P,Q
Ayşe SANDIKÇI1
Abstract. Let m be a locally integrable function on 2n . If
2n
2πi ξ η,x
mˆ ˆB f ,g x f ξ g η m ξ,η e dξdη
defines a bounded bilinear operator from n n
1 1 2 2M P ,Q M P ,Q to n
3 3M P ,Q
m is said to be a bilinear multiplier on n
of type 1 1 2 2P ,Q ;P ,Q , where nM P,Q
is
the set of all tempered distributions nf S such that the short-time Fourier transform
gV f of f is in the Lorentz mixed norm space 2nL P,Q . The study of the some properties
of these spaces is investigated and give some examples. Some key references are given below.
Keywords. Bilinear multiplier, Lorentz mixed normed modulation space.
AMS 2010. 42A45, 42B15.
References
[1] Blasco, O., Notes on the spaces of bilinear multipliers, Revista de la Union Matematica
Argentina, 50(2), 2337, 2009.
[2] Blasco, O., Bilinear multipliers and transference, International Journal of Mathematics
and Mathematical Sciences, 4, 545-554, 2005
[3] Leeuw, K., On Lp multipliers, Annals of Mathematics, Second Series, Vol.81, No.2, 364-
379, 1965.
[4] Sandıkçı, A., On Lorentz mixed normed modulation spaces, J. PseudoDi⁄er. Oper. Appl. 3,
263-281, 2012.
[5] Villarroya, F., Bilinear multipliers on Lorentz spaces, Czechoslovak Mathematical
Journal, 58(133), 1045-1057, 2008
1 Ondokuz Mayis University, Samsun, TURKEY, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
46
Reguler Equivalents of a Measure Space
Bahaettin CENGİZ1 and Banu GÜNTÜRK2
Abstract. Integration in Lebesgue sense with respect to a regular Borel measure has
considerable advantages over integration with respect to an arbitrary positive measure, and it
becomes even more advantageous if the regular measure is also perfect. In this article, for a
given finite positive measure space (𝑋, 𝒜, 𝜇) we determine all locally compact Hausdorff
spaces 𝑌 on which there is a regular Borel measure ν with support 𝑌 such that the measure
algebras 𝒜/𝜇 and ℬ/𝜈 are isometric, (and consequently their respective 𝐿𝑝 spaces are
isometric) where ℬ denotes the Borel algebra of 𝑌. And also some partial results are obtained
in the case of an infinite measure space. As a bonus, one other thing we prove in this paper is
that for an arbitrary measure 𝜇, the space 𝐿∞(𝜇) is the topological dual of 𝐿1(𝜇) if and only if
its maximal ideal space is extremally disconnected.
Keywords. Equivalent measures, maximal ideal space, perfect measure, Hyperstonean
space.
AMS 2010. 28A25, 28C15, 46J10, 28A60.
1 Baskent University, Ankara, TURKEY, [email protected] 2 Baskent University, Ankara, TURKEY, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
47
Hilbert Matrix on Spaces of Holomorphic Functions
Boban KARAPETROVIĆ1
Abstract. We prove that the Hilbert matrix operator 𝐻 is bounded on the Besov space
𝐻𝜈𝑝,𝑞,𝛼
if and only if 0 < 𝜅𝑝,𝛼,𝜈 < 1, where 𝜅𝑝,𝛼,𝜈 = 𝜈 − 𝛼 −1
𝑝+ 1. In particular, 𝐻 is
bounded on the Bergman space 𝐴𝑝,𝛼 if and only if 1 < 𝛼 + 2 < 𝑝 and is bounded on the
Dirichlet space 𝐷𝛼𝑝 = 𝐴1
𝑝,𝛼 if and only if max−1, 𝑝 − 2 < 𝛼 < 2𝑝 − 2 (see [6]). In [4] it
was shown that the norm of the Hilbert matrix operator 𝐻 on the Bergman space 𝐴𝑝 is equal
𝜋
sin2𝜋
𝑝
, when 4 ≤ 𝑝 < ∞, and it was also conjectured that ‖𝐻‖𝐴𝑝→𝐴𝑝 =𝜋
sin2𝜋
𝑝
, when 2 < 𝑝 < 4.
Following [1] we prove this conjecture.
Keywords. Hilbert matrix, Bergman spaces, Besov spaces, Dirichlet spaces.
AMS 2010. 47B35, 47B38,30H20, 30H25.
References
[1] Božin, V., Karapetrović, B., Norm of the Hilbert matrix on Bergman spaces, submitted for
publication, 2017.
[2] Diamantopoulos, E., Hilbert matrix on Bergman spaces, Illinois Journal of
Mathematics,48, no. 3, 1067-1078, 2004.
[3] Diamantopoulos, E., Siskakis, A. G., Composition operators and the Hilbert matrix,
Studia Mathematica, 140, no. 2, 191-198, 2000.
[4] Dostanić, M., Jevtić, M., Vukotić, D., Norm of the Hilbert matrix on the Bergman and
Hardy spaces and theorem of Nehari type, J. Funct. Anal., 254, 2800-2815, 2008.
[5] Galanopoulos, P., Girela, D., Peláez, J. A., Siskakis, A. G., Generalized Hilbert operators,
Ann. Acad. Sci. Fenn. Math., 39, 231-258, 2014.
[6] Jevtić, M., Karapetrović, B., Hilbert matrix on spaces of Bergman-type, J. Math. Anal.
Appl., 453, 241-254, 2017.
1 University of Belgrade, Faculty of Mathematics, Belgrade, SERBIA, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
48
Some Relation on the Poly-Genocchi Numbers and Polynomials with a q-Parameter
Burak KURT1
Abstract. Recently, Cenkci and Komatsu defined and investigated poly-Bernoulli
numbers and polynomials with a q-parameter. They proved some relations between these
polynomials and weighted Stirling numbers of the first and second kind. Also, they gave some
relations between poly-Bernoulli polynomials a q-parameter and the poly-Cauchy
polynomials of the first and the second kind with a q-parameter. Komatsu generalized the
poly-Cauchy numbers with a q-parameter.
In this work, we define poly-Genocchi numbers and polynomials with a q-parameter.
We give a recurrence relation for the poly-Genocchi polynomials. Also, we prove some
relations and closed formulae for the poly-Genocchi numbers with a q-parameter.
Keywords. Genocchi numbers and polynomials, Bernoulli numbers and polynomials,
The Stirling numbers of the second kind, Weighted Stirling numbers of the second kind..
AMS 2010. 11B68; 11B73; 11B75.
References
[1] Arakawa, T, Kaneko M., On poly-Bernoulli numbers, Commentarıı Mathematic Univ.
Sanct. Pauli, 48, 159-167, 1999.
[2] Bayad, A, Hamahata Y., Polylogarithms and poly-Bernoulli polynomials, Kyushu. J.
Math., 65, 15-24, 2011.
[3] Jolany, H, Corcino, R. B., Explicit formula for generalization of poly-Bernoulli numbers
and polynomials with a, b, c parameters, J. of classical Analysis, 6(2), 119-135, 2015.
[4] Kaneko, M., Poly-Bernoulli numbers, Journal de Théorie des Nombres de Bardeox, 9,
221-228, 1997.
[5] Kamono, K., A formula for multi-poly-Bernoulli numbers of negative index, Kyush. J.
Math., 67, 29-37, 2013.
1 Akdeniz University, Antalya, TURKEY, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
49
The Fixed Point of the Sum of Nonlinear Operators in WC-Banach Algebras Relative to
the Weak Topology
Cesim TEMEL1and Musa ÇAKIR2
Abstract. In this work we aim to investigate the existence of solutions of
Krasnoselskii type equations for weak topology in WC-Banach algebras. We also provide
some results of Schauder and Banach fixed point theorems in WC-Banach algebras for use in
the new variant of Krasnoselskii fixed theorem in WC-Banach algebras.
Keywords. Fixed point theorem, Krasnoselskii theorem, weakly sequentially
continuous operator, weakly compact operator, WC-Banach algebras.
AMS 2010. 47H10, 47H08, 47H09.
References
[1] Arino, O., Gautier, S. , Pento, J. P., A fixed point theorem for sequentially continuous
mapping with application to ordinary differential equations, Functional Ekvac. 27 (3), 273-
279, 1984.
[2] Banas, J., Taoudi, M. A., Fixed points and solutions of operator equations for the weak
topology in Banach algebras, Taiwanese J. Math. 3, 871-893, 2014.
[3] Ben Amar, A., Chouayekh, S., Jeribi, A., Fixed point theory in a new class of Banach
algebras and application, Afr. Math. 24, 705-724, 2013.
[4] Jeribi, A., Krichen, B., Mefteh, B., Fixed point theory in WC-Banach algebras, Turk. J.
Math.40, 283-291, 2016.
[5] Krasnoselskii, M. A., Some problems of nonlinear analyasis, Amer. Math. Soc. Trans. 10
(2), 345-409, 1958.
[6] Smart, D.R., Fixed points theorems, Cambridge University Press, London, New York
1974.
1Yuzuncu Yil University, Van, TURKEY, [email protected] 2Yuzuncu Yil University, Van, TURKEY, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
50
Investigating the Center of L-weakly and M-weakly Compact Operators
E. BAYRAM1, A.W. WICKSTEAD2
Abstract. An operator T on an ordered vector space E , with positive cone E , is
called central if it is bounded by a multiple of the identity operator, i.e., there exists some
scalar 0 such that - x Tx x holds for all x E . The collection of all central
operators on E is called the centre of E and denoted by Z E . Amongst many works on the
centre, especially in the context of vector lattices, there have been attempts to identify the
centre of various spaces of regular operators. Some isometric results were given by Wickstead
showing that if E and F are Banach lattices then there is an embedding of the algebraic tensor
product Z E Z F into ,rZ L E F which is an isometry when Z E Z F is given
the injective tensor product norm. He also gives some density results regarding this, and
similar, embeddings. Also, some related results are obtained for spaces of compact and
weakly compact operators.
In this study we extend and modified known results concerning the centre of spaces of
regular (resp. weakly compact or compact) operators between two Banach lattices to the
setting of L-weakly compact and M-weakly compact operators. We also show that the L-
weakly compact, M-weakly compact and compact operators lying in the centre of a Banach
lattice coincide.
Keywords: L-weakly compact operator, M-weakly compact operator, Banach lattice, Centre
of ordered vector space.
Acknowledgement: This research was supported by The Scientific and Technological
Research Council of Turkey (TUBITAK) within the context of 2219-Post Doctoral
Fellowship Program and by the Research Foundation of Namik Kemal University (Project
No. NKUBAP.01.GA.17.108)
1 Namik Kemal University, Tekirdag, TURKEY, [email protected] 2 Queen's University, Belfast, NORTHERN IRELAND, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
51
The Generating Function of ( , )p q -Bernstein Polynomials and Their Properties Based
on ( , )p q -Calculus
Erkan AGYUZ1 and Mehmet ACIKGOZ2
Abstract. In the present paper, we have proposed a new generating function for
Bernstein polynomials based on ( , )p q -calculus. From these generating functions, we derive
some new properties and results for ( , )p q - Bernstein polynomials.
Keywords. ( , )p q -calculus, ( , )p q -Bernstein polynomials, Generating functions.
AMS 2010. 05A30, 11B68.
References
[1] Acıkgoz, M., Araci, S., On the generating function of the Bernstein polynomials, in
Proceedings of the 8th International Conference of Numerical Analysis and Applied
Mathematics (ICNAAM.10),AIP,Rhodes,Greece, March 2010.
[2] Bernstein, S., Demonstration du théoréme de Weierstrass fondeé sur la calcul des
probabilités, Communications of the Mathematical Society of Charkow. Séries 2 , vol. 13, pp.
1.2, 1912-1913.
[3] Mursaleen, M, Ansari, K.J., Khan, A., On (p; q)-analogue of Bernstein operators, Appl.
Math. Comput. 266, 874-882, 2015.
[4] Goldman R., Simeonov P., Simsek Y., Generating Functions for the q-Bernstein Bases,
SIAM Journal on Discrete Mathematics, 28(3), 1009.1025, 2014.
[5] Simsek Y., Acikgoz M., .A new generating function of q-Bernstein-type polynomials and
their interpolation function, Abstract and Applied Analysis, Article ID 769095, 12 pages,
2010.
1 Gaziantep University, Gaziantep, TURKEY, [email protected] 2 Gaziantep University, Gaziantep, TURKEY, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
52
Characterizations of Matrix and Compact Operators on the Space |𝑬𝒓𝜽|𝒑
Fadime GÖKÇE1 and Mehmet Ali SARIGÖL2
Abstract: In this study, we introduce the new spaces |𝐸𝑟𝜃|
𝑝 and |𝐸𝑟
𝜃|𝑝,∞
as the
sets of all series summable by the methods |𝐸𝑟, 𝜃𝑛|𝑝 for 1 ≤ 𝑝 < ∞ and |𝐸𝑟, 𝜃𝑛|∞
for 𝑝 = ∞ and then, we mention briefly some topological and algebraic properties of
these spaces. Furthermore, we characterize certain matrix and compact operators on those
spaces and also determine their norms and Hausdroff meausures of noncompactness.
Key Words and Phrases: Absolute Euler summability, matrix transformations,
sequence spaces, inclusion relations, norms, Hausdroff meausure of noncompactness
2010 Mathematics Subject Classification: 40C05,40D25,40F05,46A45
References
[1] Altay, B., Başar, F. and Mursaleen, M., On the Euler sequence spaces which include
the spaces 𝑙𝑝 and 𝑙∞ II, Nonlinear Anal., 65, no. 3, 707—717, 2006.
[2] Altay, B., Başar, F. and Mursaleen, M., On the Euler sequence spaces which include
the spaces 𝑙𝑝 and 𝑙∞ I”, Inform. Sci, 176, no.10 1450-1462, 2005.
[3] Boos, J. and Cass, P., Classical and modern methods in summability, Oxford
University Press, New York, 2000.I. Daubechies, Comm. Pure Appl. Math. 41, 909, 1988.
[4] Mursaleen, R. M., Applied Summability Methods, Springer, 2013.
[5] Sarıgöl, M.A., Spaces of Series Summable by Absolute Cesaro and Matrix
Operators, Comm. Math Appl., 7, no. 1, 11-22, 2016.
[6] Sarıgöl, M.A., Extension of Mazhar's theorem on summability factors, Kuwait J.
Sci., 42, no. 3, 28-35, 2015.
1 Pamukkale University, Denizli, TURKEY, [email protected] 2 Pamukkale University, Denizli, TURKEY, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
53
[7] Sarıgöl, M.A., Matrix transformations on fields of absolute weighted mean
summability, Studia Sci. Math. Hungar., 48, no. 3, 331-341, 2011.
[8] Sarıgöl, M.A., On the local properties of factored Fourier series, Appl. Math.Comp.,
216, no.11, 3386-3390, 2010.
[9] Stieglitz, M., Tietz, H., Matrix transformationen von Folgenraumen Eine
Ergebnisübersicht, Mathematische Zeitschrift, 154, no. 1, 1-16, 1977.
[10] Wilansky, A., Summability Through Functional Analysis”, Mathematics Studies. 85.
North Holland, Amsterdam, 1984.
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
54
Matrix and Compact Operators on the Absolute Fibonacci Spaces
Fadime GÖKÇE1 and Mehmet Ali SARIGÖL2
Abstract: In the present study, we introduce the absolute Fibonacci space
|𝐹𝑢|𝑘 where 𝑢 is a sequence of positive numbers and 1 ≤ 𝑘 < ∞. Then, we give some
inclusion relations, investigate topological and algebraic structure such as BK-space, 𝛼-,𝛽-, 𝛾
-duals and base. Further, we characterize certain matrix and compact operators on these
spaces, also determine their norms and Hausdroff meausures of noncompactness.
Key Words and Phrases: Absolute summability,Fibonacci numbers, matrix
transformations, sequence spaces, norms, Hausdroff meausure of noncompactness
2010 Mathematics Subject Classification: 40C05,40D25,40F05,46A45
References
[1] Boos, J.and Cass, P ., Classical and modern methods in summability, Oxford
University Press, New York, 2000.I. Daubechies, Comm. Pure Appl. Math. 41, 909, 1988.
[2] Kara, E. E., & Ilkhan, M., Some properties of generalized Fibonacci sequence spaces,
Linear and Multilinear Algebra, 64(11), 2208-2223, 2016.
[3] Kara, E. E., Some topological and geometrical properties of new Banach sequence
spaces. Journal of Inequalities and Applications, 2013(1), 38, 2013.
[4] Malkowsky, E. and Rakocevic, V., On matrix domains of triangles, Appl. Math.
Comp., 189, no. 2, 1146-1163, 2007.
[5] Malkowsky, E. and Rakocevic, V., An introduction into the theory of sequence space
and measures of noncompactness, Zb. Rad.(Beogr), 9, no. 17, 2000.
[6] Sarıgöl, M.A., Spaces of Series Summable by Absolute Cesaro and Matrix
Operators, Comm. Math Appl., 7, no. 1, 11-22, 2016.
1 Pamukkale University, Denizli, TURKEY, [email protected] 2 Pamukkale University, Denizli, TURKEY, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
55
[7] Sarıgöl, M.A., Extension of Mazhar's theorem on summability factors, Kuwait J.
Sci., 42, no. 3, 28-35, 2015.
[8] Sarıgöl, M.A., On the local properties of factored Fourier series, Appl. Math.Comp.,
216, no.11, 3386-3390, 2010.
[9] Stieglitz, M. and Tietz, H., Matrix transformationen von Folgenraumen Eine
Ergebnisübersicht, Mathematische Zeitschrift, 154, no. 1, 1-16, 1977.
[10] Wilansky, A. , Summability Through Functional Analysis, Mathematics Studies. 85.
North Holland, Amsterdam, 1984.
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
56
The Theory of 𝒏-Scales
Furkan Semih DÜNDAR1
Abstract. We provide a theory of n-scales previously called as n dimensional time scales. In
previous approaches to the theory of time scales by Bohner and Guseinov, multi-dimensional
scales were taken as product space of two time scales. n-scales make the mathematical
structure more flexible and appropriate to real world applications in physics and related fields.
Here we define an n-scale as an arbitrary closed subset of R^n. Modified forward and
backward jump operators, Δ-derivatives and multiple integrals on n-scales are defined.
1 Boğaziçi University. İstanbul, TURKEY, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
57
Some Specific Properties of Concave Functions Defined by the Generalized Srivastava-
Attiya Operator
Hasan BAYRAM1 and Şahsene ALTINKAYA2
Abstract. In this paper we investigate a class Ψ𝜇,𝑏𝑚,𝑘𝐶0(𝛼) of concave functions by
using the generalized Srivastava-Attiya operator. Such as, coefficient inequalities, distortion
bounds.
Keywords. Hadamard product, concave functions, distortion theorem, Hurwitz-Lerch
Zeta functions, multiplier transformation, Srivastava-Attiya operator.
AMS 2010. 30C45, 30C50.
References
[1] Attiya, A. A., Hakami, A. H., Some subordination results associated with generalized
Srivastava-Attiya operator, Adv. Difference Equ., 105, 2013.
[2] Choi, J. and H. M. Srivastava, Certain families of series associated with the Hurwitz-
Lerch Zeta function, Appl. Math. Comput. 170, 399—409, 2005.
[3] Lopez C. Ferreira and J. L. Lopez, Asymptotic expansions of the Hurwitz-Lerch Zeta
function, J. Math. Anal. Appl., 298, 210—224, 2004.
[4] Murugusundaramoorthy, G., Coefficient estimate of bi-Bazilevic function defined by
Srivastava-Attiya operator, 69, 2, 45-56, 2014.
[5] Srivastava H. M., Some Fox-Wright generalized hypergeometric functions and associated
families of convolution operators, Appl. Anal. Discrete Math., 1, 56—71, 2007.
1 Uludağ University, Bursa, TURKEY, [email protected] 2 Uludağ University, Bursa, TURKEY, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
58
A Note On Weighted Composition Operators on Besov-Type Spaces
Hamid Vaezi1 and Sima Houdfar 2
Abstract. Let D be the open unit disc in the complex plane C. Denote by
by H(D) the class of all analytic functions on D. Given an analytic self map ᵩ
and analytic function ᴪ in the unit disc D, the weighted composition operator
ᴪ Cᵩf on H(D) is defined by
ᴪ Cᵩf (z) =ᴪ(z) f (ᵩ (z)) , zϵD.
In this article we study the boundedness and compactness of the
Weighted composition operators between Besov- type spaces, by using
the Carleson measure.
Keywords Weighted composition operator, Besovtype space, Carleson measure
AMS 2010. 47B33, 30H25.
References
[1] Colonna, F., Li, S., , Weighted composition operators from the Besove spaces into the
Bloch spaces, Bull Malays. Math. Sci. Soc., 36, 1027-1039, 2013.
[2] Cowen, C. C., Maccluer B. D., Composition operators on spaces of analytic functions,
Studies in Advanced Math., CRC Press, Boca Raton, 1995.
[3] Hassanlou, M., Vaezi, H., Wang, M., Weighted composition operators on weak vector
valued Bergman spaces and Hardy spaces, J. Math. Anal., 9, 35-43, 2015.
[4] Shapiro, J. H., Composition operators and classical function theory, Springer Verlag,
NewYork, 1993.
1 University of Tabriz, Tabriz Iran, [email protected] 2 University of Tabriz, Tabriz Iran, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
59
Generalized Lupaş Operators
Hatice Gül İNCE İLARSLAN1, Ali ARAL2 and Gülen BAŞCANBAZ-TUNCA3
Abstract. In this work, by taking a continuously differentiable, increasing and
unbounded function ρ, we consider an extension of the Lupaş operator Ln in the form
Ln (f∘ρ⁻¹)∘ρ for convenient functions f on [0,∞). We give weighted approximation,
Voronovskaya type theorem, quantitative estimates for the local approximation.
Keywords. Generalized Lupaş operator; Weighted approximation; Voronovskaya type
theorem.
AMS 2010. 41A36, 41A25.
References
[1] Agratini, O., On a sequence of linear and positive operators. Facta Univ. Ser. Math.
Inform. No. 14, 41—48, 1999.
[2] Aral, A., Inoan, D. and Raşa, I., On the generalized Szász-Mirakyan operators. Results
Math. 65, no. 3-4, 441-452, 2014.
[3] Gadjiev A. D., Aral A., The estimates of approximation by using a new type of weighted
modulus of continuity. Comp. Math. Appl., 54, 127-135, 2007.
[4] Lupas¸ A., The approximation by some positive linear operators. In: Proceedings of the
International Dortmund Meeting on Approximation Theory (M.W. Müller et al.,eds.),Ak
ademie Verlag, Berlin, 201-229, 1995.
1 Gazi University, Ankara, TURKEY, [email protected] 2 Kırıkkale University, Kırıkkale, TURKEY, [email protected] 3 Ankara University, Ankara, TURKEY, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
60
Statistical Convergence of order (β,γ) for Sequences of Fuzzy Numbers
Hıfsı ALTINOK1 and Mikail ET2
Abstract. In this paper, we introduce the concept of statistical convergence of order
(β,γ) and strongly p-Cesàro summable of order (β,γ) for sequences of fuzzy numbers and give
some inclusion relations between them.
Keywords. Sequence of fuzzy numbers, statistical convergence, Cesàro summability.
AMS 2010. 40A05; 40A25; 40A30; 40C05; 03E72.
References
[1] Çolak, R. Statistical convergence of order α, Modern Methods in Analysis and Its
Applications, New Delhi, India: Anamaya Pub, 121—129, 2010.
[2] Et, M. and Sengul, H. Some Cesaro-Type Summability Spaces of Order alpha and
Lacunary Statistical Convergence of Order alpha, FILOMAT, 28, 8, 1593-1602, 2014.
[3] Fast, H. Sur la convergence statistique, Colloq. Math. 2, 241-244, 1951.
[4] Matloka, M. Sequences of fuzzy numbers, BUSEFAL 28, 28-37, 1986.
[5] Nuray, F. and Savaş, E. Statistical convergence of sequences of fuzzy real numbers, Math.
Slovaca 45, 3, 269-273, 1995.
[6] Savas, E. A note on sequence of fuzzy numbers, Inform. Sci. 124, 1,4, 297-300, 2000.
[7] Schoenberg, IJ. The integrability of certain functions and related summability methods,
Amer. Math. Monthly 66, 361-375, 1959.
1 Firat University, Elazığ, TURKEY, [email protected] 2 Firat University, Elazığ, TURKEY, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
61
Statistical Convergence of order β for Double Sequences of Fuzzy Numbers Defined by a
Modulus Function
Hıfsı ALTINOK1, Yavuz ALTIN2 and Mahmut IŞIK3
Abstract. In the present paper, we extend the notions of statistically convergence of
order β and strong Cesàro summability of order β for sequences of fuzzy numbers and
introduce the notions of f-statistically convergence of order β and strong Cesàro summability
of order β for β∈(0,1] with respect to an unbounded modulus function f for double sequences
of fuzzy numbers and give some inclusion theorems.
Keywords. Statistical convergence, Cesàro summability, Modulus function
AMS 2010. 40A05; 40A25; 40A30; 40C05; 03E72.
References
[1] Aizpuru, A.; Listan-Garcia, M. C. and Rambla-Barreno, F. Density by moduli and
statistical convergence. Quaest. Math. 37, 525-530, 2014.
[2] Bhardwaj, V. K. and Dhawan, S. f-statistical convergence of order α and strong Cesaro
summability of order α with respect to a modulus, J. Inequal. Appl. 2015:332 DOI
10.1186/s13660-015-0850-x, 2015.
[3] Colak, R. and Altin, Y. Statistical convergence of double sequences of order α, J. Funct.
Space. Appl. 2013.
[4] Et, M. and Sengul, H. Some Cesaro-Type Summability Spaces of Order alpha and
Lacunary Statistical Convergence of Order alpha, FILOMAT, 28, 8, 1593-1602, 2014.
[5] Fast, H. Sur la convergence statistique, Colloq. Math. 2, 241-244, 1951.
[6] Matloka, M. Sequences of fuzzy numbers, BUSEFAL 28, 28-37, 1986.
[7] Savaş, E. A note on double sequences of fuzzy numbers, Tr. J. of Mathematics, 20, 175—
178, 1996.
1 Firat University, Elazığ, TURKEY, [email protected] 2 Firat University, Elazığ, TURKEY, [email protected] 3 Firat University, Elazığ, TURKEY, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
62
Δnm-lacunary Statistical Convergence of Order α
Hıfsı ALTINOK 1, Mikail ET2 and Mahmut IŞIK 3
Abstract. The purpose of this work is to introduce the concepts of Δnm-lacunary
statistical convergence of order α and Δnm -lacunary strongly convergence of order α. We
establish some connections between Δnm -lacunary strongly convergence of order α and Δn
m -
lacunary statistical convergence of order α. It is shown that if a sequence is Δnm -lacunary
strongly convergent of order α then it is Δnm -lacunary statistically convergent of order α.
Keywords. Difference sequence, statistical convergence, lacunary sequence.
AMS 2010. 40A05, 40C05, 46A45.
References
[1] Çolak, R., Statistical convergence of order α, Modern Methods in Analysis and Its
Applications, NewDelhi, India: Anamaya Pub, 121—129, 2010.
[2] Et, M. and Colak, R. : On some generalized difference sequence spaces, Soochow J. Math.
21, 4, 377-386, 1995.
[3] Freedman, A. R. Sember, J. J. and Raphael, M. : Some Cesàro-type summability spaces,
Proc. Lond. Math. Soc. 37 (1978), 508-520.
[6] Fridy, J. A. : On the statistical convergence, Analysis 5 (1985), 301 - 313.
[5] Fridy, J. A. and Orhan, C. : Lacunary statistical convergence, Pacific J. Math. 160 (1993),
43-51.
[4] Kizmaz, H. : On certain sequence spaces, Canadian Math. Bull. 24 (1981 ), 169-176.
[5] Tripathy, B. C. ; Esi, A. and Tripathy, B. K. On a new type of generalized difference
Cesaro Sequence spaces, Soochow J. Math. 31:3 (2005), 333-340.
1 Fırat University, Elazığ, TURKEY, [email protected] 2 Fırat University, Elazığ, TURKEY, [email protected] 3 Fırat University, Elazığ, TURKEY, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
63
Generalized Statistical Convergence of order β for Sequences of Fuzzy Numbers
Hıfsı ALTINOK1, Abdulkadir KARAKAŞ2 and Yavuz ALTIN3
Abstract. In the present paper, we introduce the concepts of ∆m-statistical
convergence of order β for sequences of fuzzy numbers and strongly ∆m-summable statistical
convergence of order β for sequences of fuzzy numbers by using a modulus function f and
taking supremum on metric d for 0 < β ≤ 1 and give some inclusion relations between them.
Keywords. Sequence of fuzzy numbers, statistical convergence, difference sequence.
AMS 2010. 40A05; 40A25; 40A30; 40C05; 03E72.
References
[1] Çolak, R. Statistical convergence of order α, Modern Methods in Analysis and Its
Applications, New Delhi, India: Anamaya Pub, 121-129, 2010.
[2] Et, M. and Çolak, R. On some generalized difference sequence spaces, Soochow J. Math.
21, 4, 377-386, 1995.
[3] Kızmaz, H. On certain sequence spaces, Canad. Math. Bull. 24, 2, 169-176, 1981.
[4] Matloka, M. Sequences of fuzzy numbers, BUSEFAL 28, 28-37, 1986.
[5] Nuray, F. and Savaş, E. Statistical convergence of sequences of fuzzy real numbers, Math.
Slovaca 45, 3, 269-273, 1995.
[6] Savas, E. A note on sequence of fuzzy numbers, Inform. Sci. 124, 1,4, 297-300, 2000.
[7] Schoenberg, IJ. The integrability of certain functions and related summability methods,
Amer. Math. Monthly 66, 361-375, 1959.
[8] Srivastava, P. D. and Mohanta, S. Statistical convergence of generalized difference
sequence space of fuzzy numbers. Taiwanese J. Math. 17, 5, 1659-1676, 2013.
1 Firat University, Elazığ, TURKEY, [email protected] 2 Siirt University, Elazığ, TURKEY, [email protected] 3 Firat University, Elazığ, TURKEY, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
64
A New Application of Quasi Power Increasing Sequences
Hikmet SEYHAN ÖZARSLAN1
Abstract. In the present paper, we prove a general theorem dealing with absolute
matrix summability methods of infinite series by using the concept of quasi power
increasing sequence instead of almost increasing sequence. Some important known theorems
are deduced from our theorems.
Keywords. Riesz mean, summability factor, absolute matrix summability, quasi
power increasing sequences, infinite series, Hölder inequality, Minkowski inequality.
AMS 2010. 26D15, 40D15, 40F05, 40G99.
References
[1] Bari, N. K., Stečkin, S. B., Best approximations and differential properties of two
conjugate functions, Trudy Moskov. Mat. Obšč., 5, 483-522, 1956 (in Russian).
[2] Bor, H., On two summability methods, Math. Proc. Cambridge Philos. Soc., 97, 147-149,
1985.
[3] Bor, H., On absolute Riesz summability factors, Adv. Stud. Contemp. Math. (Pusan), 3(2)
23-29 , 2001.
[4] Hardy, G. H., Divergent Series, Oxford University Press, Oxford, 1949.
[5] Leindler, L., A new application of quasi power increasing sequences, Publ. Math.
Debrecen., 58, 791-796, 2001 .
[6] Özarslan, H. S., Kandefer, T., On the relative strength of two absolute summability
methods, J. Comput. Anal. Appl., 11, 576-583, 2009.
[7] Özarslan, H. S., A new application of absolute matrix summability, C. R. Acad. Bulgare
Sci., 68, 967-972, 2015.
[8] Sulaiman, W. T., Inclusion theorems for absolute matrix summability methods of an
infinite series. IV, Indian J. Pure Appl. Math., 34 (11), 1547-1557, 2003.
This work was supported by Research Fund of the Erciyes University, Project Number: FDK-2017-6945 1 Erciyes University, Kayseri, TURKEY, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
65
Almost Increasing Sequences and Their New Applications
Hikmet SEYHAN ÖZARSLAN1 and Bağdagül KARTAL2
Abstract. In [3], Bor has proved a theorem concerning absolute summability of
infinite series. In the present paper, this known theorem has been generalized to absolute
matrix summability by using matrix transformations and almost increasing sequences. Some
results have also been obtained.
Keywords. Matrix transformations, absolute matrix summability, almost increasing
sequences, infinite series, Hölder inequality, Minkowski inequality.
AMS 2010. 26D15, 40D15, 40F05, 40G99.
References
[1] Bari, N. K., Stečkin, S. B., Best approximations and differential properties of two
conjugate functions, Trudy Moskov. Mat. Obšč., 5, 483-522, 1956 (in Russian).
[2] Bor, H., On two summability methods, Math. Proc. Cambridge Philos. Soc., 97, 147-149,
1985.
[3] Bor, H., A note on absolute Riesz summability factors, Math. Inequal. Appl., 10, 619-625,
2007.
[4] Hardy, G. H., Divergent Series, Oxford University Press, Oxford, 1949.
[5] Mazhar, S. M., A note on absolute summability factors, Bull. Inst. Math. Acad. Sinica.,
25, 233-242, 1997.
[6] Özarslan, H. S., A new application of almost increasing sequences, Miskolc Math. Notes.,
14, 201-208, 2013.
[7] Özarslan, H. S., A new application of generalized almost increasing sequences, Bull.
Math. Anal. Appl., 8, 9-15, 2016.
[8] Özarslan, H. S., Karakaş, A., A new result on the almost increasing sequences, J. Comp.
Anal. Appl., 22, 989-998, 2017.
This work was supported by Research Fund of the Erciyes University, Project Number: FDK-2017-6945 1 Erciyes University, Kayseri, TURKEY, [email protected] 2 Erciyes University, Kayseri, TURKEY, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
66
Wavelet-Type Transforms Generated by Some Semigroups and Their Applications
İlham A. ALIEV1
Abstract. We introduce wavelet-type transforms generated by some “wavelet
measures” and so-called “modified beta-semigroups”. These semigroups are natural
generalizations of the Gauss-Weierstrass and Abel-Poisson semigroups associated with the
Laplace-Bessel differential operator. We introduce also a notion of bi-parametric potential-
type operators, which generalize the Bessel and Flett potentials associated with the Laplace-
Bessel differential operator. Explicit inversion formulae for these bi-parametric potential-type
operators are obtained by making use of the aforementioned wavelet-type transforms.
Keywords. Generalized Bessel potentials, wavelet transforms, Gauss-Weierstrass
semigroup, Poisson semigroup, inversion formulas
AMS 2010. Primary 26A33; Secondary 42C40, 44A35, 45P05
1 Akdeniz University, Antalya, TURKEY; [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
67
Higher Order Fractional Boundary Value Problems with Integral Boundary Conditions
İsmail YASLAN1
Abstract. In this paper, we investigate the existence of positive solutions for higher
order multi-point nonlinear fractional boundary value problems with integral boundary
conditions. We establish the criteria for the existence of at least one, two and three positive
solutions for higher order m-point nonlinear fractional boundary value problems with integral
boundary conditions by using a result from the theory of fixed point index, Avery-Henderson
fixed point theorem and the Legget-Williams fixed point theorem, respectively.
Keywords. Boundary value problems, cone, fixed point theorems, positive solutions,
Riemann-Liouville fractional derivative, integral boundary conditions
AMS 2010. 26A33, 34B10, 34B18.
References
[1] Anco, S., Wald, R., Does there exist a sensible quantum theory of an algebravalued scalar
field, Phys. Rev., 39, 2297-2307, 1989.
[2] Avery, R. I., Henderson, J., Two positive fixed points of nonlinear operators on ordered
Banach spaces, Comm. Appl. Nonlinear Anal., 8, 27-36, 2001.
[3] Guo, D., Lakshmikantham, V., Nonlinear Problems in Abstract Cones, Academic Press,
San Diego, 1988.
[4] Lan, K. Q., Multiple positive solutions of semilinear differential equations with
singularities, J. London Math. Soc., 63, 690-704, 2001.
[5] Legget, R.W., Williams, L.R., Multiple positive fixed points of nonlinear operators on
ordered Banach space, Indiana Univ. Math. J., 28, 673-688, 1979.
[6] Sabatier, O.P., Agrawal, J.A., Machado, T., Advances in Fractional Calculus, Springer,
Dordrecht, The Netherlands, 2007.
[7] Samko, S.G., Kilbas, A.A., Marichev, O.I., Fractional Integral and Derivatives: Theory
and Applications, Gordon and Breach, Yverdon, Switzerland, 1993.
1 Pamukkale University, Denizli, TURKEY, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
68
Boundedness of Products of Weighted Composition Operators and Differentiation
Operators between Weighted Bergman Spaces and Weighted Zygmund Spaces of
Analytic Functions
Jasbir S. MANHAS1
Abstract. Let v and w be weights on the unit disc D. Let Av,p(D) be the weighted
Bergman space of analytic functions and Zw(D) be the weighted Zygmund space of analytic
functions. In this paper, we investigate the analytic mappings φ : D →D and ψ :D→₵ which
characterize the boundedness of products of weighted composition operators and
differentiation operators 𝐷𝑊𝜓,φ and 𝑊𝜓,φ𝐷 between the weighted Bergman spaces Av,p(D)
and Zygmund spaces Zw(D).
Keywords. Weighted Bergman Spaces, Weighted Zygmund Spaces, Weighted
Composition Operators, Differentiation Operators, Bounded Operators.
AMS 2010. 47B38, 47B33.
1 Sultan Qaboos University, Muscat, OMAN, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
69
The Vitali Convergence Theorem for Nonlinear Integrals
Jun KAWABE1
Abstract. The Choquet [1] integral and the Sugeno integral [5] for a measurable
function with respect to a nonadditive (also called monotone) measure may be considered as
nonlinear functionals and are widely used in application areas such as decision theory under
uncertainty, game theory, data mining and others. For those functionals, their continuity
corresponds to the convergence theorem of integrals, which means that the limit of the
integrals of a sequence of functions is the integral of the limit function. Thus many attempts
have been made to formulate the counterparts of the monotone, the bounded, and the
dominated convergence theorems for the Choquet, the Šipoš [2], the Sugeno, and the Shilkret
integrals [3].
The purpose of this talk is to present the Vitali convergence theorem for such
nonlinear integrals. A key ingredient is a perturbation of integral that manages the small
change of the integral value arising as a result of adding small amounts to a measure and an
integrand [2].
Keywords. Nonadditive measure, nonlinear integral, Vitali convergence theorem.
AMS 2010. 28A25, 28E10.
References
[1] Choquet, G., Theory of capacities, Ann. Inst. Fourier (Grenoble), 5, 131-295, 1953-1954.
[2] Kawabe, J., A unified approach to the monotone convergence theorem for nonlinear
integrals, Fuzzy Sets Syst., 304, 1-19, 2016.
[3] Shilkret, N., Maxitive measure and integration, Indag. Math., 33, 109-116, 1971.
[4] Šipoš, J., Integral with respect to a pre-measure, Math. Slovaca, 29, 141-155, 1979.
[5] Sugeno, M., Theory of fuzzy integrals and its applications, Ph.D. dissertation, Tokyo
Institute of Technology, Tokyo, 1974.
1 Shinshu University, Nagano, JAPAN, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
70
Berezin Number Inequalities and Engliš Algebras
M. GÜRDAL1, M.B. HUBAN2 and M.T. GARAYEV3
Abstract. In this paper, we consider the following natural question: when the power
inequality nn AberAber holds for any integer ?1n We solve this question in some
particular case.
Acknowledgement: This work is supported by TUBA through Young Scientist
Award Program (TUBA-GEBIP/2015).
Keywords. Berezin symbol, Berezin number, Engliš Algebras.
AMS 2010. 47A63, 47B35.
References
[1] Berezin, F.A., Covariant and contravariant symbols for operators, Math. USSR-Izv., 6,
1117-1151, 1972.
[2] Dragomir, S.S., A survey of some recent inequalities for the norm and numerical radius of
operators in Hilbert spaces, Banach J. Math. Anal., 1, 154-175, 2007.
[3] Engliś, M., Toeplitz operators and the Berezin transform on H², Linear Alg. Appl.,
223/224, 171-204, 1995.
[4] Garayev, M.T., Gürdal, M., Huban, M.B., Reproducing kernels, Engliš algebras and some
applications, Studia Mathematica, 232(2), 113-141, 2016.
[5] Hardy, G., Littlewood, J.E., Polya, G., Inequalities, 2 nd ed. Cambridge University Press,
Cambridge, 1967.
[6] Saitoh, S., Sawano, Y., Theory of reproducing kernels and applications, Springer, 2016.
1 Suleyman Demirel University, Isparta, TURKEY, [email protected] 2 Suleyman Demirel University, Isparta, TURKEY, [email protected] 3 King Saud University, Riyadh, SAUDİ ARABİA, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
71
Hadamard Type Inequalities for m-Convex and (α,m)-Convex Functions via Fractional
Integrals
Merve Avcı ARDIÇ1, Alper EKİNCİ2, Ahmet Ocak AKDEMİR3 and M. Emin ÖZDEMİR4
Abstract. In this paper, we established some new Hadamard-type integral inequalities
for functions whose derivatives of absolute values are m-convex and (α,m)-convex functions
via Riemann-Liouville fractional integrals.
Keywords. m-convex functions, (α,m)-convex functions, Riemann-Liouville
fractional integral.
AMS 2010. 26A51, 26D15.
References
[1] Bakula, M. K., Özdemir, M. E., Pečarić, J., Hadamard type inequalities for m-convex and
(α,m)-convex functions, J. Inequal. Pure Appl. Math. 9, Article 96, 2008.
[2] Bakula, M. K., Pečarić, J., Ribičić, M., Companion inequalities to Jensen's inequality for
m-convex and (α,m)-convex functions, J. Inequal. Pure Appl. Math. 7, Article 194, 2006.
[3] Dragomir, S.S., On some new inequalities of Hermite-Hadamard type for m-convex
functions, Tamkang J. Math., 3 (1) 2002.
[4] Dragomir, S.S., Agarwal, R.P., Two inequalities for differentiable mappings and
applications to special means of real numbers and to trapezoidal formula, Appl. Math. Lett.,
11 (5), 91-95,1998.
[5] Dragomir, S.S., Toader, G.H., Some inequalities for m-convex functions, Studia Univ.
Babeş-Bolyai, Math., 38 (1), 21-28, 1993.
[6] Gorenflo, R., Mainardi, F., Fractional calculus: integral and differential equations of
fractional order, Springer Verlag, Wien, 223-276, 1997.
1 Adıyaman University, Adıyaman, TURKEY, [email protected] 2 Ağrı İbrahim Çeçen University, Ağrı, TURKEY, [email protected] 3 Ağrı İbrahim Çeçen University, Ağrı, TURKEY, [email protected] 4 Uludağ University, Bursa, TURKEY, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
72
[7] Gradshteyn, I.S., Ryzhik, I.M., Table of Integrals, Series, and Products, 7th ed., Academic
Press, Elsevier Inc., 2007.
[8] Miheşan, V.G., A generalization of the convexity, Seminar of Functional Equations,
Approx. and Convex, Cluj-Napoca (Romania) , 1993.
[9] Miller S., Ross, B., An introduction to the Fractional Calculus and Fractional Differential
Equations, John Wiley and Sons, USA, p.2, 1993.
[10] Özdemir, M.E., Avcı, M., Kavurmacı, H., Hermite-Hadamard-type inequalities via
(α,m)-convexity, Comput. Math. Appl., 61, 2614-2620, 2011.
[11] Özdemir, M.E., Avcı, M., Set, E., On some inequalities of Hermite--Hadamard type via
m-convexity, Appl. Math. Lett. 23 (9), 1065-1070, 2010.
[12] Özdemir, M.E, Kavurmacı, H., Set, E., Ostrowski's type inequalities for (α,m)-convex
functions, Kyungpook Math. J. 50, 371-378, 2010.
[13] Özdemir, M. E., Set, E., Sarıkaya, M. Z., Some new Hadamard's type inequalities for co-
ordinated m-convex and (α,m)-convex functions, Hacettepe J. of. Math. and Statistics., 40,
219-229, 2011.
[14] Podlubni, I., Fractional Differential Equations, Academic Press, San Diego, 1999.
[15] Toader, G. H., Some generalisations of the convexity, Proc. Colloq. Approx. Optim, 329-
338, 1984.
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
73
On Wijsman Asymptotically Deferred Statistical Equivalence of Order α for Set
Sequences
Mikail ET1, Hıfsı ALTINOK 2 and Rifat ÇOLAK3
Abstract: In this study we introduce and examine the concepts of Wijsman
asymptotically deferred statistical equivalence of order α and Wijsman strong r-deferred
Cesàro asymptotically equivalence of order α for set sequences. Also, we give some relations
connected to these concepts.
Keywords. Asymptotically Statistical Equivalence, Deferred Statistical Convergence;
AMS 2010. 40A05, 40C05, 46A45.
References
[1] Agnew, R. P. On deferred Cesàro means, Ann. of Math. (2) 33(3), 413—421, 1932.
[2] Çolak, R. Statistical convergence of order α, Modern Methods in Analysis and Its
Applications, NewDelhi, India: Anamaya Pub, 121—129, 2010.
[3] Küçükaslan, M. and Yılmaztürk, M. On deferred statistical convergence of
sequences. Kyungpook Math. J. 56, no. 2, 357—366, 2016.
[4] Marouf, M. Asymptotic equivalence and summability. Int. J. Math. Math. Sci., 16, 755-
762, 1993.
[5] Patterson, R. F. On asymptotically statistical equivalent sequences. Demonstratio Math.
36(1), 149—153, 2003.
[6] Ulusu, U. and Nuray, F. On asymptotically lacunary statistical equivalent set sequences, J.
Math. 2013, Art. ID 310438, 5, 2013
[7] Yılmaztürk, M. and Küçükaslan, M. On strongly deferred Cesàro summability and
deferred statistical convergence of the sequences, Bitlis Eren Univ. J. Sci. and Technol. 3, 22-
25, 2011.
1 Fırat University, Elazığ, TURKEY, [email protected] 2 Fırat University, Elazığ, TURKEY, [email protected] 3 Fırat University, Elazığ, TURKEY, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
74
On Infinite Bernoulli Matrices
Murat KİRİŞCİ1
Abstract. In this work, we study some properties of infinite Bernoulli matrices.
Further, we investigate relations between infinite Bernoulli matrices and some infinite
matrices such as Fibonacci, Pascal and special matrices.
Infinite Bernoulli polynomials matrix B(x) and Bernoulli matrix B are investigated.
Linearity as an operator, regularity and invertibility of the matrix B are showed. The examples
of the matrices B(x) and B are given.
Keywords. Bernoulli matrix, Bernoulli numbers, Pascal matrix, Fibonacci matrix
AMS 2010. Primary 11B68, Secondary 11C08, 05A10.
References
[1] G.S. Call, Pascal's matrices, Amer. Math. Monthly, 100 (1993), 372-376, 1993.
[2] Gi-Snac, Cheon, A Note on the Bernoulli and Euler Polynomials, Appl. Math. Letters, 16,
365-368, 2003.
[3] E.E. Kara, M. Ilkhan, Some properties of generalized Fibonacci sequence spaces, Linear
and Multilinear Algebra, 39(2), 217-230, 2016.
[4] M. Kirisci, Fibonacci Statistical Convergence and Korovkin type Approximation
Theorems, (to appear).
[5] R. E. Powel, S.M. Shah, Summability Theory and Applications, Prentice-Hall, 1988.
[6] Z. Zhang and J. Wang, Bernoulli matrix and its algebraic properties, Discrete Appl. Math,
154, 1622-1632, 2006.
1 Istanbul University, Istanbul, TURKEY, [email protected]; [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
75
Curious Bounds for the Gamma Function
Necdet BATIR1
Abstract. We improve some inequalities proved in [1] and establish the following
new inequalities for the gamma function.
4/1
2
4/1
2
18
1
32)1(...049653963.0
32
x
xexxx
xex xxxx
ve
1
2exp)1(
)1(logexp
xxx
x
xx .
Here )(/)(')( xxx is the digamma function.
Keywords. Gamma function, digamma function, inequalities.
AMS 2010. 33B15.
References
[1] Alzer, H. and Batir, N., Monotonicity properties of the gamma function, Appl. Math.
Letters, 20, 778-781, 2007.
[2] Batir, N., Inequalities for the gamma function, Arch. Math.(Basel), (91), 554-563, 2008.
[3] Batir, N., Very accurate approximations for the factorial function, J. Math. Inequal., 3,
335-344, 2010.
[4] Borwein, J. M. and Borwein, P., Pi and the AGM, John Wiles and Sons, 1987.
[5] Chen, C-P. and Tong, W-W., Sharp inequalities and asymptotic expansions for the gamma
function, J. Number Theory, 160, 418-431, 2016.
[6] Chen, C-P. and Liu, J-Y., Inequalities and asymptotic expansions for the gamma function,
J. Number Theory, 149, 313-326, 2015.
[7] Marsden, J. E., Basic Complex Analysis, W. H. Freeman and Company, San Fransisco,
1973.
1 Nevşehir HBV University, Nevşehir, TURKEY, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
76
[8] Mortici, C., New approximations of the gamma function in terms of the digamma function,
Appl. Math. Letters, 23, 97–100, 2010.
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
77
Some Fixed-Circle Theorems and Discontinuity at Fixed Circle
Nihal Yılmaz ÖZGÜR1 and Nihal TAŞ 2
Abstract. In this talk, we give some existence and uniqueness theorems for fixed
circles of self-mappings on a metric space with some illustrative examples. Recently, real-
valued neural networks with discontinuous activation functions have been a great importance
in practice. Hence we give some new results for discontinuity at fixed circle on a metric
space.
Keywords. Fixed circle, discontinuity, activation function, metric space.
AMS 2010. 54E40, 47H09.
References
[1] Bisht, R. K., Pant, R. P., A remark on discontinuity at fixed point, J. Math. Anal. Appl.
445, 1239-1242, 2017.
[2] Bisht, R. K., Pant, R. P., Contractive definitions and discontinuity at fixed point, Appl.
Gen. Topol. 18, 1, 173-182, 2017.
[3] Forti, M., Nistri, P., Global convergence of neural networks with discontinuous neuron
activations, IEEE Trans. Circuits Syst. I, Fundam. Theory Appl. 50, 11, 1421-1435, 2003.
[4] Özdemir, N., İskender, B.B., Özgür, N.Y., Complex valued neural network with Möbius
activation function, Commun. Nonlinear Sci. Numer. Simul. 16, 12, 4698-4703, 2011.
[5] Özgür, N. Y., Taş, N., Some fixed-circle theorems on metric spaces, arXiv:1703.00771
[math.MG].
[6] Taş, N., Özgür, N.Y., A new contribution to discontinuity at fixed point, arXiv:1705.03699
[math.MG].
[7] Wang, L. L., Chen, T. P., Multistability of neural networks with Mexican-hat-type
activation functions, IEEE Trans. Neural Netw. Learn. Syst. 23, 11, 1816-1826, 2012.
1 Balikesir University, Balikesir, TURKEY, [email protected] 2 Balikesir University, Balikesir, TURKEY, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
78
Bergman Projections on the Weighted Harmonic Bloch Spaces on the Ball, Atomic
Decompositions and Gleason’s Problem
Ömer Faruk DOĞAN 1
Abstract. We study the properties of one parameter family of weighted harmonic
Bloch Spaces 𝑏𝛼, 𝛼 ∈ ℝ, on the unit ball of ℝ𝑛. We define projections and characterize those
that are bounded from 𝐿𝛼∞ onto 𝑏𝛼 by using reproducing kernels of Bergman-Besov spaces.
The projections provide integral representations for the functions in these spaces. We solve
the Gleason problem and provide atomic decomposition for all 𝑏𝛼 , 𝛼 ∈ ℝ . Finally we give an
oscillatory characterization of 𝑏𝛼 when 𝛼 > −1. This is joint work with Adem Ersin Üreyen.
Keywords. Harmonic Bloch space, Bergman space, Reproducing kernel, Radial
fractional derivative, Bergman projection, Duality, Gleason problem, Atomic decomposition,
Oscillatory characterization.
AMS 2010. 31B05, 31B10, 26A33, 46E15.
References
[1] Axler, S., Bourdon, P., Ramey,W., Harmonic Function Theory, vol. 137, 2nd edn.
Springer, New York (2001). Grad. Texts in Math
[2] Choe, B.R., Koo, H., Yi, H., Derivatives of harmonic Bergman and Bloch functions on the
ball. J. Math. Anal. Appl. 260, 100–123, 2001.
[3] Choe, B.R., Lee, Y.J., Note on atomic decompositions of harmonic Bergman functions. In:
Complex Analysis and its Applications, OCAMI Studies, vol. 2, Osaka Munic. Univ. Press,
Osaka, pp. 11–24, 2007.
[4] Coifman, R.R., Rochberg, R., Representation theorems for holomorphic and harmonic
functions in 𝐿𝑝. Astérisque 77, 12–66, 1980.
[5] Gergün, S., Kaptanoğlu, H.T., Üreyen, A.E.: Reproducing kernels for harmonic Besov
spaces on the ball. C. R. Math. Acad. Sci. Paris 347, 735–738, 2009.
[6] Gergün, S., Kaptanoğlu, H. T., Üreyen, A. E., Harmonic Besov Spaces on the Ball,
Internat. J. Math., 27(9), 165007059, 2016.
[7] Holland, F., Walsh, D., Criteria for membership of Bloch space and its subspace, BMOA.
Math. Ann. 273, 317–335, 1986.
1 Namık Kemal University, Tekirdağ, TURKEY, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
79
[8] Hu, Z., Gleason’s problem for harmonic mixed norm and Bloch spaces in convex
domains. Math. Nachr. 279(1–2), 164–178, 2006.
[9] Jevtic, M., Pavlovic, M., Harmonic Bergman functions on the unit ball in ℝ𝑛. Acta Math.
Hung. 85, 81–96, 1999.
[10] Kaptanoğlu, H.T., Tülü, S., Weighted Bloch, Lipschitz, Zygmund, Bers, and growth
spaces of the ball: Bergman projections and characterizations. Taiwan. J. Math. 15(1), 101–
127, 2011.
[11] Ligocka, E., On the reproducing kernel for harmonic functions and the space of Bloch
harmonic functions on the unit ball in ℝ𝑛. Stud. Math. 87, 23–32, 1987.
[12] Miao, J., Reproducing kernels for harmonic Bergman spaces of the unit ball. Mon. Math.
125, 25–35, 1998.
[13] Ren, G., Kähler, U., Weighted harmonic Bloch spaces and Gleason’s problem. Complex
Var. Theory Appl. 48, 235–245, 2003.
[14] Ren, G., Kähler, U., Weighted Lipschitz continuity and harmonic Bloch and Besov spaces
in the unit real ball. Proc. Edinb. Math. Soc. 48, 743–755, 2005.
[15] Ren, G., Tu, C., Bloch spaces in the unit ball of ℂ𝑛. Proc. Am. Math. Soc. 133, 719–726,
2005.
[16] Stroethoff, K., The Bloch space and Besov spaces of analytic functions. Bull. Aust. Math.
Soc. 54, 211–219, 1996.
[17] Stroethoff, K., Harmonic Bergman Spaces, Holomorphic Spaces, vol. 33. Mathematical
Sciences Research Institute Publications, Cambridge University, Cambridge, 1998.
[18] Tanaka, K., Representation theorem for harmonic Bergman and Bloch functions. Osaka
J. Math. 50, 947–961, 2013.
[19] Zhao, R., A characterization of Bloch-type spaces on the unit ball of ℂ𝑛. J. Math. Anal.
Appl. 330, 291–297, 2007.
[20] Zhao, R., Zhu, K., Theory of Bergman Spaces in the Unit Ball of ℂ𝑛, Mém. Soc. Math.
Fr., vol. 115, pp. vi+103, 2008.
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
80
Majorization Problem on a Subclass of Analytic Functions
Öznur ÖZKAN KILIÇ1, Osman ALTINTAŞ2
Abstract. The main object of this paper is to investigate a majorization problem involving
a subclass H(α,A,B) of analytic functions in the unit disk U. Relevant connection of the result
presented here with those given by earlier workers on the subject are also indicated.
Keywords. Analytic function, Subordination, Majorization.
AMS 2010. 30C45.
References
[1] Altıntaş O., On the coefficients of functions majorized by univalent functions, Hacettepe
Bulletin of Natur. Sci. and Eng., 10, 23-30, 1981.
[2] Altıntaş O., On the coefficients of certain analytic functions, Hacettepe Bulletin of Natur.
Sci. and Eng., Vol. II, 147-156, 1982.
[3] Goodman A. W., Univalent functions, Vol. I, 1983.
[4] Kakeya S., On the limits of the roots of an algebra equation with positive coefficients,
Tohoku Math. J., 140-142, 1912.
[5] Kılıç Ö. Ö., Altıntaş O.,¨ Coefficients bounds for certain subclasses of analytic functions.
[6] MacGregor T.H. , Majorization by univalent functions, Duke Math. J. , 34, 95-102, 1967.
[7] Srivastava H. M. , Altintaş O. and Serenbay S. K. , Coefficient bounds for certain
subclasses of starlike functions of complex order, Appl. Math. Lett., 24, 1359-1363, 2011.
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
81
A Note on the Extended Type Riemann-Liouville Fractional Derivative Operator
Praveen AGARWAL1, İ. Onur KIYMAZ2, Shilpi JAIN3 and Ayşegül ÇETİNKAYA4
Abstract. In this paper, by using the extended type beta function, establish the new
extended definition for the Riemann-Liouville fractional derivative operator and discuses its
properties. Also, we establish the some relations to extended special functions of two and
three variables via generating functions.
Keywords. Beta function Hypergeometric function of two and three variables
Fractional derivative operator, Generating functions
AMS 2010. Primary 33C0533C1533C20; Secondary 33C6533C99.
References
[1] Chaudhry M. A., Qadir A., Raflque M., and Zubair S. M., Extension of Euler's Beta
function, J. Comput. Appl. Math. 78, 19-32, 1997.
[2]Chaudhry M. A., Qadir A., Srivastava H. M., and Paris R. B., Extended hypergeometric
and confluent hypergeometric functions, Appl. Math. Comput. 159, 589-602, 2004.
[3] Chaudhry M. A., Temme N. M., and Veling E. J. M., Asymptotic and closed form of a
generalized incomplete gamma function, J. Comput. Appl. Math. 67, 371-379, 1996.
[4] Chaudhry M. A. and Zubair S. M., Generalized incomplete gamma functions with
applications, J. Comput. Appl. Math. 55, 99-124, 1994.
[5] Chaudhry M. A. and Zubair S. M., On the decomposition of generalized incomplete
gamma functions with applications to Fourier transforms, J. Comput. Appl. Math. 59, 253-
284, 1995.
[6] Chaudhry M. A. and Zubair S. M., Extended incomplete gamma functions with
applications, J. Math. Anal. Appl. 274, 725-745, 2002.
1 Anand International Collage of Engineering, Jaipur, INDIA, [email protected] 2,4 Ahi EvranUniversity, Kırşehir, TURKEY, [email protected] , [email protected] 3 Poornima Collage of Engineering, Jaipur, INDIA, [email protected]
Acknowledgement: This work was supported by Ahi Evran University Scientific Research Projects Unit. Project Number:
FEF.D1.16.001
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
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[7] Chaudhry M. A. and Zubair S. M., On a Class of Incomplete Gamma Functions with
Applications, CRC Press (Chapman and Hall), Boca Raton, FL, 2002.
[8] J. Choi, A.K. Rathie and Rakesh K. Parmar, Extension of extended beta, hypergeometric
and confluent hypergeometric functions, Honam Math. J. 36(2), 339-367, 2004.
[9] Ozarslan M.A., Ozergin E., Some generating relations for extended hypergeometric
function via generalized fractional derivative operator, Math. Comput. Modelling 52, 1825—
1833, 2010.
[10] Emine Ozergin, Mehmet Ali Ozarslan, Abdullah Altin, Extension of gamma, beta and
hypergeometric function, Journal of Computational and Applied Mathematics, 235 (16),
4601-4610, 2011.
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
83
On the Harmonic Averages of Sequences of Fuzzy Numbers
Rahmet SAVAŞ1 and Sefa Anıl SEZER2
Abstract. In this study, we introduce summability methods of harmonic averages of
sequences fuzzy numbers and we state necessary and sufficient conditions which enable us to
obtain convergence of a sequence of fuzzy numbers from its summability by harmonic
methods.
Keywords. Harmonic averages, Abelian theorem, Tauberian theorem, slowly
oscillating sequences.
AMS 2010. 26E50, 40E05, 40C05.
References
[1] Bede, B., Mathematics of fuzzy sets and fuzzy logic, Springer, Berlin, 2013.
[2] Moricz, F., On the harmonic averages of numerical sequences, Arch. Math., 86, 375–384,
2006.
[3] Moricz, F., On the arithmetic and harmonic averages in nondiscrete setting, Math.
Inequal. Appl., 16, 633–643, 2013.
[4] Önder, Z., Sezer, S. A., Çanak, İ., A Tauberian theorem for the weighted mean method of
summability of sequences of fuzzy numbers, J. Intell. Fuzzy Systems, 28, 1403-1409, 2015.
[5] Yavuz, E., Çoşkun, H., On the logarithmic summability method for sequences of fuzzy
numbers, Soft Comput. doi:10.1007/s00500-016-2156-4, 2016.
1 Istanbul Medeniyet University, Istanbul, TURKEY, [email protected] 2 Istanbul Medeniyet University, Istanbul, TURKEY, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
84
Malliavin Calculus of Bismut Type for an Operator of Order Four on a Lie Group
Rémi LÉANDRE1
Abstract. I adapt Bismu’t way of the Malliavin Calculus to a four order generator on a
Lie group. In such a case, the semi-group does not preserve the positivity. We apply this
machinery to get Varadhan type estimates of the associated heat-kernel.
Keywords. Malliavin Calculus. Large deviation estimates.
AMS 2010. 60H07
References
[1] Léandre R, Malliavin Calculus of Bismut type for an operator of order four on a Lie
group, To appear in “Journal of pseudodifferential operators and its applications”.
[2] Léandre R.: Varadhan estimates for an operator of order four on a Lie group. In
“Control, decision and information technologies” IEEE (IEEE X-plore), 2016.
1 Université de Bourgogne-Franche-Comté, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
85
On Convergence Properties of Sequences of k-Positive Operators in the Subspace of
Analytic Functions
Seda KARATEKE1 and Tülin COŞKUN2
Abstract. In this presentation, we put in evidence that Korovkin type theorem is true
for the space 𝐴𝑔 that consists of analytic functions whose coefficients satisfy the inequality
|𝑓𝑘| ≤ 𝑀𝑓 𝑔(𝑘) such that 𝑔(𝑘) = 1 + 𝑘2𝑝. The theorem is proved by choosing the convenient
test functions. On the other hand we see that Korovkin type theorem is not valid in the space
of analytic functions by giving an example with graphics.
Keywords. Korovkin type theorem, k-positive linear operators, analytic functions.
References
[1] Gadjiev, A. D., Hacısalihoğlu, H. H., Lineer Pozitif Operatörler Dizilerinin
Yakınsaklığı, 1. Basım A.Ü.F.F Döner Sermaye İşletmesi Yayınları:31, Ankara, s. 78-100,
1995.
[2] Korovkin, P. P., Linear Operators and Approximation Theory, Hindustan Publishing,
1-100, 1960
[3] Evgrafov, M. A., Analytic Functions, W. B. Saunders Company, Philadelphia and
London, 24-46, 1966.
[4] Başkan, T., Kompleks Fonksiyonlar Teorisi, 5. Basım Nobel Yayınları, Ankara, 26-
200, 2003.
[5] Freitag, E., Busam, R., Complex Analysis, Springer-Verlag Berlin Heidelberg, New
York, 9-244, 2005.
[6] Evgrafov, M. A., The method of near systems in the space of analytic functions and its
application to interpolation, American Mathematical Society Translations, Series:2,
Volume:16, American Mathematical Society, USA, 89-201, 1960.
[7] Gadjiev, A. D., Ghorbanalizadeh, A. M., On an approximation process in the space of
analytic functions, Cent. Eur. J. Math. 8 (2), 389-398, 2010.
1 İstanbul Arel University,, İstanbul, TURKEY, [email protected] 2 Bülent Ecevit University, Zonguldak, TURKEY, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
86
[8] Gadjiev, A. D., Ghorbanalizadeh, A. M., Approximation of analytical functions by the
sequences of k-positive linear operators, Journal of Approximation Theory, (6), 1245-1255,
2010.
[9] Gadjiev, A. D., Simultaneous statistical approximation of analytic functions and their
derivatives by k-positive linear operators, Azerbaijan Jornal of Mathematics, (1), 57-66,
2011.
[10] Gadjiev, A. D., Duman, O., Ghorbanalizadeh, A. M., Ideal convergence of k-positive
linear operators, Journal of function spaces and applications, 2012.
[11] Ispir, N., Convergence of sequences of k-positive linear operators in subspaces of the
space of analytic functions, Hacettepe Bulletin of Natural Sciences and Engineering, (28), 47-
53, 1999.
[12] Kurt, S. B., Analitik fonksiyonlar uzayında lineer k-pozitif operatör dizilerinin
yakınsaklık koşulları, (YL), Bülent Ecevit Üniversitesi, Fen Bilimleri Enstitüsü (Danışman:
Doç. Dr. Tülin COŞKUN), Zonguldak, 1-73, 2010
[13] Gadjiev, A. D. and Ghorbanalizadeh, A M., On an approximation process in the space of
analytic functions, Cent. Eur. J. Math., 8 (2), 389-398, 2010.
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
87
Some Operator Inequalities Related to Means and Entropies
Shigeru FURUICHI1
Abstract. We give some new refinements and reverses Young inequalities for two positive
operators. Our results refine the ordering relations among the arithmetic mean, the geometric
mean and the harmonic mean for two positive invertible operators.
In addition, we give the tight bounds of Tsallis relative operator entropy by the use of
Hermite-Hadamard's inequality. We also give alternative tight bounds for the Tsallis relative
operator entropy.
Furthermore, we study the properties on monotonicity for the weight of operator means,
and for the parameter of relative operator entropies.
My talk will be composed by my recent results in [1,2,3,4,5,6,7,8].
Keywords. Operator inequality, positive operator, operator mean, Young inequality,
Hermite-Hadamard's inequality and relative operator entropy.
AMS 2010. 15A39, 15A45, 47A63, 46L05 and 47A60
References
[1] Furuichi, S., On refined Young inequalities and reverse inequalities, J. Math. Inequal.,
Vol.5(2011), 21-31.
[2] Furuichi, S., Refined Young inequalities with Specht's ratio, J. Egyptian Math. Soc.,
Vol.20(2012), 46-49.
[3] Furuichi, S., Operator inequalities among arithmetic mean, geometric mean and harmonic
mean, J. Math. Inequal., Vol.8, 669-672, 2014.
[4] Furuichi, S., M.B.Ghaemi, N.Gharakhanlu, Generalized reverse Young and Heinz
inequalities, Bull.Malays.Math.Sci.Soc., DOI 10.1007/s40840-017-0483-y
[5] Moradi, H. R., Furuichi, S., Minculete, N., Estimates for Tsallis relative operator entropy,
to appear in Math. Ineq. Appl.
[6] Furuichi, S. and Moradi, H.R., Operator inequalities among arithmetic mean, geometric
mean and harmonic mean. II, preprint.
[7] Furuichi, S. and Minculete, N., Alternative estimates for Tsallis relative operator entropy,
preprint.
[8] Furuichi, S. and Minculete, N., On monotonicity of the weight of operator means and
relative operator entropies, preprint.
1 Nihon University, Tokyo, Japan, e-mail:[email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
88
On a Subclass of Harmonic Univalent Functions Involving a Linear Operator
Sibel Yalçın TOKGÖZ1 and Şahsene ALTINKAYA2
Abstract. Making use of a multiplier transformation, we introduce a new class of
complex valued harmonic functions which are orientation preserving and univalent in the
open unit disk U . Relevant connections of the results presented here with various known
results are briefly indicated.
Keywords. Harmonic functions, univalent functions, Hadamard product, modified
generalized Sălăgean operator, multiplier transformation.
AMS 2010. 30C45, 30C80.
References
[1] Cho, N.E., Srivastava, H.M. Argument estimates of certain analytic functions defined by a
class of multiplier transformations, Math. Comput. Modelling, 37, 39-49, 2003.
[2] Cho, N.E., Kim, T.H., Multiplier transformations and strongly close-to-convex functions,
Bull. Korean Math. Soc. 40, 3, 399-410, 2003.
[3] Clunie, J., Sheil-Small, T., Harmonic univalent functions, Ann. Acad. Sci. Fenn. Ser. A I
Math. 9, 3-25, 1984.
[4] Dziok, J., Classes of harmonic functions defined by subordination, Abstr. Appl. Anal.
2015, Article ID 756928, 2015.
[5] Dziok, J, On Janowski harmonic functions, J. Appl. Anal. 21, 2, 99-107, 2015.
[6] Dziok, J., Jahangiri, J., Silverman, H., Harmonic functions with varying coefficients,
Journal of Inequalities and Applications, 139, 2016.
[7] Jahangiri, J.M., Harmonic functions starlike in the unit disk, J. Math. Anal. Appl. 235,
470-477, 1999.
1 Uludag University, Bursa, TURKEY, [email protected] 2 Uludag University, Bursa, TURKEY, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
89
[8] Jahangiri, J.M., Murugusundaramoorthy, G., Vijaya, K., Salagean-type harmonic
univalent functions, South J. Pure Appl. Math. 2, 77-82, 2002.
[9] Salagean, G.S., Subclasses of univalent functions, Lecture Notes in Math. Springer-
Verlag Heidelberg 1013, 362-372, 1983.
[10] Silverman, H., Harmonic univalent functions with negative coefficients, J. Math. Anal.
Appl. 220, 283-289, 1998.
[11] Silverman, H., Silvia, E.M., Subclasses of harmonic univalent functions, N. Z. J. Math.
28, 275-284, 1999.
[12] Uralegaddi, B.A. and Somanatha, C., Certain classes of univalent functions, Current
topics in analytic function theory, (Edited by H.M. Srivastava and S. Owa), 371-374, World
Sci. Publishing, Singapore, 1992.
[13] Yasar, E., Yalcin, S., Certain properties of a subclasses of harmonic functions, Appl.
Math. Inf. Sci. 7, 5, 1749-1753, 2013.
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
90
On a Coefficient Problem for a Subclass of Bi-Univalent Functions Defined by Using a
q-Derivative Operator
Sibel Yalçın TOKGÖZ1 and Şahsene ALTINKAYA 2
Abstract. We introduce a new class of bi-univalent functions defined by using a q-
derivative operator. Moreover, using the Faber polynomials, we obtain general coefficient
estimates for functions in this class.
Keywords. Bi-univalent functions, Faber polynomials, coefficient estimates, q-
derivative.
AMS 2010. 30C45, 05A30, 33D15.
References
[1] Airault, H., Symmetric sums associated to the factorization of Grunsky coefficients, in
Conference, Groups and Symmetries, Montreal, Canada, April, 2007.
[2] Airault, H., Bouali, H., Differential calculus on the Faber polynomials, Bulletin des
Sciences Mathematiques, 130, 179-222, 2006.
[3] Airault, H., Ren, J., An algebra of differential operators and generating functions on the
set of univalent functions, Bulletin des Sciences Mathematiques, 126, 343-367, 2002.
[4] Altınkaya, Ş., Yalçın, S., Faber polynomial coefficient bounds for a subclass of bi-
univalent functions, C. R. Acad. Sci. Paris, Ser. I, 353, 12, 1075-1080, 2015.
[5] Duren, P. L., Univalent functions, Grundlehren der Mathematischen Wissenschaften,
Springer, New York, USA, 259, 1983.
[6] Gasper, G., Rahman, M., Basic Hypergeometric Series (with a Foreword by Richard
Askey), Encyclopedia of Mathematics and Its Applications, Vol. 35, Cambridge University
Press, Cambridge, New York, Port Chester, Melbourne and Sydney, 1990; Second edition,
1 Uludag University, Bursa, TURKEY, [email protected] 2 Uludag University, Bursa, TURKEY, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
91
Encyclopedia of Mathematics and Its Applications, Vol. 96, Cambridge University Press,
Cambridge, London and New York, 2004.
[7] Hamidi, S. G., Jahangiri, J. M., Faber polynomial coefficient estimates for analytic bi-
close-to-convex functions, C. R. Acad. Sci. Paris, Ser. I, 352, 17-20, 2014.
[8] Jackson, F.H., On q-functions and a certain difference operator, Transactions of the Royal
Society of Edinburgh, 46, 253-281, 1908.
[9] Srivastava, H.M., Univalent functions, fractional calculus, and associated generalized
hypergeometric functions, in Univalent Functions; Fractional Calculus; and Their
Applications (H. M. Srivastava and S. Owa, Editors), Halsted Press (Ellis Horwood Limited,
Chichester), John Wiley and Sons, New York, Chichester, Brisbane and Toronto, 1989.
[10] Zireh A., Hajiparvaneh, S., Bulut, S., Faber polynomial coefficient estimates for a
comprehensive subclass of analytic bi-univalent functions defined by subordination, Bull.
Belg. Math. Soc. Simon Stevin 23, 487-504, 2016.
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
92
On Asymptotically f-Statistical Equivalent Set Sequences
Sukran KONCA1 and Mehmet KUCUKASLAN2
Abstract. The aim of this paper is to introduce a generalization of statistical
convergence of asymptotically equivalent set sequences and examine some inclusion relations
related to a new concept of Wijsman asymptotically equivalent statistical convergence of
sequences of sets with respect to a modulus function f.
Keywords. Modulus function, asymptotically statistical equivalent set sequences,
Wijsman convergence.
AMS 2010. 46A45, 40G15, 40A05.
References
[1] Wijsman, R. A., Convergence of sequences of convex sets, cones and functions, Bull. Am.
Math. Soc. 70, 186-188, 1964.
[2] Aizpuru, A., Listan-Garcia, M. C., Rambla-Barreno, F., Density by moduli and statistical
convergence, Quaestiones Mathematicae. 37, 4, 525–530, 2014.
[3] Bhardwaj, V. K., Dhawan, S., Density by moduli and Wijsman lacunary statistical
convergence of sequences of sets, J. Ineq. Appl. 2017, 25, 2017. DOI 10.1186/s13660-017-
1294-2
[4] Seneta, E., Regularly varying functions, Lecture Notes in Mathematics, Springer-Verlag,
Berlin-Heidelberg-New York, 1976.
1 Bitlis Eren University, Bitlis, TURKEY, [email protected] 2 Mersin University, Mersin, TURKEY, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
93
Reverse Inequalities for the Berezin Numbers of Operators
U. YAMANCI1, M. GÜRDAL2 and M. T. GARAYEV3
Abstract. In this paper, we prove reverse inequalities for the so-called Berezin number
of some operators in reproducing kernel Hilbert space.
Acknowledgement: The second author is supported by TUBA through Young
Scientist Award Program (TUBA-GEBIP/2015).
Keywords. Positive operator, Berezin symbol, Berezin number.
AMS 2010. 47A63.
References
[1] Berezin, F.A., Covariant and contravariant symbols for operators, Math. USSR-Izv., 6,
1117-1151, 1972.
[2] Hardy, G., Littlewood, J.E., Polya, G., Inequalities, 2 nd ed. Cambridge University Press,
Cambridge, 1967.
[3] Dragomir, S.S., Reverse inequalities for the numerical radius of linear operators in
Hilbert spaces, Bull. Aust. Math. Soc., 73, 255-262, 2006.
[4] Saitoh, S., Sawano, Y., Theory of reproducing kernels and applications, Springer, 2016.
1 Suleyman Demirel University, Isparta, TURKEY, [email protected] 2 Suleyman Demirel University, Isparta, TURKEY, [email protected] 3 King Saud University, Riyadh, SAUDİ ARABİA, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
94
On the (p,q)-Analogue of Poly-Bernoulli Polynomials
Veli KURT 1
Abstract. One of the different generalization of the Bernoulli numbers and
polynomials are the (p,q)-analogue of Poly-Bernoulli polynomials. In this article, we
introduce and investigate a (p,q)-analogue of the poly-Bernoulli polynomials and numbers by
using the (p,q)-polylogarithm function. There new sequences are generalization of the poly-
Bernoulli numbers and polynomials. We give several combinatorial identities and properties
of these new polynomials.
Keywords. Poly-Bernoulli polynomials and numbers, Polylogarithm, Generating
Function.
AMS 2010. 11B68, 11B73, 11B83.
References
[1] Bayad, A, Hamahata, Y, Polylogarithms and poly-Bernoulli polynomials, Kyushu. J.
Math. 65, 15-24, 2011.
[2] Kim, D, Kim, T, A note on poly-bernoulli and higher order-poly-Bernoulli polynomials,
Russian J. of Math. Physics, 22, 1, 26-33, 2015.
[3] Komatsu, T, Ramirez, J, S, Sirvent, V.F, A (p,q)-Analogue of poly-Euler polynomials and
some related polynomials, Arxiv: 1604.3787 [Math.NT] 2016.
[4] Liu, H, Wang, W, Some identities on the Bernoulli,Euler and Genocchi polynomials via
power sums and alternate power sums, Discrete Math. 309, 3346-3363, 2009.
[5] Sandjang. P.N, On the fundamental theorem of (p,q)-calculus and some (p,q)-Taylor
formulas, Arxiv: 1309.3934 [math QA] 2013.
1 Antalya, TURKEY, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
95
APPLIED
MATHEMATICS
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
96
An Affine Scaling Method Using a Class of Differential Barrier Functions
Abdessamad BARBARA1
Abstract. In this paper we address a practical aspect of differential barrier penalty
functions in linear programming. In this respect we propose an affine scaling interior point
algorithm based on a large classes of differential barrier functions. The comparison of the
algorithm with a version of the classical affine scaling algorithm shows that the algorithm is
robust and efficient. We thus show that differential barrier functions open up new
perspectives in linear optimization.
1 Institut de Mathématiques de Bourgogne(IMB)- UMR 5584 CNRS, Université de Bourgogne, 9 avenue Alain Savary
BP 47870, 21078 Dijon cedex, FRANCE, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
97
Hyperbolic Tangent Solution to the Conformable Time Fractional Zakharov-Kuznetsov
Equation in 3D Space
Alper KORKMAZ1 and Ozlem Ersoy HEPSON2
Abstract. In the study, hyperbolic tangent (tanh) ansatz solution is investigated for the
conformable time fractional Zakharov-Kuznetsov Equation (fZKE) in 3D space. Transformation of the
fZKE to an ODE by the compatible wave transformation is the first step of the methodology. It is
assumed that there exists a solution of positive integer power of hyperbolic tangent form. Determining
the power of the predicted solution follows some algebra to find the relations among the other
parameters given in the solution. The final step is transforming the solution into original variables.
Keywords. Conformable fractional Zahharov-Kuznetsov Equation, exact solution, solitary
wave.
AMS 2010. 35R11, 35C07, 76B25.
References
[1] Zakharov, V., Kuznetsov, E., On three dimensional solitons, Zhurnal Eksp. Teoret. Fiz 66, 594–
597, 1974.
[2] Mace, R., Hellberg, M., The Korteweg-de Vries-Zakharov-Kuznetsov equation for electron-acoustic
waves, Physics of Plasmas 8, 2649–2656, 2001.
[3] Matebese, B., Adem, A., Khalique, C., Biswas, A., Solutions of Zakharov-Kuznetsov equation with
power law nonlinearity in (1+ 3) dimensions, Physics of Wave Phenomena 19, 148–154, 2011.
[4] Atangana, A., Baleanu, D., Alsaedi, A., New properties of conformable derivative, Open
Mathematics 13, 1–10, 2015.
1 Çankırı Karatekin University, Çankırı, TURKEY, [email protected] 2 Eskişehir Osmangazi University, Eskişehir, TURKEY, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
98
Asymptotic Properties of the Lasota Equation in Various Functional Spaces
Anna POSKROBKO1 and Antoni Leon DAWIDOWICZ2
Abstract. We study asymptotic properties of the dynamical system generated by the Lasota
equation
).,()( xuFx
uxc
t
u
This equation is interesting in the context of its properties as well as its applications.
It describes the process of reproduction and differentiation of a population of red blood cells. We give
the conditions of its stability and chaos in the sense of Devaney in various functional spaces. A
examples illustrate the criteria when the semigroup generated by the equation has not asymptotic
behaviour.
Keywords. Lasota equation, chaos, stability.
AMS 2010. 35B10, 35B35, 37C75.
This work is supported by Bialystok University of Technology (Grant No. S/WI/1/2016) and founded
by the resources for research by Ministry of Science and Higher Education.
1 Faculty of Computer Science, Bialystok University of Technology, Białystok, POLAND, [email protected] 2 Faculty of Mathematics and Computer Science, Jagiellonian University, Kraków, POLAND,
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
99
Iterative Solutions to the Systems of Linear Differential Equations
Arzu GÜLEROĞLU1
Abstract. The solutions to the systems of three first-order linear differential equations with
variable coefficients are obtained by using the asymptotic iteration method. This method is applied to
some special linear systems.
Keywords. Asymptotic iteration method, linear systems.
AMS 2010. 35A24, 34G10.
References
[1] Polyanin, A. D., F. Zaitsev, V. F., Handbook of Exact Solutions for Ordinary Differential
Equations, Chapman and Hall/CRC, America, 2003.
[2] Ciftci, H., Hall, R. L., Saad, N., Asymptotic iteration method for eigenvalue problems, J.
Phys. A: Math. Gen., 36, 47, 11807-11816, 2003.
[3] Ciftci, H., Hall, R. L., Saad, N, Iterative solutions to the Dirac equation, Phys. Rev. A, 72,
2, 022101, 2005.
[4] Robertson, K. M., Saad, N., Solvable systems of linear differential equations, J. Appl.
Math. Comput., 31, 475–494, 2009.
1Trakya University, Edirne, TURKEY, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
100
A Mathematical Model on Coral-Population Dynamics
Arzu ÜNAL
Abstract. Rapid increases in ocean temperatures are causing repeated thermal stresses to
populations of tropical corals, are causing differential coral mortality, and are consequently changing
the composition of coral-reef communities. Examining recovery rates of reef organisms after thermal
disturbances is a prerequisite to understanding population resilience. Here we use a nonlinear system of
differential equations to predict the dynamics of coral populations that are forced by the complexities of
multi-cyclic temperature cycles. We show that the intrinsic rates of recovery vary among species, from
0.4 to 2.3. Our forecasting simulations of the response of coral populations to three different climate
change scenarios predict that most population-recovery rates will dampen through time as thermal
stresses become more frequent.
Keywords. Nonlinear differential equation system, coral population, climate change.
AMS 2010. 92D25, 92D40, 60H10
References
[1] Crabbe, M. J. C., Global warming and coral reefs: modelling the effects of temperature on
Acropora palmata colony growth, Computational Biology and Chemistry, 31, 294–297, 2007.
[2] Crabbe, M. J. C., Climate change, global warming and coral reefs: modelling the effects of
temperature, Computational Biology and Chemistery, 32, 311–314, 2008.
[3] Hughes, T. P., Baird, A. H., Bellwood, D. R., Card, M., Connolly, S. R., Folke, C.,
Grosberg, R., Hoegh-Guldberg, O., Jackson, J. B. C., Kleypas, J., Lough, J. M., Marshall, P.,
Nystrom, M., Palumbi, S. R., Pandolfi, J. M., Rosen, B. and Roughtgarden, J., Climate change,
human impacts, and the resilience of coral reefs. Science, 301, 929-933, 2003.
[4] Van Woesik, R. and Koksal, S. , A coral population response (CPR) model for thermal
stress, Coral Reefs and Climate Change: Science and Management, 61, 129-144, 2006.
[5] Yee, S. H. , Santavy, D. L. and Barron, M. G., Assessing the effects of disease and
bleaching on Florida Keys corals by fitting population models to data. Ecological modelling,
222, 1323-1332, 2011.
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
101
The Class of 𝑳 ∩ 𝑫 and Its Application to Renewal Reward Process
Asli Bektaş KAMIŞLIK1, Tülay KESEMEN2, Tahir KHANIYEV3
Abstract. The class of 𝐿 ∩ 𝐷 is generated by intersection of two important subclasses of heavy
tailed distributions: The long tailed distributions and dominated varying distributions. This class itself
is also an important member of heavy tailed distributions and has some principal application areas
especially in renewal, renewal reward and random walk processes.
The aim of this study is to observe some well and less known results on renewal functions
generated by the class of 𝐿 ∩ 𝐷 and apply them into a special renewal reward process which is known in
the literature a semi Markovian inventory model of type (s,S). Especially we focused on Pareto
distribution which belongs to the 𝐿 ∩ 𝐷 subclass of heavy tailed distributions. As a first step we
obtained asymptotic results for renewal function generated by Pareto distribution from the class of 𝐿 ∩
𝐷 using some well-known results by Emrechts and Omey [3]. Then we applied the results we obtained
for Pareto distribution to renewal reward processes. As an application we investigate inventory model of
type (s,S) when demands have Pareto distribution from the class of 𝐿 ∩ 𝐷. We obtained asymptotic
expansion for ergodic distribution function and finally we reached asymptotic expansion for nth order
moments of distribution of this process.
Keywords. The class of 𝐿 ∩ 𝐷, Heavy tailed distributions, Renewal reward process, Asymptotic
expansion, Ergodic distribution, Moments for ergodic distribution.
AMS 2010. 60K05, 41A60.
Acknowledgements: The authors wish to thank to Scientific and Technological Research Council of
Turkey (TÜBİTAK), for the financial support. Project Number: 115F221
References
[1] Adler, R.J., Feldman, R.E. and Taqqu, M.S. A Practical Guide to Heavy Tails: Statistical
Techniques and Applications, Birkhäuser, Boston, 1998.
[2] Bekar, N.O., Aliyev,R. ve Khaniyev, T., Asymptotic expansions for a renewal-reward process with
Weibull distributed interference of chance, Contemporary Analysis and Applied Mathematics, 2, 1, 200-
211, 2013.
[3] Embrechts, P., Omey, E., A property of longtailed distributions, J. Appl. Prob., 21 80-87, 1984.
1 Recep Tayyip Erdoğan University, Rize, TURKEY, [email protected] 2 Karadeniz Technical University, Trabzon, TURKEY, [email protected] 3 TOBB University of Economics and Technology, Ankara, TURKEY, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
102
[4] Khaniyev, T.A., Kokangul, A. ve Aliyev, R.T., An asymptotic approach for a semi-Markovian
inventory model of type (s, S), Applied Stochastic Models in Business and Industry, 29, 5, 439 – 453,
2013.
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
103
Mathematical Modeling for Real Life Problems
Aysegul SAGLAM ARSLAN
Abstract. The aim of this study is to analyze the ability of students to solve real-life problems
requiring mathematical modeling. For this purpose, the researchers developed a data collection tool
aiming to evaluate students’ ability to create, apply and finalize models using experimental and
theoretical methods. A series of activities (such as Why does a spoon in a cup seem broken? What if an
object is threw down? Why do springs get jammed? How can I throw a ball into the basket pot? How is
a bicycle's energy calculated?) was organized with the participation of ten 9th grade students. Students'
answers were classified under five categories: prediction, observation, experience and measurement,
mathematical modeling, model testing. The problems, difficulties and mistakes that students
encountered in each of the mentioned categories have been identified.
The analysis of ongoing data has shown that students are more successful in the modeling
activities with the theoretical method than the modeling activities with the experimental method.
Accordingly, it has been determined that participant students can easily create a mathematical model of
a real-life problem they are working on by using an equation they have. On the other hand, it has been
observed that the participants experienced considerable difficulties while using the experimental method
in order to construct a mathematical model and to test the model by returning to the problem.
Keywords. Modelling, mathematical model, real life problems.
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
104
Higher Order Trigonometric B-Spline Algorithms to the Solution of Coupled Burgers’
Equation
Aysun Tok ONARCAN1 and Ozlem ERSOY HEPSON2
Abstract. Trigonometric B-spline functions of higher degrees have advantages over lower ones
since they can be used as approximate functions in the numerical methods if the differential equation
include higher order derivatives. Thus the collocation methods based on higher degree trigonometric B-
splines are set up to get numerical solutions to the Coupled Burgers’ equation. The unknowns of the
coupled Burgers equation are integrated in time by aid of the Crank-Nicolson method. Resulting time-
integrated coupled Burgers’ equation is discretized using the trigonometric B-spline collocation method.
Several initial and boundary value problems of the Coupled Burgers’ equation are studied by the
proposed technique.
Keywords. Trigonometric B-Spline, collocation method, Coupled Burgers’ Equation
AMS 2010. 41A15, 65M70.
References
[1] Esipov, S. E., Coupled Burgers Equations- A Model of Polydispersive sedimentation, James Franck
Institute and Department of Physics, University of Chicago, 1995.
[2] Mittal R. C., Arora, G., Numerical solution of the coupled viscous Burgers' equation, Commun
Nonlinear Sci Numer Simulat, 16, 1304-1313, 2011.
[3] Nikolis, A., Seimenis, I., Solving dynamical systems with cubic trigonometric splines, Applied
Mathematics E-notes, 5, 116-123, 2005
[4] Kutluay, S., Ucar, Y., Numerical solutions of the coupled Burgers' equation by the Galerkin
quadratic B-spline finite element method, Math. Meth. Appl. Sci., 36 2403-2415, 2013.
1 Eskişehir Osmangazi University, Eskişehir, TURKEY, [email protected] 2 Eskişehir Osmangazi University, Eskişehir, TURKEY, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
105
Numerical Solution of Nonlinear Parabolic Type Equation in the Class of Generalized
Functions with Degeneration
Bahaddin SINSOYSAL1, Hakan BAL2and Kubra YILMAZ3
Abstract. In this paper, a method for obtaining the numerical solution of the first type initial
boundary value problem for a degenerative second order parabolic equation is suggested. To this end, in
accordance with the smoothness of the solution of the investigated problem, we introduce two auxiliary
problems, as they have some advantages over the main problem, and are equivalent to the main problem
in a definite sense. Using the introduced auxiliary problems, we write more sensitive numerical scheme
whose solution express all of the physical properties of the problem accurately. Moreover, using the
auxiliary problems, we can write the higher order numerical schemes with respect to the time variable.
Some comparative results of the exact and numerical solutions are presented.
Keywords. Linear parabolic equation with degeneration, sensitive numerical algorithms
AMS 2010. 35K20, 35K65, 65M06.
References
[1] Antoncev, S. N., On the localization of solutions of non-linear degenerate elliptic and parabolic
equations, Soviet Mathematics. Doklady, 24, 420–424, 1981.
[2] Martinson, L. K. and Pavlov, K. B., On the problem of spatial localization of thermal perturbations
in the theory of non-linear heat conduction, USSR Computational Math and Math Physics, 12, 4, 261–
268, 1972.
[3] Rasulov, M. A., A numerical method of solving a parabolic equation with degeneration, Differential
Equations, 18, 8, 1418–1427, 1992.
[4] Sinsoysal, B., The Analytical and a Higher-Accuracy Numerical Solution of a Free
Boundary Problem in a Class of Discontinuous Functions, Mathematical Problems in
Engineering, Article ID 791026, 2012.
1 Beykent University, Department of Management Information Systems, Istanbul, TURKEY, [email protected] 2 Beykent University, Faculty of Economics and Administrative Sciences, Istanbul, TURKEY, [email protected] 3 Beykent University, Department of Mathematics and Computing, Istanbul, TURKEY, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
106
A Finite Differences Scheme for Solving the System of Differential Equations of Non-
Isentropic Flow in a Class of Discontinuous Functions
Bahaddin SINSOYSAL1, Hasan CARFI2 and Nazife KOC3
Abstract. In this paper for the Cauchy problem of system of equations
𝜌𝑡 + (𝜌𝑢)𝑥 = 0, (1)
(𝜌𝑢)𝑡 + (𝑝 + 𝜌𝑢2)𝑥 = 0, (2)
𝜖𝑡 + [𝜌𝑢𝜖 + 𝑝𝑢]𝑥 = 0, (3)
describing of non-isentropic flow in a class of discontinuous functions
𝜌(𝑥, 0) = 𝑓(𝑥), (4)
𝑢(𝑥, 0) = 𝑔(𝑥), (5)
𝜖(𝑥, 0) = ℎ(𝑥). (6)
Here 𝑓, 𝑔 and ℎ are known functions and 𝜖 = 𝜀 +𝑢2
2. The characteristic properties of the solution of
the system (1)-(3) have discontinuity points whose locations are unknown beforehand. These points do
not allow to apply classical numerical methods to the solution of problem (1)-(6). To this end, we
introduce the special auxiliary problem having some advantages over the main problem as follows:
𝑞𝑡 + 𝜌𝑢 = 0,
𝜋𝑡 + 𝑝 + 𝜌𝑢2 = 0,
𝜇𝑡 + 𝜌𝑢(𝜀 + 𝑝𝑢) = 0,
𝑞(𝑥, 0) = 𝑞(0)(𝑥),
𝜋(𝑥, 0) = 𝜋(0)(𝑥),
𝜇(𝑥, 0) = 𝜇(0)(𝑥),
where 𝑞(0)(𝑥), 𝜋(0)(𝑥) and 𝜇(0)(𝑥) are solutions of equations (𝑞(0))𝑥
= 𝑓(𝑥), (𝜋(0))𝑥
= 𝑔(𝑥), and
(𝜇(0))𝑥
= ℎ(𝑥), respectively.
Using the proposed auxiliary problem we develop a finite differences method to obtain
numerical solution of the problem (1)-(6) describing all of the physical properties.
Keywords. Non-isentropic flow, finite differences scheme in a class of discontinuous functions.
AMS 2010. 35L67, 65M06.
1 Beykent University, Department of Management Information Systems, Istanbul, TURKEY, [email protected] 2 Nisantasi University, Vocational SchoolUniversity, Istanbul, TURKEY, [email protected] 3 Beykent University, Institute of Sciences and Technology, Istanbul, TURKEY, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
107
A Finite Difference Method for Solving Generalized FitzHugh-Nagumo Equation
Bilge İNAN1
Abstract. In this paper, we propose Crank-Nicolson exponential finite difference method for
solving the generalized Fitzhugh-Nagumo equation. Numerical solutions obtained by this method when
compared with the exact solution are shown that the obtained solution produces high accurate results.
The comparisons showed that proposed method is reliable, precise and convenient alternative method
for solution of the generalized FitzHugh-Nagumo equation.
Keywords. Generalized Fitzhugh-Nagumo equation, Finite difference method, Exponential
finite difference method, Crank-Nicolson exponential finite difference method.
AMS 2010. 65N06, 35D99
References
[1] Bhrawy, A. H., A Jacobi-Gauss-Lobatto collocation method for solving generalized Fitzhugh-
Nagumo equation with time-dependent coefficients, Appl. Math. Comput., 222, 255-264, 2013.
[2] Ruiz-Ramírez, J., Macías-Díaz, J. E., A finite-difference scheme to approximate non-negative and
bounded solutions of a FitzHugh-Nagumo equation, Int. J. Comp. Math., 88, 3186-3201, 2011.
[3] Bhattacharya, M. C., An explicit conditionally stable finite difference equation for heat conduction
problems, Int. J. Num. Meth. Eng., 21, 239-265, 1985.
[4] Bahadır, A. R., Exponential finite-difference method applied to Korteweg-de Vries equation for
small times, Appl. Math. Comput., 160, 675-682, 2005.
[5] İnan, B., Bahadır, A. R., Numerical solution of the one-dimensional Burgers equation: Implicit and
fully implicit exponential finite difference methods, Pramana J. Phys., 81, 547-556, 2013.
1 Kilis 7 Aralık University, Kilis, TURKEY, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
108
Numerical Solutions of Boundary Value Problems for Nonlinear Delay Differential
Equations
Bülent YILMAZ1 and Volkan YAMAN2
Abstract. : In this paper we investigated solutions of boundary value problems for nonlinear
delay differential equations obtained by Adomian Decomposition Method, Differential Transformation
Method and Daftardar – Jafari Method. Comparison and error analysis of the solutions of 3 methods
have been given with three numerical examples.
Keywords. Delay Differential, Boundary value problems, Adomian Decomposition,
Differential Transformation, Daftardar-Jafari
AMS 2010. 34K28, 34L30, 34K06
References
[1] Adomian, G. and Rach, R., Inversion of nonlinear stochastic operators, Journal of Mathematical
Analysis and Applications, 91, 39–46, 1983.
[2] Adomian, G., Nonlinear Stochastic Operator Equations, Academic Press, New York, 1986.
[3] Adomian, G., Solving Frontier Problems of Physics: the Decomposition Method, Kluwer Academic
Publishers, 1994.
[4] Erturk, V. S., The Numerical Solution of a Second-Order Differential Equation With Neumann
Boundary Conditions via Differential Transformation Method, Anadolu University Journal of Science
and Technology-B, Vol:1, 19-29, 2011
[5] Haziqah, C., Hussin, C., Kilicman, A., On the Solutions of Nonlinear Higher-Order Boundary Value
Problems by Using Differential Transformation Method and Adomian Decomposition Method,
Mathematical Problems In Engineering, vol: 2011, 1-19, 2011.
[6] Daftardar-Gejji, V. and Jafari, H., An Iterative Method For Solving Nonlinear Functional Equations,
Journal of Mathematical Analysis and Applications 316, 753-763, 2006.
[7] Ullah, I., Khan, H., Rahim, M.T., Numerical Solutions of Fifth and Sixth Order Nonlinear Boundary
Value Problems by Daftardar Jafari Method, Journal of Computational Engineering, Vol: 2014, Article
id: 286039, 2014
1Marmara University, Istanbul, TURKEY, [email protected] 2 Marmara University, Istanbul, TURKEY, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
109
The Analytical Solutions of the Space-Time Fractional Modified Kawahara Equation
Dilek Varol BAYRAM1, Sevil ÇULHA1, Ayşegül DAŞCIOĞLU1
Abstract. In this study, the analytical exact solutions of the space-time fractional modified
Kawahara equation are obtained for the first time in the literature. Here, the fractional derivative is
considered in conformable sense. Using Jacobi elliptic function expansion method, the solutions are
found in general form containing the hyperbolic, trigonometric, and rational functions. Also, the
complex valued solutions and soliton solutions are obtained.
Keywords. Jacobi elliptic function, fractional partial differential equation, modified Kawahara
equation.
AMS 2010. 35G20, 35L05, 35R11.
References
[1] Armitage, J. V., Eberlein, W. F., Elliptic Functions, New York: Cambridge University Press, 2006.
[2] Miller, K. S., An Introduction to Fractional Calculus and Fractional Differential Equations, J.
Wiley and Sons, New York, 1993.
[3] Kilbas, A., Srivastava, H., Trujillo, J., Theory and Applications of Fractional Differential
Equations in: Math. Studies., North-Holland, New York, 2006.
[4] Podlubny, I., Fractional Differential Equations, Academic Press, USA, 1999.
[5] Khalil, R., Horani, M. A., Yousef A., Sababheh, M., A new definition of fractional derivative,
Journal of Computational and Applied Mathematics, 264, 65–70, 2014.
[6] Abdeljawad, T., On conformable fractional calculus, Journal of Computational and Applied
Mathematics, 279, 57–66, 2015.
[7] Wazwaz, A. M., New solitary wave solutions to the modified Kawahara equation, Physics Letters
A, 360, 588–592, 2007.
[8] Fu, Z., Liu, S., Liu, S., Zhao, Q., The JEFE method and periodic solutions of two kinds of
nonlinear wave equations, Communications in Nonlinear Science and Numerical Simulation 8, 67–75,
2003.
1 Pamukkale University, Denizli, TURKEY, [email protected], [email protected], [email protected].
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
110
An Approximation Method for the Fractional Linear Fredholm Integrodifferential
Equations by Laguerre Polynomials
Dilek Varol Bayram1 and Ayşegül Daşcıoğlu1
Abstract. In this study, an approximation method based on the Laguerre polynomials is
presented to obtain the solutions of the fractional linear Fredholm integrodifferential equations. Here,
the fractional derivative is considered in Caputo sense. This method transforms the integrodifferential
equation to a system of linear algebraic equations by using the collocation points. Besides, some
examples are presented to illustrate the accuracy of the method and the results are discussed.
Keywords. Fredholm integrodifferential equations, Laguerre polynomials, Fractional
integrodifferential equations.
AMS 2010. 26A33, 45J05, 65N35.
References
[1] Khader, M. M., El Danaf, T. S., Hendy, A. S., Efficient spectral collocation method for solving
multi-term fractional differential equations based on the generalized Laguerre polynomials, Journal of
Fractional Calculus and Applications, 3 (13), 1-14, 2012.
[2] Miller, K.S., An Introduction to Fractional Calculus and Fractional Differential Equations, J.
Wiley and Sons, New York, 1993.
[3] Kilbas, A., Srivastava, H., Trujillo, J., Theory and Applications of Fractional Differential
Equations in: Math. Studies., North-Holland, New York, 2006.
[4] Podlubny, I., Fractional Differential Equations, Academic Press, USA, 1999.
[5] Mahdy, A. M. S., Shwayyea, R. T., Numerical solution of fractional integro-differential equations
by least squares method and shifted Laguerre polynomials pseudo-spectral method, International
Journal of Scientific & Engineering Research, 7(4), 1589-1596, 2016.
1 Pamukkale University, Denizli, Turkey, [email protected], [email protected].
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
111
Convergence of Backward Semi-Lagrangian Scheme with BDF2 for Two-Dimensional
Burgers’ Equation
Dojin KIM1, Soyoon BAK2, Philsu Kim3,
Abstract In this paper, we present a convergence analysis of a backward semi-Lagrangian
scheme for Burgers’ equation with Dirichlet boundary condition. We apply BDF2 for the total time
derivative and adopt the second-order central finite difference for the diffusion term together with the
local Lagrangian interpolation for the spatial discretization. For the temporal discretization, the
characteristic curve imposed by the solution of the Helmholtz equation is resolved by an error correction
method with the midpoint quadrature rule. The issues of the convergence analysis are resolved using the
discrete version of integration by parts combined with a symmetry property of the second-order central
finite difference and the Gronwall’s lemma to estimate the time evolution of the errors. Finally, we
show that the proposed scheme has the convergence order O(h2 + x2 + xp+1) in the sense of the
discrete l2-norm, where h and 4x are the step sizes of the time and the spatial grid, respectively, and p is
the degree of the interpolation polynomial. Numerical tests are presented to support the theoretical
analysis.
Keywords: Burgers’ equation, Semi-Lagrangian scheme, Non-linear advection-diffusion
equation, Convergence analysis
AMS subject classifications: 35K55, 65M06, 65M12, 65M25.
References
[1] Allievi, A., Bermejo, R., Finite element modified method of characteristics for the Navier-Stokes
equations, Int. J. Numer. Methods Fluids 32(4), 439-463, 2000.
[2] Bermúdez, A. , Nogueiras, M. R. , Vázquez, C., Numerical analysis of convection-diffusion-reaction
problems with higher order characteristics/finite elements I & III: time discretization, SIAM J. Numer.
Anal. 44, 2006.
[4] Falcone, M. , Ferretti, R., Convergence analysis for a class of high-order semi-Lagrangian
advection schemes, SIAM J. Numer. Anal. 35(3), 909-940, 1998.
[5] Kim, P., Piao, X., Kim, S. D., An error corrected Euler method for solving stiff problems based on
Chebyshev collocation, SIAM J. Numer. Anal. 49, 2211-2230, 2011.
1 Department of Mathematics, Kyungpook National University, Daegu, SOUTH KOREA, [email protected], 2 Department of Mathematics, Kyungpook National University, Daegu, SOUTH KOREA, [email protected], 3 Department of Mathematics, Kyungpook National University, Daegu, SOUTH KOREA, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
112
[6] Piao, X., Bu, S., Bak, S., Kim, P., An iteration free backward semi-Lagrangian scheme for solving
incompressible Navier-Stokes Equations, J. Comput. Phys. 283, 189-204, 2015.
[7] Piao, X. , Kim, S., Kim, P., An iteration free backward semi-Lagrangian scheme for Guiding center
problems, SIAM J. Numer. Anal. 53(1), 196-643, 2015.
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
113
Compressed Sensing with Cyclic-S Hadamard Matrix for TeraHertz Imaging
Applications
Esra Şengün ERMEYDAN1 and İlyas ÇANKAYA2
Abstract. In this study, for single pixel imaging applications, Compressed Sensing(CS) with
circular Hadamard matrix is proposed. In single pixel imaging scheme,𝑁2 samples should be taken for
√𝑁𝑥√𝑁 pixel image. CS is a popular technique claiming that the sparse signals can be reconstructed
with samples under Nyquist rate [1],[2]. CS precodes the analog signal, reduces acquisition time, power
consumption and hardware complexity since the image can be reconstructed using fewer samples.
Therefore to solve the slow data acquisition problem in Terahertz (THz) single pixel imaging, CS is a
good candidate [3]. In CS single pixel imaging, the beam is modulated via a number of √𝑁𝑥√𝑁 pixel
masks which is placed after the object and the imaged is reconstructed with generally Total variation
based algorithms using the acquired modulated signal and the information of the masks. The single pixel
cameras in optical band uses spatial light modulators(SLM) to realize the masks [4], however there is no
commercial SLM for THz. Therefore, the masks are printed on PCB to have ON/OFF switches, since
PCB material is considered as transparent for the THz beam and the copper material is considered
opaque. The mechanical PCB masks should be changed manually or mechanically with automatic
translation stage. Since each mask should be printed separately, most probably it will result more than
one PCB board depending on pixel size and resolution. The change of each mask or each PCB board for
each measurement is very disadvantageous especially for real-time or commercial real world THz
applications. To solve this problem, circular masks can be designed so that for each measurement only
one pixel column shift will be enough to change the mask [5]. In this study, the CS masks are designed
using cyclic-S matrixes based on Hadamard transform for 9x7 and 15x17 pixel images. The %50
compressed images are reconstructed using total variation based TVAL3 algorithm [6]. Matlab
simulations demonstrates that cyclic-S matrixes can be used for single pixel imaging based on CS. The
circular masks have the advantage to reduce the mechanical SLMs to a single sliding strip, whereas the
CS helps to reduce acquisition time and energy since it allows to reconstruct the image from fewer
samples.
Keywords. Compressed Sensing, Hadamard Transform, TeraHertz, single-pixel Imaging,
sparsity, total variation minimization.
AMS 2010. 53A40, 20M15.
References
1 Ankara Yıldırım Beyazıt University, Ankara, TURKEY, [email protected] 2 Ankara Yıldırım Beyazıt University, Ankara, TURKEY, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
114
[1] Donoho, D., Compressed Sensing, IEEE Transactions on Information Theory, 52(4), 1289–1306,
2006.
[2] Candes, E., Romberg J., Tao T., Robust Uncertainty Principles: Exact Signal Reconstruction from
Highly Incomplete Frequency Information, IEEE Transactions on Information Theory 52, 489–509,
2006.
[3] Chan, W. L., Charan, K., Takhar, D., Kelly, K. F., Baraniuk, R. G., Mittleman, D. M., A single-pixel
terahertz imaging system based on compressed sensing, Applied Physics Letters 93, 121105, 2008.
[4] Duarte, M. F., Davenport, M. A., Takbar, D., Laska, J. N., Sun, T., Kelly, K. F., Baraniuk, R. G.,
Single-pixel imaging via compressive sampling, IEEE Signal Processing Magazine 25, 83–91, 2008.
[5] Vasile, T., Damian, V., Coltuc, D., Petrovici, M., Single pixel sensing for THz laser beam profiler
based on Hadamard Transform, Optics & Laser Technology 79, 173–178, 2016.
[6] Li, C., Yin, W., Jiang, H., Zhang, Y., An efficient augmented lagrangian method with applications to
total variation minimization, Computational Optimization and Applications 56, 507–530, 2013.
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
115
Symbolic Regression: An Application to the Development of Models for the Surface
Tension
Eva T. López SANJUÁN1 and María Isabel Parra ARÉVALO2
Abstract. Genetic programming is an optimization technique based on stochastic, evolutionary
principles that is used to find global extreme of a given function [1]. In particular, symbolic regression
works with a set of possible expressions, represented as parse trees, and evolve them in order to
summarize the data response. It has emerged as one of the dominate modeling technology in recent
years, due to algorithmic and computer advances that have extended the scope of problems to which it
can be applied.
In this work, we study the symbolic regression algorithm and apply it to find a model for a
physical measure: the surface tension of fluids. We use a set of data, corresponding to the surface
tension of alcohols, that has been carefully selected and filtered in previous works [2], and perform
symbolic regression to develop mathematical expressions for the surface tension in terms of some
variables (temperature, critical temperature, critical pressure, molecular weight, and molar volume). We
analyze the validity of the models, and select the optimum ones.
Keywords. Symbolic Regression, Genetic Programming, Surface Tension.
AMS 2010. 62P35, 74A15, 97M10.
Acknowledgements This work was supported by GR15146 project from the Gobierno de Extremadura
and FEDER.
References
[1] Koza, J.R., Genetic Programming on the Programming of Computers by Means of Natural
Selection, MIT Press, 1998.
[2] Mulero, A., Cachadiña, I. and Sanjuán, E.L. Surface Tension of Alcohols. Data Selection and
Recommended Correlations, J. of Phys. and Chem. Ref. Data 44, 33104 2015.
1 University of Extremadura, Badajoz, SPAİN, [email protected] 2 University of Extremadura, Badajoz, SPAİN, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
116
On the Stability Analysis of Sturm-Liouville Operator with Jump Conditions
Etibar S. Panakhov1 and Ahu Ercan2
Abstract. In this paper we study stability of inverse problem for discontinuous Sturm-
Liouville equation using Ryabushko’s method. We have established certain stability of the
spectral functions of two spectral problems for discontinuous Sturm-Liouville operator. The
method which used was given firstly by Ryabushko for regular Sturm-Liouville problem in [2].
Keywords. Discontinuity, stability, spektral function.
AMS 2010. 34B09, 34D20.
References
[1] Marchenko, V. A., Maslov, K. V., Stability of the problem of recovering the Sturm-
Liouville operator from the spectral function, Mathematics of the USSR Sbornik, 81(123), 475-
502, 1970.
[2] Ryabushko T. I., Stability of the reconstruction of a Sturm–Liouville operator from two
spectra, II Teor. Funkts. Anal. Prilozhen, 18, 176–85, 1973 (in Russian).
[3] Hald, O. H., Discontinuous inverse eigenvalue problems, Comm. Pure Appl. Math., 37,
539-577, 1984.
[4] Panakhov, E. S., Sat, M., Reconstruction of potential function for Sturm-Liouville operator
with Coulomb potential, Boundary Value Problems, 49, 1-9, 2013.
1 Baku State University, Baku, Aizerbaijan, [email protected] 2 Firat University, Elazig, Turkey, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
117
Inverse Nodal Problem for p-Laplacian Diffusion Equation with Polynomially Dependent
Spectral Parameter
Etibar S. PANAKHOV1,2 Emrah YILMAZ3 and Tuba GULSEN4
Abstract. We consider the below p-Laplacian diffusion eigenvalue problem
( 1) 2 ( 1)( ) ( 1) ( ) 2 ( ) , 0 1,p p
m my p q x r x y x (1)
with the boundary conditions
(0) 0, (0) 1,y y (2)
(1, ) ( ) (1, ) 0,y f y (3)
where 1p is a constant,
2
1 2( ) ... , , 0, ,m
m i mf a a a a a m
is a spectral parameter; 2 1
20,1 , 0,1m mq x L r x W are real-valued functions defined
in the interval 0 1x for all m and
( 1)( 1) sgn ,ppy y y
is considered. To find the spectral datas as eigenvalues and nodal parameters of this problem,
we used a modified Prüfer substitution. Then, reconstruction formula of the potential function
is also given by using nodal lenghts. Furthermore, this method is similar to used in [6], our
results are more general.
Keywords. Inverse Nodal Problem, Prüfer Substitution, Diffusion Equation.
AMS 2010. 34A55, 34L05, 34L20.
References
1 Firat University, Elazig, Turkey, [email protected] 2 Baku State University, Baku, Azerbaijan, [email protected]
3 Firat University, Elazig, Turkey, [email protected] 4 Firat University, Elazig, Turkey, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
118
[1] Bairamov, E., Cakar O. and Celebi, A. O., Quadratic pencil of Schrödinger operators with
spectral singularities: discrete spectrum and principal functions, J. Math. Anal. Appl., 216(1),
303-320, 1997.
[2] Chen, H. Y., On generalized trigonometric functions, Master of Science, National Sun Yat-
sen University, Kaohsiung, Taiwan, 2009.
[3] Gulsen, T., Yilmaz, E., Inverse nodal problem for p_Laplacian diffusion equation with
polynomially dependent spectral parameter, Commun., Ser. A1; Math. Statist., 65(2), 23-36,
2016.
[4] Guseinov, G. S., On the spectral analysis of a quadratic pencil of Sturm-Liouville
operators, Sov. Math. Dokl., 32, 1292-1296, 1985.
[5] Jaulent, M. and Jean, C., The inverse s-wave scattering problem for a class of potentials
depending on energy, Commun. Math. Phys., 28(3), 177-220, 1972.
[6] Koyunbakan, H., Inverse nodal problem for p-Laplacian energy-dependent Sturm-Liouville
equation, Bound. Value Probl., 2013:272, 2013. (Erratum: Inverse nodal problem for p-
Laplacian energy-dependent Sturm-Liouville equation, Bound. Value Probl., 2014: 222, 2014.
[7] Sat, M. and Panakhov, E. S., Spectral problem for diffusion operator, Appl. Anal., 93 (6),
1178-1186, 2014.
[8] Wang, W. C., Cheng, Y. H. and Lian, W. C., Inverse nodal problems for the p-Laplacian
with eigenparameter dependent boundary conditions, Math. Comput. Modelling, 54(11-12),
2718-2724, 2011.
[9] Yang, C. F. and Yang, X. P., Inverse nodal problem for the Sturm-Liouville equation with
polynomially dependent on the eigenparameter, Inverse Prob. Sci. Eng., 19(7), 951-961, 2011.
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
119
On the Stress Distribution in the Elastic Body with a Locally Curved Triple Walled
Carbon Nanotube
Fatma COBAN1and Reşat KOSKER2
Abstract. In the framework of the piecewise homogeneous body model with the use of the
three-dimensional exact equations of the theory of elasticity, the method developed for the problem
related to the determination of the stress distribution in the infinite elastic body containing a single
locally curved triple-walled carbon nanotube (TWCNT) is considered [1]. The TWCNT is modeled as
concentrically-nested three circular locally curved hollow cylinders between which there is free space
[2]. It is assumed that on the inner surface of the outer tube (cylinder) and on the outer surface of the
inner tube (cylinder) of the TWCNT full slipping conditions satisfy. In addition to this, it is assumed
that the difference between the radial displacements of the adjacent surfaces of the tubes resists with van
der Waals forces [3].
It is assumed that thicknesses and radii of tubes of the TWCNT are constants along the tubes. Under this
assumption, surface equations and the components of its normal are written. Expressions of the shear
stresses are obtained from that. The boundary value problem is obtained by using strain-displacement
relations, equilibrium equations with boundary conditions. Mentioned boundary value problem is solved
by boundary form perturbation method and shear stresses are calculated. Effect of the various
parameters on the shear stresses are investigated and interpreted.
Acknowledgement: This research has been supported by Yıldız Technical University Scientific
Research Projects Coordination Department. Project Number: 2013-07- 03-DOP01.
Keywords: Triple walled carbon nanotube, local curving, shear stress
References
[1] Akbarov, S. D., Stability Loss and Buckling Delamination: Three-Dimensional Linearized Approach
for Elastic and Viscoelastic Composites, Springer, 465 pp., 2012.
[2] Coban F., Yerel Eğrilikli İki ve Üç Duvarli Karbon Nanotüplerin Gerilme ve Stabilite Analizi,
Doktora Tezi, 2016.
[3] Akbarov, S. D., Microbuckling of a Double-Walled Carbon Nanotube Embedded in an
Elastic Matrix, International Journal of Solids and Structures, 50 (2013) 2584- 2596, 2013.
1 Yildiz Technical University, Istanbul, TURKEY, [email protected] 2 Yildiz Technical University, Istanbul, TURKEY, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
120
Construction of Exact Solutions to Partial Differential Equations via CRE Method
Filiz TAŞCAN1 and Arzu AKBULUT2
Abstract. In this paper, the consistent Riccati expansion (CRE) method [1], [2] is presented for
constructing new exact solutions of (1+1) dimensional nonlinear dispersive modified Benjamin Bona
Mahony (DMBBM) and mKdV-Burgers equations. The obtained exact solutions are consist of
hyperbolic and exponential functions [3], [4]. The results obtained confirm that the proposed method is
an efficient technique for analytic treatment of a wide variety of nonlinear partial differential equations.
Keywords. Partial differential equations, exact solution, the consistent Riccati expansion.
AMS 2010. 35R50, 83C15.
References
[1] Xiang-Li, J., Sen-Yue, L., CRE method for solving mKdV equation and new interactions between
solitons and cnoidal periodic waves, Commun. Theor. Phys., 63, 7-9, 2015.
[2] Chen, M., Hu H., Zhu H., Consistent Riccati expansion and exact solutions of the Kuramoto-
Sivashinsky equation, App. Math. Lett., 49, 147-151, 2015.
[3] Khan, K., Akbar, M. A., Raynaul Islam S. M., Exact solutions for (1 + 1)-dimensional nonlinear
dispersive modified Benjamin-Bona-Mahony equation and coupled Klein-Gordon equations,
SpringerPlus, 3, 724, 2014.
[4] Wang, M., Exact solutions for a compound KdV-Burgers equation, Phys. Lett. A., 213, 279-287,
1996.
1 Eskişehir Osmangazi University, Eskişehir, TURKEY, [email protected] 2 Eskişehir Osmangazi University, Eskişehir, TURKEY, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
121
Numerical Solution of an Inverse Problem for the Stokes Equations
Gullazata DAIRBAYEVA1
Abstract. In the domain 𝑄 = (𝑥, 𝑦) ∈ 𝑅2: −2𝜋 < 𝑥 < 2𝜋, 𝑐𝑜𝑠𝑥 + 1 < 𝑦 < 𝑐𝑜𝑠𝑥 + 3 we
consider the Cauchy problem for the Stokes system
∆𝑢 − ∇𝑝 = 0, (1)
𝑑𝑖𝑣 𝑢 = 0, (2)
𝑢 = 𝜑, (𝑥, 𝑦) ∈ 𝛾0, (3)
𝑝𝑛 −𝜕𝑢
𝜕𝑛 = 𝑓, (𝑥, 𝑦) ∈ 𝛾0, (4)
where 𝑢 = (𝑢1, 𝑢2), 𝜕𝑄 = 𝛾0 ∪ 𝛾1 is the boundary of the domain 𝑄, 𝛾1 = (2𝜋, 𝑦): 𝑐𝑜𝑠𝑥 + 1 ≤
𝑦 ≤ 𝑐𝑜𝑠𝑥 + 3, 𝑛 = (𝑛1, 𝑛2) is the outward unit normal to 𝜕𝑄, 𝜑 ∈ 𝐻00
12⁄
(𝛾0), 𝑓 ∈ (𝐻00
12⁄
(𝛾0))
∗
. This
spaces are considered in [1].
The value of the solution is not known on 𝛾1. The problem (1)-(4) is ill-posed. It can be
formulated as the inverse problem to some well-posed problem. Analogous approach for other problems
can be found in [2].
Let’s consider the problem (1)-(3) with the following condition
𝑝𝑛 −𝜕𝑢
𝜕𝑛 = 𝑞, (𝑥, 𝑦) ∈ 𝛾1, (5)
where the function 𝑞 = (𝑞1, 𝑞2) is given. The problem (1)-(3), (5) we shall call the direct problem. This
problem is well-posed. Then the initial problem (1)-(4) is reduced to the following inverse problem to
the direct problem: it is required to determine 𝑞 on 𝛾1 using the additional information (4) about the
solution of the (1)-(3), (5).
The inverse problem (1)-(5) is solved numerically on the bases combination of the finite
element method [3] end the optimization method. The estimate on convergence rate of the algorithm
with respect to the functional is obtained.
Keywords. Stokes Equations, Inverse Problem.
References
[1] Bastay, G., Johansson, T., Lesnik. D., Kozlov, K., An Alternating Method for the Stationary Stokes
System, ZAAM (Z. Angew.Math.Mech). 86, 268-280. 2006/
[2] Kabanikhin, Sr., Inverse and ill-posed problems, The Siberian scientific publishing house, 2009.
[3] Segerlind, Larry J., Applied finite element analysis, New York, 1984.
1 Al-Farabi Kazakh National University, Almaty, KAZAKHSTAN, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
122
Existence of Solution for the Problem with a Concentrated Source in a Subdiffusive
Medium
H. Terence Liu1
1 Tatung University, Taipei, TAIWAN, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
123
Numerical Solution of the Conformable Fractional Diffusion Equation
Handan Çerdik YASLAN1
Abstract. In this work, a numerical approach for solving space-time fractional diffusion
equation with variable coeffcients is proposed by using shifted Chebyshev polynomials. The fractional
derivatives are described in the conformable sense. The space-time fractional dffusion equation with
variable coeffcients is reduced to a system of ordinary differential equations by using the properties of
Chebyshev polynomials. The finite difference method is applied to solve this system of equations.
Numerical results are provided to verify the accuracy and effciency of the proposed approach.
Keywords. Space-time fractional diffusion equation, Conformable fractional derivative, Shifted
Chebyshev polynomials, Finite difference method.
AMS 2010. 26A33, 35K05, 65M70.
References
[1] Khader, M. M., Numerical treatment for solving fractional Riccati dif ferential equation, J. Egyptian
Math. Soc. 21, 32-37, 2013.
[2] Li, Y., Solving a nonlinear fractional di_erential equation using Chebyshev wavelets, Commun.
Nonlinear Sci. Numer. Simulat. 15, 2284-2292, 2010
[3] Balaji, S., Solution of nonlinear Riccati di_erential equation using Chebyshev wavelets, WSEAS
Trans. Math. 13, 441-451, 2014.
[4] Dalir, M., Bashour, M., Applications of fractional calculus, Appl. Math. Sci. 4, 1021-1032, 2010.
[5] Sweilam, N.H., Nagy, A.M., El-Sayed, A. A., Second kind shifted Chebyshev polynomials for
solving space fractional order diffusion equation. Chaos Soliton Fract 73, 141-147, 2015.
[6] Sweilam, N. H., Nagy, A. M., El-Sayed, A. A., On the numerical solution of space fractional order
di_usion equation via shifted Chebyshev polynomials of the third kind. J King Saud Univ Sci 28, 41-47,
2016.
1Pamukkale University, Sakarya, TURKEY, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
124
Oscillation Criteria Fractional Difference Equations with Mixed Nonlinearities
Hande BÜYÜKÇAVUŞOĞLU1, Mustafa Kemal YILDIZ2 and Tuğba YALÇIN UZUN3
Abstract. In this article, we consider higher order fractional nonlinear difference equation of the
type
1
1
1
1
)(
10,0),()()()()()()(
atx
qattvtxtxtgtxtptx
at
q
ta
m
i
i
q
ta
i
where Δ𝑞 is Riemann-Liouville like discrete fractional difference operator of order q. We
obtain some oscillation criteria for this equation.
Keywords. Difference Equations, Oscillation, Nonlinear, Fractional Order.
AMS 2010. 26A33, 39A11, 39A12.
References
[1] Shao, J., Zheng, Z. and Meng, F., Oscillation criteria for higher order fractional differential
equations with mixed nonlinearities, Advanced in Difference Equations, 2013:323, 2013.
[2] Chen, D. X., Qu, P. X. and Lan, Y. H., Forced oscillation of certain fractional differential
equations, Advanced in Difference Equations, 2013:125, 2013.
[3] Grace, S. R., Agarwal, R. P., Wong, P. J. Y. and Zafer, A., On the oscillation of fractional
differential equations, An International Journal for Theory and Applications, v:15, n:2, 2012.
[4] Diethelm, K., The Analysis of Fractional Differential Equations, Springer, Berlin 2010.
.
1 Afyon Kocatepe University, Afyonkarahisar, TURKEY, [email protected] 2 Afyon Kocatepe University, Afyonkarahisar, TURKEY, [email protected] 3 Afyon Kocatepe University, Afyonkarahisar, TURKEY, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
125
Free Vibration of a Tapered Euler-Bernoulli Beam with 3D Tip Mass by Multi-Step
Differential Transformation Method
Hilal Doğanay KATI1 and Hakan GÖKDAĞ2
Abstract. Free vibration of beam-rigid tip mass system has been studied by many investigators
[1-10]. The interest in the dynamic analysis of such a structure is partly because of its suitability for
providing successful design for structures such as robot arms and manipulators. In many studies the
vibration of uniform beam-tip mass system has been dealt with analytically. However, there are few
studies about non-uniform beam-tip mass systems. Besides, majority of these works assume the tip mass
as concentric and coincident with beam end point. Thus, in this work, coupled flexural and torsional free
vibration of a tapered beam with rigid 3D tip mass whose center of gravity is, in general, not at the beam
end point is considered with the assumption of the Euler-Bernoulli beam theory. First, a set of governing
differential equations and boundary conditions are obtained for the considered system by Hamilton’s
Principle. Since the exact solution of the equations of motion is not easy, a semi-analytical method
called Multi-Step Differential Transform Method (MDTM) is applied. Later, the effects of tip mass,
beam length and taper ratio on the natural frequencies are examined. In addition, the results are verified
by ANSYS, famous finite element software. It is proved that the results by MDTM and ANSYS are in
well agreement.
Keywords. Differential Transform Method, Multi-Step Differential Transform Method,
Bending Vibration, Torsional Vibration, Coupled Deformation
AMS 2010. 53A40, 20M15.
References
[1] Bhat, B. R., and Wagner, H., Natural frequencies of a uniform cantilever with a tip mass slender in
the axial direction, Journal of Sound and Vibration, 45, 2, 304-307, 1976.
[2] Auciello, N. M., Transverse Vibrations of a Linearly Tapered Cantilever Beam with Tip Mass of
Rotatory Inertia and Eccentricity, Journal of Sound and Vibration, 194, 1, 25-34, 1996.
[3] Hoa, S. V., Vibration of a Rotating Beam With Tip Mass, Journal of Sound and Vibration, 67, 3,
369-381, 1979.
[4] Oguamanam, D. C. D., Free Vibration of beams with Finite Mass Rigid Tip Load and flexural-
torsional coupling, International Journal of Mechanical Sciences, 45, 963-979, 2003.
1 Bursa Technical University, Bursa, TURKEY, [email protected] 2 Bursa Technical University, Bursa, TURKEY, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
126
[5] Gökdağ, H., Kopmaz, O., Coupled Bending and Torsional Vibration of a Beam with inspan and tip
attachments, Journal of Sound and Vibration, 287, 3, 591-610, 2005.
[6] Oguamanam, D.C.D., Arshad, M., On the natural frequencies of a flexible manipulator with a tip
load, Proceedings of the Institution of Mechanical Engineers, 219, 1199-1205, 2005.
[7] Salarieh, H., Ghorashi, M., Free Vibration of Timoshenko Beam with Finite Mass Rigid Tip Load
and Flexural Torsional Coupling, International Journal of Mechanical Science, 48, 763-779, 2006.
[8] Mabie, H. H., Rogers, C. B., Transverse vibrations of double-tapered cantilever beams with end
support and with end mass, Journal of the Acoustical Society of America, 55, 5, 986-991, 1974.
[9] Lee, T. W., Transverse Vibrations of a Tapered Beam Carrying a Concentrated Mass, Applied
Mechanics, 43, 2, 366-367, 1976.
[10] Kati H. D., Gokdag H., Free vibration of a Timoshenko beam carrying three dimensional tip mass:
Analytical solution and experimental modal testing, Material Testing, 59, 6, 591-597, 2017.
[11] Liu Z., Yin Y., Wang F., Zhao Y., Cai L., Study on modified differential transform method for free
vibration analysis of uniform Euler-Bernoulli beam, Structural Engineering and Mechanics, 48, 5, 697-
709, 2013.
[12] Özdemir Ö., Kaya M.O., Flapwise bending vibration analysis of a rotating tapered cantilever
Bernoulli-Euler Beam by differential transform method, Journal of Sound and Vibration, 289, 413-420,
2006.
[13] Torabi K., Afshari H., Zafari E., Transverse Vibration of Non-uniform Euler-Bernoulli Beam Using
Differential Transform Method (DTM), Applied Mechanics and Materials, 110-116, 2400-2405, 2012.
[14] Ertürk V. S, Odibat Z. M., Monami S., The Multi-Step Differential Transform Method and ıts
Application to Determine the Solutions of Non-Linear Oscillators, Adv. Appl. Math. Mech., 4, 4, 422-
438, 2012.
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
127
Image Segmentation with Fan-shaped Gradients
Hosook KIM1 and Hyoungseok KIM2
Abstract. Image segmentation is the process of partitioning a digital image into multiple
segments, which plays a fundamental role in computer vision [1]. The goal of segmentation is to find
out the boundaries of more meaningful regions in images. The pixels in a region are similar with respect
to some characteristic such as color, intensity, or texture. So the sudden changes occur at the pixels of
boundaries of the region.
The most important process of image segmentation is how to exactly and efficiently find out the
sudden changes. In a mathematical sense, a digital image is a two-dimensional one-valued function
f: N × N → R, where N is the set of natural numbers and R is the set of real numbers. Since the domain
is a discrete space, there are some difficulties in applying directly the concept of gradient of continuous
space to image segmentation in order to find out the sudden changes. The domain has only restricted
directions such as horizontal and vertical directions. The pixels of images may be in general grouped
because an image is acquired by taken a picture of real-world consisting of isolated objects. The
neighbor pixels have similar intensities and are influenced by each other.
In this research, we propose a new fan-shaped gradient of image which is one variety of
grouped gradients. To confirm pros and cons of our gradient, we apply the fan-shaped gradient to the
level-set algorithm proposed by Li et. al [2].
Keywords. Gradient, Image Segmentation.
AMS 2010. 68U10, 68R01.
References
[1] Szeliski, R., Computer Vision: Algorithms and Applications, Springer, New-York, 2011.
[2] Li, C., Xu, C., Cui, C., Fox, M., Distance Regularized Level Set Evolotion and Its Application to
Image Segmentation, IEEE Trans. On Image Processing, Vol. 19, No. 12, pp 3243-3254, 2010.
1 KSA of KAIST, Busan, REP. OF KOREA, [email protected] 2 Dong-Eui University, Busan, REP. OF KOREA, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
128
The Computational and Experimental Studies on (E)-4-Methyl-N'-(1-(Pyridin-2-Yl)
Ethylidene) Benzenesulfonohydrazide
Hümeyra BATI1, Erbil Murat AYDIN2, Murat ÇINARLI3, Nezihe ÇALIŞKAN4
Abstract. In this work, the single crystal structure, (E)-4-methyl-N'-(1-(pyridin-2-
yl)ethylidene)benzenesulfonohydrazide (I), has been synthesized and characterized by IR, UV spectra
and X-ray diffraction methods. The structure was solved by direct methods using SHELXS-97 and
refined by a full-matrix least-squares procedure using the SHELXL-97 program [1]. The crystal
structure of the compound (I) obtained in P 21/c space monoclinic space group, from the results of X –
ray diffraction. In addition, the optimized structure, the vibrational assignments, the molecular orbital
energies, molecular electrostatic potential maps have been investigated by using Density Functional
Theory with B3LYP/6-311G basis set. Gaussview molecular visualization program [2] and the Gaussian
03 program [3] were used to perform DFT calculations. UV–Visible spectrum of the compound was
recorded and the electronic properties HOMO and LUMO energies were measured by time-dependent
TD-DFT approach. The experimental results of the compound have been compared with theoretical
results and it is found that the experimental data show good agreement with calculated values.
Keywords. Hydrazone, DFT calculations, crystal structure analysis.
References
[1] Sheldrick, G. M., SHELXS-97 and SHELXL-97, Univ. Göttingen, Germany, 1997.
[2] Frish, A., Nielseni, A.B., Holder, A.J., Gaussview User Manuel, Gaussian Inc., Pittsburg, 2001.
[3] Frisch, M.J., Trucks, G.W., Schlegel, H.B., Scuseria, G.E., Robb, M.A., Cheeseman, J.R.,
Zakrzewski, V.G., Montgomery, J.A., Stratmann, R.E., Burant, J.C., Dapprich, S., Millam, J.M.,
Daniels, A.D., Kudin, K.N., Strain, M.C., Farkas, O., Tomasi, J., Barone, V., Cossi, M., Cammi, R.,
Mennucci, B., Pomelli, C., Adamo, C., Clifford, S., Ochterski, J., Petersson, G.A., Ayala, P.Y.,Cui, Q.,
Morokuma, K., Malick, D.K., Rabuck, A.D., Raghavachari, K., Foresman, J.B., Cioslowski, J., Ortiz,
J.V., Baboul, A.G., Stefanov, B.B., Liu, G., Liashenko, A., Piskorz, P., Komaromi, I., Gomperts, R.,
Martin, R.L., Fox, D.J., Keith, T., Al- Laham, M.A., Peng, C.Y., Nanayakkara, A., Gonzalez, C.,
Challacombe, M., Gill, P.M.W., Johnson, B., Chen, W., Wong, M.W., Andres, J.L., Gonzalez, C., Head
Gordon, M., Replogle, E.S., Pople, J.A., GAUSSIAN 03, Revision B.02, Wallingford, CT, 2004.
1 Ondokuz Mayis University, Samsun, Turkey, [email protected] 2 Gazi University, Ankara, Turkey, [email protected] 3 Ahi Evran University, Kırşehir, Turkey, [email protected] 4 Gazi University, Ankara, Turkey, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
129
An Application of Integration Interpolation Method to Cauchy Type Singular Integral
Equation System
Hüseyin OĞUZ1 and Elçin YUSUFOĞLU2
Abstract: In this study, a numerical solution method for the Cauchy type singular
integral equation system was developed by using the Gauss quadrature formula and Lagrange
interpolation polynomial.
The proposed algorithm is applied to a multi contact problem in order to show the
effectiveness of the algorithm.
Keywords: Singular Integral Equation, Plane Contact Problem, Cauchy kernel, Gauss–Jacobi
quadrature, A system of linear algebraic equation,
References
[1] Muskhelishvili, N. I, Singular İntegral Equations, Wolters-Noordhoff Publishing, Groningen, 1958.
[2] Erdoğan, F., Approximate Solutions of Systems of Singular Integral Equations, Society for Industrial
and Applied Mathematics, 17, 1041-1059, 1969.
[3] F. Erdogan and G.D. Gupta, Cook T.S, The numerical solution of singular integral equations, Q. J.
Appl. Math. 29, 525–534, 1973.
1 Dumlupınar University, Kütahya, TURKEY, [email protected] 2 Uşak University, Uşak, TURKEY, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
130
A New Analysis for Fractional Order Ebola Model with Non-Local and Non-Singular
Kernel
Ilknur KOCA
Abstract. The mathematical model portraying the spread of Ebola hemorrhagic fever is
considered here within the scope of fractional differentiation with non-local and non-singular kernel.
This fatal disease first occurred in Central Africa near tropical rainforests in 1976. Using classical
differentiation and fractional differentiation with power law kernel, the model was established. However
the model with classical differentiation does not take into account the memory effect as well as the
complexity of the dynamical system of the spread. On the other hand the model with fractional
differentiation based on power law kernel does not accurately take into account the full memory effect
due to the bad memory induced by the power law. Different from other works, we extend model by
replacing derivative with newly established derivative with fractional order called Atangana-Baleanu
derivative in Caputo sense. Detailed analysis of existence and uniqueness of exact solution is presented
using the Banach fixed point theorem
Keywords. fractional derivative, epidemic model, fixed point theorem.
AMS 2010. 34A34, 47H10, 65L07.
References
[1] http://www.who.int/mediacentre/factsheets/fs103/en/June 2017
[2] Odibat, Z. M., Momani, S., Application of variational iteration method to nonlinear di¤erential
equation of fractional order, International Journal of Nonlinear Sciences and Numerical Simulations, 7,
27-34, 2006.
[3] Atangana, A., Baleanu, D., New fractional derivatives with nonlocal and non-singular kernel:
Theory and application to heat transfer model, Thermal Science, 20(2), 763-769, 2016.
[4] Atangana, A., Koca, I., Chaos in a simple nonlinear system with Atangana-Baleanu derivatives with
fractional order, Chaos Solitons Fractals, 89, 447-454, 2016.
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
131
Linearization of the State Equation of a Nonlinearly Elastic Material of a Circular
Composite Shaft for Estimation of Natural Frequencies of Torsional Vibrations
I.A. TARASYUK1 and A.S. KRAVCHUK2
Abstract. The present work is devoted to generalization of the torsional vibrations equation of a
circular composite shaft [1, 2] to the case of nonlinearly elastic behavior of the material. The purpose of
the research is to obtain analytically approximate expression for natural frequencies depending on
deformation characteristics of material components which can be used in engineering and construction
designing.
The generalization has been carried out using new homogenization technique [3]. The method is
based on obtaining the effective characteristics of a composite, on average isotropic material from the
rule of a mixture for the state equations got by applying the Voigt’s [5] and the Reuss’ [6] hypotheses of
strains and stresses homogeneity, respectively. Nonlinearly elastic behavior of the material has been
modelled by using the bilinear Prandtl’s diagrams as the state equations of shaft components.
Relations for the effective secant and tangent shear moduli, and proportionality limits have been
obtained in the explicit form. Using the root-mean-square approximation of the effective bilinear
Prandtl’s diagram, estimated expressions for natural frequencies being functions of nonlinearly elastic
characteristics of the composite shaft have been determined in the closed form for the following
simplest cases:
1. the case of the smallness of nonlinear deformation region in comparison with elastic region;
2. the case of the small difference between the effective secant and tangent shear moduli.
The analysis of performed calculations has discovered that nonlinearity can be neglected in the
indicated simplest cases, since only pronounced nonlinearity of the effective state equation has a
significant effect on natural frequencies values.
Keywords. Torsional vibrations, composite material, nonlinear elasticity, effective
characteristics, homogenization.
AMS 2010. 74Q10, 74Q15, 74Q20.
References
1 Belarusian State University, Minsk, BELARUS, [email protected] 2 Belarusian State University, Minsk, BELARUS, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
132
[1] Kravchuk, A.S., Kravchuk, A.I., Tarasyuk, I.A., Equation of torsional vibrations of a circular
longitudinally fibrous, cross layered and structurally heterogeneous composite shaft (in Russian),
Vestnik VGU (Fiz. Mat. Ser.) 4, 53-62, 2015.
http://www.vestnik.vsu.ru/pdf/physmath/2015/04/2015-04-15.pdf
[2] Kravchuk, A.S., Kravchuk, A.I., Tarasyuk, I.A., Rheology of a circular structurally heterogeneous
composite shaft under torsion and torsional vibrations (in Russian), Zavodskaya Laboratoriya.
Diagnostika Materialov. 11, 147-159, 2015.
http://zldm.ru/content/article.php?ID=2070
[3] Tarasyuk, I.A., Kravchuk, A.S., Reducing the Voigt-Reuss range in the theory of elastic structurally
heterogeneous composite, on average isotropic solids without application of variational principles (in
Russian), APRIORI (Est. i Tehn. Nauki Ser.) 3 [online], 2014.
http://apriori-journal.ru/seria2/3-2014/Tarasyuk-Kravchuk.pdf
[4] Gorshkov, A.G., Starovoitov, E.I., Yarovaya, A.V., Mechanics of laminated viscoelastoplastic
structural elements (in Russian), Fizmatlit, Moscow, 2005.
[5] Voigt, W., Textbook of crystal physics (in German), Teubner, Berlin, 1928.
[6] Reuss, A., 1929 Calculation of the flow limit of mixed crystals due to the plasticity condition for
single crystals (in German), ZAMM 1, 49-58, 1929
http://dx.doi.org/10.1002/zamm.19290090104
[7] Peters, S.T. Handbook of Composites, Chapman & Hall, London, 1998.
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
133
Parameter Dependent Generalized Sylvester and T-Sylvester Equations
Ivana KUZMANOVIĆ1 and Ninoslav TRUHAR2
Abstract. In this talk Sherman-Morrison-Woodbury-type formulas for the solution of
generalized Sylvester equation of the form
(𝐴0 + 𝜇𝑈1𝑉1)𝑋𝐸1 + 𝐸2𝑋(𝐵0 + 𝜇𝑈2𝑉2) = 𝐸
as well as for generalized T-Sylvester equation of the form
(𝐴0 + 𝜇𝑈1𝑉1)𝑋𝐸1 + 𝐸2𝑋𝑇(𝐵0 + 𝜇𝑈2𝑉2) = 𝐸
will be presented, where 𝑈1, 𝑈2, 𝑉1 and 𝑉2 are low-rank matrices and 𝜇 ∈ 𝑅
is parameter.
Developed formulas can be used to obtain algorithms for efficient calculation of the solution of
structured generalized Sylvester and T-Sylvester equations for many values of parameter 𝜇 as well as
for optimization of the solution with respect to 𝜇.
One specific problem where optimization of parameter dependent Sylvester equation (or more
specific, Lyapunov equation) appears is a problem of damping optimization in mechanical systems (see
[1,2]).
Keywords. Matrix equations, Sylvester equation, parameter dependent problems.
AMS 2010. 65F99, 65Z05.
References
[1] Kuzmanović, I., Truhar, N., Optimization of the solution of the parameter-dependent Sylvester
equation and applications, Journal of Computational and Applied Mathematics, 237, 1, 136-144, 2013
[2] Kuzmanović, I., Truhar, N., Sherman-Morrison-Woodbury formula for Sylvester and T-Sylvester
equation with applications, International Journal of Computer Mathematics 90, 2,306-324, 2013
1 Department of Mathematics, University of Osijek, Osijek, CROATİA, [email protected] 2 Department of Mathematics, University of Osijek, Osijek, CROATİA, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
134
Generalization of the Linear Momentum Operator in Quantum Mechanics: A Position-
Dependent Effective Mass Approach
J. J. PEÑA1, J. MORALES2, J. GARCÍA-RAVELO3, Leonardo SALINAS4
Abstract. The problem of constant mass in quantum mechanics is well known, in this context,
the linear momentum operator p and the position operator x satisfy the conmutation relation [ x , p ]=iℏ,
likewise the uncertainty relation is 𝛥x𝛥p≥ℏ/2. Nevertheless, when the mass distribution m(x) depends on
the position, the linear momentum operator p no longer commute with m(x). For example, to the
particular case m(x)=(1+𝛾x)-2 [1], the modified linear momentum opertor becomes p𝛾=- iℏ(1+ 𝛾x)d/dx
which leads to [ x , p𝛾 ]=iℏ(1+𝛾x), a similar result is obtained for the uncertainty relation 𝛥x𝛥p𝛾
≥ℏ/2(1+𝛾‹x›). In this work, by using the commutation relation [d/dx , m(x)]=dm(x)/dx, we show that it is
possible to obtain a generalized structure for the linear momentum operator under the condition of the
hermiticity of the Hamiltonian [2] from a position-dependent effective mass approach.
Keywords. Position-dependent effective mass, modified linear momentum operator.
AMS 2010. 53B50, 35J10.
References
[1] Costa Filho, R. N. et.al., Displacement operator for quantum systems with position-dependent mass.
, Phys. Rev. A, 84, 050102(R), 2011.
[2] Levy-Leblond, J-M., Elementary quantum models with position-dependent mass, Eur. J. Phys. 215-
218, 1992.
1 Universidad Autonoma Metropolitana Azc, CDMX, MÉXİCO, [email protected] 2 Universidad Autonoma Metropolitana Azc, CDMX, MÉXİCO, [email protected] 3 Instituto Politécnico Nacional ESFM, CDMX, MÉXİCO, [email protected] 4 Instituto Politécnico Nacional ESFM, CDMX, MÉXİCO, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
135
On Trace Inequalities for Generalized Quasi-Metric Adjusted Skew Informations
Kenjiro YANAGI1
Abstract. In this talk we investigate some uncertainty relations for generalized quasi-metric
adjusted skew informations. We give generalized Heisenberg type or Schrodinger type uncertainty
relations. As applications we have new trace inequality related to fidelity and trace distance.
THEOREM(Extension of [3],[1]). For ,, ( )nA B M the following hold:
1 1
1 2 20 1
1( | | ) inf ( ) ( )
2A BTr A B L R I Tr A B Tr A B
2
21 11 1( ) ( ) (1 ) | | ,
2 2A BTr A B A B Tr A B Tr L R I
Where ( ) , ( )A AL X AX R X XA .
Key words and phrases: generalized quasi-metric adjusted skew information
2010 Mathematics Subject Classification: 15A45, 47A63, 91A17
The author was partially supported by JSPS KAKENHI Grant Number 26400119.
References
[1] Audenaert K. M. R., Calsamiglia J., Masancs L. I., Munnoz-Tapia R., Acin A., Bagan E. and
Verstraete F., The quantum Chernoff bound, Phys.Rev.Lett., 98, 160501, 2007.
[2] Audenaert K. M. R., Nussbaum M., Szkola A. and Verstraete F., Asymptotic error rates in quantum
hypothesis testing, Commun.Math.Phys., 279, 251-283, 2008.
[3] Powers R. T. and Stormer E., Free states of the canonical anticommutation relations, Commun.
Math. Phys., 16, 1-33, 1970.
[4] Yanagi K., Furuichi S. and Kuriyama, K., Uncertainty relations for generalized metric adjusted
skew information and generalized metric adjusted correlation measure, J.Uncertainty Analysis and
Applications, 1, 1-14, 2013.
1 Josai University, Sakado, Saitama, JAPAN, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
136
[5] Yanagi K. and Sekikawa, K., Non-hermitian extensions of Heisenberg type and Schrodinger type
uncertainty relations, J.Inequalities and Applications, 381, 1-9,2015.
[6] Yanagi, K., Generalized trace inequalities related to fidelity and trace distance, Linear and
Nonlinear Analysis, 2, 263-270, 2016.
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
137
The Solutions of Special Second Order Ordinary Differential Equations Using
Symmetries
Kısmet KASAPOĞLU1
Abstract. In this study, certain symmetries of second order ordinary differential equations 𝑦′′ +
𝑃(𝑥)𝑦′ + 𝑞(𝑥)𝑦 = 0 are found for 𝑝(𝑥) = 2
𝑥 and special cases of 𝑞(𝑥) and then these symmetries are
generalized. Also by using these symmetries, these equations are reduced to the first order ordinary
differential equations and analytic solutions of these equations are obtained and generalized.
Keywords. Symmetry, Ordinary differential equations.
AMS 2010. 76M60, 34B60, 34A05.
References
[1] Stephani, H., Differential Equations Their Solution Using Symmetries, Cambridge University Press,
New York, 2003.
[2] Polyanin, A. D., Zaitsev, V. F., Handbook of Exact Solutions for Ordinary Differential Equations,
Chapman & Hall/CRC. , New York, 2003.
[3] Starrett, J., Solving Differential Equations by Symmetry Groups, Mathematical Association of
America, 114, 9, 778-792, 2007.
[4] Oliveri, F., Lie Symmetries of Differential Equations: Classical Results and Recent Contributions,
Symmetry, 2, 8, 658-706, 2010.
[5] Robert, G., Lie Groups, Lie Algebras, and Some of Their Applications, John Wileyond. Sons, Inc.,
New York, 1974.
[6] Hydon, P. E., Symmetry Methods for Differential Equations, Cambridge University Prees, New
York, 2000.
[7] Zwillinger D., Handbook of Differential equations, Acedemic Pres, San Diego, 1989.
1Trakya University, Edirne, TURKEY, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
138
Schrödinger Equation with Variable Effective Mass: Linear and Staggered Mass
Distributions
Leonardo SALINAS1, H. LUNA-GARCÍA 2, J. GARCÍA-RAVELO3, J. J. PEÑA4
Abstract. The problem of position-dependent effective mass applied to the Schrödinger
equation has been studied under different formalisms on condition to the hermiticity of the Hamiltonian
[1]. Different approaches have been used in the treatment of this subject [2]. In this work, a point
canonical transformation method is used for transforming the Schrödinger equation with variable mass
in to a constant mass problem [3]. A combination of linear and staggered mass distribution for solving
the Schrödinger equation with position-dependent mass, has been used. Furthermore, the simplest case
of free particle leads to a new potential generated from the linearity properties of the mass distribution
Keywords. Schrödinger equation, position-dependent effective mass, point canonical
transformation.
AMS 2010. 53A40, 20M15.
References
[1] von Roos, O, Position-dependent effective mass in semiconductor theory, Phys. Rev. B, 27, 7547,
1983.
[2] Bagchi, B., Gorain, P., Quesne, C., Roychoudhury, R., A general scheme for the effective-mass
Schrödinger equation and the generation of the associated potentials, Mod Phys Lett A, 19, No. 37,7
2765-2775, 2004.
[3] Pacheco-García, C., García-Ravelo, J., Morales, J., Peña, J.J. Exactly solvable effective mass
Schrödinger equation with Coulomb-like potential, Int. J Quant. Chem. Vol 110, 2880–2885, 2010.
1 Instituto Politécnico Nacional, ESFM, CDMX, MÉXİCO, [email protected] 2 Universidad Autonoma Metropolitana Azc, CDMX, MÉXİCO, [email protected] 3 Instituto Politécnico Nacional, ESFM CDMX, MÉXİCO, [email protected] 4 Universidad Autonoma Metropolitana Azc, CDMX, MÉXİCO, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
139
Weak Solutions to Interdiffusion Models with Vegard Rule
Lucjan SAPA1, Bogusław BOŻEK2 and Marek DANIELEWSKI3
Abstract. We study the one and multidimensional models of interdiffusion. Let a region Ω ⊂
ℝ𝑛, 𝑛 ∈ ℕ with the smooth boundary 𝜕Ω, 𝑇 > 0 and 𝑟 ∈ ℕ ∖ 1 be fixed. Moreover, let diffusion
coefficients 𝐷𝑖, partial molar volumes Ω𝑖, initial concentrations 𝑐0𝑖 and flows on the boundaries 𝑗𝑖 of the
𝑖th components of a mixture, 𝑖 = 1, … , 𝑟 be given. The unknowns are concentrations 𝑐𝑖 and the potential
𝐹 of a drift velocity 𝑣𝑑.
The local mass conservation law for fluxes with the Darken drift term and the Vegard rule lead
to the parabolic-elliptic system of strongly coupled nonlinear differential equations
𝜕𝑡𝑐𝑖 = 𝑑𝑖𝑣(𝐷𝑖(𝑐1, … , 𝑐𝑟)∇𝑐𝑖 − 𝑐𝑖∇𝐹) 𝑜𝑛 [0, 𝑇] × Ω,
∆𝐹 = 𝑑𝑖𝑣(∑ Ω𝑘𝑟𝑘=1 𝐷𝑘(𝑐1, … , 𝑐𝑟)∇𝑐𝑘) 𝑜𝑛 [0, 𝑇] × Ω,
∫ 𝐹𝑑𝑥 = 0Ω
𝑜𝑛 [0, 𝑇]
(1)
with the initial condition and the coupled nonlinear boundary conditions
𝑐𝑖(0, 𝑥) = 𝑐0𝑖(𝑥) 𝑜𝑛 Ω, (2)
−𝐷𝑖(𝑐1, … , 𝑐𝑟)
𝜕𝑐𝑖
𝜕𝜈+ 𝑐𝑖
𝜕𝐹
𝜕𝜈= 𝑗𝑖(𝑡) 𝑜𝑛 [0, 𝑇] × 𝜕Ω,
𝜕𝐹
𝜕𝜈= ∑ Ω𝑘 (𝐷𝑘(𝑐1, … , 𝑐𝑟)
𝜕𝑐𝑘
𝜕𝜈+ 𝑗𝑘(𝑡)) 𝑜𝑛 [0, 𝑇] × 𝜕Ω𝑟
𝑘=1 , (3)
𝑖 = 1, … , 𝑟. System (1) in the one dimensional case can be reduced to the parabolic one
𝜕𝑡𝑐𝑖 = 𝜕𝑥 (𝐷𝑖(𝑐1, … , 𝑐𝑟)𝜕𝑥𝑐𝑖 − 𝑐𝑖(∑ Ω𝑘𝐷𝑘(𝑐1, … , 𝑐𝑟)𝜕𝑥𝑐𝑘𝑟𝑘=1 + 𝐾(𝑡))) 𝑜𝑛 [0, 𝑇] × Ω (4)
with the initial condition (2) and the boundary conditions
−𝐷𝑖(𝑐1, … , 𝑐𝑟)𝜕𝑐𝑖
𝜕𝜈+ 𝑐𝑖 (∑ Ω𝑘𝐷𝑘(𝑐1, … , 𝑐𝑟)
𝜕𝑐𝑘
𝜕𝜈𝑟𝑘=1 + 𝐾(𝑡)𝜈) = 𝑗𝑖(𝑡) 𝑜𝑛 [0, 𝑇] × 𝜕Ω, (5)
𝑖 = 1, … , 𝑟, where 𝐾 is a some calculated function.
The theorems about existence, uniqueness and properties of weak solutions in the Sobolev
spaces to the problems (4), (2), (5) and (1)-(3) are formulated and proved. The main tool used in the
proofs of the existence results is the Galerkin approximation method. The agreement between the
theoretical results, numerical simulations and experimental data is shown.
1 AGH University of Science and Technology, Faculty of Applied Mathematics, Cracow, POLAND, [email protected] 2 AGH University of Science and Technology, Faculty of Applied Mathematics, Cracow, POLAND, [email protected] 3 AGH University of Science and Technology, Faculty of Materials Science and Ceramics, Cracow, POLAND,
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
140
Keywords. Interdiffusion, Darken method, Vegard rule, parabolic-elliptic nonlinear system,
existence, uniqueness, properties of global weak solutions, Galerkin approximation.
AMS 2010. 35K61, 35K51, 35D30, 35Q70.
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
141
Self-Adjoint Extensions of Differential Operators with Potentials-Point Interactions
Manaf Dzh. MANAFOV1
Abstract. In this study, operators generated by the most general formally self-adjoint
differential expression of even order on the Hilbert space are studied and the main result consists in an
explicit construction of a boundary triplet for the associated symmetric minimal quasi-differential
operator. Notice that the quasi-differential operators were introduced first by Shin [2] and then
essentially developed by Zettl [3], see also the monograph [1] and references therein. After the
regularization of the formal differential expression using the quasi derivatives, the minimal and maximal
operators corresponding to potentials of this type on a finite interval are constructed. All self-adjoint
extensions of the minimal operator are described.
Keywords. Self-adjoint extensions of differential operators, Quasi-differential expressions,
Potentials-point interactions.
AMS 2010. 34B24, 34B05, 47N20.
References
[1] Naimark, M. A., Linear differential operators, Part II Linear differential operators, Frederick Ungar
Publ. Co., New York, 1969.
[2] Shin, D., Quasi-differential operators in Hilbert space, Math. Sb (N. S.) 13 (55), 1, 39-70, 1943 (in
Russian).
[3] Zettl, A., Formally self-adjoint quasi-differential operators, Rocky Mountain J. Math., 5, 3, 453-
474, 1975.
1 Adıyaman University, Adıyaman, TURKEY, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
142
Detecting a Hyperbolic Quadratic Eigenvalue Problem Using a Subspace Algorithm
Marija MILOLOŽA PANDUR1
Abstract. The quadratic eigenvalue problem (QEP) is to find scalars 𝛼 and nonzero vectors 𝑥
such that 𝑄(𝛼)𝑥 = 0 holds for a given quadratic matrix polynomial 𝑄(𝛼) = 𝛼2𝑀 + 𝛼𝐷 + 𝐾, where
𝑀, 𝐷, 𝐾 are complex matrices of order 𝑛. Such 𝛼 is called an eigenvalue, and 𝑥 the associated
eigenvector. A very important is hyperbolic QEP appearing for example in dynamical analysis of
structures, such as a damped mass-spring oscillator. QEP is hyperbolic if 𝑀, 𝐷, 𝐾 are Hermitian
matrices, 𝑀 is positive definite and there exists a real 𝜇 such that 𝑄(𝜇) is negative definite matrix. Any
such 𝜇 is called a definitizing shift. In this case, QEP has 2𝑛 real eigenvalues: 𝑛 neg-type eigenvalues
smaller than 𝜇 and 𝑛 pos-type eigenvalues greather than 𝜇. There is a number of algorithms that solves
the whole or partial hyperbolic QEP. Also, there is a number of algorithms that detect if a given QEP
with Hermitian matrices, is hyperbolic or not. In this talk we present a new algorithm for detecting a
hyperbolic QEP that is suited for large and sparse QEP. Our subspace algorithm is based on iterative
testing of small compressed QEPs formed using test-subspaces of small dimensions. The algorithm has
a monotonicity property based on Cauchy-type interlacing inequalities [1]. If a small nonhyperbolic
compressed QEP is find, the algorithm ends with the conclusion that the given large QEP is
nonhyperbolic. The algorithm finds a definitizing shift if the given large QEP is hyperbolic. In this case,
the definitizing shift can be used to form the preconditioner for the LOBPCG-type algorithm given in
[2].
Keywords. hyperbolic quadratic eigenvalue problem, definitizing shift, subspace algorithm.
AMS 2010. 65F30, 15A18
References
[1] K. Veselić, Note on interlacing for hyperbolic quadratic pencils, Recent Advances in Operator
Theory in Hilbert and Krein Spaces, Oper. Theory: Adv. Appl., Vol. 198, 305–307, Birkhäuser, Boston
2009.
[2] X. Liang, R.-C. Li, The hyperbolic quadratic eigenvalue problem, Forum Math. Sigma, 3, e13 (93
pages), 2015.
1University of Osijek, Osijek, CROATIA, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
143
A Generalization of Midpoint Type Inequality of m-Logarithmically Convex Functions
Mehmet Eyüp KİRİŞ 1
Abstract. In this study, we establish generalized new inequalities for differantiable
mappings that are connected with the celebrated Hermite-Hadamard integral inequality by
using m - log -convex function.The results presented here would provide extensions of those
given in earlier works.
Keywords. Hermite-Hadamard's inequalities, Riemann-Liouville fractional integral,
integral inequalities.
AMS 2010. 26B25, 35A23, 26D10, 52A41,
References
[1] Azpeitia, A.G., Convex functions and the Hadamard inequality, Rev. Colombiana Math.,
28, 7-12, 1994.
[2] Bakula M. K., and Pečarić, J., Note on some Hadamard-type inequalities, Journal of
Inequalities in Pure and Applied Mathematics, vol. 5, no. 3, article 74, 2004.
[3] Belarbi S., and Dahmani, Z., On some new fractional integral inequalities, J. Ineq. Pure
and Appl. Math., 10(3), Art. 86, 2009.
[4] Dahmani, Z., New inequalities in fractional integrals, International Journal of Nonlinear
Scinece, 9(4), 493-497, 2010.
[5] Dahmani, Z., On Minkowski and Hermite-Hadamard integral inequalities via fractional
integration, Ann. Funct. Anal. 1(1), 51-58, 2010.
[6] Dahmani, Z., Tabharit, L., Taf, S., Some fractional integral inequalities, Nonl. Sci. Lett. A,
1(2), 155-160, 2010.
[7] Dahmani, Z., Tabharit, L., Taf, S., New generalizations of Gruss inequality usin Riemann-
Liouville fractional integrals, Bull. Math. Anal. Appl., 2(3), 93-99, 2010.
1 Afyon Kocatepe University, Afyonkarahisar, Turkey, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
144
[8] Deng J., and Wang, J., Fractional Hermite-Hadamard inequalities for ( m, )-
logarithmically convex functions. J. Inequal.Appl. 2013, Article ID 364, 2013.
[9] Dragomir S. S. and Pearce, C. E. M., Selected Topics on Hermite-Hadamard Inequalities
and Applications, RGMIA Monographs, Victoria University, 2000.
[10] Dragomir S. S. and Agarwal, R.P., Two inequalities for differentiable mappings and
applications to special means of real numbers and to trapezoidal formula, Appl. Math. lett.,
11(5), 91-95, 1998.
[11] Gorenflo, R., Mainardi, F., Fractional calculus: integral and differential equations of
fractional order, Springer Verlag, Wien, 223-276, 1976.
[12] Kilbas, A. A., Srivastava H. M., and Trujillo, J. J., Theory and Applications of Fractional
Differential Equations, North-Holland Mathematics Studies, 204, Elsevier Sci. B.V.,
Amsterdam, 2006.
[13] Latif, M.A., Dragomir S.S., and Matouk, A.E., New inequalities of Ostrowski type for co-
ordinated convex functions via fractional integrals, J. Fract. Calc. Appl. 2, 1-15, 2012.
[14] Miller S. and Ross, B., An introduction to the Fractional Calculus and Fractional
Differential Equations, John Wiley & Sons, USA, 2, 1993.
[15] Pečarić, J.E., Proschan F., and Tong, Y.L., Convex Functions, Partial Orderings and
Statistical Applications, Academic Press, Boston, 1992.
[16] Podlubni, I., Fractional Differential Equations, Academic Press, San Diego, 1999.
[17] Sarikaya M.Z. and Ogunmez, H., On new inequalities via Riemann-Liouville fractional
integration, Abstract and Applied Analysis, Volume 2012, Article ID 428983, 10 pages, 2012.
[18] Sarikaya, M.Z., Set, E., Yaldiz H. and Basak, N., Hermite -Hadamard's inequalities for
fractional integrals and related fractional inequalities, Mathematical and Computer Modelling,
DOI:10.1016/j.mcm.2011.12.048, 57, 2403—2407, 2013.
[19] Sarikaya, M.Z., Filiz H. and Kiris, M.E., On some generalized integral inequalities for
Riemann-Liouville fractional integrals, Filomat, 29:6, 1307—1314, 2015.
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
145
[20] Sarikaya M.Z. and Yildirim, H., On Hermite-Hadamard type inequalities for Riemann-
Liouville fractional integrals,Miskolc Mathematical Notes, Vol. 17, No. 2, pp. 1049—1059,
2016.
[21] Zhang Y. and Wang, J., On some new Hermite-Hadamard inequalities involving Riemann-
Liouville fractional integrals, J. Inequal. Appl., Article ID 220, 2013.
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
146
Some Approximation Results for Lupaş-Kantorovich-Schurer Operators Based on (p, q )-
Calculus
Melek SOFYALIOĞLU1 and Kadir KANAT2
Abstract. The purpose of this presentation is to introduce Lupaş-Kantorovich-Schurer operators
based on (p,q)-integers. We investigate approximation properties for these operators according to
Korovkin type approximation theorem. We prove some auxiliary results which will be needed to
establish the main results. Moreover, we estimate the rate of convergence by using modulus of
continuity, with the help of functions of Lipschitz class and Peetre’s K-functionals.
Keywords. (p,q)-integers, Lupaş operators, rate of convergence.
AMS 2010. 41A10, 41M25, 41A36.
References
[1] Mursaleen, M., Nasiruzzaman, M., Nurgali, A., Some approximation results on Bernstein-Schurer
operators defined by (p,q)-integers, Journal of Inequal. Appl. (2015), 249, 2015.
[2] Khalid Khan, D.K. Loyibal, Bezier curves based on Lupaş (p,q)-analogue of Bernstein functions in
CAGD, Journal of Comp. and Applied Mathematics, Volume 317, 458-477, 2017.
[3] Mursaleen, M., Ansari, K.J., Khan, A., On(p,q)-analogue of Bernstein operators, Applied
Mathematics and Computation 266 (2015a) 874882.( Erratum to On (p,q)-analogue of Bernstein
Operators [Appl.Math.Comput.266 (2015) 874882], Applied Mathematics and Computation 278(2016)
7071.)
[4] Lupaş, A., A q-analogue of the Bernstein operator, in: Seminar on Numerical and Statistical
Calculus, No. 9, Univ. of Cluj-Napoca, 1987.
1 Gazi University, Ankara, TURKEY, [email protected] 2 Gazi University, Ankara, TURKEY, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
147
Conformable Fractional Klein-Gordon and Cahn-Allen Equations
Meryem ODABASI1
Abstract. Numerous types of physical models can be described by fractional order differential
equations. Because of the applications in chemistry, biology, elasticity, acoustics, fluid dynamics, solid
mechanics, optics, propagation of shallow water waves, quantum field theory and others, traveling wave
solutions of the fractional differential equations have become more important. Hence, a lot of methods
and different definitions of fractional derivatives have been used so far. Recently, a new definition
called conformable fractional derivative have been proposed. In the present study, conformable time-
fractional Klein-Gordon and Cahn-Allen equations are investigated in the sense of conformable
derivative. Using the modified trial equation method, some exact traveling wave solutions of these
equations that can allow us to understand the phenomena they describe have been obtained.
Keywords. Conformable fractional differential equations, traveling wave solutions.
AMS 2010. 35R11, 35C07.
References
[1] Kilbas, A.A., Srivastava, H.M. and Trujillo, J.J., Theory and applications of fractional differential
equations, Elsevier: Amsterdam, 2006.
[2] Khalil, R., Horani, M.A., Yousef, A. and Sababheh, M., A new definition of fractional derivative,
Journal of Computational and Applied Mathematics, 264, 65–701, 2014.
[3] Hosseini, K., Mayeli P. and Ansari, R., Modified Kudryashov method for solving the conformable
time-fractional Klein–Gordon equations with quadratic and cubic nonlinearities, Optik, 130, 737–742,
2017.
[4] Liu, C.S, A new trial equation method and its applications, Communications in Theoretical Physics,
45, 3, 395-397, 2006.
[5] Odabasi, M. and Misirli, E., On the solutions of the nonlinear fractional differential equations via
the modified trial equation method, Mathematical Methods in the Applied Sciences, Doi:
10.1002/mma.3533, 2015.
1 Ege University, Tire Kutsan Vocational School, İzmir, TURKEY, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
148
On the Finite Element Approximation in the Maximum Norm of Elliptic Quasi-
Variational Inequalities with Nonlinear Source Terms
Messaoud BOULBRACHENE1
Abstract. We are concerned with the standard finite element approximation of the semilinear
quasi-variational inequality (QVI) of obstacle type: Find u ∈ H¹(Ω) such that:
a(u,v-u) ≥ (f(u),v-u) ∀ v ∈ H¹(Ω)
u ≤ ψ(u), v ≤ ψ(u)
where, Ω is a bounded convex domain of RN , N ≥ 1, with sufficiently smooth boundary Γ, f is a positive
Lipschitz and nondecreasing nonlinearity with Lipschitz constant c, ψ(u) is an implicit obstacle, which
justifies the terminology quasi-variational inequality, (.,.) is the inner product in L²(Ω), a(.,.) is a
coercive bilinear form associated with linear second order elliptic differential operator, such that the
zero order term satisfies a₀(x)≧ β> 0 ∀ x ∈Ω.
In [1], by means of a fixed point approach, we derived error estimates, in the maximum norm,
between the exact solution and its finite element counterpart, under the assumption that c/β <1. In this
paper, we drop this assumption and derive error estimate in the same norm, under the sole Lipschitz
condition on the nonlinearty f(.), which enables us to encompass a larger class of semilinear QVIs. For
that, we introduce a new method that combines an iterative scheme of Bensoussan-Lions type [2] and
the concept of subsolutions for elliptic variational inequalities [3].
Keywords. Quasi-variational-inequalities, Iterative Scheme, Subsolutions, Finite Elements,
Error estimates.
AMS 2010. 65N15, 65N30.
References
[1] Boulbrachene, M., Pointwise error estimates for a class of elliptic quasi-variational inequalities
with nonlinear source terms. Applied Mathematics and Computation, 161, 129-138, 2005.
[2] Bensoussan, A., Lions, J. L., Impulse control and quasi-variational inequalities. Gauthiers-Villars,
Paris, 1984.
[3] Bensoussan, A., Lions, J.L., Applications des inequations variationnelles en controle stochastique.
North Holland, 1978.
1 Sultan Qaboos University, Muscat, OMAN, [email protected]; [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
149
Exponentially Cusped Prismatic Shells in the 0N Approximation of I. Vekua’s
Hierarchical Models
Miranda GABELAIA1
Abstract. Bending problem of prismatic shell with the thickness as follows
)(0
22
21 xx
ehh
, 0),,(,0,0 210 xxconstconsth ,
is investigated. The solution of the posed boundary value problem is given in an explicit form.
Static problem of the prismatic shell with the following thickness
20
xehh
, 0),,(,0,0 210 xxconstconsth ,
is investigated as well.
Keywords. Cusped Prismatic Shells, Hierarchical Models, Partial Differential Equations.
AMS 2010. 74K20, 74K25, 35J47.
1 Iv. Javakhishvili Tbilisi State University, Tbilisi, GEORGİA, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
150
m m1
∈
∇
. Σ
Bounds for Blow up Time in Nonlinear Pseudo-Parabolic Equations
Müge MEYVACI1
Abstract. The point of this paper is to analyze the following initial boundary value problem:
ut − ∆ut − ∆u − u ux1 + g(t, x, u, ∇u) = |u| u, x ∈ Ω, t > 0, (1)
u(x, 0) = u0(x), x ∈ Ω, (2)
u (x, t) = 0, x ∈ ∂Ω, t ≥ 0, (3)
where m and m1 are given numbers, g(t, x, u, w) satisfies the property:
|g(t, x, u, w)| ≤ M0 |u|k + |w| , (x, t) ∈ Ω × R, u ∈ R, w ∈ RN , (4)
where M0, k are given numbers and Ω RN (N > 3) is a bounded domain with sufficiently smooth
boundary ∂Ω. During the last decades, the pseudo-parabolic equations have been contemplated by
numerous authors by different aspects. In recent years an increasing number of researchers focused in
the upper and lower bound for blow up time for parabolic and pseudo-parabolic equations. So we obtain
the sufficient conditions on initial data, m, m1 and the function g(t, x, u, u) for nonblow up case.
Moreover we decide the lower and upper bounds for the blow up time, if blow up happens.
Key words. Pseudo-parabolic equation, sobolev equation, lower bound, upper bound, blow-up,
global nonexistence.
AMS 2010. 35B40, 35B44.
References
[1] Korpusov, M. O. and Sveshnikov, A. G., Blow-Up of Solutions of Nonlinear Sobolev Type
Equations with Cubic Sources. Differential Equations, 42, 3, 431–443, 2006.
[2] Lu, Y., L. Fei, L., Bounds for blow-up time in a semilinear pseudo-parabolic equation with nonlocal
source. J. Ineq. Appl., 2016:229, 2016.
[3] Luo, P., Blow-up phenomena for a pseudo-parabolic equation. Math. Meth. Appl. Sci., 38, 12,
2636-2641, 2015.
1 Mimar Sinan Fine Art University, Istanbul, TURKEY, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
151
[4] Mei, M., Lq-Decay Rates of Solutions for Benjamin-Bona-Mahony-Burgers Equations Journal of
Differential Equations 158, 314–340, 1999.
[5] Peng, X., Shang, Y., Zheng, X., Blow-up phenomena for some nonlinear pseudo-parabolic
equations. Appl. Mat. Lett. 56, 17–22, 2016.
[6] Showalter, R. E, Ting, T. W., Pseudoparabolic Partial Differential Equations. SIAM J. Math. Anal.
1, 1–26, 1970.
[7] Showalter, R. E., Weak solutions of nonlinear evolution equations of Sobolev-Galpern type. Journal
of Differential Equations, 11 2, 252–265, 1972.
[8] Xu, R., Su, J., Global existence and finite time blow-up for a class of semilinear pseudo-parabolic
equations. Journal of Functional Analysis 264 12, 2732–2763, 2013.
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
152
Oscillation Criteria for Higher Order Fractional Difference Equations with Nonlinearities
Mustafa Kemal YILDIZ 1 , Ebru YILMAZ2 and Umut Mutlu ÖZKAN3
Abstract. This paper deals with some oscillation criteria of forced nonlinear fractional difference
equations of the form
Δ∗𝛼𝑥(𝑡) − 𝑔(𝑡 + 𝛼 − 1, 𝑥(𝑡 + 𝛼 − 1)) + 𝑓1(𝑡 + 𝛼 − 1, 𝑥(𝑡 + 𝛼 − 1)) = 𝜐(𝑡)
Δ∗𝛼𝑥(𝑡) − 𝑔(𝑡 + 𝛼 − 1, 𝑥(𝑡 + 𝛼 − 1)) + 𝑓1(𝑡 + 𝛼 − 1, 𝑥(𝑡 + 𝛼 − 1)) + 𝑓2(𝑡 + 𝛼 − 1, 𝑥(𝑡 + 𝛼 − 1))
= 𝜐(𝑡)
where Δ∗𝛼 is a Caputo like discrete fractional difference operator, 𝑡 ∈ ℕ𝑎, 𝑚 − 1 < 𝛼 ≤ 𝑚,
𝑔, 𝑓𝑖: [0, +∞) × ℝ → ℝ, 𝑖 = 1,2 ve 𝜐: [0, +∞) → ℝ are continuous with respect 𝑡 and 𝑥 and 𝑁𝑎 =
𝑎, 𝑎 + 1, 𝑎 + 2, . . . .
Keywords. Difference Equations, Oscillation, Nonlinear, Fractional Order.
AMS 2010. 26A33, 39A11, 39A12.
References
[1] Marian, S. L., Sagayaraj, M. R., Maria Selvam, A. G., M. P. Loganathan Oscillation of caputo like
discrete fractional equations, International Journal of Pure and Applied Mathematics, v:89, n:5, 2013.
[2] Marian, S. L., Sagayaraj, M. R., Maria Selvam, A. G., M. P. Loganathan Oscillation of fractional
nonlinear difference equations, Mathematica Aeterna, v:2, n:9, 2012.
[3] Chen, D. X., Qu, P. X. and . Lan, Y. H., Forced oscillation of certain fractional differantial
equations, Advanced in Difference Equations, 125, 2013.
[4] Grace, S. R., Agarwal,R. P., Wong P. J. Y. and Zafer, A., On the oscillation of fractional differential
equations , An International Journal for Theory and Applications, v:15, n:2, 2012.
[5] Han, Z., Zhao, Y., Sun, Y., Zhang, C., Oscillation for a class of fractional differential equation,
Hindawi Publishing Corporation, DiscreteDynamics in Nature and Society, Volume 2013.
[6] Anastassiou, G. A., Discrete fractional calculus and inequalities, http://arxiv.org/abs/0911.3370 v:1.
[7] Chen, F., Luo, X., Zhou, Y., Existence results for nonlinear fractional difference equations,
Advances in Difference Equations,Volume, Article ID 713201, 12 pages, 2011.
[8] Atici, F. M., Eloe, P. W., Initial value problems in discrete fractional calculus, Proceedings of the
American Mathematical Society,v:137, n:3, pp:981-989, 2009.
1 Afyon Kocatepe University, Afyonkarahisar, TURKEY, [email protected] 2 Afyon Kocatepe University, Afyonkarahisar, TURKEY, [email protected] 3 Afyon Kocatepe University, Afyonkarahisar, TURKEY, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
153
[9] Alzabut, J. O., T. Abdeljawad, T., Sufficient Conditions For The Oscillation Of Nonlinear
Fractional Difference Equations, Journal of Fractional Calculus and Applications, v:5 pp:177-187,
2014.
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
154
Lyapunov Type Inequalities for nth-order Three point BVP Differential Equations with
Mixed Nonlinearities
Naceri MOSTEPHA1 and Abdullah OZBEKLER2
Abstract. In this paper, we present some new Lyapunov and Hartman type in equalities
for second order equations with mixed nonlinearities:
𝑥(𝑛)(𝑡) + 𝑝(𝑡)|𝑥(𝑡) | 𝛽−1𝑥(𝑡) + 𝑞(𝑡)|𝑥(𝑡) | 𝛾−1𝑥(𝑡) = 0,
where 𝑝(𝑡), 𝑞(𝑡) are real-valued functions and 0<𝛾<1<𝛽<2. No sign restrictions are imposed
on the potential functions 𝑝(𝑡)and 𝑞(𝑡). The inequalities obtained generalize and compliment
the existing results for the special cases of this equation in the literature.
Keywords. Lyapunov type inequality, Mixed nonlinear, Sub-linear.
AMS 2010. 34C10, 34B15, 34B40.
References
[1] A. M. Liapunov, Probleme général de la stabilité du mouvement, (French Translation of a
Russian paper dated 1893), Ann. Fac. Sci. Univ. Toulouse 2 (1907), 27-247, Reprinted as Ann.
Math. Studies, No. 17, Princeton, 1947.
[2] A. Wintner, On the nonexistence of conjugate points, Amer. J.73, 368—380, 1951.
[3] P. Hartman, Ordinary Differential Equations, Wiley, New York, 1964 and Birkhäuser,
Boston 1982.
[4] A. Beurling, Un théorème sur les fonctions bornées et uniformément continues sur l’axe
réel. Acta Math.77, 127—136, 1945.
1 Oran school of economics, Oran, ALGERİA. [email protected] 2 Atilim University 06836, Incek, Ankara, TURKEY. [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
155
3D FEM Analysis of Buckling of a Piezoelectric Rectangular Thick Plate
Nazmiye YAHNIOGLU1 and Fatih AYLIKCI2
Abstract. In this work a buckling problem of a sandwich PZT/Metal/PZT rectangular thick
plate within the scope of the 3D linearized theory of stability loss in the framework of the piecewise
homogeneous body model is studied. It is supposed that an ideal contact conditions are satisfied
between the layers and all the lateral surfaces of the plate is simply supported mechanically and also
grounded for the PZT layers' surface only. In addition, it is assumed that on the upper and lower surface
of the plate neither mechanically load nor electrically load are act. However, this plate is subjected to bi-
axial uniformly-distributed compressive mechanical forces acting on two edge surfaces.
In the analyzing procedure, before the plate is loaded (i.e. in the natural state), the free planes of
the plate have insignificant initial imperfections. Due to action of the aforementioned compressive
forces the evolution of the initial imperfections is investigated and as a result of this investigation the
values of the critical buckling forces for the considered sandwich plate are found from the criteria,
according to which, the initial imperfections grow indefinitely with the compressive forces [1-
3]. Mathematical modeling of the considered problem is formulated within the scope of the three
dimensional exact geometrically nonlinear equations of electro-elasticity in the framework of the
piecewise homogeneous body model. The corresponding boundary-value problems are solved
numerically by employing the 3D finite element method (3D-FEM).
Acknowledgement: This work was supported by Research Fund of the Yildiz Technical
University. Project Number: 2016-07-03-DOP03.
Keywords: Buckling, piezoelectric plate, 3D FEM
References
[1]. Akbarov, S. D., Stability Loss and Buckling Delamination: Three-Dimensional Linearized
Approach for Elastic and Viscoelastic Composites, Springer. Heidelberg, New York, 2013.
[2]. Akbarov, S. D., Yahnioglu, Y., Buckling delamination of a sandwich plate-strip with
piezoelectric face and elastic core layers, Appl. Math. Model. 37, 8029 – 8038, 2013.
[3]. Cafarova, F.I., Akbarov, S.D., Yahnioglu, N., Buckling delamination of the
PZT/Metal/PZT sandwich circular plate-disc with penny-shaped interface cracks, Smart
Structures and Systems, 19 (2) (2017) 163 – 179, 2017.
1 Yildiz Technical University, Istanbul, TURKEY, [email protected] 2 Yildiz Technical University, Istanbul, TURKEY, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
156
Identities for the double L22-integral transform
Neşe DERNEK1, Fatih AYLIKÇI2
Abstract. In this paper, the authors introduce the double Laplace-type integral transform L22
and its properties. Several simple theorems dealing with general properties of the L22-integral transform
are proved. The convolution, its properties and convolution theorem are given. The main focus of this
paper is to develop a method for the L22-integral transform to solve problems in applied mathematics
which involve integral equations.
Keywords: Laplace transform, Double Laplace transform, Convolution, Integral equations
2010 MSC: 44A10, 44A30, 44A35, 33B20, 45A05
References
[1] Aghili, A., Ansari, A., and Sedghi., A., An inversion technique for the L2-transform with
applications, Int. J. Contemp. Math. Sciences 2.28 1387-1394, 2007.
[2] Churchill, R., Operational mathematics 3rd edn., Mc Graw Hill, New York 1972.
[3] Debnath, L., The double Laplace transform and their properties with applications to functional,
integral and partial differential equations, Inter. J. App. Comput. Math. Vol.2 Issue 2 p.223-241, 2016.
[4] Debnath, L., Bhatta, D., Integral transforms and their applications 3rd edn., CRC Press, Chapman
& Hall, Boca Ratan, 2015.
[5] Dernek, N., Aylikci,F., Identities for the Ln-transform, the L2n-transform and the P2n-transform and
their applications, Journal of Inequalities and Special Functions Vol.5 Issue 4 1-16, 2014.
[6] Dernek, N., Aylikci,F., Kivrak, S., An alternative technique for solving ordinary differential
equations, Konuarlp Journal of Mathematics Vol.4 No.1 68-79, 2016.
[7] Erdelyi, A. et. al., Tables of integral transforms, Mc Graw Hill Book Company Vol.1 and Vol.2
(1954)
[8] Schiff, J.L., The Laplace transforms, Springer New York, 1999.
1 Marmara University, Istanbul, TURKEY, [email protected] 2 Yildiz Technical University, Istanbul, TURKEY, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
157
[9] Sneddon, T.N., Fourier transforms, Mc Graw Hill New York, 1951.
[10] Yurekli, O., New identities involving the Laplace and the L2-transform and their applications,
Applied Mathematics and Computation 99 141-151, 1999.
[11] Yurekli, O., Sayginsoy, O., A theorem on a Laplace-type integral transform and its applications,
Internat. J. Math. Ed. Sci. Tech. 29 561-567, 1998.
[12] Yurekli, O., Sadek, I., A Parseval-Goldstein type theorem on the Widder potential transform and
its applications, Internat. J. Math. Ed. Sci. Tech. 14 517-524, 1991.
[13] Yurekli, O., Scott, W., A new method of solving Bessel's di_erential equation using L2-transform,
Applied Mathematics and Computation 130 587-591, 2002.
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
158
Simplified Hirota Method for a Nonlinear Partial Differential Equation
Ömer ÜNSAL1, Ahmet BEKIR2, Murat KOPARAN3
Abstract. In the recent past, Ma introduced and obtained complexiton solutions for the
Korteweg-de Vries equation through its bilinear form [1], [2]. After finding of this novel class of
explicit exact solutions, some new exact solution methods have been developed and used to get
complexiton solutions of nonlinear partial differential equations. For instance, extended transformed
rational function method [3], multiple Riccati equations rational expansion method [4], generalized sub-
equations rational expansion method [5], generalized compound Riccati equations rational expansion
method [6].
In this paper, we apply the simplified Hirota method to a nonlinear partial differential equation
[7], [8]. Through the bilinear form, complexiton solutions and interaction solutions are obtained. Some
graphs associated with the solutions are illustrated. This approach can also be applied to other nonlinear
partial differential equations of which bilinear form is obtained.
Keywords. Simplified Hirota Method, Complexiton Solution, Nonlinear Partial Differential
Equation.
AMS 2010. 35C99, 35Q53.
Acknowledgements. This work was supported by Eskisehir Osmangazi University Scientific
Research Projects (Grant No. 2016-1079).
References
[1] Ma, W. X., Complexiton solutions to the Kortweg-de Vries equation, Phys. Lett. A, 301, 35-44,
2002.
[2] Ma, W. X., Complexiton solutions to integrable equations, Nonlinear Anal., 63 e2461-e2471, 2005.
[3] Zhang, H., Ma, W. X., Extended transformed rational function method and applications to
complexiton solutions, Applied Mathematics and Computation, 230, 509-515, 2014.
[4] Chen, Y., Wang, Q., Multiple Riccati equations rational expansion method and complexiton
solutions of the Whitham-Broer-Kaup equation, Physics Letters A, 347, 215-227, 2005.
1 Eskisehir Osmangazi University, Eskisehir, TURKEY, [email protected] 2 Eskisehir Osmangazi University, Eskisehir, TURKEY, [email protected] 3 Anadolu University, Eskisehir, TURKEY, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
159
[5] Li, W., Zhang, H., A generalized sub-equatons rational expansion method for nonlinear evolution
equations, Commun. Nonlinear. Sci. Numer. Simulat., 15, 1454-1461, 2010.
[6] Li, W., Zhang, H., A new generalized compound Riccati equations rational expansion method to
construct many new exact complexiton solutions of nonlinear evolution equations with symbolic
computation, Chaos, Solitons and Fractals, 39, 2369-2377, 2009.
[7] Wazwaz, A. M., Zhaqilao, Nonsingular complexiton solutions for two higher-dimensional fifth order
nonlinear integrable equations, Physica Scripta, 88, 025001, 2013.
[8] Wazwaz, A.M., New solutions for two integrable cases of a generalized fifth-order nonlinear
equation, Modern Physics Letters B, 29, 1550065, 2015.
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
160
Extended Transformed Rational Function Method for Some (3+1) Dimensional Nonlinear
Partial Differential Equations
Ömer ÜNSAL1, Ahmet BEKIR2, Murat KOPARAN3
Abstract. Recently, some exact solution methods have been developed and employed to obtain
complexiton solutions of nonlinear partial differential equations [1], [2], [3], [4], [5], [6], [7]. In this
paper, we apply the extended transformed rational function method to two (3+1) dimensional nonlinear
partial differential equations [8]. Obtained solutions contain trigonometric and hyperbolic functions and
it is shown that the considered transform and method are very efficient and powerful in solving other
kinds of nonlinear partial differential equations.
Keywords. Extended Transformed Rational Function Method, Complexiton Solution,
Nonlinear Partial Differential Equation.
AMS 2010. 35C99, 35Q53.
Acknowledgements. This work was supported by Eskisehir Osmangazi University Scientific
Research Projects (Grant No. 2016-1079).
References
[1] Ma, W. X., Complexiton solutions to the Kortweg-de Vries equation, Phys. Lett. A, 301, 35-44,
2002.
[2] Ma, W. X., Complexiton solutions to integrable equations, Nonlinear Anal., 63 e2461-e2471, 2005.
[3] Chen, Y., Wang, Q., Multiple Riccati equations rational expansion method and complexiton
solutions of the Whitham-Broer-Kaup equation, Physics Letters A, 347, 215-227, 2005.
[4] Li, W., Zhang, H., A generalized sub-equatons rational expansion method for nonlinear evolution
equations, Commun. Nonlinear. Sci. Numer. Simulat., 15, 1454-1461, 2010.
[5] Li, W., Zhang, H., A new generalized compound Riccati equations rational expansion method to
construct many new exact complexiton solutions of nonlinear evolution equations with symbolic
computation, Chaos, Solitons and Fractals, 39, 2369-2377, 2009.
1 Eskisehir Osmangazi University, Eskisehir, TURKEY, [email protected] 2 Eskisehir Osmangazi University, Eskisehir, TURKEY, [email protected] 3 Anadolu University, Eskisehir, TURKEY, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
161
[6] Wazwaz, A. M., Zhaqilao, Nonsingular complexiton solutions for two higher-dimensional fifth order
nonlinear integrable equations, Physica Scripta, 88, 025001, 2013.
[7] Wazwaz, A.M., New solutions for two integrable cases of a generalized fifth-order nonlinear
equation, Modern Physics Letters B, 29, 1550065, 2015.
[8] Zhang, H., Ma, W. X., Extended transformed rational function method and applications to
complexiton solutions, Applied Mathematics and Computation, 230, 509-515, 2014.
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
162
Some Results on Cooperative Grey Games
Osman PALANCI1, Rana S.M. BAKRI2, Marwah Imad Najm NAJM3, Sirma Zeynep
ALPARSLAN GOK4
Abstract. This contribution is located in the common area of Operations Research and
Economics, with a close relation and joint future potential with optimization and cooperative game
theory. We study some game theoretic solutions on the interesting class of cooperative games where the
coalitional values are interval grey numbers. These class of games are called cooperative grey games.
Further, we deal with an axiomatization of the grey Shapley value. We introduce the Banzhaf value and
the egalitarian rule by using the theory of cooperative grey games.
Keywords. Shapley value, Banzhaf value, egalitarian rule, grey uncertainty, fairness property,
economics.
1 Suleyman Demirel University, Isparta, TURKEY, [email protected] 2 Suleyman Demirel University, Isparta, TURKEY, [email protected] 3 Suleyman Demirel University, Isparta, TURKEY, [email protected] 4 Suleyman Demirel University, Isparta, TURKEY, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
163
A Cubic Subdomain Galerkin Method over the Geometrically Graded Mesh to the
Singularly Perturbed Problem
Ozlem Ersoy HEPSON1 and Idris DAG2
Abstract. In this paper, subdomain Galerkin method is set up to find solutions of singularly
perturbed boundary value problems which are used widely in many areas such as chemical reactor
theory, aerodynamics, quantum mechanics, reaction-diffusion process, optimal control, etc. A
combination of the cubic B-spline base functions as an approximation function is used to build up the
presented method over the geometrically graded mesh. Thus finer mesh can be established through the
end parts of the problem domain where steep solutions exist.
Keywords. Subdomain Galerkin, graded mesh, spline, singularly perturbed.
AMS 2010. 41A15, 65L60.
References
[1] Kumar, M., Singh, P., Mishra, H. K., A recent survey on computational techniques for solving
singularly perturbed, International Journal of Computer Mathematics, 84(10), 1439-1463, 2007.
[2] Dag, I., Sahin, A., Numerical solution of the Burgers’ equation overgeometrically graded mesh,
Kybernetes, 8, 2649-2656, 2001.
[3] Dag, I., Sahin, A., Numerical solution of singularly perturbed problems, Int. Journal of Nonlinear
Science, 8 (1), 32-39, 2009.
[4] Gupta, Y., Kumar, M., Anthology of spline based numerical tecniques for singularly perturbed
boundary value problem, International Journal of Pure and Applied Mathematics 74(4), 437-453, 2012.
1 Eskişehir Osmangazi University, Eskişehir, TURKEY, [email protected] 2 Eskişehir Osmangazi University, Eskişehir, TURKEY, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
164
Optimal Control of Distributed Systems in Problems of Quartz Optical Fiber Production
Pervadchuk V.P.1, Vladimirova D.B. 2, Gordeeva I.V.3
Abstract. The problems of optimal control of distributed systems arise in different fields of
science and technology [1]. Therefore, in the last decades the efforts of many researchers are aimed at
solving these problems. They are of special importance for high-precision processes and technologies
which include the production of special quartz optical fibers.
Fiber production involves two main stages: the first stage is quartz tube alloying (by Modified
Chemical Vapour Deposition (MCVD), as a rule), and the second one is fiber drawing from a preform.
In the MCVD process the temperature field distribution plays a decisive role. For the description of
temperature field distribution, two-dimensional and one-dimensional mathematical models of quartz
tube heating have been proposed in our research. The purpose of control was to minimize the deviation
of the actual (measured) temperature field on the outer surface of the tube from the programmed
(prescribed) temperature field. In this case, the heat flux from the moving burner was controlled. Two
problems of optimal control of distributed systems were formulated: a two-dimensional problem with
boundary control and a one-dimensional problem with distributed control. The optimality systems for
each of the problems were obtained and the algorithms for their realization were proposed [2].
A quasi-one-dimensional nonstationary model of the process was used to study fiber drawing.
The stationary mode of fiber drawing was considered as a programmed (given) movement. A
disturbance movement was the deviation of the real movement from the programmed one. The purpose
of the control was to minimize perturbations. The drawing speed (winding) of the fiber was the
controlled parameter. Thus, a problem with boundary control was formulated. The convex continuous
coercive functional of an integral form was considered as the objective functional. The optimality
system was obtained from the condition that the first variation of the considered functional was equal to
zero [3].
The optimality systems obtained are a set of partial differential equations. For some of them the
initial conditions are given at the zero point in time, and for the other equations at the final moment in
time. This feature makes it difficult to find a solution and requires special approaches to the numerical
implementation of such problems. However, the proposed method allows us to find the control functions
in an explicit form, depending on the optimality system solutions.
Keywords. Optimal control, distributed system, optimality system, moving heat source.
1 Perm national research polytechnic university, Perm, RUSSİA, [email protected] 2 Perm national research polytechnic university, Perm, RUSSİA, [email protected] 3 Perm national research polytechnic university, Perm, RUSSİA, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
165
AMS 2010. 49J20, 49K20, 93C20.
References
[1] Shumkova, D.B., Optimal control in problems with unknown boundaries and moving
sources [Text]: diss. ... PhD: 01.01.02: it is protected on December, 21st, 2006: approved
12.10.2007, 108-110, 2006.
[2] Pervadchuk, V. P., Shumkova, D. B., Optimal control of a moving heat source, St.
Petersburg State Polytechnical University Journal, 37-44, 2010.
[3] Pervadchuk, V. P., Shumkova, D. B., Gordeeva, I. V., Optimal stabilizing control in the
problems of optical fiber drawing process, Applied Mathematics and Control Sciences, 4, 125-
134, 2016.
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
166
On the Kinetics of the Dimer-Dimer Reactions over Supported Catalysts
Pranas KATAUSKIS1
Abstract. We consider a phenomenological model of the dimer-dimer reaction [1] proceeding
on the composite (supported) catalysts [2] which involves: the bulk diffusion of reactants from a
bounded vessel towards the adsorbent and the product bulk one from the catalyst surface into the same
vessel, adsorption and desorption reactants molecules, and surface diffusion of adsorbed particles. The
model is based on the Langmuir-Hinshelwood surface reaction mechanism. The bulk diffusion of both
reactants and product particles is described by the Fick law while the surface diffusion of the adsorbed
molecules is based on the particle jumping mechanism [3]. The kinetics of the reaction is described by a
coupled system of partial and ordinary differential equations. Simulations of the mean-field model were
performed using the finite difference technique. Two distinct arrangements of the adsorption sites are
used for numerical calculations: (i) the same total amount of active and inactive in the surface reaction
adsorption sites, (ii) the same concentrations of active and inactive sites. Two adsorption cases of both
reactants for each arrangement of adsorption sites are considered: (a) each reactant adsorbs on both
active and inactive sites, (ii) both reactants adsorb only on the support. The mathematical model where
concentrations of both reactants at the catalyst surface are given is also studied. The influence of the size
of the catalytic particle, surface diffusivity, adsorption rate constants, and particle jump rate constants
via the catalyst-support interface on the catalytic reactivity of the supported catalyst is studied.
Keywords. Heterogeneous reactions, spillover, surface diffusion.
AMS 2010. 00A69.
References
[1] Skakauskas, V., Katauskis, P., Modelling dimer-dimer reactions on supported catalysts, J. Math.
Chem., 53, 604-617, 2015.
[2 Zhdanov, V. P., Kasemo, B., Simulations of the reaction kinetics on nanometer supported catalyst
particles, Surf. Sci. Rep., 39, 25-104, 2000.
[3] Gorban, A. N., Sargsyan, H. P., Wahab, H. A., Quasichemical models of multicomponent nonlinear
diffusion, Math. Model. Nat. Phenom., 6, 184-262, 2011.
1 Vilnius University, Vilnius, LİTHUANİA, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
167
The Importance of Algebraic Geometry and Cusp Singularities for Phase Transition
Dynamics of Neural Populations in Cerebral Cortex
R Murat DEMİRER1 , Oya DEMİRER2
Abstract. Freeman revealed that background activity have pseudo-equlibrium states which
leads to a phase transitions at neuronal mass level. Background activity has often been modeled in
thermodynamic terms like Carnot Cycle. The distribution of phase patterns behave like from a gas-like,
disorganized phase to more organized liquid like phase at the analogical level that jumps stochastically
over time depending on its past state and environmental state. The measures were based on entropy,
gain and spectral power. Freeman showed singularity as null-spikes in theta dynamics for adaptation
from random to perceptıon. We clarify his information on thermodynamics structures in the importance
of algebraic geometry and stochastic flows. We will define null-spikes with hypersurface cusp
singularities as the germ of a holomorphic function near the origin: it is an isolated hypersurface
singularity. We show that phase transition of each amplitude modulation pattern carries distinguished
paths over electrodes on a smooth stochastic spatio-temporal surface.
Keywords. AM pattern, Hilbert transform, null spike, phase cone, phase transitions, singularity
References
[1]W. J. Freeman, “Origin, structure, and role of background EEG activity. Part 1. Analytic amplitude,”
Clin. Neurophysiol., vol. 115, no. 9, pp. 2077–2088, 2004.
[2]W. J. Freeman, “Origin, structure, and role of background EEG activity. Part 2. Analytic phase,”
Clin. Neurophysiol., vol. 115, no. 9, pp. 2089–2107, 2004.
[3] J. Bhattacharya, H. Petsche, U. Feldmann, and B. Rescher, “EEG gamma-band phase
synchronization between posterior and frontal cortex during mental rotation in humans,” Neurosci. Lett.,
vol. 311, no. 1, pp. 29–32, 2001.
[4] Ramon, C., and M. D. Holmes, M. D., “Stochastic behavior of phase synchronization index and
cross-frequency couplings in epileptogenic zones during interictal periods measured with scalp dEEG,”
Front. Neurol., vol. 4 MAY, 2013.
[5] A. Keating, A., Homological mirror symmetry for hypersurface cusp singularities, Selecta
Mathematica, New Series, pp. 1–42, 2017.
1Üsküdar University, Istanbul, TURKEY, [email protected] 2Arel University, Istanbul, TURKEY, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
168
Stress Distribution in the Infinite Elastic Body with Two Neighboring Periodical Curved
Carbon Nanotube Located on Different and Parallel Planes
Reşat KÖŞKER1 and İsmail GÜLTEN2
Abstract. Curvature of the fibers in the structure of the unidirectional fibrous composite
materials is due to design requirements or to technological processes [1], [2]. Usually the curvature
caused by the technological process is modeled as a local one, whereas the curvature caused by design
features is modeled as a periodical. It is known that, this curvature cause to arise self-balanced stresses
and the values of these stresses can pass over adhesion resistance values. The widely explanation and
interpretation of investigations carried out on this subject are given in [1], [2]. However in all
investigations detailed in the monographs [1], [2], it was assumed that the reinforcing elements of
composite materials are made of traditional materials. In the paper [3] the attempt is made for
development of the internal stability loss problems in the structure of the unidirectional fibrous
composites for the case where the reinforcing element in the composite is the double-walled carbon
nanotube (DWCNT). In [4], the case is considered where a single periodical curved carbon nanotube
(CNT) with an infinite length is contained by an infinite body with low concentration of fibers and
stress distribution in that is investigated. However investigation on how the stress distribution
mentioned above is effected by the reciprocal effect between fibers, as volume ratio of fibers get bigger
in composites, is very important. In this study, we investigate the stress distribution in an infinite elastic
body containing two neighboring fibers, which are located along two parallel lines. It is assumed that
the middle lines of the fibers are located on the different and parallel planes and the curving of these
lines is periodic and sin-phase. The stress distribution is studied when the body is loaded at infinity by
uniformly distributed normal forces with intensity p acting in the fibers direction.
We assume that on the inter-medium surfaces the completely cohesion conditions are satisfied.
For solution of this problem we use the boundary form perturbation method according to which the
unknown values are presented in series form in small parameter. So we obtain equations set for each
approximation and making some process as detailed in [1], [2], and we obtain contact conditions for
each approximation. The original of the sought values are determined numerically. In this study,
numerical results are obtained in the framework of the zeroth and the first approximations for the
normal stress and the self-equilibrium shear stresses on the contact surfaces between CNTs and matrix.
The numerical results related to the stress distribution in considered body and illustrated the influence of
the distance between the fibers to the self-equilibrium stress distributions are given. Moreover, the
influences of the geometrical and mechanical problem parameters to these distributions are also
analyzed.
1 Yildiz Technical University, İstanbul, TURKEY, [email protected] 2 Yildiz Technical University, İstanbul, TURKEY, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
169
Acknowledgement. This research has been supported by Yıldız Technical University Scientific
Research Projects Coordination Department. Project Number: 2014-07-03-DOP01.
Keywords. carbon nanotube, stress distribution, nanocomposite.
AMS 2010. 74A10, 74E30.
References
[1] Akbarov, S.D., Stability Loss and Buckling Delamination: Three-Dimensional Linearized Approach
for Elastic and Viscoelastic Composites, Springer, 2012.
[2] Akbarov, S. D. And Guz, A. N., Mechanics of Curved Composites, Kluwer Academic Publishers,
2000.
[3] Akbarov, S.D., Microbuckling of a Double-Walled Carbon Nanotube Embedded in an Elastic
Matrix, International Journal of Solids and Structures 50, 2584- 2596, 2013.
[4] Kosker R., Akbarov S.D., Gulten İ., Stress Distribution in the Infinite Elastic Body with a
Periodical Curved Carbon Nanotube, 3rd İnternational Eurasian Conference on Mathematical Sciences
and Applications (IECMSA-2014), Book of Abstract, pp:251, 25-28 August 2014.
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
170
Discrete Fractional Solutions of a Legendre Equation
Resat YILMAZER 1
Abstract. One of the most popular research interests of science and engineering is the fractional
calculus theory in recent times. Discrete fractional calculus has also an important position in fractional
calculus. In this work, we acquire new discrete fractional solutions of the homogeneous and
nonhomogeneous Legendre differential equation by using discrete fractional nabla operator.
Theorem. Let 𝜑 ∈ 𝜑: 0 ≠ |𝜑𝛼| < ∞; 𝛼 ∈ 𝑅. Then the homogeneous Legendre equation
𝜑2(1 − 𝑟2) − 2𝜑1𝑟 + 𝜑𝜀(𝜀 + 1) = 0 (1 − 𝑟2 ≠ 0)
has solution of the form;
𝜑 = 𝑘(1 − 𝑟2)−[𝜀(𝜀+1)+2]/2−[2+𝐸−1𝜀(𝜀+1)]/2
where 𝜀 > 0 are given constant.
Keywords. Discrete fractional calculus, Legendre equation, nabla operator.
AMS 2010. 26A33, 34A08.
References
[1] Sabatier, J., Agrawal, O.P., Tenreiro Machado,J.A., Advances in Fractional Calculus: Theoretical
Developments and Applications in Physics and Engineering, Springer, 2007.
[2] Yilmazer, R., N-fractional calculus operator N- method to a modified hydrogen atom equation,
Math. Commun., 15, 489-501, 2010.
[3] Atici, F.M., Eloe, P.W., A transform method in discrete fractional calculus, Int. J. Difference
Equations 2(2), 165-176, 2007.
[4] Acar, N., Atici, F.M., Exponential Functions of Discrete Fractional Calculus, Applicable Analysis
and Discrete Mathematics, Vol. 7 (2), 343—353, 2013.
[5] Yilmazer, R., Inc, M., Tchier, F., Baleanu, D. Particular Solutions of the Confluent Hypergeometric
Differential Equation by Using the Nabla Fractional Calculus Operator. Entropy, 18, 49, 2016.
1 Fırat University, Elazığ, TURKEY, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
171
Discrete Fractional Solutions of a Associated Laguerre Equation
Resat YILMAZER1 and Erdal BAS2
Abstract. In this article, we will give theorems for the discrete fractional solutions of the
homogeneous and nonhomogeneous Laguerre equation by using discrete fractional nabla operator.
Theorem. Let 𝑦 ∈ 𝑦: 0 ≠ |𝑦𝛼| < ∞; 𝛼 ∈ 𝑅. Then the homogeneous Laguerre equation
𝑦2𝑥 + 𝑦1(𝜇 + 1 − 𝑥) + 𝑦𝜆 = 0 (𝑥 ≠ 0)
has particular solution of the form;
𝑦 = ℎ𝑥−(𝜆+𝜇+1)−(1+𝑞−1𝜆)
where 𝜇, 𝜆 are given constant.
Keywords. Discrete fractional calculus, Laguerre equation, nabla operator.
AMS 2010. 26A33, 34A08.
References
[1] Podlubny, I., Fractional Differential Equations, Academic Press, San Diego, 1999.
[2] Kelley, W.G., Peterson, A.C., Difference Equations: An Introduction with Applications, Academic
Press, San Diego, 2001.
[3] Atici, F.M., Eloe, P.W., A transform method in discrete fractional calculus, Int. J. Difference
Equations 2(2), 165-176, 2007.
[4] Atıcı, F.M., Sengül, S., Modeling with fractional difference equations, J. Math. Anal. Appl. 369, 1-9,
2010.
[5] Yilmazer, R., Inc, M., Tchier, F., Baleanu, D. Particular Solutions of the Confluent Hypergeometric
Differential Equation by Using the Nabla Fractional Calculus Operator. Entropy, 18, 49, 2016.
1 Fırat University, Elazığ, TURKEY, [email protected] 2 Fırat University, Elazığ, TURKEY, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
172
Asymptotic Stability of an Evolutionary Nonlinear Boltzmann-Type Equation
Roksana Brodnicka1 and Henryk Gacki2
Abstract. In the presentation a sufficient condition for the asymptotic stability with respect to
total variation norm of semigroup generated by an abstract evolutionary non-linear Boltzmann-type
equation in the space of signed measures with the right-hand side being a collision operator is
presented. For this purpose a sufficient condition for the asymptotic stability of Markov semigroups
acting on the space of signed measures for any distance ([4]), adapted to the total variation norm,
joined with the maximum principle for this norm is used. The presentation generalizes the result in [4]
related to the same type of non-linear Boltzmann-type equation, where the asymptotic stability in the
weaker norm, Kantorovich-Wasserstein, was investigated.
Keywords. Asymptotic stability, Markov operators, maximum principle for the total variation
metric, nonlinear Boltzmann-type equation
AMS 2010. 82B40, 82D05, 35Q20.
References
[1] Barnsley, M. F. and H. Cornille, H., General solution of a Boltzmann equation and the formation
of Maxwellian tails, Proc. Roy. London A 374, 371–400, 1981.
[2] Brodnicka, R. and Gacki, H., Asymptotic stability of a linear Boltzmann-type equation,
Appl. Math., 41, 323–334, 2014.
[3] Crandall, M. G., Differential equations on convex sets, J. Math. Soc. Japan 22 (1970), 443–455,
1970.
[4] Gacki, H., Applications of the Kantorovich-Rubinstein maximum principle in the theory of Markov
semigroups, Dissertationes Math. 448, 1–59, 2007.
[5] Gacki, H. and Lasota, A., A nonlinear version of the Kantorovich-Rubinstein maximum principle,
Nonlinear Anal. 52, 117–125, 2003.
[6] Gacki, H., On the Kantorovich-Rubinstein maximum principle for the Fortet-Mourier norm, Ann.
Pol. Math. 86.2, 107–121, 2005.
[7] Lasota, A. and Mackey, M. C., Chaos, Fractals, and Noise, Springer-Verlag, Berlin 1994.
[8] Lasota, A., Invariant principle for discrete time dynamical systems, Univ. Jagellonicae Acta Math.,
111–127, 1994.
1 University of Silesia in Katowice, Poland, [email protected] 2 University of Silesia in Katowice, Poland, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
173
[9] Lasota, A., Asymptotic stability of some nonlinear Boltzmann-type equations, J. Math. Anal. Appl.
268, 291–309, 2002.
[10] Lasota, A. and J. Traple, J., An application of the Kantorovich-Rubinstein maximum principle in
the theory of the Tjon-Wu equation, J. Differential Equations 159, 578–596, 1999.
[11] Lasota, A. and J. Traple, J., Asymptotic stability of differential equations on convex sets, J.
Dynamics and Differential Equations 15, 335–355, 2003.
[12] Lasota, A. and J. Traple, J., Properties of stationary solutions of a generalized Tjon-Wu equation,
J. Math. Anal. Appl. 335 No. 1, 669–682, 2007.
[13] Rachev, S. T., Probability Metrics and the Stability of Stochastic Models, John Wiley and Sons,
New York 1991.
[14] Tjon, J. A. and WuT. T., Numerical aspects of the approach to a Maxwellian equation, Phys.
Rev. A. 19, 883–888, 1979.
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
174
Hybrid Interval Algorithms for the Simultaneous Inclusion of Polynomial Complex
Zeros: An Experimental Comparison
Roseleine Neves MACHADO1 and Luiz Guerreiro LOPES2
Abstract. The theoretical relevance and practical importance of the polynomial root
finding problem has motivated the development of many numerical floating-point and interval
algorithms for the simultaneous approximation or inclusion of polynomial complex zeros.
Although the iterative algorithms based on complex interval arithmetic [1] produce inclusion
intervals for the exact zeros, giving error bounds automatically, contrary to what happens with
the methods based on the usual floating-point arithmetic, the computational costs of the
complex interval operations are comparatively high. In order to attempt to combine the
advantages of both classes of simultaneous polynomial zero finding algorithms, several hybrid
interval methods, in the sense of [2], have also been developed (see, e.g., [3]). However,
relatively little has been done to evaluate and compare these different iterative root finding
methods in terms of their efficiency and quality of results. In this work, an extensive
computational experiment, using a large set of randomly generated polynomials, is carried out
to assess the efficiency and accuracy of different known hybrid interval algorithms for
simultaneous polynomial root finding. The obtained results provide information on the relative
performance of the hybrid algorithms and the sharpness of the interval results they produce.
Keywords. Polynomial zeros, Simultaneous iterative methods, Hybrid interval methods.
AMS 2010. 30C10, 65G20, 65H04, 65Y20.
References
[1] Petković, M. S., Petković, L. D., Complex interval arithmetic and its applications, Wiley-
VCH, Berlin, 1998.
[2] Caprani, O., Madsen, K., Iterative methods for interval inclusion of fixed points, BIT, 18,
42-51, 1978.
[3] Petković, M. S., On the efficiency of some combined methods for polynomial complex
zeros, J. Comput. Appl. Math., 30, 99-115, 1990.
1 Federal Institute of Rio Grande do Sul, IFRS, Bento Gonçalves, RS, BRAZİL, [email protected] 2 University of Madeira, Faculty of Exact Sciences and Engineering, Funchal, Madeira Is., PORTUGAL, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
175
An Investigation on the Free Vibration of a Sphere with Inhomogeneous Initial Stresses
S. D. AKBAROV3,4 , N. YAHNIOGLU5 and H. H. GULIYEV6
Abstract. In this study, an investigation of free vibration of a hollow sphere with non-
homogeneous initial stresses are made within the exact equations of three dimensional
linearized theory of elastic waves in initially stressed bodies[1]. It is assumed that the initial
stresses are caused by the uniform radial forces act on the inner or outer surfaces of the
sphere. The material of the sphere are also assumed homogeneous and isotropic. The solutions
of the corresponding eigenvalue problems for the initial stressed hollow sphere are made
using the discrete-analytical solution method proposed [2,3]. The influence of some
geometrical parameters as well as initial stresses on the spheroidal vibration of this sphere are
analyzed and discussed. Analyzing the numerical results reveal that initial stresses in the
sphere are significantly effect on the fundamental frequency of the hollow sphere.
Acknowledgement: This study was made according to the Project No. 5/3, 2015: “Complex
of theoretical and experimental investigations related to the study of the interdisciplinary
problems of the Geomechanics” of the National Academy of Sciences of Azerbaijan.
Keywords: Free vibration of a sphere, spheroidal vibration, initial stress
References
[1] A.N. Guz, Elastic Waves in Bodies with Initial (Residual) Stresses, “A.S.K.”, Kiev, 2004
[2] S.D. Akbarov, Dynamics of Pre-Strained Bi-Material Elastic Systems: Linearized Three-
Dimensional Approach, Springer – Heildelberg, New York, 2015.
[3] S.D. Akbarov, H.H. Guliyev and N. Yahnioglu, Natural vibration of the three-layered
solid sphere with middle layer made of FGM: three-dimensional approach, SEM:
Struct.Eng. Mech, 57 (2016) 239–263.
3 Yildiz Technical University, Istanbul, TURKEY, [email protected] 4 National Academy of Sciences of Azerbaijan, Baku, AZERBAIJAN, [email protected] 5 Yildiz Technical University, Istanbul, TURKEY, [email protected] 6 National Academy of Sciences of Azerbaijan, Baku, AZERBAIJAN, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
176
GDP and Efficiency of Russian Economy
Sergey M. BORODACHEV 1
Abstract. The goal is to study GDP (gross domestic product) as an unobservable
characteristic of the Russian national economy state on the basis of more reliable observed
data on gross output (system’s output) and final consumption (system’s control). To do this,
the dynamic Leontief model is presented in a system-like form and its parameters and GDP
dynamics are estimated by the Kalman filter (KF).
We consider that all previous year's investments affect the growth of the gross output
by the next year. The weights of these investments in the sum are equal to unity and decrease
in geometric progression. The estimation of the model parameters was carried out by the
maximum likelihood method. The original data on the gross output and final consumption in
the period from 1995 to 2015 where taken from the Rosstat website, where maximally
aggregated economy of Russia is reflected in the system of national accounts.The growth of
direct costs and capital expenditures at gross output increase has been discovered, which
indicates the extensive character of the development of the economy. Investments are being
absorbed 2 - 4 years; any change of them causes a surge of commissioned fixed assets
fluctuation with a period of 2 years.
Then these parameter values were used in the KF to estimate the states of the system.
The emerging tendency of the transition of GDP growth to its fall means that the rate of
growth of final consumption is higher than the rate of GDP growth. In general, the behavior
of the curve of Rosstat GDP obviously follows from the declared investments, whereas in the
present calculation it is closer to the behavior of final consumption. Estimated GDP and
investments that really increased it were significantly less after the crisis of 2008-2009 than
officially published data.
The results of the work mean that the traditional approaches to determining GDP are
too based on not always reliable declared information on investments. The proposed approach
to the model calculation of GDP uses more reliable data and provides an objective picture of
the country's economic development.
Keywords. Dynamic Leontief model, investments in production capital, Kalman filter.
AMS 2010. 62P20, 93E11.
1 Ural Federal University named after the first President of Russia Boris Yeltsin, Ekaterinburg, RUSSİA.
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
177
Stabilization of Discrete Systems by High Order Compensators
Şerife YILMAZ İRTEM1, Taner BÜYÜKKÖROĞLU2, Vakif DZHAFAROV3
Abstract. For a unstable discrete plant the problem of stabilization by affine
compensator is considered. We consider the case where the difference of the order of the
system and the dimension of the controller vector is two or three. The obtained results are
based on the investigation for a vanishing point of a vector-valued (two dimensional or three
dimensional) multilinear map defined on a box. The required assumptions allow to roughly
display of the image sets of this multilinear map. Number of illustrative examples are
provided.
Keywords. Stabilizing vector, stabilization, stability region.
AMS 2010. 93D05, 93D15.
References
[1] Barmish, B. R., New Tools for Robustness of Linear Systems, MacMillan, New York,
1994.
[2] Bhattacharya, S. P. , Chapellat, H., Keel, L., Robust Control: The Parametric Approach,
Prentice-Hall, New Jersey, 1995.
[3] Fam, A. T., Meditch J. S., A canonical parameter space for linear systems design. IEEE
Transactions on Automatic Control, 23, 3, 454–458, 1978.
[4] Nurges, Ü., Avanessov, S. Fixed-order stabilising controller design by a
mixedrandomized/deterministic method, Int. J. Control, 88, 2, 335–346, 2015.
[5] Polyak, B.T., Shcherbakov, P. S., Hard problems in linear control theory: possible
approaches to solution, Automation and Remote Control, 65, 5, 681–718, 2005.
1 Mehmet Akif Ersoy University, Burdur, TURKEY, [email protected] 2 Anadolu University, Eskişehir, TURKEY, [email protected] 3 Anadolu University, Eskişehir, TURKEY, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
178
[6] Waqar, A. M., Swaroop, D., Bhattacharyya, S. P. Synthesis of fixed structure controllers
for discrete time systems. Numerical Linear Algebra in Signals, Systems and Control. Volume
80 of the series Lecture Notes in Electrical Engineering, 367-385, 2011.
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
179
Numerical Investigation of a Steady Flow of an Incompressible Pseudoplastic Fluid in a
Lid Driven Cavity
Serpil ŞAHİN1 and Hüseyin DEMİR2
Abstract. In this paper, numerical investigation for 2-D steady-state, incompressible
pseudoplastic viscous flow is presented. Pseudo time derivative is used to solve the continuity
and momentum equations with suitable boundary conditions. Depending on high Reynolds
number, wall motions of flow are investigated with respect to nonlinear viscosity by using
Cross model. This study has been undertaken as a first step toward understanding in heat and
mass transport in solvent and polymer processing equipment. Solution to the vorticity
equation for moving top wall is obtained numerically and found to be stable and convergent
for high value of Reynolds numbers. In fact some new results, which are governed by inertia
and variable shear-rate, are obtained.
Keywords. Pseudoplastic viscous flow, Pseudo-time derivative, High Reynolds
number.
AMS 2010. 76A05, 76D05.
References
[1] Aydin, S.H., Nesliturk, A.I., Tezer-Sezgin, M., Two-level finite element method with a
stabilizing subgrid for the incompressible MHD equations, Int. J. Numer. Meth. Fluids, 62,
188-210, 2010.
[2] Benjamin, A.S., Denny, V.E., On the convergence of numerical solutions for 2-D flows in
a cavity at large Re, J. Comp. Physics, 33, 340-358, 1979.
[3] Demir, H., Sahin, S., Numerical investigation of a steady flow of an incompressible fluid
in a lid driven cavity, Turkish Journal of Mathematics and Computer Science, Article ID:
20130031, 2013.
[4] Erturk, E., Corke, T.C., Gökçöl, C., Numerical solutions of 2-D steady incompressible
driven cavity flow at high Reynolds numbers, Int. J. Numer. Meth. Fluids, 48, 747–774, 2005.
1 Amasya University, Amasya, TURKEY, [email protected] 2 Ondokuz Mayıs University, Samsun, TURKEY, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
180
Numerical Comparision of Newtonian and Dilatant Fluids in an Enclosed Cavity Region
Serpil ŞAHİN1 and Hüseyin DEMİR2
Abstract. In this study, numerical investigation for 2-D steady-state, incompressible
Newtonian and dilatant viscous flow is presented. Pseudo time derivative is used to solve the
continuity and momentum equations with suitable boundary conditions. Depending on high
Reynolds number, wall motions of flow are investigated with respect to nonlinear viscosity by
using Cross model. Therefore, the governing equations of fluid of vorticity-stream function
formulations are solved numerically using finite difference and Gauss Elimination method.
The stream function and vorticity results are obtained for Newtonian and dilatant fluids.
These results are presented both in tables and figures.
Keywords. Pseudo Time Parameter, Newtonian Fluid, Dilatant Fluid.
AMS 2010. 76A05, 76D05.
Acknowledgment
This study was supported financially by the Research Centre of Amasya University (Project
No: FMB‐BAP 16-0201).
References
[1] Barragy, E., Carey, G.F., Stream function-vorticity driven cavity solutions using p finite
elements, Computers and Fluids. 26, 453-468, 1997.
[2] Benjamin, A.S., Denny, V.E., On the convergence of numerical solutions for 2-D flows in
a cavity at large Re, J. Comp. Physics, 33, 340-358, 1979.
[3] Demir, H., Sahin, S., Numerical investigation of a steady flow of an incompressible fluid
in a lid driven cavity, Turkish Journal of Mathematics and Computer Science, Article ID:
20130031, 2013.
[4] Erturk, E., Corke, T.C., Gökçöl, C., Numerical solutions of 2-D steady incompressible
driven cavity flow at high Reynolds numbers, Int. J. Numer. Meth. Fluids, 48, 747–774, 2005.
1 Amasya University, Amasya, Turkey, [email protected] 2 Ondokuz Mayıs University, Samsun, Turkey, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
181
New Hybrid Conjugate Gradient Method as a Convex Combination of HS and FR
Conjugate Gradient Methods
Snezana S. DJORDJEVIC1
Abstract. In this paper we present a new hybrid conjugate gradient algorithm for
unconstrained optimization. This method is a convex combination of Hestenes-Stiefel
conjugate gradient method and Fletcher-Reeves conjugate gradient method. The convex
parameter is computed in such a way that the search direction satisfies the condition of the
Newton direction.
The strong Wolfe line search conditions are used. The global convergence of this new
method is proved.
Numerical comparisons show that the present hybrid conjugate gradient algorithm is
the efficient one.
1 Faculty of Technology, University of Nis, 16000 Leskovac, SERBIA, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
182
A New Formula for Higher Order Derivatives of Type xfdx
dx
n
k
and Its
Applications
Telhat ÖZDOĞAN, Melek ERASLAN, Metin ORBAY1
Abstract. We employed a new analytical formula for the higher derivatives of type
xfdx
dx
n
k
in terms of the derivatives of the function xf . Obtained formula for these
higher derivatives enable one to handle some special functions, such as spherical Bessel
functions and Hankel functions, easily [1-4]. As Its an application, the obtained formula has
been applied to spherical Bessel and Hankel functions and then analytical formulae have been
obtained for these special functions.
Keywords. Higher order derivatives, Spherical functions.
AMS 2010. 43A90.
References
[1] Bornemann, F., Laurie, D., Wagon, S., Waldvogel, J., the siam 100-digit challenge. a
study in high-accuracy numerical computing, SIAM, Philadelphia, 2004.
[2] Arfken, G.B., Weber, H. J., mathematical methods for physicists, Elsevier Academic
Press, USA, 2005.
[3] Watson, N., Treatise A., on the theory of bessel functions, Cambridge Univ. Press,
Cambridge, 1966.
[4] Jeffrey A., and Hui Dai, H.,handbook of mathematical formulas and tables, Academic
Press, London, 2008.
1 Amasya University, Amasya, TURKEY, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
183
Asymptotic Behaviour for Functional Three-Dimensional Navier-Stokes-Voigt
Equations
T. CARABALLO 1 and Antonio M. Márquez DURÁN 2
Abstract. The Navier-Stokes-Voigt (NSV) model of the viscoelastic incompresible
fluid was introduced by Oskolkov in [1], giving an approximate description of the Kelvin-
Voigt fluid, and was recently proposed as a regularization of the 3D-Navier-Stokes equation
for the purpose of direct numerical simulation.
Our aim in this paper is consider the following version of this problem for a variety of
delay terms in a unified formulation, and to study the existence of pullback attractors, which
suppose a generalization of some recent works ([2], [3]):
2
0
. , , , ,
. 0, , ,
0, , ,
,
, , , ,0 , .
u uv u u u p g t u in
t t
u in
u on
u x u x in
u t x t x t h x
where τ 3 -dependent
delay term g(t, ut ) represents, for instance, the influence of an external force with some kind
of delay, memory or hereditary characteristic. Here, ut denotes a segment of solution, in other
words, given a function :u t
we can define the mapping ut
(s,x)=u(t+s,x), s (-∞,0], x .
Keywords. Navier-Stokes-Voigt model, delay terms, pullback attractors.
AMS 2010. Primary: 35R15, 35B41, 37B55; Secondary: 47J35.
References
[1] A. Oskolkov, The uniqueness and solvability in the large of boundary value problems for
the equations of motion of aqueous of polymers, Zap. Nauc. Semin. POMI 38, 98-116, 1973.
[2] L. Haiyan and Q. Yuming, Pullback attractors for three-dimensional Navier-Stokes-Voigt
equations with delays, Boundary value problems, 2013:191, 2013.
[3] JG. Luengo, P. Marín-Rubio and J. Real, Pullback attractors for three-dimensional non-
autonomous Navier-Stokes-Voigt equations. Nonlinearity 25, 905-930, 2012.
1Universidad de Sevilla, Sevilla, ESPAÑA, [email protected] 2Universidad Pablo de Olavide, Sevilla, ESPAÑA, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
184
Mixing Problems: Associated Matrices and Stability of Linear ODE Systems
Victor MARTINEZ-LUACES1
Abstract. In this paper, matrices related to chemical mixing problems (MP-matrices) are
characterized and studied. In previous works, mixing problems were analyzed [1], [2] and it is
possible to infer that the mathematical models are always linear ODE systems. The associated
matrices have different structures depending on whether or not there is recirculation of fluids.
In the last case, i.e., if there is not recirculation, the MP-matrix is an upper matrix; otherwise
the structure of the MP-matrix is not so simple. However, it is possible to give a necessary
condition to be satisfied by any MP-matrix. Applying the Gershgorin Circles Theorem [3] it
can be observed that all the matrix eigenvalues have non-positive real part. Even more, it can
be proved that if the real part is zero, it must be the null eigenvalue. Lastly, if the mixing
problem involves three or less components, all the eigenvalues have negative real part and so,
the corresponding ODE solutions are asymptotically stable.
Finally, a comparative analysis shows similarities and differences with other matrices
(FOCKM-matrices), usually involved in Chemical Kinetics problems [4], [5], [6]. There
exists some similarities; however the differences are important enough to obtain distinct
qualitative behaviors of the ODE solutions which are showed in this article through several
examples.
Keywords. Mixing problems, associated matrices, ODE stability.
AMS 2010. 49K15, 34L15, 97M60.
References
[1] Martinez Luaces, V., Inverse-modelling problems in chemical engineering courses, in
Vision and change for a new century. ISC-Delta, Montevideo, Uruguay, 2007.
[2] Martinez-Luaces, V., Modelling and inverse-modelling: experiences with O.D.E. linear
systems in engineering courses, International Journal of Mathematical Education in Science
and Technology 40(2), 259-268, 2009.
[3] Varga, R.S., Geršgorin and His Circles. Springer-Verlag, Berlin, 2004.
1 UdelaR, Montevideo, URUGUAY, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
185
[4] Martinez-Luaces, V., Chemical Kinetics and Inverse Modelling Problems, in Chemical
Kinetics, In Tech Open Science, Rijeka, Croatia, 2012.
[5] Martinez-Luaces, V., First Order Chemical Kinetics Matrices and Stability of O.D.E.
Systems, in Advances in Linear Algebra Research, Nova Publishers New York, 2015.
[6] Martinez-Luaces, V., Qualitative Behavior of Concentration Curves in First Order
Chemical Kinetics Mechanisms, in Advances in Chemistry Research, Nova Publishers, New
York, 2017.
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
186
Backward Semi-Lagrangian Scheme for Guiding Center Problems
Xiangfan PIAO1, Dojin KIM2 and Philsu KIM3
Abstract. In this talk, we disucss how to develop a numerical scheme based on
Backward semi-Lagrangian for solving guiding center models. A main problem in this time
discretization is to find a starting position of the characteristic curves arriving at each grid
point, which is a highly nonlinear and self-consistency problem imposed by the Poisson
equation. The proposed scheme is based on the error correction method recently developed by
the authors[1,2,3]. It is shown that the proposed scheme has a good performance in
computational cost together with a superior mass and total kinetic energy conservation
compared to the convectional second-order iteration shcme and recently developed higher
order method based on the Adams-Moulton rule[4].
Keywords. Backward semi-Lagrangian method, guiding center problem.
AMS 2010. 35Q83,65M12, 65M70.
Aknowlegement. This work was supported by Basic Science Research Program
through the National Research Foundation of Korea(NRF) funded by the Ministry of Science
ICT & Future Planning(grant number NRF-2017R1C1B1002370).
References
[1] Kim, P., Piao, X. and Kim, S.D., An error corrected Euler method for solving stiff
problems based on Chebyshev collocation, SIAM J. Numer. Anal., 49, 2211-2230, 2011.
[2] Kim, S.D., Piao, X. and Kim, P., Convergence on error correction methods for solving
initial value problems, J. Comp. App. Math., 236, 4448-4461, 2012.
[3] Piao, X., Kwon, J., Yi, D., Kim, S. and Kim, P., An iteration free backward semi-
Lagrangian scheme for guiding center problems, SIAM J. Numer. Anal. 53, 619-643, 2015.
[4] Filbet, F., Prouveur, C., High order time discretization for backward semi-Lagrangian
methods, J. Comp. Appl. Math., 303, 171-188, 2016.
1 Hannam University, Daejeon, KOREA, [email protected] 2 KyungPook National University, DAEGU, KOREA, [email protected] 3 KyungPook National University, DAEGU, KOREA, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
187
A Comparison of Multi-step and Multi-stage Method
Yonghyeon JEON1 and Sunyoung BU2, Philsu KIM3
Abstract. We study a multi-stage method compared with a multi-step method for
solving a stiff initial value problem. Due to expensive computational costs of the multi-stage
methods for solving a massive linear system induced from the linearization of a highly stiff
system, stiff problems are usually solved by the multi-step method, rather than the multi-stage
method. In this work, we investigate properties of both the multi-step and the multi-stage
methods and discuss the difference between the two methods through numerical tests in
several examples. Furthermore, the advantages of multi-stage methods can be heuristically
proved even for stiff systems by the comparison of two methods with several numerical tests.
Keywords. Ordinary differential equation, Multi-step method, Multi-stage method
References
[1] Prothero, A., Robinson, A., On the stability and accuracy of one step methods for solving
stiff systems of ordinary differential equations , Math. Comput. 28 (125)145-162, 1974.
[2] Fredebeul, C., A-BDF: a generalization of the backward differentiation formulae, SIAM J.
Numer. Anal. 35 (1998) 1917–1938.
[3] Gear, C.W., Numerical Initial Value Problems in Ordinary Differential Equations,
Prentice–Hall, 1971.
[4] Hairer, E., Nørsett, S.P., Wanner, G., Solving Ordinary Differential Equations.I Nontstiff
Problems, Springer Ser. Comput. Math., Springer, 1993.
1 Kyungpook National University, REPUBLCİ OF KOREA, [email protected] 2 Hongik University, REPUBLCİ OF KOREA, [email protected] 3 Kyungpook National University, REPUBLCİ OF KOREA, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
188
Green's Functions For Problems Simulating Potential Fields in Thin-Walled Assemblies
of Irregular Configuration
Yu. A. MELNIKOV1 and V. N. BORODIN2
Abstract. Green's functions are targeted for a specific class of boundary value problems that
simulate potential fields generated in assemblies of thin-walled fragments of irregular configuration.
Computational efficiency is explored of an approach based on a Green's function version of the
classical [1] method of functional equations. It is shown that the proposed approach appears workable
for a broad range of problem settings. Representing the key component in the approach, matrices of
Green's type are analytically constructed to a number of boundary-contact value problems for
compound regions of regular configuration.
Construction of Green's functions for applied partial differential equations and incorporation of
them into numerical schemes have been a notable part in the classical boundary integral equation
method and its various implementations. An extensive database, accumulated so far in this segment of
research, convincingly illustrates the computational potential of already developed Green's function-
based numerical algorithms. The latter provide high accuracy level achieved at a low computational
cost in solving different classes of problems for partial differential equations applicable in various
areas of engineering and science.
Prior to the actual numerical work, we obtain either closed analytical forms or computer-friendly
series containing representations of required matrices of Green's type for sets of elliptic partial
differential equations. Each of such equations is written in geographical coordinates associated with a
single fragment of the given assembly. Construction of such matrices is not a trivial exercise.
Predetermining the non-triviality, is a factor related to problems set up for assemblies of elements,
where each element hosts a single governing equation. This yields a specific type of boundary-contact
value problems for sets of equations each posed in an individual region, while some contact
conditions, imposed on the interface lines, convert the problem statement into a system format. The
classical Green's function formalism appeared inapplicable to this type of problems, requiring a special
modification, which was described in detail in [2].
Keywords. Potential fields on surfaces, Green's function method
AMS 2010. 35J15
References
1 Middle Tennessee State University, Murfreesboro, USA, [email protected] 2 Tennessee Technological University, Cookeville, USA, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
189
[1] Kupradze, V. D. and Aleksidze, M. A., The method of functional equations for approximate
solution of certain boundary value problems, USSR Computational Mathematics and Mathematical
Physics, 1964, 4, 82-126.
[2] Yuri A. Melnikov and Max Y. Melnikov, Green's Functions. Construction and Applications, De
Gruyter, Berlin-Boston, 2012.
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
190
The Hurwitz’s Formula and Boundary Value Problems for Second-Order Equations
Yu. E. ANIKONOV1, İsmet GÖLGELEYEN2 and Mustafa YILDIZ3
Abstract. In this work, we consider an ill-posed boundary value problem for
multidimensional second-order evolution equations with variable coefficients. We reduce the
boundary value problem to a functional equation by using the given data, and then we obtain
the solution by means of the Hurwitz formula.
Keywords. Ill-posed problem, functional equation, second-order differential equation.
AMS 2010. 65N20, 35R10.
References
[1] Anikonov, Yu. E., Neshchadim, M. V., Representations for the Solutions and Coefficients
of Second-Order Differential Equations, Journal of Applied and Industrial Mathematics, 7
(1):15-21, 2013.
[2] Markushevich, A. I., Introduction to the classical theory of Abelian functions,
Translations of Mathematical Monographs, 96, American Mathematical Society, Providence,
R. I., 1992.
1 Sobolev Institute of Mathematics, Siberian Branch of Russian Academy of Science, Acad. Koptyug prosp., 4, Novosibirsk,
630090, RUSSİA. [email protected] 2,3 Department of Mathematics, Faculty of Arts and Sciences, Bülent Ecevit University, Zonguldak, 67100, TURKEY.
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
191
DISCRETE
MATHEMATICS
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
192
Secret Sharing Schemes Based on Extension Fields
Selda ÇALKAVUR1
Abstract. A ,(t )n secret sharing scheme is a method of distribution of information
among n participants such that 1t can reconstruct the secret but 1t cannot. There is
numerous research about secret sharing schemes. However there is little research on secret
sharing schemes based on extension fields. In this paper, we study some secret sharing
schemes based on extension fields over finite fields. We use two methods to recover the
secret. We define the access structure and the accessibility degree for these secret sharing
schemes. We also describe our theorems, definitions and a corollary.
Keywords. Secret sharing scheme, extension fields, trace and norm functions.
AMS 2010. 94A62, 12E20.
References
[1] Blakley, G.R., Safeguarding cryptographic keys, American Federation of Information
Processing Societies, National Computer Conference, pp. 313-317, 1979.
[2] Carreras, F., Magana, A. and Munuera, C., The accessibility of an access structure,
RAIRO- Theoretical Informatics and Applications, 40.04, pp. 559-567. 2006.
[3] Ding, C., Kohel, D. R. and Ling, S., Secret-sharing with a class of ternary codes,
Theoretical Computer Science, 246(1), pp.285-298, 2000.
[4] Dougherty, S. T., Mesnager, S., Solé, P., Secret sharing schemes based on self-dual codes,
Information Theory Workshop (2008), ITW'08, IEEE, 2008.
[5] Kim. J.L., Lee, N., Secret sharing schemes based on additive codes over )4(GF ,
Applicable Algebra in Engineering, Communication and Computing, pp. 1-19, 2016.
[6] R. Lidl, H. Niederreiter, Finite Fields, vol. 20, Cambridge.
[7] Li, Zhihui, Xue, Ting Xue and Lai, Hong, Secret sharing schemes from binary linear
codes, Information Sciences 180(22), pp. 4412-4419, 2010.
1 Kocaeli University, Kocaeli, TURKEY, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
193
[8] Massey, J. L., Minimal codewords and secret sharing, in Proc. 6th Joint Swedish-Russian
Workshop on Information Theory, Mölle, Sweden, pp. 276-279, 1993.
[9] McEliece R, J. and Sarwate D. V., On sharing secrets and Reed-Solomon codes, Common.
Assoc. Comp. Mach., vol. 24, pp. 583-584, 1981.
[10] Piepryzk, J. and Zhang, X. M., Ideal threshold schemes from MDS codes, Proceedings of
Information Security and Cryptology ICISC 2002, Lecture Notes in Computer Science,
vol. 2587, Springer-Verlag, Berlin, pp. 269-279, 2003.
[11] Shamir, A., How to share a secret, Comm. of the ACM 22 pp. 612-613, 1979.
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
194
GEOMETRY
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
195
Pseudo Spherical Indicatrix Elastic Curves
Ahmet YÜCESAN1 Gözde ÖZKAN TÜKEL2 and Tunahan TURHAN3
Abstract. The tangent, normal and binormal indicatrix of a curve in Minkowski 3-
space may be positioned in De Sitter 2-space, hyperbolic 2-space and 2- dimensional lightlike
cone in terms of causal character of the curve. We separately derive Euler- Lagrange
equations of pseudo spherical indicatrix elastic curves with regard to structure of the
Minkowski 3-space. Then we discuss the solution of the equations.
Keywords. Elastic curve, Euler-Lagrange equation, pseudo-spherical image
AMS 2010. 53A35, 53B30, 74B05.
References
[1] Lopez, R., Differential Geometry of Curves and Surfaces in Lorentz-Minkowski Space. Int.
Electron. J. Geom. 7, no. 1, 44-107, 2014.
[2] O’Neill B., Semi-Riemannian Geometry with Applications to Relativity, Academic Pres.,
New York, 1993.
[3] Singer, D.A., Lectures on Elastic Curves and Rods, AIP Conf. Proc., 1002, Amer. Inst.
Phys., Melville, NY, 2008.
[4] Tukel, G.O., Elastic Strips in Minkowski 3-space, Ph.D. Thesis, Suleyman Demirel
University, Isparta, 2010.
[5] Tükel, G. O., Yücesan, A., Elastic Curves in a Two-dimensional Lightlike Cone. Int.
Electron. J. Geom. 8 , no. 2, 1–8, 2015.
[6] Weinstock, R., Calculus of Variations with Application to Physics and Engineering,
Dover Publications, Inc., New York, 1974.
[7] Yücesan, A., Oral, M., Elastica on 2-dimensional de Sitter Space, V. International
Meeting on Lorentzian Geometry, July 8-11 2009, Martina Franca (Taranto), Italy.
[8] Yücesan, A., Oral, M., Elastica on 2-dimensional Anti-de Sitter Space. Int. J. Geom.
Methods Mod. Phys. 8 , no. 1, 107–113, 2011.
1 Suleyman Demirel University, Isparta, TURKEY, [email protected] 2 Suleyman Demirel University, Isparta, TURKEY, [email protected] 3 Suleyman Demirel University, Isparta, TURKEY, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
196
Problems of g- lifts
Arif SALIMOV1 and Rabia CAKAN2
Abstract. The main purpose of this paper is to transfer complete lifts from the tangent
bundle to the cotangent bundle by using a musical isomorphism between these bundles. The
g-lifts of tensor fields are described in this study newly. We study some problems of g-lifts
that are constituted by transferring the tensor fields via musical isomorphism. The results
obtained give new remarks about g-lifts on the cotangent bundle.
Keywords. Tangent bundle, cotangent bundle, complete lift, musical isomorphism, g-
lifts.
AMS 2010. 55R10, 53C15.
References
[1] Cakan, R., Akbulut, K., Salimov, A., Musical isomorphisms and problems of lifts, Chin.
Ann. Math. Ser. B., 37, 3, 323-330, 2016.
[2] Aslancı, S., Cakan, R., On a Cotangent Bundle with Deformed Riemannian Extension,
Mediterr. J. Math., 11, 4, 1251-1260, 2014,
[3] Salimov, A.A., On operators associated with tensor fields, J. Geom., 99, 1-2, 107-145,
2010.
[4] Yano, A., Ako, M, On certain operators associated with tensor fields, Kodai Math. Sem.
Rep., 20, 414-436, 1968.
[5] Yano, K., Ishihara, S., Tangent and cotangent bundles, Pure and Applied Mathematics,
Marcel Dekker, Inc., New York, 1973.
1 Atatürk University, Erzurum, TURKEY, [email protected] 2 Kafkas University, Kars, TURKEY, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
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Position Vectors of Curves with Respect to Darboux Frame in the Galilen 3-Space
Buket CEYLAN DİRİŞEN1 and Tevfik ŞAHİN2
Abstract. This article investigates the position vector of a curve on the surface in the Galilean
3-space G3. Firstly, the position vector of a curve with respect to the Darboux frame is
determined. Secondly, we obtain the standard representation of the position vector of a curve
with respect to Darboux frame in terms of the geodesic, normal curvature and geodesic
torsion. As a result of this, we define the position vectors of geodesic, asymptotic and normal
line along with some special curves with respect to Darboux frame. Finally, we elaborate on
some examples and provide their graphs.
Keywords. : Position vector, Darboux frame, Geodesic, Galilean 3-space.
AMS 2010. 53B30, 53A35.
Acknowledgment
This study was supported financially by the Research Centre of Amasya University (Project
No: FMB‐BAP 16-0213).
References
[1] Ali A. T., Position vectors of curves in the Galilean space G3, Matematicki Vesnik,
64(3), 200-210, 2012.
[2] Roschel O., Die Geometrie des Galileischen Raumes, Habilitation Schrift, Leoben, 1984.
[3] Yaglom I. M., A simple non-Euclidean geometry and its physical basis, Springer-Verlag,
New York, 1979.
[4] Dirişen B.C., Şahin T., Position vector of a curve with respect to the Darboux frame in the
Galilean space G3, arXiv:1707. 03930v1, 2017.
1 Amasya University, Amasya, Turkey, [email protected] 2 Amasya University, Amasya, Turkey, [email protected], [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
198
Polynomial Helices in 𝑬𝒏
Bülent ALTUNKAYA1 and Levent KULA2
Abstract. In En (𝑛 ≥ 4), there are only a few examples of general helices being valid
in special conditions. In this work, we introduce general helices by using polynomial curves
in En (𝑛 ≥ 4) and we present new results, respectively.
Keywords. General Helix, Polynomial Curve.
AMS 2010. 53A04, 53A05.
References
[1] Ali, Ahmad T., Turgut, M., Some Characterizations of Slant Helices in the Euclidean
Space 𝐸𝑛 , Hacettepe Journal of Mathematics and Statistics, 39, 327-336, 2010.
[2] Camcı, C., İlarslan, K., Kula, L. and Hacısalihoğlu, H. H., Harmonic curvatures and
generalized helices in 𝐸𝑛, Chaos, Solitons and Fractals, 40, 2590–2596, 2009.
[3] Hacısalihoğlu, H. H., Diferansiyel Geometri I, Ankara, 1980.
1 Ahi Evran University, Kırşehir, TURKEY, [email protected] 2 Ahi Evran University, Kırşehir, TURKEY, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
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On Normal Magnetic Curves in 3-Dimensional Heisenberg Group H3
Cihan ÖZGÜR1
Abstract. We study normal magnetic curves in 3-dimensional Heisenberg Group H3
and we give the classification of this type curves. We obtain the parametric equations of all
normal magnetic curves whose trajectories are described by the equation ' ' ( ') q in
H3. As an application, we consider normal magnetic curves in 3-dimensional Nil space Nil3.
Keywords. Magnetic curve, Legendre curve, slant curve, Heisenberg group.
AMS 2010. 53C25, 53C40, 53A05.
References
[1] Adachi T., Curvature bound and trajectories for magnetic fields on a Hadamard surface,
Tsukuba J. Math., 20, no. 1, 225-230, 1996.
[2] Cabrerizo J. L., Fernández M., Gómez J. S., On the existence of almost contact structure
and the contact magnetic field, Acta Math. Hungar., 125, no. 1-2, 191-199, 2009.
[3] Calvaruso G., Munteanu M., Perrone A., Killing magnetic curves in three-dimensional
almost paracontact manifolds, J. Math. Anal. Appl., 426, no. 1, 423-439, 2015.
[4] Druţă-Romaniuc S., Inoguchi J., Munteanu M., Nistor A., Magnetic curves in Sasakian
manifolds, J. Nonlinear Math. Phys., 22, no. 3, 428-447, 2015.
1 Balıkesir University, Balıkesir, TURKEY, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
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On Chen Invariants for Submanifolds of Riemannian Product Manifolds
Erol KILIÇ1, Mehmet GÜLBAHAR2 and Sadık KELEŞ3
Abstract. The Chen invariant for submanifolds of a Riemannian product manifold is
studied. Some relations dealing the Chen invariant are presented. With the help of these
relations, some characterizations for invariant and anti-invariant submanifolds of a
Riemannian product manifold are obtained.
Keywords. Chen invariant, submanifold, Riemannian product manifold.
AMS 2010. 53C15, 53C40, 53C42.
References
[1] Atçeken, M., Slant submanifolds of a Riemannian product manifold, Acta Math. Sci.Ser.
B. Eng. Ed. 30(1), 215-224, 2010.
[2] Chen, B.-Y., Pseudo-Riemannian geometry, invariants and applications, World
Scientific Publishing, Hackensack, NJ 2011.
[3] Kılıç, E., Tripathi, M. M., Gülbahar, M., Chen-Ricci inequalities for submanifolds of
Riemannian and Kaehlerian product manifolds, Ann. Polon. Math. 116(1), 27-56, 2016.
[4] Sahin, B., Slant submanifolds of an almost product Riemannian manifold, J. Korean Math.
Soc. 43, 717-732, 2016.
[5] Yano, K., Structures on manifolds, World Sci. Singapore, 1984.
1 İnönü University, Malatya, TURKEY, [email protected] 2 Siirt University, Siirt, TURKEY, [email protected] 3 İnönü University, Malatya, TURKEY, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
201
Special Curves in Finsler Space
Fatma ATEŞ1, Zehra ÖZDEMİR2, F. Nejat EKMEKCİ3 and Mustafa ÇALIŞKAN4
Abstract. In this study, firstly we give the necessary notions and results in Finsler
geometry. Then, we investigate some new characterizations for special Finslerian curves.
Also, we obtain some results for these curves with using the Finslerian Metrics.
Keywords. Finsler space, special curves.
AMS 2010. 53B40, 14H45.
References
[1] Bao, D., Chern, S.S., Shen, Z., An Introduction to Riemann-Finsler Geometry (Graduate
Texts in Mathematics; 200), Springer, 2000.
[2] Bucataru, R. M., Finsler-Lagrange Geometry - Applications to Dynamical Systems, Ed.
Acad. Romane, Bucharest, 2007.
[3] Mo, X., An Introduction to Finsler Geometry, World Scientific, 2006.
1 Ankara University, Ankara, TURKEY, [email protected] 2 Ankara University, Ankara, TURKEY, [email protected] 3 Ankara University, Ankara, TURKEY, [email protected] 4 Gazi University, Ankara, TURKEY, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
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A Differential Calculus on Rh(1|2)
Fatma BULUT1
Abstract. Quantum superspaces are introduced in [1] to represent quantum
supergroups. In this presentation, we set up a differential calculus on the superspace Rh(1|2)
[2]. It contains functions on Rh(1|2), their differentials and differential forms [3].
Keywords. h-deformed, superspace, differential calculus.
AMS 2010. 17B37, 81R60.
References
[1] Manin, Y. I.: Multiparametric quantum deformation of the general linear supergroups.
Commun. Math. Phys. 123, 163–175, 1989.
[2] Aizawa, N., Chakrabarti, R.: Noncommutative geometry of super-jordanian OSph(2/1)
covariant quantum space. J. Math. Phys. 45, 1623–1638, 2004.
[3] Celik, S., Bicovariant differential calculus on the superspace Rq(1|2), J. Alg. Appl., 15,
1650172 (2016) [17 pages], doi:10.1142/S0219498816501723.
1 Bitlis Eren Üniversitesi, Bitlis, TURKEY, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
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Application Example of the Processing of the Cubes in the Geometry Lesson in
Elementary Schools
Gönül TÜRKAN DEMİR1, Keziban ORBAY2 and Emine ALTUNAY ŞAM3
Abstract. Mathematics has had an important place in Turkish educational history for
long. Philosophy, mathematics, geometry, astronomy etc. collectively known as the rational
sciences had been taught in the madrasahs, which were the most important institutions for the
educational system of Classical Ottoman Period, beginning from the reign of Mehmed the
Conqueror, as well as the traditional/religious subjects. However, with madrasahs getting
corrupted during the mid-17th century, besides other rational sciences, geometry was also
abandoned.
19th century is when the Ottoman Empire had started a reformation and transition,
implementing western models in military, politics and administration. This is also the period,
during which the most significant improvements in education and science were observed.
Transferred from the western civilization, course books on fundamental sciences that were
taught only in higher at the beginning, were also published and used in primary and secondary
education as well, in the following years.
Educational Statute 1869, involved geometry in educational system. Following this
Statute, whereas in the elementary schools, only calculus was taught, geometry took its place
in the curriculums of junior high schools (rüştiye), high schools (idadi) and higher education
(sultani). In those elementary schools, called “Mekatib-iptidâiye” and today known as
“secondary school”, formed in 1870, there were no geometry courses in the beginning,
however in the following years, geometry was attached to the curriculum. This research
studies the articles “An Example Geometry Course on Cubes for 1st Grades” published on the
Volume 4 / Issue 47 of the Journal of Education in 1919. This articles define the stages of
course planning in geometry and the equipment required and present an example course.
This is a descriptive research, conducted using screening model in order to introduce
textual contents of example geometry courses in 1919. The research uses document review
method.
The text reviewed exhibits preparation phases, review of the previous courses,
introduction of new concepts, and association of those concepts with real-life and exercises
for students. All stages presented as teacher-student dialog, it allows identifying the teacher-
1 Amasya University, Amasya, TURKEY, [email protected] 2 Amasya University, Amasya, TURKEY, [email protected] 3 Amasya University, Amasya, TURKEY, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
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student interaction of that period. This research, raising awareness on the historical
development of geometry education in terms of how the courses were conducted, is expected
to contribute to the studies of educators of this field.
Key Words: Geometry course, History of Mathematic, Secondary school, Planning in
geometry.
References
[1] Abdullatif, Hendese dersi numunesi ‘devre-i aliyede’ (mika’blar hakkında). Tedrisat
Mecmuası 4, 47, 236-240, 1919.
[2] Aslan E. ve Olkun S., Elementary school mathematics in the first curricula of Turkish
Republic. Elementary Education Online, 10, 3, 991-1009, http://ilkogretim-online.org.tr.
2011.
[3] Baki, A., Matematik tarihi ve felsefesi, Pegem Akademi, Ankara, 2014.
[4] Uzunçarşılı, İ., Osmanlı Devletinin ilmiye teşkilâtı, Ankara, 1988.
[5] Çınar S., Eskişehir eğitim tarihi (1876–2004). (Yüksek Lisans Tezi). Eskişehir Osmangazi
Üniversitesi Sosyal Bilimler Enstitüsü, Eskişehir, 2005.
[6] Göker, L., Matematik tarihi ve Türk-İslâm matematikçilerinin yeri, Ankara, Gazi
Üniversitesi, 1981.
[7] İlk Mektepler Müfredat Programı, İstanbul, Matba-ı Amire, 1924.
[8] İlk Mektepler Müfredat Programı, İstanbul, Matba-ı Amire, 1927.
[9] Süveysi, M., Hendese. Diyanet İslâm Ansiklopedisi. 17, 196-199.
http://www.diyanetislamansiklopedisi.com/hendese/.
[10] Ülger, A., Matematiğin kısa bir tarihi – İkinci dönem: Eski Yunan matematiği,
Matematik Dünyası, Yaz, 49-53, 2003.
[11] Yıldırım A. ve Şimşek H., Sosyal bilimlerde nitel araştırma yöntemleri. Ankara, Seçkin.
2013.
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
205
A study on Elastic Strips
Gözde ÖZKAN TÜKEL 1 and Ahmet YÜCESAN2
Abstract. We obtain the solution of the variational problem which gives elastic strips
with spacelike directrix in Minkowski 3-space. By changing the variation, we derive two
conservation laws of elastic strips with spacelike directrix, and so we define two new types of
elastic strips. Then, we establish some connections between elastic curves on De Sitter 3-
space and hyperbolic 2-space and elastic strips with spacelike directrix. Finally, we define P-
functional by means of the tangent vector of a spacelike curve defining force-free strip.
Keywords. Elastic strip, elastic curve, Euler-Lagrange equation, conservation laws.
AMS 2010. 53A35, 53B30, 35A15.
References
[1] Chubelaschwili, D., Pinkall, U., Elastic Strips, Manuscripta Math. 133 , no. 3-4, 307–326,
2010.
[2] Lopez, R., Differential Geometry of Curves and Surfaces in Lorentz-Minkowski Space. Int.
Electron. J. Geom. 7, no. 1, 44-107, 2014.
[3] O’Neill B., Semi-Riemannian Geometry with Applications to Relativity, Academic Pres.,
New York, 1993.
[4] Tukel, G.O., Yucesan, A., Elastic Strips with Timelike Directrix, Mathematical Reports, In
Press.
[5] Tukel, G.O., Elastic Strips in Minkowski 3-space, Ph.D. Thesis, Suleyman Demirel
University, Isparta, 2010.
[6] Weinstock, R., Calculus of Variations with Application to Physics and Engineering,
Dover Publications, Inc., New York, 1974.
1 Süleyman Demirel University, Isparta, TURKEY, [email protected] 2 Süleyman Demirel University, Isparta, TURKEY, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
206
Involutes of Order k of a Space-like Curve in 4
1IE
Günay ÖZTÜRK1, İlim KİŞİ2, Sezgin BÜYÜKKÜTÜK3 and Kadri ARSLAN4
Abstract. The orthogonal trajectories of the first tangents of a curve x are called the
involutes of x. In this study, we give a characterization of involutes of order k of a space-like
curve x with time-like principal normal in Minkowski 4-space 4
1IE .
Keywords. Involute, space-like curve, W-curve, helix.
AMS 2010. 53A04, 53A05.
References
[1] Blazenka, D., Zeljka M. S., 1999. Involutes and evolutes in n-dimensional simply isotropic
space, Journal of Information and Organizational Sciences, 2(3), 71-79, 1999.
[2] Kılıc, B., Arslan, K., Öztürk, G., Tangentially cubic curves in Euclidean spaces,
Differential Geometry-Dynamical Systems, 10, 186-196, 2008.
[3] O'Neill, B., Semi-Riemannian Geometry with Applications to Relativity, Academic Press,
New York, 1983.
[4] Öztürk, G., Arslan, K., Hacisalihoglu, H. H., A characterization of ccr-curves in mIR .
Proccedings of Estonian Academy Science, 57(4), 217-224, 2008.
[4] Turgut, M., Ali, A.T., Lopez-Bonilla, J. L., Time-like involutes of a space-like helix in
Minkowski space-time, Apeiron, 17(1), 28-41, 2010.
[5] Walfare, J., Curves and surfaces in Minkowski space, PhD thesis, K.U. Leuven, Faculty of
Science, Leuven, 1995.
[6] Yılmaz, S. Turgut, M., On the differential geometry of the curves in Minkowski space-time
I, International Journal of Contemporary Mathematical Science, 3(27), 1343-1349, 2008.
1 Kocaeli University, Kocaeli, TURKEY, [email protected] 2 Kocaeli University, Kocaeli, TURKEY, [email protected] 3 Kocaeli University, Kocaeli, TURKEY, [email protected] 4 Uludag University, Bursa, TURKEY, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
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Similarity Geometry of Null Cartan Curves
Hakan ŞİMŞEK1
Abstract. In this paper, the differential geometric properties of null Cartan curves
according to the similarity map of the Minkowski spacetime are investigated. The shape
Cartan curvatures and the pseudo-de Sitter parameter of a null Cartan curve are introduced
and the fundamental theorems (uniqueness and existence) under the similarity map in
Minkowski spacetime are given. Also, self-similar null Cartan curves parameterized by the
pseudo-de Sitter parameter are obtained.
Keywords. pseudo-de Sitter parameter, null Cartan curve, p-similarity.
AMS 2010. 14H50, 14H81, 53A35, 53A55, 53B30.
References
[1] Bejancu A., Lightlike curves in Lorentz manifolds, Publ. Math. Debrecen, 44, 145--155,
1994.
[2] Berger M. Geometry I. Springer, New York 1998.
[3] Bonnor W. B., Null curves in a Minkowski space-time, Tensor N.S. 20, 229--242, 1969.
[4] Cöken A. C, Ciftci Ü., On the Cartan curvatures of a null curve in Minkowski space-time,
Geom. Dedicata, 114, 71-78, 2005.
[5] Duggal K. L., Bejancu A. Lightlike Submanifolds of Semi-Riemannian Manifolds and
Applications, volume 364 of Mathematics and its Aplications. Kluwer Academic Publishers
Group, Dordrecht, The Netherlands, 1996.
[6] Encheva R., Georgiev G., Similar Frenet curves. Results in Mathematics, vol. 55, no. 3-4,
359--372, 2009.
[7] Falconer K., Fractal Geometry: Mathematical Foundations and Applications, Second
Edition, John Wiley & Sons, Ltd., 2003.
[8] Ferrandez A., Gimenez A, Lucas P., Null helices in Lorentzian space forms, Int. J. Mod.
Phys. A 16, 4845-4863, 2001.
1Antalya International University, Antalya, TURKEY, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
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[9] Simsek H., Özdemir M., Similar and Self-similar Curves in Minkowski n-space, Bull. of
Korean Math. Soc., 52 (6), 2071-2093, 2015.
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
209
Semisimilarity and Consemisimilarity for matrices over a Commutative Quaternion
Hidayet Hüda KÖSAL1 , Mahmut AKYİĞİT2 and Murat TOSUN3
Abstract. In this paper, we investigate some equivalence relations and results over the
commutative quaternions and their matrices. In this sense, semisimilarity, and
consemisimilarity over the commutative quaternions and their matrices are established. Also,
Equalities of these equivalence relations are explicitly determined.
Keywords. Commutative quaternions, Consemisimilarity, Semisimilarity.
AMS 2010. 53A40, 20M15.
References
[1] Hamilton, W. R., Lectures on quaternions, Hodges and Smith, Dublin: 1853.
[2] Segre, C., The real representations of complex elements and extension to bicomplex,
Systems, Math. Ann., 40, 413, 1892.
[3] Huang L., Consimilarity of quaternion matrices and complex matrices, Linear Algebra
Appl. 331, 21-30, 2001.
[4] Hartwig, R. E., Putcha, M. S. Semisimilarity for matrices over a division ring. Linear
Algebra and its Applications, 39, 125-132, 1981.
[5] Bevis, J. H., Hall, F. J. Pseudo-consimilarity and semi-consimilarity of complex matrices.
Linear Algebra and its Applications, 90, 73-80, 1987.
1 Sakarya University, Sakarya, TURKEY, [email protected] 2 Sakarya University, Sakarya, TURKEY, [email protected] 3 Sakarya University, Sakarya, TURKEY, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
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On an Almost Contact Metric Manifold with a Type of Semi-symmetric Non-metric
Connection
Hulya Bagdatli YILMAZ1
Abstract. In this paper, we consider an almost contact metric manifold endowed with
a type of semi-symmetric non-metric connection. We find the expression for the curvature
tensor of an almost contact metric manifold that admits a type of semi-symmetric non-metric
connection. Furthermore, we study the properties of the curvature tensor and the projective
curvature tensor.
Keywords. Almost contact metric manifold, Semi-symmetric non-metric connection,
Curvature tensor.
AMS 2010. 53B15.
References
[1] Agashe, N. S. and Chafle, M. R., A semi-symmetric non-metric connection on a
Riemannian manifold, Indian J. Pure Appl. Math , 23,399-409, 1992.
[2] Chaubey, S. K.,On semi symmetric non-metric connection, Prog. of Math., 41-42, 11-20,
2007.
[3] Chaubey, S. K.,and Ojha, R. H., On semi symmetric non-metric connection and quarter
symmetric connections, Tensor N. S., 70 No:2, 202-213, 2008.
[4] Chaubey, S. K.,and Ojha, R. H., On semi symmetric non-metric connection, Filomat, 25:4,
19-27, 2011.
[5] Blair, D. E.,The theory of quasi-Sasakian structures, J. Diff. Geom., 1, 331-345, 1967.
[6] Blair, D. E., Contact manifolds in Riemannian geometry, Lecture Notes in math.509,
Springer-Verlag, Berlin 1976.
[7] De, U. C. and Biswas, S.C., On a type of semi symmetric non-metric connection on a
Riemannian manifold, Pub. De L'institut Math. Nouvelle serie tome 61, 90-96, 1997.
1 Marmara University, İstanbul, TURKEY, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
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[8] Friedmann, A. and Schouten, J. A., Über die Geometric der holbsymmetrischen
Übertragurgen, Math. Z., 21, 211-233, 1924.
[9] Gray,J., Some global properties of contact structures, Ann. of Math.,69, 421-450, 1959.
[10] Hayden, H. A., Subspaces of space with torsion, Proc. London Math. Soc., 34, 27- 50,
1932.
[11] Janssens, D. and Vanhecke, L., Almost contact structures and curvature tensors, Kodai
Math. J., 4, 1-27, 1981.
[12] Oubina, J., New classes of almost contact metric structures, Publicationes Mathematicae,
32, 187-193, 1985.
[13] Sasaki, S., On diff. manifolds with certain structures which are closely related to almost
contact structure I, Tohoku Math. J., 12, 459-476, 1960.
[14] Sengupta, J., De, U. C. and Binh, T. Q., On a type of semi symmetric non-metric
connection on a Riemannian manifold, Indian J. Pure Appl. Math.,31(12),1659-1670, 2000.
[15] Yano, K., On semi-symmetric metric connections, Revue Roumania De Math. Pures
ppl., 15, 1579-1586, 1970.
[16] Yılmaz, H. B., On weakly symmetric manifolds with a type of semi-symmetric non-metric
connection, Ann. Polonici Math., 102.3, 301-308, 2011.
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
212
On Invariants of a System of Vectors in Minkowski Spacetime
İdris ÖREN1
Abstract. Let M be Minkowski spacetime (or the 4- dimensional pseudo-Euclidean
space of index 1), O(3, 1) be the group of all pseudo-orthogonal transformations of M and m-
uple x1, x2, … , xm of vectors in M. According to the group O(3, 1), invariants of an m-uple
x1, x2, … , xm are investigated. We define the new O(3,1) -invariant function Lxk , which is
the k-linearly dependently ratio of an m-uple x1, x2, … , xm in M. The invariants can be
used to judge whether two m-uples x1, x2, … , xm and y1, y2, … , ym in M are equivalent.
Methods and results of this study will be useful in the geometry. Since invariants obtained in
the present paper are computable, they are useful in problems of Bézier curves, Bézier
surfaces and Bézier vector fields.
Keywords. Invariant, Minkowski spacetime.
AMS 2010. 13A50, 51B20, 53B30.
References
[1] Misiak , A., Stasiak, E., Equivariant maps between certain G-spaces with G=O(n-1,1),
Math Bohem., 126, 555-560, 2001.
[2] O'Neill, B., Semi-Riemannian geometry: with applications to relativity, Academic
Press.,1983.
[3] Stasiak, E., Scalar concomitants of a system of vectors in pseudo-Euclidean geometry of
index 1, Publ.Math.Debrecen, 57, 55-69, 2001.
[4] Gohberg, I.,Lancaster, P and Rodman, L., Invariant subspaces of matrices with
application, Siam, 2006.
[5] Bognar, J., Indefinite inner product spaces, Springer-Verlag, 1974.
[6] Höfer , R., m-point invariants of real geometries, Beitrage Algebra Geom.,40, 261-266,
1999.
[7] Greub, W, Linear Algebra, Springer-Verlag, 1967.
1 Karadeniz Technical University, Trabzon, TURKEY, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
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[8] Ören, İ., Complete system of invariants of subspaces of Lorentzian space,
Iran.J.Sci.Technol.Trans. A Sci., 2014.(in pres)
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
214
Determining by Control Points of Null Bézier Curves with Degree 2 and 3 Curves in 3-
Dimensional Minkowski Spacetime
İdris ÖREN
Abstract. Let E13 be 3-dimensional Minkowski spacetime. In this study, the following
problems are investigated:
1) Does there exists a null Bézier curve with degree 2 and 3 in E13?
2) Determination by control points of a null Bézier curve with degree 2 and 3 in E13.
Therefore, some examples to null Bézier curves with degree 3 and 4 in E13 are given.
Keywords. Bézier curve, null curve, Minkowski spacetime.
AMS 2010. 65D17, 53B30, 51B20
References
[1] Georgiev, G. H., Space-like Bézier curves in the three-dimensional Minkowski space, AIP
Conf. Proc., 1067, Amer. Inst. Phys., Melville, NY, 2008.
[2] Pekşen,Ö,Khadjiev,D., On invariants of null curves in the pseudo-Euclidean
geometry,Differential Geom. Appl., 29, 183-187, 2013.
[3] Wang, W., Zhang, H., Liu, X., Paul, J. C., Conditions for coincidence of two cubic Bézier
curves, J. Comput. Appl. Math., 235, no. 17, 5198–5202, 2011.
[4] Chen, X., Ma, W., Deng, C., Conditions for the coincidence of two quartic Bézier curves,
Appl. Math. Comput. 225, 731–736, 2013.
[5] Ören, İ., Complete system of invariants of subspaces of Lorentzian space, Iran J Sci
Technol Trans Sci. 41,401–408,2017.
[6] Ören, İ., Equivalence conditions of two Bézier Curves in the Euclidean geometry, Iran J
Sci Technol Trans Sci.(article in pres) DOI 10.1007/s40995-016-0129-1.
[7] Pokorna, B. Quadratic space-like Bézier curves in three dimensional Minkowski space,
2012, project of Dissertation Thesis.
Karadeniz Technical University, Trabzon, Turkey, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
215
The Elliptic Matrices Associated with Eliptic Biquaternions and De-Moivre’s Formula
for These Matrices
Murat Tosun1 and Kahraman Esen Özen2
Abstract. In this study, with the aid of the Hamilton operators, we obtain the 4 4 elliptic
matrix representations of elliptic biquaternions and call them as right matrix representation
and left matrix representation by analogous Hamilton. Also, we have showed that the matrix
algebra which includes the left representations is isomorphic to the algebra of elliptic
biquaternions. Because of this isomorphism, we have called especially the left matrix
representations as the elliptic matrices associated with elliptic biquaternions. Afterwards, we
have expressed the De-Moivre’s and Euler Formulas for the elliptic matrices associated with
elliptic biquaternions by writing them in the polar form.
Keywords. De-moivre’s formula, Hamilton operator, complex quaternions.
AMS 2010. 11R52, 15A99.
References
[1] van der Waerden, B. L.. Hamilton’s discovery of quaternions, Math. Magazine, 49, 227-
234, 1976.
[2] Zhang, F.. Quaternions and Matrices of Quaternions, Linear Algebra and its Applications,
251, 21-57, 1997.
[3] Grob, J.. Trenkler, G.. Troschke, S.-O, Quaternions: further contributions to a matrix
oriented approach, Linear Algebra and its Applications 326, 205-213, 2001.
[4] Farebrother, R. W.. Grob, J.. Troschke, S-O, Matrix representation of quaternions. Linear
Algebra and its Applications 362, 251-255, 2003.
[5] Jafari, M.. Mortazaasl, H.. Yaylı, Y, De Moivre’s Formula for Matrices of Quaternions,
JP Journal of Algebra, Number Theory and Applications 21, 57-67, 2011.
1 Sakarya University, Sakarya, Turkey, [email protected] 2 Sakarya University, Sakarya, Turkey, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
216
New Results for General Helix in Euclidean 3-Space
Kazım İLARSLAN1
Abstract. Finding parametric equation of a space curve is not easy task when its
curvature functions are given. It is well known that this problem is known as fundamental
theorem of a space curve. If the curvature functions are functions of arc-length, the solution of
this problem is usually impossible. In this talk, we discuss the answer of the problem when
the curve is a general helix in Euclidean 3-space. By using slope axis of general helix, we get
parametric equations of a general helix. Also, we give some examples and their figures
Keywords. General helix, Euclidean 3-space, slope axis,.
AMS 2010. 53A04
References
[1] Ali, Ahmad T., Position vectors of general helices in Euclidean 3-space. Bull. Math.
Anal. Appl. 3, no. 2, 198–205, 2011.
[2] Barros, M., General helices and a theorem of Lancret. Proc. Amer. Math. Soc. 125, no. 5,
1503–1509, 1997.
[3] İlarslan, Kazım, Characterizations of spacelike general helices in Lorentzian manifolds.
Kragujevac J. Math. 25, 209–218, 2003.
[4] Uçum, A., Camcı, Ç. and İlarslan, K., General helices with timelike slope axis in
Minkowski 3-space. Adv. Appl. Clifford Algebr. 26, no. 2, 793–807, 2016.
1 Kırıkkale University, Kırıkkale, TURKEY, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
217
On Cardan Position for the Lorentzian Plane Motion of a Rigid Body
Kemal EREN1, Soley ERSOY2 and Mahmut ERGÜT3
Abstract. Cardan motion is defined as being generated by a circle rolling within
another circle of twice its size. A moving plane is said to be at a Cardan position if there is a
Cardan motion such that instantaneously the trajectories of the two motions hyper-osculate
[1]. In this paper, we study the instantaneous geometric properties of motion of rigid bodies in
the Lorentzian plane. For this purpose, we define Lorentzian form of Bottema’s instantaneous
invariants. We obtain the necessary and sufficient condition of a Lorentzian plane to be at
Cardan position with respect to these invariants.
Keywords. Cardan position, instantaneous invariants, Lorentzian plane
AMS 2010. 53A17, 53A35.
References
[1] Bottema, O. and Roth B., Theoretical Kinematics, North-Holland, Amsterdam, 1979.
[2] Dörrie, H., 100 Great problems of elementary mathematics: their history and solutions,
New York: Dover, 1965.
[3] Rauh, K., Marks, H., Bündgens, M., Otto, K. (1938): Kardanbewegung und
Koppelbewegung (Cardan motion and coupler motion), Schriftenreihe Praktische
Getriebetechnik, Heft 2, Berlin: VDI-Verlag, 1938.
[4] Bottema, O., On Cardan Positions for the Plane Motion of a Rigid Body, Indagationes
Mathematicae, V. XI, fasc. 3, 1949.
[5] Freudenstein, F., The Cardan Positions of a Plane, Trans. Sixth Conf. Mechanisms,
Purdue University, West Lafayette, Ind., pp. 129–133, October 1960.
1 Fatsa Science High School, Ordu, TURKEY, [email protected] 2 Sakarya University, Sakarya, TURKEY, [email protected] 3 Namık Kemal University, Tekirdağ, TURKEY, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
218
[6] Ergüt, M., Aydin, A.,P. and Bildik N., The Geometry of the Canonical Relative System and
One-Parameter Motions in 2-Lorentzian Space, The Journal of Fırat University, 3(1) 113-
122, 1988.
[7] Ergin. A.A., On the one-parameter Lorentzian motion, Communications, Faculty of
Science, University of Ankara, Series A 40, 59–66, 1991.
[8] Tutar, A., Kuruoğlu, N. and Duldul, M., On the moving coordinate system and pole points
on the Lorentzian plane, International Journal of Applied Mathematics, 7(4), 439-445, 2001.
[9] Güngör, M. A., Pirdal A.Z. and M. Tosun, Euler-Savary Formula for the Lorentzian
Planar Homothetic Motions, Int. J. Math. Comb. 2, 102–111, 2010.
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
219
Some Characterizations for Screen Isotropic Leaves on Lightlike Hypersurfaces
Mehmet GÜLBAHAR1, Sadık KELEŞ2 and Erol KILIÇ3
Abstract. Isotropic leaves in the integrable screen distribution for lightlike
hypersurfaces of a Lorentzian manifold are investigated. Some examples of such leaves are
presented. Furthermore, some relations on these leaves are obtained.
Keywords. Isotropic leaf, distribution, lightlike hypersurface, Lorentzian manifold.
AMS 2010. 53C42, 53C50.
References
[1] ] Bejan C. L., Duggal K. L., Global lightlike manifolds and harmonicity, Kodai Math. J.
28, 131-145, 2005.
[2] Duggal, K. L., Sahin, B., Differential geometry of lightlike submanifolds, Birkhauser,
Basel, 2010.
[3] Duggal, K. L., Jin D. H., Null curves and hypersurfaces of semi-Riemannian manifolds,
World Scientific Publishing, Hackensack, NJ, 2007.
[4] O’Neill, B., Isotropic and Kaehler immersions, Canad. J. Math. 17, 907-915, 1965.
1 Siirt University, Siirt, TURKEY, [email protected] 2 İnönü University, Malatya, TURKEY, [email protected] 3 İnönü University, Malatya, TURKEY, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
220
Special Smarandache Curves with Respect to Darboux Frame in the Galilean 3-Space
Merve OKUR1 and Tevfik ŞAHİN2
Abstract. This article investigates special Smarandache curves of arbitrary curves respect to
Darboux frame in Galilean space. As a result of this, we define the special Smarandache
curves of geodesic, asymptotic and line of curvature respect to Darboux frame in Galilean
space.
Keywords. : Smarandache curves, Galilean space, Geodesic.
AMS 2010. 51B20, 53A35.
References
[1] Abdel-Aziz H.S., Saad M.K., Smarandache curves of some special curves in the Galilean
3-space, Honam Mathematical J., 253-264, 2015.
[2] Ali A. T., Special Smarandache Curves in the Euclidean Space, International Journal of
Mathematical Combinatorics, 2, 30-36, 2012.
[3] Bektaş Ö, Yüce S., Special Smarandache Curves According to Darboux Frame in
Euclidean3- Space, Romanian Journal of Mathematics and Computer Science, 3, 48-59, 2013.
[4] Çetin M, Tunçer Y., Karacan M. K., Smarandache Curves According to Bishop Frame in
Euclidean 3- Space, General Mathematics Notes, 20 (2) , 50-66, 2014.
[5] Dirişen B.C., Şahin T., Position vector of a curve with respect to the Darboux frame in
the Galilean space G3, arXiv:1707. 03930v1, 2017.
[6] Okur M., Şahin T., Special Smarandache Curves with Respect to Darboux Frame in the
Galilean 3-Space, arXiv: 1707.03935v1, 2017.
1 Amasya University, Amasya, Turkey, [email protected] 2 Amasya University, Amasya, Turkey, [email protected], [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
221
Elliptical Motion on a Given Ellipsoid
Mustafa ÖZDEMİR1
Abstract. The aim of this study is to explain the motion on the ellipsoid
((x²)/(a²))+((y²)/(b²))+((z²)/(c²))=1,
as a rotation, using the proper inner product, vector product and elliptical orthogonal matrices.
In this method, the elliptical inner product, the vector product and the angles are compatible
with the parameters θ and β of the parametrization ϕ(θ,β)=(acosθcosv, bcosθsinβ, csinθ). We
use the classical methods to generate elliptical rotation matrices such as Rodrigues, Cayley
and Hauseholder transformations.
Keywords. Elliptical Motion, Rotation Matrix, Elliptical Inner and Vector Product,
Rodrigues Formula, Cayley Formula.
AMS 2010. 15A63, 15A66.
References
[1] Özdemir, M., An Alternative Approach to Elliptical Motion, Adv. Appl. Clifford Algebras
26, 279–304, 2016.
[2] Rodríguez-Andrade, M. A., Aragón-González, G., Aragón, J. L., Verde-Star, L., An
algorithm for the Cartan-Dieudonné theorem on generalized scalar product spaces, Linear
Algebra and Its Applications, Vol. 434, Issue 5, 1238-1254, 2011.
[3] Mackey, D. S., Mackey, N., Tisseur, F., G-reflectors : Analogues of Householder
transformations in scalar product spaces, Linear Algebra and its Applications Vol. 385,
187—213, 2004.
[4] Özdemir, M., Ergin, A.A., Rotations with unit timelike quaternions in Minkowski 3-space
Journal of Geometry and Physics 56, 322-336, 2006.
[6] Şimşek, H, Özdemir M., Generating hyperbolical rotation matrix for a given hyperboloid,
Linear Algebra and Its Applications, Vol, 496, 221-245, 2016.
1 Akdeniz University, Antalya, TURKEY, [email protected], [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
222
[7] Şimşek H., Özdemir M., Rotations on a lightcone in Minkowski 3-space, Adv. Appl.
Clifford Algebras, 2017.
[8] M. Özdemir, M. Erdoğdu, On the Rotation Matrix in Minkowski Space-time. Reports on
Mathematical Physics 74, 27-38, 2014.
[9] M. Erdoğdu, M. Özdemir, Cayley Formula on Matrix in Minkowski Space-time, Int. J. of
Geometric Methods in Modern Physics, Vol 12, 2015.
[10] Aragón-González, G., Aragón, J. L., Rodríguez-Andrade, M. A., The decomposition of
an orthogonal transformation as a product of reflections, J. Math. Phys. 47, 2006.
[11] Bükçü, B., On the Rotation Matrices in Semi-Euclidean Space, Commun. Fac. Sci. Univ.
Ank. Series A1. 55, 7-13, 2006.
[12] Özkaldı, S., Gündoğan H., Cayley Formula, Euler Parameters and Rotations in 3-
Dimensional Lorentzian Space, Adv. in Applied Clifford Algebras 20 367-377, 2010.
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
223
Ruled Surfaces in Contact Geometry
Nural YÜKSEL1, Murat Kemal KARACAN2, Hasibe İKİZ3 and Çağlar Zeki ODABAŞI4
Abstract. In this paper, we study ruled surface in 3-dimensional almost contact metric
manifolds by using surface theory in contact geometry .We obtain the distribution parameters
of the ruled surface and then present some results and theorems with special cases. Moreover,
some relationships among asymptotic curve and striction line of the base curve of the ruled
surface were obtained.
Keywords. Sasakian manifold; contact manifold; ruled surface
AMS 2010. 53C15, 53C25.
References
[1] Baikoussis C. and Blair, D. E., Finite type integral submanifold of the contact manifold,
Bull. Math. Acad. Sinica 19, 327–350, 1991.
[2] Camcı, C., Extended cross product in a 3-dimensional almost contact metric manifold
with applications to curve theory, Turk. J. Math. 35, 1–14, 2011.
[3] Blair, D. E., Contact Manifolds in Riemannian Geometry, Lecture Notes in Mathematics,
Vol. 509 (Springer-Verlag, 1976).
[4] Gök, İ., Surfaces theory in contact geometry, PhD thesis, 2010.
1 Erciyes University, Kayseri, TURKEY, yukseln @erciyes.edu.tr 2 Uşak University, Uşak, TURKEY, [email protected] 3 Erciyes University, Kayseri, TURKEY, [email protected] 4 Erciyes University, Kayseri, TURKEY, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
224
A Note on Spherical Orthotomic of Lorentzian Spherical Spacelike Curves
Önder Gökmen YILDIZ1
Abstract. In this paper the spherical orthotomic of spacelike curve on Lorentzian
sphere is defined. Then, its local diffeomorphic image is investigated.
Keywords. Orhotomic, spherical curve, Lorentzian sphere.
AMS 2010. 53C50, 53C40.
References
[1] Bruce, J. W., Giblin, P. J., Curves and singularities: a geometrical introduction to
singularity theory, Second Edition, Cambridge, University Press, 1992.
[2] O’neill, B., Semi-Riemannian geometry with applications to relativity, Academic Press,
1983.
[3] Pekmen, U., Paşalı, S., Some characterization of Lorentzian spherical space-like curves,
Mathematica Moravica, 3, 33-37, 1999.
[4] Petrovic-Torgasev, M Sucurovic, E., Some characterizations of Lorentzian spherical
spacelike curves with the timelike and null principal normal, Mathematica Moravica, 4, 83-
92, 2000.
[5] Xiong, J. F., Spherical orthotomic and spherical antiorthotomic, Acta Mathematica
Sinica, 23, 9, 1673-1682, 2007.
1 Bilecik Şeyh Edebali University, Bilecik, TURKEY, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
225
Local Diffeomorphic Image of Spherical Antiorthotomic of Lorentzian Spherical
Timelike Curves
Önder Gökmen YILDIZ1
Abstract. In this paper, spherical antiorthotomic of Lorentzian spherical curve on the
Lorentzian sphere is defined. Then, spherical antiorthotomic is obtained as an envelope of the
family of great circles, lying on the planes that perpendicularly bisect the chords. Finally, by
using the technique in [1] its local diffeorporhic type is determined.
Keywords. Orhotomic, antiorthotomic, spherical curve, Lorentzian sphere.
AMS 2010. 53C50, 53C40.
References
[1] Bruce, J. W., Giblin, P. J., Curves and singularities: a geometrical introduction to
singularity theory, Second Edition, Cambridge, University Press, 1992.
[2] O’neill, B., Semi-Riemannian geometry with applications to relativity, Academic Press,
1983.
[3] Pekmen, U., Paşalı, S., Some characterization of Lorentzian spherical space-like curves,
Mathematica Moravica, 3, 33-37, 1999.
[4] Petrovic-Torgasev, M Sucurovic, E., Some characterizations of Lorentzian spherical
spacelike curves with the timelike and null principal normal, Mathematica Moravica, 4, 83-
92, 2000.
[4] Petrovic-Torgasev, M Sucurovic, E., Some characterizations of the Lorentzian spherical
timelike and null curves, Mathematicki Vesnik, 53.(1-2), 21-27, 2001.
[5] Xiong, J. F., Spherical orthotomic and spherical antiorthotomic, Acta Mathematica
Sinica, 23, 9, 1673-1682, 2007.
1 Bilecik Şeyh Edebali University, Bilecik, TURKEY, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
226
The Kinetic Energy Formula for the Closed Planar Homothetic Motions in Complex
Plane
Önder ŞENER1, Ayhan TUTAR2 and Serdar SOYLU3
Abstract. In this paper, the kinetic energy formula was expressed during one-
parameter closed planar homothetic motions in complex plane. Then the relation between the
kinetic energy formula and the Steiner formula was given. As an example the sagittal motion
of a telescopic crane was considered. This motion was described by a double hinge consisting
of the fixed control panel of telescopic crane and the moving arm of telescopic crane. The
results were applied to experimentally measured motion.
Keywords. kinetic energy, Steiner Formula, homothetic motions.
AMS 2010. 53A17, 70B10.
References
[1] Dathe, H., Gezzi, R., Characteristic directions of closed planar motions, Zeitschrift für
Angewandte Mathematik und Mechanik, 92, 2-13, 2012.
[2] Dathe, H., Gezzi, R., Addenda and Erratum to: Characteristic directions of closed planar
motions, Zeitschrift für Angewandte Mathematik und Mechanik, 94, 551–554, 2014.
[3] Müller, H. R., Verallgemeinerung einer Formel von Steiner, Abh. Braunschweig. Wiss.
Ges., 29, 107-113, 1978.
[4] Steiner, J., Von dem Krümmungs-Schwerpuncte ebener Curven, Journal für die reine und
angewandte Mathematik, 21, 33-63, 1840.
[5] Tutar, A., Kuruoğlu, N., The Steiner formula and the Holditch theorem for the homothetic
motions on the planar kinematics, Mechanism and Machine Theory, 34, 1-6, 1999.
[6] Tutar, A., Sener, O., The Steiner formula and the Polar moment of inertia for the closed
planar homothetic motions in complex plane, Advances In Mathematical Physics, 2015, 1-5,
2015.
1 Ondokuz Mayis University, Samsun, TURKEY, [email protected] 2 Present address: Kyrgyz-Turk Manas University, Bishkek, KYRGYZSTAN
Permanent address: Ondokuz Mayis University, Samsun, TURKEY, [email protected] 3 Giresun University, Giresun, TURKEY, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
227
New Developments in Lightlike Submanifolds Theory
Sadık KELEŞ1, Erol KILIÇ2 and Mehmet GÜLBAHAR3
Abstract. Sectional curvature map on r lightlike submanifolds of a semi-
Riemannian manifold is investigated. Some special lightlike submanifolds are mentioned.
Some inequalities for lightlike submanifolds obtained. Using these inequalities, some
characterizations are obtained for special lightlike submanifolds.
Keywords. Curvature, lightlike submanifold, semi-Riemannian manifold.
AMS 2010. 53C40, 53C42, 53C50.
References
[1] Bejan C. L., Duggal K. L., Global lightlike manifolds and harmonicity, Kodai Math. J. 28,
131-145, 2005.
[2] Chen, B.-Y., Pseudo-Riemannian geometry, invariants and applications, World
Scientific Publishing, Hackensack, NJ 2011.
[3] Duggal, K. L., Sahin, B., Differential geometry of lightlike submanifolds, Birkhauser,
Basel, 2010.
[4] Kulkarni, R. S., The values of sectional curvature in indefinite metric, Comment Math.
Helv. Phys. 54, 173-176, 1979.
1 İnönü University, Malatya, TURKEY, [email protected] 2 İnönü University, Malatya, TURKEY, [email protected] 3 Siirt University, Siirt, TURKEY, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
228
Holditch Type Theorem for Kinetic Energy of Projective Curve Under the 1-Parameter
Spatial Motion
Serdar SOYLU 1, Ayhan TUTAR2 and Önder ŞENER3
Abstract. Firstly, we obtain the kinetic energy of the projective curve for the closed
spatial motion as following,
2𝑆 = 2𝑆0 + 𝜌 ∑ 𝑥𝑖2 −
3
𝑖=1
∑ 𝑏𝑖𝑗𝑥𝑖𝑥𝑗 +
3
𝑖,𝑗=1
∑ 𝑐𝑖𝑥𝑖
3
𝑖=1
Then, we show the Holditch type theorem for kinetic energy of the projective curve
and we get some results related with that theorem.
Keywords. Kinetic, holditch, motion.
References
[1] Dathe H. and Gezzi R. Characteristic directions of closed planar motions, Zeitschrift für
Angewandte Mathematik und Mechanik, 2-13, 2012.
[2] Dathe H. and Gezzi R., Addenda and Erratum to: Characteristic Directions of Closed
Planar Motions, Zeitschrift für Angewandte Mathematik und Mechanik, 92(9), 731-748,
2014.
[3] Dathe H., Gezzi R., Kubein-Meesenburg D. and Nagerl H., Characteristic Point and
Cycles in Planar Kinematiks with Applications to Human Gait. Acta Bioengineering and
Biomechanics. Vol 17, No.1, 2015.
[4] Düldül M., Kuruoğlu N. and Tutar A., Generalization of Steiner Formula for the
homothetic motions on the planar kinematics, Applied Mathematics and Mechanics-English
Edition, 24(8), 945-949, 2003.
[5] Düldül M., Homotetik Uzay Hareketleri ve Holditch Teoremi. Doktora Tezi, Ondokuz
Mayıs Üniversitesi Fen Bilimleri Enstitüsü, Samsun, 2004.
1 Giresun University, Giresun, TURKEY, [email protected] 2 Present address: Kyrgyz-Turk Manas University, Bishkek, KYRGYZSTAN
Permanent address: Ondokuz Mayis University, Samsun, TURKEY, [email protected] 3 Ondokuz Mayis University, Samsun, TURKEY, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
229
[6] Düldül M., Kuruoğlu N. and Yüce S., The polar moment of inertia of the enveloping
curve, Novi Sad Journal of Mathematics, 38(2), 1-4, 2008.
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
230
On the Casual Characteristics of Higher Order Bertand Mate, Involute Curve,
Mannheim of a Non-Null Curve in IL³
Şeyda KILIÇOĞLU1 and Süleyman ŞENYURT2
Abstract. We have already defined and worked on the second order Bertand mate,
involute curve, Mannheim partner of a unit-speed curve in IE 3. In this paper, first we have
defined nth order Bertand mate, involute curve, Mannheim partner of a non-null curve in IL 3.
Also their casual characteristics are examined. Further, the casual characteristics of the nth
order Bertand mate, involute curve, Mannheim partner of a non-null curve is given in a table
with the number of different type of casual characteristics. The number of the different forms
of the casual characteristics of nth order involute and Mannheim mate of a space-like curve
can be given using Fibonacci sequence. A time-like Bertrand curve has 2ⁿ different forms,
also the nth order Bertrand mate of a space-like Bertrand curve with time-like principal normal
is always space-like Bertrand curve with time-like principal normal.
Keywords. Lorentz metric, casual characteristic, involute, Bertand mate, Mannheim
partner.
AMS 2010. 53A04, 53B30
References
[1]. Bilici M.., Çalışkan, M., On the Involutes of the Spacelike Curve with a Timelike
Binormal in Minkowski 3-Space. International Mathematical Forum, 4, no. 31, 1497 – 1509,
2009.
[2]. Bukcu, B, Karacan M. K., On the involute and evolute curves of the spacelike curve with
a spacelike binormal in Minkowski 3-space. Int. Journal of Contemp. Math. Sciences, Vol.
2,no. 5-8, 221-232, 2007.
[3]. Gray, A., Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd
ed. Boca Raton, FL: CRC Press, pp. 205, 1997.
1 Baskent University, Ankara, TURKEY, [email protected] 2 Ordu University, Ordu, TURKEY, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
231
[4]. Hacısalihoğlu, H.H., Differential Geometry (in Turkish), Academic Press Inc. Ankara,
1994.
[5]. Kılıçoğlu, Ş., Senyurt, S., On the second order involute curves in IE³. Commun. Fac. Sci.
Univ. Ank. Series A1, 66(2), 332-339., Doi: 10.1501/Commua1_0000000823 (Yayın No:
3472826), 2017.
[6]. Kılıçoğlu, S., Senyurt, S., On the Second Order Mannheim Partner Curve in IE³.
International J.Math. Combin., 1, 71-77, 2017 (Yayın No: 3472825) , 2017.
[7]. Lipschutz, M.M., Differential Geometry, Schaum's Outlines, 1969 . Liu H. and Wang
F.,Mannheim partner curves in 3-space, Journal of Geometry, Vol.88, No 1-2, 120-
126(7),2008.
[8]. Senyurt, S., Kılıçoğlu S., On the Differential Geometric Elements of Bertrandian
Darboux ruled surface in E3., Sakarya Üniversitesi Fen Bilimleri Enstitüsü Dergisi. Sayı 3,
Cilt 21, 2017.
[9]. Schief, W.K., On the integrability of Bertrand curves and Razzaboni surfaces. Journal of
Geometry and Physics, Volume 45, Issues 1--2, Pages 130--150, February, 2003.
[10]. O’Neil, B., Semi-Riemannian geometry with applications to relativitiy, Academic
Press, Inc. USA 1983.
[11]. Orbay K., Kasap E., On mannheim partner curves, International Journal of Physical
Sciences, Vol. 4 (5), 261-264, 2009.
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
232
On the Second Order Involute of a Space-like Curve with Time-like Binormal in IL³
Şeyda KILIÇOĞLU1 and Süleyman ŞENYURT2
Abstract. We have already defined and worked on the second order involute curve of
a unit-speed curve in IE 3. In this paper, we consider the second order involute of a space-like
curve with time-like binormal in IL³. All Frenet apparatus of them are examined in terms of
Frenet apparatus of the space-like curve with time-like binormal α.
Keywords. Lorentz metric, casual characteristic, involute, Bertand mate, Mannheim
partner.
AMS 2010. 53A04, 53B30
References
[1]. Bilici M.., Çalışkan, M., On the Involutes of the Spacelike Curve with a Timelike
Binormal in Minkowski 3-Space. International Mathematical Forum, 4, no. 31, 1497 – 1509,
2009.
[2]. Bukcu, B, Karacan M. K., On the involute and evolute curves of the spacelike curve with
a spacelike binormal in Minkowski 3-space. Int. Journal of Contemp. Math. Sciences, Vol.
2,no. 5-8, 221-232, 2007.
[3]. Gray, A., Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd
ed. Boca Raton, FL: CRC Press, pp. 205, 1997.
[4]. Hacısalihoğlu, H. H., Differential Geometry (in Turkish), Academic Press Inc. Ankara,
1994.
[5]. Kılıçoğlu, Ş., Senyurt, S., On the second order involute curves in IE³. Commun. Fac. Sci.
Univ. Ank. Series A1, 66(2), 332-339., Doi: 10.1501/Commua1_0000000823 (Yayın No:
3472826), 2017.
[6]. Kılıçoğlu, S., Senyurt, S., On the Second Order Mannheim Partner Curve in IE³.
International J.Math. Combin., 1, 71-77, 2017 (No: 3472825) , 2017.
[7]. Lipschutz, M.M., Differential Geometry, Schaum's Outlines, 1969. Liu H. and Wang F.,
Mannheim partner curves in 3-space, Journal of Geometry, Vol.88, No 1-2, 120-126(7), 2008.
1 Baskent University, Ankara, TURKEY, [email protected] 2 Ordu University, Ordu, TURKEY, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
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On Biharmonic Hypersurfaces in Semi-Euclidean Spaces
Sibel SEVİNÇ1, Gülşah AYDIN ŞEKERCİ2 and A. Ceylan ÇÖKEN3
Abstract. In this paper we study biharmonic hypersurfaces of semi-Euclidean spaces.
We consider the Chen conjecture which states that the only biharmonic submanifolds of
Euclidean spaces are the minimal ones. In particular, we obtain some conditions for
biharmonic hypersurfaces in semi-Euclidean spaces. Finally, we give some examples for
different conditions.
Keywords. Biharmonic hypersurface, Chen conjecture, semi-Euclidean space.
AMS 2010. 53C40, 53C42.
References
[1] Chen, B-Y., Ishikawa, S., Biharmonic pseudo-Riemannian submanifolds in pseudo-
Euclidean spaces, Kyushu J. Math., 52, 1, 167-185, 1998.
[2] Chen, B-Y., Ishikawa, S., Biharmonic surfaces in pseudo-Euclidean spaces, Mem. Fac.
Sci. Kyushu Univ. Ser. A, 45, 2, 323-347, 1991.
[3] Chen, B-Y., Munteanu, M. I., Biharmonic ideal hypersurfaces in Euclidean spaces,
Differential Geometry and its Applications, 31, 1-16, 2013.
[4] Turgay, N. C., A classification of biharmonic hypersurfaces in the Minkowski spaces of
arbitrary dimension, Hacettepe J. Math. and Statistics, 45, 4, 1125-1134, 2016.
[5] Gupta, R. S., Biharmonic hypersurfaces in E⁵, An. Ştiint. Univ. Al. I. Cuza Iaşi Mat. (N.
S.), 2, 2, 585-593, 2016.
[6] Gupta, R. S., Biharmonic hypersurfaces in E6 with constant scalar curvature, International
J. Geo., 5, 2, 39-50, 2016.
1 Cumhuriyet University, Sivas, TURKEY, [email protected] 2 Süleyman Demirel University, Isparta, TURKEY, [email protected] 3 Akdeniz University, Antalya, TURKEY, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
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Timelike and Spacelike Split Quaternion Ruled Surfaces in Minkowski 3-space
Kıvanç KARAKAŞ1 , Mesut ALTINOK2 and Levent KULA3
Abstract. Quaternion rational surface which generated from quaternion product of two
rational space curves is defined by Wang and Goldman in [1]. They also defined quaternion
rational ruled surface which is special quaternion rational surface. In this work, we investigate
the new surface in semi Euclidean space which generated from the split quaternion product of
a line and a space curve. We give a timelike and spacelike split quaternion ruled surface
examples by Mathematica 10. Moreover, we describe of syzygy, mu-basis for a timelike and
spacelike split quaternion ruled surfaces and give its implicit equations.
Keywords. Syzygy, mu-basis, split quaternion rational ruled surface, spacelike
quaternion ruled surface, timelike quaternion ruled surface.
This work is supported by Ahi Evran University Scientific Research Project Coordination
Unit. Project number: PYO- FEF.A3.17.002.
AMS 2010. 53A04, 53C50.
References.
[1] Wang, X. ; Goldman, Quaternion rational surfaces: Rational surfaces generated from the
quaternion product of two rational space curves, Journal of Graphical Models, 2015, no.81,
18-32.
[2] Chen, F. ; Zheng, J. ; Sederberg, T. µ-basis of a rational ruled surface, Journal of
Computer Aided Geometric Design, 2001, no.18, 61-72.
[3] Kula, L. Bölünmüş Kuaterniyonlar ve Geometrik Uygulamalari, Ph.D. Thesis, Ankara
University Institute of Science, Ankara, 2003.
1 Ahi Evran University, Kırşehir, TURKEY, [email protected] 2 Mersin University, Mersin, TURKEY, [email protected] 3 Ahi Evran University, Kırşehir, TURKEY, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
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MATHEMATICS
EDUCATION
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
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Parameterized Examination in Econometrics
Anna MALINOVA1, Vesselin KYURKCHIEV2 and Georgi SPASOV3
Abstract. This paper presents a parameterization of basic types of exam questions in
Econometrics to automate and facilitate the process of examination, assessment and self-
preparation of a large number of students. The proposed parameterization of testing questions
reduces the time required to author tests and course assignments. It enables tutors to generate
a large number of different but equivalent dynamic questions (with dynamic answers) on a
certain topic, which are automatically assessed. The presented methods are implemented in
DisPeL (Distributed Platform for e-Learning) [1], [5], [6], [7], [8] and provide questions on
the areas of filtering and smoothing time-series data, forecasting, building and analysis of
single-equation econometric models. Questions also cover elasticity, average and marginal
characteristics, product and cost functions, measurement of monopoly power, supply, demand
and equilibrium price, consumer and product surplus, etc. Several approaches are used to
enable the required numerical computations in DisPeL – integration of third-party
mathematical libraries, developing our own procedures from scratch, and wrapping our legacy
math codes in order to modernize and reuse them [2], [3], [4].
Keywords. Econometrics, parameterization, dynamic questions, automatic test
generation, e-testing,
AMS 2010. 91G70, 97D40, 97R20, 97U50.
References
[1] Arnaudova, V., T. Terzieva, A. Rahnev, A Methodological Approach for Implementation
of Adaptive e-Learning, CBU International Conference on Innovations in Science and
Education, March 23-25, 2016, Prague, Czech Republic, CBU International Conference
Proceedings, Vol. 4, 910-917, 2016.
[2] Malinova, A., Approaches and Techniques for Legacy Software Modernization, Scientific
Works, Plovdiv University, Vol. 37, Book 3, 2010 - Mathematics, 77-85, 2010.
1 University of Plovdiv Paisii Hilendarski, Plovdiv, BULGARİA, [email protected] 2 University of Plovdiv Paisii Hilendarski, Plovdiv, BULGARIA, [email protected] 3 University of Plovdiv Paisii Hilendarski, Plovdiv, BULGARIA, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
237
[3] Malinova, A., Modernizing Legacy Physics Applications for Reuse in Web and SOA,
Serdica Journal of Computing, Vol. 5, No. 1, 79-100, 2011.
[4] Malinova, A., Ivanova, V., & Rahnev, A., Computer algebra aided generation of English
language tests, The 11th Annual International Conference on Computer Science and
Education in Computer Science CSECS 2015, June 04-07 2015, Boston, MA, USA,
Computer Science Education & Computer Science Research Journal, 11, 66-74, 2015.
[5] Rahnev, А., Pavlov, N., Kyurkchiev, V., Distributed Platform for e-Learning – DisPeL,
European International Journal of Science and Technology (EIJST), 3(1), 95-109, 2014.
[6] Rahnev, A., Pavlov, N., Golev, A., Stieger, M., Gardjeva, T., New Electronic Education
Services Using the Distributed E-Learning Platform (DisPeL), International Electronic
Journal of Pure and Applied Mathematics (IEJPAM), 7(2), 63-71, 2014.
[7] Rahnev A., Rahneva O., Testing and assessment in accounting at the Distributed e-
Testing Cluster – DeTC, Scientific Works, ECEM, Book 4, 182-189, 2008.
[8] Rahneva O., Golev, A., Pavlov, N., Dynamic Generation of Testing Question in SQL in
DeTC, Cybernetics and Information Technologies, 8(1), 73-81, 2008.
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
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The Effect of Differentiated Instruction on Mathematical Attitudes of Students
Ayten Pinar BAL1 and Onur EKİNCİ2
Abstract. Differentiated teaching approach that accounts personal differences within a
class, considers different students’ different skills and learning needs, and draws on students’
strengths gains importance [1], [2]. In this study has been carried out in order to define the
effects of differentiated instruction method on the mathematical attitudes of primary school
third grade learners. It is designed for quantitative models. The quantitative data has been
designed on semi-experimental model with pretest and posttest control group. The
experimental group of the study has been formed by semi-randomly selected 20 students who
got similar points from tests and had similar learning styles. During the application, the topic
of “Fractions” was taught by Differentiated Instruction Method in the experimental group,
while it was taught by Traditional Method in the control group according to the course book
of mathematics. In the experimental group, the courses were taught by the researcher and in
the control group it was taught by the teacher of the class. The data has been collected by
using “Mathematics Attitude Scale”. “Mathematics Attitude Scale” has been applied to both
experimental and control groups as pretest, posttest and permanence test. One factor Ancova
has been used in order to find out whether the Differentiated Instruction Method has any
statistical effects on the points of “Mathematics Attitude Test” before and after the
experiment. According to the data obtained at the end of the research, there is not a significant
difference in the post test results of attitude test in favor of the experimental group.
Keywords. Mathematics education, mathemetics attitude, differentiated instruction
References
[1] Gregory, G. H., Chapman, C., Differentiated instructional strategies: One size doesn’t fit
all. Thousand Oaks, CA: Corwin Press, 2002.
[2] Sondergeld, T. A., Schultz, R. A., Science, standards, and differentiation: It really can be
fun!, Gifted Child Today, 31(1), 34-40, 2008.
1 Çukurova University, Adana, TURKEY, [email protected] 2 Çukurova University, Adana, TURKEY, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
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Investigation of the Test Anxieties of 8th Grade Students*
Bülent ALTUNKAYA1, Kamile ŞANLI KULA2 and Cahit AYTEKİN3
Abstract. Today, the guidance and psychological counseling services in our schools
spend most of their time trying to reduce students' test concerns [1]. Students who experience
high level of test anxiety cannot use their knowledge and skills effectively during the exams
and their success decreases as a result. In a study in Turkey, it is stated that the level of
anxiety for the university exam is considerably higher than that of patients waiting to be
operated [2]. For this reason, the research that will be done in order to decrease the students'
test concerns will guide the actions to be taken. If the conditions affecting students' test
concerns are well recognized, then successive approaches in all courses, especially
mathematics, can be put into practice. In this study, it is aimed to examine the test anxieties of
the 8th grade students according to some variables. A total of 952 students from 14 different
primary schools participated in the research. 468 of these students were females, while 484
were males.
According to the research findings, it was found that female students' test concerns
were significantly higher than male students. When the scores from the burnout scale are
examined, it is seen that boys burnout levels are significantly higher than girls. Children from
low income levels were found to have higher test concerns in the sub-dimension of delusions.
There is no significant difference in the test anxiety according to the mother's education level.
However, it has been determined that burnout levels of students who have mothers of
university graduates are significantly higher than those of mothers of primary and junior high
school graduates. A similar result was found for father's educational status. Burnout levels of
students whose mothers working are higher than those of mothers who do not work. It was
determined that the test anxiety of students who took support courses for the high school
placement exam were lower than the students who did not take. Burnout levels of students
who do not attend the support course are also significantly higher than those who took support
courses. It was found that there is no significant difference in the test anxiety and burnout
levels according to the situation of receiving private lessons for the high school placement
exam. It has been determined that the students who declared that they work 0-1 hours daily
have higher delusional test anxiety than the students who declare that they are working for 4-5
hours daily. A similar result is determined about burnout levels. It is seen that the students
who declared that they never worked or that they worked 0-1 hours were found to have
significantly higher burnout levels than the other students. Students who do leisure time
activities such as painting, music and sports had significantly higher test anxiety than those
who do not. There are significant differences in the test anxieties and burnout levels according
* This work was supported by the Ahi Evran University Scientific Research Projects Coordination Unit. Project
Number: FEF.A4.17.003 1 Ahi Evran University, Kırşehir, TURKEY, [email protected] 2 Ahi Evran University, Kırşehir, TURKEY, [email protected] 3 Ahi Evran University, Kırşehir, TURKEY, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
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to the perceived level of students about relation life and lesson in school. Students who say
that their school is not related to their life at all have significantly higher test anxiety and
burnout than others. The students who stated that the teacher defined himself/herself as
unsuccessful were found to have significantly higher delusional test anxiety and burnout than
other students. A similar finding was found for students who stated that their family identified
themselves as unsuccessful. The students who say that they work only on interesting topics
have higher test anxiety and burnout level than those who work for not to take a poor grade in
school. A similar result was found in burnout levels. Finally, it was determined that the
burnout levels of the students who were waiting for their teachers to teach the information to
be useful in the daily life rather than the exam had significantly higher burnout levels than the
other students.
Keywords. Test anxiety, burnout levels, high school placement exam
References
[1] Kayapınar, E. Ortaöğretim Kurumları Öğrenci Seçme Ve Yerleştirme Sınavı (Oks)’na
Hazırlanan İlköğretim 8. Sınıf Öğrencilerinin Kaygı Düzeylerinin İncelenmesi
(Afyonkarahisar İli Örneği). Yayınlanmamış Yüksek Lisans Tezi. Afyon Kocatepe
Üniversitesi Sosyal Bilimler Enstitüsü, Afyon, 2006
[2] Baltaş A., Baltaş Z. Başarılı ve Sağlıklı Olmak İçin Stresle Başa Çıkmanın Yolları, Remzi
Kitabevi Yayınları, İstanbul, 1998.
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
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Examining Pre-service Teachers’ Perceptions and Evaluations about Student-Invented
Strategies
Büşra KARTAL1 and Yasemin KIYMAZ2
Abstract. Student-invented strategies are referred to any strategies which are different
from traditional algorithm and not including manipulatives [1]. Students who invent or adopt
strategies actively engage in making sense of mathematics [2]. Evaluating and responding to
students’ thinking and strategies that they invent are considered as a major goal of effective
mathematics teaching [3]. It was aimed to examine, describe and interpret pre-service
teachers’ views about student-invented strategies in subtraction. Within the context of this
purpose, case study was utilized. 85 pre-service teachers from elementary education and
mathematics education participated in the study. A semi-conducted interview form was used
to collect data. Pre-service teachers were introduced to three different student solutions to the
problem 173-46. Two of the solutions were student-invented strategies and the last one was
traditional algorithm. Pre-service teachers were asked (i) which of the strategies they choose
for their future students, (ii) how to respond to each of these students, and (iii) how to
introduce traditional algorithm to students that invent strategies. Descriptive and content
analyses will be used in analyzing preferred strategies and the reasons of preference; pre-
service teachers’ evaluation, feedback and comparison of students’ solutions, and approaches
that pre-service teachers state they will use in introducing traditional algorithm.
Keywords. Mathematics education, teacher education, student-invented strategies.
AMS 2010. 97D99, 97B50.
Acknowledgement.This work is supported by Ahi Evran University PYO with project
number EGT.E2.17.043.
References
[1] Van de Walle, J. A., Karp, K. S., & Williams, J. M. B., Elementary and middle school
mathematics: Teaching developmentally (7th Ed.). New York: Longman, 2010.
[2] Carpenter, T. P., Fennema, E., & Franke, M. L., Cognitively guided instruction: Building
the primary mathematics curriculum on children’s informal mathematical knowledge. A
paper presented at the annual meeting of the American Educational Research Association,
1992.
[3] National Council of Teachers of Mathematics, Principles and standards for school
mathematics. Reston, VA: Author, 2000.
1 Ahi Evran University, Kırşehir, TURKEY, [email protected]
2 Ahi Evran University, Kırşehir, TURKEY, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
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Investigation of Parent Expectations of Middle School Students from Mathematics
Education*
Cahit AYTEKİN1, Serdal BALTACI2, Bülent ALTUNKAYA3,
Avni YILDIZ4, Yasemin KIYMAZ5
Abstract.
It was stated that the academic achievement and the emotional development of the
students whose family supports and actively followed their education process develop faster
than the other students [1]. Similarly, it is known that the academic achievement of students
has significantly increased in a teaching environment actively supported by the family [3]. In
one study, the students whose parents actively participated in education were examined and
found that these students were more likely to have upper education, higher grades and more
social developments [2].
In this study, middle school student parents' expectations of "conceptual understanding
and active student participation", "positive attitude and behavior", and "authority and rule-
oriented teaching" were examined. In the research, a valid and reliable three-factor instrument
developed by the researchers was used. A total of 749 parents participated in the research. As
a result of the research, it was determined that the expectation of the parents did not show any
significant difference according to the parents' type (mother, father and other) and parental
age level. It has been found out that the university graduate student parents have less authority
and rule-oriented teaching expectation than others. It has also been determined that
individuals with moderate monthly income have a lower expectation of "authority and rule-
based mathematics instruction" than those with low-gestational age groups. Moreover, it was
found that parents who love mathematics have high "conceptual understanding and active
student participation" expectation. On the other hand, it is revealed that parents who do not
like mathematics have low level "authority and rule-oriented teaching" expectation. It was
also found that parent who have a high degree of support for their child have high expectancy
of "conceptual understanding and active student participation" and have low expectancy of
"authority and rule oriented teaching". It was also found that the parents of successful in
* This work was supported by the Ahi Evran University Scientific Research Projects Coordination Unit. Project Number:
EGT.A3.16.013
1 Ahi Evran University, Kırşehir, TURKEY, [email protected] 2 Ahi Evran University, Kırşehir, TURKEY, [email protected] 3 Ahi Evran University, Kırşehir, TURKEY, [email protected] 4 Bülent Ecevit University, Zonguldak, TURKEY, [email protected]
5 Ahi Evran University, Kırşehir, TURKEY, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
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mathematics have higher "conceptual understanding and active student participation"
expectations. Based on these findings, it can be said that parents' expectations have an
important role in education.
Keywords. Parents Expectation, Mathematics Education
References
[1] Booth, A.,& Dunn, J. F., Family school links: How do they affect educational outcomes?
Hillsdale, NJ: Erlbaum, 1996.
[2] Henderson, A. T. & Berla, N., A New Generation of evidence: The family is critical to
student achievement. Washington DC: National Committee for Citizens in Education, 2004.
[3] Nyabuto, A. N.,& Njoroge P. M., Parental involvement on pupils' performance in
mathematics in public primary schools in Kenya. Journal of Educational and Social Research,
4(1), 19-26, 2014.
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
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The Impact of the Probability Explorer Software on Preparing Probability Teaching
Plans of Elementary Mathematics Teacher Candidates*
Cahit AYTEKİN1, Serdal BALTACI2, Yasemin KIYMAZ3, Avni Yıldız4
Abstract
Studies on teaching probability in Turkey draw attention finding the most effective
teaching method to effect achievements of students [2], [3], [4], [5], [6]. Many studies show
that students have difficulty in making sense of the concepts of probability [7]. Considering
the difficulties in the teaching of probability and misconceptions, studies focused on teaching
probabiliry are very important [1]. In this study, it is aimed to develop teacher candidates
about teaching probability who will start teaching as a mathematics teacher in the near future.
This research aims to develop elementary school mathematics teachers’ skills to
prepare teaching plan about of probability. For this purpose, "Probability Explorer" software
has been taught to prospective teachers. Probability Explorer" software was developed for
probabilistic teaching. The instruction plans prepared by Teacher candidates before and after
this instruction were compared. The data of the study were analyzed by qualitative research
methods.
Thirteen different themes were found in the second plan of teacher candidates, which
were not found in their initial plans. These situations are thought to be very effective in the
teaching of probability. It was seen that from these themes "observing and interpreting
experimental results", "conducting a large number of experiments on an event", "approaching
the theoretical probability as the number of experiments increases", "estimating the
experimental results", "encouraging the student to do his own probabilistic experiments",
"comparison of the theoretical probability values" were more frequent.
When the findings of the research are examined, it has been seen that simulation
softwares are very important in teaching probability. These softwares make it possible to use
different methods in teaching probability by making difficult experiments very fast. It has
been observed that there are considerable differences in teaching plans based on using a
* This work was supported by the Ahi Evran University Scientific Research Projects Coordination Unit. Project Number:
EGT.A3.17.009
1 Ahi Evran University, Kırşehir, TURKEY, [email protected] 2 Ahi Evran University, Kırşehir, TURKEY, [email protected] 3 Ahi Evran University, Kırşehir, TURKEY, [email protected] 4 Bülent Ecevit University, Zonguldak, TURKEY, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
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simulation program for elementary school mathematics teacher candidates compared to
teaching plans prepared in the traditional way.
Keywords. Teaching Probability, Preparing Teaching Plan, Simulation Softwares.
References
[1] Bulut, S., Yetkin, E.İ ve Kazak, S., Matematik Öğretmen Adaylarının Olasılık Başarısı,
Olasılık ve Matematiğe Yönelik Tutumlarının Cinsiyete Göre İncelenmesi. Hacettepe
Üniversitesi Eğitim Fakültesi Dergisi, 22: 21-28, 2002.
[2] Çubuk, Ş., Matematik Öğretiminde “Permütasyon ve Olasılık” Konusunun Bilgisayar
Destekli Öğretim Materyalleri İle Öğretilmesinin Öğrenci Başarısına Etkisi. Marmara
Üniversitesi Eğitim Bilimleri Enstitüsü Yüksek Lisans Tezi, 2005.
[3] Ercan, Ö., Çoklu Zeka Kuramına Dayalı Öğretim Etkinliklerinin 8.Sınıf Öğrencilerinin
Matematik Dersi “Permütasyon ve Olasılık” Ünitesindeki Akademik Başarıları Etkisi. Gazi
Üniversitesi Eğitim Bilimleri Enstitüsü Yüksek Lisans Tezi, 2008.
[4] Erkin Kavasoğlu, B., İlköğretim 6,7 ve 8 Matematik Dersinde Olasılık Konusunun Oyuna
Dayalı Öğretiminin Öğrenci Başarısına Etkisi. Gazi üniversitesi eğitim bilimleri enstitüsü,
2010.
[5] Esen, B., Matematik Eğitiminde İlköğretim 6.Sınıflarda Olasılık Konusunun Öğretiminde
Bilgisayar Destekli Eğitimin Rolü. Selçuk Üniversitesi Fen Bilimleri Enstitüsü Yüksek Lisans
Tezi, 2009.
[6] Gürbüz, R., Olasılık Kavramları İle İlgili Geliştirilen Öğretim Materyallerinin
Öğrencilerin Kavramsal Gelişimine Etkisi. Dokuz eylül üniversitesi buca eğitim fakültesi
dergisi, 2006.
[7] Sev Lekesiz, E.Ç., Dördüncü ve Beşinci Sınıf Öğrencilerinin Olasılık Konusunun
Öğretiminde Karşılaştıkları Zorluklar. Eskişehir Osmangazi Üniversitesi Eğitim Bilimleri
Enstitüsü yüksek lisans tezi, 2011.
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
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Proof Evaluation: Mathematicians and Prospective Mathematics Teachers’ Perspectives
Esra Selcen YAKICI-TOPBAŞ1, Elçin EMRE-AKDOĞAN2, Fatma Nur AKTAŞ3, Gönül
YAZGAN-SAĞ4 and Ziya ARGÜN5
Abstract. Abstract. Proof plays an important role in mathematics to verify whether a
statement is true or not, to explain why a statement is true, and to communicate mathematical
knowledge ([1], [5]). “Determining whether a proof is correct and valid and also how good it
is regarding a wide range of features such as clarity, logical correctness, insight,
convincingness, fluency” are fundamental features for proof evaluation [4]. Therefore, proof
evaluation provides us with an understanding of mathematicians, mathematics teachers, and
prospective mathematics teachers’ perspectives on how they construct proofs and thus it helps
to obtain an understanding of what constitutes a proof. Also, mathematical proof is critical to
constitute the undergraduate mathematics curriculum for mathematics teachers and
prospective mathematics teachers [3]. Researchers have argued that mathematicians want to
know why a statement is correct rather than whether a statement is correct or not, that is,
proof evaluation is wider than judgment and validation [2]. In this context, the current study
aims to investigate the differences and similarities between mathematicians and prospective
teachers’ perspectives while evaluating a basic abstract mathematics theorem’s proof
examples, which include standard proof methods: direct, contradiction, and contraposition.
This study is a multiple-case study that has a qualitative approach [6]. The cases of this study
are mathematicians and prospective mathematics teachers. Two mathematicians (who are
professors, who have completed their doctorate in mathematics) and six senior prospective
mathematics teachers are participants of the study. The participants were selected through
purposeful and convenience sampling. The data were collected through individual semi-
structured interviews which were recorded with a video camera. In the interviews, two tasks
were completed by participants. We gave a proposition for proving in the first task. Then in
the second task we gave propositions and its nine different proofs including constituted direct,
contradiction, contraposition standard basic proof methods and a visual proof. We asked
participants to examine nine proofs given in the second task. We deeply examined the
participants' evaluation on the proofs. We analyzed the data through content analysis. We
shared the findings in the context of a comparison of mathematicians’ and prospective
mathematics teachers’ perspectives on the evaluation and investigation of the given proof
examples. We explored that mathematicians do not accept visual proof as a mathematical
proof, contrary to the fact that prospective teachers evaluate visual proof as a mathematical
proof that is essential for their students. On the process of evaluating proofs, all participants
pay attention to the mathematical language and the beginning of the proof.
Keywords. Proof evaluation, prospective mathematics teacher, mathematics
education.
AMS 2010. 03F03, 97-XX
1 Gazi University, Ankara, Turkey, selcenyakici@gmailcom 2 Gazi University, Ankara, Turkey, elcı[email protected]
3 Gazi University, Ankara, Turkey,[email protected]
4 Gazi University, Ankara, Turkey, [email protected]
5 Gazi University, Ankara, Turkey, [email protected]
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References
[1] Hanna, G., Some pedagogical aspects of proof, Interchange, 21(1), 6–13, 1990.
[2] Hersh, R., Proving is convincing and explaining, Educational Studies in Mathematics, 24,
389–399,1993.
[3] Moore, R. C., Mathematics Professors’ Evaluation of Students’ Proofs: A Complex
Teaching Practice, International Journal of Research in Undergraduate Mathematics
Education, 2(2), 246-278, 2016.
[4] Pfeiffer, K., Features and purposes of mathematical proofs in the view of novice students:
Observations from proof validation and evaluation performances (Doctoral dissertation,
2010).
[5] Schoenfeld, A., What do we know about mathematics curricula? Journal of Mathematical
Behavior, 13(1), 55–80, 1994.
[6] Yin, R. K., Case study research: design and methods, Applied social research methods
series. Thousand Oaks, CA: Sage Publications, Inc. Afacan, 2003.
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
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New Method of Grading Exams based on Computational Statistics
Evgeny GERSHİKOV1
Abstract. The common way to grade academic exams is by dividing the maximal
number of grade points among the exam questions and their paragraphs (if relevant), later
trying to estimate the amount of points that an individual student has scored in a each
question. The problem with this process is that sometimes a question level is not assessed
correctly and the question turns out to be too hard or too easy for the examinees, which brings
forward a question mark about the correctness of the points allocation to this particular
question. In this work, we use computational statistics to provide a different method to
allocate grade points to questions and we test our method on simulated exam results and real-
life experiments. Our tests were done using mainly multiple choice questions due to the
following reasons. Multiple choice questions have a long history [1] and are very common in
academic institutions around the world as well as in psychometric evaluations. Also, we have
designed and implemented an automatic grading system for this kind of questions and a great
benefit of this system is automatic collection of the student scores in all questions [2]. Thus,
the examiner does not have to enter the scores manually into the computerized system [3].
There are many approaches available to analyze multiple choice test data [4], however none
target question efficiency explicitly. In this work we propose statistical efficiency measures
for each question. The group of examinees is divided into three subgroups: the “strong”
students, the “weak” students and the “average” ones. The initial division is according to the
regular grades. Now a question is considered efficient if most “strong” students succeed in it
while most “weak” ones fail. We provide a numeric formula to calculate the efficiencies.
Then we weight the question scores by these efficiencies to calculate weighted grades that
represent, in our opinion, a better way of student evaluation than the regular non-weighted
grades. Optionally, we discard the totally inefficient questions, so that they do not affect the
weighted grades. The process is repeated in an iterative fashion until convergence, that is, we
repeat the step of dividing the students into three subgroups, now using the weighted grades
for the division, we calculate the new efficiencies and new weighted grades and so on. Results
and simulations are provided in our work. We conclude that this new way of grading is
beneficial to evaluate the efficiency of individual exam questions as well as the whole exam.
1 Braude Academic College, Carmiel, ISRAEL, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
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Keywords. Efficiency measures, new grading method, computational statistical
measures.
References
[1] Wood R., Multiple choice: A state of the art report, Evaluation in Education, International
Progress, 1, 3, 191-280, 1977.
[2] Gershikov E., Kosolapov S., On Image Based Fast Feedback Systems for Academic
Evaluation, International Journal of Signal Processing Systems, 3, 1, 19-24, 2014.
[3] Gershikov E., Kosolapov S., Camera-Based Instant Feedback Systems, International
Journal of Advanced Computing, 47, 1, 1463-1473, 2014.
[4] Ding L., Beichner R., Approaches to data analysis of multiple-choice questions, Physical
Review Special Topics - Physics Education Research, 5, 1-17, 2009.
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
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Difference Equations in Economics
Jelena STANOJEVIC1 and Katarina KUKIC2
Abstract. In this paper we give review of some well-known models in economy
described by difference equations. We presented application of cobweb analysis on one
supply-demand model with naive and also with rational expectations for dynamical system
described by difference equations. For same examples we briefly analyze their stability
properties and through this part we emphasize the importance of studding difference
equations in basic mathematics courses for economics. That motivated us to make a little
study with two groups of students at the University of Belgrade concerning the students'
ability and will to learn about difference equations. In the second part of our paper we present
the results of that study.
1 Faculty of Economics, University of Belgrade 2 Faculty of Traffic and Transport Engineering, University of Belgrade
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
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Factors Affecting Scoring Open-Ended Questions
Kübra Akkaya1 and Selahattin Arslan2
Abstract. Although there are different tools available for assessment, written
examinations are often used in our educational system. Written examinations outclass other
assessment tool such as multiple choice questions, but differences in scoring are their main
disadvantages. There are many factors which engender the differences in scoring that occur
when the written exams are evaluated. By using the case study method in this research, it was
aimed to determine the factors that give rise to the scoring differences that arise when the
secondary school mathematics teachers evaluate written exams. For this purpose, firstly semi-
structured interviews were made with 11 randomly selected mathematics teachers and
secondly clinical interviews were made with them with the help of by purpose-selected
student examination papers. The result of the study showed that teachers gave different scores
to the same examination paper. Further results of the study are that some teachers have
focused on the solution process, and yet others were satisfied with the results. Some factors
such paper regularity, student achievement, also affect the score given.
Keywords. Measurement and evaluation, open-ended examination, grading.
1 Milli Eğitim Müdürlüğü, Trabzon, Türkiye, [email protected] 2 Karadeniz Technical University, Trabzon, Türkiye, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
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Development Process of In-Service Training Intended for Teachers to Perform Teaching
of Mathematics with Computer Algebra Systems
Mehmet Alper ARDIÇ1 and Tevfik İŞLEYEN2
Abstract. In this study, it was dealt with the development process of in-service
training activities designed in order for mathematics teachers of secondary education to
realize teaching of mathematics, utilizing computer algebra systems. In addition, the results
obtained from the researches carried out during and after the in-service training were
summarized. In the last part of the study, suggestions were made to teachers and researchers
who want to carry out activities aimed at using computer algebra systems in teaching
environments.
Keywords. Computer-assisted mathematics instruction, computer algebra system, in-
service training.
AMS 2010. 97A99, 97D40, 97U50
1 Adıyaman University, Adıyaman, TURKEY, [email protected] 2 Atatürk University, Erzurum. TURKEY, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
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Examination of the Activities about Enciphering Done with Fifth Grade Students and
Their Views about the Activities
Muhammet ŞAHAL 1 and Ahmet Şükrü ÖZDEMİR 2
Abstract. Mathematics is defined as intangible form of real life. It is assumed as a
difficult course by students due to its discrete structure and other different factors. In order to
deal with such considerations, it is believed that to utilize activities in the course, which are
attractive for students and enable them to be active through learning process and motivate
them. Middle school students are eager to learn with games, physical activities and dealing
with problems about daily life context. One of such kind of activities are enciphering ones.
Throughout history, human have been wondering about hidden things and hiding information
from others is always seemed as interesting. It is known that enciphering was used for
military purposes in the time of the Spartans and the Roman Empire. Colossus, which was
invented by Turing, helped to decipher the codes developed by Nazi armies and gave rise to
finish the Second World War in shorter time. That’s why, the victory was dedicated to
mathematicians. Enciphering was not only crucial in the past, but also it is even more
important in recent time. In this study, it was investigated that whether activities about
enciphering is appropriate for fifth grade students or not. In addition, revealing the students’
views about the activities was another goal in the study. In order to reach the goals, activities
developed by Mathematical Enhancement Programme (1995) were used after translating into
Turkish. Then students’ views about the activities were gathered and it was asked them to
produce similar activities that they had done. Case study, which is type of qualitative research
methodology, was used in the study. Participants of the study are 30 fifth grade students. It
was reached that majority of the students were accomplished the activities successfully but
they couldn’t do deciphered ones. They also produced similar enciphering activities. Besides
it was also found that enciphering activities attracted students’ attentions, increased their
motivations and interests to the course and made the course enjoyable for them.
Keywords: enciphering; deciphered mathematics; mathematics education; motivation.
1 Yıldız Technical University, Istanbul, TURKEY, [email protected] 2 Marmara University, Istanbul, TURKEY, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
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The Concept of Zero: A Comparison of Students with Learning Disability vs Non-
Learning Disability
N. Dilşad GÜVEN1 and Esra Selcen YAKICI TOPBAŞ2 and Ziya ARGÜN3
Abstract. Labeling to non-existed object is a challenging process in history ([1], [5]).
It could be difficult for students to regard zero as a number because of labeling a non-existing
object with a number. Understanding zero and its using is different than understanding other
numbers due to its abstract nature. The numbers that are used to state multitude constitute a
symbolic system. Students with learning disability (LD) could not make connection with
multitude and symbolic system [3]. On the other side, studies in the literature remark that
students have weak knowledge about zero ([2], [4], [7]). Since students with LD having
difficulty with symbolic system and students with non-LD having difficulty about zero, zero
is a problematic concept for all students. It is issue of concern how students with LD and non-
LD conceptualize zero, what are the similarities and differences. Accordingly, the current
study aims to investigate similarities and differences between zero conception of students
with LD and non-LD.
It is a multiple-case study having qualitative approach [6]. The cases of study are
students with LD and non-LD. 3 students with LD and 3 students with non-LD are
participants of the study. The data were collected through clinical interviews. The conception
of zero with different usage (as a number, as a “nothing”, as a symbol, as a cardinality of a set
etc.) is examined deeply in the interviews. The data were analyzed through content analysis
method.
The results suggest that all students are aware of the symbol using of zero, especially
zero as a place holder. While students with non-LD consider that zero is a number, students
with LD doesn’t. However, students with non-LD attach meaning of nothing to zero similar to
students with LD. Thus, all of the students have misconceptions and mistakes about the
operations involving zero, especially multiplication and division. Students' explanations about
meaning of zero in multiplication are limited by words such as “eating, swallowing, or
absorbing”. Students could not make sense of absorbing role of zero in multiplication.
Students (especially students with LD) use algorithms for operations involving zero
1 Gazi University, Ankara, TURKEY, [email protected] /[email protected] 2 Gazi University, Ankara, TURKEY, selcenyakici@gmailcom/ [email protected] 3 Gazi University, Ankara, TURKEY, [email protected]/ [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
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developed by themselves. Thus, this could be a result or a reason of the lack of understanding
about zero concept. Conversely, students' conception about operations affect their
understanding of operations involving zero.
Keywords. Zero concept, Learning Disability, Mathematics education.
AMS 2010. 97-XX
References
[1] Gullberg, J., Mathematics: From the birth of numbers. New York: Norton, 1997.
[2] Reys, R. E., & Grouws, D. A., Division involving zero: Some revealing thoughts from
interviewing children. School science and mathematics, 75(7), 593-605, 1975.
[3] Rousselle, L., & Noël, M. P., Basic numerical skills in children with mathematics learning
disabilities: A comparison of symbolic vs non-symbolic number magnitude
processing. Cognition, 102(3),361-395, 2007.
[4] Tsamir, P., Sheffer, R., & Tirosh, D., Intuitions and Undefined Operations: The Cases of
Division by Zero. Focus on Learning Problems in Mathematics, 22(1), 1-16, 2000.
[5] Seife, C., Zero: The biography of a dangerous idea. Penguin, 2000.
[6] Yin, R. K., Case study research: design and methods, Applied social research methods
series. Thousand Oaks, CA: Sage Publications, Inc. Afacan, 2003
[7] Quinn, J. R., Lamberg, T. D., & Perrin, J. R., Teacher perceptions of division by zero. The
Clearing House, 81(3), 101-104, 2008.
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
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Preservice Mathematics Teachers’ Mathematizing Skills in The Process of Solving a
Real Life Problem
Nuray Çalışkan DEDEOĞLU1
Abstract. Trends in mathematics curricula focus on concepts like mathematical
literacy, modelling and STEM which enable to work on real life applications of mathematics.
These approaches occur within the frame of problem solving which is an essential part of the
school mathematics. Beyond providing application opportunity traditionally, problem solving
improves mathematical thinking and it also help create mathematical knowledge [1-3].
According to Freudenthal who is pioneer of Realistic Mathematics Education, during the
process of problem solving which is called mathematizing, it is required to reach formal
mathematical knowledge from informal mathematical knowledge which consists of real life
situations [4].
It is important to have mathematizing skills by mathematics teachers and preservice
mathematics teachers to have an alternative approach to create knowledge. The purpose of the
study is examining the process of creating mathematical knowledge within the frame of a real
life problem of preservice elementary mathematics teachers. The solutions of a real life
problem which is individually solved by preservice teachers generate the data of the study
which is designed as a case study that is a qualitative method. Data were analyzed by content
analysis technique. Findings show that preservice teachers could not present some important
skills such as using generalization method which is necessary to create mathematical
knowledge properly and supporting the solution of problem with mathematical models and
explanations.
Keywords. Real life problem, realistic mathematics education, mathematizing,
modelling, preservice mathematics teachers.
AMS 2010. 97-XX, 68T20, 97F90 97B50.
References
[1] Polya, G., How to solve it (2nd Ed.), Princeton University Press, 1973.
1 Sakarya University, Sakarya, TURKEY, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
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[2] Schoenfeld, A. H., Learning to think mathematically: Problem solving, metacognition,
and sense-making in mathematics. In D. Grouws (Ed.), Handbook for Research on
Mathematics Teaching and Learning, MacMillan, New-York, 334-370, 1992.
[3] Tall, D., A theory of mathematical growth through embodiment, symbolism and
proof. IREM de Strasbourg, Annales de Didactique et de Sciences Cognitives, 11, 195–215,
2006.
[4] Gravemeijer, K., RME theory and mathematics teacher education, In D. Tirosh and T.
Wood (Eds.), The International Handbook of Mathematics Teacher Education: Tools and
Processes in Mathematics Teacher Education, Sense Publishers, 283–302, 2008.
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
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Learning Implementations about Cooperative Learning Method: A Case Study in
Türkiye
Perihan DINC ARTUT1 and Ayten Pınar BAL2
Abstract. Cooperative learning method is used prevalently in many subject areas and
class levels. One of the reasons of this is that it has positive effects on academic success [1],
[2]. Although the studies about this learning method were focused on elementary and
secondary education, it is expressed in discussions about the benefits of this method that it can
also be implemented in higher education classes [3], [4]. Cooperative learning has many
different forms, such as Jigsaw, Cooperative Integrated Reading and Composition (CIRC),
Teams-Games-Tournament (TGT), Learning Together (LT), Student Teams-Achievement
Divisions (STAD). In addition to many other fields, STAD can be used in mathematics. By
the help of STAD, students interact mutually, become responsible for each other’s learning
and they develop their other skills. In the practice of STAD, students start working in their
teams and attempt to complete the task given to them on their own after the teacher’s
presentation of the topic. In line with the explanations above, it was aimed to implement
Student Teams-Achievement Divisions (STAD), which is a technique in cooperative learning
method, in the teaching of limits and derivatives topic in the course of calculus of the students
who were attending the department of science and technology teaching and to evaluate the
opinions of the students and instructors about this implementation. In this research, an
interview form was used so as to determine the opinions of students about working in groups
based on cooperation while learning mathematics. The findings of this research revealed that
the implementations of STAD technique in lessons created a positive effect, provided active
participation to the lessons and made the students understand more easily by collaborating
with their friends on their learning of limits and derivatives according to the opinions of the
students.
Keywords. Calculus, case study, cooperative learning, student teams-achievement divisions
References
1 Çukurova University, Adana, TURKEY, [email protected] 2 Çukurova University, Adana, TURKEY, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
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[1] Slavin, R. E.; Madden, N.A.; Leavey, M., Effects of team-assisted individualisation on
the mathematics achievement of academically handicapped and nonhandicapped students.
Journal of Educational Psychology, 76(5), 813-819, 1984.
[2] Häsel-Weide, U. and Nührenbörger, M., Replacing counting strategies: Children’s
constructs working on number sequences. Proceedings of the Eighth Congress of the
European Society for Research in Mathematics Education. Ankara/ Turkey: Middle East
Technical University, 2013
[3] Artut, P. D., Effect of Cooperative Learning Method on Prospective Teachers’ Non-
routine Problem-Solving Skills and Their Views About the Method. US-China Education
Review, 6(4), 244-254, 2016.
[4] Johnson, D. W., Johnson, R. T., & Smith, K. A., Cooperative learning: Improving
university instruction by basing practice on validated theory, Journal on Excellence in
University Teaching, 2013.
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
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Investigation of Number Sense Strategies Used By the 8th Grade Students in Turkey
Perihan DINC ARTUT1 and Zübeyde ER2
Abstract. The concept of number sense is a significant one in mathematics education.
According to the standards of National Council of Teachers of Mathematics (NCTM) (2000,
p.32) [1], it is stated that students understand the numbers, the ways of representing the
numbers, the relationships between the numbers and number systems, the meanings of
operations, the relationships of the numbers with each other and they can do fluent
computations and appropriate predictions and it is emphasized that the concept of number
sense should have been developed based on these standards. There are various definitions
about the concept of number sense in the literature. McIntosh, Reys and Reys (1992) [2]
describe number sense as: “a person’s general understanding of number and operations along
with the ability and inclination to use this understanding in flexible ways to make
mathematical judgments and to develop useful strategies for handling numbers and
operations” (cited Morais and Serrazina, 2013). In this research aimed to investigate the
strategies which the students use while solving number sense problems. This study was
conducted in the scope of case study design which is one of the qualitative research designs.
The research was conducted on 28 students, 17 of which were girls and 11 of which were
boys, who were attending the 8th grade in three secondary school having similar
characteristics in a city centre which is located in the south of Turkey. As the data collection
tool, a number sense test (NST) which consisted of some questions in the number sense scale
of Singh (2009) [3], which was originally adapted from McIntosh, Reys, Reys Bana, and
Farrell, (1997) was used. In the test, there are 5 components about understanding the concept
of number, using the multiple expressions of numbers, understanding the effect of numbers,
using equivalent expressions and using computing and counting strategies. NST consists of 25
items about the topics of whole numbers, decimal numbers, fractions and percentages. In this
research was seen that the students used the strategies based on rules in their solutions more
when all of the solutions obtained from the students in this research were considered. When
the strategies that the students used according to the components of number sense were
analyzed in this study, it was concluded that the component of understanding the concept of
number was the one which the students used the strategies based on number sense the least.
1 Çukurova University, Adana, TURKEY, [email protected] 2 MEB, Adana, TURKEY, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
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Keywords. Number sense, number sense components, number sense strategies
References
[1] National Council of Teachers of Mathematics (NCTM)., The principles and standards for
school mathematics. Reston, VA: Author, 2000.
[2] McIntosh, A., Reys, B. J., and Reys, R. E., A proposed framework for examining basic
number sense. For the learning of mathematics, 12(3), 2-44, 1992.
[3] Singh, P., An Assessment of Number Sense among Secondary School Students,
International Journal for Mathematics Teaching and Learning, 2009.
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
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Examining Teacher Candidates’ Thinking Skills
Perihan DİNÇ ARTUT1and Ayten Pınar BAL2
Abstract
Mathematics is one of the most effective fields of thinking. This aspect of mathematics
is also being emphasized in the educational process. In this context, it is observed tha
mathematics teaching programme in our country has given more importance to basic skills in
mathematical thinking (Ministry of NationalEducation [MEB], 2013).
The goals of mathematics curriculum to develop mathematical thinking skills have led
to focus on questions about mathematical thinking and its key components. Mathematical
thinking is defined as a set of complex processes such as "guessing, induction, deduction,
description, generalization, sampling, formal and informaL reasoning, verification, and so on"
(Liu Po-Hung, 2003).
Considering that one of the most important basic components effective in mathematics
education is mathematics teachers, it is thought that teacher training process and in this
context candidate teachers should have mathematical thinking skills. In this direction, it was
aimed to determine themathematical thinking skills of the class and mathematics teachers in
this research.
This research is a descriptive study done within the technique of survey model and
conducted on mathematics and classroom teacher candidates. As a data collection tool
“Mathematical Thinking Scale” whic was developed by Ersoy and Başer (2003) was used. As
data analyses, descriptive statistic was used. The findings revaled that both classroom and
mathematic teachers’ thinking skills are similar to each other.
Keywords.Thinking mathematicaly, mathematics, pre service teacher.
References
[1] Milli Eğitim Bakanlığı [MEB], İlköğretim matematik (1-5. Sınıflar) dersi öğretim
programı. Ankara: Talim Terbiye Başkanlığı Yayınları, 2013.
1CukurovaUniversity, Adana, Turkey, [email protected] 2CukurovaUniversity, Adana, Turkey, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
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[2] Po-Hung, L., Do Teachers need to incorporate the history of mathematics in their
teaching?. The Mathematics Teacher, 96(6), 416, 2003.
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
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Difficulties of High School Students in Complex Numbers
Selahattin ARSLAN1
Abstract. Complex numbers field is considered as difficult to understand by students
and many countries high school mathematics programmes do not accord place to this field. In
our country, this concept has been removed from high school mathematics programs with the
revision made in 2017. In this study, it was aimed to reveal the difficulties in complex
numbers of high school students and to compare students in different programmes. For this
purpose, a test of 29 questions, which is available in the literature, has been applied to 70 high
school second year students (35 studying at Linguistic/Verbal programs and 35 at language
Mathematical/Numerical program. Analyses realised has been revealed that both groups’
students had difficulties with complex numbers, but numerical learners were more successful.
Keywords. Complex numbers, difficulty, high school students.
1 Karadeniz Technical University, Trabzon, TURKEY, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
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Investigation of Mathematical Viewpoints of Primary School Teacher Program Students
Tuğba BARAN KAYA1 and Ahmet IŞIK2
Abstract. In a learning environment; The teachers have important roles such as
making their lessons attractive, ensuring students’ learning, and making the relevant lesson
feel important [1]. Particularly, teacher candidates’ perceptions towards mathematics, who are
soon to be responsible for the teaching of basic mathematical concepts in the first stage of
primary education can give deep clues about this role [2]. Various researches can be done to
mention in depth about such a perception. This research is aimed to reveal the point of view
of Primary School Teaching Program students' mathematics and mathematics relations with
other sciences. In this research using the case study method of qualitative research methods,
the data were collected with the help of test items consisting of two open ended questions
directed to the students. The data obtained by this data collection tool were analyzed by
content analysis tecniques and then the frequency of the resulting themes was found. As a
result of the research, it was revealed that students perceive mathematics mostly in the context
of components of mathematics (numbers, symbols, operations). In addition, although students
in the survey wanted to explain the connection between mathematics and other sciences, the
vast majority of students avoided doing this, but went to sort out the sciences and other fields
they thought were related to mathematics.
Keywords. Mathematics, perception, primary school teaching program students.
AMS 2010. 53A40, 20M15.
References
[1] Şahin, B., Öğretmen adaylarının “matematik öğretmeni”, “matematik” ve “matematik
dersi” kavramlarına ilişkin sahip oldukları metaforik algılar, Mersin University Journal of
the Faculty of Education, 9(1), 313-321, 2013.
[2] Güveli, E., İpek, A. S., Atasoy, E., Güveli, H., Prospective primary teachers’
metaphorical perceptions towards mathematics, Turkish Journal of Computer and
Mathematics Education, 2(2), 140-159, 2011.
1 Kırıkkale University, Kırıkkale, TURKEY, [email protected] 2 Kırıkkale University, Kırıkkale, TURKEY, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
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A Framework Suggestion for the Analyze of the Solving Process of a Geometrical
Construction Problem
Tuğçe KOZAKLI ÜLGER1, Işıl BOZKURT2 and Murat ALTUN3
Abstract. Many researchers on the problem-solving process seem to refer to the
problem-solving phases of Polya and interpret these phases for the required field like
cognitive and metacognitive processes. It brings the question to the mind; could it be more
useful to provide specific problem-solving process steps for subject areas (algebra, geometry
etc.) that are at least separated from each other by this outline? In this study, the analysis of
the solving process of any geometrical construction problem using the compass- straightedge
is discussed.
Geometrical construction refers to the standard procedures required to construct
geometric structures by using a compass-straightedge and is thought to play an important role
in the development of geometric thinking. Geometrical construction activities are important
because they provide a deep insight into the properties of the geometric structure created.
There are phases (analysis, construction, evidence, discussion) suggested by Smart regarding
the solving process of a geometrical construction problem. In this study, it is aimed to
elaborate using the steps proposed by Smart (1998), in order to provide a detailed study of
mental processes. The solutions made by the participants in the study group are used as data
to detail these mental processes.
This research used a qualitative case study approach in order to enable the in-depth
analysis of problem-solving processes. In line with the aim of the research, the study group
consists of four participants who have experience on the geometrical construction studies with
the help of a compass- straightedge and know the possibilities and limitations provided by
these tools. The data collection tools of the study were video recordings taken during drawing,
documents (reports and drafts which the solutions are detailed) and unstructured interviews
after problem solving. As a result of the study, a new model has been proposed with the
addition of sub-steps and refinement to the framework of Smart, which is included in the
literature on geometrical construction problems in the solving process.
Keywords. Geometrical construction, problem solving, using compass-straightedge
1 Uludag University, Bursa, TURKEY, [email protected] 2 Uludag University, Bursa, TURKEY, [email protected] 3 Uludag University, Bursa, TURKEY, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
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AMS 2010. 53A40, 20M15.
References
[1] Polya, G., How to solve it. Princeton, NJ: Princeton University Press, 1957.
[2] Smart, J. R., Modern geometries (5th ed.). CA: Brooks/Cole Publishing Company, Pacific
Grove.
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
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Active Learning in Flipped College Algebra Class
Violeta VASILEVSKA1
Abstract. The recent popularity of flipped classrooms in undergraduate classes and
research showing the effectiveness of it on student learning [1, 2] prompted the presenter’s
teaching endeavor in that direction. Namely, in the last few years, the presenter has been
teaching a hybrid College Algebra class in a flipped classroom. In this presentation, the
specifically designed class structure will be discussed.
For the on-line learning activities students use MyMathLab (Pearson’s on-line system that
provides online homework, tutorials, and assessment tools). Students are asked to view the
video lectures on-line and solve the on-line homework before they come to class.
The in-class activities are divided in two parts. The first part is a question/answer segment.
During this part, the instructor (presenter) does not lecture, but rather just helps answer
questions about concepts/topics that caused difficulty to students at home. The second part is
a problem-solving segment. Students are given a group quiz, which is a unique way to involve
students in collaborative problem-solving active learning. They work as teams of two on the
quiz, apply the knowledge and practice the new concepts, discuss the questions, and learn
from each other.
This structure allows students to go over/review the new concepts several times, to encourage
them to ask questions, to lead them to answer their own questions, and to spark
communication between them.
During the presentation, it will be demonstrated how the question/answer part of the class is
conducted. In addition, some survey results about the structure and effectiveness of this
course will be shared. Moreover, it will be discussed how student feedback has been
incorporated in the structure of the class, and how that change has impacted the student
learning.
Keywords. Flipped classroom, hybrid class, active learning.
1 Utah Valley University, Orem, UT, USA, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
269
AMS 2010. 97D40, 97U50.
References
[1] Herreid, C. F., Schiller, N. A., Case studies and the Flipped Classroom, Journal of
College Science Teaching 42 (5), 62-66, 2013.
[2] Love, B., Hodge, A., Ernst, D. C., Inquiry-based learning and the Flipped Classroom
Model, PRIMUS (Problem, Rosources, and Issues in Undergraduate Mathematics Studies) 25
(5), 745-762, 2015.
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
270
STATISTICS
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
271
New Method for Obtaining of Weighted Distributions
Božidar V. POPOVİĆ1
Abstract. İn the real life there are many situations where symmetric distributions are
not appropriate for fitting. In that way, Azzalini’s method [1] is widely used to get
asymmetric or skewed distribution. Shortly this technique can be described as follows: a
random variable Z possesses the skew normal distribution with parameter α if its probability
density function (pdf) is 𝑓(𝑧; 𝛼) = 2𝜙(𝛼𝑧)𝜑(𝑧), where 𝜙(∙) and 𝜑(∙) are cumulative
density function (cdf) and pdf, respectively, of the standard normal random variable. In terms
of the conditional distribution Azzalini’s method consists of the following: let 𝑋𝑖 𝑖 = 1,2 are
independent random variables with pdfs 𝑓𝑖(𝑥𝑖) and cdfs 𝐹𝑖(𝑥𝑖). Then the conditional pdf of
𝑋 = 𝑋1 given 𝛼𝑋1 > 𝑋2 is
𝑓𝑋(𝑥) =𝑓1(𝑥)𝐹2(𝛼𝑥)
𝑃𝛼𝑋1 > 𝑋2, (1)
where 𝑃𝛼𝑋1 > 𝑋2 = ∫ [∫ 𝑓2(𝑥2)𝑑𝑥2𝛼𝑥1
0]𝑓1(𝑥1)𝑑𝑥1
∞
0= 𝐸𝑋1
(𝐹2(𝛼𝑥)). The Eq.(1) can be
interpreted as weighted distribution with weight 𝜔 = 𝐹2(𝛼𝑥). Described method has been
used to construct new skewed distributions from a given symmetric distribution, for example,
skew-t [2], skew-Cauchy [3], skew-Laplace [4] and skew-logistic [5].
The aim of this note is to generalize Azzalini’s method such that it is applicable in
case when 𝑋𝑖 𝑖 = 1,2 are dependent random variables. Dependence can be modelled using
copula function. In case of dependence of the random variables 𝑋𝑖 𝑖 = 1,2 the equation (1)
becomes
𝑓𝑋(𝑥) =𝑓1(𝑥) ∫ 𝑐(𝐹1(𝑥1), 𝐹2(𝑥2); 𝜃)
𝛼𝑥
−∞𝑓2(𝑥2)𝑑𝑥2
𝑃𝛼𝑋1 > 𝑋2,
where 𝑃𝛼𝑋1 > 𝑋2 =𝐸𝑋1[∫ 𝑐(𝐹1(𝑋1), 𝐹2(𝑋2); 𝜃)𝑓2(𝑥2)
𝛼𝑋1
−∞] 𝑑𝑥2 and 𝑐(∙) is the copula
function. Some statistical properties will be derived together with some particular cases.
Keywords. Weighted distributions, generalization of Azzalini’s method.
AMS 2010. 60E05, 62P10, 62G30, 62F10.
1 University of Montenegro, Faculty of Science and Mathematics, Podgorica, Montenegro, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
272
References
[1] Azzalini, A., A class of distributions which includes normal ones, Scand. J. Stat., 12, 171-
178, 1985.
[2] Gupta, A.K., Chang, F.C. and Haung, W.J., Some skew-symmetric models, Random Oper.
Stoch. Equ. 10, 133–140, 2002.
[3] Arnold, B.C., Beaver, R.J., The skew-Cauchy distribution, Statist. Probab. Lett. 49, 285–
290, 2000.
[4] Aryal, G., Nadarajah, S., On the skew-Laplace distribution, J. Inf. Optim. Sci. 26, 205–
217, 2005.
[5] Nadarajah, S., On the skew-logistic distribution, AStA Adv. Stat. Anal. 93, 187–203,
2009.
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
273
Transmuted Exponential Power Distribution and its Distributional Properties
Buğra SARAÇOĞLU1
Abstract. In this study, a new statistical distribution called “Transmuted Exponential
Power distribution” through the quadratic rank transmutation map studied by Shaw et al. [6] is
introduced. The various statistical properties of this distribution are derived. The method of
maximum likelihood estimation has been used to estimate the parameters of this distribution.
Keywords. Transmuted exponential power distribution, maximum likelihood
estimation, keyword three.
AMS 2010. 62F10, 62G30.
References
[1] Shaw, W. T., Buckley, I. R., The alchemy of probability distributions: beyond Gram-
Charlier expansions, and a skewkurtotic-normal distribution from a rank transmutation map,
arXiv preprint arXiv:0901.0434, 2009.
[2] Aryal, G. R., Tsokos, C. P., On the transmuted extreme value distribution with
application. Nonlinear Analysis: Theory, Methods & Applications, 71(12), 1401-1407, 2009.
[3] Mahmoud, M. R., & Mandouh, R. M. (2013). On the transmuted Fréchet
distribution. Journal of Applied Sciences Research, 9(10), 5553-5561, 2013.
[4] Al-Babtain, A., Fattah, A. A., Ahmed, A. H. N., & Merovci, F., The Kumaraswamy-
transmuted exponentiated modified Weibull distribution. Communications in Statistics-
Simulation and Computation, 46(5), 3812-3832, 2017.
1 Selcuk University, Konya, TURKEY, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
274
A New Bivariate Archimedean Copula Based on a Transformed Generator Function
Çiğdem TOPÇU GÜLÖKSÜZ1, Nuri ÇELİK2
Abstract. Copula functions are commonly used to model the relationships among
multivariate outcomes. A special class that is called Archimedean has many important
properties that make it very popular in applications. Archimedean copulas can be generated
by using generator functions. In literature, there are many examples of generator functions
and related Archimedean copulas. In this study, we apply a transformation to the generator
function of Gumbel copula that is one of well-known Archimedean copulas and obtain a new
generator function. Since different generator functions generate different copulas, by using
this function we construct a new bivariate Archimedean copula. Properties of these new
functions are examined. We consider some selected parameter values of Gumbel generator
function and estimate the parameter of the new copula function.
In the application part of the study, two real data set example are used to present that the new
copula function has a potential to model some dependence structures.
Keywords. Copula, Archimedean Copulas, Gumbel Copula, dependence
AMS 2010. 53A40, 20M15.
References
[1] Belgorodski, N. Selecting pair copula families for regular vines with application to the
multivariate analysis of European stock market indices, Master Thesis, Technische
Universität München, 2010.
[2] Genest, C., and Rivest, L.P., Statistical inference procedures for bivariate archimedean
copulas, American Statistician, 88(423),pp. 1034-1043, 1993.
[3] Gumbel, E.J., Bivariate exponential distributions, Journal of the American Statistical
Association., 55, pp. 698-707, 1960.
[4] Nelsen, R., An Introduction to Copulas, Springer, NewYork, 2006.
1 Bartin University, Bartin, TURKEY, [email protected] 2 Bartin University, Bartin, TURKEY, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
275
The Comparison of Population Means under Unequal Variances
Neriman AKDAM1, M. Fedai KAYA2 and Buğra SARAÇOĞLU3
Abstract. In this study, The usage of F test is not proper for testing the equality of
means of k normal distributed populations when the assumption of variances’ homogeneity is
violated. Accordingly, a test is proposed for testing the equality of means of k normal
distributed populatons under unequal variances. As regards this test, a critical values table is
constituted for certain significance levels. A simulation study based on The power values and
Type-I errors of the test is performed. In addition, The performances of this test are compared
with those of some tests proposed for comparison of population means under unequal
variances for different population sizes.
Keywords. Critical valuess, Power value, Simulation Study, Type-I error.
AMS 2010. 62F03, 62H10.
References
[1] David, H., Order Statistics. John Wiley, New York, 1970.
[2] Ekiz, M. and Gamgam, H., On The Comparison Of The Welch Test And The Single-Stage
Test: A Simulation Study ,Commun. Fac. Sci. Uni. Ank. Series, 51-61, 2007.
[3] Welch, B.L., On the comparison of several mean values: An alternative approach,
Biometrica, 38, 330-336, 1951.
[4] Chen, S. Y., & Chen, H. J. (1998). Single-stage analysis of variance under
heteroscedasticity. Communications in Statistics-Simulation and Computation, 27(3), 641-
666.
1 Selcuk University, Konya, TURKEY, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
276
A New Lifetime Distribution
Nuri CELİK1 and Cigdem TOPCU GULOKSUZ2
Abstract. Some probability distributions have been proposed to fit real life data with
both increasing and decreasing failure rates. In this study, we propose a new life time
distribution based on the generalization of exponential distribution. The distribution has both
increasing and decreasing failure rate functions. Additionally, we also define survival
function, the hazard function and the mean time to failure of this new distribution. Type II
censoring procedure is also considered for this distribution. Additionally, stress-strength
reliability and the maximum likelihood estimators (MLE) of the unknown parameters are
obtained. As an application, a real data set is used to show that the proposed distribution gives
best fit than the alternative ones.
Keywords. Exponential Distribution, Failure Rate, Type II Censoring.
AMS 2010. 62N01, 62N05.
References
[1] Gomez Y M, Bolfarine H, Gomez H W. A New Extension of Exponential Distribution.
Revista Colombiana de Estadistica 37: 25-34, 2014.
[2] Gupta R D, Kundu K. Generalized Exponential Distributions. Australian and New Zeland
Journal of Statistics 41, 2, 173-188, 1999.
[3] Kus C. A New Lifetime Distribution. Computational Statistics and Data Analysis 51, 4497-
4509, 2007.
[4] Raqab M Z, Kundu D. Comparison of different estimators of P(Y < X) for a scaled Burr
type X distribution. Communication in Statistics Simulation and Computation 34, 2, 465-483,
2005.
1 Bartin University, Bartin, TURKEY, [email protected] 2 Bartin University, Bartin, TURKEY, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
277
TOPOLOGY
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
278
A Topological Study of Harmony and Counterpoint in Music Using Quotient Orbifolds
Aditya Sivakumar1 and Dmitri Tymoczko2
Abstract: Western music is based on two concepts – harmony, which is the simultaneous
sounding of multiple notes (chords), and voice leading, which is the motion of these notes
(voices) through time. The incorporation of these two concepts is called counterpoint.
Geometry and topology are powerful tools for studying counterpoint in music. A musical
chord can be defined as an n-tuple in an n-dimensional non-Euclidean geometric quotient
space called an orbifold. The location and characteristics of a chord in the orbifold is defined
by symmetry as well as the consonance or dissonance of the chord. The voice leading from
one chord to another within the orbifold is defined by vectors in the space. In this study, a
new technique using quotient orbifolds has been developed to identify good sounding music
and the limits of playability as a guide for efficiently composing music. Over 300,000 chord
transitions were analyzed with a Python program and a random sample was graphed onto the
orbifold. Some extremely interesting geometric properties of voice leadings were revealed
visually for the first time and this innovative visual technique easily identified the ones that
violated counterpoint rules or were unplayable. In conclusion, the study describes the
development of a powerful new technique using orbifolds and assisted by computational
methods to efficiently compose new music, and this should prove extremely useful to
musicians and composers.
Keywords: Geometry, Topology, Voice Leading, Music
1 Beaverton High School, Beaverton Oregon USA. [email protected] 2 Princeton University, Princeton, New Jersey, USA.
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
279
New Functors from Fuzzy Normed Spaces Category
Deniz Pınar SUNAOĞLU 1 and Erdal GÜNER 2
Abstract. For each mathematical discipline we define at first objects and then
admissible maps for describing the objects. This procedure is formalized by the concept
‘category’. As well-known mathematical objects may be described by means of maps. There
is an analogous description of categories via so called functors. A functor is a morphism of
categories. Functors were first explicitly recognized in algebraic topology, where they arise
naturally when geometric properties are described by means of algebraic invariants. Category
theory and functors provide a tool by which many parallel techniques used in several branches
of mathematics can be linked and treated in a unified manner.
The aim of this study is to give some information about fuzzy normed spaces, then try
to find some new functors from fuzzy normed spaces category to anothers.
Keywords. Category, functor, fuzzy normed spaces.
AMS 2010. 18Axx, o3E72.
References
[1] Bag, T., Samanta, S.K.: A Comparative Study of Fuzzy Norms on a Linear Spaces. Fuzzy
Sets and Systems, 159, 673-684, 2008.
[2] Bag, T., Samanta, S.K.: Operator’s Fuzzy Norm and Some Properties. Fuzzy Information
and Engineering, 7, 151-164, 2015.
[3] Cheng, S.C., Mordeson, J.N.: Fuzzy Linear Operator and Fuzzy Normed Linear Spaces.
Bull. Calcutta Math. Soc., 86, 429-436, 1994.
[4] Nadaban,S.: Fuzzy Continuous Mappings in Fuzzy Normed Linear Spaces. İnternational
Journal Of Computers Communications and Control, ISSN 1841-9836, 10(6):834-843,
December, 2015.
[5] Preuss, G.: Theory of Topological Structures- An Approach to Categorical Topology.
D.Reidel Publishing Company, 1944.
[6] Zadeh, L. A. ,"Fuzzy Sets" , Inform. Contr., 8 : 338-353, 1965.
1 University of Ankara, Ankara, TURKEY, [email protected] 2 University of Ankara, Ankara, TURKEY, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
280
Soft Metric Spaces
Ebru Aydoğdu1, Abdülkadir Aygünoğlu2 and Halis Aygün3
Abstract. In this work, we firstly introduce the concept of soft metric and we give
relationships between soft metrics and fuzzy metrics by the sense of George, Veeramani [5].
Later we discuss the relationships between soft metrics and clasical metrics with examples.
Keywords. Soft set, Soft metric spaces
AMS 2010. 06D72, 03B52
References
[1] Atanassov, K. T., Intuitionistic fuzzy sets, Fuzzy sets and Systems 20.1, 87-96, 1986.
[2] Aygünoğlu, A., and Halis A., Some notes on soft topological spaces, Neural Computing
and Applications 21.1, 113-119, 2012.
[3] Çağman, N., and Enginoğlu S., Soft set theory and uni-int decision making, European
Journal of Operational Research 207.2, 848-855, 2010.
[4] Das S., Samanta S.K., Soft real sets,soft real numbers and their properties, J.Fuzzy Math.
20(3), 551-576,2012.
[5] George, A., P. Veeramani. On some results in fuzzy metric spaces. Fuzzy sets and
systems 64.3, 395-399, 1994.
[6] Maji, P.K., Biswas R., Roy A.R., Soft set theory, Computers & Mathematics with
Applications 45, 555-562, 2003.
[7] Maji P.K., Roy A.R, and Biswas R., An application of soft sets in a decision making
problem, Computers & Mathematics with Applications 44.8, 1077-1083, 2002.
[8] Shabir M., Naz M., On soft topological spaces., Comput Math Appl, 61,1786-1799, 2011.
1 Kocaeli University, Kocaeli, Turkey, [email protected] 2 Kocaeli University, Kocaeli, Turkey, [email protected] 3 Kocaeli University, Kocaeli, Turkey, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
281
A New Property between Compactness and Completeness in Generalized Metric Spaces
Emrah Evren KARA1 and Merve İLKHAN 2
Abstract. The main purpose of this presentation is to study on a new concept stronger
than completeness but weaker than compactness in a generalized metric space. For this aim,
we carry some basic concepts to generalized metric spaces. We define three new types of
Cauchy sequences belonging to a larger class than the class of all G-Cauchy sequences. Also,
we examine the relations between these new G-Cauchy sequences with corresponding ones in
the metric generated by G-metric. By defining three new types of G-completeness, we prove
some attracting theorems related to G-compactness, G-relative compactness, G-uniformly
locally compactness and some other concepts.
Keywords. Cauchy sequences, compactness, completeness.
AMS 2010. 54E35, 54E50, 54A20.
References
[1] Mustafa, Z. and Sims, B., A new approach to generalized metric spaces, J. Nonlinear
Convex Anal. 7(2), 289-297, 2006.
[2] Kaewcharoen, A., Common fixed point theorems for contractive mappings satisfying 𝛷-
maps in G-metric spaces, Banach J. Math. Anal. 6(1), 101-111, 2012.
[3] Beer, G., Between compactness and completeness, Topology Appl. 155(6), 503-514, 2008.
[4] Garrido, M. I. and Merono, A. S., New types of completeness in metric spaces, Ann. Acad.
Sci. Fenn. Math. 39, 733-758, 2014.
[5] Mustafa, Z., Khandagji, M. and Shatanawi, W., Fixed point results on complete G-metric
spaces, Studia Sci. Math. Hungar. 48(3), 304-319, 2011.
1 Düzce University, Düzce, TURKEY, [email protected] 2 Düzce University, Düzce, TURKEY, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
282
Extensions of T_0-quasi-metrics
Filiz YILDIZ1 and Hans-Peter KÜNZİ2
Abstract. Let ( , , )X m be a partially ordered metric space, that is, a metric space
(X,m) equipped with a partial order on X.
We say that a T_0-quasi-metric d on X is m-splitting provided that 1d d m .
Furthermore d is said to be ( , , )X m producing provided that d is m-splitting and the
specialization order of d is equal to .
It is known and easy to see that if ( , , )X m is a partially ordered metric space that is
produced by a T_0-quasi-metric and is a total order, then there exists exactly one producing
T_0-quasi-metric on X.
We first shall give an example that shows that a partially ordered metric space can be
uniquely produced by a T_0-quasi-metric although is not total.
Then we present solutions to the following two problems: Let ( , , )X m be a partially
ordered metric space and A a subset of X.
(1) Suppose that d is a T_0-quasi-metric on A which is | ( )m A A splitting . When
can d be extended to an m-splitting T_0-quasi-metric d on X?
(2) Suppose that d is a T_0-quasi-metric on A which is ( , | ( ), | ( ))A m A A A A
producing. When can d be extended to a T_0-quasi-metric d on X that produces ( , , )X m ?
References
[1] Gaba, Y.U. and Kunzi, H.-P.A., Splitting metrics by T_0-quasi-metrics, Topology
Applications 193, 84-96, 2015.
[2] Gaba, Y.U. and Kunzi, H.-P.A., Partially ordered metric spaces produced by T_0-quasi-
metrics, Topology Applications 202, 366-383, 2016.
1 Hacettepe University, Ankara, TURKEY, [email protected] 2 University of Cape Town, Rondebosch , SOUTH AFRICA, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
283
Some Categorical Approaches to the Relations between Inverse Systems of Bitopological
and Ditopological Spaces
Filiz YILDIZ1
Abstract. Followed by the constructing in [4] a detailed analysis of the theory of
inverse (projective) systems [2] and limits insofar as the subcategories of textures are
concerned, the relations between bitopological and ditopological inverse systems are
investigated and studied in a categorical context.
Especially, many interesting categorical [1] and functoral results on the
relationships between the inverse systems in a subcategory of Bitop consisting of the
bitopological spaces and the inverse systems in a corresponding subcategory of ifDitop
consisting of ditopological spaces are obtained in a natural way.
Finally, by introducing the suitable functor isomorphisms [3], we actually proved that
a natural transformation described between those particular subcategories is identity.
Keywords. Bitopology, Ditopology, Inverse System, Isomorphism Functor
AMS 2010. 46M40, 54B30, 18A22, 54E55
References
[1] Adámek J., Herrlich H, Strecker G. E., Abstract and Concrete Categories, John Wiley
& Sons, Inc., 1990.
[2] Eilenberg S, Steenrod N., Foundations of Algebraic Topology, Princeton University
Press, Princeton, New Jersey, 1952.
[3] Yıldız F., Connections between real compactifications in various categories, Quaestiones
Mathematicae, 38 (3) , 431-455, 2015.
[4] Yıldız F., Inverse Systems and Inverse Limits in the Category of Plain Textures, Topology
and Its Applications, (201), 217-234, 2016.
1 Hacettepe University, Ankara, TURKEY, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
284
Normal Product Adjacency for Simplicial Homology Groups of Digital Images
Gülseli BURAK1
Abstract: We study to compute a simplicial homology groups for Cartesian product of digital
images based on the normal product adjacency. We give some properties of the normal
product adjacency for finite for Cartesian product of digital images and give some examples
concerning homology groups of digital images.
Keywords: Digital topology, digital simplicial homology group, normal product adjacency
References
[1] Boxer, L., Generalized normal product adjacency in digital topology, arXiv.org,
arXiv:1608.03204v2, 2017.
[2] Karaca, I.,Boxer, L., Öztel, A., Topological invariants in digital images, Jour. Of
Mathematical Sciences: Advances and Applications, 11, No:2,109-140, 2011.
[3]Karaca, I., Demir, E., Simplicial homology groups of certain surfaces, Hacettepe Journal of
Mathematics and Statistic, 44(5), 1011-1022, 2015.
1 Pamukkale University, Denizli, TURKEY, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
285
Bracket Polynomials of Torus Links as Fibonacci Polynomials
Kemal TAŞKÖPRÜ1, İsmet ALTINTAŞ2 and Merve BEYAZTAŞ3
Abstract. In this paper we work the bracket polynomial of (2,n)-torus link as a
Fibonacci polynomial. We show that the bracket polynomial of (2,n)-torus link provides a
recurrence relation as similar to Fibonacci polynomial and give its some fundamental
properties. We also prove important identities, which are similar to the Fibonacci identities,
for the bracket polynomial of (2,n)-torus link and prove Fibonacci-like identities of the Jones
polynomial of (2,n)-torus link as a result of the bracket polynomial. Finally, we observe that
the bracket polynomial of (2,n)-torus link and therefore its Jones polynomial can be derived
from its Alexander-Conway polynomial or classical Fibonacci polynomial.
Keywords. Bracket polynomial, torus link, Fibonacci polynomial, Fibonacci
identities, Jones polynomial.
AMS 2010. 57M25, 11B39, 11C08.
References
[1] Jones, V. F. R., A polynomial invariant for knots via von Neumann algebras, Bull. Amer.
Math. Soc., 12, 103-111, 1985.
[2] Kauffman, L. H., State models and the Jones polynomial, Topology, 26, 395-407, 1987.
[3] Kauffman, L. H., An invariant of regular isotopy, Trans. Amer. Math. Soc., 318, 417-471,
1990.
[4] Koshy, T., Fibonacci and Lucas numbers with applications, Pure and Applied
Mathematics: A Wiley Series of Texts, Monographs and Tracts, Wiley, USA, 2011.
[5] Nalli, A., Haukkanen, P., On generalized Fibonacci and Lucas polynomials, Chaos,
Solitons and Fractals, 42, 3179-3186, 2009.
[6] Taşköprü, K., Altıntaş, İ., HOMFLY polynomials of torus links as generalized Fibonacci
polynomials, Electron. J. Combin., 22, 4.8, 2015.
1 Bilecik Şeyh Edebali University, Bilecik, TURKEY, [email protected] 2 Sakarya University, Sakarya, TURKEY, [email protected] 3 Sakarya University, Sakarya, TURKEY, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
286
Controllability of Affine Control Systems on Solvable Lie Groups
Memet KULE1
Abstract. In this paper we study controllability by taking in consideration the
eigenvalues of associated derivations. As soon as the state space is a solvable connected Lie
group, an affine control system is controllable if the induced invariant control system is
controllable and any eigenvalues of the derivations have zero real parts.
Keywords. Affine control systems; solvable Lie groups; controllability.
AMS 2010. 37N35, 22E25, 93C10.
References
[1] Ayala V, Tirao J., Linear control systems on Lie groups and Controllability, Proceedings
of Symposia in Pure Mathematics, 64, 47-64, 1999.
[2] Da Silva A.J., Controllability of linear systems on solvable Lie groups, SIAM J. Control
Opt., 54 No. 1, 372-390, 2016.
[3] Jouan, Ph., Equivalence of control systems with linear systems on Lie groups and
homogeneous spaces, ESAIM: Control Optim Calc Var. , 16, 956-973, 2010.
[4] Jurdjevic, V., Sallet, G., Controllability Properties of Affine Systems, SIAM J.Control
Opt., 22, 501-518, 1984.
[5] Kara, A., Kule, M., Controllability of Affine Control Systems on Carnot Groups, Int. J. of
Contemp. Math. Sci., 5, 2167-2172, 2010.
[6] Kara, A., San Martin. L., Controllability of Affine Control System for the Generalized
Heisenberg Lie Groups, Int. J. Pure Appl. Math., 29, 1-6, 2006.
[7] Kule, M., Controllability of Affine Control Systems on Lie Groups, Mediterr. J. Math.
3(2), 873-882, 2016.
1 1 Kilis 7 Aralık University, Kilis, TURKEY, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
287
[8] Kule, M., Controllability of affine control systems on graded Lie groups, Kuwait J. Sci.
43(1), 61-68, 2016.
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
288
Some Effective Results Related to the New Statistical Cauchy Sequence
Merve İLKHAN1 and Emrah Evren KARA2
Abstract. In this presentation, we introduce the statistical Bourbaki Cauchy sequence
as a new concept in a metric space. Whereas a Bourbaki Cauchy sequence or a statistical
Cauchy sequence is statistical Bourbaki Cauchy, we show with examples that the reverse
statements are false. Further, we give a new condition by using a statistical Bourbaki Cauchy
sequence and prove that this is equivalent to Bourbaki completeness. Moreover, we
characterize Bourbaki completeness and Bourbaki boundedness of a metric space by using
functions preserving these generalized sequences.
Keywords. Statistical Cauchy sequence, completeness, Bourbaki completeness,
Bourbaki boundedness.
AMS 2010. 54E35, 54E50, 54A20, 40A05, 11B05.
References
[1] Atsuji, M., Uniform continuity of continuous functions of metric spaces, Pacific J. Math. 8,
11-16, 1958.
[2] Beer, G., Metric spaces on which continuous functions are uniformly continuous and
Hausdorff distance, Proc. Amer. Math. Soc. 95, 653-658, 1985.
[3] Beer, G., More about metric spaces on which continuous functions are uniformly
continuous, Bull. Aust. Math. Soc. 33, 397-406, 1986.
[4] Garrido M. I. and Merono, A. S., New types of completeness in metric spaces, Ann. Acad.
Sci. Fenn. Math. 39, 733-758, 2014.
[5] Schoenberg, I. J., The integrability of certain functions and related summability methods,
Amer. Math. Monthly 66, 361-375, 1959.
1 Düzce University, Düzce, TURKEY, [email protected] 2 Düzce University, Düzce, TURKEY, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
289
Products of Hopfian Manifolds and Their Shape Fibrators' Properties
Violeta VASILEVSKA1
Abstract. Approximate fibrations were introduced and studied by Coram and Duvall
in the late 1970s [1, 2], as a generalization of the concept of Hurewicz fibrations. In the late
1980s, Daverman introduced the concept of codimension-k fibrator [3] and later the concept
of PL fibrator [4]. PL fibrators, by definition, are PL manifolds that provide detection of PL
approximate fibrations. Following this concept of Daverman, the presenter introduced a
slightly changed (‘special’) PL setting [5, 6]. In this talk, we present to what extent such
results can be obtained in the new setting. We introduce manifolds that can ‘detect’
approximate fibrations in the ‘special’ PL setting, called shape msimpl fibrators. In addition, the
shape msimpl fibrators properties of Hopfian manifolds and their products will be discussed.
Additionally, a particular type of Hopfian groups will be introduced and their properties
discussed. It will be shown how important these special groups are as fundamental groups of
the discussed shape msimpl fibrators.
Keywords. Approximate fibration, Hopfian manifolds, Hopfian groups.
AMS 2010. 57N15, 57M07.
References
[1] Coram, D. S., Duvall, P. F., Approximate fibrations, Rocky Mountain J. Math. 7, 275-288,
1970.
[2] Coram, D. S., Duvall, P.F., Approximate fibrations and movability conditions for a map,
Pacific J. Math. 72, 41-56, 1977.
[3] Daverman, R. J., Submanifold decomposition that induce approximate fibrations,
Topology Appl. 33, 173-184, 1989.
[4] Daverman, R. J., PL manifolds with manifolds fibers, J. London Math. Soc. (2) 45, 180-
192, 1992.
[5] Vasilevska, V., Homology n-spheres as codimension-(n+1) shape msimpl-fibrators,
Topology Appl. 155, 1140-1148, 2008.
1 Utah Valley University, Orem, UT, USA, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
290
[6] Vasilevska, V., Special manifolds and shape fibrators properties, Topology Appl. 153,
2765-2781, 2006.
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
291
Soft AB-Sets and Soft αAB-Sets in Soft Topological Spaces
Zehra Güzel ERGÜL1 and Naime TOZLU2
Abstract. The present study aims to give some new concepts in soft topological
spaces such as soft semi-regular sets, soft AB-sets and soft αAB-sets. We investigate many
basic properties of these concepts with the help of some counterexamples. We discuss their
relationships with different types of subsets of soft topological spaces. In addition, we
introduce soft AB-continuous and soft αAB-continuous functions and we obtain the new
decompositions of soft continuity. We introduce these concepts which are defined over an
initial universe with a fixed set of parameters.
Keywords. Soft set, soft topological space.
AMS 2010. 06D72, 54A40.
Acknowledgements: This work is supported by Ahi Evran University Scientific Research
Projects Coordination Unit. Project Number: FEF. A3.16.020.
References
[1] Akdağ, M., Özkan, A., Soft α-open sets and soft α-continuous functions, Abstr. Appl.
Anal., http://dx.doi.org/10.1155/2014/891341, 2014.
[2] Arockiarani, I., Arokialancy, A., Generalized soft gβ-closed sets and soft gsβ-closed sets
in soft topological spaces, International Journal of Mathematical Archive, 4, 2, 1-7, 2013.
[3] Aygünoğlu, A., Aygün, H., Some notes on soft topological spaces, Neural Computing and
Applications, 21, 1, 113-119, 2012.
[4] Chen, B., Soft semi-open sets and related properties in soft topological spaces, Appl.
Math. Inf. Sci., 7, 1, 287-294, 2013.
[5] Feng, F., Jun, Y. B., Zhao, X. Z., Soft semirings, Comput. Math. Appl., 56, 2621-2628,
2008.
1 Ahi Evran University, Kırşehir, TURKEY, [email protected] 2 Ömer Halisdemir University, Niğde, TURKEY, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
292
[6] Kandil, A., Tantawy, O. A. E., El-Sheikh, S. A., Abd El-Latif, A. M., γ-operation and
decompositions of some forms of soft continuity in soft topological spaces, Ann. Fuzzy Math.
Inform., 7, 2, 181-196, 2014.
[7] Kharal, A., Ahmad, B., Mappings on soft classes, New Math. Nat. Comput., 7, 3, 471-481,
2011.
[8] Mahanta, J., Das, P. K., On soft topological space via semi-open and semi-closed soft sets,
Cornell University Library, 2012.
[9] Maji, P. K., Biswas, R., Roy, A. R., Soft set theory, Comput. Math. Appl., 45, 4-5, 555-
562, 2003.
[10] Molodtsov, D., Soft set theory-first results, Comput. Math. Appl., 37, 4-5, 19-31, 1999.
[11] Shabir, M., Naz, M., On soft topological spaces, Comput. Math. Appl., 61, 7, 1786-1799,
2011.
[12] Tozlu, N., Yüksel, Ş., Soft A-sets and soft B-sets in soft topological spaces, Accepted by
Mathematical Sciences and Applications E-Notes, 2017.
[13] Yüksel, Ş., Tozlu, N., Güzel Ergül, Z., Soft regular generalized closed sets in soft
topological spaces, International Journal of Mathematical Analysis, 8, 8, 355-367, 2014.
[14] Zorlutuna, I., Akdağ, M., Min, W. K., Atmaca, S., Remarks on soft topological spaces,
Ann. Fuzzy Math. Inform., 3, 2, 171-185, 2012.
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
293
THE OTHER
AREAS
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
294
Approximation by Certain Linear Positive Operators of Three Variables
Afşin Kürşat GAZANFER1
Abstract. We introduce positive linear operators which are combined with the
Chlodowsky and Szasz type operators and study some approximation properties of these
operators in the space of continuous functions of three variables on a compact set. The
convergence rate of these operators are obtained by means of the modulus of continuity. And
we also obtain weighted approximation properties for these positive linear operators in a
weighted space of functions of three variables and find the convergence rate for these
operators by using the weighted modulus of continuity.
Keywords. Chlodowsky-Szasz operators, linear positive operators, weighted modulus
of continuity.
References
[1] Gazanfer A. K., Büyükyazıcı İ., Approximation by certain linear positive operators of two
variables, Abstract and Applied Analysis, DOI 10.1155, 2014.
[2] Agrawal P. N., İspir N., Degree of approximation for bivariate Chlodowsky-Szasz-
Charlier type operators, Results in Mathematics, DOI 10.1007/s00025-015-0495-6, 2015.
[3] İspir N., Quantitative estimates for GBS operators of Chlodowsky-Szasz type, Faculty of
Sciences and Mathematics, University of Nis, Serbia, 31:5, 1175-1184, 2017.
[4] İspir N., Büyükyazıcı, İ., Quantitative estimates for a certain bivariate Chlodowsky-Szasz-
Kantorovich type operators, Mathematical Communications, 21(1):31-44, 2016.
[5] İbikli E., On approximation for functions of two variables on a triangular domain by
Bernstein-Chlodowsky polynomials, Rocky Mountain Journal of Mathematics, 35(5):1523-
1531, 2005.
1 Bülent Ecevit University, Zonguldak, TURKEY, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
295
Extension of Silver-Meal Algorithm with Variable Demand and Delivery Time to Cope
with Bullwhip Effect in Multi-Echelon Supply Chains
Halil İbrahim CEBECİ1 and Doğan ÜNAL2
Abstract. In multi-echelon supply chains, inventory management is very important to
avoid the adverse effect known by the bullwhip effect. Therefore, dynamic stock control
policies are preferred in the ordering process. The Dynamic Inventory Control Policy (DSKP)
approach based on demand forecasting, a proactive approach, is a dynamic inventory control
method that requires recalculating (r, Q) continuous replenishment inventory control
parameters at each forecast interval. The Silver-Meal algorithm, which is generally used in
Inventory quantity calculation, works under constant delivery time and demand conditions. In
our work, our purpose to design a stochastic extension of the silver-meal algorithm for use in
dynamic inventory control policies in multi-echelon supply chains. A simulation study has
been carried out for the validity of the relevant model.
Keywords. Bullwhip Effect, Dynamic Inventory Control Policy, Silver-Meal
Alghoritm.
AMS 2010. 68W01,60G15.
References
[1] Babai, M.Z. and Dallery, Y., “Dynamic versus static control policies in single stage
production-inventory systems”, International Journal of Production Research, 47(2), pp. 415-
433, 2009.
[2] Babai, M.Z. and Dallery, Y., “Inventory management: forecast based approach vs.
standard approach”, International Conference on Industrial Engineering and Systems
Management, pp. 57-67, 2005.
[3] Bregman, R.L. and Silver, E.A., “A Modification of silver meal heuristic to handle MRP
purchase discount situations”, Journal of Operational Research Society, 44(7), pp. 717-723,
1993.
[4] Burbidge, J.L., “Period batch control (PBC) with GT – the way forward from MRP”,
BPCIS Annual Conference, 1991.
[5] Forrester J., “Industrial Dynamics”, MIT Press, 1961.
[6] Houlihan J.B., “International supply chain management”, International Journal of
Physical Distribution and Materials Management, 17(2), pp.51-66, 198.
[7] Lee, H.L., Padmanabhan, P. and Whang, S., “Information distortion in a s supply chain:
the bullwhip effect”, Management Science, 43(4), 543-558, 1997a.
[8] Lee, H.L., Padmanabhan, P. and Whang, S., “The Bullwhip effect in supply chains”, Sloan
Management Review, 38(3), pp. 93-102, 1997b.
1 Sakarya University, Sakarya, Turkey, [email protected] 2 Sakarya University, Sakarya, Turkey, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
296
[9] Omar, M. and Smith, D.K., “An optimal batch size for a production system under linearly
increasing time-varying demand process”, Computers and Industrial Engineering, 42(1), pp.
35-42, 2002.
[10] Silver, E., Pyke, F.D. and Peterson, R., “Inventory management and production planning
and scheduling”, John Wiley & Sons, New York, 1998.
[11] Şenyiğit, E., “New heuristics to stochastic dynamic lot sizing problem”, Gazi University
Journal of Science, 22(2), pp. 97-106, 2009.
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
297
Consumer and Producer Surpluses of the Quadratic Demand and Supply Functions by
Using Trapezoidal Fuzzy Numbers and Signed Distance
İsmail ÖZCAN1 and Salih AYTAR2
Abstract. In the present study, we calculate the consumer surplus of the quadratic
demand function 𝑃𝐷(𝑥) = − 𝑥 − 𝑥2 and the producer surplus of the quadratic supply
function 𝑃𝑆(𝑥) = + 𝑥 + 𝑥2, where 𝑥 is a crisp quantity and , , , , and are
trapezoidal fuzzy numbers.
Keywords. Trapezoidal Fuzzy Number, Signed Distance Defuzzification Method,
Consumer Surplus, Producer Surplus.
AMS 2010. 03E72, 91B42, 91B02
References
[1] Wu K., Consumer surplus and producer surplus in fuzzy sense, Fuzzy Sets and Systems
103, 405-419, 1999.
[2] Kaufmann, A., Gupta, M.M, Introduction to Fuzzy Arithmetic: Theory and Application,
Van Nostrand Reinhold, New York, 1992.
1 Süleyman Demirel University, Isparta, TURKEY, [email protected] 2 Süleyman Demirel University, Isparta, TURKEY, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
298
On Creating Tessellations with Using Transformational Geometry
Kübra ÖZLÜ DEĞER1 and Ali Hikmet DEĞER2
Abstract. Translation, rotation and reflection are the bases of transformational
geometry. Using these transformations we can create beautiful tessellating shapes and cover
the plane with them. These two-dimensional patterns is decorative works in art and design.
Maurits Cornelis Escher (1898-1972) was a famous graphic artist using these types of
tessellations in his works.
In the present study we use transformational geometry to design some decorative
patterns of application to different surfaces. We also offer suggestions on the use of
transformational geometry for the mathematical education of interior design students.
Keywords. Transformational geometry, Tessellation, Decorative patterns.
AMS 2010. 97B10, 97D10, 97D40, 97G40, 97G50.
References
[1] Deger, K.O., Deger, A.H., An application of mathematical tessellation method in interior
designing, Procedia-Social and Behavioral Sciences, 51, 249-256, 2012.
[2] Hilden, H.M., Montesinos, J.M., Tejada, D.M., Toro, M.M., Artifacts for stamping
symmetric designs, The American Mathematical Monthly, 118, 4, 327-343, 2015.
[3] Feijs, L., Bartneck, C., Teaching geometric principles to design students, Digital Culture
& Education, 1,2, 104-115, 2009.
[4] Toth, L.F., Tessellation of the Plane with Convex Polygons Having a Constant Number of
Neighbors, The American Mathematical Monthly, 82, 3, 273–276, 1975.
[5] Dobson, J., Gage, J., Using art to teach maths, using maths to create art, Bridges
Donostia: Mathematics, Music, Art, Architecture, Culture, 445-452, 2007.
[6] Hart, G., Heathfield, E., Making math visible, Proceedings of Bridges 2017: Mathematics,
Art, Music, Architecture, Education, Culture, 63-70, 2017.
1 Karadeniz Technical University, Trabzon, TURKEY, [email protected] 2 Karadeniz Technical University, Trabzon, TURKEY, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
299
Relationship between Electricity Consumption and Economic Growth in some
Developing Countries: MS-VECM Analysis
Melike BİLDİRİCİ1 , Fazıl KAYIKÇI2
Abstract. This study aims at analyzing the relationship between electricity
consumption and economic growth in Some Developing Countries with variety of
econometric techniques: Markov Switching – Vector Error Correction Model. Empirical
results reveal that electricity consumption and economic growth are cointegrated for these
countries. Furthermore, there is a causality between electricity consumption and economic
growth such that policymakers in these countries should consider this reality when designing
energy and growth policies.
Keywords. Electricity Consumption, Economic Growth, MS-VECM
References
[1] Lee, C.C., Chang, C.P., Energy consumption and GDP revisited: a panel analysis of
developed and developing countries, Energy Economics, 29, 1206-1223, 2007.
[2] Levin, A., Lin, C.F., Chu, C.S.J., Unit Root Tests in Panel Data: Asymptotic and Finite-
Sample properties, Journal of Econometrics, 108, 1-24, 2002.
[3] Pedroni, P., Critical Values for Cointegration Tests in Heterogeneous Panels with
Multiple Regressors, Oxford Bulletin of Economics and Statistics, 61, 653-670, 1999.
[4] Pesaran, M.H., Shin, Y., Smith, R.J., Pooled mean group estimation of dynamic
heterogeneous panels, Journal of the American Statistical Association, 94, 621-634, 1999.
1 Yıldız Technical University, İstanbul, TURKEY, [email protected] 2 Yıldız Technical University, İstanbul, TURKEY, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
300
Test of validity of Baltic Dry index with MS-ARMA and SET-ARMA models
Melike BİLDİRİCİ1 , Fazıl KAYIKÇI2 and Işıl Şahin ONAT3
Abstract. The Baltic Dry Index has become one of the foremost indicators on the cost
of shipping and an important barometer on the volume of worldwide trade and manufacturing
activity since its establishment. Global factors also play important role in supply and demand
of BDI index. Average of price of 23 different shipping routes around the world compiles
daily to form the Baltic Dry Index. Economic indicators such as unemployment rate, inflation
and oil prices that can be manipulated or influenced by governments and speculators,
however, Baltic Dry Index is difficult to manipulate because it is driven by clear forces of
supply and demand. In this paper, some econometic models such as Markov Switching Auto
Regressive Moving Average and SET- Auto Regressive Moving Average models were used
to analyze the validity of the BDI index for some developed and developing countries in the
world.
Keywords. BDI index, MS-ARMA, SET-ARMA
References
[1] Clements, M.C., Krolzig, H.-M., Modelling Business CycleFeatures Using
SwitchingRegimeModels, ," Economics Series WorkingPapers 9958, 1-17, 2002.
[2] Psaradakis, Z., Morten, O.R. and Sola M., Markov-Switching causality and the Money-
Output Relationship, Journal of Applied Econometrics, 665-83, 2005.
[3] Geman, H. and Smith, W Shipping Markets and Freight Rates: An Analysis of the Baltic
Dry Index, The Journal of Alternative Investments, 1, 98-109, 2012.
[4] Hamilton, J.D., A new approach to the economic analysis of nonstationary time series and
the business cycle, Econometrica, 57, 357-384, 1989.
1 Yıldız Technical University, İstanbul, TURKEY, [email protected] 2 Yıldız Technical University, İstanbul, TURKEY, [email protected] 3 Yıldız Technical University, İstanbul, TURKEY, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
301
Chaotic Examination of Turkish Financial Market
Melike E. BİLDİRİCİ1 , Fulya ÖZAKSOY 2 and Bahri SONÜSTÜN3
Abstract. Although chaos theory aspired great interest in physics by Lorenz attractor
in 1960s, its application to many branches of social sciences such as economics, mathematics,
psychology, behavioral sciences, neurosciences dates back to 1980s. In the regards of this
‘new economy’, ‘new concepts’ such as complexity, determinism, quantum mechanics,
relativity, multiple equilibrium, complexity, (continuously) instability, nonlinearity,
heterogeneous agents, irregularity were widely questioned in economics. It is noticed that
linear models are insufficient for analyzing unpredictable, irregular and noncyclical
oscillations of economies, and for predicting bubbles, financial crisis and business cycles in
financial markets. Therefore, economists gave great consequence to use appropriate tools for
modelling non-linear dynamical structures and chaotic behaviors of economies especially in
macro and financial economy. In this paper, we aim to model chaotic structure of exchange
rates (USD-TL and EUR-TL) in Turkish financial market. To determine non-linear pattern of
the selected financial time series, daily returns of the exchange rates were tested by BDS
during the period from January 01, 2002 to May 11, 2017 which covers after the era of 2001
financial crisis. After specifying the non-linear structure of the selected financial time series,
it was aimed to examine the chaotic characteristic in Turkish financial market for the selected
time period by Lyapunov Exponents. The findings verify the existence of the chaotic structure
of the exchange rate returns in the analyzed time period.
Keywords: Chaos, Non-linearity, BDS test, Exchange rate returns, Lyapunov
Exponent
1 Yıldız Technical University, İstanbul, TURKEY, [email protected] 2 Dogus University, İstanbul, TURKEY, [email protected] 3 Yıldız Technical University, İstanbul, TURKEY, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
302
Chaotic Structure of Oil Price
Melike E. BİLDİRİCİ1 , Fulya ÖZAKSOY 2
Abstract. The fluctuations in oil prices are very complicated and therefore, it is unable
to predict the dynamics in economies. However, there is a main constraints of linear economic
models is that they are not sufficient for modelling price movements of complex patterns of
economies such as oil prices. Thus, in recent years, economists attached great attention to
non-linear structure of oil prices. For analyzing this relationship, GARCH models are mostly
used in the papers. Distinctively from other papers, in this paper, we aim to analyze chaotic
pattern of oil prices. Thus, firstly to determine chaotic behavior of oil prices, BDS model was
used. Following, this structure was modelled by Lyapunov Exponents for the selected time
period and forecasting achievement of the models were compared with each others. The
findings confirm the existence of the non-linear and chaotic framework of oil prices.
Keywords: Oil price, BDS test, Lyapunov Exponent, Chaos, Non-linearity
1 Yıldız Technical University, İstanbul, TURKEY, [email protected] 2 Dogus University, İstanbul, TURKEY, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
303
Optimization of MANET Routing Tables with Plant Propagation Algorithm
Mevlüt ERSOY1 and Tuncay YİĞİT2
Abstract. Mobile Ad-hoc Network (MANET) is a network in which many mobile
nodes communicate with each other without any central device [1]. In these networks, the
communication between the source and the target node is achieved by routing through the
nodes in the network. The routing process of the packet sent from the source node to the
destination node is provided by the request of the nodes in the network or by the tables
maintained in their memory [2]. In MANET, routing roots can change dynamically due to
newly added nodes, battery life of the nodes, etc. Finding the optimal path in terms of the
energy efficiency and delay time of the package sent from the source to the destination is an
important and complex problem. For this reason, natural - inspired heuristic methods can be
used in MANET routing problem. Especially, there are many studies with GA to solve the
MANET routing problem [3] [4]. In recent years, many problems have been solved with the
Plant Propagation Algorithm (PPA) [5] [6]. In this work, we used PPA for finding the optimal
path of the package sent from the source to the destination. The algorithm has been
implemented by taking battery life of the nodes and the new nodes added to the network into
account. In this study, the performance of the routing tables formed with PPA and GA has
been analyzed in terms of delay times and energy costs. The results have been obtained from a
MANET consisting of 10 nodes. It has been observed that PPA is better in terms of energy
costs.
Keywords. MANET, routing optimization, naturel inspired algorithms
References
[1] Bayılmış C., Erturk İ., Çeken C., Bandırmalı N. DSR ve AODV MANET Yönlendirme
Protokollerinin Başarım Değerlendirmesi, Elektrik Mühendisleri Odası, 2–5, 2005.
[2] Doğru İA., Şimşek M., Akcayol MA. Hareketli Ad-Hoc Ağlarda Bir Hareketlilik Yönetimi
Protokolü, Politeknik Dergisi, 11, 4, 313–318, 2008.
1 Süleyman Demirel University, Isparta, TURKEY, [email protected] 2 Süleyman Demirel University, Isparta, TURKEY, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
304
[3] Lu T., Zhu J. Genetic algorithm for energy-efficient QoS multicast routing, IEEE
Communications Letters, 17, 1, 31–34, 2013.
[4] Bari A., Wazed S., Jaekel A., Bandyopadhyay S. A genetic algorithm based approach for
energy efficient routing in two-tiered sensor networks, Ad Hoc Networks, 7, 4, 665–676,
2009.
[5] Selamoglu B.I., Salhi A., Alsoufi G. Solving Container Terminal Scheduling Problems
with the Plant Propagation Algorithm, 5th Internatıonal Eurasıan Conference On
Mathematıcal Scıences And Applıcatıons, Begrad,Serbia, 82–83, 2016.
[6] Selamoğlu Bİ., Salhi A. The Plant Propagation Algorithm for Discrete Optimisation: The
Case of the Travelling Salesman Problem, Nature-Inspired Computation in Engineering,
Cham, 43–61, 2016.
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
305
The Integrability and the Zero-Hopf Bifurcation of the Three-Dimensional Lotka-
Volterra Systems
Rizgar H. SALIH1
Abstract: This paper is devoted to study the zero-Hopf bifurcation of the three-
dimensional Lotka-Volterra systems. The explicit conditions for the existence of two first
integrals for the system and a line of singularities with zero eigenvalue are given. We
characteristic the parameters for which a zero-Hopf equilibrium point takes place at any
points on the line. We prove that there are three 3-parameter families exhibiting such
equilibria. The averaging theory of the first order is also applied but any information about the
possible periodic orbits bifurcating from the zero-Hopf equilibria is not provided by this
theorem.
1 University of Raparin-College of Science, Kurdistan Region-IRAQ. [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
306
POSTERS
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
307
Estimates of Initial Coefficients of a New Analytic and Bi-Univalent Function Class
Defined by Integral Operator Involving Polylogarithm Function
Arzu AKGÜL1
Abstract. In this study, the author aims at introducing a new subclass of the function
class of bi-univalent functions defined by integral operator involving polylogarithm function.
Furthermore, it is established that bounds for the coefficients for this subclass and several
related classes are also considered and connections to earlier known results are made.
Keywords: Bi-univalent functions, bi-starlike functions, coefficient estimates,
polylogarithm function, integral operator.
Mathematics Subject Classification: 2010: 30C45, 30C50.
References
[1] Akgul, A., Identi.cation of Taylor-Maclaurin coefficients for generalizes subclass of bi-
univalent functions, Sahand Communications in Mathematical Analysis, 2017 (Accepted).
[2] Altınkaya, Ş., Yalçın, S., Second Hankel determinant for a general subclass of bi-
univalent functions, TWMS J.Pure Appl. Math. V.7, N.1, 98-104, 2016.
[3] Altınkaya, Ş., Yalçın, S., Coefficient Estimates for Two New Subclasses of Bi-univalent
Functions, Acta Universitatis Apulensis, 43, 53-63, 2015.
[4] Bateman, H., Higher transendental functions, vol. 1 of Edited by: A. Erdely, W. Mangnus,
F. Oberhettinger, F. G. Tricomi, Mc Graw-Hill, New York, NY, USA, 1953
[5] Brannan, D.A., Taha, T.S., On some classes of bi-univalent functions, in: S.M. Mazhar, A.
Hamoui, N.S. Faour (Eds.), Math. Anal. and Appl.,Kuwait; February 18.21, 1985, in: KFAS
Proceedings Series, vol. 3, Pergamon Press, Elsevier Science Limited, Oxford, 1988, pp.
53.60. see also Studia Univ. Babe¸s-Bolyai Math. 31 (2)70.77, 1986.
1 Kocaeli University, Kocaeli, TURKEY, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
308
[6] Brannan D. A. and Clunie, J. G., Aspects of comtemporary complex analysis, (Proceedings
of the NATO Advanced Study Instute Held at University of Durham: July 1-20, 1979). New
York: Academic Press, 1980.
[7] Duren, P. L., Univalent Functions, Grundlehren der Mathematischen Wissenschaften,
Springer, New York, NY, USA, 259, 1983.
[8] Frasin, B. A. and Aouf, M. K., New subclasses of bi-univalent functions, Applied
Mathematics Letters, 24, 1569-1573, 2011.
[9] Hamidi S. G. and Jahangiri, J. M., Faber polynomial coefficient estimates for analytic bi-
close-to-convex functions, C. R. Acad. Sci. Paris, Ser.I 352 (1), 17.20, 2014.
[10] Jahangiri, J. M. and Hamidi, S. G., Coefficient estimates for certain classes of bi-
univalent functions, Int. J. Math. Math. Sci., ArticleID 190560, 4 pp, 2013.
[11] Lewin, M., On a coe¢ cient problem for bi-univalent functions, Proceeding of the
American Mathematical Society, 18, 63-68, 1967.
[12] Netanyahu, E., The minimal distance of the image boundary from the orijin and the
second coe¢ cient of a univalent function in jzj < 1; Archive for Rational Mechanics and
Analysis, 32, 100-112, 1969.
[13] AL-Shaqsi, K., Strong differential subordinations obtained with new integral operator
defined by polylogarithm function, International Journal of Mathematics and Mathematical
Sciences, Vol2014,Art ID 260198, 6 pages.
[14] Srivastava, H. M., Mishra, A. K. and Gochhayat, P., Certain subclasses of analytic and
bi-univalent functions, Applied Mathematics Letters, 23, 10, 1188-1192, 2010.
[15] Xu, Q. H., Gui, Y. C., and Srivastava, H. M., Coefficient estimates for a certain subclass
of analytic and bi-univalent functions, Applied Mathematics Letters, 25, 990-994, 2012.
[16] Srivastava, H. M. and Attiya, A. A., An integral operator Associated with Hurwitz-
Lerch Zeta function and differential subordination, Integral Transforms and Special functions,
18, 3,207-216, 2007.
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
309
[18] Xu, Q. H., Xiao, H. G. and Srivastava, H. M., A certain general subclass of analytic and
bi-univalent functions and associated coefficient estimate problems, Appl. Math. Comput.,218
2012.
[19] Srivastava, H. M., Joshi, S. B., Joshi, S. S., Pawar, H., Coefficient estimates for certain
subclasses of meromorphically bi-univalent functions, Palest. J. Math., 5 (Special Issue: 1),
250-258, 2016.
[20] Taha, T. S., Topics in univalent function theory, Ph.D. Thesis, University of London,
1981.
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
310
Finding Initial Coefficients for a Class of Bi-Univalent Functions Given by q-Derivative
Operators
Arzu AKGÜL1
Abstract. In the present study, the author _nds coincident bounds for functions in a
subclasses of analytic and bi-univalent functions introduced by using q-derivative operator.
Some corollaries and consequences of the main results are also considered. The results are
presented in this paper generalize the recent work of Srivastava et all. [21].
Keywords: Analytic and univalent functions; bi-univalent functions; bi-starlike and
bi-convex functions; coefficient bounds; q-derivative operator.
References
[1] Ali, R. M., Lee, S. K., Ravichandran, V. and Supramanian, S., Coefficient estimates for bi-
univalent Ma-Minda starlike and convex functions, App. Math. Lett., 25(3), 344-351, 2012.
[2] Altınkaya, S. and Yalçın, S., Coefficient bounds for a subclass of bi-univalent functions,
TWMS J. Pure Appl. Math., 6(2), 180-185, 2015.
[3] Aral,A.,Gupta, V. and Agarwal, R. P., Applications of q-calculus in operator theory,
Springer, New York, USA, 2013.
[4] Brannan, D. A., Taha, T. S., On some classes of bi-univalent functions, Stad. Univ. Babes-
Bolyai, Math. 31(2), 70-77, 1986.
[5] Caglar, M., Orhan, H., Yagmur, N., Coe_cient bounds for new subclass of bi-univalent
functions, Filomat, 27, 1165-1171, 2013.
[6] Crisan,O., Coefficient estimates of certain subclass of bi-univalent functions, Gen. Math.
Notes, 16(2), 93-102, 2013.
[7] Duren, P. L., Grundlehren der Mathematischen Wissenchaften, Springer, New York,NY,
USA, 1983.
1 Kocaeli University, Kocaeli, TURKEY, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
311
[8] Frasin, B. A. and Aouf, M.K., New subclass of bi-univalent functions, Appl. Math.
Lett.,24, 1569-1573, 2011.
[9] Jackson, F. H., On q-functions and a certain di_erence operator, Trans. R. Soc. Edinb.,46,
253-281, 1908.
[10] Lewin, M., On a coe_cient problem of bi-univalent functions, Proc. Amer. Math. Soc.,
18, 63-68, 1967.
[11] Li, X. F. and Wang, A. P., Two new subclasses of bi-univalent functions, International
Mathematical Forum, 7, 1495-1504, 2012.
[12] Magesh, N. and Yamini, J., Coe_cient bounds for a certain subclass of bi-univalent
functions, International Mathematical Forum 8(22), 1337-1344, 2013.
[13] Netanyahu, E., The minimal distance of the image boundary from the orijin and the
second coefficient of a univalent function in mid z j< 1 , Archive for Rational Mechanics and
Analysis,32, 100-112, 1969.
[14] Pommerenke, C. H., Univalent functions, Vandenhoeck and Rupercht, Göttingen, 1975.
[15] Ponnusamy, S., Inclusion theorems for convolution product of second order
polylogariyhms and functions with the derivative in a half plane, Rocky Montain J. Math.,
28(2), 695-733, 1998.
[16] Ponnusamy, S. and Sabapathy, S., Polylogarithms in the theory of univalent functions,
Result in Mathematics, 30,136-150, 1996.
[17] Porwal, S. and Darus, M., On a new subclass of bi-univalent functions, J. Egypt. Math.
Soc.,21(13),190-193, 2013
[18] St. Ruscheweyh, New criteria for univalent functions, Proc. Amer. Math. Soc., 49,109-
115, 1975.
[19] Sâlâgean, G. S., Subclasses of univalent functions, Lecture Notes in Math., Springer
Verlag, 1013,362-372, 1983.
[20] Al, K. Shaqsi and Darus, M., An oparator de_ned by convolution involving the
polylogarithms functions, Journal of Mathematics and Statics, 4(1), 46-50, 2008.
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
312
[21] Srivastava, H. M., Mishra, A. K. and Gochhayat, P., Certain subclass of analytic and bi-
univalent functions, Appl. Math. Lett., 23,1188- 1192, 2010.
[22] Srutha Koerthi, B. and Raja, B., Coeffcient inequality for certain new subclass of analytic
bi-univalnt functions, Theoratical Mathematics and Applications, 3(1), 1-10, 2013.
[23] Xu, Q. H., Gui, Y. C. and Srivastava, H. M., Coe_cient estimates for a certain subclass
of analytic and bi-univalent functions, Appl. Math. Lett., 23,990-994, 2012.
[24] Xu, Q. H., Xiao, H. G. and Srivastava, H. M., A certain general subclass of analytic and
bi-univalent functions and associated coe_cient estimate problems, Appl. Math. Comput.,218
(2012), 11461-11465.See also Studia Univ. BabeÅ- Bolyai Math. 31(2), 70-77, 1986.
[25] Xu, Q. H. Srivastava, H. M. and Li, Z. A certain subclass of analytic and close-to-convex
functions, Appl. Math. Lett.,24, 396-401, 2011.
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
313
Dual-Hyperbolic Fibonacci and Lucas Numbers
Arzu CİHAN1, Ayşe Zeynep AZAK2 and Mehmet Ali GÜNGÖR3
Abstract: In this study, we initially defined the dual hyperbolic Fibonacci and dual
hyperbolic Lucas numbers. Then, the fundamental identities were proved for these numbers.
Additionally, we gave the identities with respect to negadual hyperbolic Fibonacci and
negadual hyperbolic Lucas numbers. At last, Binet formulas, D’cogne’s, Catalan and Cassini
identities were obtained for dual hyperbolic Fibonacci and dual hyperbolic Lucas numbers.
Keywords: Dual hyperbolic numbers, dual hyperbolic Fibonacci numbers, dual
hyperbolic Lucas numbers
Referrences
[1] Alfsmann, D., On families of 2N dimensional Hypercomplex Algebras suitable for digital
signal processing. In: Proceedings of EURASIP 14th European Signal Processing
Conference (EUSIPCO 2006), Florence, Italy 2006.
[2] Akyiğit M., Kösal H.H., Tosun M., Split Fibonacci Quaternions, Adv. Appl. Clifford
Algebras, 23, 535-545, 2013.
[3] Catoni, F., Cannata, R., Catoni, V., Zampetti, P., Hyperbolic trigonometry in two-
dimensional space-time geometry. N. Cim. B 118, 475–491, 2003.
[4] Clifford W.K., Preliminary Sketch of Biquaternions, Proc. London Mathematical Society,
4, 64, 381-395, 1873.
[5] Dunlap R.A., The Golden Ratio and Fibonacci Numbers, World Scientific, 1997.
[6] Fjelstad, P. Extending special relativity via the perplex numbers. Am. J. Phys. 54(5), 416–
422 1986.
[7] Halıcı S., On Fibonacci Quaternions, Adv. Appl. Clifford Algebras, 22, 321-327, 2012.
1 Sakarya University, Sakarya, TURKEY, [email protected] 2 Sakarya University, Sakarya, TURKEY, [email protected] 3 Sakarya University, Sakarya, TURKEY, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
314
[8] Halıcı S., On Complex Fibonacci Quaternions, Adv. Appl. Clifford Algebras, 23, 105-
112, 2013.
[9] Horadam A. F., Complex Fibonacci Numbers and Fibonacci Quaternions, Amer. Math.
Monthly, 70, 289-291, 1963.
[10] Iyer M.R., Some Results on Fibonacci Quaternions, The Fibonacci Quarterly, 7, 2, 201-
210, 1969.
[11] Kantor I. L., Solodownikow, V. N. , Hyperkomplexe Zahlen, Teubner, Leipzig 1978.
[12] Knuth D., Negafibonacci Numbers and Hyperbolic Plane, Annual Meeting of the Math.
Association of America, 2013.
[13] Koshy T., Fibonacci and Lucas Numbers with Applications, A Wiley-Intersience
Publication, USA, 2001.
[14] Majernik V., Multicompenent Number Systems, Acta Physica Polonica A, 90, 491-498,
1996.
[15] Nurkan S.K., Güven I.A., A Note on Bicomplex Fibonacci and Lucas Numbers,
https://arxiv.org/abs/1508.03972v1, 2015.
[16] Swamy M.N., On Generalized Fibonacci Quaternions, The Fibonacci Quarterly, 5, 547-
550, 1973.
[17] Vajda S., Fibonacci and Lucas Numbers and the Golden Section, Ellis Horwood Books
Math. Appl., Ellis Horwood Ltd./Halsted Press; Chichester/New York etc., 1989.
[18] Verner E., Hoggatt Jr., Fibonacci and Lucas Numbers, The Fibonacci Association, 1969.
[19] Yüce S., Aydın F.T., A New Aspect of Dual Fibonacci Quaternions, Adv. Appl. Clifford
Algebras, 26, 873-884, 2016.
[20] Güngör M.A., Azak A.Z. Investigation of Dual-Complex Fibonacci,Dual-Complex Lucas
Numbers and Their Properties, submitted.
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
315
Toeplitz Operators with Symbols in Some Function Spaces
Ayşe SANDIKÇI 1
Abstract. Toeplitz operators are introduced as a class of pseudo differential operators
which depend on a symbol and two different window functions. We investigate new
conditions for the boundedness properties on Lorentz type modulation spaces using symbols
in some function spaces. Some key references are given below.
Keywords. Modulation space, Lorentz type modulation space, Toeplitz operator.
AMS 2010. 42B10, 47B38.
References
[1] Cordero E. and Tabacco A., Localization Operators via Time-Frequency Analysis, In:
Ashino R., Boggiatto P., Wong M.W. (Eds) Advances in Pseudo-Differential Operators.
Operator Theory: Advances and Applications, vol 155. Birkhäuser, Basel, 2004.
[2] Cordero, E. and Gröchenig, K., Time-frequency analysis of localization operators, J.
Funct. Anal., 205(1) (2003), 107-131.
[3] Gröchenig,K., Foundation of Time-Frequency Analysis. Birkhäuser, Boston 2001, ISBN
0-8176-4022-3.
[4] O'Neil, R., Integral Transforms and Tensor Products on Orlicz Spaces and L(p,q) spaces,
J. d'Analyse Math., 21, 1-276, 1968.
[5] Sandıkçı, A., On Lorentz mixed normed modulation spaces, J. Pseudo-Differ. Oper. Appl.,
3 (2012), 263-281.
[6] Sandıkçı, A., Boundedness of localization operators on Lorentz mixed normed modulation
spaces, Journal of Inequalities and Applications, 2014, 2014: 430.
1 Ondokuz Mayis University, Samsun, TURKEY, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
316
A Further Generalization of Gamma, Beta and Hypergeometric Functions
Ayşegül ÇETINKAYA1, İ. Onur KIYMAZ1, Praveen AGARWAL2 and Shilpi JAIN3
Abstract. In this work, we present further generalization of gamma and beta functions
by using a confluent hypergeometric function with six parameters. Then we introduce new
generalization of Gauss and confluent hypergeometric functions by using this generalization
of beta function and systematically investigate their properties.
Keywords. Gamma function, beta function, hypergeometric functions.
AMS 2010. 33B15, 33C05, 33C15.
References
[1] Chaudhry M. A., Qadir A., Rafique M., Zubair S.M., Extension of Euler's beta function, J.
Comput. Appl. Math., 78, 19-32, 1997.
[2] Chaudhry M.A., Qadir A., Srivastava H.M., Paris R.B., Extended hypergeometric and
confluent hypergeometric functions, Appl. Math. Comput., 159 (2), 589-602, 2004.
[3] Choi J., Rathie A.K., Parmar R.K., Extension of extended beta, hypergeometric and
confluent hypergeometric functions, Honam Mathematical J., 36 (2), 357-385, 2014.
[4] Luo Min-Jie, Milovanovic G.V., Agarwal P., Some results on the extended beta and
extended hypergeometric functions, Appl. Math. Comput., 248, 631-651, 2014.
[5] Özergin E., Özarslan M.A., Altın A., Extension of gamma, beta and hypergeometric
functions, J. Comput. Appl. Math., 235, 4601-4610, 2011.
[6] Parmar R.K., A new generalization of Gamma, Beta, hypergeometric and confluent
hypergeometric functions, Matematiche (Catania) 69, 33-52, 2013.
[7] Rainville E.D., Special Functions, Macmillan Company, New York, 1960; Reprinted by
1 Ahi EvranUniversity, Kırşehir, TURKEY, [email protected] , [email protected] 2 Anand International Collage of Engineering, Jaipur, INDİA, [email protected] 3 Poornima Collage of Engineering, Jaipur, INDIA, [email protected] Acknowledgement: This work was supported by Ahi Evran University Scientific Research Projects Unit. Project Number: FEF.D1.16.001
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
317
Chelsea Publishing Company, Bronx, New York, 1971.
[8] Srivastava H.M., Agarwal P., Jain S., Generating functions for the generalized Gauss
hypergeometric functions, Appl. Math. Comput., 247, 348-352, 2014.
[9] Srivastava H.M., Jain R., Bansal M.K., A Study of the S-Generalized Gauss
Hypergeometric Function and Its Associated Integral Transforms, Turkish Journal of
Analysis and Number Theory, 3 (5), 116-119, 2015.
[10] Srivastava H.M., Karlsson P.W., Multiple Gaussian hypergeometric series, Ellis
Horwood Limited, 1985.
[11] Srivastava H.M., Manocha H.L., A treatise on generating functions, Ellis Horwood
Limited, Chichester, 1984.
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
318
Teachers’ Perceptions About Written Examination Preparation Process
Ayten Pınar BAL1 and Fatma SADIK2
Abstract. The purpose of this study is to figure out teachers’ perceptions about written
examination preparation process. The study carried out with twenty secondary school
mathematics teachers working at different public secondary schools in Adana. The data was
collected through document analysis. At the end of the research it was found out that teachers
generally use multiple choice and classic tests and ask 20 questions on average. Moreover, the
other preffered question types are True-False and Fill in the blanks. In terms of the difficulty
levels of the questions; It was seen that there are comprehension and application types of
questions on all of the exam papers. In addition, it was also found out that the teachers in the
study are tend to give equal points for the questions that they ask.
NOTE: This article was supported by Cukurova University Scientific Research Unit. Project
No: EF2013BAP11
Keywords. Assessment and evaluation, mathematics teacher, success test, written
examination.
1 Çukurova University, Adana, TURKEY, [email protected] 2 Çukurova University, Adana, TURKEY, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
319
The Analysis of Mistakes and Solution Strategies Applıed in Algebraic Verbal Problems
Ayten Pınar BAL1 and Ahmet KARACAOĞLU2
Abstract. This study is a scanning model research in order to obtain algebraic verbal
problem solution strategies of 6th, 7th and 8th class students and their mistakes during this
period. Totally 1017 students from 6th, 7th and 8th grade students educating at medium socio
economic level of central districts of Adana province, formed working group of this study. As
data collection tool “Identification Test of Algebraic Verbal Problems Solution Strategies and
Mistakes of Students” developed by researchers was used in the study. As a result of research
it was found out that secondary school students especially use systematic distribution, inverse
operation, ordering after division, empirical and equation strategies in solving of algebraic
verbal problems and as error type they frequent make logic and calculation errors.
Keywords. Algebraic verbal problem, error type, problem solving strategy
1 Cukurova University, Adana, TURKEY, [email protected] 2 Cukurova University, Adana, TURKEY, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
320
Examination of Burnout and Life Satisfaction of Nurses
Emrah GÜRLEK1 and Kamile ŞANLI KULA2 and Mehmet YETİŞ3 and Aysu YETİŞ4
Abstract. In this study, burnout and life satisfaction of nurses working at Ahi Evran
University Training and Research Hospital will be examined in terms of various variables.
The study will be conducted with the nurses working in Ahi Evran University Education and
Research Hospital, volunteers who agree to participate in the research. For this purpose,
Personal Information form developed by researchers as a data collection tool, Burnout Scale
and Life Satisfaction Scale will be used.
Keywords. Nurse, Burnout, Life Satisfaction.
MSC 2010. 62P10.
This work was supported by the Scientific Research Projects Council of Ahi Evran
University, Kırşehir, Turkey under Grant TIP.A3.17.005.
References
[1] Neugarten, B., Havighurst, R. and Tobin, S., The Measure of Life Satisfaction. Journal of
Gerontology, 16, 134-143, 1961.
[2] Çelebi, B., Workers' burnout and job satisfaction: Alanya State Hospital Nurses sample.
Unpublished Master's Thesis, Beykent University Social Sciences Institute. İstanbul, 2014.
[3] Durmuş, S. ve Günay, O., Factors affecting job satisfaction and anxiety levels in the
nurses. Erciyes Medical Journal. 29(2), 139-146, 2007.
[4] Maslach, C., Schaufeli, W. B., ve Leiter, M. P., Job Burnout. Annual Reviews of
Psychology, 52, 397-422, 2001.
[5] Kaplan, H., The relatıonshıp between job satısfactıon and lıfe satısfactıon: Denızlı
Servergazı State Hospıtal examples of nurse and mıdwıves. Unpublished Master's Thesis.
Beykent University Social Sciences Institute, Istanbul, 2014.
1 Ahi Evran University, Kırşehir, TURKEY, [email protected] 2 Ahi Evran University, Kırşehir, TURKEY, [email protected] 3 Ahi Evran University, Kırşehir, TURKEY, [email protected] 4 Ahi Evran University Education and Research Hospital, Kırşehir, TURKEY, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
321
[6] Şenyüz, Z., Burnout ın nurses and midwives working ın hospitals. Unpublished Master's
Thesis. Beykent University Social Sciences Institute, İstanbul, 2015.
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
322
Ruled Surface Pair Generated by Darboux Vectors of a Curve and Its Natural Lift in
𝐈𝐑𝟏𝟑
Evren ERGÜN1 , Mustafa ÇALIŞKAN2 and Keziban ORBAY3
Abstract. In this study, firstly, the darboux vector W of the natural lift 𝛼 of a
curve 𝛼 are calculated in terms of those of 𝛼 in IR13. Secondly, we obtained striction lines and
distribution parameters of ruled surface pair generated by Darboux Vectors of the curve 𝛼 and
its natural lift 𝛼 . Finally, for 𝛼 and 𝛼 those notions are compared with each other.
Keywords. Natural Lift, Ruled Surface, Striction Line, Distribution Parameter.
AMS 2010. 51B20, 53A15, 53A04.
References
[1] Çalışkan, M., Sivridağ, A.İ., Hacısalihoğlu, H.H, Some Characterizations for the natural
lift curves and the geodesic spray, Communications, Fac. Sci.Univ. Ankara Ser. A Math. 33,
Num. 28,235-242, 1984
[2] Çalışkan, M., Ergün, E.,On The -Integral Curves and -Geodesic Sprays In Minkowski
3-Space International Journal of Contemp. Math. Sciences,Vol. 6, no. 39, 1935-1939, 2011.
[3] Ergün, E., Çalışkan, M., On Geodesic Sprays in Minkowski 3-Space, International Journal
of Contemp. Math. Sciences, Vol. 6, no. 39, 1929-1933, 2011.
[4] Sivridağ A.İ. Çalışkan M. On the -Integral Curves and -Geodesic Sprays Erc.Uni. Fen
Bil. Derg. 7, 2, 1283-1287, 1991
[5] Thorpe, J.A., Elementary Topics In Differential Geometry,Springer-Verlag,New York,
Heidelberg-Berlin, 1979.
[6] Walrave, J., Curves and Surfaces in Minkowski Space K. U. Leuven Faculteit,Der
Wetenschappen, 1995.
1 Ondokuz Mayıs University, Samsun, TURKEY, [email protected] 2 Gazi University, Ankara, TURKEY, [email protected] 3 Amasya University, Amasya, TURKEY, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
323
Caratheodory’s Theorem in B-1-convexity
Gabil ADILOV1 and Ilknur YESILCE2
Abstract. B-1-convexity is an abstract convexity type [1, 2]. B-1-convex sets are
introduced in [3]. Separation in B-1-convexity is investigated in [4].
In this work, for continuing studies regarding B-1-convexity, we give some important
theorems like Carathedory’s Theorem for B-1-convex sets.
Keywords. Abstract convexity, B-1-convex sets, Carathedory’s Theorem.
AMS 2010. 52A20, 52A35.
References
[1] Rubinov, A., Abstract Convexity and Global Optimization, Kluwer Academic Publishers,
Boston-Dordrecht-London, 2000.
[2] Singer, I., Abstract Convex Analysis, John Wiley & Sons., New York, 1997.
[3] Adilov, G. and Yesilce, I., B-1-convex Sets and B-1-measurable Maps, Numerical
Functional Analysis and Optimization, 33, 2, 131-141, 2012.
[4] Tinaztepe, G., Yesilce, I. and Adilov, G., Separation of B-1-convex Sets by B-1-measurable
Maps, Journal of Convex Analysis 21, 2, 571-580, 2014.
1 Akdeniz University, Antalya, TURKEY, [email protected] 2 Mersin University, Mersin, TURKEY, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
324
The Simulation of the Order Parameter Probability Distribution for the Four
Dimensional Ising Model on the Creutz Cellular Automaton with the Linear Dimensions
L=24, 26 and 28
Ganimet MÜLAZIMOĞLU KIZILIRMAK1
Abstract. The aim of this study is study of the finite-size scaling relation for the order-
parameter probability distribution of the four dimensional İsing model with Creutz Cellular
Automaton. The four-dimensional İsing model is simulated on the Creutz Cellular Automaton
with the linear dimensions L=24, 26 and 28. The order parameter probability distribution are
calculated at the critical temperature values of infinitive lattice. The finite size scaling relation
for the order parameter probability distribution is tested and verified numerically by the
Creutz Cellular Automaton simulation. The constants of the analytical function are estimated
by fitting it to probability function obtained numerically at the finite size critical point [1-4].
Keywords. Ising Model, cellular automaton, critical temperature, order parameter,
magnetic susceptibility,
Acknowledgement. This work was supported by the Ahi Evran University Scientific
Research Projects Coordination Unit. Project Number: FEF.A3.17.004
References
[1] Aktekin, N., Dört Boyutlu Isıng Modelinde Düzen Parametresi İhtimaliyet Dağılımı İçin
Sonlu Örgü Ölçekleme Bağıntısı, G.Ü. Fen Bilimleri Dergisi 17(3), 59-70, 2004.
[2] Binder, K., Finite-size scaling analysis of Ising model block distribution functions, Z.
Phys. B, 43, 119- 140, 1981.
[3] Binder, K., Critical properties from Monte Carlo coarse graining and renormalization,
Phys. Rev. Lett., 47, 693-696, 1981.
[4] Brezin, E. and Zinn-Justin, J., Finite-size effects in phase transitions, Nucl. Phys. B, 257,
867- 893, 1985.
1Ahi Evran University, Kırşehir, Turkey, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
325
Some Numerical Experiments for a Sharper Version of Hölder Inequality
Gültekin TINAZTEPE1
Abstract. In this work, the sharper version of Hölder Inequality which is given [1], [2]
and its derivation is presented. Also, to show the efficiency of given sharper version, some
new numerical experiments carried out are presented.
Keywords. Hölder inequality, convexity.
AMS 2010. 26D07.
References
[1] Tınaztepe, G., The Sharpening Hölder Inequality via Abstract Convexity, Turk. J. Math.,
40, 438-444, 2016.
[2] Tınaztepe, G., Tınaztepe, R. and Kemali, S., On Some Inequalities and Their Refinements,
4th International Eurasian Conference on Mathematical Sciences and Applications, Greece,
2015.
[3] Beckenbach, E. and Bellman, R., Inequalities, Springer-Verlag, 1961.
1 Akdeniz University, Antalya, TURKEY, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
326
Conformable variational iteration method for solving the time-fractional Fornberg-
Whitham equation
Handan Çerdik YASLAN
Abstract. In this work, conformable variational iteration method (CVIM) is applied to
the time fractional Fornberg-Whitham equation in the sense of the conformable derivative.
The exact and numerical solutions obtained by the conformable variational iteration method
are compared. Numerical results show that the approach performs extremely well in terms of
efficiency and simplicity
Keywords. Time fractional FornbergWhitham equation, Conformable derivative,
Variational iteration method.
AMS 2010. 35R11, 65K15.
References
[1] He, J. H., Variational iteration method for delay differential equations,
Commun.nonlinear Sci. Numer. Simul. 2 (4), 235236, 1997.
[2] He, J. H., Variational iteration method a kind of non-linear analytical technique: some
examples, Int. J. Nonlinear Mech. 34, 699708, 1999.
[3] Prakash, A., Kumar, M., He's Variational Iteration Method for the Solution of Nonlinear
Newell-Whitehead-Segel Equation. JAAC 6, 738-748, 2016.
[4] He, J. H., Approximate analytical solution for seepage ow with fractional derivatives in
porous media, Comput. Methods Appl. Mech. Engrg. 167 110, 5768, 1998.
[5] Odibat, Z., Momani, S., The variational iteration method: an effcient scheme for handling
fractional partial di_erential equations in fluid mechanics, Comput. Math. Appl. 58,
21992208, 2009.
[6] Yulita Molliq, R., Noorani, M.S.M., Hashim, I., Ahmad, R.R., Approximate solutions of
fractional ZakharovKuznetsov equations by VIM, J. Comput. Appl. Math. 233 (2)103108,
2009.
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
327
[7] Inc, M., The approximate and exact solutions of the space- and timefractional burgers
equations with initial conditions by variational iteration method, J. Math. Anal. Appl. 345 (1)
476484, 2008.
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
328
A Further Generalization of Fractional Operators
İ. Onur KIYMAZ1, Ayşegül ÇETİNKAYA1, Praveen AGARWAL2 and Shilpi JAIN3
Abstract. The purpose of this work is to present a systematic study of a further
generalization of fractional operators by using a confluent hypergeometric function with six
parameters. We investigate some of their properties such as integral representations, Mellin
transforms, generating functions, etc.
Keywords. Generalized beta function, fractional operators, Mellin transform.
AMS 2010. 26A33, 33B15, 33C15.
References
[1] Agarwal P., Choi J., Paris R.B., Extended Riemann-Liouville fractional derivative
operator and its applications, J. Nonlinear Sci. Appl., 8, 451-466, 2015.
[2] Baleanu D., Parmar R.K., Agarwal P., Salahshourd S., Extension of the fractional
derivative operator of the Riemann-Liouville, J. Nonlinear Sci. Appl., 10, 2914–2924, 2017.
[3] Chaudhry M.A., Qadir A., Rafique M., Zubair S.M., Extension of Euler's beta function, J.
Comput. Appl. Math., 78, 19-32, 1997.
[4] Chaudhry M.A., Qadir A., Srivastava H.M., Paris R.B., Extended hypergeometric and
confluent hypergeometric functions, Appl. Math. Comput., 159 (2), 589-602, 2004.
[5] Choi, J., Parmar, R.K., Extension of extended beta, hypergeometric and confluent
hypergeometric functions, Honam Mathematical J., 36 (2), 357-385, 2014.
[6] Kıymaz İ.O., Çetinkaya A., Agarwal P., An extension of Caputo fractional derivative
operator and its applications, J. Nonlinear Sci. Appl., 9, 3611-3621, 2016.
1 Ahi EvranUniversity, Kırşehir, TURKEY, [email protected], [email protected] 2 Anand International Collage of Engineering, Jaipur, INDIA, [email protected] 3 Poornima Collage of Engineering, Jaipur, INDİA, [email protected] Acknowladgement: This work was supported by Ahi Evran University Scientific Research Projects Unit. Project Number: FEF.D1.16.001
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
329
[7] Kilbas A.A., Srivastava H.M., Trujillo J.J., Theory and Applications of Fractional
Differential Equations, Elsevier, Amsterdam etc., 2006.
[8] Özarslan M.A., Özergin E., Some generating relations for extended hypergeometric
functions via generalized fractional derivative operator, Math. Comp. Mod., 52, 1825-1833,
2011.
[9] Podlubny, I. Fractional differential equations, Academic Pres, New York, 1999.
[10] Samko, S.G., Kilbas, A.A., Marichev, O.I., Fractional Integrals and Derivatives Theory
and Applications, Gordon and Breach, Longhorne, 1993.
[11] Srivastava H.M., Karlsson P.W., Multiple Gaussian hypergeometric series, Ellis
Horwood Limited, 1985.
[12] Srivastava H. M., Manocha H.L., A treatise on generating functions, Ellis Horwood
Limited, Chichester, 1984.
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
330
Higher-Order Multi-Point Fractional Boundary Value Problems
İsmail YASLAN
Abstract. In this paper, we concerned with existence of positive solutions for higher-
order multi-point fractional boundary value problem. We establish the criteria for the
existence of at least one and three positive solutions for higher order nonlinear multi-point
fractional boundary value problem by using the Krasnosel'skii fixed point theorem and the
Legget-Williams fixed point theorem, respectively.
Keywords. Boundary value problems, cone, fixed point theorems, positive solutions,
Riemann-Liouville fractional derivative.
AMS 2010. 34B07, 34D05, 34L20, 34K37.
References
[1] Fen, F.T., Karaca, I.Y., Ozen, O.B., Positive Solutions of Boundary Value Problems for p-
Laplacian Fractional Differential Equations, Filomat, 31, 1265-1277, 2017.
[2] Günendi, M., Yaslan, I., Positive solutions of higher-order nonlinear multi-point
fractional equations with integral boundary conditions, Fract. Calc. Appl. Anal., 19, 989-
1009, 2016.
[3] Guo, D., Lakshmikantham, V., Nonlinear Problems in Abstract Cones, Academic Press,
San Diego, 1988.
[4] Kilbas, A.A., Srivastava, H.M., Trujillo, J.J., Theory and Applications of Fractional
Differential Equations, Elsevier, Amsterdam, 2006.
[5] Legget, R.W., Williams, L.R., Multiple positive fixed points of nonlinear operators on
ordered Banach space, Indiana Univ. Math. J., 28, 673-688, 1979.
[6] Sabatier, O.P., Agrawal, J.A., Machado, T., Advances in Fractional Calculus, Springer,
Dordrecht, The Netherlands, 2007.
[7] Samko, S.G., Kilbas, A.A., Marichev, O.I., Fractional Integral and Derivatives: Theory
and Applications, Gordon and Breach, Yverdon, Switzerland, 1993.
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
331
Elliptic Biquaternion Algebra
Kahraman Esen Özen1 and Murat Tosun2
Abstract. In this study, we define the elliptic biquaternion algebra and give basic properties
of elliptic biquaternions. An elliptic biquaternion is in the form 0 1 2 3A A A A i j k which is a
linear combination of 1, , ,i j k where the four components 0 1 2, ,A A A and 3A are elliptic
numbers and 1, , ,i j k is the quaternion basis which are exactly the same in the quaternion
algebra and complexified quaternion algebra and satisfies the same multiplication rules. Also,
we express the De-Moivre’s and Euler Formulas for elliptic biquaternions by writing elliptic
biquaternions in the polar form. Furthermore, with the aid of these formulas, we find powers
and roots of any elliptic biquaternions.
Keywords.Complex quaternion, De-Moivre’s formula, Elliptical numbers.
AMS 2010. 15A06, 11R52.
References
[1] van der Waerden, B. L.. Hamilton’s discovery of quaternions, Math. Magazine, 49, 227-
234, 1976.
[2] Clifford, W. K.. Mathematical Papers, edited by R. Tucker (MacMillan Co., London),
1882.
[3] Muses C.. Applied hypernumbers: Computational concepts, Appl. Math. Comput. 3, 211-
216, 1976.
[4] Muses C.. Hypernumbers and quantum field theory with a summary of physically
applicable hypernumber aritmetics and their geometries, Appl. Math. Comput. 6, 63-94,
1980.
[5] Carmody K.. Circular and hyperbolic quaternions, octanions, and sedenions, Appl. Math.
Comput. 28, 47-72, 1988.
1 Sakarya University, Sakarya, Turkey, [email protected] 2 Sakarya University, Sakarya, Turkey, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
332
Beliefs about Mathematical Problem Solving of University Students
Kamile ŞANLI KULA1 and Ezgi ÇAĞATAY İN2
Abstract. In this study, Ahi Evran University students of different faculties and colleges
to solve mathematical problems related to the beliefs examine in terms of their gender, class,
the faculties/colleges, the felt level in mathematics, and weekly schedule. For this purpose,
"Personal Information Form" developed by the researcher and developed by Kloosterman and
Stage (1992), Hacıömeroğlu (2011) by Turkish adapted to the scale of the "Belief in Relation
to Mathematical Problems Solving" will be used.
Keywords. Mathematics, Problem Solving, University, Student.
MSC 2010. 62P25, 97C20.
This work was supported by the Scientific Research Projects Council of Ahi Evran
University, Kırşehir, Turkey under Grant FEF.A3.16.036.
References
[1] Duatepe Paksu, A., Comparing Teachers’ Beliefs about Mathematics in term of their
Branches and Gender, Hacettepe University Journal of Education, 35, 87-97, 2008.
[2] Hacıömeroğlu, G, Turkish Adaptation of Beliefs about Mathematical Problem Solving
Instrument, Dicle University Journal of Ziya Gökalp Faculty of Education, 17, 119-132, 2011.
[3] Kayan, F. ve Çakıroğlu, E., Preservice Elementary Mathematics Teachers’ Mathematical
Problem Solving Beliefs, Hacettepe University Journal of Education, 35, 218-226, 2008.
[4] Kloosterman, P. and Stage, F. K., Measuring beliefs about mathematical problem solving.
School Science and Mathematics, 92(3), 109-115, 1992.
1 Ahi Evran University, Kırşehir, TÜRKİYE, [email protected] 2 Ahi Evran University, Kırşehir, TÜRKİYE, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
333
Investigation of Structural Phase Transformation in Iridium Dioxide (IrO2) under High
Pressure
Köksal KIZILIRMAK1, Ganimet M. KIZILIRMAK2 and Hülya ÖZTÜRK3
Abstract. The structural phase transformation in iridium dioxide (IrO2) has been
investigated using first-principles calculations by the density functional theory (DFT) with the
ab initio program SIESTA under the hydrostatic pressure. We used the generalized gradient
approximation (GGA) for the exchange-correlation energy. The rutile type structure of IrO2
with space group P42/mnm transformed to CaCl2 type structure with space group Pnnm. This
phase transformation is also analyzed from the total energy and enthalpy calculations.
Keywords. Structural Phase Transformation, Density Functional Theory, Metal
Dioxides
Acknowledgement. This work was supported by the Ahi Evran University Scientific
Research Projects Coordination Unit. Project Number: FEF.A4.17.015
References
[1] Hohenberg, P.; Kohn, W., Phys. Rev B 1964, 136, 864.
[2] Parr, R. G.; Yang, W. J. Am. Chem. Soc. 1980, 106, 4049-4050.
[3] Kohn, W.; Sham, L. J. Phys. Rev. 1965, 140, A1133.
[4] Ordejon, P.; Artacho, E.; Soler, J. M. Phys. Rev. B 1996, 53, (R10) 411-415.
[5] Ceperley, D. M.; Alder, M. J. Phys. Rev. Lett. 1980, 45, 566-569.
[6] Perdew, J. P.; Wang, Y. Phys. Rev. B 1992, 45, 13244-13249.
[7] Perdew, J. P.; Burke, K.; Ernzerhof, M. Phys. Rev. Lett. 1996, 77, 3865-3868.
1Ahi Evran University, Kırşehir, TURKEY, [email protected] 2Ahi Evran University, Kırşehir, TURKEY, [email protected] 3Ahi Evran University, Kırşehir, TURKEY, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
334
[8] Troullier, N.; Martins, J. L. Phys. Rev. B 1991, 43, 1993-2006.
[9] Parrinello, M.; Rahman, A. Phys. Rev. Lett. 1980, 45, 1196.
[10] Birch, F. Phys. Rev. 1947, 71, 809.
[11] Murnaghan, F. D. Proc. Natl. Acad. Sci. U.S.A. 1944, 30, 244.
[12] Kürkçü, C. MgF2'de yüksek basınç etkisiyle yapısal faz dönüşümleri ve dönüşüm
mekanizmasının incelenmesi, Yüksek Lisans Tezi, Ahi Evran Üniversitesi, Kırşehir, 73s,
2014.
[13] Bolzan, A. A., Fong, C., Kennedy, B. J. and Howard, C. J., Acta Cryst, B53, 373-380,
1997.
[14] Szpunar, B., Szpunar, J., Journal of Physics and Chemistry of Solids, 74, 1632–1639,
2013.
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
335
Helixes on Clifford Surfaces in a Hyperbolic Space of Positive Curvature
Lyudmila ROMAKINA1
Abstract. Analogs of a Clifford surface in a hyperbolic space 3 of positive curvature
[1] are investigated. It is proved that the Clifford surfaces of the space 3 are proejectively
equivalents of an oval surface and form two types [2]. A Clifford surface has two rotation
axes which are mutually polar with respect to the absolute oval surface of the space 3 and
belong to the different types. A hyperbolic (or elliptic) Clifford surface of the space 3 can be
formed by one of two rotations. These rotations are as follows: (a) the rotation of a hyperbolic
(or, respectively, elliptic) cycle of the hyperbolic plane of positive curvature around its base;
(b) the rotation of a hypercycle of the hyperbolic plane of positive curvature (or, respectively,
a circle of the elliptic plane) around its base. The Clifford surfaces of the space 3 can be
used for the construction of orthogonal coordinate systems at the volumes calculation [3]. The
volume formulae for the orthogonal layers of the Clifford surfaces are obtained. A Clifford
surface of the space 3 contains helixes which can be represented by spirals with two poles
on rotation ellipsoids in Euclidean space. All helixes on a hyperbolic Clifford surface are
elliptic. Any helix on an elliptic Clifford surface belongs to one of three types. Depending on
the type of tangents it can be an elliptic, hyperbolic or parabolic curve.
Keywords. Hyperbolic space of positive curvature, Clifford surface, helix.
AMS 2010. 51F10, 51N25, 51N30.
References
[1] Romakina, L., Dihedrons of a hyperbolic three-space of positive curvature, International
Electronic Journal of Geometry, 9, 2, 50-58, 2016.
[2] Romakina, L., Clifford surfaces in a hyperbolic space of positive curvature, Sci. articles
collection of the 5th Int. Sci. Conf. of Eurasian Sci. Association, 5(27), 30-33, 2017.
[3] Romakina, L., The volume of a monopolar tetrahedron in a hyperbolic space of positive
curvature, Sci. articles collection of the 5th Int. Sci. Conf. of Eurasian Sci. Association, 5(27),
27-30, 2017.
1 Saratov State University, Saratov, RUSSİA, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
336
A New Generalization of Whittaker Function and its Properties
M. Baki YAĞBASAN1, Ayşegül ÇETİNKAYA1 and İ. Onur KIYMAZ1
Abstract. We introduce a new generalization of the extended Whittaker function by
using a generalization of confluent hypergeometric function of the first kind and investigate
its properties such as integral representations, integral transforms, differential formula and
recurrence relations.
Keywords. Whittaker function, generalized beta function, generalized confluent
hypergeometric function, generalized Gauss hypergeometric function.
AMS 2010. 33B15, 33C05, 33C15
References
[1] Chaudhry, M. A., Qadir, A., Rafique, M., and Zubair, S. M., Extension of Euler's beta
function, J. Comput. Appl. Math., 78, no. 1, 19-32, 1997.
[2] Chaudhry, M. A., Qadir, A., Srivastava, H. M. and Paris, R. B., Extended
hypergeometric and confluent hypergeometric functions, Appl. Math. Comput., 159 no. 2,
589-602, 2004.
[3] Choi, J., Ghayasuddin, M., Khan, N., Generalized Extended Whittaker Function and
Its Properties, Applied Mathematical Sciences, Vol. 9, 2015, no. 131, 6529-6541.
[4] Erdelyi, A., Magnus, W., Oberhettinger, F., and Tricomi, F.G., Table of Integral
Transforms, Vol. 1, MaGraw-Hill, New York, 1954.
[5] Khan, N., Ghayasuddin, M., A Note on generalized extended Whittaker function,
Honam Mathematical J. 38, No. 2, pp. 325-335, 2016.
[6] Nagar, D. K., Vasquez, R. A. M. and Gupta, A. K., Properties of the extended
Whittaker function, Progr. Appl. Math., 6, no. 2, 70-80, 2013.
[7] Özergin, E., Özarslan, M. A. and Altın, A., Extension of gamma, beta and
hypergeometric functions, J. Comput. Appl. Math., 235, 4601-4610, 2011.
1 Ahi EvranUniversity, Kırşehir, Turkey, [email protected] Acknowladgement: This work was supported by Ahi Evran University Scientific Research Projects Unit. Project Number:
FEF.A4.17.002
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
337
[8] Parmar, R. K., A new generalization of gamma, beta hypergeometric and confluent
hypergeometric functions, Matematiche (Catania), LXVIII, 33-52, 2013.
[9] Rainville, E. D., Special Functions, Macmillan Company, New York, 1960,
[10] Slater, L. J., Confluent Hypergeometric Functions, Cambridge University Press,
Cambridge, London and New York, 1960.
[11] Srivastava, H. M. and Manocha, H. L., A Treatise on Generating Functions, Halsted
Press (Ellis Horwood Limited, Chichester), John Wiley and Sons, New York, Chichester,
Brisbane, and Toronto, 1984.
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
338
Classification of Homothetical Hypersurfaces and Its Applications to Production
Functions in Economics
Mahmut ERGUT 1 and Muhittin Evren AYDIN 2
Abstract. In this study, we completely classify the homothetical hypersurfaces
having null Gauss- Kronocker curvature in a Euclidean 1n space. Several applications
to the production functions in economics areal so given.
Keywords. Homothetical hypersurface, Gauss- Kronocker curvature, production
function
AMS 2010. 53B25, 91B38.
References
[1] Aydin, M. E., Ergut, M., Hessian determinants of composite functions with applications
or production functions in economics, Kragujevac J. Math. 38(2), 259–268, 2014.
[2]nAlodan, H.,Chen, B.-Y.,Deshmukh, S., Vilcu, G.E.,On some geometric properties
of quasi- produc tproduction models, arXiv:1512.05190v1.
[3] Van de Woestyne, I., Minimal homothetical hypersurfaces of a semi-Euclidian
space, Results in Mathematics 27, 333-342, 1995.
[4] Lopez, R., Moruz, M., Translation and homothetical surfaces in Euclidean space with
constant curvature, J. Korean Math. Soc. 52(3), 523---535, 2015.
[5] Saglam, D., Sabuncuoglu, A., Minimal homothetical lightlike (degenerate)
hypersurfaces of semi- Euclidean spaces, Kuwait J. Sci. Eng. 38 (1A), 1-14, 2011.
1 Namik Kemal University, Tekirdağ, TURKEY, [email protected] 2 Firat University, Elazığ, TURKEY, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
339
Generation of Pseudo-Random Numbers from Given Probabilistic Distribution with the
Use of Chaotic Maps
Marcin LAWNİK1
Abstract. The scope of the paper is the presentation of a new method of generating
numbers from a given distribution. The method uses the inverse cumulative distribution
function and a method of flattening of probabilistic distributions [1]. On the grounds of these,
a new constructions of chaotic maps were derived:
𝑥𝑘+1 = 𝐹−1(𝑈𝑛(|𝑎𝑥𝑘|)), (1)
𝑥𝑘+1 = 𝑇(𝑈𝑛(|𝑎𝑥𝑘|)), (2)
where 𝐹−1 is a inverse cumulative distribution function, 𝑇 is a given transformation, 𝑈𝑛 is the
𝑛-th iteration of the chaotic map with a uniform distribution, and 𝑎 is a normative coefficient.
The analysis of (1) and (2) was conducted on the example of maps constructed on the
basis of Box-Muller transformation and the quantile function of the exponential distribution.
The obtained results certify, that the proposed method may be successively applicable for the
construction of generators of pseudo-random numbers.
Keywords. Pseudo-random numbers, chaotic maps, standard normal distribution.
AMS 2010. 65C10, 37M25.
References
[1] Lawnik, M., Generation of numbers with the distribution close to uniform with the use of
chaotic maps, Proceedings of the 4th International Conference on Simulation and Modeling
Methodologies, Technologies and Applications, Scitepress, 451–455, 2014.
1 Silesian University of Technology, Faculty of Applied Mathematics, Gliwice, POLAND, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
340
A New Approach on Slant Curves in three Dimensional Lie Groups
Osman Zeki OKUYUCU1, Caner DEĞIRMEN2 and İsmail GÖK3
Abstract. In this paper, the alternative moving frame ℕ, ℂ, 𝕎 is obtained for a unit
speed curve given in the 3-dimensional Lie group 𝐺 and ℂ -slant helices are defined according
to this frame. In addition, some characterizations of these curves are investigated.
Keywords. General helices, Slant helices, ℂ -slant helices, Lie groups.
AMS 2010. 53A04, 22E15.
References
[1] Crouch, P., Silva, F. L., The Dynamic Interpolation Problem: On Riemannian Manifolds
Lie Groups and Symmetric Spaces, Journal of Dyn. Control Syst., 1(2), 177-202, 1995.
[2] Çiftçi, Ü., A Generalization of Lancret’s Theorem, Journal of Geometry Physics, 59,
1597-1603, 2009.
[3] Izumiya, S., Takeuchi, N., New Special Curves and Developable Surfaces, Turkish Journal
of Mathematics, 28, 153-163, 2004.
[4] Lancret, M. A., Mémoire sur les courbes à double courbure, Mémoires présentés à
I’Institut 1, 416-454, 1806.
[5] Okuyucu, O. Z., Gök, İ., Yaylı, Y., Ekmekci, N., Slant Helices in three Dimensional Lie
Groups, Appl. Math. and Comp., 221, 672-683, 2013.
[6] Uzunoğlu, B., Gök, İ., Yaylı, Y., “A New Approach on Curves of Constant Precession”,
Applied Mathematics and Computation, 275, 317-323, 2016.
1 Bilecik Şeyh Edebali University, Bilecik, TURKEY, [email protected] 2 Bilecik Şeyh Edebali University, Bilecik, TURKEY, [email protected] 3 Ankara University, Ankara, Turkey, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
341
Analysis of Antimicrobial Resistance among Clinical Isolates from Denizli
Selma KIRAÇ1, Dilek KESKIN2 and Muradiye YARAR3
Abstract. The main causes of many diseases are bacterial infections. Antibiotics are
used for the treatment of these infections. Antibiotic resistance to bacterial infections is an
important health problem. The aim of this study was to analyze the resistance status of gram
negative bacterial strains isolated from clinical specimens in Denizli.
Multiple antibiotic resistance was calculated using the mar index value = number of
antibiotics in which the strains were resistant / number of antibiotics tested. If the result is
greater than 0.2, it is evaluated as having multiple antibiotic resistance.
Multiple antibiotic resistance was detected in 88% of Proteus mirabilis strains and 59%
of Pseudomonas aeroginosa strains and %100 Acinetobacter baumanii strains. This study is
important in order to guide the correct treatment of infections caused by Gram negative
bacteria in Denizli.
Keywords. Analysis Resistance, Antibiotic, Gram negative bacteria, infection control.
References
[1] Zhao, B., Zhang, X., Mathematical analysis of multi-antibiotic resistance. International
Journal of Cardiology 219, 33–37, 2016.
[2] Chen, C.,Chen, Y., Lu, P.L., Lin, W.R., Chen,T.C., Lin,C.Y., Proteus mirabilis urinary
tract infection and bacteremia: Risk factors, clinical presentation, and outcomes. Journal of
Microbiology, Immunology and Infection 45, 228-236, 2012.
[3] M. Haber, B.R. Levin, P. Kramarz, Antibiotic control of antibiotic resistance in
hospitals:a simulation study, BMC Infect. Dis. 10 1–10, 2010.
[4] Singer, B. Mathematical Models of infectious diseases:seeking new tools for planning and
evaluating control programs. Supplement to Popul. Dev. Rev., 10:347–365, 1984.
1 Pamukkale University, Denizli, TURKEY, [email protected] 2 Adnan Menderes University Cine-Aydin, TURKEY 3 Denizli State Hospital, Denizli, TURKEY
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
342
[5] Altunsoy, A., Aypak, C., Azap, A., Ergonul, O., Balik, I., The impact of a nationwide
antibiotic restriction program on antibiotic usage and resistance against nosocomial
pathogens in Turkey. Int J Med Sci 8(4):339–344, 2011.
[6] Somily A. M., Absar M. M., Arshad M.Z., Antimicrobial susceptibility patterns of
multidrug resistant Pseudomonas aeruginosa and Acinetobacter baumannii against
carbapenems, colistin, and tigecycline. Saudi Med J.; 33(7): 750-5, 2012.
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
343
An Approach to Volterra and Fredholm Integral Equations by Homotopy Perturbation
Method
Serpil ŞAHİN 1
Abstract. In this study, an application of the homotopy perturbation method for
Volterra and Fredholm integral equations is presented. The proposed method is based on the
homotopy perturbation method, which consists in constructing the series whose sum is the
solution of the problem considered. In this method, a homotopy with an imbedding parameter
pÎ 0,1[ ] is constructed. Furthermore, these integral equations are solved by Adomian
Decomposition Method, Variational Iteration Method and Successive Approximations
Method. The effectiveness and practicality of the homotopy perturbation method is evaluated
according to analytical and numerical results.
Keywords. Integral equations, Homotopy perturbation method.
AMS 2010. 33E30
References
[1] Abbasbandy, S., Iterated He’s homotopy perturbation method for quadratic Riccati
differential equation, Applied Mathematics and Computation, 175, 581-589, 2006.
[2] Arikoğlu, A., Özkol, I., Solutions of integral and integro-differential equation systems by
using differential transform method, Computers and Mathematics with Applications, 56,
2411-2417, 2008.
1 Amasya University, Amasya, Turkey, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
344
Composition Formulae Associated Fractional Integral Operator with the Multi-Index
Mittag-Leffler Functions
Shilpi JAIN1, Praveen AGARWAL2, Ayşegül ÇETINKAYA3 and İ. Onur KIYMAZ4
Abstract. In this paper, we aim at establishing new composition formulae for the
Marichev-Saigo-Maeda (M-S-M) fractional integral operator and the multi-index Mittag-
Leffler functions. Here, we record four such new and interesting special cases of our main
results. The main results of this paper generalizes the results obtained by Choi and Agarwal
[2]. Further, we obtain Laplace transforms of these composition formulae.
Keywords. Marichev Saigo Maeda Fractional Integral Opearators, generalized multi-
index Mittag-Leffler function, generalized Wright function
AMS 2010. 26A33, 33E12, Secondary 33C60,33E20.
References
[1] Agarwal, P. and Choi, J., Fractional Calculus Operators and Their Images Formulas, J.
Korean Math. Soc., 53(5), 1183-1210, 2016.
[2] Choi, J. and Agarwal, P., A note on Fractional Integral Operator Associated with
Multiindex Mittag-Leffler Functions, Filomat, 30:7, 1931-1939, 2016.
[3] Debnath, L. and Bhatta, D., Integral Transforms and Their Applications, Chapman &
Hall/CRC, Boca Raton, London, New York, 2007.
[4] Haubold, H. J., Mathai, A. M. and . Saxena, R. K, Mittag-Leffler functions and their
applications, J. Appl. Math., 2011.
[5] Kiryakova, V., Multiple (multiindex) Mittag-Leffler functions and relations to generalized
fractional calculus, J. Comput. Appl. Math., 118, 241-259, 2000.
1 Poornima Collage of Engineering, Jaipur, INDIA, [email protected] 2 Anand International Collage of Engineering, Jaipur, INDİA, [email protected] 3 Ahi EvranUniversity, Kırşehir, TURKEY, [email protected] 4 Ahi EvranUniversity, Kırşehir, TURKEY, [email protected] Acknowledgement: This work was supported by Ahi Evran University Scientific Research Projects Unit. Project Number: FEF.D1.16.001
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
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[6] Mathai, A. M. and Saxena, R. K., The H-function with Applications in Statistics and Other
Disciplines,Halsted Press [John Wiley & Sons], New York, London, Sydney, 1978
[7] Saxena, R. K. and Nishimoto, K., N-fractional calculus of generalized Mittag- Leer
functions, J. Fract.Calc., 37, 43-52, 2010.
[8] Saxena, R. K., Pogany, T. K., Ram, J. and Daiya, J., Dirichlet averages of generalized
multi-index Mittag-Leffler functions, Armen. J. Math., 3(4), 174-187, 2010.
[9] Srivastava, H. M., Harjule, P. and Jain, R., A General Fractional Di
erential Equation Associated With an Integral Operator With the H-Function in the Kernel,
Russian Journal of Mathematical Physics, 22(1), 112-126, 2015
[10] Wright, E. M., The asymptotic expansion of the generalized hypergeometric function, J.
London Math. Soc., 10, 286-293, 1935.
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
346
On a General Subclass of Univalent Functions Based on the q-Derivative Operator
Sibel Yalçın TOKGÖZ1 and Şahsene ALTINKAYA2
Abstract. In our present investigation, we aim at investigating a general subclass of
univalent functions based on the q-derivative operator in the open unit disc
1and: zzzU C . Making use of Sigmoid function, we find estimates on the
coefficients 2a and 3a . Furthermore, upper bounds are obtained for 223 aa , where
C .
Keywords. Coefficient estimates, Sigmoid function, q-derivative operator, Fekete-
Szegö inequalities..
AMS 2010. 30C45, 33D15.
References
[1] Aydoğan, M., Kahramaner, Y., Polatoğlu, Y., Close-to-convex functions defined by
fractional operator, Applied Mathematical Sciences, 7, 56, 2769-2775, 2013.
[2] Duren, P. L., Univalent Functions, Grundlehren der Mathematischen Wissenschaften,
Springer, New York, USA, 259, 1983.
[3] Jackson, F. H., On q-functions and a certain difference operator, Transactions of the
Royal Society of Edinburgh, 46, 253-281, 1908.
[4] Fadipe-Joseph, O. A., Oladipo, A.T., Ezeafulukwe, U. A., Modified Sigmoid function in
univalent function theory, International J. of Math. Sci. & Engg. Appls. 7, 313-317, 2013.
[5] Kanas, S., Raducanu, D., Some class of analytic functions related to conic domain,
Mathematica Slovaca, 64 (5)(2014) 1183-1196.
[6] Y. Polatoğlu, Growth and distortion theorems for generalized q-starlike functions,
Advances in Mathematics: Scientific Journal, 5, 1, 7-12, 2016.
1 Uludag University, Bursa, TURKEY, [email protected] 2 Uludag University, Bursa, TURKEY, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
347
[7] Purohit, S. D., Raina, R. K., Certain subclass of analytic functions associated with
fractional q-calculus operators, Math. Scand. 109, 55-70, 2011.
[8] Robertson, M. S., On the theory of univalent functions, Ann. Math., 37, 374-408, 1936.
[9] Srivastava, H. M. Univalent functions, fractional calculus, and associated generalized
hypergeometric functions. in Univalent Functions, Fractional Calculus, and Their
Applications, (H. M. Srivastava and S. Owa, Editors), Halsted Press (Ellis Horwood Limited,
Chichester), John Wiley and Sons, New York, Chichester, Brisbane and Toronto, 1989.
[10] Seoudy, T. M., Aouf, M. K., Coefficient estimates of new classes of q-starlike and q-
convex functions of complex order, 10, 1, 135-145, 2016.
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
348
Estimates of Coefficients for a Subclass of Bi-Univalent Functions by Making Use of
Faber Polynomial Expansions
Sibel Yalçın TOKGÖZ1 and Şahsene ALTINKAYA2
Abstract. In the present investigation, we give a new subclass of bi-univalent
functions which is defined by subordinations in the open unit disk
1and: zzzU C .
Making use of Faber polynomial expansions, we determine the general coefficient bounds
na of functions belonging to this new analytic bi-univalent function class.
Keywords. Coefficient estimates, Faber polynomials, subordination.
AMS 2010. 30C45.
References
[1] Airault, H., Symmetric sums associated to the factorization of Grunsky coefficients, in
Conference, Groups and Symmetries, Montreal, Canada, April, 2007.
[2] Airault, H., Bouali, H., Differential calculus on the Faber polynomials, Bulletin des
Sciences Mathematiques, 130, 179-222, 2006.
[3] Airault, H., Ren, J., An algebra of differential operators and generating functions on the
set of univalent functions, Bulletin des Sciences Mathematiques, 126, 343-367, 2002.
[4] Altınkaya, Ş., Yalçın, S., Coefficient estimates for a subclass of analytic and bi-univalent
functions, Acta Universitatis Apulensis, 40, 347-354, 2014.
[5] Altınkaya, Ş., Yalçın, S., Faber polynomial coefficient bounds for a subclass of bi-
univalent functions, C. R. Acad. Sci. Paris, Ser. I, 353, 12, 1075-1080, 2015.
[6] Duren, P. L., Univalent functions, Grundlehren der Mathematischen Wissenschaften,
Springer, New York, USA, 259, 1983.
1 Uludag University, Bursa, TURKEY, [email protected] 2 Uludag University, Bursa, TURKEY, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
349
[7] Hamidi, G. S., Halim, S. A., Jahangiri, J. M., Faber polynomial coefficient estimates for
meromorphic bi-starlike functions, International Journal of Mathematics and Mathematical
Sciences, Article ID 498159, 4 pages, 2013.
[8] Hamidi, S. G., Jahangiri, J. M., Faber polynomial coefficient estimates for analytic bi-
close-to-convex functions, C. R. Acad. Sci. Paris, Ser. I, 352, 17-20, 2014.
[9] Lewin, M., On a coefficient problem for bi-univalent functions, Proc. Amer. Math. Soc.,
18, 63-68, 1967.
[10] Netanyahu, E., The minimal distance of the image boundary from the origin and the
second coefficient of a univalent function in 1z , Archive for Rational Mechanics and
Analysis, 32, 100-112, 1969.
[11] Salagean, G. S., Subclasses of univalent functions, in: Complex Analysis- Fifth
Romanian Finish Seminar, Bucharest, 1, 362-372, 1983.
[12] Srivastava, H. M., Mishra, A. K., Gochhayat, P., Certain subclasses of analytic and bi-
univalent functions, Applied Mathematics Letters, 23, 1188-1192, 2010.
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
350
On a Subclass of Univalent Functions Defined by Chebyshev Polynomials
Sibel Yalçın TOKGÖZ1 and Şahsene ALTINKAYA2
Abstract. Making use of Chebyshev polynomials, we get upper bound estimates for
the initial coefficients 2a , 3a and the Fekete-Szegö functional 223 aa of univalent
functions.
Keywords. Coefficient estimates, Chebyshev polynomials, Fekete-Szegö functional,
subordination.
AMS 2010. 30C45, 30C50.
References
[1] Altınkaya, Ş., Yalçın, S., On the Chebyshev polynomial bounds for classes of univalent
functions, Khayyam J. Math., 2, 1-5, 2016.
[2] Doha, E. H., The first and second kind Chebyshev coefficients of the moments of the
general-order derivative of an infinitely differentiable function, Intern. J. Comput. Math., 51,
21-35, 1994.
[3] Fekete, M., Szegö, G. Eine bemerkung über ungerade schlichte funktionen, Journal of the
London Mathematical Society, 2, 85-89, 1933.
[4] Mason, J. C., Chebyshev polynomials approximations for the L-membrane eigenvalue
problem, SIAM J. Appl. Math., 15, 172-186, 1967.
1 Uludag University, Bursa, TURKEY, [email protected] 2 Uludag University, Bursa, TURKEY, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
351
Coefficient Estimates for Analytic Bi-Bazilevi cFunctions of Order and Type
Sibel Yalçın TOKGÖZ1 and Şahsene ALTINKAYA2
Abstract. A function is said to be bi-univalent in the open unit disk
1and: zzzU C if both the function and its inverse are univalent in U . By the same
token, a function is said to be bi-Bazilevi c in U if both the function and its inverse are
Bazilevi c there. We find estimates on the Taylor-Maclaurin coefficients 2a , 3a and by
using the Faber polynomials, we obtain general coefficient bound na for functions in the
class ),( B . Furthermore, we get the upper bound for the functional 23422 )1( aaaH .
Keywords. Coefficient estimates, Hankel determinant, bi-Bazilevi c functions, Faber
polynomials.
AMS 2010. 30C45.
References
[1] Airault, H., Remarks on Faber polynomials, Int. Math. Forum, 3, 449-456 , 2008.
[2] Airault, H. and Ren, J., An algebra of differential operators and generating functions on
the set of univalent functions, Bulletin des Sciences Mathematiques, 126, 343-367, 2002.
[3] Altınkaya Ş., Yalçın S., Upper Bound of Second Hankel Determinant for Bi-Bazilevic
Functions, Mediterranean Journal of Mathematics (MJOM), 13, 4081–4090, 2016.
[4] Altınkaya Ş., Yalçın S., On the Faber polynomial coefficient bounds of bi-Bazilevic
functions, Communications Faculty of Sciences University of Ankara Series A1: Mathematics
and Statistics, 66, 289-296, 2017.
[5] Bazilevic, I.E., On a case of integrability in quadratures of the Loewner-Kufarev
equation, Matematicheskii Sbornik, 37, 471–476, 1955.
1 Uludag University, Bursa, TURKEY, [email protected] 2 Uludag University, Bursa, TURKEY, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
352
[6] Duren, P.L., Univalent Functions, Grundlehren der Mathematischen Wissenschaften,
Springer 259, New York, 1983.
[7] Noonan, J.W., Thomas, D.K., On the second Hankel determinant of areally mean p-valent
functions, Trans. Am. Math. Soc. 223, 337–346, 1976.
[8] Pommerenke, C., Univalent Functions, Vandenhoeck & Ruprecht, Göttingen, 1975.
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
353
Singularities of the Darboux Ruled Surface of a Space Curve in the Pseudo-Galilean
Space
Tevfik ŞAHİN1
Abstract. In this article, we establish the singularity theory in a pseudo-Galilean space
G
1
3, a special case of Cayley- Klein spaces. We consider the cases where the Darboux ruled
surface in G
1
3 is diffeomorphic to some surfaces in the neighbourhood of a singular point. In
addition, we investigate the relationship between singularities of discriminant, bifurcation sets
of the function, and geometric invariants of curves in G
1
3.
Keywords. Height function, singularities, Darboux ruled surface, pseudo-Galilean
space.
AMS 2010. 58K05, 53A35.
References
[1] Mond, D., Singularities of the tangent developable surface of a space curve, The
Quarterly Journal of Mathematics, 40, 79-91, 1989.
[2] Sahin, T., Yılmaz, M., On singularities of the Galilean spherical darboux ruled surface of
a space curve in G
3, Ukranian Mathematical Journal, 62(10), 1597- 1610, 2010.
[3] Sahin, T., Yılmaz, M., The rectifying developable and the tangent indicatrix of a curve in
Galilean 3-space, Acta Mathematica Hungarica, 132(1-2), 154-167, 2011.
[4] Izumiya, S., Katsumi, H., Yamasaki, T., The Rectifying developable and the spherical
darboux image of a space curve, Geometry and Topology of Caustics '98 - Banach Center
Publications., 50, 137-149,1999.
1 Amasya University, Amasya, Turkey, [email protected], [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
354
On the Hyperbolic Spinors and Split Quaternions
Tülay ERİŞİR1 and Mehmet Ali GÜNGÖR2
Abstract. In [1], the relationships between quaternions and spinors with complex
components and the kinematics of quaternion and spinor were given by J. Kronsbein. In
addition, Vivarelli offered a new approach to quaternions and spinors in the Euclidean 3-
space deriving from the vector formulation of the Euler’s theorem on the general
displacement of a rigid body with a fixed point in [2].
In this study, considering the studies mentioned above, firstly, we have introduced
hyperbolic spinors with two hyperbolic components and split quaternions. Then, we have
given the hyperbolic spinor representation of the rotations can be expressed with split
quaternions in Minkowski space. Finally, we have showed the hyperbolic spinor model of the
some characterizations of the rotations with the aid of split quaternions.
Keywords. Split quaternions, Hyperbolic spinors.
AMS 2010. 15A66, 51B20.
References
[1] Vivarelli, M. D., Development of spinors descriptions of rotational mechanics from
Euler’s rigid body displacement theorem, Celes. Mech. 32, 193-207, 1984
[2] Hacısalihoğlu, H. H., Hareket geometrisi ve kuaterniyonlar teorisi, Gazi Üniversitesi,
Fen-Edebiyat Fakultesi Yayinlari, Ankara, 2, 1983.
[3] Cartan, É., The theory of Spinors, The M.I.T. Press, Cambridge, 1966.
[4] Erisir, T., Gungor, M. A. and Tosun, M., Geometry of Hyperbolic Spinors Corresponding
to Alternative Frame, Adv. in Appl. Clifford Algebr., 25, 4, 799-810, 2015.
1 Erzincan University, Erzincan, TURKEY, [email protected] 2 Sakarya University, Sakarya, TURKEY, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
355
An Introduction to Fibonaci, Lucas and Generalized Fibonacci Commutative
Quaternions
Ömer Tetik1 and The Mahmut Akyiğit2
Abstract. In this article, we will introduce the commutative Fibonacci quaternion, the
commutative Lucas quaternion and the commutative generalized Fibonacci quaternion. We
procure the relations between the commutative Fibonacci quaternion, the commutative Lucas
quaternion and the commutative generalized Fibonacci quaternion. To find these relations we
used the well-known identities related to the Fibonacci and Lucas numbers. Furthermore we
give Binet formulas and Cassini identities for commutative quaternions.
Keywords. Keyword one, keyword two, keyword three.
AMS 2010. 53A40, 20M15.
References
[1] Horadam, A. F., A Generalized Fibonacci Sequence, Amer. Math. Monthly 68, 455-459,
1961.
[2] Catoni, F., Cannata, R. and Zampetti, P., An Introduction to Commutative Quaternions,
Adv. Appl.Clifford Algebras 16, 1-28, 2006.
[3] Kosal, H. H., Akyigit, M. and Tosun, M., Consimilarity of Commutative Quaternion
Matrices, Miskolc Math. Notes 16 (2015), 965-977.
[4] Akyiğit, M., Kosal, H. H. and Tosun, M., Fibonacci Generalized Quaternions. Adv.
Appl. Clifford Algebras 24, 631–641, 2014.
[5] Dunlap, R. A., The Golden Ratio and Fibonacci Numbers, World Scientific, 1997.
[6] Vajda, S., Fibonacci and Lucas Numbers and the Golden Section, Ellis Horwood Limited
Publ., England, 1989.
1 Sakarya university, Sakarya, Turkey, [email protected] 2 Sakarya university, Sakarya, Turkey, [email protected]
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
356
PARTICIPANTS
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
357
List of Participants of IECMSA-2017 Title Name-Surname University
Prof. Dr. Ahmet Yucesan Suleyman Demirel University
Prof. Dr. A. Ranjbari University of Tabriz,
Prof. Dr. Ayhan Tutar Ondokuz Mayis University
Prof. Dr. Ayse Nese Dernek Marmara University
Prof. Dr. Cihan Ozgur Balikesir University
Prof. Dr. Erol Kilic Inonu University
Prof. Dr. Eva López Sanjuán University of Extremadura
Prof. Dr. F. Nejat Ekmekci Ankara Univeristy
Prof. Dr. Gabil Adilov Akdeniz University
Prof. Dr. Hans-Peter Kunzi University of Cape Town
Prof. Dr. H. Hilmi Hacisalihoglu
Bilecik Seyh Edebali Univeristy
Prof. Dr. Hamid Vaezi University of Tabriz,
Prof. Dr. Hector Luna Garcia Universidad Autonoma Metropolitana
Prof. Dr. Hikmet Ozarslan Erciyes University
Prof. Dr. Ilham Aliyev Akdeniz University
Prof. Dr. Ismail Yaslan Pamukkale University
Prof. Dr. István Juhász Alfred Renyi Institute of Mathematics
Prof. Dr. Jose Juan Pena Gil Universidad Autónoma Metropolitana Azc.
Prof. Dr. Jun Kawabe Shinshu University
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
358
Prof. Dr. Kamile Sanli Kula Ahi Evran University
Prof. Dr. Karoly Bezdek University of Calgary
Prof. Dr. Kazim Ilarslan Kirikkale University
Prof. Dr. Kenjiro Yanagi Josai University
Prof. Dr. Keziban Orbay Amasya University
Prof. Dr. Léandre Rémi Université de Bourgogne-Franche-Comté
Prof. Dr. Levent Kula Ahi Evran University
Prof. Dr. Ljubisa Kocinac University of Nis
Prof. Dr. Lyudmila Romakina Saratov State University
Prof. Dr. Mahir Rasulov Beykent University
Prof. Dr. Mahmut Ergut Namik Kemal University
Prof. Dr. Manaf Manafli Adiyaman University
Prof. Dr. Mehmet Ali Gungor Sakarya University
Prof. Dr. Mehmet Gurdal Suleyman Demirel University
Prof. Dr. Melike Bildirici Yildiz Technical University
Prof. Dr. Metin Orbay Amasya University
Prof. Dr. Mikail Et Firat University
Prof. Dr. Mohammad Mursaleen
Aligarh Muslim University
Prof. Dr. Murat Altun Uludag University
Prof. Dr. Murat Tosun Sakarya University
Prof. Dr. Mustafa Caliskan Gazi University
Prof. Dr. Mustafa Kemal Yildiz Afyon Kocatepe University
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
359
Prof. Dr. Nazmiye Yahnioglu Yildiz Technical University
Prof. Dr. Necdet Batir Nevsehir HBV University
Prof. Dr. Nezihe Caliskan Gazi University
Prof. Dr. Nihal Yilmaz Ozgur Balikesir University
Prof. Dr. Perihan Dinc Artut Cukurova University
Prof. Dr. Praveen Agarwal Anand ICE
Prof. Dr. Sadik Keles Inonu University
Prof. Dr. Salih Aytar Suleyman Demirel University
Prof. Dr. Salih Rizgar University of Raparin
Prof. Dr. Sang Eon Han Chonbuk National University
Prof. Dr. Shigeru Furuichi Nihon University
Prof. Dr. Shilpi Jain Poornima College of Engineering
Prof. Dr. Sibel Yalcin Tokgoz Uludag University
Prof. Dr. Szymon Dolecki Burgundy University
Prof. Dr. Veli Kurt Akdeniz University
Prof. Dr. Victor Martinez-Luaces
Fing Udelar
Prof. Dr. Xiangfan Piao Hannam University
Prof. Dr. Yuri Melnikov Middle Tennessee State University
Assoc. Prof. Dr. Alp Arslan Kirac Pamukkale University
Assoc. Prof. Dr. Anna Malinova University of Plovdiv Paisii Hilendarski
Assoc. Prof. Dr. Antonio Miguel Márquez Durán
Pablo de Olavide University
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
360
Assoc. Prof. Dr. Ayse Sandikci Ondokuz Mayis University
Assoc. Prof. Dr. Aysegul Cetinkaya Ahi Evran University
Assoc. Prof. Dr. Aysegul Saglam Arslan
Karadeniz Technical University
Assoc. Prof. Dr. Ayten Pinar Bal Cukurova University
Assoc. Prof. Dr. Bahaddin Sinsoysal Beykent University
Assoc. Prof. Dr. Bugra Saracoglu Selcuk University
Assoc. Prof. Dr. Bulent Yilmaz Marmara University
Assoc. Prof. Dr. Cesim Temel Yuzuncu Yil University
Assoc. Prof. Dr. Cristina Flaut Ovidius University, Constanta
Assoc. Prof. Dr. Emrah Evren Kara Duzce University
Assoc. Prof. Dr. Fazil Kayikci Yildiz Technical University
Assoc. Prof. Dr. Figen Oke Trakya University
Assoc. Prof. Dr. Filiz Tascan Guney Eskisehir Osmangazi University
Assoc. Prof. Dr. Filiz Yildiz Hacettepe University
Assoc. Prof. Dr. Gul Karadeniz Gozeri Istanbul University
Assoc. Prof. Dr. Gullazata Dairbayeva Al-Farabi Kazakh National University
Assoc. Prof. Dr. Gultekin Tinaztepe Akdeniz University
Assoc. Prof. Dr. Gunay Ozturk Kocaeli University
Assoc. Prof. Dr. H. Terence Liu Tatung University
Assoc. Prof. Dr. Handan Cerdik Yaslan
Pamukkale University
Assoc. Prof. Dr. Hatice Gul Ince Ilarslan
Gazi University
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
361
Assoc. Prof. Dr. Hifsi Altinok Firat University
Assoc. Prof. Dr. Hulya Bagdatli Yilmaz
Marmara University
Assoc. Prof. Dr. I. Onur Kiymaz Ahi Evran University
Assoc. Prof. Dr. Luca Guerrini Polytechnic University of Marche
Assoc. Prof. Dr. Mahmut Akyigit Sakarya University
Assoc. Prof. Dr. Memet Kule Kilis 7 Aralik University
Assoc. Prof. Dr. Messaoud Boulbrachene
Sultan Qaboos University
Assoc. Prof. Dr. Murat Kirisci Istanbul University
Assoc. Prof. Dr. Mustafa Ozdemir Akdeniz University
Assoc. Prof. Dr. Naceri Mostepha Oran School of Economics
Assoc. Prof. Dr. Nural Yuksel Erciyes University
Assoc. Prof. Dr. Oznur Ozkan Kilic Baskent University
Assoc. Prof. Dr. Rahmet Savas Istanbul Medeniyet University
Assoc. Prof. Dr. Resat Kosker Yildiz Technical Univesity
Assoc. Prof. Dr. Resat Yilmazer Firat University
Assoc. Prof. Dr. Selahattin Arslan Karadeniz Technical University
Assoc. Prof. Dr. Sergey Borodachev Ural Federal University
Assoc. Prof. Dr. Serpil Halici Pamukkale University
Assoc. Prof. Dr. Soley Ersoy Sakarya University
Assoc. Prof. Dr. Tulay Kesemen Karadeniz Technical University
Assoc. Prof. Dr. Violeta Vasilevska Utah Valley University
Assist. Prof. Dr. Abdulkadir Karakas Siirt University
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362
Assist. Prof. Dr. Afsin Kursat Gazanfer
Bulent Ecevit University
Assist. Prof. Dr. Ali Bashan Bulent Ecevit University
Assist. Prof. Dr. Ali Hikmet Deger Karadeniz Technical University
Assist. Prof. Dr. Arzu Akgul Kocaeli University
Assist. Prof. Dr. Arzu Guleroglu Trakya University
Assist. Prof. Dr. Arzu Unal Ankara University
Assist. Prof. Dr. Banu Gunturk Baskent University
Assist. Prof. Dr. Bilge Inan Kilis 7 Aralik University
Assist. Prof. Dr. Bozidar Popovic University of Montenegro
Assist. Prof. Dr. Bulent Altunkaya Ahi Evran University
Assist. Prof. Dr. Cahit Aytekin Ahi Evran University
Assist. Prof. Dr. Cigdem Topcu Guloksuz
Bartin University
Assist. Prof. Dr. Erdal Bayram Namik Kemal University
Assist. Prof. Dr. Evrim Guven Kocaeli University
Assist. Prof. Dr. Ganimet Mulazimoglu Kizilirmak
Ahi Evran University
Assist. Prof. Dr. Gozde Ozkan Tukel Suleyman Demirel University
Assist. Prof. Dr. Gulistan Kaya Gok Hakkari University
Assist. Prof. Dr. Gulseli Burak Pamukkale University
Assist. Prof. Dr. Hakan Simsek Antalya International University
Assist. Prof. Dr. Hidayet Huda Kosal Sakarya University
Assist. Prof. Dr. Idris Oren Karadeniz Technical University
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
363
Assist. Prof. Dr. Ilknur Koca Mehmet Akif Ersoy University
Assist. Prof. Dr. Lucjan Sapa AGH University of Science and Technology
Assist. Prof. Dr. Luiz Guerreiro Lopes University of Madeira
Assist. Prof. Dr. Mehmet Baki Yagbasan
Ahi Evran University
Assist. Prof. Dr. Mehmet Eyup Kiris Afyon Kocatepe University
Assist. Prof. Dr. Mehmet Gulbahar Siirt University
Assist. Prof. Dr. Merve Avci Ardic Adiyaman University
Assist. Prof. Dr. Mevlut Ersoy Suleyman Demirel University
Assist. Prof. Dr. Muge Meyvaci Mimar Sinan Fine Arts University
Assist. Prof. Dr. Murat Demirer Uskudar University
Assist. Prof. Dr. Mustafa Yildiz Bulent Ecevit University
Assist. Prof. Dr. Neriman Akdam Selcuk University
Assist. Prof. Dr. Nuray Caliskan Dedeoglu
Sakarya University
Assist. Prof. Dr. Nuri Celik Bartin University
Assist. Prof. Dr. Omer Unsal Eskisehir Osmangazi University
Assist. Prof. Dr. Onder Gokmen Yildiz Bilecik Seyh Edebali University
Assist. Prof. Dr. Osman Palanci Suleyman Demirel University
Assist. Prof. Dr. Osman Zeki Okuyucu Bilecik Seyh Edebali University
Assist. Prof. Dr. Rabia Cakan Akpinar Kafkas University
Assist. Prof. Dr. Selda Calkavur Kocaeli University
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
364
Assist. Prof. Dr. Serife Yilmaz Irtem Mehmet Akif Ersoy University
Assist. Prof. Dr. Serpil Sahin Amasya University
Assist. Prof. Dr. Seyda Kilicoglu Baskent University
Assist. Prof. Dr. Sukran Konca Bitlis Eren University
Assist. Prof. Dr. Tevfik Sahin Amasya University
Assist. Prof. Dr. Tulay Erisir Erzincan University
Assist. Prof. Dr. Yasemen Ucan Yildiz Technical University
Assist. Prof. Dr. Yasemin Kiymaz Ahi Evran University
Lecturer Dr. Meryem Odabasi Ege University
Lecturer Dr. Seda Karateke Istanbul Arel University
Lecturer Dr. Serdar Soylu Giresun University
Lecturer Furkan Aydin Kahramanmaras Sutcu Imam University
Dr. Abdessamad Barbara IMB, Université de Bourgogne
Dr. Anna Poskrobko Bialystok University of Technology
Dr. Anna Poskrobko Bialystok University of Technology
Dr. Boban Karapetrovic University of Belgrade
Dr. Burak Kurt Akdeniz University
Dr. Dojin Kim Kyungpook National University
Dr. Evgeny Gershikov Braude Academic College
Dr. Fatma Coban Yildiz Technical Univesity
Dr. Hosook Kim Korea Science Academy of KAIST
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
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Dr. Huseyin Oguz Dumlupinar University
Dr. Ivana Kuzmanovic University of Osijek
Dr. Jasbir Singh Manhas Sultan Qaboos University
Dr. Jelena Stanojevic University of Belgrade
Dr. Jelena Stanojevic University of Belgrade
Dr. Jeongho Park UNIST
Dr. Kismet Kasapoglu Trakya University
Dr. Marcin Lawnik Silesian University of Technology
Dr. Marija Miloloza Pandur
University of Osijek
Dr. Onder Sener Ondokuz Mayis University
Dr. Pranas Katauskis Vilnius University
Dr. Selma Kirac Pamukkale University
Dr. Snezana S. Djordjevic University of Nis
Rsc. Assist. Dr. Fulya Ozaksoy Dogus University
Rsc. Assist. Dr. Mehmet Alper Ardic Adiyaman University
Rsc. Assist. Dr. Mustafa Ozkan Trakya University
Rsc. Assist. Dr. Omer Faruk Dogan Namık Kemal University
Rsc. Assist. Dr. Ozlem Ersoy Hepson Eskisehir Osmangazi University
Rsc. Assist. Dr. Tuba Gulsen Firat University
Rsc. Assist. Dr. Zehra Guzel Ergul Ahi Evran University
Rsc. Assist. Adnan Karatas Pamukkale University
Rsc. Assist. Bagdagul Kartal Erciyes University
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
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Rsc. Assist. Deniz Pinar Sunaoglu Ankara University
Rsc. Assist. Dilek Varol Bayram Pamukkale University
Rsc. Assist. Ebru Aydogdu Kocaeli University
Rsc. Assist. Emrah Gurlek Ahi Evran University
Rsc. Assist. Erkan Agyuz Gaziantep University
Rsc. Assist. Esra Sengun Ermeydan
Ankara Yildirim Beyazit University
Rsc. Assist. Fadime Gokce Pamukkale University
Rsc. Assist. Fatih Aylikci Yildiz Technical University
Rsc. Assist. Fatma Ates Ankara University
Rsc. Assist. Fatma Bulut Bitlis Eren University
Rsc. Assist. Furkan Semih Dundar
Bogazici University
Rsc. Assist. Gulsah Aydin Sekerci Suleyman Demirel University
Rsc. Assist. Hilal Doganay Kati Bursa Technical University
Rsc. Assist. Kemal Taskopru Bilecik Seyh Edebali University
Rsc. Assist. Kubra Ozlu Deger Karadeniz Technical University
Rsc. Assist. Melek Sofyalioglu Gazi University
Rsc. Assist. Merve Ilkhan Duzce University
Rsc. Assist. Muhammet Sahal Yildiz Technical Univesity
Rsc. Assist. Sahsene Altinkaya Uludag University
Rsc. Assist. Esra Selcen Yakici Topbas
Gazi University
Rsc. Assist. Sibel Koparal Kocaeli University
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
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Rsc. Assist. Sibel Sevinc Cumhuriyet University
Rsc. Assist. Tugba Baran Kaya Kirikkale University
Rsc. Assist. Zeliha Bedir Cumhuriyet University
Student Aditya Sivakumar Beaverton High School
Student Aysun Tok Onarcan Eskisehir Osmangazi University
Student Caglar Zeki Odabasi Erciyes University
Student Cemile Duygu Colak Kocaeli University
Student Gordeeva Irina Viktorovna
Perm National Research Polytechnic University
Student Ivan Tarasyuk Belarusian State Univrsity
Student Kemal Eren Sakarya Univerity
Student Koksal Kizilirmak Ahi Evran University
Student Miranda Gabelaia Iv. Javakhishvili Tbilisi State University
Student Mualla Birgul Huban Suleyman Demirel University
Student Roksana Brodnicka University of Silesia in Katowice, Poland
Student Veysel Kivanc Karakas
Ahi Evran University
Student Yonghyeon Jeon Kyungpook National University
Student Zubeyde Er Ministry of Education of Turkey
Other Hasan Bayram Uludag University
6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2017)
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