Editor - Arc Journals · 2015-01-25 · Dr. Md. Rafiqul Islam Professor Department of Population...

Editor–in-Chief

Dr.K.V.L.N.Acharyulu

Unique World Records Holder,

Associate Professor,

Department of Mathematics,

Bapatla Engineering College,

Bapatla, India.

Editorial Board

Dr.N.Ch.Pattabhi Ramacharyulu Professor (Retd.) of Mathematics,

Department of Mathematics, National Institute of Technology,

Warangal, A.P., India.

Dr.Norio Yoshida Professor of Mathematics, Department of

Mathematics, University of

Toyama,Toyama, Japan.

Dr.V.P.Saxena (Ex VC Jiwaji University, Gwalior),

Director, Sagar Institute of Research,

Technology & Science, Ayodhya Bypass Road, Bhopal, India.

Dr.S.G.Ahmed Sayed Ahmed Professor of Engineering Mathematics, Department of Engineering Mathematics,

Zagazig University, Zagazig, Egypt.

Dr.G.Sarojamma Ex.Vice-Chancellor, Professor of

Mathematics,

Department of Mathematics, S.P. Mahila

Viswa Vidyalaya, Tirupati, India.

Dr.Sergei Silvestrov Professor of Mathematics & Applied

Mathematics, The School of Education, Culture and Communication (UKK),

Mälardalen University, Sweden.

Dr.N.Phani Kumar Professor & Head, Dept.of Mathematics, Vignan Institute of echnology&Science,

Hyderabad, India.

Dr.Md.Rafiqul Islam Professor and Ex-Chairman, Department of

Population Science and Human Resource

Development, University of Rajshahi, Bangladesh.

Dr.G.Chakradhara Rao Professor & BOS, Department of

Mathematics, Andhra University, Visakhapatnam, India.

Dr.Mircea I. Cirnu Professor of Mathematics, Department of

Applied Mathematics, University Politehnica, Bucharest, Romania.

Dr.M.A.Srinivas Professor& Head, Dept. of Mathematics, JNTU, Hyderabad, India.

Dr.Huseyin Bor Prof. of Mathematics, P.O. Box 121, Bahçelievler, Ankara, Turkey.

Dr.S.Vijayakumar Varma Professor of Mathematics, Dept. of

Mathematics, S.V.U College of Science, S.V. University, Tirupati, India.

Dr.Martin Bohner Professor of Mathematics, Department of Mathematics & Statistics, Missouri

University of Science and Technology,

Rolla, Missouri, USA.

Dr.J.Venkateswara Rao Professor of Mathematics, Mekelle

University Main Campus, Mekelle, Ethiopia.

Dr.Sidi El Vally Mohameden Department of Mathematics, King Khalid

University, Kingdom of Saudi Arabia.

Dr.J.Baskar Babujee Associate Professor, Department of

Mathematics, Anna University,

Chennai, India.

Dr.Khalil Ezzinbi Professor of Mathematics, Department of

Mathematics, Cadi Ayyad University,

Morocco.

Dr.Kunderu Lakshmi Narayan Professor of Mathematics, SLC'S Institute of

Engineering & Technology, Piglipur, Hyderabad, India.

Dr.Piroska Csorgo Professor of Mathematics, Eszterhazy

Karoly College, Eger, Hungary.

Dr.S.Kalesha Vali Professor& Head, Dept. of Basic Science & HSS, JNTU(K) University College of

Engineering, Vizianagaram, India.

Dr.Themistocles M.Rassias Professor of Mathematics, Department of Mathematics, National Technical University

of Athens, Zografou Campus, Greece.

Dr.P.V.Srinivas Associate Professor, Department of Science

and Humanities, DVR & Dr. HS MIC

College of Technology, Kanchikacherla, India.

Dr.Francesco Zirilli Professor of Mathematics, Department of

Mathematics, Universita di Roma La Sapienza, Roma, Italy.

Dr.B.Ravindra Reddy Department of Science and Humanities, JNTUH College Of Engineering,

Nachupally, India.

Dr.Taekyun Kim Professor of Mathematics, Department of

Mathematics, Kwangwoon University,

Seoul, South Korea.

Dr.Noorbhasha Rafi Department of Mathematics, Bapatla

Engineering College, Bapatla, India.

Dr.Nicolae Adrian Secelean Associate Professor, Department of

Mathematics & Informatics, Lucian Blaga

University, Sibiu, Romania.

Dr.Muhammad Zaini Bin Ahmad Institute of Engineering Mathematics,

Universiti Malaysia Perlis, Perlis, Malaysia.

Dr.P.Radha Krishna Kishore Department of Mathematics, Arba Minch

University, Gamo Gofa, Ethiopia.

Dr.Sergio Amat Plata Department of Applied Mathematics and

Statistics, Universidad Politécnica de

Cartagena, Cartagena (Murcia), Spain.

Dr.Yongkun Li Professor of Mathematics, Department of

Mathematics, Yunnan University, Kunming,

Yunnan, P.R., China.

Dr.Jyotindra C.Prajapati Associate Professor and Head, Department

of Mathematical Sciences, Charotar University of Science and Technology,

Changa, Anand, Gujarat, Inida.

Dr.Edi Cahyono Department of Mathematics FMIPA, Universitas Haluoleo, Kampus Bumi

Tridharma Anduonohu, Kendari, Indonesia.

Dr.Praveen Agarwal Associate Professor and Head, Department of Mathematical Sciences, Anand

International College of Engineering, Jaipur,

Rajasthan, India.

Dr.Anil Kumar HOD Applied Science & Humanities, Dean

Diploma Engg.,World Institute of Technology, Sohna, Gurgaon.

Dr.Mustafa AVCI Dept. of Economics and Administrative

Sciences, Batman University, Batman, Turkey.

Dr.V.Srinivas Kumar Department of Mathematics, JNTUH College of Engineering, Kukatpally,

Hyderabad.

Dr.Vishnu Narayan Mishra Department of Applied Mathematics &

Humanities, Sardar Vallabhbhai National

Institute of Technology, Surat, Gujarat,

India.

Dr.G.Manimannan Department of Statistics, Madras Christian

College, Tambaram, Chennai, India.

Dr.Ritu Agarwal Dept. of Mathematics, Institute of

Engineering and Technology, JK

Lakshmipat University, Jaipur, India.

Dr.Tanmay Biswas Department of Mathematics, University of

Kalyani, West Bengal, India.

Dr.Ramesh Chand Department of Mathematics, Govt. College,

Dhaliara, Distt. Kangra, Himachal Pradesh, India.

Dr.S.V.S.S.N.V.G.Krishna Murthy Associate Professor, Department of Applied

Mathematics, Defence Institute of Advanced Technology, Girinagar, Pune, India.

Dr.Punam Gupta Department of Mathematics & Statistics, Dr.Hari Singh Gour Central University,

Sagar, Madhya Pradesh, India.

Contents

S.No. Title & Name of the Author(s) Page No.

1. Editorial


i

2. A Special Feature - Stalwarts of IJSIMR's Editorial Board

Prof. Dr. Md. Rafiqul Islam

ii

3. Some Elementary Problems From the Note Books of

Srinivasa Ramanujan-Part(II)

Prof. N.Ch. Pattabhi Ramacharyulu

1-15

4. Total Coloring of -Graph

Dr.S. Sudha, K. Manikandan

16-22

5. Extreme Value Modeling of Precipitation in Case Studies for

China

Manuela Ender, Tong Ma

23-36

6. On some generalized results on -

summability factors of Infinite Series

Aditya Kumar Raghuvanshi

37-43

7. The H-Integrability and the Weak Laws of Large Numbers

for Arrays

S. O. Edeki, S. A. Adeosun

44-50

8. Ideals of Almost Distributive Lattices with respect to a

Congruence

Noorbhasha Rafi

51-57

9. An Unsteady Viscous Incompressible Flow in a Porous

Medium Between two Impermeable Parallel Plates

Impulsively Stopped from Relatively Motion

Mohammed Sarfaraz Hussain, N.Ch.Pattabhi Ramacharyulu

58-67

10. Coincidence and Fixed Points of Nonself Maps using

Generalized T-Weak Commutativity

Praveen Kumar Shrivastava, Yogesh Kumar Vijaywar,

Ravindra Kumar Sonwane

68-78

11. Composite Mapping of Flip-Flop Poset of Join-Irreducible

Elements of Distributive Lattice

Namrata Kaushal, Madhu Tiwari, C. L. Parihar

79-82

12. Geometric Progression in Operations Research (PERT)-A

Special Case Study

Kanduri Venkata Lakshmi Narasimhacharyulu, I.Pothuraju

83-93

13. On Product Summability of Fourier

Series


94-99

14 Trignometric Inequations and Fuzzy Information Theory

P.K. Sharma, Nidhi Joshi,

100-107

15 Maximal Square Sum Subgraph of a Complete Graph

Reena Sebastian, K A Germina

108-115

16 Steady Flow of a Viscous Incompressible Fluid through Long

Tubes Employing a Complex Variable Technique

Bandi. Ravi

116-124

i

Editorial

Dear Readers/ Research Scholars,

Research is a river of innovative knowledge which flows continuously. New streams of modern

inventions come and join the river from time to time and strengthen it. Hence it is our bounded

duty to invite the new trends and encourage the activity of research. Keeping in view this

responsibility, IJISMR is extending its cooperation to all the manifold directions of research. I

humbly request the research scholars to utilize this opportunity.

Most of our readers and research scholars are of the idea to make IJSIMR a monthly journal. Even

though it is not an easy task to run a monthly journal, in order to accept the humble request of our

beloved readers, we are making our journal a monthly one from this issue. It is the equal

responsibility of our readers and research scholars also to make this attempt a successful one.

It gives me immense pleasure to inform you that the Best Research Article Award is given to

‘Praveen Kumar Shrivastava , Yogesh Kumar Vijaywar and Ravindra Kumar Sonwane’

for the article entitled “Coincidence and Fixed Points of Nonself Maps using Generalized T-

Weak Commutativity” in this issue.

The salient features of International Journal of Scientific and Innovative Mathematical

Research (IJSIMR) are:

Blind peer reviewed journal.

Eminent and great scholars in IJSIMR-Editorial Board from throughout the world

Prompt and rapid response to the authors according to the schedule.

Simple and standard Template.

Free English Language assistance from Academician's Research Center.

Online and print versions in two formats (e-journal and Print) with reasonable cost.

Issuing the Certificate of Publication to each Author with free of cost.

Best Research Article Award will be given. For this, the minimum number of articles in

that issue must be 10.

Kindly inform about IJSIMR to your Friends / Students/ Colleagues / Associates and Fellow

Researchers who may utilize this opportunity as a platform for their new research inventions.

With Best Regards


Editor-In-Chief

ii

A Special Feature

Stalwarts of IJSIMR's Editorial Board

Dr. Md. Rafiqul Islam

Professor

Department of Population Science

and Human Resource Development

University of Rajshahi

Rajshahi-6205, Bangladesh

Dr. Md. Rafiqul Islam was born on 30th October in 1968 at Gomastapur Upazila in Chapai

Nawabganj District, Bangladesh. He passed SSC examination securing 1st division from

Chowdala High School at Gomastapur Upazila in Chapai Nawabganj, Bangladesh. He passed

HSC examination securing 1st division from Chapai Nawabganj Government College at Chapai

Nawabganj District in Bangladesh. He obtained B. Sc. honours in Mathematics securing 1st class

1st position in 1989 and M. Sc. in Pure Mathematics securing 1

st class 2

nd position in 1991 from

Rajshahi University, Bangladesh. He received Rajshahi University award and Gold Medal for

outstanding results in B. Sc. honours. He also studied Statistics and Economics as related subjects

at honours level. He joined as Lecturer in 1998 in the Dept. of Population Science and Human

Resource Development, Rajshahi University, Bangladesh. Now, he has been serving as Professor

in this Dept. since January 2011. He was awarded Ph. D. degree in Demography in 2004. He has

been teaching and doing research in this department since 1998. He published more than one

hundred and thirty five research articles as author and co-author in more than seventy five

national and international journals. Furthermore, he published four research books as author and

co-author which were published in foreign countries. He had produced a good number of M. Sc,

M. Phil and Ph. D students. In addition to this, his name is included in many international journals

as a member of editorial board and reviewer. Moreover, his name is also included in the list of

Top Bangladesh Development Researcher in The Millennium survey surveyed by Bangladesh

Development Research Centre sponsored by BRAC University and USA.

IJSIMR feels it as a pride to have him as a member of Editorial Board and extends its hearty best

wishes to Prof. Dr. Md. Rafiqul Islam.

IJSIMR Team

Academicians’ Research Center

International Journal of Scientific and Innovative Mathematical Research (IJSIMR)

Volume 2, Issue 1, January - 2014, PP 1-15

ISSN 2347-307X (Print) & ISSN 2347-3142 (Online)

www.arcjournals.org

©ARC Page | 1

Some Elementary Problems From the Note Books of Srinivasa

Ramanujan-Part(II)

N.Ch. Pattabhi Ramacharyulu

Professor (Rtd) NIT Warangal

Warangal-506004

[email protected]

§§ Problem 13: SRMs (189) p and NBSR Vol. II p 193

Let a x( , ) = (1 + a)(1 + ax)(1 + ax2)(1 + ax

3)(1 + ax

4) . . . . and so on --------------------- ( 13.1 )

Then (i) n

a x

ax x

( , )

( , )

= (a + x)

n when x = 1 ( *.1 )

(ii) n n

x x

x x x1

( , )

(1 ) ( , )

= n! where x = 1 ( *.2 )

(iii) a x( , ) = a x

a x x

( , )

( , )

( *.3 )

Solution:

Note. S.R.denotes a x( , ) as the product of infinite number of terms of the type ( 1 + axk)

i.e., a x( , ) = k

k m

ax(1 )

( i) To establish the result (*.1)

nax x( , ) = ( 1 + axn) ( 1 + ax

n.x ) ( 1 + ax

n.x

2 ) ( 1 + ax

n.x

3 ) - - - - - - - - - - - -

= ( 1 + axn) ( 1 + ax

n+1 ) ( 1 + ax

n+2 ) ( 1 + ax

n+3 ) - - - - - - - - - - - -

= k

k

ax0

(1 )

(13.2)

By definition (1)

a x( , ) = (1 + a)(1 + ax)(1 + ax2)(1 + ax

3). . . . (1 + ax

n-1) (1 + ax

n)(1 + ax

n+1) . . . . . . .

= (1 + a)(1 + ax)(1 + ax2)(1 + ax

3) . . . . . . (1 + ax

n-1)

k

k m

ax(1 )

(13. 3)

= (1 + a)(1 + ax)(1 + ax2)(1 + ax

3) . . . . . . (1 + ax

n-1)

nax x( , )

n

a x

ax x

( , )

( , )

= (1 + a)(1 + ax)(1 + ax

2)(1 + ax

3) . . . . . . (1 + ax

n-1) (13.4)

When x = 1, R.H.S. of (4) = ( 1 + a ) ( 1 + a ) ( 1 + a ) ( 1 + a ) . . . . . . ( 1 + a ) (n times)

= ( 1 + a )n

This establishes the result ( *.1 )

(ii) To establish the result (*.2)

x x( , ) = ( 1 – x )(1 – x2)(1 – x

3)(1 – x

4) . . . . . . . . . . . . . . . . . . =

k

k

x1

(1 )

mailto:[email protected]


N.Ch.Pattabhi Ramacharyulu

International Journal of Scientific and Innovative Mathematical Research (IJSIMR) Page | 2

and nx x1

( , )

= ( 1 – xn+1

) ( 1 – xn+1

.x) ( 1 – xn+1

.x2) ( 1 – x

n+1.x

3) . . . . . =

k

k n

x1

(1 )

n n

x x

x x x1

( , )

(1 ) ( , )

=

2 3 4 1

1 2

(1 x)(1 x )(1 x )(1 x ).........(1 x )(1 )......

(1 ) (1 )(1 )...................

n n

n n n

x

x x x

=

2 3 4(1 x)(1 x )(1 x )(1 x ).........(1 x )

(1 )

n

nx

=

2 3(1 ) (1 ) (1 ) (1 )

. . ...........(1 ) (1 ) (1 ) (1 )

nx x x x

x x x x

= 1. (1 + x)(1 + x + x2)(1 + x + x

2 + x

3) . . . factors (13. 5)

The last factor on the R.H.S. of the above expression = 1+x+x2 +x

3+ . . . .+ x

n-1 ( n terms)

When x = 1, R.H.S. of (13.5) = 1.2.3.4. . . . . . . . . .n = n!

This establishes the result (*.2)

(iii) To establish the result (*.3) By definition (1), we have

a x( ,x) = a x x a x x a x2 3

(1 )(1 a .x)(1 . )(1 .x )..........

a x( , ) = a ax a ax a

3 512 32 2 2(1 )(1 ax )(1 )(1 x )(1 )(1 x )(1 ax )..........

a x ax x ax x ax x2 2 3{(1 a)(1 ax)(1 ax )........)}x{(1 )(1 )(1 )(1 )......}

a x( , ). a x( ,x)

a x

a x x

( , )

( , )

= a ax2 3(1 )(1 ax)(1 )(1 ax ).......... a x( , )

This result (*.3) is established.

--------------------------------------------------------------------- -----@@@@----------------------------------------------------------------------------

Problem 14: SRMs (1) P 105, NBSR Vol. I p 209 and Vol. II p 143

(*)

Note. ( i ) This fascinating representation of an integer as a nested square root expansion finds a

place in the collected works of S.R. (p.323)

( ii ) This was a problem proposed by S.R. in the Journal of Indian Mathematical Society (

problem No. 289, Vol.III ( ) p.90). The problem created an astonishment amongst the problem

solvers on the land. “An integer can be expressed as a nested square root assembly – on the face

of it an irrational ! How is this possible !!”

The problem was referred back by the Editorial Board of J.I.M.S. to S.R. for clarification and to

provide the solution. S.R. responded by giving the solution to the problem and also provided

examples such as following.

Solution: Let { L.H.S. of *}2 = f (x) (say) ----------------------------- ( 1 )

i.e., f ( x ) = ( x + n + a )2 ( Expanding the R.H.S. of (1) and rearranging the terms)

= ax + ( n + a )2 + x {( x + n ) + ( n + a ) }

Some Elementary Problems From the Note Books of Srinivasa Ramanujan-Part(II)


= ax + ( n + a )2 + x ------------------------- ( 2 )

This result ( 2 ) may be construed as a iterative formula for expressing f ( x + r )

∴ {L.H.S. of *}= = using ( 2 )

=

= ax n a x a x n n a x n a x n n a x n f x n2 2 2( ) ( ) ( ) ( ) ( 2 ) ( ) ( 2 ) ( 3 )

=

+

. . . . . . . . . and so on. R.H.S. of *.

This establishes the result *

Deductions :

( i ) When x = 1, n = 1, a = 1, we get

3=

and so on

( ii ) When x = 2, n = 1, a = 1, we get

4 = . . . .

and so on

( iii ) When a = n = x we get

3x =

. . . . . . and so on.



= and so on and many

more such results.

( iv) When x = 3, n = 1, a = 1, we get

2 2 2 2 25 (3 2 ) 3. (4 2 ) 4 (5 2 ) 5 (6 2 ) 6(7 2 ) .......... and so on.

--------------------------------------------------------------------- -----@@@@----------------------------------------------------------------------------

Problem 15 NBSR Vol. p.

§ 3 = (*)

Solution : 3 =

= 9 = 1 + 8 = 1 + 2x4 =

= 16 = 1 + 15 = 1 + 3x5 =

= 25 = 1 + 24 = 1 + 4x6 =

= 36 = 1 + 35 = 1 + 5x7 =

= 49 = 1 + 48 = 1 + 6x8 =

. . . . . . . . . . and so on.

= Result of * NOTE : (1) Observe the occurrences of square numbers at the right end of each step.

(2) This example appeared as a problem in the William Lower Putnam Competition in the year

1966 and also in many talent tests conducted at National and International levels during the last

four decades.

(3) The representation * of 3 as nested square roots is not unique. The representation changes with

the partitioning of 9 (= 32

) as illustrated here under. As a matter of fact, this non – uniqueness of

the nested root representation is true for any number.



ILLUSTRATION: I.

3 = ( ∵ 9 = 6 + 3 = 6 + )

=

=

= and so on.

ILLUSTRATION : II.

3 =

= = 9 = 3 + 6 = 3 + 2

= . . . .. . . . . . . . . . . . . . . . . . . . and so on.

ILLUSTRATION : III.

3 =

= 9 = 5 + 4 = 5 +2

= = 4 = 2 +

= . . . . .. . . . . . . . . . . . . . . . . and so on.

--------------------------------------------------------------------- -----@@@@----------------------------------------------------------------------------

Problem 16:

§ 4 = (*)

Solution: 4 = = = ( ∵ 16 = 6 + 10 = 6 +2 and 25 = 7 + 18 =7+

)



= 36 = 8 + 28 =8 +

= 49 = 9 + 40 =

= 64 =10 + 54 =

= 81 =11 + 70 =

= . . . . . . . and so on.

--------------------------------------------------------------------- -----@@@@----------------------------------------------------------------------------

Problem 17 : PROBLEM No. 289 suggested by S.R. to JIMS Vol. III p 30

§ n( n + 2 ) = n and

so on. *

Solution:

Let f(n) = n(n+2)

= n (n+2)2 = = 1 + (n+1)(n+2) = 1 + f(n+1)

= n

This relation can be taken as recurrence formula for f(n)

∴ f(n) = n

= n and so on.

Hence



n( n+2) = n and so on.

Special Cases :

(i) When n = 1 , L.H.S. * = 3, then

3 =

(ii) When n = 2, L.H.S. * = 8, then

8 =

∴ 4 =

--------------------------------------------------------------------- -----@@@@----------------------------------------------------------------------------

Problem 18 : NBSR

§ n( n + 3 ) = n and so on. *

Solution:

Let f(n) = n(n+3)

= n { ∵ (n+3)2 = (n+5) + (n+1)(n+4) = n+5 + f(n+1)}

= n

a recurrence formula for f(n) = n(n+3)

∴ n(n+3) =

= n

= n and so on.

= n and so on.

This establish the result *



Corollary: Cancelling the common factor n on both the sides of * , we have

(n+3) =

Deductions: (i) When n = 1, we get

4 = and so on.

(ii) When n =2, we get

5 = and so on.

(iii) When n = 3, we get

6 = and so on.

--------------------------------------------------------------------- -----@@@@----------------------------------------------------------------------------

Problem 19 : NBSR Vol.II P.143

§ 2 cosθ = = = *

Proof :

2 cosθ = [∵4cos2θ = 2.2cos

2θ = 2(1+ cos2θ) = 2 + 2cos2θ ]

=

= [∵4cos2 2θ = 2 + 2cos4θ ]

=

= [∵4cos2 4θ = 2 + 2cos8θ ]

= This establishes the results shown by *

NOTE:

(i) Repeating the above process n times, we get

n2cos 2 2 2 ........... 2 2cos(2 )

(ii) Repeating the process indefinitely, we get

2cos 2 2 2 2 2 2 ............ and so on

(iii) When θ = 0, we get



2 2 2 2 2 2 2 ............ and so on

--------------------------------------------------------------------- -----@@@@----------------------------------------------------------------------------

Problem 20 NBSR Vol.II P.143

§ 2 cosθ = 3 3 3 32cos3 3 2cos3 3 2cos3 3 2cos3 ...............

(*.1)

=3 3 3 36cos 6cos3 6cos9 6cos 27 ............... and so on . (*.2)

Proof:

To establish (*.1) :

2 cosθ = = (∵ 4cos3θ = cos3 θ + 3cos θ )

= . . . . . . . . . . . . . . . . . . . . . . ( 1 )

=

=

=3 3 3 32cos3 3 2cos3 3 8cos ( ∵ 8cos

3θ = 2[cos3 θ +3cos θ ] )

= 3 3 3 3 32cos3 3 2cos3 3 2cos3 3 8cos 3 ( ∵ 8cos

3θ = 6cos θ

+3 38cos 3 )

= 3 3 3 32cos3 3 2cos3 3 2cos3 3 2cos3 ............... and so on.

Note :

When θ = 0, we get 3 3 3 3 32 2 3 2 3 2 3 2 3 2 ............

--------------------------------------------------------------------- -----@@@@----------------------------------------------------------------------------

§ Problem 21 : Not from NBSR

2sinθ = [ ∵ 4sin2θ = 2.2cos

2θ = 2(1- cos2θ) = 2 - 2cos2θ]

=

=

=



=

=

=

=

= . . . . . . . . . . . . . . . . . . . . . .

n2 2 2 .............. 2 2cos(2 ) = ( n – nested square roots )

--------------------------------------------------------------------- -----@@@@----------------------------------------------------------------------------

Problem 22: NBSR Vol. II p.305

§ 2 2 2 2 2 2 2 2............ = = 2 sin 100

Proof:

= . . . . . . . . . . . .

( 1 )

=

= ( 2sin2θ = 1 – cos 2θ )

=

= ( 2cos2θ = 1 + cos 2θ )

=

= (cos = sin = sin )



= . . . . . . . . . . . . . . ( 2 )

The last term sin in (2) is the L.H.S. of ( 1 )

By repeating the steps from (1) to ( 2 ) , we obtain

= 2 2 2 2 2 2 2 2............

NOTE: The distribution of the sequence of the signs: - + + - + + - is to be noted. --------------------------------------------------------------------- -----@@@@----------------------------------------------------------------------------

Problem 23: NBSR Vol. II p 305

§ -1= -1 = 8 8 8 8 8 8 8 .. . . . and so on- - *

Proof:

-1 = 2

02 3 cos10 1

= 2 0 0

12cos 10 4 3 cos10 1

= 0 06 1 cos 20 4 3 cos10 1 [2cos

2θ = 1 + cos 2θ ]

= 0 0 0

7 4 3 cos 30 cos 20 4 3 cos10

[6= 2.3=2 = 2 (2cos300)]

= 0 0 07 2 3 2cos 30 cos 20 4 3 cos10

= 0 0 07 2 3 cos50 cos10 4 3 cos10

= 0 0

7 2 3 cos 50 2 3 cos10 [2cosAcosB= cos(A+B)+cos(A-B)]

= 0 0

7 2 3(cos50 )cos10

= 0 0

7 4 3 sin 30 sin 20

= 0

7 2 3 sin 20 0 1sin 30

2

= 0

8 (1 2 3 s )in 20



= 2

08 1 2 3 sin 20

= 0 2 0

8 1 4 3 sin 20 12sin 20

= 0 08 1 4 3 sin 20 6 1 cos 40

= 0 0

8 7 4 3 sin 20 6 40 cos

= 0 0 0

8 7 4 3 sin 20 4 3 cos 30 cos40 [6= 4 cos300]

= 0 0 08 7 4 3 cos70 2 3 cos70 cos10

[sin200 = cos70

0 and 2cos30

0cos40

0 = cos70

0 +cos10

0]

= 0 08 7 2 3 cos 70 cos10

= 0 08 7 2 3 2sin 40 sin 30

= 0

8 7 2 3 sin40

= 08 8 1 2 3 sin 40

= 2

08 8 1 2 3 sin40

= 0 2 0

8 8 1 4 3 sin40 12sin 40

= 0 08 8 1 4 3 sin40 6 1 cos80

= 0 0 08 8 7 4 3 cos50 2 3 cos110 cos50

[sin400

= cos500 and 6cos80

0 = 4 cos30

0cos80

0 = 2 (cos110

0+ cos50

0]

= 0 08 8 7 2 3 cos50 cos110



= 0 0

8 8 7 2 3 2sin80 sin30 .

= 0

8 8 7 2 3 cos10 [sin800 = cos10

0 and ]

= 08 8 8 2 3 cos10 1 . . . . . . . . . . . . ( 2 )

The term is on the L.H.S. of ( 1 ).

By continued iteration on ,

we obtain the result.

- 1 = -1 = 8 8 8 8 8 8 8 .. . . . . . and so on.

Remark: The pattern of the signs in the series - - + - - + . . . . . . . . . . is worth noting.

--------------------------------------------------------------------- -----@@@@----------------------------------------------------------------------------

§ § PROBLEM 24 : NOT FROM NBSR

3 3 3 3 3 33 11 8 5 11 8 5 11 8 5 .. . . . . . . . . . *

Solution:

33 27

3 311 8 8 27 = 11+ 8.2 = 11 + 8. 3 8

3 3 311 8 5 27 8 = 5 + 3 = 5 + 3 27

Repeating the expression for 3 27 again and again, we get

3 3 3 3 3 33 11 8 5 11 8 5 11 8 5 .. . . . . . . . . .

This establishes the result *.

--------------------------------------------------------------------- -----@@@@----------------------------------------------------------------------------

Problem 25: NBSR Vol. II p 143

§ x = a x a a a a x a a a a x a a a2 2 21 1 1 2 1 2 2 3 2 3 3 4( 2 ) 2 ( 2 ) 2 ( 2 ) ... . . . . . and so on.

Proof:

x = a1 + ( x - a1)

= x aa 112

( ) . . . . . . . . . . . . . . . ( 1 )

= a x a xa2 21 11 2

= x a a a a x aa a21 1 2 1 1 21 ( 2 ) 2 2

= x a a a aa x a21 11 2 1 2( 2 ) 2 ( )



= x a a a a x aa 2 21 1 2 1 21 ( 2 ) 2 ( )

By continued recursion., we get

x = a x a a a a x a a a a x a a a2 2 21 1 1 2 1 2 2 3 2 3 3 4( 2 ) 2 ( 2 ) 2 ( 2 ) ... . . . and so on.

Corollary:

When a = a1 = 2a2 = 22a3 =2

3a4= . . . . . . . . . . . . . . . . = 2

n-1an . . . . . . . .

We have x a x a x a x a x a2 2 2 22 .. . . and so on.

--------------------------------------------------------------------- -----@@@@----------------------------------------------------------------------------

Problem 26: § SR Type Nested square root problems not from NBSR

1. 6 =

=

= = 6 5 6 5 6 5 6 5 6........... and so on.

6 =

=

= 2 25 5 5 5 36

= 2 2 25 5 5 5 5 5 36

= 2 2 2 25 5 5 5 5 5 5 5 36........... and so on.

2. 6 =

=

=

=

= 4 4 4 4 5 20X11

= 4 4 4 4 5 20 121

= 4 4 4 4 5 20 5 4 841 [∵121=5+116 =5+4 x 29=5+ 4 ]

= 4 4 4 4 5 20 5 4 5 4 2092 ......... and so on.

7 =

=



= [42 = 6 x 7 = 6 ]

=

=

= 7 6 7 6 7 6 7 6 7 6 49 ......... and so on

7 =

=

= [∵ 39 = 3 x 13 = 3 ]

= [∵169 = 13 + 12 x 13 = 13+ 12 ]

= 10 3 13 12 13 12 13 12 13 12 169 ......... and so on.

13 =

=

=

= 13 12 13 12 13 12 13 12 13 12 169 ......... and so on.

--------------------------------------------------------------------- -----@@@@----------------------------------------------------------------------------

------------------ ( To be continued ……)

AUTHOR’S BIOGRAPHY

Prof. N. Ch. Pattabhi Ramacharyulu: He is a retired professor in

Department of Mathematics & Humanities, National Institute of

Technology, Warangal. He is a stalwart of Mathematics. His yeoman

services as a lecturer, professor, Professor Emeritus and

Deputy Director enriched the knowledge of thousands of students. He

has nearly 42 PhDs and plenty number of M.phils to his credit. His

papers more than 200 were published in various esteemed

reputable International Journals. He is a Member of Various

Professional Bodies. He published four books on Mathematics.

He received so many prestigious awards and rewards.




www.arcjournals.org

©ARC Page | 16

Total coloring of -graph

Dr.S. Sudha Professor

Ramanujan Institute

for Advanced study in Mathematics

University of Madras

Chennai, India.

[email protected]

K. Manikandan Research Scholar

Ramanujan Institute

for Advanced study in Mathematics

University of Madras

Chennai, India.

[email protected]

Abstract: In this paper, we have defined a new graph called )-graph for even and for

odd and found the lower and upper bound for the total chromatic number of -graphs. We

have also found the total chromatic number of for all and for odd .

Keywords: Total Coloring, -graphs

1. INTRODUCTION

For the past three decades many researchers have worked on total coloring of graphs. Borodin [1]

has discussed the total coloring of graphs. Sudha and K.Manikandan [3] have discussed the total

coloring and -total coloring of prisms . Prisms with nodes are characterized as

generalized Peteresen graphs H.P.Yap [4] also has defined and discussed the total

coloring of graphs. We have defined a new graph , and the definition

follows:

The graph consists of vertices denoted as The edges are defined as

follows:

(i) is adjacent to and is adjacent to

(ii) is adjacent to if

(iii) is adjacent to if .

This graph is a quartic graph and it is both Eulerian and Hamiltonian. The concept of this type of

a new graph was introduced by S.Sudha.

Definition 1: A total coloring is a coloring on the vertices and edges of a graph such that

(i) no two adjacent vertices have the same color

(ii) no two adjacent edges have the same color

(iii) no edge and its end vertices are assigned with the same color.

In this paper, we have considered the graph and obtained the upper and lower bound for

the total chromatic number.

2. TOTAL COLORING OF -GRAPHS

Theorem 1: The total-chromatic number is for and odd

Proof: Let be the vertices of the graph and its edges are defined as

(i) is adjacent to and is adjacent to

(ii) is adjacent to if

(iii) is adjacent to if .



Dr.S. Sudha & K. Manikandan


Let the coloring set of be the set

We define the function from to the set as follows:

We define the function from E(S(n, m)) to the set {1, 2, 3, . . .} as follows:

By using the above pattern of coloring, the graph S(n,m) admit total coloring. The total-chromatic

number for S(n,m), χtc(S(n,m)) = 6.

Illustration 1:

Figure 1. S(12,5)

The graph consists of vertices which are

assigned with the colors respectively. The outer

edges and ,

are assigned with

colors and theinner edges are assigned with colors

respectively. The total-chromatic number

of , .

Theorem 2: The total-chromatic number is for andis 6 for

Proof: Let be the vertices of the graph and its edges be denoted by

for and .

Let be a function that maps to the set and be a function that maps

to the set in such a way that and satisfy the condition of total coloring.

There are six cases:

(i)

Total Coloring of -Graph


(ii)

(iii)

(iv)

(v)

(vi)

Case (i): Let

By using the above pattern, the graph for admit total coloring.

The total-chromatic number of , .

Case (ii):Let

By using the above pattern, the graph admit total coloring.


Case(iii): Let





Case (iv): Let



Case(v): Let



Case(vi): Let





Illustration2:

Figure 2. S(8.2)

The graph consists of 8 vertices which are assigned with the

colors respectively. The outer edges and

are assigned with colors and the inner edges

are assigned with colors respectively.

The total-chromatic number of ,

Theorem 3: The total-chromatic number is for and is 6 for

Proof: Let be the vertices of the graph and its edges be denoted

by for and .

Let be a function that maps to the set and be a function that maps

to the set in such a way that and satisfy the condition of total coloring.

Case(i): For odd and

The edges of the form for Takes the coloring pattern as

The last three of the edges are colored given below.





Case(ii): For odd and

Now with this type of coloring, the graph is total coloring.


Illustration 3:

Figure3. S(11,3)

The graph consists of vertices which are assigned

with the colors respectively. The outer edges

and are assigned with colors

and the inner edges

are assigned with colors

respectively. The total-chromatic number of ,

3. CONCLUSION

We have found that the lower and upper bound for the total chromatic number of , in

general, satisfies . The total-chromatic number for when takes the

value 2 and 3 are also discussed.

REFERENCES

[1] O. V. Borodin, “The star coloring of graphs” , Discrete Math., 25; 211-236, 1979.

[2] J. A. Bondy, U.S.R. Murty, “Graph theory with applications”, 1982.

[3] H. P. Yap, “Total colorings of Graphs”, Lectures Notes in Mathematics 1623, Springer-

Verlag, Berlin(1996).



[4] Andrea hackmann And Arnfriedkhemnitz “Circular Total Coloring of Graphs”

[5] S.Sudha and K. Manikandan, “Total coloring and -total coloring of prisms”, accepted

for publication (2013).

AUTHORS’ BIOGRAPHY

Dr.S.Sudha has got her Ph.D., in 1984. She has got 35 yearsof teaching and

research experience. She is currently working as a Professor in Mathematics

at the Ramanujan Institute for Advanced Study in Mathematics, University of

Madras, Chennai-600005. Her fields of interest are Computational Fluid

Dynamics, Graph Theory, Fuzzy Graphs and Queueing Theory. She has

published more than 25 articles in journals. She has also published some

books.

K.Manikandan is a Ph.D. Research scholar atRamanujan Institute for

Advanced Study in Mathematics, University of Madras, Chennai-600005.He

has published one article in a journal.




www.arcjournals.org

©ARC Page | 23

Extreme Value Modeling of Precipitation in Case Studies for

China

Manuela Ender Department of Mathematical Sciences

Xi’an Jiaotong Liverpool University

Suzhou,China

[email protected]

Tong Ma Department of Mathematical Sciences

Xi’an Jiaotong Liverpool University

Suzhou,China

Abstract: This paper aims to model extreme rainfall events using 60 years of daily data based on extreme

value theory for four cities in China. The purpose is to allow decision makers to make informed decisions

and to avoid or at least reduce flood caused damage to life and property. Generalized extreme value

distributions are used for fitting monthly and semiannual maxima according to the Block Maxima

Approach. Rainfall exceeding an extremely high threshold is modelled by Generalized Pareto distributions

.The thresholds are selected based on the analysis of three methods, Hill plot, mean excess plot and

Standardized Precipitation Index. Finally, we estimate the parameters for both models and calculate return

levels for five different return periods. Statistical tests for stationarity, KPSS and Man-Kendall tests support

the study. The results show that GPD has a better fitting performance than GEV. Further, we can determine

how often a flood occur in a certain city and during which season, rainy or dry season. For example, for

Nanjing it can be neglected that a flood occurs during dry seasons. But for other cities, like Shantou, this

might be rare events but with a higher frequency of occurrence.

Keywords: Extreme value theory, precipitation, generalized extreme value distribution, generalized

Pareto distribution, return level, China

1. INTRODUCTION

During the year 2013, China experienced severe floods caused by extreme precipitation. In

August 2013 for example, the south provinces Guangdong, Guangxi and Fujian and the north

eastern provinces Heilongjiang, Jilin and Liaoning were hit by heavy rain that caused flooding

(RCSC, 2013). For the north eastern region, it was the worst flooding in 50 years (Hunt and Ke,

2013). Millions of residents were affected, many people died or went missing. Direct economic

losses solely for Guangdong province are reported to be CNY 13 billion and for Heilongjiang

province to be around CNY 7 billion where potential losses in future crop yields are not yet

included (RCSC, 2013).

Beside these recent events, China is regularly affected by extreme rainfall that has enormous

influence on social development and that threatens safety of human life and property (Feng et al.,

2007). In 2008 for example, the Spring Festival was spoiled by heavy snowstorms in many

provinces. The total costs were approximately CNY 80 billion which was mainly used for refugee

settlements and maintenance of infrastructure (BBC, 2008). In 2011, a heavy rain brought much

inconvenience for cars and pedestrians that soaked in Nanjing, Jiangsu province (CCTV, 2011).

Most floods or mud-rock slides triggered by severe rainfall impair agricultural production, public

facilities, buildings and transport (Yang, 2012).

Consequently, estimations of extreme rainfall events in China play a significant role in an

efficient risk appraisal and for the reduction of losses of the economy. To know the statistics that

a certain flood event will occur gives advice to decision makers to design efficient mitigating

measures. This covers firstly the planning and construction of water management, sewerage

systems, capacity of channels and river basins, etc. Second, it supports the decision how much

and what kind of insurance against water damage should be bought. Finally, the knowledge of


Manuela Ender & Tong Ma


return levels of floods helps to inform the citizens that they are better prepared in a case of

flooding (Overeem et al., 2008).

To contribute to the achievement of these goals, this paper focuses on the estimation of the most

appropriate distribution of extreme precipitation in four Chinese cities, Nanjing, Shantou, Urumqi

and Qiqihaer. The estimation is based on a universal principle, Extreme Value Theory (EVT).

EVT contains two fundamental distributions, Generalized Extreme Value distribution (GEV) and

Generalized Pareto distribution (GPD), which are both applied in this paper. Details for the GEV

and GPD approach are given in Section 3. We elaborate for both models the return levels of

certain flood events for 5, 10, 20, 50, and 100 years. As many regions in China have marked dry

and wet periods during the year, we distinguish further between dry and wet seasons as risk

mitigating measures may be different during each time of the year. The probability of occurrence

of extreme rainfall might be underestimated by assuming that the model is stationary (Coles et al.,

2003). Therefore, stationary tests are applied to determine whether non-stationarity influences the

application of GEV and GPD models. To sum up, this paper contributes to the existing literature

by a comprehensive statistical analysis of extreme precipitation events in four cities of China in

the north, east, south and west. Based on most recent flood events in 2013, Shantou and Qiqihaer

are included in particular to study them from a statistical perspective.

The next section of the paper gives a literature review of previous and similar work. In Section 3,

the methodology of EVT, GEV and GPD is elaborated. Section 4 presents and discusses empirical

results. Recommendations and limitations in Section 5 conclude the paper.

2. LITERATURE REVIEW

First insights in EVT are published by Fisher and Tippett in 1928. Significant contributions to the

statistical modeling of extremes followed by Jenkinson 1955 for GEV, Balkema and de Haan

1974 and Pickands 1975 for GPD. In comparison to the long history of theoretical results of EVT,

the empirical analysis of precipitation data using EVT is relatively new. In the last two decades,

researchers all over the world applied either the GEV or the GPD approach to rainfall data from

Europe (Miroslava, 1992; Bordi et al., 2007), America (Nadarajah, 2005), Oceania (Withers and

Nadarajah, 2000; Li et al., 2005) or Asia (Nadarajah and Choi, 2007; McAleer et al, 2012).

For China in particular, several papers exist dealing with the analysis of extreme precipitation and

its trends (Gemmer et al., 2004; Wang and Zhou, 2005; Liu et al., 2005; Zhai et al., 2005; Sun and

Ao, 2013). For example, Wang and Zhou (2005) stress that an obvious increasing trend of

extreme daily rainfall mainly took place in the east, southwest and northwest of China in summer

months from 1961 to 2001. However, fewer studies apply EVT techniques for the estimation of

the tail distribution and derivation of return levels. Feng et al. (2007) provide the first

comprehensive study of GEV estimation and return levels based on Chinese data. The authors

conclude that the highest return levels were found in the very Southern parts of China. In Eastern

China, high return levels are reported in comparison to low return levels in the northwest. Studies

using the GPD approach are conducted by Li (2013) for data from whole China or more locally

based by Dong et al. (2011) for Yellow-Huaihe and Yangtze-Huaihe rivers basins and Jiang et al.

(2009) for Eastern China. Jiang et al. (2009) state that the fitting performance of GPD is better

than for GEV. Their reported return levels are in line with the findings of Feng et al. (2007) and

decrease from the south to the north.

3. METHODOLOGY

3.1 Data Description

To cover locations in the north, east, south and west and in different climate zones of China, the

four cities Qiqihaer (Heilongjiang province), Nanjing (Jiangsu province), Shantou (Guangdong

province) and Urumchi (Xinjiang province) are chosen in the empirical study. Qiqihaer and

Shantou are of particular interest given most recent flood events in 2013. All data starts from year

1951 to 2010 and are obtained from the National Meteorological Information Center. The daily

precipitation amount is measured within 24 hours (8am on a particular day to 8am on next day).

The climate of Qiqihaer is humid continental and monsoon-influenced. Nanjing is located in the

subtropical monsoon climate zone. Shantou lies within the subtropical marine and monsoon

Extreme Value Modeling of Precipitation in Case Studies for China


climate zone. Urumchi belongs to the temperate zone with continental climate (Domroes and

Peng, 1988).

The descriptive statistics of the daily rainfall datasets for the four cities are given in Table 1.

According to their climate zones, Shantou is the city with most rain, followed by Nanjing and

Qiqihaer. Urumchi is the driest city.

Table 1. Descriptive statistics of daily rainfall datasets, 1951 to 2010 (unit: mm, n=21915)

Nanjing Shantou Urumchi Qiqihaer

Minimum 0 0 0 0

Maximum 207.20 297.40 57.70 135.50

Mean 2.90 4.34 0.73 1.18

Median 0 0 0 0

Mode 0 0 0 0

Standard

deviation

9.73 14.80 2.76 5.02

Range 207.20 297.40 57.70 135.50

3.2 Generalized Extreme Value Distribution (GEV) and Block Maxima

The most classical model for extreme events is the Block Maxima approach. This model is

appropriate when the maximum observations of each period or block with a predefined and fixed

length are assembled from a large number of identically and independently distributed (iid)

variables (McNeil, 1999). In this case, the asymptotic distribution of the maximum observations is

exactly one of three well known distributions (Fisher and Tippett, 1928). The cumulative

distribution function of these three distributions can be summarized by the GEV (Jenkinson,

1955) and is given by

where are the extreme values from the blocks, and are the shape, scale and location

parameters, respectively. For the Gumbel distribution is determined. For , we get the

Frechet distribution with a fat-tail. In the case of the Weibull distribution is obtained. The

parameters are estimated by Maximum likelihood estimation (MLE). When we assume

that the variables are independent, the likelihood function is given by the product of the

observations’ densities.

3.3 Generalized Pareto Distribution (GPD) and Peak over Threshold (POT)

A second methods to analyze the distribution of extreme events is called Peak over Threshold

(POT) method which considers the maximum variables exceeding a predetermined threshold.

Given a threshold , the distribution function of extreme values of over is,

represents the probability that the value of exceeds the threshold by at most amount ,

where . Balkema and de Haan (1974) and Pickands (1975) showed that the distribution

converges to GPD when the threshold is sufficiently high. The cumulative distribution

function for GPD is,



where are the exceedances, and are the shape and scale parameter, respectively. There are

three types of GPD. For we have an Exponential distribution with medium-size right tail.

If ξ < 0, H(x) is an Ordinary Pareto distribution describing a positive long tail. In the case of ξ >

0, H(x) is a Pareto (II) type distribution for a positive short tail.

The estimation of parameters is possible with the method of probability weighted moments, the so

called L-moments, or with MLE (Hosking et al., 1984). In this study, MLE is used. Before the

parameter estimation, the initial step is to choose an appropriate threshold. We are faced with the

problem to find a threshold that is sufficiently high to support the convergence. However, it

should not be too high as otherwise the sample of these extreme high observations is very small.

In other words, there is a tradeoff between bias and variance. We use three methods of threshold

selection including Hill plot (Hill, 1975), sample mean excess plots, and Standardized

Precipitation Index (SPI) (McKee et al., 1993).

The Hill plot specifies the relationship between the estimated tail index and either or the

threshold, where is the number of exceedances and is the sample size:

A threshold is chosen from the Hill plot where the tail index starts to be stable (Hill, 1975).

The mean excess function is the expectation of each observation deducted by a fixed amount

given that this observation is not smaller than that fixed amount:

Empirically, is estimated by based on the realizations of random variables, .

The SPI is used to quantify extreme wetness and dryness per month in a location (McKee et al.,

1993). McKee et al. (1993) state that the auto-correlation of individual observations is unobvious

as the monthly precipitation is usually independent. Hence, the frequency distribution of monthly

precipitation is estimated by a two-parameter Gamma distribution for every month of a year. As a

next step, the empirical Gamma probability density distribution is transferred into the Normal

distribution. Abnormality in the transformation quantifies the meaning of relatively wet and dry.

Therefore, the SPI values can be converted into Z-indexes of the Standard Normal Distribution

(Bordi et al., 2007). Based on Yuan and Zhou (2004), Table 2 lists the classification of scales for

SPI and Z-index in China. For example, the extremely wet conditions can be identified by Z-

values which are greater than 1.96.

To summarize, Hill-plot and sample mean excesses belong to the family of semi-parametric

models, while SPI fits to non-parametric model. This paper applies all three methods to select

thresholds.

Table. 2 Classification of Scales for SPI and Z-Index

Z-index SPI values Class

>1.96 >2.00 Extremely wet

1.44 to1.96 1.50 to 2.00 Moderately wet

0.84 to1.44 1.00 to 1.50 Slightly wet

-0.84 to 0.84 -1.00 to 1.00 Normal

-1.44 to -0.84 -1.50 to -1.00 Slightly dry

-1.96 to -1.44 -2.00 to -1.50 Moderately dry

<-1.96 <-2.00 Extremely dry



3.4 Extreme Quantile Estimation for GEV and GPD

A T-year return level has a probability of per cent to be exceeded once in a year. For

example, rainfall of 170 mm or larger equally occurs every 60 years in a region. The return levels

at different return periods can be evaluated through the quantile estimation of fitted GEV and

GPD, that is,

is the inverse of the GEV or GPD. For GPD, it is assumed that the number of exceedances

over the extremely large threshold is approximately close to a Poisson distribution with

parameter , which is also the rate of exceedances per year. Hence, is the number of

exceedances in the return period of T years. The can be estimated by , where is

the number of years with available data (Maraun, 2010).

3.5 Stationary Tests

Stationary tests including graphic examination, KPSS (Kwiatkowski, Philips, Schmidt and Skin)

and non-parametric Mann-Kendall tests should be carried out since the assumption of stationarity

is crucial for the application of GEV and GPD. The KPSS stationary test (Kwiatkowski, Phillips,

Schmidt and Shin) judges whether the trend is stabilized around a constant, a linear line or non-

stationary (Hasna and Chung, 2010). The test statistics are compared with critical values at

different significant levels. Thus, the null (H0) and alternative (H1) hypotheses are:

H0: Stationary around a constant or a linear trend,

H1: The trend is non-stationary.

The Mann-Kendall (MK) test determines the existence of either an increasing or decreasing

tendency in monthly and semi-annual extreme rainfall (Hasna and Chung, 2010). The p-value of

the null hypothesis is used to determine the tendency of the observations. Thus, the null (H0) and

alternative (H1) hypotheses are:

H0: There is no trend,

H1: There is an increasing/ decreasing trend.

4. RESULTS AND DISCUSSION

4.1 Modeling using GEV

For GEV estimation, the Block Maxima of monthly and half-yearly rainfall are extracted for all

four cities. Table 3 contains descriptive statistics for both periods.

Table. 3 Descriptive statistics of monthly (n=720) and half-yearly (n=120) maximum, 1951 to 2010 (unit:

mm)


Monthly Half-

yearly

Monthly Half-

yearly

Monthly Half-

yearly

Monthly Half-

yearly

Minimum 0 24.30 0 24.60 0 6.60 0 8.80

Maximum 207.20 207.20 297.40 297.40 57.70 57.70 135.50 135.50

Mean 32.07 74.81 45.50 116.21 9.72 22.23 13.42 39.89

Median 24.00 65.20 30.60 103.25 7.30 20.95 5.50 33.65

Mode 12.80 40.20 0 75.00 4.00 18.60 0 30.30

Standard

deviation

28.91 37.15 46.33 53.28 8.37 10.14 17.69 21.77

Range 207.20 182.90 297.40 272.80 57.70 51.10 135.50 126.70

Table 4 gives the parameters of GEV as results of MLE fitted to monthly and half-yearly maxima.

The first line of each parameter indicates the value and the standard errors (s.e.). The

corresponding 95% confidence intervals (CI) are included in the second line. For monthly

maxima, the shape parameter ξ is for all four cities positive and the CI do not include zero which



supports the positive sign of ξ. This means that the fat-tailed Frechet distributions are obtained.

For half-yearly maxima, zero is included in the CI. Therefore, a Gumbel distribution cannot be

excluded. These findings are in line with Feng et al. (2007).

Table. 4 GEV parameter estimates for monthly and half-yearly maxima


Parameter

s (95% CI)

Monthl

y / s.e.

Half-yearly /

s.e.

Monthly

/ s.e.

Half-

yearly /

s.e.

Month

ly /

s.e.

Half-

yearly /

s.e.

Monthl

y / s.e.

Half-

yearly /

s.e.

ξ

(95% CI)

0.319 /

0.036

0.171 / 0.089 0.497 /

0.054

0.035 /

0.073

0.285 /

0.038

0.074 /

0.074

1.245 /

0.067

0.127 /

0.083

(0.250,

0.389)

(-0.005,

0.346)

(0.391,

0.603)

(-0.108,

0.178)

(0.211,

0.358)

(-0.070,

0.219)

(1.113,

1.377)

(-0.035,

0.288)

σ

(95% CI)

14.478 /

0.534

24.510 /

2.082

21.647 /

0.973

40.135 /

3.081

4.550 /

0.167

7.248 /

0.565

3.906 /

0.263

14.693 /

1.203

(13.469,

15.563) (20.750,28.95

0)

(19.822,

23.641)

(34.529,

46.651)

(4.234,

4.889)

(6.221,

8.444)

(3.424,

4.457)

(12.515,

17.250)

μ

(95% CI)

17.880 /

0.625

56.120 /

2.622

19.776 /

1.032

91.504 /

4.163

5.543 /

0.199

17.476 /

0.753

2.606 /

0.180

29.395 /

1.553

(16.655,

19.104)

(50.981,61.25

9)

(17.753,

21.798)

(83.344,

99.664)

(5.154,

5.932)

(16.000,

18.952)

(2.254,

2.958)

(26.351,

32.439)

By checking density plots, cumulative distribution plots, probability plots and QQ plots

respectively, the fitting performance of GEV can be analyzed. The fitted lines in the density plots

have clear tails for all cities. The cumulative plots are well fitted to the empirical data for both

monthly and half-yearly maxima. In terms of probability plots, the GEV fitted lines estimate the

extreme values much more precise than the Normal distribution. For monthly and half-yearly

observations, the probability fitting curves and QQ plots are congruent in all cases. Only a few

highest points diverge from the fitted lines.

Table 5 lists the return level estimates at different return periods for monthly and half-yearly

maxima. The 95% CI are included in brackets. The estimated return levels and CI increase with

the increase of the return period. Compared with the lower bound of the confidence interval, the

upper bound is likely to be further away from the predicted return level when the return period is

longer. Recalling Table 3, the highest rainfall amount in Nanjing of the observed period from

1951 to 2010 was 207.2 mm. This value appears in CI at T=10 for monthly and at T=20 for half-

yearly samples. This pattern is similar for Urumchi. For Shantou, the highest observation is also

covered by T=20 for half-yearly samples. For monthly samples it is already included in T=5. The

maximum value of Qiqihaer lies within the CI of T=50 for semiannual values. Return levels based

on monthly maxima are not possible to compute.

According to China Meteorological Administration (2012), 24-hour rainfall amount exceeding

250 mm is considered as extreme rainfall that can cause floods in Southern China. For Nanjing,

this value is contained in the CI of T=20 for maximum monthly observations and T=50 for

semiannual maxima. In Shantou, the event of a floods occurs much more often as 250mm is

included in the CI of T=5 for monthly and T=10 for half-yearly observations. Hence, floods

might occur once in every 20 to 50 years in Nanjing and every 5 to 10 years in Shantou.

Table 5. GEV return level estimates for monthly and half-yearly maxima

Selection Period

T (years) 5 10 20 50 100

Monthly Max

Nanjing

139.438

(120.161,

166.147)

180.973

(151.041,

223.980)

232.660

(187.681,

299.699)

320.981

(247.536,

437.008)

407.135

(301.523,

578.915)



Half-yearly Max

Nanjing

123.361

(109.749,

145.588)

150.920

(129.830,

190.498)

181.447

(149.689,

247.645)

227.373

(175.645,

348.148)

267.060

(195.140,

449.122)

Monthly Max

Shantou

307.790

(242.743,

409.104)

445.022

(331.759,

631.786)

638.3502

(448.555,

970.388)

1020.600

(676.900,

1703.000)

Not

computable

Half-yearly Max

Shantou

185.452

(168.566,

210.172)

217.089

(194.011,

256.593)

248.900

(217.348,

309.097)

291.723

(245.484,

389.603)

324.8883

(264.927,

460.329)

Monthly Max

Urumchi

40.713

(35.227,

48.393)

51.949

(43.543,

64.166)

65.608

(53.158,

84.387)

88.314

(68.240,

120.160)

109.878

(81.774,

156.250)

Half-yearly Max

Urumchi

35.2310

(31.919,

40.163)

41.567

(36.916,

49.642)

48.121

(41.642,

60.700)

57.234

(47.523,

78.277)

64.521

(51.727,

94.274)

Half-yearly Max

Qiqihaer

67.655

(60.116,

79.383)

82.383

(71.345,

102.623)

98.194

(82.228,

131.153)

121.141

(96.153,

179.253)

140.281

(106.432,

225.622)

4.2 Modeling using GPD

The choice of appropriate thresholds is based on the three methods presented in section 3.3. To

examine the differences in dry and rainy seasons, the whole year’s data is divided into two

seasons: dry season from December to March and rainy season from April to November. We

select one constant threshold for the whole year, and two seasonal thresholds for dry and rainy

season solely.

The best indicated thresholds by all methods are finally chosen for the estimation of parameters.

Fig. 1 displays the Hill plots for daily precipitation of the whole year for the city with most rain,

Shantou and for Urumchi as the driest city. Hill plots for the other two cities follow a similar

pattern. Two red dashed curves around the blue curve are the bounds of 95% CI. The curve for

Shantou starts to be steady after around 100 mm for daily data, while the stability of the tail index

emerges after 20 mm for Urumchi.

0 50 100 150 200 250

0

0.5

1

1.5

2

2.5

3

3.5

Thresholds (mm)

xi

Shantou-Hillplot for Daily Rainfall (mm) from 1951 to 2010

0 5 10 15 20 25 30 35 40 45 50 55

0

0.5

1

1.5

2

2.5

3

Thresholds (mm)

xi

Urumchi-Hillplot for Daily Rainfall (mm) from 1951 to 2010

Figure 1. Hill plots for daily precipitation for Shantou (above) and Urumchi (below)



Second, the mean excess plots are plotted. The possible threshold is the point when the mean

excess plot shows linearity, and estimated parameters look stable at different thresholds (Hasan

and Chung, 2010). As an example, the sample mean excess plot for the constant threshold of

Nanjing is presented in Fig. 2. In this case the constant threshold could be chosen as 60 mm.

Mean excess plots for rainy and dry seasons for all cities can be drawn and analyzed in a similar

way.

The third method incorporates the SPI. Fig. 3 exposes tamount of rainfall against Z-indexes for

six years for Qiqihaer. The red straight line in each graph is the benchmark of 1.96. We can

choose the level as a threshold when the Z-index starts to be greater than 1.96. As an average over

the years, a constant threshold of 25 mm seems appropriate. The analysis like this is done for

other cities and for rainy and dry seasons as well.

Table 6 gives an overview of the chosen threshold after completing the analysis of the three

methods for all cities and all seasons.

0 20 40 60 80 100 120 140 160 180 2005

10

15

20

25

30

35

Threshold (mm)

Me

an

Exce

ss

Nanjing Mean Excess Plot of Daily Rainfall (mm)

Figure 2. Mean excess plot for daily precipitation for Nanjing

0 5 10 15 20 25 30 35 40 45 500

1

2

3

Rainfall (mm)

Z

July to August 1951

0 10 20 30 40 50 60-1

0

1

2

3

Rainfall (mm)

Z

July to August 1961

0 5 10 15 20 25 30 350

1

2

3

Rainfall (mm)

Z

July to August 1971

0 10 20 30 40 50 60 70-1

0

1

2

3

Rainfall (mm)

Z

July to August 1981

0 5 10 15 20 25 30 35 40 45-1

0

1

2

3

Rainfall (mm)

Z

July to August 1991

0 10 20 30 40 50 60-1

0

1

2

3

Rainfall (mm)

Z

July to August 2001

Figure 3. Z-index against daily rainfall in July and August for six years for Qiqihaer

Table 6. Constant and seasonal thresholds

Threshold (mm) Nanjing Shantou Urumchi Qiqihaer

constant

60 (n=124) 110 (n=78) 20 (n=90) 25 (n=222)

rainy season

90 150 20 45

dry season

30 60 10 10



After threshold selection, the parameters for GPD as given in equation (3) are estimated using

MLE. The results are represented in Table 7. The exceedances of Nanjing have a long tail and

follow the Ordinary Pareto distribution as the shape parameters for all seasons are negative. For

Shantou, ξ is negative for constant and rainy threshold, but not for the dry season. Data from

Urumchi follow the Ordinary Pareto distribution for rainy and dry season, but not for the whole

year. In the case of Qiqihaer, a Pareto (II) type distribution was found in all cases. As all CI

include the zero for the shape parameter, the Exponential distribution cannot be excluded.

Table 7. GPD parameter estimates for constant and seasonal thresholds

Nanjing Shantou

Parameters

(95% CI)

Constantover

60 / s.e.

Rainy over

90 / s.e.

Dry over

30 / s.e.

Constantover

110 / s.e.

Rainy over

150 / s.e.

Dry over

60 / s.e.

ξ

(95% CI)

-0.052 /

0.095

-0.159 /

0.153

-0.101/

0.108

-0.127 /

0.114

-0.232 /

0.203

0.238 /

0.189

(-0.237,

0.134)

(-0.459,

0.141)

(-0.313,

0.110)

(-0.350,

0.097)

(-0.630,

0.166)

(-0.131,

0.608)

σ

(95% CI)

31.052 /

4.049

34.746 /

7.416

9.495 /

1.449

50.443 /

8.072

54.692 /

14.983

21.269 /

5.078

(24.048,

40.094)

(22.868,

52.792)

(7.040,

12.806)

(36.863,

69.026)

(31.969,

93.566)

(13.321,

33.959)

Urumchi Qiqihaer

Parameters

(95% CI)

Constant

over 20 / s.e.

Rainy over

20/ s.e.

Dry over 10

/ s.e.

Constant

over 25 / s.e.

Rainy over

45 / s.e.

Dry over

10 / s.e.

ξ

(95% CI)

0.002 /

0.113

-0.037 /

0.117

-0.118 /

0.095

0.095 /

0.076

0.032 /

0.139

0.372 /

0.311

(-0.219,

0.223)

(-0.266,

0.193)

(-0.304,

0.069)

(-0.055,

0.244)

(-0.241,

0.304)

(-0.237,

0.981)

σ

(95% CI)

7.907 /

1.220

8.765 /

1.420

4.908 /

0.665

13.078 /

1.326

15.895 /

3.079

3.764 /

1.310

(5.843,

10.700)

(6.381,

12.040)

(3.764,

6.340)

(10.721,

15.952)

(10.874,

23.235)

(1.902,

7.447)

Similar to GEV, we examine how well the GPD model fits to the exceedances with probability

distribution plots, fitted and empirical distribution plots, probability plots and QQ plots. We can

conclude that the fitted probability density plots and the fitted cumulative plots match the

empirical data consistently. Compared to GEV, the GPD probability plots are more appreciable

for all cities, especially for higher values. From the QQ plots, the number of deviating

observations for GPD is less than the number of observations modeled by GEV in all cases. To

sum up, GPD is superior to GEV in terms of fitting which is in line with the literature (Jiang et al.,

2009).

Table 8 describes the return level estimates at different return periods of daily exceedances over

constant and seasonal thresholds. 95% CI are included in brackets. The pattern of the results is

similar to GEV. The maximum values during the observation period never appear in the CI for

dry seasons for Nanjing and Urumchi. For Shantou and Qiqihaer, the highest value of the past can

be expected to happen even in dry seasons every 50 years.

Recalling the trigger amount of rain with 250 mm that can cause a flood in Southern China

(CMA, 2012), we can conclude that a flood event does not happen during dry season in Nanjing.

Even in rainy seasons, a flood above 250 mm should happen only once in 100 years. However,

the upper bound of T=50 is already very close to this amount. For Shantou, we get a different

picture. Here, the city experience a flood every 20 years in rainy seasons and every 50 years in

dry seasons.



Table 8. GPD return level estimates for constant and seasonal thresholds

Selection

Period

T (years)

5 10 20 50 100

Constant

Nanjing

(60 mm)

128.328

(118.120,

142.419)

147.077

(133.909,

170.074)

165.170

(148.147,

201.586)

188.117

(164.424,

249.705)

204.770 (174.930,

291.557)

Rainy

Nanjing

(90 mm)

130.785

(120.245,

143.968)

149.330

(135.954,

168.371)

165.939

(150.176,

198.551)

185.265

(166.075,

248.597)

198.122 (175.882,

295.092)

Dry Nanjing

(30 mm)

46.860

(43.908,

50.630)

52.072

(48.401,

58.100)

56.9300

(52.788,

66.726)

62.849

(56.978,

79.992)

66.975 (59.868,

91.571)

Constant

Shantou

(110 mm)

194.0472

(178.836,

213.038)

220.445

(201.863,

250.044)

244.621

(222.210,

292.282)

273.485

(244.695,

356.332)

293.197 (258.519,

411.417)

Rainy

Shantou

(150 mm)

192.070

(177.516,

212.280)

220.839

(200.222,

247.139)

245.334

(221.295,

289.245)

272.222

(244.958,

364.340)

289.084 (260.912,

437.436)

Dry Shantou

(60 mm)

92.388

(82.513,

106.947)

114.232

(98.202,

145.591)

139.998

(116.803,

209.595)

181.290

(139.196,

361.692)

219.095 (156.720,

564.733)

Constant

Urumchi

(20 mm)

35.958

(33.031,

39.963)

41.4601

(39.5391,

43.8961)

46.968

(44.296,

50.777)

54.259

(50.168,

60.772)

59.782 (54.288,

69.058)

Rainy

Urumchi

(20 mm)

36.064

(34.618,

37.685)

41.661

(39.737,

44.061)

47.118

(44.513,

50.802)

54.121

(50.246,

60.348)

59.266 (54.138,

68.060)

Dry

Urumchi

(10 mm)

19.434

(18.760,

20.182)

21.964

(21.114,

23.008)

24.296

(23.208,

25.798)

27.101

(25.587,

29.436)

29.031 (27.121,

32.142)

Constant

Qiqihaer

(25 mm)

68.944

(62.709，

78.611)

81.291

(72.253，

97.540)

94.475

(81.618，

120.388)

113.284

(93.649，

157.913)

128.636 (102.475，

193.067)

Rainy

Qiqihaer

(45 mm)

69.788

(63.743，

77.935)

81.472

(73.038，

95.772)

93.415

(82.286，

119.709)

109.607

(93.368，

163.860)

122.169 (101.131，

209.891)

Dry

Qiqihaer

(10 mm)

14.949

(13.003，

17.821)

19.382

(16.032，

26.898)

25.119

(19.609，

47.758)

35.372

(24.564，

118.854)

45.815 (28.154，

251.963)

Comparing return levels estimated by GEV with the results of GPD, there are not significant

differences at short periods (T=5 and T=10). The apparent differences occurs at the long return

periods (after T=20). The confidence intervals obtained using GEV are higher compared to GPD.

Especially, the upper limits of GPD are much less than the upper limits of GEV for the long

return periods.

4.3 Stationary Tests

In this section, we present the results of the statistical tests introduced in section 3.5 for GEV and

GPD results. Just from graphical inspection, there are no explicit evidences of trends and no

changes in the pattern of variation in maximum precipitation. Further to graphic inspection, KPSS

tests are carried out to test for trends around a constant or a deterministic linearity. The test

statistics are revealed in Table 9. The critical values at 5% and 10% significant levels are 0.463



and 0.347 for trends around a constant, while 0.146 and 0.119 are for stationarity around a

deterministic trend, respectively. When the test statistics are smaller than the critical values, it

indicates insignificant evidence to reject the null hypotheses that there is stationarity around a

constant or a linear trend. This is the case for data from Nanjing. The result for Shantou is similar

with one exception for GPD with constant threshold. In the case of Urumchi, for all results from

GEV the null hypotheses can be rejected and non-stationarity can be assumed. This continues for

GPD and for stationarity around a linear trend which is rejected for constant and rainy season

threshold. Qiqihaer accepts stationarity for GEV, but rejects it for GPD with constant and rainy

season threshold. We can conclude that the assumption that data are stationary is appropriate for

Nanjing and Shantou. But there might be problems for Urumchi and also for Qiqihaer in some

cases which reveals limitations of the EVT approach.

Table 9. KPSS test statistics for GEV - monthly and half-yearly maxima and for GPD – constant and

seasonal thresholds; critical values: 5%: constant 0.463, linear 0.146, 10%: constant 0.347, linear 0.119


GEV Stat.

around

constant

Stat.

around

linear

trend

Stat.

around

constant

Stat.

around

linear

trend

Stat.

around

constant

Stat.

around

linear

trend

Stat.

around

constant

Stat.

around

linear

trend

Month 0.124 0.018 0.032 0.024 1.576 0.253 0.124 0.068

Half-year 0.118 0.024 0.215 0.049 0.675 0.169 0.233 0.111

GPD

Constant 0.263 0.025

0.494 0.064

0.154 0.154 0.484 0.250

Rainy 0.066 0.059 0.194 0.060 0.129 0.129 0.457 0.122

Dry 0.179 0.090 0.080 0.071 0.146 0.027 0.049 0.041

The p-values of MK test are showed in Table 10. When the p-value is larger than 0.05, the null

hypotheses cannot be rejected that neither increasing nor decreasing trends exist in the

exceedances. This is the case for almost all cities with only four exceptions, two for GPD with

constant threshold from Nanjing and Shantou. The other two exceptions are from Urumchi with

GEV where it is assumed that an increasing trend exists in the exceedances.

Table 10. MK p-values (5%) for GEV - monthly and half-yearly maxima and for GPD – constant and

seasonal thresholds


GEV Test for

positive

trend

Test for

negative

trend

Test for

positive

trend

Test for

negative

trend

Test for

positive

trend

Test for

negative

trend

Test for

positive

trend

Test

for

negativ

e trend

Month 0.193 0.807 0.247 0.753 <0.001 1.000 0.344 0.656

Half-year 0.191 0.809 0.632 0.358 0.009 0.991 0.243 0.757

GPD

Constant 0.029 0.971 0.987 0.013 0.698 0.302 0.212 0.788

Rainy 0.326 0.674 0.859 0.151 0.736 0.264 0.118 0.882

Dry 0.898 0.102 0.558 0.450 0.138 0.862 0.825 0.175

5. CONCLUSION

This paper applied the Block Maxima model with GEV and the POT approach with GPD on 60

years of daily rainfall data in four cities in China, Nanjing, Shantou, Urumchi and Qiqihaer. This

includes the estimation of parameters using MLE techniques and the calculation of return levels

for different return periods. The purpose is to support decision makers in these regions with



statistical knowledge about extreme precipitation that they can choose appropriate risk mitigating

measures to reduce the damage caused by floods.

The results from the parameter estimation show that the GPD approach is the preferable model as

it has a better goodness-of-fit performance. There are fewer deviations compared with GEV,

especially in the QQ plots. However for both models, there are several small deviations for

considerably large rainfall. The obtained return levels show that the return period of a flood event

in Nanjing is 20 to 50 years for GEV and 50 to 100 years for GPD. This takes place during the

rainy season from April to November, but it is highly unlikely to observe a flood in dry season

from December to March. Shantou experiences a flood event caused by heavy rain every 5 to 10

years for GEV and every 10 to 20 years in rainy seasons for GPD. Every 50 years, it is further

possible to see a flood during dry season. To sum up, extreme rainfall events can be predicted by

the analysis of EVT. GEV and GPD are proper approaches to estimate return levels. This paper

suggests using the GPD approach.

The limitations of this study are first, that it considers extreme rainfall without removing clusters.

The classical GEV and GPD models assume that the observations are independent, but fail to take

account of dependency in climate data in reality. Thus, further studies should use observations

after declustering. Furthermore, GEV and GPD models need to be refined because of the

existence of seasonality. Besides that, the spatial homogeneity of extreme rainfall for a region is

valuable to test, which will be useful for choosing the best model from GEV and GPD for each

city in a region. Instead of MLE, other estimation techniques such as L-moments should be

considered for this purpose. As the first results of this paper are promising, future research is

needed to overcome those limitations.

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[34] Withers, C. S., and Nadarajah, S., Evidence of trend in return levels for daily rainfall in

New Zealand. J. Hydrol.(NZ), 39(2), pp. 155-166, (2000).

[35] Yang, E., Extreme weather - heavy rainfall. Retrieved from:

http://www.niwa.co.nz/natural-hazards/extreme-weather-heavy-rainfall , (2012).

[36] Yuan, W.P. and Zhou, G.S., Comparison between standardized precipitation and Z-index

in China. Acta Phytoecologica Sinica. 28 (4), pp. 523-529, (2004).

[37] Zhai, P., Zhang, X., Wan, H., and Pan, X., Trends in total precipitation and frequency of

daily precipitation extremes over China. Journal of Climate, 18(7), pp. 1096-1108, (2005).




Manuela Ender works as a lecturer at Xi’an Jiaotong-Liverpool University

in Suzhou China. She got her PhD from the University of Cologne, Germany

and worked as research manager and management consultant for a software

company that is ranked among the companies leading the market for financial

software solutions in Germany. Her research interests lie in the area of

financial derivatives and risk management.

Tong Ma holds a double degree in BEcon Financial Mathematics from the

Xi’an Jiaotong-Liverpool Univeristy and in BSc in Financial Mathematics

from the University of Liverpool. Her research interests lie in the area of

mathematical finance, especially derivatives pricing and hedging.




www.arcjournals.org

©ARC Page | 37

On some generalized results on -summability

factors of Infinite Series


Department of mathematics,

IFTM University Moradabad (U.P.), India

[email protected]

Abstract: In this paper I have proved a theorem on | , , , |n kN p summability factors which

generalizes some previous known results and gives some unknown result.

Keywords: Weighted mean Summability, Summability factors, Infinite series.

AMS subject classification: 40D15, 40F05, 40G99.

1. INTRODUCTION

Let ( )n be sequence of positive real number, let na be a given infinite series with partial sums

( )ns and ( )nt denote the n-th Cesaro means of the sequence ( )nna . Then the series na is said

to be summable | ,1| , 1kC k if (Flett [3]).

1

1| |kn

n

tn

(1.1)

and it is said to be summable | ,1| , 1kC k if (Seyhan [6]).

1

1

| |k

knnk

n

tn

(1.2)

If we are taking , | ,1|n kn C -summability reduces to | ,1|kC -summability.

Let ( )np be a sequence of positive numbers such that

1

n

n v

v

P p

as n (1.3)

The sequence to sequence transformation

1

1 n

n v v

vn

u p sP

(1.4)

defines the sequence ( )nu of the ( , )nN p mean of the sequences ( )ns generated by the sequence

of coefficients ( )np (Hardy [4]).

The series na is said to be summable | , | , 1n kN p k if (Bor [1])




1

1

1

| |

k

knn

n n

Pu

p

(1.5)

and it is said to summable | , , | , 1n kN p k an d 0 if (Bor [2])

1

1

1

| |

k k

knn

n n

Pu

p

(1.6)

where 1 1 1

11

: 1n

nn n n v v

vn n

pu u u p a n

P P

and na is said to summable | , , , | , 1, 0n kN p k and 1 if

( 1)

1

1

| |

k k

knn

n n

Pu

p

(1.7)

and it is said to summable | , , , | , 1, 0, 1n kN p k

( 1)

1

1

( ) | |k k k

n n

n

u

(1.8)

If nn

n

P

p then | , , |n kN p -summability reduces to | , , , |n kN p -summability and if

n n O and 1 then | , , , |n kN p -summability reduces to | ,1|kC -summability.

2. KNOWN RESULTS

Concerning | ,1|kC -summability, Mazhar [5] has proved the following theorem.

Theorem 2.1

If d (1)m O , as m (2.1)

2

1

log | | (1)m

n

n

n n O

, as m (2.2)

1

| |(log )

kmv

v

tO m

v

as m (2.3)

then the series n na is summable | ,1| , 1kC k .

And Sulaiman (7) has proved the following theorem.

Theorem 2.2 Let ( )n and ( )nX be sequences of positive real numbers such that ( )nX is non

decreasing and condition (2.1) is satisfied

If ( ), ( )n n n nnp O P P O np , as n (2.4)

1 ( )n nO (2.5)

1( )n nO n as n (2.6)

2

1

| | (1)n n

n

nX O

(2.7)

On some generalized results on -summability factors of Infinite Series


1

11

| | | |( )

k k kmn v n

mk kn n

sO X

n X

as m (2.8)

1 1

1

1

k kmn v

k kn v n v

Ov P v P

then the series n n na is summable | , | , 1n kN p k .

3. MAIN RESULTS

The aim of this paper is to generalize the theorem (2.2), here I have proved the following theorem.

Theorem 3.1 Let ( )n and ( )nX be sequences of positive real numbers such that ( )nX is non

decreasing and if the conditions (2.1), (2.4), (2.5), (2.6), (2.7) are satisfied.

( 1)

11

( ) | | | |( )

k k k kmn v n

mk kn n

sO X

n X

as m (3.1)

( 1) ( 1)

1

1

( ) k k k kmn v

k kn v n v

Ov P v P

(3.2)

Then the series n n na is summable | , , , | , 1, 1n kN p k and 0 .

4. LEMMA

To prove the above theorem following Lemma is required.

Lemma 4.1 Sulaiman [7] The conditions (2.1) and (2.7) implies.

1

| | (1)n n

n

X O

(4.1)

| | O(1)n nnX as n (4.2)

| | (1)n nX O as n (4.3)

5. PROOF OF THE THEOREM 3.1

Let nT be the ( , )nN p mean of the series1

n n n

n

a

, we have

1 1

1

1

1

1( )

n v

n v r r r

v rn

n

v v v v v

vn

T p aP

P P aP

And hence

1 1

11

, 1n

nn n v v v v

vn n

pT T P a n

P P

.

Using Abeles transformation, we have



1

1 1

11

1

1

11

,1 ,2 ,3 ,4

( ) s

( )

nn n

n n v v v v n n n

vn n n

nn

v v v v v v v v v v v v

vn n

n n n n

p pT T s P

P P P

pp s P s P s

P P

T T T T

Since 4

1 2 3 4 1 2 3 4| | 4 (| | | | | | | | )k k k k k

n n n n n n n nT T T T T T T T

In order to complete the proof, it is sufficient to show that

( 1)

,

1

( ) | | , 1,2,3,4k k k

n n r

n

T r

Applying Hölders inequality, we have

1

( 1) ( 1)

,1

2 2 11

( 1) 1

2 11

( 1) 1 1

2 1 11 1

( ) | | ( )

( )(1) | || || |

( )(1) | | | | | |

km m n

nk k k k k

n n n v v v v

n n vn n

kk k km nn n

v v v vk kn vn n

k k km n nk k kn n v

v v v vkn v vn n n

pT p s

P P

pO p s

P P

p pO p s

P P P

1

( 1)

1 1

( 1)

1 1

( 1)

11

11

( )(1) | | | | | |

( )(1) | | | | | |

( )(1) | | | | | |

( )(1) | | | | | | | |

k

k k km mk k k n n

v v v v k kv n v n n

k km mk k k n

v v v v kv n v n

k kmk k kv v

v v vkv v

k kk k kv

v v v vk

pO p s

P P

O p sn P

pO s

v p

O sv

1

( 1)

11

( 1)

11

( 1)

11 1

1

( )(1) | | | | | |

( )(1) | | | | | |

( )(1) | | | | | |

(1) | |

(1).

m

v

k kmk kv

v v nk kv n vv

k kmk kv

v v nk kv n vv

k knk kv

n v vk kn v v

n n

n

O sv X

O sv X

O sv X

O X

O



1( 1) ( 1)

,2

2 2 11

( 1) 1

1 11

( 1) 11

1 11

( ) | | ( )

( )(1) | || || |

( )(1) | || || |

(

km m n

nk k k k k

n n n v v v v

n n vn n

kk k km nn n

v v v vk kn vn n

kk k km nn n

v v v vk kn vn n

pT p s

P P

pO p s

P P

pO v p s

P P

O

1( 1) 1 1

1

1 1 11 1

( 1)

1 1

( 1)

1 1

1) | | | | | |

( )(1) | | | | | |

( )(1) | | | | | |

((1)

k kk k km n n

k k kn n v vv v v vk

n v vn n v n

k k km mk k k n n

v v v v kv n v n n

k km mk k k n

v v v v kv n v n

p P pv p s

P P p P

pO p s

P P

O p sv P

O

( 1)

11

1

1

,1.

)| | | | | |

( )(1) | | | | | |

(1), as in the case of

k kmk k kv v

v v vkv v

k kmk k kv

v v vkv

n

ps

v P

O sv

O T

11 ( 1)

,2 1

2 2 11

( 1)1 1

1 11

( 1) 1

11 11

| | (1) ( )

( )(1) | || || |

( )(1) | | | || | |

km m n

nk k k k

n n n v v v v

n n vn n

kk k km nn n

v v v vk kn vn n

k k k km nk kn n v

v v v vk k kn vn n v

pT O p s

P P

pO p s

P P

p PO s X

P P X

1

1

1

( k k 1)1

11 1 1

( 1)1

11 1 1

( 1)

11 1

|

( )(1) | | | || |

( )(1) | | | || |

( )(1) | | | || |

kn

v

v

k km mk kv n n

v v vk k kn n vv n

k k km mk kv n

v v vk kv n vv n

k kmk kv n

v v vk kv n vv n

P pO s

X v P

PO s

X v P

PO s

X v P

1

1

( 1)

1 11

1

11

( 1)1

11 1

( )(1) | | | || |

( ) | | | |(1) | |

( ) | | | |(1) ( | |)

m

k kmk kv

v v vk kv v

k k k kmv v v

v k kv v

k k k km vv r r

v k kv r r

O sv X

sO v

v X

sO v

r X



( 1)

11

1 12

1 1

( ) | | | |(1) | |

(1) | | (1) | | (1) | |

(1)

k k k kmv v v

m k kv v

m m

v v v v m m

v v

sO m

v X

O v X O X O m X

O

( 1) ( 1)

,4

1 1 1

( 1) 1

1

( 1)

11

( k k 1)

1

( ) | | ( )

(1) ( ) | | | | | | | |

( ) | | | |(1) | |

( ) | | |(1) | |

km m

nk k k k k

n n n n n n

n n n n

km

k k k k knn n n n n

n n

k k k kmn n n

vk kn v nn

kmn n n

n

v

pT s

P P

pO s

P

sO

n X

sO

11

1

|

(1) | |

(1)

kv

k kn n

m

v v

v

n X

O X

O

This completes the proof of the theorem.

6. COROLLARY

This theorem have the following results as corollaries.

Corollary 6.1

If we are taking n

n

P

p then the infinite series n n na is | , , , |n kN p -summable

0, 1 and 1.k

Corollary 6.2

If we are taking 0, 1 then the infinite series n n na is | , |n kN p -summable, 1k .

Corollary 6.3

If we are taking 0, 1, n

n

P

p then the infinite series n n na is | , |n kN p -summable

1k .

Corollary 6.4

If we are taking n then the infinite series. n n na is | ,1, , |kC -summable 0, 1

and 1.k

Corollary 6.5

If we are taking n , 0, 1 then the infinite series n n na is | ,1|kC -summable,

1.k



7. CONCLUSION

The results of this theorem is more general rather than the results of any other previous proved

theorem, which will be enrich the literate of summability theory of infinite series.

ACKNOWLEDGEMENTS

I am very thankful to Dr. B.K. Singh (Professor and Head of the Department of Mathematics,

IFTM University Moradabad,U.P., India),whose great inspirations lead me to complete this paper.

REFERENCES

[1] Bor. H; On two summability methods. Math. Proc. Camb. Philos Soc. 97 (1985).

[2] Bor. H; On local property of | , , |n kN p -summability of factored Fourier series. J. Math.

Anal. Appl. 179 (1993).

[3] Flett. T.M; On an extension of absolute summability and some theorems of Littlewood and

Paley, Proc. London Math. Soc. 7 (1957).

[4] Hardy, G.H; Divergent Series, Oxford Univ. Press, Oxford (1949).

[5] Mazhar S.M; On | ,1|kC summability factors of infinite series. Indian J. Math 14 (1972).

[6] Seyhan H; The absolute summability methods, Ph.D. Thesis Kayseri (1995).

[7] SulaimanW.T; On some absolute summability factors of infinite series. Gen. Math. Notes.

Vol.2 (2011).


Mr. Aditya Kumar Raghuvanshi is presently a research scholar in the

department of Mathematics, IFTM university Moradabad, U.P., India. He

has completed his M.Sc.(maths) and M.A.(Economics) from MJPR

University Bareilly,U.P.,India, B.Ed. from CCS Uni. Meerut, U.P.,

India and he has also compeleted his M. Phil. (Maths) from The Global

Open University Nagaland, India. He has published fifteen Research

papers in various International journals. His fields of research are O.R.,

Summability and Approximation Theory.


Volume 2, Issue 1, January- 2014, PP 44-50


www.arcjournals.org

©ARC Page | 44

The h-Integrability and the Weak Laws of Large Numbers for

Arrays

S. O. Edeki Department of Industrial Mathematics,

Covenant University, Cannanland,

Otta, Nigeria.

[email protected]

S. A. Adeosun Department of Mathematical Sciences,

Crescent University,

Abeokuta, Nigeria.

[email protected]

Abstract: In this paper, the concept of weak laws of large numbers for arrays (WLLNFA) is studied, and a

new notion of uniform integrability referred to as h-integrability is introduced as a condition for WLLNFA

in obtaining the main results.

Keywords: Weak Laws of Large Numbers, Uniform Integrability, Boundedness, h-Integrability and

Convergence.

1. INTRODUCTION

The law of large numbers (LLN) as a theorem describes the result of performing the same

experiment a large number of times. According to this, the average of the results obtained from a

large number of trials should be close to the expected value, and have a tendency to become

closer as more trials are performed. The LLN ensures stable long-term results for the averages of

random events. It only applies when a large number of observations are considered. While the

weak law of large numbers (WLLN) states that the sample average converges in probability

towards the expected value.

The concept of uniform integrability has been a core aspect in applied probability with regard to

the theory of convergence of weak law of large numbers. Chandra [1] obtains the weak law of

large numbers under a new condition known as the Cesa´ro uniform integrability, which is weaker

than uniform integrability. Cabrera [2] studies the weak convergence of weighted sums of random

variables and introduces the condition of uniform integrability concerning the weights, which is

weaker than uniform integrability; this leads to a special case known as Cesa´ro uniform

integrability.

On the condition of -uniform integrability, Cabrera [2] obtains the weak law of large

numbers for weighted sums of pairwise independent random variables. Sung [3] introduces the

concept of Cesa´ro type uniform integrability with exponent . Chandra and Goswami [4]

introduce the concept of Cesa´ro α-integrability (α > 0), and show that Cesa´ro α-integrability for

any α > 0 is weaker than Cesa´ro uniform integrability.

Cabrera and Volodin [5] introduce the notion of h-integrability for an array of random variables

concerning an array of constant weights, and prove that this concept is weaker than Cesa´ro

uniform integrability, - uniform integrability and Cesa´ro α-integrability, and also show that

h-integrability concerning the weights is sufficient for the weak law of large numbers to hold for

weighted sums of an array of random variables, when these random variables are subject to some

special kind of rowwise dependence.

Sung et al [6] remarked that the main idea of notions of -uniform integrability introduced in

Cabrera [2] and h-integrability with respect to the array of constants introduced in Cabrera

and Volodin [5] is to deal with weighted sums of random variables. Sung et al in [7] introduce a

new concept of integrability which deals with usual normed sums of random variables. Adeosun



http://en.wikipedia.org/wiki/Convergence_in_probability

S. O. Edeki & S. A. Adeosun


and Edeki [8], on a survey of uniform integrability of sequences of random variables, noted some

new conditions required for such. In this paper, we study and present the basic theorem regarding

uniform integrability, in terms of weak laws of large numbers for arrays.

2. PRELIMINARY DEFINITIONS AND LEMMAS

In this section, the needed technical definitions and lemmas for the main results will be discussed

accordingly, with their corresponding proof, if need be.

Definition 2.1 An array of random variables is said to be

Cesa´ ro uniform integrability if

where is a sequence of positive integers such that as n → ∞.

Definition 2.2 Let be an array of random variables and

an array of constants with for all ∈ and some

constant . The array is -uniform integrable if

-uniform integrability tends to Cesaro uniform integrability when ,

,

Definition 2.3 Let } be an array of random variables and

>0. The array is said to be Cesa´ro type uniformly integrable with

exponent if

We note that the conditions of Cesa´ro uniform integrability and Cesa´ro type uniformly

integrable with exponent are equivalent when , , and . Sung [3]

obtains the weak law of large numbers for an array satisfying Cesa´ro type uniform

integrability with exponent for some .

Definition 2.4 For any , a sequence of random variables is said to be

Cesa´ro α-integrable if

Definition 2.5 Let be an array of random variables and

an array of constants with for all ∈ and some

constant . Let moreover be an increasing sequence of positive constants

with ↑ ∞ as n ↑ ∞. The array is said to be h-integrable with

respect to the array of constants if

The H-Integrability and the Weak Laws of Large Numbers for Arrays.


Definition 2.6 Let be an array of random variables and . Let

moreover, be an increasing sequence of positive constants with ↑ ∞ as n ↑ ∞.

The array is said to be h-integrable with exponent r if

Lemma 2.1 [6] If the array satisfies the condition of Cesa´ro type

uniform integrability with exponent r > 0, then it satisfies the condition of h-integrability with

exponent r.

Proof: We note that the first condition of the Cesa´ro type uniform integrability with exponent r

and the first condition of the h-integrability with exponent r are the same. Hence, it suffices to

show that the second condition of Cesa´ro type uniform integrability with exponent r implies the

second condition of h-integrability with exponent r. If satisfies the Cesa´ro type uniform

integrability with exponent r, then there exist A > 0 such that

if . Since ↑ ∞ as , such that

if . For ,

.

Hence the second condition of h-integrability with exponent r is satisfied

Note 2.1. As noted by [6], the concept of h-integrability with exponent r is strictly weaker than

the concept of Cesa´ro type uniform integrability with exponent r, i.e., there exists an array

which is h-integrability with exponent r, but not Cesa´ro type

uniform integrability with exponent r.

Remark 2.1 In considering an array of random variables defined on

a probability space and setting , and

, authors like [10], [9], [7], [5], [3] established weak laws of large

numbers. Gut [9] proved that, for some ,

if is an array of Cesa´ro uniformly

integrable random variables, where if and if

.

Lemma 2.2[3] Suppose that is an array of h-integrability with

exponent r for some , , ↑ ∞, and → 0. Then the following statements

hold.

(i) if ,

(ii) if .

3. UNIFORM LAWS OF LARGE NUMBERS AND CONVERGENCE



This section deals with uniform laws of large numbers as condition for convergence of a sample

mean of random variables on a compact space, and the general WLLNFA.

3.1 Uniform Laws of Large Numbers (ULLN) and a Compact Euclidean Space

Let and be two Euclidean sets with a Cartesian product . Define on a real-

valued function such that is Lebesgue measurable for every . Then for any

fixed , the sequence will be a sequence of independent and identically distributed (iid)

random variables on with its sample mean converges in probability to . A uniform

weak law of large numbers therefore defines a set of conditions on which

Theorem 3.1: If (a) is compact, (b) is continuous at each with probability

one, (c) , and (d) . Then:

Proof: Let . So

as since

(i) a.s. by condition (b).

(ii) by condition (c), and condition (d).

Hence, for all and , there exists such that .

Obviously, the whole parameter space , is covered by . So since, is

compact, we can find a finite subcover, such that is covered by .

We recall that

,

Note. The first equality holds by the WLLN, since , and

the last inequality follows the definition of , with arbitrarily chosen.

Whence,

.

Remark 3.1

The uniform law of large numbers states the condition under which the

convergence happens uniformly.



Uniform strong law of large numbers defines such set of conditions if the

convergence satisfies almost surely instead of in probability.

Theorem 3.2 Suppose that is an array of h-integrability with

exponent random variables, , and . Then

in and, hence, in probability as , where if

and if .

Proof: Let and . First we consider the

case . Since , it follows that

.

By the Cr-inequality, Jensen’s inequality, and Lemma 2.2 with and , we have

,

which completes the proof for the case .

Now considering the case of , it is observed that:

.

Hence, using the Cr-inequality, Burkholder’s and Davis’ inequalities [12]; [13] for

and, respectively), Jensen’s inequality, and Lemma 2.2 ( i.e. Lemma 1, Sung [3]) with

and , we obtain

.



where is a constant depending only on . Hence, the proof of the case is done

Corollary 3.1 Suppose that is an array of random variables

satisfying the Cesa´ro type uniform integrability with exponent 0 < r < 2 and → ∞. Then

in and, hence, in probability as n → ∞, where = 0 if

and if .

Proof: By lemma2.1 [7] the condition of Cesa´ro type uniform integrability with exponent r

implies the condition of h-integrability with exponent r, and so the result follows from Thrm 3.2.

4. CONCLUDING REMARKS

In conclusion, weak laws of large numbers for the array of dependent random variables satisfying

the condition of h-integrability with exponent r is obtained. The work extends and sharpens

previous results with regard to applied probability, in terms of theorems and conditions for

boundedness, and convergence of laws of large numbers with applications to stationary data.

REFERENCES

[1] T.K. Chandra, ‘‘Uniform integrability in the Cesro sense and the weak law of large

numbers’’, Sankhya Ser. A 51 (3),(1989) 309- 317.

[2] M. Cabrera, ‘‘Convergence of weighted sums of random variable and uniform integrability

concerning the weights’’, Collect. Math. 45 (2), (1994), 121 - 132.

[3] S.H. Sung, ‘‘Weak law of large numbers for arrays of random variables’’, Statist. Probab.

Lett 42 (3), (1999), 293-298.

[4] T.K. Chandra and A. Goswami, ‘‘Ces ro α- integrability and laws of large numbers Int.,J.

Theoret. Prob. 16 (3), (2003), 655- 669.

[5] M. Cabrera and A. Volodin, ‘‘Mean convergence theorems and weak laws of large numbers

for weighted sums of random variables under a condition of weighted integrability’’, J. Math.

Anal. Appl. 305(2), (2005), 644 -658.

[6] S.H. Sung, S. Lisawadi, and A. Volodin, ‘‘Weak laws of large numbers for arrays under a

condition of uniform integrability’’, J. Korean Math. Soc. 45 (1),(2008), 289-300.

[7] S.H. Sung, T.C. Hu, and A. Volodin, ‘‘On the weak laws for arrays of random variables’’,

Statist. Probab. Lett. 72 (4),(2005),291-298.

[8] S. A. Adeosun and S. O. Edeki ‘‘On a Survey of Uniform Integrability of Sequences of

Random Variables’’ Int. J. Math & Stat Stud. 2 (1),(2014),1-14.

[9] A. Gut. ‘‘The weak law of large numbers for arrays’’, Statist. Probab. Lett.14 (1), (1992),49-

52.

[10] D. H. Hong and K.S. Oh, On the weak law of large numbers for arrays, Statist. Probab.

Lett. 22 (1),(1995),55-57.

[11] D. Burkholder, ‘‘Martingales transforms’’, Ann. Math. Statist. 37 (1966), 1497-1504.

[12] B. Davis, ‘‘On the integrability of the martingale square function’’, Israel J. Math. 8

(1970),187-190.

[13] A. Leblanc and B. C. Johnson, ‘‘A family of inequalities related to binomial prob-

abilities’’. Department of Statistics, University of Manitoba. Tech. Report, 2006.

[14] Newey W.K. ‘‘Uniform Convergence in Probability and Stochastic Equicontinuity’’

Econometrica 59(4) (1991), 1161-1167.

[15] B.M. Brown, ‘‘Moments of a stopping rule related to the central limit theorem’’. Ann.

Math. Statist. 40 (2004), 1236 - 1249.

[16] D. Li, A. Rosalsky, and A. Volodin, ‘‘On the strong law of large numbers for sequences

of pairwise negative quadrant dependent random variables’’, Bull. Inst. Math. Acad. Sin.

(N.S) 1 (2), (2006), 281-305.



[17] Jennrich,R.I. ‘‘Asymptotic Properties of non-linear least squares Estimators’’,The

Annals of Mathematical Statistics 40(1969), 633-643.

[18] J. Bae, D. Jun, and S. Levental, ‘‘The uniform clt for martingale diff erence arrays under

the uniformly integrable entropy’’, Bull. Korean Math. Soc. 47(1), (2010),39-51.


Volume 2, Issue 1, January- 2014, PP 51-57


www.arcjournals.org

©ARC Page | 51

Ideals of Almost Distributive Lattices with respect to a

Congruence

Noorbhasha Rafi Department of Mathematics,

Bapatla Engineering College Bapatla,

Andhra Pradesh, India-522101

[email protected]

Abstract: The concept of -ideals is introduced in an Almost Distributive lattice(ADL) with respect to a

congruence and the properties of -ideals are studied. Derived a set of equivalent conditions for a -ideal

to become a -prime ideal.

Keywords: Almost Distributive Lattice (ADL), congruence, ideal , Prime filter, -ideal; -Prime ideal

AMS Mathematics Subject Classification (2010): 06D99,06D15

1. INTRODUCTION

After Booles axiomatization of two valued propositional calculus as a Boolean algebra, a number

of generalizations both ring theoretically and lattice theoretically have come into being. The

concept of an Almost Distributive Lattice (ADL) was introduced by Swamy and Rao [6] as a

common abstraction of many existing ring theoretic generalizations of a Boolean algebra on one

hand and the class of distributive lattices on the other. In that paper, the concept of an ideal in an

ADL was introduced analogous to that in a distributive lattice and it was observed that the set

PI(L) of all principal ideals of L forms a distributive lattice. This enables us to extend many

existing concepts from the class of distributive lattices to the class of ADLs. Swamy, G.C. Rao

and G.N. Rao introduced the concept of Stone ADL and characterized it in terms of its ideals. In

[3], N. Rafi, G.C. Rao and Ravi kumar Bandaru introduced -filters in an Almost Distributive

Lattices and proved their properties. The usual lattice theoretic duality principle doesn’t hold in

ADLs. For example, in an ADL L, ∧ is right distributive over ∨ but ∨ is not right distributive

over ∧ . So that in this paper, the concept of -ideals is introduced in an ADL and then

characterized in terms of ADL congruences. Also the concept of -prime ideals is introduced and

established a set of equivalent conditions for every -ideal to become a -prime ideal. Some

properties of -ideals and -prime ideals are studied. The class of all -ideals of an ADL can be

made into a bounded distributive lattice. Finally, the prime ideal theorem is generalized in the

case of -prime ideals in an ADL.

2. PRELIMINARIES

Definition 2.1.[6] An Almost Distributive Lattice with zero or simply ADL is an algebra (L, ∨,

∧, 0) of type (2, 2, 0) satisfying

1. (x ∨ y) ∧ z = (x ∧ z) ∨ (y ∧ z)

2. x ∧ (y ∨ z) = (x ∧ y) ∨ (x ∧ z)

3. (x ∨ y) ∧ y = y

4. (x ∨ y) ∧ x = x

5. x ∨ (x ∧ y) = x

6. 0 ∧ x = 0

7. x ∨ 0 = x, for any x, y, z ∈ L.


Noorbhasha Rafi


Every non-empty set X can be regarded as an ADL as follows. Let x0 ∈X. Define the binary

operations ∨, ∧ on X by

Then (X, ∨, ∧, x0) is an ADL (where x0 is the zero) and is called a discrete ADL. If (L, ∨, ∧, 0) is

an ADL, for any a, b ∈ L, define a ≤ b if and only if a = a ∧ b (or equivalently, a ∨ b = b), then ≤

is a partial ordering on L.

Theorem 2.2: ([6]) If (L, ∨, ∧, 0) is an ADL, for any a, b, c ∈ L, we have the following:

(1) a ∨ b = a ⇔ a ∧ b = b

(2) a ∨ b = b ⇔ a ∧ b = a

(3) ∧ is associative in L

(4) a ∧ b ∧ c = b ∧ a ∧ c

(5) (a ∨ b) ∧ c = (b ∨ a) ∧ c

(6) a ∧ b = 0 ⇔ b ∧ a = 0

(7) a ∨ (b ∧ c) = (a ∨ b) ∧ (a ∨ c)

(8) a ∧ (a ∨ b) = a, (a ∧ b) ∨ b = b and a ∨ (b ∧ a) = a

(9) a ≤ a ∨ b and a ∧ b ≤ b

(10) a ∧ a = a and a ∨ a = a

(11) 0 ∨ a = a and a ∧ 0 = 0

(12) If a ≤ c, b ≤ c then a ∧ b = b ∧ a and a ∨ b = b ∨ a

(13) a ∨ b = (a ∨ b) ∨ a.

It can be observed that an ADL L satisfies almost all the properties of a distributive lattice except

the right distributivity of ∨ over ∧, commutativity of ∨, commutativity of ∧. Any one of these

properties make an ADL L a distributive lattice.

Theorem 2.3. ([6]) Let (L, ∨, ∧, 0) be an ADL with 0. Then the following are equivalent:

(1) (L, ∨, ∧, 0) is a distributive lattice

(2) a ∨ b = b ∨ a, for all a, b ∈ L

(3) a ∧ b = b ∧ a, for all a, b ∈ L

(4) (a ∧ b) ∨ c = (a ∨ c) ∧ (b ∨ c), for all a, b, c ∈ L.

As usual, an element m ∈ L is called maximal if it is a maximal element in the partially ordered

set (L, ≤). That is, for any a ∈ L, m ≤ a ⇒ m = a.

Theorem 2.4: ([6]) Let L be an ADL and m ∈ L. Then the following are equivalent:

(1) m is maximal with respect to ≤

(2) m ∨ a = m, for all a ∈ L

(3) m ∧ a = a, for all a ∈ L

(4) a ∨ m is maximal, for all a ∈ L.

As in distributive lattices [[1], [2]], a non-empty sub set I of an ADL L is called an ideal of L if

a b I and a x I for any a, b I and x L. Also, a non-empty subset F of L is said to be a

filter of L if a b F and x a F, for a, b F and x L. The set I(L) of all ideals of L is a

bounded distributive lattice with least element {0} and greatest element L under set inclusion in

which, for any I, J ∈ I(L), I J is the infimum of I and J while the supremum is given by I J :=

{a b | a I, b J}. A proper ideal P of L is called a prime ideal if, for any x, y L, x y P

⇒ x P or y P. A proper ideal M of L is said to be maximal if it is not properly contained in

any proper ideal of L. It can be observed that every maximal ideal of L is a prime ideal. Every

proper ideal of L is contained in a maximal ideal. For any subset S of L the smallest ideal

containing S is given by

Ideals of Almost Distributive Lattices with respect to a Congruence


(S] := {1

( )n

ii

s x | is S, x L and n N}. If S = {s}, we write (s] instead of (S]. Similarly,

for any S ⊆ L, [S) := {1

( )n

ii

x s

| is S, x L and n N}. If S = {s}, we write [s) instead of

[S).

Theorem 2.5 ([6]). For any x, y in L the following are equivalent:

1). (x] (y]

2). y x = x

3). y x = y

4). [y) [x).

For any x, y L, it can be verified that (x] (y] = (x y] and (x] (y] = (x y]. Hence the set

PI(L) of all principal ideals of L is a sublattice of the distributive lattice I(L) of ideals of L.

Theorem 2.6([4]). Let I be an ideal and F a filter of L such that I F = . Then there exists a

prime ideal P such that I P and P F = .

Definition 2.7 (4]). An equivalence relation on an ADL L is called a congruence relation on L

if (a c, bd), (a c, b d) , for all (a, b), (c, d) .

Definition 2.8 ([4]). For any congruence relation on an ADL L and a L, we define [ ]a = {b

L | (a, b) } and it is called the congruence class containing a.

Theorem 2.9 ([4]). An equivalence relation on an ADL L is a congruence relation if and only

if for any (a, b) , x L, (a x, b x), (x a, x b), (a x, b x), (x a, x b) are

all in .

3. -IDEALS IN AN ADL

The concept of -filters in an Almost Distributive Lattice was given by Rafi, Rao and Ravi

Kumar [3]. The usual lattice theoretic duality principle doesn’t hold in ADLs. For example, in an

ADL L, ∧ is right distributive over ∨ but ∨ is not right distributive over ∧ . So that we introduce

the concept of -ideals in an ADL and study their important properties. Throughout this paper L

represents an ADL with 0.

Now we begin with the definition of a -ideal in an ADL L.

Definition 3.1: Let be a congruence relation on an ADL L. An ideal I of L is called a -ideal

of L, if for any a I that implies [ ]a I .

For any congruence on a ADL L, it can be easily observed that the zero ideal {0} is a -ideal

if and only if [0] = {0}.

The following lemma can be verified easily.

Lemma 3.2. Let be a congruence on L and m be any maximal element of L. For any ideal I of

L, the following hold:

1. If I is a -ideal, then [0] ⊆ I;

2. If I is a proper -ideal, then I ∩[ ]m = .

3. If is the smallest congruence then every ideal is a -ideal;

Example 3.3: Let D = {0 ' , a ' } be a discrete ADL and R a distributive lattice whose Hasse

diagram is given in the figure. Then

L = D ×R = {(0 ' , 0), (0 ' , a), (0 ' , b), (0 ' , c), (0 ' , 1), (a ' , 0), (a ' , a), (a ' , b), (a ' , c), (a ' , 1)} is

an ADL under point-wise operations.

Noorbhasha Rafi


Distributive Lattice

Take

= {((0 ' , 0), (0 ' , 0)), ((0 ' , a), (0 ' , a)), ((0 ' , b), (0 ' , b)), ((0 ' , c), (0 ' , c)),

((0 ' , 1), (0 ' , 1)), ((a ' , 0), (a ' , 0)), ((a ' , a), (a ' , a)), ((a ' , b), (a ' , b)),

((a ' , c), (a ' , c)), ((a ' , 1), (a ' , 1)), ((0 ' , c), (0 ' , 1)), ((0 ' , 1), (0 ' , c)) }. Clearly is a congruence

relation on L. Consider the ideal I = {(0 ' , 0), (0 ' , a)}. Clearly I is a -ideal of L.

But J = {(0, 0), (0, a), (0, b), (0’, c)} is not a -ideal, because [(0’, c) ] J.

Theorem3.4: Let be a congruence relation on an ADL L. For any ideal I of L, the following

conditions are equivalent:

1. I is a -ideal

2. For any , ,x y L ( , )x y and x I y I

3. [ ] .x I

I x

Proof: (1) (2) : Assume that I is a -ideal of L. Let x, y ∈ L be such that (x, y) ∈ . Suppose

x ∈ I. Therefore we get that y ∈ [ ]x ⊆ I.

(2) (3) : Assume the condition (2). Let x ∈ I. Since x ∈ [ ]x , we get I ⊆ [ ]x I

x

. Conversely,

let a ∈ [ ]x I

x

. Then (a, x) ∈ , for some x ∈ I. By the condition (2), we get that a ∈ I.

Therefore I = [ ]x I

x

.

(3) (1) : Assume that the condition (3) holds. Let a ∈ I. Then we get (x, a) ∈ , for some x

∈ I. Let t ∈ [ ]a . Then we get (t, a) ∈ . Hence (x, t) ∈ . Thus it yields that t ∈ [ ]x ⊆ I.

Therefore I is a -ideal of L. The concept of -prime ideals is now introduced in an ADL.

Defination 3.5: Let be a congruence relation on an ADL L. A proper -ideal P of an ADL L is

called a -prime ideal of L if for any ,a b L with [0]a b either a P or .b P

Now, we have the following.

Lemma3.6: If is the smallest congruence relation on an ADL, then every prime ideal of L is a

-prime ideal.

Proof: Suppose that is the smallest congruence on L. Let P be a prime ideal of L. Then by

above lemma 3.2, P is a -ideal of L. Let a, b ∈ L be such that a∧ b ∈ [0] . Then we get that



[a∧ b ] = [0] . Since -is the smallest congruence on L, it can be concluded that a ∧ b = 0 ∈ P.

Therefore P is a -prime ideal of L.

Lemma 3.7: Let be a congruence relation on an ADL L. Then every prime -ideal of L is a

-prime ideal of L.

Proof: Let P be a prime -ideal of an ADL L. Let x, y ∈ L be such that x ∧ y ∈ [0] . Since P is

a -ideal of L, we get that x ∧ y ∈ [0] ⊆ P. Since P is a prime ideal of L, we get that either x ∈

P or y ∈ P. Therefore P is a -prime ideal of L.

Theorem 3.8: Let be a congruence relation on an ADL L and P, a -ideal of L. Then the

following conditions are equivalent:

1. P is a -prime ideal of L

2. For any ideals ,I J of L with [0]I J implies that I P or J P

3. For any , ,a b L [ ] [ ] [0]a b implies that either a P or .b P

Proof: (1) (2) : Assume that P is a -prime ideal of L. Let I, J be two ideals of L such that I ∩

J ⊆ [0] . Let a ∈ I and b ∈ J. Then a ∧ b ∈ I ∩ J ⊆ [0] . Since P is -prime, we get that either a

∈ P or b ∈ P. Thus we get that either I ⊆ P or J ⊆ P.

(2) (3) : Assume the condition (2). Suppose that [a ] ∩ [b ] = [0 ] for any a, b ∈ L. Then we

get [a ∧ b ] = [0 ] . Thus it yields that a ∧ b ∈ [0 ] and hence (a] ∩ (b] ⊆ [0 ] . Therefore by the

assumed condition (2), we get that either a ∈ (a] ⊆ P or b ∈ (b] ⊆ P.

(3) (1) : Assume that the condition (3) holds. Let a, b ∈ L be such that a ∧ b ∈ [0 ] . Hence we

get [a ] ∩ [b ] = [a ∧ b ] = [0 ] . Thus by condition (3), we get that either a ∈ P or b ∈ P.

Therefore P is a -prime ideal of L.

Now, we prove the following.

Lemma 3.9: Let be a congruence relation on an ADL L and m be any maximal element of L.

Then every maximal ideal disjoint from [m ] is a -ideal of L.

Proof: Let M be a maximal ideal of L and m be any maximal element of L such that M ∩ [m ] =

. Let x, y ∈ L be such that (x, y) ∈ and x ∈ M. Suppose y M. Then M ∨ (y] = L. That

implies a∨ y is a maximal element of L for some a ∈ M. Since (x, y) ∈ , we get that (a∨ x,

a∨ y) ∈ . Thus we can obtain that a∨ x ∈ [a∨ y ] . Since a ∨ x ∈ M, we get that M ∩ [a ∨ y ] =

, which is a contradiction. Therefore y ∈ M, which yields that M is a -ideal of L.

The following Corollary is a direct consequence of the above.

Corollary 3.10: Let be a congruence relation on an ADL L and m be any maximal element of

L. If [ ] { },m m then every maximal ideal of L is a -ideal of L.

Now, we have the following definition.

Definition 3.11: Let be a congruence relation on an ADL L. For any ideal I of L, define the set

I as given by { | ( , ) , for some }I x L x a a I .

Lemma 3.12: Let be a congruence relation on an ADL L. For any ideal I of L, the set I is an

ideal of L.

Proof: Clearly, 0 ∈ I . Let x, y ∈ I . Then we get (x, a) ∈ and (y, b) ∈ , for some a, b ∈ I.

Hence we get (x ∨ y, a ∨ b) ∈ . That implies x ∨ y ∈ I . Again, let x ∈ I and r ∈ L. Then (x,

a) ∈ , for some a ∈ I. Since is a congruence, we get (x ∧ r, a ∧ r) ∈ . Since a ∧ r ∈ I, we

get x ∧ r ∈ I . Therefore I is an ideal of L.

Noorbhasha Rafi


Lemma 3.13: Let be a congruence relation on an ADL L. For any two ideals I, J of L, we have

the following:

1. I I

2. I J implies I J

3. ( )I J I J

4. ( ) .I I

Proof: 1. Let a I . We have ( , )a a and hence a I . Therefore I I .

2. Suppose that I J . Let x I . Then ,x a , for some a I . Since I J , we get

( , )x a and a J . Therefore .x J Hence .I J

3. Clearly ( ) .I J I J Conversely let .x I J This implies ( , ),( , )x a x b for some

a I and .b J So that ( , )x a b and .a b I J Implies that ( ) .x I J Therefore

( ) .I J I J

4. Clearly ( ) .I I Again, let ( ) .x I Then ( , ) ,x a for some .a I Since ,a I we have

( , ) ,a b for some .b I This implies ( , ) ,x b b I and hence .x I Therefore ( ) .I I

Thus ( ) .I I

Proposition 3.14: Let be congruence relation on an ADL L. For any ideal I of L , I is the

smallest -ideal of L such that .I I

Proof: From Lemma 3.12 and Lemma 3.13(1), we get that I is a -ideal of L containing the

ideal I. Let K be a -ideal of L such that I ⊆ K. Let x ∈ I . Then we get (x, a) ∈ for some a ∈

I ⊆ K. Hence x ∈ [x ] = [a ] ⊆K. Therefore I ⊆ K.

Let R be a distributive lattice whose Hasse diagram is given in the example 3.3. For any

congruence relation on a distributive lattice R, one can easily observe that the set (R) of all

-ideals of R is not a sublattice of the ideal lattice (R). For, consider the ideal I = {0, a} and J

= {0, b}. Now, for the congruence relation whose partition is {{0}, {a}, {b}, {c, 1}}, we can

observe that I and J are both the -ideals of the distributive lattice R. But the ideal I ∨ J is a not a

-ideal of R. Keeping in view of the operation depicted in the Definition 3.11, it can be observed

that (L) can be made into a distributive lattice with respect to the following operations for any

I, J ∈ (L), I∧ J = I∩J and I⊔J = (I∨ J )

Theorem 3.15: Let be a congruence relation on an ADL L. For any proper -ideal I of L, we

have { |I P P is a -prime ideal and }.I P

Proof: Take0 { |I P P is a -prime ideal, }.I P Clearly

0.I I Let .a I ConsiderF = { |J J

is a -ideal, I J and }.a I Clearly I F. Let{ }J be a chain of -ideals inF.

Clearly,

J is a -ideal of L such that I J

and .a J

Hence by the Zorn’s lemma, F

has a maximal element ,M say. That means M is a -ideal, I M and .a M Suppose ,x y L

such that x M and .y M Then ( ] ( ( ])M M x M x and ( ] ( ( ]) .M M y M y By

the maximality of ,M we get that ( ( ]) ( ( ]) ( ( ]).a M x M y M x y If x ∧ y ∈ [0 ] ,

then x ∧ y ∈ [0 ] ⊆ M. Hence a ∈ M,

which is a contradiction. Hence M is -prime. Therefore for any a I,

there exists a -prime ideal M of L such that I ⊆ M and a M. Thus

a 0.I Therefore

0I ⊆ I. Hence 0I = I.



Corollary 3.16: [0] { |P P is a -prime ideal}.

Corollary 3.17. If is the smallest congruence on L, then we have {0} = { |P P is a -prime

ideal}.

Corollary 3.18: Let L be a congruence relation on an ADL L. If [0]a then there exists a -

prime ideal P of L such that .a P

Finally, we conclude this paper with the following theorem.

Theorem 3.19: Let L be a congruence on L. Suppose I is a -ideal and F is a filter of L such

that .I F Then there exists a -prime ideal P of L such that I P and .F P

Proof: Let I be a -ideal and F, a filter of L such that I ∩ F = . Consider

= {J | J is a -ideal, I ⊆ J and J ∩ F = }. Clearly I ∈ . Let {iJ | i } be a chain of -

ideals in . Clearly, ii

J

is a -ideal such that I ⊆ ii

J

and ( ii

J

) ∩ F = . Let M be a

maximal element of . Suppose x, y ∈ L such that x M and y M . Then M ⊂ M ∨ (x] ⊆{M

∨ (x]} and M ⊂ M ∨ (y] ⊆ {M ∨ (y] }

. By the maximality of M, we get that

{M ∨ (x] }

∩F and {M ∨ (y] }

∩F . Choose a ∈ {M ∨ (x] }

∩F and

b ∈ {M ∨ (y] }

∩F. Hence a∧ b ∈ {M ∨ (x] }

∩{M ∨ (y] }

= {M ∨ (x∧ y] }

.

If x ∧ y ∈ [0 ] , then x ∧ y ∈ [0 ] ⊆ M. Since M is a -ideal, we get that a ∧ b ∈ M = M.

Hence a ∧ b ∈ M ∩ F, which is a contradiction. Therefore M is a -prime ideal of L.

4. CONCLUSION

Some remarkable results have been established on -ideals by using congruence in an Almost

Distributive Lattice(ADL).The change of -ideal into a -Prime ideal is achieved with the help

of a set of equivalent conditions.

REFERENCES

[1] Birkhoff, G.: Lattice Theory. Amer. Math. Soc. Colloq. Publ. XXV, Providence (1967),

U.S.A.

[2] Gratzer, G.: General Lattice Theory. Academic Press, New York, Sanfransisco (1978).

[3] Rafi, N., Rao, G.C. and Ravi Kumar Bandaru: −Filters in Almost Distributive Lattices.

Accepted for publication in International Journal of Scientific and Innovative Mathematical

Research.

[4] Rao, G.C.: Almost Distributive Lattices. Doctoral Thesis (1980), Dept. of Mathematics,

Andhra University, Visakhapatnam.

[5] Rao, G.C. and Ravi Kumar, S.: Minimal prime ideals in an ADL. Int. J. Contemp. Sciences, 4

(2009), 475-484.

[6] Swamy, U.M. and Rao, G.C.: Almost Distributive Lattices. J. Aust. Math. Soc. (Series A), 31

(1981), 77-91.


NOORBHASHA RAFI is working as a Assistant Professor in the

Department of Mathematics, Bapatla Engineering College,

Bapatla, Andhra Pradesh. He obtained his Ph.D Degree in

Mathematics from Andhra University under the able guidance of

Prof.G. Chakradhara Rao. He has participated in many seminars and

presented his papers. He honored as Chairperson in one session of

international conference on algebra conducted by Gadjah Mada

University, Indonesia, 2010. His articles more than 20 are published

in various National and International Journals.




www.arcjournals.org

©ARC Page | 58

An Unsteady Viscous Incompressible Flow In A Porous

Medium Between Two Impermeable Parallel Plates

Impulsively Stopped From Relatively Motion

Mohammed Sarfaraz Hussain Faculty in Mathematics,

Ibri College of Technology,

Ibri, Oman.

[email protected]

N.Ch.Pattabhi Ramacharyulu Retd. Prof. of Mathematics,

NIT Warangal, India.

Abstract: The problem presented in this paper is an unsteady flow of viscous incompressible

homogeneous fluid through a porous slab bounded between two impermeable parallel plates separated by a

finite gap with the neglect pressure gradient. The flow is generated by the motion of one of the plates with a

constant velocity parallel to itself while the other is kept rest. When the steady state is reached the moving

plate is suddenly stopped and the subsequent motion investigated employing the Laplace Transform

technique to obtain the fluid-velocity field. Expressions for ensuing flow-rate and skin friction on the

boundaries have been obtained.

Variations of these flow parameters with time and the porosity coefficient have been illustrated based on

which some conclusions have been reached using Matlab

1. INTRODUCTION

The viscous flow between two parallel plates is a classical problem, we know that the fluid exerts

viscosity effect when there is a tendency of shear flow of the fluid. Fluid flow through porous

media have been attracting the attention of Engineers and Applied Mathematicians for the last one

and half century as it has various applications in certain devices such as electrostatic precipitation,

magneto-hydrodynamic (MHD)power generators, MHD pumps, accelerators, petroleum industry,

polymer technology, aerodynamics heating, purification of molten metals from non-metallic

inclusions and fluid droplets-sprays.

Impulsive flows of incompressible fluids between rigid boundaries is attracting the attention of

researchers due to its wide range of application in several branches of sciences and technology.

Knowledge of the flow through porous media is immensely useful in the efficient recovery of

crude oil from the porous reservoir rocks by the displacement with immissible water

(Rudraiah.et.al [1] ). Run-up and spin-up flows belong to this category of flow. Kazakai and

Rivlin[2] investigated run-up and spin-up flows of non-Newtonian liquids in parallel plate and

circular geometries. The flow field is arising from super position of waves propagated into the

fluid and reflected back and forth at the boundaries has been investigated via the solution of initial

value problem, later by Pattabhi Ramacharyulu and Appalaraju [3], Appala Raju.K [4] for flows

through porous media and Ramakrishna [5] for second order liquids examined.T.Gnana Prasuna

,Pattabhi Ramacharyulu and Ramanamurthi [6] and T.Gnana Prasuna [7] investigated a run-up

flow of a visco elastic fluid through a porous medium in parallel plate geometry. Run up flows

were studied earlier by Ramakrishna[5] for fluids with particle suspension and Ramanamurthi [8]

for second order fluid. Forced and natural flows were discussed by Schlitchting [9],Eckert and

Drake [10] and Bansal [11].Pop and Soundalgekar [12] investigated free convection flow past an

accelerated vertical infinite plate. The present authors[13] recently investigated an impulsive flow

in porous slab bounded between two impermeable parallel plates

Here we investigate a special run-up retarding flow of a viscous incompressible homogeneous

fluid through a porous slab of finite thickness bounded between two impermeable parallel plates.


Mohammed Sarfaraz Hussain & N.Ch.Pattabhi Ramacharyulu


Following Yamamoto and Yashida Z. [14], we adopt the following momentum equation for the

flow through porous media which takes care of the fluid inertia and the viscous stress in addition

to the classical Darcy’s friction

where ρ is the fluid density

μ is the coefficient of viscosity

k is the Darcy coefficient of porosity of the medium

p is the pressure and

is the fluid velocity vector

Initially, the flow is generated by a motion of one of the plates with a constant velocity. When the

steady state is reached, the moving plate is suddenly stopped and the subsequent fluid-flow is

examined.

Analytic expressions for the Velocity, Flow rate and Skin-Friction on the two plates are obtained,

by employing Laplace Transformation technique and variation of all the flow parameters are

shown with illustration and conclusions are drawn.

2. MATHEMATICAL FORMULATION

Consider a Cartesian frame of reference O(x,y,z) with origin is on a fixed plate. x-axis parallel to

the moving plate and y-axisperpendicular to the plates

Plate in relative motion y = h

\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \

Y

U( Y,T)

/ / / / / / / / / / / / / / / x-axis

Plate at rest y = 0

In this Cartesian frame, the fluid velocity can be taken as = [u(y,t),o,o]

This evidently satisfied by the continuity equation

= 0 (1)

The momentum equation is in X-direction characterizing the flow through porous media is

u (2)

Where is the fluid density, is the coeff of viscosity and

is the Darcy coefficient of porosity of the medium

The present problem is solved in two stages. In the first stage the flow is generated by the motion

of one of the plates with the constant velocity parallel to itself while the other is kept at rest and

the second stage is concerns with the subsequent unsteady receding flow, when the moving plate

(cited in stage-1) is stopped.

Stage – 1:

This is the steady state of the fluid-flow generated by the motion of the plate y = h with a velocity

.

Let be the velocity of the fluid.

The momentum equation in this case is

An Unsteady Viscous Incompressible Flow in a Porous Medium between two Impermeable Parallel

Plates Impulsively Stopped from Relatively Motion:


0 = (3)

With the boundary conditions

(4)

The solution of the equation (3) subject to the boundary condition (4) is

(5)

Stage – 2:

After attaining the steady flow considered in stage – 1, the moving plate y = h is suddenly

stopped.

Let be the fluid velocity in the subsequent motion be given by

This satisfies the momentum equation

(6)

Where = (7)

with the boundary conditions:

(8)

and with the initial conditions:

(9)

where f(y) given in equ(5))

The following system of non-dimensional quantities is introduced for the simplicity of

presentation of the result

Y = hY; t = ; ; (10)

where is the velocity of the moving plate which is generating the motion

in the stage - 1

The momentum equation expressed in the non-dimensional form is

(11)

Where (12)

and with the boundary condition:

u(o,T) = o & u(1,T) = o (13)

and the initial condition:

u(y,o) = (14)

Let Laplace Transform of u(Y,T) be

(15)

Taking L.T. of the equation (11), we get

(16)

Where m (17)

With the boundary conditions are

(18)



The solution of (16) satisfy the boundary condition (18) is

(19)

The inverse Laplace transform of which is

(20)

Where

Is the Heaviside unit step functions

The Laplace inverse in the second term of (20) is obtained by using the Residue theorem

(vide: Appendix)

The flow variables at this stage are

Velocity:

(21)

Flow-Rate:

The flow rate Q(t) is given by (22)

the Laplace Transform of which is

(23)

Taking the inverse Laplace transform we get

Q(t) =

Shear-Stress:

The shear stress is in the flow-region is

(24)

the Laplace transfrom of which is

(25)

Taking the inverse L.T. we get

(26)

Therefore the shear stress on the bottom plate (y = 0) is

(27)

And that on the other plate (y = 1) is




(28)

3. CONCLUSION

Numerical computations have been carried out for the values of different flow variables with

diverse values of coefficient of porosity D at different time instants. It is noticed that the effect of

porosity on the velocity profiles is to flatten type with the velocity attaining maximum near the

middle of the plates and due to friction, it decrease towards the plates. This effect will be more

predominant profiles and as D increases. Further the point of maximum velocity is shifted

towards the upper plate (y=1).

(Vide Figs (1-10))

Fig 12 and Fig 13 illustrate the variation of the shear stress on the lower plate y=0 and upper plate

y=1

Fig 13 shows the variation of the flow rate with D and t. as D increases and also as t

increases.

APPENDIX

Let ( ) ( )L f t f s then

1( ) ( )

1 ( )2

f t L f s

yi stf s e dsii x

where is a vertical contour in the complex plane chosen so that all the poles of the integrand

are to the left of it [15] Eric W. Weirstein, CRC, Concise Encyclopedia of Mathematics, 1999,

page 176, [16] Sneddon I. N. , Use of Integral transforms, Tata Mac Graw-Hill, 1979.

Therefore 1 ( )L f s Sum of the residues of all the poles of f(s) ste

i.e. of f(s) inside the Bronwich Contour.

from (20)

The poles of the second term are

s = 0 and sinh(m) = 0

n=0, 1, 2, 3…

Residue at s=0:

Residue at s = :

Thus,



Fig.1. Velocity profiles at different time instants when D=0.5

Fig.2. Velocity profiles at different time instants when D=1

Fig.3. Velocity profiles at different time instants when D=2

0 0.1 0.2 0.3 0.4 0.5 0.6 0.70

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1D=0.5

---------->U

----

----

-->

y

t=0.01

t=0.02

t=0.05

0.1

0.2

0.3

0 0.1 0.2 0.3 0.4 0.5 0.6 0.70

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1D=1

---------->U

----

----

-->

y

t=0.01

t=0.02

t=0.05

0.1

0.2

0.3

0 0.1 0.2 0.3 0.4 0.5 0.6 0.70

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1D=1

---------->U

----

----

-->

y

t=0.01

t=0.02

t=0.05

0.1

0.2

0.3




Fig4. Velocity profiles at different time instants when D=10

Fig.5. Velocity profiles at different time instants when D=12.5

Fig.6. Velocity profiles verses D at t = 0.01

0 0.01 0.02 0.03 0.04 0.05 0.060

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1D=10

---------->U

----

----

-->

y

t=0.01

t=0.02

0 0.005 0.01 0.015 0.02 0.0250

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1D=12.5

---------->U

----

----

-->

y

t=0.02

t=0.02

0 0.1 0.2 0.3 0.4 0.5 0.6 0.70

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1t=0.01

------>U

----

-->

y

D=0.5

D=1

D=2

D=5





Fig.9. Velocity profiles verses D at t = 1

0 0.05 0.1 0.15 0.2 0.250

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1t=0.1

------>U

----

-->

y

D=0.5

D=1

D=2

D=5

0 1 2 3 4

x 10-3

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1t=0.5

------>U

----

-->

y

D=0.5

D=1

D=1.5

D=2

0 0.5 1 1.5 2 2.5 3

x 10-5

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1t=1

------>U

----

-->

y

D=0.5

D=1

D=1.5

D=2




Fig.10. Velocity profiles verses D at t = 2

Fig.11. Shear stress on the lower plate verses D at different t

Fig.12. Shear stress on the upper plate verses D at different t

0 0.2 0.4 0.6 0.8 1 1.2

x 10-9

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1t=2

------>U

------

>y

D=0.5

D=1

D=1.5



Fig 13. Flow rate verses t for different values of D

REFERENCES

[1] Rudraiah N, Veerabhadraiah R, Chandrasekhar B. C. and Nagaraju S. T:Some flow problem

in porous media, PGSAM Vol. 2 (1979).

[2] Kazakia Y and Rivlin R.S :Run-up and spin-up in visco-elastic fluid-І, RhealogicaActal 20,

pp.111-127 (1981).

[3] PattabhiRamacharyula N.Ch and AppalaRaju K :Run-up flow in a generalized porous

medium in the journal of pure and applied Mathematics Vol.15 (1984) pp. 665-670.

[4] And AppaluRaju. K :Some fluid flow problems in porous media, Ph.D. Thesis, Kakatiya

University, Warangal(1984).

[5] Ramakrishna D :From problems in the dynamics of fluid with the particle suspension,

Ph.D. Thesis, Kakatiya University, Warangal(1986)

[6] GnanaPrasuna.T, PattabhiRamacharyulu N.Ch. and RamanaMurthi.M.V.:A unsteady flow of

visco elastic fluid through a porous media between two impermeable parallel plates (an

initial value problem) – journal of Engineering trends in Engineering and applied sciences

Vol.1 no. 2(2010) pp.222-224.

[7] GnanaPrasuna. T :Some problem on visco-elastic fluid through porous media, Ph.D.Thesis,

Osmania University, Hyderabad(2011).

[8] RamanaMurthi :Some second order fluid flows, Ph.D. Thesis, Kakatiya University,

Warangal(1986).

[9] Schlichting, H.: Boundary layer theory, McGraw hill Book Co. Inc., New York (1996).

[10] Eckert, E. R. G. and Drake, R. M. : Heat and Mass Transfer, Tata McGraw Hill Pub. Comp.

Ltd., New Delhi.

[11] Bansal, J. L. : Viscous fluid Dynamics, Oxford & IBH Pub. Comp., Calcutta (India) (1977)

[12] Pop , I. and Soundalgekar, V. M. : ZAMM 60 , p. 167 (1980)

[13] Md. Sarfaraz Hussain and N.Ch. Pattabhi Ramacharyulu: An unsteady viscous flow through

a porous slab bounded between two impermeable parallel plates—International Journal of

Mathematical Archive Vol.4 No.3 (2013) page 260-275

[14] Yamamoto, K. and Yashida, Z. Journal of Physical society of Japan,vol.37 (1994) P774

[15] Weirstein Eric W. CRC, Concise Encyclopedia of Mathematics,1999,page 176

[16] Sneddon I. N. , Use of Integral transforms, Tata Mac Graw- Hill, 1979.




www.arcjournals.org

©ARC Page | 68

Coincidence and Fixed Points of Nonself Maps using

Generalized T-Weak Commutativity

Praveen Kumar Shrivastava Department of Mathematics,

Govt. P.G. College, Balaghat (M.P.),India

[email protected]

Yogesh Kumar Vijaywar


Govt. P.G. College, Balaghat (M.P.), India

[email protected]

Ravindra Kumar Sonwane Department of Mathematics,

Govt. P.G. College, Balaghat (M.P.), India

[email protected]

Abstract: The purpose of this paper is to introduced the concept of sequential T-weak commutativity and

generalized T-weak commutativity, which is an extension of T-weak commutativity defined by Kamran for

hybrid pair of mappings. We prove that this concept is equivalent to generalized compatibility of type(N) at

coincidence points recently introduced by us and we have shown that compatibility of type (N) is more

general than (IT)-commutativity introduced by Singh and Mishra. Using generalized T-weak commutativity

we also extend a result of Pathak and Mishra who have indicated that a result of Chang admits a counter

example and presented a corrected version of the result. We have also shown that the corrected version of

the result of Chang is also derivable from our results. We thus extend and generalize many known results in

this way.

Keywords: Coincidence and fixed point, Hausdorff metric, generalized compatibility of type(N),

generalized T-weak commuting mappings.

1. INTRODUCTION

The study of fixed points of non-self multi-valued and single valued contractions (Hybrid

contractions) is a new development in the domain of contractive type multi-valued theory (see, for

instance [1,2, 3, 6, 7, 9, 11, 13, 14, 15, 23-26]). Recently, in an attempt to improve/ generalize

certain results of Sastry, Naidu and Prasad, [18] and Naidu [15], Chang [3] obtained some fixed

point theorems for hybrid of single valued and multi-valued mappings, however as indicated by

Pathak and Mishra [17], his main theorem admits a counter example. Pathak and Mishra[17] have

also suggested modification and presented a corrected version in more general way using

sequential commutativity of hybrid pair of mappings. Our main purpose in this paper is to prove

the concept of sequential T-weak commutativity and generalized T-weak commutativity

introduced by us in this paper is more general than sequential commutativity introduced by Pathak

and Mishra [17]. We also suggest a restriction which must be imposed for defining concept of

sequential commutativity.

(AMS-2000) Sub . Classification No. 54 H 25

2. PRELIMINARIES

We generally follow the definitions and notations used in [1, 2]. Given a metric space (X, d), let

(C(X), H), (CB(X), H), and (CL(X), H) denote respectively the hyperspaces of nonempty

compact, nonempty closed bounded, and nonempty closed subsets of X, where H is the Hausdorff

metric induced by d. The space (CL(X), H) contains the other two spaces. Throughout, d(A, B)

will denote the ordinary distance between nonempty subsets A and B of X while d(x, B) stands

for d(A, B) when A = {x}, the singleton set. For any A X, (A) will denote the boundary of A.

For details of hyperspaces one may refer to [14].


Praveen Kumar Shrivastava et al.


Following Hadzic and Gajic [4] and pant [16], Singh and Mishra [24] have introduced the notion

of R-weak commutativity of a hybrid pair of single-valued and multivalued maps, as follows,

Definition 1 : Let K be a nonempty subset of a metric space X, T : K X and F : K CL(X).

Then T and F will be called pointwise R-weakly commuting on K if given x K and Tx K,

there exists R 0 such that,

d(Ty, FTx) Rd(Fx, Tx) for each y K Fx. (1)

Maps T and F will be called R-weakly commuting on K if for each x K, Tx K and (1) holds

for some R 0.

If R = 1, they get the definition of weak commutativity of F and T on K due to Hadzic and Gajic

[4] (see [1, 2]). If F : X X and T : X X then they get the definition of point-wise R-weak

commutativity and R-weak commutativity of single-valued self-maps due to Pant [16]. He has

observed that the point-wise R-weak commutativity is more general than their compatibility. For

details on compatibility of a hybrid pair, refer to [8, 9].

It appears that Ahmad and Imdad [1] have considered a hybrid pair T, F commuting in the sense

FTx = TFx, and we shall follow the same notion throughout this paper. Following Jungck [8] and

Jungck and Rhoades [9], Singh and Mishra [24] give the following definition.

Definition 2: Maps T : K X and F : K CL(X) are weakly compatible if they commute at

their coincidence points, i.e., if TFx = FTx whenever Tx Fx K

For an excellent discussion on the role of weak compatibility in fixed point considerations, one

may refer to [9] when T : X X and F : X B(X), the set of all nonempty bounded subsets of

X. We remark that

Commutativity Weak commutativity Compatibility Weak compatibility

However, the reverse implications are not true. Nevertheless, all these notions for T and F are

equivalent at a coincidence point z (that is, when Tz Fz). Further, if T and F both are single-

valued maps then weak compatibility of T and F is equivalent to R-weak commutativity of T and

F at their coincidence points. Example 1 (below) shows that an R-weakly commuting hybrid pair

T, F need not be weakly compatible. Indeed, R-weak commutativity of a hybrid pair of maps at

coincidence points is more general than their weak compatibility. Following Itoh and Takahashi

[7], Singh and Mishra [24] also give the following :

Definition 3 : Maps T : K X and F : K CL(X) are commuting at a point x K if TFx FTx

whenever Tx Fx K and Tx K. T and F are commuting on K if they are commuting at each

point x K.

From now onward, the above commutativity will be called Itoh-Takahashi commutativity or

simply (IT)-commutativity.

The example 1 of [24] shows that (IT)-commutativity of T and F at a coincidence points is indeed

more general than their weak compatibility at the same point.

Remark 1 : In view of (1), (IT)-commutativity of T, F at a coincidence point z is equivalent to

their R-weak commutativity at z.

On the other hand, we have introduced the notion of compatibility of type(N) in [19] for hybrid

pair (T, F), where T : X X and F : X CB(X) and proved that notion of compatibility of

type(N) is more general than weak compatibility of T and F. Further, in [21] we have proved that

compatibility of type (N) is more general than commutativity of T and F at their coincidence

point, while commutativity of T and F at their coincidence point is more general than weak

compatibility of T and F proved by Singh and Mishra in [23]. Recently we have proved in [22]

that compatibility of type (N) and F-weak commutativity introduced by Kamran [10] are

equivalent at coincidence points. In the paper [20] we have extend the notion of compatibility of

type (N) for non-self hybrid pair T and F and show that it is more general than (IT)-commutativity

introduced by Singh and Mishra [24]. We thus have generalized the results of Singh and Mishra

[24] and others.

Coincidence and Fixed Points of Nonself Maps using Generalized T-Weak Commutativity


Definition 4 : [20] Suppose T : K X and F : K CL(X). The pair (T, F) is called generalized

compatible of type (N) iff T(x) F(x), F(x) K implies that TT(x) FT(x).

Note : If we put K = X in definition 5 we get our definition of compatibility of type (N)

introduced in [19].

Lemma 1 : [20] (IT)-commutativity of the hybrid pair, (T, F) implies generalized compatibility of

type (N) but not conversely.

Example 1 of [20] shows that converse is not true in general.

Definition 5 : Mappings S : X CB(X) and I : X X are called compatible if Fx CB(X) for

all x X and H(SIxn, ISxn) 0, as n whenever (xn) is a seq. x in X such that Sxn M

CB(X) and Ixn l M as n .

Following Singh and Mishra [24]. Pathak and Mishra [17] have introduced the notion of R-

sequentially commuting mappings for a hybrid pair of single valued and multi-valued maps.

Definition 6 : [17] Let K be a nonempty subset of a metric space X and I : K X and S:K

CL(X) be respectively single-valued and multi-valued mappings. Then I and S will be called R-

sequentially commuting on K if for a given sequence (xn} K with lim Ixn K, there exists R 0

such that

limn D(Iy, SIxn) R limn D (Ixn, Sxn) (*)

for each y K limn Sxn

If xn = x(x K) for all n N (naturals), Ix K and (*) holds for some R 0, then I and S have

been defined to be pointwise R-weakly commuting at x K (see [Def. 1]). If it holds for all x

K, then I and S are called R-weakly commuting on K. Further, if R = 1, they we get the definition

of weak commutativity of I and S on K due to Hadzic and Gajec [4]. If Ix S: X X, then as

mentioned in [17], we recover the definitions of pointwise R-weak conmmutativity and R-

commutativity of single-valued self-maps due to Pant [16] and all the remarks as given in [17]

apply.

Pathak and Mishra [17] have also introduced the following ;

Definition 7: Maps I : K X and S : K CL(X) are to be called sequentially commuting (or s-

commuting) at a point x K if

I (limn Sxn) SIx (**)

whenever there exists a sequence {xn} K such that limn Ixn = x limn Sxn CL(X).

Definition 8 : [17] If xn = x for all n N, in Definition 7, then the maps I and S will be said to be

weakly s-commuting at a point x K.

Remark 2 : We remarked that to form I (lim Sxn),IIx, SIx the restrictions, limnSxn K, Ix K

must be included in definition 7 and 8.

Remark 3 : With this restrictions weak s-commutativity (particular case of def. 7) defined by

Pathak and Mishra [17] is equivalent to (IT) - commutativity.

The example 1 of [17] shows that s-commutativity of I and S is indeed more general than their R-

sequential commutativity (and hence their pointwise R-commutativity and compatibility).

Definition 9: ([3]) Let R+ denote the set of all non-negative real numbers, and let A R

+. A

function : A R+ is upper semi continuous from the right if limx u+ sup (x) (u) for all u

A.

A function : R+ R

+ is said to satisfy ( )-conditions if :

(i) is upper semi-continuous from the right on (0, ) with (t) t for all t 0, and

(ii) There exists a real number s 0 such that is non-decreasing on (0, s] and



n=1

n=1

n=1

n=1

n(t) for all t (0, s], where

n denotes the composition of

with itself n

Times and 0(t) = t.

Let denote the set of all functions which satisfy the ( )-condition.

The following lemmas will be useful in proving our main results.

Lemma 2 : [17] Let (X, d) be a metric space and I, J : X X and S, T : X CL(X) be such that

S(X) J(X) and T(X) I(X) and for all x, y X,

H (Sx, Ty) (aL (x, y) + ( 1 – a) N (x, y)), (2)

where a [0, 1], : R+ R

+ is upper semi-continuous from the right on (0, ) with (t) t

for all t 0, and

L(x, y) = max {d(Ix, Jy), D(Ix, Sx), D(Jy, Ty), ½ [D (Ix, Ty) + D (Jy, Sx)]},

N(x, y) = [max {d2(Ix, Jy), D (Ix, Sx) D(Jy, Ty), D(Ix, Ty) D(Jy, Sx),

½ D(Ix, Sx) D(Jy, Sx), ½ D(Jy, Ty) D(Ix, Ty)}]1/2

Then inf x X D (Ix, Sx) = 0 = infx X D (Jx, Tx)

Lemma 3 [17] : Let X, I, J, S, T and be as defined Lemma 2 such that the inequality (2) holds.

If Ix Sx for some x X, then there exists a y X such that Ix = Jy and Jy Ty.

Lemma 4 ([18]) : Let : R+ R

+ be a non-decreasing functions such that

(i) (t+) t for all t 0 and n(t) for all t 0.

Then there exists a strictly increasing function : R+ R

+ such that

(ii) (t) (t) for all t 0 and n(t) for all t 0.

Lemma 5 ([3]) : If , then there exists a function : R+ R

+ such that.

(i) is upper semi-continuous from the right with (t) (t) t for all t 0.

(ii) is strictly increasing with (t) (t) for t (0, s], s 0 and n(t)

for t (0, s].

The following theorem is the main result of Chang [3, Theorem 1].

Theorem A : Let (X, d) be a complete metric space, let I, J be two functions from X into X, and

let S, T : X CB(X) be two set-valued functions with SX JX and TX IX. If there exists

such that for all x, y in X,

H (Sx, Ty) (max {d(Ix, Jy), D(Ix, Sx), D(Jy, Ty), ½D[(Ix, Ty) + D(Jy, Sx)]}), (C)

then there exists a sequence {xn} in X such that Ix2n z and Jx2n-1 z for some z in X and

D(Ix2n, Sx2n) 0, D(Jx2n-1, Tx2n-1) 0 as n . Moreover, if Iz = z and T and J are compatible,

then z Sz and Jz Tz. That is, z is a common fixed point of I and S, and z is a coincidence

point of J and T.

The example 2 of [17] shows that Theorem A in its present form is incorrect.

Example 3 [17] : Let X = [0, 1] with absolute value metric d and let : R+ R

+ be defined by

(t) = t2 for t [0, 1) and (t) = ½ for t 1. Define I = J : X X and

S = T : X CB(X) by Ix = 1 – x, x X and Sx = {0, 1/3, 2/3, 1} for all x X. Then for each x,

y X and , we have

II(Sx, Ty) = 0 (max {d(Ix, Jy), D(Ix, Sx), D(Jy, Ty), ½D[(Ix, Ty) + D(Jy, Sx)]}),



and for all sequence {xn} X defined by xn = 1/n for all n N, we have Sxn, Txn {0, 1/3,

2/3, 1} = M, Ixn, Jxn = 1 – 1/n 1 M X, D(Ix2n, Sx2n) 0 and D(Jx2n-1, Tx2n-1) 0 as n

. Also, z = 1/2 X is such that Iz = z and for {xn} as defined above we have limn H(TJxn, JTxn)

= 0, that is, T and J are compatible. Thus, all the conditions of Theorem A are satisfied. Evidently

z Sz, Jz Tz, that is, z = 1/2 is neither a common fixed point of I and S nor it is a coincidence

point of J and T.

Following theorem will be useful for the proof our main theorem [17].

Theorem 1 : Let (X, d) be a complete metric space, and let I, J : X X, S, T : X CL(X). Let

A be a nonempty subset of X such that I(A) and J(A) are closed subsets of X, and Tx I(A) and

Sx J(A) for all x A and there exists a such that for all x, y X, (2) holds, Then

(i) F = {Ix : x X and Ix Sx} ,

(ii) G = {Jx : x X and Jx Tx} ,

(iii) F = G if A = X. and F = G is closed.

Theorem 2: Let (X, d) be a complete metric space, and let I, J : X X, S, T : X CL(X) be

such that S(X) J(X) and T(X) I(X). If there exists a such that for all x, y X, (2)

holds, then there is a sequence {xn}in X such that Ix2n z and Jx2n-1 z for some z X and

D(Ix2n, Sx2n) 0, D(Jx2n-1, Tx2n-1) 0 as n . Moreover,

(i) if Iz Sz and d(Iz, z) D(z, Sx) for all x X, then z Sz, and if d(Iz, z) D(z, Tx) for

all x X, J and T are weakly s-commuting, then Jz Tz.

(ii) if Iz Tz and d(Jz, z) D(z, Tx) for all x X. then z Tz; and if d(Jz, z) D(z, Sx) for

all x X, I and S are weakly s-commuting, then Iz Sz.

(iii) if Iz = z and J and T are weakly s-commuting, then z Sz and Jz Tz.

(iv) if Jz = z and I and S are weakly s-commuting, then z Tz and Iz Sz.

Remark 4 : Theorem 1 of Naidu [15] and Theorem 9 of Sastry, Naidu and Prasad [18] follow as

direct corollaries of Theorem 1.

Remark 5 : For a = 1, Example 10 of Sastry, Naidu and Prasad [18] shows that Theorem 1 fails if

½[D(Ix, Ty) + D(Jy, Sx)] is replaced by max {D(Ix, Ty), D(Jy, Sx)} even if S = T, I = J = id (the

identity mapping on X) and is continuous on R+.

Remark 6 : If (1) is assumed to be valid only for those x, y X for which Ix Jy,

Ix Sx and Jy Ty instead of all x, y X, then we conclude from Theorem I that: either F = {Ix

: x X and Ix Sx} or G = {Jx : x X and Jx Tx} .

3. MAIN RESULTS

We introduce:

Let (X, d) be a metric space, K X, I : K X. T : K CB(X) or CL(X) as the case may be.

Definition: Mapping I : K X is said to be sequentially T-weak commuting at x K. if

I lim Ixn TIx whenever there exists a sequence {xn} K such that

lim Ixn = x K, and lim Txn K.

If xn = x for all n N, then we have,

Definition: I : K X is said to be generalized T-weakly commuting at x K if

IIx TIx whenever Ix K , Tx K.

If K = X, we have following def. due to Kamran [10]

Definition : I : X X is said to T-weak commuting at x if IIx TIx.



F(x) =

[ ]12

, 1 [ ]12

0, if x

[ ] 12

, 1[ ]12

0, if x {

Lemma : Sequential commutativity implies sequential T-weak commutativity but not conversely.

Proof : Suppose I and T are sequentially commuting mapping, then we have,

I(lim Txn) TIx and there exists a sequence {xn} K such that

limn Ixn = x limn Txn, and limn Txn K.

Now limn Ixn lim Txn gives I(lim Ixn) I(lim Txn) and since I(limn Txn) TIx, we have I(limn

Ixn) TIxn and Hence I is sequentially T-weak commuting.

To prove the converse we provide following example.

Example 4 : Let X = [0, ), K = [0, 1] Let T : K X be defined by

T(x) = 1- x if x [0, 1] and

Take any sequence xn in in K = [0, 1]. We consider two cases,

(i) {xn} [0, ½] (ii) {xn} (½ , 1]

Case I : First suppose

xn [0, ½]. Then Ixn = 1 –xn [½, 1] and limn Ixn

= x (say) [½, 1] = limn Fxn. Now T(limn Fxn) = T [½, 1]

= [0, ½] and FTx = F(1-x) = [½, 1] since x [½, 1] then 1 – x [0, ½]. Clearly

T (limn Fxn) = [0, ½] FTx = [½, 1] for all xn [0, ½], x [½, 1]

Case II : Suppose {xn} (½, 1]. Then Txn [0, ½] and also limn Txn = x [0, 1/2 ].

Now Fxn = [0, ½] and limn Fxn = [0, ½] therefore limn Txn limn Fxn.

But T(limn Fxn) = T[0, ½] = [½, 1] and

FTx=F(1–x) = [0, ½] since 1–x (½, 1]. Obviously T (limn Fxn) FTx for any sequence xn [½,

1].

Therefore T and F are not s-commuting. However, taking the sequence xn = ½ + 1/ 2n+1

,

n N, We have lim Txn = ½ and T (lim Txn) = ½. Further, FTx =FT(½) = F(½) = [½,1] and

Hence T(lim Txn) = ½ FT(½) = [½, 1].

Therefore T and F are sequentially T-weakly and generalized T-weakly commuting at x = ½ [0,

1]

Theorem 3 : Let (X, d) be a complete metric space and K be a subset of X. Let I, J : K X. S,

T:X CL(X) be such that S(K) J(K) and T(K) I(K). If there exists a such that for all

x,y K,

H(Sx, Ty) (aL(x, y) + (1 – a)N(x, y), where a [0, 1] : R+ R

+ is upper semi-continuous

from the right on (0, ) with (t) t for all t 0 and

L(x, y) = max {d (Tx, Jy), D(Ix, Sx), D(Jy, Ty), ½[D(Ix, Ty) + D(Jy, Sx)]}

N(x, y) = [max {d2(Ix, Jy), D(Ix, Sx) D(Jy, Ty), D(Ix, Ty), D(Jy, Sx),

½D(Ix, Sx) D(Jy, Sx), ½D(Jy, Ty) D(Ix, Iy)}]1/2

(3)



Holds, then there is a sequence {xn} in X such that Ixn z, Jx2n+1 z for same z X and D(Ix2n,

Sx2n) 0, D(Jx2n-1, Tx2n-1) 0 as n Moreover,

(i) If Iz Sz and d(Iz, z) D(z, Sx) for all x K, then z Sz, and if d(Iz, z) d(z, Tx) for

all x K. J and T are generalized T-weak commuting on G = {x : Jx Tx}, then Jz Tz.

(ii) If Jz Tz and d(Jz, z) D(z, Tx) for all x K. then z Tz; and If d(Jz, z) D(z, Sx) for

all x K. and I and S are generalized T-weak commuting on F = {x : Ix Sx}, then Iz

Sz.

(iii) If Iz = z and J and T are generalized T-weak commuting on G = {x : Jx Tx}, then z

Sz and Jz Tz.

(iv) If Jz = z and I and S are generalized T-weak commuting on F = {x : Ix Sx}, then z Tz

and Iz Sz.

Proof : Although the first part of the proof is same as given by Pathak and Mishra [17], Yet we

provide it for completeness.

We can construct a sequence {xn} n=0 K such that Jx2n+1 Sx2n, Ix2n+2 Tx2n+1 (n = 0,

1, 2 .....) and the sequence {Ix2n}, {Jx2n-1} are Cauchy sequences which converge to the

same limit z X and D(Ix2n, Sx2n) 0, D(Jx2n-1, Tx2n-1) 0 as n . It then follows

that D(z, Sx2n) 0 and D(z, Tx2n-1) 0 as n .

(i) Suppose Iz Sz, d(Iz, z) D(z, Sz) and J and T are generalized T-weak commuting on {x : Jx

Tx}.

Choose m N such that

sup{d(Ix2n, z), d(Jx2n-1, z), D(z, Sx2n), D(z, Tx2n-1) : n m} s.

Then for n m we have

D(z, Sz) d(z, Ix2n) + D(Ix2n, Sz)

d(z, Ix2n) + H(Sz, Tx2n-1)

d(z, Ix2n) + (aL(z, x2n-1) + (1 – a) N (z, x2n-1)), (4)

where

L(z, x2n-1) = max {d(Iz, Jx2n-1), D(Jz, Sz), D(Jx2n-1, Tx2n-1),

½[D(Iz, Tx2n-1) + D(Jx2n-1, Sz)]}

max {d(Iz, Jx2n-1), 0, d(x2n-1, Tx2n-1)

½[d(Jz, z) + D(z, Tx2n-1), d(Jx2n-1, z) D(z, Tx2n-1)]}

max {d(Iz, z), 0, 0, ½d(Iz, z)} as n

i.e.

limnL(z, x2n-1) D(z, Sz) ;

and N(z, x2n-1) [max{d2(Iz, z), 0, 0, 0, 0}]

1/2 as n

i.e. limnN(z, x2n-1) D(z, Sz)

Hence making n in (4), we obtain,

D(z, Sz) 0 + (aD(z, Sz) + (1 – a) D(z, Sz)),

that is, D(z, Sz) = 0 and so z Sz. Choose z' X such that Jz' = z, then

D(z, Tz') H(Sz, Tz') (5)

(aL(z, z') + (1 – a) N(z, z')) where

L(z, z') = max {d(Iz, Jz'), D(Iz, Sz), D(Jz', Tz'), ½[D(Iz, Tz') + D(Jz', Sz)]}



max{d(Iz, z), D(Iz, Sz), D(z, Tz'), ½[d(Iz, z) + D(z, Tz') + D(z, Sz)]}

= maz{d(Iz, z), D(z, Tz')} D(z, Tz')

And N(z, z') [max{d2(Iz, z), 0, 0, 0, ½D(z, Tz') [d(Iz, z) + d(z, Tz')]}]

1/2 D(z, Tz')

Hence by (5) D(z, Tz') (D(z, Tz')), and so D(z, Tz') = 0; i.e., Jz' = z Tz'

Since J and T are generalized T-weak commuting at z’, Jz' Tz', we have

JJz' TJz' which implies that Jz Tz.

(ii) The proof is analogous to the proof of (i) due to symmetry.

(iii) Suppose Iz = z and J and T are generalized T-weak commuting on {x: Jx Tx}. Choose m

as in (i), then for all n m

D(z, Sz) d(z, Ix2n) + D(Ix2n, Sz) (6)

d(z, Ix2n) + H(Sz, Tx2n-1)

d(z, Ix2n) + (aL(z, x2n-1) + (1 – a) N(z, x2n-1), where

L(z, x2n-1) max{0, D(z, Sz), 0, ½D(z, Sz)} as n ,

i.e. limn L(z, x2n-1) = D(z, Sz)

and N(z, x2n-1) [max {0, 0, 0, ½D2(z, Sz), 0}]

1/2 as n ,

i.e. limn N(z, x2n-1) = D(z, Sz).

Making n in (6), we obtain

D(z, Sz) 0 + (aD(z, Sz) + [(1 – 1) / 2]D(z, Sz) D(z, Sz),

which implies D(z, Sz) = 0 and so z Sz. Choose z' X such that Jz' = z, then

D(z, Tz') H(Sz, Tz') (aL(z, z') + (1 – a)N(z, z')) where

L(z, z') = max{d(Iz, Jz'), D(Iz, Sz), D(Jz', Tz'), ½[D(Iz, Tz') + D(Jz, Sz)]} = D(z, Tz')

and

N(z, z') = [max{d2(Iz, Jz'), D(Iz, Sz) D(Jz', Tz'), D(Iz, Tz') D(Jz', Sz), ½D(Iz, Sz) D(Jz', Sz),

½D(Jz', Tz') D(Iz, Tz')}]1/2

= (1/ 2)D(z, Tz')

Hence

D(z, Tz') (aD(z, Tz') + [(1 – a) / 2] D(z, Tz') D(z, Tz')

It follows that D(z, Tz') = 0 and so Jz' = z Tz'. Since J and T are generalized T-weak commuting

at z’, Jz' Tz', we have JJz' JTz'. Hence Jz Tz.

(iv)Due to symmetry, the proof is analogous to the proof of (iii).

Remark 7: As it has been shown in [22] that the concepts compatibility of type (N) and T-weak

commutativity are equivalent at coincidence points, the phrase “I and S (resp. J and T) are

generalized T-weak commuting on {x :Ix Sx}, (resp. {x : Jx Tx})” in every part of the

theorem 3 may be elegantly replaced by the phrase “I and S(resp. J and T) are compatible of type

(N)”.

4. CONCLUSION

Taking K= X and replacing the condition of generalized T-weak

commutativity by weak s-commutativity from theorem 3, we get theorem 2. It is also notable

that generalized T-weak commutativity is required only on coincidence points of the mappings

while in result of Pathak and Mishra [17] weak s-commutativity is required on all of X. Further,

in remark 2 we have suggested two essential restrictions



To impose for defining weakly s-commuting mappings i.e. to define weakly s-commuting

mappings we have to form I(limnSxn), SIx & I(Ix) and therefore the restrictions limnSxn K &

Ix K must be included in the definitions 7 and definition 8,with these restrictions, weak s-

commutativity defined by Pathak and Mishra [16] is nothing but (IT)-commutativity defined by

Singh and Mishra[24].Finally we claim following obvious lemma ,

Lemma7: The assumption of generalized T-weak commutativity at coincidence points

(generalized compatibility of type (N)) in case of hybrid pair (I, S ) ( resp. (J, T) ) is the minimal

condition for existence of common fixed point of the hybrid pairs.

ACKNOWLEDGEMENT

First author thanks the U.G.C. Regional Office, Bhopal (M.P.) for financial support under minor

research project F No. 4S-46/2006-07/(MRP/C

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Authors’ Biography




www.arcjournals.org

©ARC Page | 79

Composite Mapping of Flip-Flop Poset of Join-irreducible

Elements of Distributive Lattice

Namrata Kaushal Department of Engineering Mathematics,

Indore Institute of Science and Technology,

Indore, MP, INDIA

[email protected]

Madhu Tiwari Department of Mathematics,

Government Girls Post Graduate College

Ujjain, MP, INDIA

C. L. Parihar Indian Academy of Mathematics,

15 - Kaushalya Puri, Chitawad Road,

Indore, MP, India

Abstract: Every dimensional distributive lattice has a “unique irreducible decomposition” as a join of

join irreducible elements. By this spirit here we show that composition mapping of distributive lattice

having unique minimal gives the inclusion order of the ideals of sub poset of join irreducible element of

distributive lattice. This distributive lattice will be obtained by construction method, as every flip-flop poset

of a digraph is distributive lattice. Every join-prime element is also join irreducible, and every meet-prime

element is also meet irreducible. The converse holds if L is distributive. In view of this in present paper we

deals with subposet of join irreducible element of a distributive lattice and its composite mapping.

Keywords: Flip-flop, Vertex Cut, C- orientations, circulation, Join irreducible elements

1. INTRODUCTION

The subject of this paper is join irreducible element of distributive lattice and its composite

mapping. Some of the past authors prompt this in their research articles [Felsner, Propp]. Felsner

constructs a distributive lattice on those orientations of a planar graph that have fixed out degree

on every α-orientation. The α-orientations of a graph generalize f factors, spanning trees, Eulerian

orientations and Schnyder woods. Propp presents a method to generate a distributive lattice on

those orientations of a graph, which have the number of forward arcs in cycles as c-orientations.

The motivation of the work is based upon the question, whether or with which obstructions

composition mapping of distributive lattice having unique minimal gives the inclusion order of

the ideals of subposet of join irreducible element of distributive lattice. It turns out that the

generalization is possible, but yields a theory which is not as nice as in past. Therefore, we focus

on point that every flip-flop poset of a digraph is distributive lattice.

Given an undirected graph [6 ] G = (V, E) we call a directed graph D = (V, A) an orientation of G

if there is a bijection f : E → A such that f ({u, v}) ∈ {(u, v), (v, u)}. If ω is an orientation of G

then G is unique up to isomorphism and we call the isomorphism class of G denoted by of ω

and for digraphs ω , ω ′ we say that ω ′ is a reorientation of ω if = Clearly, two digraphs

ω , ω ′ are orientations of isomorphic graphs if and only if ω and ω ′ have identical graphs which

by definition is equivalent to ω ′ being a reorientation of ω. Propp introduce two operations on

directed graphs to obtain a partial order on some of the reorientations of a given digraph ω in his

paper “lattice structure for orientation of graphs”. Fix an arbitrary vertex T of ω the forbidden

vertex and allow flipping and flopping all the other vertex cuts of ω. For reorientations of ω ′ and

ω′′ of ω, we define ω ′ ≤ ω ′′ if ω ′′ can be obtained from ω ′ by a sequence of flips at vertices

different from T. The binary relation ≤ is a partial order on the set of reorientations of ω that can

be reached from ω via a sequence of flips and flops. This set of reorientations of ω partially


Namrata Kaushal et al.

International Journal of Scientific and Innovative Mathematical Research (Ijsimr) Page | 80

ordered by ≤ is called the flip flop poset , T). A classical result “Every flip-flop poset of a

digraph is distributive lattice “shown by Kolja in his paper “Distributive Lattices on Graph

Orientations” [6] R. P Dilworth represents a finite distributive lattice D as the congruence lattice

of a finite lattice L. While G,Gratzer and E. T Schmidt [5] shown a finite lattice L with O( )

elements,where n is number of join irreducible elements of D whose construction technique

developed by G,Gratzer and H.Lakser [ 4]. As by Garrett Birkhoff poset of irreducible generalizes

the constructions to provide a representation for finite distributive lattice. Hence Birkhoff's

theorem establishes an isomorphism relationship between a given finite distributive lattice and

the set of order ideals of a particular subposet of join-irreducible elements of the

lattice . In general, the poset is obtained from as the set of join-irreducibles, and is

recovered from poset up to isomorphism by taking the set of ideals of elements from poset. The

significance of this theorem lies in the fact that the poset of join-irreducible of L can be

exponentially more succinct than the lattice itself. Since in a distributive lattices the poset of

meet-irreducible is isomorphic to the poset of join-irreducible so it is sufficient to work with

either the join-irreducible or meet-irreducible.

2. PRELIMINARIES

Throughout the paper will be a distributive lattice & [D] denotes the set of digraphs that can

be obtained from D by adding transitive arcs, where an arc is called transitive if there is a

directed path in D. Let denotes the sub poset of join irreducible elements of If Ø is a

mapping from to and is a mapping from flipflop poset , T) to a subset

then ,T)) gives the inclusion order on the ideals of Since V

( )= V (D) for every [D] the bijection Ø can be seen as a bijection from to . Where

T is a terminal vertex.

Join-irreducible Elements

In a lattice, an element x is join-irreducible if x is not the join of a finite set of other elements.

Equivalently, x is join-irreducible if it is neither the bottom element of the lattice (the join of zero

elements) nor the join of any two smaller elements. In any lattice, a join-prime element must be

join-irreducible. Equivalently, an element that is not join-irreducible is not join-prime. For, if an

element x is not join-irreducible, there exist smaller y and z such that x = y ∨ z. Then x ≤ y ∨ z,

and x is not less than or equal to either y or z, showing that it is not join-prime. There exist lattices

in which the join-prime elements form a proper subset of the join-irreducible elements, but in a

distributive lattice the two types of elements coincide. For, suppose that x is join-irreducible, and

that x ≤ y ∨ z. This inequality is equivalent to the statement that x = x ∧ (y ∨ z), and by the

distributive law x = (x ∧ y) ∨ (x ∧ z). However, since x is join-irreducible, at least one of the two

terms in this join must be x itself, showing that either x = x ∧ y (equivalently x ≤ y) or x = x ∧ z

(equivalently x ≤ z).

Vertex cut

We define a cut of D as an arc set S [x] C A, introduced by X C V. The cut consists of all the arcs

that are incident to X & V/X i.e. S [X] = {u, v) A: {u, v} ∩ X = 1} where maps a set to its

cardinality. A cut will be directed if all its arcs point from X either to V/X (positively directed) or

from V/X to X (negatively directed). A cut of the form S [{V}] for v V is called a vertex cut.

Flip-Flop Operation

Reversing the orientation on all the arcs c of a positively directed vertex cut is called a flip. The

universe operation i.e. reversing the orientation on a negatively directed vertex cut is called a flop.

Theorem [1.1]: Let be a connected digraph and T .Then the flip-flop poset ,

T) with forbidden vertex T is a distributive lattice.

Theorem [1.2]:Birkoff theorem: Any finite distributive lattice L is isomorphic to the lattice of

lower sets of the partial order of the join-irreducible elements of L. That is, there is a unique poset

P such that L =J (P).

Composite Mapping of Flip-Flop Poset of Join-Irreducible Elements of Distributive Lattice


3. THEOREM

Let be a distributive lattice & [D]. If Ø: and ,T) → { C

} then ,T))=

Proof: Since has no negative directed vertex cut apart from T.i.e.no flops can be performed at

,i.e. is a minimum of ,T), by definition of ,T).By theorem [1.1] ,T) is a

distributive lattice, and ,T) has a unique minimum, which must be .Thus every element of

,T) can be reached by a sequence of a flips starting from .By definition of mapping it

maps every ,T) to a vertex set whose vertex cuts {S[V]\V } can be flipped in

some order to reach from .Since flipping different vertex sets , ' C yields different

reoriented arc sets S[ ,S[ '], there is a unique C whose vertex cuts have to be flipped to

obtain ,from .for every C whose vertex cuts can be flipped there is -1

( )

,T).Every CV( ) corresponds to an C via the bijection Ø given in the

construction of [D].Thus Ø maps every set of vertices whose cuts can be flipped to reach an

element of ,T) to an order ideal C i.e. ,T))

C ,T)) C . The vertex cuts at the source of are

positively directed. However, the sources are connected to T, Which are not allowed to be flipped,

so the cuts of the sources can be flipped exactly once. Furthermore, every vertex cut S[v] can be

flipped only after the vertex cuts S[u] for u have been flipped. Iteratively every vertex cut

can be flipped at most as often as the sources. Together we have that every vertex cut can be

flipped at most once in a flipping sequence and Ø maps the set of vertices , whose cuts have

been flipped in any flip sequence to an ideal . Consider an ideal the

corresponding vertex set =Ø-1

( induces a directed cut S[ ] in . It is an elementary fact from

graph theory that a cycle in a digraph contributes as many forward acres as backward arcs to its

intersection with a directed cut. So reversing the orientation on S [ ] leaves the number of

forward arcs among the cycles of invariant. Thus reversing S [ ] yields a c-reorientation of .

Every cut in can be flipped at most once. Therefore for every arc (u, v) S[ ] exactly one of

the vertex cuts S[u] and S[v] has to be flipped. However, to flip S[v] one must have flipped S[u]

as before. This is S[u] and not S[v] must have been flipped. We have u and by construction Ø-

1 maps the vertices of incident to S [ ] to the maxima of . So to reverse S[ ] the vertex cuts in

Ø-1

( must be flipped i.e. C ,T)) Hence ,T))= .

Lemma [3.1]: Let be a distributive lattice & [D] then ,T) is isomorphic to .

Proof: By using main result of this paper and theorem [1.2] lemma can be proved.

4. CONCLUSION

We have presented an efficient method for inclusion order on the ideals of subposet of join

irreducible elements of distributive lattice using Birkoff representation theorem and have

extended the work using mapping , Ø is a mapping from sub poset of join irreducible

element of distributive lattice to vertex cut and is a mapping from flipflop poset to a subset of

vertex cut ,so that an ideal of sub poset of join irreducible element induces vertex cut on directed

cut in set of digraph.

ACKNOWLEDGEMENT

Guidance during preparation of this research paper fills me with a sense of gratitude. Firstly, I

express my gratitude to my supervisor Dr. Madhu Tiwari and co supervisor Dr. C. L Parihar for

their generous support. The contribution I received from Dr. Sonu Sen and Dr. Parimeeta

Chanchani needs to be specially mentioned for submission of this research paper.

Namrata Kaushal et al.

International Journal of Scientific and Innovative Mathematical Research (Ijsimr) Page | 82

REFERENCES

[1] G. Gratzer , H. Lakser, “On complete congruence lattices of complete lattices”, Trans.

Amer. Math. Soc. 327, 385-405, (1991).

[2] G. Gratzer., H. Lakser., “On congruence lattices of complete lattices”, J.Australian Math.

Soc. A-52, 57-87, (1992).

[3] G. Gratzer., H. Lakser., “Congruence lattices of planar lattices”, Acta Math. Hungar. 60,

251-268, (1992).

[4] G. Gratzer., E. T. Schmidt., “On congruence lattices of lattices”, Acta Math. Acad. Sci.

Hungar, 13, 179-185, (1962).

[5] Kolja, B. Knauer, “Distribitive Lattices on Graph Orientations”, Technical University

Berlin (2000).

[6] R. P. Dilworth, “lattices with unique irreducilale decomposition”, Ann of Maths, 41, 771-

777 (1940).

[7] R.P. Dilworth., “A decomposition theorem for partially ordered sets”, Ann of Math. 2, 161-

166, (1950).

[8] R.P. Dilworth, P. Crawley, “Decomposition theory for lattices without chain conditions”,

Trains Amer Math Soc. 96, 1-22, (1960).

[9] R. P. Dilworth, “A decomposition theorem for partially ordered sets”, Ann. of Math. 2,

161-166, (1950).

[10] S. P. Avann, “Application of the join-irreducible excess function to semimodular lattices”,

Mathematische Annalen, 142, 345-354.

[11] Birkhoff, Garrett, “Lattice Theory”, Third Editions American mathematical society 25,

Colloquium Publication, 1969.

[12] K. B. Knauer, “Partial orders on orientations via cycle flips”, www.math.turlin.de/ knauer/

diplom.pdf, 2007.




www.arcjournals.org

©ARC Page | 83

Geometric Progression in Operations Research (PERT) –

A Special Case Study

Dr. Kanduri Venkata Lakshmi Narasimhacharyulu

Associate Professor



Bapatla-522101,India.

[email protected]

I.Pothuraju Assistant Professor

Department of Mathematics


Bapatla, Andhra Pradesh, India.

[email protected]

Abstract: The present paper looks into the influence of Geometric Progression (G.P) which will whether

endorse the network or not in a special case. The huge network is illustrated in a systematic way with 46

nodes and 60 activities. G.P is employed on a most likely time estimate among the three time estimates

namely optimistic, most likely and pessimistic. The investigation has been done on the considered network.

Some remarkable results are found. All float values are also computed. Critical path is identified and

project analysis has been carried out. Periodical analysis is also established with standard normal

distribution curves which are illustrated wherever necessary.

Keywords: Network, Time estimates, Float, Critical path, Normal distribution.

AMS Classification: 90-08, 90B10, 90C90

1. INTRODUCTION

Project networks are best representations of reality. The networks have some difficulty to control

the realistic influences. In such case, no advantage will be occurred in their execution. In general

networks used to predict and explain phenomena with a high degree of accuracy. Very large

number of variables may be commanded to anticipate a phenomenon with consummate accuracy.

The best way is to find the correct variable and right relationship between them.

The normal distribution curve and it’s properties have been employed for the project with large

size network. It necessitates all activities of the project. The three time estimates estimate an

individual activity as optimistic time , pessimistic time and most probable time. These three time

estimates indicate the measure of uncertainty likely to be encountered in performing an individual

activity. The time estimates are mainly based on human observation, experience and knowledge

of the estimator about the performance of the particular activity.

Levin and Kirkpatrick [3] explained about planning and control of a project with the aid of PERT

and CPM in 1966.Wiest and Levy [4] looked into new ideas on Networks with management guide

to PERT/CPM for research scholars in the area of operations research. PERT algorithm was

established with various models by Billy E.Gillett [5] .He attempted different models of

operations Research in 1979. S.D Sharma [6] discussed the applications of PERT&CPM

techniques. K.V.L.N.Acharyulu et.al [1] analyzed some curious cases of PERT and Game theory

problems.

The authors examined in this paper to classify whether Geometric progression in a peculiar case

will assert a network or not. The Geometric progression is employed on most likely time estimate

among the three time estimations. Project analysis is also accomplished. Total Float, Free float

and Independent Float are calculated in the part of confirming the critical path. The standard

normal distribution curves are illustrated with the aid of Mat lab wherever essential.



Kanduri Venkata Lakshmi Narasimhacharyulu & I.Pothuraju


2. CONSTRUCTION OF NETWORK

A network is constructed with 46 nodes and 60 activities in a systematic way

for investigating the influence of Geometric Progression. G.P is applied on most likely time

estimate (m) in case (I) among the three estimates. Network does not accept any error and dummy

activity.

Fig.1. Drawn Network having 46 nodes with 60 activities

3. PRELIMINARIES AND NOTATIONS

(i).TE= Earliest excepted completion time of event (TE)

Geometric Progression in Operations Research (PERT) -A Special Case Study


Def: For the fixed value of j=TE(j)=Max[TE(i)+ET(i,j)] which ranges over all activities from i-j.

(ii).TL= Latest allowable event completion time (TL)

Def: For the fixed value of i=TL(i)=Min[TL(j)+ET(i,j)] which ranges over all activities from i-j.

(iii).ET= Excepted completion time of activity (I,J)

(iv). a = Optimistic time estimate

(v). m = Most likely time estimate

(vi). b = Pessimistic time estimate

(vii). ES = Earliest start of an activity

(viii). EF = Earliest finish of an activity

(ix). LS = Latest start of an activity

(x). LF = Latest finish of an activity

(xi). TF = Total Float

Def: TF of activity i-j = LFi-j-EFi-j (or) LSi-j-ESi-j

(xii).FF = Free Float

Def: FF of activity i-j = TF - (TL-TE) of node j

(xiii).IF = Independent Float

Def: IF of activity i-j = FF - (TL-TE) of node i

(xiv).SE=Slack event time

(xv).CPI=Critical Path Indicator

(xvi).SCT= Scheduled Time

(xvii). = Standard deviation of project length

4. MATERIAL AND METHODS

Step 1: Draw the project network completion time

Step 2: Compute the excepted duration of each activity by using the formula4

6

a m bET

.

From the time estimates a,m and p. Also calculate the excepted variance. 2 of each activity

Step 3: Calculate TE, TL

Step 4: Find Total Float, Free Float and Independent Float

Step 5: Find the critical path and identify the critical activities

Step 6: Compute project length which is a square root to sum of variance of all the critical

activities.

Step 7: From the standard normal variable SCT ET

z

, Where SCT is scheduled

Completion time of event, =standard deviation of project length. Using the standard normal

curve, we can estimate the probability of completing project within specified time.

5. RESULTS

By using CPM and PERT algorithms on the Network, the critical path is identified from the

scientific computations of the following tables from Table-1 to Table-6.The tables consists of all

activities, Time estimates, ET, Varience.ES, EF, LS, LF and all Float values. The Critical path

indicator provides at each critical Activity in each table.



Table-1. Scientific Computations of First two Levels

Activity Time

Estimates

ET

Earliest[E] Latest[L] TF F

F

IF C

P

I a m b ES EF LS LF

1--2 1 1.41

4

2 1.44

2

0.02

7

0 1.442 60.03

6

61.47

8

60.03

6

0 0

1--3 3 3.46

4

4 3.47

3

0.02

7

0 3.473 0 3.473 0 0 0 *

2--4 5 5.47

7

6 5.48

4

0.02

7

1.44

2

6.926 89.48

4

94.96

8

88.04

2

0 -

60.036

2--5 7 7.48

3

8 7.48

8

0.02

7

1.44

2

8.93 61.47

8

68.96

6

60.03

6

0 -

60.036

3--6 9 9.48

6

10 9.49 0.02

7

3.47

3

12.96

3

31.47

5

40.96

5

28.00

2

0 0

Table-2. Scientific Computations of Third Level

Activity Time

Estimates

ET


F

IF

a m b ES EF LS LF

4--8 1

3

13.49 1

4

13.49

3

0.02

7

6.926 20.41

9

106.96

9

120.46

2

100.04

3

0 -

88.0

42

4--9 1

5

15.49

1

1

6

15.49

4

0.02

7

6.926 22.42 94.968 110.46

2

88.042 0 -

88.0

42

5--10 1

7

17.49

2

1

8

17.49

4

0.02

7

8.93 26.42

4

80.968 98.462 72.038 0 -

60.0

36

5--11 1

9

19.49

3

2

0

19.49

5

0.02

7

8.93 28.42

5

68.966 88.461 60.036 0 -

60.0

36

6--12 2

1

21.49

4

2

2

21.49

6

0.02

7

12.96

3

34.45

9

53.265 74.461 40.002 0 -

28.0

02

6--13 2

3

23.49

4

2

4

23.49

6

0.02

7

12.96

3

36.45

9

40.965 64.461 28.002 0 -

28.0

02

7--14 2

5

25.49

5

2

6

25.49

6

0.02

7

14.96

5

40.46

1

26.965 52.461 12 0 0

7--15 2

7

27.49

5

2

8

27.49

6

0.02

7

14.96

5

42.46

1

14.965 42.461 0 0 0

Table-3. Scientific Computations of Fourth Level

Activity Time

Estimates

ET


F

IF

a m b ES EF LS LF

8--16 2

9

29.49

5

3

0

29.49

6

0.02

7

20.41

9

49.915 124.46

3

153.95

9

104.04

4

0 -

100

.04

3

8--17 3

1

31.49

6

3

2

31.49

7

0.02

7

20.41

9

51.916 120.46

2

151.95

9

100.04

3

0 -

100

.04

3

2

2

2



9--18 3

3

33.49

6

3

4

33.49

7

0.02

7

22.42 55.917 114.46

2

147.95

9

92.042 0 -

88.

042

9--19 3

5

35.49

6

3

6

35.49

7

0.02

7

22.42 57.917 110.46

2

145.95

9

88.042 0 -

88.

042

10--20 3

7

37.49

6

3

8

37.49

7

0.02

7

26.42

4

63.921 102.46

2

139.95

9

76.038 0 -

72.

038

10--21 3

9

39.49

6

4

0

39.49

7

0.02

7

26.42

4

65.921 98.462 137.95

9

72.038 0 -

72.

038

11--22 4

1

41.49

6

4

2

41.49

7

0.02

7

28.42

5

69.922 92.462 133.95

9

64.037 0 -

60.

036

11--23 4

3

43.49

7

4

4

43.49

8

0.02

7

28.42

5

71.923 88.461 131.95

9

60.036 0 -

60.

036

12--24 4

5

45.49

7

4

6

45.49

8

0.02

7

34.45

9

79.957 78.461 123.95

9

44.002 0 -

40.

002

12--25 4

7

47.49

7

4

8

47.49

8

0.02

7

34.45

9

81.957 74.459 121.95

7

40 0 -

40.

002

13--26 4

9

49.49

7

5

0

49.49

8

0.02

7

36.45

9

85.957 68.459 117.95

7

32 0 -

28.

002

13--27 5

1

51.49

7

5

2

51.49

8

0.02

7

36.45

9

87.957 64.461 115.95

9

28.002 0 -

28.

002

14--28 5

3

53.49

7

5

4

53.49

8

0.02

7

40.46

1

93.959 56.461 109.95

9

16 0 -12

14--29 5

5

55.49

7

5

6

55.49

8

0.02

7

40.46

1

95.959 52.461 107.95

9

12 0 -12

15--30 5

7

57.49

7

5

8

57.49

8

0.02

7

42.46

1

99.959 46.461 103.95

9

4 0 0

15--31 5

9

59.49

7

6

0

59.49

8

0.02

7

42.46

1

101.95

9

42.461 101.95

9

0 0 0

Table-4. Scientific Computations of Fifth Level

Activit

y

Time

Estimates

ET

Earliest[E] Latest[L] TF FF IF

a m b ES EF LS LF

16-32 6

1

61.49

7

6

2

61.49

8

0.02

7

49.91

5

111.4

13

153.9

59

215.4

57

104.0

44

4.00

1

-

100.0

43

17-32 6

3

63.49

8

6

4

63.49

8

0.02

7

51.91

6

115.4

14

151.9

59

215.4

57

100.0

43

0 -

100.0

43

18-33 6

5

65.49

8

6

6

65.49

8

0.02

7

55.91

7

121.4

15

147.9

59

213.4

57

92.04

2

4 -

88.04

2

19-33 6

7

67.49

8

6

8

67.49

8

0.02

7

57.91

7

125.4

15

145.9

59

213.4

57

88.04

2

0 -

88.04

2

20-34 6

9

69.49

8

7

0

69.49

8

0.02

7

63.92

1

133.4

19

139.9

59

209.4

57

76.03

8

4 -

72.03

8

21-34 7 71.49 7 71.49 0.02 65.92 137.4 137.9 209.4 72.03 0 -

2



1 8 2 8 7 1 19 59 57 8 72.03

8

22-35 7

3

73.49

8

7

4

73.49

8

0.02

7

69.92

2

143.4

2

133.9

59

207.4

57

64.03

7

4.00

1

-

60.03

6

23-35 7

5

75.49

8

7

6

75.49

8

0.02

7

71.92

3

147.4

21

131.9

59

207.4

57

60.03

6

0 -

60.03

6

24-36 7

7

77.49

8

7

8

77.49

8

0.02

7

79.95

7

157.4

55

123.9

59

201.4

57

44.00

2

4 -

40.00

2

25-36 7

9

79.49

8

8

0

79.49

8

0.02

7

81.95

7

161.4

55

121.9

59

201.4

57

40.00

2

0 -

40.00

2

26-37 8

1

81.49

8

8

2

81.49

8

0.02

7

85.95

7

167.4

55

117.9

59

199.4

57

32.00

2

4 -

28.00

2

27-37 8

3

83.49

8

8

4

83.49

8

0.02

7

87.95

7

171.4

55

115.9

59

199.4

57

28.00

2

0 -

28.00

2

28-38 8

5

85.49

8

8

6

85.49

8

0.02

7

93.95

1

179.4

49

109.9

59

195.4

57

16.00

8

4.00

8

-12

29-38 8

7

87.49

8

8

8

87.49

8

0.02

7

95.95

9

183.4

57

107.9

59

195.4

57

12 0 -12

30-39 8

9

89.49

8

9

0

89.49

8

0.02

7

99.95

9

189.4

57

103.9

59

193.4

57

4 4 0

31-39 9

1

91.49

8

9

2

91.49

8

0.02

7

101.9

59

193.4

57

101.9

59

193.4

57

0 0 0

Table-5. Scientific Computations of Sixth Level

Activi

ty

Time Estimates ET

Earliest[E] Latest[L] TF FF IF

a m b ES EF LS LF

32-40 93 93.49

8

94 93.49

8

0.02

7

115.4

14

208.9

12

215.4

57

308.9

55

100.0

43

12.0

01

-

88.0

42

33-40 95 95.49

8

96 95.49

8

0.02

7

125.4

15

220.9

13

213.4

57

308.9

55

88.04

2

0 -

88.0

42

34-41 97 97.49

8

98 97.49

8

0.02

7

137.4

19

234.9

17

209.4

57

306.9

55

72.03

8

12.0

02

-

60.0

36

35-41 99 99.49

8

10

0

99.49

8

0.02

7

147.4

21

246.9

19

207.4

57

306.9

55

60.03

6

0 -

60.0

36

36-42 10

1

101.4

98

10

2

101.4

98

0.02

7

161.4

55

262.9

53

201.4

57

302.9

55

40.00

2

12 -

28.0

02

37-42 10

3

103.4

98

10

4

103.4

98

0.02

7

171.4

55

274.9

53

199.4

57

302.9

55

28.00

2

0 -

28.0

02

38-43 10

5

105.4

98

10

6

105.4

98

0.02

7

183.4

57

288.9

55

195.4

57

300.9

55

12 12 0

39-43 10

7

107.4

98

10

8

107.4

98

0.02

7

193.4

57

300.9

55

193.4

57

300.9

55

0 0 0

Table-6. Scientific Computations of Seventh Level

2



Activ

ity

Time Estimates ET

Earliest[E] Latest[L] TF FF IF C

P

I a m b ES EF LS LF

40-44 10

9

109.4

98

11

0

109.4

98

0.0

27

220.9

13

330.4

11

308.9

55

418.4

53

88.0

42

28.0

06

-

60.03

6

41-44 11

1

111.4

98

11

2

111.4

98

0.0

27

246.9

19

358.4

17

306.9

55

418.4

53

60.0

36

0 -

60.03

6

42-45 11

3

113.4

98

11

4

113.4

98

0.0

27

274.9

53

388.4

51

302.9

55

416.4

53

28.0

02

28.0

02

0

43-45 11

5

115.4

98

11

6

115.4

98

0.0

27

300.9

55

416.4

53

300.9

55

416.4

53

0 0 0 *

44-46 11

7

117.4

98

11

8

117.4

98

0.0

27

358.4

17

475.9

15

418.4

53

535.9

51

60.0

36

60.0

36

0

45-46 11

9

119.4

98

12

0

119.4

8

0.0

27

416.4

53

535.9

51

416.4

53

535.9

51

0 0 0 *

Critical path is obtained as below

Project Length is defined as Sum of Variances of each Critical activity

i.e Project Length 0.027 0.027 0.027 0.027 0.027 0.027 0.027 0.027

=0.4647

The values of TE, TL and SE corresponding to every node are given in table (3).

The slack event time may be positive, negative or zero.

It is also observed that the values of slack event time vanish at each critical activity.

Slack event time is defined as the amount of time in which the event can be retarded with out

involving the scheduled completion time for the project. Any activity on the critical path

necessitates time in excess of its expected completion time and detains the project completion

consequently.

Table-7

Nodes TE TL SE Nodes TE TL SE

1 0 0 0 24 79.957 123.959 44.002

2 1.442 61.478 60.036 25 81.957 121.959 40.002

3 3.473 3.473 0 26 85.957 117.959 32.002

4 6.926 94.968 88.042 27 87.957 115.959 28.002

5 8.93 68.966 60.036 28 93.951 109.959 16.008

6 12.963 40.965 28.002 29 95.959 107.959 12

7 14.965 14.965 0 30 99.959 103.959 4

8 20.419 120.462 100.043 31 101.959 101.959 0

9 22.42 110.462 88.042 32 115.414 215.457 100.043

10 26.424 98.462 72.038 33 125.415 213.457 88.042

11 28.425 88.461 60.036 34 137.419 209.457 72.038

12 34.459 74.461 40.002 35 147.421 207.457 60.036

13 36.459 64.461 28.002 36 161.455 201.457 40.002

14 40.461 52.461 12 37 171.455 199.457 28.002

15 42.461 42.461 0 38 183.457 195.457 12

16 49.915 153.959 104.044 39 193.457 193.457 0

2

3.473 11.492 27.496 59.498 91.498 107.498 115.498 119.498

1 3 7 15 31 39 43 45 46



17 51.916 151.959 100.043 40 220.913 308.955 88.042

18 55.917 147.959 92.042 41 246.919 306.955 60.036

19 57.917 145.959 88.042 42 274.953 302.955 28.002

20 63.921 139.959 76.038 43 300.955 300.955 0

21 65.921 137.959 72.038 44 358.417 418.453 60.036

22 69.922 133.959 64.037 45 416.453 416.453 0

23 71.923 131.959 60.036 46 535.951 535.951 0

6. PROJECT ANALASIS

Project analysis is carried out with specific schedule times and the standard normal variables are

identified in the possible range of probability from o to 1.The percentage of possibilities

of completion of the Project are obtained and specified in the following Table-8.The graphs are

also illustrated.

Table-8

SCT ETC Z Probability Percentage of Possibility (%)

534 535.951 -4.918 0 0

535 535.951 -2.046 0.2068 20.6

536 535.951 0.105 0.53983 53.9

537 535.951 2.257 0.98778 98.7

538 535.951 4.409 1 100

The derived Standard Normal Curves are shown from Fig.2-Fig.6

-6 -4 -2 0 2 4 60

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

The aim of culmination with zero probability

Probability(0)

Z=-4.918

Fig.2



-6 -4 -2 0 2 4 60

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

The aim of culmination with paltry probability

Probability(0.2068)Z=-2.046

Fig.3

-6 -4 -2 0 2 4 60

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

The aim of culmination with partial probability

Z=-0.105

Probability(0.53983)

Fig.4

-6 -4 -2 0 2 4 60

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

The aim of culmination with most acceptable probability

Probability(0.98778)

Z=2.257

Fig.5



-6 -4 -2 0 2 4 60

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

The aim of culmination with most acceptable probability

Z=4.409

Probability(1)

Fig.6

7. CONCLUSIONS

The following conclusions are incurred in this investigation of scientific study.

(i).In Critical Path

(a).It is noted that all Total Float values of Critical activities are vanished.

(b).The value of Slack event of each node in critical path has become zero.

(c). TE and TL are same at each node in critical path.

(ii).G.P sustains consistently the Network even though the network has large size.

(iii).Variances are identical at any activity of the Network.

(iv).The case in which G.P is conceived on most likely time estimate, the expected completion

time of successive activity is gradually increased.

(v).The influence of G.P in the Network are identified as

(a).G.P supports accurately only when SCT is greater than ETC, except SCT value is nearer to

ETC.

(b).G.P does not support effectively when SCT is less than or equal to ETC.

(c). Standard Normal Distribution curves illustrate the percentage of possibilities of the Project.

REFERENCES

[1] K.V.L.N.Acharyulu and Maddi.N.Murali Krishna,(2013). Some Remarkable Results in Row

and Column both Dominance Game with Brown’s Algorithm, International Journal of

Mathematics and Computer Applications Research,Vol. 3, No.1, pp.139-150.

[2] K.V.L.N.Acharyulu and Maddi.N.Murali Krishna, (2013). A Scientific Computation On A

Peculiar Case of Game Theory in Operations Research, International Journal of Computer

Science Engineering and Information Technology Research,Vol. 3 , No.1, pp.175-190.

[3] Levin, R., and C.A. Krik Patrick,(1966).Planning and control with PERT/CPM, McGraw-Hill

Book company, New York.

[4] Wiest, J.D., and F-Levy,(1969).A management Guide to PERT/CPM, Patrick-mall, Inc. Engle

Wood Cliffs, N.J.

[5] Billy E. Gillett,(1979).Introduction to operations Research, Tata McGraw-Hill Publishing

Company limited, PP.434-453,New York.

[6] S.D.Sharma,(1999). Operations Research, PP.4.300-4.355, Kedar Nath Ram Nath & Co.




Kanduri Venkata Lakshmi Narasimhacharyulu Who is known as

Dr.K.V.L.N.Acharyulu is working sas Associate Professor in the

Department of Mathematics, Bapatla Engineering College, Bapatla which is

a prestigious institution of Andhra Pradesh. He took his M.Phil. Degree in

Mathematics from the University of Madras and stood in first Rank,R.K.M.

Vivekananda College,Chennai. Nearly for the last thirteen years he is

rendering his services to the students and he is applauded by one and all for

his best way of teaching. He has participated in some seminars and presented

his papers on various topics. More than 70 articles were published in various

International high impact factor Journals. He obtained his Ph.D from ANU

under the able guidance of Prof. N.Ch.Pattabhi Ramacharyulu,NIT,Warangal. He authored three

books. He is a Member of Various Professional Bodies and created three world records in

research field. He received so many awards and rewards for his research excellency in the field of

Mathematics.

I.Pothuraju, He is working as Assistant professor in Department of

Mathematics, Bapatla Engineering College. He has two years of teaching

experience. He is doing his M.phil under the guidance of

Dr.K.V.L.N.Acharyulu.He did M.Sc(Mathematics) in Bapatla Engineering

College. He obtained MBA from Pydah College,Andhra University. He

completed his B.Sc(M.P.C) in Bapatla College of Arts & Science. He has a

zeal to invent new findings in Mathematics.




www.arcjournals.org

©ARC Page | 94

On Product Summability of Fourier Series



IFTM University.Moradabad, U.P.,India, 244001

[email protected]

Abstract: In this paper, a theorem on ( , , )( , , )n nN p q C product summability of Fourier series has

been estabilished..

Keywords: ( , , )n nN p q -mean, ( , , )C - mean, ( , , )( , , )n nN p q C -product mean and Fourier

series.

Mathematical classification: 40B05, 42B08.

1. INTRODUCTION

Let na be a given infinite series with sequence of its nth partial sums { }ns . Let { }np be a

sequences of non-negative, non increasing real constants such that

0

n

n v

v

P p

as n . (1.1)

For a positive real sequence { }nq q , we define an increasing sequence { }nr such that

0

( * )n

n n n v v

v

r p q p q

as n (1.2)

denotes the convolution product where

0

, 0, 1n

n v i i

v

Q q Q q i

(1.3)

The sequence-to-sequence transformation

0

1 n

n n v v v

vn

t p q sr

(1.4)

defences the sequence { }nt of the | , , |n nN p q - mean of the sequence { }ns , (Borwein [2]).

If nt s as n , then the series na is said to be | , , |n nN p q -summable to s .

Again let na be a given infinite series with partial sum { }ns and ,

nt

denotes the nthcesaro

mean of order ( , ) with 1 of the sequence { }ns such that,

, 1

1

1n n v v

vn

t A sA

(1.5)





where O( )nA n and 0 1A .

If ,

nt s as n . Then the series na is said to be ( , , )C summable to s . The product

of ( , , )n nN p q -summability with ( , , )C -summability defines ( , , )n nN p q ( , , )C

summability and denoted by,

pq nN C and

If , 1

0 0

.1 n kv k k

pq n k v v v

k vn k

p qN C A A s s

r A

as n (1.6)

Then the series na is said to summable to s by ( , , )( , , )n nN p q C -summability method.

In the case when 1 and 1nq n N , then the method ( , , )( , , )n nN p q C reduces to

( , )nN p ( , )C and if 1np n N and, 1 then the method ( , , )n nN p q reduces to

( , )( , )nN q C method. it is known ( , , )n nN p q and ( , , )C methods are regular (Hardy [3]).

It is suppose that ( , , )( , , )n nN p q c is regular throughout this paper.

Let ( )f t be a periodic function with period 2 , integrable in the sence of Lebesgue over

( , ) then

0

1 0

( ) ( cos sin ) ( )2

n n n

n n

af t a nt b nt A t

(1.7)

Is the Fourier series associated with f .

We use the following notation throughout this paper.

( ) ( ) ( ) 2 ( )t f x t f x t f x

1

0

1sin( )

1 2( )2

sin2

n kn k k

n k v van v on k

v tp q

K t A Atr A

2. KNOWN RESULT

Dealing with ( , , )( , )n nN p q E z summability method of a Fourier series,Padhy et al [4] estabi -

lished the following theorem.

Theorem 2.1

Let { },{ }n np q and { }nr be sequences satisfying (1.2), (1.2) and

0( ) | ( ) | O as O

(1/ )

t tt u du t

t

(2.1)

And ( ) as a n n (2.2)

where ( )t is positive, non increasing function of t , then the Fourier series 1

( )n

n

A t

is

summable ( , , )(E, Z)n nN p q at the point t .



3. MAIN RESULT

In this paper, we have estabilished a theorem on ( , , )( , , )n nN p q C product summability of

Fourier series.

Theorem 3.1. Let { },{ }n np q and{ }nr be sequences satisfying (1.1), (1.2) and

0( ) | (u) | O , as O

(1/ )

t tt du t

t

(3.1)

And ( ) as n n (3.2)

where ( )t be a positive, non-increasing function of t .

The Fourier series 0

( )n

n

A t

is summable ( , , )n nN p q ( , , )C at the point t .

4. REQUIRED LEMMA

We have required the following lemmas to prove the theorem.

Lemma 4.1

1| ( ) | O( ),O

1nk t n t

n

Proof:

For 1

01

tn

, we have (Boose [1])

sin sinnt n t and1

0

n

k v v k

v

A A A

Then

1

0

(2 1)sin1 2| ( ) |

2sin

2

n k k v vn k k

n

k o vn k

tA A v

p qk t

tr A

1

0

1(2 1)

2

n kn k k

k v v

k o vn k

p qk A A

r A

1(2 1)

2

nn k k

k

k on k

p qk A

r A

(2 1)| |

2n k k

n

np q

r

O( )n

Lemma 4.2

| ( ) | O(1/ ), for 1/nk t t n t



Proof:

For 1

tn

, we have by Jordon's lemma,

sin( / 2) , ( / ), sin 1.t t nt

Then

11 2

0 0 2

sin( )1| ( ) |

2 sin

n kn k k

n k v v tk vn k

v tp qk t A A

r A

1

0 0

1

2

n kn k k

k v v

k vn k

p qA A

r A t

0

1

2

n k kk

kn k

p qA

r A

O(1/ )t

5. PROOF OF THE THEOREM

If ( ; )ns f x is the n-th partial sum of the Fourier series 0

( )n

n

A t

of ( )f t then by using Riemann–

Lebesgue theorem, we have (Titchmarch [5]).

12

02

sin( )1( ; ) ( ) ( )

2 sin( )n t

n ts f x f x t dt

If ,

pq nN C denote the ( , , )( , , )n nN p q C transform of ( ; )ns f x , we have

, 1 120

0 0

1 ( )( ) sin( )

2 sin( / 2)

n kn k k

pq n k v v

k vn k

p q tN C f x A A v t dt

r A t

0( ) ( )nt k t dt

In order to prove the theorem, it is sufficient to show that

0( ) ( ) O(1) as nt k t dt n

For 0 , we have

,

0( ) ( ) ( )pq n nN C f x t k t dt

1/

0 1/( ) ( )

n

nn

t k t dt

1 2 3 (say)I I I

Now

1/

10

| | ( ) ( )n

nI t k t dt



1/

0| ( ) || ( ) |

n

nt k t dt

1O( ) O

( )n

n n

1O as

( )n

n

O(1) as n

Next

21/

| | ( ) ( )nn

I t k t dt

1/| ( ) || ( ) |n

nt k t dt

1/

| ( ) |O

n

tdt

t

21/1/

( ) ( )-O

nn

t tdt

t t

1/1/

1 1O O O

(1/ ) ( )

n

n

dut u u

Where1

ut

and 0 1

1/

1 1O O

( ) ( )

n

dua n n n

Using second mean value theorem for the integral in the 2nd

term as ( )n is monotonic

O(1) O(1), as n

O(1) as n

Finally

3| | | ( ) | | ( ) |nI t k t dt

O(1) as n

by using Riemann-Lebesgue theorem and the regularity condition of the method of summability.

Thus , ( ) O(1) as pq nN C f x n

This completes the proof of the theorem.

6. CONCLUSION

In this paper a more general result for summability of Fourier series is established which will be

enrich the Literature of Fourier series.



ACKNOWLEDGEMENTS

I am very thankful to Dr. B.K. Singh, (Professor and Head of the Dept. of Mathematics, IFTM

Uni. Moradabad, U.P., India) whose great inspirations lead me to complete this paper.

REFERENCES

[1] Boose, J.; Classical and modern methods in summability, Oxford University Press, Oxford

(1949).

[2] Borwein, D.; On product of sequences, Journal of Lon. Math. Soc. 33 (1958).

[3] Hardy, G.H.; Divergent series, Uni. Press Oxford, (1959).

[4] Padhy, B.P.; On( , )(E, Z)n nN p q

product summability of Fourier series, AJCEM Vol-1 Issue

3 (2012).

[5] Titchmarch, E.C.; The theory of functions, Oxford University Press, Oxford (1939).


Mr. Aditya Kumar Raghuvanshi is presently a research scholar in the

deptt Of Mathematics, I F T M University Moradabad (U.P.), India.

He has completed his M.Sc.(Maths) and M.A. (Economics) from

MJPR University Bareilly (U.P.), B.Ed from C C S University Meerut

(U.P.), and he has also completed his M.Phill. (Maths) from The

Global Open University Nagaland, India. He has published twenty

Research papers in various International Journals. His fields of research

are O.R. , Summability and Approximation Theory.




www.arcjournals.org

©ARC Page | 100

Trignometric Inequations and Fuzzy Information Theory

P.K. Sharma,

Head, Post Graduate

Dept. of Mathematics,

Hindu College, Amritsar-143001,

[email protected]

Nidhi Joshi,

Assistant Professor,

Dept. of Mathematics,

Arya College, Ludhiana

[email protected]

Abstract: Some new trigonometric measures of fuzzy entropy involving trigonometric functions have been

provided and their validity is checked by studying their essential properties and certain inequalities

involving trigonometric angles of a convex polygon of n sides have been proved by making use of some

concepts from fuzzy information theory.

Key words: Trigonometry, fuzzy information theory, entropy, fuzzy entropy trigonometric inequalities

1. INTRODUCTION

For the probability distribution P = ( p1 , p2 , ......, pn), Shannon [8] obtained the measure of

entropy as :

1S (P) =

n

i

ii pp

1

log (1.1)

Which is a concave function and has maximum value when1

....1 2

p p pnn

Corresponding to (1.1), Deluca and Termini [2] suggested the measure of fuzzy entropy as

1H (A) = ( ) log ( ) (1 ( )) log 1 ( )1

nx x x xi i i iA A A Ai

(1.2)

Which is a concave function and has maximum value when A is most fuzzy set.

Renyi’s [7] probabilistic measure of entropy is given by

1H (P) log

1 1

npi

i, > 0, 1 (1.3)

Corresponding to (1.3), Bhandari and Pal [1] suggested measure of fuzzy entropy as

1H (A) log ( ) 1 ( )

1 1

nx xi iA Ai

; > 0, 1 (1.4)

Corresponding to Havrada and Charvat’s [3] probabilistic measure of entropy,

1H (P) 1

1 1

npi

i ; > 0, 1, (1.5)

Kapur [4] suggested the following measure of fuzzy entropy :


P.K. Sharma & Nidhi Joshi


1H (A) ( ) (1 ( )) 1

1 1

nx xi iA Ai

; > 0, 1 (1.6)

Apparently, there seems to be no relation between trigonometry and fuzzy information theory.

Nevertheless one relationship arises as measures of fuzzy entropy used in fuzzy information

theory are concave functions and some trigonometric functions are also concave functions. Our

interest is to exploit this relationship and establish some inequalities between angles of a convex

polygon.

A measure of entropy having these properties and involving trigonometric functions has been

given by Kapur and Tripathi [5 ]. This measure is given by

( ) sin2 1

nS P pi

i (1.7)

Corresponding to (1.7), Kapur [ 4] gave measure of fuzzy entropy as

( ) sin ( ) sin (1 ( ))2 1

nH A x xi iA Ai

(1.8)

2 sin ( )1

nxiAi

Another measure of entropy having same properties and involving logarithmic function is given

by

( ) log3 1

nS P pi

i (1.9)

Corresponding to (1.9), Parkash and Sharma [6] gave measure of fuzzy entropy as

( ) log ( ) log 1 ( )3 1

nH A x xi iA Ai

(1.10)

2. MEASURES OF FUZZY ENTROPY AND THEIR VALIDITY

We, now propose new measures of fuzzy entropy as follows:

( )4

H A = 2( 2) ( ) 2( 2) (1 ( ))

sin{ } sin{ }1

n n x n xi iA A

n ni

2( 2)sin sin

1

n n

ni (2.1)

And

( )5

H A =2 ( 2) ( ) 2 ( 2) (1 ( ))

tan{ } tan{ }1

n x n xn i iA A

n ni

2( 2)

tan tan{ }1

n n

ni (2.2)

(a) Differentiate (2.1) w.r.t. ( )xiA , we get

2 ( )4

2 ( )

H A

xiA



2 ( 2) ( ) 2 ( 2) (1 ( ))2 2 2( 2) sin{ } sin{ }1

n n x n xi iA An n n

i

For 2( 2) ( )

0n xiA

n and since 0 sin 1 0 1when

Thus

2 ( )4

2 ( )

H A

xiA

< 0 and ( )4

H A is a concave function of ( )xiA and its maximum arises when

( )xiA =

1

2

Thus we have the following results :

(i) ( )4

H A is a concave function of A (xi).

(ii) ( )4

H A doesn’t change when A (xi) is changed to 1– A (xi).

(iii) ( )4

H A is an increasing function of A (xi) when 0 A (xi) 1/2.

(iv) ( )4

H A is a decreasing function of A (xi) when 2

1 A (xi) 1.

(v) ( )4

H A =0 when A(xi) = 0 or 1.

Hence ( )4

H A is a valid measure of fuzzy entropy.

(b) Differentiate (2.2) w.r.t. ( )xiA , we get

2 ( )5

2 ( )

H A

xiA

= 2( 2) ( ) 2( 2) ( )2 2 2 22( 2) sec { }tan{ }

1


i

2( 2) (1 ( )) 2( 2) (1 ( ))2 2 2 22( 2) sec { }tan{ }1


i

For ( )xiA <

1

2n, we have

( 2) ( )n xiA < <

1 if

2 2

So that

2 ( )5

2 ( )

H A

xiA

< 0 and ( )5

H A is a concave function of ( )xiA

Thus we have the following results :

(i) ( )5

H A is a concave function of A (xi).

(ii) ( )5

H A doesn’t change when A (xi) is changed to 1– A (xi).



(iii) ( )5

H A is an increasing function of A (xi) when 0 A (xi) 1/2.

(iv) ( )5

H A is a decreasing function of A (xi) when 2

1 A (xi) 1.

(v) ( )5

H A = 0 when A(xi) = 0 or 1.

Hence ( )5

H A is a valid measure of fuzzy entropy.

3. BASIC TRIGNOMETRIC INQUALITIES

Let , , ,.....,1 2 3

A A A An be the angles measured in radians of a convex polygon of n sides, so that

....... ( 2)1 2 3,

A A A A nn

Let ( )xiA =

2( 2)

nAi

n , i = 1,2,3,….n.

Now consider the measure of fuzzy entropy

( )4

H A = 2( 2) ( ) 2( 2) (1 ( ))

sin{ } sin{ }n x n xi iA A

n n

2( 2)sin sin

1

n n

ni (3.1)

Where and are two parameters satisfying 0 < < and 0 < <1

Now each Ai < since each Ai is an angle of a convex polygon.

2( 2) ( )n xiA

n < or ( )xiA

< 2( 2)

n

n

2( 2) ( )n xiA

n < = ( )

Since ( )4

H A is a concave function of ( ), ( ), ( ),....., ( )1 2 3

x x x xnA A A A and has

maximum value for the most fuzzy set so that

( )4

H A 1

( )4 2

H (3.2)

( ), ( ), ( ),.... ( )4 1 2 3

H x x x xnA A A A

1 1 1 1, , ,...,

4 2 2 2 2H

2( 2) ( ) 2( 2) (1 ( ))sin{ } sin{ }

1

n n x n xi iA A

n ni

( 2) ( 2)sin{ } sin{ }1

n n nn n

i

sin( )1

nAi

i

( 2)sin( )

nn

n (3.3)



Moreover the inequality sign in (3.2) holds only when ( )xiA =

1

2 for each i so that equality

sign in (3.3) holds when = =.....=1 2 3

A A A An( 2)n

n

Inequality (3.3) is our basic inequality involving trigonometric functions of the angles

, , ,.....,1 2 3

A A A An of a convex polygon of n sides. The equality sign in (3.3) will hold when all

the angles are equal.

4. SPECIAL CASES

The inequality (3.3) represents a triple infinity of inequalities since it involves three parameters n,

and . Firstly, we can give any integral value 3 to n. Secondly, we can give any real

value to lying between o and . Thirdly, corresponding to any value of , we can give any

positive value to k less than 1- .

We can get special inequalities by giving particular values to , n, .

CASE I N = 3

For a triangle with angles , ,1 2 3

A A A , (3.3) gives

sin( ) sin( ) sin( )1 2 3

A A A 3sin( )3

; 0< 1 (3.4)

This gives the inequalities

sin sin sin1 2 3

A A A 3 3

2

1 2 3

3sin 2 sin 2 sin 2

2A A A

3 3cos 2 cos 2 cos 2

1 2 3 2A A A

CASE II N = 4

For a quadrilateral with angles , , ,1 2 3 4

A A A A , we have

sin( ) sin( ) sin( ) sin( )1 2 3 4

A A A A 4sin( )2

,

0< 1

This gives the inequalities

sin sin sin sin1 2 3 4

A A A A 4

31 2 4sin sin sin sin 2 22 2 2 2

AA A A

31 2 4sin sin sin sin 23 3 3 3

AA A A



31 2 4cos cos cos cos 2 23 3 3 3

AA A A

Since ( )5

H A is a concave function of ( ), ( ), ( ),....., ( )1 2 3

x x x xnA A A A and has

maximum value for the most fuzzy set so that

( )5

H A 1( )5 2

H

that is

( ), ( ), ( ),.... ( )5 1 2 3

H x x x xnA A A A

1 1 1 1, , ,...,

5 2 2 2 2H

so that

2 ( 2) ( )tan{ }

1

n xn iA

ni

( 2)tan{ }

nn

n

Or

tan( )1

nAi

i

( 2)tan{ }

nn

n (3.5)

This is our basic inequality involving tangents of angles of a convex polygon.

Again the inequality (3.5) gives a triple infinity of inequalities since it involves three parameters

n, and ,. The parameter n can take all integral values 3 . can take all real values less

than and can take all real values less than 1

2.

Particular case

For =0

1tan tan( 2) ; <

21

nA n ni ni

For triangle, 1 2 3

A A A , n=3, = 1

3 so that

31 2tan tan tan 3tan3 3 3 9

AA A

For quadrilateral, 31 2 4tan tan tan tan 4 tan4 4 4 4 8

AA A A

5. CONCLUSION

Minimum Area of a Triangle with Given Perimeter

Although there are many proofs of the result that the area of a triangle with given

perimeter is maximum when the triangle is equilateral. Also we have many proofs of the

result that the perimeter of a triangle with area is minimum when the triangle is

equilateral. Here we give proofs of these results by making use of Fuzzy information

theoretic approach

From Hero’s formula



( )( )( )s s a s b s c

Where a,b,c are the lengths of the sides of the triangle, so that

1log log log( ) log( ) log( )

2s s a s b s c

1 ( ) ( ) ( ) 2log log log log log log log 3log2 2 22

a s a b s b c s cs a b c s

s s s

17log log log log log log log log

2

a b c s a s b s cs abc

s s s s s s

71log log ( ) log{1 ( )} log ( ) log{1 ( )} log ( ) log{1 ( )}

1 1 2 2 3 32

sx x x x x x

A A A A A Aabc

7 31 1log log ( ) log{1 ( )}

2 2 1

sx xi iA Aabc i

71 1log ( )

32 2

sH A

abc,

where ( )3

H A is fuzzy entropy corresponding to Shannon’s[8] probabilistic measure of

entropy . and ( )1

ax

A s , ( )

2b

xA s

, ( )3

cx

A s ,

3 2( ) 2

1

a b c sxiA s si

Now if S is given log is maximum when ( )3

H A is maximum. Also ( )3

H A is

maximum when ( ) ( ) ( )1 2 3

x x xA A A

), that is, when a=b=c

Again if is given, log is minimum when ( )3

H A is maximum that is when the

triangle is equilateral.

Thus we have proved both the results stated above.Now for a general quadrilateral, a

formula like (4.1) is not available. However, for cyclic quadrilaterals, we have

Brahmagupta`s formula

= ( )( )( )( )s a s b s c s d , 2

a b c ds

log =1

log( ) log( ) log( ) log( )2

s a s b s c s d

= 1 ( ) ( ) ( ) ( ) 2log log log log log 4log

2 2 2 22

a s a b s b c s c d s dabcd s

s s s s

81log ( )

42

sH A

abcd, where ( )

4H A is fuzzy entropy corresponding to Shannon’s[8]

probabilistic measure of fuzzy entropy

That is out of all cyclic quadrilaterals with a given perimeter, the square has maximum

area and out of all cyclic quadrilaterals with a given area , the square has a minimum

perimeter.



REFERENCES

[1] Bhandari, D. and Pal, N.R. (1993), “Some new information measures of fuzzy sets,”

Information Science, 67, 209 228.

[2] Deluca, A and Termini, S. (1971), “A definition of non-probabilistic entropy in the setting of

fuzzy set theory,” Information and Control, 20, 301 312.

[3] Havrada, J.H. and Charvat, F. (1967), “Quantification methods of classification processes,

concept of structural entropy,” Kybernetika, 3, 30 35.

[4] Kapur, J.N.and Tripathi,G.P. (1990), “On Trignometric Measures of Information”, Journ.

Math. Phy. Sci.,24,1-10.

[5] Kapur, J.N. (1997), “Measures of Fuzzy Information, “ Mathematical Science Trust Society,

New Delhi.

[6] Parkash,O. and Sharma, P.K.(2004), “Measures of Fuzzy Entropy and their Relations”, Inter.

Jour. Of Mgt. and Sys., 20, 1, 65-72.

[7] Renyi, A. (1961), “Measures of entropy and information,” Proc. 4th Berkeley Symp. Prob.

Stats., 1, 547 561.

[8] Shannon, C.E. (1948), “The mathematical theory of communication,” Bell Syst. Tech. Journ.,

27, 423 467.




www.arcjournals.org

©ARC Page | 108

Maximal Square Sum Subgraph of a Complete

Graph

Reena Sebastian


S.E.S College

Sreekandapuram

India.

[email protected]

K.A. Germina

School of Mathematical and Physical

Sciences Central University of Kerala

Kasaragode

India.

[email protected]

Abstract: A (p, q) graph G is said to be square sum, if there exists a bijection f : V (G)

→{0, 1, 2, . . . , p − 1} such that the induced function f ∗ : E(G) → N defined by f ∗ (uv) = (f

(u))2 + (f (v))2, for every uv ∈ E(G) is injective. In [2] it is proved that complete graph Kn is

square sum if and only if n ≤ 5. In this paper we consider the problem of square sum labeling of

maximal square sum subgraph of Kn , n ≥ 6. The minimum number of edges to be deleted from

Kn , n ≥ 6 so that the graph is square sum is defined as critical number of Kn . We call this

maximal square sum subgraph of Kn as critically square sum subgraph of Kn. In this paper we

calculate the critical number of Kn for 6 ≤ n ≤ 50 and developed an algorithm to decide the

critical number of a critically square sum subgraph of Kn .

Key words: Square sum graphs, Critical number

1. INTRODUCTION

Graph labeling, where the vertices and edges are assigned real values or subset of a set are

subject to certain conditions, have often been motivated by their utility to various applied

fields. Several practical problems in real life situations have motivated the study of

labeling of graphs which are required to obey a variety of conditions depending on the structure

First author is indebted to the University Grants Commission(UGC) for granting her

Teacher Fellowship under UGC’s Faculty Development Programme during XI plan of

graphs. Graph l ab e l i ng has a strong communication between number theory [4] and

structure of graphs [7] and [6]. Here we are interested in the study o f vertex funct ions

f : V (G) → A, A ⊆ N for which the induced edge function f ∗(uv) = (f (u))2 + (f (v))2,

for all uv E(G) is injective.

The wide-angular history of sum of squares of numbers motivated the authors to study the

particular graphs named square sum graphs. Unless mentioned otherwise , by a graph we

shall mean in this paper a finite, undirected, connected graph without l o o p s or multiple

edges. Terms not defined here are used in the sense of Harary[7]. Square sum graphs are

vertex labeled graphs with the labels from the set {0, 1, 2, . . . , p − 1} such that the induced edge labels as the sum of the squares of the labels of the end vertices are all distinct. Not

every graph is square sum. For example, any complete graph Kn, n ≥ 6 is not square sum

[2]. It is natural to inquire the size (measured by the number of edges) of the largest square

sum subgraph of Kn. If the nodes of Kn are numbered from the consecutive integers from 0,

1, . . . , n − 1, some edges ei receives the same labels. Removing all the edges with same labels except one from each collection ei of same labels, we obtain a square sum subgraph

of Kn , n ≥ 6. The resulting graph is the maximal square sum subgraph of Kn .

We are interested to find out the minimum number of edges ei to be deleted from Kn so

that the edge labels of Kn − ei are distinct or Kn − ei is square sum. The minimum number



Reena Sebastian& K.A. Germina


of edges to be deleted from Kn so that the graph is square sum is defined as critical

number. We call this maximal square sum subgraph a s critically square sum subgraph o f

Kn . In this paper we compute the critical number of Kn , n ≥ 6 and hence determine the

size of critically square sum subgraph of Kn for n ≤ 50. We have developed an algorithm

to decide the critical number of a critically square sum subgraph of Kn and to find the

size of the critically square sum subgraph of Kn .

2. SQUARE SUM GRAPHS

Acharya and Germina [1] defined a square sum labeling of a (p, q)-graph G [2] as follows.

Definition 2.1. A (p, q) graph G is said to be square sum, if there exists a bijection

f: V (G) → {0, 1, 2, . . . , p − 1} such that induced function f ∗ : E(G) → N defined by

f∗(uv) = (f (u))2 + (f (v))2, for every uv ∈ E(G) is injective.

Theorem 2.2. [2] Complete graph Kn is square sum if and only if n ≤ 5.

Following algorithm determine the edges having same labels.

Algorithm 2.2.1.

step 1. Start

step 2. Declare and initialize the variables.

step 3. Accept the value for n ≤ 50 from user.

step 4. Begin outer loop i.

step 5. Begin inner loop j.

step 6. Compute C [i][j] = i ∗ i + j ∗ j

step 7. End inner loop.

step 8. print C [i][j] value.

step 9. End outer loop.

step 10. Begin outer loop i.

step 11. Begin first inner loop j.

step 12. Begin second inner loop p.

step 13. Begin third inner loop q.

step 14. Compute C [i][j] = C [p][q] if (i! = p) and (j! = q) and (i! = j) and (j! = p)

step 15. Print (i, j), (p, q) and C [i][j] else goto step 10.

Step 16. End third inner loop.

step 17. End second inner loop.

step 18. End first inner loop.

Step 19. End outer loop.

step 20. Stop.

The square sum labelling of the largest square sum subgraphs of K6 is shown in Figure 1

below. A programme in C ++ is developed based on Algorithm 2.2.1. The critical number

of Kn and size of the maximal square sum subgraph of Kn , 6 ≤ n ≤ 50 are given in tables

[1], [2], [3], [4], [5] and [6] respectively.

Notation: In the table MSG denote the maximal square sum subgraph o f G and Cr.No

denote the critical number.

Maximal Square Sum Subgraph of a Complete Graph


Figure 1. The Square sum labelings of maximal square sum subgraph o f K6

Table 1:

n kn Edges having

same labels Size

of MSG Cr.No.

6 K6 {(0, 5)(3, 4)} 14 1

7 K7 {(0, 5)(3, 4)} 20 1

8 K8 {(0, 5)(3, 4)} 27 1

9 K9 {(0, 5)(3, 4)}, {(1, 8)(4, 7)} 34 2

10 K10 {(0, 5)(3, 4)}, {(1, 8)(4, 7)}, {(2, 9)(6, 7)} 42 3

11 K11 {(0, 5)(3, 4)}, {(1, 8)(4, 7)}, {(2, 9)(6, 7)}

{(0, 10)(6, 8)} 51 4

12 K12 {(0, 5)(3, 4)}, {(1, 8)(4, 7)}, {(2, 9)(6, 7)}, 60 6

13 K13 {(0, 5)(3, 4)}, {(1, 8)(4, 7)}, {(2, 9)(6, 7)}, {(0, 10)(6, 8)},

{(2, 11)(10, 5)}, {(3, 11)(7, 9)}, {(1, 12)(8, 9)} 71 7

14 K14 {(0, 5)(3, 4)}, {(1, 8)(4, 7)}, {(2, 9)(6, 7)},

{(0, 10)(6, 8)}, {(2, 11)(10, 5)}, {(3, 11)(7, 9)}, {(1, 12)(8, 9)}, {(0, 13)(5, 12)}{(1, 13)(7, 11)}, {(4, 13)(11, 8)}

81 10

15 K15

{(0, 5)(3, 4)}, {(1, 8)(4, 7)}, {(2, 9)(6, 7)}, {(0, 10)(6, 8)}, {2, 11)(10, 5)}, {(3, 11)(7, 9)}, {(1, 12)(8, 9)}, {(0, 13)

(5, 12)}, {(1, 13)(7, 11)}, {(4, 13)(11, 8)}, {(5, 14)(10, 11)}, {(3, 14)(6, 13)}

93 12

16 K16 {(0, 5)(3, 4)}, {(1, 8)(4, 7)}, {(2, 9)(6, 7)},{(0, 10)(6, 8)}, {2, 11)(10, 5)}, {(3, 11)(7, 9)}, {(1, 12)(8, 9)}, {(0, 13)(5,

12)}, {(1, 13)(7, 11)}, {(4, 13)(11, 8)},

106 14

17 K17

{(0, 5)(3, 4)}, {(1, 8)(4, 7)}, {(2, 9)(6, 7)}, {(0, 10)(6, 8)}, {(2, 11)(10, 5)}, {(3, 11)(7, 9)}, {(1, 12)(8, 9)},

{(0, 13)(5, 12)}, {(1, 13)(7, 11)}, {(4, 13)(11, 8)}, {(5, 14) (10, 11)}, {(3, 14)(6, 13)}, {(0, 15)(12, 9)}, {(5, 15)(9, 13)},

{(2, 16)(8, 14)}, {(3, 16)(11, 12)}

120 16

18 K18

{(0, 5)(3, 4)}, {(1, 8)(4, 7)}, {(2, 9)(6, 7)} {(0, 10)(6, 8)}, {(2, 11)(10, 5)}, {(3, 11)(7, 9)}, {(1, 12)(8, 9)},

{(0, 13)(5, 12)}, {(1, 13)(7, 11)}{(4, 13)(11, 8)}, {(5, 14)(10,11)},

{(3, 14)(6, 13)}, {(0, 15)(12, 9)}, {(5, 15)(9, 13)}, {(2, 16)(8, 14)},{(3, 16)(11, 12)},{(0, 17)(8, 15)}, {(1, 17)

(11, 3)}, {(4, 17)(7, 16)}, {(6, 17)(10, 15)}

133 20



Table 2:

n kn Edges having

same labels

Size

of

MSG

Cr.No.

19 K19 Edges having same labels in

K18 , {(1, 18)(6, 17)(10, 15)}, {(4, 18)(14, 12)} 149 22

20 K20

Edges having same labels in

K19 , {(8, 19)(16, 13)}, {(2, 19)(13, 14)}, {(3, 19)(17, 9)},

{(4, 19)(11, 16)}, {(7, 19)(11, 17)}

163 27

21 K21


K20 , {(0, 20)(12, 16)}, {(5, 20)(8, 19)(13, 16)},

{(9, 20)(15, 16)}

180 30


K21 , {(1, 21)(9, 19)}, {(2, 21)(11, 18)}, {(8, 21)(12, 19)} 198 33

23 K23


K22 , {(1, 22)(14, 17)},{(3, 22)(13, 18)}, {(4, 22)(10, 20)},

{(6, 22)(14, 18)}

216 37

24 K24


K23 , {(1, 23)(13, 19)}, {(2, 23)(7, 22)}, {(4, 23)(16, 17)},

{(6, 23)(9, 22)}, {(9, 23)(13, 21)}, {(23, 11)(17, 19)}

233 43

25 K25


K24 , {(2, 24)(16, 18)}, {(3, 24)(21, 12)}, {(7, 24)(15, 20)},

{(11, 24)(16, 21)}

253 47

26 K26


K25 , {(0, 25)(7, 24), (15, 20)}, {(2, 25)(10, 23)},

{(5, 25)((11, 23)(17, 19)}, {(8, 25)(17, 20)}, {(10, 25)

(14, 23)}

273 52


K26 , {(0, 26)(10, 24)}, {(2, 26)(14, 22)}, {(3,26)(18, 19)},

{(8, 26)(18, 19)},{(7, 26)(10, 25)(14, 23)},{(13, 26)

(19, 22)}

293 58



Table 3:

n kn Edges having

same labels

Size

of

MSG

Cr.No.


K27 , {(1, 27)(17, 21)}, {(11, 27)(15, 25)}, {(4, 27)(13, 24)},

{(5, 27)(15, 23)}, {(6,27)(18, 21)}, {(14, 27)(21, 22)}

314 64


K28 , {(1, 28)(16, 23)},{(3, 28)(8, 27)}, {(6,28) (12, 26)},

{(9, 28)(17, 24)},{(10, 28)(22, 20)}

337 69


K29 , {(0, 29)(20, 21)}, {(12, 29)(16, 27)},{(2, 29)

(13, 26)(19, 22)}, {(3, 29)(11, 27)(15, 25)}, {(7,29)(19, 23)},

{(8, 29)(28, 11)}, {(14, 29)(19, 26)}, {(15, 29)(21, 15)}

358 77


K30 , {(0, 30)(18, 24)}, {(1, 30)(15, 26)}, {(5, 30)

(14, 27)(21, 22)}, {(7, 30)(18, 25)}, {(10, 30)(18, 26)}

383 82


K31 , {(1, 31)(11, 29)}, {(2, 31)(26, 17)}, {(3, 31)

(21, 23)}, {(5, 31)(19, 25)}, {(7, 31)(29, 13)}, {(13, 31)

(17, 29)},{(12, 31)(24, 23)}, {(8, 31)(25, 20)}

406 90

33 K33 Edges having same label in

K32 , {(1, 32)(8, 31)(25, 20)}, {(4, 32)(16, 28),

{(6, 32)(22, 24)}, {(7, 32)(28, 17)}, {(17, 32)(23, 28)},

{(11, 32)(19, 23}, {(9, 32)(24, 23)(12, 31)}

431 97


K33 , {(1, 33)(19, 27)}, {(4, 33)(9, 32)(12, 31)(24, 23)},

{(6, 33)(15, 30)}, {(9, 33)(27, 21)}, {(10, 33)(17, 30)},

{(13, 33)(23, 27)}, {(14, 33)(18, 31)}

457 104


K34 , {(0, 34)(16, 30)}, {(1, 34)(14, 31)}, {(2, 34)(22, 26)},

{(12, 34)(20, 30)}, {(3, 34)(29, 18)}, {(7, 34)(23, 26)},

{(8, 34)(14, 32)}, {(19, 34)(26, 29)},{(13, 34) (29, 22)},

{(17, 34)(22, 31)}

481 114



Table 4:

n kn Edges having

same labels

Size of MSG

Cr.No.

36 K36 Edges having same label in K35 , {(0, 35)(21, 28)}, {(4, 35)(20, 29)},

{(5, 35)(17, 31)}, {(6, 35)(19, 30)}, {(10, 35)(13, 34) (29, 22)}, {(15, 35)(33, 19)}, {(19, 35)(31, 25)},

{(28, 29)(20, 35)}

508 122


K36 , {(2, 36)(12, 34)(20, 30)}, {(3, 36)(24, 27)},

{(7, 36)(16, 33)}, {(8, 36)(24, 28)}, {(11, 36)(24, 29)}, {(13, 36)(21, 32)}

538 128


K37 , {(0, 37)(35, 12)}, {(1, 37)(29, 23)},

{(3, 37)(17,33)}, {(4, 37)(32, 19)}, {(5, 37)(35, 13)},

{(6, 37)(26, 27)}, {(9, 37)(15, 35)(19, 33)},

{(11, 37)(23, 21)}, {(12, 37)(27, 28)}, {(16, 37)

(20,35)(28, 29)}

565 138

39 K39 Edges having same label in K38 , {(1, 38)(17, 34)(22, 31)}, {(4, 38)(28, 26)},

{(5, 38)(10, 37)}, {(6, 38)(18, 34)}, {(8, 38)(22, 32)}, {(9, 38)(25, 30)}, {(11, 38)(14, 37)}, {(14, 38)(22, 34)},

{(21, 38)(27, 34)}, {(16, 38)(32, 26)}

593 148,


K39 , {(0, 39)(36, 15)}, {(2, 39)(9, 38)(25, 30)},

{(3,39)(21, 33)}, {(4, 39)(24, 31)}, {(7, 39)(27, 29)},

{(8, 39)(17, 36)}, {(12, 39)(24, 33)}, {(17, 39)(21, 37)},

{(20, 39)(25, 36)}, {(23, 29)(31, 33)}, {(13, 39)(27, 31)}

621 159


K40 , {(0, 40)(24, 32)}, {(5, 40)(16, 37)

(20, 35)(28,29)},{(7, 40)(25, 32)}, {(10, 40)(16, 38)

(32,26)},{(13, 40)(20, 37)}, {(15, 40)(23, 36)},

{(18,40)(30, 32)}

654 166


K41 , {(0, 41)(9, 40)}, {(2, 41)(23, 34)}{(6, 41)

(14,39)},{(3, 41)(13, 39)(27, 31)}, {(7, 41)(19, 37)},

{(8,41)(28, 31)}, {(10, 41)(25, 34)}, {(11, 41)(29, 31)},

{(12, 41)(15, 40)(23, 36)}, {(18, 41)(22, 39)},

{(24, 41)(31, 36)}, {(13, 41)(35, 25)},{(23, 41)(37, 29)}

682 179



Table 5:

n kn Edges having

same labels

Size of MSG

Cr.No.


K42 , {(1, 42)(33, 26)}, {(2, 42)(18, 38)}, {(4, 42)

(22,36)},{(9, 42)(18, 39)}, {(15, 42)(30, 33)},

{(16, 42)(24, 38)}, {(19, 42)(30, 35)}, {(11, 42)

(21, 38)(27, 34)}

716 187


K43 , {(1, 43)(35, 25)(13, 41)}, {(2, 43)(22, 37)},

{(4,43)(32, 29)},{(6, 43)(11, 42)(21, 38)(27, 34)},

{(7, 43)(37, 23)}, {(9, 43)(29, 33)},{(11, 43)(17, 41)},

{(14, 43)(26, 37)}, {(18, 43)(27, 38)},{(20, 43)(32, 35)},

{(23, 41)(19, 43)(37, 29)}

748 198


K44 , {(1, 44)(41, 16)}, {(2, 44)(28, 34)}, {(3, 44)

(24,37)}, {(5, 44)(19, 40)}, {(6, 44)(26, 36)},

{(7, 44)(31, 32)}, {(8, 44)(20, 40)}, {(12, 44)(28, 36)},

{(13, 44)(16, 43)}, {(25, 44)(31, 40)}, {(27, 44)(36,

37)}, {(23, 44)(28, 41)}, {(17, 44)(25, 40)}

779 211


K45 , {(0, 45)(27, 36)}, {(4, 45)(21, 40)}, {(5, 45)

(23,39)(31, 33)}, {(7, 45)(15, 43)}, {(11, 45)(25, 39),

{(10, 45)(19, 42)(30, 35)}, {(15, 45)(27, 39)}, {(17, 45)(33, 35)}, {(20, 45)(24, 43)}

815 220


K46 , {(1, 46)(31, 34)}, {(2, 46)(26, 38)}, {(4, 46)

(14,44)}, {(8, 46)(32, 34)}, {(3, 46)(10, 45)(19, 42)

(30,35)}, {(9, 46)(26, 39)}, {(12, 46)(18, 44)},

{(13, 46)(29, 38)}, {(17, 46)(31, 38)}, {(18, 46)

(42, 46)},{(7, 46)(22, 41)}

849 231


K47 , {(1, 47)(19, 43)(23, 41)(37, 29)}, {(6, 47)(34,33)},

{(4, 47)(17, 44)(25, 40)}, {(9, 47)(21, 43)}, {(11, 47) (31, 37)}, {(13, 47)(23, 43)}, {(16, 47)(23, 44)(28, 41)}, {(18, 47)(33, 38)}, {(21, 47)(25, 45)}, {(27, 47)(43,33)},

{(29, 47)(37, 41)}

886 242


K48 , {(1, 48)(28, 39)}, {(4, 48)(32, 36)}, {(5, 48)

(27,40)}, {(6, 48)(24, 42)}, {(7, 48)(12, 47)},

{(9, 48)(33, 36)}, {(11, 48)(20, 45)(24, 43)},

{(19,48)(27, 44)(36, 37)}, {(22, 48)(32, 42)},

{(29, 48)(36, 43)}

924 252



Table 6:

n kn Edges having

same labels

Size of MSG

Cr.No.

50 K50 Edges having same label in K49 , {(2, 49)(14, 47)(17, 46)(31, 38)}, {(3, 49)

(27,41)}, {(8, 49)(16, 47)(28, 41)(44, 23)}, {(9, 49) (39,31)}, {(11, 49)(29, 41)}, {(12, 49)(32, 39)},

{(12,49)(32, 39)}, {(17, 49)(29, 43)}, {(18, 49)(31, 42)}, {(22, 49)(26, 47)},{(13, 49)(47, 19)}, {(26, 49)(31, 46)},

{(24, 49)(36, 41)}

960 265

The square graphs W 2 , n ≥ 6 and K 2, m + n ≥ 6 are complete graphs, the critical number

and size of the maximal square sum subgraph of ,n and , m+n≥6 are also

obtained from the algorithm. We believe that there is great potential for developing

practical mathematics with numbered graphs .

REFERENCES

[1] B.D.Acharya, Personal Communication, September 2011.

[2] Ajitha V, Studies in Graph Theory-Labeling of Graphs, Ph.D thesis (2007), Kannur

Univeristy, Kannur.

[3] Bolomon W.Golobm, the largest graceful subgraph of the complete graph.

[4] Amer,Math,monthly B1(197-)499-501.

[5] D.M Burton, Elementary number theory, Second Edition, Wm.C.Brown Company publish-

ers, 1980.

[6] Gary S Bloom and Solomn W.Golomb, Numbered Complete graph, unusual Rulers and

As- sorted Applications.

[7] J.A.Gallian, A dynamic survey of graph labrling, The Electronic Journal of

Combinatorics (DS6),2005

[8] F. Harary, Graph Theory, Addison-Wesley Pub. Comp., Reading, Massachusetts, 1969.


Volume 2, Issue 1, February - 2014, PP 116-124


www.arcjournals.org

©ARC Page | 116

Steady Flow of a Viscous Incompressible Fluid through Long

Tubes Employing a Complex Variable Technique

Bandi. Ravi M.Sc.,NIT-Warangal,

[email protected]

Abstract: A steady flow of a viscous fluid through straight long non-circular tubes under a pressure

gradient down the tube length is discussed in the paper.

The fluid is assumed to be homogenous and incompressible. For such a flow the pressure gradient has to be

a constant. The axial velocity of the flow satisfies Poisson’s equation together with the no-slip condition on

the boundary of the tube.

A complex variable technique has been developed to solve the problem. This technique can be successfully

applied whenever the tube cross section can be expressed in the form where

x iy and x iy it’s conjugate. Farther is a regular function and is it

conjugate.

This technique is employed to find the velocity field for the following tube-cross sections is. 1. Ellipse, 2. Equilateral triangle, 3.Hyperbola bounded by a chord, 4. Two co-axial hyperbolas, 5. Two conjugate

hyperbolas and 6. Lune bounded by two circles.

1. INTRODUCTION

Consider a viscous incompressible homogenous fluid flowing steadily through a long straight

horizontal tube under a pressure gradient. Such a flow through a circular tube was considered

earlier in the year 1846 by J.L.M. Poisceuille a French physician in connection with the estimation of the flow rate of blood through arteries’. This has been treated by several authors of

books classical test books on fluid mechanics.

Consider a Cartesian System O (x, y, z) with the co -axis along the tube length and x and y axes in

a plane perpendicular to tube length with the origin ‘O’ can be taken anywhere on the axis.

The flow is under a pressure gradient down the tube length. Let fluid velocity be taken

q = (o, o, w) (1)

The velocity satisfies the equation of the continuity

Y

X

Z W (x, y)

O



Bandi. Ravi


Div 0q

0w

z (2)

.˙. W is independent of Z

W is function of x and y and t

Since the flow is steady, W is independent of time’t’ W is a function of x and y only

The momentum equation (Navier-stokes’ equation) that characterized the steady flow in the

absent of the external force

2d qp D q

dx (3)

Where q

= fluid velocity, = fluid density (assumed constant), P = pressure, µ = viscosity

coefficient

Equation of motion in the x-direction

0p

x (4)

Equation of motion in the y-direction

0p

y (5)

Equation of motion in the z –direction

2 2

2 20

p w w

z x y (6)

Differentiating with respect to ‘z ’we get

2 2 2

2 2 2

2 2 2

2 2 2

0

0

p d w w

z dz x y

p dw

z x y dz

2

20

p

z0

w

z

p

z is independent of ‘z’ (7)

From the equations (4), (5) and (7)

Y

X O

Steady Flow of a Viscous Incompressible Fluid Through Long Tubes Employing a Complex Variable

Technique


.˙. p

z = C (constant)

The equation (6) that satisfied by the axial component w(x,y) of the velocity is give by the

Poisson equation.

2w c constat (8)

with the boundary condition,

W=0 on the boundary on the tube (9)

Proposed Complex variable technique for solving the equation (8) for the velocity fluid(w)

Let us Introduce the complex variables Z and

x iy and x iy (10)

The equations can now to transformation as

2

4

p c

z (11)

Integrating (11) with Z

4

p cfuction

z (12)

This on integrating with repeat to yield ds

4

cw function fuctions (13)

This equation of W is a real function of X and Y of Z and and on the R.H.S. Z is real.

Therefore The Velocity w can be expressed as

4

cw f f (14)

where f(z) is a regular function of Z and f f is conjugate function of f(Z)

The function f (z) is to be so choose such that the no-slip condition on the 0w on the boundary

Γ

Therefore on the boundary Γ of the tube

4

cf f (15)

This method is successful whenever the equation to the boundary Γ can be expressed as

(16)

Expressed the equation we noticed that

4

cf f (17)

From (17) it can be noticed from the equation that (17)

4

cf and

4

cf (18)

Bandi. Ravi


The velocity in this given be the equation (14)

2. APPLICATIONS

A) Cross section bounded circular tube:

2 2 2: 0x y a (A.1)

2: a (A.2)

Here

2 2

2 2

a a

2

2

a (A.3)

2

8

af c f (A.4)

4

cw f f

2

4

ca

2 2 2

4

ca x y (A.5)

B) Cross section bounded Ellipse:

2 2

2 2: 1 0

x y

a b (B.1)

This can be expressed as

Y

X O

Y

X O


Technique


2 2 2 2 2 2 0b x a y a b

2 2

2 2 2 2 02 2

b a a bi

This can be written as

2 2 2 22

2

2 2 2 24 2

a b a b

a b a b

2

2 2 2 2 2 2 2 2 2

2 2

1

4a b a b a b a b

a b (B.2)

This is a second degree in Z and

2 2 2 2 2

2 2

1

4a b a b

a b (B.3)

22 2 2 2

2 2

1

4( )a b a b

a b (B.4)

4

cf and

4

cf

The velocity is the equation substituting we get

4

cw f f

2 2 2 2 2 2

4

cb x a y a b (B.5)

C) Cross section bounded Equilateral triangle tube :

: 3 3 0x a x y y x (C.1)

This can be written as 2 23 0x a x y

= 3 2

3 212 2

2a a

a (C.2)

This is a cubic in Z and

O

Y

X

Bandi. Ravi


3 212

2a

a (C.3)

3 212

2a

a (C.4)

4

cf and

4

cf (C.5)

3 228

cf a

(C.6)

3 2

28

cf a

(C.7)

The velocity is now given by 4

cw f f

3 23 22 2

4 8 8

c c cw a a

a a

2 234

cw x a x y (C.8)

D) Cross section hyperbola bounded by a chord:

2 2 2: 3 0x a x y b we get (D.1)

3 23 2 2 2 2 21

2 22

b a a b a a a ba

(D.2)

This a cubic in Z and

X O


Technique


3 2 2 212

2b a a b

a (D.3)

3 22 21

22

a a a ba

(D.4)

4

cf and

4

cf (D.5)

Values substituting get 4

cw f f 2 2 23

4

cw x a x y b

(D.6)

E) Cross section bounded by two coaxial hyperbolas

2 22 2 2 2: 0

3 2 2 3 2 2

y yx a x b (E.1)

On simplifying we get

4 24 2 2 2 2 2 2 2 2 2

2 2

12 2 2 2 2 2 2 2

2 2 1 2 1Z b a a b b a a b

b a

(E.2)

4 2 2 2 2 2

2 2

2 2 2 2

2 2 1 2 1

b a a b

b a (E.3)

4 22 2 2 2

2 2

2 2 2 2

2 2 1 2 1

b a a b

b a (E.4)

4

cf and

4

cf (E.5)

A fourth degree expression in Z and

4

cw f f

2 2 2 2 2 23 2 2 3 2 2 3 2 2 3 2 24

cw x y a x y b (E.6)

Y

X O

Bandi. Ravi


F) Cross section bounded by a hyperbola and its conjugate:

2 22 2 2 2: 0

3 2 2 3 2 2

y yx a x b (F.1)

4 24 2 2 2 2 2 2 2 2 2

2 2

12 2 2 2 2 2 2 2

2 2 1 2 1b a a b b a a b

b a

(F.2)

This is a fourth degree expression in Z and

4 2 2 2 2 2

2 2

2 2 2 2

2 2 1 2 1

b a a b

b a (F.3)

4 22 2 2 2

2 2

2 2 2 2

2 2 1 2 1

b a a b

b a (F.4)

4

cf and

4

cf (F.5)

Hence the velocity is give by4

cw f f

2 2 2 2 2 23 2 2 3 2 2 3 2 2 3 2 24

cw x y a x y b

(F.6)

G) Cross section bounded by two circles

2 2 2 2 2 2: 2 0x y a x y bx (G.1)

2 2 2x y a , 2 2 2x y bx ,

2 2 2x y b (G.2)

Y

X

O


Technique


This as expressing and 1

2 2

2 2

a a bb

2 2

2 2

a a bb (G.4)

4

cf and

4

cf (G.5)

The velocity is given by 4

cw f f

.˙. An Expression involving Z and

3. CONCLUSION

The steady flow of a viscous liquid through a straight tube of non-circular cross section can be

solved by a complex variable method.

This method is successful whenever the boundary of the tube can be expressed as

Z = function of Z + its conjugate.

ACKNOWLEDGEMENT

The author acknowledged the encouragement of Prof N.Ch. Pattabhi Ramacharyulu with utmost

gratitude.

REFERENCES

[1] Csonka : Proceedings Of 5th Congress Of Istam (1959) [2] Therotecal Hydro Dynamics By .L.M.Milne – Thomson, C.B.E, Macmillan Company, New

York.(Chapter-Xxi, 21.44, Page-[653])

[3] Fluid Dynamics By. Frank .Chorlton, C.B.S Publishers And Distributors, Delhi,(2004) (Part-

V [Chapter-14]) [4] Fluid Dynamics By .Prentice Hall, Englewood Cliffs Nj 1967)

X

Y

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