Bibliography

General Works

[I] R. Courant and H. Robbins, What is Mathematics? 4th ed., University Press, Oxford, 1979. [2] H. B. Griffith and P. J. Hilton, A comprehensive textbook of classical mathematics in a contemporary

interpretation, Van Nostrand Reinhold, New York-London, 1970. [3] The VN Encyclopedia of Mathematics (translated from the German), Van Nostrand Reinhold, New

York-London, 1973. [4] Mathematics, ils contents methods, and, meaning (translated from the Russian), 3 vols., American

Mathematical Society, Providence, RI, 1962. [S] B. H. Chirgwin and C. Plumpton, A course of mathematics for engineers and scientists, 6 vols., 2nd ed.,.

Pergamon, Oxford, 1972. [6] R. Courant and F. John, Introduction to calculus and analysis, 2 vols., Wiley-Interscience, New York­

London, 1980. [7] A. Jeffrey, Mathematics for engineers and scientists, 2nd ed., Van Nostrand Reinhold, New York­

London, 1979. [8] H. S. W. Massey and H. Kestelman, Ancillary mathematics, 2nd ed., Pitman, London, 1964. [9] V. I. Smirnov, A course of higher mathematics (translated from the Russian), 5 vols., Pergamon, Ox­

ford, 1964.

[ 1.1]

[1.2] [ 1.3]

[ 1.4] [ 1.5] [1.6] [ 1.7] [1.8] [ 1.9]

[1.10]

[ 1.11]

[1.12]

[ 1.13]

1.1 Tables

M. Abramowitz and I. A. Stegun, Handbook of mathematical functions, with formulas, graphs, and mathematical tables, 5th ed., Dover, New York, 1968. W. H. Beyer (ed.), Standard mathematical tables, 25th ed., CRC Press, Boca Raton, FL, 1978. R. S. Burington, Handbook of mathematical tables and formulas, 5th ed., McGraw-Hill, New York­London, 1977. J. R. and M. Clifford, Computer mathematical handbook, Allyn & Bacon, Boston, 1974. L. J. Comrie, Chambers' shorter 6-figure mathematical tables, new ed., Chambers, Edinburgh, 1971. H. B. Dwight, Mathematical tables, 3rd ed., Dover, New York, 1968. H. B. Dwight, Tables of integrals, Macmillan, New York-London, 1974. A. Erdelyi, Higher transcendental functions, 3 vols., new ed., McGraw-Hill, New York-London, 1980. I. S. Gradshtein and I. M. Ryzhik, Tables of integrals, series, and products (translated from the Rus­sian), Academic Press, New York, 1980. E. Jahnke and F. Emde, Tables offunctions with formulae and curves (translated from the German), 4th ed., Dover, New York, 1945. A. V. Lebedev and R. M. Fedorov, A guide to mathematical tables (translated from the Russian), Pergamon, Oxford, 1980. W. Magnus, F.Oberhettinger, and R. P. Soni, Formulas and theorems for the special functions of mathematical physics, 3rd ed., Springer-Verlag, Berlin-Heidelberg-New York, 1966. M. R. Spiegel, Mathematical handbook of formulas and tables, McGraw-Hill, New York-London, 1968.

1.2 Graphs of elementary functions

[1.14] M. Abramowitz and 1. A. Stegun, Handbook of mathematical functions with formulas, graphs, and mathematical tables, 5th ed., Dover, New York, 1968.

[1.15] G. Loria, Speziel/e algebraische und transzendente ebene Kurven, 2 vols., 2nd ed., Teubner, Leipzig­Berlin, 1911.

2 Elementary Mathematics

[2.1] H. Lenz, Grundlagen der Elementarmathematik, 3rd ed., De Gruyter, Berlin, 1975.

2.2 Combinatorics

[2.2] M. Aigner, Combinatorial theory, Springer-Verlag, Berlin-Heidelberg-New York, 1979. [2.3] 0.1. A. Cohen, Basic techniques of combinatorial theory, Wiley, New York-London, 1978. [2.4] H. J. Ryser, Combinatorial mathematics, Mathematical Association of America, New York, 1963.

[2.5]

[2.6] [2.7] [2.8] [2.9] [2.10] [2.11]

61*

2.4 Algebra

G. Birkhoff and S. MacLane, A survey of modern algebra, 5th ed., Macmillan, London-New York, 1977. N. Bourbaki, Algebra (translated from the French), Addison-Wesley, Reading, MA, 1974. P. M. Cohn, Algebra, 2 vols., 2nd ed., Wiley, New York-London, 1982. F. R. Gantmacher, Matrix theory (translated from the Russian), 2 vols., Chelsea, New York, 1960. N. Jacobson, Lectures in abstract algebra, 3 vols., 3rd ed., Van Nostrand Reinhold, New York, 1974_ N. Jacobson, Basic algebra, 2 vols., Freeman, San Francisco, 1980. A. G. Kurosh, Lectures on general algebra (translated from the Russian), Chelsea, New York, 1965-

948 Bibliography

(2.12) S. Lang, Algebra, Addison-Wesley, Reading MA, 1964. (2.13) S. Lang, Linear algebra, Addison-Wesley, Reading, MA, 1973. [2.14) M. Marcus and H. Mine, A survey of matrix theory and matrix inequalities, Allyn and Bacon, Boston,

MA, 1961. [2.15) S. Noble and I. W. Daniel, Applied linear algebra, Prentice Hall, Englewood Cliffs, NJ, 1977. [2.16) B. L. van der Waerden, Modern algebra (translated from the German), 2 vols., 3rd ed., Ungar, New

York,1967.

2.6 Geometry

[2.18) W. Abbott, Practical geometry and engineering graphics, 8th ed., Blackie, London-Glasgow, 1971. [2.19) H. G. Ayre, R. Stephens, and G. D. Mock, Analytic geometry, 2nd ed., Van Nostrand Reinhold,

New York-London, 1967. [2.20) A. Barton, An introduction to coordinate geometry, University of London Press, London, 1958. [2.21) H. S. M. Coxeter, Introduction to geometry, Wiley, New York-London, 1961. [2.22) N. V. Efimov, An elementary course in analytical geometry (translated from the Russian), 2 vols.,

Pergamon, Oxford, 1966. [2.23) L. P. Eisenhart, Coordinate geometry, Dover, New York, 1960. [2.24) M. L. Keedy and C. W. Nelson, Geometry-a modern introduction, Addison-Wesley, Reading, MA,

1965. [2.25) E. A. Maxwell, Elementary coordinate geometry, Cambridge University Press, Cambridge, 1954. [2.26) K. L. Nielsen and J. H. Vanlonkhuyzen, Plane and spherical trigonometry, Barnes and Noble, New

York,1960. [2.27) S. M. Slaby, Fundamentals of three-dimensional descriptive geometry, Harcourt-Brace, New York­

Chicago, 1966. [2.28) D. M. Y. Sommerville, Analytical geometry of three dimensions, Cambridge University Press, Cam­

bridge, 1934.

3.1 Differential and integral calculus

(3.1r T. Apostol, Mathematical analysis, Addison-Wesley, Reading, MA, 1957. [3.2) T. Apostol, Calculus, 2 vols., Blaisdell, New York, 1961. [3.3) R. G. Bartle, The elements of real analysis, Wiley, New York-London, 1964. (3.4) J. C. Burkill, A first course in mathematical analysis, Cambridge University Press, Cambridge, 1962. (3.5) R. Courant, Differential and integral calculus (translated from the German), 2 vols., Blackie and Son,

Glasgow, 1945. (3.6) H. G. Eggleston, Elementary real analysis, Cambridge University Press, Cambridge, 1962. [3.7) T. M. Flelt, Mathematical analysis, McGraw-HiII, London, 1966. [3.8) E. W. Hobson, The theory of functions of a real variable and the theory of Fourier's series, 2 vols.,

reprint, Dover, New York, 1957. (3.9) K. Knopp, Theory and application of infinite series (translated from the German), Hafner, New York,

1951. [3.10) S. Lang, Analysis. I, Addison-Wesley, Reading, MA, 1968. (3.11) A. S.-T. Lue, Basic pure mathematics, n, Van Nostrand Reinhold, New York-London, 1974. (3.12) D. B. Scolt and S. R. Tims, Mathematical analysis, Cambridge University Press, Cambridge, 1966. [3.13) G. L. Simmons, Introduction to topology and modern analysis, McGraw-Hill, New York-London, 1963. [3.14) M. Spivak, Calculus, Addison-Wesley, Reading, MA, 1973. [3.15) E. T. Whittaker and G. N. Watson, Modern analysis, 6th ed., Cambridge University Press, Cambridge,

1957.

3.2 Calculus of variations and optimal processes

(3.16) R. Courant, Differential and integral calculus (translated from the German), vol. 2, Blackie and Son, Glasgow, 1945.

[3.17) R. Courant and D. Hitbert, Methods of mathematical physics (translated from the German), vol. I, Witey, New York-London, 1953.

(3.18) I. Gumowski and C. Mira, Optimization in control theory and practice, Cambridge University Press, Cambridge, 1968.

[3.19) H. and B. S. Jeffreys, Methods of mathematical physics, Cambridge University Press, Cambridge, 1946. [3.20) A. G. J. MacFarlane, Dynamical system models, Harrap, London, 1970. [3.21) L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze, and E. F. Mishchenko, The mathematical

theory of optimal processes (translated from the Russian), Witey, New York-London, 1962. [3.22) G. Stephenson, Inequalities and optimal problems in mathematics and the sciences, Longman, London,

1971.

3.3 Differential equations

(3.23) T. Apostol, Calculus, 2 vols., Blaisdell, New York, 1961. (3.24) J. C. Burkill, The theory of ordinary differential equations, Oliver and Boyd, Edinburgh-London, 1962. [3.25) R. Courant, Differential and integral calculus (translated from the German), 2 vols., Blackie, Glasgow,

1945. (3.26) R. Courant and D. Hitbert, Methods of mathematical physics (translated from the German), vol. 2,

Wiley, New York-London, 1953. [3.27) B. Friedman, Principles and techniques of applied mathematics, Wiley, New York-London, 1956. (3.28) S. H. Gould, Variational methods for eigenvalue problems, University of Toronto Press, Toronto, 1957. [3.29) E. L. Ince, Ordinary differential equations (reprint), Dover, New York, 1956.

Bibl iography 949

[3.30] A. Jeffrey, Mathematics for engineers and scientists, 2nd ed., Van Nostrand Reinhold, New York-Lon-don, 1979.

[3.31] H. and B. S. Jeffreys, Methods of mathematical physics, Cambridge University Press, Cambridge, 1946. [3.32] S. Lang, Analysis, I, Addison-Wesley, Reading, MA, 1968. [3.33] P. M. Morse and H. Feshbach, Methods of theoretical phys ~S, ;, MoG.aw-Hill, New York-London,

1953. [3.34] G. E. H. Reuter, Elementary differential equations and operato", Routledge-Kegan Paul, London,

1958. [3.35] Sommerfeld, A., Partial differential equations in physics (translated from the German), Academic Press,

New York-London, 1967. [3.36] G. N. Watson, A treatise on the theory of Bessel functions, 2nd ed., Cambridge University Press, Cam­

bridge, 1944. [3.37] A. J. White, Real analysis: an introduction, Addison-Wesley, Reading, MA, 1968. [3.38] E. T. Whittaker and G. N. Watson, Modern analysis, 6th ed., Cambridge University Press, Cambridge,

1957.

3.4 Theory of functions of a complex variable

[3.39] L. V. Ahlfors, Complex analysis, McGraw-Hill, New York-London, 1953. [3.40] T. M. Apostol, Mathematical analysis, Addison-Wesley, Reading, MA, 1957. [3.41] c. Caratheodory, Theory of functions of a complex ,·ariable (translated from the German), 2 vols.,

Chelsea, New York, 1954. [3.42] E. T. Copson, An introduction to the theory of functions of a complex variable, Oxford University Press,

Oxford, 1935. [3.43] H. and B. S. Jeffreys, Methods of mathematical physics, Cambrid~cVniversity Press, Cambridge, 1946. [3.44] K. Knopp, Theory of functions (translated from the German), 2 vols., Dover, New York, 1947. [3.45] N. W. McLachlan, Complex variable theory and transform calculus (with technical applications), 2nd ed.,

Cambridge University Press, Cambridge, 1963. [3.46] E. G. Phillips, Functions of a complex variable, Oliver and Boyd, Edinburgh, 1940. [3.47] W. Rudin, Real and complex analysis, McGraw-Hill, New York-London, 1970. [3.48] E. C. Titchmarsh, The theory of functions, Oxford University Press, Oxford, 1939. [3.49] E. T. Whittaker and G. N. Watson, Modern analysis, 6th ed., Cambridge University Press, Cambridge,

1957.

4.1 Sets, relations, mappings

[4.1] J. C. Abbott, Sets,lattices, and Boolean algebras, Allyn and Bacon, Boston, 1969. [4.2] P. Bernays and A. A. Fraenkel, Axiomatic set theory, North Holland, Amsterdam, 1958. [4.3] N. Bourbaki, Theory of sets (translated from the French), Addison-Wesley, Reading, MA, 1968. [4.4] P. R. Halmos. Naive set theory, Van Nostrand Reinhold, New York-London 1960. [4.5] P. R. Halmos, Lectures on Boolean algebras, Van Nostrand Reinhold, New York-London, 1963. [4.6] R. R. Stoll, S,t theory and logic, Freeman, San Francisco, 1963. [4.7] A. Tarski, Introduction to logic, Oxford University Press, Oxford, 1941. [4.8] R. L. Wilder, Introduction to the foundations of mathematics, Wiley, New York-London, 1965.

4.2 Vector calculus

[4.9] D. E. Bourne and P: C. Kendall, Vector analysis, Oldbourne, London, 1967. [4.10] R. Braae, Vector analysis, Pitman, London, 1968. [4.11] J. W. Gibbs, Vector analysis, 2nd ed., Dover, New York, 1960. [4.12] N. Kemmer, Vector analysis, Cambridge University Press, Cambridge, 1977. [4.13] A. M. Macbeath, Elementary vector algebra, Oxford University Press, Oxford, 1964. [4.14] B. Spain, Vector analysis, Van Nostrand Reinhold, New York-London, 1965. [4.151 M. R. Spiegel, Vector analysis, McGraw-HiII, New York, 1959. [4.16] C. E. Weatherburn, Elementary vector analysis, 2nd ed., Bell, London, 1959.

4.3 Differential geometry

[4.17] L. P. Eisenhart, An introduction to differential geometry, Princeton University Press, Princeton, NJ r

1940. [4.18] R. H. Fowler, The elementary differential geometry of plane curves, Cambridge University Press, Cam-

bridge, 1920. [4.19] M. M. Lipschutz, Differential geometry, McGraw-HiII, New York, 1969. [4.20] A. V. Pogorelov, Differential geometry (translated from the Russian), Noordhof, Groningen, 1959. [4.21] J. J. Stoker, Differential geometry, Wiley, New York-London, 1969. [4.22] D. J. Struik, Differential geometry, Addison-Wesley, Reading, MA, 1957. [4.23] C. E. Weatherburn, Differential geometry, I, Cambridge University Press, Cambridge, 1961.

4.4 Fourier series, Fourier integrals, and Laplace transform [4.24] H. S. Carslaw, Fourier's series and integrals, 3rd ed., Macmillan, London, 1930. [4:25] R. V. Churchill, Operational mathematics, 3rd ed., McGraw-Hill, New York-London, 1972. [4.26] A. Erdelyi, Tables of integral transforms, 2 vols., McGraw-HiII, New York-London, 1954. [4.27] P. Franklin, An introduction to Fourier methods and the Laplace transform, Dover, New York, 1958. [4.28] J. C. Jaeger and G. Newstead, An introduction to the Laplace transformation, 3rd ed., Methuen, London,

1969. [4.29] C. Lanczos, Discourse on Fourier series, Oliver and Boyd, Edinburgh, 1966.

950 Bibliography

[4.30] Open University course team, Laplace transforms, Open University Press, Milton Keynes, 1975. [4.31] E. D. Rainville, The Laplace transform, Macmillan, New York-London, 1963. [4.32] P. D. Robinson, Fourier and Laplace transforms, Routledge-Kegan Paul, London, 1968. [4.33] R. T. Seeley, An introduction to Fourier series and integrals, Benjamin, New York-Amsterdam, 1966. [4.34] R. D. Strum and J. R. Ward, Laplace transform solution of differential equations, Prentice-Hall, Ensle-

wood Cliffs, NJ, 1968. [4.35] R. D. Stuart, An introduction to Fourier analysis, Chapman and Hall, London, 1966. [4..36] W. T. Thomson, Laplace transformation, Longmans-Green, London-New York, 1957. [4.37) G. P. Tolstov, Fourier series (translated from the Russian), Prentice-Hall, London, 1962.

5 Probability theory and mathematical statistics

[5.1] T. S. Arthanasi and Y. Dodge, Mathematical programming in statistics, Wiley, New York-London, 1981.

[5.2) 1. F. Blake, An introduction to applied probability, Wiley, New York-London, 1979. [5.3) G. E. P. Box and W. G. Hunter, Statistics for experimenters, Wiley, New York-London, 1978. [5.4] S. Brandt, Statistical and computational methods in data analysis, 2nd ed., North-Holland, Amsterdam­

Oxford, 1976. [5.5) M. G. Bulmer, Principles of statistics (reprint), Dover, New York, 1979. [5.6) Kai Lai Chung, Elementary probability theory with stochastic processes, 3rd ed., Springer-VerIag, Berlin­

Heidelberg-New York, 1979. [5.7] B. E. Cooper, Statistics for experimentalists, Pergamon, Oxford, 1969. [5.8] M. Fisz, Probability theory and mathematical statistics (translated from the Polish), Wiley, New York­

London, 1963. [5.9) W. Freiberger and U. Grenander, A course in computational probability and statistics, Springer-Verlag,

Berlin-Heidelberg-New York, 1971. [5.10) R. V. Hogg and A. T. Craig, Introduction to mathematical statistics, 4th ed., Macmillan, New York-

London, 1975. [5.11) M. Kendall, Multivariate analysis, 2nd ed., Macmillan, New York-London, 1980. [5.12) W. J. Kennedy and J. E. Gentle, Statistical computing, Marcel Dekker, New York, 1980. [5.13) B. W. Lindgren, G. W. McElrath, and D. A. Berg, Introduction to probability and statistics, 4th ed.,

Macmillan, New York-London, 1978. [5.14) J. Mandel, The statistical analysis of experimental data, Wiley, New York-London, 1964. [5.15) K. V. Mardian, J. T. Kent, and J. M. Bibby, Multivariate analysis, Academic Press, New York-London,

1979. [5.16) J. L. Melsa and D. L. Cohn, Decision and estimation theory, McGraw-Hill, New York-London, 1975. [5.17] P. L. Meyer, Introductory probability and statistical applications, Addison-Wesley, Reading, MA, 1965. [S.IS] 1. Olkin, L. J. Gleser, and C. Derman, Probability models and applications, Macmillan, New York-

London, 19S0. [5.19) C. G. Paradine and B. H. P. Rivett, Statistics for technologists, English Universities Press, London,

1969. {5.20] E. Parzen, Stochastic processes, Holden-Day, San Francisco, 1962. [5.21] A. Renyi, Probability theory (translated from the Hungarian), North-Holland, Amsterdam-London,

1976. [5.22] A. P. Sage and J. L. Melsa, Estimation theory with applications to communications and control, McGraw-

Hill, New York-London, 1971. . [5.23] W. M. Smart, Combination of observations, Cambridge University Press, Cambridge, 1958. [5.24) G. W. Snedecor and W. C. Cochran, Statistical methods, 7th ed., Iowa State University Press, Ames,

19S0. [5.25) A. M. Yaglom, An introduction to the theory of stationary functions (translated from the Russian),

Prentice-Hall, Englewood Cliffs, NJ, 1965. 15.26) S. Zubrzycki, Probability theory and mathematical statistics (translated from the Polish), American

Elsevier, New York, 1972.

[6.1] [6.2]

[6.3) [6.4)

[6.5]

[6.6] )6.7) [6.S) [6.9) [6.10)

6 Linear optimization E. M. L. Beale, Mathematical programming in practice, Pitman, London, 1968. S.1. Gass, Linear programming. Methods and applications, 4th ed., McGraw-Hill, New York-London, 1975. A. J. Jones, Game theory: Mathematical models of conflict, Wiley, New York-London, 1980. A. Kaufmann, Graphs, dynamic programming, and finite games (translated from the French), Academic Press, New York-London, 1967. B. Kolman and R. E. Beck, Elementary linear programming with applications, Academic Press, New York­London, 19S0. K. G. Murty, Linear and combinatorial programming, Wiley, New York-London, 1976. R. J. Rothenberg, Linear programming, North Holland, Amsterdam-New York, 1979. J. F. Shapiro, Mathematical programming: Structures and algorithms, Wiley, New York-London, 1970. G. E. Thompson, Linear programming. An elementary introduction, Macmillan, New York-London, 1971. N. N. Vorob'ev, Game theory. Lectures for economists and systems scientists (translated from the Rus~ sian), Springer-Verlag, Heidelberg-New York-Berlin, 1977.

7.1 Numerical mathematics [7.1) N. S. Bahvalov, Numerical methods (translated from the Russian), Mir, Moscow, 1976. [7.2) J. C. P. Bus, Numerical solution of systems of non-linear equations, Math. Centre Tracts 122, Amsterdam,

1980.

Bibliography 951

[7.3) A. J. Davies, The finite element method, Oxfor:! University Press, Oxford, 1980. [7.4) B. P. Demidovich and I. A. Maron, Computational mathematics (translated from the Russian), Mir,

Moscow, 1973. [7.5) V. F. Dyachenko, Basic computational mathematics (translated from the Russian), Mir, Moscow, 1979. [7.6) C. F. Gerald, Applied numerical analysis, 2nd ed., Addison-Wesley, Reading, MA, 1978. [7.7) E. Isaacson and R. B. Keller, Analysis 0/ numerical methods, Wiley, New York-London, 1966. [7.8) M. K. Jain, Numerical solution 0/ differential equations, Wiley, New York-London, 1979. [7.9) J. M. Pollard, Numerical and statistical techniques, Cambridge University Press, Cambridge, 1979. [7.10) C. E. Roberts, Jr., Ordinary differential equations. A computational approach, Prentice-Hall, Englewood

Cliffs, NJ, 1979. [7.11) G. D. Smith, Numerical solution 0/ partial differe.'tial equations (reprint), Oxford University Press,

Oxford, 1975. [7.12) R. Wait, The numerical solution 0/ algebraic equations, Wiley, New York-London, 1979.

7.2 Computer science

[7.\3) L. I. Kronsjo, Algorithms: their complexity and efficiency, Wiley, New York-London, 1979. [7.14) Z. Manna, Lectures on the logic 0/ computer programming, SIAM, Philadelphia, PA, 1980. [7.15) M. Macktey and P. Young, An introduction to the general theory 0/ algorithms, North-Holland, New

York-London, 1978. [7.16) E. S. Page and L. B. Wilson, An introduction to computational combinatorics, Cambridge University

Press, Cambridge, 1979. [7.17) I. Poh1 and A. C. Shaw, The nature 0/ computation: an introduction to computer science, Computer

Science Press,.Rockville, MD, 1981. [7.18) T. Rus, Data structures and operating systems (translated from the Roumanian), Wiley, New York­

London, 1979. [7.19) M. H. Weik (ed.), Computer dictionary, IEEE Computer Society, New York, 1979.

8.1 Functional analysis

[8.1) N.1. Akhizer and I. M. Glazman, Theory o/linear operators in HUbert space (translated from the Rus­sian), Ungar, New York, 1963.

[8.2) L. Collatz, Functional analysis and numerical mathematics (translated from the German), Academic Pre .. , New York-London, 1966.

[8.3) N. Dunford and J. T. Schwarz, Linear operators, Interscience, New York-London, 1955. [8.4) L. V. Kantorovich and G. P. Akilov, Functional analysis in normed spaces (translated from the Russian),

Pergamon, Oxford, 1964. [S.5) M. A. Krasnosel'skii, G. M. Vainikko, P. P. Zabreiko, Ya. B. Rutitskii and V. Ya. Stetsenko, Approxi­

mate solution %perator equations (translated from the Russian), Wolters-Noordhoff, Groningen, 1972. [8.6) L. A. Lyusternik and V. J. Sobolev, Elements for functional analysis (translated from the Russian),

Ungar, New York, 1961. [8.7) M. A. Naimark, Normed algebras (translated from the Russian), 2nd ed., Wolters-Noordhoff, Gronin­

gen, 1972. [S.S) F. Riesz and B. Sz.-Nagy, Functional analysis (translated from the French), Blackie, London-Glasgow,

1956. [S.9) V. I. Smirnov, A course 0/ higher mathematics (translated from the Russian), vol. 5, Pergamon, Oxford,

1964. 8.2 Measure theory and the Lehesgue-Stieltjes integral

[S.10) M. Fisz, Probability theory and mathematical statistics (translated from the Polish), Wiley, New York-London, 1963.

[S.ll) P. R. Halmos, Measure theory, Van Nostrand Reinhold, New York-London, 1950. [S.12) M. E. Munroe, Introduction to measure and integration, Addison-Wesley·, Cambridge, MA, 1953. [S.13) I. P. Natanson, Theory of/unctions o/a real variable (translated from the Russian), Ungar, New York,

1970. [S.14) A. Renyi, Probability theory (translated from the Hungarian), North-Holland, Amsterdam-London,

1970. [S.15) E. C. Titchmarsh, The theory of/unctions, Oxford University Press, Oxford, 1932. [S.16) J. H. Williamson, Lebesgue integration, Holt, Rinehart and Winston, New York, 1962. [S.17) A. C. Zaanen, An introduction to the theory o/integration, North-Holland, Amsterdam-London, 1955.

8.3 Tensor calculus

[8.IS) R. M. Bowen and C.-C. Wang, Introduction to vectors and tensors, vol. 2, Plenum Press, New York­London, 1976.

[S.19) L. P. Eisenhart, An introduction to differential geometry, Princeton University Press, Princeton, NJ, 1940.

[S.20) A. E. Green and W. Zerna, Theoretical elasticity, Oxford University Press, Oxford, 1954. [S.2I) W. V. D. Hodge, The theory and applications 0/ harmonic integrals, Cambridge University Press, Cam­

bridge, 1941. [S.22) D. F. Lawden, An introduction to tensor calculus and relativity, Methuen, London, 1962. [S.23) I. S. Sokolnikoff, Tensor analysis, theory and applications to geometry and mechanics 0/ continua, 2nd ed.,

Wiley, New York-London, 1964. [S.24) B. Spain, Tensor calculus, Oliver and Boyd, Edinburgh-London, 1953. [S.25) R. C. Tolman, Relativity, thermodynamics, and cosmology, Oxford University Press, Oxford, 1934.

952 Bibliography

[S.26) C. E. Weatherburn, An introduction to Riemannian geometry and the tensor calculus, Cambridge Uni­versity Press, Cambridge, 1935.

[S.27) T. J. Willmore, An introduction to differential geometry, Oxford University Press, Oxford, 1959. [S.2S) E. C. Young, Vector and tensor analysis, Marcel Dekker, New York, 1975.

S.4 Integral equations

[S.29) J. A. Cochran, The analysis 0/ linear integral equations, McGraw-Hill, New York-London, 1972. [S.30) L. V. Kantorovich and V. I. Krylov, Approximate methods 0/ higher analysis (translated from the Rus­

sian), Noordhoff, Groningen, 1955. [S.31) S. G. Mikhlin, Integral equations and thei,. application to problems in mechanics, mathematical physics

and technology (translated from the Russian), Pergamon, Oxford, 1957. [S.32) S. G. Mikhlin and K. L. Smolitskii, Approximate methods for the solution 0/ differential and integral

equations (translated from the Russian), Elsevier, New York, 1967. [8.33) V. I. Smirnov, A course o/higher mathematics (translated from the Russian), vol. 4, Pergamon, Oxford,

1964.

[9.1) [9.2)

[9.3)

[9.4) [9.5) [9.6)

[9.7)

[9.S)

[9.9) [9.10)

[9.11)

[9.12)

[9.13)

[9.14) [9.15)

[9.16) [9.17)

[9.1S)

[9.19)

[10.1)

[10.2) [10.3] [10.4) [10.5)

[10.6]

[10.7)

9 Mathematical methods of operations research

M. S. Bazaraa and J. J. Jarvis, Linear programming and network flows, Witey, New York-London, 1977. V. Chachra, P. M. Ghane, and J. M. Moore, Applications of graph theory algorithms, North-Holland, New York-Oxford, 1979. Sven Dane, Linear programming in industry, 4th cd., Springer-Verlag, Berlin-Hcidelberg-New York, 1974. D. Gale, The theory 0/ linear economic models, McGraw-HiII, New York-London, 1960. R. S. Garfinkel and G. L. Nemhauser, Integer programming, Witey, New York-London, 1972. A. Ghosal, S. G. Loo~ and N. Singh, Examples and exercises in operations research, Gordon & Breach, New York-London, 1975. S. K. Gupta and J. M. Cozzolino, Fundamentales of operations research for management. Holden-Day, San Francisco-London, 1975. G. Y. Handler and P. B. Mirchandani, Location in network theory and algorithms, MIT Press, Cambridge, MA, London, 1979. N. A. J. Hastings, Dynamic programming, Gordon & Breach, New York-London, 1975. J. P. Ignizio and J. N. D. Gupta, Operations research in decision making, Crane, Russak & Co., New York,1975. J. L. Kennington and R. v. Helgason, Algorithms/or network programming, Wiley, New York-London, 19S0. E. L. Lowler, Combinatorial optimization: networks and matroids, Holt, Rinehart and Winston, New York-London, 1976. V. L. Makarov and A. M. Rubinov, Mathematical theory 0/ economic dynamics and equilibrium (trans­lated from the Russian), Springer-Verlag, Berlin-Heidelberg-New York, 1977. R. M. Nauss, Parametric integer programming, University of Missouri Press, Columbia, MO, 1979. R. C. Pfaffenberger and D. A. Walker, Mathematical programming in economics and business, Iowa Uni­versity Press, Ames, 10, 1976. J. Ponstein, Approaches to the theory 0/ optimization, Cambridge University Press. Cambridge, 19S0. C. van der Panne, Methods for linear and quadratic programming, North-Holland, Amsterdam-London, 1975. D. A. Wismer and R. Chattergy, Introduction to non-linear optimization: a problem solving approach, North-Holland, Amsterdam-London, 1975. G. Zoutendijk, Mathematical programming methods, North-Holland, Amsterdam-London, 1976.

10.2 Automata. 10.3 Algorithms

Z. Bubnicki, Identification 0/ control plants (translated from the Polish), Elsevier, Amsterdam-New York,1972. S. Eilenberg, Automata, languages, and machines, I. Academic Press, New York-London, 1974. S. Eilenberg, Automata, languages, and machines. 11, Academic Press, New York-London, 1976. R. J. Nelson, Introduction to automata, Wiley, New York-London, 1965. P. H. Starke, Abstract automata (translated from the German), North-Holland, Amsterdam-London, 1972. B. A. Trachtenbrot and N. E. Kobrinskii, Introduction to the theory 0/ finite automata (translated from the Russian), North-Holland, Amsterdam-London, 1965. A. S. Willsky, Digital signal processing and control and estimation theory, MIT Press, Cambridge, MA, London, 1979.

10.4 Elementary switching algebra

[IO.S) [10.9) [10.10)

P. R. Adby, Applied circuit theory: matrix and computer methods, Witey, New York-London, 19S0. R. K. Bray ton and R. Spence, Sensitivity and optimization, Elsevier, Amsterdam-New York, 19S0. W. Keister, A. E. Ritchie, and S. H. Washburn, The design of switching circuits, Van Nostrand Rein­hold, New York-London, 19S0.

10.5 Simulation and statistical design and optimization of experiments

[10.11] E. S. Edgington, Randomization tests, Marcel Dekker, New York, 19S0. [10.12) J. M. Hammersley and D. C. Handscomb, Monte Carlo methods, Methuen, London, 1964. [10.13) O. Kempthorne, The design and analysis 0/ experiments, Witey, New York-London, 1952. [10.14) H. B. Mann, Analysis and design 0/ experiments, Dover, New York, 1949.

Index

Abelian group 745 Abel's integral equation 829 - limit theorem 340 - test for convergence 332 - - - uniform convergence 336 Abscissa 192, 194,225 Absolute value 218, 472 Absolutely continuous function 235, 750, 806, 807 - convergent improper integral 285 - - series 333 Abstract deterministic automaton 912 Acceleration vector 524 Access time 731 Accessible state of automaton 917 Accumulation point 220, 742 Action space 898 Addition, axioms of 215 -, associative law of 21 5 -, commutative law of 215 - theorems of hyperbolic functions 172, 482 - - - trigonometric functions 163,482 - of vectors 577 Address 731, 927 -, part 732 -, relative 733 -, symbolic 733 Adjoint differential expression 454 - operator 768 - system 380 - vector function 374 Admissible boundary 874 - control 372, 373 - direction 874 - domain 639 Affine coordinates 518 - space 198 d' Alembert's operator 443 - ratio test 330 - solution 457 Algebra 104 -, Banach 761 -, normed 761 -, partial 515 -, universal 515 Algebraic curve 76, 557 - equation 116, 118 - function 153,479 - -, integrals of 60 - -, graphs of 61 - -, power series of 22 - multiplicity 785 Algorithm 924 -, Euclid's 113 -, Ford's 892 -, Lang and Doig's 860 -, golden ratio 871 Alignment line 724 - plane 724 Almost everywhere 804 Alphabet input 913 - output 913 Alternating differential form 817

Alternating group 97 - product of tensors 811, 818 - series 332 Altitude 173, 184 Analogies, Napier's 188 Analogue computers 735 Analysis of switchwork 930 Analytic continuation 491, 494 - function 342, 482 - geometry, plane 198 - -, space 206 Angle between curves 565 - - vectors 133 - bisector 173 - cosine theorem 185, 188 Annulus 176 Antisymmetric dyad 547 - relation 511 - inequalities 114 Antitone operator 781 Antizigzag precautions 875 ApoUonius, theorem of 202 AppeU's function 414 Appropriate differentiation 707 - value 87 Approximation, calculus of 86 - method of Nystriim 844 - problem 689 -, successive 836 - theorem of Weierstrass 693 Arc of graph 891 - length in plane 553 - - in space 559 - - on surface 565 - measure of angle 183 Archimedean axiom 217 - spiral 83 Area 288 - of circle 175 - - domain 298 - - eUipse 200 - - polygon 199 - - surface 315, 565 - - revolution 289 - - triangle 174, 185 - - -, spherical 187 - - preserving map 566 Argument of complex number 474 Arithmetic expression 104 - mean 103,218 - sequence 102 Arzela, theorem of 850 Arzela-Ascoli, theorem of 337, 744 Assembly code 733 Assignment 108, 503 - problem 903 Asymmetry 511 Asymptote to curve 557 - of graph of rational function 125 - of hyperbola 204 Attainability, set of 373 Attribute 504

954 Index

Automaton 912 -, complete 9 I 3 -, deterministic 912, 913 -, finite 913 -, graph of 913,914 -, initial 9 I 3 -, Mealy 912 -, minimal 917 -, Moore 914 -, non-deterministic 921 -, partial 9 I 4 -, reduced 916 -, state of 913 -, stochastic 921 Autonomous linear system 373 Axiom of Archimedes 217 - - choice 515 - - Dedekind cut 216 - - mathematical induction 217 Axioms of addition 215 - - multiplication 215 - - order 216 - - Kolmogorov 599 Axis imaginary 203, 205 - intercept equation 199 -, major 201 -, minor 20t -, real 203

Backus notation 102 Backward solution 884 Baire's theorem 743 Ball, closed 743 -, open 743 Banach algebra 761 - space 748 - -, reflexive 762 Banach-Steinhaus theorem 763 Banach's closed graph theorem 767 - - range theorem 77 I - fixed point theorem 778,789 Barrel 182 Barrier function 882 Barycentric coordinate system 191 Base-line of trapezium 174 Basic conjunction 502 - disjunction 502 - list 620 Basis of vector space 131, 746 -, inverse 641 -, matrix 64 t - solution 642 -, exchange in 643 Bayes' formula 602 Bellman's functional equation 885 - optimality principle 884 Bernoulli differential equation 391 - numbers 345 - -, table 22 Bernoulli's inequality 2 I 8 - method of iteration 686 Bessel functions 34 I, 4 I 0 - -, modified 4 I I - -, spherical 4 I 2 - -, table 3 Bessel's differential equation 410 - inequality 753 Beta function 303 B ijection 5 I 4 Bilinear function 500 Binary coding 920 - system 86 Binomial coefficients 92

Binominal distribution 604 - equation 112 - integral 274 - series 424 - -, table 22 - theorem 94, 474 Binormal vector 560 Biorthogonal system 759 Biquadratic equation 122 Bit 730 Bite 730 Bi-unique correspondence 513 Block of programme 733 Bolza, problem of 363 Bolzano theorem 234 Bolzano-Weierstrass theorem 220 Boolean function 503 - normal form 502 - optimization 857 - set 506 Borel field 802 - measure 802 Bound (lower, upper) 219,513 - of error 88 - renaming 505 - variable 504 - vector 517 Boundary of domain 477 - condition, natural 356 - method 714 - point 221 - value problem 383, 386,421,446,469 Bounded (above, below) 219 - complex function 478 - linear functional 762 - operator 756 - point sequence 224 - - set 220 - sequence 220 - set 744 - variation, function of.235, 750 Box-Wilson optimization 944 Brachystochrone 350, 354 Branch-and-bound algorithms 907 Branch manifold 433 - point 497 - of root 496 Branching procedure 860 Brown-Robinson iteration procedure 901 Bunyakovskii inequality 133 Buffer times 895

Calculus of variations 349 - - -, inverse pro blem of 364 Canonical basis of vector space 131 - bracketing 101, 502 - form 639 - equation of dynamical system 374 - normal form 502 - system of differential equations 439 - transformation 440 Cantor's diagonal method 516 Cardano's formula 120 Cardinality 506, 515 Cardioid 78 Carrier of line element 388 Cart an calculus 817 Cartesian coordinate system 191, 194,518 - product of sets 508 Cassinian oval 79 Catenary 85 Cauchy-Hadamard theorem 338 Cauchy-Kovalevskii theorem 429, 43 I

Cauchy-Riemann differential equation 483 Cauchy-Schwarz-Bunyakovskii inequality, 114,

133,472 Cauchy principal value 283 - problem 386,429,446,470 - sequence 742 - test for convergence 223, 429, 328,742 - - - existence of limit 228 - transform 769 - uniform convergence 355 Cauchy's integral theorem 486 - formulae 487 - method for inhomogeneous differential 401 - theorem for product of series 334 Central differences 697 - limit theorem 619 - moment 608 - storage unit 730,732 Centrally symmetric vector field 524 Centre of curvature 554, 560 - ~ mass290,298,310,314,320 Centroid 173,198,206 Cesaro summation method 764 Chain 513 - rule 243, 483 - -, generalized 253 -, Sturm 124 Characteristic cone 444 - curve 432 - equation 148,443 - function 803 - manifold 433, 435 - polynomial 148 - strip 432 - surface 443 - system of differential equations 432 Chebyshev, altemant theorem of 693 - approximation 693 - -, discrete 694 - inequality 114,617 - polynomials 692, 752 -, theorem of 617 Christoffel symbols 569, 815 chi-square distribution 622 - test of fit 631 Choice, axiom of 515 - function 515 Chord of circle 175 Circle 175 - chain method 495 - of convergence of power series 489 - of curvature 554 Circumscribed circle 173 - -, radius of 184, 189 - quadrilateral 174 circumference of circle 175 - - ellipse 202 Cissoid 76 Closed ball 743 - graph theorem of Banach 767 - interval 220 - point set 220 - range theorem of Banach 771 - set 743 Closure of set 220, 743 - - -, convex 747 - - -, linear 747 Clothoid 84 Cofactor 135 Coefficient of polynomial 11 I Collinear vectors 517 Column matrix 135 Combination 99

Index 955

Combinatorial optimization problems 902 Combinatorics 91 Comparable elements 512 Comparison function 324., 351 - test 330 Compact operator 772 - set 744 Compatibility 110 - of inequalities 115 Complete automaton 913 - integral of differential equation 436 - metric space 742 - orthonormal system 571 - system of functions 368 Completely additive set function 810 - continuous operator 772 Completing the square 119 Completion of metric space 742 - - normed space 748 Complement of set 220, 507 - - -,orthogonal 134,754 Complementary events 599 - subspaces 770 Complex function 477 - -, bounded 478 - -, zeros of 478 - number 471 - -, absolute value 472 - -, argument 474 - -, length 473 - -, modulus 472 - -, principal value 476 - -, trigonometric form 474 - plane, closed 476 - -, finite 476 - variable 477 Composite function 226, 239 Computing automata 927 Concave 246 Conchoid 77 Concentrated parameters 380 Condition of Jacobi 354 - - Legendr. 354 - - Weierstrass 355 Conditional convergence 333 - distribution 614 - gradient method 384 - probability 601 Confluent hypergeometric function 414 Confidence estimator 626 - interval 628 Conformal mapping 498, 566 Cone 181 Conical surface 181 Conics 200 Conjugate complex number 472 - diameters 202, 204 - hyperbolas 204 - point 354 - space 762 Conjunction 501 Connected automaton 907 - point set 221 - relation 51 I Conservative field 531 Constraint 636 Constant coefficients, linear differential equation

with 402 - function 61 Constants, table 1 Contact transformation 438 Content 800 -, elementary 801

956 Index

Content, Peano-Jordan 801 -, a-finite 802 -, weakly finite 802 Continuation by power series 490 -, analytic 491 Continuous dynamical system 372 - complex function 478 - curve 477 - function, at point 230, 238 - -, on set 231, 239 - -, piecewise 233 - operator 755 - spectrum 773 Continuously differentiable 242, 249 Continuity equation 536 Contour integral 486 Contragredient transformation 132 Contravariant coordinates 191, 195,518,520 - tensor 562, 809 Contractive operator 778 Contradiction 109 Control device 730 - domain 732 - space 732 - vector 732 Controllability 373 Convergence, almost surely 618 -, Cauchy's test for 223, 225 - of improper integrals 280, 282, 323 - in norm 762 - of point sequence 274 - in probability 617 - of sequence 222, 488, 742 - - series 328, 488 -, strong 762 - test for infinite integrals 284 -, uniform 301, 334, 762 -, weak 762 Convex closure 747 - function 246 - optimization 863, 868 Convolution theorem 582 Coordinate angle 192, 194 - axis 194 - function 368 - system 190 - -, barycentric 191

. - -, Cartesian 191, 194 - -, contravariant 191 - -, covariant 192, 195 - -, curvilinear 192, 195 - -, cylindrical 195 - -, homogeneous 191 - -, parallel 191 - - in plane 191 - -, polar 193, 195 - -, rectangular 192, 194 - -, rotation of 196 - -, transformation 196 - -, translation 196 Coordinates, affine 518 -, contragredient 132, 518 - transformation 132 - in vector space 132, 518 Coplanar vectors 517 Correlated random variables 614 Correlation coefficient 614, 633 Correspondence 513 -, inverse 514 Cosine formula 183 - function 68,161,165,482 - -, hyperbolic 74, 169,482 - theorem, plane 183

Cosine formula, spherical 188 Cotransitivity 511 Countable set 219,516 Covariance matrix 614 - coordinates 195,520 - differentiation 815 - tensor 809 Cover of space 144 Covering theorem of Heine-Borel 222 Cramer's rule 145 Criterion, see also test - of Dirichlet-Jordan 581 - - Routh-Hurwitz 419 -, row sum 779 Critical path 893 Cube 178 Cubic curve 76 - equation 119 - function 61 Cuboid 178 Curvature of plane curve 553 - - space curve 560 - - surface, Gaussian 567 - - -, mean 567 -, lines of 568 - tensor 569, 816 Curve, algebraic 76 -, elementary 75 -, discussion of 248 -, length of 288 - of second order 200 - theory, fundamental theorem 561 Curvilinear coordinate system 192, 195,814 - integral 530 Curl of tensor field 816 Cut method of Gomory 858 Cycle 892 - of permutation 98 Cycloids 80 Cylinder 180 - functions 410 - -, table 3 Cylindrical coordinate system 195, 526 - surface 180

Dantzig-Wolfe decomposition 657 Data processing 730 Deadline 895 Decimal point 86 - system 86 Decimals 87 Defect index 769 - subs pace 769 Definite divergence of point sequence 225 - - - sequence 224 - integral, tables 56 - -, geometric interpretation 264 - operator 770 Deformation tensor 549 Degree measure of angle 183 - of polynomial I11 Delambre's formulae 188 De Moivre's formulae 122, 164, 171,474 Dense set. 143 Density of distribution 606 - - -, table 9 Denominator 216 Dependent functions 256, 399 - subset 130 - variable 225, 477 - vectors 130, 404 Derivative 240 - of complex function 483

Derivative, directional 252 - formulae of Gauss and Weingarten 569 -, Frechet 371 -, logarithmic 242 -, partial 249 Derived set 220 Descartes' rule of sign 125 Design of experiments, statistical 941 Detection of errors 613 Determinant 135 -, functional 255 -, Jacobian 255 Deterministic automaton 912 Developable surface 586 Diameters of conic 202 - - -, conjugate 204, 205 Difference 215 -, brckward 697 -, central 697 -, divided 696 -, forward 697 - procedures 115 - quotient 240 -, rising 696 - of sets 507 - - vectors 517 Differentiable function 240, 249, 483 - -', continuously 242, 249 - -, n times 242 -, partially 254 .-, totally 251 - mapping 254 Differential of arc length 553 -, Gateaux 371 -, total 251 - equation 385 - -, BernoulJi 391 - -,Bessel410 - -, Clairaut 393,431 - - with constant coefficients 402 - -, elliptic 445, 446, 460, 534 - -, Euler 403 - -, Euler-Lagrange 352 ..., -, exact 389 - -, explicit 385, 387 - -, Hamilton-Jacobi 364, 439 - -, homogeneous 388 - -, hyperbolic 445, 446, 453 - -, hypergeometric 413, 414 - -, implicit 385, 392 - -, Kummer 414 - -, Lagrange 393 - -, Laplace 538 - -, Legendre 414 - -, non-linear 417 - -, ordinary 385 - -, -, first order 431 - -, -, second order 442 - -, parabolic 445, 441, 469 - -, partial 385, 427 - -, Poisson 365 - -, Riccati 391 - - with separated variables 388 - -, singular point 395 Differentiation, term-by-term 490 - of vector function !23 Diffusion equation 443 Dimension of vector space 131 - - theorems I 32 Direct numerical methods 366, 383 Directed simulation 936 Direction field of differential equation 387 Directional derivative 252, 528

Directly proportional 61 Directrix 201, 203, 205 Dirichlet conditions 581 - function 153 Dirichlet-lordan criterion 581

Index 957

- problems 447, 452, 467, 716, 796 - test 332 -, theorem of 572 Dirac measure 80 I Discrete Chebyshev approximation 694 - random variable 603 - system 382 Discontinuity of function 231 - - first or second kind 232 -,jump 232 -, removable 232 Discriminant 119, 120 - locus 394 Discussion of curves 248 Disjoint set 509 Disjunction 501 -, basic 502 -, elementary 502 -, normal form of 502, 932 Displacement vector 549 Distance between complex numbers 413 - - lines 208 - - line and plane 201 - -, in metric space 741 - - points 198 Distributed parameters 380 Distribution, binomial 604 -, chi-square 622 -, conditional 614 - density 606 -, exponential 601 -, Fisher's Z- 623 - function 603, 610, 620 -, Gaussian 607 -, hypergeometric 605 -, marginal 612 -, moments of 608 -, normal 605 - -, table 9 -, Poisson 605 -, -, table 8 -, rectangular 606 -, Snedecor's F- 623 -, Student'S f- 623 -, Weibull 607 Distributive law 215 Divergence, definite 223 - of dyad 549 - - tensor field 816 - - vector field 535 Oivergent improper integral 280, 282, 323 - point sequence 224 - sequence 222, 328, 488 - series 228, 488 Division 216 - of polynomials 112 Domain 221, 477 _, admissible 636, 638, 639, 863 - of definition 225, 236, 477 - - dependence 454 - differentiation 306 - method 714 - of operator 755 -, star shaped 776 - of variability 504 Double integral 305, 806 - - transformation of variables 301 - point 556

958 Index

Du Bois-Reymond, theorem of 353 Dual operator 767 - simplex algorithm 651 - space 762 Duality in linear optimization 650 - theorems 650, 871 Duration 893 Dynamical model 889 - optimization 364, 882 - system 372 Dyad 546 - fields 548 -, nabla calculos 550 Dyadic product 546

Eberlein-Shmul'yan theorem 766 Eccentricity 201,203, 205 Echelon form 143 Edge 177 - of graph 891 Efficient estimator 624 E-function of Weierstrass 355 Eigenfunction 425,460 Eigenspace 149 -, generalized 773, 785 Eigenvalue 149, 425 - of operator 151 - problem 425 - -, linear 681 - of tensor surface 548 Eigenvector 149, 151 Einstein summation convention 562, 808 Electronic digital computer 730 Element of set 505 -, maximal, minimal 513 Elementary conjunction 502 - content 80 I - discussion of curves 248 - disjunction 502 - event 598, 918 - function 153, 479 Ellipse 201 Ellipsoid 211 Elliptic differential equation 442, 445, 446, 460 - integral 267 - -, tables 6 - paraboloid 213 Empirical distribution function 621 - variance 622 Empty set 506 End-times, free 378 Envelope of family of curves 559 Epicycloid 81 Equality between arithmetic expressions 109 Equally possible event 600 Equating coefficients, method of 339 Equation 116 -, algebraic 116 -, binomial 122 -, biquadratic 122 -, cubic 119 - of curve 75

function 225 - - higher degree 122 - - heat conduction 444, 469, 470 - - line 199 -, linear 118 -, logarithmic 126 - of plane curve 551 -, quadratic 119 -, quartic 121 -, soluble 116 -, solution of 116

Equation, transcendental 126 - of vibration 442 Equicontinuous function 337, 744 Eq uilateral triangle 173 Equipotent sets 515 Equivalence 501 - class 511 - relation 511 - transformation 116, 117 Equivalent assignments 110 - automata 915 - dynamical systems 372 - eq uations I 17 - functions 775 - inequalities 114 - matrices 148 - norms 775 - sets 515 - states 915 Errors, absolute 88 -, bound of 88 -, calculus of 88 -, detection of 673 - of first kind 629 -, input 674 -, random 675 -, relative 88 - of second kind 629 -, squared, method of 714 -, true 88, 673 Essential spectrum 774 Estimator, confidence 626 -, efficient 626 -, maximum likelihood 625 -, point 623 -, unbiased 623 Euclidean algorithm 113 - vector space 133 - space 741 Euler-Cauchy polygonal arc 709 Euler-Lagrange differential equation 352, 360, 364 Euler numbers 21 - -, tables 22 - substitution 275 - triangle 187 Euler-Venn diagram 508 Euler's formula 474 - integral of the first kind 303 - - - - second kind 304 - method 367 - theorem on normal sections 568 - - - polyhedra 180 - transformation 721 Evaluation of arithmatric expression 108 Evolute 558 Event, critical 893 -, elementary 598, 918 -, impossible 598 -, indicator of 604 -, random 599 -, regular 918 -, sure 598 Events, complementary 599 -, equally possible 600 -, incompatible 599 -, product of 598 -, sum of 598 Excess, spherical 187 Exchange lemma of Steinitz 131 Excluded middle, principle of 501 Existence and uniqueness theorem 386 Existential quantifier 504 Explicit differential equations 385, 387

Expansion, Laplace 136 -, Taylor 245 -, -, Lagrange's form of remainder 245, 254 Expectation 608 Exponential distribution 609 - equation 126 - form of complex number 474 - function 70, 168,479 - integrals 51, 56 - -, tables 7, 24 Expression, arithmetical 104 -, functional 105 -. predicative 504 -, propositional 501 Extensionality, principle of 50 I, 505 Extended matrix 142 Exterior point 221 Extremals 352 Extremum 246, 259, 350 -, relative, necessary and sufficient conditions for

247, 260

Face 177 Factor 215 Factorial 91 Factorization I 19 Family of sets 509, 514 Farkas, theorem of 869 Fatou's lemma 806 F-distribution, Snedecor's 623 - -, table 14 Feasible domain 636 - equality 109 - point 636 Field 524 - lines 526 - theory 360 Finite automaton 913 - complex plane 476 - function 774 - geometric series 102 - plane 476 - point 476 - sequence 514 - set 219, 515 - variation 750 Finitely additive measure 800 - generated vector space 131 First fundamental form 564 - mean value theorem of integral calculus 265 - order differential equation 387 - rule of Guldin (Pappus) 290 - variation 350 Fisher's Z-distribution 623 - -, table 18 Fixed point of linear map 500 - - - permutation 97 - - theorem of Banach 780 - - - - Brouwer 780 - - - - Schauder 780 Flux of scalar field 533 - - vector field 533, 535 Focal strip 431 Focus 201, 203,205 Folium of Descartes 76 Ford's algorithm 892 Form, canonical 639 Formula of Bayes 602 - - Euler 474 - - Green, first 460, 541 - - -, second 460, 542 - - I'Huilier 190 - - Kirchhoff 4~8

Formula ofLiouville 400, 404 - - Maclaurin 245 - - de Moivre 171,474 - - Poisson 457, 468

Index 959

Formulae of Delambre (Gauss) 188 -, empirical, establishing 726 - of Hamilton 363 - - Mollweide 183, 188 - - de Morg.n 510 Forward solution 884 Fourier coefficients 571, 753, 854 - expansions, tables 573 - integral 581 - series 571,753, 854 - transform 582 - -, rapid 580 - -, tables 583 Fourier's method 447 Fractional line equation 118 - - function 500 - rational functions 63 Frechet derivative 371,791 - differentiable 791 Fredholm alternative 773, 783, 846 - integral equation 826, 836 - - -, second kind 780, 840 - method 840 - operator 772 - point 774 - radius 783 Free buffer time 895 - end-time 318 - optimization problem 871 - renaming 505 Freely controlled integrator 737 Frenet formulae 561 Frequency, relative 599 Fresnel integral 591 F-test 629 Fubini's theorem 805 Fundamental form, first 564 - -, second 566 - lemma of calculus of variations 352 - sequence 742 - system of solutions of differential equation 400 - theorem of algebra 122, 154

calculus 268 - - - curve theory 561 - - - integral calculus 486 - - - surface theory 569 Fully free variable 504 Function 513 -, absolutely continuous 235, 150, 806, 807 -, algebraic 153,419 -, analytic 342, 484 -, Bessel 410 -, -, tables 3 -, bounded 227, 237, 478 -, -, uniformly 337, 744 - of bounded variation 235, 750 - - complex variable 477 -, composite 226, 239 -, constant 61 -, continuous 230, 231, 238, 239, 478 -, cubic 61 -, differentiable 241 , 250, 483 -, -, continuously 242, 249 -, Dirichlet's 153 -, discontinuous 231 -, element 495 -, elementary 153,479 -, exponential 70, 168,479 -, finite 774

960 Index

Function, graph of 225, 257 -, Green's 422,467,468 -, Hamiltonian 363, 374 -, Hankel 411 -, harmonic 462, 484 - homogenuous 238 -, hyperbolic 74, 169, 482 -, hypergeometric 413, 414 -, implicitly defined 256 -, integer rational 61, 123, 153,479 -, inverse 226, 495 -, irrational 479 -, Kummer's 414 - ladder 722 -, Lagrange's 357, 360 -, Legendre's 416 -, limit of 227,478 -,linear 61,155,499 -,logarithmic 168,479 -, Macdonald's 411 -, measurable 803 -, modulus of 478 -, monotonic 234, 245 -, one-sheeted 495 -, periodic 235 -, piecewise continuous 233 -, presentation of 719 -, quadratic 61, 155, 500 -, rational 63, 155,227,479,500 -, real-valued 225 -, - -, of several variables 236 -, regular 484 -, Riemann's 455 - scale 722 -, schlicht 495 -, semicontinuous 236 - sequence 334 - series 335, 489 -, totally bounded 744 -, transcendental 161, 479 -, trigonometric 68,161,482 -, uniformly bounded 337, 744 -, - continuous 235, 238 -, Weber's 410 -, Weierstrass' E- 355 Functional 35 1,755 -, bilinear bounded 762 - determinant 255 - equations, Bellman's 885 - expression 105 -, linear 755 - matrix 255

Galerkin's method 799 Game, fair 900 -, matrix 899 - theory 898 - -, main theorem of 900 -, value of 899 -, - -, theorem of shifting 900 Game-theoretical model 939 Gamma function 91,153,304,349 - -, tables 2 Gateaux differential 371 Gauss-Banachiewicz method 677 Gauss-Bonnet theorem 571 Gauss-Chebyshev quadrature 705 Gauss-Jordan method 676 Gauss-Legendre quadrature 704 Gauss-Seidel iteration 790 - - - method 680 Gaussian algorithm 138, 143 - curvature 567

Gaussian distribution 607 - integral 296 - plane 413 - quadrature 703 Gauss's bell-shaped curve 71, 607 - derivative formulae 569 - formulae 188 - method of elimination 675 - product for the Gamma function 304 - test 331 - theorem for integrals 322, 541, 825, 826 - theorem a egregium 569 General integral of differential equation 387 - quadrilateral 174 - solubility theorem 267 - solution of differential equation 386 Generalized chain rule 253 - Chebychev inequalities 114 - derivative 774 - eigenspace 785 - mean-value theor.em 244 - product 515 - triangle inequality 218 Generating set 129 Generator of quadric 214 Geodesic 186, 570 - curvature 570 Geometric interpretation of complex numbers 473 - - - convexity 246 - - - derivative 240 - - - partial derivative 249 - - - total differential 25 I - mean 103,218 - multiplicity 773 - representation of complex 473 - - real numbers 217 - sequence 102 - series 102, 489 Glivenko's theorem 621 Golden ratio algorithm 871 - section 104 Gomory's cut method 858 Gradient 252 - function 361 - method 369, 383 - -, conditional384 - -, projected 384, 877 - procedure for problems with constraints 874 - of scalar field 528 Graeffe's method 685 Gram determinant 752 - matrix 689 Graph 891 - of automaton 913 - - elementary function 61 - - operator 767 - theory 891 Graphical differentiation 90 - integration 91 - representation of function 225, 237 Gravitational attraction 315, 320 Great circle 181, 186 Greatest common divisor 113 - lower bound 513 Green's formulae 320,423,460, 541 - function 421, 422, 467 .- -, table 424 Group, Abelian 745 Guldin's rules 290

Haar's theorem 362 Hahn-Banach theorem 758 Half-angle theorem 183, 188

Half-open interval 220 Half-side theorem 188 Hamilton-Jacobi differential equation 364, 439 - - theory 363 - - -, fundamental theorem of 364 Hamiltonian 363 - formulae 363 - function 374, 382 Hamilton's principle 357 Hankel function 411 Hansen's problem 185 Harmonic analysis 571, 689 - -, numerical 579 - function 462, 784 - -, spherical 464 - mean 218 - series 329 Harnack's inequality 467 Hasse diagram 572 Hausdorff's principle 513 - theorem 744 Heat conduction, equation of 443,452,469 - - - -, infinite bar 450 Heine-Bovel covering theorem 222 Helmholtz' equation 443, 464 Hermitian matrix 141 - operator 460 - polynomials 753 Heron's formula 174, 185 Hessian normal form of line 199 - - - - plane 208 Higher derivatives 242, 250 Hilbert's method 850 Hilbert-Schmidt kernel 757 - - norm 758 - - operator 758 Hilbert space 753 Hildreth-d'Esopo procedure 866 Histogram 620 Hodograph 522 HoerI's ridge line analysis 944 Holder's inequality 775 Holomorphic function 484 Homogeneous coordinate system 191 - differential equation 388 - function 238 - integral equation 826 - system of differential equations 404 - - - linear equations 141 Hooke-Jeves, method of 872 Horner's scheme 719 I'Hospital's rule 230 Hull, linear 129 Hungarian method 903 Hyperbola 203 -, equilateral 63 Hyperbolic differential equation 442, 445, 446, 453 - - -, homogeneous 447 - - -, inhomogeneous 448 - functions 169, 482 - -, graphs 74 - -, inverse 171 - -, power series 25 - paraboloid 213 - spiral 83 Hypergeometric differential equation 413 - function 413, 482 - -, confluent 414 - -, generalized 414 Hyperplane 198 Hypocycloids 82

62 Bronsteill, englisch

Index 961

Identity of arithmetic expressions 109 - map 514 - operator 149 - permutation 97 - theorem for power series 339, 490 Image 146, 514 -, inverse 146,514 Imaginary axis 473 - number 471 - part 471 - unit 471 Immediate suc·cessor 512 - predecessor 512 Implication 501 Implicit differential equation 385 - - -, first order 392 - differentiation 253 Implicitly defined function 256 Impossible event 598 Improper integral 280, 323, 494 - limit 228 - parameter integral 30 I Inclusion, strict 505 Incomparable elements 512 Imcompatible events 599 Indefinite integral 266, 486 - -, tables 26 Independence of path 296, 486 Independent functions 399 - random events 602 - solutions of differential equation 404 - variable 477 - vectors 130 Indeterminate I11 Index of operator 772 Indicator of event 604 Indicial equation 409 Indirect numerical method 383 Individual constant 504 Induction, mathematical 217 Inductive order 513 Inequality of arithmetic expressions 113 -, Bernoulli's 218 -, Bessel's 753 -, Cauchy-Schwarz-Bunyakovskii 114 -, Chebyshev's 114,617 -, Harnack's 467 -, Holder's 753, 175 -, Minkowski's 749, 753 -, Poincare's 777 -, Schwarz' 218,751,753,849 -, triangle 114, 133, 187,218,472 Inferior limit 223 Infimum of function 227, 237 - - point set 219 - - set 513 Infinite point 232 - product 347 - sequence 514 - series 328 - set 219,515 Infinitesimal rotation 545 Infinity, point at 476 Inflection, point of 247,554 Information presentation 730 - processing 911 Inhomogeneous integral equation 826 _ system of differential equations 406 ___ linear equations 141 Initial automaton 913 _ value problem 386 - vertex 891 Injective linear map 147

962 Index

Injective map 514 Input alphabet 913 - device 730 Inscribed circle 173 - -, radius of 183, 189 - quadrilateral 174 Inteler 217 -, rational function 61, 1S3, 479 Intelrability condition 296,297,531,569 - criterion, Riemann's 263 Intelral calculus 263 - -, fundamental theorem 268, 486 - -, mean value theorems 265 - -, substitution rule 269, 287 -. Cauchy principal value 283 -, complete 436 - of complex function 484 -, contour 486 - curve of dilferential equation 386 -, definite 263 -, -, tables 56 -, double 305, 301, 323, 324 -, elliptic 214 -, -, tables 6 - equation 826 - -, Abel 829 - -, Fredholm 829, 836, 840, 846 - -. homoleneous 826 - -. inhomoaeneoul 826 - -, kernel of 826 - -, linear 826 - -, Volterra 826, 828, 830, 839, 840 - exponential function 52 - formulae of Cauchy 487 -, Gaussian 296 -. leneral, of dilferential equation 387, 388 -, Lebeslue 804 -, Lebeslue-Stieltjes 804 -, improper 280, 323, 494 -, -, absolute converaence 285 -, indefinite 266, 486 -, -, tables 21, 267 -,line 291 - linear optimization 857 - 10larithm 53 -, parameter 299 -, Poisson 470 -, Schwarz-Christolfel 499 - test 331 - theorem of Cauchy 486 - - - Gauss 322 - - - Stokes 323 -, triple 310 Intelration of alaebraic function 214 -, Iraphical 91 - over infinitive interval 282 - method, Riemann's 454 - by parts 269, 286 - of rational function 270 - by substitution 269, 287 -, term-by-term 490 - of transcendental function 217 Intelrator 737 Intercept equation of line 199 - - - plane 208 Interior of domain 417 - point 221, 743, 869 Intermediary lanluale 133 Intermediate value theorem 234 Interpolation polynomial 694 - -, Lalranae's 695 - -. Newton's 695 Intersectinl planes, method of 819

In,ersection of family of sets 509 - - relations 5 10 - - sets 506 Interval 220 Inverse correspondence '14 - function 226 - hyperbolic functions 74, 172, 483 - - -, table of intelrals 55 - - -, table of power series 25 - imaae 146. 574 - matrix 140, 679 - operator 149 - permutation 97 - problem of calculus of variations 364 - relation 510 - trilonometric functions 64, 166, 483 - - -, table of intelrals 54 - - -, table of power series 24 Inversion of matrix 679 - in permutation 97 - - unit circle 417, 500 Involute 558 - of circle 84 Irrational alaebraic functions 160,479 - - -, Iraphs 66 - - -, tables of intelrals 32 - number 218 Irreducible polynomial 1S4 I rreflexity 54 Irrotational vector field S43 Isolated point 220 - sinaular point 4114 Isometric map 566 - operator 769 - spaces 742 Isoperimetric problem 360 .. otone operator 781 Iteration method of Picard-lndel6f 779 - procedure 867 - - of Brown-Robinson 901 -, whole-step 779

Jacobi condition 354 - identity 441 - method 680, 683 - polynomials 752 -, theorem of 439 Jacobian determinant 255, 301 - matrix 255 Join of relations '10 Jordan curve 477, 493 - measurable 802 Jonkowlki function 497

Karnaulh table 932 Kelvin transformation 466 Kepler's rule 102 Kernel, delenerate 831 -, Hilbert-Schmidt 757 - of intelral equation 826 - - linear map 146 - - operator 157 -, product 831 -, resolvent 837 Kolmolorov axioms 600 Kolmollorov-Smirnov distribution, table 20 - - teS! of fit 633 Kronecker symbol 134, 521 Kuhn-Tucker conditions 810 - - theorem 869 Kummer's dilferential equation 414 - function 414

Lagrange bracket 441 - multipliers 360, 371 Lagrange's form of remainder 245, 254 - identity 519 - interpolation polynomial 695 - problem 351 Lagrangian function 351, 360, 311 Laguerre polynomials 153 Lang and Doig, algorithm of 860 Landau symbols 0 and 0 233 Laplace equation 443, 538 - expansion 136 - operator 443, 538 - transform 419, 452, 592 Laurent series 491 Law for arithmetical expression 109 - oflarge numbers, strong 618 - of large numbers, weak 617 Lax-M ilgram theorem 761 Least squares, method of 261,635,689, 121 Lebesgue integral 804 - measure 802 Lebesgue-Stieltjes integral 804, 806 Lebesgue-Stieltjes measure 803 Lebesgue theorem of 750, 806 Left half-open interval 220 Left-hand continuous 232 - - derivative 241 - - differentiable 240 - - limit 228, 240 Left-handed system 518 Left range of relation 5 10 Left-unique correspondence 513 Legendre functions 416 - polynomials 415, 691, 752 - -, tables 5 - transformation 430, 439 Legendre's differential equation 414 - duplication formula 304 - optimization condition 354 - theorem 190 Leibniz' product rule 244 - test 332 Lemke's procedure 866 Lemma, Fatou's 806 -, Ricci's 816 -, Zorn's 513 Lemniscate 19 Lelfgth of complex number 413 - - curve 288, 553 - - interval 220 - - vector I 33 Level curve 231 - line 524 - surface 231, 524 B. Levi's theorem 806 Levi~Civita tensors S6S Lexicographic order 648 Lima~on 18 Limit, Cauchy's test for existence of 228 - of complex function 428 - function 221, 238, 418, 489 - -, table 229 -, lower. upper; inferior, superior 228 - of point sequence 223 - of sequence 222 - set of seq uence 223 - theorem of Abel 340 - -, central 619 - - de Moivre-Laplace 618 - theorems for integrals 806 - - of probability theory 611 Lindeberg condition 619

62*

Line of curvature 568 - element 388, 394 - integral, first kind 291 - -, second kind 293 - -, independent of path 296 - in plane 199 - in space 177,207 Lines, skew 177 Linear algebra 127

I n de x 963

- approximation in Hilbert space 689 - closure 747 - combination 129 - complementarity problem 868 - dependence and independence 130, 399,404 - differential equation 385, 390, 399 - - -, constant coefficients 402 - - -, second order 407 - equation 118 - eigenvalue problem 681 - function 61, 155,499 - -, fractional 63, 500 - functional 762 - hull 129, 745 - integral optimization 857 - map 145 - operator 147 - optimization 636 - -, applications 665 - -, parametric 668 - order 512 - quotient optimization 863 - space 127, 745 - span 129 - subspace 129, 198 - system of equations 141 - - - differential equations 404 - vector function S45 - transformation 145 Linearity 511 Liouville's formula 400, 404 - theorem 484 Lipschitz condition 236, 381 - constant 236, 387 Local extremum 246, 259 - Kuhn-Tucker conditions 870 Logarithm, natural 479, 500 Logarithmic branch point 497 - derivative 242 - eq uation 126 - function 168 - -, definite integrals 58 - -, graphs 71 - -, indefinite integrals 53 - slide rule, power series 24 - "Viral 84, 123 Logic, mathematical SO 1 -, propositional 501 Logically true 504 Loop of graph 891 lp-spaces 749 Lower bound 219, 513 - -, greatest S 13 - limit 219, 223 - triangular matrix 135 Lune 187 Lyapunov stable 418

McClusky's method 933 Macdonald function 411 M aclaurin formula 245 . - series 245 Machine staffing problem 903 - programme 132

964 Index

Machine word 730 Macro 732 Macrocommand 732 Magnetic dISC, - drum, - tape 731 Magnitude of vector 516 Main diagonal of matrix 135 - theorem of mathematical statistics 621 Mainardi-Codazzi equations 569 Manifold, branch 433 -, characteristic 433 -, initial 432 -, strip 434 Mantissa 731 Map, mapping 477,514 -, area· preserving 566 -, bijective 514 . -, conformal 498, 566 -, differentiable 254 -, identity 514 -, injective 514 -, isometric 566 -, kernel of linear 146 -, linear 145 - -, representation by matrices 147 -, nullity of 146 -, rank of 146 -, surjective 514 - theorem of Riemann 498 Marginal distribution 612 Mass 298, 310, 314 -, centre of 290, 298, 310, 314 Mathematical equipment of computer 734 - induction 217 - logic 501 - statistics 620 - -, main theorem of 621 Mathematics, numerical 673 Matrix 135 - algebra 138 -, characteristic equation of 149 -, eigenvalue and eigenvector of 149 -, functional (Jacobian) 255 -, game 899 -, gram 689 -, Hermitian 141 -, inverse 140 - inversion 679 -,orthogonal 141 -, rank of 137 -, regular 137 - representation-of operator 157 - ring 140 -, singular 137 -, skew-Hermitian 141 -, skew-symmetric 141 -, square 135 -, symmetric 141 -, transposed 136 Maximal element 513 - sequence 795 Maximum 219,227,237,313,513 -, absolute 351 -, relative 246, 259, 351 Maximum-likelihood estimator 625 - - method 625 Maximum-minimum principle, theorem 469 Maximum principle 484 - - of Pontryagin 364, 373, 374, 380 Mayer and Bolza, problems of 362 Mealy automaton 912 Mean, arithmetic 103,218 - curvature 567 -, geometric 103,218

Mean, harmonic 218 - pay-off 900 - square, root 103 - value theorem of differential calculus 244 - - - - integral calculus 265, 306, 805 Measurable 800, 803 Measure 800 -, Borel 802 -, complete 802 -, con vergence in 804 -, Dirac 801 -, finitely additive 800 -, Lebesgue 802 -, Lebesgue-Stieltjes 803 -, space 801 -, totally additive 801 Median 123, 185 Membership 505 Method, see also procedure - of artificial variables 646 -, Cauchy's 401 -, Fourier's 441 -, Galerkin's 799 - of gradients 369, 383 - - -, conditional 384 - - -, projected 384, 877 -, Hilbert's 850 - of Hooke-Jeeves 872 -, Hungarian 903 - of intersecting planes 879 - - least squares 261,635,727 -, McCluskey's 933 -, maximum-likelihood 625 - of moments 624 -, Nelson's 934 -, Newton's 791 - of Newton-Kantorovich 687, 688 - - partial domains 714 - - penalty function 881 -, Quine's 933 -, Ritz 796 - of steepest descent 873 - - successive approximation 398 -, Trefftz 798 Metric coefficients 520 - normal form of quadratic form 152 - properties of surface 564 - space 741 - -, separable 743 - tensor 565, 812, 815 Meusnier's theorem 569 Mid-line 173, 174 - - point 198,206 Minimal automaton 917 - disjunctive normal form 932 - element 513 - normal form 934 - sequence 366, 383 - surface 568 Minimax theorem 869 Minimizing an automaton 916 Minimum 219,221,237,313,513 -, absolute 351 -, relative 246, 259, 351 Minkowski inequality 149 Minor of matrix 135 Von Mises procedure 683 Mixed problem 469 - strategy 899 Miibius strip 315 Model, game-theoretical 939 -, numerical, realization of 118 - sensitivity 940

Model, servicing 939 -, stochastic-dynamic 889 -, strategic 898 Module of programme 733 Modulus of complex function 478 - - - number 472 - - vector 133 De Moivre, formula of 164,171,474 De Moivre-Laplace, limit theorem of 618 Mollweide's formulae 183, 188 Moment, central 608 - of distribution 608 - - inertia 291, 310, 314 Moments, method of 624 Monge axis 431 - cone 431 Monic equation 118 Monitor 734 Monodromy theorem 495 Monotonic (ascending or descending) function

234,245 - sequence 223 Monotonicity 216, 886 Monte-Carlo method 937 Moore automaton 914 Moreva, theorem of 486 De Morgan's formulae 507, 510 Moving trihedron 560, 563 Multi-dimensional parameter integral 325 Multinomial coefficients 94 - theorem 95 Multiple products of vectors 519 Multiplication, axions (laws) of 215 -,limits of 101 - of matrices 139 - of vectors 517, 518 - theorem for Jacobian determinants 255 Multiplier, Lagrange's 360, 371 Multiplicity, algebraic 785 -, geometric 773 - of root 122 - of zero 154 Multiply-connected domain 221, 477 Multi-programme processing 731 - - working 733

Nabla calculus 539 - - for dyads 550 - operator 539 Napier's analogies 188 - rule 190 Natural boundary condition 356 - logarithm 479 - number 217 Negation 105, 510 Negative definite operator 770 - - quadratic form 260 . - number 216 Neill's parabola 76 Neighbourhood 220, 477 Nelson's method 934 Net plan 889, 892 - - matrix 892 Net, plane 725 Neumann approximation 836 - problem 447,467,468,797 - series 831 Newton-Cotes formulae 102 Newton-Kantorovich inethod 687 Newton-Raphsan method 90 Newton's interpolation polynomial 695 - method 684, 687, 191 - -, simplified 793

Newton's rule 124 - 3/8-rule 702 Noetherian operator 772 Nanography 722

Index 965

Node of curve 395 Non-deterministic automaton 921 - - differential equation 417 - - linear equation 684 - - - -, system of 687 - - - optimization 863 - - significant digits 88 Norm of bounded sesquilinear form 761 - - element 746 -, energy 797 - of operator 156 - - space 746 - - vector 133 Normal distribution 607 - -, tables 9, 10 - form, Boolean 502 - -, canonical 512 - - of conic 200 .- - - conjunction and disjunction 502, 923 - - - ellipse 201 - -, Hessian 199, 208 - - of hyperbola 203 - - - linear operator ISO - -, minimal 934 - -, optimal 932 - form of parabola 205 - intercept 552 - operator 764 - plane 560 - section 568 - system of differential equations 439 - vector to plane curve 852 - - - surface 563 Normed algebra 161 - connection plan 139 - space 146 - -, completion of 148 Norms, equivalent 175 Null 215 - matrix 135 - relation 510 - sequence 222 - space 129 - - of operator 156 - vector 128;516 Nullity of linear map 146 Numerator 216 Numerical algorithm 673 _ harmonic analysis 579 - mathematics 613 - methods 366, 675 - quatrature 701 - transformation to principal axes 681 Nystrom approximation method 844

Obelisk 119 Object function 636 - programme 732 Oblique section 569 Octant 195 One-sheeted domain 495 - sided surface 315 .- to-one correspondence 513 Open ball 743 - interval 220 _ neighbourhood 477 - point set 220,743 Operand 732 Operation 515

966 Index

Operation code 732 - system 730, 134 Operational method for differential equations 419 - research 857 Operator 155 -, additive 755 -, adjoint 768 -. d' Alembert 443 -, antitone 781 -, bounded 756 -, closed 767 -, compact 112 -, completely continuous 771 -, continuous 7SS -, contractive 178 -, definite, positive or negative 770 -, domain of 755 -, dual 767 -, finite-dimensional 772, 782 -, Fredholm 172 -, Hilbert-Schmidt kernel of 751 -, homogeneous 755 -, identity 149 -, inverse 149, 756 -, isometric 769 -, isotone 781 -, kernel of 751 -, Laplace 259, 443 -, linear 146, 755 -, -, eigenvalue and eigenvector of 151 -, -, normal form of 150 -, maximal 769 -, negative 770 -, Noetherian 772 -, norm of 756 -, negative 770 -, normal 768 -, - soluble 772, 775 -, null space of 756 -, positive 770 -, pseudo-inverse 172 -, range of 755 -, regular 147 -, self-adjoint 768 -, semi-bounded 769 -, sequence 762 -, singular 147 -, symmetric 168 -, unitary 769 - of wave propagation 443 Optimal control 373 - normal form 932 - point 637 - policy 883 - position 262 - process 364, 372 - step length 874 - trajectory 372, 373 - value function 669 Optimality principle, Bellman's 373, 884 Optimization, Boolean 857 -, Box-Witson 944 -, combinatorial 902 -, condition, Legendre's 354 -, convex 863, 868 -, discrete 902 -, dynamic 882 -, free 811 -, integral linear 857 -, linear 636 -, -, applications 665 -, -, parametric 668 -, - quotient 863

Optimization, non-linear 863 -, quadratic 863 -, statistical 941 Order, axioms of 216 - of differential equation 385 -, inductive 513 -, linear 512 - of magnitude 233 -, monotonicity of 216 -, partial 512 - of pole 233 - - system of differential equations 386 -, total 512 -, transitivity of 216 - of zero 233 Ordered pair 508 Ordinary differential equation 385 Ordinate 192, 225 Orientation of surface 565 Oriented surface 532 Orthocentre 173 Orthogonal complement 134, 154 - elements, mutually 751 - family of curves 498 - matrix 141 - projection 134,753,770 - subspaces 134, 770 - sum 751 - system 134 - vectors 134,517 Orthogonality condition 357 Orthogonalizing (orthonormalizing) process of

Schmidt 134,751,849 Orthonormal basis 134 - system 134, 751, 854 Oscillation 450 Osculating plane 560 Output 673 - alphabet 913 - device 730

Pappus' rules 290 Parabola 205 -, cubical 62 -, Neill's 76 - of order n 63 -, semicubical 76 Parabolic arc 206 - differential equation 445, 447, 449, 469 - segment 206 Paraboloid 213 Parallel 177 - coordinate system 191, 195 Parallelepiped 178 Parallelogram 174 Parameter of conic 201, 203, 205 - integral 299 - -, improper 301 - -, multidimensional 325 Parameters, concentrated or distributed 380 Parametric optimization 668 - representation, problems in 359 - - of surface 534 Parseval's equality 754 Partial algebra 515 - automaton 913 - derivatin 249 - differential equation 385, 427 - - -, first order 431 - - -, second order 442 - fractions 158, 271 - -, tables :n - -, integration of 272

Partial operation SIS - order SI2 - sum of infinite series 328 Partition of set 511 Pascal's hma~on 78 - triangle 93 Path 891 -, critical 893 -, shortest 896 Pay-off function 898 Peano, theorem of 387, 745 Peano-Jordan content 80 I Penalty function 881 Periodic function 235 Periodicity of trigonomeiric functions 162 Peripheral device 731 - storage unit 731 Permanent summation method 765 Permutation 96 -, cycle of 98 -, parity (odd or even) 97 Permutations, group of 97 Perpendicular 134 - bisector 173 PERT 895 Perturbation function 399, 402, 826 Phase coordinates 372 - shift 166 - space 372 - vector 372 Physical coordinates 814 Picard-Lindelijf iteration method 779 Picard's method of successive approximation 398,

836 - iheorem 492 Piecewise continuous function 233 Pivot 682 - element 643, 676 - row and column 643 Pivotization 676 Planar graph 892 Plane 208 -, complex or Gaussian 473 -, coordinate systems in 191 - curves 5~1 - geometry 173 - net 72S -, normal, rectifying, osculating 560 - in space 173 - trigonometry 183 - vector field 525 Planned simulation 936 Poincarc's inequality 777 - theorem 820 Point of accumulation 220 -, boundary 221 -, conjugate 354 - coordinates 518 - - direction equation 199 - estimator 623 -, exterior 221 -, feasible b36 -, finite 476 - at infinity 476 - of innection 247, 554 -, isolated 220, 556 -, regular 408, 484 -, singular 484 -, -, weakly 408 - sequence 224 - set 219 - -, bounded 220 - -, closed 220, 743

Point set, compact 222 - -, connected 221 - -, convex 222 - -, open 220, 743 - spectrum 773 Points, chosen, method of 729 Poisson formula 457 - bracket 441

Index 967

- differential equation 365,443,467 - distribution 60S - -, table 8 - formula 457, 468 - 'integral 470 Polar coordinates 193, 195,255,474 - equation of conic 201 Pole 193 - of complex function 418, 492 - - rational function 156 - - resolvent 785 Polya's theorem 764 Polygon 175 Polygonal arc, Euler-CauchY 709 Polyhedral set 639 Polyhedron 118 -, convex 639 -, regular 180 , Polynomial Ill, 153, 227, 419 -, graphs 61 -, reducible and irreducible 154 -, trigonometric 571 Polynomials, Chebyshev 752 -, Hermite 753 -, Jacobi 752 -, Laguerre 753 -, Legendre 415, 752 Pontryagin's maximum principle 364, 373, 374 Positional system 86 Positive definite operator 170 - - quadratic form 260 - number 216 Potential, retarded 459 Power of complex number 474, 482 - function 15S - method 683 - series 338, 489 _ -, expansions, tables 22 - -, continuation by 490 - -, identity theorem 339, 490 - -, radius of convergence 338; 489 _ -, transformation of 490 - of set 506 - set 506 Predecessor vertex 891 Predicate 50 I , 504 Predicative expression 504 Predicior-corrector procedure 712 Preorder 511 Prime conjunction 932 Prilnitive of function 486 Primordial object 505 Principal axes, transformation to 152, 209, 547 - -, - -, numerical 681 - branch of root 496 - connecting symbol 102 - curvature 566 - direction 568 - normal 560 - part of Laurent series 491 - radii of curvature S67 - value of complex number 470 - - of integral, Cauchy 283 Principle of direct attack 681, 718 -, Duhamel's 458

968 Index

Principle of excluded middle 501 - - extensionality 501, 505 -, Hamilton's 357 - of mathematical induction 215 - - reflection, Schwarz's 495 Prism 178 Problem of brachystochrone 350, 354 -, Dirichlet's 447, 467, 468, 796, 797 -, isoperirnetric 360 -, Neumann's 447, 468, 469, 797 Problems of Mayer and Bolza 362 - in parameter representation 359 Probable time, most 895 Probabilities, multiplication rule 601 Probability, conditional 601 -, function 603 -, theorem on total 602 - theory 598 - -, axioms of 600 Procedure, see also method -, Adams-Bashforth 711 -, Adams-Moulton 711 -, difference 715 -, gradient 873, 874 -, Hildreth-d'Esopo 866 -, Henrici-Milne 711 -, iteration 867 -, Lemke 868 -, Nystrom 711 -, predictor·corrector 712 Process, critical 893 -, N-step 382 -, optimal 372, 373 Processor 730 Product, Cartesian 508 -, dyadic 546 - of events 599 - - functions 226 -, generalized 515 - index 101 -, infinite 347 -, -. absolute convergence 349 - of infinite series 334 - kernel 831 - matrix 139 - of relations 510 - rule for differentiation 243 - -, Leibniz' 244 Programme 731 -, blocks of 733 - library 734 -, modules of 733 -, running in of 735 Programming 732 - language 733 - system 733 Projected gradients, method of 384, 877 Projection, orthogonal 134, 770 -, stereographic 477 - theorem 183, 770 Proper subset 505 Propositional expression 501,503 - logic 501 Pseudotensor 813 Purely imaginary number 471 Pyramid 178 -, truncated 179 Pythagoras, theorem of 174

Quadrant 192 Quadratic equation 119 - form 152, 260 - -, transformation to principal axes 152

Quadratic function 61, 155, 500 - interpolation 872 - optimization 864 quadrature, numerical 70 I, 764 quadrilateral 174 quantifier, existential and universal 504 -, restricted 505 Quartic curves 77 - equation 121 Quasi-order 511 Quotient of functions 226 - rule for differentiation 243

Radicals, solution by 118 Radius of circumscribed circle 184, 189 - - convergence 338,489 - - curvature 554, 560 -, Fredholm 183 - of inscribed circle 124, 189 -, spectral 773 - vector 518 Radon-Nikodyn theorem 806 Random event 599 - -, independent 602 - experiment 598 - variable 603 - -, absolutely continuous 606 - -, correlated 614 - -, discrete 603 - vector 610 - -; absolutely continuous 611 - -, discrete 611 Range of correspondence 513 - - function 225, 236

operator 755 - - quantifier 504 ~ - relation 510 - - values of complex function 477 Rank of linear map 146 - - matrix 137 Rational function 153, 155,227,479 - -, integration of 270 - number 218 Real axis 473 - line 217 - numbers, axioms for 215 - part 471 Real-valued function 225, 236 Realization of automaton 920 - - numerical model 718 'Reciprocal basis 520, 812 Rectangle 174 Rectangular coordinate system 102,194,518 - distribution 606 - hyperbola 204 Rectifying plane 560 Reducible polynomial 154 Redundant constraint 630 Reflection, Schwarz' principle of 495 Reflexive Banach space 762 Reflexivity 109,511 - of ineq uality 114 Region 221 Register 732 Regression coefficients 633 - curves and lines 616 - quantities 6 I 5 Regula falsi 90 Regular event 918 - function 484 - linear map 147 - matrix 157 - point 408, 484

Regular point of Laurent series 491 - polygon 175 - polyhedron 180 - term918 Relation 510 Relative address 733 - compactness 744 - error 88 - extremum 246,259, 351 - frequency 599 Relaxation method 366 - procedure 902 Reliable digits 88 Remainder of Taylor expansion, Lagrange's form

245, 254 Remes algorithm 693 Removable discontinuity 232 - singularity 491 Renaming 505 Replacement 108, 110 Residue 492 - of resolvent 785 - theorem 493 Resolvent kernel 837 -, pole of 785 -, residue of 785 - set 773 Restriction of function 514 Ricci's lemma 816 Ridge line analysis, Hoeri's 944 Riemann content 801 - curvature tensor 569 - double integral 305 - integrable function, one variable 263 - - -, two variables 305 - integral 263, 805 - mapping theorem 498 - measurable 802 - sphere 476 Riemann-Stieltjes integral 787 Riemann surface 495, 496, 497 Riemann's criterion for integrability 263 - function 455 - integration method 454 - theorem on series 333 Riesz-Fischer theorem 754 F. Riesz's. theorem 760 Right-hand continuous 240

derivative 241 - - differentiable 240 - - limit 228 - - system 518 Right range of relation 510 - unique correspondence S13 Ritz's method 368 Romberg integration 705 Rotation 549 - of coordinate system 193, 196 - of dyad 549 -, infinitesimal 545 - of vector field 537 Rounding up and down, rules for 87 Routh-Hurwitz criterion 419 Row matrix 135 - sum criterion 779 Runge-Kutta procedures 710 Runge's principle 710

Saddle point 396, 869 Sample 620 - function 622 - mean 622 Sarrus' rule 136

Scalar 127,516,809 - field 524 - flux 533

Index 969

- product 133, 518, 751,767,848 - - space 751 - triple product 519 Schlicht domain 495 Schmidt orthogonalizing process 134,751,849,

851, 854 - series 854 Schwarz-Christoffel integral 499 - inequality 133,751,805,849 - reflection principle 495 Secant of circle 175 - function 69, 162 - method 685 Second fundamental form 566 - mean value theorem of integral calculus 265 - variation 350 Section, oblique 569 Sector of circle 176 Secular equation 681 Segment of circle 176 Selection 99 Self-adjoint boundary-value problem 423 - - differential expression 423, 455 - - operator 768 Semi bounded operator 769 Semicontinuous function 236 Separation of variables 388, 429 Sequence, arithmetic 102 -, bounded 222 -, Cauchy 742 - of complex numbers 487 -, convergent 222, 742 -, divergent 222, 488,514 -, finite 100,514 -, fundamental 742 -, geometric 102 -, infinite 514 -, limit of 222 -, monotonic 222 -, null 222 -, sum of 514 -, term of 514 Sequential product 514 - switching 921 - word function 915 Series, alternating 332 - of complex numbers 487 -, finite geometric 102, 474 -, generalized sum of 764 -, geometric 489 -, harmonic 329 -, infinite, convergent and divergent 328 _, -, conditionally convergent 330 -, Laurent 491 -, McLaurin 245 -, Neumann 837 Series-parallel switch work 929 Series, Schmidt 854 -, summable 764 -, sums of, table 21 -, Taylor 245, 342, 491 Servicing model 939 Sesquilinear form 761 Set of attainability 623 -, bounded 744 -, -, totally 744 -, closed 743 -, compact 744 -, -, relatively 744 -, convex 222

970 Index

Set, countable 219,516 -, finite 219,515 - function 801 Set-power 514 Set, polyhedral 629 -, uncountable 219, 516 St.ts, equipotent 515 -, equivalent 515 -, family or system of 509,514 Sharing-out problem 888 Shortest path 896 Side cosine theorem 188 a-additive set function 802 ... -finite set function 802 Similar triangles 174 Simplex algorithm 642 - -, dual 651 - -, revised 652 - -,-,dual 655 - multiplier 662 - procedure 865 - step 643 - tableau 640 Simply-connected domain 221 Simpson's rule 102 Simulation 936 Simultaneous substitution 108, 110 SIne function 68,161,165,482 - -, hyperbolic 74, 169, 482 - theorem 183, 188 Single-valued correspondence 513 Singular figure of quadric 209 - integral 394 - linear map 147 - line element 394 - matrix 137 - operator 147 - point 395, 484, 555, 556, 551 - -, isolated 484 - - on surface 564 - solution 438 Singularity, removable 491 -, essential 492 -, isolated 484 -, removable 491 Sink of graph 891 - - vector field 544 Skew lines 177 Slide rule 723 Sliding vector 517 Smooth curve 315 Sobolev embedding theorem 726 - spaces 774 Solenoidal vector field 543

.Solid angle 178 - of revolution, volume of 289 Soluble equation 116 Solubility theorem, general 257 Solution of differential equation 385 - - - -, general 386 - - - -, singular 386, 394,438 - - equation 116 - - linear equations 143 - - triangle, plane 183 - - -, spherical 188 Source of graph 891 - - vector field 544 Space, affine 198 -, complete 742 -, conjugate 762 -, coordinate system in 194 - curve 288, 560 -, dual 76?

Space, energy 797 -, Euclidean 741 -, line· in 207 -, linear 127, 745 -, metric 141 -, normed 746 -, plane in 208 -, scalar product 751 Spaces, isometric 742 -, Sobolev 724 -, unitary 751 Span, linear 129 Spanning set 129 Spectral family 787 - radius 773 Spectrum 773 -, essential 774 Sphere 181,211 Spherical coordinates 527 - excess 187, 190 - functions 412 - -, table 5 - harmonics 464 - sector 182 - segment 182 - triangle 187 - trigonometry 186 - zone 182 Spirals 83 Spline function 699 - interpolation 698 Stability 418 -, domain of 669 Standard basis of vector space 131 - integrals 266 Star-shaped domain 776 State of automaton 9 I3 - coordinates 372 - domain 885 - function 883 - vector 882 Stationary process 884 Statistical design of experiments 941 - optimization 941 Statistics; mathematical 620 Steering of electronic digital computer 733 Steinitz' exchange lemma 131 Steklov's theorem 764 Stepwise approximation 836 Stereographic projection 477 Stieltjes integral 759 Stirling's formula 92 Stochastic automaton 922 - dynamical model 889 Stokes' theorem 323, 543, 825 Storage problem 887 - unit 730 - -, peripheral 731 Strain tensor 545, 549 Stretch tensor 545 Stretched string 457 Strict inclusion 505 Strictly convex 246 - monotone 223, 234, 245 Strip manifold 434 Strong convergence 762 - extremum 352 - law of large numbers 618 Student's I-distribution 62; - - -, table 13 Sturm boundary condition 423 - chain 124 Sturm-LiouviUe operator 460

Sturm-Liouville problem 426 Sturm, theorem of 124 Subnormal 552 Subroutine 733 Subsequence 223 Subset 505 Subspace 129, 198 -, complementary 110 Substitution 110,258 - in inequality 115 -, integration by 269, 287 -, simultaneous 108 Subtangent 552 Successor vertex 891 Sum 215 - of events 598 - - functions 226 Summable series 764 Summand 215 Summation convention, Einstein's 562, ·808 - index 101 - method 764 - -, Cesaro's 765 - -, permanent 765 Sums of series, tables 21 Superior limit 223 Superposition of solutions 430 Supervisor 734 Support of correspondence 573 Supremum 219, 227, 237, 513 Surface, area of 315 -, characteristic 443 -, curvature of, Gaussian 567 -, - -, mean 567 -, - -, principal 566 -, - -, total 571 -, developable 568 - element 308 -, first fundamental form of 564 -, generator of 214 -, geodesics on 570 - integrals 316, 533 - -, improper 323 -, metric properties of 564 -, moving trihedron 563 -, normal vector to 563 -, oblique section of 569 -, one-sided 315 -, orientation of coordinate system 565 -, oriented 532 -, Riemann 495, 496, 497 -, ruled 568 -, second fundamental form of 566 -, smooth 315 Surjective map 514 Surveying 185 Switch algebra 929 - function 929 Switching, sequential 921 Switch work analysis and synthesis 930 Synthesis of automaton 918 - - switch work 931 System,adjoint 380 - of algebraic equations 125 -, complete 747 - with concentrated or distributed parameters 380 - of differential equations 386, 404 -, digital, deterministic 910 -, discrete 382 -, dynamical 372 -, left-handed 518 -, programming 132 -, right-handed 518

System, slochastic 910 Symmetric difference 507 - dyad 541 - group 91 - matrix 141 - operator 168 Symmetry, 110,511

Table of values 225. 231 Tangent 240, 55 I - to conic 201, 203, 206 - formula 184 - function 68, 161, 482 - -. hyperbolic 14, 169, 482 - intercept 552 - plane 251 , 563 - theorem 183 - vector 552, 560 Taylor expansion 245, 254 - series 245, 254, 342, 343, 491 - theorem 245 I-distribution, Student's 623 - -, table 13 tensor 545 -, absolute differential of 816 -, alternating product 811 -, contraction of 81 0 -, contra variant 809 -, covariant 809 -, density 813 -, deformation 549 -, dual complement of 813 - field 562. 814 - -. contravariant 562 - -. covariant 562 - -, curl of 816 - -, divergence of 816 - function 812 -. inner multiplication of 811 -, Levi-Civita 812 -, metric 812. 815 -, skew-symmetric 811 - strain 549 - surface 541 -, symmetric 811 -, trace of 810 Term 116,.918 -, regular 918 - of sequence 514 -, value of919 term-by-term-differentiation 337 - - - - of power series 340 - - - integration 336 Test, s"" also criterion

Index 971

- for convergence. Abel's 332, 336 - - -, d' Alembert's 330 - - -, Cauchy's 223, 225. - - -, Dirichlet's 332. 336 - - -, Dirichlet-Jordan 581 - - -, Gauss's 331 - - -, integral 331 - of fit, Kolmogorov-Smirnov 633 - - -, chi-square 631 Testing of hypotheses 628 Tetrahedron 179 -, volume of 207 Theorem, Abel's 340 - of Apollonius 202 -, Arzela's 850 -, Arzela-Ascoli 337, 144 -, Baire's 743 -, Banach-Steinhaus 163 -, Bernoulli's 617

972 Index

Theorem, binomial 94 -, du Bois-Reymond's 353 -, Bolzano's 234 -, Bolzano-Weierstrass 220 -, Cauchy's 386 -, Cauchy's integral 486 -, Cauchy-Kovalevskii 429 -, Chebyshev's 617 -, Dirichlet's 572 - of Eberlein-Shmul'yan 766 -, Euler's 568 - of Farkas 869 -, Fubini's 805 -, Gauss's 541, 825, 826 -, Gauss-Bonnet 571 - of Glivenko 621 -, Haar's 362 -, Hahn-Banach 758 -, Hausdorff's 744 -, Heine-Borel 222 -, intermediate value 234 -, Jacobi's 439 -, Kolmogorov's 618 -, Kuhn-Tucker 869 -, Lax-Milgram 761 -, Lebesgue's 750, 806 -, Legendre's 190 -, Leibniz' 332 - of B. Levi 806 -, Lindelof's 386 -, Liouville's 484 - on maximum-minimum principle 469 -, Mazur's 766 -, mean-value, of differential calculus 244 -, mean-value, of integral calculus 265 -, Meusnier's 569 - of de Moivre-Laplace 618 -, monodromy 495 -, Morera's 486 -, multinomial 95 -, Peano's 387, 745 -, Picard's 386,492 -, Poincare's 820 -, Polya's 764 -, Pythagoras' 174 -, Radon-Nikodyn 806 -, residue 493 -, Riemann's 333 -, - mapping 498 - of F. Riesz 760 -, Riesz-Fischer 754 -, Schauder's 772 -, Steklov's 764 -, Stokes 543, 825 -, Sturm's 124 -, Szego's 764 -, Taylor's 245 -, Toeplitz' 765 - on total probability 602 -, Vieta's 121, 123 -, Weierstrass' 234 Theorema egregium of Gauss 569 Theoretical regression quantities 615 Theory of games 898 Theory, Hamilton-Jacobi 363 Time-optimal problem 378 - sharing 733 Torus 182 . Total curvature 571 - differential 25 I, 296, 297 - order 512 - probability 602

. - variation 735, 750

Totally additive set function 801 - bounded set 744 - differentiable function 25 I - independent random events 602 Tractrix 85 Trajectory 372 -, optimal 373 Transcendental functions 68, 161,479 - -, table of integrals 50 - equation 118, 126 Transfer channel 731 Transformation of coordinate system 132, 193,

196 - to principal axes 152,209, 681 Translation 193, 196 Transitivity 110, 115, 511 Transportation problem 658, 903 - -, algorithm 662 Transposed matrix 136 Transversality condition 357, 375 Trapezium 174 Travelling salesman problem 903 Tree 892 Trial 598 Triangle 173 -, spherical 187 - inequality 113, 133, 187,218,472,741,746 Triangular matrix 135 Trichotomy 511 Tricomi's equation 445 Trigonometric equations 127 - functions 68,161,482 - -, integrals 42, 57 - -, power series 23 - polynomial 571 Trigonometry, plane 183 -, spherical 186 Trihedron, moving 560, 563 Triple integral 310,312 Trochoids 80 Truncation error 87 Truth value 501 - function 503 I-test 629 Turing machine 924 - computable 926 - decidable 926 Two-body problem 440 Two-point equation 199 - - boundary-value problem 383 Two-sided surface 315 Type of an algebra 515

Umbilic 568 Unbiased estimator 623 Unbounded interval 220 Uncountable set 219,516 Uniform convergence 301, 326, 334, 762 - continuity 235, 239 Uniformly bounded set of functions 337 Unimodal871 Union of sets 506, 509 Unit vector 133, 516 Unitary matrix 141 - operator 769 - space 751 Uniqueness theorem for analytic functions 484 Universal algebra 515 - equivalence 110 - quantifier 504 - relation 510 - validity 109

Upper bound 219, 513 - limit 219, 223 - triangular matrix 135

Value of arithmetic expression 108 - - game 900 - - term 919 Variable, bound 504 -,dependent 225,477 -, fully free 504 -, independent 225. 477 -, random 603 Variability, domain of 116 Variance 608 -, empirical 622 Variation 235 -, finite 235, 750 -, first 350 -, function of bound~d 235, 750 - of parameter 390,401,406,408 -, second 350 -, total 750 Variational principle 357 Variations, calculus of 349 -, - -, inverse problem 364 -, - -, main lemma 352 Vector 127,516 -, algebra 516 -, adjoint 383 - calculus 522 -, contra variant 809 -, coordinates of 132 -, covariant 809 - field 525 - -, axially symmetric 526 - -, centrally symmetric 526 - -, circulation of 537 - -, conservative 531 - -, coordinates of, contravariant 814 - -, - -, covariant 815 - -,- -,phy~c~ 814 - -, divergence of 535, 537 - -, irrotational 543 - -, Laplace operator of 538 - -, plane 525 - -, rotation of 537 - -, rotation Hnes 537 - -, sink of 544 - -, solenoidal 543 - -, source of 544 - function 522 - -, adjoint 374 - -, linear 545 - gradient 539, 548 - iteration 683 -, magnitude of 516 -,norm of 133, 746 -, null 516 _. product 518 - -, multiple 519 - space 127,146 - -, basis of 132 - -, dimension of 132

Vector space, Euclidean 133 - -, finitely generated 131 - -, normed 746 -, unit 133,516 Vectors, collinear 517 -, conjugate 873 -, coplanar 517 -, linear combination of 517

Index 973

-, - dependence and independence of 130, 517 -, orthogonal 517 Velocity vector 523 Vertex 111,201,203,205 - of admissible domain 639 - - graph 891 - - plane curve 555 Vibrating string 456 Vibration of clamped membrane 45 I -, equation of 442 Vieta, theorem of 121, 123 Volume derivative 535 - of solid 313,320 - - - of revolution 289 - - sphere 182 - - tetrahedron 207 Volterra integral equation 826 Vortex 397

Wave equation 443, 456, 451, 458 - -, inhomogeneous 458 - propagation operator 443 Weak convergence 762, 165 - .-convergence 763 - law of large numbers 617 - relative extremum 35 I - *-.. lative sequential compactness 163 - singular point 408 Wedge 179 Weber function 410 Weibull distribution 601 Weierstrass approximation theorem 693 - condition 355 - criterion for uniform convergence 335 Weierstrass-Erdmann corner condition 355 -, theorem of 234 Weingarten, derivative formulae of 569 Well-ordering 512 Well-posed problem 447 Whirl 397 Whole·step iteration 779 Wielandt's iteration 683 Wilcoxon test 630 - -, table 19 Witch of Agnesi 76 Wolfe's procedure 864 Word in alphabet 914 - function 914 - -, sequential 915 Work done by force 298 Wronskian determinant 400

Zero divisor 139 - of function 154 - - complex function 478 Zorn's lemma 513