Mathematics

A. N. Kolmogorov (1903-87) was a prominent Russian mathematician who made significant contributions to many areas of math, including probability theory, topology, logic, turbulence, and mechanics. In his many years on the faculty of Moscow State University, Dr. Kolmogorv's doctoral students included several who became prominent 20th-century mathematicians.

• Volume 1. Part 1Chapter 1. A general view of mathematics (A.D. Aleksandrov)1. The characteristic features of mathematics2. Arithmetic3. Geometry4. Arithmetic and geometry5. The age of elementary mathematics6. Mathematics of variable magnitudes7. Contemporary mathematicsSuggested readingChapter 2. Analysis (M.A. Lavrent'ev and S.M. Nikol'skii)1. Introduction2. Function3. Limits4. Continuous functions5. Derivative6. Rules for differentiation7. Maximum and minimum; investigation of the graphs of functions8. Increment and differential of a function9. Taylor's formula10. Integral11. Indefinite integrals; the technique of integration12. Functions of several variables13. Generalizations of the concept of integral14. SeriesSuggested readingPart 2.Chapter 3. Analytic Geometry (B. N. Delone)1. Introduction2. Descartes' two fundamental concepts3. Elementary problems4. Discussion of curves represented by first- and second-degree equations5. Descartes' method of solving third- and fourth-degree algebraic equations6. Newton's general theory of diameters7. Ellipse, hyperbola, and parabola8. The reduction of the general second-degree equation to canonical form9. The representation of forces, velocities, and accelerations by triples of numbers; theory of vectors10. Analytic geometry in space; equations of a surface in space and equations of a curve11. Affine and orthogonal transformations12. Theory of invariants13. Projective geometry14. Lorentz transformationsConclusions; Suggested readingChapter 4. Algebra: Theory of algebraic equations (B. N. Delone)1. Introduction2. Algebraic solution of an equation3. The fundamental theorem of algebra4. Investigation of the distribution of the roots of a polynomial on the complex plane5. Approximate calculation of rootsSuggested readingChapter 5. Ordinary differential equations (I. G. Petrovskii)1. Introduction2. Linear differential equations with constant coefficients3. Some general remarks on the formation and solution of differential equations4. Geometric interpretation of the problem of integrating differential equations; generalization of the problem5. Existence and uniqueness of the solution of a differential equation; approximate solution of equations6. Singular points7. Qualitative theory of ordinary differential equationsSuggested readingVolume 2 Part 3Chapter 6. Partial differential equations (S. L. Sobolev and O. A. Ladyzenskaja)1. Introduction2. The simplest equations of mathematical physics3. Initial-value and boundary-value problems; uniqueness of a solution4. The propagation of waves5. Methods of constructing solutions6. Generalized solutionsSuggested readingChapter 7. Curves and surfaces (A. D. Aleksandrov)1. Topics and methods in the theory of curves and surfaces2. The theory of curves3. Basic concepts in the theory of surfaces4. Intrinsic geometry and deformation of surfaces5. New Developments in the theory of curves and surfacesSuggested readingChapter 8. The calculus of variations (V. I. Krylov)1. Introduction2. The differential equations of the calculus of variations3. Methods of approximate solution of problems in the calculus of variationsSuggested readingChapter 9. Functions of a complex variable (M. V. Keldys)1. Complex numbers and functions of a complex variable2. The connection between functions of a complex variable and the problems of mathematical physics3. The connection of functions of a complex variable with geometry4. The line integral; Cauchy's formula and its corollaries5. Uniqueness properties and analytic continuation6. ConclusionSuggested readingPart 4.Chapter 10. Prime numbers (K. K. Mardzanisvili and A. B. Postnikov)1. The study of the theory of numbers2. The investigation of problems concerning prime numbers3. Chebyshev's method4. Vinogradov's method5. Decomposition of integers into the sum of two squares; complex integersSuggested readingChapter 11. The theory of probability (A. N. Kolmogorov)1. The laws of probability2. The axioms and basic formulas of the elementary theory of probability3. The law of large numbers and limit theorems4. Further remarks on the basic concepts of the theory of probability5. Deterministic and random processes6. Random processes of Markov typeSuggested readingChapter 12. Approximations of functions (S. M. Nikol'skii)1. Introduction2. Interpolation polynomials3. Approximation of definite integrals4. The Chebyshev concept of best uniform approximation5. The Chebyshev polynomials deviating least from zero6. The theorem of Weierstrass; the best approximation to a function as related to its properties of differentiability7. Fourier series8. Approximation in the sense of the mean squareSuggested readingChapter 13. Approximation methods and computing techniques (V. I. Krylov)1. Approximation and numerical methods2. The simplest auxiliary means of computationSuggested readingChapter 14. Electronic computing machines (S. A. Lebedev and L. V. Kantorovich)1. Purposes and basic principles of the operation of electronic computers2. Programming and coding for high-speed electronic machines3. Technical principles of the various units of a high-speed computing machine4. Prospects for the development and use of electronic computing machinesSuggested readingVolume 3. Part 5.Chapter 15. Theory of functions of a real variable (S. B. Stechkin)1. Introduction2. Sets3. Real Numbers4. Point sets5. Measure of sets6. The Lebesque integralSuggested readingChapter 16. Linear algebra (D. K. Faddeev)1. The scope of linear algebra and its apparatus2. Linear spaces3. Systems of linear equations4. Linear transformations5. Quadratic forms6. Functions of matrices and some of their applicationsSuggested readingChapter 17. Non-Euclidean geometry (A. D. Aleksandrov)1. History of Euclid's postulate2. The solution of Lobachevskii3. Lobachevskii geometry4. The real meaning of Lobachevskii geometry5. The axioms of geometry; their verification in the present case6. Separation of independent geometric theories from Euclidean geometry7. Many-dimensional spaces8. Generalization of the scope of geometry9. Riemannian geometry10. Abstract geometry and the real spaceSuggested readingPart 6.Chapter 18. Topology (P. S. Aleksandrov)1. The object of topology2. Surfaces3. Manifolds4. The combinatorial method5. Vector fields6. The development of topology7. Metric and topological spaceSuggested readingChapter 19. Functional analysis (I. M. Gelfand)1. n-dimensional space2. Hilbert space (Infinite-dimensional space)<4. Integral equations5. Linear operators and further developments of functional analysisSuggested readingChapter 20. Groups and other algebraic systems (A. I. Malcev)1. Introduction2. Symmetry and transformations3. Groups of transformations4. Fedorov groups (crystallographic groups)5. Galois groups6. Fundamental concepts of the general theory of groups7. Continuous groups8. Fundamental groups9. Representations and characters of groups10. The general theory of groups11. Hypercomplex numbers12. Associative algebras13. Lie algebras14. Rings15. Lattices16. Other algebraic systemsSuggested readingIndex