Volume 1. Part 1 Chapter 1. A general view of mathematics (A.D. Aleksandrov) 1. The characteristic features of mathematics 2. Arithmetic 3. Geometry 4. Arithmetic and geometry 5. The age of elementary mathematics 6. Mathematics of variable magnitudes 7. Contemporary mathematics Suggested reading Chapter 2. Analysis (M.A. Lavrent'ev and S.M. Nikol'skii) 1. Introduction 2. Function 3. Limits 4. Continuous functions 5. Derivative 6. Rules for differentiation 7. Maximum and minimum; investigation of the graphs of functions 8. Increment and differential of a function 9. Taylor's formula 10. Integral 11. Indefinite integrals; the technique of integration 12. Functions of several variables 13. Generalizations of the concept of integral 14. Series Suggested reading Part 2. Chapter 3. Analytic Geometry (B. N. Delone) 1. Introduction 2. Descartes' two fundamental concepts 3. Elementary problems 4. Discussion of curves represented by first- and second-degree equations 5. Descartes' method of solving third- and fourth-degree algebraic equations 6. Newton's general theory of diameters 7. Ellipse, hyperbola, and parabola 8. The reduction of the general second-degree equation to canonical form 9. The representation of forces, velocities, and accelerations by triples of numbers; theory of vectors 10. Analytic geometry in space; equations of a surface in space and equations of a curve 11. Affine and orthogonal transformations 12. Theory of invariants 13. Projective geometry 14. Lorentz transformations Conclusions; Suggested reading Chapter 4. Algebra: Theory of algebraic equations (B. N. Delone) 1. Introduction 2. Algebraic solution of an equation 3. The fundamental theorem of algebra 4. Investigation of the distribution of the roots of a polynomial on the complex plane 5. Approximate calculation of roots Suggested reading Chapter 5. Ordinary differential equations (I. G. Petrovskii) 1. Introduction 2. Linear differential equations with constant coefficients 3. Some general remarks on the formation and solution of differential equations 4. Geometric interpretation of the problem of integrating differential equations; generalization of the problem 5. Existence and uniqueness of the solution of a differential equation; approximate solution of equations 6. Singular points 7. Qualitative theory of ordinary differential equations Suggested re Volume 2 Part 3 Chapter 6. Partial differential equations (S. L. Sobolev and O. A. Ladyzenskaja) 1. Introduction 2. The simplest equations of mathematical physics 3. Initial-value and boundary-value problems; uniqueness of a solution 4. The propagation of waves 5. Methods of constructing solutions 6. Generalized solutions Suggested reading Chapter 7. Curves and surfaces (A. D. Aleksandrov) 1. Topics and methods in the theory of curves and surfaces 2. The theory of curves 3. Basic concepts in the theory of surfaces 4. Intrinsic geometry and deformation of surfaces 5. New Developments in the theory of curves and surfaces Suggested reading Chapter 8. The calculus of variations (V. I. Krylov) 1. Introduction 2. The differential equations of the calculus of variations 3. Methods of approximate solution of problems in the calculus of variations Suggested reading Chapter 9. Functions of a complex variable (M. V. Keldys) 1. Complex numbers and functions of a complex variable 2. The connection between functions of a complex variable and the problems of mathematical physics 3. The connection of functions of a complex variable with geometry 4. The line integral; Cauchy's formula and its corollaries 5. Uniqueness properties and analytic continuation 6. Conclusion Suggested reading Part 4. Chapter 10. Prime numbers (K. K. Mardzanisvili and A. B. Postnikov) 1. The study of the theory of numbers 2. The investigation of problems concerning prime numbers 3. Chebyshev's method 4. Vinogradov's method 5. Decomposition of integers into the sum of two squares; complex integers Suggested reading Chapter 11. The theory of probability (A. N. Kolmogorov) 1. The laws of probability 2. The axioms and basic formulas of the elementary theory of probability 3. The law of large numbers and limit theorems 4. Further remarks on the basic concepts of the theory of probability 5. Deterministic and random processes 6. Random processes of Markov type Suggested reading Chapter 12. Approximations of functions (S. M. Nikol'skii) 1. Introduction 2. Interpolation polynomials 3. Approximation of definite integrals 4. The Chebyshev concept of best uniform approximation 5. The Chebyshev polynomials deviating least from zero 6. The theorem of Weierstrass; the best approximation to a function as related to its properties of differentiability 7. Fourier series 8. Approximation in the sense of the mean square Suggested reading Chapter 13. Approximation methods and computing techniques (V. I. Kr 1. Approximation and numerical methods 2. The simplest auxiliary means of computation Suggested reading Chapter 14. Electronic computing machines (S. A. Lebedev and L. V. Kantorovich) 1. Purposes and basic principles of the operation of electronic computers 2. Programming and coding for high-speed electronic machines 3. Technical principles of the various units of a high-speed computing machine 4. Prospects for the development and use of electronic computing machines Suggested reading Volume 3. Part 5. Chapter 15. Theory of functions of a real variable (S. B. Stechkin) 1. Introduction 2. Sets 3. Real Numbers 4. Point sets 5. Measure of sets 6. The Lebesque integral Suggested reading Chapter 16. Linear algebra (D. K. Faddeev) 1. The scope of linear algebra and its apparatus 2. Linear spaces 3. Systems of linear equations 4. Linear transformations 5. Quadratic forms 6. Functions of matrices and some of their applications Suggested reading Chapter 17. Non-Euclidean geometry (A. D. Aleksandrov) 1. History of Euclid's postulate 2. The solution of Lobachevskii 3. Lobachevskii geometry 4. The real meaning of Lobachevskii geometry 5. The axioms of geometry; their verification in the present case 6. Separation of independent geometric theories from Euclidean geometry 7. Many-dimensional spaces 8. Generalization of the scope of geometry 9. Riemannian geometry 10. Abstract geometry and the real space Suggested reading Part 6. Chapter 18. Topology (P. S. Aleksandrov) 1. The object of topology 2. Surfaces 3. Manifolds 4. The combinatorial method 5. Vector fields 6. The development of topology 7. Metric and topological space Suggested reading Chapter 19. Functional analysis (I. M. Gelfand) 1. n-dimensional space 2. Hilbert space (Infinite-dimensional space)< 4. Integral equations 5. Linear operators and further developments of functional analysis Suggested reading Chapter 20. Groups and other algebraic systems (A. I. Malcev) 1. Introduction 2. Symmetry and transformations 3. Groups of transformations 4. Fedorov groups (crystallographic groups) 5. Galois groups 6. Fundamental concepts of the general theory of groups 7. Continuous groups 8. Fundamental groups 9. Representations and characters of g 10. The general theory of groups 11. Hypercomplex numbers 12. Associative algebras 13. Lie algebras 14. Rings 15. Lattices 16. Other algebraic systems Suggested reading Index