In this post we will see Selected Problems and Theorems in
Elementary Mathematics – Arithmetic and Algebra by D. O. Shklyarsky,  N. N. Chentsov and I. M. Yaglom. This is another book in the  Problems and Solutions series.

This book contains the conditions of problems, the answers and hints  to them and the solutions of the problems. The conditions of the most difficult problems are marked by stars.

We recommend the reader to start with trying to solve without
assistance the problem he is interested in. In case this attempt
fails he can read the hint or the answer to the problem, which may
facilitate the solution, Finally, if this does not help, the
solution of the problem given in the book should be studied. However, for the starred problems it may turn out to be appropriate to begin with reading the hints or the answers before proceeding to solve the problems.

Most of the problems in the book are independent of one another
except those in the last two sections (“Complex Numbers” and
“Several Problems in Number Theory”) where the problems are more  closely interrelated.

It is advisable to choose a definite section ot the book and to spend some time on solving the problems of that section. Only alter that (this does not of course mean that all the problems or most of the problems must necessarily be solved) should the reader pass to another section and so on. However, the order in which the sections are arranged in the book may not be followed. The solutions of some
problems include indications concerning possible generalisations of the conditions of the problems. The reader is advised to think of similar generalizations for other problems; it is also interesting to try to state new problems akin to those collected in this book.

This book was translated from the Russian by V. M. Volosov and
I. G. Volosova. The book was published by first Mir Publishers in
1978.

All credits to the original uploader.

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Table of Contents
Preface 5
Instructions 8
Problems 9
1. Introductory Problems (I-29) 9
2. Permutation of Digits in a Number (30-45) 16
3. Problems in Divisibility of Numbers (46-96) 17
4. Miscellaneous Problems in Arithmetic (97-149) 23
5. Finding Integral Solutions of Equations (150.171) 31
6. Matrices, Sequences and Functions (172-210) 84
7. Estimating Sums snd Products (211-234) 44
8. Miscellaneous Problems ln Algebra (235-291 ) 50
9. Algebra of Polynomia!s (292-320) 59
10. Complex Numbers (321-339) 63
11. Several Problems in Number Theory (340.350) 67
Solutions 71
Answers and Hints 412

In this post we will see A Problem Book in Algebra by   V. A. Krechmar.

This book contains 486 problems in various fields of algebra with
solutions for the problems.

This book was translated from the Russian by Victor Shiffer and the
translation was edited by Leonid Levant. The book was published by
first Mir Publishers in 1974 and reprinted in 1978.

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Table of Contents
1. Whole Rational Expressions 7
Solutions to Section 1 117
2. Rational Fractions 15
Solutions to Section 2 136
3. Radicals. Inverse Trigonometric Functions.Logarithms 28
Solutions to Section 3 174
4. Equations and Systems of Equations of the First Degree 40
Solutions to Section 4 208
5. Equations and Systems of Equations of the Second Degree 53
Solutions to Section 5 247
6. Complex Numbers and Polynomials 64
Solutions to Section 6 285
7. Progressions and Sums 83
Solutions to Section 7 361
8. Inequalities 93
Solutions to Section 8 396
9. Mathematical Induction 104
Solutions to Section 9 450
10. Limits 110
Solutions to Section 10 480

In this post we will see Mathematical Logic by Yu. L. Ershov,  E. A. Palyutin.

This book presents in a systematic way a number of topics in modern
mathematical logic and the theory of algorithms. It can be used as
both a text book on mathematical logic for university students and
a text for specialist courses. The sections corresponding to the
obligatory syllabus (Sections 1 to 9 of Chapter 1,without the small
type, Sections 10 and 11 of Chapter 2, Sections 15 and 16 of Chapter
3,Sections 18 to 20, 22 and 23 of Chapter 4 and Section 35of Chapter 7) are written more thoroughly and in more detail than the sectionsrelating to more special questions.
The exposition of the propositional calculus and the calculus of predicates is not a conventional one, beginning as it does with a study of sequential variants of the calculi of natural deduction( although the traditional calculi, referred to as Hilbertian ,also appears here). The reasons for this are:
A) the possibility of providing a good explanation of the meaning of all the rules of inference;
B) the possibility of acquiring more rapidly the knack of making formal proofs;
C) a practical opportunity of making all the formal proofs necessary in the course for these calculi.

This book was translated from the Russian by Vladimir Shokurov. The book was published by first Mir Publishers in 1984.

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Table of Contents

Preface 7
INTRODUCTION 9
Chapter 1.
THE PROPOSITIONAL CALCULUS 15
1. Sets and words 15
2. The language of the propositional calculus 21
3. Axiom system and rules of inference 25
4. The equivalence of formulas 32
5. Normal forms35
6. Semantics of the propositional calculus 43
7. Characterization of provable formula 48
8. Hilbertian propositional calculus 52
9. Conservative extension of calculi 56

Chapter 2.
SET THEORY 65

10. Predicates and mappings 65
11. Partially ordered sets 70
12. Filters of Boolean algebra 78
13.The power of a set 82
14.The axiom of choice 90

Chapter 3.
TRUTH ON ALGEBRAIC SYSTEMS 96

15. Algebraic systems96
16. Formulas of the signature $\Sigma$ 102
17. Compactness theorem110

Chapter 4.
THE CALCULUS OF PREDICATES 117
18. Axioms and rules of inference 117
19. The equivalence of formulas 126
20. Normal forms 130
21. Theorem on the existence of a model 132
22. Hilbertian calculus of predicates 139
23. Pure calculus of predicates 144

Chapter 5.
MODEL THEORY 149

24. Elementary equivalence 149
25. Axiomatizable classes 157
26. Skolem functions 165
27. Mechanism of compatibility 168
28. Countable homogeneity and universality 181
29. Categoricity 188

Chapter6.
PROOF THEORY 198

30. The Gentzen system G 198
31. The invertibility of rules 204
32. Comparison of the calculi $CP^\Sigma$ and G 210
33. Herbrand theorem 217
34. The calculi of resolvents 228

Chapter7.
ALGORITHMS AND RECURSIVE FUNCTIONS236

35. Normal algorithms and Turing machines 236
36. Recursive functions 247
37. Recursively enumerable predicates 264
38. Undecidability of the calculus of predicates and Godel’s incompleteness theorem 276
List of symbols 292
Subject Index  295

We now come to Problems in Mathematical Analysis edited by B. P. Demidovich. The list of authors is G. Baranenkov, B. Demidovich, V. Efimenko, S. Kogan, G. Lunts, E. Porshneva, E. Sychera, S. Frolov, R. Shostak and A.  Yanpolsky.

This collection of problems and exercises in mathematical analysis
covers the maximum requirements of general courses in higher
mathematics for higher technical schools. It contains over 3,000
problems sequentially arranged in Chapters I to X covering branches
of higher mathematics (with the exception of analytical geometry)
given in college courses. Particular attention is given to the most
important sections of the course that require established skills
(the finding of limits, differentiation techniques, the graphing of
functions, integration techniques, the applications all of definite
integrals, series, the solution of differential equations).

Since some institutes have extended courses of mathematics, the
authors have included problems on field theory, method, and the
Fourier approximate calculaiions. Experience shows that problems
given in this book not only fully satisfies the number of the
requirements of the student, as far as practical mastering of the
various sections of the course goes, but also enables the instructor
to supply a varied choice of problems in each section to select
problems for tests and examinations.

Each chapter begins with a brief theoretical introduction that
covers the basic definitions and formulas of that section of the
course. Here the most important typical problems are worked out in
full. We believe that this will greatly simplify the work of the
student. Answers are given to all computational problems; one
asterisk indicates that hints to the solution are given in the
answers, two asterisks, that the solution is given. The are
frequently illustrated by drawings.

This collection of problems is the result of many years of teaching
higher mathematics in the technical schools of the Soviet Union. It
includes, in addition to original problems and examples, a large
number of commonly used problems.

This book was translated from the Russian by George Yankovsky. The  book was published by first Mir Publishers in 19??.

All credits to the original uploader.

Thanks Siddharth for providing the link.

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Table of Contents

Preface 9

Chapter I
INTRODUCTION TO ANALYSIS 

Sec. 1. Functions 11
Sec. 2. Graphs of Elementary Functions 16
Sec. 3 Limits 22
Sec. 4 Infinitely Small and Large Quantities 33
Sec. 5. Continuity of Functions 36

Chapter II
DIFFERENTIATION OF FUNCTIONS

Sec. 1. Calculating Derivatives Directly 42
Sec. 2. Tabular Differentiation 46
Sec. 3 The Derivatives of Functions Not Represented Explicitly 56
Sec. 4. Geometrical and Mechanical Applications of the Derivative 60
Sec. 5. Derivatives of Higher Orders 66
Sec. 6. Differentials of First and Higher Orders 71
Sec. 7. Mean Value Theorems 75
Sec. 8. Taylor’s Formula 77
Sec. 9. The L’Hospital-Bernoulli Rule for Evaluating Indeterminate
Forms 78

Chapter III
THE EXTREMA OF A FUNCTION AND THE GEOMETRIC
APPLICATIONS OF A DERIVATIVE

Sec. 1. The Extrema of a Function of One Argument 83
Sec. 2. The Direction of Concavity. Points of Inflection 91
Sec. 3. Asymptotes 93
Sec. 4. Graphing Functions by Characteristic Points 96
Sec. 5. Differential of an Arc Curvature 101

Chapter IV
INDEFINITE INTEGRALS
Sec. 1. Direct Integration 107
Sec. 2. Integration by Substitution 113
Sec. 3. Integration by Parts 116
Sec. 4. Standard Integrals Containing a Quadratic Trinomial 118
Sec. 5. Integration of Rational Functions 121
Sec. 6. Integrating Certain Irrational Functions 125
Sec. 7. Integrating Trigoncrretric Functions 128
Sec. 8. Integration of Hyperbolic Functions 133
Sec. 9. Using Ingonometric and Hyperbolic Substitutions for
Finding integrals of the Form $\int R(x, \sqrt{ax^2 + bx + c}) dx$ R Where R
is a Rational Function
Sec. 10. Integration of Various Transcendental Functions 135
Sec. 11. Using Reduction Formulas 135
Sec. 12. Miscellaneous Examples on Integration 136

Chapter V
DEFINITE INTEGRALS

Sec. 1. The Definite Integral as the Limit of a Sum 138
Sec. 2. Evaluating Definite Integrals by Means of Indefinite Integrals 140
Sec. 3 Improper Integrals 143
Sec. 4. Change of Variable in a Definite Integral 146
Sec. 5. Integration by Parts 149
Sec. 6. Mean-Value Theorem 150
Sec. 7. The Areas of Plane Figures 153
Sec 8. The Arc Length of a Curve 158
Sec 9 Volumes of Solids 161
Sec 10 The Area of a Surface of Revolution 166
Sec. 11. Moments. Centres of Gravity. Guldin’s Theorems 168
Sec. 12. Applying Definite Integrals to the Solution of Physical
Problems 173

Chapter VI.
FUNCTIONS OF SEVERAL VARIABLES
Sec. 1. Basic Notions 180
Sec. 2. Continuity 184
Sec. 3. Partial Derivatives 185
Sec. 4. Total Differential of a Function 187
Sec. 5. Differentiation of Composite Functions 190
Sec. 6. Derivative in a Given Direction and the Gradient of a Function 193
Sec. 7. Higher -Order Derivatives and Differentials 197
Sec. 8. Integration of Total Differentials 202
Sec. 9. Differentiation of Implicit Functions 205
Sec. 10. Change of Variables 211
Sec. 11. The Tangent Plane and the Normal to a Surface 217
Sec. 12. Taylor’s Formula for a Function of Several Variables 220
Sec. 13. The Extremum of a Function of Several Variables 222
Sec. 14. Finding the Greatest and smallest Values of Functions 227
Sec. 15. Singular Points of Plane Curves 230
Sec. 16. Envelope 232
Sec. 17. Arc Length of a Space Curve 234
Sec. 18. The Vector Function of a Scalar Argument 235
Sec. 19. The Natural Trihedron of a Space Curve 238
Sec. 20. Curvature and Torsion of a Space Curve 242

Chapter VII.
MULTIPLE AND LINE INTEGRALS

Sec. 1. The Double Integral in Rectangular Coordinates 246
Sec. 2. Change of Variables in a Double Integral 252
Sec. 3. Computing Areas 256
Sec. 4. Computing Volumes 258
Sec. 5. Computing the Areas of Surfaces 259
Sec. 6 Applications of the Double Integral in Mechanics 260
Sec. 7. Triple Integrals 262
Sec. 8. Improper Integrals Dependent on a Parameter. Improper Multiple Integrals  269
Sec. 9. Line Integrals 273
Sec. 10. Surface Integrals 284
Sec. 11. The Ostrogradsky-Gauss Formula 286
Sec. 12. Fundamentals of Field Theory 288

Chapter VIII.
SERIES
Sec. 1. Number Series 293
Sec. 2. Functional Series 304
Sec. 3. Taylor’s Series 318
Sec. 4. Fourier’s Series 311

Chapter IX
DIFFERENTIAL EQUATIONS

Sec. 1. Verifying Solutions. Forming Differential Equations of Families of
Curves. Initial Conditions 322
Sec. 2. First-Order Differential Equations 324
Sec. 3. First-Order Diflerential Equations with Variables
Separable. Orthogonal Trajectories 327
Sec. 4. First-Order Homogeneous Differential Equations 330
Sec. 5. First-Order Linear Differential Equations. Bernoulli’s
Equation 332
Sec. 6 Exact Differential Equations. Integrating Factor 335
Sec 7 First-Order Differential Equations not Solved for the Derivative 337
Sec. 8. The Lagrange and Clairaut Equations 339
Sec. 9. Miscellaneous Exercises on First-Order Differential Equations 340
Sec. 10. Higher-Order Differential Equations 345
Sec. 11. Linear Differential Equations 349
Sec. 12. Linear Differential Equations of Second Order with Constant
Coefficients 351
Sec. 13. Linear Differential Equations of Order Higher than Two with
Constant Coefficients 356
Sec. 14. Euler’s Equations 357
Sec. 15. Systems of Differential Equations 359
Sec. 16. Integration of Differential Equations by Means of Power Series 361
Sec. 17. Problems on Fourier’s Method 363

Chapter X.
APPROXIMATE CALCULATIONS

Sec. 1. Operations on Approximate Numbers 367
Sec. 2. Interpolation of Functions 372
Sec. 3. Computing the Real Roots of Equations 376
Sec. 4. Numerical Integration of Functions 382
Sec. 5. Numerical Integration of Ordinary Differential Equations 384
Sec. 6. Approximating Fourier’s Coefficients 393

ANSWERS 396
APPENDIX 475
I. Greek Alphabet 475
II. Some Constants 475
III. Inverse Quantities, Powers, Roots, Logarithms 476
IV. Trigonometric Functions 478
V. Exponential, Hyperbolic and Trigonometric Functions 479
VI. Some Curves 480

In this post we see Higher Algebra by A. Kurosh.

The education of the mathematics major begins with the
study of three basic disciplines: mathematical analysis, analytic
geometry and higher algebra. These disciplines have a number of
points of contact, some of which overlap; together they constitute
the foundation upon which rests the whole edifice of modern
mathematical science.

Higher algebra – the subject of this text – is a far-reaching and
natural generalization of the basic school course of elementary
algebra. Central to elementary algebra is without doubt the problem
of solving equations. The study of equations begins with the very
simple case of one equation of the first degree in one unknown. From
there on, the development proceeds in two directions: to systems of
two and three equations of the first degree in two and, respectively,
three unknowns, and to a single quadratic equation in one unknown and
also to a few special types of higher-degree equations which readily
reduce to quadratic equations (quartic equations, for example). Both
trends are further developed in the course of higher algebra, thus
determining its two large areas of study. One – the foundations of
linear algebra – starts with the study of arbitrary systems of
equations of the first degree (linear equations). When the number of
equations equals the number of unknowns, solutions of such systems
are obtained by means of the theory of determinants.

The second half of the course of higher algebra, called the algebra
of polynomials, is devoted to the study of a single equation in one
unknown but of arbitrary degree. Since there is a formula for solving
quadratic equations, it was natural to seek similar formulas for
higher-degree equations. That is precisely how this division of
algebra developed historically. Formulas for solving equations of
third and fourth degree were found in the sixteenth century. The
search was then on for formulas capable of expressing the roots of
equations of fifth and higher degree in terms -of the coefficients of
the equations by means of radicals, even radicals within radicals. It
was futile, though it continued up to the beginning of the nineteenth
century, when it was proved that no such formulas exist and that for
all degrees beyond the fourth there even exist specific examples of
equations with integral coefficients whose roots cannot be written
down by means of radicals.

This book was translated from the Russian by George Yankovsky. The  book was published by first Mir Publishers in 1972, with reprints in  1975, 1980 and 1984. The book below is from the 1984 reprint.

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Table of Contents

Introduction 7
Chapter 1.
Systems of linear equations. Determinants 15

1. The Method of Successive Elimination of Unknowns 15
2. Determinants of Second and Third Order. 22
3. Arrangements and Permutations 27
4. Determinants of nth Order 36
5. Minors and Their Cofactors 43
6. Evaluating Determinants 46
7. Cramer’s Rule 53

Chapter 2.
Systems of linear equations ( general theory) 59

8. n-Dimensional Vector Space 59
9. Linear Dependence of Vectors 62
10. Rank of a Matrix 69
11. Systems of Linear Equations. 76
12. Systems of Homogeneous Linear Equations 82

Chapter 3.
The algebra of matrices 87

13. Matrix Multiplication 87
14. Inverse Matrices 93
15. Matrix Addition and Multiplication of a Matrix by a Scalar 99
16. An Axiomatic Construction of the Theory of Determinants 103

Chapter 4.
Complex numbers 110

17. The System of Complex Numbers 110
18. A Deeper Look at Complex Numbers 112
19. Taking Roots of Complex Numbers 120

Chapter 5.
Polynomials and their roots 126

20. Operations on Polynomials 126
21. Divisors. Greatest Common Divisor 131
22. Roots of Polynomials. 139
23. Fundamental Theorem 142
24. Corollaries to the Fundamental Theorem 151
25. Rational Fractions 156

Chapter 6.
Quadratic forms 161

26. Reducing a Quadratic Form to Canonical Form 161
27. Law of Inertia. 169
28. Positive Definite Forms 174

Chapter 7
Linear spaces 178

29. Definition of a Linear Space. An Isomorphism 178
30. Finite-Dimensional Spaces. Bases 182
31. Linear Transformations 188
32. Linear Subspaces. 195
33. Characteristic Roots and Eigenvalues 199

Chapter 8
Euclidean spaces204
34. Definition of a Euclidean Space. Orthonormal Bases 204
35. Orthogonal Matrices, Orthogonal Transformations. 210
36. Symmetric Transformations. 215
37. Reducing a Quadratic Form to Principal Axes. Pairs of Forms 219

Chapter 9.
Evaluating roots of polynomials 225

38. Equations of Second, Third and Fourth Degree 225
39. Bounds of Roots 232
40. Sturm’s Theorem 238
41. Other Theorems on the Number of Real Roots 244
42. Approximation of Roots 250

Chapter 10.
Fields and polynomials 257

43. Number Rings and Fields 257
44. Rings 260
45. Fields 267
46. Isomorphisms of Rings (Fields). The Uniqueness of the Field of Complex Numbers 272
47. Linear Algebra and the Algebra of Polynomials over an Arbitrary Field 276
48. Factorization of Polynomials into Irreducible Factors 281
49. Theorem on the Existence of a Root 290
50. The Field of Rational Fractions 297

Chapter 11.
Polynomials in several unknowns 303

51. The Ring of Polynomials in Several Unknowns 303
52. Symmetric Polynomials. 312
53. Symmetric Polynomials Continued 319
54. Resultant. Elimination of Unknown. Discriminant 329
55. Alternative Proof of the Fundamental Theorem of the Algebra of
Complex Numbers 337

Chapter 12.
Polynomials with rational coefficients 341

56. Reducibility of Polynomials over the Field of Rationals 341
57. Rational Roots of Integral Polynomials 345
58. Algebraic Numbers 349

Chapter 13.
Normal form of a matrix 355

59. Equivalence of $\lambda$-matrices.  355
60. Unimodular $\lambda$-matrices. Relationship Between Similarity of
Numerical Matrices and the Equivalence of their Characteristic Matrices 362
61. JordanNormalForm  370
62. MinimalPolynomials  377

Chapter 14.
Groups. 382

63. Definition of a Group. Examples 382
64. Subgroups 388
65. Normal Divisors, Factor Groups, Homomorphisms 394
66. Direct Sums of Abelian Groups 399
67. Finite Abelian Groups 406

Bibliography 414
Index 416

We now come to Analytical Geometry by A. V. Pogorelov.

  Analytical geometry has no strictly defined contents. It is the   method but not the subject under investigation, that constitutes the   leading feature of this branch of geometry.   The essence of this method consists in that geometric objects are   associated in some standard way with equations (or systems of   equations) so that geometric relations of figures are expressed   through properties of their equations.   For instance, in case of Cartesian coordinates any straight line in   the plane is uniquely associated with a linear equation ax+by+ c =   0.   The intersection of three straight, lines at one point is     expressed by the condition of compatibility of a system of three     equations which specify these lines.

Due to a multi purpose approach to solving various problems, the     method of analytic geometry has become the leading method in     geometric investigations and is widely applied in other fields of     exact natural sciences, such as mechanics and physics.     Analytical geometry joined geometry with algebra and analysis –     the fact which has told fruitfully on further development of     these three subject of mathematics.     The principal ideas of analytical geometry are traced back to the     French mathematician, Rene Descartes (1595-1650), who in 1637     described the fundamentals of its method in his famous work     “Geometric”.

The present book, which is a course of lectures, treats the
fundamentals of the method of analytic geometry as applied to the
simplest geometric objects. It is designed for the university
students majoring in physics and mathematics,

This book was translated from the Russian by Leonid Levant and
was first published by Mir Publishers in 1980.

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 Table of Contents

Chapter 1
Rectangular Cartesian Coordinates In a Plane 11

Sec. 1-1. Introducing Coordinate in a Plane 11
Sec. 1-2. The Distance Between Points 15
Sec. 1-3. Dividing n Line Segment in a Given Ratio 17
Sec. 1-4. The Notion of the Equation of a Curve The Equation of a
Circle 21
Sec. 1-5. The Equation of a Curve Represented Parametrically 25
Sec. 1-6. The Points of Intersection of Curves 28

Chapter 2
The Straight Line 32

Sec. 2-1. The General Equation of the Straight Line 32
Sec. 2-2. Particular Cases of the Equation of a Straight Line 35
Sec. 2-3. The Equation of a Straight. Line in the Form Solved
with Respect to y. The Angle Between Two Straight Lines 38
Sec. 2-4. The Parallelism and Perpenrlicularitp Conditions of Two
Straight Lines 40
Sec. 2-5. The Mutual Positions of a Straight Line and a Point 43
Sec. 2-6. Basic Problems on the Straight Line 47
Sec. 2-7. Transformation of Coordinates 49

Chapter 3
Conic Sections 55

Sec. 3-1. Polar Coordinates 55
Sec. 3-2. Conic Sections and Their Equations in Polar Coordinates 58
Sec. 3-3. The Equations of Conic Sections in Rectangular Cartesian
Coordinates in Canonical Form 62
Sec. 3-4. Studying the Shape of Conic Sections 64
Sec. 3-5. A Tangent Line to a Conic Section 70
Sec 3-6. The Focal Properties of Conic Sections 74
Sec. 3-7. The Diameters of a Conic Section 78
Sec. 3-8. Second-Order Curves (Quadric Curves) 82

Chapter 4
 Vectors 87

Sec. 4-1. Addition and Subtraction of Vector’s 87
Sec. 4-2. Multiplication of a Vector by a Number 90
Sec. 4-3. Scalar Product of Vectors 93
Sec. 4-4. The Vector Product of Vector 96
Sec. 4-5. The Triple Product of Vectors 98
Sec. 4-6. The Coordinates of a Vector Relative to a Given Basis 101

Chapter 5
Rectangular Cartesian Coordinates in Space 106

Sec. 5-1. Cartesian Coordinates 106
Sec. 5-2. Elementary Problems of Solid Analytic Geometry 108
Sec. 5-3. Equations of a Surf’ace aml a Curve in Space 111
Sec. 5-4. Transformation of Coordinates 115

Chapter 6
A Plane and a Straight Line 119

Sec. 6-1.  The Equation of a Plane 119
Sec. 6-2. Special Cases of the Position of a Plane Relative to a Coordinate System 121
Sec. 6-3. The Normal Form of the Equation of a Plane 123
Sec. 6-4. Relative Position of Planes 125
Sec. 6-5. Equations of the Straight Line 129
Sec. 6-6. Relative Position of a Straight Line and a Plane, of Two
Straight Lines 131
Sec. 6.7 Basic Problems on the Straight Line and the Plane 134

Chapter 7
Surfaces of the Second Order (Quadric Surfaces)

Sec. 7-1.  A Special System of Coordinates 139
Sec. 7-2. Quadric Surfaces Classified 142
Sec. 7-3. The Ellipsoid 145
Sec. 7-4. Hyperboloids 148
Sec. 7-5. Paraboloids 150
Sec. 7-6. The Cone and Cylinders 152
Sec. 7-7. Rectilinear Generators on Quadric Surfaces 155
Sec. 7-8. Diameters and Diametral Planes of a Quadric Surface 157

Chapter 8
Investigating Quadric Curves and Surfaces Specified by Equations of
the General Form 160

Sec. 8-1. Transformation of the Quadratic Form to New Variables 160
Sec. 8-2. lnvariants of the Equations of Quadric Curves and Surfaces
with Reference to Transformation of Coordinates 162
Sec. 8-3. Investigating a Quadric Curve by Its Equation in Arbitrary
Coordinates 165
Sec. 8-4. Investigating s Quadric Surface Specified by an Equation in Arbitrary Coordinates 168
Sec. 8-5. Diameters of a Curve, Diametral Planes of a Surface. The Centre of a Curve and a Surface 171
Sec. 8-6. Axes of Symmetry of a Curve. Planes of Symmetry of a Surface 173
Sec. 8-7. The Asymptotes of a Hyperbola. The Asymptotic Cone of a Hyperboloid 175
Sec. 8-8. Tangent Line to a Curve. A Tangent Plane to a Surface  176

Chapter 9
Linear Transformations 180

Sec. 9-1. Orthogonal Transformation 180
Sec. 9-2. Affine Transformations 183
Sec. 9-3. The Affine Transformation of a Straight Line and a Plane 185
Sec. 9-4.  The Principal Invariant of the Affine Transformation 187
Sec. 9-5. Affine Transformations of Quadric Curves and Surfaces 188
Sec. 9-6. Projective Transformations 192
Sec. 9-7. Homogeneous Coordinates. Supplementing a Plane and a Space with Elements at
Infinity 195
Sec. 9-8.  The Projective Transformations of Quadric, Curves and Surfaces 198
Sec. 9-9.  The Pole and Polar 201
Sec. 9-10. Tangential Coordinates 206

Answers to the Exercises,Hints and Solutions 211

We now come to Computational Mathematics by B. P. Demidovich,
  I. A. Maron.

The basic aim of this book is to give as far as possible a
systematic and modern presentation of the most important methods and  techniques of computational mathematics on the basis of the general  course of higher mathematics taught in higher technical schools. The  book has been arranged so that the basic portion constitutes a  manual for the first cycle of studies in approximate computations  for higher technical colleges. The text contains supplementary  material which goes beyond the scope of the ordinary college course,  but the reader can select those sections which interest him and omit  any extra material without loss of continuity. The chapters and  sections which may be dropped out in a first reading are marked with an asterisk.

For a full comprehension of the contents of this book, the reader
should have a background of linear algebra and the theory of linear
vector spaces. With the aim of making the text as self-contained as
possible, the authors have included all the necessary starting
material in these subjects. The appropriate chapter are completely
independent of the basic text and can be omitted by readers who have  already studied these sections.

A few words about the contents of the book. In the main it is
devoted to the following problems: operations involving approximate  numbers, computation of functions by means of series and iterative  processes, approximate and numerical solution of algebraic and  transcendental equations, computational methods of linear algebra,  interpolation of functions, numerical differentiation and  integration of functions, and the Monte Carlo method.

This book was translated from the Russian by George Yankovsky. The book was  published by first Mir Publishers in 1973, with reprints in 1976,
and 1981. The book below is from the 1981 reprint.

All credits to the original uploader.

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Table of Contents
PREFACE

INTRODUCTION.

GENERAL RULES OF COMPUTATIONAL WORK

CHAPTER 1
APPROXIMATE  NUMBERS 19

1.1 Absolute and relative errors 19
1.2 Basic sources of errors  22
1.3 Scientific notation. Significant digits, The number of correct
digits 23
1.4 Rounding of numbers  26
1.5 relationship between the relative error of an approximate number
and the number of correct digits  27
1.6 Tables for determining the limiting relative error from the number
of correct digits and vice versa 30
1.7 The error of a sum 33
1.8 The error of a difference 35
1.9 The error of a product 37
1.10 The number of correct digits in a product 39
1.11 The error of a quotient 40
1.12 The number of correct digits in a quotient 41
1.13 The relative error of a power 41
1.14 The relative error of a root 41
1.15 Computations in which errors are not taken into exact account 42
1.16 General formula for errors 42
1.17 The inverse problem of the theory of errors 44
1.18 Accuracy in the determination of arguments from a tabulated
function 48
1.19 The method of bounds 50
1.20 The notion of a probability error estimate 52
References for Chapter 1 54

CHAPTER 2
SOME FACTS FROM THE THEORY OF CONTINUOUS FRACTIONS 55

2.1 The definition of a continued fraction 55
2.2 Converting a continued fraction to a simple fraction and vice
versa 56
2.3 Convergents 58
2.4 Nonterminating continued fractions 66
2.5 Expanding functions into continued fractions 72
References for Chapter 2 76

CHAPTER 3
COMPUTING THE VALUES OF FUNCTIONS 77

3.1 Computing the values of a polynomial. Horner’s scheme  77
3.2 The generalized Horner scheme 80
3.3 Computing the values of rational fractions 82
3.4 Approximating the sums of numerical series 83
3.5 Computing the values of an analytic function 89
3.6 Computing the values of exponential functions 91
3.7 Computing the values of a logarithmic function 95
3.8 Computing the values of trigonometric functions 98
3.9 Computing the values of hyperbolic functions 101
3.10 Using the method of iteration for approximating the values of
function 103
3.11 Computing reciprocals 104
3.12 Computing square roots 107
3.13 Computing the reciprocal of a square root 111
3.14 Computing cube roots 112
References for Chapter 3 114

CHAPTER 4
APPROXIMATE SOLUTIONS OF ALGEBRAIC AND TRANSCENDENTAL EQUATIONS 115

4.1 Isolation of roots 115
4.2 Graphical solution of equations 119
4.3 The halving method 121
4.4 The method of proportional parts (method of chords) 122
4.5 Newton’s method {method of tangents) 127
4.6 Modified Newton method 135
4.7 Combination method 136
4.8 The method of iteration 138
4.9 The method of iteration for a system of two equations 152
4.10 Newton’s method for a’system of two equations 156
4.11 Newton’s method for the case of complex roots 157
References for Chapter 5 161

CHAPTER 5
SPECIAL TECHNIQUES FOR APPROXIMATE SOLUTION OF EQUATIONS 162

5.1 General properties of algebraic equations 162
5.2 The bounds of real roots of algebraic equations 167
5.3 The method of alternating sums 169
5.4 Newton’s method 171
5.5 The number of real roots of a polynomial 173
5.6 The theorem of Budan-Fourier 175
5.7 The underlying principle of the method of Lobachevsky-Graeife 179
5.8 The root-squaring process 182
5.9 The Lobachevsky-Graeffe method for the case of real and distinct
roots 184
5.10 The Lobachevsky-Graeife method for the case of complex roots 187
5.11 The case of a pair of complex roots 190
5.12 The case of two pairs of complex roots 194
5.13 Bernoulli’s method 198
References for Chapter 5 202

CHAPTER 6
ACCELERATING THE CONVERGENCE OF SERIES 203

6.1 Accelerating the convergence of numerical series 203
6.2 Accelerating the convergence of power series by the Euler-Abel
method 209
6.3 Estimates of Fourier coefficient 213
6.4 Accelerating the convergence of Fourier trigonometric series by
the method of A, N. Krylov 217
6.5 Trigonometric approximation 225
References for Chapter 6 228

CHAPTER 7
MATRIX ALGEBRA 229

7.1 Basic definitions 229
7.2 Operations involving matrices 230
7.3 The transpose of a matrix 234
7.4 The inverse matrix 236
7.5 Powers of a matrix 240
7.6 Rational functions of a matrix 241
7.7 The absolute value and norm of a matrix 242
7.8 The rank of a matrix 248
7.9 The limit of a matrix 249
7.10 Series of matrices 251
7.11 Partitioned matrices 256
7.12 Matrix inversion by partitioning 260
7.13 Triangular matrices 265
7.14 Elementary transformations of matrices 268
7.15 Computation of determinants 269
References for Chapter 7 272

CHAPTER 8
SOLVING  SYSTEMS OF LINEAR EQUATIONS 273

8.1 A general description of methods of solving systems of linear
equations 273
8.2 Solution by inversion of matrices. Cramer’s rule 273
8.3 The Gaussian method 277
8.4 Improving roots  284
8.5 The method of principal elements 287
8.6 Use of the Gaussian method in computing determinants 288
8.7 Inversion of matrices by the Gaussian method 290
8.8 Square-root method 293
8.9 The scheme of Khaletsky 296
8.10 The method of iteration 300
8.11 Reducing a linear system to a form convenient for iteration 307
8.12 The Seidel method 309
8.13 The case of a normal system 311
8.14 The method of relaxation 313
8.15 Correcting elements of an approximate inverse matrix 316
References for Chapter 8 321

CHAPTER 9
THE CONVERGENCE OF ITERATION PROCESSES FOR SYSTEMS OF LINEAR EQUATIONS 322

9.1 Sufficient conditions for the convergence of the iteration process 322
9.2 An estimate of the error of approximations in the iteration
process 324
9.3 First sufficient condition for convergence of the Seidel process 327
9.4 Estimating the error of approximations in the Seidel process by the m-norm 330
9.5 Second sufficient condition for convergence of the Seidel process 330
9.6 Estimating the error of approximations in the Seidei process by
the l-norm 332
9.7 Third sufficient condition for convergence of the Seidel process 333
References for Chapter 9 335

CHAPTER 10
ESSENTIALS OF THEORY OF LINEAR VECTOR SPACES 336

10.1 The concept of a linear vector space 336
10.2 The linear dependence of vectors 337
10.3 The scalar product of vectors 343
10.4 Orthogonal systems of vectors 345
10.5 Transformations of the coordinates of a vector the basis 348
10.6 Orthogonal matrices 350
10.7 Orthogonalization of matrices 351
10.8 Applying orthogonalixation methods to the solutions of linear
equations 358
10.9 The solution space of a homogeneous system 364
10.10 Linear transformations of variables 367
10.11 Inverse transformation 373
10.12 Eigenvectors and eigenvalues of a matrix 375
10.13 Similar matrices 380
10.14 Bilinear form of a matrix 384
10.15 Properties of symmetric matrices 384
10.16 Properties of matrices with real elements 389
References for Chapter 10 393

CHAPTER 11
ADDITIONAL FACTS ABOUT THE CONVERGENCE OF ITERATION PROCESSES FOR
SYSTEMS OF LINEAR EQUATIQHS 394

11.1 The convergence of matrix power series 394
11.2 The Cayley-Hamilton theorem 397
11.3 Necessary and sufficient conditions for the convergence of the
process of iteration for a system of linear equations 398
11.4 Necessary and sufficient conditions for the convergence of the
Seidel process for a system of linear equations 400
11.5 Convergence of the Seidel process for a normal system 403
11.6 Methods for effectively checking the conditions of convergence 405
References for Chapter 11 409

CHAPTER 12
FINDING THE EIGENVALUES AND EIGENVECTORS OF A MATRIX 410

12.1 Introductory remarks 410
12.2 Expansion of secular determinants 410
12.3 The method of Danilevsky 412
12.4 Exceptional cases in the Danilevsky method 418
12.5 Computation of eigenvectors by the Danilevsky method 420
12.6 The method of Krylov 421
12.7 Computation of eigenvectors by the Krylov method 424
12.8 Leverrier’s method 426
12.9 On the method of undetermined coefficients 428
12.10 A comparison of different methods of expanding a secular
determinant 429
12.11 Finding the numerically largest eigenvalue of a matrix and the
corresponding eigenvector 430
12.12 The method of scalar products for finding the first eigenvalue
of a real matrix 436
12.13 Finding the second eigenvalue of a matrix and the second
eigenvector 439
12.14 The method of exhaustion 443
12.15 Finding the eigenvalues and eigenvectors of a positive definite
symmetric matrix 445
12.16 Using the coefficients of the characteristic polynomial of a
matrix for matrix inversion 450
12.17 The method of Lyusternik for accelerating the convergence of the
iteration process in the solution of a system of linear equation 453
References for Chapter 12  458

CHAPTER 13
APPROXIMATE SOLUTION OF SYSTEMS OF NOHLINEAR EQUATIONS  459

13.1 Newton’s method 459
13.2 General remarks on the convergence of the Newton process 465
13.3 The existence of roots of a system and the convergence of the
Newton process 469
13.4 The rapidity of convergence of the Newton process 474
13.5 Uniqueness of solution 475
13.6 Stability of convergence of the Newton process under variations
of the initial approximation 478
13.7 The modified Newton method 481
13.8 The method of iteration 484
13.9 The notion of a contraction mapping 487
13.10 First sufficient condition for the convergence of the process of
iteration 491
13.11 Second sufficient condition for the convergence of the process
of iteration 493
13.12 The method of steepest descent (gradient method) 496
13.13 The method of steepest descent for the case of a system of
linear equations 501
13.14 The method of power series 504
References for Chapter 13 506

CHAPTER 14
THE INTERPOLATION OF FUNCTIONS 507

14.1 Finite differences of various orders 507
14.2 Difference table 510
14.3 Generalized power 517
14.4 Statement of the problem of interpolation 518
14.5 Newton’s first interpolation formula 519
14.6 Newton’s second interpolation formula 526
14.7 Table of central differences 530
14.8 Gaussian interpolation formulas 531
14.9 Stirling’s interpolation formula 533
14.10 Bessel’s interpolation formula 534
14.11 General description of interpolation formulas with constant
interval 536
14.12 Lagrange’s interpolation formula 539
14.13 Computing Lagrangian coefficients 543
14.14 Error estimate of Lagrange’s interpolation formula 547
14.15 Error estimates of Newton’s interpolation formulas 550
14.16 Error estimates of the central interpolation formulas 552
14.17 On the best choice of interpolation points 553
14.18 Divided differences 554
14.19 Newton’s interpolation formula for unequally spaced values of
the argument 556
14.20 Inverse interpolation for the case of equally spaced points 559
14.21 Inverse interpolation for the case of unequally spaced points 562
14.22 Finding the roots of an equation by inverse interpolation 564
14.23 The interpolation method for expanding a secular determinant 565
14.24 Interpolation of functions of two variables 567
14.25 Double differences of higher order 570
14.26 Newton’s interpolation formula for a function of two variables 571
References for Chapter 14 573

CHAPTER 15
APPROXIMATE DIFFERENTIATION 574

15.1 Statement of the problem 574
15.2 Formulas of approximate differentiation based on Newton’s first
interpolation formula 575
15.3 Formulas of approximate differentiation based on Stirling’s
formula 580
15.4 Formulas of numerical differentiation for equally spaced points 583
15.5 Graphical differentiation 586
15.6 On the approximate calculation of partial derivatives 588
References for Chapter 15  589

CHAPTER 16
APPROXIMATE INTEGRATION OF FUNCTIONS 590

16.1 General remarks 590
16.2 Newton-Cotes quadrature formulas 593
16.3 The trapezoidal formula and its remainder term 595
16.4 Simpson’s formula and its remainder term 596
16.5 Newton-Cotes formulas of higher orders 599
16.6 General trapezoidal formula (trapezoidal rule) 601
16.7 Simpson’s general formula (parabolic rule) 603
16.8 On Chebyshev’s quadrature formula 607
16.9 Gaussian quadrature formula 611
16.10 Some remarks on the accuracy of quadrature formulas 618
16.11 Richardson extrapolation 622
16.12 Bernoulli numbers 625
16.13 Euler-Maclaurin formula 628
16.14 Approximation of improper integrals 633
16.15 The method of Kantorovich for isolating singularities 635
16.16 Graphical integration 639
16.17 On cubature formulas 641
16.18 A cubature formula of Simpson type 644
References for Chapter 16 648

CHAPTER 17
THE MONTE CARLO METHOD 649

17.1 The idea of the Monte Carlo method 649
17.2 Random numbers 650
17.3 Ways of generating random numbers 653
17.4 Monte Carlo evaluation of multiple integrals 656
17.5 Solving systems of linear algebraic equations method by the Monte
Carlo method 666
References for Chapter 17 674

COMPLETE LIST OF REFERENCES 675

INDEX 679 

We now come to Fundamental of Quantum Mechanics by V. A. Fock.

Vladimir Aleksandrovich Fock was one of the group of brilliant physics theoreticians whose work built the magnificent edifice of the quantum theory.

From the vast subject of the quantum theory the author has chosen material limited in two respects. First, the book considers none but the main principles and simplest applications of quantum mechanics, It concerns itself exclusively with the one-body problem. It does not deal with the many-body problem or the Pauli exclusion principle, basic to that problem. Second, the author has sought to confine himself to that part of the theory that is’ considered proved, that is, quantum mechanics proper. He has not examined quantum electrodynamics since this theory has yet to be fully elaborated.

The author’s main purpose is to introduce the reader to a new set of ideas differing greatly from the classical theory. He has endeavoured to avoid using images from the classical theory as being inapplicable to quantum physics. Rather, he has attempted to familiarize the reader with the basic concepts underlying a quantum description of the states of atomic systems.

The second edition of this book, unlike the first, devotes a separate chapter to the nonrelativistic theory of the electron spin (Pauli’s theory of the electron) and contains a chapter on the many-electron problem of quantum mechanics. In addition, some of the author’s findings have been incorporated as separate sections. Otherwise, the subject matter of the book (both the mathematical theory and its physical interpretation) remains the same, except for certain new formulations of an epistemological character (the concepts of relativity with respect to the means of observation and of potential possibility), which has necessitated changing the expression “the statistical interpretation of quantum mechanics” to “the probabilistic interpretation”. The new formulations are more precise than the previous ones.

The title of the book speaks for itself. The word “fundamentals” can be understood as “basic principles” or as “introductory facts”.

This book was translated from the Russian by Eugene Yankovsky. The book was published by first Mir Publishers in 1978 with a reprint
in 1982.

All credits to the original uploader.

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Table of Contents
Foreword
Preface to the Second Russian Edition
Preface to the First Russian Edition

PART I
BASIC CONCEPTS OF QUANTUM MECHANICS

Chapter I. The physical and epistemological bases of quantum mechanics 13

1. The need for new methods and concepts in describing atomic
phenomena 13
2. The classical description of phenomena 13
3. Range of application of the classical way of describing phenomena
Heisenberg’s and Bohr’s uncertainty relations 15
4. Relativity with respect to the means of observation as the basis
for the quantum way of describing phenomena 11
5. Potential possibility in quantum mechanics 19

Chapter II. The mathematical apparatus of quantum mechanics 22

1. Quantum mechanics and the linear-operator problems 22
2. The operator concept and examples 23
3. Hermitian conjugate. Hermiticity 24
4. Operator and matrix multiplication 21
5. Eigenvalues and eigenfunctions 30
6. The Stieltjes integral and the operator corresponding to
multiplication into the independent variable 32
7. Orthogonality of eigenfunctions and normalization 34
8. Expansion in eigenfunctions. Completeness property of eigenfunctions 37

Chapter III. Quantum mechanical operators 41

1. Interpretation of the eigenvalues of an operator 41
2. Poisson brackets 42
3. Position and momentum operators 45
4. Eigenfunctions and eigenvalues of the momentum operator 48
5. Quantum description of systems 51
6. Commutativity of operators 52
7. Angular momentum 54
8. The energy operator 57
9. Canonical transformation 59
10. An example of canonical transformation 63
11. Canonical transformation as an operator 64
12. Unitary invariants 66
13. Time evolution of systems. Time dependence of operators 69
14. Heisenberg’s matrices
15. Semiclassical approximation
16. Relation between canonical transformation and the contact
transformation of classical mechanics

Chapter IV. The probabilistic Interpretation of quantum mechanics

1. Mathemalical expectation in the probability theory
2. Mathematical expectation In quantum mechanics
3. The probability formula
4. Time dependence of mathematical expectation
5. Correspondence between the theory of linear operators and the quantum theory
6. The concept of statistical ensemble In quantum mechanics

PART II
SCHRODINGER’S THEORY

Chapter I. The Schrodinger equation. The harmonic oscillator

1. Equations of motion and the wave equation
2. Constants of the motion
3. The Schrodinger equation for the harmonic oscillator
4. The one-dimensional harmonic oscillator
5. Hermite polynomials
6. Canonical transformation a Iillustrated by the harmonic-oscillator
problem
7. Heisenberg’s uncertainty relations
8. The time dependence of matrices. A comparison with the classical theory
9. An elementary criterion for the applicability of the formuIas of
classical mechanics

Chapter II. Perturbation theory

I. Statement of the problem
2. Solution of the nonhomogeneous equation
3. Nondegenerate eigenvalues
4. Degenerate eigenvalues. Expansion in powers of the smallness parameter
5. The eigenfunctions in the zeroth-order approximation
6. The first and higher approxirnatlcns
7. The case of adjacent eigenvalues
8. The anharmonic oscillator

Chapter III. Radiation, the theory of dispersion, and the law of decay

I. Classical formulas
2. Charge density and current density
3. Frequencies and intensities
4. Intensities in a continuous spectrum
5. Perturbation of an atom by a tight wave
6. The dispersion formula
7. Penetration of a potential barrier by a particle
8. The law of decay of a quasi-stationary state

Chapter IV. An electron In a central Ileld
1. General remarks
2. Conservation of angular momentum
3. Operators in spherical coordinates. Separation of variables
4. Solution of the differential equation for spherical harmonics
5. Some properties of spherical harmonics
6. Normalized spherical harmonics
7. The radial functions. A general survey
8. Description of the states of a valence electron. Quantum numbers
9. The selection rule

Chapter V. The Coulomb field

1. General remarks
2. The radial equation for the hydrogen atom. Atomic units
3. Solution of an auxiliary problem
4. Some properties of generalized Laguerre polynomials
5. Eigenvalues and eigenfunctions of the auxiliary problem
6. Energy levels and radial functtons for the discrete hydrogen spectrum
7. Solution of the differential equation for the continuous spectrum
in the form of a definite integral
8. Derivation of the asymptotic expression
9. Radial functions for the continuous hydrogen spectrum
10. Intensities in the hydrogen spectrum
11. The Stark effect. General remarks
12. The SchrOdinger equation in parabolic coordinates
13. Splitting of energy levels in an electric field
14. Scattering of \aplha -particles. Statement of the problem
I5. Solution of equations
16. The Rutherford scattering law
17. The virial theorem in classical and in quantum mechanics
18. Some remarks concerning the superposition principle and the
probabilstic interpretation of the wave function

PART III
PAULI’S THEORY OF THE ELECTRON
1. The electron angular momentum
2. The operators of total angular momentum in spherical coordinates
3. Spherical harmonics with spin
4. Some properties of spherical harmonica with spin
5. The Pauli wave equation
6. Operator P in spherical and cylindrical coordinates and its
relation R
7. An electron In a magnetic field

PART IV
THE MANY-ELECTRON PROBLEM OF QUANTUM MECHANICS
AND THE STRUCTURE OF ATOMS

1. Symmetry properties of the wave function
2. The Hamiltonian and Its symmetry
3. The self -conslstent Held method
4. The equation for the valence electron and the operator of quantum exchange
5. The self-consistent field method in the theory of atoms
6. The symmetry of the Hamiltonian of a hydrogen-like atom

PART V
DIRAC’S THEORY OF THE ELECTRON

Chapter I. The Dirac equation 2811

1. Quantum mechanics and the theory of relativity 281
2. Classical equations of motion 281
3. Derivation of the wave equation 283
4. The Dirac matrices 284
5. The Dirac equation lor a Free electron 288
6. Lorentz translormations
7. Form of matrix S for spatial rotations of axes and for Lorentz
transformations 293
8. Current density
9. The Dirac equation in the case of a field. Equations of motion 298
10. Angular momentum and the spin vector in Dirac’s theory 301
11. The kinetic energy of an electron 304
12. The second intrinsic degree of freedom of the electron 305
13. Second-order equations

Chapter II. The use of the Dirac equation In physical problems

1. The Free electron
2. An electron in a homogeneous magnetic field
3. Constants of the motion in the problem with spherical
4. Generalized spherical harmonics
5. The radial equation
6. Comparison with the Schrodinger equation
7. General investigation of the radial equations
8. Quantum numbers
9. Heisenberg’s matrices and the selection rule
10. Alternative derivation of the selection rule
11. The hydrogen atom. Radial funclions
12. Fine-structure levels of hydrogen
13. The Zeeman ellect. Statement of the problem
14. Calculatlon of the perturbation matrix
15. Splitting of energy levels in a magnetic field

Chapter III. On the theory of positrons

1. Charge conjugation
2. Basic ideas 01 positron theory
3. Positrons as unfilled slates

Index

We will now see Theory of Probability by B. V. Gnedenko.

This book aims to give an exposition of the fundamentals of the
theory of probability, a mathematical science that treats of the
regularities of random phenomena.

This book was translated from the Russian by George Yankovsky. The
book was published by first Mir Publishers in 1969, with reprints in
1973, 1976 and 1978. The book below is from the 1978 reprint.

All credits to the original uploader.

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Table of Contents

Introduction 7
Chapter 1.
THE CONCEPT OF PROBABILITY 13

Sec.1. Certain, Impossible, and Random Events 13
Sec.2. Different Approaches to the Definition of Probability 16
Sec.3. The Sample Space 19
Sec.4. The Classical Definition of Probability 23
Sec.5. The Classical Definition of Probability. Examples. 26
Sec.6. Geometrical Probability 33
Sec.7. Frequency and Probability 39
Sec.8. An Axiomatic Construction of the Theory of Probability 45
Sec.9. Conditional ProbabiHty and the Most Elementary Basic Formulas 51
Sec.10. Examples 59
Exercises 67

Chapter 2.
SEQUENCES OF INDEPENDENT TRIALS 70
Sec.11. Independent Trials. Bernoulli’s Formulas 70
Sec.12. The Local Limit Theorem 76
Sec.13. The Integral Limit Theorem 85
Sec.14. Applications of the Integral Theorem of DeMoivre-Laplace 92
Sec.15. Poisson’s Theorem 97
Sec.16. An Illustration of the Scheme of Independent Trials 102
Exercises 104

Chapter 3.
MARKOV CHAINS 107

Sec.17. Markov Chains Defined. Transition Matrix 107
Sec.18. Classification of Possible States 111
Sec.19. Theorem on Limiting Probabilities 113
Sec.20. Generalizing the DeMoivre-Laplace Theorem to a Sequence of
Chain-Dependent Trials 116

Exercises 123

Chapter 4.
RANDOM VARIABLES AND DISTRIBUTION FUNCTIONS 124

Sec.21. Basic Properties of Distribution Functions 124
Sec.22. Continuous and Discrete Distributions 130
Sec.23. Multidimensional Distribution Functions 134
Sec.24. Functions of Random Variables 142
Sec.25. The Stieltjes Integral 155
Exercises 160

Chapter 5.
NUMERICAL CHARACTERISTICS OF RANDOM VARIABLES 164

Sec.26. Mathematical Expectation 164
Sec.27. Variance 169
Sec.28. Theorems on Expectation and Variance 176
Sec.29. Mathematical Expectation Defined in the Axiomatics of Kolmogorov 182
Sec.30. Moments 185
Exercises 191

Chapter6.
THE LAW OF LARGE NUMBERS 195

Sec.31. Mass-Scale. Phenomena and the Law of Large Numbers 195
Sec.32. Chebyshev’s Form of the Law of Large Numbers 198
Sec.33. A Necessary and Sufficient Condition for the Law of Large Numbers 206
Sec.34. The Strong Law of Large Numbers 209
Exercises 218

Chapter 7.
CHARACTERISTIC FUNCTIONS 219

Sec.35. Definition and Elementary Properties of Characteristic Functions 219
Sec.36. The Inversion Formula and the Uniqueness Theorem 224
Sec.37. Helly’s Theorems 230
Sec.38. Limit Theorems for Characteristic Functions 235
Sec.39. Positive Definite Functions 239
Sec.40. Characteristic Functions of Multidimensional Random Variables  243
Exercises 248

Chapter 8.
THE CLASSICAL LIMIT THEOREM 251

Sec.41. Statement of the Problem 251
Sec.42. Lyapunov’s Theorem 254
Sec.43. The Local Limit Theorem 259
Exercises 266

Chapter 9.
THE THEORY OF INFINITELY DIVISIBLE DISTRIBUTION LAWS 267

Sec.44. Infinitely Divisible Laws and Their Basic Properties 268
Sec.45. The Canonical Representation of Infinitely Divisible Laws 270
Sec.46: A Limit Theorem for Infinitely Divisible Laws 275
Sec.47. Statement of the Problem of Limit Theorems for Sums 278
Sec.48. Limit Theorems for Sums 279
Sec.49. Conditions for Convergence to the Normal and Poisson Laws 282
Exercises 285

Chapter10.
THE THEORY OF STOCHASTIC PROCESSES. 287
Sec.50. Introductory Remarks. 287
Sec.51. The Poisson Process 291
Sec.52. Conditional Distribution Functions and Bayes’ Formula. 298
Sec.53. Generalized Markov Equation 302
Sec.54. Continuous Stochastic Processes. Kolmogorov’s Equations 303
Sec.55. Purely Discontinuous Stochastic Processes. The Kolmogorov-Feller Equations 311
Sec.56. Homogeneous Stochastic Processes with Independent Increments 318
Sec.57. The Concept of a Stationary Stochastic Process. Khinchin’s
Theorem on the Correlation Coefficient 323
Sec.58. The Concept of a Stochastic Integral. The Spectral
Decomposition of Stationary Processes 331
Sec.59. The Birkhoff-Khinchin Ergodic Theorem 334

Chapter11.
ELEMENTS OF QUEUEING THEORY 339

Sec.60. A General Description of the Problems of the Theory 339
Sec.61. Birth and Death Processes 346
Sec.62. Single-Server Queueing System 355
Sec.63. A Limit Theorem for Flows 361
Sec.64. Elements of the Theory of Stand by Systems 367

APPENDIX376
BIBLIOGRAPHY.382
SUBJECT INDEX388

In this post we will see the second part of Course in Mathematical
Analysis by S. M. Nikolsky.

The major part of this two-volume textbook stems from the
course in mathematical analysis given by the author for many
years at the Moscow Physico-technical Institute.

The first volume consisting of eleven chapters includes an
introduction (Chapter 1)which treats offundamental notions of
mathematical analysis using an intuitive concept ofa limit. With
the aid of visual interpretation and some considerations of a
physical character it establishes the relationship between the
derivative and the integral and gives some elements of differentiation
and integration techniques necessary to those readers
who are simultaneously studying physics.

The notion of a real number is interpreted in the first volume
(Chapter 2) on the basis of its representation as an infinite decimal.
Chapters 3-11 contain the following topics: Limit of Sequence,
Limit of Function, Functions of One Variable, Functions
of Several Variables, Indefinite Integral, Definite Integral,
Some Applications of Integrals, Series.

This book was translated from the Russian by V. M. Volosov. The
book was published by first Mir Publishers in 1977 with reprints in
1981, 1985 and 1987. The copy below is from the 1987 print.

All credits to the original uploader.

DJVU | 7.5 MB | Pages: 446 | Cover

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Table of Contents

Chapter 12. Multiple Integrals 9

§ 12.1. Introduction 9
§ 12.2. Jordan Squarable Sets 11
§ 12.3. Some Important Examples of Squarable Sets 17
§ 12.4. One More Test for Measurability of a Set. Area in Polar Coordinates. 19
§ 12.5. Jordan Measurable Three-dimensional and n-dimensional Sets. 20
§ 12.6. The Notion of Multiple Integral  24
§ 12.7. Upper and Lower Integral Sums. Key Theorem 27
§ 12.8. Integrability of a Continuous Function on a Measurable Closed Set.
Some Other Integrability Conditions    32
§ 12.9. Set of Lebesgue Measure Zero  34
§ 12.10. Proof ofLebesgue’s Theorem. Connection Between Integrability and
Boundedness of a Function 35
§ 12.11. Properties of Multiple Integrals 38
§ 12.12. Reduction of Multiple Integral to Iterated Integral 41
§ 12.13. Continuity of Integral Dependent on Parameter 48
§ 12.14. Geometrical Interpretation of the Sign of a Determinant 51
§ 12.15. Change of Variables in Multiple Integral. Simplest Case  54
§ 12.16. Change of Variables in Multiple Integral. General Case  56
§ 12.17. Proof of Lemma 1, § 12.16 59
§ 12.18. Double Integral in Polar Coordinates. 63
§ 12.19. Triple Integral in Spherical Coordinates 65
§ 12.20. General Properties of Continuous Operators 67
§ 12.21. More on Change of Variables in Multiple Integral        68
§ 12.22. Improper Integral with Singularities on the Boundary of the Domain
of Integration. Change of Variables 71
§ 12.23. Surface Area  73

Chapter 13. Scalar and Vector Fields. Differentiation and Integration
of Integral
with Respect to Parameter. Improper Integrals     80

§ 13.1. Line Integral of the First Type  80
§ 13.2. Line Integral of the Second Type            81
§ 13.3. Potential of a Vector Field  83
§ 13.4. Orientation of a Domain in the Plane 91
§ 13.5. Green’s Formula. Computing Area with the Aid of Line Integral  92
§ 13.6. Surface Integral of the First Type  96
§ 13.7. Orientation of a Surface  98
§ 13.8. Integral over an Oriented Domain in the Plane          102
§ 13.9. Flux of a Vector Through an Oriented Surface 104
§ 13.10. Divergence. Gauss-Ostrogradsky Theorem 107
§ 13.11. Rotation of a Vector. Stokes’ Theorem. 114
§ 13.12. Differentiation of Integral with Respect to Parameter 118
§ 13.13. Improper Integrals 121
§ 13.14. Uniform Convergence of Improper Integrals 128
§ 13.15. Uniformly Convergent Integral over Unbounded Domain. 135
§ 13.16. Uniformly Convergent Improper Integral with Variable Singularity 140

Chapter 14. Normed Linear Spaces. Orthogonal Systems 147

§ 14.1. Space C of Continuous Functions. 147
§ 14.2. Spaces L’, L’_p and l_p 149
§ 14.3. Spaces L_2 and L’_2  154
§ 14.4. Approximation with Finite Functions      156
§ 14.:5. Linear Spaces. Fundamentals ofthe Theory ofNormed Linear Spaces 163
§ 14.6. Orthogonal Systems in Space with Scalar Product 170
§ 14.7. Orthogonalization Process            181
§ 14.8. Properties of Spaces L’_2(\Omega) and L_2(\Omega) . 185
§ 14.9. Complete Systems of Functions in the Spaces C, L’_2 and L’ (L_2, L) 187

Chapter 15. Fourier Series. Approximation of Functions with Polynomials   188

§ 15.1. Preliminaries   188
§ I5.2. Dirichlet’s Sum 195
§ 15.3. Formulas for the Remainder of Fourier’s Series 197
§ 15.4. Oscillation Lemmas  199
§ 15.5. Test for Convergence of Fourier Series. Completeness of Trigonometric
System of Functions 203
§ 15.6. Complex Form of Fourier Series 211
§ 15.7. Differentiation and Integration of Fourier Series  213
§ 15.8. Estimating the Remainder of Fourier’s Series 216
§ 15.9. Gibbs’ Phenomenon                   217
§ 15.10. Fejer’g Sums               221
§ 15.11. Elements of the Theory of Fourier Series for Functions of Several
Variables. 225
§ 15.12. Algebraic Polynomials. Chebyshev’s Polynomials     235
§ 15.13. Weierstrass’ Theorem   236
§ 15.14. Legendre’s Polynomials 237

Chapter 16. Fourier Integral. Generalized Functions 240
§ 16.1. Notion of Fourier Integral 240
§ 16.2. Lemma on Change of Order of Integration 243
§ 16.3. Convergence of Fourier’s Single Integral 245
§ 16.4. Fourier Transform and Its Inverse. Iterated Fourier
Integral. Fourier Cosine and Sine Transforms 247
§ 16.5. Differentiation and Fourier Transformation It 249
§ 16.6. Space S 250
§ 16.7. Space S’ of Generalized Functions 255
§ 16.8. Many-dimensional Fourier Integrals and Generalized Functions  265
§ 16.9. Finite Step Functions. Approximation in the Mean Square 273
§ 16.10. Plancherel’s Theorem. Estimating Speed of Convergence of Fourier’s
Integrals 278
§ 16.11. Generalized Periodic Functions 283

Chapter 17. Differentiable Manifolds and Differential Forms 289
§ 17.1. Differentiable Manifolds 289
§ 17.2. Boundary of a Differentiable Manifold and Its Orientation 299
§ 17.3. Differential Forms. 310
§ 17.4. Stokes’ Theorem 220

Chapter 18. Supplementary Topics 326
§ 18.1. Generalized Minkowski’s Inequality 326
§ 18.2. Sobolev’s Regularization of Function 329
§ 18.3. Convolution 333
§ 18.4. Partition of Unity 335

Chapter 19. Lebesgue Integral  338

§ 19.1. Lebesgue Mea.sure  338
§ 19.2. Measurable Functions  348
§ 19.3. Lebesgue lntegral 35S
§ 19.4. Lebesgue Integral on Unbounded Set 388
§ 19.5. Sobolev’s Generalized Derivative  392
§ 19.6. Space D’ of Generalized Functions 404
§ 19.7. Incompleteness of Space L 407
§ 19.8. Generalization of Jordan Measure 408
§ 19.9. Riemann-Stieltjes Integral  414
§ 19.10. Stieltjes Integral  415
§ 19.11. Generalization of Lebesgue Integral 423
§ 19.12 Lebesgue-StieJtjes Integral 424
§ 19.13. Extension of Functions. Weierstrass’ Theorem 433
Name Index 437
Subject Index  438

In this post we will see Problems in Higher Algebra – Faddeev, Sominsky  

This book of problems in higher algebra grew out of a course
of instruction at the Leningrad State University and the Herzen
Pedagogical Institute. It is designed for students of universities
and teacher’s colleges as a problem book in higher algebra.

The problems included here are of two radically different types. On
the one hand, there are a large number of numerical examples aimed at developing computational skills and illustrating the basic propositions of the theory. The authors believe that the number of
problems is sufficient to cover work in class, at home and for tests.

On the other hand, there are a rather large number of problems
of medium difficulty and many which will demand all the initiative and ingenuity of the student. Many of the problems of
this category are accompanied by hints and suggestions to be
found in Part II. These problems are starred.

Answers are given to all problems, some of the problems are
supplied with detailed solutions.

This book was translated from the Russian by George Yankovsky. The  book was published by first Mir Publishers in 1978.

All credits to the original uploader.

PDF | 3.4 MB | Pages: 318 | Bookmarked | OCR | Cover
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Table of Contents

Contents
Part I
PROBLEMS

CHAPTER I. COMPLEX NUMBERS 11

1. Operations on Complex Numbers 11
2. Complex Numbers in Trigonometric Form 13
3. Equations of Third and Fourth Degree 19
4. Roots of Unity 21

CHAPTER 2. EVALUATION OF DETERMINANTS 25

1. Determinants of Second and Third Order 25
2. Permutations 26
3. Definition of a Determinant 27
4. Basic Properties of Determinants 29
5. Computing Determinants 31
6. Multiplication of Determinants 51
7. Miscellaneous Problems 56

CHAPTER 3. SYSTEMS OF LINEAR EQUATIONS 61

1. Cramer’s Theorem 61
2. Rank of a Matrix 64
3. Systems of Linear Forms 66
4. Systems of Linear Equations 68

CHAPTER 4. MATRICES 76

1. Operations on Square Matrices 76
2. Rectangular Matrices. Some Inequalities 83

CHAPTER 5. POLYNOMIALS AND RATIONAL FUNCTIONS OF ONE VARIABLE 88

1. Operations on Polynomials. Taylor’s Formula. Multiple Roots 88
2. Proof of the Fundamental Theorem of Higher Algebra and Allied
Questions 92
3. Factorization into Linear Factors. Factorization into Irreducible
Factors in the Field of Reals. Relationships Between Coefficients
and Roots 93
4. Euclid’s Algorithm  97
5. The Interpolation Problem and Fractional Rational Functions 100
6. Rational Roots of Polynomials. Reducibility and Irreducibility over
the Field of Rationals 103
7. Bounds of the Roots of a Polynomial 107
8. Sturm’s Theorem 108
9. Theorems on the Distribution of Roots of a Polynomial 111
10. Approximating Roots of a Polynomial 115

CHAPTER 6. SYMMETRIC FUNCTIONS 116
1. Expressing Symmetric Functions in Terms of Elementary Symmetric
Functions. Computing Symmetric Functions of the Roots of an
Algebraic Equation 116
2. Power Sums 121
3. Transformation of Equations 123
4. Resultant and Discriminant 124
5. The Tschirnhausen Transformation and Rationalization of the
Denominator 129
6. Polynomials that Remain Unchanged under Even Permutations of the Variables. Polynomials that Remain Unchanged under Circular
Permutations of the Variables 130

CHAPTER 7. LINEAR ALGEBRA 133
1. Subspaces and Linear Manifolds. Transformation of Coordinates 133
2. Elementary Geometry of n-Dimensional Euclidean Space 135
3. Eigenvalues and Eigenvectors of a Matrix 139
4. Quadratic Forms and Symmetric Matrices 141
5. Linear Transformations. Jordan Canonical Form 146
PART II

HINTS TO SOLUTIONS

CHAPTER 1. COMPLEX NUMBERS 151
CHAPTER 2. EVALUATION OF DETERMINANTS 153
CHAPTER 4. MATRICES 159
CHAPTER 5. POLYNOMIALS AND RATIONAL FUNCTIONS OF ONE VARIABLE 160
CHAPTER 6. SYMMETRIC FUNCTIONS 164
CHAPTER 7. LINEAR ALGEBRA 166

PART III

ANSWERS AND SOLUTIONS
CHAPTER 1. COMPLEX NUMBERS 168
CHAPTER 2. EVALUATION OF DETERMINANTS 186
CHAPTER 3. SYSTEMS OF LINEAR EQUATIONS 196
CHAPTER 4. MATRICES 203
CHAPTER 5. POLYNOMIALS AND RATIONAL FUNCTIONS OF ONE VARIABLE 221
CHAPTER 6. SYMMETRIC FUNCTIONS 261
CHAPTER 7. LINEAR  ALGEBRA 286
INDEX 313

In this post we will see another problem book in mathematics titled Problems in Elementary Mathematics by V. Lidsky, L. Ovsyannikov, A. Tuliakov and M. Shabunin.

This book is written by a group of Soviet mathematicians under the guidance of Professor Victor Lidsky D.Sc. (Phys. & Maths). It includes advanced problems in elementary mathematics with hints and solutions.

In each section – algebra, geometry and trigonometry – the problems are arranged in the order of increasing difficulty. There are 658 problems in all.

The text can be used in mathematical schools and school mathematical societies.

This book was translated from the Russian by V. Vosolov and was first published by Mir Publishers in 1973.

All credits to the original uploader.

A note on the quality of the book: When the book was picked up from the net, it needed cleaning. We did some cleaning, OCR and bookmarking.

PDF | OCR | Cover | Bookmarked | 25.5 MB (20.5 MB Zipped) | 382 pages | Paginated

Get the book here. (Password, if needed: mirtitles)

For magnet links / torrents go here.

See the FAQs for extraction problems.

Contents

CONTENTS
Algebra
Problems / Solutions
1. Arithmetic and Geometric Progressions (1-23) 7/87
2. Algebraic Equations and Systems of Equations (24-95) 10/95
3. Algebraic Inequalities (96-123) 20/134
4. Logarithmic and Exponential Equations, Identities and Inequalities ( 124-169) 24/142
5. Combinatorial Analysis and Newton’s Binomial Theorem (170-188) 29/157
6. Problems in Forming Equations ( 189-228) 32/162
7. Miscellaneous Problems (229-291) 38/180
Geometry
A. Plane Geometry
1. Computation Problems (292-324) 47/202
2. Construction Problems (325-338) 51/217
3. Proof Problems (339-408) 52/223
4. Loci of Points (409-420) 59/254
5. The Greatest and Least Values (421-430) 61/260
B. Solid Geometry
1. Computation Problems (431-500) 62/256
2. Proof Problems (501-523) 70/209
3. Loci of Points (524-530) 72/322
4. The Greatest and Least Values (531-532) 72/325
Trigonometry

1. Transforming Express ions Containing Trigonometric Functions (533-554) 74/327
2. Trigonometric Equations and Systems of Equations (555-618) 77/333
3. Inverse Trigonometric Functions (619-628) 82/363
4. Trigonometric Inequalities (629-645) 83/366
5. Miscellaneous Problems (646-658) 85/372

In an earlier post we have seen a wonderful book on special relativity titled Relativity and Man by V. Smilga. In that post, it was pointed out another book by this wonderful author titled In theSearch for Beauty, which has a very artful cover. I had first seen this book on the wonderful Soviet Books blog.

We were in process of searching this beauty, literally and figuratively both (see my and s.sanjay’s comments in Relativity and Man post), until we got this message from Boris Smilga the grandson of V. Smilga!

Hello. I have stumbled upon this excellent site somewhat accidentally, and was happy to find granddad’s footprints, so to say, on Indian soil. As the keeper of his estate, I happen to have the English edition of “In the Search for Beauty”, and, since you are interested in it, I have scanned it for the common good. You’ll find the file at https://dpworks.net/files/search-for-beauty.djvu (size≈23MB at 300 dpi, md5=718dd7f03a78325e82f0527854a9a394); please contact me if you have any trouble downloading it.

This is something that has really made me happy. I cannot thank Boris enough for this. Thanks once again Boris!!

Now something about the book:

The book takes us on a epic journey on the origins of non-Euclidean geometry  based on the parallel lines postulate and its  culmination in General theory of relativity of Einstein. It starts with era of Greek geometers, Euclid and his fifth postulate. Also the mathematical genius of the poet, polymath Omar Khayyam is given. Then the pages turn to mathematical giants like Bolyai, Lobachevsky, Gauss and their contributions to the development of non-Euclidean geometry. Then we come to general relativity and the final chapter on Einstein. The book is illustrated with witty cartoons throughout. (I have started but not yet finished the book, so the synopsis is not complete or exhaustive, please feel free to add your own review!).

The book was translated from the Russian by George Yankovsky. The book was published by Mir Publishers in 1970.

All credits to Boris Smilga.

You can get the book here (Boris’ link) and here (filecloud)  .

Djvu | OCR | 300 dpi| Covers | 175 pages (2-in-1 scan)| 23 MB

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Contents

1. Before Euclid – Prehistoric Times. 7
2. Euclid. 26
3. The Fifth Postulate. 57
4. The Age of Proofs. The Beginning. 81
5. Omar Khayyam. 92
6. The Age of Proofs. Continued. 129
7. Non-Euclidean Geometry. The Solution. 155
8. Nikolai Ivanovich Lobachevsky. 198
9. Non-Euclidean Geometry. Some Illustrations. 230
10. New Ideas. Riemann. Non-contradictoriness. 246
11. An Unexpected Finale. The General Theory of Relativity. 269
12. Einstein 301

We now come to two volume set on Introduction to Topology by Yu. Borisovich, N. Bliznyakov, Ya. Izrailevich, T. Fomenko.

This is a two volume book set, which has 5 sections in all. It is based on lectures delivered by Yu. G. Borisovich at Mathematics Department of Voronezh University. Each of
the section is preceeded by an illustration which has a lot of mathematical content which are by Prof. A. T. Fomenko.

About the book

This textbooks is one of the many possible variants of a first course in topology and is written in accordance with both the author’s preferences and their experience as lecturers and researchers. It deals with those areas of topology that are most closely related to fundamental courses in general mathematics and applications. The material leaves a lecturer a free choice as to how he or she may want to design his or her own topology course and seminar classes.

The books were translated from the Russian by Oleg Efimov and was
first published by Mir Publishers in 1980.

Thanks to the original uploader for the scan.

Note: The scan quality is poor and text is barely readable at
times. We have added OCR, which is not reliable, reduced file size, combined 2 pdfs into
one, bookmarked and paginated and pdfs.  Vol. 1 and Vol. 2 which are
physically separate books have been combined in one single pdf. The
page numbering is continuous between the two volumes. Page 147 onwards are
the contents of Vol. 2. We have access to the hard copies, and might see a better version in the future.

PDF | Bookmarked | Paginated | OCR | Cover | 324 Pages |  24.2 MB

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 Contents

FIRST NOTIONS OF TOPOLOGY

1. What is topology? 11
2. Generalization of the concepts of space and function 15
3. From a metric to topological space 18
4. The notion of Riemann surface 28
5. Something about knots 34
Further Reading 37

GENERAL TOPOLOGY

1. T opological spaces and continuous mappings 41
2. Topology and continuous mapping of metric spaces. Spaces R^n,
S^(n-1) and D^n 46
3. Factor space and quotient topology 52
4. Classification of Surfaces 57
5 . Orbit Spaces. Projective and Iens spaces 67
6. Operations over sets in topological space 70
7. Operations over sets in metric spaces. Spheres and
balls. Completeness 73
8. Properties of continuous Mappings 76
9. Products of topological spaces 80
10. Connectedness of topological spaces 84
11. Countability and separation axioms 88
12. Normal spaces and functional separability 92
13. Compact spaces and their mappings 97
14. Compactifications of topological spaces. Metrization 105
Further Reading 107

HOMOTOPY THEORY

1. Mapping space. Homotopies, retractions, and deformations 111
2. Category, functor and algebraization of topological problems 118
3. Functors of homotopy grous 121
4. Computing the fundamental homotopy groups of some space 131
Further Reading 146

MANIFOLDS AND FIBRE BUNDLES

1. Basic notions of differential calculus in n-dimensional space 149
2. Smooth submanifolds in Euclidean space 157
3. Smooth Manifolds 161
4. Smooth functions in a manifold and smooth partition of unity 173
5. Mapping of manifolds 180
6. Tangent bundle and tangential map 188
7.  Tangent vector as differential operator. Differential of function
and cotangent bundle 199
8. Vector fields on smooth manifolds 208
9. Fibre  bundles and covering  213
10. Smooth function on manifold and cellular structure of manifold (example) 235
11. Non-degenrate ciritcal point and its index 240
12. Describing homotopy type of manifold by means of critical values 244
Further Reading 249

HOMOLOGY THEORY

1. Preliminary Notes 253
2. Homology groups of chain complexes 255
3. Homology groups of simplical complexes 257
4. Singular Homology Theory 268
5. Homology theory axioms 278
6. Homology groups of spheres. Degree of mapping 281
7. Homology groups of cell complexes 289
8. Euler Characteristic and Lefschetz number 292
Further reading 299

Illustrations 302

References 303

Name Index 306

Subject Index 308

In this post we will see This Chancy, Chancy, Chancy World By Leonard Rastrigin.

Have you ever sat down and thought about how often chance affects your life? If you have, then you probably realize that chance literally hits us from every side. We live in a world more vulnerable to the vicissitudes of chance than the wildest imagination could devise.Chance abounds in an endless variety of forms. Some darken our existence, confound our plans and prevent us from realizing our most cherished ambitions. Others do not affect us, while others still illuminate our lives with all the colours of the rainbow and bring happiness and success (eureka!).

But is it really worth talking about chance? What is there to say about it? Chance is chancy, and that’s that.
In fact there is a great deal we can say about chance and there is even more we can ask about it. For example: how does chaos arise? What is control? How should we act in circumstances involving chance? How can we come to terms with the difficulties that arise from chance obstacles in our lives? What is the Monte Carlo method? Why is learning necessary? What role does chance play in evolution and progress? How is it that our chancy, chancy, chancy world gets along quite well? Is it possible to make it better still? Answers to these and many other questions will be found in this book.

About the Author
LEONARD RASTRIGIN graduated in aircraft design from the Moscow Aeronautical Institute and, in 1960, presented his Ph.D. thesis on mechanics. He then made a 179-degree turn and ‘retreated’ into cybernetics, where he studied random search a new technique for finding optimum solutions to complex problems. Cybernetics brings him both joy and sorrow. His work in this field has gained him his doctorate and a professorship, and he is now Director of the only random search laboratory in the world. Here his task is to vindicate the claims of random search and to demonstrate its advantages in practical applications. Professor Rastrigin is a very busy man. Yet no sooner does he have a day off duty than he reaches for his pen. In the space of a few years he has written two monographs and over a hundred scientific articles. This Chancy, Chancy, Chancy World is his first book devoted to acquainting the general reader with his special field of study.

The book was translated from the Russian by R. H. M. Woodhouse and was first published by Mir Publishers in 1973, with a second reprint in1984. The present version is the 1984 one.

Many thanks to gnv64 for this book. And thanks to Biju for making this post.

PDF | Paginated | Bookmarked | 287 pages | 9.36 mb

You can get the book here and here (thanks Siddharth for this link).
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Contents:

INTRODUCTION
CONTENTS    4
What is chance?    10

 PART-1

CHANCE THE OBSTACLE        34

1. Change at the cradle of cybernetics    34
2. Control    38
3. The history of control stage one    60
4. The battle with chance interface    82
5. Alternatives risk and decision    126

PART-2-
WELCOME CHANCE    152

1. Sherlock holmes speak his mind at last    152
2. The monte carlo method    158
3. Chance in games.    176
4. Learning conditioned relexes and chance    184
5. Chance and recongnition    206
6. Chance selection and evolution    230
7. Self-adjustment    248
8. Seabch(path and wanderings)    266

Conclusion    284

In this post we see a book on neuroscience titled On The Neuronal Organization of The Brain by G. I. Poliakov. Though much of the material in the book may be dated.

This monograph is devoted to a description of certain processesrelating to the establishment, development, complication, andperfection of the orgnniza tion of reflex mechanisms in the evolutionof animal organisms. The material outlined in our present work is, of course, only afragment of the future theory of the neuronal organization of thebrain.

The book was translated from the Russian by H. C. Creighton and was first published by Mir Publishers in 1971.

PDF | OCR | Cover | Bookmarked | 600 dpi | Paginated | 14.9 MB| 177 pages

(Note: I did not have access to original cover, so created one. If you have access to the cover please post a link in the comments. Some foldout illustrations maybe missing from the scan.)

You can get the book here.
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Contents:
Introduction

Chapter I
Reflex Mechanism of the Brain

1. Organs of Signalling Activity 1
2. Basic Divisions of the Nervous System 15
3. The Coordination Mechanism 30
4. The Analyser-coordination Mechanism 47
5. The Analyser Systems 54

Chapter II
Regulation, Control and Direction in the Animal Organism

1. The Problem 61
2. The Functional Significance and Interconnections of Reflex Mechanisms of Different Levels of Organization 62
3. Auto-regulation and Regulation 73
4. Auto-control and Control 75
5. General Character of Auto-direction and Direction 78
6. Auto-direction (‘Unfree’ or Automatic Direction) 80
7. Direction Proper CFree’ or Voluntary and Automatized Direction) 82
8 Psychophysiological Aspects of the Problem of Direction 93
9. The Relation of the Functions of Auto/regulation, Auto/control, and Auto/direction from the Standpoint of Evolution 98

Chapter III
The Neuronal Network

1. The Origin and Complication of the Neuronal Network 103
2. Progressive Differentiation of Neurons 106
3. Internuncial Neurons and Their Role in Reflex Activity 111
4. Forms of Contacts and Functional Interconnections between Neurons 120

Chapter IV
The Basic Scheme of Switches in the Neuronal Network

1. The Central Switching Apparatus in Analysers 127
2 Central Nervous Apparatuses for Perception and Imprinting 135
3. Structural Basis of Functional Localization in the Cortex 140
4 General Scheme of Interconnections between the Various Switching Levels in the Analysers 148

Conclusion 159

References 165

YOU WON'T REGRET READING THIS BOOK

We now come to much awaited book by Lev Tarasov The World Is Built On
Probability.

And this post also is the 200th post on Mirtitles!
Thanks to all who helped and encouraged!!

From the Back Cover:

This text is divided into two major parts.  The aim of the first part
is to convince the reader that the random world begins directly in his
or her own living room because, in fact, all modern life is based on
probability. The first part is on the concept of probability and
considers making decisions in conflict situations, optimizing queues,
games, and the control of various processes, and doing random
searches.

The second part of this text shows how fundamental chance is in nature using the probabilistic laws of modern physics and biology as
examples. Elements of quantum mechanics are also involved, and this
allows the author to demonstrate how probabilistic laws are basic to
microscopic phenomena. The idea is that the reader, passing from the
first part of the book to the second one, would see that probability
is not only around us but is at the basis of everything.

The Russian edition ran into 230,000 copies and is sold out.

YOU WON’T REGRET READING THIS BOOK

The book was translated from the Russian by Michael Burov and was
first published by Mir Publishers in 1988.

Thanks to Karen for getting this book from Amazon of Amerika (and I had to pay heftily in U$D, but that’s okay, I think, for a book like this). It is pretty ironic that we have now to get Soviet books from Amerika by paying in U$D.

Occasionally, some of your visitors may see an advertisement here.

Now, you will be seeing ads in the posts, by WordPress, which I do not like personally. For this particular reason, we may shift to a different server. They are asking for 30 U$D per year to keep it ad-free. With some more money, I think it is better to migrate to our own.

PDF | OCR | Pagination | Bookmarked | Cover | 600 dpi | 200 Pages |
22.8 MB

Note the files on Internet Archive may have different parameters.

Get the book here or here or here.

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Contents

Introduction 9

Part One Tamed Chance 17

Chapter 1
Mathematics of Randomness 18

Probability 18
Random Numbers 26
Random Events 30
Discrete Random Variables 34
Continuous Random Variables 42

Chapter 2
Decision Making 46

These Difficult Decisions 46
Random Processes with Discrete States 50
Queueing Systems 55
Method of Statistical Testing 63
Games and Decision Making 69

Chapter 3
Control and Self-Control 79

The Problem of Control 79
From the “Black Box” to Cybernetics 81
Information 84
Selection of Information from Noise 94
On the Way to a Stochastic Model of the Brain 99

Part Two
Fundamentality of the Probability Laws 105

Chapter 4
Probability in Classical Physics 106
Thermodynamics and Its Puzzles 106
Molecules in a Gas and Probability 115
Pressure and Temperature of an Ideal Gas 124
Fluctuations 127
Entropy and Probability 132
Entropy and Information 138

Chapter 5
Probability in the Microcosm 142
Spontaneous Microprocesses 142
From Uncertainty Relations to the Wave Function 149
Interference and Summing Probability Amplitudes 154
Probability and Causality 160

Chapter 6
Probability in Biology 164
Introduction 164
The Patterns After the Random Combination of Genes in Crossbreeding 169
Mutations 176
Evolution Through the Eyes of geneticists 179

A concluding conversation 184
Recommended Literature 189

We now come to Equations of Mathematical Physics by A. V. Bitsazde.

About the book:

The present book consists of an introduction and six chapters. The introduction discusses basic notions and definitions of the traditional course of mathematical physics and also mathematical models of some phenomena in physics and engineering. Chapters 1 and 2
are devoted to elliptic partial differential equations. Here much emphasis is placed on the Cauchy- Riemann system of partial differential equations, that is on fundamentals of the theory of analytic functions, which facilitates the understanding of the role played in mathematical physics by the theory of functions of a complex variable.

In Chapters 3 and 4 the structural properties of the solutions of hyperbolic and parabolic partial differential equations are studied and much attention is paid to basic problems of the theory of wave equation and heat conduction equation.

In Chapter 5 some elements of the theory of linear integral equations are given. A separate section of this chapter is devoted to singular
integral equations which are frequently used in applications. Chapter 6 is devoted to basic practical methods for the solution of partial differential equations. This chapter contains a number of typical examples demonstrating the essence of the Fourier method of separation of variables, the method of integral transformations, the finite difference method, the melthod of asymptotic expansions and also the variational methods.

To study the book it is sufficient for the reader to be familiar with an ordinary classical course on mathematical analysis studied in colleges. Since such a course usually does not involve functional analysis, the embedding theorems for function’ spaces are not included in the present book.

The book was translated from the Russian by V. M. Volosov and I. G. Volosova and was first published by Mir Publishers in 1980.

Scan credits to the original uploader. We have cleaned the 2-in-1 copy to single page format and added the cover.

You can get the book here and here.

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Contents
Preface 5
Introduction 13

1. Basic Notions and Definitions 13

1. The Notion of a Partial Differential Equation and Its Solution 13
2. Characteristic Form of a Linear Partial Differential Equation. Classification of Linear Partial Differential 15
Equations of the Second Order by Type 15
3. Classification of Higher-Order Partial Differential Equations 17
4. Systems of Partial Differential Equations 18

2. Normal Form of Linear Partial Differential Equations of the Second
Order in Two Independent Variables 20

1. Characteristic Curves and Characteristic Directions 20
2. Transformation of Partial Differential Equations of the Second
Order in Two Independent Variables into the Normal Form 22

3. Simplest Examples of the Three Basic Types of Second-Order Partial Differential Equations 26

1. The Laplace Equation 26
2. Wave Equation 29
3. Heat Conduction Equation 32
4. Statement of Some Problems for Partial Differential Equations 33

4. The Notion of an Integral Equation 35

1. Notation and Basic Definitions 35
2. Classification of Linear Integral Equations 36

5. Simplified Mathematical Models for Some Phenomena in Physics and Engineering 37

1. Electrostatic Field 37
2. Oscillation of a Membrane 40
3. Propagation of Heat 43
4. The Motion of a Material Point under theAction of the Force of Gravity 44

Chapter 1.
Elliptic Partial Differential Equations 47

1. Basic Properties of Harmonic Functions 47

1. Definition of a Harmonic Function and Some of Its Basic Properties 47
2. Integral Representation of Harmonic Functions 50
3. Mean-Value Formulas 51
4. The Extremum Principle for the Dirichlet Problem. Uniqueness of the Solution 53

2. The Notion of Green’s Function. Solution of the Dirichlet Problem for a Ball and for a Half-Space 55

1. Green’s Function of the Dirichlet Problem for the Laplace Equation 55
2. Solution of the Dirichlet Problem for a Ball. Poisson’s Formula 57
3. Verification of Boundary Conditions 60
4. Solution of the Dirichlet Problem for a Half-Space 61
5. Some Important Consequences of Poisson’s Formula. Theorems of Liouville and Harnack 63

3. Potential Function for a Volume Distribution of Mass 65

1. Continuity of Volume Potential and Its Derivatives of the First Order 65
2. Existence of the Derivatives of the Second Order of Volume Potential 67
3. The Poisson Equation 69
4. Gauss Formula 72

4. Double-Layer and Single-Layer Potentials 74

1. Definition of a Double-Layer Potential 74
2. Formula for the Jump of a Double-Layer Potential Reduction of the Dirichlet Problem to an Integral Equation 77
3. Single-Layer Potential. The Neumann Problem 81
4. The Dirichlet Problem and the Neumann Problem for Unbounded Domains 84

5. Elements of the General Theory of Elliptic Linear Partial
Differential Equations of the Second Order 86

1. Adjoint Operator. Green’s Theorem 86
2. Existence of Solutions of Elliptic Linear Partial Differential Equations of the Second Order 88
3. Boundary-Value Problems 90
4. The Extremum Principle. The Uniqueness of the Solution of the Dirichlet Problem 92
5. Generalized Single-Layer and Double-Layer Potentials 94

Chapter 2.
Cauchy-Riemann System of Partial Differential Equations. Elements of the Theory of Analytic Functions 97

1.The Notion of an Analytic Function of a Complex Variable 97

1. Cauchy-Riemann System of Partial Differential Equations 97
2. The Notion of an Analytic Function 98
3. Examples of Analytic Functions 102
4. Conformal Mapping 104
5. Conformal Mappings Determined by Some Elementary Functions. Inverse Functions. The Notion of a Riemann Surface 109

2. Complex Integrals 116

1. Integration along a Curve in the Complex Plane 116
2. Cauchy’s Theorem 118
3. Cauchy’s Integral Formula 121
4. The Cauchy-Type Integral 124
5. Conjugate Harmonic Functions. Morera’s Theorem 125

3. Some Important Consequences of Cauchy’s Integral Formula 127

1. Maximum Modulus Principle for Analytic Functions 127
2. Weierstrass’ Theorems 129
3. Taylor’s Series 131
4. Uniqueness Theorem for Analytic Functions. Liouville’s Theorem 133
5. Laurent Series 134
6. Singular Points and Residues of an Analytic Function t38
7. Schwarz’s Formula. Solution of Dirichlet Problem 143

4. Analytic Continuation 146

1. The Notion of Analytic Continuation 146
2. The Continuity Principle 146
3. The Riemann-Schwarz Symmetry Principle 148

5. Formulas for Limiting Values of Cauchy-Type Integral and Their Applications 149

1. Cauchy’s Principal Value of a Singular Integral 149
2. Tangential Derivative of a Single-Layer Potential 151
3. Limiting Values of Cauchy-Type Integral 154
4. The Notion of a Piecewise Analytic Function 156
5. Application to Boundary-Value Problems 157

6. Functions of Several Variables 163

1. Notation and Basic Notions 163
2. The Notion of an Analytic Function of Several Variables 164
3. Multiple Power Series 166
4. Cauchy’s Integral Formula and Taylor’s Theorem 168
5. Analytic Functions of Real Variables 170
6. Conformal Mappings in Euclidean Spaces 172

Chapter 3.
Hyperbolic Partial Differential Equations 176

1. Wave Equation 176

1. Wave Equation with Three Spatial Variables. Kirchhoff’s Formula 176
2. Wave Equation with Two Spatial Variables. Poisson’s Formula 178
3. Equation of Oscillation of a String. D’Alemhert’s Formula 179
4. The Notion of the Domains of Dependence, Influence and Propagation 181

2. Non-Homogeneous Wave Equation 183

1. The Case of Three Spatial Variables. Retarded Potential 183
2. The Case of Two, or One, Spatial Variables 184

3. Well-Posed Problems for Hyperbolic Partial Differential Equations 186

1. Uniqueness of the Solution of the Cauchy Problem 186
2. Correctness of the Cauchy Problem for Wave Equation 187
3. General Statement of the Cauchy Problem 188
4. Goursat Problem 191
5. Some Improperly Posed Problems 192

4. General Linear Hyperbolic Partial Differential Equation of the Second Order in Two Independent Variables 193

1. Riemann’s Function 193
2. Goursat Problem 196
3. Cauchy Problem 198

Chapter 4.
Parabolic Partial Differential Equations 200

1. Heat Conduction Equation. First Boundary-Value Problem 200
1. Extremum Principle 200
2. First Boundary-Value Problem for Heat Conduction Equation 202

2. Cauchy-Dirichlet Problem 204

1. Statement of Cauchy-Dirichlet Problem and the Proof of the Existence of Its Solution 204
2. Uniqueness and Stability of the Solution of Cauchy-Dirichlet Problem 206
3. Non-Homogeneous Heat Conduction Equation 208

3. On Smoothness of Solutions of Partial Differential Equations 208

1. The Case of Elliptic and Parabolic Partial Differential Equations 208
2. The Case of Hyperbolic Partial Differential Equations 209

Chapter 5.
Integral Equations 210

1. Iterative Method for Solving Integral Equations 210

1. General Remarks 210
2. Solution of Fredholm Integral Equation of the Second Kind for Small Values of the Parameter Using Iterative Method 211
3. Volterra Integral Equation of the Second Kind 213

2. Fredholm Theorems 215

1. Fredholm Integral Equation of the Second Kind with Degenerate Kernel 215
2. The Notions of Iterated and Resolvent Kernels 219
3. Fredholm Integral Equation of the Second Kind with an Arbitrary Continuous Kernel 220
4. The Notion of Spectrum 224
5. Volterra Integral Equation of the Second Kind with Multiple Integral 225
6. Volterra Integral Equation of the First Kind 226

3. Applications of the Theory of Linear Integral Equations of the Second Kind 228

1. Application of Fredholm Alternative to the Theory of Boundary-Value Problems for Harmonic Functions 228
2. Reduction of Cauchy Problem for an Ordinary Linear Differential Equation to a Volterra Integral Equation
of the Second Kind 231
3. Boundary-Value Problem for Ordinary Linear Differential Equations of the Second Order 233

4. Singular Integral Equations 236

1. The Notion of a Singular Integral Equation 236
2. Hilbert’s Integral Equation 237
3. Hilbert Transformation 240
4. Integral Equation of the Theory of the Wing of an Airplane 241
5. Integral Equation with a Kernel Having Logarithmic Singularity 244

Chapter 6.
Basic Practical Methods for the Solution of Partial Differential Equations 246

1. The Method of Separation of Variables 246

1. Solution of Mixed Problem for Equation of Oscillation of a String 246
2. Oscillation of a Membrane 251
3. The Notion of a Complete Orthonormal System of Functions 254
4. Oscillation of Circular Membrane 257
5. Some General Remarks on the Method of Separation of Variables 260
6. Solid and Surface Spherical Harmonics 262
7. Forced Oscillation 264

2. The Method of Integral Transformation 266

1. Integral Representation of Solutions of Ordinary Linear Differential Equations of the Second Order 266
2. Laplace, Fourier and Mellin Transforms 272
3. Application of the Method of Integral Transformations to Partial Differential Equations 275
4. Application of Fourier Transformation to the Solution of Cauchy Problem for the Equation of Oscillation of a String 278
5. Convolution 281
6. Dirac’s Delta Function 284

3. The Method of Finite Differences 286

1. Finite-Difference Approximation of Partial Differential Equations 286
2. Dirichlet Problem for Laplace’s Equation 287
3. First Boundary-Value Problem for Heat Conduction Equation 289
4. Some General Remarks on Finite-Difference Method 290

4. Asymptotic Expansions 291

1. Asymptotic Expansion of a Function of One Variable 291
2. Watson’s Method for Asymptotic Expansion 296
3. Saddle-Point Method 299

5. Variational Methods 303

1. Dirichlet Principle 303
2. Eigenvalue Problem 305
3. Minimizing Sequence 307
4. Ritz Method 308
5. Approximate Solution of Eigenvalue Problem. Bubnov-Galerkin Method 309

Name Index 312
Subject Index 313

In this post we will see a wonderful book on mathematics  titled Straight Lines and Curves by N. Vasilyev and V. Gutenmacher.

About the book:

The authors N. B. Vasilyev and V. L. Gutenmacher are professors of mathematics at Moscow University. N. B. Vasilyev works in the field of the application of mathematical methods to biology, while V. I. Gutenmacher works in the field of mathematical methods used in the analysis of economic models.

In addition to their scientific work, they have both written many articles and books for high school and university students, and have worked with the Correspondence Mathematics School, which draws its pupils from all over the Soviet Union. They have worked on the committee organizing the “mathematical olympiad” problem competitions, which have greatly stimulated interest in mathematics among young people in the Soviet Union. They regularly contribute to the magazine “Kvant” (“Quantum”), a remarkable educational magazine devoted to mathematics and physics. This book contains a wealth of material usually found in geometry courses, and takes a new look at some of the usual theorems.

It deals with paths traced out by moving points, sets of points satisfying given geometrical conditions, and problems on finding maxima and minima. The book contains more than 200 problems which lead the reader towards some important areas of modern mathematics, and will interest a wide range of readers whether they be high school or university students, teachers, or simply lovers of mathematics.

The book was translated from the Russian by Anjan Kundu ans was published by Mir Publishers in 1980. It was also republished by Birkhauser in 2004. The link below is for the Mir Edition.

For those who really want to explore the world of mathematics, I would highly recommend using dynamic mathematics software GeoGebra. It is a Free and Open Source Software. Exploring the problems and exercises in this particular book (and of course otherwise also) with help of GeoGebra, has been for me, an exceptionally rewarding experience. This will reveal to you many subtle points and lead you to mathematical depths which you may have not thought of (or could not) diving into.

And I couldn’t but resist to add this quote:

… the feeling for beauty in mathematics is infectious.
It is caught not taught. It affects those with a flair for the subject.

- H. E. Huntley from The Divine Proportion

I hope you too get infected and infect others as well.

PDF | 11.3 MB | OCR | Cover | 600 dpi | Bookmarked | Paginated

(Some parameters on IA maybe different.)

You can get the book here (IA) and here (filecloud).

Password, if needed: mirtitles

Table of Contents

Preface (7)

INTRODUCTION (9)

Introductory Problems (9)
Copernicus’ Theorem (13)

1. SET OF POINTS (17)

A Family of Lines and Motion (23)
Construction Problems (25)
Set of Problems (30)

2. THE ALPHABET (35)

A Circle and a Pair of Arcs (38)
Squares of Distances (42)
Distances from Straight Lines (51)
The Entire Alphabet (57)

3. LOGICAL COMBINATIONS (60)

Through a Single Point (60)
Intersection and Union (67)
The “Cheese” Problem (74)

4. MAXIMUM AND MINIMUM (78)

Where to Put the Point (82)
The “Motor-Boat” Problem (84)

5. LEVEL CURVES (90)

The “Bus” Problem (90)
Functions on a Plane (93)
Level Curves (94)
Graph of a Function (94)
The Map of a Function (100)
Boundary Lines (101)
Extrema of Functions (103)

6. QUADRATIC CURVES (108)

Ellipses, Hyperbolas, Parabolas (108)
Foci and Tangents (113)
Focal Property of a Parabola (117)
Curves as the Envelopes of Straight Lines (121)
Equations of Curves (124)
The Elimination of the Radicals (129)
The End of Our Alphabet (130)
Algebraic Curves (138)

7. ROTATIONS AND TRAJECTORIES (140)

The Cardioid (141)
Addition of Rotations (142)
A Theorem on Two Circles (153)
Velocities and Tangents (157)
Parametric Equations (166)
Conclusion (170)

ANSWERS, HINTS, SOLUTIONS (172)
APPENDIX I. Method of Coordinates (181)
APPENDIX II. A Few Facts from School Geometry (183)
APPENDIX III. A Dozen Assignments (187)

Notation (196)

In this post we will see Problems in Linear Algebra by I. V. Proskuryakov.

From the Preface:

In preparing this book of problems the author attempted firstly, to give a sufficient number of exercises for developing skills in the solution of typical problems (for example, the computing of determinants with numerical elements, the solution of systems of linear equations with numerical coefficients, and the like), secondly, to provide problems that will help to clarify basic concepts and their interrelations (for example, the connection between the properties of matrices and those of quadratic forms, on the one hand and those of linear transformations, on the other), thirdly to provide for a set of problems that might supplement the course of lectures and help to expand the mathematical horizon of the student (instances are the properties of the Pfaffian of the skew-symmetric determinant, the properties of associated matrices, and so on).

Compared with other problem book, this one has few new basic features. They include problems dealing with polynomial matrices (Sec. 13), linear transformations of affine and metric spaces (Secs. 18 and 19), and a supplement devoted to group rings, and fields. The problems of the supplement deal with the most elementary portions of the theory. Still and all, I think it can be used in pre-seminar discussions in the first and second years of study.

Starred numbers indicated problems that have been worked out or provided with hints. Solutions are given for a small number of problems.

The book was translated from the Russian by George Yankovsky and was first published by Mir Publishers in 1978.

Note: Though the file size is large (~ 2^ M) the scan quality is really poor and is barely readable at times. 2-in-1 page scan with lot of warping. We tried to rectify this but were unable to do so. There is a weird colour hue (pink and blue) on many of the pages. If any one has access to a better copy please let us know.

All credits to original uploader.

You can get the book here.

Password, if needed: mirtitles

Contents

Preface 5

Chapter I
DETERMINANTS
Sec. 1. Second and third-order determinants 9
Sec. 2. Permutations and substitutions 17
Sec. 3. Definition and elementary properties of determinants of any order 22
Sec. 4. Evaluating determinants with numerical elements 31
Sec. 5. Methods of computing determinants of the th order 33
Sec. 6. Monirs, cofactors and the Laplace theorem 65
Sec. 7. Multiplication of determinants 74
Sec. 8. Miscellaneous problems 86

Chapter II
SYSTEMS OF LINEAR EQUATIONS

Sec. 9. Systems of equation solved by the Cramer rule 95
Sec. 10. The rank of a matrix. The linear dependence of vectors and linear forms 105
Sec. 11. Systems of linear equations 115

Chapter III
MATRICES AND QUADRATIC FORMS

Sec. 12. Operations involving matrices 131
Sec. 13. Polynomial matrices 155
Sec. 14. Similar matrices, characteristic and minimal polynomials. Jordan and diagonal forms of a matrix. Functions of matrices. 166
Sec. 15. Quadratic forms 182

Chapter IV
VECTOR SPACES AND THEIR LINEAR TRANSFORMATIONS

Sec. 16. Affine vector spaces 195
Sec. 17. Euclidean and unitary vector spaces 205
Sec. 18. Linear transformations of arbitrary vector spaces 220
Sec. 19. Linear transformations of Euclidean and unitary vector spaces 236
Sec. 20. Groups 251
Sec. 21. Rings and fields 265
Sec. 22. Modules 275
Sec. 23. Linear spaces and linear transformations (appendices to Secs. 10 and 16 to 19) 280
Sec. 24. Linear, bilinear, and quadratic functions and forms (appendix to Sec. 15) 284
Sec. 25. Affine (or point-vector) spaces 288
Sec. 26. Tensor algebra 295

ANSWERS

Chapter I. Determinants 312
Chapter II. Systems of linear equations 342
Chapter III. Matrices and quadratic forms 359
Chapter IV. Vector spaces and their linear transformations 397
Supplement 427

Index 449