Limit Distributions For Sums Of Independent Random Variables – Gnedenko, Kolmogorov

Posted on March 13, 2022 by The Mitr

In this post, we will see the book Limit Distributions For Sums Of Independent Random Variables – B. V. Gnedenko, A. N. Kolmogorov.

About the book

This is a translation of the Russian book (1949). There are various points of contact with the treatises by P. Levy [76] and by H. Cramer [21], but much of the material in the book has been hitherto available only in periodical articles, many of which are in Russian. The systematic account presented here combines generality with simplicity, making some of the most important and difficult parts of the theory of probability easily accessible to the reader. Beyond a knowledge of the calculus on the level of, say, Hardy’s Pure Mathematics, the book is formally self-contained. However, a certain amount of mathemati­cal maturity, perhaps a touch of single-minded perfectionism, is needed to penetrate the depth and appreciate the classic beauty of this definitive work.

The book was translated from Russian by K. L. Chung with Appendix by J. L. Doob. The book was published in 1954.

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Contents

PREFACE

PART I. INTRODUCTION

CHAPTER l. PROBABILITY DISTRIBUTIONS. RANDOM VARIABLES
AND MATHEMATICAL EXPECTATIONS 13

§ 1. Preliminary remarks 13
§ 2. Measures 16
§ 3. Perfect measures 18
§ 4. The Lebesgue integral 19
§ 5. Mathematical foundations of the theory of probability 20
§ 6. Probability distributions in R and in R^{1} 22
§ 7. Independence. Composition of distributions 26
§ 8. The Stieltjes integral 29

CHAPTER 2. DISTRIBUTIONS IN R^{1} AND THEIR CHARACTERISTIC FUNCTIONS 32

§ 9. Weak convergence of distributions 32
§ 10. Types of distributions 39
§ 11. The definition and the simplest properties of the characteristic function 44
§ 12. The inversion formula. and the uniqueness theater 48
§ 13. Continuity of the correspondence between distribution and characteristic functions 52
§ 14. Some special theorems about characteristic functions 55
§ 15. Moments and semi-invariants 61

CHAPTER 3. INFINITELY DIVISIBLE DISTRIBUTIONS 67

$ 16. Statement of the problem. Random functions with independent increments 67
§ 17. Definition and basic properties 71
§ 18. The canonical representation 76
§ 19. Conditions for convergence of infinitely divisible distributions 87

PART II. GENERAL LIMIT THEOREMS

CHAPTER 4. GENERAL LIMIT THEOREMS FOR SUMS OF INDEPENDENT SUMMANDS 94

§ 20. Statement of the problem. Sums of infinitely divisible summands 94
§ 21. Limit distributions with finite variances 97
§ 22. Law of large numbers 105
§ 23. Two auxiliary theorems 109
§ 24. The general form of the limit theorems: The accompanying infinitely divisible laws 112
§ 25. Necessary and sufficient conditioning for convergence 116

CHAPTER 5. CONVERGENCE TO NORMAL, POISSON, AND UNITARY DISTRIBUTIONS 125

§ 26. Conditions for convergence to normal and Poisson laws 125
§ 27. The law of large numbers 133
§ 28. Relative stability 139

CHAPTER 6. LIMIT THEOREMS FOR CUMULATIVE SUMS 145

§ 29. Distributions of the class L 145
§ 30. Canonical representation of distributions of the L 149
§ 31. Conditions for convergence 152
§ 32. Unimodality of distributions of the Glass L 157

PART III. IDENTICALLY DISTRIBUTED SUMMANDS

CHAPTER 7. FUNDAMENTAL LIMIT THEOREMS 162

§ 33. Statement of the problem. Stable laws 162
§ 34. Canonical representation of stable laws 164
§ 35. Domains of attraction for stable laws 171
§ 36. Properties of stable laws 182
§ 37. Domains of partial attraction 183

CHAPTER 8. IMPROVEMENT OF THEOREMS ABOUT THE CONVERGENCE TO THE NORMAL LAW 191

§ 38. Statement of the problem 191
§ 39. Two auxiliary theorems 196
§ 40. Estimation of the remainder term in a Lyapunov’s Theorem 201
§ 41. An auxiliary theorem 204
§ 42. Improvement of Lyapunov’s Theorem for nonlattice distribution 208
§ 43. Deviation from the limit law in the case of a lattice distribution 212
§ 44. The extremal character of the Bernoulli case 217
§ 45. Improvement of Lyapunov’s Theorem with higher monena {ог the continuous case 220
§ 46. Limit theorem for densities 222
§ 47. Improvement of the limit theorem for densities 228

CHAPTER 9. LOCAL LIMIT THEOREMS FOR LATTICE DISTRIBUTIONS 231

§ 48. Statement of the problem 231
§ 49. A local theorem for the normal limit distribution 232
§ 50. A local limit theorem for non-normal stable limit distributions 235
§ 51. Improvement of the limit theorem in the case of convergence to the normal distribution 240

APPENDIX I. NOTES ON CHAPTER 245
APPENDIX II. NOTES ON § 32 252
BIBLIOGRAPHY 257
INDEX 262

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Unsolved And Unsolvable Problems In Geometry – Meschkowski

Posted on March 12, 2022 by The Mitr

In this post, we will see the book Unsolved And Unsolvable Problems In Geometry
by H. Meschkowski.

About the book

We shall discuss some famous problems which occupied the attention of mathematicians for thousands of years and which have been shown to be unsolvable in recent decades and, in addition, some very old classical problems which, up till now, have scarcely been mentioned in books on the subject. We have endeavoured to vary well-known methods of proof for classical problems and to generalise their formulation.

Thus, the object of this book is not the systematic representation of a well-defined section of geometry. But rather our aim is of a methodological nature: we want to develop the kind of arguments which are suitable for the solution of the geometrical problems considered and to discuss the questions implied by the existence of unsolvable problems.

In the experience of mathematical institutes and authors of mathematical publications attempts are still made to solve problems which are known to be unsolvable. And so we fear that there will be some readers who, after reading this book, will believe that, despite everything, they have found a solution to the problem of trisecting an angle. We should like to advise these readers as follows: Read through the proof showing that such a solution is impossible three times and then try to find the mistake in your argument. Better still, give up such attempts since they really cannot lead anywhere. There are plenty of open questions—many of which are stated in this book—to which mathematical intuition may be applied with real prospect of success. However, it should be noted that it is not particularly easy to solve the open questions stated. It is sometimes much simpler to discover new results in a new branch of mathematics rather than to solve one of the problems left open in elementary geometry. However, according to Erhard Schmidt, it is better to solve old problems with new methods than to solve new problems with old. Therefore we feel justified in encouraging work on the “ elementary ” problems stated in this book.

For the reader’s convenience there is, at the end of the Table of Contents, a diagram showing the interrelationships of the chapters.

The book was translated from Russian by Jane A. C. Burlak and was published in 1966.

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Contents

PREFACE v

Chapter I. THE PROBLEMS 1

Chapter II. REGULAR PACKINGS IN THE PLANE AND ON THE SPHERE 5

1. Problems on the density of packing 5
2. Lemmas 8
3. Packing problems in the plane 13
4. Packing problems on the sphere 18
5. Further problems 25

Chapter III. IRREGULAR PACKINGS AND COVERINGS 29

1. The existence theorem 29
2. The packing problem graph 30
3. A point system for n = 7 32
4. The packing problem for x = 7 34
5. The covering problem graph 38
6. The covering problem for n = 5 40
7. The covering problem for 1 = 7 45
8. An Outline of further problems 47

IV. PACKING OF CONGRUENT SPHERES 50

1. The densest lattice packing 50
2. The finest rigid packing 54

V. LEBESGUE’S “TILE” PROBLEM 57

1. Formulation of the problem 57
2. The hexagonal tile 58
3. The tiles of Pal and Sprague 60
4. Curves and domains of constant width 62
5. Applications 65
6. The covering problem in ℛ3 67

VI. ON THE EQUIVALENCE OF POLYHEDRA BY DISSECTION 71

1. The plane problem 71
2. Dehn’s theorem 73
3. Examples on Dehn’s theorem 75
4. The algebra of polyhedra 82
5. The basis polyhedra 84

VII. THE DECOMPOSITION OF RECTANGLES INTO INCONGRUENT SQUARES 91

1. The problem 91
2. The definition of a graph 95
3. Some calculations for decompositions 97
4. The interpretation in electrical engineering 100
5. The decomposition of rectangles with commensurable sides 101

 

VIII. UNSOLVABLE EXTREMAL PROBLEMS

1. A theorem on the triangle 103
2. Besicovitch’s theorem 103
3. Sets of points at an integral distance apart 107

IX. CONSTRUCTIONS WITH RULER AND COMPASSES 113

1. Some comments on the use of ruler and compasses 113
2. Some construction problems 117
3. The proof of the impossibility of (8a) and (8b) 119
4. The quinquesection of the angle 123
5. The uniqueness of the construction 126

X. CONSTRUCTIONS ON THE SPHERE 128

1. Stereographic projection 128
2. Constructions with spherical compasses and spherical ruler 130
3. Squaring the circle on a sphere 133
4. A lemma 135

XI. PROBLEMS IN THE GEOMETRY OF SETS 137

1. A new definition of equivalence by dissection 137
2. Hausdorff’s theorem 141
3. Paradoxical decompositions in #3 150
4. Paradoxical decompositions in the plane 155

XII. THE METHODOLOGICAL SIGNIFICANCE OF “UNSOLVABLE” PROBLEMS 159

BIBLIOGRAPHY 163
INDEX 167

 

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The USSR Olympiad Problem Book – Shklarsky, Chentzov, Yaglom

Posted on March 11, 2022 by The Mitr

In this post, we will see the book The USSR Olympiad Problem Book by D. O. Shklarsky, N. N. Chentzov and I. M. Yaglom.

About the book

This book contains 320 unconventional problems in algebra, arithme­tic, elementary number theory, and trigonometry. Most of these problems first appeared in competitive examinations sponsored by the School Mathematical Society of the Moscow State University and in the Mathematical Olympiads held in Moscow. The book is designed for students having a mathematical background at the high school level ;T very many of the problems are within reach of seventh and eighth grade students of outstanding ability. Solutions are given for all the problems. The solutions for the more difficult problems are especially detailed.

The book was translated from Russian by John Maykovich and edited by Irving Sussman and was published in 1962.

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Contents

 

Foreword to the Third (Russian) Edition v
Preface to the Second (Russian) Edition vii
Editor’s Foreword to the English Edition xi

From the Authors 1
Suggestions for Using this Book 3
Numerical Reference to the Problems Given in the Moscow Mathematical Olympiads 5

1. Introductory Problems (1-14) 6
2. Alterations of Digits in Integers (15-26) 11
3. The Divisibility of Integers (27-71) 13
4. Some Problems from Arithmetic (72-109) 20
5. Equations Having Integer Solutions (110-130) 27
6. Evaluating Sums and Products (131-159) 30
7. Miscellaneous Problems from Algebra (160-195) 38
8. The Algebra of Polynomials (196-221) 45
9. Complex Numbers (222-239) 50
10. Some Problems of Number Theory (240-254) 56
11. Some Distinctive Inequalities (255-308) 61
12. Difference Sequences and Sums (309-320) 74

Solutions 80

Answers and Hints 423

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Problems In Probability Theory, Mathematical Statistics And Theory Of Random Functions – Sveshnikov

Posted on March 10, 2022 by The Mitr

In this post, we will see the book Problems In Probability Theory, Mathematical Statistics And Theory Of Random Functions by A. A. Sveshnikov.

About the book

Students at all levels of study in the theory of probability and in the theory of statistics will find in this book a broad and deep cross-section of problems (and their solutions) ranging from the simplest combinatorial probability problems in finite sample spaces through information theory, limit theorems and the use of moments.
The introductions to the sections in each chapter establish the basic formulas and notation and give a general sketch of that part of the theory that is to be covered by the problems to follow. Preceding each group of problems, there are typical examples and their solutions carried out in great detail. Each of these is keyed to the problems themselves so that a student seeking guidance in the solution of a problem can, by checking through the examples, discover the appropriate technique required for the solution.

The book was translated from Russian by Scripta Technica and edited by Bernard Gelbaum and was published in 1968.

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Contents

 

I. RANDOM EVENT 1

1. Relations among random events l
2. A direct method for evaluating probabilities 4
3. Geometric probabilities 6
4. Conditional probability. The multiplication theorem for probabilites 12
5. The addition theorem for probabilities 16
6. The total probability formula 22
7. Computation of the probabilities of hypotheses after a trial (Bayes formula) 26
8. Evaluation of probabilities of occurrence of an event in repeated independent trials 30
9. The multinomial distribution. Recursion formulas. Generating functions 36

II. RANDOM: VARIABLES 43

10. The probability distribution series, the distribution polygon and the distribution function of a discrete random variable 43
11. The distribution function and the probability density function of a continuous random variable 48
12. Numerical characteristics of discrete random variables 54
13. Numerical characteristics of continuous random variables 62
14. Poissons law 67
15. The normal distribution law 70
16. Characteristic functions 74
17. The computation of the total probability and the probability density in terms of conditional probability 80

III. SYSTEMS OF RANDOM VARIABLES 84

18. Distribution laws and numerical characteristics of systems of random variables 84
19. The normal distribution law in the plane and in space. The multidimensional normal distribution 91

20. Distribution laws of subsystems of continuous random variables and conditional distribution laws 99

IV. NUMERICAL CHARACTERISTICS AND DISTRIBUTION LAWS OF FUNCTIONS OF RANDOM VARIABLES 107

21. Numerical characteristics of functions of random variables .107
22. The distribution laws of functions of random variables. 115
23. The characteristic functions of systems and functions of random Variables 124
24. Convolution of distribution laws 128
25. The linearization of functions of random variables 136
26. The convolution of two-dimensional and three-dimensional normal distribution laws by use of the notion of deviation 145

V. ENTROPY AND INFORMATION 157

27. The entropy of random events and variables 157
28. The quantity of information 163

VI. THE LIMIT THEOREMS 171

29. The law of large numbers 171
30. The de Moivre-Laplace and Lyapunov theorems 176

 

VII. THE CORRELATION THEORY OF RANDOM FUNCTIONS 181

31. General properties of correlation functions and distribution laws-of random functions 181
32. Linear operations with random functions. 185
33. Problems on Passages 192
34. Spectral decomposition of stationary random functions. 198
35. Computation of probability characteristics of random functions at the output of dynamical systems 205
36: Optimal-Dynamical systems 216
37: The method of envelopes 226

VIII. MARKOV PROCESSES 231

38. Markov Chains 231
39. The Markov processes with a discrete number of states 246
40. Continuous Markov processes 256

IX. METHODS OF DATA PROCESSING 275

41. Determination of the moments of random variables from experimental data 275
42. Confidence levels and confidence intervals 286
43; “Tests of goodness-of-fit 300
44. Data processing by the method of least squares 325
45. Statistical methods of quality control 346
46. Determination of probability characteristics of random functions from experimental data 368

ANSWERS: AND SOLUTIONS 375
SOURCES OF TABLES REFERRED TO IN THE TEXT 471
BIBLIOGRAPHY 475

 

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Combinatorial Topology (Vol 1, 2, 3) – Aleksandrov

Posted on March 9, 2022 by The Mitr

In this post, we will see the three volume set of Combinatorial Topology by P. S. Aleksandrov.

Vol. 1: Introduction. Complexes. Coverings. Dimension.

Vol. 2: The Betti Groups

Vol. 3: Homological Manifolds. The Duality Theorems. Cohomology Groups of Compacta. Continuous Mappings of Polyhedra.

 

 

About the books

Volume 1 is a translation of the first third of P. S. Aleksandrov’s Kombinatornaya Topologiya. An appendix on the analytic geometry of Euclidean n-space is also included. The volume, complete in itself, deals with certain classical problems such as the Jordan curve theorem and the classification of closed surfaces without using the formal techniques of homology theory. The elementary but rigorous treatment of these problems, the introductory chapters on complexes and coverings and their applications to dimension theory, and the large number of examples and pictures should provide an excellent intuitive background for further study in combinatorial topology.
In Chapter I the references have been expanded to include a number of standard works in English. References to these and to the books and papers cited in Chapter I of the original are listed at the end of the chapter and correspond to the numbers enclosed in brackets in the body of the text. References in the remaining chapters are enclosed in brackets, capital letters referring to books and lower case letters to papers. These refer to the bibliography at the end of the book. The bibliography includes all papers mentioned in the original edition and a few which have been added by the translator. Volume 2 of this three volume set on Combinatorial Topology covers Betti Groups and Delta-Groups comprehensively. Third volume covers Homological manifolds, the duality theorems, cohomology groups of compacta, continuous mappings of polyhedra.

The books were translated from Russian by Horace Komm was first published in 1956 and third reprint by 1969.

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Contents

VOLUME 1

PART ONE
INTRODUCTION

Chapter I. SURVEY OF THE ELEMENTARY PROPERTIES OF TOPOLOGICAL SPACES 2
Chapter II. THE JORDAN THEOREM 39
Chapter III. SURFACES 66

PART TWO
COMPLEXES. COVERINGS. DIMESION

Chapter IV. COMPLEXES 116

Chapter V. SPERNER’S LEMMA AND ITS CORROLORIES 156

CHAPTER VI. INTRODUCTION TO DIMENSION THEORY 170

APPENDIX 1 202
BIBLIOGRAPHY 216
INDEX 218

VOLUME 2

PART THREE
THE BETTI GROUPS

CHAPTER VII. CHAINS. THE OPERATOR 𝝙 2

CHAPTER VIII. 𝝙-GROUPS OF COMPLEXES (LOWER BETTI OR HOMOLOGY GROUPS) 50

CHAPTER IX. THE OPERATOR 𝝙 AND THE GROUPS 𝝙^r. CANOICAL BASES. CALCULATIONS OF THE GROUPS 𝝙^r (𝕽,𝖀) and 𝝙^r (𝕽,𝖀) BY MEANS OF GROUPS 𝝙^r_0 (𝕽) 90

CHAPTER X. INVARIANCE OF THE BETTI GROUPS 125

CHAPTER XI. THE 𝛥-GROUPS OF COMPACTA 158

CHAPTER XII. THE RELATIVE CYCLES AND THEIR APPLICATIONS 178

APPENDIX 2 210
LIST OF SYMBOLS 238
INDEX 241

VOLUME 3

PART FOUR
HOMOLOGICAL MANIFOLDS. THE DUALITY THEOREMS. COHOMOLOGY GROUPS OF COMPACTA

CHAPTER XIII. HOMOLOGICAL MANIFOLDS (h-MANIFOLDS) 4

CHAPTER XIV. COHOMOLOGY GROUPS OF COMPACTA AND THE ALEXANDER PONTRYAGIN DUALITY 41

XV. LINKING. THE LITTLE ALEXANDER DUALITY 73

PART FIVE
INTRODUCTION TO THE THEORY OF CONTINUOUS MAPPINGS OF POLYHEDRA

XVI. THE BROUWER THEORY OF CONTINUOUS MAPPING IN R^n AND S^n 100

XVII. FIXED POINTS OF CONTINUOUS MAPPINGS OF POLYHEDRA 128

REFERENCES 146

INDEX 147

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Basic Equations And Special Functions Of Mathematical Physics – Arsenin

Posted on March 8, 2022 by The Mitr

In this post, we will see the book Basic Equations And Special Functions Of Mathematical Physics by V. Ya. Arsenin.

About the book

This book consists of two parts. Part 1 gives an account of the methods for solving typical problems in mathematical physics and an introduction to integral equations. Part 2 deals with the appli­cations of these methods to problems requiring the use of special functions. Extensive use is made in this book of the Dirac (𝛿-function. Generalised functions are introduced and their applications are described. Each chapter concludes with a number of problems illustrating the main text (altogether 150 problems with answers are given). This text was designed for students of physics and engineering and was based on a course given at the Department of Theoretical and Experimental Physics of the Moscow Engineering-Physics Institute.

The book was translated from the Russian by S. Chomet was published in 1968.

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Contents

PART I.

Chapter 1 Linear Equations with Two Independent Variables 3
Chapter 2 Partial Differential Equations in Physics. Boundary-value Problems 13
Chapter 3 The Method of Characteristics 32
Chapter 4 Separation of Variables (Fourier Method) 59
Chapter 5 The Method of Green’s Functions for Parabolic Equations 107
Chapter 6 The Method of Green’s Functions for Elliptical Equations 124
Chapter 7 Potentials 146
Chapter 8 Integral Equations 168
Chapter 9 Integral Equations with Symmetric Kernels 188

PART II

Chapter 10 Gamma Function 215
Chapter 11 Cylindrical Functions 223
Chapter 12 Spherical Harmonics 268
Chapter 13 Chebyshev-Hermite and Chebyshev—Laguerre Polynomials 291

Appendix Definition of Generalised Functions. The 𝛿 Function 303
Answers and Solutions 321
Bibliography 355
Index 357

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Partial Differential Equations Of Mathematical Physics (Vols. 1 and 2) – Tychonov, Samarski

Posted on March 7, 2022 by The Mitr

In this post, we will see the two set volume of Partial Differential Equations Of Mathematical Physics by A. N. Tychonov; A. A. Samarski.

About the books

This text reflects the authors’ unique approach to the study of the basic types of partial differential equations of mathematical physics. The system­atic presentation of the material offers the reader a natural entree to the subject. Each of the basic types of equations which are to be studied is motivated by its physical origins. The derivation of an equation from the physics to its final mathematical structure is very instructive to the student.
The authors have gone to great length to make clear the meaning of a solution to an initial value or boundary-value problem. Various methods of solving such problems are treated in great detail, as are the questions of existence and uniqueness of solutions. Thus, the student gains an apprecia­tion of the theoretical foundations of the subject and simultaneously acquires the manipulative skills for solving such problems.
The exercises which accompany each chapter have been selected to test the student’s ability both to formulate the correct mathematical statement of the problem and to apply the appropriate method for its solution. The applications treated by the authors are non-trivial and are completely worked out in detail.
The first volume covers the two dimensional class of partial differential equations of mathematical physics and is well suited as a basic text for both the undergraduate and graduate level at the university. The second volume will cover the three dimensional counterparts of the present volume and contain an additional chapter on the special functions which arise in mathe­matical physics.

The second volume covers the three-dimensional aspects of the material contained in the first volume. There is an additional chapter on the special functions that arise in the solution of the problems treated; the basic properties and representations of these functions are derived in a simple and straight­-forward manner. As in the first volume, the topics in each chapter are motivated by their physical origins and are supplemented by examples, as well as by exercises. A few additional but pertinent references have also been included.

The book were translated from the Russian by S. Radding. Volume 1 was published in 1964 and Volume 2 was published in 1967.

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Contents

1. CLASSIFICATIONS OF PARTIAL DIFFERENTIAL EQUATIONS OF THE SECOND ORDER

1-1. Differential equations of the second order with two independent variables 1
1-2. Differential equations of the second order with several independent variables 7
1-3. The canonical forms of linear equations with constant coefficients 9

2. HYPERBOLIC DIFFERENTIAL EQUATIONS

2-1. Simple problems which lead to hyperbolic differential equations and boundary-value problems 12
2-2. Wave-propagation method 36
2-3. Separation of variables 66
2-4. Problems with auxiliary conditions on the characteristics 101
2-5. Solutions of general linear hyperbolic differential equations 107
2-6. Applications to Chapter 2 117

3. PARABOLIC DIFFERENTIAL EQUATIONS

3-1. Simple problems which lead to parabolic differential equations 153
3-2. The method of separation of variables 171
3-3. Problems for the infinite straight line 188
3-4. Problems without initial conditions 210
3.5. Applications to Chapter 3 215

4. ELLIPTIC DIFFERENTIAL EQUATIONS

4-1. Problems which lead to Laplace’s differential equation 241
4-2. General properties of harmonic functions 252
4-3. Solutions of the boundary-value problems for simple regions by separation of variables 270
4-4, Green’s function (source function) 279
4-5. Potential theory 288
4-6. The difference method 325
4-7. Applications to Chapter 4 332

APPENDIX
Tables of error integrals and some cylindrical functions 363

Literature references of the editor 371

5. SPATIAL WAVE PROPAGATION 381

5-1. Initial Value Problems 381
1. The mean value method 381
2. The method of descent 383
3. Physical interpretation 385
4. The method of images 387

5-2. The Kirchhoff Formula 388
1. Derivation of the Kirchhoff formula 388
2. Consequences of the Kirchhoff wave formula 392

5-3. Vibrations of a Bounded Spatial Region 394
1. General scheme for the method of separation of variables.
Standing waves 394
2. Vibrations of a rectangular membrane 399
3. Vibrations of a circular membrane Problems 402

Problems 407
5-4. Applications of Chapter 5 408
1. Reduction of the equations of the theory of elasticity to wave equations 408

2. The equations of the electromagnetic field 410
1. The equations of electromagnetic fields and boundary conditions 410
2. Potential of an electromagnetic field 414
3. The electromagnetic field of an oscillator 416

6. SPATIAL HEAT PROPAGATION 422

6-1. Heat Propagation in an Unbounded Space 422
1. Green’s function 422
2. Heat propagation in an unbounded space 426

6-2. Heat Propagation in Bounded Regions 429
1. Scheme of the method of separation of variables
2. Cooling of a circular cylinder 429
3. Determination of the critical dimensions 432

6-3. Boundary-Valve Problems for Regions With Variable Boundaries 434
1. Green’s formula for the heat-conduction equation and
Green’s function 436
2. Solution of the boundary-value problem 441
3. Green’s function for an interval 443

6-4. Heat Potentials 445
1. Properties of heat potentials of single and double layers 445
2. Solution of boundary-value problems 446
Problems 449

6-5. Applications of Chapter 6 450
1. Diffusion of a cloud 450
2. The demagnetization of cylinders 453
3. The difference method for the heat-conduction equation 457

7. ELLIPTIC DIFFERENTIAL EQUATIONS (Continued) 465

7-1. Some Fundamental Problems which Lead to the Differential Equation 𝛥v+cv=0 465
1. Forced vibrations 465
2. Diffusion of gases with decomposition phenomena and with chain reactions 466
3. Diffusion in a moving medium 466
4. Formulation of the interior boundary-value problem for the equation 𝛥v+cv=0 467

7-2. Green’s Function 468
1. Green’s function 468
2. Integral representation of the solution 470
3. Potentials 473

7-3. Problems for an Unbounded Region: Radiation Principle 476
1. The equation Ju+cv=—f in an unbounded space 476
2. The principle of limiting absorption 477
3. The principle of limiting amplitude 478
4. Radiation conditions 480

7-4. Problems of the Mathematical Theory of Diffraction 484
1. Statement of the problem 484
2. Uniqueness of the solutions 485
3. Diffraction by a sphere 488
Problems 495

7-5. Applications of Chapter 7 497
1. Waves in conducting tubes (wave guides) 497
2. Electromagnetic vibrations in cavity resonators 507
1. The eigenvibrations of cylindrical endovibrators 507
2. Electromagnetic energy of eigenvibrations 511
3. Generation of vibrations in an endovibrator 513
3. Skin effect 515
4. The propagation of radio waves on the surface of the earth 519

APPENDIX: SPECIAL FUNCTIONS 525

A-1 Introduction 525

1. Differential equations of special functions 525
2. Formulation of the boundary value problems in the case k(a)=0 526

A-2 Cylindrical Functions 526

1. The cylindrical functions 532
1. Power series 532
2. Recursion formulas 532
3. The Bessel function of the first kind of order n + 1/2 539
4. Asymptotic representation of cylindrical functions for
large x 540

2. Boundary-value problems for the Bessel differential equation 542

3. The different types of cylindrical functions 545

1. Hankel functions 545
2. Hankel and Neumann functions 546
3. Bessel functions with imaginary arguments 548
4. The function K_0(x) 549

4. Integral representations and asymptotic representations of the Bessel functions 553

1. Integral representations for the Bessel functions of
integral order 553
2. Asymptotic representations of Bessel functions of the
first kind 556

5. The Fourier-Bessel integral and some integrals which contain Bessel functions 559

1. Fourier-Bessel integral 559
2. Some integrals whose integrands contain Bessel functions 559

6. Representation of cylindrical functions by contour integrals 563

1. Representation of cylindrical functions by contour integrals 563
2. The saddle point method. Asymptotic representations 568

A-3 Spherical Functions 571

1. Legendre polynomials 571
1. The generating function and Legendre polynomials 571
2. A recursion formula 573
3. The Legendre differential equation 573
4. The orthogonality of the Legendre polynomials 574
5. The norm of the Legendre polynomials 576
6. A differential relation for the Legendre polynomials 577
7. An integral formula. The boundedness of the Legendre polynomials
8. The associated Legendre polynomials 578
9. The closedness of systems of associated Legendre polynomials 580

2. Harmonic polynomials and spherical functions 584
1. Harmonic polynomials 584
2. Spherical functions 585
3. The orthogonality of the system of spherical functions 588
4. The completeness of systems of spherical functions 590
5. The expansion in spherical functions 591
3. Some examples of the application of spherical functions 594
1. The polarization of a sphere in a homogeneous field 594
2. The eigenvibrations of a sphere 597
3. The exterior boundary-value problem for the sphere 599

A4 The Tschebyscheff-Hermite and the Tschebyscheff-Laguerre
Polynomials 601

1. The Tschebyscheff-Hermite polynomial 601
2. The Tschebyscheff-Laguerre polynomials 604
3. Simple problems for the Schrédinger equation 610
1. The Schrédinger equation 610
2. The harmonic oscillator 611
3. The eigenvalue problem of rotators 613
4. Motion of electrons in a Coulomb field 614

Index 619

 

 

 

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Multicoloured Fins – Datskevich

Posted on March 6, 2022 by The Mitr

In this post, we will see the book Multicoloured Fins by V. Datskevich.

About the book

A little book describing various colourful fish and habits and habitats. It describes various tropical and non-tropical fish which are reared in aquariums.

The book was translated from Russian by was published in  by Publishers.

You can get the book here.

PS: This book brings back special memories for me. Till some years back I had two planted aquariums which hosted a variety of fish. One of the tanks had wild guppies and other fish collected from various streams and lakes. I used river sand as a substrate and plants would be very healthy. I have had vallisneria, hydrilla, ludwigia, amazon swords, cadomba, water ferns along with several other aquatic plants, though crown were water lilies which would continuously flower in the tanks. I have had almost all fish mentioned in the book at some point of time, so it was refreshing to know a bit more about them and their scientific names as well.

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Contents

 

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Challenging Mathematical Problems With Elementary Solution (Vols. 1 & 2) – Yaglom, Yaglom

Posted on March 5, 2022 by The Mitr

In this post, we will see the two volume set of Challenging Mathematical Problems With Elementary Solution by A. M. Yaglom; I. M. Yaglom .

Volume 1: Combinatorial Analysis and Probability Theory

Volume 2: Problems From Various Branches of Mathematics

About the books

This book is the first of a two-volume translation and adaptation of a well-known Russian problem book entitled Non-Elementary Problems in an Elementary Exposition The first part of the original, Problems on Combinatorial Analysis and Probability Theory, appears as Volume I, and the second part, Problems from Various Branches of Mathematics, as Volume II. The authors, Akiva and Isaak Yaglom, are twin brothers, prominent both as mathematicians and as expositors, whose many excellent books have been exercising considerable influence on mathematics education in the Soviet Union.

This adaptation is designed for mathematics enthusiasts in the upper grades of high school and the early years of college, for mathematics instructors or teachers and for students in teachers’ colleges, and for all lovers of the discipline; it can also be used in problem seminars and mathematics clubs. Some of the problems in the book were originally discussed in sections of the School Mathematics Circle (for secondary school students) at Moscow State University; others were given at Moscow Mathematical Olympiads, the mass problem-solving contests held annually for mathematically gifted secondary school students.
The chief aim of the book is to acquaint the reader with a variety of new mathematical facts, ideas, and methods. The form of a problem book has been chosen to stimulate active, creative work on the materials presented.

The first volume contains 100 problems and detailed solutions to them. Although the problems differ greatly in formulation and method of solution, they all deal with a single branch of mathematics: combinatorial analysis. While little or no work on this subject is done in American high schools, no knowledge of mathematics beyond what is imparted in a good high school course is required for this book. The authors have tried to outline the elementary methods of combinatorial analysis with some completeness, however. Occasionally, when needed, additional explanation is given before the statement of a problem.

Designed for advanced high school students, undergraduates, graduate students, mathematics teachers and any lover of mathematical challenges, this two-volume set offers a broad spectrum of challenging problems—ranging from relatively simple to extremely difficult. Indeed, some rank among the finest achievements of outstanding mathematicians.

Translated from a well-known Russian work entitled Non-Elementary Problems in an Elementary Exposition, the chief aim of the book is to acquaint the reader with a variety of new mathematical facts, ideas and methods. And while the majority of the problems represent questions in higher (“non-elementary”) mathematics, most can be solved with elementary mathematics. In fact, for the most part, no knowledge of mathematics beyond a good high school course is required.

Volume Two contains 74 problems from various branches of mathematics, dealing with such topics as points and lines, lattices of points in the plane, topology, convex polygons, distribution of objects, non-decimal counting, theory of primes and more. In both volumes the statements of the problems are given first, followed by a section giving complete solutions. Answers and hints are given at the end of the book

Ideal as a textbook, for self-study, or as a working resource for a mathematics club, this wide-ranging compilation offers 174 carefully chosen problems that will test the mathematical acuity and problem-solving skills of almost any student, teacher or mathematician.

Volume 1 was translated from Russian by James McCawley Jr. was published in 1964.

Volume 2 was translated from Russian by James McCawley Jr. was published 1967.

Credits to original uploader.

You can get Volume 1 here.

You can get Volume 2  here.

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Contents

Volume 1

PROBLEMS

I. Introductory problems 4
II. The representation of integers as sums and products 5
III. Combinatorial problems on the chessboard 10
IV. Geometric problems on combinatorial analysis 12
V. Problems on the binomial coefficients 15
VI. Problems on computing probabilities 20
VII. Experiments with infinitely many possible outcomes 27
VIII. Experiments with a continuum of possible outcomes 30

SOLUTIONS

I. Introductory problems 39
II. The representation of integers as sums and products 52
III. Combinatorial problems on the chessboard 76
IV. Geometric problems on combinatorial analysis 102
V. Problems on the binomial coefficients 125
VI. Problems on computing probabilities 141
VII. Experiments with infinitely many possible outcomes 194
VIII. Experiments with a continuum of possible outcomes 211

Volume 2

Preface to the American Edition v

Suggestions for Using the Book vii

Problems 3
I. Points and Lines 3
II. Lattices of Points in the Plane 5
III. Topology 7
IV. A Property of the Reciprocals of Integers 11
V. Convex Polygons 11
VI. Some Properties of Sequences of Integers 12
VII. Distribution of Objects 13
VIII. Nondecimal Counting 13
IX. Polynomials with Minimum Deviation from Zero (Tchebychev Polynomials) 20
X. Four Formulas for 𝜋 22
XI. The Calculation of Areas of Regions Bounded by Curves 33
XII. Some Remarkable Limits 38
XIII. The Theory of Primes 45

Solutions 45

Hints and Answers 199

Bibliography 213

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Mechanics Of Gyroscopic Systems – Ishlinskii

Posted on March 4, 2022 by The Mitr

In this post, we will see the book .

About the book

This book discusses a fairly wide range of problems in mechanics con­nected with the practical application of gyroscopes. The classical studies of A.N. Krylov and B. V. Bulgakov on the theory of gyroscopes are insufficient for solving the problems encountered in the development of new gyroscopic systems. Stricter standards of accuracy have made it necessary to take into account factors formerly neglected and to explain previously undetected experimental facts. New problems in kine­matics, the applied theory of elasticity, the theory of oscillations and sta­bility, and the theory of gyroscopes proper have thus arisen.

Several new papers on the theory of gyroscopic systems have been pub­lished by the author since this book was written (the present monograph is a second slightly revised edition of the book which was first printed in 1952 in a limited issue). Three of them are given here as appendixes.

The book was translated from Russian by Israel Program for Scientific Translations and was published in 1965.

Original scan by NASA TechDocs.

You can get the book here.

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Contents

 

FOREWORD 1

Chapter I. GEOMETRY AND KINEMATICS OF GYROSCOPIC SYSTEMS 5

Chapter II. ORIENTATION OF GYRO-CONTROLLED OBJECTS 46

Chapter III. PHENOMENA CONNECTED WITH THE ELASTICITY OF GYRO-SYSTEM ELEMENTS 75

Chapter IV. LINEAR THEORY OF GYROSCOPIC SYSTEMS 105

Chapter V. NONLINEAR PROBLEMS IN THE THEORY OF GYROSCOPES 178

Chapter VI. VARIOUS PROBLEMS IN GYRO-SYSTEM MECHANICS 221

APPENDIX I. THEORY OF COMPLEX GYROSCOPIC STABILIZATION SYSTEMS
APPENDIX II. THEORY OF THE GYROHORIZONCOMPASS
APPENDIX III. DETERMINING THE POSITION OF A MOVING OBJECT BY GYROS AND ACCELEROMETERS

BIBLIOGRAPHY 311

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