A Method For Studying Model Hamiltonians – A Minimax Principle For Problems In Statistical Physics – Bogolyubov Jr

Posted on April 28, 2022 by The Mitr

In this post, we will see the book A Method For Studying Model Hamiltonians – A Minimax Principle For Problems In Statistical Physics by N. N. Bogolyubov Jr..

About the book

In this book methods are proposed for solving certain problems in statistical physics which contain four-fermion interaction.

It has been possible, by means of “approximating (trial) Hamiltonians”, to distinguish a whole class of exactly soluble model systems. An essential difference between the two types of problem with positive and negative four-fermion interaction is discovered and examined. The determination of exact solutions for the free energies, single-time and many-time correlation functions, T-products and Green’s functions is treated for each type of problem.

The more general problem for which the Hamiltonian contains some terms with positive and others with negative four-fermion interaction is also investigated. On the basis of analysing and general­izing the results of Chapters 1 to 3, it becomes possible to formulate and develop a new principle, the minimax principle, for problems in statistical physics (Chapter 4).

The book was translated from Russian by P. J. Shepherd and was published in 1972.

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Contents

INTRODUCTION 1

§ I. GENERAL REMARKS 1
§ II. REMARKS ON QUASI AVERAGES 16

CHAPTER 1
PROOF OF THE ASYMPTOTIC RELATIONS FOR THE MANY-TIME CORRELATION FUNCTIONS

§ 1. GENERAL TREATMENT OF THE PROBLEM. SOME PRELIMINARY RESULTS AND FORMULATION OF THE PROBLEM 25

§ 2. EQUATIONS OF MOTION AND AUXILIARY OPERATOR INEQUALITIES 33

§ 3. ADDITIONAL INEQUALITIES 37

§ 4. BOUNDS FOR THE DIFFERENCE OF THE SINGLE-TIME AVERAGES 40

§ 5. REMARK (1) 47

§ 6. PROOF OF THE CLOSENESS OF AVERAGES CONSTRUCTED ON THE BASIS OF MODEL AND TRIAL HAMILTONIANS FOR “NORMAL” ORDERING OF THE OPERATORS IN THE AVERAGES 50

§ 7. PROOF OF THE CLOSENESS OF THE AVERAGES FOR ARBITRARY ORDERING OF THE OPERATORS IN THE AVERAGES 54

§ 8. ESTIMATES OF THE ASYMPTOTIC CLOSENESS OF THE MANY-TIME CORRELATION AVERAGES 57

CHAPTER 2
CONSTRUCTION OF A PROOF OF THE GENERALIZED ASYMPTOTIC RELATIONS FOR THE MANY-TIME CORRELATION AVERAGES 65

§ 1. SELECTION RULES AND CALCULATION OF THE AVERAGES 65

§ 2. GENERALIZED CONVERGENCE 70

§ 3. REMARK 74

§ 4. PROOF OF THE ASYMPTOTIC RELATIONS 76

§ 5. REMARK ON THE CONSTRUCTION OF UNIFORM BOUNDS 79

§ 6. GENERALIZED ASYMPTOTIC RELATIONS FOR THE GREEN’S FUNCTIONS 82

§ 7. THE EXISTENCE OF GENERALIZED LIMITS 85

CHAPTER 3
CORRELATION FUNCTIONS FOR SYSTEMS WITH FOUR-FERMION NEGATIVE INTERACTION 90

§ 1. CALCULATION OF THE FREE ENERGY FOR MODEL SYSTEMS WITH ATTRACTION 90

§ 2. FURTHER PROPERTIES OF THE EXPRESSIONS FOR THE FREE ENERGY 101

§ 3. CONSTRUCTION OF ASYMPTOTIC RELATIONS FOR THE FREE ENERGY 105

§ 4. ON THE UNIFORM CONVERGENCE WITH RESPECT TO 𝛳 OF THE FREE ENERGY FUNCTION AND ON BOUNDS FOR THE QUANTITIES 𝛿_{V} 111

§ 5. PROPERTIES OF PARTIAL DERIVATIVES OF THE FREE ENERGY FUNCTION. THEOREM 3.III 114

§ 6. RIDER TO THEOREM 3.III AND CONSTRUCTION OF AN AUXILIARY INEQUALITY 117

§ 7. ON THE DIFFICULTIES OF INTRODUCING QUASI-AVERAGES 120

§ 8. A NEW METHOD OF INTRODUCING QUASI-AVERAGES 124

§ 9. THE QUESTION OF THE CHOICE OF SIGN FOR THE SOURCE-TERMS 30

§ 10. THE CONSTRUCTION OF UPPER-BOUND INEQUALITIES IN THE CASE WHEN C = 0 131

CHAPTER 4
MODEL SYSTEMS WITH POSITIVE AND NEGATIVE INTERACTION COMPONENTS 137

§ 1. HAMILTONIAN WITH NEGATIVE COUPLING CONSTANTS (REPULSIVE INTERACTION) 137

§ 2. FEATURES OF THE ASYMPTOTIC RELATIONS FOR THE FREE ENERGIES IN THE CASE OF SYSTEMS WITH POSITIVE INTERACTION 141

§ 3. BOUNDS FOR THE FREE ENERGIES AND CORRELATION FUNCTIONS 143

§ 4. EXAMINATION OF AN AUXILIARY PROBLEM 146

§ 5. SOLUTION OF THE QUESTION OF UNIQUENESS 150

§ 6. HAMILTONIANS WITH COUPLING CONSTANTS OF DIFFERENT SIGNS. THE MINIMAX PRINCIPLE 154

REFERENCES 164

INDEX 169

 

 

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Calculus Of Variations – Gelfand, Fomin

Posted on April 27, 2022 by The Mitr

In this post, we will see the book Calculus Of Variations by I.M. Gelfand; S.V. Fomin.

About the book

This book is a modern introduction to the calculus of variations and certain of its ramifications, and I trust that its fresh and lively point of view will serve to make it a welcome addition to the English-language literature on the subject. The present edition is rather different from the Russian original. With the authors’ consent, I have given free rein to the tendency of any mathematically educated translator to assume the functions of annotator and stylist.

The problems appearing at the end of each of the eight chapters and two appendices were made specifically for the English edition, and many of them comment further on the corresponding parts of the text.

The book was translated from Russian by Richard Silverman and was published in 1963.

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Contents

1. ELEMENTS OF THE THEORY 1.

1: Functionals. Some Simple Variational Problems, 1.
2: Function Spaces, 4.
3: The Variation of a Functional. A Necessary Condition for an Extremum, 8.
4: The Simplest Variational Problem. Euler’s Equation, 14.
5: The Case of Several Variables, 22.
6: A Simple Variable End Point Problem, 25.
7: The Variational Derivative, 27.
8: Invariance of Euler’s Equation, 29. Problems, 31.

2. FURTHER GENERALIZATIONS 34.

9: The Fixed End Point Problem for n Unknown Functions, 34.
10. Variational Problems in Parametric Form, 38.
11: Functionals Depending on Higher-Order Derivatives, 40.
12: Variational Problems with Subsidiary Conditions, 42.
Problems, 50.

3. THE GENERAL VARIATION OF A FUNCTIONAL 54.

13: Derivation of the Basic Formula, 54.
14: End Points Lying on Two Given Curves or Surfaces, 59.
15: Broken Extremals. The Weierstrass-Erdmann Conditions, 61.
Problems, 63.

4. THE CANONICAL FORM OF THE EULER EQUATIONS AND RELATED TOPICS 67.

16: The Canonical Form of the Euler Equations, 67.
17: First Integrals of the Euler Equations, 70.
18: The Legendre Transformation, 71.
19: Canonical Transformations, 77. 20: Noether’s Theorem, 79.
21: The Principle of Least Action, 83.
22: Conservation Laws, 85.
23: The Hamilton-Jacobi Equation. Jacobi’s Theorem, 88.
Problems, 94,

5. THE SECOND VARIATION. SUFFICIENT CONDITIONS FOR A WEAK EXTREMUM 97.

24: Quadratic Functionals. The Second Variation of a Func-tional, 97.
25: The Formula for the Second Variation. Legendre’s Condition, 101. 26: Analysis of the Quadratic Functional | (Ph’? + Qh?) dx, 105.
27: Jacobi’s Necessary Condition. More on Conjugate Points, 111.
28: Sufficient Conditions for a Weak Extremum, 115.
29: Generalization to n Unknown Functions, 117.
30: Connection Between Jacobi’s Condition and the Theory of Quadratic Forms, 125.
Problems, 129.

6. FIELDS. SUFFICIENT CONDITIONS FOR A STRONG EXTREMUM 131.

31: Consistent Boundary Conditions. General Definition of a Field, 131.
32: The Field of a Functional, 137.
33: Hilbert’s Invariant Integral, 145.
34: The Weierstrass E-Function. Sufficient Conditions for a Strong Extremum, 146.
Problems, 150.

7. VARIATIONAL PROBLEMS INVOLVING MULTIPLE INTEGRALS 152.

35: Variation of a Functional Defined on a Fixed Region, 152.
36: Variational Derivation of the Equations of Motion of Continuous Mechanical Systems, 154.
37: Variation of a Functional
Defined on a Variable Region, 168.
38: Applications to Field Theory, 180.
Problems, 190.

8. DIRECT METHODS IN THE CALCULUS OF VARIATIONS 192.

39: Minimizing Sequences, 193.
40: The Ritz Method and the Method of Finite Differences, 195.
41: The Sturm-Liouville Problem, 198.
Problems, 206.

APPENDIX I PROPAGATION OF DISTURBANCES AND THE CANONICAL EQUATIONS 208.

APPENDIX II VARIATIONAL METHODS IN PROBLEMS OF OPTIMAL CONTROL 218.

BIBLIOGRAPHY 227.

INDEX 228.

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Forging Practice – Kamenshchikov, Koltun, Naumov, Chernobrovkin

Posted on April 26, 2022 by The Mitr

In this post, we will see the book Forging Practice by G. Kamenshchikov; S. Koltun; V. Naumov; B. Chernobrovkin.

About the book

All machines are built up of parts made of different materials and by various manufacturing processes. Some parts are cast from metals; some are forged, while others are produced by machining on different kinds of machine tools. Castings and forgings have to be machined before they acquire their proper shape, exact dimensions and surface finish. Forged parts, whether they are to be machined or not, are called forgings. This book describes various technologies used in the forgings.

The book was translated from Russian (translator’s name is not mentioned) was published  by Peace Publishers in 1960.

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Contents

Introduction 7
The Importance of Forging in Machine Building7

Chapter I. Bench Operations 8
Chapter II. Introduction to Forging Practice 18
Chapter III. Fuel and Its Combustion 36
Chapter IV. Heating Devices. 60
Chapter V. Heating Steel for Forging 101
Chapter VI. Chief Hand-Forging Operations 115
Chapter VII. The Influence of Deformation on Forgings and the Calculation of Forgings 161
Chapter VIII. Hammers for Hammer Forging 190
Chapter IX. Forging Operations and Hammer Forging Tools 232
Chapter X. The Technological Process and Examples of Hammer Forging 259
Chapter XI. Forging presses and Their Operation 278
Chapter XII. Automatic Forging and Stamping Machines 313
Chapter XIII. Drop-Forging (Hot Stamping) 322
Chapter XIV. Special Features of Forging Non-Ferrous Metals and Their Alloys 366
Chapter XV. Heat Treatment, Defects and Inspection of Forgings 368
Chapter XVI. Organisation of Work and of the Working Place 376
Chapter XVII. Safety Engineering 398

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Quantum Mechanics – Davydov

Posted on April 25, 2022 by The Mitr

In this post, we will see the book Quantum Mechanics by A. S. Davydov.

About the book

The present book is an extended exposition of a course of lectures in quantum mechanics given by the author over several years to students of the Physics Depart­ment of Moscow University. This course is given after the sections on atomic and nuclear physics of the general course in physics. In those sections of the course a historical outline of the development of contemporary ideas about the structure of atoms and of atomic nuclei is given, as well as the experimental data on which quantum mechanics is based. The present book therefore does not touch at all upon the historical development of quantum theory.
The main emphasis in the present book is upon the physical ideas and the mathematical formalism of the quantum theory of the non-relativistic and quasi-relativistic (up to terms of order v2lc2) motion of a single particle in an external field. In particular, we show the inapplicability of the concept of an essentially relativistic motion of a single particle. We put great emphasis upon representation theory, the theory of canonical transformations, scattering theory, and quantum transitions. A relatively detailed exposition is given of the theory of systems consisting of identical bosons or fermions. We also devote several sections to the theory of molecules, the theory of chemical binding, and solid state theory.
An important role is played in this book by the theory of second quantisation as a method to study systems consisting of a large number of identical particles. In parti­cular, we give the basic ideas of the theories of superconductivity and of superfluidity. The basic ideas are given of the methods for quantising the meson field, the electro­magnetic field (without charges) and the electron-positron field, neglecting diver­gencies and renormalisation, as these topics are dealt with in special books which are studied after quantum mechanics.
The present book can be used as an introduction to a study of quantum elctrodynamics, nuclear theory, or solid state theory. To read it, it is necessary to be fami­liar with the usual contents of university courses in mathematics, classical mechanics, and electrodynamics. For reference purposes we give at the end of the book some mathematical appendices about special functions, matrices, and group theory.
In this book we mainly refer to review or original papers when we want to indicate where the reader can study a more detailed discussion of a topic. These references do not pretend to be complete.
Although we do not consider in this book special methodological problems, the exposition is based upon dialectic materialism, that is, we start from the idea that the regularities of atomic and nuclear physics which are studied in quantum mechanics are objective regularities of nature.

The book is intended for students of physics, studying quantum mechanics. It can also be used as a reference book for teachers and other scientists.

The book was translated from Russian by D. Ter Haar was published in 1965.

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Contents

PREFACE

PREFACE TO THE ENGLISH EDITION

CHAPTER I. THE BASIC CONCEPTS OF QUANTUM MECHANICS 1
CHAPTER II. CHANGE OF QUANTUM STATES IN TIME 45
CHAPTER III. THE CONNEXION BETWEEN QUANTUM MECHANICS AND CLASSICAL MECHANICS 69
CHAPTER IV. ELEMENTARY REPRESENTATION THEORY 85
CHAPTER V. THE SIMPLEST APPLICATIONS OF QUANTUM MECHANICS 106
CHAPTER VI. THE MOTION OF A PARTICLE IN A CENTRAL FIELD OF FORCE 123
CHAPTER VII. APPROXIMATE METHODs FOR EVALUATING EIGENVALUES AND EIGENFUNCTIONS 169
CHAPTER VIII. THE FOUNDATIONS OF A QUASI-RELATIVISTIC QUANTUM THEORY OF THE MOTION OF A PARTICLE IN AN EXTERNAL FIELD 189
CHAPTER IX. THE THEORY OF QUANTUM TRANSITIONS UNDER THE INFLUENCE OF AN EXTERNAL PERTURBATION 285
CHAPTER X. QUANTUM THEORY OF SYSTEMS CONSISTING OF IDENTICAL PARTICLES 336
CHAPTER XI. QUANTUM THEORY OF SCATTERING 374
CHAPTER XII. ELEMENTARY THEORY OF MOLECULES AND CHEMICAL BONDS 472
CHAPTER XIII. Basic IDEAS OF THE QUANTUM THEORY OF THE SOLID STATE 526
CHAPTER XIV. SECOND QUANTISATION OF SYSTEMS OF IDENTICAL BOSONS 553
CHAPTER XV. SECOND QUANTISATION OF SYSTEMS OF IDENTICAL FERMIONS 617

MATHEMATICAL APPENDICES
A. Some properties of the Dirac delta-function 654
B. The angular momentum operators in spherical coordinates 657
C. Linear operators in a vector space; matrices 658
D. Confluent hypergeometric functions; Bessel functions 664
E. Group theory 670

INDEX 675

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Diseases Of The Thyroid Gland – Khavin, Nikolayev

Posted on April 24, 2022 by The Mitr

In this post, we will see the book Diseases Of The Thyroid Gland by I. Khavin and O. Nikolayev.

About the book

A book about various diseases of the thyroid gland and their treatments. For each disease the pathogenesis, pathohistology and aetiology along with clinical picture is provided. Finally the ways of treating these diseases are discussed.

The book was translated from Russian (translators name is not mentioned) and was published in 1955 by Foreign Languages Publishing House.

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Contents

 

Chapter I. Structure and Function of the Thyroid Gland 5

Chapter II. Thyrotoxicosis 15

Pathogenesis and Aetiology 15
Clinical Picture 28
Treatment 68

Chapter III. Hypothyroidism and Myxoedema 107

Pathogenesis and Aetiology 107
Clinical Picture 110
Treatment 119
Dysthyroidism 122

Chapter IV. Thyroiditis (Strumitis) 125

Pathogenesis and Aetiology 125
Clinical Picture 126
Treatment 130

Chapter V. Endemic Goitre 132

Definition 132
History 134
Distribution of Endemic Goitre in the World and Pathohistology 136
Aetiology and Pathogenesis 141
Clinical Picture 176
Prophylaxis and Treatment 191

Chapter VI. Malignant Neoplasms of the Thyroid 204

Chapter VII. Surgical Treatment of Thyroid Diseases 220

History 220
Surgical Treatment of Endemic Goitre 224
Surgical Treatment of Basedow’s Disease (Thyrotoxicosis) 238

 

 

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A World We Do Not See – Lev Potkov

Posted on April 23, 2022 by The Mitr

In this post, we will see the book A World We Do Not See by Lev Potkov.

About the book

The reading public has always been keenly interested in the world of microscopic beings. And it is not difficult to explain why this is so, for the organisms which can
be seen only through the microscope play an exceeding­ly important role in Nature and in the life of man. It is micro-organisms that are responsible for the rotation of various substances in soil and to a great extent de­termine its productivity. Therefore, it is only natural that microbes occupy such a prominent place in the works of the great Russian soil scientists V. Dokuchayev, P. Kostychev and V. Williams.
The activity of microbes in water reservoirs is also very important as they are the direct producers of fodder for aquatic fauna. Today we know that a number of geological formations owe their existence to microbes, which, consequent­ly, are a potent geological factor. Microbiological methods are now widely used in geological survey, par­ticularly, in petroleum prospecting. Many industries, especially those dealing with fermen­tation and food processing, are based on the action of micro-organisms. Further development of such industries is wholly dependent on the progress of microbiology. The readers hardly need being reminded of the fact that micro-organisms are the cause of various diseases of man, animals and plants. Mankind cannot success­fully control disease without a thorough study of its agents.

Recently, microbes have been discovered which produce therapeutic substances—antibiotics—whose prac­tical application has proved highly effective, so that a
number of diseases formerlv considered incurable no longer constitute a threat to humanity. Antibiotics can be used for the treatment not only of humans, but also of animals and plants. The production of antibiotics is an extensive branch of industry.
We could give many more examples illustrating the importance of microbes, but even the few already mentioned are sufficient to draw attention to this group of organisms and make them the object of thorough study. Russian scientists have made a valuable contribution to microbiology, and the history of the science shall preserve for ever the names of I. Mechnikov, N. Gama- leya, I). Zabolotny, S. Vinogradsky, V. Omelyansky, B. Isachenko, and many other workers. Soviet microbiologists are today further developing the achievements of their teachers on the basis of up-to-date techniques.
It is not easy to describe in popular form the develop­ment and successes of microbiology. The book by L. Potkov is an attempt to solve this task. The author had worked long and assiduously, and we hope his effort will not prove in vain and that the book will meet the requirements of the readers.

The book was translated from Russian by W. Perelman and was published in 1953 by Foreign Languages Publishing House.

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Contents

EDITOR’S NOTE 5

INTRODUCTION 7

CHAPTER I. AN HISTORICAL ESSAY 10
CHAPTER IT. MICROBES 56
CHAPTER II. SOIL, PLANTS AND MICRO-ORGANISMS 81
CHAPTER IV. THE EARHVS HISTORY AND MICRO-ORGANISMS 103
CHAPTER V. MICROBES IN FOOD AND FODDER 117

Wine-Making 117
Production of Alcohol 119
Brewing 121
Kvass 122
Production of Vinegar 123
Bacteria in Milk and Dairy Desine 125
Cheese 128
Bread-Making 131
Sauerkraut and Pickled Cucumbers 133
Silage 134

CHAPTER VI. FOODSTUFFS AND FODDER SPOILT BY MICROORGANISMS 136

Bread 136
Spontaneous Heating in Grain, Flour, Silage and Hay 138
Spoilage Organisms in Milk and Dairy Produce 139
Putrefactive Microbes 141
Diseases of Wine 144

CHAPTER VII. PROTECTION OF FOODSTUFFS FROM SPOILAGE MICRO-ORGANISMS 146

Pickling 148
Canning 148
Cold Storage 149

CHAPTER VIIT. MICROBES IN CHEMICAL INDUSTRY 152

Butyric Acid 152
Citric Acid 152
Lactic Acid 153
Glycerine 153
Acetone and Butyl Alcohol 154
Microbes and the Textile Industry 154
Finishing Textiles 155
Tanning Leather 156
Fermentation of Cellulose 157

CHAPTER IX. MICROBES HARMFUL. TO FUEL, METAL 159

CHAPTER X. MICROBES CAUSING INFECTIOUS DISEASES IN MAN AND ANIMALS AND WOOD 163

Intestinal Diseases (Typhoid Fever and Dysentery) 166
Asiatic Cholera 168
Brucellosis 170
Botulism 172
Smallpox or Variola 173
Diphtheria 175
Typhus 177
Recurrent (Relapsing) Fever 178
Injections by Pyogenic Micro-Organisms 179
Porest-Spring Encephalitis 181
Plague 184
Anthrax 187
Tetanus 192
Rabies (Hydophobia) 194
Tuberculosis 196
Leprosy 199

CHAPTER XI. FIGHT AGAINST PATHOGENIC MICROBES 201

CHAPTER XII. DIRECTED VARIABILITY IN MICROBES 212

EPILOGUA 127

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Computational Methods Of Linear Algebra – Faddeeva

Posted on April 22, 2022 by The Mitr

In this post, we will see the book Computational Methods Of Linear Algebra by V. N. Faddeeva.

About the book

English-speaking physicists, mathematicians, and engineers will welcome this first English translation of a unique and valuable Russian work. Translated especially for this edition by Curtis D. Benster, it is a basic work in English that presents a systematic exposition of computational methods of linear algebra— the classical ones, as well as those developed quite recently in Russia and elsewhere, by A. N. Krylov, A. M. Danilevsky, D. K. Faddeev, and others.

This unusual computer’s guide shows in detail how to derive numerical solutions of problems in mathematical physics which are frequently connected with the numerical solution of basic problems of linear algebra. Theory as well as individual practices are given.

The book is divided into three long chapters, with numerous sub-chapters. The first chapter provides the basic material from linear algebra (matrices, linear transformations, the Jordan canonical form, etc.) that is indispensable to what follows. The second chapter describes methods of numerical solution of systems of linear equations. The third chapter provides methods of computing the proper numbers and proper vectors of a matrix.

One of the outstanding and valuable features of this work is the care which has been taken in the preparation of the twenty-three tables which accompany chapters II and III. These tables have been specially rechecked and corrected by the translator, and carefully set (with uniform double-spacing) so as to allow the user to follow the computations throughout with case.

The book was translated from Russian by Curtis D. Benster and was published in 1959.

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Contents

Chapter 1. Basic material from linear algebra 1

§1. Matrices 1
§2. n-Dimensional vector space 23
§3. Linear transformations 33
§4. The Jordan canonical form 49
§5. The concept of limit for vectors and matrices 54

Chapter 2. Systems of linear equations 63

§6. Gauss’s method 65
§7. The evaluation of determinants 72
§8. Compact arrangements for the solution of sacteenogeneyaes linear systems 75
§9. The connection of Gauss’s method with the decomposition of a matrix into factors 79
§10. The square-root method 81
§11. The inversion of a matrix 85
§12. The problem of elimination 90
§13. Correction of the elements of the inverse matrix 99
§14. The inversion of a matrix by partitioning 102
§15. The bordering method 105
§16. The escalator method 111
§17. The method of iteration 117
§18. The preparatory conversion of a system of linear equations into form suitable for the method of iteration 127
§19. Scidel’s method 131
§20. Comparison of the methods 142

Chapter 3. The proper numbers and proper vectors of a matrix 147

§2l. The method of A. N. Krylov 149
§22. The determination of proper vectors by the method of A. N. Krylov 159
§23. Samuelson’s method 161
§24. The method of A. M. Danthcsky 166
§25. Leverrier’s method in D. K. Faddeev’s modification 177
§26. ‘The escalator method 183
§27. ‘The method of interpolation 192
§28. Comparison of the methods 201
§29. Determination of the first proper number of a matrix, First case 202
§30. Improving the convergence of the iterative process 211
§31. Finding the proper numbers next in line 219
§32. Determination of the proper numbers next in line and their proper vectors as well 222
§ 33. Determination of the first proper number 234
§ 34. The case of a matrix with nonlinear elementary divisors 235
§ 35. Improving the convergence of the iterative process for solving
systems of linear equations 239

Bibliography 243
Index 247

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Collection of Problems on Classical Mechanics – Kotkin, Serbo

Posted on April 21, 2022 by The Mitr

In this post, we will see the book Collection Of Problems In Classical Mechanics
by G. L. Kotkin; V. G. Serbo.

About the book

This collection is meant for physics students. Its contents correspond roughly to the mechanics course in the textbooks by Landau and Lifshitz (I960), Goldstein (1950), or ter Haar (1964). We hope that the reading of this collection will give pleasure not only to students studying mechanics, but also to people who already know it. We follow the order in which the material is presented by Landau and Lifshitz, except that we start using the Lagrangian equations in § 4. The problems in §§ 1-3 can be solved using the Newtonian equations of motion together with the energy, linear momentum and angular momen­tum conservation laws. As a rule, the solution of a problem is not finished with obtaining the required formulae. It is necessary to analyse the results and this is of great interest and by no means a “mechanical” part of the solution. In particular, it is very desirable to study limiting cases. This is useful not only for checking purposes and for an understanding of the solution obtained, but also for a preliminary analysis of the problem which can be used to learn how to find the motion of a system by intuition. It is also very useful to investigate what happens to a solution, if the conditions of the problem are varied. We have, therefore, suggested further problems at the end of several solutions.

 

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Contents

PREFACE

PROBLEMS

1. Integration of One-dimensional Equations of Motion 3
2. Motion of a Particle in Three-dimensional Potentials 6
3. Scattering in a Given Field. Collisions between Particles 10
4. Lagrangian Equations of Motion. Conservation Laws 13
5. Small Oscillations of Systems with One Degree of Freedom 19
6. Small Oscillations of Systems with Several Degrees of Freedom 24
7. Oscillations of Linear Chains 34
8. Non-linear Oscillations 36
9. Rigid-body motion. Non-inertial Coordinate Systems 38
10. The Hamiltonian Equations of Motion 42
11. Poisson Brackets. Canonical Transformations 44
12. The Hamilton-Jacobi Equation 50
13. Adiabatic Invariants 53

ANSWERS AND SOLUTIONS

1. Integration of One-dimensional Equations of Motion 63
2. Motion of a Particle in Three-dimensional Potentials 73
3. Scattering in a Given Field. Collisions between Particles 110
4. Lagrangian Equations of Motion. Conservation Laws 125
5. Small Oscillations of Systems with One Degree of Freedom 137
6. Small Oscillations of Systems with Several Degrees of Freedom 152
7. Oscillations of Linear Chains 183
8. Non-linear Oscillations 197
9. Rigid-body motion. Non-inertial Coordinate Systems 206
10. The Hamiltonian Equations of Motion 219
11. Poisson Brackets. Canonical Transformations 221
12. The Hamilton-Jacobi Equation 236
13. Adiabatic Invariants 253

REFERENCES 275

INDEX 277

 

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Problems in Electrodynamics – Batygin, Toptygin

Posted on April 20, 2022 by The Mitr

In this post, we will see the book Problems In Electrodynamics by V.V. Batygin; I.N. Toptygin
.

About the book

This book contains about 750 problems on classical electrodynamics and its more important applications, including over 150 problems on the special theory of relativity, and about 70 prob­lems on vector and tensor analysis.
In addition to problems illustrating fundamental concepts and laws of electrodynamics, which can be solved by purely mathematical methods, the collection includes a large number of more complicated problems (these are indicated by asterisks). Some of the solutions involve a considerable amount of effort, while others are purely theoretical in nature and follow from a lecture course on electrodynamics (propagation of waves in anisotropic and gyrotropic media, motion of charged particles in the electro­magnetic field, representation of the electromagnetic field by a set of oscillators, and so on). Finally, there are problems which are concerned with topics which are not well covered by existing texts, for example, interaction of charged particles with matter (Chapter XIII), applications of conservation laws to the analysis of collision processes and particle disintegration (Chapter XI), ferro­magnetic resonance (Chapter VI), and so on. The second part of the book gives answers and solutions to a large number of these problems.
Each section is prefaced by a short theoretical introduction in which the necessary formulae are given. These short introductions do not pretend to be complete; more extensive treat­ments will be found in the books listed in the bibliography.

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

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Contents

Preface iii

Problems Solutions
Chapter I. Vector and tensor calculus 1 185

Chapter II. Electrostatics in vacuum 15 194
Chapter III. Electrostatics of conductors and dielectrics 27 207
Chapter IV. Steady currents 49 240
Chapter V. Magnetostatics 56 247
Chapter VI. Electrical and magnetic properties of matter 68 266
Chapter VII. Quasi-stationary electromagnetic fields 82 286
Chapter VIII. Propagation of electromagnetic waves 93 313
Chapter IX. Electromagnetic oscillations in bounded bodies 113 358
Chapter X. Special theory of relativity 120 375
Chapter XI. Relativistic mechanics 135 391
Chapter XII. Emission of electromagnetic waves 153 420
Chapter XIII. The radiation emitted during the interaction of charged particles with matter 177 462
Appendix I. The 𝛿-function 481
Appendix II. Spherical Legendre functions 234
Appendix III. Cylindrical functions 487

Index 491

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Problems In Quantum Mechanics – Gol’dman, Krivchenkov, Kogan, Galitskii

Posted on April 19, 2022 by The Mitr

In this post, we will see the book Problems In Quantum Mechanics
by I.I. Gol’dman, V.D. Krivchenkov, V.I. Kogan, V.M. Galitskii and edited by D. Ter Haar.

About the book

A comprehensive collection of problems of varying degrees of difficulty in nonrelativistic quantum mechanics, with answers and completely worked-out solutions. Among the topics: one-dimensional motion, transmission through a potential barrier, commutation relations, angular momentum and spin, and motion of a particle in a magnetic field. An ideal adjunct to any textbook in quantum mechanics, useful in courses in atomic and nuclear physics, mathematical methods in physics, quantum statistics and applied differential equations.

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

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You can get the book here.

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Contents

Preface
One dimensional motion (Problems 3, Solutions 81)
Tunnel effect 10, 129
Commutation relations; Heisenberg relations; spreading of wave packets; operators 15, 155
Angular momentum; spin 24, 216
Central field of force 34, 233
Motion of particles in a magnetic field 38, 254
Atoms 42, 270
Molecules 53, 350
Scattering 58, 383
Creation and annihilation operators; density matrix 67, 430
Relativistic wave equations 74, 443
Subject index

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