About the book:

The book is aimed at engineers with a strong mathematical background, scientists working in mechanics and applied mathematics, and undergraduate and postgraduate students of Applied Physics and Physics and Mathematics departments. The book is based on a course of lectures presented by the author to engineering students at the Mechanics and Mathematics Department of Moscow University in 1956-1976.

The book has two parts

Part One of the book is devoted to the combination of the Lyapunov, Poincare, and averaging methods as applied to the analysis of oscillations in Lyapunov and nearly Lyapunov systems.

The second part of the book is also based on the results achieved in one of the classical methods developed in the years spanning the late 19th and early 20th centuries, the theory of normal forms (Poincare, Lyapunov, Dulac, Siegel, Moser, Arnold, Pliss, and others).

The book requires considerable mathematical background and is not an easy read for those who are not thorough with quite advanced and topical stuff regarding solving equations.

The book was translated from the Russian by V. I. Kisin and was first published by Mir Publishers in 1980.

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Contents

PART ONE

OSCILLATIONS IN LYAPUNOV SYSTEMS

Chapter I. Introduction (13)

§ 1. Transformation of Lyapunov Systems (13)

1.1. General case (13).

1.2. Systems of second-order equations (16).

§ 2. On the Poincare Method of Finding Periodic Solutions of Non-autonomous

Quasilinear Systems (19)

2.1. Differential equations of the generating solution and first corrections (19).

2.2. Non-resonant case (20).

2.3. Resonant case (22).

2.4. Variational equations for periodic unperturbed motion (24).

2.5. Case of distinct multipliers of unperturbed system of variational equations (25).

2.6. Case of multiple multipliers (27).

2.7. Examples (28).

§ 3. Forced Vibrations of Centrifuges Used for Spinning (33)

3.1. Statement of the problem and equations of motion (33).

3.2. Determination of a periodic solution (35).

3.3. Stability analysis (37)

Chapter II. Oscillatory Chains (40)

§ 1. Completely Elastic Free Oscillatory Chains (40)

1.1. Definition of an oscillatory chain (40).

1.2. Determination of equilibrium positions (43).

1.3. Asymptotic stability in the large of the lower equilibrium position for distinct resistance forces (46).

1.4. Variational equations for Vertical oscillations of the system (47).

1.5. Conservative case (49).

1.6. Stability of vertical vibrations of a spring-loaded pendulum (50).

§ 2. Partly Elastic Free Oscillatory Chains (55)

2.1. Statement of the problem (55).

2.2. Kinetic and potential energies (57).

2.3. Example (59).

2.4. Pendulum subject to elastic free suspension (62).

2.5. Pendulum subject to elastic guided suspension (65).

Chapter III. Application of the Methods of Small Parameter to Oscillations in

Lyapunov Systems (67)

§ 1. Loss of Stability of Vertical Vibrations of a Spring-Loaded Pendulum (67)

1.1. Step 1 (68).

1.2. Step 2 (69).

1.3. Step 3 (72).

§ 2. On Coupling of Radial and Vertical Oscillations of Particles in Cyclic

Accelerators (75)

2.1. Step 1 (75).

2.2. Step 2 (77).

2.3. Step 3 (78).

§ 3. Loss of Stability of Vertical Oscillations of a Pendulum Subject to Elastic Guided suspension (79)

3.1. Determination of nontrivial periodic modes (Step 2) (79).

3.2. Transient process (Step 3) (80).

§ 4. Periodic Modes of a Pendulum Subject to Elastic Free Suspension (82)

4.1. Transformation of equations of motion (82).

4.2. Periodic solution (83).

Chapter IV. Oscillations in Modified Lyapunov Systems (84)

§ 1. Lyapunov Systems with Damping (84)

1.1. Transformation of Equations of motion (84).

1.2. Complete system of variational Equations in the Poincare parameter and its solution (86).

1.3. Vibration in mechanical systems with one degree of freedom and different types of nonlinearity (89).

1.4. The Duffing equation with linear damping (92).

1.5. Spring-loaded pendulum with linear damping (95).

§ 2. On Lyapunov Type Systems (!)8)

2.1. Statement of the problem (98).

2.2. Transformation of Lyapunov systems (100).

PART TWO

APPLICATION OF THE THEOHY OF NORMAL FORMS TO OSCILLATION PROBLEMS

Chapter V. Elements of the Theory of Normal Forms of Real Autonomous Systems of Ordinary Differential Equations (103)

§ 1. Introductory Information (103)

1.1. Statement of the problem (103).

1.2. The fundamental Brjuno theorem (144).

1.3. The Poincare theorem (106).

§ 2. Additional Information (107)

2.1. Some properties of normalizing transformations (107).

2.2. Classification of normal forms; integrable normal forms (107).

2.3. Concept of power transformations (109).

2.4. The Brjuno theorem on convergence and divergence of normalizing transformations ( 111).

§ 3. Practical Calculation of Coefficients of Normalizing Transformation and Normal Form. ( 112)

3.1. Fundamental identities (112).

3.2. Computational alternative (114).

3.3. Fundamental identities in general form and their transformation (116).

3.4. Computational alternative in general case (120).

3.5. Remark: on the transition from symmetrized coefficients to ordinary <Jill’S (122).

3.6. Formulas for coefficients of fourth-power Variables (123).

3.7. Case of composite elementary divisors of the matrix of the linear part (123).

Chapter VI. Normal Forms of Arbitrary-Order Systems in the Cast of Asymptotic Stability in Linear Approximation ( **128)**

§ 1. Damped Oscillatory Systems (128)

1.1. Reduction to diagonal form (128).

1.2. Calculation of coefficients of normalising transformation (129).

1.3. General solution of the initial system (general solution of the Cauchy problem) (130).

§ 2. Examples (132)

2.1. A system with one degree of freedom (132).

2.2. Oscillations of a spring suspended mass with linear damping (133).

Chapter VII. Normal Forms of Third-Order Systems (136)

§ 1. Case of Two Pure Imaginary Eigenvalues of the Matrix of the Linear Part (136)

1.1. Reduction to normal form (136).

1.2. Calculation of coefficients of normalizing transformation and normal form ( 138).

1.3. Application of power transformation (140).

1.4. Free oscillations of an electric servodrive (142).

§ 2. Case of Neutral Linear Approximation (146)

2.1. Normal form (146).

2.2. Calculation of coefficients of normalizing transformation and normal form (148).

2.3. Remark on convergence (150).

2.4. Conclusions on stability (15U).

2.5. Integration of normal form in Quadratic approximation (152).

2.6. Example (155).

§ 3. Case of a Zero Eigenvalue of the Matrix of the Linear Part (156)

3.1. Normal form and normalizing transformation (156).

3.2. Integration of normal form (158).

3.3. Remark on convergence (159).

3.4. Free oscillations in a tracking system with a TV sensor (159).

Chapter VIII. Normal Forms of Fourth- and Six-Order Systems in Neutral** **Linear Approximation ( 165)

§ 1. Fourth-Order Systems (165)

1.1. Remark on coefficients of systems of diagonal form (16:i).

1.2. Reduction to normal form (166).

1.3. Calculation of coefficients of normalizing transformation and normal forms (168).

1.4. The Molchanov criterion of oscillation stability (170).

1.5. The Bibikov-Pliss criterion (173).

§ 2. The Ishlinskii Problem (173)

2.1. Reduction of equations of mot ion to tho Lyapunov form (173).

2.2. Transformation of systems similar to Lyapunov (176).

2.3. Determination of periodic solutions (178).

2.4. Reduction of equations of motion to diagonal form and transformation to normal form (180).

2.5. General solution of the Cauchy problem (182).

2.6. Preliminary conclusions on stability (184).

2.7. Construction of tho Lyapunov function (185).

§ 3. The Trajectory Described by the Centre of a Shaft’s Cross Section in One Revolution (186)

3.1. Statement of tho problem and equations of motion (186).

3.2. Reduction to diagonal form (190).

3.3. Reduction to normal form (193).

3.4. General solution of the Cauchy problem (194).

§ 4. Sixth-Order Systems** **(196)

4.1. Solutions of the resonant equation (197).

4.2. Normal forms (200).

4.3. Calculation of coefficients of normalizing transformation and normal forms (201).

4.4. Stability in the third approximation. The Molchanov** **criterion (205).

Chapter IX. Oscillations of a Heavy Solid Body with a Fixed Point About the Lower Equilibrium Position (208)

§ 1. Case of Centroid Located in a Principal Plane of the Ellipsoid of Inertia with respect** **to a Fixed Point (208)

1.1. Reduction to diagonal form (208).

1.2. Reduction to the Lyapunov form (211).

1.3. Resonances (212).

1.4. Simplest motions (213).

1.5. Transformation of equations of diagonal form (214).

1.6. Possible generalizations (215).

1.7. Situation similar to the Kovalevskaya case (216).

1.8. Application of the method of successive approximations (218).

1.9. Remarks on the determination of tho position of a solid body with a fixed point (219).

§ 2. The General Case (219)

2.1. Base reference frame (220).

2.2. Special reference frame (222).

2.3. Equations of motion of a heavy solid body in the special reference frame (223).

2.4. Reduction to the Lyapunov form (226).

2.5. Resonances (228).

2.6. Application of the method of successive approximations (229).

Brief Bibliographical Notes (232)

References (236)

Subject Index (262)

]]>About the book

A collection of 928 problems in arithmetic, algebra, geometry and trigonometry (with answers) prepared for home study by correspondence students and others studying or brushing supplementary mathematics without a teacher. Problems with similar solutions are grouped together with a detailed example of the solution of the first problem in the group. Will be found a useful resource of questions for revision, tests, and examination papers. Has had 17 large editions in Russian.

The book was translated from the Russian by Leonid Levant and was published by Mir Publishers in 1982.

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Contents

Formulas for Reference

**PART ONE ARITHMETIC AND ALGEBRA Problems | Answers**

Chapter I. Arithmetic Calculations *11 | 89*

Chapter II. Algebraic Transformations *14 | 90*

Chapter Ill. Algebraic Equations *22 | 112*

Chapter IV. Logarithmic and Exponential Equations *29 | 142*

Chapter V. Progression *32 | 159*

Chapter VI. Combinatorics and Newton’s Binomial Theorem *36 | 169*

Chapter VII. Algebraic and Arithmetic Problems *39 | 177*

**PART TWO GEOMETRY AND TRIGONOMETRY Problems | Answers**

Chapter VIII. Plane Geometry *55 | 208*

Chapter IX. Polyhedrons *62 | 246*

Chapter X. Solids of Revolution *76 | 327*

Chapter X I. Trigonometric Transfonnation *81 | 363*

Chapter X II. Trigonometric Equations *84 | 372*

Chapter X Ill. Inverse Trigonometric Functions *87 | 395*

About the book

The primary purpose of this book is to provide readers with concrete

facts, drawn from present-day biology, to serve as proofs in the

materialist theory of anthropogenesis. These include the most important

information on the living anthropoid apes necessary to make a correct

study of the fossil remains of their extinct ancestors, to find among

them the immediate precursors of man and to discover the main features

of their palaeobiology.The second task which the author has set himself is to outline the

more significant stages in the development of fossil man.The third task is to explain the anthropological viewpoint of the

way in which fossil man developed, using for this purpose the labour

theory of anthropogenesis, and also to criticize the idealist concepts

of the formation of man and the races of mankind.

The book was translated from the Russian by George H. Hanna and was first published by Foreign Languages Publishing House in 1959.

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Contents

Preface 5

PART ONE

THE DARWIN AND OTHER HYPOTHESES CONCERNING

ANTHROPOGENESIS

Chapter One. DARWIN ON THE ORIGIN OF MAN

1. Anthropogenesis Before Darwin

2. Darwin on the Evolution of the Animal Kingdom

3. Darwin’s Genealogy of Man

Chapter Two. THE ANTHROPOID APES AND THEIR ORIGIN

1. Living Anthropoids

2. Extinct Anthropoids

Chapter Three. CRITICISM OF THE LATER HYPOTHESES CONCERNING THE ORIGIN OF MAN

1. Religious Explanations of Anthropogenesis

2. The Tarsier Hypothesis

3. Some Simian Hypotheses

4. Osborne’s Hypothesis

5. Weidenreich’s Hypothesis of Anthropogenesis

PART TWO

PHYSICAL PECULIARITIES OF THE HUMAN BODY AND THE

EMERGENCE OF MAN

Chapter One. THE ROLE OF WORK AND ERECT LOCOMOTION IN ANTHROPOGENESIS

1. The Role of Work

2. Methods of Locomotion of the Great Apes

3. The Weight of the Body and the Centre of Gravity in Man and the Apes

4. The Inferior Extremities

5. The Pelvis, Spine and Thorax

6. The Superior Extremities

7. The Proportions of the Body and Asymmetry

8. The Skull

Chapter Two. THE BRAIN AND HIGHER NERVOUS ACTIVITY IN MAN AND THE APES

1. The Brain and Analysers of Man and the Apes

2. Development of the Peripheral Regions of the Analysers

3. Higher Nervous Activity in Monkeys

4. The Second Signalling System-the Distinguishing Feature or Human

Thought

Chapter Three. THE HERD INSTINCf IN MONKEYS AND RUDIMENTARY FORMS OF LABOUR

1. The Herd Instinct in Monkeys

2. Inceptual Forms of Labour

PART THREE

PALAEANTHROPOLOGICAL DATA ON THE MAKING OF MAN

Chapter One. THE FIRST STAGE: THE EARLIEST MEN (PITHECANTHROPI)

1. The Java Pithecanthropus

2. The Sinanthropus

3. The Heidelberg Man

Chapter Two. THE SECOND STAGE: EARLY MEN (PALAEANTHROPI)

1. The Ice Age

2. Neanderthalers and Their Physical Type

3. Neanderthalers on U.S.S.R. Territory

4. The Palestine Neanderthalers

5. Primitive Man’s Way of Life

6. The Development of Fossil Man’s Brain

Chapter Three. MODERN MAN (NEANTHROPUS)

1. Upper Palaeolithic Man

2. False Hypotheses Concerning the Origin of Modem Man and Their

Criticism

3. The Races of Mankind

4. Science Against Racism

Bibliography

]]>

About the book:

This book by I. R. Lavretsky, Dr. Sc. (Hist.), Is concerned with the life and activities of the outstanding revolutionary and fighter for the national liberation of the Latin American peoples Ernesto Che Guevara. The author makes use of numerous documents, press Items, notes from personal conversations with friends, relatives and comrades-in-arms of Che Guevara, as well as a wealth of photographs.

The book was translated from the Russian by A. B. Eklof and was designed by V. An. The book was first published by Progress Publishers in 1976.

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Contents

**THE ROAD TO THE “GRANMA”**

First Steps

Character Formation

A Lost Battle

The “Granma”

**SIERRA MAESTRA**

Fighting in the Mountains

The Daily Life of a Guerrilla

Through Santa Clara to Havana

**“PATRIA 0 MUERTE”**

In the Whirlwind of Revolution

The World of Socialism

A Shock Worker for Communism

“Cuba Si, Yanqui No!”

**“BOLIVIAN DIARY”**

A Mysterious Disappearance

The Camp on the Nancahuasu River

And Again the Thunder of Battle

On the Other Side of the Barricades

The Immortal Cause of Revolution

**LANDMARKS IN THE LIFE OF ERNESTO CHE GUEVARA**

About the book:

The structure of this book permits the student to read and master its matter in varying order according to the aims and purposes he or she is pursuing. The introduction “What Is Philosophy?” provides basic information about philosophy, its subject-matter and methods, the main points that distinguish it from other disciplines, and its place in the system of Marxism-Leninism. This chapter also contains information on the origin and various stages in the evolution of philosophy, and singles out the main matters that will subsequently be discussed. These problems will be treated in more detail in the following chapters, the material being arranged in order of increasing difficulty. Each of the successive chapters depends on the preceding ones. For the reader’s better assimilation of the proofs and arguments by which the superiority of materialism over idealism, of dialectics over metaphysics, and of Marxist-Leninist philosophy over other philosophical schools and currents is demonstrated, the text includes dialogues and talks between imaginary persons who express different points of view. These dialogues should be read and studied as attentively as the basic text.

The book was translated from the Russian by H. Campbell Creighton and was first published in English by Progress Publishers in 1989. This book is a part of the series *Student’s Library *in which many books were published especially pertaining to philosophy and sociology within the framework of dialectical-materialism.

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Contents

How to Use This Book 7

Introduction 9

**What Is Philosophy 9**

**Who Needs Philosophy? And Why? 9**

001 Man in the Modern World 9

002 “The Intellectual Quintessence of Its Time” 10

003 Philosophy and World Outlook 12

004 Philosophy and the General Methodology of Activity and Cognition 13

005 Philosophy and Ideology 16

006 The Main Task of Marxist-Leninist Philosophy 20

**The Basic Question. The Subject Matter and Method of Philosophy 22**

007 The Basic Question of Philosophy 22

008 The ‘ First Aspect of the Basic Question of Philosophy. Idealism and Materialism 24.

009 A Dialogue of a Materialist and an Idealist 26

010 The Second Aspect of the Basic Question of Philosophy 30

011 A Dialogue about the Knowability of the World 31

012 The Method of Philosophy; the Preliminary Concept of Dialectics and Metaphysics 34.

013 The Subject Matter of Marxist-Leninist Philosophy 36

014 The Principle of Partisanship in Philosophy 37

**The Origin and Development of Philosophy 39**

015 The Philosophy of Antiquity 39

016 The Philosophy of the Orient 41

017 The Philosophy of the Middle Ages 42

018 The Philosophy and Culture of the Renaissance 43

019 The Philosophy of Bourgeois Society 44

020 The Philosophical, Social, and Scientific Prerequisites of Marxian Philosophy 46

021 The Rise of Dialectical Materialism: a Radical Turn in the Development of Philosophy 49

022 A New Stage in the Development of Marxist Philosophy 51

**Chapter I**

**Matter and Consciousness**

**Matter and the Picture of the World 56**

101 Notion and Category 56

102 What Is Matter? 57

103 How Views of Matter Developed 59

104 The Contemporary Scientific Picture of the World 62

105 The Material Unity of the World 65

106 System, Structure, Element 65

107 Necessity and Chance 68

108 Laws of the Objective World 70.

**Motion, Time and Space 74**

109 Matter and Motion 74

110 Dialogue on Motion and Rest 76

111 Form and Content 77

112 The Forms of the Motion of Matter 79

113 Time and Space 81

114 The Irreconcilability of the Idealist and Materialist Conceptions of Time and Space 82

115 Modern Scientific Notions of Time and Space 84

116 Cause and Effect 85

**Reflection as a General Property of Matter 88**

117 The Basic Question of Philosophy in the “Computer Age” 88

118 What Is Reflection? 89

119 Reflection in the Inorganic World 90

120 The Complication of Reflection during the Transition to Animate Nature 92

121 The Evolution of Life and Origin of the Nervous System 93

122 Active and Passive Reflection of Reality 95

123 The Psychic and the Physical, the Ideal and the Material 97

**Human Consciousness 101**

124 The Brain as the Material Organ of Mental Activity 10l

125 Work as the Basis of Consciousness 102

126 Language and Thought 104

127 On the Relative Character of the Opposition of Matter and Consciousness 106

128 Can Computers Think? 108

129 Some Conclusions. The Synthesising Function of Philosophy 111

**Chapter II**

**Social Being and Social Consciousness 114**

**The Materialist Conception of Society and Its History 114**

201 A Talk about the Idealist and Materialist Conceptions of Society 114

202 Man and Activity. Preconditions for the Materialist Conception of History 116

203 The Development of Society &s a NaturalHistorical Process 118

204 The Mode of Production as the Basis of the Development and Functioning of Society 120

205 Basis and Superstructure 125

206 Classes and Class Struggle 127

207 The State in the System of the Superstructure 130

208 Political Parties in the System of the Superstructure 134

209 Social Organisations in the System of the Superstructure 137

210 Social Being and Social Consciousness 139

211 The Basic Principle of Historical Materialism 142

**The Theory of SocioEconomic Formation 144**

212 The Individual, Particular, and Universal 144

213 What Is a SocioEconomic Formation? 146

214 Social Revolution 148

215 The Structure of a Social Revolution 149

216 The Forming of Human Society 151

217 The Primitive Communal Formation 152

218 The SlaveOwning Formation 154

219 The Feudal Formation 156

220 The Capitalist Formation 158

221 The Communist Formation 161

222 The Category “SocioEconomic Formation” and Historical Reality 164

**The Functions and Forms of Social Consciousness 166**

223 Social Consciousness and the Development of Society 166

224 Ideology in the System of Social Consciousness 168

225 Social Psychology and Everyday Consciousness 170

226 Political Consciousness and Politics 172

227 Legal Consciousness and Law 174

228 Morality as a Form of Social Consciousness 176

229 Economic Consciousness 179

230 Religion as a Form of Social Consciousness 181

231 Artistic Consciousness and Art 183

232 Individual and Social Consciousness 187

233 On the Relative Independence of Social Consciousness 189

234 Growth of the Role of the Subjective Factor under Socialism 192

**Chapter III**

**Nature and Society 196**

**On the Relationship of Nature and Society 196**

301 Nature and Society 196

302 Dialogue about Nature and Society 198

303 PreMarxian Views on Nature and Society 201

304 Dialectical Materialism on the Relation of Nature and Society 203.

**The Environment. The Biological and Social in Social Development 205**

305 The Structure of the Environment 205

306 Mankind and the Natural Environment 206

307 The Biological and Social in Man 209

308 Races and Nations 211

309 The Role of Population in the Development of Society 216

310 The Artificial Habitat 220

**Nature and Society in the Age of Scientific and Technological Progress 222**

311 What Is Scientific and Technological Progress or the Scientific and Technical Revolution? 222

312 Scientific and Technical Progress and Its Consequences under Capitalism and Socialism 226

313 Ecological Consciousness and Ideological Struggle 230

**Chapter IV**

**The Main Laws of Dialectics 234**

**The Sources of Development 234**

401 The Idea of Development 234

402 What Is the Source of Development? 237

403 The Categories of “Opposition” and “Contradiction” 239

404 The Unity and Mutual Conversion of Opposites 241

405 The Struggle of Opposites and Resolution of Contradictions: the Source of Development 244

406 Forms of Contradictions 247

407 The Resolution of Contradictions in Socialist Society 252

408 The Law of the Unity and Struggle of Opposites: the Essence and Core of Dialectics 255

**Forms of Development 257**

409 On the Form of Development 257

410 A Dialogue about the Continuous and Intermittent, the Gradual and Sudden in the Process of Development 257

411 Quantity, Quality, Measure, and Leap 260

412 Evolution and Revolution 263

413 The Dialectic Connection between Quantitative and Qualitative Changes 266

414 The Law of the Transition of Quantitative Changes into Qualitative, and Vice Versa 270

415 Quantitative and Qualitative Changes in the Structure of the Socialist Revolution 271

416 The Dialectic of Quantity and Quality in the Present Stage of the Development of Socialism 274

**The Direction of Development 417**

A Dialogue on the Direction of Development 277

418 The Spiral Like Character of Development 278

419 Dialectical Negation and Continuity 279

420 Possibility and Actuality 283

421 The Dialectic of the Possible and the Real in a Revolutionary Situation 285

422 What Is Social Progress? 286

423 The Dialectical Law of the Negation of Negation 289

**Chapter V**

**The Theory of Knowledge 291**

**The Dialectics of Knowing 291**

501 What Does It Mean to Know? 291. 502 Cognition as Reflection 293. 503 A TalJ… about the Sources of Knowledge · 294. 504 The Role of Sensation in Knowing 298. 505 The Role of Abstraction in Knowing. The Method of the Ascent from the Abstract to the Concrete 300. 506 The Epistemological Roots of Idealism 304. 507 What Is Truth? 305. 508 The Role of Practice in Knowing 309. 509 Appearance and Essence. The Dialectics of Knowing 311.

**The Forms and Methods of Scientific Cognition.**

510 Theory and Hypothesis 314

511 Experiment and Observation in Scientific Cognition 318

512 Certain General Scientific Methods of Cognition 320

513 Models and Modelling in Scientific Cognition 326

514 The Application of Mathematics and Modern Science 327

515 Science and Society 329.

**Chapter VI**

**Man and Society**

601 A Chat on the Essence of Man and the Sense of Life 333

602 Freedom and Necessity 338

603 The Role of the Individual and of the Masses in the Development and Life of Society 342

604 The Individual and the Masses in Socialist Society 346

605 Socialist Democracy and Communist Education 348

606 Acceleration of Socio-Economic Progress. Reorganisation (Perestroika) and the Human Factor 351

607 The Road to a New Civilisation 353

608 The Struggle for Peace and the Destiny of Humankind 356

609 Predicting the Future 359

**A Last Chat with the Reader 363**

About the book:

The present collection of problems in vector analysis contains the required minimum of problems and exercises for the course of vector analysis of engineering colleges.

Each section starts with a brief review of theory and detailed solutions of a sufficient number of typical problems. The text contains 100 worked problems and there are 314 problems left to the student. There are also a certain number of problems of an applied nature that have been chosen so that their analysis does not require supplementary information in specialized fields. The material of the sixth chapter is devoted to curvilinear coordinates and the basic operations of vector analysis in curvilinear coordinates. Its purpose is to give the reader at least a few problems to develop the necessary skills.

The exposition in this text follows closely the lines currently employed at the chair of higher mathematics of the Moscow Power Institute.

The present text may be regarded as a short course in vector analysis in which the basic facts are given without proof but with illustrative examples of a practical nature. Hence this problem book may be used in a recapitulation of the essentials of vector analysis or as a text for readers who wish merely to master the techniques of vector analysis, while dispensing with the proofs of propositions and theorems.

This collection of problems is designed for students of day and evening departments at engineering colleges and also for correspondence students with a background of vector algebra and calculus as given in the first two years of college study.

The book was translated from the Russian by George Yankovsky and was first published by Mir in 1983.

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Contents

Preface 7

CHAPTER I. THE VECTOR FUNCTION OF A SCALAR ARGUMENT

Sec. 1. The hodograph of a vector function 9

Sec. 2. The limit and continuity of a vector function of a scalar argument 11

Sec. 3. The derivative of a vector function with respect to a scalar argument 14

Sec. 4, Integrating a vector function of a scalar argument 18

Sec. 5. The first and second derivatives of a vector with

respect to the arc length of a curve. The curvature of a curve. The principal normal. 27

Sec. 6. Osculating plane. Binormal. Torsion. The Frenet formulas. 31

CHAPTER II. SCALAR FIELDS

Sec. 7. Examples of scalar fields. Level surfaces and level linea 35

Sec. 8. Directional derivative 39

Sec. 9. The gradient of a scalar field 44

CHAPTER III. VECTOR FIELDS

Sec. 10. Vector linea. Differential equations of vector linea 52

Sec. 11. The flux of a vector field. Methods of calculating flux 58

Sec. 12. The flux of a vector through a closed surface. The Gauss-Ostrogradsky Theorem. 89

Sec. 13. The divergence of a vector field. Solenoidal fields. 89

See. 14. A line integral in a vector field. The circulation of a vector field 96

Sec. 15. The curl (rotation) of a vector field 108

Sec. 16. Stokes’ theorem 111

Sec. 17. The independence of a line integral of the path

of integration. Green’s formula 115

CHAPTER IV. POTENTIAL FIELDS

See. 18. The criterion for the potentiality of a vector field t2t

See. 19. Computing a line integral in a potential field 124

CHAPTER V. THE HAMILTONIAN OPERATOR. SECOND-ORDER DIFFERENTIAL OPER~

ATIONS. THE LAPLACE OPERATOR

See. 20. The Hamiltonian operator del 130

See. 21. Second-order differential operations. The Laplace operator 135

See. 22. Vector potential 146

CHAPTER VI. CURVILINEAR COORDINATES. BASIC OPERATIONS OF VECTOR ANALYSIS IN CURVILINEAR COORDINATES

See. 23. Curvilinear coordinates 152

See. 24. Basic operations of vector analysis in curvilinear coordinates 156

See. 25. The Laplace operator in orthogonal coordinates 174

ANSWERS 177

APPENDIX I 184

APPENDIX II 186

BIBLIOGRAPHY 187

INDEX 188

About the book:

This book is a popular description of the unity of the forces of nature for the general reader.

At present, all interactions between bodies in nature are thought to be ultimately due to the interaction of elementary particles and involve only four types of forces: universal gravitation, or gravitational forces, electromagnetic forces, nuclear forces and the so- called weak interactions.

This book describes the principal properties of these four types of force and their “sphere of action”, that is, the part they play in diverse natural processes that range from galaxies to the atomic nucleus and the mutual transformations of elementary particles. It includes the latest achievements of physics and gives a picture of the unresolved problems that confront science today.

About the authors:

The authors of this book are Candidates of Physico-mathematical sciences Vladimir Grigoryev and Gennady Myakishev. In 1948 they graduated from the Physics Department of Moscow University and completed their graduate studies in 1951. At present both are Associate Professors of the Moscow University Physics Department.

V. Grigoryev has published over twenty papers on problems of quantum field theory and has been particularly interested in particle formation via high- energy collisions. G. Myakishev has written a number of works on electronics and problems of the methodology of science, several elementary numerous textbooks on physics and articles for popular-science magazines.

There are some amazing line drawings to illustrate the topics.

The book was translated from the Russian by George Yankovsky and was first published by Mir in 1967.

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Contents

Chapter One IN LIEU OF AN INTRODUCTION

1. A Word About the Word “Force” 9

2. Forces in Mechanics 10

3. Is It Always Possible to Describe an Interaction by Forces? 15

4. The Unity of the Forces of Nature 22

Chapter Two GRAVITATIONAL FORCES

1. From Anaxagoras to Newton 27

2. The Law of Universal Gravitation 31

3. Gravitation in Action 41

4. Geometry and Gravitation 53

Chapter Three ELECTROMAGNETIC FORCES

I. What Forces Are Called Electromagnetic? 89

2. What Is an Electric Charge? 93

3. The Interaction of Stationary Electric Charges 99

4. The Interaction of Moving Electric Charges 107

5. Close-range Action or Action at a Distance? 116

6. What Is an Electric Field and a Magnetic Field? 123

7. Relationships Between Electric and Magnetic Fields 131

8. Electromagnetic Waves 143

Chapter with No Number ELECTROMAGNETIC FORCES IN ACTION

1. How Do Electromagnetic Forces Manifest Themselves? 153

2. Forces, the Structure of Matter, the Equations of Motion 158

3. Electromagnetic Forces in Electrically Neutral Bodies 163

4. FreeChargesandCurrentsinNature 191

5. Electromagnetic Waves in Nature 213

6. Why Electromagnetic Interactions Take Up Most of This Book? 226

7. An Insertion with All the Rights of a Real Chapter .228

Chapter Four NUCLEAR FORCES

1. The Nucleus and Elementary Particles 249

2. Nuclear Interactions and How They Occur 258

3. The Transformation of Atomic Nuclei 268

Chapter Five WEAK INTERACTIONS

1. The Disintegration of Elementary Particles and the Neutrino 285

2. The Charge and the Transformations of Elementary Particles 300

3. The Neutrino and the Evolution of the Universe 308

4. An Early Summary of What We Have Learned 314

Chapter Six IN LIEU OF AN APPENDIX

1. What Are the Resonance Particles? .319

2. Systematics of the Elementary Particles 328

]]>About the book (from the Preface):

This present book, by Professor M. F. Nesturkh, is based on the methodological principles of Soviet anthropology and the factual data obtained by that science. The author connects the origin of the races with the origin of mankind as a whole and acquaints the reader with the present status of these two problems; he deals at length with the history of the formation, dissemination and mingling of individual anthropological (racial) types and their groups, uses facts to expose the reactionary nature of racism and proves that it has no foundation in science.

Professor Nesturkh naturally devotes greater attention to anthropology proper, but he makes extensive use of other natural and social sciences-comparative anatomy, physiology, palaeontology, archaeology, ethnography, psychology, and linguisticsin accordance with Frederick Engels’s well-known postulate that anthropology is the transition from the morphology and physiology of man and his races to history.

It must not be thought that Professor Nesturkh’s book is limited to an exposure of racism. The problems dealt with are of a much broader cognitive significance. Among other things the reader will find the latest information on the anthropoids of the Tertiary period (the distant ancestors of man and the modern anthropoid apes), the earliest hominids (Pithecanthropus and Sinanthropus), Neanderthal man and the fossil men of the modern type. The author also deals with the natural selection of the earliest men, geographical isolation, inter-racial crossing, the times and places in which the great races were formed, the ways in which they became disseminated and the relation of tribes, nationalities and nations to the races. He takes the reader on a journey through the animal kingdom to the world of man and introduces him to the beginnings of human history, in the course of which the laws of evolution that apply to the organic world have been replaced by the qualitatively new laws of social development.

In the final chapter Professor Nesturkh, in addition to his exposureof racism, touches on such important scientific problems as “Race and Language”, “Race and Mentality”. He adduces convincing evidence of the absence of any causal relation between the racial groups and language groups of mankind; he also shows that all modern races and nations are identical in their mental abilities. Stress is properly laid on the tremendous successes of communist construction in the U.S.S.R. and socialist construction of the People’s Democracies of Europe and Asia. The experience of these countries has completely destroyed the reactionary myth that mankind is divided into “higher” and “lower” races and shown that all peoples, irrespective of their racial make-up, are capable of developing genuinely progressive culture and science. The unscientific concept of racism is also refuted by the economic, political and cultural development of the young states in Asia, Africa and Latin America, recently liberated from the yoke of colonialism.

A Western reviewer Coon has this to say about the book:

Its bibliogarphy of 43 sources include only two non-Soviet works, one by Charles Darwin and the other by Marx and Engels. All of the photographs have been retouched, and none accredited to its proper source; many are easily recognizable.

The text is divided into four sections : Definition of the Races of

Mankind, Races and Origins of Man, The Origin of Races, and Races and Racism. In Section One, Nesturkh follows the threefold classification of ” great races ” devised by the aforementioned Cheboksarov in 1956: (1) Negroid or Afro-Asian, (2) Europeoid or Eurasian, and (3) Mongoloid or Asio- American. Under the first category he lumps Negroes, Pygmies, Bushmen, Negritos, Australian aborigines, Tasmanians, and Melanesians.Genetics is studiously avoided, except once, apparently a slip. Classifications of race by blood groups are passed by in silence.

He uses the word mutation once (the previously mentioned slip) to explain the parallel evolution of the African and Australoid dwarfs, and in this I agree.

The last Section is on “racists” whom he blasts but does not name, and on “racism ” which he says can be explained very simply, as follows : “The theory of ‘ higher’ and ‘ lower 5 races, of the right of one race to dominate over another, justifies war between nations – it is the ideological mask concealing imperialistic politics” (p. 98).

Having explained “racism” he then tries to disprove it. This too is very simple. According to him, the ” racists ” claim that rich people are dolichocephalic and poor people mesocephalic or brachycephalic. But in Sweden the “bourgeois,” workers, and peasants all have cephalic indices of 77.0. So much for “racism.”

He disproves Aryanism by showing, like Boas, that language is acquired independently of racial features. As a final blow, he discredits intelligence tests. ” Bourgeois ” scientists, he says, believe in intelligence tests, while “genuine scientists, of course, display a sharply negative attitude toward them” (p. 102).

He ends with a pean of praise for the elysian treatment of minorities

in the U. S. S. R., overlooking the forced migrations of the Kalmucks and others, the recurrent waves of anti-Semitism, and the recent troubles with African students.It is important for educated Americans to read this book, alongside some of our domestic books and articles on the same subject.

Carleton S. Coon *Human Biology*, Vol. 37, No. 1 (February, 1965), pp. 57-59

The era when the book was published (in the 1960s) the Soviet science was just coming out of the pseudoscientific theories of Lysenko, who was a leading figure in the biological sciences in Soviet Russian from 1940s-50s. After he became director of the Institute of Genetics within the USSR’s Academy of Sciences, he used this position of power along with his political connections to push for** anti-Mendelian doctrines** in Soviet science and education. Soviet scientists who refused to renounce genetics were dismissed from their posts and left destitute. Hundreds if not thousands of others were imprisoned. Several were sentenced to death as enemies of the state, including the botanist Nikolai Vavilov. Scientific dissent from Lysenko’s theories of environmentally acquired inheritance was formally outlawed in the Soviet Union in 1948. Though Lysenko remained at his post in the Institute of Genetics until 1965,^{} his influence on Soviet agricultural practice had declined by the 1950s. [some text from his wikipage] So it is no wonder, that Nesturkh’s book makes scant references to the idea of genetics and is driven by a larger agenda of the ideology of the state. The state of Soviet biology never fully recovered from this setback. Meanwhile, mainly due to efforts of Sakhrov (I may be wrong here), the physical sciences could actually work mostly without much interference from the state ideology.

The book was translated from the Russian by George Hanna and was designed by Vladimir An. The book was first published by Foreign Languages Publishing House in 1964/1966 (as no date is printed on the book).

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Contents

CONTENTS

PREFACE 5

INTRODUCTION 8

DEFINITION OF THE RACES OF MANKIND 11

I. Racial Characteristics and Their Study 11

2. The Negroid Great Race 18

3. The Europeoid Great Race 23

4· The Mongoloid Great Race 25

5· Features Common to All Races 29

RACES AND THE ORIGIN OF MAN 31

1. Fossil Men of the Modern Type 31

2. Neanderthal Man-the Ancestor of Modern Man 33

3. Earliest Man-the Ancestor of the Neanderthaler 36

4. The Anthropoids-the Ancestors of the Earliest Men 41

5. The Racial Peculiarities of Man and the Anthropoid Type of

Structure 46

6. The Main Features of the Structure of the Human Body: Band,

Foot, Brain 52

THE ORIGIN OF THE RACES 57

1. The Races of Mankind-the Result of Historical Development 57

2. Geographical and Social Isolation 59

3. Natural Selection 61

4. Intermarriage 63

5. The Formation of the Great Races 66

6. The Europeoid Great Race 69

7. The Negroid Great Race 75

8. The Mongoloid Great Race 89

RACES AND RACISM 96

1. The Essence of Racism 96

2. Race and Language 99

3· Race and Mentality 101

4· Equality of Races and Nations in the U.S.S.R 105

]]>About the book:

The present book closely follows the structure of the book by V. Voyevodin with some insignificant deviations demanded by the particulars of the course of study. Thus, since the corresponding topic of the course of lectures is studied at the very end of the first term, seminar classes cannot keep up with the course and so the section devoted to metric spaces is included in Chapter 8.

It is a basic requirement that any problem book should contain a sufficient number of useful and comprehensive problems for seminar classes, home-assignments, tests and examinations. The author hopes that this requirement has been fulfilled. Moreover, he has attempted to supply the strongest students with a material for personal study, and to lead them to problems currently faced in computational algebra.

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Contents

Front Cover 1 ,-7

Title Page 4 ,-72

Contents 6 ,-161

Preface 8 ,-197

CHAPTER 1 Linear Spaces 12 42

1.0. Terminology and General Notes 12 ,42

1.1. Definition of Linear Space 18 ,457

1.2. Linear Dependence 20 ,372

1.3. Spans. Rank of Vector Sets 23 ,75

1.4. Basis and Dimension of Space 27 ,-144

1.5. Sum and Intersection of Subspaces 30 ,281

CHAPTER 2 Euclidean and Unitary Spaces 34 45

2.0. Terminology .and General Notes 34 ,45

2.1. Definition of Euclidean Space 36 ,405

2.2. Orthogonality, Orthonormal Basis, Orthogonalization Procedure 39 ,238

2.3. Orthogonal Complement, Orthogonal Sums of Subspaces 42 ,202

2.4. Lengths, Angles, Distances 46 ,232

2.5. Unitary Spaces 49 ,392

CHAPTER 3 Determinants 53 -105

3.0. Terminology and General Notes 53 ,-105

3.1. Evaluation and the Simplest Properties of Determinants 57 ,389

3.2. Minors, Cofactors and the Laplace Theorem 64 ,297

3.3. Determinants and the Volume of a Parallelepiped in a Euclidean Space 70 ,291

3.4. Computing the Determinants by the Elimination Method 75 ,72

CHAPTER 4 Systems of Linear Equations 82 -298

4.0. Terminology and General Notes 82 ,130

4.1. The Rank of a Matrix 83 ,526

4.2. Planes in a Linear Space 87 ,438

4.3. Planes in a Euclidean Space 90 ,170

4.4. Homogeneous Systems of Linear Equations 93 ,382

4.5. Nonhomogeneous Systems of Linear Equations 100 ,143

CHAPTER 5 Linear Operators and Matrices 108 598

5.0. Terminology and General Notes 109 ,85

5.1. The Definition of a Linear Operator, the Image and Kernel of an Operator 113 ,212

5.2. Linear Operations over Operators 118 ,291

5.3. Multiplication of Operators 120 ,490

5.4. Operations over Matrices 125 ,-43

5.5. The Inverse of a Matrix 138 ,147

5.6. The Matrix of a Linear Operator, Transfer to Another Basis, Equivalent and Similar Matrices 147 ,-43

CHAPTER 6 Linear Operator Structure 153 -213

6.0. Terminology and General Notes 153 ,-213

6.1. Eigenvalues and Eigenvectors 154 ,127

6.2. The Characteristic Polynomial 157 ,287

6.3. Invariant Subspaces 162 ,415

6.4. Root Subspaces and the Jordan Form 167 ,245

CHAPTER 7 Unitary Space Operators 179 -229

7.0. Terminology and General Notes 179 ,-148

7.1. Conjugate Operator. Conjugate Matrix 183 ,-174

7.2. Normal Operators and Matrices 188 ,238

7.3. Unitary Operators and Matrices 192 ,150

7.4. Hermitian Operators and Matrices 197 ,-331

7.5. Positive-Semi definite and Positive-Definite Operators and Matrices 202 ,470

7.6. Singular Values and the Polar Representation 209 ,232

7.7. Hermitian Decomposition 214 ,441

7.8. Pseudosolutions and Pseudoinverse Operators 217 ,143

7.9. Quadratic Forms 222 ,88

CHAPTER 8 Metric Problems in Linear Space 228 -331

8.0. Terminology and General Notes 228 ,-49

8.1. Normed Linear Space 231 ,274

8.2. Norms of Operators and Matrices 236 ,-112

8.3. Matrix Norms and Systems of Linear Equations 240 ,611

8.4. Matrix Norms and Eigenvalues 245 ,343

Hints 254 ,-229

Answers and Solutions 267 ,-59

Index 325 ,-23

About the book:

This textbook is a comprehensive united course in linear algebra and analytic geometry based on lectures read by the author for many years at various institutes to future specialists in computational mathematics.

It is intended mainly for those in whose education computational mathematics is to occupy a substantial place. Much of the instruction in this speciality is connected with the traditional mathematical courses. Nevertheless the interests of computational mathematics make it necessary to introduce large enough changes in both the methods of presentation of these courses and their content.

The book was translated from the Russian by Vladimir Shokurov and was first published by Mir Publishers in 1983.

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Contents

Front Cover 1

Title Page 4

Contents 5

Preface 9

PART I Vector Spaces 11 396

CHAPTER 1 Sets, Elements, Operations 11 396

1. Sets and elements 12 107

2. Algebraic operation 14 104

3. lnverse operation 18 588

4. Equivalence relation 20 95

5. Directed line segments 22 41

6. Addition of directed line segments 25 452

7. Groups 28 540

8. Rings and fields 31 85

9. Multiplication of directed line segments by a number 35 531

10. Vector spaces 38 517

11. Finite sums and products 41 21

12. Approximate calculations 44 311

CHAPTER 2 The Structure of a Vector Space 45 329

13. Linear combinations and spans 46 225

14. Linear dependence 48 79

15. Equivalent systems of vectors 51 465

16. The basis 54 146

17. Simple examples of vector spaces 56 143

18. Vector spaces of directed line segments 57 299

19. The sum and intersection of subspaces 61 113

20. The direct sum of subspaces 64 403

21. Isomorphism of vector spaces 66 90

22. Linear dependence and systems of linear equations 70 324

CHAPTER 3 Measurements in Vector Space 75 452

23. Affine coordinate systems 75 452

24. Other coordinate systems 80 429

25. Some problems 82 317

26. Scalar product 89 310

27. Euclidean space 92 67

28. Orthogonality 95 128

29. Lengths, angles, distances 99 109

30. Inclined line, perpendicular, projection 102 577

CHAPTER 4 The Volume of a System of Vectors in Vector Space 110 44

31. Euclidean isomorphism 106 533

32. Unitary spaces 107 223

33. Linear dependence and orthonormal systems 108 213

34. Vector and triple scalar products 110 171

35. Volume and oriented volume of a system of vectors 115 9

36. Geometrical and algebraic properties of a volume 118 299

37. Algebraic properties of an oriented volume 122 0

38. Permutations 125 595

39. The existence of an oriented volume 126 95

40. Determinants 128 333

41. Linear dependence and determinants 133 27

42. Calculation of determinants 136 137

CHAPTER 5 The Straight Line and the Plane in Vector Space 137 104

43. The equations of a straight line and of a plane 137 104

44. Relative positions 142 220

45. The plane in vector space 146 164

46. The straight line and the hyperplane 149 229

47. The half space 154 230

CHAPTER 6 The Limit in Vector Space 161 146

49. Metric spaces 161 146

50. Complete spaces 163 359

51. Auxiliary inequalities 166 542

52. Normed spaces 168 53

53. Convergence in the norm and coordinate convergence 170 116

54. Completeness of normed spaces 173 76

55. The limit and computational processes 175 35

PART II Linear Operators 177 197

CHAPTER 7 Matrices and Linear Operators 177 400

56. Operators 177 400

57. The vector space of operators 181 115

58. The ring of operators 183 232

59. The group of nonsingular operators 185 248

60. The matrix of an operator 188 21

61. Operations on matriees 192 139

62. Matrices and determinants 196 449

63. Change of basis 199 72

64. Equivalent and similar matrices 202 345

CHAPTER 8 The Characteristic Polynomial 205 424

65. Eigenvalues and eigenvectors 205 424

66. The characteristic polynomial 208 271

67. The polynomial ring 210 238

68. The fundamental theorem of algebra 214 318

69. Consequences of the fundamental theorem 218 5

CHAPTER 9 The Structureof a Linear Operator 223 530

70. Invariant subspaees 223 530

71. The operator polynomial 225 384

72. The triangular form 228 211

73. A direct sum of operators 229 277

74. The Jordan canonical form 232 291

75. The adjoint operator 236 326

76. The normal operator 240 329

77. Unitary and Hermitian operators 243 546

78. Operators A*A and AA* 246 282

79. Decomposition of an arbitrary operator 249 169

80. Operators in the real space 251 304

81. Matrices of a special form 254 39

CHAPTER 10 Metric Properties of an Operator 257 454

82. The continuity and boundedness of an operator 257 454

83. The norm of an operator 259 23

84. Matrix norms of an operator 263 283

85. Operator equations 266 243

86. Pseudosolutions andthe pseudoinverse operator 268 412

87. Perturbation and nonsingularity of an operator 271 16

88. Stable solution of equations 275 276

89. Perturbation and eigenvalues 280 104

PART III Bilinear Forms 283 296

CHAPTER 11 Bilinear and Quadratic Forms 284 107

90. General properties of bilinearand quadratic forms 284 107

91. The matrices of bilinear and quadratic forms 290 359

92. Reduction to canonical form 296 331

93. Congruence and matrix decompositions 304 368

94. Symmetric bilinear forms 309 240

95. Second degree hypersurfaces 316 273

96. Second degree curves 321 109

97. Second degree surfaces 328 331

CHAPTER 12 Bilinear Metric Spaces 333 227

98. The Gram matrix and determinant 334 4

99. Nonsingular subspaces 340 248

100. Orthogonality in bases 344 509

101. Operators and bilinear forms 350 352

102. Bilinear metric Isomorphism 355 480

CHAPTER 13 Bilinear Forms in Computational Processes 358 213

103. Orthogonalization processes 358 213

104. Orthogonalizatio of a power sequence 363 322

105. Methods of conjugate directions 368 289

106. Main variants 374 71

107. Operator equations and pseudoduality 377 313

108. Bilinear forms in spectral problems 382 248

Conclusion 388 498

INDEX 390 336

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