Similarity And Dimensional Methods In Mechanics – Sedov

Posted on March 26, 2022 by The Mitr

In this post, we will see the book Similarity And Dimensional Methods In Mechanics by L.I. Sedov.

About the book

This book contains a complete development of the fundamental concepts of Dimensional Analysis and Similarity Methods, illustrated by applications to a wide variety of problems in mechanics, and particu­larly in fluid dynamics. The subject is developed from first principles and can be understood with the aid of an elementary knowledge of mathematical analysis and fluid dynamics. More advanced physical concepts are explained in the book itself. The first three chapters describe the basic ideas of the subject with illustrations from familiar problems in mechanics. The last two chapters show the power of Dimensional and Similarity Methods in solving new problems in the theory of explosions and astrophysics.

The book should be of interest to students who wish to learn dimen­sional analysis and similarity methods for the first time and to students of fluid dynamics who should gain further insight into the subject by following the presentation given here. The book as a whole and particularly the application to recent problems should appeal to all those connected with the many present-day aspects of gas dynamics including astrophysics, space technology and atomic energy.

The basic ideas behind Similarity and Dimensional Methods are given in the first chapter, which is general and descriptive in character. The second chapter consists of a series of examples of application of the methods, to familiar problems such as the motion of a simple pendulum, modelling in ship design, and the many scaling effects which arise in wind tunnel or water tank testing. Chapter III shows the use of Similarity and Dimensional Analysis in developing fundamental contributions to viscous fluid theory, such as the Blasius flat plate solution, and the various theories of isotropic turbulence.

The book was translated from Russian by Morris Friedman and edited by Maurice Holt and was published in 1959.

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Contents

I. General Dimensional Theory 1

1. Introduction 1
2. Dimensional and Nondimensional Quantities 2
3. Fundamental and Derived Units of Measurement 3
4. Dimensional Formulas 8
5. On Newton’s Second Law 10
6. Nature of the Functional Relations between Physical Quantities 16
7. Parameters defining a Class of Phenomena 20

II. Similarity, Modelling and Examples of the Application of Dimensional Analysis 24

1. Motion of a Simple Pendulum 24
2. Flow of a Heavy Liquid through a Spillway 27
3. Fluid Motion in Pipes 28
4. Motion of a Body in a Fluid 33
5. Heat Transfer from a Body in a Fluid Flow Field 40
6. Dynamic Similarity and Modelling of Phenomena 43
7. Steady Motion of a Solid Body in a Compressible Fluid 52
8. Unsteady Motion of a Fluid 56
9. Ship Motion 61
10. Planing over the Water Surface 69
11. Impact on Water 75
12. Entry of a Cone and Wedge at Constant Speed into a Fluid 83
13. Shallow Waves on the Surface of an Incompressible Fluid 85
14. Three-dimensional Self-Similar Motions of Compressible
Median 93

III. Application to the Theory of Motion of a Viscous Fluid and to the Theory of Turbulence 97

1. Diffusion of Vorticity in a Viscous Fluid 97
2. Exact Solutions of the Equations of Motion of a Viscous Incompressible Fluid 99
3. Boundary Layer in the Flow of a Viscous Fluid past a Flat
Plate 106
4. Isotropic Turbulent Motion of an Incompressible Fluid 110
5. Steady Turbulent Motion 133

IV. One-Dimensional Unsteady Motion of a Gas 146

1. Self-similar Motion of Spherical, Cylindrical and Plane Waves in a Gas 146
2. Ordinary Differential Equations and the Shock Conditions for Self-similar Motions 155
3. Algebraic Integrals for Self-similar Motions 166
4. Motions which are Self-similar in the Limit 174
5. Investigation of the Family of Integral Curves in the z, V Plane 177
6. The Piston Problem 187
7. Problem of Implosion and Explosion at a Point 191
8. Spherical Detonation 193
9. Flame Propagation 200
10. Collapse of an Arbitrary Discontinuity in a Combustible Mixture 206
11. The Problem of an Intense Explosion 210
12. Point Explosion taking Counter Pressure into Account 238
13. On Simulation and on Formulas for the Peak Pressure and Impulse in Explosions
14. Problem of an Intense Explosion in a Medium with Variable 260
15. Unsteady Motion of a Gas, when the Velocity is Proportional to Distance from the Centre of Symmetry 271
16. On the General Theory of One-dimensional Motion of a Gas 281
17. Asymptotic Laws of Shock Wave Decay 295

V. Application to Astrophysical Problems 305

1. Certain Observational Results 305
2. On the Equations of Equilibrium and Motion of a Gaseous Mass Simulating a Star 315
3. Theoretical Formulas relating Luminosity with Mass and Radius with Mass 321
4. Certain Simple Solutions of the System of Equations of Stellar Equilibrium 325
5. On the Relation between the Period of Variation of the Brightness and the Average Density for Cepheids 331
6. On the Theory of the Flare-Ups of Novae and Supernovae 334

REFERENCES 355
AUTHOR INDEX 359
SUBJECT: INDEX 361

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Steel Foundry Practice – Bidulya

Posted on March 25, 2022 by The Mitr

In this post, we will see the book Steel Foundry Practice by P.N. Bidulya.

About the book

A book describing various aspects of steel foundries.

The book was translated from Russian was published in 1960 by Peace Publishers.

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Contents

Chapter I. Properties of Steel Castings 7

1. Products of Steel Foundry 7
2. Classes of Steel Castings 8
3. Casting Methods 10
4. Steel Casting Process 11
5. Carbon Steel Castings 13
6. Castings of Alloy Structural Steel 15

Chapter II. Casting and Other Engineering Properties of Steel 23

7. Fluidity of Steel and Mould Filling Ability 23
8. Saturation of Steel with Gases. Gas Cavities and Flakes in Castings 33
9. Casting Defects Related to Moulding Sands 40
10. The Effect of Nonmetallic Inclusions on Cast Steel 48
11. Solidification of Steel in a Foundry Mould 56
12. Shrinkage, After effects and Prevention 70
13. Directional Solidification 91
14. Hot Cracking, Causes and Prevention 100
15. Residual Stresses, Warping and Annealing Cracks 115

Chapter III. Manufacture of Steel Castings 126

16. Process Designing 128
17. Moulding and Core Sands for Steel Castings 135
18. Gating Systems for Steel Castings 141
19. Risers. Design, Shape and Size. Calculation Methods 167
20. Heating the Risers 180
21. Pressure in the Risers 184
22. Moulding and Mould Assembly 186
23. External and Internal Chills. Application and Calculation 193
24. Necked-down Risers 208
25. Drying the Moulds and Cores Used for Steel Castings 210
26. Steel Pouring Theory and Practice 212
27. Cooling of Castings. Cooling Rates. Calculation 218
28. Cleaning. Removal of Gates and Risers. 225
29. Heat Treatment of Castings 226
30. Finishing and Inspection 231

Chapter IV. Operating mers. of Cast Steel 233

31. Carbon Steel Bae 233
32. Manganese Steel 235
33. Copper Steel 237
34. Silico-manganese Steel 238
35. Multicomponent Alloy Steels 238
36. High-alloy Steels with Special Properties 245
37. Austenitic Manganese Steel 246
38. Stainless Steel 249
39. Iron-base and Other Heat-Resistant Alloys 259
40. Tools Cast from High-speed Steel 266
41. The Properties of Magnetic and Nonmagnetic Alloys 267

Chapter V. Smelting Steel for Shaped Castings 269

42. Side-blown Converter Process 269
43. Open-hearth Process 284
44. Electric Arc Process 294
45. Combined Processes 308
46. Vacuum Treatment of Steel 311

Index 317

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A Course Of Mineralogy – Betekhtin

Posted on March 24, 2022 by The Mitr

In this post, we will see the book A Course Of Mineralogy by A. Betekhtin.

About the book

Mineralogy is one of the geological sciences concerned with the study of the earth’s crust. The term literally means the science of minerals and embraces all aspects of minerals including their genesis. The word mineral comes from “minera”, which once meant an ore specimen, which shows that it dates back to the beginnings of mining.

This book introduces the reader to various types of minerals and their classification, their composition, their physical and chemical properties.

The book was translated from Russian by Translated from the Russian by
V. Agol and edited by A. Gurevich. The book was published in 1966 by Peace Publishers.

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Contents

PART ONE
Introduction 11

CHAPTER I. The Earth’s Crust. Structure and Composition 29
CHAPTER II. Properties of Minerals 37
CHAPTER III. Methods for Detailed Study of Minerals 88
CHAPTER IV. Formation of Minerals in Nature 100

PART TWO
Classification of Minerals 135

SECTION I. Native Elements and Intermetallic Compounds 140
SECTION II. Sulphides, Sulphosalts, and Similar Compounds 169
SECTION III. Halides 236
SECTION. IV, Oxides 252
SECTION V. Oxygen Salts (Oxysalts)

Anhydrous Phosphates, Arsenates, and Vanadates 402
Hydrous Phosphates, Arsenates, and Vanadates 412

Class 8, Silicates 425

PART THREE

CHAPTER I. Minerals of the Earth’s Crust 585

CHAPTER II. Mineral Associations in Rocks and Ore Deposits 590
1. List of the Most Important Minerals Grouped According to Principal Metals (Elements) 622
Index 633

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A Theory Of Earth’s Origin – Schmidt

Posted on March 23, 2022 by The Mitr

In this post, we will see the book A Theory Of Earth’s Origin by Otto Schmidt.

About the book

During the six years that followed the publication of the Second Edition of this book, Otto J. Schmidt, despite his serious illness. continued to develop .his cosmogonic theory, taking advantage of every brief respite the sickness allowed him. In those years he published articles on the origin of asteroids and on the role of solid particles in planet cosmogony; he also prepared some chapters of a capital work on his theory. His untimely death prevented him from completing his work. He left behind him material for his book and many other manuscripts on various problems as well as
working notes and calculations, the majority of which were written between 1951 and 1955.

In a draft foreword for his fundamental work on tHe theory, Schmidt wrote: “The theory haS continually developed and grown richer. In the course of that development, preliminary tentative ideas have gradually been replaced by precise and concrete tenets, gaps have been filled in and the number of phenomena that can be explained by the theory has increased. This development was due to three factors: the Appearance of new facts and more profound generalizations in many branches of science, criticism and numerous discussions and, finally, its internal growth, i.e., the further exten”Sion of work on the theory. Some erroneous details have now been dropped but, on the whole, the theory proved capable of development and its ‘basic tenets have been proved sound. There can and must be further development and greater precision.”

Owing to the considerable development of the theory since the Second Edition of the Four Lectures it was not thought advisable to reprint them in their previous form. By making use of articles published by Otto J. Schmidt between 1951 and 1955 and unpublished work, we have been able to follow the author’s plan for the revision of the hook; the present Third Edition, therefore, reflects the present state of the theory. The addition of new material l.ed to a certain disproportion in the lectures and in the space allotted to the problems they cover.

The book was translated from Russian by George H. Hanna and was published in 1958 by Foreign Languages Publishing House.

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Contents

Preface to the Third Edition 5
Author’s Preface to the Second Edition 7

Lecture 1. Present State of the Problem — Formulation of the Problem – Fundamental Ideas and Facts 9

Lecture 2 Fundamental Regularities of the Planetary System—the Result of Gas-Dust Cloud Evolution 37

Lecture 3. The Problem of the Origin of the Gas-Dust Cloud 79

Lecture 4. The Planet Earth 101

Appendix I. 125
Appendix II 128
Appendix III. 129
Bibliography of Papers on the Schmidt Theory 131

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Applications of Functional Analysis in Mathematical Physics – Sobolev

Posted on March 22, 2022 by The Mitr

In this post, we will see the book Applications of Functional Analysis in Mathematical Physics – S. L. Sobolev.

About the book

The present book arose as a result of revising a course of lectures given by the writer at the Leningrad State University. The notes for the lectures were taken and revised by H. L. Smolicki and I. A. Jakovlev, who contributed to them a series of valuable remarks and additions. Several additions, arising naturally during the lectures, were also made by the author himself.
In this fashion there came into being this monograph, a unifying treatment from a single point of view of a number of problems in the theory of partial differential equations. There are considered in it variational methods with applications to the Laplace equation and the polyharmonic equations as well as the Cauchy problem for linear and quasi-linear hyperbolic equations. The presentation of the problems of mathematical physics demands a suitable consideration of some new results and methods in functional analysis, which constitute in themselves the basis of all the later material. The first part is concerned with this basis. The material indicated above, the particular problems posed, and the methods for their investigation are not to be found in the ordinary course in mathematical physics and in particular, they are not in my book Equations of mathemati­cal physics. The present book is of value for graduate students and re­search workers.

The book was translated from Russian by F. E. Browder was published in 1963.

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Contents

CHAPTER 1. SPECIAL PROBLEMS OF FUNCTIONAL ANALYSIS 1

§1. INTRODUCTION 1

1. Summable functions (1).
2. The Hélder and Minkowski inequalities (3).
3. The reverse of the Holder and Minkowski inequalities (7).

§2. BASIC PROPERTIES OF THE SPACES L_{p} 9

1. Norms. Definitions (9).
2. The Riesz-Fischer Theorem (11).
3. Continuity in the large of functions in L_{p} (11).
4. Countable dense nets (13).

§3. LINEAR FUNCTIONALS ON L_{p}

1. Definitions. Boundedness of linear functionals (16).
2. Clarkson’s inequalities (17).
3. Theorem on the general form of linear functionals (22).
4. Convergence of functionals (25).

$4. COMPACTNESS OF SPACES 28

1. Definition of compactness (28).
2. A theorem on weak compactness (29).
3. A theorem on strong compactness (30).
4. Proof of the theorem on strong compactness (31).

§5. GENERALIZED DERIVATIVES 33

1. Basic definitions (33).
2. Derivatives of averaged functions (35).
3. Rules for differentiation (37).
4. Independence of the domain (39).

§6. PROPERTIES OF INTEGRALS OF POTENTIAL TYPE 42

1. Integrals of potential type. Continuity (42).
2. Membership in L_{q} (43).

§7. THE SPACES L^{l}_{p} AND W^{l}_{p} 45

1. Definitions (45).
2. The norms in te (46).
3. Decompositions of wi and its norming (48).
4. Special decompositions of wi (50).

§8. IMBEDDING THEOREMS 56

1. The imbedding of W^{l}_{p} in C (56).
2. Imbedding of W^{l}_{p} in L_q (57).
3. Examples (58).

§9. GENERAL METHODS OF NORMING w? AND COROLLARIES OF THE IMBEDDING
THEOREM 60

1. A theorem on equivalent norms (60).
2. The general form of norms
equivalent to a given one (62).
3. Norms equivalent to the special norm (64).
4. Spherical projection operators (64).
5. Nonstar-like domains (66).
6. Examples (67).

§10. SOME CONSEQUENCES OF THE IMBEDDING THEOREM 68

1. Completeness of the space W,. (68).
2. The imbedding of We in Why (69).
3. Invariant norming of W¢) (72).

§11. THE COMPLETE CONTINUITY OF THE IMBEDDING OPERATOR (KONDRASEV’S
THEOREM) 74

1. Formulation of the problem (74). 2. A lemma on the compactness of the special integrals in C (75).
3. A lemma on the compactness of integrals in L_{q} (77).
4. Complete continuity of the imbedding operator in C (82).
5. Complete continuity of the operator of imbedding in L_{q} (84).

CHAPTER II. VARIATIONAL METHODS IN MATHEMATICAL PHYSICS

§12. THE DIRICHLET PROBLEM 87

1. Introduction (87).
2. Solution of the variational problem (88). 3. Solution of the Dirichlet problem (91).
4. Uniqueness of the solution of the Dirichlet problem (94).
5. Hadamard’s example (97).

§13. THE NEUMANN PROBLEM 99
1. Formulation of the problem (99).
2. Solution of the variational problem (100).
3. Solution of the Neumann problem (101).

§14. POLYHARMONIC EQUATIONS 103

1. The behaviour of functions from W3$” on boundary manifolds of various dimensions (103).
2. Formulation of the basic boundary value problem (105).
3. Solution of the variational problem (106).
4. Solution of the basic boundary value problem (108).

§15. UNIQUENESS OF THE SOLUTION OF THE BASIC BOUNDARY VALUE PROBLEM
FOR THE POLYHARMONIC EQUATION 112

1. Formulation of the problem (112).
2. Lemma (112). 3. The structure of the domains 𝛺_{h} 𝛺_{3h} (115).
4. Proof of the lemma for k<|s/2| (116).
5. Proof of the lemma for k= [s/2]+1 (118).
6. Proof of the lemma for [s 2] + 2 ≤ k ≤ m (120).
7. Remarks on the formulation of the boundary conditions (122).

§16. THE EIGENVALUE PROBLEM 123

1. Introduction (123).
2. Auxiliary inequalities (124).
3. Minimal sequences and the equation of variations (126).
4. Existence of the first eigenfunction (128).
5. Existence of the later eigenfunctions (131).
6. The infinite sequence of eigenvalues (134).
7. Closedness of the set of eigenfunctions (136).

 

 

CHAPTER III. THE THEORY OF HYPERBOLIC PARTIAL
DIFFERENTIAL EQUATIONS

§17. SOLUTION OF THE WAVE EQUATION WITH SMOOTH INITIAL CONDITIONS 139

1. Derivation of the basic inequality (139).
2. Estimates for the growth of the solution and its derivatives (142).
3. Solutions for special initial data (144).
4. Proof of the basic theorem (146).

§18. THE GENERALIZED CAUCHY PROBLEM FOR THE WAVE EQUATION 148

1. Twice differentiable solutions (148).
2. Example (150).
3. Generalized solutions (152).
4. Existence of initial data (153). 5. Solutions of the generalized Cauchy problem (155).

§19. LINEAR EQUATIONS OF NORMAL HYPERBOLIC TYPE WITH VARIABLE COEF-
FICIENTS (BASIC PROPERTIES) 157

1. Characteristics and bicharacteristics (157).
2. The characteristic conoid (164).
3. Equations in canonical coordinates (166).
4. The basic operators M^{(0)} and L^{(0)} in polar coordinates (168).
5. The system of basic relations on the cone (178).

§20. THE CAUCHY PROBLEM FOR LINEAR EQUATIONS WITH SMOOTH COEFFICIENTS 175

1. The operators adjoint to the operators of the basic system (175). 2. The construction of the functions 𝜎 (177).
3. Investigation of the properties of the functions 𝜎 (179). 4. Derivation of the basic integral identity Bu=SF (181).
5. The inverse integral operator B^{-1} and the method of successive
approximations (183).
6. The adjoint integral operator B* (187).
7. The adjoint integral operator S* (191).
8. Solution of the Cauchy problem for an even number of variables (192).
9. The Cauchy problem for an odd number of variables (195).

§21. INVESTIGATION OF LINEAR HYPERBOLIC EQUATIONS WITH VARIABLE COEFFICIENTS 196

1. Simplification of the equation (196).
2. Formulation of the Cauchy problem for generalized solutions (198).
3. Basic inequalities (200).
4. A lemma on estimates for approximating solutions (204).
5. Solution of the generalized problem (209).
6. Formulation of the classical Cauchy problem (210).
7. A lemma on estimates for derivatives (213).
8. Solution of the classical Cauchy problem (215).

§22. QUASILINEAR EQUATIONS 217

1. Formulation of functions of functions (217).
2. Basic inequalities (221).
3. Petrovsky’s functional equation (224).
4. The Cauchy problem with homogeneous initial conditions (229).
5. Properties of averaged functions (231).
6. Transformation of initial conditions (235).
7. The general case for the Cauchy problem for quasilinear equations (237).

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Partial Differential Equations of Mathematical Physics – Sobolev

Posted on March 21, 2022 by The Mitr

In this post, we will see the book Partial Differential Equations of Mathematical Physics – S. L. Sobolev.

About the book

The classical partial differential equations of mathematical physics, for­mulated and intensively studied by the great mathematicians of the nineteenth century, remain the foundation of investigations into waves, heat conduction, hydrodynamics, and other physical problems. These equations, in the early twentieth century, prompted further mathematical researches, and in turn themselves benefited by the application of new methods in pure mathematics. The theories of sets and of Lebesgue integration enable us to state conditions and to characterize solutions in a much more precise fashion; a differential equation with the boundary conditions to be imposed on its solution can be absorbed into a single formulation as an integral equation; Green’s function permits a formal explicit solution; eigenvalues and eigenfunctions generalize Fourier’s analysis to a wide variety of problems.
All these matters are dealt with in Sobolev’s book, without assumption of previous acquaintance. The reader has only to be familiar with element­ary analysis; from there he is introduced to these more advanced concepts, which are developed in detail and with great precision as far as they are re­quired for the main purposes of the book. Care has been taken to render the exposition suitable for a novice in this field: theorems are often approach­ed through the study of special simpler cases, before being proved in their full generality, and are applied to many particular physical problems.

The book was translated from Russian by E. R. Dawson was published in 1964.

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Contents

TRANSLATION EDITOR’S PREFACE ix
AUTHOR’S PREFACES TO THE FIRST AND THIRD EDITIONS x

LECTURE 1. DERIVATION OF THE FUNDAMENTAL EQUATIONS 1

§ 1. Ostrogradski’s Formula 1
§ 2. Equation for Vibrations of a String 3
§ 3. Equation for Vibrations of a Membrane 6
§ 4. Equation of Continuity for Motion of a Fluid. Laplace’s Equation 8
§ 5. Equation of Heat Conduction 13
§ 6. Sound Waves 17

LECTURE 2. THE FORMULATION OF PROBLEMS OF MATHEMATICAL PHYSICS.
HADAMARD’S EXAMPLE 22

§ 1. Initial Conditions and Boundary Conditions 22
§ 2. The Dependence of the Solution on the Boundary Conditions. Hadamard’s Example 26

LECTURE 3. THE CLASSIFICATION OF LINEAR EQUATIONS OF THE SECOND ORDER 33

§ 1. Linear Equations and Quadratic Forms. Canonical Form of an Equation 33
§ 2. Canonical Form of Equations in Two Independent Variables 38
§ 3. Second Canonical Form of Hyperbolic Equations in Two Independent Variables 42
§ 4. Characteristics 43

LECTURE 4. THE EQUATION FOR A VIBRATING STRING AND ITS SOLUTION BY D’ALEMBERT’S METHOD 46

§ 1. D’Alembert’s Formula. Infinite String 46
§ 2. String with Two Fixed Ends 49
§ 3. Solution of the Problem for a Non-Homogeneous Equation and for More General Boundary Conditions 51

LECTURE 5. RIEMANN’S METHOD 58

§ 1. The Boundary-Value Problem of the First Kind for Hyperbolic Equations 58
§ 2. Adjoint Differential Operators 62
§ 3. Riemann’s Method 65
§ 4. Riemann’s Function for the Adjoint Equation 68
§ 5. Some Qualitative Consequences of Riemann’s Formula 71

 

LECTURE 6. MULTIPLE INTEGRALS: LEBESGUE INTEGRATION 72

§ 1. Closed and Open Sets of Points 73
§ 2. Integrals of Continuous Functions on Open Sets 79
§ 3. Integrals of Continuous Functions on Bounded Closed Sets 85
§ 4. Summable Functions 92
§ 5. The Indefinite Integral of a Function of One Variable. Examples 99
§ 6. Measurable Sets. Egorov’s Theorem 103
§ 7. Convergence in the Mean of Summable Functions 111
§ 8. The Lebesgue—Fubini Theorem 121

LECTURE 7. INTEGRALS DEPENDENT ON A PARAMETER 126

§ 1. Integrals which are Uniformly Convergent for a Given Value of Parameter 126
§ 2. The Derivative of an Improper Integral with respect to a Parameter 129

LECTURE 8. THE EQUATION OF HEAT CONDUCTION 133

§ 1. Principal Solution 133
§ 2. The Solution of Cauchy’s Problem 139

LECTURE 9. LAPLACE’S EQUATION AND POISSON’s EQUATION 146

§ 1. The Theorem of the Maximum 146
§ 2. The Principal Solution. Green’s Formula 148
§ 3. The Potential due to a Volume, to a Single Layer, and to a Double Layer 150

 

LECTURE 10. SOME GENERAL CONSEQUENCES OF GREEN’S FORMULA 155

§ 1. The Mean-Value Theorem for a Harmonic Function 155
§ 2. Behaviour of a Harmonic Function near a Singular Point 158
§ 3. Behaviour of a Harmonic Function at Infinity. Inverse Points 162

LECTURE 11. POISSON’S EQUATION IN AN UNBOUNDED MEDIUM. NEWTONIAN
POTENTIAL 166

LECTURE 12. THE SOLUTION OF THE DIRICHLET PROBLEM FOR A SPHERE 171

LECTURE 13. THE DIRICHLET PROBLEM AND THE NEUMANN PROBLEM FOR A
HALF-SPACE 180

LECTURE 14. THE WAVE EQUATION AND THE RETARDED POTENTIAL 188

§ 1. The Characteristics of the Wave Equation 188
§ 2. Kirchhoff’s Method of Solution of Cauchy’s Problem 189

LECTURE 15. PROPERTIES OF THE POTENTIALS OF SINGLE AND DOUBLE LAYERS 202

§ 1. General Remarks 202
§ 2. Properties of the Potential of a Double Layer 203
§ 3. Properties of the Potential of a Single Layer 210
§ 4. Regular Normal Derivative 217

§ 5. Normal Derivative of the Potential of a Double Layer 218
§ 6. Behaviour of the Potentials at Infinity 220

LECTURE 16. REDUCTION OF THE DIRICHLET PROBLEM AND THE NEUMANN
PROBLEM TO INTEGRAL EQUATIONS 222

§ 1. Formulation of the Problems and the Uniqueness of their Solutions 222
§ 2. The Integral Equations for the Formulated Problems 225

 

LECTURE 17. LAPLACE’S EQUATION AND POISSON’S EQUATION IN A PLANE 228

§ 1. The Principal Solution 228
§ 2. The Basic Problems 230
§ 3. The Logarithmic Potential 234

LECTURE 18. THE THEORY OF INTEGRAL EQUATIONS 237

§ 1. General Remarks 237
§ 2. The Method of Successive Approximations 238
§ 3. Volterra Equations 242
§ 4. Equations with Degenerate Kernel 243
§ 5. A Kernel of Special Type. Fredholm’s Theorems 248
§ 6. Generalization of the Results 253
§ 7. Equations with Unbounded Kernels of a Special Form 256

 

LECTURE 19. APPLICATION OF THE THEORY OF FREDHOLM EQUATIONS TO THE
SOLUTION OF THE DIRICHLET AND NEUMANN PROBLEMS 258

§ 1. Derivation of the Properties of Integral Equations 258
§ 2. Investigation of the Equations 260

LECTURE 20. GREEN’S FUNCTION 265

§ 1. The Differential Operator with One Independent Variable 265
§ 2. Adjoint Operators and Adjoint Families 268
§ 3. The Fundamental Lemma on the Integrals of Adjoint Equations 271
§ 4. The Influence Function 275
§ 5. Definition and Construction of Green’s Function 278
§ 6. The Generalized Green’s Function for a Linear Second-Order Equation 281
§ 7. Examples 286

 

LECTURE 21. GREEN’S FUNCTION FOR THE LAPLACE OPERATOR 291

§ 1. Green’s Function for the Dirichlet Problem 291
§ 2. The Concept of Green’s Function for the Neumann Problem 296

LECTURE 22. CORRECTNESS OF FORMULATION OF THE BOUNDARY-VALUE
PROBLEMS OF MATHEMATICAL PHYSICS 301

§ 1. The Equation of Heat Conduction 301
§ 2. The Concept of the Generalized Solution 304
§ 3. The Wave Equation 307
§ 4. The Generalized Solution of the Wave Equation 311
§ 5. A Property of Generalized Solutions of Homogeneous Equations 317
§ 6. Bunyakovski’s Inequality and Minkovski’s Inequality 322
§ 7. The Riesz—Fischer Theorem 324

LECTURE 23. FOURIER’S METHOD 327

§ 1. Separation of the Variables 327
§ 2. The Analogy between the Problems of Vibrations of a Continuous Medium and Vibrations of Mechanical Systems with a Finite Number of Degrees of Freedom 334
§ 3. The Inhomogeneous Equation 336
§ 4. Longitudinal Vibrations of a Bar 339

LECTURE 24. INTEGRAL EQUATIONS WITH REAL, SYMMETRIC KERNELS 342

§ 1. Elementary Properties. Completely Continuous Operators 342
§ 2. Proof of the Existence of an Eigenvalue 354

LECTURE 25. THE BILINEAR FORMULA AND THE HILBERT—SCHMIDT THEOREM 357

§ 1. The Bilinear Formula 357
§ 2. The Hilbert-Schmidt Theorem 364
§ 3. Proof of the Fourier Method for the Solution of the Boundary-Value Problems of Mathematical Physics 367
§ 4. An Application of the Theory of Integral Equations with Symmetric Kernel 375

LECTURE 26. THE INHOMOGENEOUS INTEGRAL EQUATION WITH A SYMMETRIC
KERNEL 376

§ 1. Expansion of the Resolvent 376
§ 2. Representation of the Solution by means of Analytical Functions 378

LECTURE 27. VIBRATIONS OF A RECTANGULAR PARALLELEPIPED 382

LECTURE 28. LAPLACE’S EQUATION IN CURVILINEAR COORDINATES. EXAMPLES OF THE USE OF FOURIER’S METHOD 388

§ 1. Laplace’s Equation in Curvilinear Coordinates 388
§ 2. Bessel Functions 394
§ 3. Complete Separation of the Variables in the Equation V? u = 0 in Polar Coordinates 397

LECTURE 29. HARMONIC POLYNOMIALS AND SPHERICAL FUNCTIONS 401

§ 1. Definition of Spherical Functions 401
§ 2. Approximation by means of Spherical Harmonics 404
§ 3. The Dirichlet Problem for a Sphere 407
§ 4. The Differential Equations for Spherical Functions 408

LECTURE 30. SOME ELEMENTARY PROPERTIES OF SPHERICAL FUNCTIONS 414

§ 1. Legendre Polynomials 414
§ 2. The Generating Function 415
§ 3. Laplace’s Formula 418

INDEX 421

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A Collection of Problems in The Theory of Numbers – Sierpinski

Posted on March 20, 2022 by The Mitr

In this post, we will see the book A Collection of Problems in The Theory of Numbers by Waclaw Sierpinski.

 

About the book

A Selection of Problems in the Theory of Numbers focuses on mathematical problems within the boundaries of geometry and arithmetic, including an introduction to prime numbers. This book discusses the conjecture of Goldbach; hypothesis of Gilbreath; decomposition of a natural number into prime factors; simple theorem of Fermat; and Lagrange’s theorem. The decomposition of a prime number into the sum of two squares; quadratic residues; Mersenne numbers; solution of equations in prime numbers; and magic squares formed from prime numbers are also elaborated in this text. This publication is a good reference for students majoring in mathematics, specifically on arithmetic and geometry.

The book was translated from Polish by A. Sharma and was published in  1964.

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Contents

On the Borders of Geometry and Arithmetic

What We Know and What We Do Not Know about Prime Numbers

1. What are Prime Numbers?

2. Prime Divisors of a Natural Number

3. How Many Prime Numbers are There?

4. How to Find All the Primes Less than a Given Number

5. Twin Primes

6. Conjecture of Goldbach

7. Hypothesis of Gilbreath

8. Decomposition of a Natural Number into Prime Factors

9. Which Digits Can There Be at the Beginning and at the End of a Prime Number?

10. Number of Primes Not Greater than a Given Number

11. Some Properties of the n-th Prime Number

12. Polynomials and Prime Numbers

13. Arithmetic Progressions Consisting of Prime Numbers

14. Simple Theorem of Fermat

15. Proof That There is an Infinity of Primes in the Sequences 4k+1, 4k+3 and 6k+5

16. Some Hypotheses about Prime Numbers

17. Lagrange’s Theorem

18. Wilson’s Theorem

19. Decomposition of a Prime Number into the Sum of Two Squares

20. Decomposition of a Prime Number into the Difference of Two Squares and Other Decompositions

21. Quadratic Residues

22. Fermat Numbers

23. Prime Numbers of the Form nn + 1, nnn + 1 etc.

24. Three False Propositions of Fermat

25. Mersenne Numbers

26. Prime Numbers in Several Infinite Sequences

27. Solution of Equations in Prime Numbers

28. Magic Squares Formed from Prime Numbers

29. Hypothesis of A. Schinzel

One Hundred Elementary but Difficult Problems in Arithmetic

References

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A History Of Psychology – Yaroshevsky

Posted on March 19, 2022 by The Mitr

In this post, we will see the book A History Of Psychology by Mikhail Yaroshevsky.

About the book

The book provides a comprehensive history of development of psychological thought through the ages. Starting from ancient times, the Eastern, Greek and Roman ideas about mind and behavior, the book takes us through medieval and renaissance. The further development of psychology after Kant and earlier century is presented in later chapters. The rise of empiricism and associationism in Eighteenth-Century is dealt in chapter 6. Further the development of theory of reflexes and theory of brain in later nineteenth century is presented. This concludes the first part of the book.

The second part of the book traces the development of psychology as an independent science. The rise of experimental doctrine and its various branches in late 19th and early 20th century are discussed in chapters 9-10. Chapter 11 discusses various schools of thought in psychology that arose in the first half of 20th century such as Titchener’s Structuralistic School, The Würzburg School, Functionalism, Behaviourism, Gestalt Psychology, Levin’s School and the The Freudian School. Chapter 12 discusses further growth of of behaviorism and Piaget’s ideas. The last chapter discusses the history and state of psychology in Russia.

The book was translated from Russian by Ruth English and was published in 1990 by Progress Publishers.

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Contents

Introduction. History as the Self-Cognition of Science 5

PART I PSYCHOLOGY AS AN ELEMENT WITHIN PHILOSOPHY AND NATURAL
SCIENCES

Chapter 1. The Beginnings of Psychological Thought in the
Countries of the East 19

Chapter 2. Psychology in Classical Times 30

The Ideas of Natural Scientists in Ancient Greeks Concerning the Soul 30
Psychological Ideas in the Hellenic Period 62
Psychology on Ancient Rome 69
The Decline of Ancient Psychology 71

Chapter 3. Theories of the Soul in the Feudal epoch 76

Development of Psychological Ideas in Arabic Science 77
The Psychological Ideas in Mediaeval Europe 83
Nominalism 88

Chapter 4. Psychological thought in the Renaissance 90

Psychology in the Period of the Italian Renaissance 90
The Empirical Trend in Psychology in Spain 95

Chapter 5. Psychological Doctrines of the Seventeenth Century 100

Outcome of the Development of ‘Psychological “Thought in the Seventeenth Century 126

Chapter 6. The Supremacy of Empiricism and Associationism in Eighteenth-Century 131

Associative Psychology 131
The Psychology of Abilities 137
Materialist Psychology in France 139
The Rise of a Materialist Trend in Russian Psychology 145
Dawn of the Idea of Cultural-Historical Laws Governing Spiritual Life 149
The Significance of Immanuel Kant’s Doctrine for the Development of Psychology 152
Summary of Development of Psychology in the Eighteenth Century 156

Chapter 7. Psychology in the First Half of the Nineteenth Century 159

Theory of Reflexes 159
Theories of the Brain 167
Associative Psychology 169

PART II DEVELOPMENT OF PSYCHOLOGY AS AN INDEPENDENT SCIENCE

Chapter 8. Preconditions for the Psychology Becoming an Independent Science 174

Philosophical Doctrines 174
Premises Provided by Natural Sciences 179

Chapter 9. Programmes for Developing Psychology into an Experimental Science 196

Psychology as the Doctrine of Intentional Acts of Consciousness. F. Brentano 202
Psychology as a Doctrine of the Performance of Psychic Activities. I. M. Sechenov 205

Chapter 10. Development of Branches of Psychology in the Late Nineteenth and Early Twentieth Century 213

Experimental Psychology 213
Differential Psychology 223
Child Psychology and Educational: Psychology 234
Zoopsychology 241
Social and Cultural- Historical Psychology 246
Psychotechnics 254

Chapter 11. Schools of Thought in Psychology 256

Titchener’s Structuralistic School 258
The Wurzburg School 262
Functionalism in American Psychology 267
Behaviourism 275
Gestalt Psychology 284
Kurt Levin and His School 294
The Freudian School 299

Chapter 12. Psychology in the Capitalist Countries in the
1930s and 1940s of the Twentieth Century 316

The Evolution of Behaviourism 318
Cognitive Behaviourism319
Neo-Freudianism 332
Jean Piaget’s Doctrine of Intellectual Development 335

Chapter 13. Psychology in Russia 340

A Concise History of Russian Psychological Thought 340
Soviet Psychology. 357

Conclusion. 409
Name Index. 410

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Mechanics of Gyroscopic Systems – Ishlinskii

Posted on March 18, 2022 by The Mitr

In this post, we will see the book Mechanics of Gyroscopic Systems by A. Yu. Ishlinskii.

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.

 

 

Translated from Russian under Israel Program for Scientific Translations Jerusalem 1965

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Contents

FOREWORD. 1

Chapter I. GEOMETRY AND KINEMATICS OF GYROSCOPIC SYSTEMS 5

§ 1. Geometry of gimbal suspension systems. Determination of a
ship’s pitch and roll angles and its course. Gimbal error. Bicardan suspensions. 5
§ 2. Relative rotation of two stabilized systems during ship’s rolling 12
§ 3. Stabilization errors caused by inaccurate mounting of the
gimbal systems (geometry of two bicardan suspensions) 17
§ 4. Horizontal stabilization errors of combinations of different types of gimbal systems 24
§ 5. Variation of the polar coordinates of a fixed point caused by
horizontal stabilization errors (analytical treatment) 30
§ 6. Geometric determination of the stabilization errors by the theory of infinitesimal rotations of a rigid body 35
§ 7. Variation of the ship’s roll and pitch angles and of its course caused by a finite rotation of the ship about an arbitrary axis 40

Chapter II. ORIENTATION OF GYRO-CONTROLLED OBJECTS 46

§ 1. The orientation accuracy of an object launched from an
inclined base 46
§ 2. Deviation of a self- guiding missile from the specified direction during flight 55
§ 3. Some general considerations on methods for solving problems on the geometry of stabilization systems 62
§ 4. Nonholonomic motions of gyroscopic systems 67

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

§ 1. Elastic deformations of the gyro rotor under the influence of centrifugal forces 75
§ 2. Deformation of the gyro housing 79
§ 3. The rigidity of the gimbal rings 82
§ 4. Discontinuous motion of insufficiently rigid kinematic transmissions 86
§ 5. Influence of the rigidity of the gyroscopic-system elements on the frequency of nutations 93
§ 6 The damping of gyroscopic and other devices mounted on objects moving at high accelerations 97

Chapter IV. LINEAR THEORY OF GYROSCOPIC SYSTEMS 105

§ 1. The equations of gyroscopic systems 105
§ 2. Theory of the gyrovertical with aerodynamic suspension and its possible improvements 115
§ 3. Gyrovertical with auxiliary gyro 134
§ 4. Theory of the gyroscopic heel equalizer 148
§ 5. The gyroscopic frame 169

Chapter V. NONLINEAR PROBLEMS IN THE THEORY OF GYROSCOPES 178

§ 1. Sliding motions of gyroscopic systems aera 178
§ 2. Energy method for investigating the stability of gyroscopic systems 192
§ 3. Forced oscillations of a gyroscopic frame (monoaxial stabilizer) 203
§ 4. Behavior of a directional gyro on a rolling base 211

Chapter VI. VARIOUS PROBLEMS IN GYRO-SYSTEM MECHANICS 221

§ 1. Application of probability methods to determining the errors of a gyrohorizon with contact correction during rolling 221
§ 2. The effect of the ship’s yaw on the accuracy of the gyrohorizon readings 232
§ 3. The gyro Top Bow 237
§ 4. The errors of the gyroscopic apparent-velocity meter 243
§ 5. Precessional oscillations of a gyroscope acted upon by a load 245
§ 6. Influence of vibrations on the accuracy of gyroscopic instrument readings 250
§ 7. Theory of follow-up systems 251

 

APPENDIX I. THEORY OF COMPLEX GYROSCOPIC STABILIZATION SYSTEMS 261
APPENDIX II. THEORY OF THE GYROHORIZONCOMPASS 275
APPENDIX III. DETERMINING THE POSITION OF A MOVING OBJECT BY GYROS AND ACCELEROMETERS 288

BIBLIOGRAPHY 304
LIST OF RUSSIAN ABBREVIATIONS 311

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Limits and Continuity – Korovkin

Posted on March 17, 2022 by The Mitr

In this post, we will see the book Limits and Continuity by P. P. Korovkin.

About the book

The present volume in The Pocket Mathematical Library stems in large part from Chapters 1-4 of P. P. Korovkin’s Mathema­tical Analysis, Moscow (1963). The material has been heavily
rewritten and supplemented by 21 problem sets, one after each section. The result is a succinct but remarkably complete introduction to the theory o f limits and continuity. The book may also be thought of as a “precalculus” text in that it deals with those properties of functions which can be successfully discussed short of introducing the notion of a derivative.

Answers to the even-numbered problems will be found at the end of the book.

The book was translated from Russian by was published in 1969. The book is a part of the Pocket Mathematical Library Series.

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Contents

Chapter 1. Functions 1

1. Variables and Functions. Intervals and Sequences 1
2. Absolute Values. Neighborhoods 9
3. Graphs and Tables 12
4. Some Simple Function Classes 14
5. Real Numbers and Decimal Expansions 21

Chapter 2. Limits 24

6. Basic Concepts 24
7. Algebraic Properties of Limits 32
8. Limits Relative to a Set. One-Sided Limits 37
9. Infinite Limits. Indeterminate Forms 43
10. Limits at Infinity 49
11. Limits of Sequences. The Greatest Lower Bound Property 53
12. The Bolzano-Weierstrass Theorem. The Cauchy Convergence Criterion 58
13. Limits of Monotonic Functions. The Function a^{x} and
the Number e 62

Chapter 3. Continuity 73

14. Continuous Functions 72
15. One-Sided Continuity. Classification of Discontinuities 79
16. The Intermediate Value Theorem. Absolute Extrema 83
17. Inverse Functions 89
18. Elementary Functions 92
19. Evaluation of Limits 101
20. Asymptotes 105
21. The Modulus of Continuity. Uniform Continuity 111

Answers to Even-Numbered Problems 116
Index 123

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