## Computational Mathematics – Demidov, Maron

In this post, we will see the bookComputational Mathematics by B. P. Demidovich and I. A. Maron.

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

This text makes wide use of matrix calcu]us. The concepts of a vector, matrix, inverse matrix, eigenvalue and eigenvector of a matrix, etc. are workaday tools. The use of matrices offers a number of advantages in presenting the subject matter since they greatly facilitate an understanding of the development of many computations. In this sense a particular gain is achieved in the proofs of the convergence theorems of various numerical processes. Also, modern high-speed computers are nicely adapted to the performance of the basic matrix operations.
For a full comprehension of the contents of this. book, the reader should have a background of linear algebra and the theory of linear vector spaces. With the aim of making the text as self-contained as possible, the authors have included all the necessary starting material in these subjects. The appropriate chapter~ are completely independent of the basic text and can be omitted by readers who have already studied these sections.

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

A great deal of attention is devoted to methods of error estimation. Nearly all processes are provided with proofs of convergence theorems, and the presentation is such that the proofs may be omitted if one wishes to confine himself to the technical aspects of the matter. In. certain case?, in order to pictorialize and lighten the presentation, the computational techniques are given as simple recipes.

The basic methods are carried to numerical applications that include computational schemes and numerical examples with de- tailed step.s of solution. To facilitate understanding the essence of the matter at hand, most of the problems are stated in simple form and are of an illustrative nature. References are given at the. end of each chapter and the complete list (in alphabetical order) is given at the end of the book.

The present text offers selected methods in computational mathematics and does not include material that involves empirical formulas, quadratic approximation of functions, approximate solutions of differential equations, etc. Likewise, the book does not include material on programming and the technical aspects of solving mathematical problems on computers. The interested reader must consult the special literature on these subjects.

The book was translated from Russian by George Yankovsky and was published by Mir in 1981.

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Contents

PREFACE 5

INTRODUCTION. GENERAL RULES OF COMPUTATIONAL WORK 15

CHAPTER 1 APPROXIMATE NUMBERS 19

1.1 Absolute and relative errors 19
1.2 Basic sources of errors 22
1.3 Scientific notation. Significant digits. The number of correct digits 23
1.4 Rounding of numbers 26
1.5 Relationship between the relative error of an approximate number and the number of correct digits 27
1.6 Tables for determining the limiting relative error from the number of correct digits and vice versa 30
1.7 The error of a sum 33
1.8 The error of a difference 35
1.9 The error of a product 37
1.10 The number of correct digits in a product 29
1.11 The error of a quotient 40
1.12 The number of correct digits in a quotient 41
1.13 The relative error of a power 41
1.14 The relative error of a root 41
1.15 Computations in which errors are not taken into exact account 42
1.16 General formula for errors 42
1.17 The inverse problem of the theory of errors 44
1.18 Accuracy in the determination of arguments from a tabulated function 48
1.19 The method of bounds 50
1.20 The notion of a probability error estimate 52

References for Chapter 1 54

CHAPTER 2 SOME FACTS FROM THE THEORY OF CONTINUED FRACTIONS 55

2.1 The definition of a continued fraction 55
2.2 Converting a continued fraction to a simple fraction and vice versa 56
2.3 Convergents 58
2.4 Nonterminating continued fractions 66
2.6 Expanding functions into continued fractions 72

References for Chapter 2 76

CHAPTER 3 COMPUTING THE VALUES OF FUNCTIONS 77

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

CHAPTER 4 APPROXIMATE SOLUTIONS OF ALGEBRAIC AND TRANSCENDENTAL EQUATIONS 115

4.1 Isolation of roots 115
4,2 Graphical solution of equations 119
4.3 The halving method 121
4.4 The method of proportional parts (method of chords) 122
4.5 Newton’s method (method of tangents) 127
4.6 Modified Newton method 135
4,7 Combination method 136
4.8 The method of iteration 138
4.9 The method of iteration for a system of two equations 152
4.10 Newton’s method for a system of two equations 156
4.11 Newton’s method for the case of complex roots 157

References for Chapter 4 161

CHAPTER 5 SPECIAL TECHNIQUES FOR APPROXIMATE SOLUTION OF ALGEBRAIC EQUATIONS 162

5.1 General properties of algebraic equations 162
5.2 The bounds of real roots of algebraic equations 167
5.3 The method of alternating sums 169
5,4 Newton’s method 171
5.5 The number of real roots of a polynomial 173
5.6 The theorem of Budan-Fourier 175
5.7 The underlying principle of the method of Lobachevsky-Graeffe 179
5.8 The root-squaring process 182
5.9 The Lobachevsky-Graeffe method for the case of real and distinct roots 184
5.10 The Lobachevsky-Graeffe method for the case of complex foots 187
5.11 The case of a pair of complex roots 190
5.12 The case of two pairs of complex roots 194
5.13 Bernoulli’s method 198
References for Chapter 5 202

CHAPTER 6 ACCELERATING THE CONVERGENCE OF SERIES 203

6.1 Accelerating the convergence of numerical series 203
6.2 Aecelerating the convergence of power series by the Euler-Abel methods 209
6.3 Estimates of Fourier coefficients 213
6.4 Accelerating the convergence of Fourier trigonometric series by the method of A. N. Krylov 217
6.5 Trigonometric approximation 225

References for Chapter 6 228

CHAPTER 7 MATRIX ALGEBRA 229

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

References for Chapter 7 272

CHAPTER 8 SOLVING SYSTEMS OF LINEAR EQUATIONS 273

8.1 A general description of methods of solving systems of linear equations 273
8.2 Solution by inversion of matrices, Cramer’s rule 273
8.3 The Gaussian method 277
8.4 Improving roots 284 287 288 290 293 296 300 307 309 311 313 316 321 322 322 324 327
8.5 The method of principal elements 287
8.6 Use of the Gaussian method in computing determinants 288
8.7 Inversion of matrices by the Gaussian method 290
8.8 Square-root method 293
8.9 The scheme of Khaletsky 296
8.10 The method of iteration 300
8.11 Reducing a linear system to a form convenient for iteration 307
8.12 The Seidel method 309
8.13 The case of a normal system 311
8.14 The method of relaxation 313
8.15 Correcting elements of an approximate inverse matrix 316

References for Chapter 8 321

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

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

References for Chapter 9 335

CHAPTER 10 ESSENTIALS OF THE THEORY OF LINEAR VECTOR SPACES 336

10.1 The concept of a linear vector space 336
10.2 The linear dependence of vectors 337
10.3 The scalar product of vectors 343
10,4 Orthogonal systems of vectors 345
10.5 Transformations of the coordinates of a vector under changes in the basis 348
10.6 Orthogonal matrices 350
10.7 Orthogonalization of matrices 351
10.8 Applying orthogonalization methods to the solution of systems of linear equations 358
10.9 The solution space of a homogeneous system 364
10.10 Linear transformations of variables 367
10.11 Inverse transformation 373
10.12 Eigenvectors and eigenvalues of a matrix 375
10.13 Similar matrices 380
10.14 Bilinear form of a matrix 384
10.15 Properties of symmetric matrices 384
10.16 Properties of matrices with real elements 389

References for Chapter 19 393

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

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

References for Chapter 11 409

CHAPTER 12 FINDING THE EIGENVALUES AND EIGENVECTORS OF A MATRIX 410

12.1 Introductory remarks 410
12.2 Expansion of secular determinants 410
12,3 The method of Danilevsky 412
12.4 Exceptional cases in the Danilevsky method 418
12.5 Computation of eigenvectors by the Danilevsky method 420
12.6 The method of Krylov 421
12.7 Computation of eigenvectors by the Krylov method 424
12.8 Leverrier’s method 426
12.9 On the method of undetermined coefficients 428
12.10 A comparison of different methods of expanding a secular determinant 429
12.11 Finding the numerically largesi eigenvalue of a matrix and the corresponding eigenvector 430
12.12 The method of ‘scalar products for finding the first eigenvalue of a real matrix 436
12.13 Finding the second eigenvalue of a matrix and the second eigenvector 439
12.14 The method of exhaustion 443
12.15 Finding the EES and eigenvectors of a positive definite symmetric matrix 445
12.16 Using the coefficients of the characteristic polynomial of a matrix for matrix inversion 450
12.17 The method of Lyusternik for accelerating the convergence of the iteration process in the solution of a system of linear equations 453

References for Chapter 12 458

CHAPTER 13 APPROXIMATE SOLUTION OF SYSTEMS OF NONLINEAR EQUATIONS 459

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

References for Chapter 13 506

CHAPTER 14 THE INTERPOLATION OF FUNCTIONS 507

14.1 Finite differences of various orders 507
14.2 Difference table 510
14.3 Generalized power 517
14.4 Statement of the problem of interpolation 518
14.5 Newton’s first interpolation formula 519
14,6 Newton’s second interpolation formula 526
14.7 Table of central differences 530
14.8 Gaussian interpolation formulas 531
14.9 Stirling’s interpolation formula 533
14.10 Bessel’s interpolation formula 534
14.11 General description of interpolation formulas with constant interval 536
14.12 Lagrange’s interpolation formula 539
14.13 Computing Lagrangian coefficients 543
14.14 Error estimate of Lagrange’s interpolation formula 547
14.15 Error estimates of Newton’s interpolation formulas 550
14.16 Error estimates of the centrai interpolation formulas 552
14.17 On the best choice of interpolation points 553
14.18 Divided differences 554
14.19 Newton’s interpolation formula for unequally spaced values of the argument 556
14.20 Inverse interpolation for the case of equally spaced points 559
14.21 Inverse interpolation for the case of unequally spaced points 562
14.22 Finding the roots of an equation by inverse interpolation 564
14.23 The interpolation method for expanding a secular determinant 565
14.24 Interpolation of functions of two variables 567
14.25 Double differences of higher order 570
14.26 Newton’s interpolation formula for a function of two variables 571

References for Chapter 14 573

CHAPTER 15 APPROXIMATE DIFFERENTIATION 574

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

References for Chapter 15 FBO

CHAPTER 16 APPROXIMATE INTEGRATION OF FUNCTIONS 590

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

References for Chapter 16 648

CHAPTER 17 THE MONTE CARLO METHOD 649

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

References for Chapter 17 674

Complet list of references 675

INDEX 679

## Lectures on Nuclear Theory – Landau, Smorodinsky

In this post, we will see the book Lectures On Nuclear Theory by L. D. Landau; Ya. Smorodinsky.

This book is based on a series of lectures delivered to experimental physicists by one of the authors (L. Landau) in Moscow in 1954.

In maintaining the lecture form in the printed edition we are emphasizing the fact that the presentation makes no pretense at completeness and that the choice of subject matter is purely arbitrary.

Since there is, at the present time, no rational theory of nuclear forces, we have limited our conclusions concerning nuclear structure to those which can be reached from an analysis of the available experimental data, using only general quantum-mechanical relations.

No attempt has been made to give a bibliography of the literature; rather we have indicated only new experimental results.

The book was translated from Russian and was published in 1959.

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Contents

Foreword V

LECTURE ONE: Nuclear Forces 1

LECTURE TWO: Nuclear Forces (Scattering of Nucleons by Nucleons) 13

LECTURE THREE: Nuclear Forces (Scattering of Nucleons at High Energies) 23

LECTURE FOUR: Nuclear Structure (Independent Particle Model) 33

LECTURE FIVE: Structure of the Nucleus (Light Nuclei) 43

LECTURE SIX: Structure of the Nucleus (Heavy Nuclei) 53

LECTURE SEVEN: Nuclear Reactions (Statistical Theory) 69

LECTURE EIGHT: Nuclear Reactions (Optical Model. Deuteron Reactions) 77

LECTURE NINE: 𝜋-Mesons 87

LECTURE TEN: Interaction of 𝜋-Mesons with Nucleons 99

## Space and Time in the Microworld – Blokhintsev

In this post, we will see the book Space and Time in the Microworld by D. I. Blokhintsev  .

In comprehending the physical content of dynamic variables which have geometric meaning, for example, the space-time particle coordinates x, y, z, t it is often necessary to have recourse to gedanken experiments which, although not feasible in practice, can nevertheless be compatible with the basic principles of geometry and quantum mechanics. In a desert sea of abstract constructions there is a still larger distance between macroscopic concepts of space-time and the way of employing the coordinates x, y, z and t in relativistic quantum field theory.

It is shown in this monograph that if elementary particles have a structure it is doubtful whether the coordinates of elementary particles, x, y, z, t, can even be defined exactly, let alone the coordinates of the elements which make up these particles (if they do not exist only in our imagination). This important fact is revealed in even the most favourable gedanken experiments.

From this fact, doubt arises about the logical validity of using the symbols x, y, z, t as the space-time coordinates to describe phenomena inside elementary particles. This allows theoreticians a certain freedom of choice of space-time and causal relationships within elementary particles; in other words, an arbitrariness of choice of the geometry in
the small.

The last chapters of this book describe some models used to illustrate the situation described above. In concluding, experimental data and experimental possibilities relating to geometric and causal problems in the microworld are discussed.

The book was translated from Russian by Zdenka Smith and was published in 1970.

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Contents

INTRODUCTION

I. GEOMETRICAL MEASUREMENTS IN THE MACROWORLD 1

1. The Arithmetization of Space-Time 1
2. The Physical Methods of Arithmetization of Space-Time 3
3. On Dividing the Manifold of Events into Space and Time 10
4. The Affine Manifold 19
5. The Riemann Manifold 23
6. The Physics of Arithmetization of the Space-Time Manifold 28
7. Arithmetization of Events in the Case of the Non-Linear Theory of Fields 33
8. The General Theory of Relativity and the Arithmetization of Space-Time 37
9. Chronogeometry 42

II. GEOMETRICAL MEASUREMENTS IN THE MICROWORLD 44

10. Some Remarks on Measurements in the Microworld 44
11. The Measurement of Coordinates of the Microparticles 46
12. The Mechanics of Measuring Coordinates of Microparticles 52
13. Indirect Measurement of a Microparticle Coordinates at a Given Instant in Time 64

III. GEOMETRICAL MEASUREMENTS IN THE MICROWORLD IN THE RELATIVISTIC CASE 69

14. The Fermion Field 69
15. The Uncertainty Relation for Fermions 74
16. The Boson Field 77
17. The Localization of Photons 82
18. The Diffusion of Relativistic Packets 86
19. The Coordinates of Newton and Wigner 89
20. The Measurement of a Microparticle’s Coordinates in the Relativistic Case

IV. THE ROLE OF FINITE DIMENSIONS OF ELEMENTARY PARTICLES 95

21. The Polarization of Vacuum. The Dimensions of an Electron 95
22. The Electromagnetic Structure of Nucleons 99
23. The Meson Structure of Nucleons 108
24. The Structure of Particles in Quantized Field Theory 114

V. CAUSALITY IN QUANTUM THEORY 124

25. A Few Remarks on Causality in the Classical Theory of Fields 124
26. Causality in Quantum Field Theory 132
27. The Propagation of a Signal “Inside” a Microparticle 141
28. Microcausality in the Quantum Field Theory 147
29. Microcausality in the Theory of Scattering Matrices 153
30. Causality and the Analytical Properties of the Scattering Matrix 159

VI. MACROSCOPIC CAUSALITY 173

31. Formal S-matrix Theory 173
32. Space-Time Descriptions Using the S-matrix 182
33. The Scale for the Asymptotic Time T 187
34. Unstable Particles (Resonances) 191
35. Conditions of Macroscopic Causality for the S-matrix 200
36. Examples of Acausal Influence Functions 207
37. An Example of Constructing an Acausal Scattering Matrix 211
38. The Dispersion Relation for the Acausal S_{a},-Matrix 219

VII. A GENERALIZATION OF CAUSAL RELATIONSHIPS AND GEOMETRY 226

39. Two Possible Generalizations 226
40. Euclidean Geometry in the Microworld 232
41. Stochastic Geometry 237
42. Discrete Space-Time 243
43. Quasi-Particles in Quantized Space 250
44. Fluctuations of the Metric 255
45. Nonlinear Fields and the Quantization of Space-Time 261

VIII. EXPERIMENTAL QUESTIONS 269

46. Concluding Remarks on the Theory 269
47. Experimental Consequences of Local Acausality 270
48. Experimental Results of Models with the “External” Vector 278

APPENDICES 282

BIBLIOGRAPHY 326

## Configuration Theorems – Argunov, Skornyakov

In this post, we will see the book Configuration Theorems by B. I. Argunov, L. A. Skornyakov. The book is a part of Topics in Mathematics series. We will see some more books in this series in the future.

This booklet presents several important configuration theorems, along with their applications to the study of the properties of figures and to the solutions of several practical problems. In doing this, the authors introduce the reader to some fundamental concepts of projective geometry-central projection and ideal elements of space. Only the most elementary knowledge of plane and solid geometry is presupposed.

Chapters 2 and 3 are devoted to the two most important configuration theorems, the Pappus-Pascal theorem and that of Desargues. The chapters which follow present applications of these theorems. Chapter 6 touches upon the algebraic interpretation of configuration theorems and the general method of arriving at such theorems.

The book was translated from Russian by Edgar E. Enochs and Robert B. Brown and was published in 1963.

You can get the book here.

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Contents

Introduction 1

1. What is a configuration theorem? 1
2. Example of a configuration theorem 2

Chapter 1 Central Projection and Ideal Elements 4

3. Central projection in a plane; ideal elements
4. Elimination of exceptional cases
5. Basic theorems about projective lines
6. Central projection in space

Chapter 2 Theorem of Pappus and Pascal

7. The Pappus-Pascal Theorem 9
8. Introduction to the proof of Pappus-Pascal Theorem 11
9. Completion of the proof of Pappus-Pascal Theorem 13
10. Brianchon’s Theorem 14

Chapter 3 Desargue’s Theorem 16

11. Desargue’s Theorem 16
12. Alternative proofs of Desargues’s theorem 19
13. The converse of Desargue’s Theorem 23

Chapter 4 Some Properties of Polygons 24

14. Some properties of quadrilaterals 24
15. Some properties of pentagons 25
16. More properties of quadrilaterals 26

Chapter 5. Problems 29

17. Inaccessible points or lines 29
18. Constructions involving inaccessible points or lines30
19. Problems for solutions by the reader 34

Chapter 6. The Algebraic Meaning of Configuration Theorems 37

20. Algebraic Identities as Configuration Theorems 37
21. Schematic notation of Configuration Theorems 38

Bibliography 41

## Design of Automotive Engines – Kolchin, Demidov

In this post, we will see the book Design of Automotive Engines by A. Kolchin, V. Demidov.

The book contains the necessary information and systematized methods for the design of motor vehicle and tractor engines.
Assisting the students in assimilating the material and gaining deep knowledge, this work focuses on the practical use of the knowledge in the design and analysis of motor vehicle and tractor engines.

This educational aid includes many reference data on modern engines and tables covering the ranges in changing the basic mechanical parameters, permissible stresses and strains, etc.

The book was translated from Russian by P. Zabolotnyi and was published in 1984.

You can get the book here.

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Contents

Preface

Part One WORKING PROCESSES AND CHARACTERISTICS

Chapter 1. FUEL AND CHEMICAL REACTIONS 9

1.1 General 9
1.2 Chemical Reactions in Fuel Combustion 12
1.3 Heat of Combustion of Fuel and Fuel-Air Mixture 19
1.4 Heat Capacity of Gases 21

Chapter 2. THEORETICAL CYCLES OF PISTON ENGINES 24

2.1. General 24
2.2. Closed Theoretical Cycles 31
2.3. Open Theoretical Cycles 41

Chapter 3. ANALYSIS OF ACTUAL CYCLE 47

3.1 Induction Process 47
3.2 Compression Process 54
3.3 Combustion Process 58
3.4 Expansion Process 63
3.5 Exhaust process and Methods of pollution control 67
3.6 Indicated Parameters of Working Cycle 69
3.7 Engine Performance Figures 74
3.8 Indicator Diagram 78

Chapter 4. HEAT ANALYSIS AND HEAT BALANCE 82

4.1 General 82
4.2 Heat Analysis and Heat Balance of a Carburettor Engine 86
4.3 Heat Analysis and Heat Balance of Diesel Engine 105

Chapter 5. SPEED CHARACTERISTICS 116

5.1 General 116
5.2 Plotting External Speed Characteristic 118
5.3 Plotting External Speed Characteristic of Carburettor Engine 121
5.4 Plotting External Speed Characteristic of Diesel Engine 123

Part Two KINEMATICS AND DYNAMICS

Chapter 6. KINEMATICS OF CRANK MECHANISM 127

6.1 General 127
6.2 Piston Stroke 130
6.3 Piston Speed 132
6.4 Piston Acceleration 134

Chapter 7. DYNAMICS OF CRANK MECHANISM 137

7.1 General 137
7.2 Gas Pressure Forces 137
7.3 Referring Masses of Crank Mechanism Parts 139
7.4 Inertial Forces 141
7.5 Total Forces Acting in Crank Mechanism 142
7.6 Forces Acting on Crankpins 147
7.7 Forces Acting on Main Journals 152
7.8 Crankshaft Journals and Pins Wear 157

Chapter 8. ENGINE BALANCING 158

8.1 General 158
8.2 Balancing Engines of Different Types 160
8.3 Uniformity of Engine torque and Run 167
8.4 Design of Flywheel 170

Chapter 9. ANALYSIS OF ENGINE KINEMATICS AND DYNAMICS 171

9.1 Design of an In-Line Carburettor Engine 171
9.2 Design of V-Type Four-Stroke Diese Engine 191

Part Three DESIGN OF PRINCIPAL PARTS

Chapter 10. PREREQUISITE FOR DESIGN AND DESIGN CONDITIONS 215

10.1 General 215
10.2 Design Conditions 216
10.3 Design of Parts Working Under Alternating Loads 217

Chapter 11. DESIGN OF PISTON ASSEMBLY 226

11.1. Piston 226
11.2. Piston Rings 234
11.3. Piston Pin 238

Chapter 12. DESIGN OF CONNECTING ROD ASSEMBLY 244

12.1. Connecting Rod Small End 244
12.2. Connecting Rod Big End 259
12.3. Connecting Rod Shank 261
12.4. Connecting hod Bolts 266

Chapter 13. DESIGN OF CRANKSHAFT 268

13.1 General 268
13.2 Unit Area Pressures on Crankpins and Journals 271
13.3 Design of Journals and Crankpins 272
13.4 Design of Crankwebs 277
13.5 Design of In-Line Engine Crankshaft 278
13.6 Design of V-Type Engine Crankshaft 286

Chapter 14. DESIGN OF ENGINE STRUCTURE 296

14.1 Cylinder Block and Upper Craakease 296
14.2 Cylinder Liners 298

Chapter 15. DESIGN OF VALVE GEAR 308

15.1 General 308
15.2 Cam Profile Construction 310
15.3 Shaping Harmonic Cams 314
15.4 Time-Section of Valve 320
15.5 Design of the Valve Gear for a Carburettor Engine 321
15.6 Design of Valve Spring 331
15.7 Design of the Camshaft 339

Part Four ENGINE SYSTEMS

Chapter 16. SUPERCHARGING 343

16.1 General 343
16.2 Supercharging Units and Systems 344
16.3 Turbo-Supercharger Design Fundamentals 348
16.4 Approximate Computation of a Compressor and a Turbine 362

Chapter 17. DESIGN OF FUEL SYSTEM ELEMENTS 372

17.1 General 372
17.2 Carburettor 373
17.3 Design of Carburettor 380
17.4 Design of Diesel Engine Fuel System Elements 385

Chapter 18. DESIGN OF LUBRICATING SYSTEM ELEMENTS 390

18.1 Oil Pump 390
18.2 Centrifugal Oil Filter 394
18.3 Oil Cooler 397
18.4 Design of Bearings 400

Chapter 19. DESIGN OF COOLING SYSTEM COMPONENTS 403

19.1 General 403
19.2 Water Pump 404
19.4 Cooling Fan 412
19.5 Computation of Air Cooling Surface 415

Appendices 417
References 424
Index 426

## A Collection of Problems on the Equations of Mathematical Physics – Vladimirov

In this post, we will see the book A Collection of Problems on the Equations of Mathematical Physics edited by V. S. Vladimirov. The contributors to the book include V .S. Vladimirov, V .P . Mikhailov, A. A. Vasharin, Kh. Kh.Karimova, Yu. V. Sidorov, and M.I. Shabunin. This is the associated problem book for Equations of Mathematical Physics and Partial Differential Equations which we have seen earlier.

The extensive application of modern mathematical techniques to
theoretical and mathematical physics requires a fresh approach to the course of equations of mathematical physics. This is especially true with regards to such a fundamental concept as the solution of a boundary value problem. The concept of a generalized solution considerably broadens the field of problems and enables solving from a unified position the most interesting problems that cannot be solved by applying classical methods. To this end two new courses have been written at the Department of Higher Mathematics at the Moscow Physics and Technology Institute, namely, “Equations of Mathematical Physics” by V.S. Vladimirov and “Partial Differential Equations” by V.P. Mikhailov (both books have been translated into English by Mir Publishers, the first in 1984 and the second
in 1978).

The present collection of problems is based on these courses and
amplifies them considerably. Besides the classical boundary value problems, we have included a large number of boundary value problems that have only generalized solutions. Solution of these requires using the methods and results of various branches of modern analysis. For this reason we have included problems in Lebesgue integration, problems involving function spaces (especially spaces of generalized differentiable functions) and generalized functions (with Fourier and Laplace transforms), and integral equations.

The book is aimed at undergraduate and graduate students in the physical sciences, engineering, and applied mathematics who have taken the typical “methods” course that includes vector analysis, elementary complex variables, and an introduction to Fourier series and boundary value problems. Asterisks denote the more difficult problems.

The book was translated from Russian by Eugene Yankovsky and was published in 1986.

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Contents

Preface 5

Symbols and Definitions 9

Chapter I Statement of Boundary Value Problems in Mathematical Physics 12

1 Deriving Equations of Mathematical Physics 12
2 Classification of Second-order Equations 35

Chapter II Function Spaces and Integral Equations 41

3 Measurable Functions. The Lebesgue Integral 41
4 Function Spaces 48
5 Integral Equations 67

Chapter III Generalized Functions 88

6 Test and Generalized Functions 88
7 Differentiation of Generalized Functions 95
8 The Direct Product and Convolution of Generalized Functions 104
9 The Fourier Transform of Generalized Functions of Slow Growth 114
10 The Laplace Transform of Generalized Functions 121
11 Fundamental Solutions of Linear Differential Operators 125

Chapter IV The Cauchy Problem 134

12 The Cauchy Problem for Second-order Equations of Hyperbolic Type 134
13 The Cauchy Problem for the Heat Conduction Equation 157
14 The Cauchy Problem for Other Equations and Goursat’s Problem 167

Chapter V Boundary Value Problems for Equations of Elliptic Type 180

15 The Sturm-Liouville Problem 181
16 Fourier’s Method for Laplace’s and Poisson’s Equations 190
17 Green’s Functions of the Dirichlet Problem 205
18 The Method of Potentials 211
19 Variational Methods 230

Chapter VI Mixed Problems 239

20 Fourier’s Method 239
21 Other Methods 269

Appendix Examples of Solution Techniques for Some Typical Problems 277

A1 Method of Characteristics 277
A2 Fourier’s Method 279
A3 Integral Equations with a Degenerate Kernel 281
A4 Variational Problems 283

References 284

Subject Index 287

## Equations of Mathematical Physics – Vladimirov

In this post, we will see the book Equations of Mathematical Physics by V. S. Vladimirov.

This book examines classical boundary value problems for differentia equations of mathematical physics. Instead of the traditional means of presentation, we use the concept of the generalized solution. Generalized solutions arise in solving integral equations of the local balance type and the calculation of these solutions leads to generalized formulations of the problems of mathematical physics.

Many sections contain problems for exercises. A number of problems are given in the form of theorems which are an important addition to the basic material. For further exercises we recommend the books of B. M. Budak et al. (1) and M. M. Smirnov (1).
This book is a fuller version of the lectures which I have given over the years to the students of the Moscow Physical and Technical Institute. It is intended for students, physicists, and mathematicians who have already mastered the basis of mathematical analysis during the first two courses at university.

The book was translated from the Russian by Audrey Littlewood and edited by ALan Jeffrey and was published in 1971.

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## The Differential Equations of Thermodynamics – Sychev

In this post, we will see the book The Differential Equations of Thermodynamics by V. V. Sychev.

Thermodynamics, as is known, is constructed quite simply. Two of its main laws have been established experimentally, and by applying mathematical tools to them we can obtain the range of conclusions in which thermodynamics is so rich.

The mathematical tools of thermodynamics are simple but in
certain aspects at the same time quite sophisticated. Neglecting
some of these sophisticated “trifles” often results in crude mistakes, even in reputable works on thermodynamics.

The restricted size of the usual textbooks on thermodynamics does not permit discussing more extensively these important questions concerning the mathematical tools. For this reason it was felt necessary to consider these problems in a special book, which though limited in size would at the same time go into details.

Naturally, the author does not aim at a presentation of thermo-
dynamics and its physical, chemical, and technical applications.
These have been sufficiently discussed in the existing textbooks and monographs. The purpose of this book is more modest—to deepen the reader’s knowledge of the mathematical tools of thermodynamics, to systematize them, and at the same time to emphasize questions that are often a source of error in thermodynamic calculations. The book is therefore designed to meet the needs of students and graduates majoring in thermal physics, physical engineering, and physico- technical specialities who already have a background in general thermodynamics. I hope that the book may also prove useful to scientists, engineers, and teachers specializing in thermodynamics.

The book was translated from the Russian by Eugene Yankovsky and second edition was published by Mir in 1983.

You can get the book here.

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## The Theory of Space, Time and Gravitation – Fock

In this post we will see the book The Theory of Space, Time and Gravitation by V. A. Fock.

The aim of this book is threefold. Firstly, we intended to give a text-book on Relativity Theory and on Einstein’s Theory of Gravitation. Secondly, we wanted to give an exposition of our own researches on these subjects. Thirdly, our aim was to develop a new, non-local, point of view on the theory and to correct a widespread misinterpretation of the Einsteinian Gravitation Theory as some kind of general relativity.

The second edition differs from the first by some additions and reformulations. The question of the uniqueness of the mass tensor is treated in more detail (Section 31*) and is illustrated by two examples (Appendices B and C). The notion of conformal space is introduced and used as a basis for the treatment of Einsteinian statics (Sections 56 and 57). Greatest care has been applied to the formulation of the basic ideas of the theory and to the elucidation of those points on which the author’s views differ from the traditional (Einsteinian) ones. Thus, in order to discuss the general aspects of the relativity principle Section 49* has been added.

The author’s views on the theory are explicitly formulated in different parts of the book and are implicit in the reasoning throughout the whole text. Their general trend is to lay stress on the Absolute rather than on the Relative.

The basic ideas of Einstein’s Theory of Gravitation are considered to be:
(a) the introduction of a space-time manifold with an indefinite metric,
(b) the hypothesis that the space-time metric is not rigid but can be influenced by physical
processes and
(c) the idea of the unity of metric and gravitation.

On the other hand, the principles of relativity and of equivalence are of limited application and, notwithstanding their heuristic value, they are not unrestrictedly part of Einstein’s Theory of Gravitation as expressed by the gravitational equations.

The book was translated from the Russian by N. Kemmer and was published in 1964.

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## An A-Z Of Cosmonautics – Gor’kov, Avdeev

In this post we will see the book An A-Z Of Cosmonautics by V.  Gor’kov and Yu. Avdeev.

In the year marking the 20th anniversary of Yuri Gagarin’s flight into space, I was responsible for running the “Small Inter- cosmos’’competition. It was given this title because with the So­viet children taking part in this competition were others from Bulgaria, Hungary, Vietnam, the GDR, Laos, Mongolia and Czechoslovakia. Each o f the competitors had to devise an expe­riment which in his or her opinion might be carried out in space. These children sent in more than two thousand suggestions and amongst these were some that, in my opinion, most children would be capable o f inventing. Take a few examples. A cat, as we know, is one o f the most agile o f all creatures. But how would it behave in a state of weightlessness?

Other children suggested taking ants and bees on board a spaceship. These insects have an excellent sense o f orientation when on Earth. The children proposed an investigation as to whe­ther this sense would be quite so keen if the insects were inside an orbital space station.

There were many suggestions and projects and it would be im­possible to list them all here. I can say only one thing— that cosmonautics has ceased to be the domain o f adults alone. That is not really surprising. Many schoolchildren know more about outer space now than the first cosmonauts themselves knew when preparing for their flights.

It was by chance that I began here by mentioning the “Small Intercosmos’’. This competition underlined the fact that child­ren need good and attractive books for their studies in cosmo­nautics. An A-Z of Cosmonautics is just such a book.

The book was translated from the Russian by K. Ford and was published by Mir in 1989. The design and illustrations were done by V. Stulikov, E. Ilatovsky.

The state of Soviet space technology depicted here is perhaps in its prime (the Russian edition was in 1984). Now in 2021 it seems much of the technology and institutes depicted in this book might be in ruins, and they definitely are not in the best form as shown here. This book provides an unique, and perhaps the last (1989), insight into the Soviet space venture, detailing the history and state of the art that time in a way that is enchanting for young readers.

Though the file size is large (~130M), it does justice to fantastic colour images in the book. I will perhaps post an optimised file sometime later.

You can get the book here.

POST SCRIPT

This book holds a special significance for me. It is the only book that my father gifted to me about two and a half decades back (mid nineties). This book was source of endless fascination about outer space, rockets and space ships. Not to mention the amazing, yet simple drawings depicting the various aspects of space travel. They inspired me to do my own drawings.. In today’s an hypermedia era when you can get any images and information with a simple search, such books may not hold that magic. But back in those days, this book (and books like this) provided the much needed fuel for flights of fantasy! You could model your space station along these lines, and imagine the kind of food cosmonauts eat, how they live, how they train.. I hope others also find this book as fascinating as I did in my younger years.

I came to know about Laika and Yuri Gagarin from this book.

Space port Baikonur!

Training hard!

And we have liftoff!

How does inside of the space station looks like?

Weightlessness you say??

Welcome back home!

The multi-page spanner internal structure of Soyuz is highly fascinating.

And also of various other vehicles

Also Sputniks which started it all!

The book also has some famous illustrations from the past, for example the Flammarion engraving.

The book took time to clean as you can see in the comparison images below. I had to  manually clean almost all the pages with images, but the result was worth it.

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