The book has been written by a group of scientists working

for many years at Higher Educational Institutions of the Soviet

Union and is a textbook for students both of Higher and of

Secondary Schools.The book contains the theoretical fundamentals of thermal

engineering (engineering thermodynamics and heat transfer), contains

characteristics of fuels, describes combustion processes, boiler

units and heat engines, such as steam engines, internal-combustion

engines, steam and gas turbines and steam power plants.

Besides being a textbook for students, the book will also be of

interest to specialists in the field of thermal engineering.

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

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Foreword 11

**Chapter I. Basic Concepts of Thermodynamics 13**

1. Parameters of Gases and Relation between Them. The Equation of State for Gases 13

2. Mixtures of Gases 19

3. Thermodynamic Processes bn hake wave 28

4. Work and Heat of a Process. Heat Capacity 204

5. Internal Energy of a Gas 33

6. Enthalpy of a Gas 35

7. Entropy 36

**Chapter II. The First Law of Thermodynamics and Investigation of Thermodynamic Processes 37**

1. The Law of Conservation and Conversion of Energy 37

2. The First Law of Thermodynamics 38

3. Investigation of the Basic Thermodynamic Processes for Ideal Gases

**Chapter III. Water Vapour and Steam 55**

1. Process of Vaporization in p-v and T-s Diagrams 55

2. Determining the Parameters of Steam of Different States 58

3. The i-s (Mollier) Diagram for Steam 61

4. Thermodynamic Processes of Water Vapour (Steam) 62

5. Humid Air Sad Bice 65

6. Flow and Throttling of Gases and Vapours. 68

**Chapter IV. The Second Law of Thermodynamics 76**

1. Cyclic Processes and the Carnot Cycle 76

2. The Second Law of Thermodynamics 82

3. Mathematical Expression of the Second Law of Thermodynamics and Change in the Entropy of an Isolated System 83

**Chapter V. Ideal Cycles of Heat Engines 89**

1. Cycles of Internal-combustion Engines View 89

2. Gas Turbine Cycles 93

3. Compressor Processes 95

4. Basic Cycle of a Steam Power Plant 98

5. Methods of Improving the Efficiency of the Basic Cycle 101

6. Heating and Power Systems 103

7. Regenerative Cycle 105

8. Steam-gas Cycle 107

9. Refrigeration Cycles 108

**Chapter I. Kinds of Heat Transfer 111**

1. Conduction 111

2. Convection 112

3. Heat Transfer upon Change in Aggregate State 126

4. Radiation 130

**Chapter II. Heat-exchange Equipment 140**

1. Combined Heat Transfer 140

2. Calculation of Heat-exchange Equipment 145

**Chapter I. Properties of Fuel 150**

1. General Information 150

2. Composition 151

3. Basic Specifications of Fuel 152

**Chapter II. Kinds of Fuels and Processing Thereof 159**

1. Fuels 159

2. Liquid Fuels 164

3. Gaseous Fuels 167

4. Processing of Solid Fuels 173

**Chapter I. General Information on Boiler Installations 179**

1. Classification of Boiler Installations 179

2. General Information on Boiler Units 180

**Chapter II. Combustion Processes 182.**

1. Combustion of Fuels and Ignition Temperature 182

2. Air Required for Combustion 184

3. Excess-air Coefficient 184

4. Volumes of Combustion Products Calculated from the Elementary Composition of Fuel 184

5. Volumes of Dry Combustion Products Determined by Flue

Gas Analysis 188

6. Volumes of Combustion Products oi siens Fuels 189

7. Enthalpy of Fuel Combustion Products 190

**Chapter III. Heat Balance of a Boiler Unit 192**

1. General Equation 192

2. Available Heat 192

3. Heat Utilized in Boiler Unit 192

4. Heat Losses with Flue Gases 193

5. Heat Losses due to Chemically Incomplete Combustion 194

6. Heat Losses due to Mechanically Incomplete Combustion 195

7. Heat Losses due to External Cooling of Boiler Unit 195

8. Heat Losses due to Physical Heat of Slags and for Cooling of Beams and Panels not Included into Boiler Circulation Systems 196

9. Boiler Unit Efficiency 196

10. Fuel Consumption 196

11. Evaporative Capacity of Fuel 197

**Chapter IV. Temperatures and Heat Transfer in Furnace 198**

1. Temperatures in Furnace 198

2. Heat Transfer in Furnace 199

**Chapter V. Furnaces 201**

1. Classification. of Furnaces 201

2. Thermal Characteristics of Furnaces 205

3. Waterwalls 206

4. Gas-fired Furnaces 206

5. Furnaces for Fuel Oil 211

6. Pulverized-coal Furnaces 216

7. Grate-fired Furnaces for Solid Fuel 225

**Chapter VI. Boiler Units 241**

1. General Information and Parameters 241

2. Development of Natural Circulation Boilers 252

3. Forced Circulation Boilers 260

**Chapter VII. Superheaters, Water Economizers, Air Heaters 263**

1. Superheaters 268

2. Water Economizers 266

3. Air Heaters 268

**Chapter VIII. Heat Transfer in Convective Passes of Boiler Units 270**

**Chapter IX. Auxiliary Equipment, Settings and Frame 272**

1. Draught Production Equipment 272

2. Equipment for Flue Gas Purification 275

3. Boiler Unit Settings and Frames 279

4. Feedwater Pumps and Piping 280

**Chapter X Conditions in Boiler Units and Feedwater Treatment 281**

1. Boiler Water Conditions 281

2. Characteristics of Initial Water 282

3. Methods of Feedwater Treatment апа Equipment Used 282

**Chapter I. Steam Engines 286**

1. Classification of Steam 288

2. Working Cycle 289

3. Determination of Engine Power 290

4. Steam Engine Efficiencies 294

5. Steam Consumption 297

6. Valve Gear 297

7. Methods of Governing Engine Power 303

8. Types of Engines 305

**Chapter II. Reciprocating Compressors 309**

**Chapter III. Internal-combustion Engines 314**

1. Operating Principles and Classification of Engines 314

2. Working Cycles of Engines 316

3. Utilization of Heat in Engines 322

4. Indicated Efficiency (were ag the cus 323

5. Mechanical Efficiency 329

6. Brake Thermal Efficiency and Parameters of Working Cycle 330

7. Comparison of Theoretical and Actual Indicator Diagrams 335

8. Determination of Principal Dimensions of Engines 336

9. Heat Balance of Engine 338

10. Mixture Formation 340

11. Fuel Supply Equipment 343

12. Flywheel and Unbalanced Forces of Inertia 349

13. Governing Systems and Governors 350

14. Timing 354

15. Fuel System 356

16. Lubrication 359

17. Cooling Systems 361

18. Electric Ignition 364

19. Starting Devices 367

20. Types of Engines 368

21. Further Development of Internal combustion Engines 374

**Chapter I. Steam Turbines 36**

1. General Information oe 976

2. Processes in Turbine Nozzles and Moving Blades 378

3. Heat Losses and Stage Efficiency 388

4. Turbine Heat Calculations 392

5. Turbine Governing 402

6. Steam Turbine Design 405

7. Condenser Units 413

**Chapter II. Gas Turbines 419**

1. Development of Gas Turbines A19

2. Fundamentals of Theory of Heat Processes. in 1 Gas Turbines 423

3. Heat. Cycles 428

4. Efficiencies of Gas Turbine Installations 435

5. Constant-volume Combustion Gas Turbines 436

6. Closed-cycle Gas Turbine Installations 438

7. Cycle Diagram Calculations 44l

8. Compressors 446

9. Gas Turbine Design 448

10. Materials for and Cooling of Turbine Blades and Disks 452

I]. Regenerators 454

12. Fuel and Combustion Chambers 455

**Chapter I. Heat Electric Generating Plants 458**

1. Types of Plants 458

2. Layout and Equipment of Steam Turbine Plants 463

3. Fuel Facilities 469

4. Power Plant Economy 476

**Chapter II. Automation in Power Plants 480**

]]>

This book is intended for pupils in the top classes in high schools and for students in mathematics departments of universities and teachers’ colleges. It may also be useful in the work of mathematical societies and may be of interest to teachers of mathematics in junior high and high schools.

The subject matter is concerned with both algebra and geometry. There are many useful connections between these two disciplines. Many applications of algebra to geometry and of geometry to algebra were known in antiquity; nearer to our time there appeared the important subject of analytical geometry, which led to algebraic geometry, a vast and rapidly developing science, concerned equally with algebra and geometry. Algebraic methods are now used in projective geometry, so that it is uncertain whether projective geometry should be called a branch of geometry or algebra. In the same way the study of complex numbers, which arises primarily within the bounds of algebra, proved to be very closely connected with geometry; this can be

seen if only from the fact that geometers, perhaps, made a greater contribution to the development of the theory than algebraists.The book is intended for quite a wide circle of readers. The early sections of each chapter may be used in mathematical classes in secondary schools, and the later sections are obviously intended for more advanced students (this has necessitated a rather complicated system of notation to distinguish the various parts of the book).

The book was translated from Russian by Eric Primrose and was published in 1968 from a 1963 Russian edition.

Credits to the original uploader.

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Translator’s Note v

Preface vii

**Chapter I: Three Types of Complex Numbers**

1. Ordinary Complex Numbers 1

2. Generalized Complex Numbers 7

3. The Most General Complex Numbers 10

5. Dual Numbers 14

5. **Double Numbers 18

6. **Hypercomplex Numbers 22

**Chapter II: Geometrical Interpretation of Complex Numbers**

7. Ordinary Complex Numbers as Points of a Plane 26

8. *Applications and Examples 34

9. Dual Numbers as Oriented Lines of a Plane 80

10. *Applications and Examples 95

11. **Interpretation of Ordinary Complex Numbers in the Lobachevskii Plane 108

12. **Double Numbers as Oriented Lines of the Lobachevskii Plane 118

**Chapter III: Circular Transformations and Circular Geometry**

13. Ordinary Circular Transformations (Mobius Transformations) 130

14. *Applications and Examples 145

15. Axial Circular Transformations (Laguerre Transformations) 157

16. *Applications and Examples 161

17. **Circular Transformations of the Lobachevskii Plane 179

18. **Axial Circular Transformations of the Lobachevskii Plane 188

Appendix: Non-Euclidean Geometries in the Plane and Complex Numbers

A1. Non-Euclidean Geometries in the Plane 195

A2. Complex Coordinates of Points and Lines of the plane Non-Euclidean Geometries 205

A3. Cycles and Circular Transformations 212

Addenda 220

Index 241

]]>Sorry for being erratic in the posts in the last few months.

Thank you for the fantastic response on the reprinting idea. We have 400+ people who have shown interest in buying the entire set. We will soon start the pre-order on books, hopefully start shipping sometime at end of October. Sorry for all the delays in the printing, most of it my procrastination and some of it the pandemic.

Keep the fingers crossed!!

Recently got a nice haul of books. Many thanks to @desperadomar and @imsmam for making this possible.

In coming couple of months we will see a lot of new titles being added. The first batch will be from this set:

There are few titles already on the Internet Archive which I not posted on the blog so far. You can check them at https://archive.org/details/@mirtitles

Plus we will see some children’s books in Indic languages as well: Marathi, Bengali and Hindi

]]>The theory of elasticity is concerned with the mechanics of deformable media which, after the removal of the forces producing deformation, completely recover their original shape and give up all the work expended in the deformation.

The first attempts to develop the theory of elasticity on the basis of the concept of a continuous medium, which enables one to ignore its molecular structure and describe macroscopic phenomena by the methods of mathematical analysis, date back to the first half of the eighteenth century.

The fundamental contribution to the classical theory was made by R. Hooke, C. L. M. H. Navier, A. L. Cauchy, G. Lame, G. Green, B. P. E. Clapeyron. In 1678 Hooke established a law linearly con- necting stresses and strains.

After Navier established the basic equations in 1821 and Cauchy developed the theory of stress and strain, of great importance in the development of elasticity theory were the investigations of B. de Saint Venant. In his classical work on the theory of torsion and bending Saint Venant gave the solution of the problems of torsion and bending of prismatic bars on the basis of the general equations of the theory of elasticity. In these investigations Saint Venant devised a semi-inverse method for the solution of elasticity problems, formulated the famous Saint Venant’s principle, which enables one to obtain the solution of elasticity problems. Since then much effort has been made to develop the theory of elasticity and its applications, a number of general theorems have been proved, the general methods for the integration of differential equations of equilibrium and motion have been proposed, many special problems of fundamental interest have been solved. The development of new fields of engineering demands deeper and more extensive studies of the theory of elasticity. High velocities call for the formulation and solution of complex vibrational problems. Lightweight metallic structures draw particular attention to the question of elastic stability. The concentration of stress entails dangerous consequences, which cannot safely be ignored.

The book was translated from Russian by M. Konyaeva and was published in 1979 by Mir.

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Notation 9

Introduction 13

Chapter I. ELEMENTS OF TENSOR CALCULUS 15

1. Scalars, Vectors, and Tensors 16

2. Addition, Multiplication, and Contraction of Tensors. The Quotient Law of Tensors 19

3. The Metric Tensor 22

4. Differentiation of Base Vectors. The Christoffel Symbols 28

5. A Parallel Field of Vectors 30

6. The Riemann-Christoffel Tensor. Derivative of a Veetor. The Gauss-Ostrogradsky Formula. The 𝜀-tensor 32

Chapter II. THEORY OF STRESS 39

7. Types of External Forces 39

8. The Method of Sections. The Stress Vector 41

9. The Stress Tensor 43

10. Equations of Motion and Equilibrium in Terms of the Components of the Stress Tensor 44

11. Surface Conditions 47

12. Equations of Motion and Equilibrium Referred to a Cartesian Co-ordinate System 48

13. Equations of Motion and Equilibrium Referred to Cylindrical and Spherical Co-ordinates 49

14. Determination of the Principal Normal Stresses 52

Chapter III. THEORY OF STRAIN 55

15. The Finite Strain Tensor 55

16. The Small Strain Tensor 59

17. Strai Compatibility Equations 60

18. The Strain Tensor Referred to a Cartesian Co-ordinate System 61

19. Components of the Small Strain and Rotation Tensors Referred to Cylindrical and Spherical Co-ordinates 62

20. Principal Extensions 64

21. Strain Compatibility Equations in Some Co-ordinate Systems (Saint Venant’s Conditions) 65

22. Determination of Displacements from the Components of the Small Strain Tensor 66

Chapter IV. STRESS-STRAIN RELATIONS 69

23. Generalized Hooke’s Law 69

24. Work Done by External Forces 70

25. Stress Tensor Potential 71

26. Potential in the Case of a Linearly Elastic Body 75

27. Various Cases of Elastic Symmetry of a Body 75

28. Thermal Stresses 80

29. A Energy Integral for the Equations of Motion of an Elastic Body 80

30. Betti’s Identity 82

31. Clapeyron’s Theorem 82

Chapter V. COMPLETE SYSTEM OF FUNDAMENTAL EQUATIONS IN THE THEORY OF ELASTICITY 84

32. Equations of Elastic Equilibrium and Motion in Terms of Displacements 84

33. Equations in Terms of Stress Components 90

34. Fundamental Boundary Value Problems in Elastostatics. Uniqueness of Solution 93

35. Fundamental Problems in Elastodynamics 95

36. Saint Venant’s Principle (Principle of Softening of Boundary Conditions) 96

37. Direct and Inverse Solutions of Elasticity Problems. Saint Venant’s Semi-inverse Method 98

38. Simple Problems of the Theory of Elasticity 99

Chapter VI. THE PLANE PROBLEM IN THE THEORY OF ELASTICITY 108

39. Plane Strain 108

40. Plane Stress 111

41. Generalized Plane Stress 113

42. Airy’s Stress Function 115

43. Airy’s Function in Polar Co-ordinates. Lamé’s Problem 120

44. Complex Representation of a Biharmonic Function, of the Components of the Displacement Vector and the Stress Tensor 127

45. Degree of Determinancy of the Introduced Functions and Restrictions Imposed on Them 132

46. Fundamental Boundary Value Problems and Their Reduction to Problems of Complex Function Theory 138

47. Maurice Lévy’s Theorem 141

48. Conformal Mapping Method 142

49. Cauchy-type Integral 145

50. Harnack’s Theorem 151

51. Riemann Boundary Value Problem 151

52. Reduction of the Fundamental Boundary Value Problems to Functional Equations 154

53. Equilibrium of a Hollow Circular Cylinder 155

54. Infinite Plate with an Elliptic Hole 159

55. Solution of Boundary Value Problems for a Half-plane 164

56. Some Information on Fourier Integral Transformation 170

57. Infinite Plane Deformed Under Body Forces 174

58. Solution of the Biharmonic Equation for a Weightless Half-plane 177

Chapter VII. TORSION AND BENDING OF PRISMATIC BODIES 182

59. Torsion of a Prismatic Body of Arbitrary Simply Connected Cross Section 182

60. Some Properties of Shearing Stresses 187

61. Torsion at Hollow Prismatic Bodies 188

62. Shear Circulation Theorem 190

63. Analogies in Torsion 191

64. Complex Torsion Function 196

65. Solution of Special Torsion Problems 198

66. Bending of a Prismatic Body Fixed at One End 206

67. The Centre of Flexure 211

68. Bending of a Prismatic Body of Elliptical Cross Section 216

Chapter VIII. GENERAL THEOREMS OF THE THEORY OF ELASTICITY. VARIATIONAL METHODS 219

69. Betti’s Reciprocal Theorem 219

70. Principle of Minimum Potential Energy 220

71. Principle of Minimum Complementary Work—Castigliano’s Principle 222

72. Rayleigh-Ritz Method 224

73. Reissner’s Variational Principle 228

74. Equilibrium Equations and Boundary Conditions for a Geometrically Non-linear Body 230

Chapter IX. THREE-DIMENSIONAL STATIC PROBLEMS 232

75. Kelvin’s and the Boussinesq-Papkovich Solutions 232

76. Doursinesa’s Elementary Solutions of the First and Second Kind 236

77. Pressure on the Surface of a Semi-infinite Body 238

78. Hertz’s Problem of the Pressure Between Two Bodies in Contact 240

79. Symmetrical Deformation of a Bedy of Revolution 246

80. Thermal Stresses 256

Chapter X. THEORY OF PROPAGATION OF ELASTIC WAVES 258

81. Two Types of Waves 258

82. Rayleigh Surface Waves 262

83. Love Waves 265

Chapter XI. THEORY OF THIN PLATES 268

84. Differential Equation for Bending of Thin Plates 268

85. Boundary Conditions 271

86. Bending Equation for a Plate Referred to Polar Co-ordinates 274

87. Symmetrical Bending of a Circular Plate 276

Literature 278

Subject Index 279

]]>

These days radio engineering has become a very important branch of science solving a large number of problems associated with economic, technological and cultural progress. Every year, it finds ever increasing application and the number of people using radio equipment constantly grows. Many of these people have only rudimentary or no knowledge of radio engineering, although modern radio equipment is often so complicated that its effective use is impossible without some training.

The wide sphere of radio application in different branches of science and technology, as well as its close connexion with art and sport has also created a great number of radio amateurs in all countries. Some build radio receivers, tape recorders and TV sets, others design radio controlled models, short and ultra shortwave transmitters or equipment for a fascinating game called “hunting for a fox”, etc.

All this increases the interest in radio engineering knowledge on the part of an ever growing number of people. The study of radio is, however, made more difficult for the majority of readers as it is usually explained with the use of higher mathematics. On the other hand, when higher mathematics is not used, many important problems are often oversimplified and treated without sufficient explanation and demonstration. Moreover radio engineering is a coherent science in which everything is interrelated and interdependent; therefore lack of understanding of fundamental phenomena and laws prevents the reader from fully understanding further problems. It is far from clear what one should understand under the name of “radio engineering,’ and its fundamentals since this branch has been extended, diversified and become interwoven with many other branches of science and technology. Under “radio engineering” proper one usually understands the use of electromagnetic radiation for the obtaining of information from a distant source. This is effected through the use of a transmitting (radiating) device and a receiving device provided conditions for propagation of radio waves are favourable. In accordance with this, the book describes the operating principles of radio transmitters, radio receivers and radiating devices, as well as radio wave propagation. It goes without saying that one book cannot exhaustively deal with all the varieties of existing radio circuits and devices; therefore, we concentrate our attention only on the most important and representative types.

The book was translated from Russian by A. Ulyanov and was published by Mir in 1976.

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Foreword 9

**Chapter I. PRINCIPLES OF RADIO COMMUNICATION 11**

1-1. Basic Properties of Electromagnetic Waves 11

1-2. General Principles of Radio Communication 15

1-3. Electromagnetic Waves Used in Radio Communication 28

1-4. A Brief History of Radio in the USSR 20

1-5. Components Used in Radio Engineering 23

**Chapter II. AC CIRCUITS 26**

2-1. Sinusoidal Quantities and Their Vector Representation 26

2-2. Basic Components of Radio Circuits and Alternating Currents in Them 28

2-3. AC Power 43

2-4. Steady-State and Transient Processes in Electric Circuits Including Capacitors or Coils 48

2-5. Nonsinusoidal Currents and Their Spectra 55

2-6. Free Oscillations in a Circuit 63

2-7. Forced Oscillations in a Series-Connected Circuit 66

2-8. Forced Oscillations in a Parallel-Connected Circuit 72

2-9. Transient Processes in Oscillatory Circuits 81

2-10. Parallel Circuits with Reactive Elements of Both Types in One of Their Branches 86

**Chapter III. COUPLED CIRCUITS 92**

3-1. Oscillations in Two Coupled Circuits 92

3-2. Tuning a System of Two Coupled Circuits 104

**Chapter IV. ELECTRICAL FILTERS 106**

4-1. Purpose of Filters 106

4-2. Filters for DC Supplies 106

4-3. Low-Pass Filters 109

4-4 High-Pass Filters 114

4-5. Bandpass and Band-Elimination Filters 117

**Chapter V. TRANSMISSION LINES 120**

5-1. Electrical Signals in Ideal Infinitely Long Lines 120

5-2. Signals in Finite Length Lines with Far End Open-Circuited 128

5-3. Signals in Finite Length Lines with Far End Short-Circuited 134

5-4. Signals in Lines with a Reactive Load 136

5-5. Signals in Lines with a Resistive Load 138

5-6. Signals in Lines with a Combined Load 143

5-7. Actual Lines with Losses 145

5-8. Transmission Lines as Reactive Elements and Impedance Transformers 161

**Chapter VI. ANTENNAS 169**

6-1. Radiating Systems 169

6-2. Double-Dipole Antennas 172

6-3. Effect of the Ground on Antenna Radiation. Asymmetrical Dipoles 187

6-4. Antenna Resonant Frequencies. Harmonic Antennas 192

6-5. Inphased and Antiphased Antennas. Reflectors and Directors 195

6-6. Ground Effect on Antenna Radiation Patterns 203

6-7. Complex Dipoles 209

6-8. Loop Antennas 212

6-9. Long- and Medium-Wave Antennas 214

6-10.Short-Wave Antennas 217

6-11.Ultrashort-Wave Antennas 225

**Chapter VII. RADIO WAVE PROPAGATION 233**

7-1. Properties of Atmosphere and Ground Affecting Radio Wave Propagation 233

7-2. Radio Waves Propagating in Atmosphere. General Regularities 245

7-3. Long-Wave Propagation 252

7-4. Medium-Wave Propagation 253

7-5. Short-Wave Propagation 256

7-6. Ultrashort-Wave Propagation 265

7-7. Electromagnetic Waves in Outer Space 272

**Chapter VIII. VACUUM AND SEMICONDUCTOR DEVICES 277**

8-1. Modern Electronics 277

8-2. Motion of Electrons in Vacuum. Cathodes of Electron Valves 279

8-3. Diodes 290

8-4. Triodes 302

8-5. Multigrid Electron Valves 324

8-6. Electric Conduction in Semiconductors 337

8-7. P-N Junction and Crystal Diodes 343

8-8. Transistors 351

8-9. Miniaturization of Electronic Devices 363

8-10.Cathode-Ray Tubes 365

**Chapter IX. PRIMARY-SIGNAL AMPLIFIERS 372**

9-1. Purpose and Classification 372

9-2. Audio-Frequency Amplifiers. General 378

9-3. Audio-Frequency Small-Signal Amplifiers 393

9-4. Audio-Frequency Output Amplifiers 403

9-5, Driver Stages. Feedback in Amplifiers 415

9-6. Video Amplifiers 421

**Chapter X. WAVE GENERATION 428**

10-1. Operating Principles of Valve Oscillators 428

10-2. Separately Excited Oscillators (Amplifiers) 435

10-3. Self-Excited Oscillators 449

10-4. Ultrahigh-Frequency Valve Oscillators 473

10-5. Klystron Amplifiers and Oscillators. 481

10-6. Travelling-Wave Oscillators 488

10-7. Backward-Wave Oscillators 492

10-8. M-type Travelling-Wave Oscillators 496

10-9. Transistor Oscillators and Amplifiers 507

10-10. Negative-Resistance Oscillators 510

10-11. Sinewave RC Oscillators 511

10-12. Frequency Pulling in Self-Oscillators 513

10-13. Self-Oscillator Lock-in 517

10-14. Nonsinewave Oscillators 528

**Chapter XI. CONVERSION OF ELECTRIC SIGNALS 532**

11-1. Concept of Signal Conversion 532

11-2. Amplitude Modulation 533

11-3. Frequency and Phase Modulation 545

11-4. Pulse Modulation 555

11-5. Detection of Radio Signals 561

11-6. Frequency Converters 565

11-7. Conversion of Electric Pulses 571

**Chapter XII. RECEIVERS 579**

12-1. Purpose and Basic Characteristics 579

12-2. Receiver Input Circuits 587

12-3. High-Frequency Amplifiers 595

12-4. Intermediate-Frequency Amplifiers 607

12-5. Radio Interference 611

12-6. Frequency Converters 618

12-7. Receiver Detector Stages 633

12-8. Controls and Adjustments in Receivers 647

12-9. Examples of Receiver Circuitry 653

Reference Data 659

Index 664

In the present book, we shall show how it is possible, by beginning with other curves (such as the equilateral hyperbola or Bernoulli’s lemniscate (a curve having the form of a figure- eight), to define interesting and important functions analogous to the trigonometric functions, similar to them in some respects but possessing certain new characteristics. These functions are called respectively hyperbolic and lemniscate functions. In analogy with them, we shall refer to the trigonometric functions as circular functions.

The reader is assumed to have a familiarity with the ele ments of analytic geometry and differential and integral calculus. The necessary material on integration in the complex plane will be given in the present book though proofs will be omitted.

The ultimate purpose of the book is to acquaint the reader not possessing an extensive knowledge of the theory of functions of a complex variable with the simplest representatives of the class of elliptic functions, namely, lemniscate functions and the somewhat more general Jacobian elliptic functions.

In conclusion, we warn the reader that this book is not in tended for light reading. He must read it with his pencil in his hand.

The book was translated from Russian by Scripta Technica and was published in 1966.

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Preface

1. Geometric Definition of Circular, Hyperbolic And Lemniscate Functions 1

2. Generalized Sines 13

3. Integration in the Complex Plane 25

4. Euler’s Method of Deriving the Addition Theorems 41

5. Further Study of Complex Values of the Argument 49

6. Zeros and Poles. Simple and Double Periodicity. The Concept of an Elliptic Function 73

Index 99

]]>

In elementary mathematics there are many difficult and interesting problems not connected with the name of an individual, but rather possessing the character of a kind of “mathematical folklore”. Such problems are scattered throughout the wide literature of popular (or, simply, entertaining!) mathematics, and often it is very dif ficult to establish the source of a particular problem.

These problems often circulate in several versions. Sometimes several such problems combine into a single, more complex, one, sometimes the opposite happens and one problem splits up into several simple ones: thus it is often difficult to distinguish between the end of one

problem and the beginning of another. We should consider that in each of these problems we are dealing with little mathematical theories, each with its own history, its own complex of problems and its own characteristic methods, all, however, closely connected with the history and methods of “great mathematics”.The theory of Fibonacci numbers is just such a theory. Derived from the famous “rabbit problem”, going back nearly 750 years, Fibonacci numbers, even now, provide one of the most fascinating chapters of elementary mathe matics. Problems connected with Fibonacci numbers occur in many popular books on mathematics, are discussed at meetings of school mathematical societies, and feature in mathematical competitions.

The present booklet contains a set of problems which were the themes of several meetings of the school children’s mathematical club of Leningrad State University in the academic year 1949-50. In accordance with the wishes of those taking part, the questions discussed at these meetings were mostly number-theoretical, a theme which is developed in greater detail here.

This book is designed to appeal basically to pupils of 16 or 17 years of age in a high school. The concept o a limit is met with only in examples 7 and 8 in chapter III. The reader who is not acquainted with this concept can omit these without prejudice to his understanding of what follows. That applies also to binomial coefficients (I, example 8) and to trigonometry (IV, examples 2 & 3). The elements which are presented of the theory of divisibility and of the theory of continued fractions do not presuppose any knowledge beyond the limits of a school course.Those readers who develop an interest in the principle of constructing recurrent series are recommended to read the small but full booklet of A.I. Markushevich, “Recurrent Sequences” (Vozvratnyye posledovatel’ nosti) (Gostekhizdat, 1950). Those who become interested in facts relating to the theory of numbers are referred to textbooks in this subject*.

The book was translated from Russian by Halina Moss (edited by Ian Sneddon) and was published in 1961.

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Foreword vii

Introduction 1

І. The simplest properties оf Fibonacci numbers 6

II. Number-theoretic properties of Fibonacci numbers 25

III. Fibonacci numbers and continued fractions 36

IV. Fibonacci numbers and geometry 55

V. Conclusion 65

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The theory of convex figures and polyhedra provides an excellent example of a body of mathematical knowledge that offers theorems with elementary formulations and vivid geometric meaning. Despite this simplicity of formulation, the proofs are often not elementary. Thus the area presents a particular challenge to mathematicians, who have investigated convex figures and polyhedra for millenia, and yet have by far not exhausted the subject. Many of the theorems in this volume were in fact proved only a few years ago.

The material in this book will be suitable for study in mathematics clubs or by readers with a background of secondary school mathematics only. The topics considered are stimulating and chal lenging, and moreover, convexity ideas are valuable in the study of modem higher mathematics. Mathematical analysis, higher geome try, and topology each use convexity notions in an essential way.

The book was translated from Russian by Donald L. Barnett and was published in 1966.

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**CHAPTER l. Convex Figures 1**

1. Plane convex figures 1

2. Intersections and partitions of plane convex figures 5

3. Supporting lines for two-dimensional convex figures 8

4. Directed convex curves and directed supporting lines 10

5. Vectors; external normals to plane convex figures 13

6. Circuit of a polygon; length of a convex curve 14

7. Convex solids 17

8. Supporting planes and external normals for convex solids 20

9. Central projection; cones 23

10. Convex spherical figures 26

11. Greatest and least widths of convex figures 28

12. Ovals of constant width; Barbier’s Theorem 34

CHAPTER 2. Central-symmetric Convex Figures 39

13. Central symmetry and (parallel) translation 39

14. Partitioning central-symmetric polyhedra 42

15. The greatest central-symmetric convex figure in a lattice of integers; Minkowski’s Theorem 44

16. Filling the plane and space with convex figures 51

CHAPTER 3. Networks and Convex Polyhedra 58

17. Vertices (nodes), faces (regions), and edges (lines); Euler’s Theorem 58

18. Proof of the theorem for connected networks 61

19. Disconnected networks; inequalities 64

20. Congruent and symmetric polyhedra; Cauchy’s Theorem 66

21. Proof of Cauchy’s Theorem 71

22. Steinitz’ correction of Cauchy’s proof 73

23. Abstract and convex polyhedra; Steinitz Theorem 81

24. Development of a convex polyhedron; Aleksandrov’s Theorem 95

CHAPTER 4. Linear Systems of Convex Figures 97

25. Linear operations on points

26. Linear operations on figures; “mixing” figures

27. Linear systems of convex polygons; areas and “mixed areas”

28. Applications

29. Schwarz inequality; other inequalities

30. Relation between areas of Q, Q_{1}, and Q_{S},; the Brunn-Minkowski inequality

31. Relation between areas of plane sections of convex solids

32. Greatest area theorems

CHAPTER 5. Theorems of Minkowski and Aleksandrov for Congruent Convex Polyhedra 132

33. Formulation of the theorems 132

34. A theorem about convex polygons 134

35. Mean polygons and polyhedra 141

36. Proof of Aleksandrov’s Theorem 146

CHAPTER 6. Supplement 150

37. Precise definition of a convex figure 150

38. Continuous mapping and functions 152

39. Regular networks; regular and semiregular polyhedra 153

40. The isoperimetric problem 164

41. Chords of arbitrary continua; Levi’s Theorem 166

42. Figures in a lattice of integers; Blichfeldt’s Theorem 172

43. Topological theorems of Lebesgue and Bol’-Brouwer 175

44. Generalization to n dimensions 182

45. Convex figures in normed spaces 185

Bibliography 191

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Volume 1 Convective Heat Exchange in a Homogeneous Medium edited by: B.S. Petukhov, I.P. Ginzburg, and А. S. Kasperovich

About the book

This book comprises reports and communications dealing with the problems of convective heat transfer in a homogeneous medium, and with the heat and mass transfer during the interaction of bodies with liquid and gas streams. Most papers deal with studies based on the boundary layer theory. The papers include theoretical and experimental works on unsteady-state heat transfer, on heat transfer at variable physical properties of liquids and gases, and heat transfer during supersonic flows in dense and rarefied media.

The book was translated from Russian by A. Aladjem and was edited by R. Kondor published in 1967 by Israel Program for Scientific Translations.

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Contents

B.S. PETUKHOV. Convective Heat Transfer in a Homogeneous Medium 1

I. HEAT EXCHANGE AND FRICTION RESISTANCE DURING SUBSTANTIAL CHANGES IN THE PHYSICAL PROPERTIES OF LIQUIDS AND GASES AS A FUNCTION OF TEMPERATURE AND PRESSURE 7

A. A. Gukhman, A. F. Gandel’Sman, Y. Y. Usanov, AND G. N. Shorin.

New Data on the Properties of Trans-sonic Flows 7

V. L. Lel’chuk, G. I. Elfimov, AND Yu. P. Fedotov.

Experimental Study of the Heat Transfer from Pipe Walls to Mono-, Di-, and Triatomic Gases at Large Temperature Gradients 11

L. S. Sterman AND V. V. Petukhov.

Investigation of the Heat Transfer to Organic Liquids 19

E. A. Krasnoshchekov, V. S. Protopopov, Wang Feng, AND I. V. Kuraeva.

Experimental Study of the Heat Transfer in The Supercritical Region of Carbon Dioxide 25

E. N. Dubrovina AND V. P. Skripov.

Convective Heat Transfer In The Supercritical Region of Carbon Dioxide 32

V. N. Popov.

Theoretical Calculation of the Heat Transfer And Friction Resistance for Carbon Dioxide in The Supercritical Region 41

I. Т. Alad’ev, P. I. Povarnin, L. I. Malkina, AND E. Yu. Merkel’.

Investigation of the Cooling Properties of Ethanol At Pressures up to 800.9.8-10^4 N/m^2 49

D. M. Kalachev, I. S. Kudryavtsev, B. L. Paskar’, AND I. I. Yakubovich.

Application of the Method of High-frequency Heating To Liquid- Metal Heat Transfer Media 53

II. HEAT EXCHANGE AND FRICTION RESISTANCE IN PIPES AND CHANNELS OF VARIOUS GEOMETRICAL SHAPES 56

B. S. PETUKHOV AND L.I. Roizen.

Heat Exchange During gas Flow In Pipes With an Annular Cross Section 56

P. I. Puchkov AND O. S. Vinogradov.

Heat Transfer and Hydraulic Resistance in Annular Channels With Smooth and Rough Heat Transfer Surfaces 65

L. M. Zysina-Molozhen AND I. B. Uskov.

Experimental Investigation Of the Heat Transfer on the end Wall of a Blade Channel [In Turbines] 79

Yu. P. Finat’ev.

Calculation of the Hydraulic Resistance Of Annular Channels 88

B. P. Ustimenko, K. A. Zhurgembaev, AND D. A. Nusupbekova.

Calculation of the Convective Heat Transfer for An Incompressible Liquid in Channels With Complicated Shapes 99

I.S. Kochenov, L.I. Baranova, AND V. V. Vasil’ev.

Flow in Channels With Permeable Walls 113

М. E. Podol’Skii.

Attractive Action of a Non-isothermal Lubricating Layer 117

V. N. Zmeikov АND B. P. Ustimenko.

Hydrodynamics and Heat Transfer in a Convoluted Stream Between two Coaxial Cylinders 127

P.N. Romanenko AND A. N. Oblivin.

Experimental Study of The Friction and Heat Transfer During gas Flow in a Diffuser

Channel With Cooled Walls, During Combustion 140

III. INVESTIGATION OF THE HEAT TRANSFER AND [HYDRODYNAMIC] RESISTANCE IN THE ENTRY SECTIONS OF TUBES AND CHANNELS 148

B.S. Petukhov AND Chang-Chéng Yung.

Heat Transfer in The Hydrodynamic Entry Section of a Round Tube During Laminar Liquid Flow 148

A.A. Zhukauskas AND I.I. Zhyugzhda.

Experimental Study of The Heat Transfer and Hydraulic Resistance in the Entry Section of a Flat Channel During Laminar Flow of A

Viscous Liquid 158

E. E. Solodkin AND A. S. Ginevskii.

Turbulent Non-isothermal Flow Of a Viscous Compressible gas in the Entry Sections Of Axisymmetrical and Flat Widening Channels With Zero Pressure Gradient 163

P. N. Romanenko AND N. V. Krylova.

Effect of the Entry Conditions on the Heat Transfer in the Entry Section Of a Tube With Turbulent air Flow 175

IV. STUDIES OF THE INTENSIFICATION OF CONVECTIVE HEAT TRANSFER PROCESSES 184

A. V. Ivanova.

Intensification of the Heat Transfer in An Air-cooled Round Tube 184

E. K. Karasev.

Investigation of the Hydrodynamics And Heat Transfer in a Channel With Turbulizers on The

Heat Transfer Surface 190

A. S. Nevskii, A. V. Arseev, L. A. Chukanova, A. I. Malysehva, AND

T. V. Sharova.

Convective Heat Transfer in Cylindrical Chambers With Recirculation 198

К. Rybáček.

Certain Characteristics of Heat Transfer And Friction in the Case of Longitudinal Flow Around [Fuel] Element 206

V. F. Yudin AND L. S. Tokhtarova,

Investigation of the Heat Transfer and Resistance of Finned, Staggered Banks With Fins of Different Shapes: 215

I. Vampola.

Generalization of the Laws Governing Heat Transfer and Pressure Drop During Transverse Flow Of Gases in Finned Tube Banks 224

A. I. Mitskevich

Efficiency of Heat Transfer Surfaces 232

V. CONVECTIVE HEAT TRANSFER UNDER UNSTEADY-STATE CONDITIONS 239

Yu. L. Rozenshtok.

The Unsteady Laminar Thermal Boundary Layer on a Semi-infinite Plate in a Viscous Liquid Flow 239

Е.K. Kalinin.

Determination of the Stream Temperature and Friction Coefficient in Channels During Unsteady Nonisothermal Flow of a Heat-transfer Medium 249

L. I. Kudryashev and A. A. Smirnov

Accounting for the Effect of Thermal Unsteady State on the Coefficient of Convective Heat Transfer During Flow Round Spherical Bodies at Small Reynolds Numbers 258

I. S. Kochenov AND Yu. N. Kuznetsov.

Unsteady Flow in Tubes 266

Explanatory List of Abbreviations 274

]]>**About the book**

The purpose of this book is to systematize the available material and to delineate the principal trends in the investigation of those closely interrelated properties—thermal (heat capacity, thermal conductivity, thermal expansion), acoustical, dielectric, visco- elastic, etc.—which govern the entire set of important physical properties of polymers at low temperatures. An attempt is made to show how the chemical constitution and the supermolecular

structure influence the physical properties of polymers in the low- temperature region·All the chapters of this book, except the last two, are organized according to the same plan. Each chapter, devoted to a single physical property, consists'”of three sections. First, the theory of the property under discussion and the related physical phenomena are briefly considered. This is followed by a description of methods of investigation of these properties and phenomena at low temperatures. Finally, systematized data from experimental investigations are presented.

The last three chapters are concerned mainly with studies of the properties of polymers at low temperatures which have been carried out in recent years by the author and his collaborators P .D. Golub and V .E. Sorokin.

The book was translated from Russian by A. Beknazarov and was published in .

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**Contents**

Chapter 1 Heat Capacity of Polymers at low Temperatures 1

1.1. Basic Concepts of the Heat Capacity of Solids 1

1.2. Basic Concepts of the Heat Capacity of Polymers 8

1.3. Methods of Measuring Heat Capacity 17

1.4. The Heat Capacity of Polymers 20

Chapter 2 Thermal Conductivity of Polymers at low Temperatures 45

2.1. Basic Concepts of the Thermal Conductivity of Dielectrics 45

2.2. Methods of Measuring the Thermal Conductivity of Polymers 56

2.3. Thermal Conductivity of Polymers at low Temperatures 57

Chapter 3 Thermal Expansion of Polymers at low Temperatures 82

3.1. Thermal Expansion of Solids 82

3.2. Equations of State for Solids 83

3.3. Basic Concepts of the Thermal Expansion of Polymers 86

3.4. Methods of Measuring Thermal Expansion Coefficients of Polymers 95

3.5. Thermal Expansion Coefficients of Polymers 97

Chapter 4 Electrical Properties of Polymers at low Temperatures 110

4.1. Basic Concepts of the Electrical Properties of Polymers 110

4.2 Page 112 is Missing

4.3. Phenomenological Relaxation Theory of the Dielectric Properties of Polymers 113

4.4. The Mechanism of Dielectric Relaxation 120

4.5. Methods of Studying the Dielectric Properties of Polymers 124

4.6. Dielectric Properties of Polymers 126

Chapter 5 Nuclear Magnetic Resonance in Polymers at low Temperatures 147

5.1. Basic Concepts of Nuclear Magnetic Resonance in Polymers 147

5.2. Effect of the Structure and Composition of Polymers on Nuclear Magnetic Resonance 154

Page 155 Missing

5.3. Investigation of the Molecular Motion in Polymers at low Temperatures by the nmr Method 164

Chapter 6 Dynamic Mechanical Properties of Polymers at low Temperatures 178

6.1. Effect of Chemical Constitution, Structure and Composition on the Dynamic Mechanical Properties of Polymers. Basic Concepts of Acoustic Spectroscopy of Polymers 178

6.2. Methods of Investigating the Dynamic Mechanical Properties of Polymers 183

6.3. Relaxation Processes in Polymers at low Temperatures 184

Chapter 7 the Acoustical Properties of Polymers at low Temperatures 206

7.1. Propagation of Ultrasonic Waves in Dielectrics 206

7.2. The Phenomenological Theory of Sound Propagation in Polymers 208

7.3. Experimental Methods of Acoustical Measurements in Polymers 211

7.4. Ultrasonic Velocity and Relaxation Processes in Polymers 216

7.5. Ultrasonic Velocity and Relaxation Processes in Linear Crystalline Polymers at Helium Temperatures. the Low-temperature Plateau 218

7.6. Ultrasonic Velocity and Relaxation Processes in Polymers Containing Methyl Groups 228

7.7. Ultrasonic Velocity and Relaxation Processes in Polymers With an Asymmetric Potential Barrier 232

7.8. Effect of Structure on Acoustical Properties 236

Chapter 8 Viscoelastic Parameters of Polymers at low Temperatures 241

8.1. Determination of the Main Viscoelastic Parameters of Polymers from Acoustic Measurements 241

8.2. Dynamic Elastic Moduli of Polymers 243

8.3. Poisson’s Ratio for Polymers Near the Liquid-helium Temperature 249

Chapter 9 Determining the Thermophysical Characteristics of Polymers by Acoustic Measurements at Helium Temperatures

9.1. Debye Temperatures and Heat Capacities Determined by Acoustic Measurements 255

9.2. Determining the Thermal Expansion Coefficients and Grüneisen Constants from Acoustic Measurements 261

Appendix 268

References 275

Name Index 285

Subject Index 297

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