Multicolor Problems ( Mathematical Conversations Part 1) – Dynkin, Uspenskii

Posted on February 25, 2022 by The Mitr

In this post, we will see the book Multicolor Problems ( Mathematical Conversations Part 1) by E. B. Dynkin and V. A. Uspenskii.

About the book

Multicolor Problems is a translation of Part One of Mathe­matical Conversations by E. B. Dynkin and V. A. Uspenskii, which was published in the Russian series. Library of the Mathematics Circle. The originality of the exposition and the variety of the problems presented here make this booklet especially useful in stimulating an inventive approach to mathematics.
This booklet deals with several of the classical map-coloring problems. The technique is one of developing an ordered presentation of problems and extensive solutions to them. A discussion of the famous four-color problem, which has puz­zled mathematicians for nearly a century, is included.
The booklet is designed for the reader’s active participation, as the problems are carefully integrated with the text and should be solved in sequence. The reader should have a back­ ground of high school algebra and should also be acquainted with the method of mathematical induction.
E. B. DYNKIN, a Professor at Moscow State University, is an eminent mathematician and author, whose specialties are higher algebra, topology, and probability theory. V. A. USPENSKII, a Lecturer at Moscow State University, spe­ cializes in mathematical logic.

The book was translated from Russian by was published in 1962. There is a recent volume by Dover which has all three parts in one book.

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Contents

Introduction 1

CHAPTER 1. Coloring with Two Colors 3

1. Simple two-color problems 3
2. Problems on square boards 5
3. Problems involving even and odd numbers 6
4. Networks and maps 7
5. General two-color problems 9

CHAPTER 2. Coloring with Three Colors 12

6. A simple three-color problem 12
7. Problems on hexagonal boards 12
8. Dual diagrams 14
9. Triangulation 16
10. Dual maps 19
11. Normal maps in three colors 23

CHAPTER 3. The Four-Color Problem 24

12. Normal maps in four colors 24
13. Volynskii’s theorem 25

CHAPTER 4. The Five-Color Theorem 27

14. Euler’s theorem 27
15. The five-color theorem 32

Concluding Remarks 33

Appendix 34
Coloring a sphere with three colors 34

Solutions to Problems 40

Bibliography 66

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Random Walks ( Mathematical Conversations Part 3) – Dynkin, Uspenskii

Posted on February 24, 2022 by The Mitr

In this post, we will see the book Random Walks ( Mathematical Conversations Part 3) by E. B. Dynkin and V. A. Uspenskii.

About the book

Random Walks is a translation of Part Three of Mathematical Conversations by E. B. Dynkin and V. A. Uspenskii, which was published in the Russian series, Library of the Mathematics Circle. The originality of the exposition and the variety of the problems presented here make this booklet especially useful in stimulating an inventive approach to mathematics. Extensive solutions to all problems are provided.

This booklet deals with some of the more elementary problems in probability theory. The exposition ranges from the simplest examples of a random walk on a line to such more complex examples as random walks through a city, and Markov chains.

The booklet is designed for the reader’s active participation, as the problems are carefully integrated with the text and should be solved in sequence. The reader should have a background of high school algebra.

E. B. DYNKIN, a Professor at Moscow State University, is an eminent mathematician and author, whose specialties are higher algebra, topology, and probability theory. V. A. USPENSKII, a Lecturer at Moscow State University, specializes in mathematical logic.

The book was translated from Russian by Norman D. Whaland, Jr., Olga A. Titelbaum was published in 1963. There is a recent reprint of all three parts combined into one by Dover.

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Contents

Introduction 1

CHAPTER 1. Probability 5

1. Fundamental properties of probability 5
2. Conditional probability 8
3. The formula for complete probability 13

CHAPTER 2. Problems Concerning a Random Walk on an Infinite Line 17

4. Graph of coin tosses 17
5. The triangle of probabilities 18
6. Central elements of the triangle of probabilities 21
7. Estimation of arbitrary elements of the triangle 25
8. The law of the square root of n 26
9. The law of large numbers 34

CHAPTER 3. Random Walks with Finitely Many States 37

10. Random walks on a finite line 37
11. Random walks through a city 39
12. Markov chains 46
13. The meeting problem 47

CHAPTER 4. Random Walks with Infinitely Many States 50

14. Random walks on an infinite path 51
15. The meeting problem 53
16. The infinitely large city with a checkerboard pattern 57

Concluding Remarks 62

Solutions to Problems 64

 

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Quantum Electrodynamics – Akhiezer, Berestetskii

Posted on February 23, 2022 by The Mitr

In this post, we will see the book Quantum Electrodynamics by A. I. Akhiezer; V. B. Berestetskii.

About the book

At the present time a number of particles is known which correspond to various quantum fields interacting with each other. However, of the many types of physical interactions existing in nature the only one, apart from gravitation, that has been studied in sufficient detail is the electromagnetic interaction. The theory of the latter interaction is the subject of quantum electrodynamics, to which the systematic exposition of this book is devoted.

Since the electromagnetic interaction is the fundamental one in the case of electrons and photons, quantum electrodynamics enables us to explain and to predict a wide range of phenomena related to the behavior of these particles.

In preparing the second edition we have subjected the book to extensive revision. The principal aim and the contents of the book have not been altered; the book is devoted to the systematic presentation of electromagnetic processes only. Only some general theorems and methods go beyond the framework of electrodynamics proper. In the second edition the space devoted to these has been increased

(reflection properties, Green’s functions, functional methods, etc.).
In the presentation of the principles of quantum electrodynamics the theory of renormalizations has been subjected to the most extensive revision. Without claiming complete mathematical rigor we have attempted to present the concept of renormalization from a single simple physical point of view, avoiding purely prescription-like methods of eliminating divergences, and making maximum use of the general properties of quantum mechanical systems.

The book was translated from Russian by G. M. Volkoff  was published in 1965.

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Contents

CHAPTER I
QUANTUM MECHANICS OF THE PHOTON

§ 1. The Photon Wave Function 1

1. Introduction.
2. The Photon Wave Function in k-Space.
3. Energy.
4. Normalization of the Photon Wave Function.

§ 2. Photon States of Definite Momentum 9

1. Photon Momentum Operator.
2. Impossibility of Introducing a Photon Wave Function in the Coordinate
Representation.
3. Plane Waves.
4. Polarization Density Matrix for the Photon.

§ 3. Angular Momentum. Photon Spin 17

1. Angular Momentum Operator.
2. Photon Spin Operator.
3. Photon Spin Wave Functions.

§ 4. Photon States of Definite Angular Momentum and Parity

1. Eigenfunctions of the Photon Angular Momentum Operator.
2. Longitudinal and Transverse Vector Spherical Harmonics.
3. Parity of Photon States.
4. Expansion in Spherical Waves.
5. Expressions for the Electric and Magnetic Fields.

§ 5. Scattering of Photons by a System of Charges 36

1. Incoming and Outgoing Waves.
2. Effective Scattering Cross Section.
3. The Optical Theorem. 4. Dispersion Relations.

§ 6. The Photon Field Potentials 46

1. Transverse, Longitudinal, and Scalar Potentials.
2. Longitudinally Polarized “Photon.”
3. Potentials for Plane and Spherical Waves.

§ 7. System of Photons 52

1. Wave Function for a System of Two Photons.
2. Even and Odd States of Two Photons.
3. Classification of the States of Two Photons of Definite Angular Momentum.
4. Wave Function for a System of an Arbitrary Number of Photons.

§ 8. L-Vectors and Spherical Harmonics 62

1. Irreducible Tensors.
2. The Algebra of L-Vectors.
3. Spherical Harmonics.

CHAPTER II
RELATIVISTIC QUANTUM MECHANICS OF THE ELECTRON

§ 9. The Dirac Equation 73

1. Spinors. Pauli Matrices.
2. Dirac Equations. Dirac Matrices.
3. Unitary Transformations of Bispinors.
4. The Necessity for Four-Component Electron Wave Functions.
5. Symmetric Form of the Dirac Equation. Equation of Continuity.
6. Invariance of the Dirac Equation.
7. Bilinear Combinations of the Components of the Wave Function.

§ 10. Electron and Positron States. States of Definite
Momentum and Polarization 86

1. Solutions with Positive and Negative Frequencies.
2. The Charge Conjugation Transformation.
3. The Positron Wave Function.
4. Plane Waves.
5. Polarization of a Plane Wave.
6. Polarization Density Matrix for the Electron.
7. Averaging over Polarization States.

§ 11. Electron States of Definite Angular Momentum and Parity 105

1. Orbital and Spin Functions. Spherical Spinors.
2. Wave Function of a State of Definite Angular Momentum.
3. Parity of a State.
4. Expansion in Spherical Waves.

§ 12. Electron in an External Field 122

1. The Dirac Equation with an External Field.
2. Separation of Variables in a Central Field.
3. Asymptotic Behavior of the Radial Functions.
4. Behavior of Energy Levels as Functions of the Potential Well Depth.
5. Electron in a Constant Homogeneous Magnetic Field.

§ 13. Motion of an Electron in the Field of a Nucleus 131

1. Solution of the Radial Equations for the Coulomb Field.
2. Wave Functions for the Continuous Spectrum.
3. Isotopic Level Shift.
4. General Investigation of the Effect of Finite Nuclear Size.

 

§ 14. Electron Scattering 144

1. Spinor Scattering Amplitude.
2. Expression for the Cross Section in Terms of Phases.
3. Polarization and Azimuthal Asymmetry.
4. Scattering by a Coulomb Field.
5. Small Angle Scattering.

§ 15. Nonrelativistie Approximation 153

1. Transition to the Pauli Equation.
2. Second Approximation.
3. Application of the Dirac Equation to Nucleons.

CHAPTER III
QUANTIZED ELECTROMAGNETIC AND ELECTRON-POSITRON FIELDS

§ 16. Quantization of the Electromagnetic Field 153

1. Four-Dimensional Form of the Field Equations.
2. Variational Principle. Energy-Momentum Tensor of the Electromagnetic Field.
3. Expansion of the Potentials into Plane Waves.
4. Quantization of the Electromagnetic Field.
5. Use of the Indefinite Metric.

§ 17. Commutators of the Electromagnetic Field 174

1. Commutation Relations for the Potentials and the Field Components.
2. Chronological and Normal Products of Components of the Potential. 3. Singular Functions Associated with the Operators ▢ and (▢^2 — m^2).

§ 18. Quantization of the Electron-Positron Field 195

1. Variational Principle for the Dirac Equation. Energy-Momentum Tensor of the Electron-Positron Field.
2. Quantization Rules for the Electron-Positron Field.

 

§ 19. Anticommutators of the Electron-Positron Field. Chronological and Normal Products of Field Components. Current Density 205

1. Commutation Relations for Field Components.
2. Chronological and Normal Products of Operators of
the Electron-Positron Field. 3. Electric Current Density.

§ 20. General Properties of Wave Fields 214

1. Wave Functions of a Field and the Lorentz Group.
2. Irreducible Finite-Dimensional Representations of the Lorentz Group.
3. Energy-Momentum Tensor and Angular Momentum Tensor.
4. Current Density Vector.
5. Relativistically Invariant Field Equations.
6. Wave Equations for Particles of Spin Zero and Unity.

 

§ 21. Quantization of Fields. Connection between Spin and
Statistics 237

1. Nondefiniteness of the Charge in the Case of Integral Spin and of the Energy in the Case of Half-Integral Spin.
2. Quantization of Fields for Integral and Half-Integral Spin. Pauli’s Theorem.
3. Inversion of Coordinates and Time Reversal.

CHAPTER IV
FUNDAMENTAL EQUATIONS OF QUANTUM ELECTRODYNAMICS

§ 22. Interacting Electromagnetic and Electron-Positron
Fields 253

1. System of Equations for Interacting ) Fields.
2. Lagrangian. Energy-Momentum Tensor.
3. Field Equations in Poisson Bracket Form.
4. Invariance Properties of the Equations of Quantum Electrodynamics.

 

§ 23. Equations of Quantum Electrodynamics in the Interaction Picture. Invariant Perturbation Theory 268

1. Heisenberg and Schrödinger Pictures. Interaction Picture.
2. Transition to the Interaction Picture in Quantum Electrodynamics. 3. Charge Conjugation Operator.
4. Perturbation Theory.

§ 24. The Scattering Matrix 290

1. The Scattering Problem and the Definition of the Scattering Matrix.
2. Matrix Elements of Field Operators.
3. Representation of the Scattering Matrix as a Sum of Normal Products.
4. General Relation between T- and N-Orderings.
5. Symmetry of the Scattering Matrix under Time Reversal.

§ 25. Graphical Representation of the Elements of the Scattering Matrix. The Scattering Matrix in Momentum Space 307

1. Graphical Representation of Normal Products.
2. Various Interaction Processes between Fields.
3. Transition to Momentum Space.
4. Closed Electron Loops with an Odd Number of Vertices.
5. Rules for Writing Down Matrix Elements.

§ 26. Probabilities of Various Processes 327

1. General Formula for the Probability.
2. Effective Cross Section.
3. Summation and Averaging over Polarization States of Electrons and Photons.
4. Probabilities of Processes Involving Polarized Particles.
5. Probabilities of Processes in the Presence of an External Field.
6. Feynman’s Notation.

CHAPTER V
INTERACTION OF ELECTRONS WITH PHOTONS

§ 27. Emission and Absorption of a Photon 345

1. General Expression for the Matrix Element.
2. Electric Multipole Radiation.
3. Magnetic Multipole Radiation.
4. Selection Rules.
5. Angular Distribution and Polarization of the Radiation.

§ 28. Scattering of a Photon by a Free Electron 363

1. Scattering Matrix Element.
2. Application of Conservation Laws.
3. Differential Cross Section for Unpolarized Particles.
4. Angular Distribution and Total Cross Section.
5. Distribution of Recoil Electrons.
6. Scattering of Polarized Photons.
7. Scattering of Photons by Polarized Electrons.

§ 29. Bremsstrahlung. 378

1. Perturbation Theory for an Electron Wave Function in the Continuum. Incoming and Outgoing Waves.
2. Effective Cross Section for Bremsstrahlung.
3. Angular Distribution of the Radiation in a Coulomb Field.
4. Polarization of the Radiation.
5. Spectrum of the Radiation.
6. Screening.
7. Radiative Energy Losses.
8. Exact Theory of Bremsstrahlung in the Nonrelativistic Domain.
9. Exact Theory of Bremsstrahlung in the Extreme Relativistic Domain.
10. Radiation Emitted in Electron-Electron and Electron-Positron Collisions.

§ 30. Emission of Photons of Long Wavelength 413

1. “The Infrared Catastrophe.”
2. Investigation of the Divergence in the Low Frequency Domain by Means of the Scattering Matrix.
3. Relation between the Photon “Mass” and the Minimum Frequency.

§ 31. Photoeffect 429

1. Photoeffect in the Nonrelativistic Domain.
2. Photoeffect in the Relativistic Domain.

§ 32. Production of Electron-Positron Pairs 438

1. Production of an Electron-Positron Pair by a Photon in the Field of a Nucleus.
2. Exact Theory of Pair Production by a Photon in the Field of a Nucleus in the Nonrelativistic and Extreme Relativistic Cases.
3. Pair Production by Two Photons.
4. Pair Production in a Photon-Electron Collision.
5. Pair Production in a Collision of Two Fast Charged Particles.

§ 33. Annihilation of Electron-Positron Pairs into Photons 457

1. Annihilation of a Pair into Two Photons.
2. Polarization Effects in the Two-Photon Annihilation of a Pair.
3. Annihilation of a Pair into One Photon.
4. Positronium Decay.
5. Three-Photon Decay of Orthopositronium.
6. Multiple Photon Production Accompanying the Annihilation of a Pair.

§ 34. The Method of Equivalent Photons 473

1. The Number of Equivalent Photons.
2. Bremsstrahlung from a Fast Electron in the Field of a Nucleus.
3. Radiation Emitted in an Electron-Electron Collision.
4. Pair Production by a Photon in the Field of a Nucleus.
5. Pair Production in a Collision of Two Fast Particles.

§ 35. Scattering of a Photon by a Bound Electron. Emission of Two Photons 484

1, The Dispersion Formula.
2. Resonance Scattering.
3. Compton Scattering by Bound Electrons.
4. Emission of Two Photons. The Metastable 2s;, State of the Hy-
drogen Atom.

§ 36. Electron-Electron and Positron-Electron Scattering 499

1. Electron-Electron Scattering.
2. Positron-Electron Scattering.
3. Scattering of Polarized Electrons and Positrons.
4. Annihilation of an Electron-Positron Pair into a 𝜋-Meson Pair.

CHAPTER VI
RETARDED INTERACTION BETWEEN TWO CHARGES

§ 37. Retarded Potentials 509

1. Interaction Function for Two Charges.
2. General Form of the Matrix Element.
3. Retarded Potentials and Transition Currents.

§ 38. Interaction Energy of Two Electrons to Terms of Order v2/c? 517

1. The Breit Formula.
2. Schrödinger Equation for a Two-Electron System.
3. Interaction between an Electron and a Positron.
4. Exchange Interaction between an Electron and a Positron.

§ 39. Positronium. 527

1. Hamiltonian Operator and the Unperturbed Equation.
2. Perturbation Operator.
3. Fine Structure.
4. Zeeman Effect.

§ 40. Internal Conversion of Gamma-Rays 537

1. Expansion of Retarded Potentials in Spherical Waves.
2. Conversion Coefficient.
3. Conversion in the K-Shell.
4. Effect of Finite Nuclear Size.
5. Effect of Electron Shells on Radiation from the Nucleus.

§ 41. Conversion Accompanied by Pair Production. Excitation of Nuclei by Electrons 554

1. Conversion of Magnetic Multipole Radiation.
2. Conversion of Electric Multipole Radiation.
3. Excitation of Nuclei by Electrons.
4. Monoenergetic Positrons.

§ 42. Coulomb (Monopole) Transitions 565

1. Reduction to the Static Interaction.
2. Conversion and Nuclear Excitation in the Case of an E0-Transition.

CHAPTER VII
INVESTIGATION OF THE SCATTERING MATRIX

§ 43. Properties of Exact Solutions of the Equations of Quantum Electrodynamics. Propagators 571

1. Stationary States of a System of Interacting Fields.
2. Propagators and Their Spectral Representation.
3. Connection between Propagators and the Scattering
Matrix. Integral Equations for Propagators.
4. Electromagnetic Mass of the Electron.

§ 44. Structure of the Scattering Matrix 593

1. Self-Energy Parts of Diagrams.
2. Vertex Parts of Diagrams.
3. Renormalization of Electron Mass.

§ 45. Renormalization of Electron Charge 605

1. Physical Charge of the Electron.
2. Renormalization of Propagators and Vertex Parts.
3. Three-Photon Vertex Parts.
4. Renormalization of Matrix Elements.
5. Formulation of Perturbation Theory as an Expansion of Powers of e_c.

 

§ 46. Divergences in the Scattering Matrix and their Removal 619

1. Divergences in Irreducible Diagrams.
2. Introduction of a Cut-Off Momentum.
3. Convergence of Regularized Expressions for Irreducible Vertex Parts and Self-Energy Parts.
4. Convergence of Regularized Quantities in the Case of Reducible Diagrams.

§ 47. Evaluation of Self-Energy and Vertex Parts 631

1. Evaluation of Integrals over Four-Dimensional Regions.
2. Second Order Electron Self-Energy Part.
3. Second Order Photon Self-Energy Part.
4. Third Order Vertex Part in the Case of External Electron Lines.
5. Third Order Vertex Part in the Case of One External Electron Line.

§ 48. Functional Properties of Green’s Functions. Limits of
Applicability of Quantum Electrodynamics 657

1. Expansion Parameters of Perturbation Theory.
2. Zero Order Approximation in the Expansion in Powers of e_c.
3. Integral Equations for the Zero Order Approximation.
4. The Renormalization Group. 5. Derivation of Asymptotic Expressions for the Green’s Functions with the Aid of Differential Equations of the Renormalization Group.
6. The Problem of Closure of Quantum Electrodynamics.

§ 49. Generalized Green’s Functions 676

1. Green’s Functions in the Presence of External Fields.
2. Green’s Function for Two Electrons. Equation for Bound States of the Electron-Positron System.
3. Equations for Green’s Functions in Terms of Variational Derivatives.
4. Expressions for Green’s Functions in Terms of Functional Integrals.

CHAPTER VIII
RADIATION CORRECTIONS TO ELECTROMAGNETIC PROCESSES

§ 50. Effective Potential Energy of the Electron. Radiation Corrections to the Electron Magnetic Moment and to Coulomb’s Law 693

1. Energy of Interaction of the Electron with the Electromagnetic Field Taking into Account Corrections of Order 𝛼.
2. Radiation Corrections to the Electron Magnetic Moment.
3. Radiation Corrections to Coulomb’s Law.

§ 51. Radiation Corrections to Electron Scattering 705

1. Electron Scattering by the Coulomb Field of a Nucleus in the Second Born Approximation.
2. Differential Cross Section for the Scattering of an Electron by the Coulomb Field of a Nucleus taking into Account Radiation Corrections of Order 𝛼.
3. Elimination of the Photon “Mass” from the Scattering Cross Section.
4. Removal of the Infrared Divergence for an Arbitrary Scattering Process.
5. Scattering of High Energy Electrons by an External Field.
6. Radiation Corrections to Electron-Electron and Electron-Positron Scattering.

§ 52. Radiation Corrections to Photon-Electron Scattering, to Pair Creation and Annihilation, and to Bremsstrahlung. 731

1. Radiation Corrections to the Compton Effect.
2. Limiting Cases of Low and High Energies.
3. Radiation Corrections to Two-Photon Pair Annihilations.
4. Radiation Corrections to Bremsstrahlung.
5. Radiation Corrections to Photon Production and Single Photon Annihilation of Pairs.

§ 53. Radiation Corrections to Atomic Levels 751

1. Radiation Shift of Atomic Levels.
2. Radiation Shift of the Levels of 𝜇-Mesohydrogen.
3. Natural Line Widths.
4. Photon Scattering near Resonance.

§ 54. Photon-Photon Scattering and the Lagrangian for the
Electromagnetic Field 764

1. Photon-Photon Scattering Tensor of the Fourth Rank.
2. Photon-Photon Scattering.
3. Connection between the Photon-Photon Scattering Cross Section and the Radiation Corrections to the Lagrangian of the Electromagnetic Field.
4. Exact Expressions for the Lagrangian of the Electromagnetic Field.

§ 55. Photon Scattering by the Coulomb Field of a Nucleus 792

1. General Expression for the Cross Section for Photon Scattering by a Constant Electromagnetic Field.
2. Relation between the Forward Scattering Amplitude for a Photon and Pair-Production by a Photon in the Field of a Nucleus.
3. Momentum Distribution of Recoil Nuclei Accompanying Pair Production by a Photon in the Field of a Nucleus.
4. Angular Distribution of Recoil Nuclei and Total Cross Section for Pair Production by a Photon in the Coulomb Field of a Nucleus.
5. Small Angle Coherent Scattering of Photons by the Field of a Nucleus.

CHAPTER IX
ELECTRODYNAMICS OF PARTICLES OF SPIN ZERO

§ 56. Field Equations for Scalar Particles 819

1. First Order Equations.
2. Quantization of the Free Scalar Field.
3. Commutators of the Field. Vacuum Expectation Values of Products of Field Components.

 

§ 57. The Scattering Matrix in Scalar Electrodynamics 827

1. The Interaction Picture.
2. Rules for Calculating Elements of the Scattering Matrix.
3. Divergences of the Scattering Matrix.

§ 58. Scattering of Scalar Particles 835

1. Scattering of Scalar Particles by the Coulomb Field of a Nucleus. 2. Scattering of a Charged Scalar Particle by a Scalar Particle.

§ 59. Scattering of a Photon by a Scalar Particle. Bremsstrahlung Photons from a Scalar Particle 838

1. Scattering of a Photon by a Scalar Particle.
2. Bremsstrahlung from Scalar Particles.

§ 60. Production and Annihilation of Pairs of Scalar Particles. 842

1. Production of Pairs of Scalar Particles by a Photon in the Coulomb Field of a Nucleus.
2. Production of a Pair of Scalar Particles by Two Photons.
3. Two-Photon Annihilation of a Pair of Scalar Particles.
4. Annihilation of Pairs of Scalar Particles into Electron-Positron Pairs and the Inverse Process.

§ 61. Polarization of the Vacuum in the Case of Charged Scalar Particles 847

1. Vacuum Polarization Tensor for Scalar Particles.
2. Correction to Coulomb’s Law.
3. Photon-Photon Scattering. Radiation Corrections to the Lagrangian of the Electromagnetic Field.

Concluding Remarks 852

References 855

Subject Index 863

 

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The ABC of Chess – Grishin, Ilyin

Posted on February 22, 2022 by The Mitr

In this post, we will see the book The ABC of Chess by V. Grishin and E. llyin.

About the book

An introductory book for children to learn chess. Teaches the basics of the game. There are a lot of children between the ages of 5 – 7 who are interested in chess and want to learn to play. This book is written to help younger children take their first steps on the chessboard. As a rule the first books a child comes to know are read aloud to him by his parents and this one need be no exception. Don’t hurry when you read “The ABC of Chess” to your child, don’t cover more than one chapter a week. In your instruction pay more attention to the principle of play.

The book was translated from Russian by Vivienne Burdon was published in 1986 by Raduga Publishers.

All credits to gnv64   and Guptaji.

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Contents

Foreword for Parents. 3

A War of Wood. 7

A Revealing Story. 15

The Battlefield. 23

Only Straightforward!. 29

Who Has Been Placed in the Corner?. 35

Leaping Horses. 41

“Lightweight” Bishops. 47

The Most Powerful Piece. 51

Watch Out, Your Majesty!. 57

Touch — Move. 63

Who’s Won? IPs a Stalemate!. 69

Not Numbers but Know-How. 75

The King Goes on the March. 79

All Children Should Learn. 83

Appendix. 86

Answer These Questions Without Your Parents Help.87

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How The Machine Learned The Alphabet – Zubkov

Posted on February 21, 2022 by The Mitr

In this post, we will see the book .

About the book

A little book describing how machines (computers) a can learn various things and think.

The book was translated from Russian by Vladimir Korotky and illustrated by Boris Kyshtymov. The book was published in 1985 by Malysh Publishers.

All credits to Guptaji.

You can get the book here (cleaned optimised) and here.

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Contents

How to make a machine that can read and why we write postcodes on envelopes 3

The importance of “Yes” and “No” and How People speak with lifts, mills, steam engines and rockets 8

How machines learned that a cat was a cat and why people began calling machines intelligent 12

Iron ducks and Trumpeters, What is automation and how many automatons there are around 16

Why a robot is called a robot, One scientific experiment and what kind of arms and legs robots have 20

 

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Tomorrow? … Yesterday? – Zubkov

Posted on February 20, 2022 by The Mitr

In this post, we will see the book Tomorrow? … Yesterday? by Boris Zubkov.

About the book

A book describing history and evolution of measuring time at different epochs. Discusses various natural phenomena which were used to keep calendars and inventions of clocks to measure time. The book also touches upon the circadian rhythm present in all living organisms. Final part discusses aspects of time management in factories.

The book was translated from Russian by Eleanor Yankovskaya and illustrated by I. Kabakov. The book was published in 1978 by Malysh Publishers.

You can get the book here (hi-res and cleaned). Another copy from Guptaji here.

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Contents

What happens everyday? 3

How People Learned to Measure Time? (and about clocks that travel in the sky, fire a gun and even race around a city) 6

There Has Always Been Time (and travels to distant places, a faithful keeper of Time in Leningrad, what Tomorrow means, and why a lazy person is bored) 13

The Clock Inside Us (also how oysters and bean plants  remember Time, and who night owls and early birds are) 17

Every Minute is Precious (What does time mean for factories? How does Time help people work better? 22

 

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Foundations Of Combinatorial Topology – Pontryagin

Posted on February 19, 2022 by The Mitr

In this post, we will see the book Foundations Of Combinatorial Topology by L. S. Pontryagin.

About the book

This book represents essentially a semester course in combinatorial topology which I have given several times at the Moscow National University. It contains a very rigorous but concise presentation of homology theory. The formal prerequisites are merely a few simple facts about functions of a real variable, matrices, and commutative groups. Actually, how­ ever, considerable mathematical maturity is required of the reader. An essential defect in the book is its complete omission of examples, which are so indispensable for clarifying the geometric content of combinatorial topology. In this sense a good complementary volume would be Sketch of the Fundamental Notions of Topology by Alexandrov and Efremovitch, in which the attention is focused on the geometric content rather than on the completeness and rigor of proofs. In spite of this shortcoming, it seems to me that the present work has certain advantages over the existing voluminous treatises, especially in view of its brevity. It can be used as a reference for obtaining preliminary information required for participation in a serious seminar on combinatorial topology. It is convenient in preparing for an examination in a course, since the proofs are carried out in the book with sufficient detail. For a more qualified reader, e.g., an aspiring mathematician, it can also serve as a source of basic information on combinatorial
topology.

The present book makes use of a few facts concerning metric spaces which are now ordinarily included in a course in the theory of functions of a real variable, and which can be found in the sixth chapter of Hausdorff’s Mengenlehre or in the third chapter of Alexandrov and Kolmogorov’s Theory of Functions of a Real Variable. Information concerning commutative groups may be found in the fifth chapter (see §21 and §22) of Kurosh’s Theory of Groups.

The book was translated from Russian by F. Bagemihl, H. Komm and W. Seidel. The book was published in 1952.

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You can get the book here.

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Contents

Preface vii
Introduction ix
Notation xi

CHAPTER I COMPLEXES AND THEIR BETTI GROUPS

§ 1. Euclidean Space. I
§ 2. Simplex. Complex. Polyhedron 9
§ 3. Application to Dimension Theory. 16
§ 4. The Betti Groups. 23
§ 5. Decomposition into Components. The Zero-dimensional Betti Group. 26
§ 6. The Betti Numbers. The Euler-Poincaré Formula. 30

CHAPTER II THE INVARIANCE OF THE BETTI GROUPS

§ 7. Simplicial Mappings and Approximations. 36
§ 8. The Cone Construction. 43
§ 9. Barycentric Subdivision of a Complex. 48
§ 10. A Lemma on the Covering of a Simplex, and its Application. 53
§ 11. The Invariance of the Betti Groups under Barycentric Sub-division. 58
§12. The Invariance of the Betti Groups. 61

CHAPTER III CONTINUOUS MAPPINGS AND FIXED POINTS

§13. Homotopic Mappings. 69
§14. The Cylinder Construction. 72
§15. Homology Invariants of Continuous Mappings. 79
§16. The Existence Theorem for Fixed Points. 84

List of Definitions. 94
List of Theorems. 95

Basic Literature on Combinatorial Topology. 96

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Mathematical Problems And Puzzles from the Polish Mathematical Olympiads (Popular Lectures in Mathematics Vol 10) – Straszewicz

Posted on February 18, 2022 by The Mitr

In this post, we will see the book Mathematical Problems And Puzzles from the Polish Mathematical Olympiads by S. Straszewicz. This book is volume 10 of Popular Lectures in Mathematics series.

About the book

THIS book is a translation of the second Polish edition, published in 1960, in which various improvements were made.

The problems set at the contests require only a knowledge of school mathematics (i.e. elementary algebra, geometry and trigonometry) but are on the whole more difficult than the usual school exercises. Their degree of difficulty, however, is not uniform, for it is considered desirable that not only the most gifted pupils but also those of average ability should —with a certain effort— manage to solve some of the problems and gain a number of points at the contests.

This book contains the problems set at the first five Olympiads. It has been prepared in order to provide secondary school pupils with suitable topics to be worked out on their own, individually or collectively. It aims at extending their knowledge of mathematics and training them in mathematical thinking. Accordingly, the solutions of most of the problems have been given in an extended form, the readers’ attention being drawn to various details of the reasoning. In addition, several problems have been provided with commentaries containing generalizations or further development of the topics in question, including various supplementary data of elementary mathematics outside the scope of the school syllabus.

The book was translated from Polish by J. SMÓLSKA by was published in 1965.

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Contents

PART I ARITHMETIC AND ALGEBRA  2

PROBLEMS 3

SOLUTIONS 10

 

PART 2 GEOMETRY AND TRIGONOMETRY 112

PROBLEMS 113

SOLUTIONS 123

 

 

 

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One Hundred Problems In Elementary Mathematics (Popular Lectures in Mathematics Vol. 7)- Steinhaus

Posted on February 17, 2022 by The Mitr

In this post, we will see the book One Hundred Problems In Elementary Mathematics by Hugo Steinhaus. his book is volume 7 of Popular Lectures in Mathematics series.

About the book

This booklet is an answer to a challenge: a few years after the war the inadequacy of mathematical education in our high schools became evident to the staffs of universities and technological institutes. Some responsible people felt that a closer collaboration between mathematicians and school teachers could no longer be postponed. A few scientists were among those who did their best to stimulate interest in mathematics by means of elementary problems published in an educational journal. Here the reader will find one hundred elementary problems and their solutions. Some of them are familiar to students in high schools, but it was by no means my intention to provide the teacher with questions he could find in every textbook.

The book was translated from Polish and published in 1964.

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You can get the book here.

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Contents

Foreword
Preface

Chapter I PROBLEMS ON NUMBERS, EQUATIONS AND INEQUALITIES

Problem Solution

1. Exercise on the multiplication table 11 55
2. An interesting property of numbers 11 55
3. Division by 12 58
4. The divisibility of numbers 12 59
5. A simplified form of Fermat’s theorem 12 60
6. Distribution of numbers 12 61
7. Generalization 13 61
8. Proportions 13 65
9. Irrationality of the root 13 65
10: Inequality 13 65
11. A sequence of numbers 13 67

Chapter II PROBLEMS ON POINTS, POLYGONS, CIRCLES AND ELLIPSES

12. Points in a plane 14 68
13. Examination of an angle 14 69
14. Area of a triangle 14 69
15. Triple halving of the perimeter of a triangle 15 70
16. Division of a triangle 15 75
17. Triangles 15 77
18. Triangular network 15 77
19. Triangular network 16 78
20. What is left from a rectangle? 16 79
21. Division of a square 16 81
22. Square network 16 83
23. Lattice points 17 84
24. Lattice points inside a circle 17 84
25. 14 = 15 17 85
26. Polygon 17 85
27. Points and a circle 18 86
28. Geometrical problem 18 87

Chapter III PROBLEMS ON SPACE, POLYHEDRA AND SPHERES

29. Division of space 19 89
30. Two projections 19 89
31. Cube 19 90
32. Geodesics 20 92
33. Motion of a particle 20 95
34. Diagrams of the cube 20 96
35. Cubes 21 96
36. Hexahedron 21 98
34. Tetrahedra 21 98
38. Tetrahedron with congruent faces 21 99
39. Octahedron 21 101
40. Distance on a surface 21 103
4l. The wandering of a fly 22 104
42. Regular dodecahedron 22 105
43. Polyhedron 22 108
44. Non-convex polyhedron 23 109
45. Problem from Wonderland 23 110
46. Three spheres and a line 24 113
47. A property of the sphere 24 113

Chapter IV PRACTICAL AND NON-PRACTICAL PROBLEMS

48. Puzzle 25 115
49. Picnic ham 25 115
50. Quartering of a pie 26 116
51. Another pie 26 117
52. Weighings 26 118
55. Calibration of rollers 27 121
56. 120 ball-bearings 28 121
57. Ribbon on the roll 28 122
58. Watch with both hands identical 28 123
59. Problems of giants and midgets 29 125
60. Acks and backs 29 127
61. Statistics 30 127
62. Blood groups 31 128
63. Blood groups again 32 129
64. Excess of labour 32 130
65. Diagonal of a wooden block 32 133
66. The tying of boxes 33 133
67. A primitive device 33 134
68. The minimal length 33 136
69. Division of plots 34 136
70. A practical problem 34 143
71. Neighbouring towns 35 144
72. Railway lines (I) 35 145
73. Railway lines (II) 35 146
74. Test Flight 35 149
75. Sun and Moon 36 150
76. Cosmography 36 150

Chapter V PROBLEMS ON CHESS, VOLLEYBALL AND PURSUIT

77. Chessboard 37 153
78. Chessboard revisited 37 153
79. Rooks on the chessboard 37 157
80. Elliptical billiards 38 161
81. A sports problem (I) 38 162
82. A sports problem (II) 38 162
83. Theory of sport eliminations 38 163
84. Volleyball league 39 164
85. Tournaments 39 165
86. Bicyclist and walkers 40 165
87. Four dogs 40 166
88. Chase (I) 41 166
89. Chase (II) 41 167
90. Incomplete data 41 168
91. Motorboat (I) 41 168
92. Motorboat (II) 42 169

Chapter VI MATHEMATICAL ADVENTURES OF DR. ABRACADABRUS

93. The strange number 43 171
94. The tailor’s tape 43 171
95. Word quiz 43 171
96. Student debts 44 172
97. A strange social set 44 172
98. ABACUS 44 173
99. Washing the streets 45 174
100. French cities 45 174

 

Chapter VII PROBLEMS WITHOUT SOLUTION

Plus and minus 47
Triangle in a triangle 48
Parts of a square 48
Division of a circle 49
Radii in space 49
Unlimited chessboard 49
The abacus again 49
Tins in a drawer 50
Bacilli 50
The circus is coming 50
Three Cowboys 51
Investigation 51
Arrows on a dodecahedron 52

 

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Elements of Number Theory – Vinogradov

Posted on February 16, 2022 by The Mitr

In this post, we will see the book Elements Of Number Theory by I. M. Vinogradov.

About the book (from the author)

In my book I present a systematic exposition of the funda­ mentals of number theory within the scope of a university course. A large collection of problems introduces the reader to some of the new ideas in number theory.
This fifth edition of my book differs considerably from the fourth. A series of changes, allowing a simpler exposition, have been made in all the chapters of the book. The most important changes are the merging of the old chapters IV and V into one chapter IV (reducing the number of chapters to six) and the new, simpler proof of the existence of primitive roots.
The problems at the end of each chapter have been essentially revised. The order of the problems is now in complete correspondence with the order of the presentation of the theoretical material. Some new problems have been added; but the number of numbered problems has been substantially reduced. This was accomplished by the unification, under the letters a, b, c, . . ., of previously separate problems which were related by the method of solution or by content. All the solutions of the problems have been reviewed; in many cases these solutions have been simplified or replaced by better ones. Particularly essential changes have been made in the solutions of the problems relating to the distribution of n-th power residues and non-residues, and primitive roots, as well as in the estimations of the corresponding trigonometric sums.

The book was translated from Russian by Saul Kravetz was published in 1954.

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Contents

Preface vii

Chapter I DIVISIBILITY THEORY

§1. Basic Concepts and Theorems (1).
§2. The Greatest Common Divisor (2).
§3. The Least Common Multiple (7).
§4. The Relation of Euclid’s Algorithm to Continued Fractions (8). §5. Prime Numbers (14).
§6. The Unicity of Prime Decomposition (15),
Problems for Chapter I (17).
Numerical Exercises for Chapter I (20).

Chapter II IMPORTANT NUMBER-THEORETICAL FUNCTIONS

§1. The Function’s {x},x (21).
§2. Sums Extended over the Divisors of a Number (22),
§3. The Mdbius Function (24).
§4, The Euler Function (26).
Problems for Chapter II (28).
Numerical Exercises for Chapter II (40).

Chapter III. CONGRUENCES

§1. Basic Concepts (41).
§2. Properties of Congruences Similar to those of Equations (42). §3. Further Properties of Congruences (44).
§4. Complete Systems of Residues (45).
§5. Reduced Systems of Residues (47).
§6. The Theorems of Euler and Fermat (48).
Problems for Chapter III (49).
Numerical Exercises for Chapter III (58).

Chapter IV CONGRUENCES IN ONE UNKNOWN

§1. Basic Concepts (59).
§2. Congruences of the First Degree (60).
§3. Systems of Congruences of the First Degree (63).
§4. Congruences of Arbitrary Degree with Prime Modulus (65).
§5. Congruences of Arbitrary Degree with Composite Modulus (66). Problems for Chapter IV (71).
Numerical Exercises for Chapter IV (77).

Chapter V CONGRUENCES OF SECOND DEGREE

§1. General Theorems (79).
§2. The Legendre Symbol (81).
§3. The Jacobi Symbol (87).
§4. The Case of Composite
Moduli (91).
Problems for Chapter V (95).
Numerical Exercises for Chapter V (103).

Chapter VI PRIMITIVE ROOTS AND INDICES

§1. General Theorems (105).
§2. Primitive Roots Modulo p* and 2p* (106).
§3. Evaluation of Primitive Roots for the Moduli p* and 2p* (108). §4. Indices for the Moduli p and 2p* (110).
§5. Consequences of the Preceding Theory (113).
§6. Indices Modulo 2% (116).
§7. Indices for Arbitrary Composite Modulus (119).
Problems for Chapter VI (121).
Numerical Exercises for Chapter VI (130).

SOLUTIONS OF THE PROBLEMS

Solutions for Chapter I (133).
Solutions for Chapter II (139).
Solutions for Chapter III (161).
Solutions for Chapter IV (178).
Solutions for Chapter V (187).
Solutions for Chapter VI (202).

ANSWERS TO THE NUMERICAL EXERCISES

Answers for Chapter I (217).
Answers for Chapter II (217).
Answers for Chapter III (218).
Answers for Chapter IV (218).
Answers for Chapter V (218).
Answers for Chapter VI (219).

TABLES OF INDICES 220

TABLES OF PRIMES <4000 AND THEIR LEAST PRIMITIVE 226

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