Geometric Transformations 2 by I. M. Yaglom

Posted on June 12, 2024 by The Mitr

In this post, we will see the book Geometric Transformations 2 by I. M. Yaglom.

About the book

This volume is Part II of “Geometric Transformations” by I. M. Yaglom, following the previously published Part I. The original Russian edition (1955) combined Parts I and II, while Part III was a separate volume. An English translation of Part III is forthcoming. This book is not an introduction to plane geometry; it assumes the reader has some prior knowledge and focuses on similar figures and transformations that preserve similarity.

Part II contains 83 problems, central to the book, which readers should attempt before consulting the provided solutions. The English edition renumbers the problems for clarity, with a chart included to compare with the original Russian numbering.

This work explores elementary geometry, emphasizing two foundational ideas: the deductive method and geometric transformations with their group-theoretic basis. These ideas are pivotal for advancing into non-Euclidean geometry. The book aims for simplicity and clarity, avoiding rigorous definitions and complex terminology to aid less experienced readers, though this sacrifices some precision for well-prepared readers.

Problems throughout the book help readers test their understanding, focusing on methods rather than results. Readers are encouraged to solve at least one problem from each group and then study the solutions. The book includes many construction problems, valued for their logical interest rather than practical execution.

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Geometric Transformations 1 by I. M. Yaglom

Posted on June 11, 2024 by The Mitr

In this post, we will see the book Geometric Transformations 1 by I. M. Yaglom.

About the book

This book, part of the New Mathematical Library series, aims to make key mathematical concepts accessible to high school students and general readers. These volumes often cover topics outside the standard high school curriculum, varying in difficulty. Readers may need to focus more on certain sections but generally require minimal technical knowledge.

Mathematical books can’t be read quickly or fully understood on the first pass. Readers should feel free to skip and return to complex parts. Learning mathematics effectively involves solving problems, so readers are encouraged to engage actively with the material using paper and pencil.

This work focuses on elementary geometry, exploring significant ideas like the deductive method and geometric transformations, which lead to non-Euclidean geometry. The book prioritizes simplicity and clarity over rigor, avoiding complex definitions and terms to aid understanding.

Problems in the book test comprehension and encourage practice. Readers should attempt at least one problem from each group, consulting provided solutions. The book emphasizes methods over results, using problems to illustrate concepts rather than focusing on three-dimensional propositions, keeping the primary ideas intact.

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Geometric Transformations Volume 1 Euclidean And Affine Transformations by P.S. Modenov; A.S. Parkhomenko

Posted on June 10, 2024 by The Mitr

In this post, we will see the book Geometric Transformations Volume 1 Euclidean And Affine Transformations by P.S. Modenov; A.S. Parkhomenko.

About the book

This is the first volume of a two-volume translation of the Russian book Geometric Transformations, by Modenov and Parkhomenko. This volume embraces Chapters I—IV; Vol­ume 2 (Projective Transformations) contains the translation of the original Chapters V and VI.
The treatment is elementary, and should be accessible to the high school senior. The prerequisites amount to some famil­ iarity with Euclidean geometry, including the use of coordi­ nates, elementary trigonometry, and linear equations (up to determinants). A little knowledge of vectors and conics might also be helpful. However, the material covered or referred to ranges much further, and should be of interest to a very broad spectrum of readers, from high school senior to college teacher.
This book is not designed to be a standard text. As will be seen from the introduction, the material covered is not usually included in the curriculum, and its style is more suitable for browsing than for systematic class study. The purpose of the book is rather to introduce the reader to a fascinating and not at all difficult area of geometry, at the same time acquainting him painlessly with some of the simpler methods and concepts of advanced mathematics. Since the topic is one for which everyone will have some intuitive feeling, and the exposition is consistently straightforward, there is no danger that the reader will find himself suddenly out of his depth.
This book is intended for use in geometry seminars in universities and teacher-training colleges. It may also be used as supplementary reading by high school teachers who wish to extend their range of knowledge. Finally, many sections may be used as source material for school mathematics clubs under the guidance of a teacher.
The subject matter is those transformations of the plane that preserve the fundamental figures of geometry: straight lines and circles. In particular, we discuss orthogonal, affine, pro­ jective, and similarity transformations, and inversions.
The treatment is elementary, though in a number of in­ stances (where a synthetic treatment seems more cumbersome) coordinate methods are used. A little use is also made of vec­ tor algebra, but the text here is self-contained.
In order to clarify a number of points, we give some elemen­ tary facts from projective geometry; also, in the addendum to Chapter I of Volume 2 (the topology of the projective plane), the structure of the projective plane is examined in greater detail.

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Geometric Transformations Volume 2 Projective Transformations by P.S. Modenov; A.S. Parkhomenko

Posted on June 9, 2024 by The Mitr

In this post, we will see the book Geometric Transformations Volume 2 Projective Transformations by P.S. Modenov; A.S. Parkhomenko.

About the book

This book is intended for use in geometry seminars at universities and teacher-training colleges. It may also serve as supplementary reading for high school teachers seeking to extend their knowledge. Additionally, many sections can be used as source material for school mathematics clubs under a teacher’s guidance.

The subject matter covers transformations of the plane that preserve fundamental geometric figures: straight lines and circles. Specifically, it discusses orthogonal, affine, projective, and similarity transformations, as well as inversions. The treatment is elementary, though coordinate methods are used where a synthetic approach is more cumbersome. A little vector algebra is also employed, but the text is self-contained.

This is the second volume of a two-volume translation of the Russian book *Geometric Transformations* by Modenov and Parkhomenko. It contains translations of Chapters V (with Appendix) and VI of the original, presented here as Chapters I and II, respectively.

The greater portion of this volume focuses on projective transformations, which are collinearity-preserving transformations of the projective plane. Many mappings of the ordinary plane that preserve collinearity are best understood as defined on an extended object called the projective plane. Notably, the affine transformations discussed in Chapter IV of the first volume (*Euclidean and Affine Transformations*) are conveniently regarded as projective transformations that fix the ideal line.

Chapter I begins with the motivation for constructing the projective plane, followed by various alternative constructions. It then proves most of the basic facts and outlines some applications. Chapter II addresses an independent topic at the same level of sophistication. However, the appendices to Chapter I introduce more advanced concepts without detailed motivation or treatment. Readers who are unprepared should not hesitate to read through them, absorbing what they can and taking the rest on faith. These sections provide glimpses into the methods and concerns of modern geometry and algebra.

Overall, this book is probably suitable for undergraduate readers. However, interest and ability are more important for benefiting from it than the quantity of previous knowledge.

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Mathematical Aspects Of The Three Body Problem In The Quantum Scattering Theory by L. D. Faddeev

Posted on June 8, 2024 by The Mitr

In this post, we will see the book Mathematical Aspects Of The Three Body Problem In The Quantum Scattering Theory by L. D. Faddeev.

About the book

The book consists of eleven sections and four appendices. In § 1 the operators in question are rigorously defined in Hilbert space, i. e. , their domain of definition is specified and their self-adjointness proved. Here are also formulated the basic conditions imposed on the potentials which in the rest of the book are tacitly assumed to be fulfilled. §§ 2 and 4, and §§ 3, 5, 6, 7, respectively, deal with the resolvents of energy operators for two and three bodies; the resolvents are expressed in term s of integral operators, and integral equations are set up and investigated in order to derive estimates for the kernels and examine their singularities in the variable z near the real axis. The obtained results are applied in §§ 8 and 9 to the proof of eigenfunction expansion theorems, and in §§10 and 11 to the time- dependent formulation of the scattering problem and the construction of the scattering operator. Appendix I gives the derivation of some properties of functions which satisfy the Holder condition, and of singular integrals containing these functions. Appendices II and III give proofs of estimates of some integrals applied in the text. Appendix IV contains remarks and references to the literature which are not mentioned in the main text.

This book is a translation of
MATEMATICHESKIE VOPROSY KVANTOVOI TEORII RASSEYANIYA
DLYA SISTEMY TREKH CHASTITS In: Trudy Matematicheskogo Instituta
imeni V. A. Steklova. LXIX
Izdatel’stvo Akademii Nauk SSSR Moskva-Leningrad
1963

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Hydrodynamics For Aeronautical Engineers by B. Nekrasov

Posted on June 7, 2024 by The Mitr

In this post, we will see the book Hydrodynamics For Aeronautical Engineers by B. Nekrasov.

About the book

 Boris Nekrasov is an eminent Soviet expert in hydraulics and its industrial applications. An excellent textbook originally written for aeronau­tical engineering students, it also presents the main general hydraulic problems as well as fundamentals of the theory of hydraulic machinery employed in aircraft systems.
The practical applications of hydraulic theory being the author’s main concern, the book contains many worked problems for simple hydraulic sys­tems and machinery. One hundred and seventy- eight drawings and diagrams amply illustrate the principles and their practical applications dis­cussed in the book.

Translated from the Russian by V. Talmy

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Theory Of Turbulent Plasma by A. A. Vedenov

Posted on June 6, 2024 by The Mitr

In this post, we will see the book Theory Of Turbulent Plasma by A. A. Vedenov.

About the book

This monograph reviews the theory of turbulent plasma. Chapter I outlines methods for describing laminar and weakly turbulent plasma dynamics, which inform the discussion of collective oscillations (Chapter 2) and laminar plasma stability (Chapter 3), with experimental confirmation. Chapter 4 examines hysteresis in transitions between laminar and turbulent states. Chapter 5 details the quasi-linear theory of tenuous plasma, predicting particle distribution plateaus, confirmed experimentally. Chapters 6 and 7 address interactions between plasma oscillations and kinetic equations for wave distribution functions. Chapter 8 proposes a principle for spectral energy density in strongly turbulent plasma, validated by experiments on turbulent gravitational waves. Chapter 9 discusses transport coefficients in turbulent plasma, and Chapter 10 explores electromagnetic wave scattering in turbulent plasma, pertinent to ionospheric and laboratory studies.

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The Theory Of Optimum Noise Immunity by V. A. Kotel’nikov

Posted on June 5, 2024 by The Mitr

In this post, we will see the book The Theory Of Optimum Noise Immunity by V. A. Kotel’nikov.

About the book

This book is the author’s doctoral dissertation, presented in January, 1947, before the academic council of the Molotov Energy Institute in Moscow. Despite the fact that many works devoted to noise immunity have appeared in the time that has elapsed since the writing of this dissertation, not all of the topics considered in it have as yet appeared in print. Considering the great interest shown in these matters, and also the number of references made to this work in the literature, the author has deemed it appropriate to publish it, without introducing any supplementary material. However, in preparing the manuscript for publication, it was somewhat condensed, at the expense of material of secondary interest. Moreover, Chapter 2, which contains auxiliary mathematical material, has been revised somewhat, to make it easier reading, and some of the material has been relegated to the appendices.
This investigation of the performance of various communications systems in the presence of additive Gaussian noise is a basic work in information theory. In it, Dr.
Kotel’nikov, one of the few electronics engineers to be elected an Academician of the Academy of Sciences of the U.S.S.R., provides remarkably clear expositions of his important contributions, many of which anticipated similar developments in the West by a considerable period of time.
To enable the reader to use the book more readily, the author first sets the frame­ work of the problems to be considered and helps the reader to sharpen his mathe­matical tools for the rest of the book. The second part then treats of the transmission of discrete messages, while the third and fourth parts cover transmission of a continuous parameter value and continuous time functions. Each of these parts contains examples of the application of general theory to special types of com­munications systems: AM, FM, PAM, PPM, SSB, and others.
Idle author’s statistical analysis of his problems invokes what are now called decision theory techniques. His approach, however, is that of the practical engineer, necessitating only elementary notions of probability theory. The reader with some familiarity with Fourier series, discrete and continuous probabilities and probability densities will have little trouble in following the text. Extensive use-of geometrical models of the signalling and detection process is very helpful to the student.

Translated from the Russian by R. A. Silverman

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Linear Differential Operators: Part 1 – Elementary Theory of Linear Differential Operators by M. A. Naimark

Posted on June 4, 2024 by The Mitr

In this post, we will see the book Linear Differential Operators: Part 1 – Elementary Theory of Linear Differential Operators by M. A. Naimark.

About the book

In the present book, the theory of linear, ordinary, differential operators of arbitrary order is developed. Extensive use is made of the ideas and theorems of functional analysis, especially those of the theory of linear operators in Hilbert space. For the reader’s convenience all the necessary theorems in functional analysis are developed in the text itself, to make the book, as the author hopes, self-contained and accessible to a wide range of readers-. Many topics in the theory oflinear differential operators can also be discussed without using functional analysis, as Titchmarsh’s treatment [112a] shows. However, the author does not consider such a treatment to be appropriate, since it is only by using the ideas and methods of functional analysis that a deeper understanding of the theory, and its most general results, can be achieved.
The book is in two parts. In the first part, which may be described as an elementary theory of differential operators, the use of the methods of functional analysis is kept to a minimum. This part presents the theory of differential operators defined on a finite interval, including the case of differential opera­ tors which are not self-adjoint, the theory being developed on the assumption that the coefficients of the operators are sufficiently smooth, i.e., sufficiently differentiable. To understand Part I the reader needs only an elementary know­ ledge of the theory of ordinary differential and integral equations and of the theory of functions.
Part II develops the theory of differential operators, using Hilbert space methods. Here, in addition to the subjects already mentioned, the reader will be expected to have some knowledge of the basic results in the theory of the Lebesgue integral.

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The Theory Of Electron Atom Collisions by G. F. Drukarev

Posted on June 3, 2024 by The Mitr

In this post, we will see the book The Theory Of Electron Atom Collisions by G. F. Drukarev.

About the book

The present book is largely devoted to recent developments in the theory of electron-atom collisions. It expounds both the general theory and its applications to specific problems. Chapters 1 and 2 are devoted to central force problems. The general theory of electron-atom collisions is discussed in Chapter 3 in the simplest case, i.e. the case of the hydrogen atom. Equations describing the collision of an electron with an arbitrary atom or ion are considered in Chapter 8. The remaining chapters are concerned with concrete calculations.
“The Theory of Electron-Atom Collisions” should find a place not only with the professional quantum theorists but with those physicists who require merely to understand the effects produced by elastic and inelastic collisions, without personally manipulating the mathematics. No longer is it possible for the more classically inclined ionized gas physicist to aver that the quantum theory methods only nibble at the collision problem. Consider the achieve­ments as of 1964: the ability to predict elastic scattering and excitation functions for simple atoms, with pointers in the direction of success for many-electron atoms; the prediction of the existence of virtual and real energy levels, and the scattering effects they produce; the establishment of upper limits with confidence which is far greater than that usually applied to theoretical cross-sections; the prediction of scattering cross-sections for electrons with ions and with excited atoms, collisions which have hardly yet been attempted in experiment. Only in ionizing collisions is the theoretical position rather less satisfactory.

Translated from the Russian by S. Chomet
Translation edited by J. B. Hasted

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