## Similar Ebooks

Although addressed primarily to the researcher, parts of this monograph can be used as a basis for a well-balanced, one-semester graduate course.

This book aims to strengthen the foundations in its presentation of mathematical approaches to neural networks. It is through these that a suitable explanatory framework is expected to be found. The approaches span a broad range, from single neuron details to numerical analysis, functional analysis and dynamical systems theory. Each of these avenues provides its own insights into the way neural networks can be understood, both for artificial ones and simplified simulations. As a whole, the publication underlines the importance of the ever-deepening mathematical understanding of neural networks.

This book contains a new theory developed by the authors to deal with problems occurring in diffentiable dynamics that are within the scope of general topology. To follow it, the book provides an adequate foundation for topological theory of dynamical systems, and contains tools which are sufficiently powerful throughout the book.

Graduate students (and some undergraduates) with sufficient knowledge of basic general topology, basic topological dynamics, and basic algebraic topology will find little difficulty in reading this book.

This practical, friendly guide focuses on critical conceptstaught in a typical geometry course, from the properties oftriangles, parallelograms, circles, and cylinders, to the skillsand strategies you need to write geometry proofs. GeometryEssentials For Dummies is perfect for cramming or doing homework,or as a reference for parents helping kids study for exams.

Get down to the basics — get a handle on the basics ofgeometry, from lines, segments, and angles, to vertices, altitudes,and diagonals

Conquer proofs with confidence — follow easy-to-graspinstructions for understanding the components of a formal geometryproof

Take triangles in strides — learn how to take in atriangle's sides, analyze its angles, work through an SAS proof,and apply the Pythagorean Theorem

Polish up on polygons — get the lowdown on quadrilateralsand other polygons: their angles, areas, properties, perimeters,and much more

Open the book and find:

Plain-English explanations of geometry terms

Tips for tackling geometry proofs

The seven members of the quadrilateral family

Straight talk on circles

Essential triangle formulas

The lowdown on 3-D: spheres, cylinders, prisms, and pyramids

Ten things to use as reasons in geometry proofs

Learn to:

Core concepts about the geometry of shapes and geometryproofs

Critical theorems, postulates, and definitions

The principles and formulas you need to know

Scientists involved in discrete mathematics, combinatorics, computer science, information theory, geometry, algebra or number theory will find the book of particular significance. It is designed both as an introductory textbook for the beginner and as a reference book for the expert mathematician and engineer.

A number of unsolved problems suitable for research projects are also discussed.

This book focuses on keystream sequences which can be analysed using number theory. It turns out that a great deal of information can be deducted about the cryptographic properties of many classes of sequences by applying the terminology and theorems of number theory. These connections can be explicitly made by describing three kinds of bridges between stream ciphering problems and number theory problems. A detailed summary of these ideas is given in the introductory Chapter 1.

Many results in the book are new, and over seventy percent of these results described in this book are based on recent research results.

The algorithms form a new calculus which allows to make local and asymptotical analysis of solutions to those systems.

The efficiency of the calculus is demonstrated with regard to several complicated problems from Robotics, Celestial Mechanics, Hydrodynamics and Thermodynamics. The calculus also gives classical results obtained earlier intuitively and is an alternative to Algebraic Geometry, Differential Algebra, Lie group Analysis and Nonstandard Analysis.

The book offers a first complete treatment of the mathematical theory of codes on Euclidean spheres. Many new results are published here for the first time. Engineering applications are emphasized throughout the text. The theory is illustrated by many examples. The book also contains an extensive table of best known spherical codes in dimensions 3-24, including exact constructions.

About the Book

Established as a successful practical workbook series with more than 20 titles in the language learning category, Practice Makes Perfect now provides the same clear, concise approach and extensive exercises to key fields within mathematics. The key to the Practice Makes Perfect series is the extensive exercises that provide learners with all the practice they need for mastery.

Techniques from general topology, infinite-dimensional topology, functional analysis and descriptive set theory

are primarily used for the study of these spaces. The mix of

methods from several disciplines makes the subject

particularly interesting. Among other things, a complete and self-contained proof of the Dobrowolski-Marciszewski-Mogilski Theorem that all function spaces of low Borel complexity are topologically homeomorphic, is presented.

In order to understand what is going on, a solid background in

infinite-dimensional topology is needed. And for that a fair amount of knowledge of dimension theory as well as ANR theory is needed. The necessary material was partially covered in our previous book `Infinite-dimensional topology, prerequisites and introduction'. A selection of what was done there can be found here as well, but completely revised and at many places expanded with recent results. A `scenic' route has been chosen towards the

Dobrowolski-Marciszewski-Mogilski Theorem, linking the

results needed for its proof to interesting recent research developments in dimension theory and infinite-dimensional topology.

The first five chapters of this book are intended as a text for

graduate courses in topology. For a course in dimension theory, Chapters 2 and 3 and part of Chapter 1 should be covered. For a course in infinite-dimensional topology, Chapters 1, 4 and 5. In Chapter 6, which deals with function spaces, recent research results are discussed. It could also be used for a graduate course in topology but its flavor is more that of a research monograph than of a textbook; it is therefore

more suitable as a text for a research seminar. The book

consequently has the character of both textbook and a research monograph. In Chapters 1 through 5, unless stated

otherwise, all spaces under discussion are separable and

metrizable. In Chapter 6 results for more general classes of spaces are presented.

In Appendix A for easy reference and some basic facts that are important in the book have been collected. The book is not intended as a basis for a course in topology; its purpose is to collect knowledge about general topology.

The exercises in the book serve three purposes: 1) to test the reader's understanding of the material 2) to supply proofs of statements that are used in the text, but are not proven there

3) to provide additional information not covered by the text.

Solutions to selected exercises have been included in Appendix B.

These exercises are important or difficult.

* It contains a variety of inequalities which find numerous applications in various branches of mathematics.

* It contains many inequalities which have only recently appeared in the literature and cannot yet be found in other books.

* It will be a valuable reference for someone requiring a result about inequalities for use in some applications in various other branches of mathematics.

* Each chapter ends with some miscellaneous inequalities for futher study.

* The work will be of interest to researchers working both in pure and applied mathematics, and it could also be used as the text for an advanced graduate course.

This major survey of mathematics, featuring the work of 18 outstanding Russian mathematicians and including material on both elementary and advanced levels, encompasses 20 prime subject areas in mathematics in terms of their simple origins and their subsequent sophisticated developement. As Professor Morris Kline of New York University noted, "This unique work presents the amazing panorama of mathematics proper. It is the best answer in print to what mathematics contains both on the elementary and advanced levels."

Beginning with an overview and analysis of mathematics, the first of three major divisions of the book progresses to an exploration of analytic geometry, algebra, and ordinary differential equations. The second part introduces partial differential equations, along with theories of curves and surfaces, the calculus of variations, and functions of a complex variable. It furthur examines prime numbers, the theory of probability, approximations, and the role of computers in mathematics. The theory of functions of a real variable opens the final section, followed by discussions of linear algebra and nonEuclidian geometry, topology, functional analysis, and groups and other algebraic systems.

Thorough, coherent explanations of each topic are further augumented by numerous illustrative figures, and every chapter concludes with a suggested reading list. Formerly issued as a three-volume set, this mathematical masterpiece is now available in a convenient and modestly priced one-volume edition, perfect for study or reference.

"This is a masterful English translation of a stupendous and formidable mathematical masterpiece . . ." — Social Science

Key features

- Theory of connections of vectors and directions on the unitary tangent fibre bundle.

- Complete list of Bianchi identities for a regular conection of directions.

- Geometry of generalized Einstein manifolds.

- Classification of Finslerian manifolds.

- Affine, isometric, conformal and projective vector fields on the unitary tangent fibre bundle.Theory of connections of vectors and directions on the unitary tangent fibre bundle.Complete list of Bianchi identities for a regular conection of directions.Geometry of generalized Einstein manifolds.Classification of Finslerian manifolds.Affine, isometric, conformal and projective vector fields on the unitary tangent fibre bundle.

Hollywood actress and math whiz Danica McKellar has completely shattered the “math nerd” stereotype. For years, she’s been showing girls how to feel confident and ace their math classes – with style! With Girls Get Curves, she applies her winning techniques to high school geometry, giving readers the tools they need to feel great and totally “get” everything from congruent triangles to theorems, and more. Inside you’ll find:

· Time-saving tips and tricks for homework and tests

· Illuminating practice problems (and proofs!) with detailed solutions

· Totally relatable real-world examples

· True stories from Danica’s own life as an actress and math student

· A Troubleshooting Guide, for getting unstuck during even the trickiest proofs!

With Danica as a coach, girls everywhere can stop hiding from their homework and watch their scores rise!

Key features:

* New the Hardy – Friedrichs – Wirtinger type inequalities as well as new integral inequalities related to the Cauchy problem for a differential equation.

* Precise exponents of the solution decreasing rate near boundary singular points and best possible conditions for this.

* The question about the influence of the coefficients smoothness on the regularity of solutions.

* New existence theorems for the Dirichlet problem for linear and quasilinear equations in domains with conical points.

* The precise power modulus of continuity at singular boundary point for solutions of the Dirichlet, mixed and the Robin problems.

* The behaviour of weak solutions near conical point for the Dirichlet problem for m – Laplacian.

* The behaviour of weak solutions near a boundary edge for the Dirichlet and mixed problem for elliptic quasilinear equations with triple degeneration.

* Precise exponents of the solution decreasing rate near boundary singular points and best possible conditions for this.

* The question about the influence of the coefficients smoothness on the regularity of solutions.

* New existence theorems for the Dirichlet problem for linear and quasilinear equations in domains with conical points.

* The precise power modulus of continuity at singular boundary point for solutions of the Dirichlet, mixed and the Robin problems.

* The behaviour of weak solutions near conical point for the Dirichlet problem for m - Laplacian.

* The behaviour of weak solutions near a boundary edge for the Dirichlet and mixed problem for elliptic quasilinear equations with triple degeneration.

An Introduction to Numerical Methods and Analysis, SecondEdition reflects the latest trends in the field, includesnew material and revised exercises, and offers a unique emphasis onapplications. The author clearly explains how to both construct andevaluate approximations for accuracy and performance, which are keyskills in a variety of fields. A wide range of higher-level methodsand solutions, including new topics such as the roots ofpolynomials, spectral collocation, finite element ideas, andClenshaw-Curtis quadrature, are presented from an introductoryperspective, and theSecond Edition also features:ulstyle="line-height: 25px; margin-left: 15px; margin-top: 0px; font-family: Arial; font-size: 13px;"Chapters and sections that begin with basic, elementarymaterial followed by gradual coverage of more advancedmaterialExercises ranging from simple hand computations to challengingderivations and minor proofs to programming exercisesWidespread exposure and utilization of MATLAB®An appendix that contains proofs of various theorems and othermaterial

The Appendix sketches some of the most essential new results, constructions and ideas, including the solutions of the Luroth and Zariski problems, the theory of the descent and obstructions to the Hasse principle on rational varieties, and recent applications of K-theory to arithmetic.

Based on Mlodinow's extensive historical research; his studies alongside colleagues such as Richard Feynman and Kip Thorne; and interviews with leading physicists and mathematicians such as Murray Gell-Mann, Edward Witten, and Brian Greene, Euclid's Window is an extraordinary blend of rigorous, authoritative investigation and accessible, good-humored storytelling that makes a stunningly original argument asserting the primacy of geometry. For those who have looked through Euclid's Window, no space, no thing, and no time will ever be quite the same.

The book will be invaluable for researchers and PhD students in differential geometry, global analysis, differential equations on manifolds, and mathematical physics, and for the readers who wish to undertake further rigorous study in this broad interdisciplinary field.

Featured topics

- Analysis on manifolds

- Differential forms on jet spaces

- Global variational functionals

- Euler-Lagrange mapping

- Helmholtz form and the inverse problem

- Symmetries and the Noether’s theory of conservation laws

- Regularity and the Hamilton theory

- Variational sequences

- Differential invariants and natural variational principles

- First book on the geometric foundations of Lagrange structures

- New ideas on global variational functionals

- Complete proofs of all theorems

- Exact treatment of variational principles in field theory, inc. general relativity

- Basic structures and tools: global analysis, smooth manifolds, fibred spaces

The second part of the book is at a more advanced level and includes a treatment of random walks on general locally compact abelian groups. Further chapters develop renewal theory, an introduction to Martin boundary and the study of chains recurrent in the Harris sense. Finally, the last chapter deals with the construction of chains starting from a kernel satisfying some kind of maximum principle.

Can geometry be exciting?

It can for those who can appreciate the beauty of numbers and their relationships.

This book contains a meticulous geometric investigation of the 5 Platonic Solids and 5 other important polyhedra, as well as reference charts for each solid.

Poly (many) hedron (face) means "many faces." Polyhedra are 3 dimensional figures with 4 or more faces, or sides. These polyhedra are reflections of Nature herself, and a study of them provides insight into the way the world is structured. Nature is not only beautiful, but highly intelligent. As you explore the polyhedra in this book, this will become apparent over and over again.

The book contains a geometric explanation of the Phi Ratio and Fibonacci series, and a detailed analysis of the pentagon, which forms the basis for many of these solids. The pentagon is composed entirely of Phi relationships, and is integral to a proper understanding of sacred geometry.

With over 140 full-color illustrations, this book is perfect for teachers and students of geometry alike. It is a must for those who are serious about sacred geometry.

This book is dedicated to those who can appreciate the logic of numbers and the beauty of nature, for they are both aspects of the same unifying principle.

Prerequisites: knowledge of simple algebra and elementary trigonometry. No brainiac math skills required! Only the ability to appreciate nature's own logic.

Book #1 in the Geometric Explorations Series

From Loving Healing Press (www.LovingHealing.com)

The publication first deals with arithmetical semigroups, arithmetical functions, and enumeration problems. Discussions focus on special functions and additive arithmetical semigroups, enumeration and zeta functions in special cases, infinite sums and products, double series and products, integral domains and arithmetical semigroups, and categories satisfying theorems of the Krull-Schmidt type. The text then ponders on semigroups satisfying Axiom A, asymptotic enumeration and "statistical" properties of arithmetical functions, and abstract prime number theorem. Topics include asymptotic properties of prime-divisor functions, maximum and minimum orders of magnitude of certain functions, asymptotic enumeration in certain categories, distribution functions of prime-independent functions, and approximate average values of special arithmetical functions.

The manuscript takes a look at arithmetical formations, additive arithmetical semigroups, and Fourier analysis of arithmetical functions, including Fourier theory of almost even functions, additive abstract prime number theorem, asymptotic average values and densities, and average values of arithmetical functions over a class.

The book is a vital reference for researchers interested in the abstract analytic number theory.

The first chapters of the book focus on the basic concepts and facts of analytic geometry, the theory of space curves, and the foundations of the theory of surfaces, including problems closely related to the first and second fundamental forms. The treatment of the theory of surfaces makes full use of the tensor calculus.

The later chapters address geodesics, mappings of surfaces, special surfaces, and the absolute differential calculus and the displacement of Levi-Cività. Problems at the end of each section (with solutions at the end of the book) will help students meaningfully review the material presented, and familiarize themselves with the manner of reasoning in differential geometry.

The book will be invaluable for researchers and PhD students in differential geometry, global analysis, differential equations on manifolds, and mathematical physics, and for the readers who wish to undertake further rigorous study in this broad interdisciplinary field.

Featured topics

- Analysis on manifolds

- Differential forms on jet spaces

- Global variational functionals

- Euler-Lagrange mapping

- Helmholtz form and the inverse problem

- Symmetries and the Noether’s theory of conservation laws

- Regularity and the Hamilton theory

- Variational sequences

- Differential invariants and natural variational principles

- First book on the geometric foundations of Lagrange structures

- New ideas on global variational functionals

- Complete proofs of all theorems

- Exact treatment of variational principles in field theory, inc. general relativity

- Basic structures and tools: global analysis, smooth manifolds, fibred spaces

In addition to providing a historical perspective on plane geometry, this text covers non-Euclidean geometries, allowing students to cultivate an appreciation of axiomatic systems. Additional topics include circles and regular polygons, projective geometry, symmetries, inversions, knots and links, graphs, surfaces, and informal topology. This republication of a popular text is substantially less expensive than prior editions and offers a new Preface by the author.

The publication first ponders on the foundations of shape theory and shape invariants. Discussions focus on the stability and movability of spaces, homotopy and homology pro-groups, shape dimension, inverse limits and shape of compacta, topological shape, and absolute neighborhood retracts. The text then takes a look at a survey of selected topics, including basic topological constructions and shape, shape dimension of metric compacta, complement theorems of shape theory, shape fibrations, and cell-like maps.

The text ponders on polyhedra and Borsuk's approach to shape. Topics include shape category of metric compacta and metric pairs, homotopy type of polyhedra, and topology of simplicial complexes.

The publication is a valuable source of data for researchers interested in the inverse system approach.

The principal aim of analysis of tensors is to investigate those relations which remain valid when we change from one coordinate system to another. This book on Tensors requires only a knowledge of elementary calculus, differential equations and classical mechanics as pre-requisites. It provides the readers with all the information about the tensors along with the derivation of all the tensorial relations/equations in a simple manner. The book also deals in detail with topics of importance to the study of special and general relativity and the geometry of differentiable manifolds with a crystal clear exposition. The concepts dealt within the book are well supported by a number of solved examples. A carefully selected set of unsolved problems is also given at the end of each chapter, and the answers and hints for the solution of these problems are given at the end of the book. The applications of tensors to the fields of differential geometry, relativity, cosmology and electromagnetism is another attraction of the present book.

This book is intended to serve as text for postgraduate students of mathematics, physics and engineering. It is ideally suited for both students and teachers who are engaged in research in General Theory of Relativity and Differential Geometry.

The variety of topics covered here includes divisibility, diophantine equations, prime numbers (especially Mersenne and Fermat primes), the basic arithmetic functions, congruences, the quadratic reciprocity law, expansion of real numbers into decimal fractions, decomposition of integers into sums of powers, some other problems of the additive theory of numbers and the theory of Gaussian integers.

The reader will find many new topics in chapters IV-VIII, e.g. theory of Wallmann-Shanin's compactification, realcompact space, various generalizations of paracompactness, generalized metric spaces, Dugundji type extension theory, linearly ordered topological space, theory of cardinal functions, dyadic space, etc., that were, in the author's opinion, mostly special or isolated topics some twenty years ago but now settle down into the mainstream of general topology.

1,001 Geometry Practice Problems For Dummies gives you 1,001 opportunities to practice solving problems that deal with core geometry topics, such as points, lines, angles, and planes, as well as area and volume of shapes. You'll also find practice problems on more advanced topics, such as proofs, theorems, and postulates. The companion website gives you free online access to 500 practice problems and solutions. You can track your progress and ID where you should focus your study time. The online component works in conjunction with the book to help you polish your skills and build confidence.

As the perfect companion to Geometry For Dummies or a stand-alone practice tool for students, this book & website will help you put your geometry skills into practice, encouraging deeper understanding and retention. The companion website includes:

Hundreds of practice problemsCustomizable practice sets for self-directed studyProblems ranked as easy, medium, and hardFree one-year access to the online questions bankWith 1,001 Geometry Practice Problems For Dummies, you'll get the practice you need to master geometry and gain confidence in the classroom.

The next chapter is an introduction to real and complex manifolds. It contains an exposition of the theorem of Frobenius, the lemmata of Poincaré and Grothendieck with applications of Grothendieck's lemma to complex analysis, the imbedding theorem of Whitney and Thom's transversality theorem.

Chapter 3 includes characterizations of linear differentiable operators, due to Peetre and Hormander. The inequalities of Garding and of Friedrichs on elliptic operators are proved and are used to prove the regularity of weak solutions of elliptic equations. The chapter ends with the approximation theorem of Malgrange-Lax and its application to the proof of the Runge theorem on open Riemann surfaces due to Behnke and Stein.

The last part of the book is devoted to a mathematical interpretation of the main features of Dirac's formalism. It involves a pairing between distributional bras and kets, continuum expansions and continuum matrices.

This second edition was prepared jointly by P.M. Gruber and the author of the first edition. The authors have retained the existing text (with minor corrections) while adding to each chapter supplementary sections on the more recent developments. While this method may have drawbacks, it has the definite advantage of showing clearly where recent progress has taken place and in what areas interesting results may be expected in the future.

The book gathers results concerning linear operators defined in general spaces of a certain kind, principally in Banach spaces, examples of which are: the space of continuous functions, that of the pth-power-summable functions, Hilbert space, etc. The general theorems are interpreted in various mathematical areas, such as group theory, differential equations, integral equations, equations with infinitely many unknowns, functions of a real variable, summation methods and orthogonal series.

A new fifty-page section (``Some Aspects of the Present Theory of Banach Spaces'') complements this important monograph.

- Basics of Geometry

- Reasoning and Proof

- Perpendicular and Parallel Lines

- Congruent Triangles

- Properties of Triangles

- Quadrilaterals

- Transformations

- Similarity

- Right Triangles and Trigonometry

- Circles

- Area of Polygons and Circles

- Surface Area and Volume

The first part contains a detailed treatment of the fundamentals of Geometric Logic, which combines four central ideas: natural transformations, sheaves, adjoint functors, and topoi.

A special feature of the work is a general calculus of relations presented in the second part. This calculus offers another, often more amenable framework for concepts and methods discussed in part one. Some aspects of this approach find their origin in the relational calculi of Peirce and Schroeder from the last century, and in the 1940's in the work of Tarski and others on relational algebras. The representation theorems discussed are an original feature of this approach.

The papers cover basic fields such as metrization and extension of maps, as well as newly-developed fields like categorical topology and topological dynamics. Each chapter may be read independently of the others, with a few exceptions. It is assumed that the reader has some knowledge of set theory, algebra, analysis and basic general topology.