Tag Archives: Geometry

Relativity, Gravitation and Cosmology: A Basic Introduction

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Relativity, Gravitation and Cosmology: A Basic Introduction By Ta-Pei Cheng
2009 | 355 Pages | ISBN: 0199573638 , 0199573646 | PDF | 6 MB

Relativity, Gravitation and Cosmology: A Basic Introduction
Einstein's general theory of relativity is introduced in this advanced undergraduate and beginning graduate level textbook. Topics include special relativity, in the formalism of Minkowski's four-dimensional space-time, the principle of equivalence, Riemannian geometry and tensor analysis, Einstein field equation, as well as many modern cosmological subjects, from primordial inflation and cosmic microwave anisotropy to the dark energy that propels an accelerating universe.
The author presents the subject with an emphasis on physical examples and simple applications without the full tensor apparatus. The reader first learns how to describe curved spacetime. At this mathematically more accessible level, the reader can already study the many interesting phenomena such as gravitational lensing, precession of Mercury's perihelion, black holes, and cosmology. The full tensor formulation is presented later, when the Einstein equation is solved for a few symmetric cases. Many modern topics in cosmology are discussed in this book: from inflation, cosmic microwave anisotropy to the "dark energy" that propels an accelerating universe.
Mathematical accessibility, together with the various pedagogical devices (e.g., worked-out solutions of chapter-end problems), make it practical for interested readers to use the book to study general relativity and cosmology on their own.
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Mean Curvature Flow and Isoperimetric Inequalities

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Manuel Ritoré, Carlo Sinestrari, Vicente Miquel, Joan Porti, "Mean Curvature Flow and Isoperimetric Inequalities"
2010 | pages: 121 | ISBN: 303460212X | PDF | 1,6 mb

Mean Curvature Flow and Isoperimetric Inequalities
Geometric flows have many applications in physics and geometry. The mean curvature flow occurs in the description of the interface evolution in certain physical models. This is related to the property that such a flow is the gradient flow of the area functional and therefore appears naturally in problems where a surface energy is minimized. The mean curvature flow also has many geometric applications, in analogy with the Ricci flow of metrics on abstract riemannian manifolds. One can use this flow as a tool to obtain classification results for surfaces satisfying certain curvature conditions, as well as to construct minimal surfaces. Geometric flows, obtained from solutions of geometric parabolic equations, can be considered as an alternative tool to prove isoperimetric inequalities. On the other hand, isoperimetric inequalities can help in treating several aspects of convergence of these flows. Isoperimetric inequalities have many applications in other fields of geometry, like hyperbolic manifolds.
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Differential Geometry and Topology, Discrete and Computational Geometry

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Differential Geometry and Topology, Discrete and Computational Geometry By M. Boucetta, J.M. Morvan
English | 2005 | 385 Pages | ISBN: 158603507X | DJVU | 3 MB

Differential Geometry and Topology, Discrete and Computational Geometry
The aim of this volume is to give an introduction and overview to differential topology, differential geometry and computational geometry with an emphasis on some interconnections between these three domains of mathematics.

The chapters give the background required to begin research in these fields or at their interfaces. They introduce new research domains and both old and new conjectures in these different subjects show some interaction between other sciences close to mathematics. Topics discussed are; the basis of differential topology and combinatorial topology, the link between differential geometry and topology, Riemanian geometry (Levi-Civita connextion, curvature tensor, geodesic, completeness and curvature tensor), characteristic classes (to associate every fibre bundle with isomorphic fiber bundles), the link between differential geometry and the geometry of non smooth objects, computational geometry and concrete applications such as structural geology and graphism.
IOS Press is an international science, technical and medical publisher of high-quality books for academics, scientists, and professionals in all fields.
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Entropy, Compactness and the Approximation of Operators

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Bernd Carl, "Entropy, Compactness and the Approximation of Operators"
English | ISBN: 0521330114, 0521090946 | 1990 | 287 pages | PDF | 10 MB

Entropy, Compactness and the Approximation of Operators
Entropy quantities are connected with the 'degree of compactness' of compact or precompact spaces, and so are appropriate tools for investigating linear and compact operators between Banach spaces. The main intention of this Tract is to study the relations between compactness and other analytical properties, e.g. approximability and eigenvalue sequences, of such operators. The authors present many generalized results, some of which have not appeared in the literature before. In the final chapter, the authors demonstrate that, to a certain extent, the geometry of Banach spaces can also be developed on the basis of operator theory. All mathematicians working in functional analysis and operator theory will welcome this work as a reference or for advanced graduate courses.
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D-Modules, Perverse Sheaves, and Representation Theory

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Ryoshi Hotta, Kiyoshi Takeuchi, Toshiyuki Tanisaki, "D-Modules, Perverse Sheaves, and Representation Theory"
2007 | pages: 408 | ISBN: 081764363X | PDF | 3,4 mb

D-Modules, Perverse Sheaves, and Representation Theory
D-modules continues to be an active area of stimulating research in such mathematical areas as algebraic, analysis, differential equations, and representation theory. Key to D-modules, Perverse Sheaves, and Representation Theory is the authors' essential algebraic-analytic approach to the theory, which connects D-modules to representation theory and other areas of mathematics.

To further aid the reader, and to make the work as self-contained as possible, appendices are provided as background for the theory of derived categories and algebraic varieties. The book is intended to serve graduate students in a classroom setting and as self-study for researchers in algebraic geometry, representation theory.
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Barycentric Calculus in Euclidean and Hyperbolic Geometry: A Comparative Introduction

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Ungar Abraham Albert, "Barycentric Calculus in Euclidean and Hyperbolic Geometry: A Comparative Introduction"
2010 | ISBN-10: 981430493X | 360 pages | PDF | 3 MB

Barycentric Calculus in Euclidean and Hyperbolic Geometry: A Comparative Introduction
The word barycentric is derived from the Greek word barys (heavy), and refers to center of gravity. Barycentric calculus is a method of treating geometry by considering a point as the center of gravity of certain other points to which weights are ascribed. Hence, in particular, barycentric calculus provides excellent insight into triangle centers. This unique book on barycentric calculus in Euclidean and hyperbolic geometry provides an introduction to the fascinating and beautiful subject of novel triangle centers in hyperbolic geometry along with analogies they share with familiar triangle centers in Euclidean geometry. As such, the book uncovers magnificent unifying notions that Euclidean and hyperbolic triangle centers share.

In his earlier books the author adopted Cartesian coordinates, trigonometry and vector algebra for use in hyperbolic geometry that is fully analogous to the common use of Cartesian coordinates, trigonometry and vector algebra in Euclidean geometry. As a result, powerful tools that are commonly available in Euclidean geometry became available in hyperbolic geometry as well, enabling one to explore hyperbolic geometry in novel ways. In particular, this new book establishes hyperbolic barycentric coordinates that are used to determine various hyperbolic triangle centers just as Euclidean barycentric coordinates are commonly used to determine various Euclidean triangle centers.

The hunt for Euclidean triangle centers is an old tradition in Euclidean geometry, resulting in a repertoire of more than three thousand triangle centers that are known by their barycentric coordinate representations. The aim of this book is to initiate a fully analogous hunt for hyperbolic triangle centers that will broaden the repertoire of hyperbolic triangle centers provided here.
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Who Gave you the Epsilon?: & Other Tales of Mathematical History

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Marlow Anderson, "Who Gave you the Epsilon?: & Other Tales of Mathematical History"
English | ISBN: 0883855690 | 2009 | 433 pages | PDF | 11 MB

Who Gave you the Epsilon?: & Other Tales of Mathematical History
This book picks up the history of mathematics from where Sherlock Holmes in Babylon left it. The forty articles of Who Gave You the Epsilon? continue the story of the development of mathematics into the nineteenth and twentieth centuries.

The articles have all been published in the Mathematical Association of America journals and are in many cases written by distinguished mathematicians such as G. H. Hardy and B. van der Waerden. The articles are arranged thematically to show the development of analysis, geometry, algebra and number theory through this period of time. Each chapter is preceded by a foreword, giving the historical background and setting and the scene, and is followed by an afterword, reporting on advances in our historical knowledge and understanding since the articles first appeared. This book is ideal for anyone wanting to explore the history of mathematics.
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Valuation Theory in Interaction

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Antonio Campillo Lopez, Franz-Viktor Kuhlman, "Valuation Theory in Interaction"
2014 | ISBN-10: 303719149X | 670 pages | PDF | 12 MB

Valuation Theory in Interaction
For more than a century, valuation theory has had its classical roots in algebraic number theory, algebraic geometry and the theory of ordered fields and groups. In recent decades it has seen an amazing expansion into many other areas. Moreover, having been dormant for a while in algebraic geometry, it has now been reintroduced as a tool to attack the open problem of resolution of singularities in positive characteristic and to analyze the structure of singularities. Driven by this topic, and by its many new applications in other areas, the research in valuation theory itself has also been intensified, with a particular emphasis on the deep open problems in positive characteristic. The multifaceted development of valuation theory has been monitored by two International Conferences and Workshops: the first in 1999 in Saskatoon, Canada, and the second in 2011 in Segovia and El Escorial in Spain. This book grew out of the second conference and presents high quality papers on recent research together with survey papers that illustrate the state of the art in several areas and applications of valuation theory. This book is addressed to researchers and graduate students who work in valuation theory or the areas where it is applied, as well as a general mathematical audience interested in the expansion and usefulness of the valuation theoretical approach, which has been called the "most analytic" form of algebraic reasoning. For young mathematicians who want to enter these areas of research, it provides a valuable source of up-to-date information. A publication of the European Mathematical Society (EMS). Distributed within the Americas by the American Mathematical Society.
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Eighth Grade Common Core Math: 16 Days to Mastering

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Eighth Grade Common Core Math: 16 Days to Mastering by John D. Forlini
English | December 9, 2014 | ISBN: 1505452384 | 146 Pages | EPUB/MOBI/AZW3/PDF (Converted) | 4 MB
This book is a Common Core course for students taking 8th grade math. It can be used prior or during the 8th grade to help students excel. It teaches students the material that will be covered in the classrooms that follow Common Core curriculum.

Eighth Grade Common Core Math: 16 Days to Mastering
Besides including a complete Common Core curriculum, there are just the right amount of problems and solutions provided to orient the student for this course. This book is divided into 16 distinct days for learning. A student and teacher can develop their own pace. At the conclusion of the 16th day there is a chapter titled "Beyond Common Core Math (Prep for 9th grade Algebra"). The book is designed for self-teaching by a student or-for a teacher or parent working with the student. The ideal time to read this material would be the summer before taking the actual 8th grade course. However, it can be used any time during the 8th grade. In addition, there is a method revealed in the book that will truly show students how to get the most out of their study time and classroom learning experience towards obtaining an A. It's like getting a 10 minute head start in a 5 k race. This 5 k race is the basis for the rest of a student's life. As mentioned, it also includes a final chapter that is titled "Beyond Common Core". While Common Core does a good job of covering basics, and including introduction to geometry and statistics, it is weak in introducing algebra. Since algebra precedes geometry in high school, this section will introduce the student to algebra with emphasis on word problems. This book is also valuable for preparing for the SAT and ACT tests. It is a solid refresher for these.
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