A thorough analysis of the fundamentals of plane geometry The reader is provided with an abundance of geometrical facts such as the classical results of plane Euclidean and non-Euclidean geometry, congruence theorems, concurrence theorems, classification of isometries, angle addition, trigonometrical formulas, etc.

The aim of Summable Spaces and Their Duals, Matrix Transformations and Geometric Properties is to discuss primarily about different kinds of summable spaces, compute their duals and then characterize several matrix classes transforming one summable space into other. The book also discusses several geometric properties of summable spaces, as well as dealing with the construction of summable spaces using Orlicz functions, and explores several structural properties of such spaces. Each chapter contains a conclusion section highlighting the importance of results, and points the reader in the direction of possible new ideas for further study. Features Suitable for graduate schools, graduate students, researchers and faculty, and could be used as a key text for special Analysis seminars Investigates different types of summable spaces and computes their duals Characterizes several matrix classes transforming one summable space into other Discusses several geometric properties of summable spaces Examines several possible generalizations of Orlicz sequence spaces

The series is aimed specifically at publishing peer reviewed reviews and contributions presented at workshops and conferences. Each volume is associated with a particular conference, symposium or workshop. These events cover various topics within pure and applied mathematics and provide up-to-date coverage of new developments, methods and applications.

These 25 papers from a conference held in August 1998 at Pusan National U. provide a broad overview of contemporary group theory, with a particular emphasis on geometric and topological methods. Topics covered include: deformations and rigidity, combinatorial group theory and wild metric complexes, generalized triangle groups, HNN extensions, Eilenberg-Ganea Conjecture, cyclically presented groups, Takahashi manifolds, wreath products, reduction formulae, group actions on graphs and designs, Grushko-Neumann theorem, and variations on a theme of Higman and Conder. Includes a list of the authors and participants with contact information. Conference sponsors included the Korea Science and Engineering Foundation and International Mathematical Union Commission on Development and Exchange. Lacks an index.

The volume consists of invited refereed research papers. The contributions cover a wide spectrum in algebraic geometry, from motives theory to numerical algebraic geometry and are mainly focused on higher dimensional varieties and Minimal Model Program and surfaces of general type. A part of the articles grew out a Conference in memory of Paolo Francia (1951-2000) held in Genova in September 2001 with about 70 participants.

The aim of the Expositions is to present new and important developments in pure and applied mathematics. Well established in the community over more than two decades, the series offers a large library of mathematical works, including several important classics. The volumes supply thorough and detailed expositions of the methods and ideas essential to the topics in question. In addition, they convey their relationships to other parts of mathematics. The series is addressed to advanced readers interested in a thorough study of the subject. Editorial Board Lev Birbrair, Universidade Federal do Ceara, Fortaleza, Brasil Walter D. Neumann, Columbia University, New York, USA Markus J. Pflaum, University of Colorado, Boulder, USA Dierk Schleicher, Jacobs University, Bremen, Germany Katrin Wendland, University of Freiburg, Germany Honorary Editor Victor P. Maslov, Russian Academy of Sciences, Moscow, Russia Titles in planning include Yuri A. Bahturin, Identical Relations in Lie Algebras (2019) Yakov G. Berkovich, Lev G. Kazarin, and Emmanuel M. Zhmud', Characters of Finite Groups, Volume 2 (2019) Jorge Herbert Soares de Lira, Variational Problems for Hypersurfaces in Riemannian Manifolds (2019) Volker Mayer, Mariusz Urbanski, and Anna Zdunik, Random and Conformal Dynamical Systems (2021) Ioannis Diamantis, Bostjan Gabrovsek, Sofia Lambropoulou, and Maciej Mroczkowski, Knot Theory of Lens Spaces (2021)

Geometry is both elegantly simple and infinitely profound. Many professionals find they need to be able to draw geometric shapes accurately, and this unique book shows them how. It provides step-by-step instructions for constructing two-dimensional geometric shapes, which can be readily followed by a beginner, or used as an invaluable source book by students and professionals.

The aim of this monograph is to give an overview of various classes of in?ni- dimensional Lie groups and their applications, mostly in Hamiltonian - chanics, ?uid dynamics, integrable systems, and complex geometry. We have chosen to present the unifying ideas of the theory by concentrating on speci?c typesandexamplesofin?nite-dimensionalLiegroups. Ofcourse, theselection of the topics is largely in?uenced by the taste of the authors, but we hope thatthisselectioniswideenoughtodescribevariousphenomenaarisinginthe geometry of in?nite-dimensional Lie groups and to convince the reader that they are appealing objects to study from both purely mathematical and more applied points of view. This book can be thought of as complementary to the existing more algebraic treatments, in particular, those covering the str- ture and representation theory of in?nite-dimensional Lie algebras, as well as to more analytic ones developing calculus on in?nite-dimensional manifolds. This monograph originated from advanced graduate courses and mi- courses on in?nite-dimensional groups and gauge theory given by the ?rst author at the University of Toronto, at the CIRM in Marseille, and at the Ecole Polytechnique in Paris in 2001-2004. It is based on various classical and recentresultsthathaveshapedthisnewlyemergedpartofin?nite-dimensional geometry and group theory. Our intention was to make the book concise, relatively self-contained, and useful in a graduate course. For this reason, throughout the text, we have included a large number of problems, ranging from simple exercises to open questions

This volume contains original research articles, survey articles and lecture notes related to the Computations with Modular Forms 2011 Summer School and Conference, held at the University of Heidelberg. A key theme of the Conference and Summer School was the interplay between theory, algorithms and experiment. The 14 papers offer readers both, instructional courses on the latest algorithms for computing modular and automorphic forms, as well as original research articles reporting on the latest developments in the field. The three Summer School lectures provide an introduction to modern algorithms together with some theoretical background for computations of and with modular forms, including computing cohomology of arithmetic groups, algebraic automorphic forms, and overconvergent modular symbols. The 11 Conference papers cover a wide range of themes related to computations with modular forms, including lattice methods for algebraic modular forms on classical groups, a generalization of the Maeda conjecture, an efficient algorithm for special values of p-adic Rankin triple product L-functions, arithmetic aspects and experimental data of Bianchi groups, a theoretical study of the real Jacobian of modular curves, results on computing weight one modular forms, and more.

This book is a comprehensive collection of known results about the Lozi map, a piecewise-affine version of the Henon map. Henon map is one of the most studied examples in dynamical systems and it attracts a lot of attention from researchers, however it is difficult to analyze analytically. Simpler structure of the Lozi map makes it more suitable for such analysis. The book is not only a good introduction to the Lozi map and its generalizations, it also summarizes of important concepts in dynamical systems theory such as hyperbolicity, SRB measures, attractor types, and more.

Appliies variational methods and critical point theory on infinite dimenstional manifolds to some problems in Lorentzian geometry which have a variational nature, such as existence and multiplicity results on geodesics and relations between such geodesics and the topology of the manifold.

This introductory textbook describes fundamental groups and their topological soul mates, the covering spaces. The author provides several illustrative examples that touch upon different areas of mathematics, but in keeping with the books introductory aim, they are all quite elementary. Basic concepts are clearly defined, proofs are complete, and no results from the exercises are assumed in the text.

Written by researchers who have helped found and shape the field, this book is a definitive introduction to geometric modeling. The authors present all of the necessary techniques for curve and surface representations in computer-aided modeling with a focus on how the techniques are used in design. They achieve a balance between mathematical rigor and broad applicability. Appropriate for readers with a moderate degree of mathematical maturity, this book is suitable as an undergraduate or graduate text, or particularly as a resource for self-study.

This book is a comprehensive tool both for self-study and for use as a text in classical geometry. It explains the concepts that form the basis for computer-aided geometric design.

This research monograph in the field of algebraic topology contains many thought-provoking discussions of open problems and promising research directions.

Investigations by Baire, Lebesgue, Hausdorff, Marczewski, and othes have culminated invarious schemes for classifying point sets. This important reference/text bringstogether in a single theoretical framework the properties common to these classifications.Providing a clear, thorough overview and analysis of the field, Point Set Theoryutilizes the axiomatically determined notion of a category base for extending generaltopological theorems to a higher level of abstraction ... axiomatically unifies analogiesbetween Baire category and Lebesgue measure . .. enhances understanding of thematerial with numerous examples and discussions of abstract concepts ... and more.Imparting a solid foundation for the modem theory of real functions and associated areas,this authoritative resource is a vital reference for set theorists, logicians, analysts, andresearch mathematicians involved in topology, measure theory, or real analysis. It is anideal text for graduate mathematics students in the above disciplines who havecompleted undergraduate courses in set theory and real analysis.

This book proposes a new approach which is designed to serve as an introductory course in differential geometry for advanced undergraduate students. It is based on lectures given by the author at several universities, and discusses calculus, topology, and linear algebra.

Prevalent in animation movies and interactive games, subdivision methods allow users to design and implement simple but efficient schemes for rendering curves and surfaces. Adding to the current subdivision toolbox, Wavelet Subdivision Methods: GEMS for Rendering Curves and Surfaces introduces geometry editing and manipulation schemes (GEMS) and covers both subdivision and wavelet analysis for generating and editing parametric curves and surfaces of desirable geometric shapes. The authors develop a complete constructive theory and effective algorithms to derive synthesis wavelets with minimum support and any desirable order of vanishing moments, along with decomposition filters. Through numerous examples, the book shows how to represent curves and construct convergent subdivision schemes. It comprehensively details subdivision schemes for parametric curve rendering, offering complete algorithms for implementation and theoretical development as well as detailed examples of the most commonly used schemes for rendering both open and closed curves. It also develops an existence and regularity theory for the interpolatory scaling function and extends cardinal B-splines to box splines for surface subdivision. Keeping mathematical derivations at an elementary level without sacrificing mathematical rigor, this book shows how to apply bottom-up wavelet algorithms to curve and surface editing. It offers an accessible approach to subdivision methods that integrates the techniques and algorithms of bottom-up wavelets.

The literature on the spectral analysis of second order elliptic differential operators contains a great deal of information on the spectral functions for explicitly known spectra. The same is not true, however, for situations where the spectra are not explicitly known. Over the last several years, the author and his colleagues have developed new, innovative methods for the exact analysis of a variety of spectral functions occurring in spectral geometry and under external conditions in statistical mechanics and quantum field theory. Spectral Functions in Mathematics and Physics presents a detailed overview of these advances. The author develops and applies methods for analyzing determinants arising when the external conditions originate from the Casimir effect, dielectric media, scalar backgrounds, and magnetic backgrounds. The zeta function underlies all of these techniques, and the book begins by deriving its basic properties and relations to the spectral functions. The author then uses those relations to develop and apply methods for calculating heat kernel coefficients, functional determinants, and Casimir energies. He also explores applications in the non-relativistic context, in particular applying the techniques to the Bose-Einstein condensation of an ideal Bose gas. Self-contained and clearly written, Spectral Functions in Mathematics and Physics offers a unique opportunity to acquire valuable new techniques, use them in a variety of applications, and be inspired to make further advances.

The series is aimed specifically at publishing peer reviewed reviews and contributions presented at workshops and conferences. Each volume is associated with a particular conference, symposium or workshop. These events cover various topics within pure and applied mathematics and provide up-to-date coverage of new developments, methods and applications.

This textbook provides a thorough introduction to the differential geometry of parametrized curves and surfaces, along with a wealth of applications to specific architectural elements. Geometric elements in architecture respond to practical, physical and aesthetic needs. Proper understanding of the mathematics underlying the geometry provides control over the construction. This book relates the classical mathematical theory of parametrized curves and surfaces to multiple applications in architecture. The presentation is mathematically complete with numerous figures and animations illustrating the theory, and special attention is given to some of the recent trends in the field. Solved exercises are provided to see the theory in practice. Intended as a textbook for lecture courses, Parametric Geometry of Curves and Surfaces is suitable for mathematically-inclined students in engineering, architecture and related fields, and can also serve as a textbook for traditional differential geometry courses to mathematics students. Researchers interested in the mathematics of architecture or computer-aided design will also value its combination of precise mathematics and architectural examples.

The central theme of this book is the theorem of Ambrose and Singer, which gives for a connected, complete and simply connected Riemannian manifold a necessary and sufficient condition for it to be homogeneous. This is a local condition which has to be satisfied at all points, and in this way it is a generalization of E. Cartan's method for symmetric spaces. The main aim of the authors is to use this theorem and representation theory to give a classification of homogeneous Riemannian structures on a manifold. There are eight classes, and some of these are discussed in detail. Using the constructive proof of Ambrose and Singer many examples are discussed with special attention to the natural correspondence between the homogeneous structure and the groups acting transitively and effectively as isometrics on the manifold.