Topics in Hyperplane Arrangements, Polytopes and Box-Splines

Author: Corrado De Concini,Claudio Procesi

Publisher: Springer Science & Business Media

ISBN: 0387789634

Category: Mathematics

Page: 381

View: 6044

Topics in Hyperplane Arrangements, Polytopes and Box-Splines brings together many areas of research that focus on methods to compute the number of integral points in suitable families or variable polytopes. The topics introduced expand upon differential and difference equations, approximation theory, cohomology, and module theory. This book, written by two distinguished authors, engages a broad audience by proving the a strong foudation. This book may be used in the classroom setting as well as a reference for researchers.

Topics in Hyperplane Arrangements

Author: Marcelo Aguiar,Swapneel Mahajan

Publisher: American Mathematical Soc.

ISBN: 1470437112

Category: Algebraic spaces

Page: 611

View: 5859

This monograph studies the interplay between various algebraic, geometric and combinatorial aspects of real hyperplane arrangements. It provides a careful, organized and unified treatment of several recent developments in the field, and brings forth many new ideas and results. It has two parts, each divided into eight chapters, and five appendices with background material. Part I gives a detailed discussion on faces, flats, chambers, cones, gallery intervals, lunes and other geometric notions associated with arrangements. The Tits monoid plays a central role. Another important object is the category of lunes which generalizes the classical associative operad. Also discussed are the descent and lune identities, distance functions on chambers, and the combinatorics of the braid arrangement and related examples. Part II studies the structure and representation theory of the Tits algebra of an arrangement. It gives a detailed analysis of idempotents and Peirce decompositions, and connects them to the classical theory of Eulerian idempotents. It introduces the space of Lie elements of an arrangement which generalizes the classical Lie operad. This space is the last nonzero power of the radical of the Tits algebra. It is also the socle of the left ideal of chambers and of the right ideal of Zie elements. Zie elements generalize the classical Lie idempotents. They include Dynkin elements associated to generic half-spaces which generalize the classical Dynkin idempotent. Another important object is the lune-incidence algebra which marks the beginning of noncommutative Möbius theory. These ideas are also brought upon the study of the Solomon descent algebra. The monograph is written with clarity and in sufficient detail to make it accessible to graduate students. It can also serve as a useful reference to experts.

Combinatorial Methods in Topology and Algebra

Author: Bruno Benedetti,Emanuele Delucchi,Luca Moci

Publisher: Springer

ISBN: 3319201557

Category: Mathematics

Page: 227

View: 626

Combinatorics plays a prominent role in contemporary mathematics, due to the vibrant development it has experienced in the last two decades and its many interactions with other subjects. This book arises from the INdAM conference "CoMeTA 2013 - Combinatorial Methods in Topology and Algebra,'' which was held in Cortona in September 2013. The event brought together emerging and leading researchers at the crossroads of Combinatorics, Topology and Algebra, with a particular focus on new trends in subjects such as: hyperplane arrangements; discrete geometry and combinatorial topology; polytope theory and triangulations of manifolds; combinatorial algebraic geometry and commutative algebra; algebraic combinatorics; and combinatorial representation theory. The book is divided into two parts. The first expands on the topics discussed at the conference by providing additional background and explanations, while the second presents original contributions on new trends in the topics addressed by the conference.

Configuration Spaces

Geometry, Topology and Representation Theory

Author: Filippo Callegaro,Frederick Cohen,Corrado De Concini,Eva Maria Feichtner,Giovanni Gaiffi,Mario Salvetti

Publisher: Springer

ISBN: 3319315803

Category: Mathematics

Page: 379

View: 3831

This book collects the scientific contributions of a group of leading experts who took part in the INdAM Meeting held in Cortona in September 2014. With combinatorial techniques as the central theme, it focuses on recent developments in configuration spaces from various perspectives. It also discusses their applications in areas ranging from representation theory, toric geometry and geometric group theory to applied algebraic topology.

Lie Groups

An Approach through Invariants and Representations

Author: Claudio Procesi

Publisher: Springer Science & Business Media

ISBN: 0387289291

Category: Mathematics

Page: 600

View: 9733

Lie groups has been an increasing area of focus and rich research since the middle of the 20th century. In Lie Groups: An Approach through Invariants and Representations, the author's masterful approach gives the reader a comprehensive treatment of the classical Lie groups along with an extensive introduction to a wide range of topics associated with Lie groups: symmetric functions, theory of algebraic forms, Lie algebras, tensor algebra and symmetry, semisimple Lie algebras, algebraic groups, group representations, invariants, Hilbert theory, and binary forms with fields ranging from pure algebra to functional analysis. By covering sufficient background material, the book is made accessible to a reader with a relatively modest mathematical background. Historical information, examples, exercises are all woven into the text. This unique exposition is suitable for a broad audience, including advanced undergraduates, graduates, mathematicians in a variety of areas from pure algebra to functional analysis and mathematical physics.

Harmonic Analysis of Operators on Hilbert Space

Author: Béla Sz Nagy,Ciprian Foias,Hari Bercovici,László Kérchy

Publisher: Springer Science & Business Media

ISBN: 1441960937

Category: Mathematics

Page: 478

View: 1877

The existence of unitary dilations makes it possible to study arbitrary contractions on a Hilbert space using the tools of harmonic analysis. The first edition of this book was an account of the progress done in this direction in 1950-70. Since then, this work has influenced many other areas of mathematics, most notably interpolation theory and control theory. This second edition, in addition to revising and amending the original text, focuses on further developments of the theory, including the study of two operator classes: operators whose powers do not converge strongly to zero, and operators whose functional calculus (as introduced in Chapter III) is not injective. For both of these classes, a wealth of material on structure, classification and invariant subspaces is included in Chapters IX and X. Several chapters conclude with a sketch of other developments related with (and developing) the material of the first edition.

An Introduction to Manifolds

Author: Loring W. Tu

Publisher: Springer Science & Business Media

ISBN: 1441974008

Category: Mathematics

Page: 410

View: 991

Manifolds, the higher-dimensional analogs of smooth curves and surfaces, are fundamental objects in modern mathematics. Combining aspects of algebra, topology, and analysis, manifolds have also been applied to classical mechanics, general relativity, and quantum field theory. In this streamlined introduction to the subject, the theory of manifolds is presented with the aim of helping the reader achieve a rapid mastery of the essential topics. By the end of the book the reader should be able to compute, at least for simple spaces, one of the most basic topological invariants of a manifold, its de Rham cohomology. Along the way, the reader acquires the knowledge and skills necessary for further study of geometry and topology. The requisite point-set topology is included in an appendix of twenty pages; other appendices review facts from real analysis and linear algebra. Hints and solutions are provided to many of the exercises and problems. This work may be used as the text for a one-semester graduate or advanced undergraduate course, as well as by students engaged in self-study. Requiring only minimal undergraduate prerequisites, 'Introduction to Manifolds' is also an excellent foundation for Springer's GTM 82, 'Differential Forms in Algebraic Topology'.

Moduli of Double EPW-Sextics

Author: Kieran G. O'Grady

Publisher: American Mathematical Soc.

ISBN: 1470416964

Category: Equations, Sextic

Page: 172

View: 9439

The author studies the GIT quotient of the symplectic grassmannian parametrizing lagrangian subspaces of ⋀3C6 modulo the natural action of SL6, call it M. This is a compactification of the moduli space of smooth double EPW-sextics and hence birational to the moduli space of HK 4-folds of Type K3[2] polarized by a divisor of square 2 for the Beauville-Bogomolov quadratic form. The author will determine the stable points. His work bears a strong analogy with the work of Voisin, Laza and Looijenga on moduli and periods of cubic 4-folds.

Box Splines

Author: Carl de Boor,Klaus Höllig,Sherman Riemenschneider

Publisher: Springer Science & Business Media

ISBN: 1475722443

Category: Mathematics

Page: 201

View: 3400

Compactly supported smooth piecewise polynomial functions provide an efficient tool for the approximation of curves and surfaces and other smooth functions of one and several arguments. Since they are locally polynomial, they are easy to evaluate. Since they are smooth, they can be used when smoothness is required, as in the numerical solution of partial differential equations (in the Finite Element method) or the modeling of smooth sur faces (in Computer Aided Geometric Design). Since they are compactly supported, their linear span has the needed flexibility to approximate at all, and the systems to be solved in the construction of approximations are 'banded'. The construction of compactly supported smooth piecewise polynomials becomes ever more difficult as the dimension, s, of their domain G ~ IRs, i. e. , the number of arguments, increases. In the univariate case, there is only one kind of cell in any useful partition, namely, an interval, and its boundary consists of two separated points, across which polynomial pieces would have to be matched as one constructs a smooth piecewise polynomial function. This can be done easily, with the only limitation that the num ber of smoothness conditions across such a breakpoint should not exceed the polynomial degree (since that would force the two joining polynomial pieces to coincide). In particular, on any partition, there are (nontrivial) compactly supported piecewise polynomials of degree ~ k and in C(k-l), of which the univariate B-spline is the most useful example.

Survival and Event History Analysis

A Process Point of View

Author: Odd Aalen,Ornulf Borgan,Hakon Gjessing

Publisher: Springer Science & Business Media

ISBN: 038768560X

Category: Mathematics

Page: 540

View: 7324

The aim of this book is to bridge the gap between standard textbook models and a range of models where the dynamic structure of the data manifests itself fully. The common denominator of such models is stochastic processes. The authors show how counting processes, martingales, and stochastic integrals fit very nicely with censored data. Beginning with standard analyses such as Kaplan-Meier plots and Cox regression, the presentation progresses to the additive hazard model and recurrent event data. Stochastic processes are also used as natural models for individual frailty; they allow sensible interpretations of a number of surprising artifacts seen in population data. The stochastic process framework is naturally connected to causality. The authors show how dynamic path analyses can incorporate many modern causality ideas in a framework that takes the time aspect seriously. To make the material accessible to the reader, a large number of practical examples, mainly from medicine, are developed in detail. Stochastic processes are introduced in an intuitive and non-technical manner. The book is aimed at investigators who use event history methods and want a better understanding of the statistical concepts. It is suitable as a textbook for graduate courses in statistics and biostatistics.

Formal Power Series and Algebraic Combinatorics

12th International Conference, FPSAC’00, Moscow, Russia, June 2000, Proceedings

Author: Daniel Krob,Alexander A. Mikhalev,Alexander V. Mikhalev

Publisher: Springer Science & Business Media

ISBN: 3662041669

Category: Mathematics

Page: 808

View: 6296

The Riemann-Hilbert Problem

A Publication from the Steklov Institute of Mathematics Adviser: Armen Sergeev

Author: D. V. Anosov,A. A. Bolibruch

Publisher: Springer Science & Business Media

ISBN: 3322929094

Category: Mathematics

Page: 193

View: 8394

The Riemann-Hilbert problem (Hilbert's 21st problem) belongs to the theory of linear systems of ordinary differential equations in the complex domain. The problem concerns the existence of a Fuchsian system with prescribed singularities and monodromy. Hilbert was convinced that such a system always exists. However, this turned out to be a rare case of a wrong forecast made by him. In 1989 the second author (A. B.) discovered a counterexample, thus obtaining a negative solution to Hilbert's 21st problem in its original form.

Configuration Spaces

Geometry, Combinatorics and Topology

Author: Anders Björner,Fred Cohen,Corrado De Concini,Claudo Procesi,Mario Salvetti

Publisher: Springer

ISBN: 8876424318

Category: Mathematics

Page: 536

View: 2851

These proceedings contain the contributions of some of the participants in the "intensive research period" held at the De Giorgi Research Center in Pisa, during the period May-June 2010. The central theme of this research period was the study of configuration spaces from various points of view. This topic originated from the intersection of several classical theories: Braid groups and related topics, configurations of vectors (of great importance in Lie theory and representation theory), arrangements of hyperplanes and of subspaces, combinatorics, singularity theory. Recently, however, configuration spaces have acquired independent interest and indeed the contributions in this volume go far beyond the above subjects, making it attractive to a large audience of mathematicians.

Combinatorial and Computational Geometry

Author: Jacob E. Goodman,Janos Pach,Emo Welzl

Publisher: Cambridge University Press

ISBN: 9780521848626

Category: Computers

Page: 616

View: 1707

This 2005 book deals with interest topics in Discrete and Algorithmic aspects of Geometry.

Design for a Vulnerable Planet

Author: Frederick Steiner

Publisher: University of Texas Press

ISBN: 0292723857

Category: Architecture

Page: 288

View: 6061

We inhabit a vulnerable planet. The devastation caused by natural disasters such as the southern Asian tsunami, Hurricanes Katrina and Ike, and the earthquakes in China's Sichuan province, Haiti, and Chile—as well as the ongoing depletion and degradation of the world's natural resources caused by a burgeoning human population—have made it clear that "business as usual" is no longer sustainable. We need to find ways to improve how we live on this planet while minimizing our impact on it. Design for a Vulnerable Planet sounds a call for designers and planners to go beyond traditional concepts of sustainability toward innovative new design that fosters regeneration and resilience. Drawing on his own and others' experiences across three continents, Frederick Steiner advocates design practice grounded in ecology and democracy and informed by critical regionalism and reflection. He begins by establishing the foundation for a more ecological approach to planning and design, adopting a broad view of ecology as encompassing human and natural, urban and wild environments. Steiner explores precedents for human ecological design provided by architect Paul Cret, landscape architect Ian McHarg, and developer George Mitchell while discussing their planning for the University of Texas campus, the Lake Austin watershed, and The Woodlands. Steiner then focuses on emerging Texas urbanism and extends his discussion to broader considerations beyond the Lone Star State, including regionalism, urbanism, and landscape in China and Italy. He also examines the lessons to be learned from human and natural disasters such as 9/11, Hurricane Katrina, and the BP oil spill. Finally, Steiner offers a blueprint for designing with nature to help heal the planet's vulnerabilities.

Lectures on Polytopes

Author: Günter M. Ziegler

Publisher: Springer Science & Business Media

ISBN: 9780387943657

Category: Mathematics

Page: 370

View: 2339

Based on a graduate course at the Technische Universität, Berlin, this book presents a wealth of material on the modern theory of convex polytopes. With linear algebra as a prerequisite, the text moves quickly from the basics to topics of recent research.

Arrangements and Spreads

Author: Branko GrŸnbaum

Publisher: American Mathematical Soc.

ISBN: 0821816594

Category: Algebraic topology

Page: 114

View: 8699

Beauville Surfaces and Groups

Author: Ingrid Bauer,Shelly Garion,Alina Vdovina

Publisher: Springer

ISBN: 3319138626

Category: Mathematics

Page: 183

View: 8233

This collection of surveys and research articles explores a fascinating class of varieties: Beauville surfaces. It is the first time that these objects are discussed from the points of view of algebraic geometry as well as group theory. The book also includes various open problems and conjectures related to these surfaces. Beauville surfaces are a class of rigid regular surfaces of general type, which can be described in a purely algebraic combinatoric way. They play an important role in different fields of mathematics like algebraic geometry, group theory and number theory. The notion of Beauville surface was introduced by Fabrizio Catanese in 2000 and after the first systematic study of these surfaces by Ingrid Bauer, Fabrizio Catanese and Fritz Grunewald, there has been an increasing interest in the subject. These proceedings reflect the topics of the lectures presented during the workshop ‘Beauville surfaces and groups 2012’, held at Newcastle University, UK in June 2012. This conference brought together, for the first time, experts of different fields of mathematics interested in Beauville surfaces.

RNA Technologies in Cardiovascular Medicine and Research

Author: Volker A. Erdmann (Deceased),Wolfgang Poller,Jan Barciszewski

Publisher: Springer Science & Business Media

ISBN: 3540787097

Category: Science

Page: 358

View: 4862

In this book the authors review the field and explore the potential role of RNAi and other RNA technologies in cardiovascular medicine and research. They highlight the impressive recent progress but also the hurdles that still must be overcome before this promising technology is finally ready for translation and clinical use.