Space-Filling Curves

Author: Hans Sagan

Publisher: Springer Science & Business Media

ISBN: 1461208718

Category: Mathematics

Page: 194

View: 9835

The subject of space-filling curves has fascinated mathematicians for over a century and has intrigued many generations of students of mathematics. Working in this area is like skating on the edge of reason. Unfortunately, no comprehensive treatment has ever been attempted other than the gallant effort by W. Sierpiriski in 1912. At that time, the subject was still in its infancy and the most interesting and perplexing results were still to come. Besides, Sierpiriski's paper was written in Polish and published in a journal that is not readily accessible (Sierpiriski [2]). Most of the early literature on the subject is in French, German, and Polish, providing an additional raison d'etre for a comprehensive treatment in English. While there was, understandably, some intensive research activity on this subject around the turn of the century, contributions have, nevertheless, continued up to the present and there is no end in sight, indicating that the subject is still very much alive. The recent interest in fractals has refocused interest on space filling curves, and the study of fractals has thrown some new light on this small but venerable part of mathematics. This monograph is neither a textbook nor an encyclopedic treatment of the subject nor a historical account, but it is a little of each. While it may lend structure to a seminar or pro-seminar, or be useful as a supplement in a course on topology or mathematical analysis, it is primarily intended for self-study by the aficionados of classical analysis.
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Space-Filling Curves

An Introduction With Applications in Scientific Computing

Author: Michael Bader

Publisher: Springer Science & Business Media

ISBN: 3642310451

Category: Computers

Page: 278

View: 5095

The present book provides an introduction to using space-filling curves (SFC) as tools in scientific computing. Special focus is laid on the representation of SFC and on resulting algorithms. For example, grammar-based techniques are introduced for traversals of Cartesian and octree-type meshes, and arithmetisation of SFC is explained to compute SFC mappings and indexings. The locality properties of SFC are discussed in detail, together with their importance for algorithms. Templates for parallelisation and cache-efficient algorithms are presented to reflect the most important applications of SFC in scientific computing. Special attention is also given to the interplay of adaptive mesh refinement and SFC, including the structured refinement of triangular and tetrahedral grids. For each topic, a short overview is given on the most important publications and recent research activities.
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Space-Filling Curves

An Introduction with Applications in Scientific Computing

Author: Michael Bader

Publisher: Springer Science & Business Media

ISBN: 364231046X

Category: Computers

Page: 285

View: 7000

The present book provides an introduction to using space-filling curves (SFC) as tools in scientific computing. Special focus is laid on the representation of SFC and on resulting algorithms. For example, grammar-based techniques are introduced for traversals of Cartesian and octree-type meshes, and arithmetisation of SFC is explained to compute SFC mappings and indexings. The locality properties of SFC are discussed in detail, together with their importance for algorithms. Templates for parallelisation and cache-efficient algorithms are presented to reflect the most important applications of SFC in scientific computing. Special attention is also given to the interplay of adaptive mesh refinement and SFC, including the structured refinement of triangular and tetrahedral grids. For each topic, a short overview is given on the most important publications and recent research activities.
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Geometry

Author: Michele Audin

Publisher: Springer Science & Business Media

ISBN: 3642561276

Category: Mathematics

Page: 361

View: 2409

Geometry, this very ancient field of study of mathematics, frequently remains too little familiar to students. Michle Audin, professor at the University of Strasbourg, has written a book allowing them to remedy this situation and, starting from linear algebra, extend their knowledge of affine, Euclidean and projective geometry, conic sections and quadrics, curves and surfaces. It includes many nice theorems like the nine-point circle, Feuerbach's theorem, and so on. Everything is presented clearly and rigourously. Each property is proved, examples and exercises illustrate the course content perfectly. Precise hints for most of the exercises are provided at the end of the book. This very comprehensive text is addressed to students at upper undergraduate and Master's level to discover geometry and deepen their knowledge and understanding.
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Algorithms - ESA 2008

16th Annual European Symposium, Karlsruhe, Germany, September 15-17, 2008, Proceedings

Author: Dan Halperin

Publisher: Springer Science & Business Media

ISBN: 3540877436

Category: Computers

Page: 844

View: 301

This book constitutes the refereed proceedings of the 16th Annual European Symposium on Algorithms, ESA 2008, held in Karlsruhe, Germany, in September 2008 in the context of the combined conference ALGO 2008. The 67 revised full papers presented together with 2 invited lectures were carefully reviewed and selected: 51 papers out of 147 submissions for the design and analysis track and 16 out of 53 submissions in the engineering and applications track. The papers address all current subjects in algorithmics reaching from design and analysis issues of algorithms over to real-world applications and engineering of algorithms in various fields. Special focus is given to mathematical programming and operations research, including combinatorial optimization, integer programming, polyhedral combinatorics and network optimization.
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Algebraic Geometry

An Introduction

Author: Daniel Perrin

Publisher: Springer Science & Business Media

ISBN: 9781848000568

Category: Mathematics

Page: 263

View: 7803

Aimed primarily at graduate students and beginning researchers, this book provides an introduction to algebraic geometry that is particularly suitable for those with no previous contact with the subject; it assumes only the standard background of undergraduate algebra. The book starts with easily-formulated problems with non-trivial solutions and uses these problems to introduce the fundamental tools of modern algebraic geometry: dimension; singularities; sheaves; varieties; and cohomology. A range of exercises is provided for each topic discussed, and a selection of problems and exam papers are collected in an appendix to provide material for further study.
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Continuous Nowhere Differentiable Functions

The Monsters of Analysis

Author: Marek Jarnicki,Peter Pflug

Publisher: Springer

ISBN: 3319126709

Category: Mathematics

Page: 299

View: 8786

This book covers the construction, analysis, and theory of continuous nowhere differentiable functions, comprehensively and accessibly. After illuminating the significance of the subject through an overview of its history, the reader is introduced to the sophisticated toolkit of ideas and tricks used to study the explicit continuous nowhere differentiable functions of Weierstrass, Takagi–van der Waerden, Bolzano, and others. Modern tools of functional analysis, measure theory, and Fourier analysis are applied to examine the generic nature of continuous nowhere differentiable functions, as well as linear structures within the (nonlinear) space of continuous nowhere differentiable functions. To round out the presentation, advanced techniques from several areas of mathematics are brought together to give a state-of-the-art analysis of Riemann’s continuous, and purportedly nowhere differentiable, function. For the reader’s benefit, claims requiring elaboration, and open problems, are clearly indicated. An appendix conveniently provides background material from analysis and number theory, and comprehensive indices of symbols, problems, and figures enhance the book’s utility as a reference work. Students and researchers of analysis will value this unique book as a self-contained guide to the subject and its methods.
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Differentiable Manifolds

A First Course

Author: Lawrence Conlon

Publisher: Springer Science & Business Media

ISBN: 1475722842

Category: Mathematics

Page: 395

View: 5628

This book is based on the full year Ph.D. qualifying course on differentiable manifolds, global calculus, differential geometry, and related topics, given by the author at Washington University several times over a twenty year period. It is addressed primarily to second year graduate students and well prepared first year students. Presupposed is a good grounding in general topology and modern algebra, especially linear algebra and the analogous theory of modules over a commutative, unitary ring. Although billed as a "first course" , the book is not intended to be an overly sketchy introduction. Mastery of this material should prepare the student for advanced topics courses and seminars in differen tial topology and geometry. There are certain basic themes of which the reader should be aware. The first concerns the role of differentiation as a process of linear approximation of non linear problems. The well understood methods of linear algebra are then applied to the resulting linear problem and, where possible, the results are reinterpreted in terms of the original nonlinear problem. The process of solving differential equations (i. e., integration) is the reverse of differentiation. It reassembles an infinite array of linear approximations, result ing from differentiation, into the original nonlinear data. This is the principal tool for the reinterpretation of the linear algebra results referred to above.
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Patterns of the Universe

A Coloring Adventure in Math and Beauty

Author: Alex Bellos,Edmund Harriss

Publisher: The Experiment

ISBN: 1615193235

Category: Art

Page: 144

View: 654

"A coloring book that reveals math's hidden beauty and contemplative power as never before with 78 coloring designs and games that explore symmetry, fractals, tessellations, randomness, and more."--
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Lineability

The Search for Linearity in Mathematics

Author: Richard M. Aron,Luis Bernal-Gonzalez,Daniel M. Pellegrino,Juan B. Seoane Sepulveda

Publisher: CRC Press

ISBN: 1482299100

Category: Mathematics

Page: 308

View: 767

Renewed interest in vector spaces and linear algebras has spurred the search for large algebraic structures composed of mathematical objects with special properties. Bringing together research that was otherwise scattered throughout the literature, Lineability: The Search for Linearity in Mathematics collects the main results on the conditions for the existence of large algebraic substructures. It investigates lineability issues in a variety of areas, including real and complex analysis. After presenting basic concepts about the existence of linear structures, the book discusses lineability properties of families of functions defined on a subset of the real line as well as the lineability of special families of holomorphic (or analytic) functions defined on some domain of the complex plane. It next focuses on spaces of sequences and spaces of integrable functions before covering the phenomenon of universality from an algebraic point of view. The authors then describe the linear structure of the set of zeros of a polynomial defined on a real or complex Banach space and explore specialized topics, such as the lineability of various families of vectors. The book concludes with an account of general techniques for discovering lineability in its diverse degrees.
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Mathematica® in Action

Problem Solving Through Visualization and Computation

Author: Stan Wagon

Publisher: Springer Science & Business Media

ISBN: 9780387754772

Category: Mathematics

Page: 580

View: 3778

Plenty of examples and case studies utilize Mathematica 7's newest tools, such as dynamic manipulations and adaptive three-dimensional plotting. Emphasizes the breadth of Mathematica and the impressive results of combining techniques from different areas. Whenever possible, the book shows how Mathematica can be used to discover new things. Striking examples include the design of a road on which a square wheel bike can ride, the design of a drill that can drill square holes, and new and surprising formulas for p. Visualization is emphasized throughout, with finely crafted graphics in each chapter.
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A Geometric Approach to Differential Forms

Author: David Bachman

Publisher: Springer Science & Business Media

ISBN: 0817683046

Category: Mathematics

Page: 156

View: 8818

This text presents differential forms from a geometric perspective accessible at the undergraduate level. It begins with basic concepts such as partial differentiation and multiple integration and gently develops the entire machinery of differential forms. The subject is approached with the idea that complex concepts can be built up by analogy from simpler cases, which, being inherently geometric, often can be best understood visually. Each new concept is presented with a natural picture that students can easily grasp. Algebraic properties then follow. The book contains excellent motivation, numerous illustrations and solutions to selected problems.
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Logic and Structure

Author: Dirk van Dalen

Publisher: Springer Science & Business Media

ISBN: 3662029626

Category: Mathematics

Page: 220

View: 8141

New corrected printing of a well-established text on logic at the introductory level.
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From Calculus to Analysis

Author: Steen Pedersen

Publisher: Springer

ISBN: 3319136410

Category: Mathematics

Page: 342

View: 4365

This textbook features applications including a proof of the Fundamental Theorem of Algebra, space filling curves, and the theory of irrational numbers. In addition to the standard results of advanced calculus, the book contains several interesting applications of these results. The text is intended to form a bridge between calculus and analysis. It is based on the authors lecture notes used and revised nearly every year over the last decade. The book contains numerous illustrations and cross references throughout, as well as exercises with solutions at the end of each section.
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Hacker's Delight

Author: Henry S. Warren

Publisher: Pearson Education

ISBN: 0321842685

Category: Computers

Page: 494

View: 7295

Compiles programming hacks intended to help computer programmers build more efficient software, in an updated edition that covers cyclic redundancy checking and new algorithms and that includes exercises with answers.
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Newton Methods for Nonlinear Problems

Affine Invariance and Adaptive Algorithms

Author: Peter Deuflhard

Publisher: Springer Science & Business Media

ISBN: 3642238998

Category: Mathematics

Page: 424

View: 4752

This book deals with the efficient numerical solution of challenging nonlinear problems in science and engineering, both in finite dimension (algebraic systems) and in infinite dimension (ordinary and partial differential equations). Its focus is on local and global Newton methods for direct problems or Gauss-Newton methods for inverse problems. The term 'affine invariance' means that the presented algorithms and their convergence analysis are invariant under one out of four subclasses of affine transformations of the problem to be solved. Compared to traditional textbooks, the distinguishing affine invariance approach leads to shorter theorems and proofs and permits the construction of fully adaptive algorithms. Lots of numerical illustrations, comparison tables, and exercises make the text useful in computational mathematics classes. At the same time, the book opens many directions for possible future research.
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An Introduction to Homological Algebra

Author: Joseph J. Rotman

Publisher: Springer Science & Business Media

ISBN: 0387683240

Category: Mathematics

Page: 710

View: 7382

Graduate mathematics students will find this book an easy-to-follow, step-by-step guide to the subject. Rotman’s book gives a treatment of homological algebra which approaches the subject in terms of its origins in algebraic topology. In this new edition the book has been updated and revised throughout and new material on sheaves and cup products has been added. The author has also included material about homotopical algebra, alias K-theory. Learning homological algebra is a two-stage affair. First, one must learn the language of Ext and Tor. Second, one must be able to compute these things with spectral sequences. Here is a work that combines the two.
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Partial Differential Equations in Action

From Modelling to Theory

Author: Sandro Salsa

Publisher: Springer

ISBN: 3319312383

Category: Mathematics

Page: 686

View: 7869

The book is intended as an advanced undergraduate or first-year graduate course for students from various disciplines, including applied mathematics, physics and engineering. It has evolved from courses offered on partial differential equations (PDEs) over the last several years at the Politecnico di Milano. These courses had a twofold purpose: on the one hand, to teach students to appreciate the interplay between theory and modeling in problems arising in the applied sciences, and on the other to provide them with a solid theoretical background in numerical methods, such as finite elements. Accordingly, this textbook is divided into two parts. The first part, chapters 2 to 5, is more elementary in nature and focuses on developing and studying basic problems from the macro-areas of diffusion, propagation and transport, waves and vibrations. In turn the second part, chapters 6 to 11, concentrates on the development of Hilbert spaces methods for the variational formulation and the analysis of (mainly) linear boundary and initial-boundary value problems.The third edition contains a few text and formulas revisions and new exercises.
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A Course on Finite Groups

Author: H.E. Rose

Publisher: Springer Science & Business Media

ISBN: 9781848828896

Category: Mathematics

Page: 311

View: 1891

Introduces the richness of group theory to advanced undergraduate and graduate students, concentrating on the finite aspects. Provides a wealth of exercises and problems to support self-study. Additional online resources on more challenging and more specialised topics can be used as extension material for courses, or for further independent study.
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Nonlinear Methods in Riemannian and Kählerian Geometry

Author: J. Jost

Publisher: Birkhäuser

ISBN: 3034876904

Category: Science

Page: 156

View: 4548

In this book, I present an expanded version of the contents of my lectures at a Seminar of the DMV (Deutsche Mathematiker Vereinigung) in Diisseldorf, June, 1986. The title "Nonlinear methods in complex geometry" already indicates a combination of techniques from nonlinear partial differential equations and geometric concepts. In older geometric investigations, usually the local aspects attracted more attention than the global ones as differential geometry in its foundations provides approximations of local phenomena through infinitesimal or differential constructions. Here, all equations are linear. If one wants to consider global aspects, however, usually the presence of curvature leads to a nonlinearity in the equations. The simplest case is the one of geodesics which are described by a system of second order nonlinear ODE; their linearizations are the Jacobi fields. More recently, nonlinear PDE played a more and more prominent role in geometry. Let us list some of the most important ones: - harmonic maps between Riemannian and Kahlerian manifolds - minimal surfaces in Riemannian manifolds - Monge-Ampere equations on Kahler manifolds - Yang-Mills equations in vector bundles over manifolds. While the solution of these equations usually is nontrivial, it can lead to very signifi cant results in geometry, as solutions provide maps, submanifolds, metrics, or connections which are distinguished by geometric properties in a given context. All these equations are elliptic, but often parabolic equations are used as an auxiliary tool to solve the elliptic ones.
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