Space-Filling Curves

Author: Hans Sagan

Publisher: Springer Science & Business Media

ISBN: 1461208718

Category: Mathematics

Page: 194

View: 6320

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: 3930

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: 8373

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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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: 3226

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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Continuous Nowhere Differentiable Functions

The Monsters of Analysis

Author: Marek Jarnicki,Peter Pflug

Publisher: Springer

ISBN: 3319126709

Category: Mathematics

Page: 299

View: 8615

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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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: 3144

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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Differential Geometry

Author: Wolfgang Kühnel

Publisher: American Mathematical Soc.

ISBN: 1470423200

Category: Curves

Page: 403

View: 2633

This carefully written book is an introduction to the beautiful ideas and results of differential geometry. The first half covers the geometry of curves and surfaces, which provide much of the motivation and intuition for the general theory. The second part studies the geometry of general manifolds, with particular emphasis on connections and curvature. The text is illustrated with many figures and examples. The prerequisites are undergraduate analysis and linear algebra. This new edition provides many advancements, including more figures and exercises, and--as a new feature--a good number of solutions to selected exercises.
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From Calculus to Analysis

Author: Steen Pedersen

Publisher: Springer

ISBN: 3319136410

Category: Mathematics

Page: 342

View: 2557

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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Geometry

Author: Michele Audin

Publisher: Springer Science & Business Media

ISBN: 3642561276

Category: Mathematics

Page: 361

View: 5813

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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Computing and Combinatorics

Author: Wen-Lian Hsu,Ming-Yang Kao

Publisher: Springer Verlag

ISBN: N.A

Category: Computer science

Page: N.A

View: 8578

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The Joy of Sets

Fundamentals of Contemporary Set Theory

Author: Keith Devlin

Publisher: Springer Science & Business Media

ISBN: 146120903X

Category: Mathematics

Page: 194

View: 8259

This text covers the parts of contemporary set theory relevant to other areas of pure mathematics. After a review of "naïve" set theory, it develops the Zermelo-Fraenkel axioms of the theory before discussing the ordinal and cardinal numbers. It then delves into contemporary set theory, covering such topics as the Borel hierarchy and Lebesgue measure. A final chapter presents an alternative conception of set theory useful in computer science.
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Introduction to Differentiable Manifolds

Author: Louis Auslander,Robert E. MacKenzie

Publisher: Courier Corporation

ISBN: 048615808X

Category: Mathematics

Page: 224

View: 2386

This text presents basic concepts in the modern approach to differential geometry. Topics include Euclidean spaces, submanifolds, and abstract manifolds; fundamental concepts of Lie theory; fiber bundles; and multilinear algebra. 1963 edition.
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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: 3707

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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Differentiable Manifolds

A First Course

Author: Lawrence Conlon

Publisher: Springer Science & Business Media

ISBN: 1475722842

Category: Mathematics

Page: 395

View: 6571

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: Games & Activities

Page: 144

View: 841

"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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Differential Forms and Applications

Author: Manfredo P. Do Carmo

Publisher: Springer Science & Business Media

ISBN: 3642579515

Category: Mathematics

Page: 118

View: 7966

An application of differential forms for the study of some local and global aspects of the differential geometry of surfaces. Differential forms are introduced in a simple way that will make them attractive to "users" of mathematics. A brief and elementary introduction to differentiable manifolds is given so that the main theorem, namely Stokes' theorem, can be presented in its natural setting. The applications consist in developing the method of moving frames expounded by E. Cartan to study the local differential geometry of immersed surfaces in R3 as well as the intrinsic geometry of surfaces. This is then collated in the last chapter to present Chern's proof of the Gauss-Bonnet theorem for compact surfaces.
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Lectures on Hyperbolic Geometry

Author: Riccardo Benedetti,Carlo Petronio

Publisher: Springer Science & Business Media

ISBN: 3642581587

Category: Mathematics

Page: 330

View: 8866

Focussing on the geometry of hyperbolic manifolds, the aim here is to provide an exposition of some fundamental results, while being as self-contained, complete, detailed and unified as possible. Following some classical material on the hyperbolic space and the Teichmüller space, the book centers on the two fundamental results: Mostow's rigidity theorem (including a complete proof, following Gromov and Thurston) and Margulis' lemma. These then form the basis for studying Chabauty and geometric topology; a unified exposition is given of Wang's theorem and the Jorgensen-Thurston theory; and much space is devoted to the 3D case: a complete and elementary proof of the hyperbolic surgery theorem, based on the representation of three manifolds as glued ideal tetrahedra.
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Geometry of Surfaces

Author: John Stillwell

Publisher: Springer Science & Business Media

ISBN: 1461209293

Category: Mathematics

Page: 236

View: 3330

The geometry of surfaces is an ideal starting point for learning geometry, for, among other reasons, the theory of surfaces of constant curvature has maximal connectivity with the rest of mathematics. This text provides the student with the knowledge of a geometry of greater scope than the classical geometry taught today, which is no longer an adequate basis for mathematics or physics, both of which are becoming increasingly geometric. It includes exercises and informal discussions.
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Advanced Analysis

on the Real Line

Author: R. Kannan,Carole K. Krueger

Publisher: Springer Science & Business Media

ISBN: 1461384745

Category: Mathematics

Page: 260

View: 1675

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