A Geometry Of Approximation

Author : Piero Pagliani,Mihir Chakraborty
Publisher : Springer Science & Business Media
Page : 771 pages
File Size : 41,7 Mb
Release : 2008-10-09
Category : Philosophy
ISBN : 9781402086229

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A Geometry of Approximation by Piero Pagliani,Mihir Chakraborty Pdf

'A Geometry of Approximation' addresses Rough Set Theory, a field of interdisciplinary research first proposed by Zdzislaw Pawlak in 1982, and focuses mainly on its logic-algebraic interpretation. The theory is embedded in a broader perspective that includes logical and mathematical methodologies pertaining to the theory, as well as related epistemological issues. Any mathematical technique that is introduced in the book is preceded by logical and epistemological explanations. Intuitive justifications are also provided, insofar as possible, so that the general perspective is not lost. Such an approach endows the present treatise with a unique character. Due to this uniqueness in the treatment of the subject, the book will be useful to researchers, graduate and pre-graduate students from various disciplines, such as computer science, mathematics and philosophy. It features an impressive number of examples supported by about 40 tables and 230 figures. The comprehensive index of concepts turns the book into a sort of encyclopaedia for researchers from a number of fields. 'A Geometry of Approximation' links many areas of academic pursuit without losing track of its focal point, Rough Sets.

Author : Sariel Har-Peled
Publisher : American Mathematical Soc.
Page : 378 pages
File Size : 41,5 Mb
Release : 2011
Category : Computers
ISBN : 9780821849118

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Geometric Approximation Algorithms by Sariel Har-Peled Pdf

Exact algorithms for dealing with geometric objects are complicated, hard to implement in practice, and slow. Over the last 20 years a theory of geometric approximation algorithms has emerged. These algorithms tend to be simple, fast, and more robust than their exact counterparts. This book is the first to cover geometric approximation algorithms in detail. In addition, more traditional computational geometry techniques that are widely used in developing such algorithms, like sampling, linear programming, etc., are also surveyed. Other topics covered include approximate nearest-neighbor search, shape approximation, coresets, dimension reduction, and embeddings. The topics covered are relatively independent and are supplemented by exercises. Close to 200 color figures are included in the text to illustrate proofs and ideas.

Author : John A. Parker
Publisher : Unknown
Page : 222 pages
File Size : 52,9 Mb
Release : 1851
Category : Circle-squaring
ISBN : NYPL:33433069092132

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The Quadrature of the Circle by John A. Parker Pdf

Author : Bernd Gärtner,Jiri Matousek
Publisher : Springer Science & Business Media
Page : 253 pages
File Size : 48,7 Mb
Release : 2012-01-10
Category : Mathematics
ISBN : 9783642220159

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Approximation Algorithms and Semidefinite Programming by Bernd Gärtner,Jiri Matousek Pdf

Semidefinite programs constitute one of the largest classes of optimization problems that can be solved with reasonable efficiency - both in theory and practice. They play a key role in a variety of research areas, such as combinatorial optimization, approximation algorithms, computational complexity, graph theory, geometry, real algebraic geometry and quantum computing. This book is an introduction to selected aspects of semidefinite programming and its use in approximation algorithms. It covers the basics but also a significant amount of recent and more advanced material. There are many computational problems, such as MAXCUT, for which one cannot reasonably expect to obtain an exact solution efficiently, and in such case, one has to settle for approximate solutions. For MAXCUT and its relatives, exciting recent results suggest that semidefinite programming is probably the ultimate tool. Indeed, assuming the Unique Games Conjecture, a plausible but as yet unproven hypothesis, it was shown that for these problems, known algorithms based on semidefinite programming deliver the best possible approximation ratios among all polynomial-time algorithms. This book follows the “semidefinite side” of these developments, presenting some of the main ideas behind approximation algorithms based on semidefinite programming. It develops the basic theory of semidefinite programming, presents one of the known efficient algorithms in detail, and describes the principles of some others. It also includes applications, focusing on approximation algorithms.

Author : Lior Fishman,David Simmons,Mariusz Urbański
Publisher : American Mathematical Soc.
Page : 137 pages
File Size : 52,8 Mb
Release : 2018-08-09
Category : Electronic
ISBN : 9781470428860

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Diophantine Approximation and the Geometry of Limit Sets in Gromov Hyperbolic Metric Spaces by Lior Fishman,David Simmons,Mariusz Urbański Pdf

In this paper, the authors provide a complete theory of Diophantine approximation in the limit set of a group acting on a Gromov hyperbolic metric space. This summarizes and completes a long line of results by many authors, from Patterson's classic 1976 paper to more recent results of Hersonsky and Paulin (2002, 2004, 2007). The authors consider concrete examples of situations which have not been considered before. These include geometrically infinite Kleinian groups, geometrically finite Kleinian groups where the approximating point is not a fixed point of any element of the group, and groups acting on infinite-dimensional hyperbolic space. Moreover, in addition to providing much greater generality than any prior work of which the authors are aware, the results also give new insight into the nature of the connection between Diophantine approximation and the geometry of the limit set within which it takes place. Two results are also contained here which are purely geometric: a generalization of a theorem of Bishop and Jones (1997) to Gromov hyperbolic metric spaces, and a proof that the uniformly radial limit set of a group acting on a proper geodesic Gromov hyperbolic metric space has zero Patterson–Sullivan measure unless the group is quasiconvex-cocompact. The latter is an application of a Diophantine theorem.

Author : John a Parker
Publisher : Sagwan Press
Page : 218 pages
File Size : 46,5 Mb
Release : 2015-08-21
Category : Electronic
ISBN : 1297911334

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The Quadrature of the Circle by John a Parker Pdf

This work has been selected by scholars as being culturally important, and is part of the knowledge base of civilization as we know it. This work was reproduced from the original artifact, and remains as true to the original work as possible. Therefore, you will see the original copyright references, library stamps (as most of these works have been housed in our most important libraries around the world), and other notations in the work. This work is in the public domain in the United States of America, and possibly other nations. Within the United States, you may freely copy and distribute this work, as no entity (individual or corporate) has a copyright on the body of the work.As a reproduction of a historical artifact, this work may contain missing or blurred pages, poor pictures, errant marks, etc. Scholars believe, and we concur, that this work is important enough to be preserved, reproduced, and made generally available to the public. We appreciate your support of the preservation process, and thank you for being an important part of keeping this knowledge alive and relevant.

Author : Oleg Karpenkov
Publisher : Springer Science & Business Media
Page : 409 pages
File Size : 51,7 Mb
Release : 2013-08-15
Category : Mathematics
ISBN : 9783642393686

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Geometry of Continued Fractions by Oleg Karpenkov Pdf

Traditionally a subject of number theory, continued fractions appear in dynamical systems, algebraic geometry, topology, and even celestial mechanics. The rise of computational geometry has resulted in renewed interest in multidimensional generalizations of continued fractions. Numerous classical theorems have been extended to the multidimensional case, casting light on phenomena in diverse areas of mathematics. This book introduces a new geometric vision of continued fractions. It covers several applications to questions related to such areas as Diophantine approximation, algebraic number theory, and toric geometry. The reader will find an overview of current progress in the geometric theory of multidimensional continued fractions accompanied by currently open problems. Whenever possible, we illustrate geometric constructions with figures and examples. Each chapter has exercises useful for undergraduate or graduate courses.

Author : Fabio Cuzzolin
Publisher : Springer Nature
Page : 850 pages
File Size : 50,8 Mb
Release : 2020-12-17
Category : Computers
ISBN : 9783030631536

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The Geometry of Uncertainty by Fabio Cuzzolin Pdf

The principal aim of this book is to introduce to the widest possible audience an original view of belief calculus and uncertainty theory. In this geometric approach to uncertainty, uncertainty measures can be seen as points of a suitably complex geometric space, and manipulated in that space, for example, combined or conditioned. In the chapters in Part I, Theories of Uncertainty, the author offers an extensive recapitulation of the state of the art in the mathematics of uncertainty. This part of the book contains the most comprehensive summary to date of the whole of belief theory, with Chap. 4 outlining for the first time, and in a logical order, all the steps of the reasoning chain associated with modelling uncertainty using belief functions, in an attempt to provide a self-contained manual for the working scientist. In addition, the book proposes in Chap. 5 what is possibly the most detailed compendium available of all theories of uncertainty. Part II, The Geometry of Uncertainty, is the core of this book, as it introduces the author’s own geometric approach to uncertainty theory, starting with the geometry of belief functions: Chap. 7 studies the geometry of the space of belief functions, or belief space, both in terms of a simplex and in terms of its recursive bundle structure; Chap. 8 extends the analysis to Dempster’s rule of combination, introducing the notion of a conditional subspace and outlining a simple geometric construction for Dempster’s sum; Chap. 9 delves into the combinatorial properties of plausibility and commonality functions, as equivalent representations of the evidence carried by a belief function; then Chap. 10 starts extending the applicability of the geometric approach to other uncertainty measures, focusing in particular on possibility measures (consonant belief functions) and the related notion of a consistent belief function. The chapters in Part III, Geometric Interplays, are concerned with the interplay of uncertainty measures of different kinds, and the geometry of their relationship, with a particular focus on the approximation problem. Part IV, Geometric Reasoning, examines the application of the geometric approach to the various elements of the reasoning chain illustrated in Chap. 4, in particular conditioning and decision making. Part V concludes the book by outlining a future, complete statistical theory of random sets, future extensions of the geometric approach, and identifying high-impact applications to climate change, machine learning and artificial intelligence. The book is suitable for researchers in artificial intelligence, statistics, and applied science engaged with theories of uncertainty. The book is supported with the most comprehensive bibliography on belief and uncertainty theory.

Author : Alexey R. Alimov,Igor’ G. Tsar’kov
Publisher : Springer Nature
Page : 523 pages
File Size : 40,8 Mb
Release : 2022-03-29
Category : Mathematics
ISBN : 9783030909512

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Geometric Approximation Theory by Alexey R. Alimov,Igor’ G. Tsar’kov Pdf

This monograph provides a comprehensive introduction to the classical geometric approximation theory, emphasizing important themes related to the theory including uniqueness, stability, and existence of elements of best approximation. It presents a number of fundamental results for both these and related problems, many of which appear for the first time in monograph form. The text also discusses the interrelations between main objects of geometric approximation theory, formulating a number of auxiliary problems for demonstration. Central ideas include the problems of existence and uniqueness of elements of best approximations as well as properties of sets including subspaces of polynomials and splines, classes of rational functions, and abstract subsets of normed linear spaces. The book begins with a brief introduction to geometric approximation theory, progressing through fundamental classical ideas and results as a basis for various approximation sets, suns, and Chebyshev systems. It concludes with a review of approximation by abstract sets and related problems, presenting novel results throughout the section. This text is suitable for both theoretical and applied viewpoints and especially researchers interested in advanced aspects of the field.

Author : John A. Parker
Publisher : Unknown
Page : 230 pages
File Size : 52,5 Mb
Release : 2020-04-24
Category : Electronic
ISBN : 0461832690

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The Quadrature of the Circle by John A. Parker Pdf

This is a reproduction of the original artefact. Generally these books are created from careful scans of the original. This allows us to preserve the book accurately and present it in the way the author intended. Since the original versions are generally quite old, there may occasionally be certain imperfections within these reproductions. We're happy to make these classics available again for future generations to enjoy!

Author : Jayanta Mukhopadhyay
Publisher : Springer Nature
Page : 159 pages
File Size : 41,5 Mb
Release : 2020-12-02
Category : Computers
ISBN : 9789811599019

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Approximation of Euclidean Metric by Digital Distances by Jayanta Mukhopadhyay Pdf

This book discusses different types of distance functions defined in an n-D integral space for their usefulness in approximating the Euclidean metric. It discusses the properties of these distance functions and presents various kinds of error analysis in approximating Euclidean metrics. It also presents a historical perspective on efforts and motivation for approximating Euclidean metrics by digital distances from the mid-sixties of the previous century. The book also contains an in-depth presentation of recent progress, and new research problems in this area.

Author : Giorgio Ausiello,Pierluigi Crescenzi,Giorgio Gambosi,Viggo Kann,Alberto Marchetti-Spaccamela,Marco Protasi
Publisher : Springer Science & Business Media
Page : 536 pages
File Size : 45,8 Mb
Release : 2012-12-06
Category : Computers
ISBN : 9783642584121

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Complexity and Approximation by Giorgio Ausiello,Pierluigi Crescenzi,Giorgio Gambosi,Viggo Kann,Alberto Marchetti-Spaccamela,Marco Protasi Pdf

This book documents the state of the art in combinatorial optimization, presenting approximate solutions of virtually all relevant classes of NP-hard optimization problems. The wealth of problems, algorithms, results, and techniques make it an indispensible source of reference for professionals. The text smoothly integrates numerous illustrations, examples, and exercises.

Author : Bas Edixhoven,Jan-Hendrik Evertse
Publisher : Springer
Page : 136 pages
File Size : 55,8 Mb
Release : 2009-02-05
Category : Mathematics
ISBN : 9783540482086

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Diophantine Approximation and Abelian Varieties by Bas Edixhoven,Jan-Hendrik Evertse Pdf

The 13 chapters of this book centre around the proof of Theorem 1 of Faltings' paper "Diophantine approximation on abelian varieties", Ann. Math.133 (1991) and together give an approach to the proof that is accessible to Ph.D-level students in number theory and algebraic geometry. Each chapter is based on an instructional lecture given by its author ata special conference for graduate students, on the topic of Faltings' paper.

Author : Alastair Aitchison
Publisher : Apress
Page : 553 pages
File Size : 52,8 Mb
Release : 2012-07-25
Category : Computers
ISBN : 9781430234920

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Pro Spatial with SQL Server 2012 by Alastair Aitchison Pdf

Microsoft SQL Server implements extensive support for location-based data. Pro Spatial with SQL Server 2012 introduces SQL Server’s spatial feature set, and covers everything you'll need to know to store, manipulate, and analyze information about the physical location of objects in space. You’ll learn about the geography and geometry datatypes, and how to apply them in practical situations involving the spatial relationships of people, places, and things on Earth. Author Alastair Aitchison first introduces you to SQL Server’s spatial feature set and the fundamental concepts involved in working with spatial data, including spatial references and co-ordinate systems. You’ll learn to query, analyze, and interpret spatial data using tools such as Bing Maps and SQL Server Reporting Services. Throughout, you'll find helpful code examples that you can adopt and extend as a basis for your own projects. Explains spatial concepts from the ground up—no prior knowledge is necessary Provides comprehensive guidance for every stage of working with spatial data, from importing through cleansing and storing, to querying, and finally for retrieval and display of spatial data in an application layer Brilliantly illustrated with code examples that run in SQL Server 2012, that you can adapt and use as the basis for your own projects.

Author : George A. Anastassiou,Sorin G. Gal
Publisher : Springer Science & Business Media
Page : 520 pages
File Size : 40,9 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9781461213604

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Approximation Theory by George A. Anastassiou,Sorin G. Gal Pdf

We study in Part I of this monograph the computational aspect of almost all moduli of continuity over wide classes of functions exploiting some of their convexity properties. To our knowledge it is the first time the entire calculus of moduli of smoothness has been included in a book. We then present numerous applications of Approximation Theory, giving exact val ues of errors in explicit forms. The K-functional method is systematically avoided since it produces nonexplicit constants. All other related books so far have allocated very little space to the computational aspect of moduli of smoothness. In Part II, we study/examine the Global Smoothness Preservation Prop erty (GSPP) for almost all known linear approximation operators of ap proximation theory including: trigonometric operators and algebraic in terpolation operators of Lagrange, Hermite-Fejer and Shepard type, also operators of stochastic type, convolution type, wavelet type integral opera tors and singular integral operators, etc. We present also a sufficient general theory for GSPP to hold true. We provide a great variety of applications of GSPP to Approximation Theory and many other fields of mathemat ics such as Functional analysis, and outside of mathematics, fields such as computer-aided geometric design (CAGD). Most of the time GSPP meth ods are optimal. Various moduli of smoothness are intensively involved in Part II. Therefore, methods from Part I can be used to calculate exactly the error of global smoothness preservation. It is the first time in the literature that a book has studied GSPP.