Intersecting Convex Sets By Rays
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Intersecting Convex Sets By Rays
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Author : Radoslav Fulek
language : en
Publisher:
Release Date : 2008
Intersecting Convex Sets By Rays written by Radoslav Fulek and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2008 with Convex sets categories.
What is the smallest number tau=tau_d(n) such that for any collection C of n pairwise disjoint compact convex sets in Rd̂, there is a point such that any ray (half-line) emanating from it meets at most tau sets of the collection? We show an upper and several lower bounds on the value tau_d(n), and thereby we completely answer the above question for R2̂, and partially for higher dimensions. We show the order of magnitude for an analog of tau_2(n) for collections of fat sets with bounded diameter. We conclude the thesis with some algorithmic solutions for finding a point p that minimizes the maximum number of sets in C we are able to intersect by a ray emanating from p in the plane, and for finding a point that basically witnesses our upper bound on tau_d(n) in any dimension. However, the latter works only for restricted sets of objects.
Join Geometries
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Author : W. Prenowitz
language : en
Publisher: Springer Science & Business Media
Release Date : 2012-12-06
Join Geometries written by W. Prenowitz and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012-12-06 with Mathematics categories.
The main object of this book is to reorient and revitalize classical geometry in a way that will bring it closer to the mainstream of contemporary mathematics. The postulational basis of the subject will be radically revised in order to construct a broad-scale and conceptually unified treatment. The familiar figures of classical geometry-points, segments, lines, planes, triangles, circles, and so on-stem from problems in the physical world and seem to be conceptually unrelated. However, a natural setting for their study is provided by the concept of convex set, which is compara tively new in the history of geometrical ideas. The familiarfigures can then appear as convex sets, boundaries of convex sets, or finite unions of convex sets. Moreover, two basic types of figure in linear geometry are special cases of convex set: linear space (point, line, and plane) and halfspace (ray, halfplane, and halfspace). Therefore we choose convex set to be the central type of figure in our treatment of geometry. How can the wealth of geometric knowledge be organized around this idea? By defini tion, a set is convex if it contains the segment joining each pair of its points; that is, if it is closed under the operation of joining two points to form a segment. But this is precisely the basic operation in Euclid.
Restricted Orientation Convexity
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Author : Eugene Fink
language : en
Publisher: Springer Science & Business Media
Release Date : 2012-12-06
Restricted Orientation Convexity written by Eugene Fink and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012-12-06 with Computers categories.
Restricted-orientation convexity is the study of geometric objects whose intersections with lines from some fixed set are connected. This notion generalizes standard convexity and several types of nontraditional convexity. The authors explore the properties of this generalized convexity in multidimensional Euclidean space, and describ restricted-orientation analogs of lines, hyperplanes, flats, halfspaces, and identify major properties of standard convex sets that also hold for restricted-orientation convexity. They then introduce the notion of strong restricted-orientation convexity, which is an alternative generalization of convexity, and show that its properties are also similar to that of standard convexity.
The Foundations Of Geometry And The Non Euclidean Plane
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Author : G.E. Martin
language : en
Publisher: Springer Science & Business Media
Release Date : 2012-12-06
The Foundations Of Geometry And The Non Euclidean Plane written by G.E. Martin and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012-12-06 with Mathematics categories.
This book is a text for junior, senior, or first-year graduate courses traditionally titled Foundations of Geometry and/or Non Euclidean Geometry. The first 29 chapters are for a semester or year course on the foundations of geometry. The remaining chap ters may then be used for either a regular course or independent study courses. Another possibility, which is also especially suited for in-service teachers of high school geometry, is to survey the the fundamentals of absolute geometry (Chapters 1 -20) very quickly and begin earnest study with the theory of parallels and isometries (Chapters 21 -30). The text is self-contained, except that the elementary calculus is assumed for some parts of the material on advanced hyperbolic geometry (Chapters 31 -34). There are over 650 exercises, 30 of which are 10-part true-or-false questions. A rigorous ruler-and-protractor axiomatic development of the Euclidean and hyperbolic planes, including the classification of the isometries of these planes, is balanced by the discussion about this development. Models, such as Taxicab Geometry, are used exten sively to illustrate theory. Historical aspects and alternatives to the selected axioms are prominent. The classical axiom systems of Euclid and Hilbert are discussed, as are axiom systems for three and four-dimensional absolute geometry and Pieri's system based on rigid motions. The text is divided into three parts. The Introduction (Chapters 1 -4) is to be read as quickly as possible and then used for ref erence if necessary.
Convex Sets
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Author : Frederick Albert Valentine
language : en
Publisher:
Release Date : 1964
Convex Sets written by Frederick Albert Valentine and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 1964 with Convex sets categories.
Convex Surfaces
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Author : Herbert Busemann
language : en
Publisher: Courier Corporation
Release Date : 2013-11-07
Convex Surfaces written by Herbert Busemann and has been published by Courier Corporation this book supported file pdf, txt, epub, kindle and other format this book has been release on 2013-11-07 with Mathematics categories.
This exploration of convex surfaces focuses on extrinsic geometry and applications of the Brunn-Minkowski theory. It also examines intrinsic geometry and the realization of intrinsic metrics. 1958 edition.
Quantum Theory From First Principles
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Author : Giacomo Mauro D'Ariano
language : en
Publisher: Cambridge University Press
Release Date : 2017-01-28
Quantum Theory From First Principles written by Giacomo Mauro D'Ariano and has been published by Cambridge University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2017-01-28 with Science categories.
A new presentation of quantum theory and quantum information based on fundamental principles, for anyone seeking a deeper understanding of the subject.
Convex Sets And Their Applications
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Author : Ky Fan
language : en
Publisher:
Release Date : 1959
Convex Sets And Their Applications written by Ky Fan and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 1959 with Convex bodies categories.
An Intersection Property For Cones In A Linear Space
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Author : R. E. Fullerton
language : en
Publisher:
Release Date : 1957
An Intersection Property For Cones In A Linear Space written by R. E. Fullerton and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 1957 with Cone categories.
Real Analysis Measures Integrals And Applications
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Author : Boris Makarov
language : en
Publisher: Springer Science & Business Media
Release Date : 2013-06-14
Real Analysis Measures Integrals And Applications written by Boris Makarov and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2013-06-14 with Mathematics categories.
Real Analysis: Measures, Integrals and Applications is devoted to the basics of integration theory and its related topics. The main emphasis is made on the properties of the Lebesgue integral and various applications both classical and those rarely covered in literature. This book provides a detailed introduction to Lebesgue measure and integration as well as the classical results concerning integrals of multivariable functions. It examines the concept of the Hausdorff measure, the properties of the area on smooth and Lipschitz surfaces, the divergence formula, and Laplace's method for finding the asymptotic behavior of integrals. The general theory is then applied to harmonic analysis, geometry, and topology. Preliminaries are provided on probability theory, including the study of the Rademacher functions as a sequence of independent random variables. The book contains more than 600 examples and exercises. The reader who has mastered the first third of the book will be able to study other areas of mathematics that use integration, such as probability theory, statistics, functional analysis, partial probability theory, statistics, functional analysis, partial differential equations and others. Real Analysis: Measures, Integrals and Applications is intended for advanced undergraduate and graduate students in mathematics and physics. It assumes that the reader is familiar with basic linear algebra and differential calculus of functions of several variables.