Boolean Metric Spaces
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Boolean Metric Spaces
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Author : Christiaan Jan Penning
language : en
Publisher:
Release Date : 1960
Boolean Metric Spaces written by Christiaan Jan Penning and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 1960 with Algebra, Boolean categories.
Boolean Valued Probabilistic Metric Spaces
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Author : Mickey Charles Holsey
language : en
Publisher:
Release Date : 1989
Boolean Valued Probabilistic Metric Spaces written by Mickey Charles Holsey and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 1989 with categories.
In this paper the Boolean valued method is used to develop a theory closely resembling the theory of probabilistic metric spaces. In this development the complete Boolean algebra used must have the form of the quotient algebra of some atomless probability space ($Omega,{cal A},P$) modulo its class of null sets. In this setting the real numbers in the Boolean valued model, $Vsp{cal B}$, correspond to random variables with domain $Omega$, and the sets which act as binary operations on $Re$ in the model correspond to binary operations on the class of random variables with domain $Omega$. The next step is to generalize the theory of metric spaces by requiring the triangle inequality be satisfied for some generalized addition $oplus$ (i.e. some commutative, associative, nondecreasing binary operation on $Resp+$ which has 0 as identity). If the generalized addition is continuous then the generalized metric space has a natural topology on it which satisfies the same general properties that metric topologies satisfy. A Boolean valued probabilistic metric space is a triple ($tilde S,{cal D},{cal G}$) which corresponds to some triple ($S,d,oplus$) in $Vsp{cal B}$ which acts as a generalized metric space in the nonstandard model. Thus $tilde S$ is a subset of $Vsp{cal B},{cal D}$ is a function which maps ordered pairs of elements of $tilde S$ to a.e. nonnegative random variables, and ${cal G}$ is a binary operation on the class of a.e. nonnegative random variables. The topology on the generalized metric space in $Vsp{cal B}$ corresponds to a topology on $tilde S$ which is uniform. There are a large number of probabilistic metric spaces which can be isomorphically injected into some Boolean valued probabilistic metric space. This mapping is generally not a surjection due to the richer structure of the Boolean valued probabilistic metric space. There are also probabilistic metric spaces which cannot be isomorphically injected into any Boolean valued probabilistic metric space. This paper leads to several interesting lines of inquiry. One of these is whether all uniform topological spaces may be obtained from some generalized metric space by the Boolean valued method and another is what systems in the sciences can Boolean valued probabilistic metric spaces be used to model.
Compact Zero Dimensional Metric Spaces Of Finite Type
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Author : Richard Scott Pierce
language : en
Publisher: American Mathematical Soc.
Release Date : 1972
Compact Zero Dimensional Metric Spaces Of Finite Type written by Richard Scott Pierce and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 1972 with Algebra, Boolean categories.
This monograph is devoted to the investigation of metric Boolean spaces that are of finite type. The first part contains uniqueness and existence theorems that reduce the study of the spaces under consideration to an investigation of the corresponding algebras [Fraktur capital]A(X). These algebras are characterized by identities, and they are referred to as topological Boolean algebras. In part two, the decompositions of a space X (of the kind under consideration) into disjoint unions and topological products are considered. These decompositions are related to corresponding algebraic decompositions of [Fraktur capital]A(X). Part three is devoted to the development of a duality theory for topological Boolean algebras.
Sheaves Of Algebras Over Boolean Spaces
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Author : Arthur Knoebel
language : en
Publisher: Springer Science & Business Media
Release Date : 2011-12-16
Sheaves Of Algebras Over Boolean Spaces written by Arthur Knoebel 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 2011-12-16 with Mathematics categories.
This unique monograph building bridges among a number of different areas of mathematics such as algebra, topology, and category theory. The author uses various tools to develop new applications of classical concepts. Detailed proofs are given for all major theorems, about half of which are completely new. Sheaves of Algebras over Boolean Spaces will take readers on a journey through sheaf theory, an important part of universal algebra. This excellent reference text is suitable for graduate students, researchers, and those who wish to learn about sheaves of algebras.
Fundamentals Of Signal Processing In Metric Spaces With Lattice Properties
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Author : Andrey Popoff
language : en
Publisher: CRC Press
Release Date : 2017-11-03
Fundamentals Of Signal Processing In Metric Spaces With Lattice Properties written by Andrey Popoff and has been published by CRC Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2017-11-03 with Mathematics categories.
Exploring the interrelation between information theory and signal processing theory, the book contains a new algebraic approach to signal processing theory. Readers will learn this new approach to constructing the unified mathematical fundamentals of both information theory and signal processing theory in addition to new methods of evaluating quality indices of signal processing. The book discusses the methodology of synthesis and analysis of signal processing algorithms providing qualitative increase of signal processing efficiency under parametric and nonparametric prior uncertainty conditions. Examples are included throughout the book to further emphasize new material.
Probabilistic Metric Spaces
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Author : B. Schweizer
language : en
Publisher: Courier Corporation
Release Date : 2011-10-14
Probabilistic Metric Spaces written by B. Schweizer and has been published by Courier Corporation this book supported file pdf, txt, epub, kindle and other format this book has been release on 2011-10-14 with Mathematics categories.
This distinctly nonclassical treatment focuses on developing aspects that differ from the theory of ordinary metric spaces, working directly with probability distribution functions rather than random variables. The two-part treatment begins with an overview that discusses the theory's historical evolution, followed by a development of related mathematical machinery. The presentation defines all needed concepts, states all necessary results, and provides relevant proofs. The second part opens with definitions of probabilistic metric spaces and proceeds to examinations of special classes of probabilistic metric spaces, topologies, and several related structures, such as probabilistic normed and inner-product spaces. Throughout, the authors focus on developing aspects that differ from the theory of ordinary metric spaces, rather than simply transferring known metric space results to a more general setting.
Compact Zero Dimensional Metric Spaces Of Finite Type
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Author :
language : en
Publisher:
Release Date : 1969
Compact Zero Dimensional Metric Spaces Of Finite Type written by and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 1969 with categories.
Algebra And Geometry
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Author : R. V. Gamkrelidze
language : en
Publisher: Springer Science & Business Media
Release Date : 2013-03-09
Algebra And Geometry written by R. V. Gamkrelidze 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 2013-03-09 with Mathematics categories.
This volume contains five review articles, three in the Al gebra part and two in the Geometry part, surveying the fields of ring theory, modules, and lattice theory in the former, and those of integral geometry and differential-geometric methods in the calculus of variations in the latter. The literature covered is primarily that published in 1965-1968. v CONTENTS ALGEBRA RING THEORY L. A. Bokut', K. A. Zhevlakov, and E. N. Kuz'min § 1. Associative Rings. . . . . . . . . . . . . . . . . . . . 3 § 2. Lie Algebras and Their Generalizations. . . . . . . 13 ~ 3. Alternative and Jordan Rings. . . . . . . . . . . . . . . . 18 Bibliography. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 MODULES A. V. Mikhalev and L. A. Skornyakov § 1. Radicals. . . . . . . . . . . . . . . . . . . 59 § 2. Projection, Injection, etc. . . . . . . . . . . . . . . . . . . 62 § 3. Homological Classification of Rings. . . . . . . . . . . . 66 § 4. Quasi-Frobenius Rings and Their Generalizations. . 71 § 5. Some Aspects of Homological Algebra . . . . . . . . . . 75 § 6. Endomorphism Rings . . . . . . . . . . . . . . . . . . . . . 83 § 7. Other Aspects. . . . . . . . . . . . . . . . . . . 87 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . , 91 LATTICE THEORY M. M. Glukhov, 1. V. Stelletskii, and T. S. Fofanova § 1. Boolean Algebras . . . . . . . . . . . . . . . . . . . . . " 111 § 2. Identity and Defining Relations in Lattices . . . . . . 120 § 3. Distributive Lattices. . . . . . . . . . . . . . . . . . . . . 122 vii viii CONTENTS § 4. Geometrical Aspects and the Related Investigations. . . . . . . . . . . . • . . • . . . . . . . . . • 125 § 5. Homological Aspects. . . . . . . . . . . . . . . . . . . . . . 129 § 6. Lattices of Congruences and of Ideals of a Lattice . . 133 § 7. Lattices of Subsets, of Subalgebras, etc. . . . . . . . . 134 § 8. Closure Operators . . . . . . . . . . . . . . . . . . . . . . . 136 § 9. Topological Aspects. . . . . . . . . . . . . . . . . . . . . . 137 § 10. Partially-Ordered Sets. . . . . . . . . . . . . . . . . . . . 141 § 11. Other Questions. . . . . . . . . . . . . . . . . . . . . . . . . 146 Bibliography. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 GEOMETRY INTEGRAL GEOMETRY G. 1. Drinfel'd Preface . . . . . . . . .
Category Theory 1991 Proceedings Of The 1991 Summer Category Theory Meeting Montreal Canada
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Author : Robert Andrew George Seely
language : en
Publisher: American Mathematical Soc.
Release Date : 1992
Category Theory 1991 Proceedings Of The 1991 Summer Category Theory Meeting Montreal Canada written by Robert Andrew George Seely and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 1992 with Mathematics categories.
Representing this diversity of the field, this book contains the proceedings of an international conference on category theory. The subjects covered here range from topology and geometry to logic and theoretical computer science, from homotopy to braids and conformal field theory. Although generally aimed at experts in the various fields represented, the book will also provide an excellent opportunity for nonexperts to get a feel for the diversity of current applications of category theory.
Encyclopedia Of Distances
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Author : Michel Marie Deza
language : en
Publisher: Springer Science & Business Media
Release Date : 2012-10-28
Encyclopedia Of Distances written by Michel Marie Deza 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 2012-10-28 with Mathematics categories.
This updated and revised second edition of the leading reference volume on distance metrics includes a wealth of new material that reflects advances in a developing field now regarded as an essential tool in many areas of pure and applied mathematics. Its publication coincides with intensifying research efforts into metric spaces and especially distance design for applications. Accurate metrics have become a crucial goal in computational biology, image analysis, speech recognition and information retrieval. The content focuses on providing academics with an invaluable comprehensive listing of the main available distances. As well as standalone introductions and definitions, the encyclopedia facilitates swift cross-referencing with easily navigable bold-faced textual links to core entries, and includes a wealth of fascinating curiosities that enable non-specialists to deploy research tools previously viewed as arcane. Its value-added context is certain to open novel avenues of research.