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# Topic: Category of metric spaces

 Theory and Applications of Categories Pullback and finite coproduct preserving functors between categories of permutation representations: Corrigendum and Addendum Pullback and finite coproduct preserving functors between categories of permutation representations On the monadicity of categories with chosen colimits www.tac.mta.ca /tac   (1373 words)

 METRIC SPACE FACTS AND INFORMATION In mathematics, a metric space is a set where a notion of distance between elements of the set is defined. Similarly, in 3D, the metrics on the surface of a polyhedron include the ordinary metric, and the distance over the surface; a third metric on the edges of a polyhedron is one where the "paths" are the edges. An important consequence is that every metric space admits partitions_of_unity and that every continuous real-valued function defined on a closed subset of a metric space can be extended to a continuous map on the whole space (Tietze_extension_theorem). www.gottagetflowers.com /metric_space   (1609 words)

 wikien.info: Main_Page   (Site not responding. Last check: 2007-10-27) In metric spaces, one can talk about limits of sequences; a metric space in which every Cauchy sequence has a limit is said to be complete. The original space M is thus isometrically isomorphic to a subspace of a complete metric space, and it is usually identified with this subspace. Other embedding constructions show that every metric space is isometrically isomorphic to a closed subset of some normed vector space and that every complete metric space is isometrically isomorphic to a closed subset of some Banach space. pardus.info /index.php?title=Metric_space   (1652 words)

 Metric_space   (Site not responding. Last check: 2007-10-27) The geometry of the space depends on the metric chosen and by using a different metric we can construct interesting non-Euclidean geometries which are used in the theory of general relativity. A metric space is a 2-tuple (X,d) where X is a set and d is a metric on X. So to decide in what sense two metrics spaces are equivalent we have to discuss continuous functions between them (morphisms preserving the topology of the metric spaces). www.apawn.com /search.php?title=Metric_space   (1431 words)

 Metric space - Wikipedia, the free encyclopedia   (Site not responding. Last check: 2007-10-27) A metric d on M is called intrinsic if any two points x and y in M can be joint by a curve with length arbitrarily close to d(x, y). A metric space M is called bounded if there exists some number r > 0 such that d(x,y) ≤ r for all x and y in M (not to be confused with "finite", which refers to the number of elements, not to how far the set extends; finiteness implies boundedness, but not conversely). Every such metric can be rescaled to a finite metric (using d'(x, y) = d(x, y) / (1 + d(x, y)) or d''(x, y) = min(1, d(x, y))) and the two concepts of metric space are therefore equivalent as far as notions of topology (such as continuity or convergence) are concerned. xahlee.org /_p/wiki/Metric_spaces.html   (1390 words)

 PH. D. THESIS In parallel, with the aim of completing for frames a picture analogous to the one for spaces, we characterize uniform structures in terms of "gauge structures", that is, families of metric diameters satisfying certain axioms. The language of category theory has proved to be an adequate tool in the approach to the type of conceptual problems we are interested in, namely in the selection of the best axiomatizations for some frame structures and in the study of the relationship between them. Category theory provides a language to formulate such questions with the kind of precision needed to analyse advantages and disadvantages of various alternatives. www.mat.uc.pt /~picado/publicat/Intr.html   (2247 words)

 Journal of Vision - Metric category spaces of biological motion, by Giese, Thornton, & Edelman   (Site not responding. Last check: 2007-10-27) Converging experimental evidence suggests that shape categories are mentally represented as continuous spaces that preserve the parametric similarities among individual object shapes. The recovered 2D configurations in the reconstructed perceptual space were quantitatively compared with the configurations in the original linear morphing space. The recovered metric structure closely matched the original metric structure in the morphing space. www.journalofvision.org /3/9/83   (320 words)

 [No title] In particular: F is a class, hence {F} is a conglomerate.] A metacategory_is defined in the same way as a category except that the obj* *ects and the morphisms are allowed to be conglomerates and the requirement that the conglomerate of mo* *rphisms between two objects be a set is dropped. In a* *ny category, a morphism is an isomorphism iff it is both a monomorphism and an ext* *remal epimorphism iff it is both an extremal monomorphism and an epimorphism. A category C that has a zero object is abelian iff it has pullbacks, pushouts,* * and ev- ery monomorphism (epimorphism) is the kernel (cokernel) of a morphism. hopf.math.purdue.edu /WarnerG/warner-book.txt   (13171 words)

 A relation between Linear Metric Spaces and Normed Spaces   (Site not responding. Last check: 2007-10-27) ro^.ng ra affine spaces with a metric derived from an > arbitrary norm. :-) Thu+.c ra ma` no'i the category of linear metric spaces cha(?ng kha'c gi` the category of normed spaces. Young students dda~ bie^'t la` each normed space is linear metric space, ro~ nhu+ ban nga`y, indeed the metric is defined by, for example, d(x,y) := x - y. www-users.cs.umn.edu /~mnguyen/vnsa/Feb-1999/read.cgi?00203   (254 words)

 54E: Spaces with richer structures especially metric spaces A set X with a metric on it is called a metric space since the collection of unions of balls is a topology on X. A major theme in research is to investigate the influence a metric has on the underlying topology. Among those mentioned in the MSC we observe semimetric spaces (topological spaces whose topology is given by the balls with respect to a semimetric -- a distance function not meeting the triangle inequality); cosmic spaces (continuous image of a separable metric space), and probabilistic metric spaces. The theory of metric spaces is almost always presented with an eye towards its connections either with general topology or with analysis; this is true both at the beginning undergraduate level and at advanced levels. www.math.niu.edu /~rusin/known-math/index/54EXX.html   (1288 words)

 [No title]   (Site not responding. Last check: 2007-10-27) Remined that the collection of all metric spaces is a category and Lipschitz maps are morphisms in this category (bi-Lipschitz maps are isomorphisms). In general, the question of Lipschitz classification with respect to the induced metric is more complicated then the corresponding question with respect to the length metric. We prove in \S2-\S4 that the bi-Lipschitz equivalence in the category of germs of semialgebraic curves corresponds to the isomorphism in the category of H\"{o}lder Semicomplexes. home.imf.au.dk /esn/preprints/152   (1972 words)

 Functional Analysis   (Site not responding. Last check: 2007-10-27) All these constructs are treated as initial or final objects in some categories of diagrams, in order to see how this trick gives a trivial proof of uniqueness. The relevance of this to functional analysis is that projective and direct limits really do play a much larger role than often imagined, and this viewpoint can be a big help in understanding proofs. [ Catalogue of topological vector spaces ] This is a collection of some important and `popular' topological vectorspaces which arise in practice. www.math.umn.edu /~garrett/m/fun/syl.shtml   (581 words)

 categorytheory.html The Category or Groups, Rings, Fields etc - The Category of Sets - onto is equivalent to epimorphism, and one to one is equivalent to monomorphism. One can find examples of monomorphisms that are not one to one as Set maps. www.umsl.edu /~siegelj/SetTheoryandTopology/categorytheory.html   (67 words)

 Cauchy.html We would like to discuss the Category of Complete Metric Spaces. There is an important sub-class of continuous functions which do preserve Cauchy sequences and, in fact, are the continous functions on an important sub-Category of Complete Metric Spaces. To verify that complete metric spaces and uniformly continuous maps form a category we need to check the the composition of uniformly continuous maps is uniformly continuous. www.umsl.edu /~siegel/SetTheoryandTopology/Cauchy.html   (236 words)

 Short map   (Site not responding. Last check: 2007-10-27) In mathematics, a short map, or nonexpansive map, is a special kind of continuous function between metric spaces. Specifically, suppose that X and Y are metric spaces and f is a function from X to Y. Thus metric spaces and short maps form a category Met; Met is a subcategory of the category of metric spaces and Lipschitz maps, and the isomorphisms in Met are the isometries. www.worldhistory.com /wiki/S/Short-map.htm   (267 words)

 Nhu Nguyen   (Site not responding. Last check: 2007-10-27) [4] Shape of a metric space in the category of metric spaces and uniformly continuous maps, Bull. [5] Fundamental deformation retracts and weak deformation retracts in the category of metric spaces and uniformly continuous maps, Bull. [9] Lipchitz extensions and Lipchitz retractions in metric spaces, (with N. Khue). www.math.nmsu.edu /~nnguyen/VITAE3.htm   (825 words)

 De Bakker-Zucker Processes Revisited (ResearchIndex)   (Site not responding. Last check: 2007-10-27) Abstract: The sets of compact and of closed subsets of a metric space endowed with the Hausdorff metric are studied. Both give rise to a functor on the category of 1-bounded metric spaces and nonexpansive functions. It is shown that the former functor has a terminal coalgebra and that the latter does not. citeseer.ist.psu.edu /breugel99de.html   (442 words)

 A Fixed Point Theorem in a Category of Compact Metric Spaces   (Site not responding. Last check: 2007-10-27) Various results appear in the literature for deriving existence and uniqueness of fixed points for endofunctors on categories of complete metric spaces. All these results are proved for contracting functors which satisfy some further requirements, depending on the category in question. Following a new kind of approach, based on the notion of eta-isometry, we show that the sole hypothesis of contractivity is enough for proving existence and uniqueness of fixed points for endofunctors on the category of compact metric spaces and embedding-projection pairs. www.di.unipi.it /~baldan/Papers/Abstract/Fixed.html   (94 words)

 [No title]   (Site not responding. Last check: 2007-10-27) Included are several foundational and introductory papers developing the methodology of metric semantics, studies on the comparative semantics of parallel object-oriented and logic programming, and papers on full abstraction and transition system specifications. The approach is flexible in that both linear time, branching time (or bisimulation) and intermediate models can be handled, as well as schematic and interpreted elementary actions. The reprints are preceded by an extensive introduction surveying related work on metric semantics. www.worldscibooks.com /compsci/1720.txt   (170 words)

 Oxford University Press This is the first book to treat all three concepts as a special case of the concept of approach spaces. This theory provides an answer to natural questions in the interplay between topological and metric spaces by introducing a uniquely well suited supercategory of TOP and MET. The book explains the richness of approach structures in great detail; it provides a comprehensive explanation of the categorical set-up, develops the basic theory and provides many examples, displaying links with various areas of mathematics such as approximation theory, probability theory, analysis and hyperspace theory. www.oup.com /ca/isbn/0-19-850030-0   (243 words)

 OUP: Approach Spaces: Lowen In topology the three basic concepts of metrics, topologies and uniformities have been treated so far as separate entities by means of different methods and terminology. This is the first book to treat all three concepts as a special case of the concept of approach spaces developed previously by the author in a number of research articles. The specification in this catalogue, including without limitation price, format, extent, number of illustrations, and month of publication, was as accurate as possible at the time the catalogue was compiled. www.oup.co.uk /isbn/0-19-850030-0   (296 words)

 Table of contents for Library of Congress control number 99025755   (Site not responding. Last check: 2007-10-27) Table of contents for Introduction to the analysis of normed linear spaces / J.R. Giles. The spectral theorem for compact normal operators on Hilbert space 20. The spectral theorem for compact operators on Hilbert space Appendices. www.loc.gov /catdir/toc/cam027/99025755.html   (106 words)

 Practical Sailing Solutions - Products Really a very fine polish, actually designed to be a finishing material or compound that would typically be used after more agressive rubbing compounds. Category: Skin Care » Bite & Sting Relief I was attracted at first because these wrenches were constructed of stainless steel, but I was skeptical at first of the utility of the articulating mechanism. shop.goebms.com /ps/search.html   (1205 words)

 Information and Computation Bibliography   (Site not responding. Last check: 2007-10-27) We further strengthen these logical principles to deal with contexts and prove that such strengthening is valid when the (abstract) logic we consider is contextually functionally complete. The abstract is also available as a LaTeX file, a DVI file, or a PostScript file. Solving reflexive domain equations in a category of complete metric spaces. theory.lcs.mit.edu /~iandc/References/hermidaj1998:107.html   (305 words)

 Windows Live Gallery - Author Page Use the arrow keys to move your ship and press the space bar to shoot. The object of the game is to force all the moving balls into the blue area, while using the minimum steps possible. You can remove any obstacle by clicking on it or create new ones by clicking on an empty space. gallery.live.com /author.aspx?a=9e14ee0c-87a1-4a17-bb20-7b6e7929fe72   (1017 words)

 On the definition of the strong shape category   (Site not responding. Last check: 2007-10-27) On the definition of the strong shape category In this paper we show how it is possible to define the strong shape category for general topological spaces, using only the strong shape category of metric spaces, and the so called generalized shape theories. Proepireflector, reflective functor, generalized shape theory, homotopy preservation, strong shape category. www.math.hr /glasnik/vol_32/no1_14.html   (73 words)

 Top 1-50 Business & Finance programs at GoldSofts.com The unique space optimizer designed to help you plan quickly and easily the best compact arrangement of a number of different size 3D rectangular objects within one or more 3D rectangular enclosures: containers, trucks, trailers, pallets, boxes. MITCalc support 2D and 3D CAD systems, Imperial and Metric units and many international standards (ANSI, ISO, DINâ¦.). Application supports Imperial and Metric units, is based on ANSI/AGMA and ISO/DIN standards and support many 2D and 3D CAD systems www.goldsofts.com /category/2/0/1/0.html   (1645 words)

 MATHEMATICAL STRUCTURES RESEARCH Research topics include mathematical models and theories in the empirical sciences, models and theories in mathematics, category theory, and the use of mathematical structures in theoretical computer science. Cambridge University Press, 1994.[The Language of Categories, Limits, Adjoint Functors, Generatorsand Projectives, Categories of Fractions, Flat Functors and CauchyCompleteness, Bicategories and Distributors, Internal CategoryTheory] Herrlich, Horst and Strecker, George E. Category Theory.Allyn and Bacon, 1973 www.mmsysgrp.com /mathstrc.htm   (365 words)

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