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# Topic: Metric space

###### In the News (Tue 23 Apr 19)

 Metric space - Wikipedia, the free encyclopedia 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. Metric spaces are paracompact Hausdorff spaces and hence normal (indeed they are perfectly normal). en.wikipedia.org /wiki/Metric_space   (1957 words)

 Complete space - Wikipedia, the free encyclopedia In mathematical analysis, a metric space M is said to be complete (or Cauchy) if every Cauchy sequence of points in M has a limit that is also in M. For any metric space M, one can construct a complete metric space M' (which is also denoted as M with a bar over it), which contains M as a dense subspace. Completely metrizable spaces can be characterized as those spaces which can be written as an intersection of countably many open subsets of some complete metric space. en.wikipedia.org /wiki/Complete_space   (1204 words)

 Metric space : Pseudometric space A metric space in which every Cauchy sequence has a limit is said to be complete. If M is a connected Riemannian manifold, then we can turn M into a metric space by defining the distance of two points as the infimum of the lengths of the paths (continuously differentible curves) connecting them. 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). www.fastload.org /ps/Pseudometric_space.html   (1193 words)

 PlanetMath: sphere (metric space) This generalizes the notion of spheres to metric spaces. Note that the sphere in a metric space need not look like a sphere in Euclidean space. This is version 3 of sphere (metric space), born on 2004-11-04, modified 2005-05-02. planetmath.org /encyclopedia/SphereMetricSpace.html   (113 words)

 PlanetMath: metric space formed by these open sets is called the metric topology, and in fact the open sets form a basis for this topology (proof). More generally, any normed vector space has an underlying metric space structure; when the vector space is finite dimensional, the resulting metric space is isomorphic to Euclidean space. This is version 9 of metric space, born on 2001-10-25, modified 2005-11-28. planetmath.org /encyclopedia/MetricSpace.html   (195 words)

 Temporal coding: metric space analysis   (Site not responding. Last check: 2007-11-02) The foundation of the approach is the construction of several families of novel distances (metrics) between neuronal impulse trains. Rather, the proposed metrics formalize physiologically-based hypotheses for what aspects of the firing pattern might be stimulus-dependent, and make essential use of the point process nature of neural discharges. The correct information, as calculated via the spike metrics, should be approximately twice as high for the parameters given in the text, which specified phases of {0, pi/2, pi, and 3*pi/2}. www-users.med.cornell.edu /~jdvicto/vipu97.html   (344 words)

 Similarity Search - The Metric Space Approach Similarity Search-The Metric Space Approach focuses on efficient ways to locate user-relevant information in collections of objects, the similarity of which is quantified using a pairwise distance measure. After describing the most popular centralized disk-based metric indexes, we present approximation techniques are presented as a way to significantly speed up search time at the expense of some imprecision in query results, and finally we discuss parallel and distributed disk-based metric indexes. After defining a metric space we show examples of several distance measures which are used for searching in diverse data collections. www.nmis.isti.cnr.it /amato/similarity-search-book   (929 words)

 MoBIoS and Metric Space Indexing   (Site not responding. Last check: 2007-11-02) It is widely understood that the growth of biological data demands that O(log n) indexing structures be developed in order to attain scalable performance of biological databases. Metric space indexing techniques are often cited as a method to achieve that goal [ The MoBIoS metric space index requires a distance function to be defined over the search space. www.cs.utexas.edu /users/smriti/ch391l/project/final_project/node3.html   (176 words)

 Is Perceptual Color Space a Metric Space?   (Site not responding. Last check: 2007-11-02) The three formal conditions for a metric space can be reformulated as the following assumptions for the existence of a perceptual color metric space. A stronger form of a metric space is a Euclidean metric space. A weaker form of a metric space is a Riemann space. www.nd.edu /~sboker/ColorVision2/node12.html   (284 words)

 Analysis WebNotes: Chapter 05, Class 21 We now turn to a number of examples, which relate the modes of convergence from the examples of the last chapter to metric spaces. The following proposition (as well as being an important fact) is a useful exercise in how to use the axioms of a metric space in proofs. You'll find that it plays much the same role in studying convergence in general metric spaces, as the Sandwich Lemma did for convergence in the real numbers. www.math.unl.edu /~webnotes/classes/class21/class21.htm   (347 words)

 Convergence in a metric space Just as a convergent sequence in R can be thought of as a sequence of better and better approximtions to a limit, so a sequence of "points" in a metric space can approximate a limit here. For R with its usual metric this is the same as before. The situation for infinite-dimensional spaces of sequences or functions is different as we will see in the next section. www-groups.dcs.st-and.ac.uk /~john/analysis/Lectures/L16.html   (247 words)

 Continuous function on compact metric space   (Site not responding. Last check: 2007-11-02) Borel Measures on Compact Metric Spaces are Regular... 54E: Spaces with richer structures especially metric spaces... Metric space -- Facts, Info, and Encyclopedia article... www.scienceoxygen.com /math/366.html   (112 words)

 Citebase - Space of spaces as a metric space   (Site not responding. Last check: 2007-11-02) Based on the scheme of the spectral representation of geometry, we construct a space of all compact Riemannian manifolds equipped with the spectral measure of closeness. We first introduce the spectral distance, which is a measure of closeness between spaces defined in terms of the spectra of the Laplacian. We derive the evolution equations for the spectra of the Universe. citebase.eprints.org /cgi-bin/citations?id=oai:arXiv.org:gr-qc/9908078   (1324 words)

 Metric spaces The technical term for a distance function is a metric, and a space equipped with a metric is called a metric space. We may quickly prove that with this distance function code-space qualifies as a metric space. Once we have a metric it is possible to reproduce many of the concepts that occur in calculus, in particular the central idea of limit. www.math.okstate.edu /mathdept/dynamics/lecnotes/node33.html   (554 words)

 The Completion of a Metric Space   (Site not responding. Last check: 2007-11-02) Given a metric space s, let t be the set of Cauchy sequences taken from s. Turn t into a proper metric space by clumping equivalent sequences together. The result is a metric space, the completion of s. www.mathreference.com /top-ms,comp.html   (179 words)

 Citebase - Geometry in Urysohn's universal metric space   (Site not responding. Last check: 2007-11-02) Citebase - Geometry in Urysohn's universal metric space Recently, much interest was devoted to the Urysohn universal metric space U and its isometries; this paper is a contribution to this field of research. We also answer in the negative a question of Clemens, proving that any Polish metric space is isometric to the set of fixed points of some isometry of U. Comment: v2: corrected some grammatical errors and typos, and added a part dealing with properties of the sets of fixed points of isometries citebase.eprints.org /cgi-bin/citations?id=oai:arXiv.org:math/0505508   (403 words)

 www.myspace.com/metricband I'm not sure where I heard this but Metric makes smoking weed better, not vice versa. Metric has received the official "Caleb Danger Dunn" seal of aproval. My friend and I Emily and the lead singer from The Sounds divided between the two of us...the claim has been made, and now if any other man tries to take either, there gonna be some popped caps. www.myspace.com /metricband   (654 words)

 Pattern Recognition Method for Metric Space by Four Points Embedding (ResearchIndex)   (Site not responding. Last check: 2007-11-02) On the other hand in the metric space, Nearest Neighbor method and K-Nearest Neighbor method are frequently used without defining any prototypes. In this paper, we propose a new pattern recognition method for the metric space that can use prototypes which are the centroid of any three patterns in a class. This method is based on the theorem that four points of any metric space can be... citeseer.ist.psu.edu /28039.html   (318 words)

 Clustering Proteins in Non-metric Space   (Site not responding. Last check: 2007-11-02) Typical clustering algorithms assume that the measure of distance between proteins satisfies the axioms of a metric. We examine hierarchical clustering in a non-metric space with multiple membership. The goal of clustering in a non-metric space and allowing each protein the possibility of membership in multiple clusters challenges common assumptions and influences the effectiveness of typical techniques. www.iscb.org /ismb2003/posters/poulinATcs.ualberta.ca_118.html   (131 words)

 Probabilistic Tracking in a Metric Space   (Site not responding. Last check: 2007-11-02) Mouth tracking, train and test on same person (different sequences), metric is L-2 image distance: ml2.mpg. Mouth tracking, train and test on same person (different sequences), metric is shuffle distance: mshuffle.mpg. Tracking a ballet dancer (training and test sequence are the same): ballet0.mpg. research.microsoft.com /vision/papers/ICCV2001ToyamaBlake   (77 words)

 Hilbert Space   (Site not responding. Last check: 2007-11-02) As the example above shows, the space of rational numbers, with the usual notion of distance, is not a complete metric space. I1, as a metric space with a "distance between functions f and g" defined by You should now compare these representations with those for a finite dimensional vector space, and convince yourself that these two sets are formally identical. jcbmac.chem.brown.edu /baird/QuantumPDF/Tan_on_Hilbert_Space.html   (1211 words)

 PRM  operating in a metric space The path tiling result given in the previous (sub?)section can be generalized to hold for a general metric space. We also require that paths can be defined in the space. Definition 2.0.1 A geodesic in a metric space is a length minimizing curve. www.cs.rice.edu /CS/Robotics/robotics/analysis/node2.html   (174 words)

 Unsupervised Clustering using Metric Space Connectedness (ResearchIndex)   (Site not responding. Last check: 2007-11-02) Abstract: In metric space theory connectedness can be described in terms of a mapping of sets onto the real axis. This function is essentially a labelling function which clustering methods approximate. With the use of metric space theory a proposition and a conjecture are given which are implemented to perform unsupervised clustering which lacks many of the limitations of other, model-based, methods. citeseer.ist.psu.edu /191894.html   (214 words)

 Estimating Relative Density on a Metric Space ,..., be stationary and ergodic random variables with values in a metric space M with distance d, let P(A) = P(X This kinds of measures may be of considerable interest and examples where they arise can be given. James MacQueen, "Estimating Relative Density on a Metric Space" (January 1, 2003). repositories.cdlib.org /uclastat/papers/2003010110   (291 words)

 Sublinear Time Algorithms for Metric Space Problems - Indyk (ResearchIndex)   (Site not responding. Last check: 2007-11-02) The key property of our algorithms is that their running time is linear in the number of metric space points. As the full specification o`f an n-point metric space is of size \Theta(n 2), the complexity of our algorithms is... On Approximate Nearest Neighbors in Non-Euclidean Spaces - Indyk (1998) sherry.ifi.unizh.ch /202192.html   (476 words)

 Nearest Neighbor Search and Metric Space Dimensions A metric space `(U,D)` has `D(x,y) ge 0` and `D(x,x)=0` for all `x,y in U`, and also: Suppose there is also a measure `mu`, so have a metric measure space `(U,D,mu)` In Euclidean space, fast estimates of integral can be done with bucketing [BF98] cm.bell-labs.com /who/clarkson/nn_survey/t/t.xml   (927 words)

 metric space from FOLDOC   (Site not responding. Last check: 2007-11-02) A set of points together with a function, d, called a metric function or distance function. The sum of the lengths of two sides of a triangle is equal to or exceeds the length of the third side. Nearby terms: me too « metre « metric « metric space » Metropolitan Area Network » M-expression LISP » MFC ftp.sunet.se /foldoc/foldoc.cgi?metric+space   (133 words)

 Metric Space Embeddings   (Site not responding. Last check: 2007-11-02) There is a rich theory of metric space embeddings; these methods are now being applied to algorithms that deal with super-fast data streams. I will describe such applications in spaces that deal with vectors, strings and trees. His current interest is in massive data set analysis and data streaming algorithms. www.cs.uvm.edu /research/researchday-03/Muthukrishnan.html   (115 words)

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