Results 1 to 10 of about 70 (69)
Union of Euclidean Metric Spaces is Euclidean [PDF]
Discrete Analysis, 2016 Suppose that a metric space $X$ is the union of two metric subspaces $A$ and $B$ that embed into Euclidean space with distortions $D_A$ and $D_B$, respectively. We prove that then $X$ embeds into Euclidean space with a bounded distortion (namely, with distortion at most $7D_A D_B + 2(D_A+D_B)$).
Konstantin Makarychev, Yury Makarychev
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On Coverings of Ellipsoids in Euclidean Spaces [PDF]
IEEE Transactions on Information Theory, 2004 The thinnest coverings of ellipsoids are studied in the Euclidean spaces of an arbitrary dimension n. Given any ellipsoid, the main goal is to find its /spl epsiv/-entropy, which is the logarithm of the minimum number of the balls of radius /spl epsiv/ needed to cover this ellipsoid.
Ilya Dumer +2 more
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The Space of Clouds in Euclidean Space
Experimental Mathematics, 2004 We study the space {\small $\nua{m}{d}$} of clouds in {\small $\bbr^d$} (ordered sets of m points modulo the action of the group of affine isometries). We show that {\small $\nua{m}{d}$} is a smooth space, stratified over a certain hyperplane arrangement in {\small $\bbr^m$}.
Hausmann, Jean-Claude +1 more
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Embeddings of graphs in euclidean spaces [PDF]
Discrete & Computational Geometry, 1989 zbMATH Open Web Interface contents unavailable due to conflicting licenses.
Reiterman, J., Rödl, V, Sinajová, E.
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Partitioning Euclidean space [PDF]
Discrete & Computational Geometry, 1993 The author proves the following theorem: For any fixed \(n\)-simplex \(S\) in \(\mathbb{R}^ n\) there exists a partition of \(\mathbb{R}^ n\) into countably many pieces none of which contains an \(n\)-simplex similar to \(S\). The proof uses the Axiom of Choice.
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On Distance Mapping from non-Euclidean Spaces to Euclidean Spaces [PDF]
, 2017 Most Machine Learning techniques traditionally rely on some forms of Euclidean Distances, computed in a Euclidean space (typically $$\mathbb {R}^{d}$$). In more general cases, data might not live in a classical Euclidean space, and it can be difficult (or impossible) to find a direct representation for it in $$\mathbb {R}^{d}$$.
Wei Ren +5 more
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A characterization of light open maps of Euclidean spaces into Euclidean spaces [PDF]
Proceedings of the American Mathematical Society, 1958 We will use the term open map to mean a map f of X into Y such that f(U) is open in Y for every open set U in X. A map f of Rn (Euclidean n-space) into Rm is pseudo-monotone if and only if Rn_ -f(X) has no bounded component for every closed set X in Rn such that Rm -X has no bounded component.
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A nonamenable “factor” of a Euclidean space
The Annals of Probability, 2021 23 pages, 4 ...
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Equiangular Subspaces in Euclidean Spaces [PDF]
Discrete & Computational Geometry, 2018 A set of lines through the origin is called equiangular if every pair of lines defines the same angle, and the maximum size of an equiangular set of lines in $\mathbb{R}^n$ was studied extensively for the last 70 years. In this paper, we study analogous questions for $k$-dimensional subspaces.
Balla, Igor, Sudakov, Benny
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On a class of surfaces in the euclidean space
Communications, Faculty Of Science, University of Ankara Series A1Mathematics and Statistics, 2002 This work represents a brief report on a special class of 2-D submanifolds of Euclidean spaces. For a good understanding of this paper, the reader is expected to be familiar with semiparallel \(n\)-dimensional submanifolds in \((n+d)\)-dimensional Euclidean spaces. In this sense, the reviewer recommends Thm. 2.2. of [\textit{J.
Özgür, Cihan +2 more
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