Topic

# Euclidean space

About: Euclidean space is a research topic. Over the lifetime, 12344 publications have been published within this topic receiving 221173 citations. The topic is also known as: Euclidean vector space & finite-dimensional real vector space.

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01 Nov 1971

TL;DR: In this paper, the authors present a unified treatment of basic topics that arise in Fourier analysis, and illustrate the role played by the structure of Euclidean spaces, particularly the action of translations, dilatations, and rotations.

Abstract: The authors present a unified treatment of basic topics that arise in Fourier analysis. Their intention is to illustrate the role played by the structure of Euclidean spaces, particularly the action of translations, dilatations, and rotations, and to motivate the study of harmonic analysis on more general spaces having an analogous structure, e.g., symmetric spaces.

5,406 citations

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TL;DR: In this paper, the authors derived necessary and sufficient conditions for a solution to exist in real Euclidean space for a multivariate multivariate sample of size n as points P1, P2,..., PI in a Euclidian space and discussed the interpretation of the distance A(Pi, Pj) between the ith and jth members of the sample.

Abstract: SUMMARY This paper is concerned with the representation of a multivariate sample of size n as points P1, P2, ..., PI in a Euclidean space. The interpretation of the distance A(Pi, Pj) between the ith andjth members of the sample is discussed for some commonly used types of analysis, including both Q and R techniques. When all the distances between n points are known a method is derived which finds their co-ordinates referred to principal axes. A set of necessary and sufficient conditions for a solution to exist in real Euclidean space is found. Q and R techniques are defined as being dual to one another when they both lead to a set of n points with the same inter-point distances. Pairs of dual techniques are derived. In factor analysis the distances between points whose co-ordinates are the estimated factor scores can be interpreted as D2 with a singular dispersion matrix.

3,559 citations

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TL;DR: In this article, the authors give a simple technique for verifying the restricted isometry property for random matrices that underlies compressive sensing, and obtain simple and direct proofs of Kashin's theorems on widths of finite balls in Euclidean space.

Abstract: We give a simple technique for verifying the Restricted Isometry Property (as introduced by Candes and Tao) for random matrices that underlies Compressed Sensing. Our approach has two main ingredients: (i) concentration inequalities for random inner products that have recently provided algorithmically simple proofs of the Johnson–Lindenstrauss lemma; and (ii) covering numbers for finite-dimensional balls in Euclidean space. This leads to an elementary proof of the Restricted Isometry Property and brings out connections between Compressed Sensing and the Johnson–Lindenstrauss lemma. As a result, we obtain simple and direct proofs of Kashin’s theorems on widths of finite balls in Euclidean space (and their improvements due to Gluskin) and proofs of the existence of optimal Compressed Sensing measurement matrices. In the process, we also prove that these measurements have a certain universality with respect to the sparsity-inducing basis.

2,472 citations

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TL;DR: In this article, the authors find regular solutions of the four dimensional euclidean Yang-Mills equations and show that these solutions minimize locally the action integrals, which is finite in this case.

Abstract: We find regular solutions of the four dimensional euclidean Yang-Mills equations. The solutions minimize locally the action integrals which is finite in this case. The topological nature of the solutions is discussed.

2,241 citations

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09 May 1983

TL;DR: In this article, the authors describe the geometric theory of discrete groups and associated tesselations of the underlying space, and develop the theory of Mobius transformations in n-dimensional Euclidean space.

Abstract: Describing the geometric theory of discrete groups and the associated tesselations of the underlying space, this work also develops the theory of Mobius transformations in n-dimensional Euclidean space. These transformations are discussed as isometries of hyperbolic space and are then identified with the elementary transformations of complex analysis. A detailed account of analytic hyperbolic trigonometry is given, and this forms the basis of the subsequent analysis of tesselations of the hyperbolic plane. Emphasis is placed on the geometrical aspects of the subject and on the universal constraints which must be satisfied by all tesselations.

1,780 citations