Topic
Seven-dimensional space
About: Seven-dimensional space is a research topic. Over the lifetime, 801 publications have been published within this topic receiving 15708 citations. The topic is also known as: 7D.
Papers published on a yearly basis
1 / 2
Papers
TL;DR: A result of Johnson and Lindenstrauss shows that a set of n points in high dimensional Euclidean space can be mapped into an O(log n/e2)-dimensional Euclidesan space such that the distance between any two points changes by only a factor of (1 ± e).
Abstract: A result of Johnson and Lindenstrauss [13] shows that a set of n points in high dimensional Euclidean space can be mapped into an O(log n/e2)-dimensional Euclidean space such that the distance between any two points changes by only a factor of (1 ± e). In this note, we prove this theorem using elementary probabilistic techniques.
1,261 citations
1 May 2001
TL;DR: This work gives a novel construction of the embedding of k-dimensional Euclidean space, suitable for database applications, which amounts to computing a simple aggregate over k random attribute partitions.
Abstract: A classic result of Johnson and Lindenstrauss asserts that any set of n points in d-dimensional Euclidean space can be embedded into k-dimensional Euclidean space where k is logarithmic in n and independent of d so that all pairwise distances are maintained within an arbitrarily small factor. All known constructions of such embeddings involve projecting the n points onto a random k-dimensional hyperplane. We give a novel construction of the embedding, suitable for database applications, which amounts to computing a simple aggregate over k random attribute partitions.
1,055 citations
TL;DR: In this article, necessary and sufficient conditions are given for a set of numbers to be the mutual distances of real points in Euclidean space, and matrices are found whose ranks determine the dimension of the smallest space containing such points, and methods for determining the configuration of these points and for approximating to them by points in a space of lower dimensionality.
Abstract: Necessary and sufficient conditions are given for a set of numbers to be the mutual distances of a set of real points in Euclidean space, and matrices are found whose ranks determine the dimension of the smallest Euclidean space containing such points. Methods are indicated for determining the configuration of these points, and for approximating to them by points in a space of lower dimensionality.
845 citations
488 citations
TL;DR: In this paper, the authors considered a class of fully nonlinear parabolic evolution equations for hypersurfaces in Euclidean space and proved that any strictly convex compact initial hypersurface contracts to a point in finite time, becoming spherical in shape as the limit is approached.
Abstract: We consider a class of fully nonlinear parabolic evolution equations for hypersurfaces in Euclidean space. A new geometrical lemma is used to prove that any strictly convex compact initial hypersurface contracts to a point in finite time, becoming spherical in shape as the limit is approached. In the particular case of the mean curvature flow this provides a simple new proof of a theorem of Huisken.
377 citations
Related Topics (5)
Performance Metrics
| Year | Papers |
|---|---|
| 2025 | 3 |
| 2024 | 12 |
| 2023 | 3 |
| 2022 | 9 |
| 2021 | 1 |
| 2020 | 1 |