Search results for "Hyperplane"
showing 10 items of 23 documents
Large Scale Knowledge Matching with Balanced Efficiency-Effectiveness Using LSH Forest
2017
Evolving Knowledge Ecosystems were proposed to approach the Big Data challenge, following the hypothesis that knowledge evolves in a way similar to biological systems. Therefore, the inner working of the knowledge ecosystem can be spotted from natural evolution. An evolving knowledge ecosystem consists of Knowledge Organisms, which form a representation of the knowledge, and the environment in which they reside. The environment consists of contexts, which are composed of so-called knowledge tokens. These tokens are ontological fragments extracted from information tokens, in turn, which originate from the streams of information flowing into the ecosystem. In this article we investigate the u…
Computation of the Multivariate Oja Median
2003
The multivariate Oja median (Oja, 1983) is an affine equivariant multivariate location estimate with high efficiency. This estimate has a bounded influence function but zero breakdown. The computation of the estimate appears to be highly intensive. We consider different, exact and stochastic, algorithms for the calculation of the value of the estimate. In the stochastic algorithms, the gradient of the objective function, the rank function, is estimated by sampling observation. hyperplanes. The estimated rank function with its estimated accuracy then yields a confidence region for the true sample Oja median, and the confidence region shrinks to the sample median with the increasing number of…
Transition state ensemble optimization for reactions of arbitrary complexity.
2015
In the present work, we use Variational Transition State Theory (VTST) to develop a practical method for transition state ensemble optimization by looking for an optimal hyperplanar dividing surface in a space of meaningful trial collective variables. These might be interatomic distances, angles, electrostatic potentials, etc. Restrained molecular dynamics simulations are used to obtain on-the-fly estimates of ensemble averages that guide the variations of the hyperplane maximizing the transmission coefficient. A central result of our work is an expression that quantitatively estimates the importance of the coordinates used for the localization of the transition state ensemble. Starting fro…
Expecting the unexpected: Quantifying the persistence of unexpected hypersurfaces
2021
If $X \subset \mathbb P^n$ is a reduced subscheme, we say that $X$ admits an unexpected hypersurface of degree $t$ for multiplicity $m$ if the imposition of having multiplicity $m$ at a general point $P$ fails to impose the expected number of conditions on the linear system of hypersurfaces of degree $t$ containing $X$. Conditions which either guarantee the occurrence of unexpected hypersurfaces, or which ensure that they cannot occur, are not well understand. We introduce new methods for studying unexpectedness, such as the use of generic initial ideals and partial elimination ideals to clarify when it can and when it cannot occur. We also exhibit algebraic and geometric properties of $X$ …
Pappus type theorems for hypersurfaces in a space form
2002
In order to get further insight on the Weyl’s formula for the volume of a tubular hypersurface, we consider the following situation. Letc(t) be a curve in a space formM λ n of sectional curvature λ. LetP 0 be a totally geodesic hypersurface ofM λ n throughc(0) and orthogonal toc(t). LetC 0 be a hypersurface ofP 0. LetC be the hypersurface ofM λ n obtained by a motion ofC 0 alongc(t). We shall denote it byC PorC Fif it is obtained by a parallel or Frenet motion, respectively. We get a formula for volume(C). Among other consequences of this formula we get that, ifc(0) is the centre of mass ofC 0, then volume(C) ≥ volume(C),P),and the equality holds whenC 0 is contained in a geodesic sphere or…
Global properties of codimension two spacelike submanifolds in Minkowski space
2009
Abstract We consider codimension two spacelike submanifolds with a parallel normal field (i.e. vanishing normal curvature) in Minkowski space. We use the analysis of their contacts with hyperplanes and hyperquadrics in order to get some global information on them. As a consequence we obtain new versions of Carathéodory's and Loewner's conjectures on spacelike surfaces in 4-dimensional Minkowski space and 4-flattenings theorems for closed spacelike curves in 3-dimensional Minkowski space.
Removable sets for Sobolev spaces
1999
We study removable sets for the Sobolev space W1,p. We show that removability for sets lying in a hyperplane is essentially determined by their thickness measured in terms of a concept of p-porosity.
Optimal signed-rank tests based on hyperplanes
2005
Abstract For analysing k -variate data sets, Randles (J. Amer. Statist. Assoc. 84 (1989) 1045) considered hyperplanes going through k - 1 data points and the origin. He then introduced an empirical angular distance between two k -variate data vectors based on the number of hyperplanes (the so-called interdirections ) that separate these two points, and proposed a multivariate sign test based on those interdirections. In this paper, we present an analogous concept (namely, lift-interdirections ) to measure the regular distances between data points. The empirical distance between two k -variate data vectors is again determined by the number of hyperplanes that separate these two points; in th…
Three-qutrit entanglement and simple singularities
2016
In this paper, we use singularity theory to study the entanglement nature of pure three-qutrit systems. We first consider the algebraic variety $X$ of separable three-qutrit states within the projective Hilbert space $\mathbb{P}(\mathcal{H}) = \mathbb{P}^{26}$. Given a quantum pure state $|\varphi\rangle\in \mathbb{P}(\mathcal{H})$ we define the $X_\varphi$-hypersuface by cutting $X$ with a hyperplane $H_\varphi$ defined by the linear form $\langle\varphi|$ (the $X_\varphi$-hypersurface of $X$ is $X\cap H_\varphi \subset X$). We prove that when $|\varphi\rangle$ ranges over the SLOCC entanglement classes, the "worst" possible singular $X_\varphi$-hypersuface with isolated singularities, has…
An algebraic representation of Steiner triple systems of order 13
2021
Abstract In this paper we construct an incidence structure isomorphic to a Steiner triple system of order 13 by defining a set B of twentysix vectors in the 13-dimensional vector space V = GF ( 5 ) 13 , with the property that there exist precisely thirteen 6-subsets of B whose elements sum up to zero in V , which can also be characterized as the intersections of B with thirteen linear hyperplanes of V .