scholarly journals Distribution of Distances between Elements in a Compact Set

Stats ◽  
2019 ◽  
Vol 3 (1) ◽  
pp. 1-15
Author(s):  
Solal Lellouche ◽  
Marc Souris
Keyword(s):  
Vector Space ◽  
Statistical Physics ◽  
Space Dimension ◽  
Compact Set ◽  
Random Set ◽  
Compact Sets ◽  
Homogeneous Poisson Process ◽  
Resolution Method ◽  
Typical Shape ◽  
Odd Dimensions

In this article, we propose a review of studies evaluating the distribution of distances between elements of a random set independently and uniformly distributed over a region of space in a normed R -vector space (for example, point events generated by a homogeneous Poisson process in a compact set). The distribution of distances between individuals is present in many situations when interaction depends on distance and concerns many disciplines, such as statistical physics, biology, ecology, geography, networking, etc. After reviewing the solutions proposed in the literature, we present a modern, general and unified resolution method using convolution of random vectors. We apply this method to typical compact sets: segments, rectangles, disks, spheres and hyperspheres. We show, for example, that in a hypersphere the distribution of distances has a typical shape and is polynomial for odd dimensions. We also present various applications of these results and we show, for example, that variance of distances in a hypersphere tends to zero when space dimension increases.

A generalization of the Pompeiu mean-value theorem to compact sets

2019 ◽  
Vol 35 (2) ◽  
pp. 147-152
Author(s):  
LARISA CHEREGI ◽  
VICUTA NEAGOS ◽  
◽  
Keyword(s):  
Continuous Function ◽  
Mean Value ◽  
Mean Value Theorem ◽  
Compact Set ◽  
Compact Sets

We generalize the Pompeiu mean-value theorem by replacing the graph of a continuous function with a compact set.

A Class of Spaces in Which Compact Sets are Finite

1981 ◽  
Vol 24 (3) ◽  
pp. 373-375 ◽  
Author(s):  
P. L. Sharma
Keyword(s):  
Hausdorff Space ◽  
Compact Set ◽  
Compact Sets

AbstractIt is shown that in a dense-in-itself Hausdorff space if every set having a dense interior is open, then every compact set is finite.

scholarly journals Spaces of Whitney Functions on Cantor-Type Sets

2002 ◽  
Vol 54 (2) ◽  
pp. 225-238 ◽  
Author(s):  
Bora Arslan ◽  
Alexander P. Goncharov ◽  
Mefharet Kocatepe
Keyword(s):  
Extension Property ◽  
Compact Set ◽  
Compact Sets ◽  
Whitney Functions ◽  
Type Sets

AbstractWe introduce the concept of logarithmic dimension of a compact set. In terms of this magnitude, the extension property and the diametral dimension of spaces Ɛ(K) can be described for Cantor-type compact sets.

scholarly journals Invariance of green equilibrium measure on the domain

Filomat ◽  
2013 ◽  
Vol 27 (4) ◽  
pp. 593-600 ◽  
Author(s):  
Stamatis Pouliasis
Keyword(s):  
Level Set ◽  
Wide Class ◽  
Equilibrium Measure ◽  
The Other ◽  
Equilibrium Potential ◽  
Compact Set ◽  
Compact Sets ◽  
Equilibrium Measures ◽  
The One

We prove that the Green equilibrium measure and the Green equilibrium energy of a compact set K relative to the domains D and ? are the same if and only if D is nearly equal to ?, for a wide class of compact sets K. Also, we prove that equality of Green equilibrium measures arises if and only if the one domain is related with a level set of the Green equilibrium potential of K relative to the other domain.

scholarly journals A New Hybrid Summarizer Based on Vector Space Model, Statistical Physics and Linguistics

2007 ◽  
pp. 872-882 ◽  
Author(s):  
Iria da Cunha ◽  
Silvia Fernández ◽  
Patricia Velázquez Morales ◽  
Jorge Vivaldi ◽  
Eric SanJuan ◽  
...  
Keyword(s):  
Vector Space ◽  
Statistical Physics ◽  
Vector Space Model ◽  
Space Model ◽  
And Linguistics

scholarly journals $$\omega ^\omega $$-Base and infinite-dimensional compact sets in locally convex spaces

2021 ◽  
Author(s):  
Taras Banakh ◽  
Jerzy Ka̧kol ◽  
Johannes Philipp Schürz
Keyword(s):  
Convex Space ◽  
Locally Convex Space ◽  
Compact Set ◽  
Compact Sets ◽  
Infinite Dimensional ◽  
Neighborhood Base ◽  
Remarkable Result ◽  
Locally Convex Topology ◽  
Locally Convex ◽  
Compact Subsets

AbstractA locally convex space (lcs) E is said to have an $$\omega ^{\omega }$$ ω ω -base if E has a neighborhood base $$\{U_{\alpha }:\alpha \in \omega ^\omega \}$$ { U α : α ∈ ω ω } at zero such that $$U_{\beta }\subseteq U_{\alpha }$$ U β ⊆ U α for all $$\alpha \le \beta $$ α ≤ β . The class of lcs with an $$\omega ^{\omega }$$ ω ω -base is large, among others contains all (LM)-spaces (hence (LF)-spaces), strong duals of distinguished Fréchet lcs (hence spaces of distributions $$D^{\prime }(\Omega )$$ D ′ ( Ω ) ). A remarkable result of Cascales-Orihuela states that every compact set in an lcs with an $$\omega ^{\omega }$$ ω ω -base is metrizable. Our main result shows that every uncountable-dimensional lcs with an $$\omega ^{\omega }$$ ω ω -base contains an infinite-dimensional metrizable compact subset. On the other hand, the countable-dimensional vector space $$\varphi $$ φ endowed with the finest locally convex topology has an $$\omega ^\omega $$ ω ω -base but contains no infinite-dimensional compact subsets. It turns out that $$\varphi $$ φ is a unique infinite-dimensional locally convex space which is a $$k_{\mathbb {R}}$$ k R -space containing no infinite-dimensional compact subsets. Applications to spaces $$C_{p}(X)$$ C p ( X ) are provided.

scholarly journals Functions with bounded variation in locally convex space

2011 ◽  
Vol 49 (1) ◽  
pp. 89-98
Author(s):  
Miloslav Duchoˇn ◽  
Camille Debiève
Keyword(s):  
Vector Space ◽  
Banach Spaces ◽  
Bounded Variation ◽  
Convex Space ◽  
Locally Convex Space ◽  
Vector Spaces ◽  
Compact Set ◽  
Locally Convex Spaces ◽  
Vector Valued ◽  
Locally Convex

ABSTRACT The present paper is concerned with some properties of functions with values in locally convex vector space, namely functions having bounded variation and generalizations of some theorems for functions with values in locally convex vector spaces replacing Banach spaces, namely Theorem: If X is a sequentially complete locally convex vector space, then the function x(・) : [a, b] → X having a bounded variation on the interval [a, b] defines a vector-valued measure m on borelian subsets of [a, b] with values in X and with the bounded variation on the borelian subsets of [a, b]; the range of this measure is also a weakly relatively compact set in X. This theorem is an extension of the results from Banach spaces to locally convex spaces.

scholarly journals Near-optimal analysis of Lasserre’s univariate measure-based bounds for multivariate polynomial optimization

2020 ◽  
Author(s):  
Lucas Slot ◽  
Monique Laurent
Keyword(s):  
Polynomial Optimization ◽  
Convex Bodies ◽  
Sums Of Squares ◽  
Geometric Condition ◽  
Compact Set ◽  
Compact Sets ◽  
Optimal Analysis ◽  
Forward Measure ◽  
Convergence Results ◽  
Push Forward

Abstract We consider a hierarchy of upper approximations for the minimization of a polynomial f over a compact set $$K \subseteq \mathbb {R}^n$$ K ⊆ R n proposed recently by Lasserre (arXiv:1907.097784, 2019). This hierarchy relies on using the push-forward measure of the Lebesgue measure on K by the polynomial f and involves univariate sums of squares of polynomials with growing degrees 2r. Hence it is weaker, but cheaper to compute, than an earlier hierarchy by Lasserre (SIAM Journal on Optimization 21(3), 864–885, 2011), which uses multivariate sums of squares. We show that this new hierarchy converges to the global minimum of f at a rate in $$O(\log ^2 r / r^2)$$ O ( log 2 r / r 2 ) whenever K satisfies a mild geometric condition, which holds, eg., for convex bodies and for compact semialgebraic sets with dense interior. As an application this rate of convergence also applies to the stronger hierarchy based on multivariate sums of squares, which improves and extends earlier convergence results to a wider class of compact sets. Furthermore, we show that our analysis is near-optimal by proving a lower bound on the convergence rate in $$\varOmega (1/r^2)$$ Ω ( 1 / r 2 ) for a class of polynomials on $$K=[-1,1]$$ K = [ - 1 , 1 ] , obtained by exploiting a connection to orthogonal polynomials.

Degenerate Cases of Uniform Approximation by Solutions of Systems with Surjective Symbols

1993 ◽  
Vol 45 (4) ◽  
pp. 740-757 ◽  
Author(s):  
P. M. Gauthier ◽  
N. N. Tarkhanov
Keyword(s):  
Sobolev Space ◽  
Differential Equations ◽  
Uniform Approximation ◽  
Compact Set ◽  
Compact Sets ◽  
System Of Differential Equations ◽  
Vector Valued Function ◽  
Vector Valued ◽  
Degenerate Cases

AbstractWe prove that each (vector-valued) function in Sobolev space on a compact set K, which in the interior K0 of K satisfies a system of differential equations, can be approximated by solutions in a neighbourhood of K plus sums of potentials of measures supported on the boundary of K. We discuss the particular case where, for all compact sets K, one can dispense with potentials in such approximations

scholarly journals Some Remarks on Symmetric and Frobenius Algebras

1960 ◽  
Vol 16 ◽  
pp. 65-71 ◽  
Author(s):  
J. P. Jans
Keyword(s):  
Vector Space ◽  
Space Dimension ◽  
Symmetric Algebra ◽  
Dimensional Case ◽  
Infinite Dimensional ◽  
Symmetric Algebras ◽  
Finite Dimensional ◽  
Frobenius Algebras ◽  
Finite Dimensional Case

In [5] we defined the concepts of Frobenius and symmetric algebra for algebras of infinite vector space dimension over a field. It was shown there that with the introduction of a topology and the judicious use of the terms continuous and closed, many of the classical theorems of Nakayama [7, 8] on Frobenius and symmetric algebras could be generalized to the infinite dimensional case. In this paper we shall be concerned with showing certain algebras are (or are not) Frobenius or symmetric. In Section 3, we shall see that an algebra can be symmetric or Frobenius in “many ways”. This is a problem which did not arise in the finite dimensional case.

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