Mean and variance of vacancy for distribution of k-dimensional spheres within k-dimensional space

1984 ◽  
Vol 21 (04) ◽  
pp. 738-752
Author(s):  
Peter Hall
Keyword(s):  
Dimensional Space ◽  
Sufficient Conditions ◽  
Unit Cube ◽  
Necessary And Sufficient Conditions ◽  
Dimensional Sphere ◽  
Asymptotic Formulae ◽  
Dimensional Unit ◽  
Mean And Variance ◽  
The Mean ◽  
Necessary And Sufficient

Let n points be distributed independently within a k-dimensional unit cube according to density f. At each point, construct a k-dimensional sphere of content an. Let V denote the vacancy, or ‘volume' not covered by the spheres. We derive asymptotic formulae for the mean and variance of V, as n → ∞and an → 0. The formulae separate naturally into three cases, corresponding to nan → 0, nan → a (0 < a <∞) and nan →∞, respectively. We apply the formulae to derive necessary and sufficient conditions for V/E(V) → 1 in L2.

Mean and variance of vacancy for distribution of k-dimensional spheres within k-dimensional space

1984 ◽  
Vol 21 (4) ◽  
pp. 738-752 ◽  
Author(s):  
Peter Hall
Keyword(s):  
Dimensional Space ◽  
Sufficient Conditions ◽  
Unit Cube ◽  
Necessary And Sufficient Conditions ◽  
Dimensional Sphere ◽  
Asymptotic Formulae ◽  
Dimensional Unit ◽  
Mean And Variance ◽  
The Mean ◽  
Necessary And Sufficient

Let n points be distributed independently within a k-dimensional unit cube according to density f. At each point, construct a k-dimensional sphere of content an. Let V denote the vacancy, or ‘volume' not covered by the spheres. We derive asymptotic formulae for the mean and variance of V, as n → ∞and an → 0. The formulae separate naturally into three cases, corresponding to nan → 0, nan → a (0 < a <∞) and nan →∞, respectively. We apply the formulae to derive necessary and sufficient conditions for V/E(V) → 1 in L2.

A general canonical expression

1933 ◽  
Vol 29 (4) ◽  
pp. 465-469 ◽  
Author(s):  
J. Bronowski
Keyword(s):  
Linear Space ◽  
Dimensional Space ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
General Point ◽  
Linear Forms ◽  
Powers Of Linear Forms ◽  
Necessary And Sufficient

1. In a recent paper I established new conditions for a form φ of order n, homogeneous in r + 1 variables, to be expressible as the sum of nth powers of linear forms in these variables; and for this expression, if it exists, to be unique. These conditions, I further showed, may be stated as general theorems regarding the secant spaces of manifolds Mr in higher space, namely:Necessary and sufficient conditions that through a general point of a space N, of h (r + 1) − 1 dimensions, there passes (i) no, (ii) a unique (h − 1)-dimensional space containing h points of a manifold Mr lying in N are that(i) the space projecting a general point of Mr from the join of h − 1 general r-dimensional tangent spaces of Mr meets Mr in a curve, so that Mr cannot be so projected upon a linear space of r dimensions;(ii) the space projecting a general point of Mr from the join of h − 1 general r-dimensional tangent spaces of Mr does not meet Mr again, so that Mr can be so projected, birationally, upon a linear space of r dimensions..

scholarly journals Necessary and Sufficient Conditions for Circle Formations of Mobile Agents with Coupling Delay via Sampled-Data Control

2016 ◽  
Vol 2016 ◽  
pp. 1-8
Author(s):  
Jianwei Zhao ◽  
Hongxiang Hu ◽  
Chen Wang ◽  
Guangming Xie
Keyword(s):  
Mobile Agents ◽  
Dimensional Space ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Order System ◽  
Sampled Data ◽  
Coupling Delay ◽  
Data Control ◽  
Sampled Data Control ◽  
Necessary And Sufficient

A circle forming problem for a group of mobile agents governed by first-order system is investigated, where each agent can only sense the relative angular positions of its neighboring two agents with time delay and move on the one-dimensional space of a given circle. To solve this problem, a novel decentralized sampled-data control law is proposed. By combining algebraic graph theory with control theory, some necessary and sufficient conditions are established to guarantee that all the mobile agents form a pregiven circle formation asymptotically. Moreover, the ranges of the sampling period and the coupling delay are determined, respectively. Finally, the theoretical results are demonstrated by numerical simulations.

ON THE TEMPLATES CORRESPONDING TO CYCLE-SYMMETRIC CONNECTIVITY IN CELLULAR NEURAL NETWORKS

2002 ◽  
Vol 12 (12) ◽  
pp. 2957-2966 ◽  
Author(s):  
CHIH-WEN SHIH ◽  
CHIH-WEN WENG
Keyword(s):  
Neural Networks ◽  
Dimensional Space ◽  
Sufficient Conditions ◽  
Cellular Neural Networks ◽  
Necessary And Sufficient Conditions ◽  
Local Interaction ◽  
Two Dimensional ◽  
Linear Coupling ◽  
Symmetric Connectivity ◽  
Necessary And Sufficient

In the architecture of cellular neural networks (CNN), connections among cells are built on linear coupling laws. These laws are characterized by the so-called templates which express the local interaction weights among cells. Recently, the complete stability for CNN has been extended from symmetric connections to cycle-symmetric connections. In this presentation, we investigate a class of two-dimensional space-invariant templates. We find necessary and sufficient conditions for the class of templates to have cycle-symmetric connections. Complete stability for CNN with several interesting templates is thus concluded.

scholarly journals MEAN VALUE TYPE INEQUALITIES FOR QUASINEARLY SUBHARMONIC FUNCTIONS

2013 ◽  
Vol 55 (2) ◽  
pp. 349-368 ◽  
Author(s):  
OLEKSIY DOVGOSHEY ◽  
JUHANI RIIHENTAUS
Keyword(s):  
Sufficient Conditions ◽  
Continuous Functions ◽  
Mean Value ◽  
Necessary And Sufficient Conditions ◽  
Subharmonic Functions ◽  
Mean Value Inequality ◽  
The Mean ◽  
Mean Value Inequalities ◽  
Necessary And Sufficient ◽  
Bounded Sets

AbstractThe mean value inequality is characteristic for upper semi-continuous functions to be subharmonic. Quasinearly subharmonic functions generalise subharmonic functions. We find the necessary and sufficient conditions under which subsets of balls are big enough for the characterisation of non-negative, quasinearly subharmonic functions by mean value inequalities. Similar result is obtained also for generalised mean value inequalities where, instead of balls, we consider arbitrary bounded sets, which have non-void interiors and instead of the volume of ball some functions depending on the radius of this ball.

scholarly journals Necessary and Sufficient Conditions for Complementary Stochastic Quadratic Operators of Finite-Dimensional Simplex

2017 ◽  
Vol 1 (1) ◽  
pp. 22 ◽  
Author(s):  
Rawad Abdulghafor ◽  
Sherzod Turaev ◽  
Akram Zeki
Keyword(s):  
Dimensional Space ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Finite Dimensional Space ◽  
Quadratic Operator ◽  
Dimensional Simplex ◽  
Quadratic Stochastic Operator ◽  
Finite Dimensional ◽  
Stochastic Operator ◽  
Necessary And Sufficient

We define a complementary stochastic quadratic operator on finite-dimensional space as a new sub-class of quadratic stochastic operator. We give necessary and sufficient conditions for complementary stochastic quadratic operator.  

SURFACES IN Sn WITH PRESCRIBED GAUSS MAP AND MEAN CURVATURE VECTOR FIELD

2006 ◽  
Vol 17 (10) ◽  
pp. 1127-1143
Author(s):  
AYAKO TANAKA
Keyword(s):  
Vector Field ◽  
Mean Curvature ◽  
Sufficient Conditions ◽  
Curvature Vector ◽  
Gauss Map ◽  
Necessary And Sufficient Conditions ◽  
Mean Curvature Vector ◽  
The Mean ◽  
Necessary And Sufficient

We give relations between the Gauss map and the mean curvature vector field of a surface in the Euclidean unit n-sphere Sn. These relations are necessary and sufficient conditions for the existence of a surface in Sn with prescribed Gauss map and mean curvature vector field. We show that such surfaces can be expressed explicitly by using given data.

Sign in / Sign up

Export Citation Format

Share Document