Cell Complexes, Valuations, and the Euler Relation

1970 ◽  
Vol 22 (2) ◽  
pp. 235-241 ◽  
Author(s):  
M. A. Perles ◽  
G. T. Sallee
Keyword(s):  
Euclidean Space ◽  
Hausdorff Metric ◽  
Dimensional Euclidean Space ◽  
Cell Complexes ◽  
Finite Dimensional ◽  
Finite Set ◽  
Euler Relation ◽  
Set Of Points ◽  
The Relationship ◽  
Dimensional Face

1. Recently a number of functions have been shown to satisfy relations on polytopes similar to the classic Euler relation. Much of this work has been done by Shephard, and an excellent summary of results of this type may be found in [11]. For such functions, only continuity (with respect to the Hausdorff metric) is required to assure that it is a valuation, and the relationship between these two concepts was explored in [8]. It is our aim in this paper to extend the results obtained there to illustrate the relationship between valuations and the Euler relation on cell complexes.To fix our notions, we will suppose that everything takes place in a given finite-dimensional Euclidean space X.A polytope is the convex hull of a finite set of points and will be referred to as a d-polytope if it has dimension d. Polytopes have faces of all dimensions from 0 to d – 1 and each of these is in turn a polytope. A k-dimensional face will be termed simply a k-face.

ON THE STEINER RATIO IN $\mathcal{R}_{n}$

2011 ◽  
Vol 03 (04) ◽  
pp. 473-489
Author(s):  
HAI DU ◽  
WEILI WU ◽  
ZAIXIN LU ◽  
YINFENG XU
Keyword(s):  
Euclidean Space ◽  
Spanning Tree ◽  
Minimum Spanning Tree ◽  
Dimensional Euclidean Space ◽  
Minimum Length ◽  
Steiner Ratio ◽  
Line Segments ◽  
Finite Set ◽  
Steiner Minimum Tree ◽  
Set Of Points

The Steiner minimum tree and the minimum spanning tree are two important problems in combinatorial optimization. Let P denote a finite set of points, called terminals, in the Euclidean space. A Steiner minimum tree of P, denoted by SMT(P), is a network with minimum length to interconnect all terminals, and a minimum spanning tree of P, denoted by MST(P), is also a minimum network interconnecting all the points in P, however, subject to the constraint that all the line segments in it have to terminate at terminals. Therefore, SMT(P) may contain points not in P, but MST(P) cannot contain such kind of points. Let [Formula: see text] denote the n-dimensional Euclidean space. The Steiner ratio in [Formula: see text] is defined to be [Formula: see text], where Ls(P) and Lm(P), respectively, denote lengths of a Steiner minimum tree and a minimum spanning tree of P. The best previously known lower bound for [Formula: see text] in the literature is 0.615. In this paper, we show that [Formula: see text] for any n ≥ 2.

On Vitali's Theorem for Groups of Motions of Euclidean Space

1999 ◽  
Vol 6 (4) ◽  
pp. 323-334
Author(s):  
A. Kharazishvili
Keyword(s):  
Euclidean Space ◽  
Dimensional Euclidean Space ◽  
Finite Dimensional

Abstract We give a characterization of all those groups of isometric transformations of a finite-dimensional Euclidean space, for which an analogue of the classical Vitali theorem [Sul problema della misura dei gruppi di punti di una retta, 1905] holds true. This characterization is formulated in purely geometrical terms.

On inequalities of the Tchebychev type

1963 ◽  
Vol 59 (1) ◽  
pp. 135-146 ◽  
Author(s):  
J. F. C. Kingman
Keyword(s):  
Probability Theory ◽  
Euclidean Space ◽  
Random Variable ◽  
Dimensional Euclidean Space ◽  
Real Random Variable ◽  
Finite Dimensional

1. A type of problem which frequently occurs in probability theory and statistics can be formulated in the following way. We are given real-valued functions f(x), gi(x) (i = 1, 2, …, k) on a space (typically finite-dimensional Euclidean space). Then the problem is to set bounds for Ef(X), where X is a random variable taking values in , about which all we know is the values of Egi(X). For example, we might wish to set bounds for P(X > a), where X is a real random variable with some of its moments given.

scholarly journals A Helly-type theorem on a sphere

1967 ◽  
Vol 7 (3) ◽  
pp. 323-326 ◽  
Author(s):  
M. J. C. Baker
Keyword(s):  
Euclidean Space ◽  
Convex Sets ◽  
Type Theorem ◽  
Dimensional Euclidean Space ◽  
Dimensional Sphere ◽  
Strongly Convex ◽  
Set Of Points

The purpose of this paper is to prove that if n+3, or more, strongly convex sets on an n dimensional sphere are such that each intersection of n+2 of them is empty, then the intersection of some n+1 of them is empty. (The n dimensional sphere is understood to be the set of points in n+1 dimensional Euclidean space satisfying x21+x22+ …+x2n+1 = 1.)

scholarly journals On the construction of resolving control in the problem of getting close at a fixed time moment

2021 ◽  
Vol 58 ◽  
pp. 73-93
Author(s):  
V.N. Ushakov ◽  
A.V. Ushakov ◽  
O.A. Kuvshinov
Keyword(s):  
Euclidean Space ◽  
Compact Space ◽  
Time Moment ◽  
Fixed Time ◽  
Dimensional Euclidean Space ◽  
Controlled System ◽  
Finite Dimensional ◽  
Solvability Set

The problem of getting close of a controlled system with a compact space in a finite-dimensional Euclidean space at a fixed time is studied. A method of constructing a solution to the problem is proposed which is based on the ideology of the maximum shift of the motion of the controlled system by the solvability set of the getting close problem.

scholarly journals Constructions of Maximum Few-Distance Sets in Euclidean Spaces

10.37236/8565 ◽  
2020 ◽  
Vol 27 (1) ◽  
Author(s):  
Ferenc Szöllősi ◽  
Patric R.J. Östergård
Keyword(s):  
Euclidean Space ◽  
Gröbner Basis ◽  
Dimensional Euclidean Space ◽  
Combined Approach ◽  
Euclidean Spaces ◽  
Grobner Basis ◽  
Finite Set ◽  
Distance Set ◽  
Distance Sets ◽  
Basis Computation

A finite set of vectors $\mathcal{X}$ in the $d$-dimensional Euclidean space $\mathbb{R}^d$ is called an $s$-distance set if the set of mutual distances between distinct elements of $\mathcal{X}$ has cardinality exactly $s$. In this paper we present a combined approach of isomorph-free exhaustive generation of graphs and Gröbner basis computation to classify the largest $3$-distance sets in $\mathbb{R}^4$, the largest $4$-distance sets in $\mathbb{R}^3$, and the largest $6$-distance sets in $\mathbb{R}^2$. We also construct new examples of large $s$-distance sets in $\mathbb{R}^d$ for $d\leq 8$ and $s\leq 6$, and independently verify several earlier results from the literature.

scholarly journals On targeting an integral funnel of control system at a target set in the phase space

2020 ◽  
Vol 56 ◽  
pp. 79-101
Author(s):  
V.N. Ushakov ◽  
A.V. Ushakov
Keyword(s):  
Exact Solution ◽  
Control System ◽  
Phase Space ◽  
Euclidean Space ◽  
Time Moment ◽  
Dimensional Euclidean Space ◽  
Time Interval ◽  
Finite Dimensional ◽  
Approximate Construction ◽  
Target Set

A control system in finite-dimensional Euclidean space is considered. On a given time interval, we investigate the problem of constructing an integral funnel for which a section corresponding to the last time moment of interval is equal to a target set in a phase space. Since the exact solution of such a funnel is possible only in rare cases, the question of the approximate construction of an integral funnel is being studied.

scholarly journals On the complexity of some partition problems of a finite set of points in Euclidean space into balanced clusters

2019 ◽  
Vol 488 (1) ◽  
pp. 16-20
Author(s):  
A. V. Kel’manov ◽  
A. V. Pyatkin ◽  
V. I. Khandeev
Keyword(s):  
Euclidean Space ◽  
Cluster Size ◽  
Np Completeness ◽  
Finite Set ◽  
Set Of Points

We consider some problems of partitioning a finite set of N points in d-dimension Euclidean space into two clusters balancing the value of (1) the quadratic variance normalized by a cluster size, (2) the quadratic variance, and (3) the size-weighted quadratic variance. We have proved the NP-completeness of all these problems.

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