scholarly journals On Euclidean Designs and Potential Energy

10.37236/8 ◽  
2012 ◽  
Vol 19 (1) ◽  
Author(s):  
Tsuyoshi Miezaki ◽  
Makoto Tagami
Keyword(s):  
Linear Programming ◽  
Lower Bound ◽  
Euclidean Space ◽  
Potential Energy ◽  
Harmonic Analysis ◽  
Upper Bound ◽  
Type Inequality ◽  
Linear Programming Bound ◽  
Concentric Spheres ◽  
Finite Set

We study Euclidean designs from the viewpoint of the potential energy. For a finite set in Euclidean space, we formulate a linear programming bound for the potential energy by applying harmonic analysis on a sphere. We also introduce the concept of strong Euclidean designs from the viewpoint of the linear programming bound, and we give a Fisher type inequality for strong Euclidean designs. A finite set on Euclidean space is called a Euclidean $a$-code if any distinct two points in the set are separated at least by $a$. As a corollary of the linear programming bound, we give a method to determine an upper bound on the cardinalities of Euclidean $a$-codes on concentric spheres of given radii. Similarly we also give a method to determine a lower bound on the cardinalities of Euclidean $t$-designs as an analogue of the linear programming bound.

scholarly journals Matrix Games with Interval-Valued 2-Tuple Linguistic Information

Games ◽  
2018 ◽  
Vol 9 (3) ◽  
pp. 62 ◽  
Author(s):  
Anjali Singh ◽  
Anjana Gupta
Keyword(s):  
Linear Programming ◽  
Lower Bound ◽  
Real World ◽  
Upper Bound ◽  
Duality Theorem ◽  
Matrix Game ◽  
Matrix Games ◽  
Linguistic Information ◽  
The Real ◽  
Interval Valued

In this paper, a two-player constant-sum interval-valued 2-tuple linguistic matrix game is construed. The value of a linguistic matrix game is proven as a non-decreasing function of the linguistic values in the payoffs, and, hence, a pair of auxiliary linguistic linear programming (LLP) problems is formulated to obtain the linguistic lower bound and the linguistic upper bound of the interval-valued linguistic value of such class of games. The duality theorem of LLP is also adopted to establish the equality of values of the interval linguistic matrix game for players I and II. A flowchart to summarize the proposed algorithm is also given. The methodology is then illustrated via a hypothetical example to demonstrate the applicability of the proposed theory in the real world. The designed algorithm demonstrates acceptable results in the two-player constant-sum game problems with interval-valued 2-tuple linguistic payoffs.

scholarly journals Upper bounds for packings of spheres of several radii

2014 ◽  
Vol 2 ◽  
Author(s):  
DAVID DE LAAT ◽  
FERNANDO MÁRIO DE OLIVEIRA FILHO ◽  
FRANK VALLENTIN
Keyword(s):  
Linear Programming ◽  
Euclidean Space ◽  
Semidefinite Programming ◽  
Unit Sphere ◽  
Upper Bound ◽  
Convex Bodies ◽  
Classical Problem ◽  
Upper Bounds ◽  
Sphere Packings ◽  
Spherical Caps

AbstractWe give theorems that can be used to upper bound the densities of packings of different spherical caps in the unit sphere and of translates of different convex bodies in Euclidean space. These theorems extend the linear programming bounds for packings of spherical caps and of convex bodies through the use of semidefinite programming. We perform explicit computations, obtaining new bounds for packings of spherical caps of two different sizes and for binary sphere packings. We also slightly improve the bounds for the classical problem of packing identical spheres.

scholarly journals Large Equiangular Sets of Lines in Euclidean Space

10.37236/1533 ◽  
2000 ◽  
Vol 7 (1) ◽  
Author(s):  
D. De Caen
Keyword(s):  
Lower Bound ◽  
Positive Integer ◽  
Euclidean Space ◽  
Upper Bound ◽  
Equiangular Lines

A construction is given of ${{2}\over {9}} (d+1)^2$ equiangular lines in Euclidean $d$-space, when $d = 3 \cdot 2^{2t-1}-1$ with $t$ any positive integer. This compares with the well known "absolute" upper bound of ${{1}\over {2}} d(d+1)$ lines in any equiangular set; it is the first known constructive lower bound of order $d^2$ .

An Expansion Property of Boolean Linear Maps

2016 ◽  
Vol 31 ◽  
pp. 381-407 ◽  
Author(s):  
Yaokun Wu ◽  
Zeying Xu ◽  
Yinfeng Zhu
Keyword(s):  
Dynamical System ◽  
Lower Bound ◽  
Upper Bound ◽  
Linear Dynamical Systems ◽  
Linear Map ◽  
Expansion Property ◽  
Linear Maps ◽  
Multilinear Maps ◽  
Finite Set ◽  
Upper Bound Estimate

Given a finite set $K$, a Boolean linear map on $K$ is a map $f$ from the set $2^K$ of all subsets of $K$ into itself with $f(\emptyset )=\emptyset$ such that $f(A\cup B)=f(A)\cup f(B)$ holds for all $A,B\in 2^K$. For fixed subsets $X, Y$ of $K$, to predict if $Y$ is reachable from $X$ in the dynamical system driven by $f$, one can assume the existence of nonnegative integers $h$ with $f^h(X)=Y$, find an upper bound $\alpha$ for the minimum of all such assumed integers $h$, and test if $Y$ really appears in $f^0(X), \ldots, f^\alpha(X)$. In order to get such an upper bound estimate, this paper establishes an expansion property for the Boolean linear map $f$. Namely, the authors find a lower bound on the size of $f^h(X)$ for any nonnegative integer $h$. Besides presenting several direct applications of the derived expansion property, this paper collects some related problems on Boolean linear dynamical systems, including problems on primitive multilinear maps and inhomogeneous topological Markov chains.

On some percolation results of J. M. Hammersley

1979 ◽  
Vol 16 (03) ◽  
pp. 526-540 ◽  
Author(s):  
J. G. Oxley ◽  
D. J. A. Welsh
Keyword(s):  
Lower Bound ◽  
Upper Bound ◽  
Percolation Theory ◽  
Fkg Inequality ◽  
Percolation Probability ◽  
Finite Set ◽  
The Fkg Inequality ◽  
Arbitrary Graphs

We examine how much classical percolation theory on lattices can be extended to arbitrary graphs or even clutters of subsets of a finite set. In the process we get new short proofs of some theorems of J. M. Hammersley. The FKG inequality is used to get an upper bound for the percolation probability and we also derive a lower bound. In each case we characterise when these bounds are attained.

On some percolation results of J. M. Hammersley

1979 ◽  
Vol 16 (3) ◽  
pp. 526-540 ◽  
Author(s):  
J. G. Oxley ◽  
D. J. A. Welsh
Keyword(s):  
Lower Bound ◽  
Upper Bound ◽  
Percolation Theory ◽  
Fkg Inequality ◽  
Percolation Probability ◽  
Finite Set ◽  
The Fkg Inequality ◽  
Arbitrary Graphs

We examine how much classical percolation theory on lattices can be extended to arbitrary graphs or even clutters of subsets of a finite set. In the process we get new short proofs of some theorems of J. M. Hammersley. The FKG inequality is used to get an upper bound for the percolation probability and we also derive a lower bound. In each case we characterise when these bounds are attained.

scholarly journals An estimate of sectional curvatures of hypersurfaces with positive Ricci curvatures

1995 ◽  
Vol 38 (1) ◽  
pp. 167-170
Author(s):  
Ju Seon Kim ◽  
Sang Og Kim
Keyword(s):  
Lower Bound ◽  
Euclidean Space ◽  
Sectional Curvature ◽  
Ricci Curvature ◽  
Upper Bound ◽  
Mean Curvature ◽  
Nonnegative Constant ◽  
Sectional Curvatures ◽  
Bounded Below

Let M be a hypersurface in Euclidean space and let the Ricci curvature of M be bounded below by some nonnegative constant. In this paper, we estimate the sectional curvature of M in terms of the lower bound of Ricci curvature and the upper bound of mean curvature.

scholarly journals Multiple closed orbits for N-body-type problems

1998 ◽  
Vol 58 (1) ◽  
pp. 1-13 ◽  
Author(s):  
Shiqing Zhang
Keyword(s):  
Lower Bound ◽  
Periodic Solutions ◽  
Upper Bound ◽  
Critical Value ◽  
Body Type ◽  
Fixed Energy ◽  
Closed Orbits

Using the equivariant Ljusternik-Schnirelmann theory and the estimate of the upper bound of the critical value and lower bound for the collision solutions, we obtain some new results in the large concerning multiple geometrically distinct periodic solutions of fixed energy for a class of planar N-body type problems.

scholarly journals Approximation by Boolean sums of linear operators: Telyakovskiǐ-type estimates

1990 ◽  
Vol 42 (2) ◽  
pp. 253-266 ◽  
Author(s):  
Jia-Ding Cao ◽  
Heinz H. Gonska
Keyword(s):  
Upper Bound ◽  
Algebraic Polynomial ◽  
Present Note ◽  
Type Inequality ◽  
Linear Operators ◽  
Polynomial Operators ◽  
Boolean Sums ◽  
General Conditions

In the present note we study the question: “Under which general conditions do certain Boolean sums of linear operators satisfy Telyakovskiǐ-type estimates?” It is shown, in particular, that any sequence of linear algebraic polynomial operators satisfying a Timan-type inequality can be modified appropriately so as to obtain the corresponding upper bound of the Telyakovskiǐ-type. Several examples are included.

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