Universal upper and lower bounds on energy of spherical designs
نویسندگان
چکیده
Linear programming (polynomial) techniques are used to obtain lower and upper bounds for the potential energy of spherical designs. This approach gives unified bounds that are valid for a large class of potential functions. Our lower bounds are optimal for absolutely monotone potentials in the sense that for the linear programming technique they cannot be improved by using polynomials of the same or lower degree. When additional information about the structure (upper and lower bounds for the inner products) of the designs is known, improvements on the bounds are obtained. Furthermore, we provide ‘test functions’ for determining when the linear programming lower bounds for energy can be improved utilizing higher degree polynomials. We also provide some asymptotic results for these energy bounds.
منابع مشابه
On Linear Programming Bounds for Spherical Codes and Designs
We investigate universal bounds on spherical codes and spherical designs that could be obtained using Delsarte’s linear programming methods. We give a lower estimate for the LP upper bound on codes, and an upper estimate for the LP lower bound on designs. Specifically, when the distance of the code is fixed and the dimension goes to infinity, the LP upper bound on codes is at least as large as ...
متن کاملLinear Programming Bounds for Spherical Codes and Designs
We describe linear programming (LP) techniques used for obtaining upper/lower bounds on the size of spherical codes/spherical designs. A survey of universal bounds is presented together with description of necessary and sufficient conditions for their optimality. If improvements are possible, we describe methods for finding new bounds. In both cases we are able to find new bounds in great range...
متن کاملOn the Riesz Energy of Spherical Designs
We show how polynomial techniques can be applied for obtaining upper and lower bounds on the Riesz energy of spherical designs.
متن کاملOn Zagreb Energy and edge-Zagreb energy
In this paper, we obtain some upper and lower bounds for the general extended energy of a graph. As an application, we obtain few bounds for the (edge) Zagreb energy of a graph. Also, we deduce a relation between Zagreb energy and edge-Zagreb energy of a graph $G$ with minimum degree $delta ge2$. A lower and upper bound for the spectral radius of the edge-Zagreb matrix is obtained. Finally, we ...
متن کاملA note on the bounds of Laplacian-energy-like-invariant
The Laplacian-energy-like of a simple connected graph G is defined as LEL:=LEL(G)=∑_(i=1)^n√(μ_i ), Where μ_1 (G)≥μ_2 (G)≥⋯≥μ_n (G)=0 are the Laplacian eigenvalues of the graph G. Some upper and lower bounds for LEL are presented in this note. Moreover, throughout this work, some results related to lower bound of spectral radius of graph are obtained using the term of ΔG as the num...
متن کامل