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Register a new userUniversal lower bounds for potential energy of spherical codes
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We derive and investigate lower bounds for the potential energy of finite spherical point sets (spherical codes). Our bounds are optimal in the following sense -- they cannot be improved by employing polynomials of the same or lower degrees in the Delsarte-Yudin method. However, improvements are sometimes possible and we provide a necessary and sufficient condition for the existence of such better bounds. All our bounds can be obtained in a unified manner that does not depend on the potential function, provided the potential is given by an absolutely monotone function of the inner product between pairs of points, and this is the reason for us to call them universal. We also establish a criterion for a given code of dimension $n$ and cardinality $N$ not to be LP-universally optimal, e.g. we show that two codes conjectured by Ballinger et al to be universally optimal are not LP-universally optimal.
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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.
In this paper we prove a new asymptotic lower bound for the minimal number of simplices in simplicial dissections of $n$-dimensional cubes. In particular we show that the number of simplices in dissections of $n$-cubes without additional vertices is at least $(n+1)^{frac {n-1} 2}$.
In this paper we present the results of computer searches using a variation of an energy minimization algorithm used by Kottwitz for finding good spherical codes. We prove that exact codes exist by representing the inner products between the vectors as algebraic numbers. For selected interesting cases, we include detailed discussion of the configurations. Of particular interest are the 20-point code in $mathbb{R}^6$ and the 24-point code in $mathbb{R}^7$, which are both the union of two cross polytopes in parallel hyperplanes. Finally, we catalogue all of the codes we have found.
Thermal energy can be conducted by different mechanisms including by single particles or collective excitations. Thermal conductivity is system-specific and shows a richness of behaviors currently explored in different systems including insulators, strange metals and cuprate superconductors. Here, we show that despite the seeming complexity of thermal transport, the thermal diffusivity $alpha$ of liquids and supercritical fluids has a lower bound which is fixed by fundamental physical constants for each system as $alpha_m=frac{1}{4pi}frac{hbar}{sqrt{m_em}}$, where $m_e$ and $m$ are electron and molecule masses. The newly introduced elementary thermal diffusivity has an absolute lower bound dependent on $hbar$ and the proton-to-electron mass ratio only. We back up this result by a wide range of experimental data. We also show that theoretical minima of $alpha$ coincide with the fundamental lower limit of kinematic viscosity $ u_m$. Consistent with experiments, this points to a universal lower bound for two distinct properties, energy and momentum diffusion, and a surprising correlation between the two transport mechanisms at their minima. We observe that $alpha_m$ gives the minimum on the phase diagram except in the vicinity of the critical point, whereas $ u_m$ gives the minimum on the entire phase diagram.
We prove $L^p$ lower bounds for Coulomb energy for radially symmetric functions in $dot H^s(R^3)$ with $frac 12 <s<frac{3}{2}$. In case $frac 12 <s leq 1$ we show that the lower bounds are sharp.
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