Contact graphs of ball packings

A contact graph of a packing of closed balls is a graph with balls as vertices and pairs of tangent balls as edges. We prove that the average degree of the contact graph of a packing of balls (with possibly different radii) in $\mathbb{R}^3$ is not greater than $13.955$. We also find new upper bounds for the average degree of contact graphs in $\mathbb{R}^4$ and $\mathbb{R}^5$

Repeated minimizers of pp-frame energies

For a collection of $N$ unit vectors $\mathbf{X}=\{x_i\}_{i=1}^N$, define the $p$-frame energy of $\mathbf{X}$ as the quantity $\sum_{i\neq j} |\langle x_i,x_j \rangle|^p$. In this paper, we connect the problem of minimizing this value to another optimization problem, so giving new lower bounds for such energies. In particular, for $p<2$, we prove that this energy is at least $2(N-d) p^{-\frac p 2} (2-p)^{\frac {p-2} 2}$ which is sharp for $d\leq N\leq 2d$ and $p=1$. We prove that for $1\leq m<d$, a repeated orthonormal basis construction of $N=d+m$ vectors minimizes the energy over an interval, $p\in[1,p_m]$, and demonstrate an analogous result for all $N$ in the case $d=2$. Finally, in connection, we give conjectures on these and other energies

Lower bounds for the simplexity of the n-cube

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 $(n+1)^{\frac {n-1} 2}$.Comment: 10 page

Revisiting spin state crossover in (MgFe)O by means of high resolution X-ray diffraction from a single crystal

(MgFe)O is a solid solution with ferrous iron undergoing the high to low spin state (HS-LS) crossover under high pressure. The exact state of the material in the region of the crossover is still a mystery, as domains with different spin states may coexist over a wide pressure range without changing the crystal structure neither from the symmetry nor from the atomic positions point of view. At the conditions of the crossover, (MgFe)O is a special type of microscopic disorder system. We explore the influences of (a) stress-strain relations in a diamond anvil cell, (b) time relaxation processes, and (c) the crossover itself on the characteristic features of a single crystal (111) Bragg spot before, during and after the transformation. Using high resolution X-ray diffraction as a novel method for studies of unconventional processes at the conditions of suppressed diffusion, we detect and discuss subtle changes of the (111) Bragg spot projections which we measure and analyze as a function of pressure. We report changes of the spot shape which can be correlated with the HS-LS relative abundance. In addition, we report the formation of structural defects as an intrinsic material response. These static defects are accumulated during transformation of the material from HS to LS.Comment: 28 pages, 11 Figure