Laplacian Distribution and Domination

Let $m_G(I)$ denote the number of Laplacian eigenvalues of a graph $G$ in an interval $I$, and let $\gamma(G)$ denote its domination number. We extend the recent result $m_G[0,1) \leq \gamma(G)$, and show that isolate-free graphs also satisfy $\gamma(G) \leq m_G[2,n]$. In pursuit of better understanding Laplacian eigenvalue distribution, we find applications for these inequalities. We relate these spectral parameters with the approximability of $\gamma(G)$, showing that $\frac{\gamma(G)}{m_G[0,1)} \not\in O(\log n)$. However, $\gamma(G) \leq m_G[2, n] \leq (c + 1) \gamma(G)$ for $c$-cyclic graphs, $c \geq 1$. For trees $T$, $\gamma(T) \leq m_T[2, n] \leq 2 \gamma(G)$

Prediction of Novel High Pressure H2O-NaCl and Carbon Oxide Compounds with Symmetry-Driven Structure Search Algorithm

Crystal structure prediction with theoretical methods is particularly challenging when unit cells with many atoms need to be considered. Here we employ a symmetry-driven structure search (SYDSS) method and combine it with density functional theory (DFT) to predict novel crystal structures at high pressure. We sample randomly from all 1,506 Wyckoff positions of the 230 space groups to generate a set of initial structures. During the subsequent structural relaxation with DFT, existing symmetries are preserved, but the symmetries and the space group may change as atoms move to more symmetric positions. By construction, our algorithm generates symmetric structures with high probability without excluding any configurations. This improves the search efficiency, especially for large cells with 20 atoms or more. We apply our SYDSS algorithm to identify stoichiometric (H2O)_n-(NaCl)_m and C_nO_m compounds at high pressure. We predict a novel H2O-NaCl structure with Pnma symmetry to form at 3.4 Mbar, which is within the range of diamond anvil experiments. In addition, we predict a novel C2O structure at 19.8 Mbar and C4O structure at 44.0 Mbar with Pbca and C2/m symmetry respectively.Comment: 8 pages,8 figures, 3 table, Physical Review B, 201

A maximum entropy approach to H-theory: Statistical mechanics of hierarchical systems

A novel formalism, called H-theory, is applied to the problem of statistical equilibrium of a hierarchical complex system with multiple time and length scales. In this approach, the system is formally treated as being composed of a small subsystem---representing the region where the measurements are made---in contact with a set of `nested heat reservoirs' corresponding to the hierarchical structure of the system. The probability distribution function (pdf) of the fluctuating temperatures at each reservoir, conditioned on the temperature of the reservoir above it, is determined from a maximum entropy principle subject to appropriate constraints that describe the thermal equilibrium properties of the system. The marginal temperature distribution of the innermost reservoir is obtained by integrating over the conditional distributions of all larger scales, and the resulting pdf is written in analytical form in terms of certain special transcendental functions, known as the Fox $H$-functions. The distribution of states of the small subsystem is then computed by averaging the quasi-equilibrium Boltzmann distribution over the temperature of the innermost reservoir. This distribution can also be written in terms of $H$-functions. The general family of distributions reported here recovers, as particular cases, the stationary distributions recently obtained by Mac\^edo {\it et al.} [Phys.~Rev.~E {\bf 95}, 032315 (2017)] from a stochastic dynamical approach to the problem.Comment: 20 pages, 2 figure