Structural anomalies for a three dimensional isotropic core-softened potential

Using molecular dynamics simulations we investigate the structure of a system of particles interacting through a continuous core-softened interparticle potential. We found for the translational order parameter, t, a local maximum at a density $\rho_{t-max}$ and a local minimum at $\rho_{t-min} > \rho_{t-max}$. Between $\rho_{t-max}$ and $\rho_{t-min}$, the $t$ parameter anomalously decreases upon pressure. For the orientational order parameter, $Q_6$, was observed a maximum at a density $\rho_{t-max}< \rho_{Qmax} < \rho_{t-min}$. For densities between $\rho_{Qmax}$ and $\rho_{t-min}$, both the translational (t) and orientational ($Q_6$) order parameters have anomalous behavior. We know that this system also exhibits density and diffusion anomaly. We found that the region in the pressure-temperature phase-diagram of the structural anomaly englobes the region of the diffusion anomaly that is larger than the region limited by the temperature of maximum density. This cascade of anomalies (structural, dynamic and thermodynamic) for our model has the same hierarchy of that one observed for the SPC/E water.Comment: 19 pages, 8 figure

Saturation numbers for Ramsey-minimal graphs

Given graphs $H_1, \dots, H_t$, a graph $G$ is $(H_1, \dots, H_t)$-Ramsey-minimal if every $t$-coloring of the edges of $G$ contains a monochromatic $H_i$ in color $i$ for some $i\in\{1, \dots, t\}$, but any proper subgraph of $G$ does not possess this property. We define $\mathcal{R}_{\min}(H_1, \dots, H_t)$ to be the family of $(H_1, \dots, H_t)$-Ramsey-minimal graphs. A graph $G$ is \dfn{$\mathcal{R}_{\min}(H_1, \dots, H_t)$-saturated} if no element of $\mathcal{R}_{\min}(H_1, \dots, H_t)$ is a subgraph of $G$, but for any edge $e$ in $\overline{G}$, some element of $\mathcal{R}_{\min}(H_1, \dots, H_t)$ is a subgraph of $G + e$. We define $sat(n, \mathcal{R}_{\min}(H_1, \dots, H_t))$ to be the minimum number of edges over all $\mathcal{R}_{\min}(H_1, \dots, H_t)$-saturated graphs on $n$ vertices. In 1987, Hanson and Toft conjectured that $sat(n, \mathcal{R}_{\min}(K_{k_1}, \dots, K_{k_t}) )= (r - 2)(n - r + 2)+\binom{r - 2}{2}$ for $n \ge r$, where $r=r(K_{k_1}, \dots, K_{k_t})$ is the classical Ramsey number for complete graphs. The first non-trivial case of Hanson and Toft's conjecture for sufficiently large $n$ was setteled in 2011, and is so far the only settled case. Motivated by Hanson and Toft's conjecture, we study the minimum number of edges over all $\mathcal{R}_{\min}(K_3, \mathcal{T}_k)$-saturated graphs on $n$ vertices, where $\mathcal{T}_k$ is the family of all trees on $k$ vertices. We show that for $n \ge 18$, $sat(n, \mathcal{R}_{\min}(K_3, \mathcal{T}_4)) =\lfloor {5n}/{2}\rfloor$. For $k \ge 5$ and $n \ge 2k + (\lceil k/2 \rceil +1) \lceil k/2 \rceil -2$, we obtain an asymptotic bound for $sat(n, \mathcal{R}_{\min}(K_3, \mathcal{T}_k))$.Comment: to appear in Discrete Mathematic

Existence of temperature on the nanoscale

We consider a regular chain of quantum particles with nearest neighbour interactions in a canonical state with temperature $T$. We analyse the conditions under which the state factors into a product of canonical density matrices with respect to groups of $n$ particles each and under which these groups have the same temperature $T$. In quantum mechanics the minimum group size $n_{min}$ depends on the temperature $T$, contrary to the classical case. We apply our analysis to a harmonic chain and find that $n_{min} = const.$ for temperatures above the Debye temperature and $n_{min} \propto T^{-3}$ below.Comment: Version that appeared in PR

Determination of superconducting anisotropy from magnetization data on random powders as applied to LuNi2_2B2_2C, YNi2_2B2_2C and MgB2_2

The recently discovered intermetallic superconductor MgB2 appears to have a highly anisotopic upper critical field with Hc2(max)/Hc2(min} = \gamma > 5. In order to determine the temperature dependence of both Hc2(max) and Hc2(min) we propose a method of extracting the superconducting anisotropy from the magnetization M(H,T) of randomly oriented powder samples. The method is based on two features in dM/dT the onset of diamagnetism at Tc(max), that is commonly associated with Hc2, and a kink in dM/dT at a lower temperature Tc(min). Results for LuNi2B2C and YNi2B2C powders are in agreement with anisotropic Hc2 obtained from magneto-transport measurements on single crystals. Using this method on four different types of MgB2 powder samples we are able to determine Hc2(max)(T) and Hc2(min)(T) with \gamma \approx 6

Bisection of Bounded Treewidth Graphs by Convolutions

In the Bisection problem, we are given as input an edge-weighted graph G. The task is to find a partition of V(G) into two parts A and B such that ||A| - |B|| <= 1 and the sum of the weights of the edges with one endpoint in A and the other in B is minimized. We show that the complexity of the Bisection problem on trees, and more generally on graphs of bounded treewidth, is intimately linked to the (min, +)-Convolution problem. Here the input consists of two sequences (a[i])^{n-1}_{i = 0} and (b[i])^{n-1}_{i = 0}, the task is to compute the sequence (c[i])^{n-1}_{i = 0}, where c[k] = min_{i=0,...,k}(a[i] + b[k - i]). In particular, we prove that if (min, +)-Convolution can be solved in O(tau(n)) time, then Bisection of graphs of treewidth t can be solved in time O(8^t t^{O(1)} log n * tau(n)), assuming a tree decomposition of width t is provided as input. Plugging in the naive O(n^2) time algorithm for (min, +)-Convolution yields a O(8^t t^{O(1)} n^2 log n) time algorithm for Bisection. This improves over the (dependence on n of the) O(2^t n^3) time algorithm of Jansen et al. [SICOMP 2005] at the cost of a worse dependence on t. "Conversely", we show that if Bisection can be solved in time O(beta(n)) on edge weighted trees, then (min, +)-Convolution can be solved in O(beta(n)) time as well. Thus, obtaining a sub-quadratic algorithm for Bisection on trees is extremely challenging, and could even be impossible. On the other hand, for unweighted graphs of treewidth t, by making use of a recent algorithm for Bounded Difference (min, +)-Convolution of Chan and Lewenstein [STOC 2015], we obtain a sub-quadratic algorithm for Bisection with running time O(8^t t^{O(1)} n^{1.864} log n)