Twin inequality for fully contextual quantum correlations

Quantum mechanics exhibits a very peculiar form of contextuality. Identifying and connecting the simplest scenarios in which more general theories can or cannot be more contextual than quantum mechanics is a fundamental step in the quest for the principle that singles out quantum contextuality. The former scenario corresponds to the Klyachko-Can-Binicioglu-Shumovsky (KCBS) inequality. Here we show that there is a simple tight inequality, twin to the KCBS, for which quantum contextuality cannot be outperformed. In a sense, this twin inequality is the simplest tool for recognizing fully contextual quantum correlations.Comment: REVTeX4, 4 pages, 1 figur

The Waldschmidt constant for squarefree monomial ideals

Given a squarefree monomial ideal $I \subseteq R =k[x_1,\ldots,x_n]$, we show that $\widehat\alpha(I)$, the Waldschmidt constant of $I$, can be expressed as the optimal solution to a linear program constructed from the primary decomposition of $I$. By applying results from fractional graph theory, we can then express $\widehat\alpha(I)$ in terms of the fractional chromatic number of a hypergraph also constructed from the primary decomposition of $I$. Moreover, expressing $\widehat\alpha(I)$ as the solution to a linear program enables us to prove a Chudnovsky-like lower bound on $\widehat\alpha(I)$, thus verifying a conjecture of Cooper-Embree-H\`a-Hoefel for monomial ideals in the squarefree case. As an application, we compute the Waldschmidt constant and the resurgence for some families of squarefree monomial ideals. For example, we determine both constants for unions of general linear subspaces of $\mathbb{P}^n$ with few components compared to $n$, and we find the Waldschmidt constant for the Stanley-Reisner ideal of a uniform matroid.Comment: 26 pages. This project was started at the Mathematisches Forschungsinstitut Oberwolfach (MFO) as part of the mini-workshop "Ideals of Linear Subspaces, Their Symbolic Powers and Waring Problems" held in February 2015. Comments are welcome. Revised version corrects some typos, updates the references, and clarifies some hypotheses. To appear in the Journal of Algebraic Combinatoric

The combinatorics of resource sharing

We discuss general models of resource-sharing computations, with emphasis on the combinatorial structures and concepts that underlie the various deadlock models that have been proposed, the design of algorithms and deadlock-handling policies, and concurrency issues. These structures are mostly graph-theoretic in nature, or partially ordered sets for the establishment of priorities among processes and acquisition orders on resources. We also discuss graph-coloring concepts as they relate to resource sharing.Comment: R. Correa et alii (eds.), Models for Parallel and Distributed Computation, pp. 27-52. Kluwer Academic Publishers, Dordrecht, The Netherlands, 200

On the chromatic number of random geometric graphs

Given independent random points X_1,...,X_n\in\eR^d with common probability distribution $\nu$, and a positive distance $r=r(n)>0$, we construct a random geometric graph $G_n$ with vertex set $\{1,...,n\}$ where distinct $i$ and $j$ are adjacent when \norm{X_i-X_j}\leq r. Here \norm{.} may be any norm on \eR^d, and $\nu$ may be any probability distribution on \eR^d with a bounded density function. We consider the chromatic number $\chi(G_n)$ of $G_n$ and its relation to the clique number $\omega(G_n)$ as $n \to \infty$. Both McDiarmid and Penrose considered the range of $r$ when $r \ll (\frac{\ln n}{n})^{1/d}$ and the range when $r \gg (\frac{\ln n}{n})^{1/d}$, and their results showed a dramatic difference between these two cases. Here we sharpen and extend the earlier results, and in particular we consider the `phase change' range when $r \sim (\frac{t\ln n}{n})^{1/d}$ with $t>0$ a fixed constant. Both McDiarmid and Penrose asked for the behaviour of the chromatic number in this range. We determine constants $c(t)$ such that $\frac{\chi(G_n)}{nr^d}\to c(t)$ almost surely. Further, we find a "sharp threshold" (except for less interesting choices of the norm when the unit ball tiles $d$-space): there is a constant $t_0>0$ such that if $t \leq t_0$ then $\frac{\chi(G_n)}{\omega(G_n)}$ tends to 1 almost surely, but if $t > t_0$ then $\frac{\chi(G_n)}{\omega(G_n)}$ tends to a limit $>1$ almost surely.Comment: 56 pages, to appear in Combinatorica. Some typos correcte

Even cycle creating paths

We say that two graphs H1, H2 on the same vertex set are G-creating if the union of the two graphs contains G as a subgraph. Let H (n, k) be the maximum number of pairwise Ck-creating Hamiltonian paths of the complete graph Kn. The behavior of H (n, 2k + 1) is much better understood than the behavior of H (n, 2k), the former is an exponential function of n whereas the latter is larger than exponential, for every fixed k. We study H (n, k) for fixed k and n tending to infinity. The only nontrivial upper bound on H (n, 2k) was proved by Cohen, Fachini, and Körner in the case of k = 2: : (Formula presented.) In this paper, we generalize their method to prove that for every k ≥ 2, (Formula presented.) and a similar, slightly better upper bound holds when k is odd. Our proof uses constructions of bipartite, regular, C2k-free graphs with many edges given in papers by Reiman, Benson, Lazebnik, Ustimenko, and Woldar. © 2019 Wiley Periodicals, Inc

A Separation Bound for Real Algebraic Expressions

Real algebraic expressions are expressions whose leaves are integers and whose internal nodes are additions, subtractions, multiplications, divisions, k-th root operations for integral k, and taking roots of polynomials whose coefficients are given by the values of subexpressions. We consider the sign computation of real algebraic expressions, a task vital for the implementation of geometric algorithms. We prove a new separation bound for real algebraic expressions and compare it analytically and experimentally with previous bounds. The bound is used in the sign test of the number type leda real.