Birthday Inequalities, Repulsion, and Hard Spheres

We study a birthday inequality in random geometric graphs: the probability of the empty graph is upper bounded by the product of the probabilities that each edge is absent. We show the birthday inequality holds at low densities, but does not hold in general. We give three different applications of the birthday inequality in statistical physics and combinatorics: we prove lower bounds on the free energy of the hard sphere model and upper bounds on the number of independent sets and matchings of a given size in d-regular graphs. The birthday inequality is implied by a repulsion inequality: the expected volume of the union of spheres of radius r around n randomly placed centers increases if we condition on the event that the centers are at pairwise distance greater than r. Surprisingly we show that the repulsion inequality is not true in general, and in particular that it fails in 24-dimensional Euclidean space: conditioning on the pairwise repulsion of centers of 24-dimensional spheres can decrease the expected volume of their union

Gray codes and symmetric chains

We consider the problem of constructing a cyclic listing of all bitstrings of length~$2n+1$ with Hamming weights in the interval $[n+1-\ell,n+\ell]$, where $1\leq \ell\leq n+1$, by flipping a single bit in each step. This is a far-ranging generalization of the well-known middle two levels problem (the case~$\ell=1$). We provide a solution for the case~$\ell=2$ and solve a relaxed version of the problem for general values of~$\ell$, by constructing cycle factors for those instances. Our proof uses symmetric chain decompositions of the hypercube, a concept known from the theory of posets, and we present several new constructions of such decompositions. In particular, we construct four pairwise edge-disjoint symmetric chain decompositions of the $n$-dimensional hypercube for any~$n\geq 12$

Tight upper bound on the maximum anti-forcing numbers of graphs

Let $G$ be a simple graph with a perfect matching. Deng and Zhang showed that the maximum anti-forcing number of $G$ is no more than the cyclomatic number. In this paper, we get a novel upper bound on the maximum anti-forcing number of $G$ and investigate the extremal graphs. If $G$ has a perfect matching $M$ whose anti-forcing number attains this upper bound, then we say $G$ is an extremal graph and $M$ is a nice perfect matching. We obtain an equivalent condition for the nice perfect matchings of $G$ and establish a one-to-one correspondence between the nice perfect matchings and the edge-involutions of $G$, which are the automorphisms $\alpha$ of order two such that $v$ and $\alpha(v)$ are adjacent for every vertex $v$. We demonstrate that all extremal graphs can be constructed from $K_2$ by implementing two expansion operations, and $G$ is extremal if and only if one factor in a Cartesian decomposition of $G$ is extremal. As examples, we have that all perfect matchings of the complete graph $K_{2n}$ and the complete bipartite graph $K_{n, n}$ are nice. Also we show that the hypercube $Q_n$, the folded hypercube $FQ_n$ ($n\geq4$) and the enhanced hypercube $Q_{n, k}$ ($0\leq k\leq n-4$) have exactly $n$, $n+1$ and $n+1$ nice perfect matchings respectively.Comment: 15 pages, 7 figure

Weighted counting of solutions to sparse systems of equations

Given complex numbers $w_1, \ldots, w_n$, we define the weight $w(X)$ of a set $X$ of 0-1 vectors as the sum of $w_1^{x_1} \cdots w_n^{x_n}$ over all vectors $(x_1, \ldots, x_n)$ in $X$. We present an algorithm, which for a set $X$ defined by a system of homogeneous linear equations with at most $r$ variables per equation and at most $c$ equations per variable, computes $w(X)$ within relative error $\epsilon >0$ in $(rc)^{O(\ln n-\ln \epsilon)}$ time provided $|w_j| \leq \beta (r \sqrt{c})^{-1}$ for an absolute constant $\beta >0$ and all $j=1, \ldots, n$. A similar algorithm is constructed for computing the weight of a linear code over ${\Bbb F}_p$. Applications include counting weighted perfect matchings in hypergraphs, counting weighted graph homomorphisms, computing weight enumerators of linear codes with sparse code generating matrices, and computing the partition functions of the ferromagnetic Potts model at low temperatures and of the hard-core model at high fugacity on biregular bipartite graphs.Comment: The exposition is improved, a couple of inaccuracies correcte

The asymptotic induced matching number of hypergraphs: balanced binary strings

We compute the asymptotic induced matching number of the $k$-partite $k$-uniform hypergraphs whose edges are the $k$-bit strings of Hamming weight $k/2$, for any large enough even number $k$. Our lower bound relies on the higher-order extension of the well-known Coppersmith-Winograd method from algebraic complexity theory, which was proven by Christandl, Vrana and Zuiddam. Our result is motivated by the study of the power of this method as well as of the power of the Strassen support functionals (which provide upper bounds on the asymptotic induced matching number), and the connections to questions in tensor theory, quantum information theory and theoretical computer science. Phrased in the language of tensors, as a direct consequence of our result, we determine the asymptotic subrank of any tensor with support given by the aforementioned hypergraphs. In the context of quantum information theory, our result amounts to an asymptotically optimal $k$-party stochastic local operations and classical communication (slocc) protocol for the problem of distilling GHZ-type entanglement from a subfamily of Dicke-type entanglement

European Journal of Combinatorics Index, Volume 27

BACKGROUND: Diabetes is an inflammatory condition associated with iron abnormalities and increased oxidative damage. We aimed to investigate how diabetes affects the interrelationships between these pathogenic mechanisms. METHODS: Glycaemic control, serum iron, proteins involved in iron homeostasis, global antioxidant capacity and levels of antioxidants and peroxidation products were measured in 39 type 1 and 67 type 2 diabetic patients and 100 control subjects. RESULTS: Although serum iron was lower in diabetes, serum ferritin was elevated in type 2 diabetes (p = 0.02). This increase was not related to inflammation (C-reactive protein) but inversely correlated with soluble transferrin receptors (r = - 0.38, p = 0.002). Haptoglobin was higher in both type 1 and type 2 diabetes (p &lt; 0.001) and haemopexin was higher in type 2 diabetes (p &lt; 0.001). The relation between C-reactive protein and haemopexin was lost in type 2 diabetes (r = 0.15, p = 0.27 vs r = 0.63, p &lt; 0.001 in type 1 diabetes and r = 0.36, p = 0.001 in controls). Haemopexin levels were independently determined by triacylglycerol (R(2) = 0.43) and the diabetic state (R(2) = 0.13). Regarding oxidative stress status, lower antioxidant concentrations were found for retinol and uric acid in type 1 diabetes, alpha-tocopherol and ascorbate in type 2 diabetes and protein thiols in both types. These decreases were partially explained by metabolic-, inflammatory- and iron alterations. An additional independent effect of the diabetic state on the oxidative stress status could be identified (R(2) = 0.5-0.14). CONCLUSIONS: Circulating proteins, body iron stores, inflammation, oxidative stress and their interrelationships are abnormal in patients with diabetes and differ between type 1 and type 2 diabetes</p

Random sampling of lattice configurations using local Markov chains

Algorithms based on Markov chains are ubiquitous across scientific disciplines, as they provide a method for extracting statistical information about large, complicated systems. Although these algorithms may be applied to arbitrary graphs, many physical applications are more naturally studied under the restriction to regular lattices. We study several local Markov chains on lattices, exploring how small changes to some parameters can greatly influence efficiency of the algorithms. We begin by examining a natural Markov Chain that arises in the context of "monotonic surfaces", where some point on a surface is sightly raised or lowered each step, but with a greater rate of raising than lowering. We show that this chain is rapidly mixing (converges quickly to the equilibrium) using a coupling argument; the novelty of our proof is that it requires defining an exponentially increasing distance function on pairs of surfaces, allowing us to derive near optimal results in many settings. Next, we present new methods for lower bounding the time local chains may take to converge to equilibrium. For many models that we study, there seems to be a phase transition as a parameter is changed, so that the chain is rapidly mixing above a critical point and slow mixing below it. Unfortunately, it is not always possible to make this intuition rigorous. We present the first proofs of slow mixing for three sampling problems motivated by statistical physics and nanotechnology: independent sets on the triangular lattice (the hard-core lattice gas model), weighted even orientations of the two-dimensional Cartesian lattice (the 8-vertex model), and non-saturated Ising (tile-based self-assembly). Previous proofs of slow mixing for other models have been based on contour arguments that allow us prove that a bottleneck in the state space constricts the mixing. The standard contour arguments do not seem to apply to these problems, so we modify this approach by introducing the notion of "fat contours" that can have nontrivial area. We use these to prove that the local chains defined for these models are slow mixing. Finally, we study another important issue that arises in the context of phase transitions in physical systems, namely how the boundary of a lattice can affect the efficiency of the Markov chain. We examine a local chain on the perfect and near-perfect matchings of the square-octagon lattice, and show for one boundary condition the chain will mix in polynomial time, while for another it will mix exponentially slowly. Strikingly, the two boundary conditions only differ at four vertices. These are the first rigorous proofs of such a phenomenon on lattice graphs.Ph.D.Committee Chair: Randall, Dana; Committee Member: Heitsch, Christine; Committee Member: Mihail, Milena; Committee Member: Trotter, Tom; Committee Member: Vigoda, Eri