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Register a new userTight bounds on the coefficients of partition functions via stability
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Partition functions arise in statistical physics and probability theory as the normalizing constant of Gibbs measures and in combinatorics and graph theory as graph polynomials. For instance the partition functions of the hard-core model and monomer-dimer model are the independence and matching polynomials respectively. We show how stability results follow naturally from the recently developed occupancy method for maximizing and minimizing physical observables over classes of regular graphs, and then show these stability results can be used to obtain tight extremal bounds on the individual coefficients of the corresponding partition functions. As applications, we prove new bounds on the number of independent sets and matchings of a given size in regular graphs. For large enough graphs and almost all sizes, the bounds are tight and confirm the Upper Matching Conjecture of Friedland, Krop, and Markstrom and a conjecture of Kahn on independent sets for a wide range of parameters. Additionally we prove tight bounds on the number of $q$-colorings of cubic graphs with a given number of monochromatic edges, and tight bounds on the number of independent sets of a given size in cubic graphs of girth at least $5$.
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Let $G$ be a simple graph with $2n$ vertices and a perfect matching. The forcing number of a perfect matching $M$ of $G$ is the smallest cardinality of a subset of $M$ that is contained in no other perfect matching of $G$. Let $f(G)$ and $F(G)$ denote the minimum and maximum forcing number of $G$ among all perfect matchings, respectively. Hetyei obtained that the maximum number of edges of graphs $G$ with a unique perfect matching is $n^2$ (see Lov{a}sz [20]). We know that $G$ has a unique perfect matching if and only if $f(G)=0$. Along this line, we generalize the classical result to all graphs $G$ with $f(G)=k$ for $0leq kleq n-1$, and obtain that the number of edges is at most $n^2+2nk-k^2-k$ and characterize the extremal graphs as well. Conversely, we get a non-trivial lower bound of $f(G)$ in terms of the order and size. For bipartite graphs, we gain corresponding stronger results. Further, we obtain a new upper bound of $F(G)$. Finally some open problems and conjectures are proposed.
The quantum adversary method is a versatile method for proving lower bounds on quantum algorithms. It yields tight bounds for many computational problems, is robust in having many equivalent formulations, and has natural connections to classical lower bounds. A further nice property of the adversary method is that it behaves very well with respect to composition of functions. We generalize the adversary method to include costs--each bit of the input can be given an arbitrary positive cost representing the difficulty of querying that bit. We use this generalization to exactly capture the adversary bound of a composite function in terms of the adversary bounds of its component functions. Our results generalize and unify previously known composition properties of adversary methods, and yield as a simple corollary the Omega(sqrt{n}) bound of Barnum and Saks on the quantum query complexity of read-once functions.
Given a hypergraph $H$ and a weight function $w: V rightarrow {1, dots, M}$ on its vertices, we say that $w$ is isolating if there is exactly one edge of minimum weight $w(e) = sum_{i in e} w(i)$. The Isolation Lemma is a combinatorial principle introduced in Mulmuley et. al (1987) which gives a lower bound on the number of isolating weight functions. Mulmuley used this as the basis of a parallel algorithm for finding perfect graph matchings. It has a number of other applications to parallel algorithms and to reductions of general search problems to unique search problems (in which there are one or zero solutions). The original bound given by Mulmuley et al. was recently improved by Ta-Shma (2015). In this paper, we show improved lower bounds on the number of isolating weight functions, and we conjecture that the extremal case is when $H$ consists of $n$ singleton edges. When $M gg n$ our improved bound matches this extremal case asymptotically. We are able to show that this conjecture holds in a number of special cases: when $H$ is a linear hypergraph or is 1-degenerate, or when $M = 2$. We also show that it holds asymptotically when $M gg n gg 1$.
The Jacobian elliptic functions are standard forms of elliptic functions, and they were independently introduced by C.G.J. Jacobi and N.H. Abel. In this paper, we study the unimodality of Taylor expansion coefficients of the Jacobian elliptic functions sn(u,k) and cn(u,k). By using the theory of gamma-positivity, we obtain that the Taylor expansion coefficients of sn(u,k) are symmetric and unimodal, and that of cn(u,k) are unimodal and alternatingly increasing.
If A is infinite and well-ordered, then |2^A|<=|Part(A)|<=|A^A|.
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