Abstract:
Let k>=3 be an integer, we prove that the maximum induced density of the k-vertex directed star in a directed graph is attained by an iterated blow-up construction. This confirms a conjecture by Falgas-Ravry and Vaughan, who proved this for k=3, 4. This question provides the first known instance of density problem for which one can prove extremality of an iterated blow-up construction. We also study the inducibility of complete bipartite digraphs and discuss other related problems.

Abstract:
In this note, we report the noise properties in La0.67Ca0.33MnOx. material., The noise spectral density was found as evident l/f behavior. By comparing some kinds of samples, our results suggest that the noise lcvel in GMR oxide is largely attributable to behavior of oxygen content for single-crystal thin films and grain boundary for polycrystalline bulk.

Abstract:
An operator algebra $\mathcal{A}$ acting on a Hilbert space is said to have the closability property if every densely defined linear transformation commuting with $\mathcal{A}$ is closable. In this paper we study the closability property of the von Neumann algebra consisting of the multiplication operators on $L^2(\mu)$, and give necessary and sufficient conditions for a normal operator $N$ such that the von Neumann algebra generated by $N$ has the closability property. We also give necessary and sufficient conditions for an operator $T$ of class $C_0$ such that the algebra generated by $T$ in the weak operator topology and the algebra $H^\infty(T)=\{u(T):u\in H^\infty\}$ have the closability property.

Abstract:
Let $\mathcal{M}$ be the set of Borel probability measures on $\mathbb{R}$. We denote by $\mu^{\mathrm{ac}}$ the absolutely continuous part of $\mu\in\mathcal{M}$. The purpose of this paper is to investigate the supports and regularity for measures of the form $(\mu^{\boxplus p})^{\uplus q}$, $\mu\in\mathcal{M}$, where $\boxplus$ and $\uplus$ are the operations of free additive and Boolean convolution on $\mathcal{M}$, respectively, and $p\geq1$, $q>0$. We show that for any $q$ the supports of $((\mu^{\boxplus p})^{\uplus q})^{\mathrm{ac}}$ and $(\mu^{\boxplus p})^{\mathrm{ac}}$ contain the same number of components and this number is a decreasing function of $p$. Explicit formulas for the densities of $((\mu^{\boxplus p})^{\uplus q})^{\mathrm{ac}}$ and criteria for determining the atoms of $(\mu^{\boxplus p})^{\uplus q}$ are given. Based on the subordination functions of free convolution powers, we give another point of view to analyze the set of $\boxplus$-infinitely divisible measures and provide explicit expressions for their Voiculescu transforms in terms of free and Boolean convolutions.

Abstract:
We consider two models for biopolymers, the $\nabla$ interaction and the $\Delta$ one, both with the Gaussian potential in the random environment. A random field $\varphi:{0,1,...,N}\rightarrow \Bbb{R}^d$ represents the position of the polymer path. The law of the field is given by $\exp(-\sum_i\frac{|\nabla\varphi_i|^2}{2})$ where $\nabla$ is the discrete gradient, and by $\exp(-\sum_i\frac{|\Delta\varphi_i|^2}{2})$ where $\Delta$ is the discrete Laplacian. For every Gaussian potential $\frac{|\cdot|^2}{2}$, a random charge is added as a factor: $(1+\beta\omega_i)\frac{|\cdot|^2}{2}$ with $\Bbb{P}(\omega_i=\pm 1)=1/2$ or $\exp(\beta\omega_i)\frac{|\cdot|^2}{2}$ with $\omega_i$ obeys a normal distribution. The interaction with the origin in the random field space is considered. Each time the field touches the origin, a reward $\epsilon\geq 0$ is given. Although these models are quite different from the pinning models studied in Giacomin (2007), the result about the gap between the annealed critical point and the quenched critical point stays the same.

Abstract:
In this paper, we study the supports of measures in the free additive convolution semigroup $\{\mu^{\boxplus t}:t>1\}$, where $\mu$ is a Borel probability measure on $\mathbb{R}$. We give a formula for the density of the absolutely continuous part of $\mu^{\boxplus t}$ and use this formula to obtain certain regularizing properties of $\mu^{\boxplus t}$. We show that the number $n(t)$ of the components in the support of $\mu^{\boxplus t}$ is a decreasing function of $t$ and give equivalent conditions so that $n(t)=1$ for sufficiently large $t$. Moreover, a measure $\mu$ so that $\mu^{\boxplus t}$ has infinitely many components in the support for all $t>1$ is given.

Abstract:
An r-unform n-vertex hypergraph H is said to have the Manickam-Mikl\'os-Singhi (MMS) property if for every assignment of weights to its vertices with nonnegative sum, the number of edges whose total weight is nonnegative is at least the minimum degree of H. In this paper we show that for n>10r^3, every r-uniform n-vertex hypergraph with equal codegrees has the MMS property, and the bound on n is essentially tight up to a constant factor. This result has two immediate corollaries. First it shows that every set of n>10k^3 real numbers with nonnegative sum has at least $\binom{n-1}{k-1}$ nonnegative k-sums, verifying the Manickam-Mikl\'os-Singhi conjecture for this range. More importantly, it implies the vector space Manickam-Mikl\'os-Singhi conjecture which states that for n >= 4k and any weighting on the 1-dimensional subspaces of F_q^n with nonnegative sum, the number of nonnegative k-dimensional subspaces is at least ${n-1 \brack k-1}_q$. We also discuss two additional generalizations, which can be regarded as analogues of the Erd\H{o}s-Ko-Rado theorem on k-intersecting families.

Abstract:
We consider a statistical mechanics model for biopolymers. Sophisticated polymer chains, such as DNA, have stiffness when they stretch chains. The Laplacian interaction is used to describe the stiffness. Also, the surface between two media has an attraction force, and the force will pull the chain back to the surface. In this paper, we deal with the random potentials when the monomers interact with the random media. Although these models are different from the pinning models studied before, the result about the gap between the annealed critical point and the quenched critical point stays the same.

Abstract:
Consider a graph obtained by taking edge disjoint union of $k$ complete bipartite graphs. Alon, Saks and Seymour conjectured that such graph has chromatic number at most $k+1$. This well known conjecture remained open for almost twenty years. In this paper, we construct a counterexample to this conjecture and discuss several related problems in combinatorial geometry and communication complexity.

Abstract:
In this paper, we study the k-neighbor bootstrap percolation process on the d-dimensional grid [n]^d, and show that the minimum number of initial vertices that percolate is (1-d/k)n^d + O(n^{d-1})$ when d<=k<=2d. This confirms a conjecture of Pete.