Counting independent sets in hypergraphs

Let $G$ be a triangle-free graph with $n$ vertices and average degree $t$. We show that $G$ contains at least $$e^{(1-n^{-1/12})\frac{1}{2}\frac{n}{t}\ln t (\frac{1}{2}\ln t-1)}$$ independent sets. This improves a recent result of the first and third authors \cite{countingind}. In particular, it implies that as $n \to \infty$, every triangle-free graph on $n$ vertices has at least $e^{(c_1-o(1)) \sqrt{n} \ln n}$ independent sets, where $c_1 = \sqrt{\ln 2}/4 = 0.208138..$. Further, we show that for all $n$, there exists a triangle-free graph with $n$ vertices which has at most $e^{(c_2+o(1))\sqrt{n}\ln n}$ independent sets, where $c_2 = 1+\ln 2 = 1.693147..$. This disproves a conjecture from \cite{countingind}. Let $H$ be a $(k+1)$-uniform linear hypergraph with $n$ vertices and average degree $t$. We also show that there exists a constant $c_k$ such that the number of independent sets in $H$ is at least $$e^{c_{k} \frac{n}{t^{1/k}}\ln^{1+1/k}{t}}.$$ This is tight apart from the constant $c_k$ and generalizes a result of Duke, Lefmann, and R\"odl \cite{uncrowdedrodl}, which guarantees the existence of an independent set of size $\Omega(\frac{n}{t^{1/k}} \ln^{1/k}t)$. Both of our lower bounds follow from a more general statement, which applies to hereditary properties of hypergraphs

Renormalization study of two-dimensional convergent solutions of the porous medium equation

In the focusing problem we study a solution of the porous medium equation $u_t=\Delta (u^m)$ whose initial distribution is positive in the exterior of a closed non-circular two dimensional region, and zero inside. We implement a numerical scheme that renormalizes the solution each time that the average size of the empty region reduces by a half. The initial condition is a function with circular level sets distorted with a small sinusoidal perturbation of wave number $k\geq 3$. We find that for nonlinearity exponents m smaller than a critical value which depends on k, the solution tends to a self-similar regime, characterized by rounded polygonal interfaces and similarity exponents that depend on m and on the discrete rotational symmetry number k. For m greater than the critical value, the final form of the interface is circular.Comment: 26 pages, Latex, 13 ps figure

Direct Estimation of Information Divergence Using Nearest Neighbor Ratios

We propose a direct estimation method for R\'{e}nyi and f-divergence measures based on a new graph theoretical interpretation. Suppose that we are given two sample sets $X$ and $Y$, respectively with $N$ and $M$ samples, where $\eta:=M/N$ is a constant value. Considering the $k$-nearest neighbor ($k$-NN) graph of $Y$ in the joint data set $(X,Y)$, we show that the average powered ratio of the number of $X$ points to the number of $Y$ points among all $k$-NN points is proportional to R\'{e}nyi divergence of $X$ and $Y$ densities. A similar method can also be used to estimate f-divergence measures. We derive bias and variance rates, and show that for the class of $\gamma$-H\"{o}lder smooth functions, the estimator achieves the MSE rate of $O(N^{-2\gamma/(\gamma+d)})$. Furthermore, by using a weighted ensemble estimation technique, for density functions with continuous and bounded derivatives of up to the order $d$, and some extra conditions at the support set boundary, we derive an ensemble estimator that achieves the parametric MSE rate of $O(1/N)$. Our estimators are more computationally tractable than other competing estimators, which makes them appealing in many practical applications.Comment: 2017 IEEE International Symposium on Information Theory (ISIT

Comparing the Ca II H and K Emission Lines in Red Giant Stars

Measurements of the asymmetry of the emission peaks in the core of the Ca II H line for 105 giant stars are reported. The asymmetry is quantified with the parameter V/R, defined as the ratio between the maximum number of counts in the blueward peak and the redward peak of the emission profile. The Ca II H and K emission lines probe the differential motion of certain chromospheric layers in the stellar atmosphere. Data on V/R for the Ca II K line are drawn from previous papers and compared to the analogous H line ratio, the H and K spectra being from the same sets of observations. It is found that the H line V/R value is +0.04 larger, on average, than the equivalent K line ratio, however, the difference varies with B-V color. Red giants cooler than B-V = 1.2 are more likely to have the H line V/R larger than the K line V/R, whereas the opposite is true for giants hotter than B-V = 1.2. The differences between the Ca II H and K line asymmetries could be caused by the layers of chromospheric material from which these emission features arise moving with different velocities in an expanding outflow.Comment: 36 pages, 12 figures, 2 tables. Accepted to PASP. Corrected a typo in Table

On Multiple Pattern Avoiding Set Partitions

We study classes of set partitions determined by the avoidance of multiple patterns, applying a natural notion of partition containment that has been introduced by Sagan. We say that two sets S and T of patterns are equivalent if for each n, the number of partitions of size n avoiding all the members of S is the same as the number of those that avoid all the members of T. Our goal is to classify the equivalence classes among two-element pattern sets of several general types. First, we focus on pairs of patterns {\sigma,\tau}, where \sigma\ is a pattern of size three with at least two distinct symbols and \tau\ is an arbitrary pattern of size k that avoids \sigma. We show that pattern-pairs of this type determine a small number of equivalence classes; in particular, the classes have on average exponential size in k. We provide a (sub-exponential) upper bound for the number of equivalence classes, and provide an explicit formula for the generating function of all such avoidance classes, showing that in all cases this generating function is rational. Next, we study partitions avoiding a pair of patterns of the form {1212,\tau}, where \tau\ is an arbitrary pattern. Note that partitions avoiding 1212 are exactly the non-crossing partitions. We provide several general equivalence criteria for pattern pairs of this type, and show that these criteria account for all the equivalences observed when \tau\ has size at most six. In the last part of the paper, we perform a full classification of the equivalence classes of all the pairs {\sigma,\tau}, where \sigma\ and \tau\ have size four.Comment: 37 pages. Corrected a typ