On constant-multiple-free sets contained in a random set of integers

For a rational number $r>1$, a set $A$ of positive integers is called an $r$-multiple-free set if $A$ does not contain any solution of the equation $rx = y$. The extremal problem on estimating the maximum possible size of $r$-multiple-free sets contained in $[n]:={1,2,...,n}$ has been studied for its own interest in combinatorial number theory and application to coding theory. Let $a$, $b$ be positive integers such that $a<b$ and the greatest common divisor of $a$ and $b$ is 1. Wakeham and Wood showed that the maximum size of $(b/a)$-multiple-free sets contained in $[n]$ is $\frac{b}{b+1}n+O(\log n)$. In this paper we generalize this result as follows. For a real number $p\in (0,1)$, let $[n]_p$ be a set of integers obtained by choosing each element $i\in [n]$ randomly and independently with probability $p$. We show that the maximum possible size of $(b/a)$-multiple-free sets contained in $[n]_p$ is $\frac{b}{b+p}pn+O(\sqrt{pn}\log n \log \log n)$ with probability that goes to 1 as $n\to \infty$.Comment: 9 pages, 1 figure, Abstract was modifie

A unique pure mechanical system revealing dipole repulsion

We study multiple elastic collisions of a block and a ball against a rigid wall in one dimension. The complete trajectory of the block is solved as an analytic function of time. Near the turning point of the block the force carried by the ball is proportional to 1/x^3, where x is the distance between the wall and the block, in the limit that the block is sufficiently heavier than the ball. This is a unique pure mechanical system that reveals dipole-like repulsion.Comment: 20 pages, 3 figures, 1 table, version published in Am. J. Phy

Magic mass ratios of complete energy-momentum transfer in one-dimensional elastic three-body collisions

We consider the one-dimensional scattering of two identical blocks of mass $M$ that exchange energy and momentum via elastic collisions with an intermediary ball of mass $m=\alpha M$. Initially, one block is incident upon the ball with the other block at rest. For $\alpha<1$, the three objects will make multiple collisions with one another. In our analysis, we construct a Euclidean vector $\textbf{V}_n$ whose components are proportional to the velocities of the objects. Energy-momentum conservation then requires a covariant recurrence relation for $\textbf{V}_n$ that transforms like a pure rotation in three dimensions. The analytic solutions of the terminal velocities result in a remarkable prediction for values of $\alpha$, in cases where the initial energy and momentum of the incident block are completely transferred to the scattered block. We call these values for $\alpha$ "magic mass ratios."Comment: 32 pages, 6 figures, 2 table

The independence number of non-uniform uncrowded hypergraphs and an anti-Ramsey type result

We prove the following: Fix an integer $k\geq 2$, and let $T$ be a real number with $T\geq 1.5$. Let \cH=(V,\cE_2\cup \cE_3\cup\dots\cup\cE_k) be a non-uniform hypergraph with the vertex set $V$ and the set \cE_i of edges of size $i=2,\ldots , k$. Suppose that \cH has no $2$-cycles (regardless of sizes of edges), and neither contains $3$-cycles nor $4$-cycles consisting of $2$-element edges. If the average degrees t_i^{i-1} := i |\cE_i|/ |V| satisfy that $t_i^{i-1} \leq T^{i-1} (\ln T)^{\frac{k-i}{k-1}}$ for $i= 2, \dots , k$, then there exists a constant $C_k > 0$, depending only on $k$, such that \alpha(\cH)\geq C_k \frac{|V|}{T} (\ln T)^{\frac{1}{k-1}}, where \alpha(\cH) denotes the independence number of \cH. This extends results of Ajtai, Koml\'os, Pintz, Spencer and Szemer\'edi and Duke, R\"odl and the second author for uniform hypergraphs. As an application, we consider an anti-Ramsey type problem on non-uniform hypergraphs. Let \cH=\cH(n;2,\ldots,\ell) be the hypergraph on the $n$-vertex set $V$ in which, for $s=2,\ldots,\ell$, each $s$-subset of $V$ is a hyperedge of \cH. Let $\Delta$ be an edge-coloring of \cH satisfying the following: (a) two hyperedges sharing a vertex have different colors; (b) two hyperedges with distinct size have different colors; (c) a color used for a hyperedge of size $s$ appears at most $u_s$ times. For such a coloring $\Delta$, let $f_{\Delta}(n;u_2,\ldots,u_{\ell})$ be the maximum size of a subset $U$ of $V$ such that each hyperedge of \cH[U] has a distinct color, and let $f(n;u_2,\ldots,u_{\ell}):=\min_{\Delta} f_{\Delta}(n;u_2,\ldots,u_{\ell}).$ We determine $f(n;u_2,\ldots,u_{\ell})$ up to a multiplicative logarithm factor.Comment: 17 page

Higher order operator splitting Fourier spectral methods for the Allen-Cahn equation

The Allen-Cahn equation is solved numerically by operator splitting Fourier spectral methods. The basic idea of the operator splitting method is to decompose the original problem into sub-equations and compose the approximate solution of the original equation using the solutions of the subproblems. Unlike the first and the second order methods, each of the heat and the free-energy evolution operators has at least one backward evaluation in higher order methods. We investigate the effect of negative time steps on a general form of third order schemes and suggest three third order methods for better stability and accuracy. Two fourth order methods are also presented. The traveling wave solution and a spinodal decomposition problem are used to demonstrate numerical properties and the order of convergence of the proposed methods

Towards extending the Ahlswede-Khachatrian theorem to cross t-intersecting families

Ahlswede and Khachatrian's diametric theorem is a weighted version of their complete intersection theorem, itself an extension of the $t$-intersecting Erd\H{o}s-Ko-Rado theorem. Their intersection theorem says that the maximum size of a family of subsets of $[n] = \{1, \dots, n\}$, every pair of which intersects in at least $t$ elements, is the size of certain trivially intersecting families proposed by Frankl. We address a cross intersecting version of their diametric theorem. Two families $\mathcal{A}$ and $\mathcal{B}$ of subsets of $[n]$ are {\em cross $t$-intersecting} if for every $A \in \mathcal{A}$ and $B \in \mathcal{B}$, $A$ and $B$ intersect in at least $t$ elements. The $p$-weight of a $k$ element subset $A$ of $[n]$ is $p^{k}(1-p)^{n-k}$, and the weight of a family $\mathcal{A}$ is the sum of the weights of its sets. The weight of a pair of families is the product of the weights of the families. The maximum $p$-weight of a $t$-intersecting family depends on the value of $p$. Ahlswede and Khachatrian showed that for $p$ in the range $[\frac{r}{t + 2r - 1}, \frac{r+1}{t + 2r + 1}]$, the maximum $p$-weight of a $t$-intersecting family is that of the family $\mathcal{F}^t_r$ consisting of all subsets of $[n]$ containing at least $t+r$ elements of the set $[t+2r]$. In a previous paper we showed a cross $t$-intersecting version of this for large $t$ in the case that $r = 0$. In this paper, we do the same in the case that $r = 1$. We show that for $p$ in the range $[\frac{1}{t + 1}, \frac{2}{t + 3}]$ the maximum $p$-weight of a cross $t$-intersecting pair of families, for $t \geq 200$, is achieved when both families are $\mathcal{F}^t_1$. Further, we show that except at the endpoints of this range, this is, up to isomorphism, the only pair of $t$-intersecting families achieving this weight.Comment: 22 pages (18 plus appendix), 3 figure

On the Heterogeneous Distributions in Paper Citations

Academic papers have been the protagonists in disseminating expertise. Naturally, paper citation pattern analysis is an efficient and essential way of investigating the knowledge structure of science and technology. For decades, it has been observed that citation of scientific literature follows a heterogeneous and heavy-tailed distribution, and many of them suggest a power-law distribution, log-normal distribution, and related distributions. However, many studies are limited to small-scale approaches; therefore, it is hard to generalize. To overcome this problem, we investigate 21 years of citation evolution through a systematic analysis of the entire citation history of 42,423,644 scientific literatures published from 1996 to 2016 and contained in SCOPUS. We tested six candidate distributions for the scientific literature in three distinct levels of Scimago Journal & Country Rank (SJR) classification scheme. First, we observe that the raw number of annual citation acquisitions tends to follow the log-normal distribution for all disciplines, except for the first year of the publication. We also find significant disparity between the yearly acquired citation number among the journals, which suggests that it is essential to remove the citation surplus inherited from the prestige of the journals. Our simple method for separating the citation preference of an individual article from the inherited citation of the journals reveals an unexpected regularity in the normalized annual acquisitions of citations across the entire field of science. Specifically, the normalized annual citation acquisitions have power-law probability distributions with an exponential cut-off of the exponents around 2.3, regardless of its publication and citation year. Our results imply that journal reputation has a substantial long-term impact on the citation.Comment: 8 pages, 7 figure

On the total variation distance between the binomial random graph and the random intersection graph

When each vertex is assigned a set, the intersection graph generated by the sets is the graph in which two distinct vertices are joined by an edge if and only if their assigned sets have a nonempty intersection. An interval graph is an intersection graph generated by intervals in the real line. A chordal graph can be considered as an intersection graph generated by subtrees of a tree. In 1999, Karo\'nski, Scheinerman and Singer-Cohen [Combin Probab Comput 8 (1999), 131--159] introduced a random intersection graph by taking randomly assigned sets. The random intersection graph $G(n,m;p)$ has $n$ vertices and sets assigned to the vertices are chosen to be i.i.d. random subsets of a fixed set $M$ of size $m$ where each element of $M$ belongs to each random subset with probability $p$, independently of all other elements in $M$. Fill, Scheinerman and Singer-Cohen [Random Struct Algorithms 16 (2000), 156--176] showed that the total variation distance between the random graph $G(n,m;p)$ and the Erd\"os-R\'enyi graph $G(n,\hat{p})$ tends to $0$ for any $0 \leq p=p(n) \leq 1$ if $m=n^{\alpha}$, $\alpha >6$, where $\hat{p}$ is chosen so that the expected numbers of edges in the two graphs are the same. In this paper, it is proved that the total variation distance still tends to $0$ for any $0 \leq p=p(n) \leq 1$ whenever $m \gg n^4$.Comment: revised version of the 1st draft "On a phase transition of the random intersection graph: Supercritical region

Dynamic coloring of graphs having no K5K_5 minor

We prove that every simple connected graph with no $K_5$ minor admits a proper 4-coloring such that the neighborhood of each vertex $v$ having more than one neighbor is not monochromatic, unless the graph is isomorphic to the cycle of length 5. This generalizes the result by S.-J. Kim, S. J. Lee, and W.-J. Park on planar graphs.Comment: Rewriting with a major change (14 pages, 1 figure

A fast direct solver for scattering from periodic structures with multiple material interfaces in two dimensions

We present a new integral equation method for the calculation of two-dimensional scattering from periodic structures involving triple-points (multiple materials meeting at a single point). The combination of a robust and high-order accurate integral representation and a fast direct solver permits the efficient simulation of scattering from fixed structures at multiple angles of incidence. We demonstrate the performance of the scheme with several numerical examples.Comment: 19 Pages. 8 Figure