Solitary waves of a generalized Ostrovsky equation

We consider the existence and stability of traveling waves of a generalized Ostrovsky equation ${(u_t-\beta u_{xxx}-f{\left(u\right)}_x)}_x=\gamma u$, where the nonlinearity $f\left(u\right)$ satisfies a power-like scaling condition. We prove that there exist ground state solutions which minimize the action among all nontrivial solutions and use this variational characterization to study their stability. We also introduce a numerical method for computing ground states based on their variational properties. The class of nonlinearities considered includes sums and differences of distinct powers.     

Asymptotic stability of superposition of stationary solutions and rarefaction waves for 1D Navier–Stokes/Allen–Cahn system

In this paper, we investigate the large time behavior of the solutions to the inflow problem for the one-dimensional Navier–Stokes/Allen–Cahn system in the half space. First, we assume that the space-asymptotic states $({\rho }_{+},u_{+},{\chi }_{+})$ and the boundary data $({\rho }_b,u_b,{\chi }_b)$ satisfy some conditions so that the time-asymptotic state of solutions for the inflow problem is a nonlinear wave which is the superposition of a stationary solution and a rarefaction wave. Then, we show the existence of the stationary solution by the center manifold theorem. Finally, we prove that the nonlinear wave is asymptotically stable when the initial data is a small perturbation of the nonlinear wave. The proof is mainly based on the energy method by taking into account the effect of the concentration χ and the complexity of nonlinear wave.     

Relation between the H-rank of a mixed graph and the rank of its underlying graph

Given a simple graph $G=(V_G,E_G)$ with vertex set $V_G$ and edge set $E_G$, the mixed graph $\stackrel{?}{G}$ is obtained from $G$ by orienting some of its edges. Let $H\left(\stackrel{?}{G}\right)$ denote the Hermitian adjacency matrix of $\stackrel{?}{G}$ and $A\left(G\right)$ be the adjacency matrix of $G$. The $H$-rank (resp. rank) of $\stackrel{?}{G}$ (resp. $G$), written as $rk\left(\stackrel{?}{G}\right)$ (resp. $r\left(G\right)$), is the rank of $H\left(\stackrel{?}{G}\right)$ (resp. $A\left(G\right)$). Denote by $d\left(G\right)$ the dimension of cycle space of $G$, that is $d\left(G\right)=\left|E_G\right|?\left|V_G\right|+\omega \left(G\right)$, where $\omega \left(G\right)$ denotes the number of connected components of $G$. In this paper, we concentrate on the relation between the $H$-rank of $\stackrel{?}{G}$ and the rank of $G$. We first show that $?2d\left(G\right)?rk\left(\stackrel{?}{G}\right)?r\left(G\right)?2d\left(G\right)$ for every mixed graph $\stackrel{?}{G}$. Then we characterize all the mixed graphs that attain the above lower (resp. upper) bound. By these obtained results in the current paper, all the main results obtained in Luo et al. (2018); Wong et al. (2016) may be deduced consequently.     

Singular properties of solutions for a parabolic equation with variable exponents and logarithmic source

This paper deals with a parabolic $p\left(x\right)$-Laplace equation with logarithmic source $u^{q\left(x\right)}logu$. The singular properties of solutions are determined completely by classifying the initial energy. Moreover, we obtain a new extinction rate of solutions, where the order of the extinction rate is greater than the maximum of variable exponent $q\left(x\right)$. This kind of extinction rate could reflect the influence of logarithmic functions on the extinction of solutions more reasonably.     

No-three-in-line problem on a torus: Periodicity

Let ${\tau }_{m,n}$ denote the maximal number of points on the discrete torus (discrete toric grid) of sizes $m\times n$ with no three collinear points. The value ${\tau }_{m,n}$ is known for the case where $gcd(m,n)$ is prime. It is also known that ${\tau }_{m,n}\le 2gcd(m,n)$. In this paper we generalize some of the known tools for determining ${\tau }_{m,n}$ and also show some new. Using these tools we prove that the sequence ${\left({\tau }_{z,n}\right)}_{n\in \mathbb{N}}$ is periodic for all fixed $z>1$. In general, we do not know the period; however, if $z=p^a$ for $p$ prime, then we can bound it. We prove that ${\tau }_{p^a,p^{(a?1)p+2}}=2p^a$ which implies that the period for the sequence is $p^b$, where $b$ is at most $(a?1)p+2$.     

Degree powers in graphs with a forbidden forest

Given a positive integer $p$ and a graph $G$ with degree sequence $d_1,\dots ,d_n$, we define $e_p\left(G\right)={\sum }_{i=1}^n{d}_i^p$. Caro and Yuster introduced a Turán-type problem for $e_p\left(G\right)$: Given a positive integer $p$ and a graph $H$, determine the function ${\text{ex}}_p(n,H)$, which is the maximum value of $e_p\left(G\right)$ taken over all graphs $G$ on $n$ vertices that do not contain $H$ as a subgraph. Clearly, ${\text{ex}}_1(n,H)=2\text{ex}(n,H)$, where $\text{ex}(n,H)$ denotes the classical Turán number. Caro and Yuster determined the function ${\text{ex}}_p(n,P_{?})$ for sufficiently large $n$, where $p\ge 2$ and $P_{?}$ denotes the path on $?$ vertices. In this paper, we generalise this result and determine ${\text{ex}}_p(n,F)$ for sufficiently large $n$, where $p\ge 2$ and $F$ is a linear forest. We also determine ${\text{ex}}_p(n,S)$, where $S$ is a star forest; and ${\text{ex}}_p(n,B)$, where $B$ is a broom graph with diameter at most six.     

A generalization of Schönemann’s theorem via a graph theoretic method

Recently, Grynkiewicz et al. (2013), using tools from additive combinatorics and group theory, proved necessary and sufficient conditions under which the linear congruence $a_1{x}_1+?+a_k{x}_k\equiv b(modn)$, where $a_1,\dots ,a_k,b,n$ ($n\ge 1$) are arbitrary integers, has a solution $〈x_1,\dots ,x_k〉\in {\mathbb{Z}}_n^k$ with all $x_i$ distinct. So, it would be an interesting problem to give an explicit formula for the number of such solutions. Quite surprisingly, this problem was first considered, in a special case, by Schönemann almost two centuries ago(!) but his result seems to have been forgotten. Schönemann (1839), proved an explicit formula for the number of such solutions when $b=0$, $n=p$ a prime, and ${\sum }_{i=1}^k{a}_i\equiv 0(modp)$ but ${\sum }_{i\in I}{a}_i?\equiv 0(modp)$ for all $0?\ne I??\{1,\dots ,k\}$. In this paper, we generalize Schönemann’s theorem using a result on the number of solutions of linear congruences due to D. N. Lehmer and also a result on graph enumeration. This seems to be a rather uncommon method in the area; besides, our proof technique or its modifications may be useful for dealing with other cases of this problem (or even the general case) or other relevant problems.     

A time-periodic reaction–diffusion–advection equation with a free boundary and sign-changing coefficients

We consider a reaction–diffusion–advection equation of the form: $u_t=u_{xx}-\beta \left(t\right)u_x+f(t,u)$ for $x\in [0,h\left(t\right))$, where $\beta \left(t\right)$ is a $T$-periodic function, $f(t,u)$ is a $T$-periodic Fisher–KPP type of nonlinearity with $a\left(t\right)≔f_u(t,0)$ changing sign, $h\left(t\right)$ is a free boundary satisfying the Stefan condition. We study the long time behavior of solutions and find that there are two critical numbers $\bar{c}$ and $B\left(\tilde{\beta }\right)$ with $B\left(\tilde{\beta }\right)>\bar{c}>0$, $\bar{\beta }≔\frac{1}{T}{\int }_0^T\beta \left(t\right)dt$ and $\tilde{\beta }\left(t\right)≔\beta \left(t\right)-\bar{\beta }$, such that a vanishing–spreading dichotomy result holds when $\left|\bar{\beta }\right|<\bar{c}$; a vanishing–transition–virtual spreading trichotomy result holds when $\bar{\beta }\in [\bar{c},B\left(\tilde{\beta }\right))$; all solutions vanish when $\bar{\beta }⩾B\left(\tilde{\beta }\right)$ or $\bar{\beta }⩽-\bar{c}$.     

On a problem of Specker about Euclidean representations of finite graphs

Say that a graph $G$ is representable in ${\mathbb{R}}^n$ if there is a map $f$ from its vertex set into the Euclidean space ${\mathbb{R}}^n$ such that $6f\left(x\right)?f\left(x^{\prime }\right)6=6f\left(y\right)?f\left(y^{\prime }\right)6$ iff $\{x,x^{\prime }\}$ and$\{y,y^{\prime }\}$ are both edges or both non-edges in $G$. The purpose of this note is to present the proof of the following result, due to Einhorn and Schoenberg in Einhorn and Schoenberg (1966): if $G$ finite is neither complete nor independent, then it is representable in ${\mathbb{R}}^{\left|G\right|?2}$. A similar result also holds in the case of finite complete edge-colored graphs.     

Initial boundary value problem for a class of strongly damped semilinear wave equations with logarithmic nonlinearity

This paper deals with the initial boundary value problem for strongly damped semilinear wave equations with logarithmic nonlinearity $u_{tt}-\Delta u-\Delta u_t={\phi }_p\left(u\right)log\left|u\right|$ in a bounded domain $\Omega \subset {\mathbb{R}}^n$. We discuss the existence, uniqueness and polynomial or exponential energy decay estimates of global weak solutions under some appropriate conditions. Moreover, we derive the finite time blow up results of weak solutions, and give the lower and upper bounds for blow-up time by the combination of the concavity method, perturbation energy method and differential–integral inequality technique.