Positive solutions for a weighted fractional system

In this article, we study positive solutions to the system{A_αu(x) = C_(n,α)PV∫_(Rn)(a1(x-y)(u(x)-u(y)))/(|x-y|~(n+α))dy = f(u(x), B_βv(x) = C_(n,β)PV ∫_(Rn)(a2(x-y)(v(x)-v(y))/(|x-y|~(n+β))dy = g(u(x),v(x)).To reach our aim, by using the method of moving planes, we prove a narrow region principle and a decay at infinity by the iteration method. On the basis of these results, we conclude radial symmetry and monotonicity of positive solutions for the problems involving the weighted fractional system on an unit ball and the whole space. Furthermore, non-existence of nonnegative solutions on a half space is given.     

Chaos in rational systems in the plane containing quadratic terms

Consider the following system of rational equations containing quadratic termsChaos in the sense of Li–Yorke is considered. This is based on the Marotto’s theorem via obtaining a snap-back repeller. In fact, first in a special case when $F_1=F_2=0$, we show that origin is a snap-back repeller and so the system has chaotic behavior in the sense of Li–Yorke under some conditions. Then in a more general case, we prove that existence of chaos in the sense of Li–Yorke for the above system.     

Blow-up for coupled nonlinear wave equations with damping and source

We consider the system of nonlinear wave equations $\{\begin{array}{cc}{u}_{tt}+u_t+{\left|u_t\right|}^{m?1}{u}_t=\mathrm{div}\left({\rho }_1\left({\left|?u\right|}^2\right)?u\right)+f_1(u,v)\text{,}\hfill & (x,t)\in \Omega \times (0,T)\text{,}\hfill \\ v_{tt}+v_t+{\left|v_t\right|}^{r?1}{v}_t=\mathrm{div}\left({\rho }_2\left({\left|?v\right|}^2\right)?v\right)+f_2(u,v)\text{,}\hfill & (x,t)\in \Omega \times (0,T)\text{,}\hfill \end{array}$ with initial and Dirichlet boundary conditions. Under some suitable assumptions on the functions$f_1$, $f_2$, ${\rho }_1$, ${\rho }_2$, parameters $r,m$ and the initial data, the result on blow-up of solutions and upper bound of blow-up time are given.     

Sharp threshold of blow-up and scattering for the fractional Hartree equation

We consider the fractional Hartree equation in the $L^2$-supercritical case, and find a sharp threshold of the scattering versus blow-up dichotomy for radial data: If $M{[u_0]}^{\frac{s?s_c}{s_c}}E[u_0]<M{[Q]}^{\frac{s?s_c}{s_c}}E[Q]$ and $M{[u_0]}^{\frac{s?s_c}{s_c}}{6u_{0}6}_{{\dot{H}}^s}^2<M{[Q]}^{\frac{s?s_c}{s_c}}{6Q6}_{{\dot{H}}^s}^2$, then the solution $u(t)$ is globally well-posed and scatters; if $M{[u_0]}^{\frac{s?s_c}{s_c}}E[u_0]<M{[Q]}^{\frac{s?s_c}{s_c}}E[Q]$ and $M{[u_0]}^{\frac{s?s_c}{s_c}}{6u_{0}6}_{{\dot{H}}^s}^2>M{[Q]}^{\frac{s?s_c}{s_c}}{6Q6}_{{\dot{H}}^s}^2$, the solution $u(t)$ blows up in finite time. This condition is sharp in the sense that the solitary wave solution $e^{it}Q(x)$ is global but not scattering, which satisfies the equality in the above conditions. Here, Q is the ground-state solution for the fractional Hartree equation.     

Expected number of real roots of random trigonometric polynomials

We investigate the asymptotics of the expected number of real roots of random trigonometric polynomials

$X_n\left(t\right)=u+\frac{1}{\sqrt{n}}\sum _{k=1}^n(A_{k}cos\left(kt\right)+B_{k}sin\left(kt\right))\text{,}t\in [0,2\pi ]\text{,}u\in \mathbb{R}$

whose coefficients $A_k,B_k$, $k\in \mathbb{N}$, are independent identically distributed random variables with zero mean and unit variance. If $N_n[a,b]$ denotes the number of real roots of $X_n$ in an interval $[a,b]?[0,2\pi ]$, we prove that

$\underset{n\to \infty }{lim}\frac{\mathbb{E}{N}_n[a,b]}{n}=\frac{b?a}{\pi \sqrt{3}}exp(?\frac{u^2}{2})\text{.}$

     

Powerful numbers in (1ℓ + qℓ)(2ℓ + qℓ)⋯(nℓ + qℓ)

Let q be a positive integer. Recently, Niu and Liu proved that, if $n\ge \max ?\{q,1198?q\}$, then the product $(1^3+q^3)(2^3+q^3)?(n^3+q^3)$ is not a powerful number. In this note, we prove (1) that, for any odd prime power ? and $n\ge \max ?\{q,11?q\}$, the product $(1^{?}+q^{?})(2^{?}+q^{?})?(n^{?}+q^{?})$ is not a powerful number, and (2) that, for any positive odd integer ?, there exists an integer $N_{q,?}$ such that, for any positive integer $n\ge N_{q,?}$, the product $(1^{?}+q^{?})(2^{?}+q^{?})?(n^{?}+q^{?})$ is not a powerful number.     

General convergence analysis for two-step projection methods and applications to variational problems

First a general model for two-step projection methods is introduced and second it has been applied to the approximation solvability of a system of nonlinear variational inequality problems in a Hilbert space setting. Let $H$ be a real Hilbert space and $K$ be a nonempty closed convex subset of $H$. For arbitrarily chosen initial points $x^0,y^0\in K$, compute sequences $\left\{x^k\right\}$ and $\left\{y^k\right\}$ such that $x^{k+1}=(1-a^k)x^k+a^k{P}_K[y^k-\rho T\left(y^k\right)]\text{for }\rho >0$$y^k=(1-b^k)x^k+b^k{P}_K[x^k-\eta T\left(x^k\right)]\text{for }\eta >0\text{,}$ where $T:K\to H$ is a nonlinear mapping on $K,P_K$ is the projection of $H$ onto $K$, and $0\le a^k,b^k\le 1$. The two-step model is applied to some variational inequality problems.     

Degree Ramsey numbers for even cycles

Let $H\stackrel{s}{?}G$ denote that any $s$-coloring of $E\left(H\right)$ contains a monochromatic $G$. The degree Ramsey number of a graph $G$, denoted by $R_{\Delta }(G,s)$, is $min\{\Delta \left(H\right):H\stackrel{s}{?}G\}$. We consider degree Ramsey numbers where $G$ is a fixed even cycle. Kinnersley, Milans, and West showed that $R_{\Delta }(C_{2k},s)\ge 2s$, and Kang and Perarnau showed that $R_{\Delta }(C_4,s)=\Theta \left(s^2\right)$. Our main result is that $R_{\Delta }(C_6,s)=\Theta \left(s^{3∕2}\right)$ and $R_{\Delta }(C_{10},s)=\Theta \left(s^{5∕4}\right)$. Additionally, we substantially improve the lower bound for $R_{\Delta }(C_{2k},s)$ for general $k$.