For graphs $G$, $G_1$ and $G_2$, let $G \to (G_1,G_2)$ signify that any red/blue edge-coloring of $G$ contains a red $G_1$ or a blue $G_2$, and let $f(G_1,G_2)$ be the minimum $N$ such that there is a graph $G$ of order $N$ with $\omega(G) = \max \{\omega(G_1),\omega(G_2)\}$ and $G \to (G_1,G_2)$. It is shown that $c_1(n/\!\log n)^{(m+1)/2} \leq f(K_m,K_{n,n}) \leq c_2 n^{m-1}$, where $c_i = c_i(m) \gt 0$ are constants. In particular, $cn^2/\log n \leq f(K_3,K_{n,n}) \leq 2n^2+2n-1$. Moreover, $f(K_m,T_n) \leq m^2(n-1)$ for all $n \geq m \geq 2$, where $T_n$ is a tree on $n$ vertices.

Let $R$ be a ring, $(S,\leq)$ a strictly ordered monoid and $\omega \colon S \to \operatorname{End}(R)$ a monoid homomorphism. The skew generalized power series ring $R[[S,\omega]]$ is a common generalization of (skew) polynomial rings, (skew) power series rings, (skew) Laurent polynomial rings, (skew) group rings, and Mal'cev-Neumann Laurent series rings. In this paper, we continue the study of skew generalized power series ring $R[[S,\omega]]$. It is shown that under suitable conditions, if $R$ has a (flat) projective socle, then so does $R[[S,\omega]]$. Necessary and sufficient conditions are obtained for $R[[S,\omega]]$ to satisfy a certain ring property which is among being local, semilocal, semiperfect, semiregular, left quasi-duo, clean, exchange, right stable range one, projective-free, and $I$-ring, respectively.

Starting with the pioneering work of Ein and Lazarsfeld [9] restrictions on values of Seshadri constants on algebraic surfaces have been studied by many authors [2,5,10,12,18,20,22,24]. In the present note we show how approximation involving continued fractions combined with recent results of Küronya and Lozovanu on Okounkov bodies of line bundles on surfaces [13,14] lead to effective statements considerably restricting possible values of Seshadri constants. These results in turn provide strong additional evidence to a conjecture governing the Seshadri constants on algebraic surfaces with Picard number $1$.

In an earlier paper [23] the authors introduced a new ellipsoid associated with a submanifold, and established an integral formula for the total mean curvature matrix of hypersurfaces. In the present paper a kinematic formula for the total mean curvature matrix of submanifolds in $\mathbb{R}^{n}$ is proved.

In this paper we investigate the relation between the number of rational points over a finite field $\mathbb{F}_{p^n}$ on a family of higher genus curves and their periods in terms of hypergeometric functions. For the case $y^\ell = x(x-1)(x-\lambda)$ we find a closed form in terms of hypergeometric functions associated with the periods of the curve. For the general situation $y^\ell = x^{a_1}(x-1)^{a_2}(x-\lambda)^{a_3}$ we show that the number of rational points is a linear combination of hypergeometric series, and we provide an algorithm to determine the coefficients involved.

This paper is devoted to the study of the mathematical model involving a frictionless contact between an electro-elasto-viscoplastic body and a conductive adhesive foundation. The process is mechanically dynamic and electrically static. The contact is modeled with a normal compliance where the adhesion is taken into account and a regularized electrical conductivity condition. We derive a variational formulation of the problem and prove its unique weak solution. The proof is based on nonlinear evolution equations with monotone operators, differential equations and fixed point arguments.

In this paper, we consider the wave equation with internal time delay and source terms\[ u_{tt}(x,t) - \triangle u(x,t) + \mu_1 u_t(x,t) + \mu_2 u_t(x,t-\tau) + f(x,u) = h(x)\]in a bounded domain. By virtue of Galerkin method combined with the priori estimates, we prove the existence and uniqueness of global solution under initial-boundary data for the above equation. Moreover, under suitable conditions on the forcing term $f(x,u)$ and $\mu_1$, $\mu_2$, the existence of a compact global attractor is proved. Further, the asymptotic behavior and the decay property of global solution are discussed.

This work is concerned with the blow-up phenomena for a nonlocal reaction-diffusion equation with null Robin boundary conditions. We establish sufficient conditions to guarantee the solution exists globally or blows up at finite time under appropriate measure sense. Moreover, upper and lower bounds for the blow-up time are derived in higher dimensional spaces. Finally, some application examples are presented.

In this paper, we study the following Klein-Gordon-Maxwell system\[\begin{cases} -\Delta u + V(x)u - (2\omega + \phi)\phi u = f(x,u), &x \in \mathbb{R}^3, \\ \Delta \phi = (\omega + \phi) u^2, &x \in \mathbb{R}^3,\end{cases}\]where the nonlinearity $f$ and the potential $V$ are allowed to be sign-changing. Under some appropriate assumptions on $V$ and $f$, we obtain the existence of two different solutions of the system via the Ekeland variational principle and the Mountain Pass Theorem.

The finiteness for an inverse three spectra Sturm-Liouville problem with potential $q$ on the interval $[0,1]$ and boundary parameters $h_{0}$, $h_{1}$ is studied in this paper. Under condition that two boundary conditions at a fixed internal rational point $a$ of $(0,1)$ are different and known a priori, we show that there exist at most a finite number of triplets $(q;h_{0},h_{1})$ corresponding to the three spectra of a Sturm-Liouville equation defined on $[0,1]$, $[0,a]$ and $[a,1]$, respectively, with the same boundary conditions at two endpoints $0$ and $1$.

In this paper, we study the following fractional Schrödinger equations\[ (-\Delta)^{\alpha}u + \lambda V(x)u = f(x,u) + \mu \xi(x) |u|^{p-2}u, \quad x \in \mathbb{R}^{N},\]where $\lambda \gt 0$ is a parameter, $V \in C(\mathbb{R}^{N})$ and $V^{-1}(0)$ has nonempty interior. Under some mild assumptions, we establish the existence of two different nontrivial solutions. Moreover, the concentration of these solutions is also explored on the set $V^{-1}(0)$ as $\lambda \to \infty$. As an application, we also give the similar results and concentration phenomenons for the above problem with concave and convex nonlinearities.

We study $L^p \to L^r$ estimates for restricted averaging operators related to algebraic varieties $V$ of $d$-dimensional vector spaces over finite fields $\mathbb{F}_q$ with $q$ elements. We observe properties of both the Fourier restriction operator and the averaging operator over $V \subset \mathbb{F}_q^d$. As a consequence, we obtain optimal results on the restricted averaging problems for spheres and paraboloids in dimensions $d \geq 2$, and cones in odd dimensions $d \geq 3$. In addition, when the variety $V$ is a cone lying in an even dimensional vector space over $\mathbb{F}_q$ and $-1$ is a square number in $\mathbb{F}_q$, we also obtain sharp estimates except for two endpoints.

The numerical approximation of the nonlinear time fractional Klein-Gordon equation in a bounded domain is considered. The time fractional derivative is described in the Caputo sense with the order $\gamma$ ($1 \lt \gamma \lt 2$). A fully discrete spectral scheme is proposed on the basis of finite difference discretization in time and Legendre spectral approximation in space. The stability and convergence of the fully discrete scheme are rigorously established. The convergence rate of the fully discrete scheme in $H^1$ norm is $\mathrm{O}(\tau^{3-\gamma} + N^{1-m})$, where $\tau$, $N$ and $m$ are the time-step size, polynomial degree and regularity in the space variable of the exact solution, respectively. Numerical examples are presented to support the theoretical results.