In this paper, we complete a classification of pentavalent arc-transitive graphs of order $2p^{2}q$, where $p$ and $q$ are distinct odd primes. This result involves a subclass of pentavalent arc-transitive graphs of cube-free order.

Let $N$ be a sufficiently large integer. In this paper, it is proved that with at most $O(N^{13/16+\varepsilon})$ exceptions, all even positive integers up to $N$ can be represented in the form $p_1^2 + p_2^2 + p_3^3 + p_4^3 + p_5^4 + p_6^4$, where $p_1$, $p_2$, $p_3$, $p_4$, $p_5$, $p_6$ are prime numbers.

In this paper, we study the behavior of the sets of volumes of the form $\operatorname{vol}(X,K_X+B+M)$, where $(X,B)$ is a log canonical pair, and $M$ is a nef $\mathbb{R}$-divisor. After a first analysis of some general properties, we focus on the case when $M$ is $\mathbb{Q}$-Cartier with given Cartier index, and $B$ has coefficients in a given DCC set. First, we show that such sets of volumes satisfy the DCC property in the case of surfaces. Once this is established, we show that surface pairs with given volume and for which $K_X+B+M$ is ample form a log bounded family. These generalize results due to Alexeev [1].

We prove the global existence and uniqueness of weak solutions to the time dependent Ginzburg-Landau system in superconductivity with Coulomb gauge.

The purpose of this paper is to study a three species competition model with diffusion. It is well known that there exists a family of traveling wave solutions connecting two equilibria $(0,1,1)$ and $(1,0,0)$. In this paper, we first establish the exact asymptotic behavior of the traveling wave profiles at $\pm \infty$. Then, by constructing a pair of explicit upper and lower solutions via the combination of traveling wave solutions, we derive the existence of some new entire solutions which behave as two traveling fronts moving towards each other from both sides of $x$-axis. Such entire solution provides another invasion way of the stronger species to the weak ones.

The paper constructs the solution of a forced capillary-gravity waves problem outside a cylindrical wavemaker of finite depth under Hocking's edge condition. After introducing the setting of the problem, it obtains the solution and proves its uniqueness. Possible applications of the result for future researches, including the application of the model to earthquakes, are also discussed at the end.

The inverses of Toeplitz-Hessenberg matrices are investigated. It is known that each inverse of such a matrix is a sum of a lower triangular matrix $L$ and a matrix $R$ of rank $1$. The formulas of $L$ and $x$, $y$ such that $xy^T = R$ are derived. Using this result we propose an algorithm for inverting Toeplitz-Hessenberg matrices. Moreover, from the expression of the inverse a formula for the determinant is deduced.

In this article we are interested in the nonlocal regional Schrödinger equation with critical exponent \[ \epsilon^{2\alpha} (-\Delta)_{\rho}^{\alpha} u + u = \lambda u^q + u^{2_{\alpha}^{*}-1} \quad \textrm{in $\mathbb{R}^{n}$}, \quad u \in H^{\alpha}(\mathbb{R}^{n}), \] where $\epsilon$ is a small positive parameter, $\alpha \in (0,1)$, $q \in (1,2_{\alpha}^{*}-1)$, $2_{\alpha}^{*} = 2n/(n-2\alpha)$ is the critical Sobolev exponent, $\lambda \gt 0$ is a parameter and $(-\Delta)_{\rho}^{\alpha}$ is a variational version of the regional Laplacian, whose range of scope is a ball with radius $\rho(x) \gt 0$. We study the existence of a ground state and we analyze the behavior of semi-classical solutions as $\varepsilon \to 0$.

In this paper, we consider the viscoelastic equation with variable coefficients and Balakrishnan-Taylor damping and source terms. This work is devoted to prove, under suitable conditions on the initial data, the asymptotic stability without imposing any restrictive growth assumption on the damping term and weakening of the usual assumptions on the relaxation function.

In this paper, a family of new finite difference (NFD) methods for solving the convection-diffusion equation with singularly perturbed parameters are considered. By taking account of infinite terms in the Taylor's expansions and using the triangle function theorem, we construct a series of NFD schemes for the one-dimensional problems firstly and derive the error estimates as well. Then, applying the ADI technique, the idea is extended to two dimensional equations. Besides no numerical oscillation, there are mainly three advantages for the proposed methods: one is that the schemes can achieve the predicted convergence orders on uniform mesh regardless of the perturbed parameter for 1D equations; Secondly, no matter which convergence order the scheme is, the generated linear systems have diagonal structures; Thirdly, the methods are easily expanded to the special mesh technique such as Shishkin mesh. Some numerical experiments are shown to verify the prediction.

In this paper, we present a Clenshaw-Curtis-Filon-type method for the weakly singular oscillatory integral with Fourier and Hankel kernels. By interpolating the non-oscillatory and nonsingular part of the integrand at $(N+1)$ Clenshaw-Curtis points, the method can be implemented in $O(N \log N)$ operations, which requires the accurate computation of modified moments. We first give a method for the derivation of recurrence relation for the modified moments, which can be applied to the derivation of recurrence relation for the modified moments corresponding to other type oscillatory integrals. By using the recurrence relation, special functions and classic quadrature methods, the modified moments can be computed accurately and efficiently. Then, we present the corresponding error bound in inverse powers of frequencies $k$ and $\omega$ for the proposed method. Numerical examples are provided to support the theoretical results and show the efficiency and accuracy of the method.

In this paper we investigate second-order Karush-Kuhn-Tucker multipliers for both local nondominated and local minimal points of set-valued optimization with variable ordering structures. We prove calculus rules of second-order contingent derivatives of index $\gamma \in \{0,1\}$ and use them to establish improved Karush-Kuhn-Tucker multiplier rules of nonclassical forms which involve separately such derivatives of the objective, constraint and ordering maps. The equivalence between the nonemptiness and boundedness of the multiplier sets in these rules and second-order constraint qualifications of the Kurcyusz-Robinson-Zowe and Mangasarian-Fromovitz types is demonstrated.