The adjacent vertex-distinguishing total chromatic number of a graph $G$, denoted by $\chi_{at}(G)$, is the smallest $k$ for which $G$ has a proper total $k$-coloring such that any two adjacent vertices have distinct sets of colors appearing on the vertex and its incident edges. In regard of this number, there is a famous conjecture (AVDTCC) which states that for any simple graph $G$, $\chi_{at}(G) \leq \Delta(G)+3$. In this paper, we study this number for the generalized Mycielski graph $\mu_m(G)$ of a graph $G$. We prove that the satisfiability of the conjecture AVDTCC in $G$ implies its satisfiability in $\mu_m(G)$. Particularly we give the exact values of $\chi_{at}(\mu_m(G))$ when $G$ is a graph with maximum degree less than $3$ or a complete graph. Moreover, we investigate $\chi_{at}(G)$ for any graph $G$ with only one maximum degree vertex by showing that $\chi_{at}(G) \leq \Delta(G)+2$ when $\Delta(G) \leq 4$.

We show that for any ample line bundle on a smooth complex projective variety with nonnegative Kodaira dimension, the semistability of co-Higgs bundles of implies the semistability of bundles. Then we investigate the criterion for surface $X$ to have $H^0(T_X) = H^0(S^2 T_X) = 0$, which implies that any co-Higgs structure of rank two is nilpotent.

The notion of an ideal submanifold was introduced by Chen at the end of the last century. A survey of recent results in this area can be found in his book [9]. Recently, in [10], an optimal collection of Chen's inequalities was obtained for Lagrangian submanifolds in complex space forms. As shown in [2], these inequalities have an immediate counterpart in centroaffine differential geometry. Centroaffine hypersurfaces realising the equality in one of these inequalities are called ideal centroaffine hypersurfaces.

So far, most results in this area have only been related with $3$- and $4$-dimensional $\delta^{\sharp}(2)$-ideal centroaffine hypersurfaces. The purpose of this paper is to classify $\delta^{\sharp}(2,2)$-ideal hypersurfaces of dimension~$5$ in centroaffine differential geometry.

This paper focuses on the study of three dimensional real hypersurfaces in non-flat complex space forms whose $^{*}$-Ricci tensor satisfies conditions of parallelism. More precisely, results concerning real hypersurfaces with vanishing, semi-parallel and pseudo-parallel $^{*}$-Ricci tensor in complex hyperbolic space are provided. Furthermore, new results concerning $\xi$-parallelism of $^{*}$-Ricci tensor of real hypersurfaces in non-flat complex space forms are presented.

This paper is devoted to the electronic polarization effect of dielectric materials in plasma physics. Using the two-scale convergence method introduced by G. Nguetseng and G. Allaire, we study the electronic polarization effect induced by two-scale homogenization. The microscopic properties of electrons in dielectric materials are characterized by Vlasov-Poisson system. The homogenized equations describing the mean behaviors of the microscopic equations are obtained. We also derive the modified Gauss law through the polarization effect. From the homogenized Poisson equation, the dielectric function is also acquired.

In this paper, we consider the quantum Zakharov system in one spatial dimension. We prove the global well-posedness of the system with $L^2$-Schrödinger data and some wave data. The regularity of the wave data is in the largest set. We give counterexamples for the boundary of the set. As the quantum parameter tends to zero, we formally recover the result of Colliander-Holmer-Tzirakis for the classical Zakharov system.

This paper is dedicated to studying the following Schrödinger-Poisson system\[ \begin{cases} -\Delta u + V(x)u + K(x) \phi(x)u = f(x,u), &x \in \mathbb{R}^{3}, \\ -\Delta \phi = K(x) u^2, &x \in \mathbb{R}^{3}, \end{cases}\]where $V(x)$, $K(x)$ and $f(x,u)$ are periodic or asymptotically periodic in $x$. We use the non-Nehari manifold approach to establish the existence of the Nehari type ground state solutions in two cases: the periodic one and the asymptotically periodic case, by introducing weaker conditions $\lim_{|t| \to \infty} \left( \int_0^t f(x,s) \, \mathrm{d}s \right)/|t|^3 = \infty$ uniformly in $x \in \mathbb{R}^3$ and\[ \left[ \frac{f(x,\tau)}{\tau^3} - \frac{f(x,t\tau)}{(t\tau)^3} \right] \operatorname{sign}(1-t) + \theta_0 V(x) \frac{|1-t^2|}{(t\tau)^2} \geq 0, \quad \forall\, x \in \mathbb{R}^3, \; t \gt 0, \; \tau \neq 0\]with constant $\theta_0 \in (0,1)$, instead of $\lim_{|t| \to \infty} \left( \int_0^t f(x,s) \, \mathrm{d}s \right)/|t|^4 = \infty$ uniformly in $x \in \mathbb{R}^3$ and the usual Nehari-type monotonic condition on $f(x,t)/|t|^3$.

In this paper, we prove the existence and uniform attractivity of the solutions of a class of functional integral equations which contain a number of classical nonlinear integral equations as special cases. Our investigations will be carried out in the space of continuous and bounded functions on an unbounded interval. The main tools here are the measure of noncompactness and the suitable fixed point theorem. We introduce also some examples and remarks showing the difference between our main result and some previous results.

In this paper, we deal with the second order discrete Hamiltonian system $\Delta[p(n) \Delta u(n-1)] - L(n) u(n) + \nabla W(n,u(n)) = 0$, where $L\colon \mathbb{Z} \to \mathbb{R}^{N \times N}$ is positive definite for sufficiently large $|n| \in \mathbb{Z}$ and $W(n,x)$ is indefinite sign. By using critical point theory, we establish some new criteria to guarantee that the above system has infinitely many nontrivial homoclinic solutions under the assumption that $W(n,x)$ is asymptotically quadratic and supquadratic, respectively. Our results generalize and improve some existing results in the literature.

In this paper, we study some properties about measure-theoretic entropy for a free semigroup action. We show some properties like conjugacy, power rule and affinity about the measure-theoretic entropy for a free semigroup action.

Using the notion of realizations, the realized homogeneous Besov spaces $\dot{\widetilde{B}}{^{s}_{p,q}}(\mathbb{R}^n)$ are subsets of tempered distributions. Then we will study the boundedness of some pseudodifferential operators on $\dot{\widetilde{B}}{^{s}_{p,q}}(\mathbb{R}^n)$, in the cases either $s \lt n/p$ or $s = n/p$ and $q = 1$.

Let $F^{p}(\phi)$ be the weighted Fock space on the complex plane $\mathbb{C}$, where $\phi$ is subharmonic with $\Delta \phi \, dA$ a doubling measure. In this paper, we characterize the positive Borel measure $\mu$ on $\mathbb{C}$ for which the induced Toeplitz operator $T_\mu$ is bounded (or compact) from one weighted Fock space $F^{p}(\phi)$ to another $F^{q}(\phi)$ for $0 \lt p, q \lt \infty$.

In this paper, we study duality theorems for quasiconvex programming with a reverse quasiconvex constraint. We introduce quasiaffine transformation methods for a reverse quasiconvex constraint. Especially, we show a linear characterization of a reverse quasiconvex constraint. By using transformation methods, we show necessary optimality conditions in terms of Greenberg-Pierskalla subdifferential. In addition, we introduce a surrogate duality theorem for quasiconvex programming with a reverse quasiconvex constraint.