In this paper, we study the following first-order nonperiodic Hamiltonian system $$\dot{z} = \mathcal {J}H_{z}(t,z),$$ where $H \in C^{1}(\mathbb{R} \times \mathbb{R}^{2N}, \mathbb{R})$ is the form $H(t,z) = \frac{1}{2} L(t)z \cdot z + R(t,z)$. Under weak superquadratic condition on the nonlinearitiy. By applying the generalized Nehari manifold method developed recently by Szulkin and Weth, we prove the existence of homoclinic orbits, which are ground state solutions for above system.

In this paper, we obtain some stable Poisson Convergence Theorems for arrays of integer-valued dependent random variables. We prove that the limiting distribution is a mixture of Poisson distribution when the conditional second moments on a given $\sigma$-algebra of the sequence converge to some positive random variable. Moreover, we apply the main results to the indicator functions of rowise interchangeable events and obtain some interesting stable Poisson convergence theorems.

Complex projection bodies were introduced by Abardia and Bernig, recently. In this paper some geometric inequalities for mixed complex projection bodies which are analogs of inequalities for mixed real projection bodies are established.

Let $X$ be a subgroup of a Coxeter group $W$. In [5], the authors developed the notion of $X$-posets, which are defined on certain equivalence classes of the (right) cosets of $X$ in $W$. These posets can be thought of as a generalization of the well-known Bruhat order of $W$. This article provides a catalogue of all the $X$-posets for various small Coxeter groups.

We study the nonlinear viscoelastic wave equation $$ u_{tt} - k_{0} \Delta u + \int_{0}^{t} g(t-s) \mathrm{div} \bigr[a(x) \nabla u(s) \bigr] ds + \bigr(k_{1} + b(x) |u_{t}|^{m-2}\bigr) u_{t} = |u|^{p-2}u $$ with dissipative boundary conditions. Under some restrictions on the initial data and the relaxation function and without imposing any restrictive assumption on $a(x)$, we show that the rate of decay is similar to that of $g$. We also prove the blow-up results for certain solutions in two cases. In the case $k_{1} = 0$, $m=2$, we show that the solutions blow up in finite time under some restrictions on initial data and for arbitrary initial energy. In another case, $k_{1}\geq 0$, $m\geq 2$, we prove a nonexistence result when the initial energy is less than potential well depth.

In this paper, we are concerned with the boundedness of all the solutions for a kind of second order differential equations with p-Laplacian and an oscillating term $(\phi_p(x'))'+a\phi_p(x^+)-b\phi_p(x^-)=G_x(x,t)+f(t)$, where $x^+ = \max (x,0)$, $x^- = \max(-x,0)$, $\phi_p(s) = |s|^{p-2}s$, $p \geq 2$, $a$ and $b$ are positive constants $(a\not=b)$, the perturbation $f(t) \in {\cal C}^{23}(\mathbb{R}/2\pi_p \mathbb{Z})$, the oscillating term $G\in {\cal C}^{21}(\mathbb{R} \times \mathbb{R}/2\pi_p \mathbb{Z})$ satisfying $\label{G} |D_x^iD_t^jG(x,t)|\le C,\quad 0\le i+j\le 21$.

The monotonicity of the least squares mean on the Riemannian manifold of positive definite matrices, conjectured by Bhatia and Holbrook and one of key axiomatic properties of matrix geometric means, was recently established based on the Strong Law of Large Number [14, 4]. A natural question concerned with the S.L.L.N is so called the no dice conjecture. It is a problem to make a construction of deterministic sequences converging to the least squares mean without any probabilistic arguments. Very recently, Holbrook [7] gave an affirmative answer to the conjecture in the space of positive definite matrices. In this paper, inspired by the work of Holbrook [7] and the fact that the convex cone of positive definite matrices is a typical example of a symmetric cone (self-dual homogeneous convex cone), we establish the no dice theorem on general symmetric cones.

In this paper we propose an algorithm for solving systems of coupled monotone inclusions in Hilbert spaces. The operators arising in each of the inclusions of the system are processed in each iteration separately, namely, the single-valued are evaluated explicitly (forward steps), while the set-valued ones via their resolvents (backward steps). In addition, most of the steps in the iterative scheme can be executed simultaneously, this making the method applicable to a variety of convex minimization problems. The numerical performances of the proposed splitting algorithm are emphasized through applications in average consensus on colored networks and image classification via support vector machines.

For a simple undirected graph $G=(V(G),E(G))$, a subset $I$ of $V(G)$ is said to be an independent set of $G$ if any two vertices in $I$ are not adjacent in $G$. An empty set is also an independent set in $G$. The set of all independent sets of a graph $G$ is denoted by $I(G)$ and its cardinality by $i(G)$ (known as the Merrifield-Simmons index in mathematical chemistry). Let $h(n,x)$ be the $x$-th largest number of independent sets among all connected $n$-vertex graphs. In this paper, we determine the numbers $h(n,x)$ for $1\le x\le \lfloor\frac{n}{2}\rfloor ^2-3\cdot\lfloor\frac{n}{2}\rfloor+3$. Besides, we also characterize the connected $n$-vertex graphs achieving these values.

In this paper, the authors use the discrete Littlewood-Paley-Stein theory to develop a theory of multi-parameter Triebel-Lizorkin and Besov spaces associated with Zygmund dilation. They also obtain the boundedness of Ricci-Stein singular integral operators on multi-parameter Triebel-Lizorkin and Besov spaces associated with Zygmund dilation.

This article is concerned with the coupled system of fractional differential equations with nonlocal integral boundary conditions. The existence results are obtained by applying some standard fixed point theorems. Finally, an example is also provided to illustrate the availability of our main results.

In this paper, we consider the existence of multiple solutions for the following $p(x)$-Laplacian equations with critical Sobolev growth conditions \[ \begin{cases} -div(|\nabla u|^{p(x)-2} \nabla u) + |u|^{p(x)-2} u = f(x,u) \; \textrm{in } \Omega, \\ u = 0 \; \textrm{on } \partial \Omega.\end{cases} \]

We show the existence of infinitely many pairs of solutions by applying the Fountain Theorem and the Dual Fountain Theorem respectively. We also present a variant of the concentration-compactness principle, which is of independent interest.

In this paper, we establish some inequalities involving $k$-Ricci curvature, $k$-scalar curvature, the screen scalar curvature on a screen homothetic lightlike hypersurface of a Lorentzian manifold. We compute Chen-Ricci inequality and Chen inequality on a screen homothetic lightlike hypersurface of a Lorentzian manifold. We give an optimal inequality involving the $\delta(n_{1},\ldots ,n_{k})$-invariant and some characterizations (totally umbilicity, totally geodesicity, minimality, etc.) for lightlike hypersurfaces.

In this paper we consider the weak viscoelastic equation with an internal time-varying delay term $$u_{tt}(x,t)-\Delta u(x,t)+\alpha(t)\int_{0}^{t}g(t-s)\Delta u(x,s)\, {\rm d}s +a_{0}u_{t}(x,t)+a_{1}u_{t}(x,t-\tau(t))=0$$ in a bounded domain. By introducing suitable energy and Lyapunov functionals, under suitable assumptions, we establish a general decay rate estimate for the energy, which depends on the behavior of both $\alpha$ and $g$.

In this paper, we continue to investigate the inexact hybrid proximal point algorithm proposed by Mashreghi and Nasri for equilibrium problems in Banach spaces. Under some classes of generalized monotone conditions, we prove that the sequence generated by the method is strongly convergent to a solution of the problem, which is closest to the initial iterate, in the sense of Bregman distance. As an application, we obtain some analogues for some classes of generalized monotone variational inequalities. The results presented in this paper generalize and improve some recent results in literatures.

In this paper we devote to generalizing some results of componentwise linear modules over a polynomial ring to the ones over a Koszul algebra. Among other things, we show that the $i$-linear strand of the minimal free resolution of a componentwise linear module is the minimal free resolution of some module which is described explicitly for any $i\in Z$. In addition we present some theorems about when graded modules with linear quotients are componentwise linear.

In this paper, by using Mawhin's continuation theorem of coincidence degree theory, we obtain some sufficient conditions for the existence of homoclinic solutions for a class of nonlinear second-order differential equations with time-varying delays. Moreover, we give an example to illustrate the feasibility of obtained results. Our results are completely new.

In this paper, we deal with the existence, asymptotic behavior and uniqueness of traveling waves for nonlocal diffusion systems with delay and global response. We first obtain the existence of traveling wave front by using upper-lower solutions method and Schauder's fixed point theorem for $c\gt c_*$ and using a limiting argument for $c=c_*$. Secondly, we find a priori asymptotic behavior of (monotone or non-monotone) traveling waves with the help of Ikehara's Theorem by constructing a Laplace transform representation of a solution. Thirdly, we show that the traveling wave front for each given wave speed is unique up to a translation. Last, we apply our results to two models with delayed nonlocal response.

This paper deals with one-dimensional (1D) nonlinear Schrödinger equation with a multiplicative potential, subject to Dirichlet boundary conditions. It is proved that for each prescribed integer $b\gt 1$, the equation admits small-amplitude quasi-periodic solutions, whose $b$-dimensional frequencies are small dilation of a given Diophantine vector. The proof is based on a modified infinite-dimensional KAM theory.