The center and the periphery of a graph is the set of vertices with minimum resp. maximum eccentricity in it. A graph is almost self-centered (ASC) if it contains exactly two non-central vertices and is almost peripheral (AP) if all its vertices but one lie in the periphery. Answering a question from (Taiwanese J. Math. 18 (2014), 463--471) it is proved that for any integer $r \geq 1$ there exists an $r$-AP graph of order $4r-1$. Using this result it is proved that any graph $G$ can be embedded into an $r$-AP graph by adding at most $4r-2$ vertices to $G$. A construction of ASC graphs from (Acta Math. Sin. (Engl. Ser.) 27 (2011), 2343--2350) is corrected and refined. Two new constructions of ASC graphs are also presented. Strong product graphs that are AP graphs are also characterized and it is shown that there are no strong product graphs that are ASC graphs. We conclude with some related open problems.

Let $G$ be a group. The question of how certain arithmetical conditions on the sizes of the conjugacy classes of $G$ influence the group structure has been studied by many authors. In this paper, we investigate the influence of conjugacy class sizes of primary and biprimary elements on the structure of $G$. A criterion for a group to have abelian Sylow subgroups is given and some well-known results on Baer-groups are generalized.

We give a characterization of $\tau$-rigid modules over Auslander algebras in terms of projective dimension of modules. Moreover, we show that for an Auslander algebra $\Lambda$ admitting finite number of non-isomorphic basic tilting $\Lambda$-modules and tilting $\Lambda^{\operatorname{op}}$-modules, if all indecomposable $\tau$-rigid $\Lambda$-modules of projective dimension $2$ are of grade $2$, then $\Lambda$ is $\tau$-tilting finite.

In this paper, we find all the solutions of the Diophantine equation $P_\ell + P_m + P_n = 2^a$, in nonnegative integer variables $(n,m,\ell,a)$ where $P_k$ is the $k$-th term of the Pell sequence $\{P_n\}_{n \geq 0}$ given by $P_0 = 0$, $P_1 = 1$ and $P_{n+1} = 2P_{n} + P_{n-1}$ for all $n \geq 1$.

Let $(M,\omega,\tau)_A$ be a $2n$-dimensional smooth manifold with a pair of symplectic forms $\omega$ and $\tau$ intertwined by a recursion operator $A \in \operatorname{End}(TM)$. We consider a codimension two submanifolds $Q \subset M$ with those restricted symplectic forms $(\omega|_Q,\tau|_Q)$. Assume that $TQ$ is $A$-invariant. We call the tuple $(M,\omega,\tau,Q)_A$ symplectic-recursion data. In this paper, we consider the problem of fibre connected sum of such two symplectic-recursion data $(M_0,\omega_0,\tau_0,Q_0)_{A_0}$ and $(M_1,\omega_1,\tau_1,Q_1)_{A_1}$. It is interesting to consider potential applications of this result to integrable systems and mathematical string theory.

In this paper we investigate blow up property of solutions for a system of nonlinear higher order Kirchhoff equations with nonlinear dissipations and positive initial energy. Some estimates for lower bound of the blow up time are also given. This improves and extends the blow up results in [16] by Liu and Wang (2006) and Gao et al. [7] (2011).

In this paper, we study the existence of stable and unstable periodic solutions of the forced pendulum of variable length. The proof is based on a stability criterion which was obtained in [11] by using the third order approximation method and a generalized version of the Poincaré-Birkhoff fixed point theorem.

In this paper, we consider the wave 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 uniform decay rates of the energy without imposing any restrictive growth near zero assumption on the damping term.

The existence of pullback exponential attractors for a nonautonomous semilinear parabolic equation with dynamical boundary condition is proved when the time-dependent forcing terms are translation bounded or even grow exponentially in the past and in the future.

Using a novel approach, we get some new explicit criteria for uniform asymptotic stability and exponential asymptotic stability of linear Volterra time-varying integro-differential equations of non-convolution type. Some examples are given to illustrate the obtained results.

In this paper, we consider a two-species Lotka-Volterra competition model in one-dimensional spatially inhomogeneous environments. It is assumed that two competitors have the same movement strategy but slightly differing in their inter- and intra-specific competition rates. By using the Lyapunov-Schmidt reduction technique as well as some analytic skills, we find that the existence and stability of coexistence states can be determined by some scalar functions, and hence the unique coexistence state of the system is established in certain cases.

In this paper, we investigate the following nonlinear and non-homogeneous elliptic system \[ \begin{cases} -\operatorname{div}(a_1(|\nabla u|) \nabla u) = \lambda_1 F_u(x,u,v) - \lambda_2 G_u(x,u,v) - \lambda_3 H_u(x,u,v) &\textrm{in $\Omega$}, \\ -\operatorname{div}(a_2(|\nabla v|) \nabla v) = \lambda_1 F_v(x,u,v) - \lambda_2 G_v(x,u,v) - \lambda_3 H_v(x,u,v) &\textrm{in $\Omega$}, \\ u = v = 0 &\textrm{on $\partial \Omega$}, \end{cases} \] where $\Omega$ is a bounded domain in $\mathbb{R}^N$ ($N \geq 1$) with smooth boundary $\partial \Omega$, $\lambda_1$, $\lambda_2$, $\lambda_3$ are three parameters, $\phi_i(t) = a_i(|t|)t$ ($i = 1,2$) are two increasing homeomorphisms from $\mathbb{R}$ onto $\mathbb{R}$, and functions $F$, $G$, $H$ are of class $C^1(\Omega \times \mathbb{R}^2, \mathbb{R})$ and satisfy some reasonable growth conditions. By using a three critical points theorem due to B. Ricceri, we obtain that system has at least three solutions. With some additional conditions, by using a four critical points theorem due to G. Anello, we obtain that system has at least four solutions.

In this paper, we are concerned with the existence of mild solutions for the initial value problem to a new class of abstract evolution equations with non-instantaneous impulses on ordered Banach spaces. The existence and uniqueness theorem of mild solution for the associated linear evolution equation with non-instantaneous impulses is established. With the aid of this theorem, the existence of mild solutions for nonlinear evolution equation with non-instantaneous impulses is obtained by using perturbation technique and iterative method under the situation that the corresponding solution semigroup $T(\cdot)$ and non-instantaneous impulsive function $g_k$ are compact, $T(\cdot)$ is not compact and $g_k$ is compact, $T(\cdot)$ and $g_k$ are not compact, respectively. The results obtained in this paper essentially improve and extend some related conclusions on this topic. Two concrete examples to parabolic partial differential equations with non-instantaneous impulses are given to illustrate that our results are valuable.

Let $\zeta = (\zeta_{0},\zeta_{1},\ldots)$ be a sequence of independent and identically distributed random variables. For $r \geq 2$, let $\mu_r$ be Mandelbrot's (limit) measure of multiplicative cascades defined with positive weights indexed by nodes of a regular $r$-ary tree, and let $Z^{(r)}$ be the mass of $\mu_r$. We study asymptotic properties of $Z^{(r)}$ and the sequence of random measures $(\mu_r)_{r}$ as $r \to \infty$. We obtain some laws of large numbers and a central limit theorem. The results extend ones established by Liu and Rouault (2000) and by Liu, Rio and Rouault (2003).