Given a connected graph $G$, a function $f$ mapping the vertex set of $G$ into the set of all integers is a coloring of $G$. For any subgraph $H$ of $G$, we denote as $f(H)$ the sum of the values of $f$ on the vertices of $H$. If for any integer $k \in \{1,2,\ldots,f(G)\}$, there exists an induced connected subgraph $H$ of $G$ such that $f(H) = k$, then the coloring $f$ is called an IC-coloring of $G$. The IC-index of $G$, written $M(G)$, is defined to be the maximum value of $f(G)$ over all possible IC-colorings $f$ of $G$. In this paper, we give a lower bound on the IC-index of any complete $\ell$-partite graph for all $\ell \geq 3$ and then show that, when $\ell = 3$, our lower bound also serves as an upper bound. As a consequence, the exact value of the IC-index of any tripartite graph is determined.

We give an overpartition analogue of Bressoud's combinatorial generalization of the Göllnitz-Gordon theorem for even moduli in general case. Let $\widetilde{O}_{k,i}(n)$ be the number of overpartitions of $n$ whose parts satisfy certain difference condition and $\widetilde{P}_{k,i}(n)$ be the number of overpartitions of $n$ whose non-overlined parts satisfy certain congruence condition. We show that $\widetilde{O}_{k,i}(n) = \widetilde{P}_{k,i}(n)$ for $1 \leq i \lt k$.

A bicoloring of a Steiner triple system $\operatorname{STS}(n)$ on $n$ vertices is a coloring of vertices in such a way that every block receives precisely two colors. The maximum (resp. minimum) number of colors in a bicoloring of an $\operatorname{STS}(n)$ is denoted by $\overline{\chi}$ (resp. $\chi$). All bicolorable $\operatorname{STS}(2^h-1)$s have upper chromatic number $\overline{\chi} \leq h$; also, if $\overline{\chi} = h \lt 10$, then lower and upper chromatic numbers coincide, namely, $\chi = \overline{\chi} = h$. In 2003, M. Gionfriddo conjectured that this equality holds whenever $\overline{\chi} = h \geq 2$.

In this paper we discuss some extensions of bicolorings of $\operatorname{STS}(v)$ to bicoloring of $\operatorname{STS}(2v+1)$ obtained by using the ‘doubling plus one construction’. We prove several necessary conditions for bicolorings of $\operatorname{STS}(2v+1)$ provided that no new color is used. In addition, for any natural number $h$ we determine a triple system $\operatorname{STS}(2^{h+1}-1$) which admits no extended bicolorings.

In this note we provide a counterexample to two questions on quasi-Baer modules raised recently by Lee and Rizvi in [5].

Let $R$ and $S$ be rings and $_S\omega_R$ a semidualizing bimodule. For a subclass $\mathcal{T}$ of the class of $\omega$-coreflexive modules and $n \geq 1$, we introduce and study modules of $\omega$-$\mathcal{T}$-class $n$. By using the properties of such modules, we get some equivalent characterizations for $\omega_S$ having finite projective dimension. In particular, we prove that the projective dimension of $\omega_S$ is at most $n$ if and only if any module of $\omega$-$\mathcal{T}$-class $n$ is $\omega$-coreflexive. Moreover, we get some equivalent characterizations for $\omega_S$ having finite projective dimension at most two or one in terms of the properties of (adjoint) $\omega$-coreflexive and $\omega$-cotorsionless modules. Finally, we give some partial answers to the Wakamatsu tilting conjecture.

In this work, we study the Schrödinger operator on the half-line with a selfadjoint boundary condition. We prove that a particular set of eigenvalues can uniquely determine the potential. The reconstruction algorithm for recovering the potential from the particular data is provided.

While one-dimensional cellular automata have been well studied, there are relatively few results about multidimensional cellular automata; the investigation of cellular automata defined on Cayley trees constitutes an intermediate class. This paper studies the reversibility of linear cellular automata defined on Cayley trees with periodic boundary condition, where the local rule is given by $f(x_0,x_1,\ldots,x_d) = b x_0 + c_1 x_1 + \cdots + c_d x_d \pmod{m}$ for some integers $m,d \geq 2$. The reversibility problem relates to solving a polynomial derived from a recurrence relation, and an explicit formula is revealed; as an example, the complete criteria of the reversibility of linear cellular automata defined on Cayley trees over $\mathbb{Z}_2$, $\mathbb{Z}_3$, and some other specific case are addressed. Further, this study achieves a possible approach for determining the reversibility of multidimensional cellular automata, which is known as a undecidable problem.

We study the nonlinear Neumann boundary value problem with a $p(x)$-Laplacian operator \[ \begin{cases} \Delta_{p(x)}u + a(x)|u|^{p(x)-2}u = f(x,u) &\textrm{in $\Omega$}, \\ |\nabla u|^{p(x)-2} \dfrac{\partial u}{\partial\nu} = |u|^{q(x)-2}u + \lambda |u|^{w(x)-2}u &\textrm{on $\partial \Omega$}, \end{cases} \] where $\Omega \subset \mathbb{R}^N$, with $N \geq 2$, is a bounded domain with smooth boundary and $q(x)$ is critical in the context of variable exponent $p_*(x) = (N-1)p(x)/(N-p(x))$. Using the variational method and a version of the concentration-compactness principle for the Sobolev trace immersion with variable exponents, we establish the existence and multiplicity of weak solutions for the above problem.

This paper is concerned with the nonlinear stability of traveling waves of a delayed monostable epidemic model with quasi-monotone condition. We prove that the traveling wave front is exponentially stable by means of the weighted-energy method and the comparison principle to perturbation in some exponentially weighted $L^{\infty}$ spaces, when the difference between initial data and traveling wave front decays exponentially as $x \to -\infty$, but the initial data can be suitable large in other locations. Finally, we present two examples to support our theoretical results.

This paper considers a stochastic Lotka-Volterra competitive model with infinite delay and impulsive perturbations. This model is new, more feasible and more accordance with the actual. The aim is to analyze what happens under the impulsive perturbations. With space $C_{g}$ as phase space, sufficient conditions for permanence in time average are established as well as extinction, stability in time average and global attractivity of each population. Numerical simulations are also exhibited to illustrate the validity of the results in this paper. In addition, a knowledge is given to illustrate that the statement in [21] is incorrect by choosing space $C_{g}$ as phase space. Our results demonstrate that impulsive perturbations which may represent human factor play a key role in protecting the population when environmental noise and interaction rates are disadvantageous to population survival.

In this paper, we study the extension of the minimizing equal mass parallelogram solutions which was derived by Chen in 2001 [2]. Chen's solution was minimizing for one quarter of the period $[0,T]$, where numerical integration had been used in his proof. In this paper we extend Chen's solution in the reduced space to $[0,4T]$ and we show that this extension is also minimizing over the intervals $[0,2T]$ and $[0,4T]$. The minimizing property of the extension is proved without using numerical integration.

The purpose of this article is to study the optimal control problem of second order stochastic evolution hemivariational inequalities with Poisson jumps by virtue of cosine operator theory in the Hilbert space. Initially, the sufficient conditions for existence of mild solution of the proposed system are verified by applying properties of Clarke's subdifferential operator and fixed point theorem in multivalued maps. Further, we formulated and proved the existence results for optimal control of the proposed system with corresponding cost function by using Balder theorem. Finally an example is provided to illustrate the main results.