A $b$-coloring of a graph $G$ with $k$ colors is a proper coloring of $G$ using $k$ colors in which each color class contains a color dominating vertex, that is, a vertex which has a neighbor in each of the other color classes. The largest positive integer $k$ for which $G$ has a $b$-coloring using $k$ colors is the $b$-chromatic number $b(G)$ of $G$. The $b$-spectrum $S_b(G)$ of a graph $G$ is the set of positive integers $k$, $\chi(G) \leq k \leq b(G)$, for which $G$ has a $b$-coloring using $k$ colors. A graph $G$ is $b$-continuous if $S_b(G) = \{\chi(G), \ldots, b(G)\}$. It is known that for any two graphs $G$ and $H$, $b(G \Box H) \geq \max \{b(G), b(H)\}$, where $\Box$ stands for the Cartesian product. In this paper, we determine some families of graphs $G$ and $H$ for which $b(G \Box H) \geq b(G)+b(H)-1$. Further if $T_i$, $i = 1, 2, \ldots, n$, are trees with $b(T_i) \geq 3$, then $b(T_1 \Box \cdots \Box T_n) \geq \sum_{i=1}^{n} b(T_i)-(n-1)$ and $S_b(T_1 \Box \cdots \Box T_n) \supseteq \{2, \ldots, \sum_{i=1}^n b(T_i)-(n-1)\}$. Also if $b(T_i) = \Delta(T_i)+1$ for each $i$, then $b(T_1 \Box \cdots \Box T_n) = \Delta(T_1 \Box \cdots \Box T_n)+1$, and $T_1 \Box \cdots \Box T_n$ is $b$-continuous.

The classical Euler decomposition theorem expressed a product of two Riemann zeta values $\zeta(p) \zeta(q)$ as a sum of $\binom{p+q}{p}$ Euler double sums of weight $p+q$.

As a generalization of Euler decomposition theorem, we shall perform the shuffle product of two multiples of height-one multiple zeta values\[ \binom{j+m}{m} \zeta(\{1\}^{j+m-1}, r-\ell+2) \quad \textrm{and} \quad \binom{k-j+n}{n} \zeta(\{1\}^{k-j+n-1}, \ell+2)\]with positive integers $m, n$ and integers $k, j, r, \ell$ such that $0 \leq j \leq k$, $0 \leq \ell \leq r$. Then we applied the resulted shuffle relation to produce weighted sum formulas such as\begin{align*} & \quad (k+1) \sum_{|\boldsymbol{\alpha}|=k+r+2} \zeta(\alpha_0+1, \ldots, \alpha_{k+1}+1) 2^{\alpha_{k+1}} \\ & \quad + 2\sum_{|\boldsymbol{\alpha}|=k+r+1} \zeta(1, \alpha_0+1, \ldots, \alpha_k+1) 2^{\alpha_k - \delta_{0k}} \\ & = \frac{1}{2} \sum_{j=0}^k \sum_{\substack{\ell=0 \\ \ell: \textrm{even}}}^r (-1)^j (j+1) (k-j+1) \zeta(\{1\}^j, r-\ell+2) \zeta(\{1\}^{k-j}, \ell+2)\end{align*}when both $k$ and $r$ are even. Here $\delta_{mn} = 0$ unless $m=n$ and $\delta_{mm=1}$.

In this note we find a bound for the so-called linear Harbourne constants for smooth hypersurfaces in $\mathbb{P}_{\mathbb{C}}^{3}$.

An additive group semiautomaton or, in brief, GS-automaton is a generalization of the well known linear state machines. The purpose of this paper is to study the near-ring of endo-transition preserving functions of additive GS-automata. This class of near-rings is a subclass of the celebrated centralizer near-rings, and includes near-rings of infra-endomorphisms. Complete characterizations using both algebraic and graphical properties of the additive GS-automaton such that the near-ring is $0$-symmetric or constant are given. Conditions such that this near-ring being simple or being a ring are also provided.

In this paper, we investigate the existence of traveling wave solutions for three components temporally discrete reaction-diffusion systems with delays by using the cross iteration method and Schauder's fixed point theorem. The obtained results are well applied to a temporally discrete three-species Lotka-Volterra competitive systems with delays.

The paper presents new existence, localization and multiplicity results for positive solutions of ordinary differential equations or systems of the form $(\phi(u'))' + f(t, u) = 0$, where $\phi : (-a, a) \to (-b, b)$, $0 \lt a, b \leq \infty$, is some homeomorphism such that $\phi(0) = 0$. Our approach is based on Krasnosel'skiĭ type compression-expansion arguments and on a weak Harnack type inequality for positive supersolutions of the operator $(\phi(u'))'$. In the case of the systems, the localization of solutions is obtained in a component-wise manner. The theory applies in particular to equations and systems with $p$-Laplacian, bounded or singular homeomorphisms.

In this paper we study a new class of boundary value problemsconsisting of a fractional differential inclusion of Riemann-Liouville typeand Hadamard fractional integral conditions. Some new existence results forconvex as well as non-convex multivalued maps are obtained by using standardfixed point theorems. Some illustrative examples are also presented.

This paper investigates the multiplicity of solutions for quasilinear elliptic equations with $p(x)$-Laplacian in $\mathbb{R}^{N}$ by using the nonsmooth critical point theory. We obtain the existence of critical points for nondifferentiable functionals.

In this paper we consider a viscoelastic problem withBalakrishnan-Taylor damping\[ u_{tt} - \left(a + b \left\|\nabla u \right\|^2 + \sigma(\nabla u, \nabla u_t) \right) \Delta u + \int^t_{0} g(t-s) \Delta u(s) \, ds = 0\]with Dirichlet boundary condition. We establish a decay result of the energy ofsolutions for the problem without imposing the usual relation between therelaxation function $g$ and its derivative. This result generalizes earlierones to an arbitrary rate of decay, which is not necessarily of exponential orpolynomial decay.

By using the critical point theory, the existence of periodic solutions for a $2n$th-order nonlinear difference equation is obtained. The main approaches used in our paper are variational techniques and the Saddle Point Theorem. The problem is to solve the existence of periodic solutions for a $2n$th-order nonlinear difference equation. Results obtained successfully complement the existing one.

In this paper we study algebraic properties of singular integral operators with Cauchy kernel on the $L^{2}$ space of the unit circle. We give an operator equation characterization for this class of Cauchy singular integral operators. This characterization provides a direct connection between the singular integral operators and multiplication operators. We then use this characterization to study when two Cauchy singular integral operators commute. Our approach also leads to generalizations of several results on normal Cauchy singular integral operators obtained recently by Nakazi and Yamamoto.

In a general symmetric cone, we show that certain sequence of the Karcher means converges to the Log-Euclidean mean by using the fact that the Karcher mean is the limit of inductive means. One can see this as a generalization of the Lie-Trotter formula of positive definite matrices into a symmetric cone setting via the least squares mean.

Let $M^{n}(c)$ be an $n$-dimensional space form with constant sectional curvature $c$. Alencar-do Carmo-Tribuzy [5] classified all parallel mean curvature (abbrev. PMC) surfaces with non-negative Gaussian curvature $K$ in $M^n(c) \times \mathbb{R}$ with $c \lt 0$. Later on, Fetcu-Rosenberg [28] generalized their results for $c \neq 0$. However, the classification to PMC surfaces in $M^n(c) \times \mathbb{R}$ with $K \equiv 0$ is still open. In this paper, we give a complete classification to the PMC surfaces in $M^n(c) \times \mathbb{R}$ with $K \equiv 0$ whose tangent plane spans the constant angle with factor $\mathbb{R}$.

The notion of pointwise $1$-type Gauss map was derived from the ordinary finite type Gauss map and it gives an interesting geometric properties on surfaces of $3$-dimensional Euclidean space. In particular, the helicoid and the right cone of $3$-dimensional Euclidean space are characterized by pointwise $1$-type Gauss map. Inspired by such a study, in this paper, we completely classify ruled submanifolds of Euclidean space with pointwise $1$-type Gauss map.