Let $\mathcal{W}$ be an additive full subcategory of the category R-Mod of left R-modules. We provide a method to construct a proper ${\mathcal{W}}^H_C$-resolution (resp. coproper ${\mathcal{W}}^T_C$-coresolution) of one term in a short exact sequence in R-Mod from those of the other two terms. By using these constructions, we introduce and study the stability of the Gorenstein categories ${\mathcal{G}}_C({\mathcal{W}}{\mathcal{W}}^T_C)$ and ${\mathcal{G}}_C({\mathcal{W}}^H_C{\mathcal{W}})$ with respect to a semidualizing bimodule C, and investigate the 2-out-of-3 property of these categories of a short exact sequence by using these constructions. Also we prove how they are related to the Gorenstein categories ${\mathcal{G}}((R{\ltimes}C){\otimes}_R{\mathcal{W}})_C$ and ${\mathcal{G}}(Hom_R(R{\ltimes}C,{\mathcal{W}}))_C$ over $R{\ltimes}C$.

In this paper, some omissions in Mishra and Lai [13], have been pointed out and their corrective measures have been discussed briefly.

This paper deals with a degenerate parabolic system with coupled nonlinear localized sources subject to weighted nonlocal Dirichlet boundary conditions. We obtain the conditions for global and blow-up solutions. It is interesting to observe that the weight functions for the nonlocal Dirichlet boundary conditions play substantial roles in determining not only whether the solutions are global or blow-up, but also whether the blowing up occurs for any positive initial data or just for large ones. Moreover, we establish the precise blow-up rate.

Let {$X_n,n{\geq}1$} be a sequence of negatively superadditive dependent random variables. In the paper, we study the strong law of large numbers for general weighted sums ${\frac{1}{g(n)}}{\sum_{i=1}^{n}}{\frac{X_i}{h(i)}}$ of negatively superadditive dependent random variables with non-identical distribution. Some sufficient conditions for the strong law of large numbers are provided. As applications, the Kolmogorov strong law of large numbers and Marcinkiewicz-Zygmund strong law of large numbers for negatively superadditive dependent random variables are obtained. Our results generalize the corresponding ones for independent random variables and negatively associated random variables.

We establish maximal moment inequalities of partial sums under ${\psi}$-weak dependence, which has been proposed by Doukhan and Louhichi [P. Doukhan and S. Louhichi, A new weak dependence condition and application to moment inequality, Stochastic Process. Appl. 84 (1999), 313-342], to unify weak dependence such as mixing, association, Gaussian sequences and Bernoulli shifts. As an application of maximal moment inequalities, a functional central limit theorem is developed for linear processes with ${\psi}$-weakly dependent innovations.

We consider ${\mathcal{T}}_{\infty}(F)$ - the space of all innite upper triangular matrices over a eld F. We give a description of all linear maps that satisfy the property: if rank(x) = 1, then $rank({\phi}(x))=1$ for all $x{\in}{\mathcal{T}}_{\infty}(F)$. Moreover, we characterize all injective linear maps on ${\mathcal{T}}_{\infty}(F)$ such that if rank(x) = k, then $rank({\phi}(x))=k$.

In this paper, we introduce and study an explicit hybrid relaxed extragradient iterative method to approximate a common solution to generalized mixed equilibrium problem and fixed point problem for a nonexpansive semigroup in Hilbert space. Further, we prove that the sequence generated by the proposed iterative scheme converges strongly to the common solution to generalized mixed equilibrium problem and fixed point problem for a nonexpansive semigroup. This common solution is the unique solution of a variational inequality problem and is the optimality condition for a minimization problem. The results presented in this paper are the supplement, improvement and generalization of the previously known results in this area.

Let $E{\subset}{\mathbb{F}}^d_q$, the d-dimensional vector space over the finite field with q elements. Construct a graph, called the distance graph of E, by letting the vertices be the elements of E and connect a pair of vertices corresponding to vectors x, y 2 E by an edge if ${\parallel}x-y{\parallel}:=(x_1-y_1)^2+{\cdots}+(x_d-y_d)^2=1$. We shall prove that the non-overlapping chains of length k, with k in an appropriate range, are uniformly distributed in the sense that the number of these chains equals the statistically correct number, $1{\cdot}{\mid}E{\mid}^{k+1}q^{-k}$ plus a much smaller remainder.

We consider a coupled system of Keller-Segel type equations and the incompressible Navier-Stokes equations in spatial dimension two. We show temporal decay estimates of solutions with small initial data and obtain their asymptotic profiles as time tends to infinity.

In this paper, subordination results of analytic function $f{\in}{\mathcal{A}}_p$ involving linear operator ${\mathcal{K}}^{{\delta},{\lambda}}_{c,p}$ are obtained. By applying the differential subordination method, results are derived under some sufficient subordination conditions. On using some hypergeometric identities, corollaries of the main results are derived. Furthermore, convolution preserving properties for a class of multivalent analytic function associated with the operator ${\mathcal{K}}^{{\delta},{\lambda}}_{c,p}$ are investigated.

We consider a mathematical model which describes the quasistatic frictional contact between a piezoelectric body and an electrically conductive obstacle, the so-called foundation. A nonlinear electro-viscoelastic constitutive law is used to model the piezoelectric material. Contact is described with Signorini's conditions and a version of Coulomb's law of dry friction in which the adhesion of contact surfaces is taken into account. The evolution of the bonding field is described by a first order differential equation. We derive a variational formulation for the model, in the form of a system for the displacements, the electric potential and the adhesion. Under a smallness assumption which involves only the electrical data of the problem, we prove the existence of a unique weak solution of the model. The proof is based on arguments of time-dependent quasi-variational inequalities, differential equations and Banach's fixed point theorem.

The r-ary number sequences given by $$(b^{(r)}_n)_{n{\geq}0}=\Large{\frac{1}{(r-1)n+1}}(^{rn}_n)$$ are analogs of the sequence of the Catalan numbers ${\frac{1}{n+1}}(^{2n}_n)$. Their history goes back at least to Lambert [8] in 1758 and they are of considerable interest in sequential testing. Usually, the sequences are considered separately and the generalizations can go in several directions. Here we link the various r first by introducing a new combinatorial structure related to GR trees and then algebraically as well. This GR transition generalizes to give r-ary analogs of many sequences of combinatorial interest. It also lets us find infinite numbers of combinatorially defined sequences that lie between the Catalan numbers and the Ternary numbers, or more generally, between $b^{(r)}_n$ and $b^{(r+1)}_n$.

In this paper, we consider the following $Schr{\ddot{o}}dinger$-Kirchhoff-type equations $\[a+b\({\int}_{{\mathbb{R}}^N}({\mid}{\nabla}u{\mid}^2+V(x){\mid}u{\mid}^2)dx\)\][-{\Delta}u+V(x)u]=f(x,u)$, in ${\mathbb{R}}^N$. Under certain assumptions on V and f, some new criteria on the existence and multiplicity of nontrivial solutions are established by the Morse theory with local linking and the genus properties in critical point theory. Some results from the previously literature are significantly extended and complemented.

We make a study of two generalizations of regular rings, concentrating our attention on the structure of idempotents. A ring R is said to be right attaching-idempotent if for $a{\in}R$ there exists $0{\neq}b{\in}R$ such that ab is an idempotent. Next R is said to be generalized regular if for $0{\neq}a{\in}R$ there exist nonzero $b{\in}R$ such that ab is a nonzero idempotent. It is first checked that generalized regular is left-right symmetric but right attaching-idempotent is not. The generalized regularity is shown to be a Morita invariant property. More structural properties of these two concepts are also investigated.

In this paper, we introduce the class of analytic extensions of M-hyponormal operators and we study various properties of this class. We also use a special Sobolev space to show that every analytic extension of an M-hyponormal operator T is subscalar of order 2k + 2. Finally we obtain that an analytic extension of an M-hyponormal operator satisfies Weyl's theorem.