For $\xi \in [1/2,3/4] $, the existence of the global attractor for the evolutionary equation corresponding to the following strongly damped nonlinear beam equation $(1+\beta A^{1/2}) u_{tt}+\delta A^{1/2}u_{t}+\alpha Au+g(\Vert u \Vert _{\xi -1/4}^{2}) A^{1/2}u=f$, $t>0$, has been studied in $\mathcal {D}_{H}(A^{\xi }) \times \mathcal {D}_{H}(A^{\xi -1/4})$. Such an equation is related to a nonlinear beam equation as well as Timoshenko's equation.

The main difficulty of our work comes from the terms $\beta A^{1/2}u_{tt}$ and $g(\Vert u \Vert _{\xi -1/4}^{2}) A^{1/2}u$, representing the rotational inertia of the beam and the tension within the beam due to its extensibility, respectively. We overcome the difficulty of introducing the solution, bounded absorbing set, and $\kappa $-contracting property by carefully using the fractional power theory and suitable time-uniform a priori estimates.

We present some reverse Young-type inequalities for the Hilbert-Schmidt norm as well as any unitarily invariant norm. Furthermore, we give some inequalities dealing with operator means. More precisely, we show that if $A, B\in {\mathfrak B}(\mathcal {H})$ are positive operators and $r\geq 0$, $A\nabla _{-r}B+2r(A\nabla B-A\sharp B)\leq A\sharp _{-r}B$. We also prove that equality holds if and only if $A=B$. In addition, we establish several reverse Young-type inequalities involving trace, determinant and singular values. In particular, we show that if $A$ and $B$ are positive definite matrices and $r\geq 0$, then $\label {reverse_trace} \mbox {tr\,}((1+r)A-rB)\leq \mbox {tr}|A^{1+r}B^{-r} |-r( \sqrt {\mbox {tr\,} A} - \sqrt {\mbox {tr} B})^{2}$.

In this paper, we study the necessary and sufficient conditions for the partial crossed product to be a fully weakly prime ring. Moreover, we give a description of the prime radical of the partial crossed product when the base ring is a fully weakly prime ring.

In the first part of this paper we give a survey of classical results on Kummer surfaces with Picard number~17 from the point of view of lattice theory. We prove ampleness properties for certain divisors on Kummer surfaces, and we use them to describe projective models of Kummer surfaces of $(1,d)$-polarized abelian surfaces for $d=1,2,3$. As a consequence, we prove that, in these cases, the N\'eron-Severi group can be generated by lines.

In the second part of the paper we use Kummer surfaces to obtain results on K3 surfaces with a symplectic action of the group $(\mathbb{Z} /2\mathbb{Z} )^4$. In particular, we describe the possible N\'eron-Severi groups of the latter in the case that the Picard number is $16$, which is the minimal possible. We also describe the N\'eron-Severi groups of the minimal resolution of the quotient surfaces which have 15 nodes. We extend certain classical results on Kummer surfaces to these families.

This paper continues the author's program of investigating the question of when a homotopy of 2-cocycles $\Omega = \{\omega _t\}_{t \in [0,1]}$ on a locally compact Hausdorff groupoid $\mathcal{G} $ induces an isomorphism of the $K$-theory groups of the twisted groupoid $C^*$-algebras: \[ K_*(C^*(\mathcal{G} , \omega _0)) \cong K_*(C^*(\mathcal{G} , \omega _1)). \] Building on our earlier work in \cite {eag-kgraph, transf-gps}, we show that, if $\pi : \mathcal{G} \to M$ is a locally trivial bundle of amenable groups over a locally compact Hausdorff space $M$, a homotopy $\Omega = \{\omega _t\}_{t \in [0,1]}$ of 2-cocycles on $\mathcal{G} $ gives rise to an isomorphism: \[ K_*(C^*(\mathcal{G} , \omega _0)) \cong K_*(C^*(\mathcal{G} , \omega _1)). \]

\textit {Uniformity} and \textit {proximity} are two different ways of defining small scale structures on a set. \textit {Coarse structures} are large scale counterparts of uniform structures. In this paper, motivated by the definition of proximity, we develop the concept of asymptotic resemblance as a relation between subsets of a set to define a large scale structure on it. We use our notion of asymptotic resemblance to generalize some basic concepts of coarse geometry. We introduce a large scale compactification which, in special cases, agrees with the \textit {Higson compactification}. At the end of the paper we show how the \textit {asymptotic dimension} of a metric space can be generalized to a set equipped with an asymptotic resemblance relation.

In this article, we consider $r-1$ disjoint lines given in a del~Pezzo surface $S_{r}$ and study how to determine if a contraction given by the lines produces a map to $S_{1}$ (one point blow up of $\mathbb {P}^{2}$) or $\mathbb {P}^{1}\times \mathbb {P}^{1}$ by checking only the configuration of lines. Here, we show that we can determine if the disjoint lines produce a contraction to $\mathbb {P}^{1}\times \mathbb {P}^{1}$ by combining a quartic rational divisor class to them. We also study the quartic rational divisor classes along the configuration of lines in del Pezzo surfaces.

The matching energy of a graph was introduced by Gutman and Wagner in 2012 and defined as the sum of the absolute values of zeros of its matching polynomial. Let $\theta (r,s,t)$ be the graph obtained by fusing two triples of pendant vertices of three paths $P_{r+2}$, $P_{s+2}$ and $P_{t+2}$ to two vertices. The graph obtained by identifying the center of the star $S_{n-g}$ with the degree~3 vertex $u$ of $\theta (1,g-3,1)$ is denoted by $S_{n-g}(u)\theta (1,g-3,1)$. In this paper, we show that, $S_{n-g}(u)\theta (1,g-3,1)$ has minimum matching energy among all bicyclic graphs with order $n$ and girth $g$.

We consider zero-cycles on algebraic varieties defined over number fields. The Hasse principle and weak approximation property are obstructed by the Brauer group of the variety and it is conjectured to be the only obstruction for all proper smooth varieties by Colliot-Th\'el\`ene, Sansuc, Kato and Saito. We summarize the progress related to this conjecture, particularly for higher dimensional varieties such as fibrations and homogeneous varieties.

Recently there has been a flurry of research on generalized factorization techniques in both integral domains and rings with zero-divisors, namely, $\tau $-factorization. There are several ways that authors have studied factorization in rings with zero-divisors. This paper focuses on the method of regular factorizations introduced by Anderson and Valdes-Leon. We investigate how one can extend the notion of $\tau $-factorization to commutative rings with zero-divisors by using the regular factorization approach. The study of regular factorization is particularly effective because the distinct notions of associate and irreducible coincide for regular elements. We also note that the popular U-factorization developed by Fletcher also coincides since every regular divisor is essential. This will greatly simplify many of the cumbersome finite factorization definitions that exist in the literature when studying factorization in rings with zero-divisors.

We count the number of graph compositions of suspended $Y$-trees via flats of the cycle matroid of the suspended $Y$-trees.

The purpose of this paper is to investigate two sided $\alpha $-derivations satisfying certain differential identities on 3-prime near-rings. Some well-known results characterizing commutativity of 3-prime near-rings by derivations (semi-derivations) have been generalized. Furthermore, examples proving the necessity of the 3-primeness hypothesis are given.