A fourth order differential operator with eigenvalue parameter dependent and periodic boundary conditions is considered. This problem is realized by an operator polynomial with self-adjoint operators. The eigenvalues are investigated and their asymptotics are provided.

We compare pairs of modules that are isomorphic to some submodules of each other. These modules are often defined over finite dimensional algebras, but some results hold also for modules over left hereditary rings or left noetherian rings or left noetherian left hereditary rings.

We provide a new construction, starting from a flat torus, of a locally CAT(0) space whose fundamental group is a finite-index subgroup of ${\rm Aut}(F_2)$, the group of automorphisms of the free group on two generators. We prove that the universal cover of this space is isometrically isomorphic to the standard CAT(0) space $X_0$ on which ${\rm Aut}(F_2)$ acts geometrically (known as the Brady complex). We prove a local rigidity theorem, according to which every CAT(0) space ``locally isomorphic'' to $X_0$ must be isometric to $X_0$. Finally, we explain the relation between these constructions and known surgery constructions for groups acting on CAT(0) spaces, from which they derive.

The set of zero divisors of an $MV$-algebra is investigated. It is proved that the set of all zero divisors of an $MV$-algebra $A$ is the union of all prime ideals of $A$. Next, $Z^{\circ}$-ideals and $Z$-ideals are introduced and their properties are studied. We examine the relationship between them and their relationship with the minimal prime ideals. We study their behavior under homomorphism of $ MV$-algebras. Also, we investigate in what conditions the annihilator of an ideal is a $Z^{\circ}$-ideal and $Z$-ideal.

Let $G = (V, E)$ be a connected graph and $d(x, y)$ be the distance between the vertices $x$ and $y$ in $V$. A subset of vertices $W = \{w_1, w_2,\ldots, w_k\}$ is called a locating set for $G$ if for every two distinct vertices $x, y\in V$, there is a vertex $w_i \in W$ such that $d(x, w_i) \neq d(y, w_i)$ for $i = 1, 2, \ldots, k$. A locating set containing the minimum number of vertices is called a metric basis for $G$ and the number of vertices in a metric basis is its metric dimension, denoted by ${\rm dim}(G)$. For a commutative ring $R$, let $\Gamma(R)$ denote the total graph of $R$, the undirected graph with vertex set $R$ and two distinct vertices $x$ and $y$ being adjacent if and only if $x+y\in Z(R)$, where $Z(R)$ is the set of zero-divisors of $R$. In this paper, we compute some types of locating parameters, namely metric dimension, edge metric dimension, fractional and strong metric dimension for the total graph $\Gamma(\Bbb{Z}_{2^{n}p^{m}})$ where $p$ is a prime number greater than two and $m, n$ are positive integers.

Motivated by the study of the asymptotic behavior of solutions to a difference equation of accretive type with a periodic forcing and the method of cyclic projections, we investigate the asymptotic behavior of a periodic sequence of nonexpansive mappings in Banach spaces.

Let $p$ be a prime number, $\mathbb{Z}_p$ the ring of $p$-adic integers, $\mathbb{Q}_p$ the field of $p$-adic numbers. Let $K$ be a complete valued field extension of $\mathbb{Q}_p$ and $\Lambda_K$ the valuation ring of $K$. Let $\mathcal{D}$ be a closed unitary subring of the ring $\Lambda_K$. Let $n\geg 2$ be an integer and let $\mathcal{H}(2n+1,\mathcal{D})$ be the $(2n+1)$-dimensional Heisenberg group with entries in $\mathcal{D}$. We have studied ultrametric Schrödinger representations of the Heisenberg group $\mathcal{H}(3, \mathcal{D})$ in two kinds of function spaces. In this paper we extend the study to the $(2n + 1)$-dimensional Heisenberg group with entries in $\mathcal{D}$. We study these representations in the case when the ring $\mathcal{D}$ is compact and in the general case their restriction to the algebra of analytic functions on $\mathcal{D}^n$ identified with the Tate algebra $T_n(K)$ in $n$ variables. These representations are topologically irreducible. From the last representations one obtains bounded linear operators satisfying Heisenberg commutation relations and the Weyl algebra $A_n(K)$ as subalgebra of the Banach algebra $\mathcal{L}(T_n(K))$ of bounded linear operators of $T_n(K)$. Its closure $\widetilde{A_n}(K)$ in $\mathcal{L}(T_n(K))$ is described and is seen to be a central simple algebra. Notice that this algebra is different from those considered by Pangalos.

For any suitable monoidal category $\mathcal{V}$, we find that $\mathcal{V}$-fully faithful lax epimorphisms in $\mathcal{V} \dash \mathcal{V}$ are precisely those $\mathcal{V}$-functors $F: \mathcal{A} \to \mathcal{B}$ whose induced $\mathcal{V}$-functors ${\mathfrak C} F: {\mathfrak C} \mathcal{A} \to {\mathfrak C} \mathcal{B}$ between the Cauchy completions are equivalences. For the case $\mathcal{V}= {\rm Set}$, this is equivalent to requiring that the induced functor $F^*:{\rm CAT}({\mathcal A},{\rm Cat}) \to {\rm CAT}({\mathcal B}, {\rm Cat})$ is an equivalence. By reducing the study of effective descent functors with respect to the indexed category of split (op)fibrations ${\mathcal F}$ to the study of the codescent factorization, we find that the observations above on fully faithful lax epimorphisms provide us with a characterization of (effective) ${\mathcal F}$-descent morphisms in the category of small categories ${\rm Cat}$; namely, we find that they are precisely the (effective) descent morphisms with respect to the indexed categories of \textit{discrete} opfibrations,previously studied by Sobral. We include some comments on the Beck-Chevalley condition and future work.