In this paper, we establish the greatest lower bound for the product $B_{i}(K)B_{i}(K^{*})$ of the width-integrals of index $i$ of convex body $K$. Further, the greatest lower bound for the product of the mixed width-integrals $A_{p,i}(K)A_{p',i}(K^{*})$ for the mixed width-integrals is given.

A space $X$ is {\it star-K-Menger} if for each sequence $(\mathcal U_n:n\in\Bbb N)$ of open covers of $X$ there exists a sequence $(K_n:n\in N)$ of compact subsets of $X$ such that $\{St(K_n,\mathcal U_n):n\in\Bbb N\}$ is an open cover of $X$. In this paper, we construct an example of a Hausdorff star-Menger space that is not star-K-Menger which gives an answer to a question of Song [12], and continue to investigate topological properties of star-K-Menger spaces.

We deal with $n$-dimensional complete submanifolds immersed with parallel nonzero mean curvature vector ${\bf H}$ in the hyperbolic space $\mathbb{H}^{n+p}$. In this setting, we establish sufficient conditions to guarantee that such a submanifold $M^n$ must be pseudo-umbilical, which means that ${\bf H}$ is an umbilical direction. In particular, we conclude that $M^n$ is a minimal submanifold of a small hypersphere of $\mathbb{H}^{n+p}$.

Let $(R,\mathfrak{m})$ be a commutative Noetherian local ring. Let $I$ be an ideal of $R$ and let $M$ be a finitely generated $R$-module of dimension $d\geq 1$. In this paper we consider the $I$-cofiniteness property of the local cohomology module $H^{d-1}_I(M)$. More precisely, we prove that the $R$-module $H^{d-1}_I(M)$ is $I$-cofinite if and only if the $R$-module $\Hom_R(R/I,H^{d-1}_I(M))$ is finitely generated. As an immediate consequence of this result, we prove that if $(R,\operatorname{\frak m})$ is a regular local ring of dimension $d\geq 2$ and $I$ is an ideal of $R$ with $\dim R/I\neq 1$, then $H^{d-1}_I(R)=0$ if and only if the $R$-module $\Hom_R(R/I,H^{d-1}_I(R))$ is finitely generated.

For modules over a finite-dimensional algebra, there is a canonical one-to-one correspondence between the projective indecomposable modules and the simple modules. In this self-contained, purely expository note, we take a straight-line path from the basic definitions to a proof of this correspondence.

First it is shown that the Nörlund matrix associated with the sequence of positive integers is a coposinormal operator on $\ell^2$. This fact then turns out to be useful for showing that this operator is also posinormal and hyponormal. In contrast with the analogous weighted mean matrix result [\textbf{6}], the proof of hyponormality is accomplished without resorting to determinants or Sylvester's criterion.

In this paper all rings are commutative with nonzero identity. Let $M$ be an $R$-module. A proper submodule $N$ of $M$ is called a {\it classical primary submodule}, if for each $m\in M$ and elements $a,b\in R$, $abm\in N$ implies that either $am\in N$ or $b^tm\in N$ for some $t\geq1$. We introduce the notion of ``weakly classical primary submodules''. A proper submodule $N$ of $M$ is a {\it weakly classical primary submodule} if whenever $a,b\in R$ and $m\in M$ with $0\neq abm\in N$, then either $am\in N$ or $b^tm\in N$ for some $t\geq1$.

In this article, using the concept of measure of noncompactness, we give some results concerning the compactness and continuity of the nonlinear Volterra and Fredholm integral operators on the space $ BC(\Omega)$ ($\Omega$ is an unbounded subset of the Euclidean space $\Bbb{R}^n$). Then, we prove an existence result for a functional integral equation which includes several classes of nonlinear integral equations. Our results generalize and improve some previous works. We will also include some examples which show that our results are applicable where the previous ones are not.

Following a previous paper by Jacqueline Ojeda and the first author, here we examine the number of possible branched values and branched functions for certain $p$-adic and complex meromorphic functions where numerator and denominator have different kind of growth, either when the denominator is small comparatively to the numerator, or vice-versa, or (for p-adic functions) when the order or the type of growth of the numerator is different from this of the denominator: this implies that one is a small function comparatively to the other. Finally, if a complex meromorphic function $\displaystyle{f\over g}$ admits four perfectly branched small functions, then $T(r,f)$ and $T(r,g)$ are close. If a p-adic meromorphic function $\displaystyle{f\over g}$ admits four branched values, then $f$ and $g$ have close growth. We also show that, given a p-adic meromorphic function $f$, there exists at most one small function $w$ such that $f-w$ admits finitely many zeros and an entire function admits no such a small function.

The aim of this paper is to discuss on some generalizations of the celebrate Darbo and Sadovskii fixed point theorems without using the maximum property. Later on, we give an existence principle for a nonlinear integral equation.

In [8] the collineation groups of some known 5-, 4- and 3-dimensional topological regular parallelisms of $PG(3,\mathbb R)$ were determined. In the present article we concentrate on 3-dimensional regular parallelisms and prove: the 3-dimensional regular parallelisms are exactly those which can be constructed from generalized line stars, see [3]. We determine the collineation groups of 3-dimensional regular parallelisms and show that only group dimension 1 or 2 is possible. If the collineation group is 2-dimensional, then the parallelism is rotational which means that there is a rotation group $SO_2(\mathbb R)$ about some axis leaving the parallelism invariant. We give a construction method for the generalized line stars which induce these parallelisms.

We provide an alternative proof for the extreme amenability of the unitary group of the hyperfinite II${}_1$-factor von Neumann algebra, endowed with the strong operator topology.