In this paper we investigate the geometry of conformal Killing graphs in a Riemannian manifold $\overline{M}_f^{ n+1}$ endowed with a weight function $f$ and having a closed conformal Killing vector field $V$ with conformal factor $\psi_V$, that is, graphs constructed through the flow generated by $V$ and which are defined over an integral leaf of the foliation $V^{\perp}$ orthogonal to $V$. For such graphs, we establish some rigidity results under appropriate constraints on the $f$-mean curvature. Afterwards, we obtain some stability results for $f$-minimal conformal Killing graphs of $ \overline{M}_f^{ n+1}$ according to the behavior of $ \psi_V$. Finally, related to conformal Killing graphs immersed in $\overline{M}_f^{n+1}$ with constant $f$-mean curvature, we study the strong stability.

Consider the Mann iteration $x_{n+1} = ( 1 - \alpha_n ) x_n + \alpha_n Tx_n$ for a nonexpansive mapping $T\colon K \to K$ defined on some subset $K$ of the normed space $X$. We present an innovative proof of the Ishikawa almost fixed point principle for nonexpansive mapping that reveals deeper aspects of the behavior of the process. This fact allows us, among other results, to derive convergence of the process under the assumption of existence of an accumulation point of $\{ x_n \}$.

We generalize a parametrization obtained by A.\,V. Corro in (2006) in the three-dimensional Euclidean space. Using this parametrization we study a class of oriented hypersurfaces $M^n$, $n\geq 2$, in Euclidean space satisfying a relation $\sum_{r=1}^{n} (-1)^{r+1}rf^{r-1} { n \choose r}H_r=0,$ where $H_r$ is the $r$th mean curvature and $f\in C^{\infty}(M^n;\mathbb{R})$, these hypersurfaces are called Weingarten hypersurfaces of the spherical type. This class of hypersurfaces includes the surfaces of the spherical type (Laguerré minimal surfaces). We characterize these hypersurfaces in terms of harmonic applications. Also, we classify the Weingarten hypersurfaces of the spherical type of rotation and we give explicit examples.

summary:Let $C$ be an associative commutative ring with 1. If $a\in C$, then $aC$ denotes the principal ideal generated by $a$. Let $l, m, n$ be nonzero elements of $C$ such that $mn \in lC$. The set of matrices $\left( \begin{smallmatrix} a_{11} & a_{12} a_{21} & -a_{11} \end{smallmatrix} \right) $, where $a_{11}\in lC$, $a_{12}\in mC$, $a_{21}\in nC$, forms a Lie ring under Lie multiplication and matrix addition. The paper studies properties of these Lie rings.

summary:It is shown that there exists a $\sigma $-compact topological group which cannot be represented as a continuous image of a Lindelöf $p$-group, see Example 2.8. This result is based on an inequality for the cardinality of continuous images of Lindelöf $p$-groups (Theorem 2.1). A closely related result is Corollary 4.4: if a space $Y$ is a continuous image of a Lindelöf $p$-group, then there exists a covering $\gamma $ of $Y$ by dyadic compacta such that $|\gamma |\leq 2^\omega $. We also show that if a homogeneous compact space $Y$ is a continuous image of a $cdc$-group $G$, then $Y$ is a dyadic compactum (Corollary 3.11).

We investigate the question of whether or not an amenable subgroup of the permutation group on $\mathbb{N}$ can have a unique invariant mean on its action. We extend the work of M. Foreman (1994) and show that in the Cohen model such an amenable group with a unique invariant mean must fail to have slow growth rate and a certain weakened solvability condition.

Hardy inequalities for the Hardy-type operators are characterized in the amalgam space which involves Banach function space and sequence space.

We consider a mathematical model which describes a contact between an elastic body and a foundation. The contact is frictionless with normal compliance. The goal of this paper is to study an optimal control problem which consists of leading the stress tensor as close as possible to a given target, by acting with a control on the boundary of the body. We state an optimal control problem which admits at least one solution. Next, we establish an optimality condition corresponding to a regularization of the model. We also introduce the regularized control problem for which we study the convergence when the regularization parameter tends to zero.

A space $X$ is {functionally countable} if $f(X)$ is countable for any continuous function $f\colon X \to {\mathbb{R}}$. We will call a space $X$ {exponentially separable} if for any countable family ${\mathcal{F}}$ of closed subsets of $X$, there exists a countable set $A\subset X$ such that $A\cap \bigcap {\mathcal{G}}\neq\emptyset$ whenever ${\mathcal{G}}\subset {\mathcal{F}}$ and $\bigcap {\mathcal{G}}\neq\emptyset$. Every exponentially separable space is functionally countable; we will show that for some nice classes of spaces exponential separability coincides with functional countability. We will also establish that the class of exponentially separable spaces has nice categorical properties: it is preserved by closed subspaces, countable unions and continuous images. Besides, it contains all Lindelöf $P$-spaces as well as some wide classes of scattered spaces. In particular, if a scattered space is either Lindelöf or ${\omega}$-bounded, then it is exponentially separable.

Let $G$ be a finite group. The main supergraph $\mathcal{S}(G)$ is a graph with vertex set $G$ in which two vertices $x$ and $y$ are adjacent if and only if $o(x) \mid o(y)$ or $o(y)\mid o(x)$. In this paper, we will show that $G\cong ^{2}G_{2}(3^{2n+1})$ if and only if $\mathcal{S}(G)\cong \mathcal{S}(^{2}G_{2}(3^{2n+1}))$. As a main consequence of our result we conclude that Thompson's problem is true for the small Ree group $^{2}G_{2}(3^{2n+1})$.