The paper presents various results studying the asymptotic behavior of a sequence of lower semicontinuous functions on a metric space. In particular, different coercivity properties are obtained extending and refining previous results. The specific features and the structure of the terms of the sequence are used to construct appropriate quantities relevant in the verification of Palais-Smale compactness type conditions.

A twisted sum in the category of topological Abelian groups is a short exact sequence $0\to Y\to X\to Z\to 0$ where all maps are assumed to be continuous and open onto their images. The twisted sum splits if it is equivalent to $0\to Y\to Y\times Z\to Z\to 0$. We study the class $S_{\text{T}\text{G}}\left(𝕋\right)$ of topological groups G for which every twisted sum $0\to 𝕋\to X\to G\to 0$ splits. We prove that this class contains Hausdorff locally precompact groups, sequential direct limits of locally compact groups, and topological groups with ${ℒ}_{\infty }$ topologies. We also prove that it is closed by taking open and dense subgroups, quotients by dually embedded subgroups, and coproducts. As means to find further subclasses of $S_{\text{T}\text{G}}\left(𝕋\right)$, we use the connection between extensions of the form $0\to 𝕋\to X\to G\to 0$ and quasi-characters on G, as well as three-space problems for topological groups. The subject is inspired on some concepts known in the framework of topological vector spaces such as the notion of $𝒦$-space, which were interpreted for topological groups by Cabello.

We propose a new method to construct a real piecewise algebraic hypersurface of a given degree with a prescribed smoothness and topology. The method is based on the smooth blending theory and the Viro method for construction of Bernstein-Bézier algebraic hypersurface piece on a simplex.

Consider a nonself-mapping $T:A\to B$, where $(A,B)$ is a pair of nonempty subsets of a modular space $X_{\rho }$. A best proximity point of $T$ is a point $z\in A$ satisfying the condition: $\rho (z-Tz)=\text{inf}\{\rho (x-y):(x,y)\in A\times B\}$. In this paper, we introduce the class of proximal quasicontraction nonself-mappings in modular spaces with the Fatou property. For such mappings, we provide sufficient conditions assuring the existence and uniqueness of best proximity points.

We show that the Smale spaces from self-similar groups are topologically mixing and their stable algebra and stable Ruelle algebra are strongly Morita equivalent to groupoid algebras of Anantharaman-Delaroche and Deaconu. And we show that $C^{\ast }(R_{\infty })$ associated to a postcritically finite hyperbolic rational function is an AT-algebra of real-rank zero with a unique trace state.

We continue our work on endpoints and startpoints in $T_0$-quasimetric spaces. In particular we specialize some of our earlier results to the case of two-valued $T_0$-quasimetrics, that is, essentially, to partial orders. For instance, we observe that in a complete lattice the startpoints (resp., endpoints) in our sense are exactly the completely join-irreducible (resp., completely meet-irreducible) elements. We also discuss for a partially ordered set the connection between its Dedekind-MacNeille completion and the $q$-hyperconvex hull of its natural $T_0$-quasimetric space.

Ran and Reurings (2004) established an interesting analogue of Banach Contraction Principle in a complete metric space equipped with a partial ordering and also utilized the same oneto discuss the existence of solutions to matrix equations. Motivated by this paper, we prove results on coincidence points for a pair of weakly increasing mappings satisfying a nonlinear contraction condition described by a rational expression on an ordered complete metric space. The uniqueness of common fixed point is also discussed. Some examples are furnished to demonstrate the validity of the hypotheses of our results. As an application, we derive an existence theorem for the solution of an integral equation.