On soft quasi-pseudometric spaces

<p>In this article, we introduce the concept of a soft quasi-pseudometric space. We show that every soft quasi-pseudometric induces a compatible quasi-pseudometric on the collection of all soft points of the absolute soft set whenever the parameter set is finite. We then introduce the concept of soft Isbell convexity and show that a self non-expansive map of a soft quasi-metric space has a nonempty soft Isbell convex fixed point set.</p>

New classes of local almost contractions

Contractions represents the foundation stone of nonlinear analysis. That is the reason why we propose to unify two different type of contractions: almost contractions, introduced by V. Berinde in [2] and local contractions (Martins da Rocha and Filipe Vailakis in [7]). These two types of contractions operate in different space settings: in metric spaces (almost contractions) and semimetric spaces (for local contractions). That new type of contraction was built up in a new space setting, which is the pseudometric space. The main results of this paper represent the extension for various type of operators on pseudometric spaces, such as: generalized ALC, Ćirić-typeALC, quasi ALC, Ćirić-Reich-Rustype ALC. We propose to study the existence and uniqueness of their fixed points, and also the continuity in their fixed points, with a large number of examples for ALC-s.

Invariant Graph Partition Comparison Measures

Symmetric graphs have non-trivial automorphism groups. This article starts with the proof that all partition comparison measures we have found in the literature fail on symmetric graphs, because they are not invariant with regard to the graph automorphisms. By the construction of a pseudometric space of equivalence classes of permutations and with Hausdorff’s and von Neumann’s methods of constructing invariant measures on the space of equivalence classes, we design three different families of invariant measures, and we present two types of invariance proofs. Last, but not least, we provide algorithms for computing invariant partition comparison measures as pseudometrics on the partition space. When combining an invariant partition comparison measure with its classical counterpart, the decomposition of the measure into a structural difference and a difference contributed by the group automorphism is derived.

Finding a measure, a ring theoretical approach

Let \(S\) be a nonempty set and \(F\) consists of all \(Z_{2}\) characteristic functions defined on \(S\). We are supposed to introduce a ring isomorphic to \((P(S),\triangle,\cap)\), whose set is \(F\). Then, assuming a finitely additive function $m$ defined on \(P(S)\), we change \(P(S)\) to a pseudometric space \((P(S),d_{m})\) in which its pseudometric is defined by \(m\). Among other things, we investigate the concepts of convergence and continuity in the induced pseudometric space. Moreover, a theorem on the measure of some kinds of elements in \((P(S),m)\) will be established. At the end, as an application in probability theory, the probability of some events in the space of permutations with uniform probability will be determined. Some illustrative examples are included to show the usefulness and applicability of results.

Best Proximity Point Theorem in Quasi-Pseudometric Spaces

In quasi-pseudometric spaces (not necessarily sequentially complete), we continue the research on the quasi-generalized pseudodistances. We introduce the concepts of semiquasiclosed map and contraction of Nadler type with respect to generalized pseudodistances. Next, inspired by Abkar and Gabeleh we proved new best proximity point theorem in a quasi-pseudometric space. A best proximity point theorem furnishes sufficient conditions that ascertain the existence of an optimal solution to the problem of globally minimizing the errorinf⁡{d(x,y):y∈T(x)}, and hence the existence of a consummate approximate solution to the equationT(X)=x.

LINEAR AND PROJECTIVE BOUNDARY OF NILPOTENT GROUPS

We define a pseudometric on the set of all unbounded subsets of a metric space. The Kolmogorov quotient of this pseudometric space is a complete metric space. The definition of the pseudometric is guided by the principle that two unbounded subsets have distance 0 whenever they stay sublinearly close. Based on this pseudometric we introduce and study a general concept of boundaries of metric spaces. Such a boundary is the closure of a subset in the Kolmogorov quotient determined by an arbitrarily chosen family of unbounded subsets. Our interest lies in those boundaries which we get by choosing unbounded cyclic sub(semi)groups of a finitely generated group (or more general of a compactly generated, locally compact Hausdorff group). We show that these boundaries are quasi-isometric invariants and determine them in the case of nilpotent groups as a disjoint union of certain spheres (or projective spaces). In addition we apply this concept to vertex-transitive graphs with polynomial growth and to random walks on nilpotent groups.

Startpoints and(α,γ)-Contractions in Quasi-Pseudometric Spaces

We introduce the concept ofstartpointandendpointfor multivalued maps defined on a quasi-pseudometric space. We investigate the relation between these new concepts and the existence of fixed points for these set valued maps.