Connectedness and Local Connectedness on Infra Soft Topological Spaces

This year, we introduced the concept of infra soft topology as a new generalization of soft topology. To complete our analysis of this space, we devote this paper to presenting the concepts of infra soft connected and infra soft locally connected spaces. We provide some descriptions for infra soft connectedness and elucidate that there is no relationship between an infra soft topological space and its parametric infra topological spaces with respect to the property of infra soft connectedness. We discuss the behaviors of infra soft connected and infra soft locally connected spaces under infra soft homeomorphism maps and a finite product of soft spaces. We complete this manuscript by defining a component of a soft point and establishing its main properties. We determine the conditions under which the number of components is finite or countable, and we discuss under what conditions the infra soft connected subsets are components.

Genre Features and Types of Adyghe Toponymic Legends

The article examines the Adyghe toponymic traditions. The relevance of the work is that it addresses one of the complex theoretical problems of modern Russian folklore — the definition of differentiating genre features of works of oral folk prose. The authors proceed from the fact that the identification of genre characteristics of Adyghe toponymic legends should contribute to the creation of a generalizing genre system of Adyghe folk non-fairy prose. The novelty of the research consists in determining the genre features of Adyghe toponymic legends and their classification. In the continuum of Adyghe toponymic traditions, two groups are distinguished: the traditions of the first group are archaic, they tell about the events of the distant past and are known throughout the entire ethnic territory. It is shown that the plot of this type of legend is expanded, complicated and includes a number of additional motives — the history of migration and settlement of the current territories by the people, genealogical excursions. It is noted that another group of toponymic legends is associated with events of the recent past and has a narrow local connectedness. The marker of historical authenticity of the described facts and events is an indication of real historical persons who act as characters in the legend. It is argued that the plot of this type of legend is not developed, they are usually one-episode, and the area of distribution of this group of legends is usually limited to one locality.

Text Summarisation Using Laplacian Centrality-Based Minimum Vertex Cover

Outdegree Centrality (OC) is a graph-based centrality measure that captures local connectedness of a node in a graph. The measure has been used in the literature to highlight key sentences in a graph-based optimisation method for summarisation. It is observed in resultant summaries that OC tends to be biased towards selecting introductory sentences of the document producing only generic summaries. The different graph centrality measures lead to different interpretations of a summary. Therefore, the authors propose to use another suitable centrality measure in order to generate more specific summary rather than a generic summary. Such a summary is expected to be highly informative covering all the subtopics of the source document. This requirement has instigated the authors to use Laplacian Centrality (LC) measure to find the significance of the nodes. The essence of this measure lies in highlighting central nodes from subgraphs which contribute non-uniformly towards the common goal of the graph. The modified method has shown significant improvement in informativeness and coherence of summaries and outperformed state-of-the-art results.

Sequential decreasing strong size properties

Let X be a continuum. The n-fold hyperspace Cn(X), n < ∞, is the space of all nonempty closed subsets of X with at most n components. A topological property $ \mathcal{P} $ is said to be a (an almost) sequential decreasing strong size property provided that if μ is a strong size map for Cn(X, $ \{t_{j}\}_{j=1}^{\infty} $ is a sequence in the interval (t,1) such that lim tj = t ∈ [0,1) (t ∈ (0,1)) and each fiber μ−1(tj) has property $ \mathcal{P} $, then so does μ−1(t). In this paper we show that the following properties are sequential decreasing strong size properties: being a Kelley continuum, local connectedness, continuum chainability and, unicoherence. Also we prove that indecomposability is an almost sequential decreasing strong size property.

An example of a non-Borel locally-connected finite-dimensional topological group

According to a classical theorem of Gleason and Montgomery, every finite-dimensional locally path-connected topological group is a Lie group. In the paper for every $n\ge 2$ we construct a locally connected subgroup $G\subset{\mathbb R}^{n+1}$ of dimension $\dim(G)=n$, which is not locally compact. This answers a question posed by S. Maillot on MathOverflow and shows that the local path-connectedness in the result of Gleason and Montgomery can not be weakened to the local connectedness.