Topical Collection "Topology and Foundations"

A topical collection in Mathematics (ISSN 2227-7390). This collection belongs to the section "Algebra, Geometry and Topology".

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Editor

Topical Collection Information

Dear Colleagues,

A foundational example in topology leading to graph theory as a new branch of mathematics is when Leonhard Euler demonstrated that it was impossible to find a route through the town of Königsberg (now Kaliningrad) that would cross each of its seven bridges exactly once (with the result depending only on connectivity properties—which bridges connect to which islands or riverbanks).

Subjects included in topology are graph theory and algebraic topology. Topology is foundational in number theory, algebraic geometry, category theory and homological algebra, K-theory, group theory and generalizations, topological groups and Lie groups, dynamical systems and ergodic theory, functional analysis, quantum theory, game theory, etc. Topology is in all fields of engineering, physical sciences, life sciences, social sciences, medicine and even arts, economics, finance, and finally mathematics-related sciences: informatics, physics, chemistry and biology.

Foundations extend beyond topology as the basis or groundwork of anything. Foundations are under algebraic geometry, tropical geometry, logic and deductive systems, functions, algebraic topology, homotopy theory, probability theory, stochastic processes, statistics, physics (fluid mechanics, optics, electromagnetic theory, thermodynamics, heat transfer, equilibrium and time-dependent statistical mechanics) and finally quantum information and its processing.

The aim of this Topical Collection is to bring together recent scientific advances, reviews, communications and short notes dealing with topology and foundations.

Prof. Dr. Lorentz Jäntschi
Collection Editor

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Published Papers (2 papers)

2023

Jump to: 2021

16 pages, 337 KiB  
Article
On Bishop–Phelps and Krein–Milman Properties
Mathematics 2023, 11(21), 4473; https://doi.org/10.3390/math11214473 - 28 Oct 2023
Viewed by 332
Abstract
A real topological vector space is said to have the Krein–Milman property if every bounded, closed, convex subset has an extreme point. In the case of every bounded, closed, convex subset is the closed convex hull of its extreme points, then we say [...] Read more.
A real topological vector space is said to have the Krein–Milman property if every bounded, closed, convex subset has an extreme point. In the case of every bounded, closed, convex subset is the closed convex hull of its extreme points, then we say that the topological vector space satisfies the strong Krein–Milman property. The strong Krein–Milman property trivially implies the Krein–Milman property. We provide a sufficient condition for these two properties to be equivalent in the class of Hausdorff locally convex real topological vector spaces. This sufficient condition is the Bishop–Phelps property, which we introduce for real topological vector spaces by means of uniform convergence linear topologies. We study the inheritance of the Bishop–Phelps property. Nontrivial examples of topological vector spaces failing the Krein–Milman property are also given, providing us with necessary conditions to assure that the Krein–Milman property is satisfied. Finally, a sufficient condition to assure the Krein–Milman property is discussed. Full article

2021

Jump to: 2023

25 pages, 764 KiB  
Communication
Figures of Graph Partitioning by Counting, Sequence and Layer Matrices
Mathematics 2021, 9(12), 1419; https://doi.org/10.3390/math9121419 - 18 Jun 2021
Cited by 17 | Viewed by 2525
Abstract
A series of counting, sequence and layer matrices are considered precursors of classifiers capable of providing the partitions of the vertices of graphs. Classifiers are given to provide different degrees of distinctiveness for the vertices of the graphs. Any partition can be represented [...] Read more.
A series of counting, sequence and layer matrices are considered precursors of classifiers capable of providing the partitions of the vertices of graphs. Classifiers are given to provide different degrees of distinctiveness for the vertices of the graphs. Any partition can be represented with colors. Following this fundamental idea, it was proposed to color the graphs according to the partitions of the graph vertices. Two alternative cases were identified: when the order of the sets in the partition is relevant (the sets are distinguished by their positions) and when the order of the sets in the partition is not relevant (the sets are not distinguished by their positions). The two isomers of C28 fullerenes were colored to test the ability of classifiers to generate different partitions and colorings, thereby providing a useful visual tool for scientists working on the functionalization of various highly symmetrical chemical structures. Full article
Show Figures

Figure 1

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