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[3] viXra:2011.0164 [pdf] submitted on 2020-11-21 17:32:58
Authors: Alexander Chang
Comments: 6 Pages.
Recent advances have begun to blur the lines between theoretical mathematics and applied mathematics. Oftentimes, in a variety of fields, concepts from not only applied mathematics but theoretical mathematics have been employed to great effect. As more and more researchers come to utilize, deploy, and develop both abstract and concrete mathematical models (both theoretical and applied), the demand for highly generalizable, accessible, and versatile mathematical models has increased drastically (Rosen, 2011). Specifically in the case of Complex Systems and the accompanying field of Complex Systems Analysis, this phenomenon has had profound effects. As researchers, academics, and scholars from these fields turn to mathematical models to assist in their scientific inquiries (specifically, concepts and ideas taken from various subsets of graph theory), the limitations of our current mathematical frameworks becomes increasingly apparent. To remedy this, we present the Chang Graph, a simple graph defined by an n-sided regular polygon surrounding a 2n-sided regular polygon. Various properties and applications of this graph are discussed, and further research is proposed for the study of this mathematical model.
Category: Topology
[2] viXra:2011.0043 [pdf] submitted on 2020-11-06 09:04:24
Authors: Yu-Lin Chou
Comments: 3 Pages. expository article
That every Euclidean subset homeomorphic to the ambient Euclidean space is open, a version of invariance of domain, is a relatively deep result whose typical proof is far from elementary. When it comes to the real line, the version of invariance of domain admits a simple proof that depends precisely on some elementary results of ``common sense''. It seems a pity that an elementary proof of the version of invariance of domain for the real line is not well-documented in the related literature even as an exercise, and it certainly deserves a space. Apart from the main purpose, as we develop the ideas we also make present some pedagogically enlightening remarks, which may or may not be well-documented.
Category: Topology
[1] viXra:2011.0042 [pdf] submitted on 2020-11-06 08:47:36
Authors: Yu-Lin Chou
Comments: 4 Pages. expository article
We wish to demystify the concept of a homeomorphism for anyone who finds the idea ``intangible'', by showing that one can construct a homeomorphism out of any given bijection in a natural way. Some interdisciplinary examples are discussed for concreteness.
Category: Topology
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