Dynamic Ramsey Theory of Mechanical Systems Forming a Complete Graph and Vibrations of Cyclic Compounds

Shvalb, Nir; Frenkel, Mark; Shoval, Shraga; Bormashenko, Edward

doi:10.3390/dynamics3020016

Cited by 4 publications

(4 citation statements)

References 23 publications

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“…Let us discuss the physical/chemical meaning/interpretation of the introduced Shannon entropies S α , S β . The complete bi-colored graph, presented in Figure 5, may be interpreted as a scheme of a cyclic molecule, in which two kinds of chemical bonds depicted with green ( α) links and red (β) links/edges are present [12]. In this case, the Shannon entropies S α , S β are seen as averaged uncertainties to find α or β cyclic sub-structures within the molecule.…”

Section: Discussionmentioning

confidence: 99%

“…In this case, the Shannon entropies S α , S β are seen as averaged uncertainties to find α or β cyclic sub-structures within the molecule. The presence of cyclic structures will influence the vibrational spectrum of the molecule [12].…”

Section: Discussionmentioning

confidence: 99%

“…Complete graphs emerge in various fields of economic science, physics, and engineering. For example, cyclic molecules may be seen as a system of particles interconnected with springs of various kinds [12]. Sets of thermodynamic states form complete graphs [13].…”

Section: Introductionmentioning

confidence: 99%

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Shannon Entropy of Ramsey Graphs with up to Six Vertices

Frenkel,

Shoval,

Bormashenko

2023

Entropy

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Shannon entropy quantifying bi-colored Ramsey complete graphs is introduced and calculated for complete graphs containing up to six vertices. Complete graphs in which vertices are connected with two types of links, labeled as α-links and β-links, are considered. Shannon entropy is introduced according to the classical Shannon formula considering the fractions of monochromatic convex α-colored polygons with n α-sides or edges, and the fraction of monochromatic β-colored convex polygons with m β-sides in the given complete graph. The introduced Shannon entropy is insensitive to the exact shape of the polygons, but it is sensitive to the distribution of monochromatic polygons in a given complete graph. The introduced Shannon entropies Sα and Sβ are interpreted as follows: Sα is interpreted as an average uncertainty to find the green α−polygon in the given graph; Sβ is, in turn, an average uncertainty to find the red β−polygon in the same graph. The re-shaping of the Ramsey theorem in terms of the Shannon entropy is suggested. Generalization for multi-colored complete graphs is proposed. Various measures quantifying the Shannon entropy of the entire complete bi-colored graphs are suggested. Physical interpretations of the suggested Shannon entropies are discussed.

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Section: Discussionmentioning

confidence: 99%

Section: Discussionmentioning

confidence: 99%

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Shannon Entropy of Ramsey Graphs with up to Six Vertices

Frenkel,

Shoval,

Bormashenko

2023

Entropy

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“…A Ramsey theory of financial distortions was reported recently [20]. The dynamic Ramsey theory of mechanical systems forming complete graphs in its relation to the analysis of vibrations of cyclic compounds was addressed [21]. Successful application of the Ramsey theory for the formulation of axiomatic thermodynamics was reported [22].…”

Section: Introductionmentioning

confidence: 99%

Fermat Principle, Ramsey Theory and Metamaterials

Frenkel,

Shoval,

Bormashenko

2023

Materials

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Reinterpretation of the Fermat principle governing the propagation of light in media within the Ramsey theory is suggested. Complete bi-colored graphs corresponding to light propagation in media are considered. The vertices of the graphs correspond to the points in real physical space in which the light sources or sensors are placed. Red links in the graphs correspond to the actual optical paths, emerging from the Fermat principle. A variety of optical events, such as refraction and reflection, may be involved in light propagation. Green links, in turn, denote the trial/virtual optical paths, which actually do not occur. The Ramsey theorem states that within the graph containing six points, inevitably, the actual or virtual optical cycle will be present. The implementation of the Ramsey theorem with regard to light propagation in metamaterials is discussed. The Fermat principle states that in metamaterials, a light ray, in going from point S to point P, must traverse an optical path length L that is stationary with respect to variations of this path. Thus, bi-colored graphs consisting of links corresponding to maxima or minima of the optical paths become possible. The graphs, comprising six vertices, will inevitably demonstrate optical cycles consisting of the mono-colored links corresponding to the maxima or minima of the optical path. The notion of the “inverse graph” is introduced and discussed. The total number of triangles in the “direct” (source) and “inverse” Ramsey optical graphs is the same. The applications of “Ramsey optics” are discussed, and an optical interpretation of the infinite Ramsey theorem is suggested.

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Shannon Entropy of Ramsey Graphs

Bormashenko¹,

Frenkel²,

Shoval³

2023

Preprint

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Shannon entropy quantifying bi-colored Ramsey complete graphs is introduced. Complete graphs in which vertices are connected with two types of links, labeled as α-links and β-links are considered. Shannon-entropy is introduced according to the classical Shannon formula considering the fractions of monochromatic convex α-colored polygons with n α-sides or edges, and the fraction of monochromatic β-colored convex polygons with m β-sides in the given complete graph. Introduced Shannon entropy is insensitive to the exact shape of the graph, but it is sensitive to the distribution of monochromatic polygons in a given graph. The introduced Shannon Entropies Sα and Sβ are interpreted as follows: Sα is interpreted as an average uncertainty to find the green α-polygon in the given graph, Sβ is, in turn, an average uncertainty to find the red β-polygon in the same graph. The re-shaping of the Ramsey theorem in terms of the Shannon Entropy is suggested. Various measures quantifying the Shannon Entropy of the entire complete bi-colored graphs are suggested. Physical interpretations of the suggested Shannon Entropies are discussed.

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Dynamic Ramsey Theory of Mechanical Systems Forming a Complete Graph and Vibrations of Cyclic Compounds

Cited by 4 publications

References 23 publications

Shannon Entropy of Ramsey Graphs with up to Six Vertices

Shannon Entropy of Ramsey Graphs with up to Six Vertices

Fermat Principle, Ramsey Theory and Metamaterials

Shannon Entropy of Ramsey Graphs

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