Universe as a Graph (Ramsey Approach to Analysis of Physical Systems)

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

doi:10.20944/preprints202211.0277.v2

Cited by 4 publications

(6 citation statements)

References 36 publications

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“…The Ramsey theory/approach shows potential for a diversity of mathematical and physical applications, including graph theory [7], ergodic theory [7][8][9], unimaginable numbers, Goodstein sequences, Knuth powers, Planar geometry [16], labeled graphs [17], theory of dynamic billiards [12][13][14]18], statistical physics [19], axiomatic thermodynamics [20] and analysis of many-body interactions [21]. We demonstrate that the Ramsey theory enables the instructive analysis of a discrete sets of points located on closed curves.…”

Section: Discussionmentioning

confidence: 99%

A Note on the Geometry of Closed Loops

et al. 2023

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In this paper, we utilize the Ramsey theory to investigate the geometrical characteristics of closed contours. We begin by examining a set of six points arranged on a closed contour and connected as a complete graph. We assign the downward-pointing edges a red color, while coloring the remaining edges green. Our analysis establishes that the curve must contain at least one monochromatic triangle. This finding has practical applications in the study of dynamical billiards. Our second result is derived from the Jordan curve theorem and the Ramsey theorem. Finally, we discuss Ramsey constructions arising from differential geometry. Applications of the Ramsey theory are discussed.

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Section: Discussionmentioning

confidence: 99%

A Note on the Geometry of Closed Loops

et al. 2023

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“…Thus, the 𝑿𝟎𝒀 plane may be seen, for the example, as maps of thermodynamic states or volt-ampere characteristics. The suggested coloring procedure will supply the predictions related to emerging of cycles in these maps [16][17][18][19][20][21].…”

Section: Discussionmentioning

confidence: 99%

Ramsey Theory and Transformations of Coordinate Systems

Frenkel¹,

Shoval²,

Bormashenko³

2023

Preprint

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Application of the Ramsey Theory to the set of the points placed in the plane is discussed. Consider a set of N points located in the same plane. The straight lines connecting these points are yikx=αikx+βik i,k=1...N. Listed below values of slopes αik are possible: αik&gt;0; αik=0; αik is not defined; αik&lt;0. Following coloring procedure is introduced: we connect the pairs of points for which αik&gt;0, αik=0 or αik is not defined, with the red links, and the pairs of points for which αik&lt;0 place with green links. The suggested coloring procedure enables building of the complete bi-colored graph for any set containing N points located in the same plane. We apply the Ramsey theorem to the complete graph emerging from the suggested coloring. For the set containing N=6 points at least one monochromatic triangle will necessarily appear in the graph. The values of the slopes αik depend on the chosen coordinate system. The rotation of coordinate axes changes the coloring of the graph; however, at least one monochromatic triangle will be present for N=6. The introduced coloring procedure diminishes the order of the symmetry group of the regular hexagon, irrespectively to the orientation of the coordinate axes. We hypothesized that this will be true for arbitrary regular n-polygon, independently on the orientation of coordinate axes. The inverse bi-color Ramsey graphs arise, when we replace red links appearing in the source graph with red ones, and vice versa. The total number of triangles in the “direct” and “inverse” Ramsey graphs is the same. We considered the particular case of the set of points for which αik&gt;0 or αik&lt;0 is true in the given coordinates. In this particular case, the rotation of the Cartesian coordinate axes to the angle θ=π2 yields the inverse complete graph. Generalization of this coloring for an arbitrary number and arbitrary location of the source points is introduced.

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“…The application of the Ramsey theory to physical systems, in which attraction and repulsion forces act between, the bodies constituting the system is discussed in ref. 9. The Ramsey theory explains why nature prefers cubic lattices over hexagonal ones for systems built of electric or magnetic dipoles [9].…”

Section: Introductionmentioning

confidence: 99%

“…9. The Ramsey theory explains why nature prefers cubic lattices over hexagonal ones for systems built of electric or magnetic dipoles [9].…”

Section: Introductionmentioning

confidence: 99%

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

Shvalb¹,

Frenkel²,

Shoval³

et al. 2023

Preprint

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Ramsey theory influences the dynamics of mechanical systems, which may be described as abstract complete graphs. We address a mechanical system which is completely interconnected with the two kinds of ideal Hookean springs. The suggested system mechanically corresponds to the cyclic molecules, in which functional groups are interconnected with two kinds of chemical bonds, represented mechanically with two springs k1 and k2. In this paper, we consider a Cyclic system (molecule) built of six equal masses m and two kinds of springs. We pose the following question: what is the minimal number of masses in the such a system in which three masses are constrained to be connected with spring k1 or three masses to be connected with spring k2? The answer to this question is supplied by the Ramsey theory, and it is formally stated as follows: what is the minimal number R3,3? The result emerging from the Ramsey theory is R3,3=6. Thus, in the aforementioned interconnected mechanical system will be necessarily present the triangles (at least one triangle), built of masses and springs. This prediction constitutes the vibrational spectrum of the system. Thus, the Ramsey Theory supplies the selection rules for the vibrational spectra of the cyclic molecules. Symmetrical system built of six vibrating entities is addressed. The Ramsey approach works for 2D and 3D molecules, which may be described as abstract complete graphs. The extension of the proposed Ramsey approach to the systems, partially connected by ideal springs, viscoelastic systems and systems, in which elasticity is of an entropy nature is discussed. “Multi-color systems” built of three kinds of ideal springs are addressed. The notion of the inverse Ramsey network is introduced and analyzed.

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Universe as a Graph (Ramsey Approach to Analysis of Physical Systems)

Cited by 4 publications

References 36 publications

A Note on the Geometry of Closed Loops

A Note on the Geometry of Closed Loops

Ramsey Theory and Transformations of Coordinate Systems

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

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