ARTICLE | doi:10.20944/preprints202309.1370.v1
Subject: Physical Sciences, Mathematical Physics Keywords: three-body problem; Lagrange triangle; continuous measure of symmetry; gravity; Coulomb interaction; asymptotic value
Online: 20 September 2023 (10:38:50 CEST)
The time evolution of the continuous measure symmetry for the system built of the three bodies interacting via the potential U(r)~1/r is reported. Gravitational and electrostatic interactions between the point bodies were addressed. In the case of the pure gravitational interaction the three-body-system deviated from its initial symmetrical location, described by the Lagrange equilateral triangle, comes to collapse, accompanied by the growth of the continuous measure of symmetry. When three point bodies interact via the Coulomb repulsive interaction, the time evolution of CMS is quite different. CMS calculated for all of studied initial configurations of the point charges and all of their charge-to-mass ratios always comes to its asymptotic value with time, evidencing the stabilization of the shape of the triangle, constituted by the interacting bodies.
ARTICLE | doi:10.20944/preprints202307.1696.v1
Subject: Computer Science And Mathematics, Discrete Mathematics And Combinatorics Keywords: Complete graph; Shannon Entropy; bi-colored graph; Ramsey Theorem; Ramsey Number; Voronoi tessellation.
Online: 25 July 2023 (10:27:56 CEST)
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.
ARTICLE | doi:10.20944/preprints202306.0685.v1
Subject: Computer Science And Mathematics, Discrete Mathematics And Combinatorics Keywords: Ramsey theory; pairs of points; slope; complete graph; symmetry; inverse graph.
Online: 9 June 2023 (09:48:51 CEST)
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>0; αik=0; αik is not defined; αik<0. Following coloring procedure is introduced: we connect the pairs of points for which αik>0, αik=0 or αik is not defined, with the red links, and the pairs of points for which αik<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>0 or αik<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.
Subject: Chemistry And Materials Science, Materials Science And Technology Keywords: metamaterials; negative effective mass; plasma oscillations; low frequency plasmons; optical and acoustical branches.
Online: 21 July 2020 (13:38:18 CEST)
We report the negative effective mass metamaterials based on the electro-mechanical coupling exploiting plasma oscillations of a free electron gas. The negative mass appears as a result of vibration of a metallic particle with a frequency of ω which is close to the frequency of the plasma oscillations of the electron gas m_2 relatively to the ionic lattice m_1. The plasma oscillations are represented with the elastic spring k_2=ω_p^2 m_2, where ω_p is the plasma frequency. Thus, the metallic particle vibrated with the external frequency ω is described by the effective mass m_eff=m_1+(m_2 ω_p^2)/(ω_p^2-ω^2 ) , which is negative when the frequency ω approaches ω_p from above. The idea is exemplified with two conducting metals, namely Au and Li embedded into various matrices. The one-dimensional lattice built of the identical metallic micro-elements m_eff connected by ideal springs k_1 representing various media such as polydimethylsiloxane and soda-lime glass is treated. The optical and acoustical branches of longitudinal modes propagating through the lattice are elucidated for various ratios ω_1/ω_p . The 1D lattice built of the thin metallic wires giving rise to the low frequency plasmons is treated. The possibility of the anti-resonant propagation, strengthening the effect of the negative mass occurring under = p = 1 is addressed.
ARTICLE | doi:10.20944/preprints201908.0153.v1
Subject: Chemistry And Materials Science, Polymers And Plastics Keywords: honeycomb polymer films; breath figures self-assembly; oil-lubricated substrates; Voronoi entropy; Cassie wetting regime
Online: 13 August 2019 (09:57:51 CEST)
Hierarchical honeycomb patterns were manufactured with the breath-figures self-assembly by drop-casting on the silicone-oil lubricated glass substrates. Silicone oil promoted spreading of the polymer solutions. The process was carried out with the industrial grade polystyrene and polystyrene with the molecular weight Mw=35.000. Both of polymers gave rise to the patterns, built from micro- and nano-scaled pores. Ordering of the pores was quantified with the Voronoi tessellations and calculating the Voronoi entropy. Measurement of the apparent contact angles evidenced the Cassie - Baxter wetting regime of the porous films.
ARTICLE | doi:10.20944/preprints201904.0336.v1
Subject: Computer Science And Mathematics, Mathematics Keywords: Voronoi entropy; Voronoi tessellation; symmetry; ordering; Shannon measure of information
Online: 30 April 2019 (11:51:44 CEST)
Voronoi entropy for the random patterns and patterns demonstrating various elements of symmetry are calculated. The symmetric patterns are characterized by the values of the Voronoi entropy very close to those inherent to random ones. This contradicts the idea that the Voronoi entropy quantifies the ordering of the seed points, constituting the pattern. The extension of the Shannon-like formula embracing symmetric patterns is suggested. Analysis of Voronoi diagrams enables revealing of the elements of symmetry of the pattern.
ARTICLE | doi:10.20944/preprints202209.0331.v2
Subject: Physical Sciences, Mathematical Physics Keywords: Ramsey theory; complete graph; vibrational spectrum; eigenfrequency; selection rule; cyclic molecule; viscoelasticity; entropic elasticity
Online: 16 February 2023 (09:55:15 CET)
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.
ARTICLE | doi:10.20944/preprints202302.0201.v1
Subject: Computer Science And Mathematics, Mathematics Keywords: Ramsey theory; closed contour; Jordan theorem; complete graph.
Online: 13 February 2023 (06:56:36 CET)
We apply the Ramsey theory to the analysis of geometrical properties of closed contours. Consider a set of six points placed on a closed contour. The straight lines connecting these points are yikx=αikx+βik i,k=1...6, αik≠0. We color the edges connecting the points for which αik>0 holds with red, and the edges for which αik<0 with green with red. At least one monochromic triangle should necessarily appear within the curve (according to the Ramsey number R3,3=6). This result is immediately applicable for the analysis of dynamical billiards. The second theorem emerges from the combination of the Jordan curve and Ramsey theorem. The closed curve is considered. We connect the points located within the same region with green links and the points placed within the different regions with red links. In this case, transitivity/intransitivity of the relations between the points should be considered. Ramsey constructions arising from the differential geometry of closed contours are discussed.
ARTICLE | doi:10.20944/preprints202211.0277.v2
Subject: Physical Sciences, Theoretical Physics Keywords: physical system; attraction; repulsion; Ramsey theory; transitivity; complete graph; dipole-dipole interaction; relativity; Hamiltonian path.
Online: 20 January 2023 (07:50:16 CET)
Application of the Ramsey graph theory to the analysis of physical systems is reported. Physical interactions may be very generally classified as attractive and repulsive. This classification creates the premises for the application of the Ramsey theory to the analysis of physical systems built of electrical charges, electric and magnetic dipoles. The notions of mathematical logic, such as transitivity and intransitivity relations, become crucial for understanding of the behavior of physical systems. The Ramsey theory explains why nature prefers cubic lattices over hexagonal ones for systems built of electric or magnetic dipoles. The Ramsey approach may be applied to the analysis of mechanical systems when actual and virtual paths between the states in the configurational space are considered. Irreversible mechanical and thermodynamic processes are seen within the reported approach as directed graphs. Chains of irreversible processes appear as transitive tournaments. These tournaments are acyclic; the transitive tournaments necessarily contain the Hamiltonian path. The set of states in the phase space of the physical system, between which irreversible processes are possible, is considered. The Hamiltonian path of the tournament emerging from the graph uniting these states is a relativistic invariant. Applications of the Ramsey theory to the general relativity become possible when the discrete changes in the metric tensor are assumed. Reconsideration of the concept of “simultaneity” within the Ramsey approach is reported.
ARTICLE | doi:10.20944/preprints202211.0077.v1
Subject: Physical Sciences, Thermodynamics Keywords: thermodynamics; Ramsey theory; graph theory; directed graph; irreversible process.
Online: 3 November 2022 (11:36:33 CET)
Re-shaping of thermodynamics with the graph theory and Ramsey theory is suggested. Maps built of thermodynamic states are addressed. Thermodynamic states may be attainable and non-attainable by the thermodynamic process in the system of constant mass. We address the following question how large should be a graph describing connections between discrete thermodynamic states to guarantee the appearance of thermodynamic cycles? The Ramsey theory supplies the answer to this question. Direct graphs emerging from the chains of irreversible thermodynamic processes are considered. In any complete directed graph, representing the thermodynamic states of the system the Hamiltonian path is found. Transitive thermodynamic tournaments are addressed. The entire transitive thermodynamic tournament built of irreversible processes does not contain a cycle of length 3, or in other words, the transitive thermodynamic tournament is acyclic and contains no directed thermodynamic cycles.
ARTICLE | doi:10.20944/preprints202301.0307.v1
Subject: Computer Science And Mathematics, Mathematics Keywords: Archimedes Spiral; Voronoi tessellation; Voronoi entropy; surface patterns; aesthetic attraction; phyllotaxis; golden ratio.
Online: 17 January 2023 (10:07:18 CET)
Voronoi mosaics inspired by the seed points placed on the Archimedes Spirals are reported. Voronoi entropy was calculated for these patterns. Equidistant and non-equidistant patterns are treated. Voronoi mosaics built from a sells of equal size which are of a primary importance for decorative arts are reported. The pronounced prevalence of hexagons is inherent for the patterns with an equidistant and non-equidistant distribution of points, when the distance between the seed points is of the same order of magnitude as the distance between the turns of the spiral. Penta- and heptagonal “defected” cells appeared in the Voronoi diagrams due to the finite nature of the pattern. The ordered Voronoi tessellations demonstrating the Voronoi entropy larger than 1.71, reported for the random 2D distribution of points, were revealed. The dependence of the Voronoi entropy on the total number of the seed points located on the Archimedes Spirals is reported. Voronoi tessellations generated by the phyllotaxis-inspired patterns are addressed. The aesthetic attraction of the Voronoi mosaics arising from seed points placed on the Archimedes Spirals is discussed.
ARTICLE | doi:10.20944/preprints202212.0012.v1
Subject: Physical Sciences, Mathematical Physics Keywords: Shannon entropy; pentagonal tiling; Marjorie Rice tiling; Voronoi tessellation; iso-symmetrical transitions
Online: 1 December 2022 (06:42:44 CET)
We used the complete set of convex pentagons enabling filing the plane without any overlaps or gaps (including the Marjorie Rice tiles) as generators of Voronoi tessellations. Shannon entropy of the tessellations was calculated. Some of the basic mosaics are flexible and give rise to a diversity of Voronoi tessellations. The Shannon entropy of these tessellations varied in a broad range. Voronoi tessellation, emerging from the basic pentagonal tiling built from hexagons only, was revealed (the Shannon entropy of this tiling is zero). Decagons and hendecagon did not appear in the studied Voronoi diagrams. The most abundant Voronoi tessellations are built from three different kinds of polygons. The most widespread is the combination of pentagons, hexagons and heptagons. The most abundant polygons are pentagons and hexagons. No Voronoi tiling built only of pentagons was registered. Flexible basic pentagonal mosaics give rise to a diversity of Voronoi tessellations, which are characterized by the same symmetry group; however, the coordination number of the vertices is variable. These Voronoi tessellations may be useful for the interpretation of the iso-symmetrical phase transitions.
ARTICLE | doi:10.20944/preprints202109.0347.v1
Subject: Physical Sciences, Thermodynamics Keywords: informational measure of symmetry; 1D shapes; 2D shapes; fractal patterns; time evolution; symmetry; pattern
Online: 20 September 2021 (16:39:57 CEST)
Informational (Shannon) measures of symmetry are introduced and analyzed for the patterns built of 1D and 2D shapes. The informational measure of symmetry Hsym (G) characterizes the an averaged uncertainty in the presence of symmetry elements from the group G in a given pattern; whereas the Shannon-like measure of symmetry Ωsym (G) quantifies averaged uncertainty of appearance of shapes possessing in total n elements of symmetry belonging to group G in a given pattern. Hsym(G1)=Ωsym(G1)=0 for the patterns built of irregular, non-symmetric shapes. Both of informational measures of symmetry are intensive parameters of the pattern and do not depend on the number of shapes, their size and area of the pattern. They are also insensitive to the long-range order inherent for the pattern. Informational measures of symmetry of fractal patterns are addressed. The mixed patterns including curves and shapes are considered. Time evolution of the Shannon measures of symmetry is treated. The close-packed and dispersed 2D patterns are analyzed.
ARTICLE | doi:10.20944/preprints202109.0076.v1
Subject: Physical Sciences, Condensed Matter Physics Keywords: symmetry; informational measure; penrose tiling; Voronoi entropy; continuous symmetry measure; ordering
Online: 3 September 2021 (21:36:08 CEST)
The notion of the informational measure of symmetry is introduced according to: HsymG=-i=1kPGilnPGi, where PGi is the probability of appearance of the symmetry operation Gi within the given 2D pattern. HsymG is interpreted as an averaged uncertainty in the presence of symmetry elements from the group G in the given pattern. The informational measure of symmetry of the “ideal” pattern built of identical equilateral triangles is established as HsymD3=1.792. The informational measure of symmetry of the random, completely disordered pattern is zero, Hsym=0. Informational measure of symmetry is calculated for the patterns generated by the P3 Penrose tessellation. Informational measure of symmetry does not correlate neither with the Voronoi entropy of the studied patterns nor with the continuous measure of symmetry of the patterns.
ARTICLE | doi:10.20944/preprints202011.0590.v1
Subject: Chemistry And Materials Science, Surfaces, Coatings And Films Keywords: cold plasma; polyvinylchloride; wettability; hydrophilization; apparent contact angle; contact angle hysteresis; hydrophobic recovery.
Online: 23 November 2020 (14:31:54 CET)
Impact of the Corona, dielectric barrier discharge and low pressure radiofrequency air plasmas on the chemical composition and wettability of the medical grade polyvinylchloride was investigated. Corona plasma treatment exerted the most pronounced increase in the hydrophilization of polyvinylchloride. The specific energy of adhesion of the pristine and plasma treated PVC tubing is reported. The kinetics of hydrophobic recovery following the plasma treatment was explored. The time evolution of the apparent contact angle under the hydrophobic recovery is satisfactorily described by the exponential fitting. Energy-dispersive X-ray spectroscopy of the chemical composition of the near-surface layers of the plasma treated catheters revealed their oxidation. The effect of the hydrophobic recovery is hardly correlated with oxidation of the polymer surface, which is irreversible.
REVIEW | doi:10.20944/preprints202204.0252.v1
Subject: Physical Sciences, Thermodynamics Keywords: Voronoi tessellation; Voronoi entropy; random set of points; ordering; lamellae; spherulite; continuous measure of symmetry
Online: 27 April 2022 (08:29:01 CEST)
Properties of the Voronoi tessellations arising from the random 2D distribution points are reported. We applied the procedure of dividing the sides of Voronoi cells into equal or random parts to Voronoi diagrams generated by a set of randomly placed on the plane points. The dividing points were then used to construct the following Voronoi diagram. Repeating this procedure led to a surprising effect of positional ordering of Voronoi cells, reminiscent of the formation of lamellae and spherulites in linear semi-crystalline polymers and metallic glasses. Thus, we can conclude, that by applying even a simple set of rules to a random set of seeds we introduce order into an initially disordered system. At the same time, the Voronoi entropy showed a tendency to values typical for completely random patterns and did not distinguish the short-range ordering. The Voronoi entropy and the continuous measure of symmetry of the patterns demonstrated the distinct asymptotic behavior, while approaching the close saturation values with the increase of the number of the iteration steps. Voronoi entropy grew, with the number of iterations, whereas the continuous measure of symmetry of the same patterns demonstrated the opposite asymptotic behavior. The Voronoi entropy is not an unambiguous measure of order in the 2D patterns. The more symmetrical patterns may demonstrate the higher values of the Voronoi entropy.
REVIEW | doi:10.20944/preprints201811.0535.v1
Subject: Physical Sciences, Condensed Matter Physics Keywords: Voronoi entropy; surface patterns; Lewis law; Aboav law; droplet cluster; self-assembly
Online: 22 November 2018 (04:49:13 CET)
The Voronoi entropy is a mathematical tool for quantitative characterization of the orderliness of points distributed on a surface. The tool is useful to study various surface self-assembly processes. We provide the historical background, from Kepler and Descartes to our days, and discuss topological properties of the Voronoi tessellation, upon which the entropy concept is based, and its scaling properties, known as the Lewis and Aboav-Weaire laws. The Voronoi entropy has been successfully applied to recently discovered self-assembled structures, such as patterned micro-porous polymer surfaces obtained by the breath figure method and levitating ordered water micro-droplet clusters.