Search Results (10)

The graph theory-based approach to the three-body problem is introduced. Vectors of linear and angular momenta of the particles form the vertices of the graph. Scalar products of the vectors of the linear and angular momenta define the colors of the links connecting the vertices. The bi-colored, complete graph emerges. This graph is called the “momenta graph”. According to the Ramsey theorem, this graph contains at least one mono-chromatic triangle. This is true even for chaotic motion of three bodies; thus, illustrating the idea supplied by the Ramsey theory, total chaos is impossible. Coloring of the graph is independent on the rotation of frames; however, it is sensitive to Galilean transformations. The coloring of the momenta graph remains the same for general linear transformations of vectors with a positive-definite matrix. For a given motion, changing the order of the vertices does not change the number and distribution of monochromatic triangles. Symmetry of the momenta graph is addressed. The symmetry group remains the same for general linear transformation of vectors of the linear and angular momenta with a positive-definite matrix. Conditions defining conservation of the coloring of the momenta graph are addressed. The notion of the stereographic momenta graph is introduced. Shannon entropy of the momenta graph is calculated. The particular configurations of bodies are addressed, including the Lagrange configuration and the figure eight-shaped motion. The suggested approach is generalized for the quantum field theory with the Pauli–Lubanski pseudo-vector. The suggested coloring procedure is the Lorenz invariant. Full article

We introduce a synchronization procedure for clocks based on the Einstein–Landauer framework. Clocks are modeled as discrete, macroscopic devices operating at a thermal equilibrium temperature T. Synchronization is achieved by transmitting photons from one clock to another; the absorption of a photon by a clock reduces the uncertainty in its timekeeping. The minimum energy required for this reduction in uncertainty is determined by the Landauer bound. We distinguish between the time-bearing and non-time-bearing degrees of freedom of the clocks. A reduction in uncertainty under synchronization in the time-bearing degrees of freedom necessarily leads to heat dissipation in the non-time-bearing ones. The minimum energy dissipation in these non-time-bearing degrees of freedom is likewise given by the Landauer limit. The same is true for mechanical synchronization of clocks. We also consider lattices of clocks and analyze synchronization using a Ramsey graph approach. Notably, clocks operating at the same temperature may be synchronized using photons of different frequencies. Each clock is categorized as either synchronized or non-synchronized, resulting in a bi-colored complete graph of clocks. By Ramsey’s theorem, such a graph inevitably contains a triad (or loop) of clocks that are either all synchronized or all non-synchronized. The extension of the Ramsey approach to infinite lattices of clocks is reported. Full article

The Ramsey approach is applied to analyses of the kinematics of systems built of non-relativistic, motile point masses/particles. This approach is based on colored graph theory. Point masses/particles serve as the vertices of the graph. The time dependence of the distance between the particles determines the coloring of the links. The vertices/particles are connected with orange links when particles move away from each other or remain at the same distance. The vertices/particles are linked with violet edges when particles converge. The sign of the time derivative of the distance between the particles dictates the color of the edge. Thus, a complete, bi-colored Ramsey temporal graph emerges. The suggested coloring procedure is not transitive. The coloring of the links is time-dependent. The proposed coloring procedure is frame-independent and insensitive to Galilean transformations. At least one monochromatic triangle will inevitably appear in the graph emerging from the motion of six particles due to the fact that the Ramsey number $R\left(\mathrm{3,3}\right)=6.$ This approach is extended to the analysis of systems containing an infinite number of moving point masses. An infinite monochromatic (violet or orange) clique will necessarily appear in the graph. Applications of the introduced approach are discussed. The suggested Ramsey approach may be useful for the analysis of turbulence seen within the Lagrangian paradigm. Full article

Ramsey’s theorem states that for any natural numbers n, m there exists a natural number N such that any red–blue coloring of the graph $K_N$ contains either a red $K_n$ or blue $K_m$ as a subgraph. The smallest such N is called the Ramsey number, denoted as $R(n,m)$. In this paper, we reformulate this theorem and present a proof of Ramsey’s theorem that is novel as far as we are aware. Full article

We propose a Ramsey-theory-based approach for the analysis of the behavior of isolated mechanical systems containing interacting particles. The total momentum of the system in the frame of the center of masses is zero. The mechanical system is described by a Ramsey-theory-based, bi-colored, complete graph. Vectors of momenta of the particles ${\overrightarrow{p}}_i$ serve as the vertices of the graph. We start from the graph representing the system in the frame of the center of masses, where the momenta of the particles in this system are ${\overrightarrow{p}}_{cmi}.$ If ${(\overrightarrow{p}}_{cmi}(t)·{\overrightarrow{p}}_{cmj}(t))\ge 0$ is true, the vectors of momenta of the particles numbered i and j are connected with a red link; if ${(\overrightarrow{p}}_{cmi}(t)·{\overrightarrow{p}}_{cmj}(t))<0$ takes place, the vectors of momenta are connected with a green link. Thus, the complete, bi-colored graph emerges. Considering an isolated system built of six interacting particles, according to the Ramsey theorem, the graph inevitably comprises at least one monochromatic triangle. The coloring procedure is invariant relative to the rotations/translations of frames; thus, the graph representing the system contains at least one monochromatic triangle in any of the frames emerging from the rotation/translation of the original frame. This gives rise to a novel kind of mechanical invariant. Similar coloring is introduced for the angular momenta of the particles. However, the coloring procedure is sensitive to Galilean/Lorenz transformations. Extensions of the suggested approach are discussed. Full article

We applied the Ramsey analysis to the sets of points belonging to Riemannian manifolds. The points are connected with two kinds of lines: geodesic and non-geodesic. This interconnection between the points is mapped into the bi-colored, complete Ramsey graph. The selected points correspond to the vertices of the graph, which are connected with the bi-colored links. The complete bi-colored graph containing six vertices inevitably contains at least one mono-colored triangle; hence, a mono-colored triangle, built of the green or red links, i.e., non-geodesic or geodesic lines, consequently appears in the graph. We also considered the bi-colored, complete Ramsey graphs emerging from the intersection of two Riemannian manifolds. Two Riemannian manifolds, namely $\left(M_1,g_1\right)$ and $\left(M_2,g_2\right)$, represented by the Riemann surfaces which intersect along the curve $\left(M_1,g_1\right){\displaystyle \cap }\left(M_2,g_2\right)=ℒ$ were addressed. Curve $ℒ$ does not contain geodesic lines in either of the manifolds$\left(M_1,g_1\right)$ and $\left(M_2,g_2\right)$. Consider six points located on the $ℒ:$ $\left\{1,\dots 6\right\}\subset ℒ$. The points $\left\{1,\dots 6\right\}\subset ℒ$ are connected with two distinguishable kinds of the geodesic lines, namely with the geodesic lines belonging to the Riemannian manifold $\left(M_1,g_1\right)$/red links, and, alternatively, with the geodesic lines belonging to the manifold $\left(M_2,g_2\right)$/green links. Points $\left\{1,\dots 6\right\}\subset ℒ$ form the vertices of the complete graph, connected with two kinds of links. The emerging graph contains at least one closed geodesic line. The extension of the theorem to the Riemann surfaces of various Euler characteristics is presented. Full article

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. Full article

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 $\alpha$-colored polygons with n α-sides or edges, and the fraction of monochromatic $\beta$-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 $S_{\alpha }$ and $S_{\beta }$ are interpreted as follows: $S_{\alpha }$ is interpreted as an average uncertainty to find the green $\alpha -$polygon in the given graph; $S_{\beta }$ is, in turn, an average uncertainty to find the red $\beta -$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. Full article

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. Full article

Given a dataset, we quantify the size of patterns that must always exist in the dataset. This is done formally through the lens of Ramsey theory of graphs, and a quantitative bound known as Goodman’s theorem. By combining statistical tools with Ramsey theory of graphs, we give a nuanced understanding of how far away a dataset is from correlated, and what qualifies as a meaningful pattern. This method is applicable to a wide range of datasets. As examples, we analyze two very different datasets. The first is a dataset of repeated voters ( $n=435$ ) in the 1984 US congress, and we quantify how homogeneous a subset of congressional voters is. We also measure how transitive a subset of voters is. Statistical Ramsey theory is also used with global economic trading data ( $n=214$ ) to provide evidence that global markets are quite transitive. While these datasets are small relative to Big Data, they illustrate the new applications we are proposing. We end with specific calls to strengthen the connections between Ramsey theory and statistical methods. Full article