Search Results (25)

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

The Ramsey number $R\left(F\right)$ of a graph F without isolated vertices is the smallest positive integer n such that every red–blue coloring of $K_n$ produces a subgraph isomorphic to F all of whose edges are colored the same. Let $\mathcal{F}$ be a set of graphs without isolated vertices. For a positive integer t, the vertex-disjoint Ramsey number $V{R}_t\left(\mathcal{F}\right)$ is the smallest positive integer n such that every red–blue coloring of the complete graph $K_n$ of order n results in at least t pairwise vertex-disjoint monochromatic graphs in $\mathcal{F}$; while the edge-disjoint Ramsey number $E{R}_t\left(\mathcal{F}\right)$ is the smallest positive integer n such that every red–blue coloring of $K_n$ produces at least t pairwise edge-disjoint monochromatic graphs in $\mathcal{F}$. If $t=1$ and $\mathcal{F}$ consists of a single graph F, then $V{R}_1\left(\mathcal{F}\right)=E{R}_1\left(\mathcal{F}\right)=R\left(F\right)$ is the Ramsey number of the graph F. Thus, the concepts of vertex-disjoint and edge-disjoint Ramsey numbers provide a generalization of the standard Ramsey number. Upper and lower bounds for $V{R}_t\left(\mathcal{F}\right)$ and $E{R}_t\left(\mathcal{F}\right)$ are established for sets $\mathcal{F}$ of graphs without isolated vertices and the sharpness of these bounds is discussed. The primary goal of this paper is to investigate the values of $V{R}_t\left(\mathcal{F}\right)$ and $E{R}_t\left(\mathcal{F}\right)$ for sets $\mathcal{F}$ of graphs of size 2 or 3 without isolated vertices. The exact values of $V{R}_t\left(\mathcal{F}\right)$ are determined for all such sets $\mathcal{F}$ and all integers $t\ge 2$. The exact values of $E{R}_t\left(\mathcal{F}\right)$ of certain such sets $\mathcal{F}$ with prescribed conditions for all integers $t\ge 2$ are determined. For some special sets $\mathcal{F}$ of graphs of size 2 or 3 without isolated vertices, the exact values of $E{R}_t\left(\mathcal{F}\right)$ are determined for $2\le t\le 4$. Additional results, problems, and conjectures are also presented dealing with these two Ramsey concepts for graphs in general. Full article

Ramsey analysis is applied to the problem of the relativistic and quantum synchronization of clocks. Various protocols of synchronization are addressed. Einstein and Eddington special relativity synchronization procedures are considered, and quantum synchronization is discussed. Clocks are seen as the vertices of the graph. Clocks may be synchronized or unsynchronized. Thus, introducing complete, bi-colored, Ramsey graphs emerging from the lattices of clocks becomes possible. The transitivity of synchronization plays a key role in the coloring of the Ramsey graph. Einstein synchronization is transitive, while general relativity and quantum synchronization procedures are not. This fact influences the value of the Ramsey number established for the synchronization graph arising from the lattice of clocks. Any lattice built of six clocks, synchronized with quantum entanglement, will inevitably contain the mono-chromatic triangle. The transitive synchronization of logical clocks is discussed. Interrelation between the symmetry of the clock lattice and the structure of the synchronization graph is addressed. Ramsey analysis of synchronization is important for the synchronization of computers in networks, LIGO, and Virgo instruments intended for the registration of gravitational waves and GPS tame-based synchronization. Full article

This paper explores various Ramsey numbers associated with cycles with pendant edges, including the classical Ramsey number, the star-critical Ramsey number, the Gallai–Ramsey number, and the star-critical Gallai–Ramsey number. These Ramsey numbers play a crucial role in combinatorial mathematics, determining the minimum number of vertices required to guarantee specific monochromatic substructures. We establish upper and lower bounds for each of these numbers, providing new insights into their behavior for cycles with pendant edges—graphs formed by attaching additional edges to one or more vertices of a cycle. The results presented contribute to the broader understanding of Ramsey theory and serve as a foundation for future research on generalized Ramsey numbers in complex graph structures. 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

It is well known that the famous Ramsey number $R(K_3,K_3)=6$. That is, the minimum positive integer n for which every red-blue coloring of the edges of the complete graph $K_n$ results in a monochromatic triangle $K_3$ is 6. It is also known that every red-blue coloring of $K_6$ results in at least two monochromatic triangles, which need not be vertex-disjoint or edge-disjoint. This fact led to an extension of Ramsey numbers. For a graph F and a positive integer t, the vertex-disjoint Ramsey number $V{R}_t\left(F\right)$ is the minimum positive integer n such that every red-blue coloring of the edges of the complete graph $K_n$ of order n results in t pairwise vertex-disjoint monochromatic copies of subgraphs isomorphic to F, while the edge-disjoint Ramsey number $E{R}_t\left(F\right)$ is the corresponding number for edge-disjoint subgraphs. Since $V{R}_1\left(F\right)$ and $E{R}_1\left(F\right)$ are the well-known Ramsey numbers of F, these new Ramsey concepts generalize the Ramsey numbers and provide a new perspective for this classical topic in graph theory. These numbers have been investigated for the two connected graphs $K_3$ and the path $P_3$ of order 3. Here, we study these numbers for the remaining connected graphs, namely, the path $P_4$ and the star $K_{1,3}$ of size 3. We show that $V{R}_t\left(P_4\right)=4t+1$ for every positive integer t and $V{R}_t\left(K_{1,3}\right)=4t$ for every integer $t\ge 2$. For $t\le 4$, the numbers $E{R}_t\left(K_{1,3}\right)$ and $E{R}_t\left(P_4\right)$ are determined. These numbers provide information towards the goal of determining how the numbers $V{R}_t\left(F\right)$ and $E{R}_t\left(F\right)$ increase as t increases for each graph $F\in \{K_{1,3},P_4\}$. Full article

In viticulture, choosing the most suitable rootstock for a specific scion cultivar is an efficient and cost-effective way to increase yield and enhance the physicochemical characteristics of the fruit. The objective of this study was to evaluate the agronomic performance of the ‘BRS Tainá’ grapevine on different rootstocks under the conditions of the Sub-Middle São Francisco Valley. The main experimental factor consisted of eight rootstocks (IAC 313, IAC 572, IAC 766, 101-14 MgT, Paulsen 1103, Ramsey, SO4, and Teleki 5C), arranged in randomized blocks with four replicates. The experiment was conducted over four production cycles, from 2021 to 2023, in a commercial crop area in Petrolina, PE, Brazil. There were significant effects of rootstocks for the yield and number of bunches per plant, as well as berry length and firmness. ‘BRS Tainá’ achieved the highest yield (22.2 kg per plant) when grafted onto the Paulsen 1103 rootstock, which was superior to the yield on 101-14 MgT, IAC 313, and IAC 572 rootstocks. The highest number of bunches (88) was obtained with ‘BRS Tainá’ grafted on Paulsen 1103, while the lowest number (63) was obtained on IAC 572; both these rootstocks were not significantly different from the other rootstocks. For all scion–rootstock combinations, the mean values for soluble solid (SS) content, titratable acidity (TA), and the SS/TA ratio were similar to those previously described for ‘BRS Tainá’, meeting the commercialization standard. The results for the yield and number of bunches per plant indicate the suitability of grafting ‘BRS Tainá’ on Paulsen 1103 under the semi-arid tropical conditions of the São Francisco Valley. Full article

A mathematical procedure enabling the transformation of an arbitrary tessellation of a surface into a bi-colored, complete graph is introduced. Polygons constituting the tessellation are represented by vertices of the graphs. Vertices of the graphs are connected by two kinds of links/edges, namely, by a green link, when polygons have the same number of sides, and by a red link, when the polygons have a different number of sides. This procedure gives rise to a semi-transitive, complete, bi-colored Ramsey graph. The Ramsey semi-transitive number was established as $R_{trans}\left(3,3\right)=5$ Shannon entropies of the tessellation and graphs are introduced. Ramsey graphs emerging from random Voronoi and Poisson Line tessellations were investigated. The limits $\zeta =\underset{N\to \infty }{\lim }{\displaystyle \frac{N_g}{N_r}}$, where N is the total number of green and red seeds, $N_g$ and $N_r$, were found $\zeta =$ 0.272 ± 0.001 (Voronoi) and $\zeta =$ 0.47 ± 0.02 (Poisson Line). The Shannon Entropy for the random Voronoi tessellation was calculated as $S=$ 1.690 ± 0.001 and for the Poisson line tessellation as S = 1.265 ± 0.015. The main contribution of the paper is the calculation of the Shannon entropy of the random point process and the establishment of the new bi-colored Ramsey graph on top of the tessellations. 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

We studied the thermodynamic properties such as the entropy, heat ($J_Q$), and work ($J_W$) rates involved when an atom passes through a Ramsey zone, which consists of a mode field inside a low-quality factor cavity that behaves classically, promoting rotations on the atomic state. Focusing on the atom, we show that $J_W$ predominates when the atomic rotations are successful, maintaining its maximum purity as computed by the von Neumann entropy. Conversely, $J_Q$ stands out when the atomic state ceases to be pure due to its entanglement with the cavity mode. With this, we interpret the quantum-to-classical transition in light of the heat and work rates. Besides, we show that, for the cavity mode to work as a Ramsey zone (classical field), several photons (of the order of ${10}^6$) need to cross the cavity, which explains its classical behavior, even when the inside average number of photons is of the order of unity. 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 our previous work, we observed that Oncidesa Gower Ramsey ‘Honey Angel’ (HA) plants became stunned on hot summer afternoons, and the seasonal trend in solar radiation affected its production schedule by limiting flower yield and quality. The objective of this work was to study the growth and flowering characteristics of HA pseudobulbs at three stages of growth (G2–G4) in response to three types of light-emitting diode (LED) lighting treatments, including full spectrum (FS), deep red/white-medium blue (DR/W-MB), and deep red/white-low blue (DR/W-LB), for two additional time intervals. The supplementary LED lighting time intervals (S) applied daily were carried out for 1 h (4:00~5:00 a.m., as S-1) or 2 h (4:00~6:00 a.m., as S-2) from March to September, 2022. Natural light without supplemental lighting was the control. The length of pseudobulb (PL), width of pseudobulb (PW), thickness of pseudobulb (PT), length of inflorescence (FL), number of branches (FB), number of florets (FN), and days to flowering (FD) per plant were recorded andcalculated when 80% of florets became mature. Light treatments significantly affected all pseudobulb growth and flower quality traits at different Gs, especially pseudobulb length (PL) and flower number (FN) under different LED types and lighting time intervals. MB-1 treatment promoted PT at both G3 and G4, whereas MB-2 treatment increased PW at both G2 and G4. Both MB-1 and LB-1 treatments had augmented effects on PL, respectively, at G2 and G3. The PW, FL, FB, and FN increased with additional light time and reached maxima under MB-2 treatment at G4 compared to other treatments and controls. Early flowering and an increased number of flowers at G4 were observed in plants grown under MB-2 treatment. Controlling light quality and supplementary light time intervals enables the production of HA plants with the desired growth and flowering quality characteristics of the pseudobulbs. Full article