Finite Number Of Colors Research Articles

Optimal [formula omitted]-algorithms for rendezvous of asynchronous mobile robots with external-lights

We study the problem Rendezvous for two autonomous mobile robots in asynchronous settings with persistent memory called light. It is well known that Rendezvous is impossible in a basic model when robots have no lights, even if the system is semi-synchronous. On the other hand, Rendezvous is possible if robots have lights of various types with a constant number of colors [10,21]. With external-lights, robots can only observe the state of the lights of other robots. With internal-lights, robots can only observe their own light, and full-lights combine both. This paper focuses on robots with external-lights in asynchronous settings and considers a particular class of algorithms called L-algorithms, where an L-algorithm computes a destination based only on the current colors of observable lights. When considering L-algorithms, Rendezvous can be solved by robots with full-lights and 3 colors in general asynchronous settings (called ASYNC) and the number of colors is optimal under these assumptions. In contrast, there exist no L-algorithms in ASYNC with external-lights regardless of the number of colors [10].This paper, extending the impossibility result, shows that there exists no L-algorithm in so-called LC-1-Bounded ASYNC with external-lights regardless of the number of colors, where LC-1-Bounded ASYNC is a proper subset of ASYNC in which no robot can execute more than 1 Look operation between the Look and its subsequent Compute operations of another robot. We also show that LC-1-Bounded ASYNC is the minimal subclass in which no L-algorithms with external-lights exist. That is, Rendezvous can be solved by L-algorithms using external-lights with a finite number of colors in LC-0-Bounded ASYNC (equivalently LC-atomic ASYNC). Furthermore, we show that the algorithms are optimal in the number of colors they use.

Delocalized SYZ mirrors and confronting top\u2013down SU(3)-structure holographic meson masses at finite g and N_c with P(article) D(ata) G(roup) values

Meson spectroscopy at finite gauge coupling – whereat any perturbative QCD computation would break down – and finite number of colors, from a top–down holographic string model, has thus far been entirely missing in the literature. This paper fills this gap. Using the delocalized type IIA SYZ mirror (with SU(3) structure) of the holographic type IIB dual of large-N thermal QCD of Mia et al. (Nucl Phys B 839:187. arXiv:0902.1540 [hep-th], 2010) as constructed in Dhuria and Misra (JHEP 1311:001. arXiv:1306.4339 [hep-th], 2013) at finite coupling and number of colors (N_c = number of D5(overline{D5})-branes wrapping a vanishing two-cycle in the top–down holographic construct of Mia et al. (Nucl Phys B 839:187. arXiv:0902.1540 [hep-th], 2010) = mathcal{O}(1) in the IR in the MQGP limit of Dhuria and Misra (JHEP 1311:001. arXiv:1306.4339 [hep-th], 2013) at the end of a Seiberg-duality cascade), we obtain analytical (not just numerical) expressions for the vector and scalar meson spectra and compare our results with previous calculations of Sakai and Sugimoto (Prog Theor Phys 113:843. doi:10.1143/PTP.113.843. arXiv:hep-th/0412141, 2005) and Dasgupta et al. (JHEP 1507:122. doi:10.1007/JHEP07(2015)122. arXiv:1409.0559 [hep-th], 2015), and we obtain a closer match with the Particle Data Group (PDG) results of Olive et al. (Particle Data Group) (Chin Phys C 38:090001, 2014). Through explicit computations, we verify that the vector and scalar meson spectra obtained by the gravity dual with a black hole for all temperatures (small and large) are nearly isospectral with the spectra obtained by a thermal gravity dual valid for only low temperatures; the isospectrality is much closer for vector mesons than scalar mesons. The black-hole gravity dual (with a horizon radius smaller than the deconfinement scale) also provides the expected large-N suppressed decrease in vector meson mass with increase of temperature.

Uniform Sampling of Subshifts of Finite Type on Grids and Trees

Let us color the vertices of the grid ℤd or the infinite regular tree 𝕋d, using a finite number of colors, with the constraint that some predefined pairs of colors are not allowed for adjacent vertices. The set of admissible colorings is called a nearest-neighbor subshift of finite type (SFT). We study “uniform” probability measures on SFT, with the motivation of having an insight into “typical” admissible configurations. We recall the known results on uniform measures on SFT on grids and we complete the picture by presenting some contributions to the description of uniform measures on SFT on 𝕋d. Then we focus on the problem of uniform random sampling of configurations of SFT. We propose a first method based on probabilistic cellular automata, which is valid under some restrictive conditions. Then we concentrate on the case of SFT on ℤ for which we propose several alternative sampling methods.

Nonrepetitive and pattern-free colorings of the plane

A classic result of Thue states that using 3 symbols one can construct an infinite nonrepetitive sequence, that is, a sequence without two identical adjacent blocks. In this paper, we present Thue-type results concerning coloring of the Euclidean plane, related to the famous Hadwiger–Nelson problem. This old problem asks for the chromatic number of the unit distance graph of the plane, the infinite graph with the set of vertices of R2 and edges between points at distance 1.We solve a problem posed by Grytczuk (2008) by showing that any coloring of the unit distance graph of R2 in which the sequence of colors of every simple path is nonrepetitive needs more than countable number of colors. Grytczuk, Kosiński and Zmarz (2016) introduced a relaxation of this problem, by restricting the nonrepetitiveness condition to a certain class of paths. Let a path in the unit distance graph of R2 be called line path if it consists of collinear points. A coloring of R2 is line nonrepetitive if the sequence of colors on every line path is nonrepetitive. We present a line nonrepetitive 18-coloring of R2, which improves the result of Grytczuk, Kosiński and Zmarz with 36 colors. Furthermore, we construct a line nonrepetitive coloring, which additionally avoids palindromes and uses 32 colors.We also consider a natural generalization of this problem to other patterns. We say that a patternq (finite sequence over a set of variables E) occurs in a sequence w (over an alphabet A) if there exists a substitution f from E to the set of nonempty sequences over A such that f(q) is a block in w. A sequence wavoids a pattern q if q does not occur in w. For a pattern q, a coloring of R2 is called linep-free if the sequence of colors on every line path avoids q. We present a 9-coloring of R2 which is line ααα-free and line αβαβ-free at once. Moreover, we give a non-constructive result for a class of patterns. Namely, let q be a pattern with no variable occurring exactly once or satisfying ℓ⩾2d, where ℓ is the length of q and d is the number of variables in q. Then there exists a line q-free coloring of R2 using a finite number of colors. In fact, this result holds even if (instead of taking into account only line paths) we consider all sequences of points in R2 with consecutive distances in some fixed interval [a,b] and without any pair of vertices at distance smaller than a fixed ε>0.

Local Finiteness, Distinguishing Numbers, and Tucker's Conjecture

A distinguishing colouring of a graph is a colouring of the vertex set such that no non-trivial automorphism preserves the colouring. Tucker conjectured that if every non-trivial automorphism of a locally finite graph moves infinitely many vertices, then there is a distinguishing 2-colouring. We show that the requirement of local finiteness is necessary by giving a non-locally finite graph for which no finite number of colours suffices.

Ramsey’s theorem for colors from a metric space

The Ramsey theorem says that for any countably infinite undirected clique whose edges are colored by a finite number of colors, there is an infinite subclique whose edges are colored by a single color. In this note, we generalize the theorem to a situation where the colors form a compact metric space.

Percolation Transition in Yang-Mills Matter at a Finite Number of Colors

We examine baryonic matter at a quark chemical potential of the order of the confinement scale μ(q)∼Λ(QCD). In this regime, quarks are supposed to be confined but baryons are close to the "tightly packed limit" where they nearly overlap in configuration space. We show that this system will exhibit a percolation phase transition when varied in the number of colors N(c): at high N(c), large distance correlations at the quark level are possible even if the quarks are essentially confined. At low N(c), this does not happen. We discuss the relevance of this for dense nuclear matter, and argue that our results suggest a new "phase transition," varying N(c) at constant μ(q).

A $$ \mathcal{N} = 8 $$ action for multiple M2-branes with an arbitrary number of colors

We obtain a U(M) action for supermembranes with central charges in the Light Cone Gauge (LCG). The theory realizes all of the symmetries and constraints of the supermembrane together with the invariance under a U(M) gauge group with M arbitrary. The worldvolume action has (LCG) N=8 supersymmetry and it corresponds to M parallel supermembranes minimally immersed on the target M9xT2 (MIM2). In order to ensure the invariance under the symmetries and to close the corresponding algebra, a star-product determined by the central charge condition is introduced. It is constructed with a nonconstant symplectic two-form where curvature terms are also present. The theory is in the strongly coupled gauge-gravity regime. At low energies, the theory enters in a decoupling limit and it is described by an ordinary N=8 SYM in the IR phase for any number of M2-branes. We analyze also other limits of the theory: when M goes to we obtain a condensate of M2-branes. If we consider, instead of the exact action, its matrix regularized version, for suitable large N but with a finite number of colors, the algebra of the first class-constraints also closes with the ordinary product and we obtain a regularized 2+1D SU(M) SYM theory in the IR phase that is coupled to the SU(N) regularized supermembrane action. The supersymmetric regularized theory with the ordinary product has purely discrete spectrum. Our theory with the star-product is a full-fledged description of multiple M2-branes minimally immersed in distinction with other constructions that represent multiple M2-branes in the low energy approximation.

Large Deviations Principle for Occupancy Problems with Colored Balls

A large deviations principle (LDP), demonstrated for occupancy problems with indistinguishable balls, is generalized to the case in which balls are distinguished by a finite number of colors. The colors of the balls are chosen independently from the occupancy process itself. There are r balls thrown into n urns with the probability of a ball entering a given urn being 1/n (i.e. Maxwell-Boltzmann statistics). The LDP applies with the scale parameter, n, tending to infinity and r increasing proportionally. The LDP holds under mild restrictions, the key one being that the coloring process by itself satisfies an LDP. This includes the important special cases of deterministic coloring patterns and colors chosen with fixed probabilities independently for each ball.

Large Deviations Principle for Occupancy Problems with Colored Balls

A large deviations principle (LDP), demonstrated for occupancy problems with indistinguishable balls, is generalized to the case in which balls are distinguished by a finite number of colors. The colors of the balls are chosen independently from the occupancy process itself. There arerballs thrown intonurns with the probability of a ball entering a given urn being 1/n(i.e. Maxwell-Boltzmann statistics). The LDP applies with the scale parameter,n, tending to infinity andrincreasing proportionally. The LDP holds under mild restrictions, the key one being that the coloring process by itself satisfies an LDP. This includes the important special cases of deterministic coloring patterns and colors chosen with fixed probabilities independently for each ball.

SU(Nc→∞)lattice data and degrees of freedom of the QCD string

Lattice simulation data on the critical temperature and long-distance potential, that probe the degrees of freedom of the QCD string, are critically reviewed. It is emphasized that comparison of experimental or SU(N_c) lattice data, at finite number of colors N_c, with free string theory can be misleading due to string interactions. Large-N_c extrapolation of pure lattice gauge theory data, in both 3 and 4 dimensions, indicates that there are more worldsheet degrees of freedom than the purely massless transverse ones of the free Nambu-Goto string. The extra variables are consistent with massive modes of oscillation that effectively contribute like c ~ 1/2 conformal degrees of freedom to highly excited states. As a concrete example, the highly excited spectrum of the Chodos-Thorn relativistic string in 1+1 dimensions is analyzed, where there are no transverse oscillations. It is found that the asymptotic density of states for this model is characteristic of a c=1/2 conformal worldsheet theory. The observations made here should also constrain the backgrounds of holographic string models for QCD.

A Characterization of Model Multi-colour Sets

Model sets are always Meyer sets, but not vice-versa. This article is about characterizing model sets (general and regular) amongst the Meyer sets in terms of two associated dynamical systems. These two dynamical systems describe two very different topologies on point sets, one local and one global. In model sets these two are strongly interconnected and this connection is essentially definitive. The paper is set in the context of multi-colour sets, that is to say, point sets in which points come in a finite number of colours, that are loosely coupled together by finite local complexity.

On a metric generalization of ramsey’s theorem

An increasing sequence of reals $x=\langle x_i: i > \omega\rangle$ is simple if all ``gaps'''' $x_{i+1}-x_i$are different. Two simple sequences $x$ and $y$ are distance similar if $x_{i+1}-x_i > x_{j+1}-x_j$ if and only if $y_{i+1}-y_i > y_{j+1}-y_j$ for all $i$ and $j$. Given any bounded simple sequence $x$ and any coloring of the pairs of rational numbers by a finite number of colors, we prove that there is a sequence $y$ distance similar to $x$ all of whose pairs are of the same color. We also consider many related problems and generalizations.

Topological and metric properties of infinite clusters in stationary two-dimensional site percolation

A dependent percolation model is a random coloring of the two-lattice. It is assumed that there are a finite number of colors and that the coloring is translation invariant. Each color defines a random subset of the lattice. Connected components of this subset are called clusters. This paper gives a classification of the infinite behavior of these clusters. In particular, it is shown that the lattice is divided into disjoint infinite strips, lying adjacently. Each strip is either composed of an infinite cluster together with isolated finite clusters or else is entirely composed of finite clusters. Examples of the various types of behavior are constructed.