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Brownian Web
In probability theory, the Brownian web is an uncountable collection of one-dimensional coalescing Brownian motions, starting from every point in space and time. It arises as the diffusive space-time scaling limit of a collection of coalescing random walks, with one walk starting from each point of the integer lattice Z at each time. History and Basic Description What is now known as the Brownian web was first conceived by Arratia in his Ph.D. thesis and a subsequent incomplete and unpublished manuscript. Arratia studied the voter model, an interacting particle system that models the evolution of a population's political opinions. The individuals of the population are represented by the vertices of a graph, and each individual carries one of two possible opinions, represented as either 0 or 1. Independently at rate 1, each individual changes its opinion to that of a randomly chosen neighbor. The voter model is known to be dual to coalescing random walks (i.e., the random ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Probability Theory
Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set of axioms. Typically these axioms formalise probability in terms of a probability space, which assigns a measure taking values between 0 and 1, termed the probability measure, to a set of outcomes called the sample space. Any specified subset of the sample space is called an event. Central subjects in probability theory include discrete and continuous random variables, probability distributions, and stochastic processes (which provide mathematical abstractions of non-deterministic or uncertain processes or measured quantities that may either be single occurrences or evolve over time in a random fashion). Although it is not possible to perfectly predict random events, much can be said about their behavior. Two major results in probability ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Brownian Motion
Brownian motion, or pedesis (from grc, πήδησις "leaping"), is the random motion of particles suspended in a medium (a liquid or a gas). This pattern of motion typically consists of random fluctuations in a particle's position inside a fluid sub-domain, followed by a relocation to another sub-domain. Each relocation is followed by more fluctuations within the new closed volume. This pattern describes a fluid at thermal equilibrium, defined by a given temperature. Within such a fluid, there exists no preferential direction of flow (as in transport phenomena). More specifically, the fluid's overall linear and angular momenta remain null over time. The kinetic energies of the molecular Brownian motions, together with those of molecular rotations and vibrations, sum up to the caloric component of a fluid's internal energy (the equipartition theorem). This motion is named after the botanist Robert Brown, who first described the phenomenon in 1827, while looking throu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Random Walk
In mathematics, a random walk is a random process that describes a path that consists of a succession of random steps on some mathematical space. An elementary example of a random walk is the random walk on the integer number line \mathbb Z which starts at 0, and at each step moves +1 or −1 with equal probability. Other examples include the path traced by a molecule as it travels in a liquid or a gas (see Brownian motion), the search path of a foraging animal, or the price of a fluctuating stock and the financial status of a gambler. Random walks have applications to engineering and many scientific fields including ecology, psychology, computer science, physics, chemistry, biology, economics, and sociology. The term ''random walk'' was first introduced by Karl Pearson in 1905. Lattice random walk A popular random walk model is that of a random walk on a regular lattice, where at each step the location jumps to another site according to some probability distribution. In a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Voter Model Graphical Construction
Voting is a method by which a group, such as a meeting or an electorate, can engage for the purpose of making a collective decision or expressing an opinion usually following discussions, debates or election campaigns. Democracies elect holders of high office by voting. Residents of a jurisdiction represented by an elected official are called "constituents," and the constituents who choose to cast a ballot for their chosen candidate are called "voters." There are different systems for collecting votes, but while many of the systems used in decision-making can also be used as electoral systems, any which cater for proportional representation can only be used in elections. In smaller organizations, voting can occur in many different ways. Formally via ballot to elect others for example within a workplace, to elect members of political associations or to choose roles for others. Informally voting could occur as a spoken agreement or as a verbal gesture like a raised hand or ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Richard Arratia
Richard Alejandro Arratia is a mathematician noted for his work in combinatorics and probability theory. Contributions Arratia developed the ideas of interlace polynomials with Béla Bollobás and Gregory Sorkin,. found an equivalent formulation of the Stanley–Wilf conjecture as the convergence of a limit, and was the first to investigate the lengths of superpatterns of permutations. He has also written highly cited papers on the Chen–Stein method on distances between probability distributions,.. on random walks with exclusion,. and on sequence alignment In bioinformatics, a sequence alignment is a way of arranging the sequences of DNA, RNA, or protein to identify regions of similarity that may be a consequence of functional, structural, or evolutionary relationships between the sequences. Alig ...... He is a coauthor of the book ''Logarithmic Combinatorial Structures: A Probabilistic Approach''.. Education and employment Arratia earned his Ph.D. in 1979 from the U ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Voter Model
In the mathematical theory of probability, the voter model is an interacting particle system introduced by Richard A. Holley and Thomas M. Liggett in 1975. One can imagine that there is a "voter" at each point on a connected graph, where the connections indicate that there is some form of interaction between a pair of voters (nodes). The opinions of any given voter on some issue changes at random times under the influence of opinions of his neighbours. A voter's opinion at any given time can take one of two values, labelled 0 and 1. At random times, a random individual is selected and that voter's opinion is changed according to a stochastic rule. Specifically, for one of the chosen voter's neighbors is chosen according to a given set of probabilities and that individual's opinion is transferred to the chosen voter. An alternative interpretation is in terms of spatial conflict. Suppose two nations control the areas (sets of nodes) labelled 0 or 1. A flip from 0 to 1 at a given ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Interacting Particle System
In probability theory, an interacting particle system (IPS) is a stochastic process (X(t))_ on some configuration space \Omega= S^G given by a site space, a countable-infinite graph G and a local state space, a compact metric space S . More precisely IPS are continuous-time Markov jump processes describing the collective behavior of stochastically interacting components. IPS are the continuous-time analogue of stochastic cellular automata. Among the main examples are the voter model, the contact process, the asymmetric simple exclusion process (ASEP), the Glauber dynamics and in particular the stochastic Ising model. IPS are usually defined via their Markov generator giving rise to a unique Markov process using Markov semigroups and the Hille-Yosida theorem. The generator again is given via so-called transition rates c_\Lambda(\eta,\xi)>0 where \Lambda\subset G is a finite set of sites and \eta,\xi\in\Omega with \eta_i=\xi_i for all i\notin\Lambda. The rates describe ex ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Donsker's Theorem
In probability theory, Donsker's theorem (also known as Donsker's invariance principle, or the functional central limit theorem), named after Monroe D. Donsker, is a functional extension of the central limit theorem. Let X_1, X_2, X_3, \ldots be a sequence of independent and identically distributed (i.i.d.) random variables with mean 0 and variance 1. Let S_n:=\sum_^n X_i. The stochastic process S:=(S_n)_ is known as a random walk. Define the diffusively rescaled random walk (partial-sum process) by : W^(t) := \frac, \qquad t\in ,1 The central limit theorem asserts that W^(1) converges in distribution to a standard Gaussian random variable W(1) as n\to\infty. Donsker's invariance principle extends this convergence to the whole function W^:=(W^(t))_. More precisely, in its modern form, Donsker's invariance principle states that: As random variables taking values in the Skorokhod space \mathcal ,1/math>, the random function W^ converges in distribution to a standard Brownian moti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Coalescing Random Walks
Coalescence may refer to: * Coalescence (chemistry), the process by which two or more separate masses of miscible substances seem to "pull" each other together should they make the slightest contact * Coalescence (computer science), the merging of adjacent blocks of memory to fill gaps caused by deallocated memory * Coalescence (genetics), the merging of genetic lineages backwards to a most recent common ancestor * Coalescence (linguistics), the merging of two or more phonological segments into one * Coalescence (physics), the merging of two or more droplets, bubbles or particles into one * COALESCE, a SQL command that selects the first non-null from a range of values * In geography, the process by which urban sprawl produces a linear conurbation A conurbation is a region comprising a number of metropolises, cities, large towns, and other urban areas which through population growth and physical expansion, have merged to form one continuous urban or industrially developed are ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bálint Tóth
Bálint Tόth (born 1955, Cluj/Kolozsvár/Klausenburg) is a Hungarian mathematician whose work concerns probability theory, stochastic process and probabilistic aspects of mathematical physics. He obtained PhD in 1988 from the Hungarian Academy of Sciences, worked as senior researcher at the Institute of Mathematics of the HAS and as professor of mathematics at TU Budapest. He holds the Chair of Probability at the University of Bristol and is a research professor at the Alfréd Rényi Institute of Mathematics, Budapest. He has worked on microscopic models of Brownian motion, quantum spin systems, limit theorems for random walks with long memory, and non-conventional stochastic processes, hydrodynamic limits, etc. In particular, Tόth contributed to the theory of self-interacting motions, that is, motions that are "reinforced", "self-avoiding" or "self-repellent". In collaboration with Wendelin Werner he constructed the random geometric object later called the Brownian web. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wendelin Werner
Wendelin Werner (born 23 September 1968) is a German-born French mathematician working on random processes such as self-avoiding random walks, Brownian motion, Schramm–Loewner evolution, and related theories in probability theory and mathematical physics. In 2006, at the 25th International Congress of Mathematicians in Madrid, Spain he received the Fields Medal "for his contributions to the development of stochastic Loewner evolution, the geometry of two-dimensional Brownian motion, and conformal field theory". He is professor at ETH Zürich. Biography Werner was born on 23 September 1968 in Cologne, West Germany. His parents moved to France when he was nine months old and he became a French citizen in 1977. After a '' classe préparatoire'' at Lycée Hoche in Versailles, he studied at École Normale Supérieure from 1987 to 1991. His 1993 doctorate was written at the Université Pierre-et-Marie-Curie and supervised by Jean-François Le Gall. Werner was a researcher at the C ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
True Self-repelling Motion
True most commonly refers to truth, the state of being in congruence with fact or reality. True may also refer to: Places * True, West Virginia, an unincorporated community in the United States * True, Wisconsin, a town in the United States * True, a townland in County Tyrone, Northern Ireland People * True (singer) (stylized as TRUE), the stage name of Japanese singer Miho Karasawa * True (surname) * True O'Brien (born 1994), an American model and actress Arts, entertainment, and media Music Albums * ''True'' (Avicii album), 2013 * ''True'' (EP), a 2012 EP by Solange Knowles * ''True'' (L'Arc-en-Ciel album), 1996 * ''True'' (Roy Montgomery and Chris Heaphy album), 1999 * ''True'' (Mika Nakashima album), 2002 * ''True'' (Spandau Ballet album), 1983 * ''True'' (TrinityRoots album), 2001 * ''True'' (TRU album), 1995 Songs * "True" (Brandy song), by Brandy Norwood from ''Human'' (2008) * "True" (Concrete Blonde song), 1987 * "True" (Ryan Cabrera song), 2004 * "True" ( ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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