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Markov Chains And Mixing Times
''Markov Chains and Mixing Times'' is a book on Markov chain mixing times. The second edition was written by David A. Levin, and Yuval Peres. Elizabeth Wilmer was a co-author on the first edition and is credited as a contributor to the second edition. The first edition was published in 2009 by the American Mathematical Society, with an expanded second edition in 2017. Background A Markov chain is a stochastic process defined by a set of states and, for each state, a probability distribution on the states. Starting from an initial state, it follows a sequence of states where each state in the sequence is chosen randomly from the distribution associated with the previous state. In that sense, it is "memoryless": each random choice depends only on the current state, and not on the past history of states. Under mild restrictions, a Markov chain with a finite set of states will have a stationary distribution that it converges to, meaning that, after a sufficiently large number of steps, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Markov Chains And Mixing Times
''Markov Chains and Mixing Times'' is a book on Markov chain mixing times. The second edition was written by David A. Levin, and Yuval Peres. Elizabeth Wilmer was a co-author on the first edition and is credited as a contributor to the second edition. The first edition was published in 2009 by the American Mathematical Society, with an expanded second edition in 2017. Background A Markov chain is a stochastic process defined by a set of states and, for each state, a probability distribution on the states. Starting from an initial state, it follows a sequence of states where each state in the sequence is chosen randomly from the distribution associated with the previous state. In that sense, it is "memoryless": each random choice depends only on the current state, and not on the past history of states. Under mild restrictions, a Markov chain with a finite set of states will have a stationary distribution that it converges to, meaning that, after a sufficiently large number of steps, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graph (discrete Mathematics)
In discrete mathematics, and more specifically in graph theory, a graph is a structure amounting to a Set (mathematics), set of objects in which some pairs of the objects are in some sense "related". The objects correspond to mathematical abstractions called ''Vertex (graph theory), vertices'' (also called ''nodes'' or ''points'') and each of the related pairs of vertices is called an ''edge'' (also called ''link'' or ''line''). Typically, a graph is depicted in diagrammatic form as a set of dots or circles for the vertices, joined by lines or curves for the edges. Graphs are one of the objects of study in discrete mathematics. The edges may be directed or undirected. For example, if the vertices represent people at a party, and there is an edge between two people if they shake hands, then this graph is undirected because any person ''A'' can shake hands with a person ''B'' only if ''B'' also shakes hands with ''A''. In contrast, if an edge from a person ''A'' to a person ''B'' m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Coupling From The Past
Among Markov chain Monte Carlo (MCMC) algorithms, coupling from the past is a method for sampling from the stationary distribution of a Markov chain. Contrary to many MCMC algorithms, coupling from the past gives in principle a perfect sample from the stationary distribution. It was invented by James Propp and David Wilson in 1996. The basic idea Consider a finite state irreducible aperiodic Markov chain M with state space S and (unique) stationary distribution \pi (\pi is a probability vector). Suppose that we come up with a probability distribution \mu on the set of maps f:S\to S with the property that for every fixed s\in S, its image f(s) is distributed according to the transition probability of M from state s. An example of such a probability distribution is the one where f(s) is independent from f(s') whenever s\ne s', but it is often worthwhile to consider other distributions. Now let f_j for j\in\mathbb Z be independent samples from \mu. Suppose that x is chosen randomly ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jim Propp
James Gary Propp is a professor of mathematics at the University of Massachusetts Lowell. Education and career In high school, Propp was one of the national winners of the United States of America Mathematical Olympiad (USAMO), and an alumnus of the Hampshire College Summer Studies in Mathematics. Propp obtained his AB in mathematics in 1982 at Harvard. After advanced study at Cambridge, he obtained his PhD from the University of California at Berkeley. He has held professorships at seven universities, including Harvard, MIT, the University of Wisconsin, and the University of Massachusetts Lowell. Mathematical research Propp is the co-editor of the book ''Microsurveys in Discrete Probability'' (1998) and has written more than fifty journal articles on game theory, combinatorics and probability, and recreational mathematics. He lectures extensively and has served on the Mathematical Olympiad Committee of the Mathematical Association of America, which sponsors the USAMO. In the e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Critical Point (thermodynamics)
In thermodynamics, a critical point (or critical state) is the end point of a phase equilibrium curve. The most prominent example is the liquid–vapor critical point, the end point of the pressure–temperature curve that designates conditions under which a liquid and its vapor can coexist. At higher temperatures, the gas cannot be liquefied by pressure alone. At the critical point, defined by a ''critical temperature'' ''T''c and a ''critical pressure'' ''p''c, phase boundaries vanish. Other examples include the liquid–liquid critical points in mixtures, and the ferromagnet–paramagnet transition (Curie temperature) in the absence of an external magnetic field. Liquid–vapor critical point Overview For simplicity and clarity, the generic notion of ''critical point'' is best introduced by discussing a specific example, the vapor–liquid critical point. This was the first critical point to be discovered, and it is still the best known and most studied one. The figu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Martingale (probability Theory)
In probability theory, a martingale is a sequence of random variables (i.e., a stochastic process) for which, at a particular time, the conditional expectation of the next value in the sequence is equal to the present value, regardless of all prior values. History Originally, '' martingale'' referred to a class of betting strategies that was popular in 18th-century France. The simplest of these strategies was designed for a game in which the gambler wins their stake if a coin comes up heads and loses it if the coin comes up tails. The strategy had the gambler double their bet after every loss so that the first win would recover all previous losses plus win a profit equal to the original stake. As the gambler's wealth and available time jointly approach infinity, their probability of eventually flipping heads approaches 1, which makes the martingale betting strategy seem like a sure thing. However, the exponential growth of the bets eventually bankrupts its users due to f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Earth Mover's Distance
In statistics, the earth mover's distance (EMD) is a measure of the distance between two probability distributions over a region ''D''. In mathematics, this is known as the Wasserstein metric. Informally, if the distributions are interpreted as two different ways of piling up a certain amount of earth (dirt) over the region ''D'', the EMD is the minimum cost of turning one pile into the other; where the cost is assumed to be the amount of dirt moved times the distance by which it is moved. The above definition is valid only if the two distributions have the same integral (informally, if the two piles have the same amount of dirt), as in normalized histograms or probability density functions. In that case, the EMD is equivalent to the 1st Mallows distance or 1st Wasserstein distance between the two distributions. Theory Assume that we have a set of points in \mathbb^d (dimension d). Instead of assigning one distribution to the set of points, we can cluster them and represe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Expander Graph
In graph theory, an expander graph is a sparse graph that has strong connectivity properties, quantified using vertex, edge or spectral expansion. Expander constructions have spawned research in pure and applied mathematics, with several applications to complexity theory, design of robust computer networks, and the theory of error-correcting codes. Definitions Intuitively, an expander graph is a finite, undirected multigraph in which every subset of the vertices that is not "too large" has a "large" boundary. Different formalisations of these notions give rise to different notions of expanders: ''edge expanders'', ''vertex expanders'', and ''spectral expanders'', as defined below. A disconnected graph is not an expander, since the boundary of a connected component is empty. Every connected graph is an expander; however, different connected graphs have different expansion parameters. The complete graph has the best expansion property, but it has largest possible degree. Informal ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lamplighter Group
In mathematics, the lamplighter group ''L'' of group theory is the restricted wreath product \mathbf_2 \wr \mathbf Z Introduction The name of the group comes from viewing the group as acting on a doubly infinite sequence of street lamps \dots,l_,l_,l_0,l_1,l_2,l_3,\dots each of which may be on or off, and a lamplighter standing at some lamp l_k. An equivalent description for this, called the base group B of L is :B=\bigoplus_^\mathbf_2, an infinite direct sum of copies of the cyclic group \mathbf Z_2 where 0 corresponds to a light that is off and 1 corresponds to a light that is on, and the direct sum is used to ensure that only finitely many lights are on at once. An element of \mathbf Z gives the position of the lamplighter, and B to encode which bulbs are illuminated. There are two generators for the group: the generator ''t'' increments ''k'', so that the lamplighter moves to the next lamp (''t'' -1 decrements ''k''), while the generator ''a'' means that the state of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Asymmetric Simple Exclusion Process
In probability theory, the asymmetric simple exclusion process (ASEP) is an interacting particle system introduced in 1970 by Frank Spitzer. Many articles have been published on it in the physics and mathematics literature since then, and it has become a "default stochastic model for transport phenomena". The process with parameters p, q \geqslant 0,\, p + q = 1 is a continuous-time Markov process on S = \lbrace 0, 1\rbrace^, the 1s being thought of as particles and the 0s as holes. Each particle waits a random exponent mean one amount of time and then attempts a jump, one site to the right with probability p and one site to the left with probability q. However, the jump is performed only if there is no particle at the target site. Otherwise, nothing happens and the particle waits another exponential time. All particles are doing this independently of each other. The model is related to the Kardar–Parisi–Zhang equation in the weakly asymmetric limit, i.e. when p-q tends to z ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chromosomal Rearrangement
In genetics, a chromosomal rearrangement is a mutation that is a type of chromosome abnormality involving a change in the structure of the native chromosome. Such changes may involve several different classes of events, like deletions, duplications, inversions, and translocations. Usually, these events are caused by a breakage in the DNA double helices at two different locations, followed by a rejoining of the broken ends to produce a new chromosomal arrangement of genes, different from the gene order of the chromosomes before they were broken. Structural chromosomal abnormalities are estimated to occur in around 0.5% of newborn infants. Some chromosomal regions are more prone to rearrangement than others and thus are the source of genetic diseases and cancer. This instability is usually due to the propensity of these regions to misalign during DNA repair, exacerbated by defects of the appearance of replication proteins (like FEN1 or Pol δ) that ubiquitously affect the integri ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ising Model
The Ising model () (or Lenz-Ising model or Ising-Lenz model), named after the physicists Ernst Ising and Wilhelm Lenz, is a mathematical model of ferromagnetism in statistical mechanics. The model consists of discrete variables that represent magnetic dipole moments of atomic "spins" that can be in one of two states (+1 or −1). The spins are arranged in a graph, usually a lattice (where the local structure repeats periodically in all directions), allowing each spin to interact with its neighbors. Neighboring spins that agree have a lower energy than those that disagree; the system tends to the lowest energy but heat disturbs this tendency, thus creating the possibility of different structural phases. The model allows the identification of phase transitions as a simplified model of reality. The two-dimensional square-lattice Ising model is one of the simplest statistical models to show a phase transition. The Ising model was invented by the physicist , who gave it as a prob ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
