|
Percolation Theory
In statistical physics and mathematics, percolation theory describes the behavior of a network when nodes or links are added. This is a geometric type of phase transition, since at a critical fraction of addition the network of small, disconnected clusters merge into significantly larger connected, so-called spanning clusters. The applications of percolation theory to materials science and in many other disciplines are discussed here and in the articles network theory and percolation. Introduction A representative question (and the source of the name) is as follows. Assume that some liquid is poured on top of some porous material. Will the liquid be able to make its way from hole to hole and reach the bottom? This physical question is modelled mathematically as a three-dimensional network of vertices, usually called "sites", in which the edge or "bonds" between each two neighbors may be open (allowing the liquid through) with probability , or closed with probability , and th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Statistical Physics
Statistical physics is a branch of physics that evolved from a foundation of statistical mechanics, which uses methods of probability theory and statistics, and particularly the Mathematics, mathematical tools for dealing with large populations and approximations, in solving physical problems. It can describe a wide variety of fields with an inherently stochastic nature. Its applications include many problems in the fields of physics, biology, chemistry, and neuroscience. Its main purpose is to clarify the properties of matter in aggregate, in terms of physical laws governing atomic motion. Statistical mechanics develop the Phenomenology (particle physics), phenomenological results of thermodynamics from a probabilistic examination of the underlying microscopic systems. Historically, one of the first topics in physics where statistical methods were applied was the field of classical mechanics, which is concerned with the motion of particles or objects when subjected to a force. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kolmogorov's Zero–one Law
In probability theory, Kolmogorov's zero–one law, named in honor of Andrey Nikolaevich Kolmogorov, specifies that a certain type of event, namely a ''tail event of independent σ-algebras'', will either almost surely happen or almost surely not happen; that is, the probability of such an event occurring is zero or one. Tail events are defined in terms of countably infinite families of σ-algebras. For illustrative purposes, we present here the special case in which each sigma algebra is generated by a random variable X_k for k\in\mathbb N. Let \mathcal be the sigma-algebra generated jointly by all of the X_k. Then, a tail event F \in \mathcal is an event which is probabilistically independent of each finite subset of these random variables. (Note: F belonging to \mathcal implies that membership in F is uniquely determined by the values of the X_k, but the latter condition is strictly weaker and does not suffice to prove the zero-one law.) For example, the event that ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Front De Percolation
Front may refer to: Arts, entertainment, and media Films * ''The Front'' (1943 film), a 1943 Soviet drama film * ''The Front'', 1976 film Music *The Front (band), an American rock band signed to Columbia Records and active in the 1980s and early 1990s *The Front (Canadian band), a Canadian studio band from the 1980s Periodicals * ''Front'' (magazine), a British men's magazine * '' Front Illustrated Paper'', a publication of the Yugoslav People's Army Television * Front TV, a Toronto broadcast design and branding firm * "The Front" (''The Blacklist''), a 2014 episode of the TV series ''The Blacklist'' * "The Front" (''The Simpsons''), a 1993 episode of the TV series ''The Simpsons'' Military * Front (military), a geographical area where armies are engaged in conflict * Front (military formation), roughly, an army group, especially in eastern Europe Places * Front, California, former name of Brown, California * Front, Piedmont, an Italian municipality * The Front, now part ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tree (graph Theory)
In graph theory In mathematics, graph theory is the study of ''graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of '' vertices'' (also called ''nodes'' or ''points'') which are conne ..., a tree is an undirected graph in which any two Vertex (graph theory), vertices are connected by ''exactly one'' Path (graph theory), path, or equivalently a Connected graph, connected Cycle (graph theory), acyclic undirected graph. A forest is an undirected graph in which any two vertices are connected by ''at most one'' path, or equivalently an acyclic undirected graph, or equivalently a Disjoint union of graphs, disjoint union of trees. A polytreeSee . (or directed tree or oriented treeSee .See . or singly connected networkSee .) is a directed acyclic graph (DAG) whose underlying undirected graph is a tree. A polyforest (or directed forest or oriented forest) is a directed acyclic graph whose underlying undirecte ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Coordination Number
In chemistry, crystallography, and materials science, the coordination number, also called ligancy, of a central atom in a molecule or crystal is the number of atoms, molecules or ions bonded to it. The ion/molecule/atom surrounding the central ion/molecule/atom is called a ligand. This number is determined somewhat differently for molecules than for crystals. For molecules and polyatomic ions the coordination number of an atom is determined by simply counting the other atoms to which it is bonded (by either single or multiple bonds). For example, r(NH3)2Cl2Br2sup>− has Cr3+ as its central cation, which has a coordination number of 6 and is described as ''hexacoordinate''. The common coordination numbers are 4, 6 and 8. Molecules, polyatomic ions and coordination complexes In chemistry, coordination number, defined originally in 1893 by Alfred Werner, is the total number of neighbors of a central atom in a molecule or ion. The concept is most commonly applied to coordin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bethe Lattice
In statistical mechanics and mathematics, the Bethe lattice (also called a regular tree) is an infinite connected cycle-free graph where all vertices have the same number of neighbors. The Bethe lattice was introduced into the physics literature by Hans Bethe in 1935. In such a graph, each node is connected to ''z'' neighbors; the number ''z'' is called either the coordination number or the degree, depending on the field. Due to its distinctive topological structure, the statistical mechanics of lattice models on this graph are often easier to solve than on other lattices. The solutions are related to the often used Bethe approximation for these systems. Basic Properties When working with the Bethe lattice, it is often convenient to mark a given vertex as the root, to be used as a reference point when considering local properties of the graph. Sizes of Layers Once a vertex is marked as the root, we can group the other vertices into layers based on their distance from the ro ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Harry Kesten
Harry Kesten (November 19, 1931 – March 29, 2019) was an American mathematician best known for his work in probability, most notably on random walks on groups and graphs, random matrices, branching processes, and percolation theory. Biography Kesten grew up in the Netherlands, where he moved with his parents in 1933 to escape the Nazis. He received his PhD in 1958 at Cornell University under supervision of Mark Kac. He was an instructor at Princeton University and the Hebrew University before returning to Cornell in 1961. Kesten died on March 29, 2019, in Ithaca at the age of 87. Mathematical work Kesten's work includes many fundamental contributions across almost the whole of probability, including the following highlights. *''Random walks on groups.'' In his 1958 PhD thesis, Kesten studied symmetric random walks on countable groups ''G'' generated by a jump distribution with support ''G''. He showed that the spectral radius equals the exponential decay rate of the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Square Lattice
In mathematics, the square lattice is a type of lattice in a two-dimensional Euclidean space. It is the two-dimensional version of the integer lattice, denoted as . It is one of the five types of two-dimensional lattices as classified by their symmetry groups; its symmetry group in IUC notation as , Coxeter notation as , and orbifold notation as . Two orientations of an image of the lattice are by far the most common. They can conveniently be referred to as the upright square lattice and diagonal square lattice; the latter is also called the centered square lattice.. They differ by an angle of 45°. This is related to the fact that a square lattice can be partitioned into two square sub-lattices, as is evident in the colouring of a checkerboard. Symmetry The square lattice's symmetry category is wallpaper group . A pattern with this lattice of translational symmetry cannot have more, but may have less symmetry than the lattice itself. An upright square lattice can be viewed ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
John Hammersley
John Michael Hammersley, (21 March 1920 – 2 May 2004) was a British mathematician best known for his foundational work in the theory of self-avoiding walks and percolation theory. Early life and education Hammersley was born in Helensburgh in Dunbartonshire, and educated at Sedbergh School. He started reading mathematics at Emmanuel College, Cambridge but was called up to join the Royal Artillery in 1941. During his time in the army he worked on ballistics. He graduated in mathematics in 1948. He never studied for a PhD but was awarded an ScD by Cambridge University and a DSc by Oxford University in 1959. Academic career With Jillian Beardwood and J.H. Halton, Hammersley is known for the Beardwood-Halton-Hammersley Theorem. Published by the Cambridge Philosophical Society in a 1959 article entitled “The Shortest Path Through Many Points,” the theorem provides a practical solution to the “traveling salesman problem.” He held a number of positions, both in and outs ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Monte Carlo Method
Monte Carlo methods, or Monte Carlo experiments, are a broad class of computational algorithms that rely on repeated random sampling to obtain numerical results. The underlying concept is to use randomness to solve problems that might be deterministic in principle. They are often used in physical and mathematical problems and are most useful when it is difficult or impossible to use other approaches. Monte Carlo methods are mainly used in three problem classes: optimization, numerical integration, and generating draws from a probability distribution. In physics-related problems, Monte Carlo methods are useful for simulating systems with many coupled degrees of freedom, such as fluids, disordered materials, strongly coupled solids, and cellular structures (see cellular Potts model, interacting particle systems, McKean–Vlasov processes, kinetic models of gases). Other examples include modeling phenomena with significant uncertainty in inputs such as the calculation of ris ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rosalind Franklin
Rosalind Elsie Franklin (25 July 192016 April 1958) was a British chemist and X-ray crystallographer whose work was central to the understanding of the molecular structures of DNA (deoxyribonucleic acid), RNA (ribonucleic acid), viruses, coal, and graphite. Although her works on coal and viruses were appreciated in her lifetime, her contributions to the discovery of the structure of DNA were largely unrecognized during her life, for which she has been variously referred to as the "wronged heroine", the "dark lady of DNA", the "forgotten heroine", a "feminist icon", and the "Sylvia Plath of molecular biology". She graduated in 1941 with a degree in natural sciences from Newnham College, Cambridge, and then enrolled for a PhD in physical chemistry under Ronald George Wreyford Norrish, the 1920 Chair of Physical Chemistry at the University of Cambridge. Disappointed by Norrish's lack of enthusiasm, she took up a research position under the British Coal Utilisation Research Ass ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
British Coal Utilisation Research Association
British Coal Utilisation Research Association (BCURA) was a non-profit association of industrial companies, incorporated 23 April 1938 and dissolved 24 February 2015. History It was founded in 1938, with an assured income of £25000 per year for five years, supplied by the Mining Association of Great Britain and a grant from the government Department of Scientific and Industrial Research, establishing a research station in West Brompton. It was formed from the research department of the Combustion (''formerly Coal-burning'') Appliance Manufacturer's Association becoming a separate entity. Laboratories were also later established in Leatherhead. The first Director was John G. Bennett. During the Second World War it developed small units for the manufacture of producer gas from coal to use in vehicles in place of petrol. A £1000,000 five-year programme was also begun with a view not only to the needs of wartime but also for industry afterwards with fuels and chemicals from coal ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
