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Schramm–Loewner Evolution
In probability theory, the Schramm–Loewner evolution with parameter ''κ'', also known as stochastic Loewner evolution (SLE''κ''), is a family of random planar curves that have been proven to be the scaling limit of a variety of two-dimensional lattice models in statistical mechanics. Given a parameter ''κ'' and a domain in the complex plane ''U'', it gives a family of random curves in ''U'', with ''κ'' controlling how much the curve turns. There are two main variants of SLE, ''chordal SLE'' which gives a family of random curves from two fixed boundary points, and ''radial SLE'', which gives a family of random curves from a fixed boundary point to a fixed interior point. These curves are defined to satisfy conformal invariance and a domain Markov property. It was discovered by as a conjectured scaling limit of the planar uniform spanning tree (UST) and the planar loop-erased random walk (LERW) probabilistic processes, and developed by him together with Greg Lawler and Wendel ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Open Set
In mathematics, open sets are a generalization of open intervals in the real line. In a metric space (a set along with a distance defined between any two points), open sets are the sets that, with every point , contain all points that are sufficiently near to (that is, all points whose distance to is less than some value depending on ). More generally, one defines open sets as the members of a given collection of subsets of a given set, a collection that has the property of containing every union of its members, every finite intersection of its members, the empty set, and the whole set itself. A set in which such a collection is given is called a topological space, and the collection is called a topology. These conditions are very loose, and allow enormous flexibility in the choice of open sets. For example, ''every'' subset can be open (the discrete topology), or no set can be open except the space itself and the empty set (the indiscrete topology). In practice, however, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cardy Formula
In physics, the Cardy formula gives the entropy of a two-dimensional conformal field theory (CFT). In recent years, this formula has been especially useful in the calculation of the entropy of BTZ black holes and in checking the AdS/CFT correspondence and the holographic principle. In 1986 J. L. Cardy derived the formula: :S=2\pi\sqrt, Here c is the central charge, L_0 = ER is the product of the total energy and radius of the system, and the shift of c/24 is related to the Casimir effect. These data emerge from the Virasoro algebra of this CFT. The proof of the above formula relies on modular invariance of a Euclidean CFT on the torus. The Cardy formula is usually understood as counting the number of states of energy \Delta = L_0 + \bar_0 of a CFT quantized on a circle. To be precise, the microcanonical entropy (that is to say, the logarithm of the number of states in a shell of width \delta \lesssim 1) is given by :S_\delta(\Delta) = 2\pi \sqrt + O(\ln \Delta) in the limit \Delt ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Itô's Lemma
In mathematics, Itô's lemma or Itô's formula (also called the Itô-Doeblin formula, especially in French literature) is an identity used in Itô calculus to find the differential of a time-dependent function of a stochastic process. It serves as the stochastic calculus counterpart of the chain rule. It can be heuristically derived by forming the Taylor series expansion of the function up to its second derivatives and retaining terms up to first order in the time increment and second order in the Wiener process increment. The lemma is widely employed in mathematical finance, and its best known application is in the derivation of the Black–Scholes equation for option values. Motivation Suppose we are given the stochastic differential equation dX_t = \mu_t\ dt + \sigma_t\ dB_t, where is a Wiener process and the functions \mu_t, \sigma_t are deterministic (not stochastic) functions of time. In general, it's not possible to write a solution X_t directly in terms of B_t. Howeve ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gamma Function
In mathematics, the gamma function (represented by , the capital letter gamma from the Greek alphabet) is one commonly used extension of the factorial function to complex numbers. The gamma function is defined for all complex numbers except the non-positive integers. For every positive integer , \Gamma(n) = (n-1)!\,. Derived by Daniel Bernoulli, for complex numbers with a positive real part, the gamma function is defined via a convergent improper integral: \Gamma(z) = \int_0^\infty t^ e^\,dt, \ \qquad \Re(z) > 0\,. The gamma function then is defined as the analytic continuation of this integral function to a meromorphic function that is holomorphic in the whole complex plane except zero and the negative integers, where the function has simple poles. The gamma function has no zeroes, so the reciprocal gamma function is an entire function. In fact, the gamma function corresponds to the Mellin transform of the negative exponential function: \Gamma(z) = \mathcal M \ (z ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hausdorff Dimension
In mathematics, Hausdorff dimension is a measure of ''roughness'', or more specifically, fractal dimension, that was first introduced in 1918 by mathematician Felix Hausdorff. For instance, the Hausdorff dimension of a single point is zero, of a line segment is 1, of a square is 2, and of a cube is 3. That is, for sets of points that define a smooth shape or a shape that has a small number of corners—the shapes of traditional geometry and science—the Hausdorff dimension is an integer agreeing with the usual sense of dimension, also known as the topological dimension. However, formulas have also been developed that allow calculation of the dimension of other less simple objects, where, solely on the basis of their properties of scaling and self-similarity, one is led to the conclusion that particular objects—including fractals—have non-integer Hausdorff dimensions. Because of the significant technical advances made by Abram Samoilovitch Besicovitch allowing computation of di ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Central Charge
In theoretical physics, a central charge is an operator ''Z'' that commutes with all the other symmetry operators. The adjective "central" refers to the center of the symmetry group—the subgroup of elements that commute with all other elements of the original group—often embedded within a Lie algebra. In some cases, such as two-dimensional conformal field theory, a central charge may also commute with all of the other operators, including operators that are not symmetry generators. Overview More precisely, the central charge is the charge that corresponds, by Noether's theorem, to the center of the central extension of the symmetry group. In theories with supersymmetry, this definition can be generalized to include supergroups and Lie superalgebras. A central charge is any operator which commutes with all the other supersymmetry generators. Theories with extended supersymmetry typically have many operators of this kind. In string theory, in the first quantized formali ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
Gaussian Free Field
In probability theory and statistical mechanics, the Gaussian free field (GFF) is a Gaussian random field, a central model of random surfaces (random height functions). gives a mathematical survey of the Gaussian free field. The discrete version can be defined on any graph, usually a lattice in ''d''-dimensional Euclidean space. The continuum version is defined on R''d'' or on a bounded subdomain of R''d''. It can be thought of as a natural generalization of one-dimensional Brownian motion to ''d'' time (but still one space) dimensions: it is a random (generalized) function from R''d'' to R. In particular, the one-dimensional continuum GFF is just the standard one-dimensional Brownian motion or Brownian bridge on an interval. In the theory of random surfaces, it is also called the harmonic crystal. It is also the starting point for many constructions in quantum field theory, where it is called the Euclidean bosonic massless free field. A key property of the 2-dimensional GFF is ... [...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] |
Bieberbach Conjecture
In complex analysis, de Branges's theorem, or the Bieberbach conjecture, is a theorem that gives a necessary condition on a holomorphic function in order for it to map the open unit disk of the complex plane injectively to the complex plane. It was posed by and finally proven by . The statement concerns the Taylor coefficients a_n of a univalent function, i.e. a one-to-one holomorphic function that maps the unit disk into the complex plane, normalized as is always possible so that a_0=0 and a_1=1. That is, we consider a function defined on the open unit disk which is holomorphic and injective ('' univalent'') with Taylor series of the form :f(z)=z+\sum_ a_n z^n. Such functions are called ''schlicht''. The theorem then states that : , a_n, \leq n \quad \textn\geq 2. The Koebe function (see below) is a function in which a_n=n for all n, and it is schlicht, so we cannot find a stricter limit on the absolute value of the nth coefficient. Schlicht functions The normalizations : ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
