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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] |
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] |
Random Matrix
In probability theory and mathematical physics, a random matrix is a matrix-valued random variable—that is, a matrix in which some or all elements are random variables. Many important properties of physical systems can be represented mathematically as matrix problems. For example, the thermal conductivity of a lattice can be computed from the dynamical matrix of the particle-particle interactions within the lattice. Applications Physics In nuclear physics, random matrices were introduced by Eugene Wigner to model the nuclei of heavy atoms. Wigner postulated that the spacings between the lines in the spectrum of a heavy atom nucleus should resemble the spacings between the eigenvalues of a random matrix, and should depend only on the symmetry class of the underlying evolution. In solid-state physics, random matrices model the behaviour of large disordered Hamiltonians in the mean-field approximation. In quantum chaos, the Bohigas–Giannoni–Schmit (BGS) conjecture asserts ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dirichlet Energy
In mathematics, the Dirichlet energy is a measure of how ''variable'' a function is. More abstractly, it is a quadratic functional on the Sobolev space . The Dirichlet energy is intimately connected to Laplace's equation and is named after the German mathematician Peter Gustav Lejeune Dirichlet. Definition Given an open set and a function the Dirichlet energy of the function is the real number :E = \frac 1 2 \int_\Omega \, \nabla u(x) \, ^2 \, dx, where denotes the gradient vector field of the function . Properties and applications Since it is the integral of a non-negative quantity, the Dirichlet energy is itself non-negative, i.e. for every function . Solving Laplace's equation -\Delta u(x) = 0 for all x \in \Omega, subject to appropriate boundary conditions, is equivalent to solving the variational problem of finding a function that satisfies the boundary conditions and has minimal Dirichlet energy. Such a solution is called a harmonic funct ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Distribution (mathematics)
Distributions, also known as Schwartz distributions or generalized functions, are objects that generalize the classical notion of functions in mathematical analysis. Distributions make it possible to differentiate functions whose derivatives do not exist in the classical sense. In particular, any locally integrable function has a distributional derivative. Distributions are widely used in the theory of partial differential equations, where it may be easier to establish the existence of distributional solutions than classical solutions, or where appropriate classical solutions may not exist. Distributions are also important in physics and engineering where many problems naturally lead to differential equations whose solutions or initial conditions are singular, such as the Dirac delta function. A function f is normally thought of as on the in the function domain by "sending" a point x in its domain to the point f(x). Instead of acting on points, distribution theory reinterpr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Probability Distribution
In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon in terms of its sample space and the probabilities of events (subsets of the sample space). For instance, if is used to denote the outcome of a coin toss ("the experiment"), then the probability distribution of would take the value 0.5 (1 in 2 or 1/2) for , and 0.5 for (assuming that the coin is fair). Examples of random phenomena include the weather conditions at some future date, the height of a randomly selected person, the fraction of male students in a school, the results of a survey to be conducted, etc. Introduction A probability distribution is a mathematical description of the probabilities of events, subsets of the sample space. The sample space, often denoted by \Omega, is the set of all possible outcomes of a random phe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Green's Function
In mathematics, a Green's function is the impulse response of an inhomogeneous linear differential operator defined on a domain with specified initial conditions or boundary conditions. This means that if \operatorname is the linear differential operator, then * the Green's function G is the solution of the equation \operatorname G = \delta, where \delta is Dirac's delta function; * the solution of the initial-value problem \operatorname y = f is the convolution (G \ast f). Through the superposition principle, given a linear ordinary differential equation (ODE), \operatorname y = f, one can first solve \operatorname G = \delta_s, for each , and realizing that, since the source is a sum of delta functions, the solution is a sum of Green's functions as well, by linearity of . Green's functions are named after the British mathematician George Green, who first developed the concept in the 1820s. In the modern study of linear partial differential equations, Green's functions are s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Covariance
In probability theory and statistics, covariance is a measure of the joint variability of two random variables. If the greater values of one variable mainly correspond with the greater values of the other variable, and the same holds for the lesser values (that is, the variables tend to show similar behavior), the covariance is positive. In the opposite case, when the greater values of one variable mainly correspond to the lesser values of the other, (that is, the variables tend to show opposite behavior), the covariance is negative. The sign of the covariance therefore shows the tendency in the linear relationship between the variables. The magnitude of the covariance is not easy to interpret because it is not normalized and hence depends on the magnitudes of the variables. The normalized version of the covariance, the correlation coefficient, however, shows by its magnitude the strength of the linear relation. A distinction must be made between (1) the covariance of two random ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Harmonic Function
In mathematics, mathematical physics and the theory of stochastic processes, a harmonic function is a twice continuously differentiable function f: U \to \mathbb R, where is an open subset of that satisfies Laplace's equation, that is, : \frac + \frac + \cdots + \frac = 0 everywhere on . This is usually written as : \nabla^2 f = 0 or :\Delta f = 0 Etymology of the term "harmonic" The descriptor "harmonic" in the name harmonic function originates from a point on a taut string which is undergoing harmonic motion. The solution to the differential equation for this type of motion can be written in terms of sines and cosines, functions which are thus referred to as ''harmonics''. Fourier analysis involves expanding functions on the unit circle in terms of a series of these harmonics. Considering higher dimensional analogues of the harmonics on the unit ''n''-sphere, one arrives at the spherical harmonics. These functions satisfy Laplace's equation and over time "harmonic" ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Expected Value
In probability theory, the expected value (also called expectation, expectancy, mathematical expectation, mean, average, or first moment) is a generalization of the weighted average. Informally, the expected value is the arithmetic mean of a large number of independently selected outcomes of a random variable. The expected value of a random variable with a finite number of outcomes is a weighted average of all possible outcomes. In the case of a continuum of possible outcomes, the expectation is defined by integration. In the axiomatic foundation for probability provided by measure theory, the expectation is given by Lebesgue integration. The expected value of a random variable is often denoted by , , or , with also often stylized as or \mathbb. History The idea of the expected value originated in the middle of the 17th century from the study of the so-called problem of points, which seeks to divide the stakes ''in a fair way'' between two players, who have to end th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lebesgue Measure
In measure theory, a branch of mathematics, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of ''n''-dimensional Euclidean space. For ''n'' = 1, 2, or 3, it coincides with the standard measure of length, area, or volume. In general, it is also called ''n''-dimensional volume, ''n''-volume, or simply volume. It is used throughout real analysis, in particular to define Lebesgue integration. Sets that can be assigned a Lebesgue measure are called Lebesgue-measurable; the measure of the Lebesgue-measurable set ''A'' is here denoted by ''λ''(''A''). Henri Lebesgue described this measure in the year 1901, followed the next year by his description of the Lebesgue integral. Both were published as part of his dissertation in 1902. Definition For any interval I = ,b/math>, or I = (a, b), in the set \mathbb of real numbers, let \ell(I)= b - a denote its length. For any subset E\subseteq\mathbb, the Lebesgue oute ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Probability Density Function
In probability theory, a probability density function (PDF), or density of a continuous random variable, is a function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) can be interpreted as providing a ''relative likelihood'' that the value of the random variable would be close to that sample. Probability density is the probability per unit length, in other words, while the ''absolute likelihood'' for a continuous random variable to take on any particular value is 0 (since there is an infinite set of possible values to begin with), the value of the PDF at two different samples can be used to infer, in any particular draw of the random variable, how much more likely it is that the random variable would be close to one sample compared to the other sample. In a more precise sense, the PDF is used to specify the probability of the random variable falling ''within a particular range of values'', as opposed to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gibbs Measure
In mathematics, the Gibbs measure, named after Josiah Willard Gibbs, is a probability measure frequently seen in many problems of probability theory and statistical mechanics. It is a generalization of the canonical ensemble to infinite systems. The canonical ensemble gives the probability of the system ''X'' being in state ''x'' (equivalently, of the random variable ''X'' having value ''x'') as :P(X=x) = \frac \exp ( - \beta E(x)). Here, is a function from the space of states to the real numbers; in physics applications, is interpreted as the energy of the configuration ''x''. The parameter is a free parameter; in physics, it is the inverse temperature. The normalizing constant is the partition function. However, in infinite systems, the total energy is no longer a finite number and cannot be used in the traditional construction of the probability distribution of a canonical ensemble. Traditional approaches in statistical physics studied the limit of intensive properties as t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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