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Laplace
Pierre-Simon, marquis de Laplace (; ; 23 March 1749 – 5 March 1827) was a French scholar and polymath whose work was important to the development of engineering, mathematics, statistics, physics, astronomy, and philosophy. He summarized and extended the work of his predecessors in his five-volume ''Mécanique céleste'' (''Celestial Mechanics'') (1799–1825). This work translated the geometric study of classical mechanics to one based on calculus, opening up a broader range of problems. In statistics, the Bayesian interpretation of probability was developed mainly by Laplace. Laplace formulated Laplace's equation, and pioneered the Laplace transform which appears in many branches of mathematical physics, a field that he took a leading role in forming. The Laplacian differential operator, widely used in mathematics, is also named after him. He restated and developed the nebular hypothesis of the origin of the Solar System and was one of the first scientists to suggest ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Laplace Transform
In mathematics, the Laplace transform, named after its discoverer Pierre-Simon Laplace (), is an integral transform In mathematics, an integral transform maps a function from its original function space into another function space via integration, where some of the properties of the original function might be more easily characterized and manipulated than in ... that converts a Function (mathematics), function of a Real number, real Variable (mathematics), variable (usually t, in the ''time domain'') to a function of a Complex number, complex variable s (in the complex frequency domain, also known as ''s''-domain, or s-plane). The transform has many applications in science and engineering because it is a tool for solving differential equations. In particular, it transforms ordinary differential equations into algebraic equations and convolution into multiplication. For suitable functions ''f'', the Laplace transform is the integral \mathcal\(s) = \int_0^\infty f(t)e^ \, dt. H ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Laplace Distribution
In probability theory and statistics, the Laplace distribution is a continuous probability distribution named after Pierre-Simon Laplace. It is also sometimes called the double exponential distribution, because it can be thought of as two exponential distributions (with an additional location parameter) spliced together along the abscissa, although the term is also sometimes used to refer to the Gumbel distribution. The difference between two independent identically distributed exponential random variables is governed by a Laplace distribution, as is a Brownian motion evaluated at an exponentially distributed random time. Increments of Laplace motion or a variance gamma process evaluated over the time scale also have a Laplace distribution. Definitions Probability density function A random variable has a \textrm(\mu, b) distribution if its probability density function is :f(x\mid\mu,b) = \frac \exp \left( -\frac \right) \,\! Here, \mu is a location parameter and b > 0, which ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Laplace's Equation
In mathematics and physics, Laplace's equation is a second-order partial differential equation named after Pierre-Simon Laplace, who first studied its properties. This is often written as \nabla^2\! f = 0 or \Delta f = 0, where \Delta = \nabla \cdot \nabla = \nabla^2 is the Laplace operator,The delta symbol, Δ, is also commonly used to represent a finite change in some quantity, for example, \Delta x = x_1 - x_2. Its use to represent the Laplacian should not be confused with this use. \nabla \cdot is the divergence operator (also symbolized "div"), \nabla is the gradient operator (also symbolized "grad"), and f (x, y, z) is a twice-differentiable real-valued function. The Laplace operator therefore maps a scalar function to another scalar function. If the right-hand side is specified as a given function, h(x, y, z), we have \Delta f = h. This is called Poisson's equation, a generalization of Laplace's equation. Laplace's equation and Poisson's equation are the simplest exa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Laplace Operator
In mathematics, the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a scalar function on Euclidean space. It is usually denoted by the symbols \nabla\cdot\nabla, \nabla^2 (where \nabla is the nabla operator), or \Delta. In a Cartesian coordinate system, the Laplacian is given by the sum of second partial derivatives of the function with respect to each independent variable. In other coordinate systems, such as cylindrical and spherical coordinates, the Laplacian also has a useful form. Informally, the Laplacian of a function at a point measures by how much the average value of over small spheres or balls centered at deviates from . The Laplace operator is named after the French mathematician Pierre-Simon de Laplace (1749–1827), who first applied the operator to the study of celestial mechanics: the Laplacian of the gravitational potential due to a given mass density distribution is a constant multiple of that densi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Laplace's Demon
In the history of science, Laplace's demon was a notable published articulation of causal determinism on a scientific basis by Pierre-Simon Laplace in 1814. According to determinism, if someone (the demon) knows the precise location and momentum of every atom in the universe, their past and future values for any given time are entailed; they can be calculated from the laws of classical mechanics. This idea states that “free will” is merely an illusion, and that every action previously taken, currently being taken, or that will take place was destined to happen from the instant of the big bang. Discoveries and theories in the decades following suggest that some elements of Laplace's original writing are wrong or incompatible with our universe. For example, irreversible processes in thermodynamics suggest that Laplace's "demon" could not reconstruct past positions and momenta from the current state. English translation This intellect is often referred to as ''Laplace's d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Laplace's Method
In mathematics, Laplace's method, named after Pierre-Simon Laplace, is a technique used to approximate integrals of the form :\int_a^b e^ \, dx, where f(x) is a twice-differentiable function, ''M'' is a large number, and the endpoints ''a'' and ''b'' could possibly be infinite. This technique was originally presented in . In Bayesian statistics, Laplace's approximation can refer to either approximating the posterior normalizing constant with Laplace's method or approximating the posterior distribution with a Gaussian centered at the maximum a posteriori estimate. Laplace approximations play a central role in the integrated nested Laplace approximations method for fast approximate Bayesian inference. The idea of Laplace's method Suppose the function f(x) has a unique global maximum at ''x''0. Let M>0 be a constant and consider the following two functions: :\begin g(x) &= Mf(x) \\ h(x) &= e^ \end Note that ''x''0 will be the global maximum of g and h as well. Now observe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Young–Laplace Equation
In physics, the Young–Laplace equation () is an algebraic equation that describes the capillary pressure difference sustained across the interface between two static fluids, such as water and air, due to the phenomenon of surface tension or wall tension, although use of the latter is only applicable if assuming that the wall is very thin. The Young–Laplace equation relates the pressure difference to the shape of the surface or wall and it is fundamentally important in the study of static capillary surfaces. It's a statement of normal stress balance for static fluids meeting at an interface, where the interface is treated as a surface (zero thickness): \begin \Delta p &= -\gamma \nabla \cdot \hat n \\ &= -2\gamma H_f \\ &= -\gamma \left(\frac + \frac\right) \end where \Delta p is the Laplace pressure, the pressure difference across the fluid interface (the exterior pressure minus the interior pressure), \gamma is the surface tension (or wall tension), \hat n is the unit norm ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Inverse Laplace Transform
In mathematics, the inverse Laplace transform of a function ''F''(''s'') is the piecewise-continuous and exponentially-restricted real function ''f''(''t'') which has the property: :\mathcal\(s) = \mathcal\(s) = F(s), where \mathcal denotes the Laplace transform. It can be proven that, if a function ''F''(''s'') has the inverse Laplace transform ''f''(''t''), then ''f''(''t'') is uniquely determined (considering functions which differ from each other only on a point set having Lebesgue measure zero as the same). This result was first proven by Mathias Lerch in 1903 and is known as Lerch's theorem. The Laplace transform and the inverse Laplace transform together have a number of properties that make them useful for analysing linear dynamical systems. Mellin's inverse formula An integral formula for the inverse Laplace transform, called the ''Mellin's inverse formula'', the '' Bromwich integral'', or the '' Fourier–Mellin integral'', is given by the line integral: :f(t) = \mathca ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Siméon Denis Poisson
Baron Siméon Denis Poisson FRS FRSE (; 21 June 1781 – 25 April 1840) was a French mathematician and physicist who worked on statistics, complex analysis, partial differential equations, the calculus of variations, analytical mechanics, electricity and magnetism, thermodynamics, elasticity, and fluid mechanics. Moreover, he predicted the Poisson spot in his attempt to disprove the wave theory of Augustin-Jean Fresnel, which was later confirmed. Biography Poisson was born in Pithiviers, Loiret district in France, the son of Siméon Poisson, an officer in the French army. In 1798, he entered the École Polytechnique in Paris as first in his year, and immediately began to attract the notice of the professors of the school, who left him free to make his own decisions as to what he would study. In his final year of study, less than two years after his entry, he published two memoirs, one on Étienne Bézout's method of elimination, the other on the number of integrals of a finite di ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Laplace Limit
In mathematics, the Laplace limit is the maximum value of the eccentricity for which a solution to Kepler's equation, in terms of a power series in the eccentricity, converges. It is approximately : 0.66274 34193 49181 58097 47420 97109 25290. Kepler's equation ''M'' = ''E'' − ε sin ''E'' relates the mean anomaly ''M'' with the eccentric anomaly ''E'' for a body moving in an ellipse with eccentricity ε. This equation cannot be solved for ''E'' in terms of elementary functions, but the Lagrange reversion theorem gives the solution as a power series in ε: : E = M + \sin(M) \, \varepsilon + \tfrac12 \sin(2M) \, \varepsilon^2 + \left( \tfrac38 \sin(3M) - \tfrac18 \sin(M) \right) \, \varepsilon^3 + \cdots or in general : E = M \;+\; \sum_^ \frac \sum_^ (-1)^k\,\binom\,(n-2k)^\,\sin((n-2k)\,M) Laplace realized that this series converges for small values of the eccentricity, but diverges for any value of ''M'' other than a multiple of π ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Laplace Invariant
In differential equations, the Laplace invariant of any of certain differential operators is a certain function of the coefficients and their derivatives. Consider a bivariate hyperbolic differential operator of the second order :\partial_x \, \partial_y + a\,\partial_x + b\,\partial_y + c, \, whose coefficients : a=a(x,y), \ \ b=c(x,y), \ \ c=c(x,y), are smooth functions of two variables. Its Laplace invariants have the form :\hat= c- ab -a_x \quad \text \quad \hat=c- ab -b_y. Their importance is due to the classical theorem: Theorem: ''Two operators of the form are equivalent under gauge transformations if and only if their Laplace invariants coincide pairwise.'' Here the operators :A \quad \text \quad \tilde A are called ''equivalent'' if there is a gauge transformation that takes one to the other: : \tilde Ag= e^A(e^g)\equiv A_\varphi g. Laplace invariants can be regarded as factorization "remainders" for the initial operator ''A'': :\partial_x\, \partial_y + ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bayesian Probability
Bayesian probability is an Probability interpretations, interpretation of the concept of probability, in which, instead of frequentist probability, frequency or propensity probability, propensity of some phenomenon, probability is interpreted as reasonable expectation representing a state of knowledge or as quantification of a personal belief. The Bayesian interpretation of probability can be seen as an extension of propositional logic that enables reasoning with Hypothesis, hypotheses; that is, with propositions whose truth value, truth or falsity is unknown. In the Bayesian view, a probability is assigned to a hypothesis, whereas under frequentist inference, a hypothesis is typically tested without being assigned a probability. Bayesian probability belongs to the category of evidential probabilities; to evaluate the probability of a hypothesis, the Bayesian probabilist specifies a prior probability. This, in turn, is then updated to a posterior probability in the light of new, re ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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