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Pierre-Simon Laplace
Pierre-Simon, Marquis de Laplace (; ; 23 March 1749 – 5 March 1827) was a French polymath, a scholar whose work has been instrumental in the fields of physics, astronomy, mathematics, engineering, statistics, and philosophy. He summarized and extended the work of his predecessors in his five-volume Traité de mécanique céleste, ''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. Laplace also popularized and further confirmed Isaac Newton, Sir Isaac Newton's work. In statistics, the Bayesian probability, 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 Laplace operator, Laplacian differential operator, widely used in mathematic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Senate
A senate is a deliberative assembly, often the upper house or chamber of a bicameral legislature. The name comes from the ancient Roman Senate (Latin: ''Senatus''), so-called as an assembly of the senior (Latin: ''senex'' meaning "the elder" or "old man") and therefore considered wiser and more experienced members of the society or ruling class. However the Roman Senate was not the ancestor or predecessor of modern parliamentarism in any sense, because the Roman senate was not a de jure legislative body. Many countries have an assembly named a ''senate'', composed of ''senators'' who may be elected, appointed, have inherited the title, or gained membership by other methods, depending on the country. Modern senates typically serve to provide a chamber of "sober second thought" to consider legislation passed by a lower house, whose members are usually elected. Most senates have asymmetrical duties and powers compared with their respective lower house meaning they have sp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
The Large Scale Structure Of Space-Time
''The'' is a grammatical article in English, denoting nouns that are already or about to be mentioned, under discussion, implied or otherwise presumed familiar to listeners, readers, or speakers. It is the definite article in English. ''The'' is the most frequently used word in the English language; studies and analyses of texts have found it to account for seven percent of all printed English-language words. It is derived from gendered articles in Old English which combined in Middle English and now has a single form used with nouns of any gender. The word can be used with both singular and plural nouns, and with a noun that starts with any letter. This is different from many other languages, which have different forms of the definite article for different genders or numbers. Pronunciation In most dialects, "the" is pronounced as (with the voiced dental fricative followed by a schwa) when followed by a consonant sound, and as (homophone of the archaic pronoun ''thee' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Laplace Principle (large Deviations Theory)
In mathematics, Laplace's principle is a basic theorem in large deviations theory which is similar to Varadhan's lemma. It gives an asymptotic expression for the Lebesgue integral of exp(−''θφ''(''x'')) over a fixed set ''A'' as ''θ'' becomes large. Such expressions can be used, for example, in statistical mechanics to determining the limiting behaviour of a system as the temperature tends to absolute zero. Statement of the result Let ''A'' be a Lebesgue-measurable subset of ''d''-dimensional Euclidean space R''d'' and let ''φ'' : R''d'' → R be a measurable function with :\int_A e^ \,dx < \infty. Then : where ess inf denotes the . Heuristically, this may be read as saying that for large ''θ'' ... [...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] |
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 o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Laplace Number
The Laplace number (), also known as the Suratman number (), is a dimensionless number used in the characterization of free surface fluid dynamics. It represents a ratio of surface tension to the momentum-transport (especially dissipation) inside a fluid. It is named after Pierre-Simon Laplace and Indonesian physicist P. C. Suratman. It is defined as follows: :\mathrm = \mathrm = \frac where: * σ = surface tension * ρ = density * L = length * μ = liquid viscosity Laplace number is related to Reynolds number (Re) and Weber number The Weber number (We) is a dimensionless number in fluid mechanics that is often useful in analysing fluid flows where there is an interface between two different fluids, especially for multiphase flows with strongly curved surfaces. It is named ... (We) in the following way: :\mathrm = \frac See also * Ohnesorge number - There is an inverse relationship, \mathrm = \mathrm^, between the Laplace number and the Ohnesorge number. References { ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Young–Laplace Equation
In physics, the Young–Laplace equation () is an 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 is 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 normal poi ... [...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 particle in the universe, their past and future values for any given time are entailed; they can be calculated from the laws of classical mechanics. English translation This intellect is often referred to as ''Laplace's demon'' (and sometimes ''Laplace's Superman'', after Hans Reichenbach). Laplace himself did not use the word "demon", which was a later embellishment. As translated into English above, he simply referred to: ''"Une intelligence ... Rien ne serait incertain pour elle, et l'avenir, comme le passé, serait présent à ses yeux."'' This idea seems to have been widespread around the time that Laplace first expressed it in 1773, particularly in France. Variations can be found in Maupertuis (1756), Nicolas de Condo ... [...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 x-axis, although the term is also sometimes used to refer to the Gumbel distribution. The difference between two Independent identically-distributed random variables, 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 \operatorname(\mu, b) distribution if its probability density function is : f(x \mid \mu, b) = \frac \exp\left( -\frac \rig ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Inverse Laplace Transform
In mathematics, the inverse Laplace transform of a function F(s) is a real function f(t) that is piecewise- continuous, exponentially-restricted (that is, , f(t), \leq Me^ \forall t \geq 0 for some constants M > 0 and \alpha \in \mathbb) and 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– Me ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Laplace Transform
In mathematics, the Laplace transform, named after Pierre-Simon Laplace (), is an integral transform 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-valued frequency domain, also known as ''s''-domain, or ''s''-plane). The transform is useful for converting derivative, differentiation and integral, integration in the time domain into much easier multiplication and Division (mathematics), division in the Laplace domain (analogous to how logarithms are useful for simplifying multiplication and division into addition and subtraction). This gives the transform many applications in science and engineering, mostly as a tool for solving linear differential equations and dynamical systems by simplifying ordinary differential equations and integral equations into algebraic equation, algebraic polynomial equations, and by simplifyin ... [...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 field, scalar function on Euclidean space. It is usually denoted by the symbols \nabla\cdot\nabla, \nabla^2 (where \nabla is the Del, 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 coordinates, 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 distributio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
