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Pierre-Simon 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 sugges ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
First French Empire
The First French Empire, officially the French Republic, then the French Empire (; Latin: ) after 1809, also known as Napoleonic France, was the empire ruled by Napoleon Bonaparte, who established French hegemony over much of continental Europe at the beginning of the 19th century. It lasted from 18 May 1804 to 11 April 1814 and again briefly from 20 March 1815 to 7 July 1815. Although France had already established a colonial empire overseas since the early 17th century, the French state had remained a kingdom under the Bourbons and a republic after the French Revolution. Historians refer to Napoleon's regime as the ''First Empire'' to distinguish it from the restorationist ''Second Empire'' (1852–1870) ruled by his nephew Napoleon III. The First French Empire is considered by some to be a " Republican empire." On 18 May 1804, Napoleon was granted the title Emperor of the French (', ) by the French and was crowned on 2 December 1804, signifying the end of the French ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bayesian Inference
Bayesian inference is a method of statistical inference in which Bayes' theorem is used to update the probability for a hypothesis as more evidence or information becomes available. Bayesian inference is an important technique in statistics, and especially in mathematical statistics. Bayesian updating is particularly important in the dynamic analysis of a sequence of data. Bayesian inference has found application in a wide range of activities, including science, engineering, philosophy, medicine, sport, and law. In the philosophy of decision theory, Bayesian inference is closely related to subjective probability, often called "Bayesian probability". Introduction to Bayes' rule Formal explanation Bayesian inference derives the posterior probability as a consequence of two antecedents: a prior probability and a "likelihood function" derived from a statistical model for the observed data. Bayesian inference computes the posterior probability according to Bayes' theorem: ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Principle Of Indifference
The principle of indifference (also called principle of insufficient reason) is a rule for assigning epistemic probabilities. The principle of indifference states that in the absence of any relevant evidence, agents should distribute their credence (or 'degrees of belief') equally among all the possible outcomes under consideration. In Bayesian probability, this is the simplest non-informative prior. The principle of indifference is meaningless under the frequency interpretation of probability, in which probabilities are relative frequencies rather than degrees of belief in uncertain propositions, conditional upon state information. Examples The textbook examples for the application of the principle of indifference are coins, dice, and cards. In a macroscopic system, at least, it must be assumed that the physical laws that govern the system are not known well enough to predict the outcome. As observed some centuries ago by John Arbuthnot (in the preface of ''Of the Laws of ... [...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 of π ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Laplace Number
The Laplace number (La), also known as the Suratman number (Su), 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 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 (We) in the following way: :\mathrm = \frac See also * Ohnesorge number The Ohnesorge number (Oh) is a dimensionless number that relates the viscous forces to inertial and surface tension forces. The number was defined by Wolfgang von Ohnesorge in his 1936 doctoral thesis. It is defined as: : \mathrm = \frac = \frac ... - There is an inverse relationship, \mathrm = \mathrm^, between the Laplace number and the Ohnesorge number. {{DEFAULTSORT:Laplace Number Dimensionless numbers of fluid mechanics Fluid dynamics ... [...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] |
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 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] |
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] |
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] |
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