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Variational Theory
Variational may refer to: *Calculus of variations, a field of mathematical analysis that deals with maximizing or minimizing functionals *Variational method (quantum mechanics), a way of finding approximations to the lowest energy eigenstate or ground state in quantum physics *Variational Bayesian methods, a family of techniques for approximating integrals in Bayesian inference and machine learning *Variational properties, properties of an organism relating to the production of variation among its offspring in evolutionary biology *Variationist sociolinguistics or variational sociolinguistics, the study of variation in language use among speakers or groups of speakers See also *List of variational topics in mathematics and physics *Variation (other) Variation or Variations may refer to: Science and mathematics * Variation (astronomy), any perturbation of the mean motion or orbit of a planet or satellite, particularly of the moon * Genetic variation, the difference in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Calculus Of Variations
The calculus of variations (or variational calculus) is a field of mathematical analysis that uses variations, which are small changes in Function (mathematics), functions and functional (mathematics), functionals, to find maxima and minima of functionals: Map (mathematics), mappings from a set of Function (mathematics), functions to the real numbers. Functionals are often expressed as definite integrals involving functions and their derivatives. Functions that maximize or minimize functionals may be found using the Euler–Lagrange equation of the calculus of variations. A simple example of such a problem is to find the curve of shortest length connecting two points. If there are no constraints, the solution is a straight line between the points. However, if the curve is constrained to lie on a surface in space, then the solution is less obvious, and possibly many solutions may exist. Such solutions are known as ''geodesics''. A related problem is posed by Fermat's principle: li ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Variational Method (quantum Mechanics)
In quantum mechanics, the variational method is one way of finding approximations to the lowest energy eigenstate or ground state, and some excited states. This allows calculating approximate wavefunctions such as molecular orbitals. The basis for this method is the variational principle. The method consists of choosing a "trial wavefunction" depending on one or more parameters, and finding the values of these parameters for which the expectation value of the energy is the lowest possible. The wavefunction obtained by fixing the parameters to such values is then an approximation to the ground state wavefunction, and the expectation value of the energy in that state is an upper bound to the ground state energy. The Hartree–Fock method, density matrix renormalization group, and Ritz method apply the variational method. Description Suppose we are given a Hilbert space and a Hermitian operator over it called the Hamiltonian H . Ignoring complications about continuous spectr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Variational Bayesian Methods
Variational Bayesian methods are a family of techniques for approximating intractable integrals arising in Bayesian inference and machine learning. They are typically used in complex statistical models consisting of observed variables (usually termed "data") as well as unknown parameters and latent variables, with various sorts of relationships among the three types of random variables, as might be described by a graphical model. As typical in Bayesian inference, the parameters and latent variables are grouped together as "unobserved variables". Variational Bayesian methods are primarily used for two purposes: # To provide an analytical approximation to the posterior probability of the unobserved variables, in order to do statistical inference over these variables. # To derive a lower bound for the marginal likelihood (sometimes called the ''evidence'') of the observed data (i.e. the marginal probability of the data given the model, with marginalization performed over unobserv ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Variational Properties
In evolutionary biology, the variational properties of an organism are those properties relating to the production of variation among its offspring. In a broader sense variational properties include phenotypic plasticity. Wagner, G. P. and Altenberg, L. 1996. Complex adaptations and the evolution of evolvability. ''Evolution'' 50 (3): 967-976. Variational properties contrast with functional properties. While the functional properties of an organism determine is level of adaptedness to its environment, it is the variational properties of the organisms in a species that chiefly determine its evolvability and genetic robustness. Variational properties group together many classical and more recent concepts of evolutionary biology. It includes the classical concepts of pleiotropy, canalization, developmental constraints, developmental bias, morphological integration, developmental homeostasis and later concepts such as robustness, neutral networks, modularity Modularity is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Variationist Sociolinguistics
Variation is a characteristic of language: there is more than one way of saying the same thing in a given language. Variation can exist in domains such as pronunciation (e.g., more than one way of pronouncing the same phoneme or the same word), lexicon (e.g., multiple words with the same meaning), grammar (e.g., different syntactic constructions expressing the same grammatical function), and other features. Different communities or individuals speaking the same language may differ from each other in their choices of which of the available linguistic features to use, and how often (inter-speaker variation), and the same speaker may make different choices on different occasions (intra-speaker variation). While diversity of variation exists, there are also some general boundaries on variation. For instance, speakers across distinct dialects of a language tend to preserve the same word order or fit new sounds into the language's established inventory of phonemes (the study of such res ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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List Of Variational Topics
{{Short description, none This is a list of variational topics in from mathematics and physics. See calculus of variations for a general introduction. * Action (physics) * Averaged Lagrangian * Brachistochrone curve * Calculus of variations * Catenoid * Cycloid * Dirichlet principle * Euler–Lagrange equation cf. Action (physics) * Fermat's principle * Functional (mathematics) * Functional derivative * Functional integral * Geodesic * Isoperimetry * Lagrangian * Lagrangian mechanics * Legendre transformation * Luke's variational principle * Minimal surface * Morse theory * Noether's theorem * Path integral formulation * Plateau's problem * Prime geodesic * Principle of least action * Soap bubble * Soap film * Tautochrone curve A tautochrone curve or isochrone curve () is the curve for which the time taken by an object sliding without friction in uniform gravity to its lowest point is independent of its starting point on the curve. The curve is a cycloid, and the time . ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |