The research team calls their new algorithm "factorized variational autoencoders" (FVAEs).
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They use variational autoencoders (VAEs) and generative adversarial networks (GANs) to build a framework for the algorithm to learn on.
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An adaptive variational finite difference framework for efficient symmetric octree viscosity (this white paper is not yet online) allows programmers to realistically melt chocolate bunnies.
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A team of researchers including Aspuru-Guzik has developed an algorithm that they call the variational quantum eigensolver (VQE), which can efficiently find the lowest-energy states of molecules even with noisy qubits.
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It uses what's called a factorized variational autoencoder — the math of it I am not even going to try to explain, but it's better than existing methods at capturing the essence of complex things like faces in motion.
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For instance, when given a prompt to draw a mosquito, you just need to draw what looks like a thorax or abdomen and Sketch-RNN will take it from there while showing you how else it predicts the image could be completed: There are two other demos, titled "Interpolation" and "Variational Auto-Encoder," that will have Sketch-RNN try to move between two different types of similar drawings in real time and also try to mimic your drawing with slight tweaks it comes up with on its own: The whole set of programs is a fascinating look underneath the hood of modern computer vision and image and object recognition tool sets tech companies have at their disposal.
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Particle filtering is a sampling-based scheme that relaxes assumptions about the form of the variational or approximate posterior density. The corresponding generalized filtering scheme is called variational filtering.K J Friston, "Variational filtering," Neuroimage, vol. 41, no.
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Any form of TST, such as microcanonical variational TST, canonical variational TST, and improved canonical variational TST, in which the transition state is not necessarily located at the saddle point, is referred to as generalized transition state theory.
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Variational integrators are numerical integrators for Hamiltonian systems derived from the Euler–Lagrange equations of a discretized Hamilton's principle. Variational integrators are momentum-preserving and symplectic.
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Cohomology of the variational bicomplex leads to the global first variational formula and first Noether's theorem. Extended to Lagrangian theory of even and odd fields on graded manifolds, the variational bicomplex provides strict mathematical formulation of classical field theory in a general case of reducible degenerate Lagrangians and the Lagrangian BRST theory.
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"On the variational theory of traffic flow: well-posedness, duality and applications". Networks and Heterogeneous Media. 1 (4). This article shows how Newell's method fits in the context of variational theory.
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Systems of PDEs often arise as the Euler–Lagrange equations for a variational problem. Systems of this form can sometimes be solved by finding an extremum of the original variational problem.
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Newell's method was developed before the variational theory of traffic flow was proposed to deal systematically with vehicle counts.Daganzo, Carlos F. 2005. "A variational formulation of kinematic waves: solution methods". Transportation Research.
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Special cases include variational filtering, dynamic expectation maximization and generalized predictive coding.
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The theory allows the construction of embedded minimal hypersurfaces though variational methods.
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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.Altenberg, L. 1994. The evolution of evolvability in genetic programming.
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Ekeland has contributed to mathematical analysis, particularly to variational calculus and mathematical optimization.
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Variational methods in general relativity refers to various mathematical techniques that employ the use of variational calculus in Einstein's theory of general relativity. The most commonly used tools are Lagrangians and Hamiltonians and are used to derive the Einstein field equations.
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Both one-dimensional and multi- dimensional eigenvalue problems can be formulated as variational problems.
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Typical generative model approaches include naive Bayes classifiers, Gaussian mixture models, variational autoencoders and others.
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Schwinger variational principle is a variational principle which expresses the scattering T-matrix as a functional depending on two unknown wave functions. The functional attains stationary value equal to actual scattering T-matrix. The functional is stationary if and only if the two functions satisfy the Lippmann-Schwinger equation. The development of the variational formulation of the scattering theory can be traced to works of L. Hultén and J. Schwinger in 1940s.
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DNA thus has both a generative role in the organism, and variational role in the lineage.
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In 2003, he invented dynamic causal modelling (DCM), which is used to infer the architecture of distributed systems like the brain. Mathematical contributions include variational (generalised) filtering and dynamic expectation maximisation (DEM), which are Variational Bayesian methods for time-series analysis. Friston currently works on models of functional integration in the human brain and the principles that underlie neuronal interactions. His main contribution to theoretical neurobiology is a variational Free energy principle (active inference in the Bayesian brain).
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In systems where an exact analytical solution may not be feasible, one can make a variational approximation. The basic idea is to make a variational ansatz for the wavefunction with free parameters, plug it into the free energy, and minimize the energy with respect to the free parameters.
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Variational free energy is a function of observations and a probability density over their hidden causes. This variational density is defined in relation to a probabilistic model that generates predicted observations from hypothesized causes. In this setting, free energy provides an approximation to Bayesian model evidence.Beal, M. J. (2003).
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"Chapter 2: Variational Calculus and Its Application to Mechanics." Analytical Mechanics. Cambridge: Cambridge UP, 1998. 45, 70. Print.
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Extended to graded manifolds, the variational bicomplex provides description of graded Lagrangian systems of even and odd variables.
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In 2019 molecules generated with a special type of variational autoencoders were validated experimentally all the way into mice.
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In the Mathematics Subject Classification scheme (MSC2010), the field of "Set-valued and variational analysis" is coded by "49J53".
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Adrian Stephen Lewis (born 1962 in England) is a British-Canadian mathematician, specializing in variational analysis and nonsmooth optimization.
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Using vibrational perturbation theory, effects such as tunnelling and variational effects can be accounted for within the SCTST formalism.
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In science and especially in mathematical studies, a variational principle is one that enables a problem to be solved using calculus of variations, which concerns finding such functions which optimize the values of quantities that depend upon those functions. For example, the problem of determining the shape of a hanging chain suspended at both ends—a catenary—can be solved using variational calculus, and in this case, the variational principle is the following: The solution is a function that minimizes the gravitational potential energy of the chain.
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Numerical analysis of finite difference methods and general variational approximation methods: In his doctoral theses and early publications, Philippe Ciarlet made innovative contributions to the numerical approximation by variational methods of problems with non-linear monotonous boundaries,Ciarlet, P.G. ; Schultz, M.H. ; Varga, R.S., « Numerical methods of high-order accuracy for nonlinear boundary value problems. I. One dimensional problem », Numer. Math., 9 (1967), p. 394–430 and introduced the concepts of discrete Green functions and the discrete maximum principle,Ciarlet, P.G., « Discrete variational Green’s function. I », Aequationes Math.
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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.
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The factor is frequently constant in the complete conditionals used in Gibbs sampling and the optimal distributions in variational methods.
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He won the 1999 Fermat Prize, jointly with Fabrice Bethuel, for several important contributions to the theory of variational calculus.
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In mathematics, the term variational analysis usually denotes the combination and extension of methods from convex optimization and the classical calculus of variations to a more general theory.Rockafellar RT, Wets R (2005) Variational analysis. Springer, New York This includes the more general problems of optimization theory, including topics in set-valued analysis, e.g. generalized derivatives.
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Valletta, 2016. PDF Since MCMC imposes significant computational burden, in cases where computational scalability is also of interest, one may alternatively resort to variational approximations to Bayesian inference, e.g. Indeed, approximate variational inference offers computational efficiency comparable to expectation-maximization, while yielding an accuracy profile only slightly inferior to exact MCMC-type Bayesian inference.
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Terada, Yoshitaka. "Variational and Improvisational Techniques of Gandingan Playing in the Maguindanaon Kulintang Ensemble." Asian Music XXVII.2 (1996): 53–79.
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"Variational and Improvisational Techniques of Gandingan Playing in the Maguindanaon Kulintang Ensemble." Asian Music XXVII.2 (1996): 53-79.Benitez, Kristina.
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Part B, Methodological. 39B (10).Daganzo, Carlos F. 2005. "A variational formulation of kinematic waves: basic theory and complex boundary conditions".
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This is a list of mathematical topics in classical mechanics, by Wikipedia page. See also list of variational topics, correspondence principle.
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171–302 that Ponte Castañeda's (1991) variational method is a secant method using the second moment by phase of local fields.
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A variational implementation was suggested by Korringa and by Kohn and Rostocker, and is often referred to as the KKR model.
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Free energy minimisation is equivalent to maximising the mutual information between sensory states and internal states that parameterise the variational density (for a fixed entropy variational density). This relates free energy minimization to the principle of minimum redundancyBarlow, H. (1961). Possible principles underlying the transformations of sensory messages . In W. Rosenblith (Ed.), Sensory Communication (pp. 217-34).
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A classical result is that a lower semicontinuous function on a compact set attains its minimum. Results from variational analysis such as Ekeland's variational principle allow us to extend this result of lower semicontinuous functions on non-compact sets provided that the function has a lower bound and at the cost of adding a small perturbation to the function.
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Choreographies can be discovered using variational methods, and more recently, topological approaches have been used to attempt a classification in the planar case.
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The Method of Weighed Residuals and Variational Principles. Academic Press. # Durran, D. R., 1999. Numerical Methods for Wave Equations in Geophysical Fluid Dynamics.
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In mathematics, a variational inequality is an inequality involving a functional, which has to be solved for all possible values of a given variable, belonging usually to a convex set. The mathematical theory of variational inequalities was initially developed to deal with equilibrium problems, precisely the Signorini problem: in that model problem, the functional involved was obtained as the first variation of the involved potential energy. Therefore it has a variational origin, recalled by the name of the general abstract problem. The applicability of the theory has since been expanded to include problems from economics, finance, optimization and game theory.
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In fluid dynamics, Luke's variational principle is a Lagrangian variational description of the motion of surface waves on a fluid with a free surface, under the action of gravity. This principle is named after J.C. Luke, who published it in 1967. This variational principle is for incompressible and inviscid potential flows, and is used to derive approximate wave models like the mild-slope equation, or using the averaged Lagrangian approach for wave propagation in inhomogeneous media. Luke's Lagrangian formulation can also be recast into a Hamiltonian formulation in terms of the surface elevation and velocity potential at the free surface.
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A Lagrangian density (or, simply, a Lagrangian) of order is defined as an -form, , on the -order jet manifold of . A Lagrangian can be introduced as an element of the variational bicomplex of the differential graded algebra of exterior forms on jet manifolds of . The coboundary operator of this bicomplex contains the variational operator which, acting on , defines the associated Euler–Lagrange operator .
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Variational message passing (VMP) is an approximate inference technique for continuous- or discrete-valued Bayesian networks, with conjugate-exponential parents, developed by John Winn. VMP was developed as a means of generalizing the approximate variational methods used by such techniques as Latent Dirichlet allocation and works by updating an approximate distribution at each node through messages in the node's Markov blanket.
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The variational functional must thus include additional terms to account for boundary conditions, since the assumed solution field only satisfies the governing differential equation.
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The notion of the angles and some of the variational properties can be naturally extended to arbitrary inner products and subspaces with infinite dimensions.
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Variational Asymptotic Method (VAM) is a powerful mathematical approach to simplify the process of finding stationary points for a described functional by taking an advantage of small parameters. VAM is the synergy of variational principles and asymptotic approaches, variational principles are applied to the defined functional as well as the asymptotes are applied to the same functional instead of applying on differential equations due to its less prone to errors. This methodology is applicable for whole range of physics problems, where the problem has to be defined in a variational form and should be able to identify the small parameters within the problem definition. In other words, VAM can be applicable where the functional is so complex in determining the stationary points either by analytical or by computationally expensive numerical analysis with an advantage of small parameters.
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However, the limitation of this method is that it can not be used for non-concave fundamental diagrams. Newell proposed the method, but Daganzo Daganzo, C.F. On the Variational Theory of Traffic Flow: Well-Posedness, Duality and Applications. UC Berkeley: UC Berkeley Center for Future Urban Transport: A Volvo Center of Excellence, 2006 using variational theory proved that the lower envelop is the unique solution.
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His work in elasticity theory includes the paper , where Fichera proves the "Fichera's maximum principle", his work on variational inequalities. The work on this last topic started with the paper , where he announced the existence and uniqueness theorem for the Signorini problem, and ended with the following one ,See also its English translation . where the full proof was published: those papers are the founding works of the field of variational inequalities, as remarked by Stuart Antman in .These are his only papers in the field of variational inequalities: see the article "Signorini problem" for a discussion of the reasons why he left this field of research.
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Mechanization in problem solving. Psychological Monographs, 54(248).Lurchins, A. S., & Lurchins, E. H. (1959). Rigidity of behaviour: A variational approach to the effects of Einstellung.
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An, J., & Cho, S. (2015). Variational autoencoder based anomaly detection using reconstruction probability. Special Lecture on IE, 2, 1-18.Zhou, C., & Paffenroth, R. C. (2017, August).
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759 Schaeffer worked with Donald Spencer at Stanford University on variational problems of conformal mapping, e.g. coefficient ranges for schlicht functions (functions analytic and one-to-one).
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The density matrix renormalization group (DMRG) is a numerical variational technique devised to obtain the low-energy physics of quantum many-body systems with high accuracy. As a variational method, DMRG is an efficient algorithm that attempts to find the lowest-energy matrix product state wavefunction of a Hamiltonian. It was invented in 1992 by Steven R. White and it is nowadays the most efficient method for 1-dimensional systems.
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The variational characterization of singular values and vectors implies as a special case a variational characterization of the angles between subspaces and their associated canonical vectors. This characterization includes the angles 0 and \pi/2 introduced above and orders the angles by increasing value. It can be given the form of the below alternative definition. In this context, it is customary to talk of principal angles and vectors.
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In this context, it can be used for example as a tool to interpolate pre-calculated interatomic potentials or directly solving the Schrödinger equation with a variational method.
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Wachs earned her doctorate in 1977 from the University of California, San Diego, under the supervision of Adriano Garsia. Her dissertation was Discrete Variational Techniques in Finite Mathematics.
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Rockafellar’s research is motivated by the goal of organizing mathematical ideas and concepts into robust frameworks that yield new insights and relations. This approach is most salient in his seminal book “Variational Analysis” (1998, with Roger J-B Wets), where numerous threads developed in the areas of convex analysis, nonlinear analysis, calculus of variation, mathematical optimization, equilibrium theory, and control systems were brought together to produce a unified approach to variational problems in finite dimensions. These various fields of study are now referred to as variational analysis. In particular, the text dispenses of differentiability as a necessary property in many areas of analysis and embraces nonsmoothness, set-valuedness, and extended real- valuedness, while still developing far-reaching calculus rules.
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With an appropriate variational principle, one could deduce the equations of motion for a given mechanical or optical system. Soon, scientists worked out the variational principles for the theory of elasticity, electromagnetism, and fluid mechanics (and, in the future, relativity and quantum theory). Whilst variational principles did not necessarily provide a simpler way to solve problems, they were of interest for philosophical or aesthetic reasons, though scientists at this time were not as motivated by religion in their work as their predecessors. In 1828, miller and autodidactic mathematician George Green published An Essay on the Application of Mathematical Analysis to the Theories of Electricity and Magnetism, making use of the mathematics of potential theory developed by Continental mathematicians.
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Areas of mathematics and science that contributed to the development of complementarity theory include: optimization, equilibrium problems, variational inequality theory, fixed point theory, topological degree theory and nonlinear analysis.
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Many of the Sobolev embedding theorems require that the domain of study be a Lipschitz domain. Consequently, many partial differential equations and variational problems are defined on Lipschitz domains.
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In this paper, we study synchronal and cyclic algorithms for finding a common fixed point x ∗ of a finite family of strictly pseudocontractive mappings, which solve the variational inequality.
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In mathematical analysis, Ekeland's variational principle, discovered by Ivar Ekeland, is a theorem that asserts that there exists nearly optimal solutions to some optimization problems. Ekeland's variational principle can be used when the lower level set of a minimization problems is not compact, so that the Bolzano-Weierstrass theorem cannot be applied. Ekeland's principle relies on the completeness of the metric space. Ekeland's principle leads to a quick proof of the Caristi fixed point theorem.
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Endre Dudich (20 March 1895 – 5 February 1971) was a Hungarian Kossuth Prize- winning professor, academician, and zoologist, noted for his application of mathematical methods for variational study of insects.
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X = (1, 0) and Y = (0, 2) are complementary, but X = (1, 1) and Y = (2, 0) are not. A complementarity problem is a special case of a variational inequality.
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Some Variational Formulations for minimum surface. Acta Mechanica, vol.89/1–4, 1991, pp. 33–43. The alternative approximated approach to the minimum surface problem solution is based on SGM.
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Karl Friedrich Gauss The principle of least constraint is one variational formulation of classical mechanics enunciated by Carl Friedrich Gauss in 1829, equivalent to all other formulations of analytical mechanics.
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The work of a force on a particle along a virtual displacement is known as the virtual work. Historically, virtual work and the associated calculus of variations were formulated to analyze systems of rigid bodies,C. Lánczos, The Variational Principles of Mechanics, 4th Ed., General Publishing Co., Canada, 1970 but they have also been developed for the study of the mechanics of deformable bodies.Dym, C. L. and I. H. Shames, Solid Mechanics: A Variational Approach, McGraw-Hill, 1973.
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For the ideal diode systems, the computations are considerably more difficult, but provided some generally valid conditions hold, the differential variational inequality can be shown to have index one. Differential variational inequalities with index greater than two are generally not meaningful, but certain conditions and interpretations can make them meaningful (see the references Acary, Brogliato and Goeleven, and Heemels, Schumacher, and Weiland below). One crucial step is to first define a suitable space of solutions (Schwartz' distributions).
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The variational Bayesian methods used for model estimation in DCM are based on the Laplace assumption, which treats the posterior over parameters as Gaussian. This approximation can fail in the context of highly non-linear models, where local minima may preclude the free energy from serving as a tight bound on log model evidence. Sampling approaches provide the gold standard; however, they are time consuming and have typically been used to validate the variational approximations in DCM.
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In mathematical analysis, Ekeland's variational principle, discovered by Ivar Ekeland, is a theorem that asserts that there exist a nearly optimal solution to a class of optimization problems. Ekeland's variational principle can be used when the lower level set of a minimization problems is not compact, so that the Bolzano-Weierstrass theorem can not be applied. Ekeland's principle relies on the completeness of the metric space. Ekeland's principle leads to a quick proof of the Caristi fixed point theorem.
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A method that avoids making the variational overestimation of HF in the first place is Quantum Monte Carlo (QMC), in its variational, diffusion, and Green's function forms. These methods work with an explicitly correlated wave function and evaluate integrals numerically using a Monte Carlo integration. Such calculations can be very time-consuming. The accuracy of QMC depends strongly on the initial guess of many-body wave-functions and the form of the many-body wave-function.
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Any physical law which can be expressed as a variational principle describes a self-adjoint operator. These expressions are also called Hermitian. Such an expression describes an invariant under a Hermitian transformation.
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Full configuration interaction (or full CI) is a linear variational approach which provides numerically exact solutions (within the infinitely flexible complete basis set) to the electronic time-independent, non-relativistic Schrödinger equation.
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The formalization of projected dynamical systems began in the 1990s. However, similar concepts can be found in the mathematical literature which predate this, especially in connection with variational inequalities and differential inclusions.
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Variational free energy is an information theoretic functional and is distinct from thermodynamic (Helmholtz) free energy.Evans, D. J. (2003). A non-equilibrium free energy theorem for deterministic systems. Molecular Physics , 101, 15551–4.
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In probability theory, the total variation distance is a distance measure for probability distributions. It is an example of a statistical distance metric, and is sometimes called the statistical distance or variational distance.
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In mathematics, the Lagrangian theory on fiber bundles is globally formulated in algebraic terms of the variational bicomplex, without appealing to the calculus of variations. For instance, this is the case of classical field theory on fiber bundles (covariant classical field theory). The variational bicomplex is a cochain complex of the differential graded algebra of exterior forms on jet manifolds of sections of a fiber bundle. Lagrangians and Euler–Lagrange operators on a fiber bundle are defined as elements of this bicomplex.
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In particular, whereas Monte Carlo techniques provide a numerical approximation to the exact posterior using a set of samples, Variational Bayes provides a locally-optimal, exact analytical solution to an approximation of the posterior. Variational Bayes can be seen as an extension of the EM (expectation-maximization) algorithm from maximum a posteriori estimation (MAP estimation) of the single most probable value of each parameter to fully Bayesian estimation which computes (an approximation to) the entire posterior distribution of the parameters and latent variables. As in EM, it finds a set of optimal parameter values, and it has the same alternating structure as does EM, based on a set of interlocked (mutually dependent) equations that cannot be solved analytically. For many applications, variational Bayes produces solutions of comparable accuracy to Gibbs sampling at greater speed.
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He also characterized the class of coordinate functions which give the best order of approximation, and has studied the stability of the variational-difference process and the growth of the condition number of the variation-difference matrix. Mikhlin also studied the finite element approximation in weighted Sobolev spaces related to the numerical solution of degenerate elliptic equations. He found the optimal order of approximation for some methods of solution of variational inequalities. The fourth branch of his research in numerical mathematics is a method for the solution of Fredholm integral equations which he called resolvent method: its essence rely on the possibility of substituting the kernel of the integral operator by its variational-difference approximation, so that the resolvent of the new kernel can be expressed by simple recurrence relations.
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Lagrangian mechanics can be applied to geometrical optics, by applying variational principles to rays of light in a medium, and solving the EL equations gives the equations of the paths the light rays follow.
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Information Theory and Statistical Mechanics. Physical Review Series II, 106 (4), 620–30. Finally, because the time average of energy is action, the principle of minimum variational free energy is a principle of least action.
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FABIA utilizes well understood model selection techniques like variational approaches and applies the Bayesian framework. The generative framework allows FABIA to determine the information content of each bicluster to separate spurious biclusters from true biclusters.
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He derived an existence and regularity theory for a class of constrained variational problems. Parks has discovered, and characterized, a type of minimal surface with surprising properties, defined in terms of the Jacobi elliptic functions.
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It was first designed as a model for brain functioning using variational Bayesian learning. After that, the algorithm was adapted to machine learning. It can be viewed as a way to train a Helmholtz Machine.
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Illumination can also be performed with two (pivoted) lightsheets (see above) to further reduce these artifacts. Alternatively, an algorithm called VSNR (Variational Stationary Noise Remover) has been developed and is available as a free Fiji plugin.
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Guido Stampacchia (26 March 1922 – 27 April 1978) was a 20th-century Italian mathematician, known for his work on the theory of variational inequalities, the calculus of variation and the theory of elliptic partial differential equations..
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Over a 20-year period, he conducted 3 ongoing seminars: with Yakov Sinai on dynamical systeme, with V. A. Egorov on celestial mechanics, and with M. Zelikin and V. M. Tikhomirov on variational problems and optimal control.
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Berlin: Springer Verlag. and embodied cognition. Because free energy can be expressed as the expected energy of observations under the variational density minus its entropy, it is also related to the maximum entropy principle.Jaynes, E. T. (1957).
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Weil's proof relies on conformal maps and harmonic analysis, Croke's proof is based on an inequality of Santaló in integral geometry, while Kleiner adopts a variational approach which reduces the problem to an estimate for total curvature.
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Mansoori completed his PhD at the University of Oklahoma in 1969 with a dissertation on "A Variational Approach to the Equilibrium Thermodynamic Properties of Simple Liquids and Phase Transitions". Mansoori did post-doctoral work at Rice University.
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DIVADIVA homepage (Data-Interpolating Variational Analysis) allows the spatial interpolation/gridding of data (analysis) in an optimal way, comparable to optimal interpolation (OI), taking into account uncertainties on observations. In comparison to standard OI, used in Data assimilation, DIVA, when applied to ocean data, takes into account coastlines, sub-basins and advection because of its variational formulation on the real domain.DIVA formulation Calculations are highly optimized and rely on a finite element resolution. Tools to generate the finite element mesh are provided as well as tools to optimize the parameters of the analysis.
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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, the G-matrix and distribution of fitness effects. Variational properties also include the production of DNA sequence variation, epigenetic variation, and phenotypic variation. While the genome is typically thought of as the storehouse of information that generates the organism, it can also be seen as the set of heritable degrees of freedom for varying the organism.
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In 2003 he became a professor at Carnegie Mellon University. He works on partial differential equations, minimal surfaces, and variational inequalities, with mathematical applications to the microstructure of biological materials, to solid state physics, and to materials science, including crystalline microstructure, liquid crystals, molecular mechanisms of intracellular transport, and models of ion transport. In 2012 Kinderlehrer was elected a Fellow of the American Mathematical Society.AMS – List of Fellows of the American Mathematical Society In 1974 in Vancouver he was an invited speaker (Elliptic Variational Inequalities) at the International Mathematical Congress.
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A metric theory of finite-dimensional epigraphical convergence ("cosmic convergence") appears in Variational analysis. Wets and his coauthor R. Tyrrell Rockafellar were awarded the 1997 Frederick W. Lanchester Prize by the Institute for Operations Research and the Management Sciences (INFORMS) for their monograph Variational Analysis, which was published in November 1997 and copyrighted in 1998. With Rockafellar, Wets proposed, studied, and implemented the progressive-hedging algorithm for stochastic programming. Besides his theoretical and computational contributions, Wets has worked with applications on lake ecology (IIASA), finance (Frank Russel investment system), and developmental economics (World Bank).
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In linear algebra and functional analysis, the min-max theorem, or variational theorem, or Courant-Fischer-Weyl min-max principle, is a result that gives a variational characterization of eigenvalues of compact Hermitian operators on Hilbert spaces. It can be viewed as the starting point of many results of similar nature. This article first discusses the finite-dimensional case and its applications before considering compact operators on infinite-dimensional Hilbert spaces. We will see that for compact operators, the proof of the main theorem uses essentially the same idea from the finite-dimensional argument.
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Variational transition-state theory is a refinement of transition-state theory. When using transition-state theory to estimate a chemical reaction rate, the dividing surface is taken to be a surface that intersects a first- order saddle point and is also perpendicular to the reaction coordinate in all other dimensions. When using variational transition-state theory, the position of the dividing surface between reactant and product regions is variationally optimized to minimize the reaction rate. This minimizes the effects of recrossing, and gives a much more accurate result.
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He won the 1999 Fermat Prize, jointly with Frédéric Hélein, for several important contributions to the theory of variational calculus. He also won the 2003 for his fundamental discoveries at the interface between analysis, topology, geometry, and physics.
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The idea of VMS turbulence modeling for Large Eddy Simulations(LES) of incompressible Navier–Stokes equations was introduced by Hughes et al. in 2000 and the main idea was to use - instead of classical filtered techniques - variational projections.
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He had applied his method to a big number of pivot and plate analysis problems. Some time before I.G.Bubnov developed a similar approach for the variational problem solution, which he interpreted as a variant of Ritz method algorithm.
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Department of Physics and Astronomy, University of New Mexico. April 1, 2018. Accessed January 18, 2019. It treats spinors, the variational- principle formulation, the initial-value formulation, (exact) gravitational waves, singularities, Penrose diagrams, Hawking radiation, and black-hole thermodynamics.
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One of the important developments arising from the geometric approach to mechanics is the incorporation of the geometry into numerical methods. In particular symplectic and variational integrators are proving particularly accurate for long-term integration of Hamiltonian and Lagrangian systems.
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In 2019 a variational autoencoder framework was used to do population synthesis by approximating high-dimensional survey data. By sampling agents from the approximated distribution new synthetic 'fake' populations, with similar statistical properties as those of the original population, were generated.
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Coonce completed his PhD in 1969 at the University of Delaware with a dissertation on A Variational Method for Functions of Bounded Boundary Rotation.Harry Bernand Coonce – Mathematics Genealogy Project. Coonce presently is a retired mathematics professor of Minnesota State University, Mankato.
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The Redfield equation and Lindblad equation are examples of approximate quantum master equations assumed to be Markovian. More accurate quantum master equations for certain applications include the polaron transformed quantum master equation, and the VPQME (variational polaron transformed quantum master equation).
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Periodic minimal surfaces can be constructed in S3 and H3.K. Polthier. New periodic minimal surfaces in h3. In G. Dziuk, G. Huisken, and J. Hutchinson, editors, Theoretical and Numerical Aspects of Geometric Variational Problems, volume 26, pages 201–210.
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The evolutionary biologist Günter P. Wagner described Goodwin's structuralism as "a fringe movement in evolutionary biology".Wagner, Günter P., Homology, Genes, and Evolutionary Innovation. Princeton University Press. 2014. Chapter 1: The Intellectual Challenge of Morphological Evolution: A Case for Variational Structuralism.
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A finite element method is characterized by a variational formulation, a discretization strategy, one or more solution algorithms, and post-processing procedures. Examples of the variational formulation are the Galerkin method, the discontinuous Galerkin method, mixed methods, etc. A discretization strategy is understood to mean a clearly defined set of procedures that cover (a) the creation of finite element meshes, (b) the definition of basis function on reference elements (also called shape functions) and (c) the mapping of reference elements onto the elements of the mesh. Examples of discretization strategies are the h-version, p-version, hp-version, x-FEM, isogeometric analysis, etc.
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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.Lorentz Trial Function for the Hydrogen Atom: A Simple, Elegant Exercise Thomas Sommerfeld Journal of Chemical Education 2011 88 (11), 1521–1524 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.
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For high energies and/or weak potential it can also be solved perturbatively by means of Born series. The method convenient also in the case of many-body physics, like in description of atomic, nuclear or molecular collisions is the method of R-matrix of Wigner and Eisenbud. Another class of methods is based on separable expansion of the potential or Green's operator like the method of continued fractions of Horáček and Sasakawa. Very important class of methods is based on variational principles, for example the Schwinger-Lanczos method combining the variational principle of Schwinger with Lanczos algorithm.
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Hilbert's definition of a regular variational problem is stronger than the currently used one, found, for example, in . property means that such kind of variational problems are minimum problems, property is the ellipticity condition on the Euler–Lagrange equations associated to the given functional, while property is a simple regularity assumption the function .Since Hilbert considers all derivatives in the "classical", i.e. not in the weak but in the strong, sense, even before the statement of its analyticity in , the function is assumed to be at least , as the use of the Hessian determinant in implies.
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Variational circuits are a family of algorithms which utilize training based on circuit parameters and an objective function. Variational circuits are generally composed of a classical device communicating input parameters (random or pre-trained parameters) into a quantum device, along with a classical Mathematical optimization function. These circuits are very heavily dependent on the architecture of the proposed quantum device because parameter adjustments are adjusted based solely on the classical components within the device. Though the application is considerably infantile in the field of quantum machine learning, it has incredibly high promise for more efficiently generating efficient optimization functions.
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In fluid dynamics and plasma physics, the Clebsch representation provides a means to overcome the difficulties to describe an inviscid flow with non-zero vorticity – in the Eulerian reference frame – using Lagrangian mechanics and Hamiltonian mechanics. At the critical point of such functionals the result is the Euler equations, a set of equations describing the fluid flow. Note that the mentioned difficulties do not arise when describing the flow through a variational principle in the Lagrangian reference frame. In case of surface gravity waves, the Clebsch representation leads to a rotational-flow form of Luke's variational principle.
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In 1962, Dreyfus simplified the Dynamic Programming-based derivation of backpropagation (due to Henry J. Kelley and Arthur E. Bryson) using only the chain rule.Stuart Dreyfus (1962). The numerical solution of variational problems. Journal of Mathematical Analysis and Applications, 5(1), 30-45.
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The work of De Donder on the other hand started from the theory of integral invariants of Élie Cartan.Roger Bielawski, Kevin Houston, Martin Speight: Variational Problems in Differential Geometry, London Mathematical Society Lecture Notes Series, no. 394, University of Leeds, 2009, , p.
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Hilbert's twentieth problem is one of the 23 Hilbert problems set out in a celebrated list compiled in 1900 by David Hilbert. It asks whether all boundary value problems can be solved (that is, do variational problems with certain boundary conditions have solutions).
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Aimed at beginning graduate students, it covers spinors, the variational-principle formulation, the initial-value formulation, (exact) gravitational waves, singularities, Penrose diagrams, Hawking radiation, and black-hole thermodynamics.A Guide to Relativity Books. John C. Baez et al. University of California, Riverside. September 1998.
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Due to the above-mentioned Serre- Swan theorem, odd classical fields on a smooth manifold are described in terms of graded manifolds. Extended to graded manifolds, the variational bicomplex provides the strict mathematical formulation of Lagrangian classical field theory and Lagrangian BRST theory.
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Ivar Ekeland is an expert on variational analysis, which studies mathematical optimization of spaces of functions. His research on periodic solutions of Hamiltonian systems and particularly to the theory of Kreĭn indices for linear systems (Floquet theory) was described in his monograph.
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The notion that self-organising biological systems – like a cell or brain – can be understood as minimising variational free energy is based upon Helmholtz’s work on unconscious inferenceHelmholtz, H. (1866/1962). Concerning the perceptions in general. In Treatise on physiological optics (J. Southall, Trans.
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Smith, Alice E.; Coit David W. Penalty functions Handbook of Evolutionary Computation, Section C 5.2. Oxford University Press and Institute of Physics Publishing, 1996. Courant, R. Variational methods for the solution of problems of equilibrium and vibrations. Bull. Amer. Math. Soc., 49, 1-23, 1943.
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A variational principle in physics is an alternative method for determining the state or dynamics of a physical system, by identifying it as an extremum (minimum, maximum or saddle point) of a function or functional. This article describes the historical development of such principles.
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In the Lagrangian formalism, homogeneity in space implies conservation of momentum, and homogeneity in time implies conservation of energy. This is shown, using variational calculus, in standard textbooks like the classical reference text of Landau & Lifshitz. This is a particular application of Noether's theorem.
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3, pp. 747-66, 2008. In variational filtering, an ensemble of particles diffuse over the free energy landscape in a frame of reference that moves with the expected (generalized) motion of the ensemble. This provides a relatively simple scheme that eschews Gaussian (unimodal) assumptions.
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Milne, E.A. (1929). The effect of collisions on monochromatic radiative equilibrium, Monthly Notices of the Royal Astronomical Society, 88: 493–502.Gyarmati, I. (1970). Non-equilibrium Thermodynamics. Field Theory and Variational Principles, translated by E. Gyarmati and W.F. Heinz, Springer, Berlin, pp. 63–66.
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Hans Hagen has published over four hundred articles in journals, books and conference proceedings. His research interests are deep and very broad. His work concentrates on physically based modeling, curve and surface interrogation and topology-based visualization. Particular emphasis is placed on variational design.
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The Zener ratio is only applicable to cubic crystals. To overcome this limitation, a 'Universal Elastic Anisotropy Index (AU)' was formulated from variational principles of elasticity and tensor algebra. The AU is now used to quantify the anisotropy of elastic crystals of all classes.
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Walther Heinrich Wilhelm Ritz (22 February 1878 – 7 July 1909) was a Swiss theoretical physicist. He is most famous for his work with Johannes Rydberg on the Rydberg–Ritz combination principle. Ritz is also known for the variational method named after him, the Ritz method.
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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.
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Daley took on a project, in close partnership with Ed Barker, to design and construct a three-dimensional variational data assimilation system specifically meant to serve the needs of the US Navy. Over a period of 5 years, he and Ed have carefully built and tested this system. It is known as the NRL Atmospheric Variational Data Assimilation System, or NAVDAS, and went operational at FNMOC and at Navy regional METOC centers in 2003.Navel Research Lab, Monterey NAVDAS is designed to meet data assimilation needs of both global models and regional nested models and holds great promise to provide a substantial increase in Navy model prediction accuracy.
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Feynman was also interested in the relationship between physics and computation. He was also one of the first scientists to conceive the possibility of quantum computers. In the 1980s he began to spend his summers working at Thinking Machines Corporation, helping to build some of the first parallel supercomputers and considering the construction of quantum computers. In 1984–1986, he developed a variational method for the approximate calculation of path integrals, which has led to a powerful method of converting divergent perturbation expansions into convergent strong-coupling expansions (variational perturbation theory) and, as a consequence, to the most accurate determination of critical exponents measured in satellite experiments.
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Variational Algorithms for Approximate Bayesian Inference. Ph.D. Thesis, University College London. Therefore, its minimisation can be seen as a Bayesian inference process. When a system actively makes observations to minimise free energy, it implicitly performs active inference and maximises the evidence for its model of the world.
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1-27, Arxiv (An algorithmic version of the Fekete problem is problem number 7 of Smale's problems.) In 2018 Boucksom was an Invited Speaker with talk Variational and non-Archimedean aspects of the Yau-Tian-Donaldson conjecture at the International Congress of Mathematicians in Rio de Janeiro.
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The term was introduced by Cornelius Lanczos in his book The Variational Principles of Mechanics (1970). Monogenic systems have excellent mathematical characteristics and are well suited for mathematical analysis. Pedagogically, within the discipline of mechanics, it is considered a logical starting point for any serious physics endeavour.
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She determines the length of each rendition and could change the rhythm at any time, speeding up or slowing down, accord to her personal taste and the composition she plays.Terada, Yoshitaka. "Variational and Improvisational Techniques of Gandingan Playing in the Maguindanaon Kulintang Ensemble." Asian Music XXVII.
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Since 2003, J.C. Michel and P. SuquetMichel J.C., Suquet P., « Nonuniform Transformation Field Analysis », Int. J. Solids and Struct., 40, 2003, pp. 6937–6955Michel JC. and P. Suquet, « A model-reduction approach in micromechanics of materials preserving the variational structure of constitutive relations », J. Mech. Phys.
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Many of his early papers were written in German and are now being translated.Hinz DF and Fried E (2014) Translation of Michael Sadowsky’s Paper "An Elementary Proof for the Existence of a Developable Möbius Band and the Attribution of the Geometric Problem to a Variational Problem", Journal of Elasticity .
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The FEniCS Project is a collection of free and open-source software components with the common goal to enable automated solution of differential equations. The components provide scientific computing tools for working with computational meshes, finite-element variational formulations of ordinary and partial differential equations, and numerical linear algebra.
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In real, complex, and functional analysis, derivatives are generalized to functions of several real or complex variables and functions between topological vector spaces. An important case is the variational derivative in the calculus of variations. Repeated application of differentiation leads to derivatives of higher order and differential operators.
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The principle may be used for the calculation of the scattering amplitude in the similar way like the variational principle for bound states, i.e. the form of the wave functions \psi, \psi' is guessed, with some free parameters, that are determined from the condition of stationarity of the functional.
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Bliss's Lectures more or less constitutes the culmination of the classic calculus of variations of Weierstrass, Hilbert, and Bolza. Subsequent work on variational problems would strike out in new directions, such as Morse theory, optimal control, and dynamic programming. Bliss also studied singularities of real transformations in the plane.
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The second definition clarified the meaning of the topological entropy: for a system given by an iterated function, the topological entropy represents the exponential growth rate of the number of distinguishable orbits of the iterates. An important variational principle relates the notions of topological and measure-theoretic entropy.
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The second square bracketed term is an expression of free surface boundary conditions; \vec n is the unit vector normal to S. For a free body (as we assume it), the latter term sums to zero and can be ignored. Thus the set of u_i that satisfies the previously mentioned conditions are those displacements that correspond to ω being a normal mode frequency of the system. This suggests that the normal vibrations of an object (Fig. 1) may be calculated by applying a variational method (in our case the Rayleigh-Ritz variational method, explained in the next paragraph) to determine both the normal mode frequencies and the description of the physical oscillations.
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In physics, Hamilton's principle is William Rowan Hamilton's formulation of the principle of stationary action. (See that article for historical formulations.) It states that the dynamics of a physical system are determined by a variational problem for a functional based on a single function, the Lagrangian, which contains all physical information concerning the system and the forces acting on it. The variational problem is equivalent to and allows for the derivation of the differential equations of motion of the physical system. Although formulated originally for classical mechanics, Hamilton's principle also applies to classical fields such as the electromagnetic and gravitational fields, and plays an important role in quantum mechanics, quantum field theory and criticality theories.
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Lord Kelvin and Dirichlet suggested a solution to the problem by a variational method based on the minimization of "Dirichlet's energy". According to Hans Freudenthal (in the Dictionary of Scientific Biography, vol 11), Bernhard Riemann was the first mathematician who solved this variational problem based on a method which he called Dirichlet's principle. The existence of a unique solution is very plausible by the 'physical argument': any charge distribution on the boundary should, by the laws of electrostatics, determine an electrical potential as solution. However, Karl Weierstrass found a flaw in Riemann's argument, and a rigorous proof of existence was found only in 1900 by David Hilbert, using his direct method in the calculus of variations.
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The corresponding variational problem is a max-min problem: one looks for a contour that minimizes the "equilibrium" measure. The study of the variational problem and the proof of existence of a regular solution, under some conditions on the external field, was done in ; the contour arising is an "S-curve", as defined and studied in the 1980s by Herbert R. Stahl, Andrei A. Gonchar and Evguenii A Rakhmanov. An alternative asymptotic analysis of Riemann–Hilbert factorization problems is provided in , especially convenient when jump matrices do not have analytic extensions. Their method is based on the analysis of d-bar problems, rather than the asymptotic analysis of singular integrals on contours.
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Many density-functional tight-binding methods, such as DFTB+, Fireball, and Hotbit, are built based on the Harris energy functional. In these methods, one often does not perform self-consistent Kohn–Sham DFT calculations and the total energy is estimated using the Harris energy functional, although a version of the Harris functional where one does perform self-consistency calculations has been used. These codes are often much faster than conventional Kohn–Sham DFT codes that solve Kohn–Sham DFT in a self-consistent manner. While the Kohn–Sham DFT energy is a variational functional (never lower than the ground state energy), the Harris DFT energy was originally believed to be anti- variational (never higher than the ground state energy).
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A few days later, during a conversation with his family Doctor Damiano Aprile, Signorini told him: reports the episode following the remembraces of Mauro Picone: see the entry "Antonio Signorini" for further details. According to the solution of the Signorini problem coincides with the birth of the field of variational inequalities.
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The Global model provides boundary information for the now retired North Atlantic European (NAE) model, for which additional shorter runs (48 hours) are produced twice a day. The model is kept close to the real atmosphere using hybrid 4D-Var data assimilationHybrid variational/ensemble data assimilation . Met Office 2011. of observations.
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619-622, October 2012. Note that alternative multi-stream data fusion strategies have also been proposed in the recent literature, e.g.Sotirios P. Chatzis, Dimitrios Kosmopoulos, "Visual Workflow Recognition Using a Variational Bayesian Treatment of Multistream Fused Hidden Markov Models," IEEE Transactions on Circuits and Systems for Video Technology, vol. 22, no.
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Classically, the action is a functional on the configuration space. The on-shell solutions are given by the variational problem of extremizing the action subject to boundary conditions. While the boundary is often ignored in textbooks, it is crucial in the study of flows. Suppose we have a "flow", i.e.
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The coordinate-wise mean of a point set is the centroid, which solves the same variational problem in the plane (or higher-dimensional Euclidean space) that the familiar average solves on the real line -- that is, the centroid has the smallest possible average squared distance to all points in the set.
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Cambridge University Press, , pp. 117–119. In classical mechanics for instance, in the action formulation, extremal solutions to the variational principle are on shell and the Euler–Lagrange equations give the on-shell equations. Noether's theorem regarding differentiable symmetries of physical action and conservation laws is another on-shell theorem.
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Exact maximum likelihood learning in this model is intractable, but approximate learning of DBMs can be carried out by using a variational approach, where mean-field inference is used to estimate data-dependent expectations and an MCMC based stochastic approximation procedure is used to approximate the model’s expected sufficient statistics.
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Nikolsky made fundamental contributions to functional analysis, approximation of functions, quadrature formulas, enclosed functional spaces and their applications to variational solutions of partial differential equations. He created a large scientific school of functions' theory and its applications. He authored over 100 scientific publications, including 3 monographs, 2 college textbooks and 7 school textbooks.
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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.
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CRTM contains forward, tangent linear, adjoint and K (full Jacobian matrices) versions of the model; the latter three modules are used in inversion methods, including variational assimilation and satellite retrievals. One of several applications of CRTM are retrievals of brightness temperature and sea surface temperature from Advanced Very High Resolution Radiometer sensor.
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Morrey worked on numerous fundamental problems in analysis, among them, the existence of quasiconformal maps, the measurable Riemann mapping theorem, Plateau's problem in the setting of Riemannian manifolds, and the characterization of lower semicontinuous variational problems in terms of quasiconvexity. He greatly contributed to the solution of Hilbert's nineteenth and twentieth problems.
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He married fellow student Sara Naldini in October 1948. Children Mauro, Renata, Giulia, and Franca were born in 1949, 1951, 1955 and 1956 respectively. Stampacchia was active in research and teaching throughout his career. He made key contributions to a number of fields, including calculus of variation, variational inequalities and differential equations.
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A consequence is that can be obtained from the functional determinant of the harmonic oscillator. This functional determinant can be computed via a product expansion, and is equivalent to the Wallis product formula. The calculation can be recast in quantum mechanics, specifically the variational approach to the spectrum of the hydrogen atom.
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An algebraic proof, based on the variational interpretation of eigenvalues, has been published in Magnus' Matrix Differential Calculus with Applications in Statistics and Econometrics. From the geometric point of view, B'AB can be considered as the orthogonal projection of A onto the linear subspace spanned by B, so the above results follow immediately.
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The method was initially developed by Fritz Coester and Hermann Kümmel in the 1950s for studying nuclear-physics phenomena, but became more frequently used when in 1966 Jiří Čížek (and later together with Josef Paldus) reformulated the method for electron correlation in atoms and molecules. It is now one of the most prevalent methods in quantum chemistry that includes electronic correlation. CC theory is simply the perturbative variant of the many-electron theory (MET) of Oktay Sinanoğlu, which is the exact (and variational) solution of the many-electron problem, so it was also called "coupled-pair MET (CPMET)". J. Čížek used the correlation function of MET and used Goldstone-type perturbation theory to get the energy expression, while original MET was completely variational.
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Euler continued to write on the topic; in his Reflexions sur quelques loix generales de la nature (1748), he called the quantity "effort". His expression corresponds to what we would now call potential energy, so that his statement of least action in statics is equivalent to the principle that a system of bodies at rest will adopt a configuration that minimizes total potential energy. The full importance of the principle to mechanics was stated by Joseph Louis Lagrange in 1760, although the variational principle was not used to derive the equations of motion until almost 75 years later, when William Rowan Hamilton in 1834 and 1835 applied the variational principle to the function L=T-V to obtain what are now called the Lagrangian equations of motion.
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In mathematics, calculus on finite weighted graphs is a discrete calculus for functions whose domain is the vertex set of a graph with a finite number of vertices and weights associated to the edges. This involves formulating discrete operators on graphs which are analogous to differential operators in calculus, such as graph Laplacians (or discrete Laplace operators) as discrete versions of the Laplacian, and using these operators to formulate differential equations, difference equations, or variational models on graphs which can be interpreted as discrete versions of partial differential equations or continuum variational models. Such equations and models are important tools to mathematically model, analyze, and process discrete information in many different research fields, e.g., image processing, machine learning, and network analysis.
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The second branch deals with the notion of stability of a numerical process introduced by Mikhlin himself. When applied to the variational method, this notion enables him to state necessary and sufficient conditions in order to minimize errors in the solution of the given problem when the error arising in the numerical construction of the algebraic system resulting from the application of the method itself is sufficiently small, no matter how large is the system's order. The third branch is the study of variational-difference and finite element methods. Mikhlin studied the completeness of the coordinate functions used in this methods in the Sobolev space }, deriving the order of approximation as a function of the smoothness properties of the functions to be approximation of functions approximated.
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Felix Klein's Erlangen program attempted to identify such invariants under a group of transformations. In what is referred to in physics as Noether's theorem, the Poincaré group of transformations (what is now called a gauge group) for general relativity defines symmetries under a group of transformations which depend on a variational principle, or action principle.
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A continuous-time optimal control problem is information rich. A number of interesting properties of a given problem can be derived by applying the Pontryagin's minimum principle or the Hamilton–Jacobi–Bellman equations. These theories implicitly use the continuity of time in their derivation.B. S. Mordukhovich, Variational Analysis and Generalized Differentiation: Basic Theory, Vol.
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It is based on a variational principle of least action, formulated in generalized coordinates.B Balaji and K Friston, "Bayesian state estimation using generalized coordinates," Proc. SPIE, p. 80501Y , 2011 Note that the concept of "generalized coordinates" as used here differs from the concept of generalized coordinates of motion as used in (multibody) dynamical systems analysis.
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There are mainly two kinds of methods to model the unilateral constraints. The first kind is based on smooth contact dynamics, including methods using Hertz's models, penalty methods, and some regularization force models, while the second kind is based on the non- smooth contact dynamics, which models the system with unilateral contacts as variational inequalities.
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His dissertation, written under the supervision of Gheorghe Călugăreanu, was titled Variational methods in the theory of univalent functions. He continued as faculty at Babeș-Bolyai University, rising to the rank of Professor in 1970. Mocanu was an invited professor at the University of Conakry in 1966–1967, and the Ohio State University in 1992.
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The simple equations that model these finite elements are then assembled into a larger system of equations that models the entire problem. The FEM then uses variational methods from the calculus of variations to approximate a solution by minimizing an associated error function. Studying or analyzing a phenomenon with FEM is often referred to as finite element analysis (FEA).
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However, they alter the probabilities of generating different offspring under the variation operators, and thus alter the individual's variational properties. Experiments seem to show faster convergence when using program representations that allow such non-coding genes, compared to program representations that do not have any non-coding genes. Julian F. Miller. "Cartesian Genetic Programming". p. 19.
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Various models have been proposed, introducing interactions between instantons or using variational methods (like the "valley approximation") endeavoring to approximate the exact multi-instanton solution as closely as possible. Many phenomenological successes have been reached. Whether an instanton liquid can explain confinement in 3+1 dimensional QCD is not known, but many physicists think that it is unlikely.
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Boris Mordukhovich is an American mathematician recognized for his research in the areas of nonlinear analysis, optimization, and control theory. Mordukhovich is one of the founders of modern variational analysis and generalized differentiation. Currently he is Distinguished University Professor and Lifetime Scholar of the Academy of Scholars at Wayne State University (Vice President, 2009-2010 and President, 2010-2011).
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As for Hermitian matrices, the key point is to prove the existence of at least one nonzero eigenvector. One cannot rely on determinants to show existence of eigenvalues, but one can use a maximization argument analogous to the variational characterization of eigenvalues. If the compactness assumption is removed, it is not true that every self-adjoint operator has eigenvectors.
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Hans Hagen (born 1953) is a professor of computer science at the University of Kaiserslautern.Curriculum vitae from Kaiserslautern, retrieved 2011-05-08. One of his major research contributions were geometric modeling techniques called "Variational Design" of curves and surfaces. His curve and surface interrogation methods are among his many contributions to topological and geometric aspects of scientific visualization.
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Gradient descent can be extended to handle constraints by including a projection onto the set of constraints. This method is only feasible when the projection is efficiently computable on a computer. Under suitable assumptions, this method converges. This method is a specific case of the forward-backward algorithm for monotone inclusions (which includes convex programming and variational inequalities).
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MFO workshop Mathematical and Numerical Aspects of Quantum Chemistry Problems Maria J. Esteban (born at Alonsotegi, 1956)Curriculm vitae, accessed 2018-10-13 is a Basque-French mathematician. In her research she studies nonlinear partial differential equations, mainly by the use of variational methods, with applications to physics and quantum chemistry. She has also worked on fluid-structure interaction.
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In 1997, she received her PhD in computer science from the University of Siegen. The title of her PhD thesis was "Variational Object-Oriented Programming Beyond Classes and Inheritance". From 1999 she was assistant professor for three years at Northeastern University. In 2002, she became Professor of Computer Science at the Department of Computer Science of TU Darmstadt.
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Lord Rayleigh published a generalization of the virial theorem in 1903. Henri Poincaré applied a form of the virial theorem in 1911 to the problem of determining cosmological stability. A variational form of the virial theorem was developed in 1945 by Ledoux. A tensor form of the virial theorem was developed by Parker, Chandrasekhar and Fermi.
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Mathematical programming with equilibrium constraints (MPEC) is the study of constrained optimization problems where the constraints include variational inequalities or complementarities. MPEC is related to the Stackelberg game. MPEC is used in the study of engineering design, economic equilibrium, and multilevel games. MPEC is difficult to deal with because its feasible region is not necessarily convex or even connected.
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Claudia Alejandra Sagastizábal is an applied mathematician known for her research in convex optimization and energy management, and for her co- authorship of the book Numerical Optimization: Theoretical and Practical Aspects. She is a researcher at the University of Campinas in Brazil. Since 2015 she has been editor-in-chief of the journal Set-Valued and Variational Analysis.
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The problem solved by Fermat is mathematically equivalent to the following: given two points in different media with different densities, minimize the density-weighted length of the path between the two points. In Louvain, in 1634 (by which time Willebrord Snellius had rediscovered Ibn Sahl's law, and Descartes had derived it but not yet published it), the Jesuit professor Wilhelm Boelmans gave a correct solution to this problem, and set its proof as an exercise for his Jesuit students (Ziggelaar, 1980). Fermat's solution was a landmark in that it unified the then-known laws of geometrical optics under a variational principle or action principle, setting the precedent for the principle of least action in classical mechanics and the corresponding principles in other fields (see History of variational principles in physics).Chaves, 2016, chapters 14,19.
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The Korringa–Kohn–Rostoker method or KKR method is used to calculate the electronic band structure of periodic solids. In the derivation of the method using multiple scattering theory by Jan Korringa and the derivation based on the Kohn and Rostoker variational method, the muffin-tin approximation was used. Later calculations are done with full potentials having no shape restrictions.
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Paul Dedecker ( Bruxelles, 1921 – Caracas, 2007) was a Belgian mathematician who worked primarily in topology on the subjects of nonabelian cohomology, general category theory, variational calculus and its relations to homological algebra, exterior calculus on manifolds and mathematical physics. He graduated in mathematics in 1948 at the Free University of Brussels, where he was a student of van den Dungen.
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He has experience in mathematics, with emphasis on differential geometry, working mainly on the following topics: calculus of variations and variational geometric problems, Riemannian and global Lorentzian geometry, Morse theory, symplectic geometry and Hamiltonian systems. Piccione develops research that deals with topics with possible applications to physics. The main results achieved in the area of Lorentzian geometry have an interpretation within General Relativity.
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The finite element method was designed to deal with problem with complicated computational regions. The PDE is first recast into a variational form which essentially forces the mean error to be small everywhere. The discretization step proceeds by dividing the computational domain into elements of triangular or rectangular shape. The solution within each element is interpolated with a polynomial of usually low order.
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Christa Dürscheid Christa Dürscheid (born October 4, 1959 in Kehl-Kork, Germany), is a German linguist and professor at the University of Zurich, Switzerland. Her main research interests include grammar, variational linguistics, didactics of language, writing systems, and media linguistics. In the English speaking research community she is best known for her publications about language use in the New Media.
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The main problem of quantum many-body physics is the fact that the Hilbert space grows exponentially with size. For example, a spin-1/2 chain of length L has 2L degrees of freedom. The DMRG is an iterative, variational method that reduces effective degrees of freedom to those most important for a target state. The target state is often the ground state.
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Generalized filtering furnishes posterior densities over hidden states (and parameters) generating observed data using a generalized gradient descent on variational free energy, under the Laplace assumption. Unlike classical (e.g. Kalman-Bucy or particle) filtering, generalized filtering eschews Markovian assumptions about random fluctuations. Furthermore, it operates online, assimilating data to approximate the posterior density over unknown quantities, without the need for a backward pass.
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The irregularly spaced observations are processed by data assimilation and objective analysis methods, which perform quality control and obtain values at locations usable by the model's mathematical algorithms (usually an evenly spaced grid). The data are then used in the model as the starting point for a forecast.University Corporation for Atmospheric Research (August 14, 2007). "The WRF Variational Data Assimilation System (WRF-Var)".
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The action principle can be extended to obtain the equations of motion for fields, such as the electromagnetic field or gravity. The Einstein equation utilizes the Einstein–Hilbert action as constrained by a variational principle. The path of a body in a gravitational field (i.e. free fall in space time, a so-called geodesic) can be found using the action principle.
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In 1962, on the recommendation of Hu Haichang, Zhong was transferred to the Dalian University of Technology (DUT) to work under Qian Lingxi. Their collaboration soon bore fruit. They published two papers in Science in China and Acta Mechanica Sinica, on the "general variational theory of limit analysis and plasticity". The research was used in submarine design and was awarded national prizes.
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Jon T. Pitts (born 1948) is an American mathematician working on geometric analysis and variational calculus. He is a professor at Texas A&M; University. Pitts obtained his Ph.D. from Princeton University in 1974 under the supervision of Frederick Almgren, Jr., with the thesis Every Compact Three- Dimensional Manifold Contains Two-Dimensional Minimal Submanifolds. He received a Sloan Fellowship in 1981.
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Functional optimization for fair surface design, Moreton, H. P. and Séquin, C. H., Proc. of ACM SIGGRAPH 1992; Variational Surface Modeling, Welch, W. and Witkin, A., Proc. of ACM SIGGRAPH 1992 This can be modeled by a suitable energy that penalizes unaesthetic behavior of the surface. A minimization of this fairness energy—subject to user-defined constraints—eventually yields the desired shape.
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The first problem involving a variational inequality was the Signorini problem, posed by Antonio Signorini in 1959 and solved by Gaetano Fichera in 1963, according to the references and : the first papers of the theory were and , . Later on, Guido Stampacchia proved his generalization to the Lax–Milgram theorem in in order to study the regularity problem for partial differential equations and coined the name "variational inequality" for all the problems involving inequalities of this kind. Georges Duvaut encouraged his graduate students to study and expand on Fichera's work, after attending a conference in Brixen on 1965 where Fichera presented his study of the Signorini problem, as reports: thus the theory become widely known throughout France. Also in 1965, Stampacchia and Jacques-Louis Lions extended earlier results of , announcing them in the paper : full proofs of their results appeared later in the paper .
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In this technique, we approximate the variational problem and end up with a finite dimensional problem. So let us start with the problem of seeking a function y(x) that extremizes an integral I[y(x)]. Assume that we are able to approximate y(x) by a linear combination of certain linearly independent functions of the type: y(x)\approx \varphi_0 (x) + c_1 \varphi_1 (x) + c_2 \varphi_2 (x)+ \cdots + c_N \varphi_N (x) where c_1,c_2,\cdots,c_N are constants to be determined by a variational method, such as one which will be described below. The selection of which approximating functions \varphi_i (x) to use is arbitrary except for the following considerations: a) If the problem has boundary conditions such as fixed end points, then \varphi_0 (x) is chosen to satisfy the problem’s boundary conditions, and all other \varphi_i (x) vanish at the boundary.
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Quality control of data can be performed and error fields can be calculated.Troupin, C, Barth, A, Sirjacobs, D, Ouberdous, M, Brankart, J.-M, Brasseur, P, Rixen, M, Alvera Azcarate, A, Belounis, M, Capet, A, Lenartz, F, Toussaint, M.-E, & Beckers, J.-M. (2012). Generation of analysis and consistent error fields using the Data Interpolating Variational Analysis (Diva). Ocean Modelling, 52-53, 90-101. doi:10.1016/j.ocemod.2012.05.
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Examples are the regularized autoencoders (Sparse, Denoising and Contractive autoencoders), proven effective in learning representations for subsequent classification tasks, and Variational autoencoders, with their recent applications as generative models. Autoencoders are effectively used for solving many applied problems, from face recognitionHinton GE, Krizhevsky A, Wang SD. Transforming auto-encoders. In International Conference on Artificial Neural Networks 2011 Jun 14 (pp. 44-51). Springer, Berlin, Heidelberg.
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Following mathematics: the theory of partial differential equation, variational calculus, Fourier analysis, potential theory, and vector analysis are perhaps most closely associated with mathematical physics. These were developed intensively from the second half of the 18th century (by, for example, D'Alembert, Euler, and Lagrange) until the 1930s. Physical applications of these developments include hydrodynamics, celestial mechanics, continuum mechanics, elasticity theory, acoustics, thermodynamics, electricity, magnetism, and aerodynamics.
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Assume that q(\mu,\tau) = q(\mu)q(\tau), i.e. that the posterior distribution factorizes into independent factors for \mu and \tau. This type of assumption underlies the variational Bayesian method. The true posterior distribution does not in fact factor this way (in fact, in this simple case, it is known to be a Gaussian-gamma distribution), and hence the result we obtain will be an approximation.
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The original article by Alain Chenciner and Richard Montgomery. Annals of Mathematics, 152 (2000), 881–901. One such orbit is a circular orbit, with equal masses at the corners of an equilateral triangle; another is the figure-8 orbit, first discovered numerically in 1993 by Cristopher Moore Moore's numerical discovery of the figure-8 choreography using variational methods. and subsequently proved to exist by Chenciner and Montgomery.
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Wriggers, P., 2006, Computational Contact Mechanics, 2nd ed., Springer, Heidelberg An example of the latter category is Kalker’s CONTACT model. A drawback of the well-founded variational approaches is their large computation times. Therefore, many different approximate approaches were devised as well. Several well-known approximate theories for the rolling contact problem are Kalker’s FASTSIM approach, the Shen-Hedrick-Elkins formula, and Polach’s approach.
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In 1961 Toponogov became a professor at a newly created Institute of Mathematics and Computing in Novosibirsk affiliated with the state university. Toponogov's scientific interests were influenced by his advisor Abram Fet, who taught at Tomsk and later at Novosibirsk. Fet was a well-recognized topologist and specialist in variational calculus in the large. Toponogov's work was also strongly influenced by the work of Aleksandr Danilovich Aleksandrov.
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In recent years a number of neural and deep-learning techniques have been proposed. Some generalize traditional Matrix factorization algorithms via a non-linear neural architecture, or leverage new model types like Variational Autoencoders. While deep learning has been applied to many different scenarios: context-aware, sequence-aware, social tagging etc. its real effectiveness when used in a simple collaborative recommendation scenario has been put into question.
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LaplacesDemon is an open-source statistical package that is intended to provide a complete environment for Bayesian inference. LaplacesDemon has been used in numerous fields. The user writes their own model specification function and selects a numerical approximation algorithm to update their Bayesian model. Some numerical approximation families of algorithms include Laplace's method (Laplace approximation), numerical integration (iterative quadrature), Markov chain Monte Carlo (MCMC), and Variational Bayes.
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In the normal state of a metal, electrons move independently, whereas in the BCS state, they are bound into Cooper pairs by the attractive interaction. The BCS formalism is based on the reduced potential for the electrons' attraction. Within this potential, a variational ansatz for the wave function is proposed. This ansatz was later shown to be exact in the dense limit of pairs.
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Each discretization strategy has certain advantages and disadvantages. A reasonable criterion in selecting a discretization strategy is to realize nearly optimal performance for the broadest set of mathematical models in a particular model class. Various numerical solution algorithms can be classified into two broad categories; direct and iterative solvers. These algorithms are designed to exploit the sparsity of matrices that depend on the choices of variational formulation and discretization strategy.
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Jane Ye （叶娟娟）is a Chinese-Canadian mathematician who works as a professor of mathematics at University of Victoria. Her interests include variational analysis and optimization constraint problems. She is the 2015 winner of the Krieger–Nelson Prize, given annually by the Canadian Mathematical Society to an outstanding female researcher in mathematics. Ye was born in China and received her B.Sc from Xiamen University in 1982.
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Economic dynamics allows for changes in economic variables over time, including in dynamic systems. The problem of finding optimal functions for such changes is studied in variational calculus and in optimal control theory. Before the Second World War, Frank Ramsey and Harold Hotelling used the calculus of variations to that end. Following Richard Bellman's work on dynamic programming and the 1962 English translation of L. Pontryagin et al.
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Optimal decision problems (usually formulated as partially observable Markov decision processes) are treated within active inference by absorbing utility functions into prior beliefs. In this setting, states that have a high utility (low cost) are states an agent expects to occupy. By equipping the generative model with hidden states that model control, policies (control sequences) that minimise variational free energy lead to high utility states.Friston, K., Samothrakis, S. & Montague, R., (2012).
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Consider a simple non-hierarchical Bayesian model consisting of a set of i.i.d. observations from a Gaussian distribution, with unknown mean and variance.Based on Chapter 10 of Pattern Recognition and Machine Learning by Christopher M. Bishop In the following, we work through this model in great detail to illustrate the workings of the variational Bayes method. For mathematical convenience, in the following example we work in terms of the precision — i.e.
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As a result, such a manifold is necessarily a (pseudo-)Riemannian manifold.Jurgen Jost, Riemannian Geometry and Geometric Analysis, (2002) Springer-Verlag David Bleeker, Gauge Theory and Variational Principles (1991) Addison-Wesely Publishing Company The Christoffel symbols provide a concrete representation of the connection of (pseudo-)Riemannian geometry in terms of coordinates on the manifold. Additional concepts, such as parallel transport, geodesics, etc. can then be expressed in terms of Christoffel symbols.
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From the above formulation, one can compute the ray paths using the Euler-Lagrange formulation; alternatively, one can compute the wave fronts by solving the Hamilton-Jacobi equation. Knowing one leads to knowing the other. The above duality is very general and applies to all systems that derive from a variational principle: either compute the trajectories using Euler-Lagrange equations or the wave fronts by using Hamilton-Jacobi equation.
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The 2014 research paper on "Variational Recurrent Auto-Encoders" attempted to generate music based on songs from 8 different video games. This project is one of the few conducted purely on video game music. The neural network in the project was able to generate data that was very similar to the data of the games it trained off of. The generated data did not translate into good quality music.
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Roger Jean-Baptiste Robert Wets (born February 1937) is a "pioneer" in stochastic programming and a leader in variational analysis who publishes as Roger J-B Wets. His research, expositions, graduate students, and his collaboration with R. Tyrrell Rockafellar have had a profound influence on optimization theory, computations, and applications. Since 2009, Wets has been a distinguished research professor at the mathematics department of the University of California, Davis.
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Gyarmati (1967/1970)Gyarmati, I. (1970). Non-equilibrium Thermodynamics: Field Theory and Variational Principles, Springer, Berlin; translated, by E. Gyarmati and W.F. Heinz, from the original 1967 Hungarian Nemegyensulyi Termodinamika, Muszaki Konyvkiado, Budapest. gives a systematic presentation, and extends Onsager's principle of least dissipation of energy, to give a more symmetric form known as Gyarmati's principle. Gyarmati (1967/1970) cites 11 papers or books authored or co-authored by Prigogine.
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The mild-slope equation can be derived by the use of several methods. Here, we will use a variational approach. The fluid is assumed to be inviscid and incompressible, and the flow is assumed to be irrotational. These assumptions are valid ones for surface gravity waves, since the effects of vorticity and viscosity are only significant in the Stokes boundary layers (for the oscillatory part of the flow).
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He also showed that, contrary to what had been expected, singularities which are not hidden in a black hole also occur. However, he then showed that such "naked singularities" are unstable. In 2000, Christodoulou published a book on general systems of partial differential equations deriving from a variational principle (or "action principle"). In 2007, he published a book on the formation of shock waves in 3-dimensional fluids.
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He was a visiting professor at the Tata Institute of Fundamental Research (in 1994 and again in 2004), at the Max Planck Institute for Mathematics in the Sciences in Leipzig (in 1998), at Caltech, at the Centre Emile Borel in Paris, at the Isaac Newton Institute, at the University of Paris VI and the University of Paris XIII, at Carnegie-Mellon University, at Stanford University (Timoshenko scholar) and at the department of aerospace engineering at the University of Minnesota. He was a one-year visiting fellow at Mansfied College and visiting professor at the Mathematical Institute in Oxford in 2013-14. Braides has done research on the calculus of variations, Gamma convergence, asymptotic homogenization, discrete variational problems, percolation, fracture mechanics, image processing, free-discontinuity problems, and geometric measure theory. In 2014 he was an Invited Speaker at the International Congress of Mathematicians in Seoul with talk Discrete-to-continuum variational methods for lattice systems.
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In mathematics, the Caristi fixed-point theorem (also known as the Caristi–Kirk fixed-point theorem) generalizes the Banach fixed-point theorem for maps of a complete metric space into itself. Caristi's fixed-point theorem modifies the ε-variational principle of Ekeland (1974, 1979). The conclusion of Caristi's theorem is equivalent to metric completeness, as proved by Weston (1977). The original result is due to the mathematicians James Caristi and William Arthur Kirk.
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Chapter 1: The Intellectual Challenge of Morphological Evolution: A Case for Variational Structuralism. Page 7 Another possibility is that a trait may have been adaptive at some point in an organism's evolutionary history, but a change in habitats caused what used to be an adaptation to become unnecessary or even maladapted. Such adaptations are termed vestigial. Many organisms have vestigial organs, which are the remnants of fully functional structures in their ancestors.
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There are a large number of different valence bond methods. Most use n valence bond orbitals for n electrons. If a single set of these orbitals is combined with all linear independent combinations of the spin functions, we have spin-coupled valence bond theory. The total wave function is optimized using the variational method by varying the coefficients of the basis functions in the valence bond orbitals and the coefficients of the different spin functions.
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Roland Glowinski (born 9 March 1937) is a French-American mathematician. He obtained his PhD in 1970 from Jacques-Louis Lions and is known for his work in applied mathematics, in particular numerical solution and applications of partial differential equations and variational inequalities. He is member of the French Academy of Sciences and since 1985 has held an endowed chair at the University of Houston. Roland has written many books on the subject of mathematics.
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However, the complexity term of variational free energy shares the same fixed point as Helmholtz free energy (under the assumption the system is thermodynamically closed but not isolated). This is because if sensory perturbations are suspended (for a suitably long period of time), complexity is minimised (because accuracy can be neglected). At this point, the system is at equilibrium and internal states minimise Helmholtz free energy, by the principle of minimum energy.Jarzynski, C. (1997).
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For N = 2 this Lagrangian will generate exactly the same equations of motion of L_1 and L_2. Therefore, from the point of view of an outside observer, everything is causal. This formulation reflects particle-particle symmetry with the variational principle applied to the N-particle system as a whole, and thus Tetrode's Machian principle. Only if we isolate the forces acting on a particular body do the advanced potentials make their appearance.
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Lagrange is one of the founders of the calculus of variations. Starting in 1754, he worked on the problem of the tautochrone, discovering a method of maximizing and minimizing functionals in a way similar to finding extrema of functions. Lagrange wrote several letters to Leonhard Euler between 1754 and 1756 describing his results. He outlined his "δ-algorithm", leading to the Euler–Lagrange equations of variational calculus and considerably simplifying Euler's earlier analysis.
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A variational explanation for the main ingredient of the Canny edge detector, that is, finding the zero crossings of the 2nd derivative along the gradient direction, was shown to be the result of minimizing a Kronrod–Minkowski functional while maximizing the integral over the alignment of the edge with the gradient field (Kimmel and Bruckstein 2003). See article on regularized Laplacian zero crossings and other optimal edge integrators for a detailed description.
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Kalman Varga is a Hungarian-American physicist, currently at Vanderbilt University. He researches computational nanoscience, focusing on developing novel computational methods for electronic structure calculations. He is an Elected Fellow of the American Physical Society. He was accredited to co- writing Computational Nanoscience: Applications for Molecules, Clusters, and Solids in 2011, as well as Structure and Reactions of Light Exotic Nuclei (2003), and Stochastic Variational Approach to Quantum-Mechanical Few-Body Problems (1998).
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152, 154106 (2020). is a quantum Monte Carlo program that was originally developed in the Theory of Condensed Matter group at the Cavendish Laboratory in Cambridge. CASINO can be used to perform variational quantum Monte Carlo and diffusion quantum Monte Carlo simulations to calculate the energy and distribution of electrons in atoms, molecules and crystals. The principal authors of this program are R. J. Needs, M. D. Towler, N. D. Drummond and P. Lopez Rios.
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Věra Kůrková (born 1948) is a Czech mathematician and computer scientist, affiliated with the Institute of Computer Science of the Czech Academy of Sciences. Her research interests include neural networks, computational learning theory, and nonlinear approximation theory. She formulated the abstract concept of a variational norm in 1997 which puts ideas of Maurey, Jones, and Barron into the context of functional analysis. See V. Kůrková, Dimension-independent rates of approximation by neural networks.
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Two papers of 1913 and 1914 are particularly important. The first Problem mit gemischten Bedingungen und variablen Endpunkten formulated a new type of variational problem now called "the Bolza problem of Bolza" after him and the second studied variations for an integral problem involving inequalities. This latter work was to become important in control theory. Bolza returned to Chicago for part of 1913 giving lecturers during the summer on function theory and integral equations.
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Marian P. Roque is a Filipina mathematician. She is the president of the Mathematical Society of the Philippines, a professor in the Institute of Mathematics of the University of the Philippines Diliman, and head of the Institute of Mathematics. Her mathematical specialty is the theory of partial differential equations. With Doina Cioranescu and Patrizia Donato, she is the author of An Introduction to Second Order Partial Differential Equations: Classical and Variational Solutions (World Scientific, 2018).
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In 1842, Carl Gustav Jacobi tackled the problem of whether the variational principle always found minima as opposed to other stationary points (maxima or stationary saddle points); most of his work focused on geodesics on two-dimensional surfaces.G.C.J. Jacobi, Vorlesungen über Dynamik, gehalten an der Universität Königsberg im Wintersemester 1842–1843. A. Clebsch (ed.) (1866); Reimer; Berlin. 290 pages, available online Œuvres complètes volume 8 at Gallica-Math from the Gallica Bibliothèque nationale de France.
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In 2005 he became a professor at Leiden University. His research deals with probability theory (e.g. theory of large deviations, potential theory methods, and systems of interacting particles), statistical physics (including applications of variational methods to phase transitions), ergodic theory, population genetics, and complex networks. Den Hollander has been a visiting professor at several academic institutions around the world, including a visit from August 1998 to January 1999 at the Fields Institute in Toronto.
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Variational Bayesian learning is based on probabilities. There is a chance that an approximation is performed with mistakes, damaging further data representations. Another downside pertains to complicated or corrupted data samples, making it difficult to infer a representational pattern. The wake-sleep algorithm has been suggested not to be powerful enough for the layers of the inference network in order to recover a good estimator of the posterior distribution of latent variables.
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A number of important problem classes can be solved. Specific examples are variational inequalities, Nash equilibria, disjunctive programs and stochastic programs. EMP is independent of the modeling language used but currently it is implemented only in GAMS. The new types of problems modeled with EMP are reformulated with the GAMS solver JAMS to well established types of problems and the reformulated models are passed to a suitable GAMS solver to be solved.
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Equilibrium problems model questions arising in the study of economic equilibria in a mathematically abstract form. Equilibrium problems include Variational Inequalities, problems with Nash Equilibria, and Multiple Optimization Problems with Equilibrium Constraints (MOPECs). Use EMP's keywords to reformulate these problems as mixed complementarity problems (MCPs), a class of problems for which mature solver technology exists. Solve the newly reformulated EMP keyword version of the problem with the PATH solver or other GAMS MCP solvers.
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The tax instrument is modeled in the upper level and the clearing market is modeled in the lower level. In general, the lower level problem may be an optimization problem or a variational inequality. Several keywords are provided to facilitate reformulating hierarchical optimization problems. Bilevel optimization problems modeled with EMP are reformulated to mathematical programs with equilibrium constraints (MPECs) and then they are solved with one of the GAMS MPEC solvers (NLPEC or KNITRO).
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A modification of canonical variational transition state theory in which, for energies below the threshold energy, the position of the dividing surface is taken to be that of the microcanonical threshold energy. This forces the contributions to rate constants to be zero if they are below the threshold energy. A compromise dividing surface is then chosen so as to minimize the contributions to the rate constant made by reactants having higher energies.
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Model based inpainting follows the Bayesian approach for which missing information is best fitted or estimated from the combination of the models of the underlying images, as well as the image data actually being observed. In deterministic language, this has led to various variational inpainting models. Manual computer methods include using a clone tool to copy existing parts of the image to restore a damaged texture. Texture synthesis may also be used.
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The optimal solution may also be obtained by Gauss elimination using other sparse-matrix techniques or some iterative methods based e.g. on Variational Calculus. However, these latter methods may solve the large matrix of all the error variances and covariances only approximately and the data fusion would not be performed in a strictly optimal fashion. Consequently, the long-term stability of Kalman filtering becomes uncertain even if Kalman's observability and controllability conditions were permanently satisfied.
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The first variational principle in physics was articulated by Euclid in his Catoptrica. It says that, for the path of light reflecting from a mirror, the angle of incidence equals the angle of reflection. Hero of Alexandria later showed that this path gave the shortest length and the least time. Fermat refined and generalized this to "light travels between two given points along the path of shortest time" now known as the principle of least time.
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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 function and such solutions are the topic of study in potential theory.
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More recently, Zipf's law has been shown to arise as the variational solution of the MFI when scale invariance is introduced in the measure, leading for the first time an explanation of this regularity from first principles. It has been also shown that MFI can be used to formulate a thermodynamics based on scale invariance instead of translational invariance, allowing the definition of the Scale-Free Ideal Gas, the scale invariant equivalent of the Ideal Gas.
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Professor Charles Keil classified forms and formal detail as "sectional, developmental, or variational." ;Sectional form: This form is built from a sequence of clear-cut units that may be referred to by letters but also often have generic names such as introduction and coda, exposition, development and recapitulation, verse, chorus or refrain, and bridge. Introductions and codas, when they are no more than that, are frequently excluded from formal analysis. All such units may typically be eight measures long.
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The preceding reasoning is not valid if \sigma vanishes identically on C. In such a case, we could allow a trial function \varphi \equiv c, where c is a constant. For such a trial function, : V[c] = c\left[ \iint_D f \, dx\,dy + \int_C g \, ds \right]. By appropriate choice of c, V can assume any value unless the quantity inside the brackets vanishes. Therefore, the variational problem is meaningless unless : \iint_D f \, dx\,dy + \int_C g \, ds =0.
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The virtual displacements of coordinates retain path of the light-like particle to be null in the pseudo-Riemann space-time, i.e. not lead to the Lorentz-invariance violation in locality and corresponds to the variational principles of mechanics. The equivalence of the solutions given by the first principle, to the geodesics, means that the using the second also turns out geodesics. The stationary energy integral principle gives a system of equations that has one equation more.
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Schwinger was a physics professor at several universities. Schwinger is recognized as one of the greatest physicists of the twentieth century, responsible for much of modern quantum field theory, including a variational approach, and the equations of motion for quantum fields. He developed the first electroweak model, and the first example of confinement in 1+1 dimensions. He is responsible for the theory of multiple neutrinos, Schwinger terms, and the theory of the spin-3/2 field.
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Complementarity problems were originally studied because the Karush–Kuhn–Tucker conditions in linear programming and quadratic programming constitute a linear complementarity problem (LCP) or a mixed complementarity problem (MCP). In 1963 Lemke and Howson showed that, for two person games, computing a Nash equilibrium point is equivalent to an LCP. In 1968 Cottle and Dantzig unified linear and quadratic programming and bimatrix games. Since then the study of complementarity problems and variational inequalities has expanded enormously.
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In 1967 he became the finalist of the 18th Mathematical Olympiad. In 1972, he graduated in mathematics at Adam Mickiewicz University in Poznań. He obtained his doctorate in Institute of Mathematics of the Polish Academy of Sciences in 1977, based on the work Lefschetz Numbers of Maps Commuting with an Action of a Group written under the direction Kazimierz Gęba. He got habilitation there in 1991, based on the work Invariant topology methods used in variational problems.
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Examples of this work are expectation optimization of L2 f-divergence for stochastic variational Bayes inference, Gaussianized bridge sampling for Bayesian evidence, and BayesFast, a surrogate model based Hamiltonian Monte Carlo sampler. Seljak is developing machine learning methods with applications to cosmology, astronomy, and other sciences. Examples are Fourier based Gaussian process for analysis of time and/or spatially ordered data, generative models with explicit physics symmetries (translation, rotation), and sliced iterative transport methods for density estimation and sampling.
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Techniques like EXIT charts can provide an approximate visualization of the progress of belief propagation and an approximate test for convergence. There are other approximate methods for marginalization including variational methods and Monte Carlo methods. One method of exact marginalization in general graphs is called the junction tree algorithm, which is simply belief propagation on a modified graph guaranteed to be a tree. The basic premise is to eliminate cycles by clustering them into single nodes.
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Among them APW method, the linear muffin-tin orbital method (LMTO) and various Green's function methods. One application is found in the variational theory developed by Jan Korringa (1947) and by Walter Kohn and N. Rostoker (1954) referred to as the KKR method. This method has been adapted to treat random materials as well, where it is called the KKR coherent potential approximation. In its simplest form, non-overlapping spheres are centered on the atomic positions.
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The ideas are closely related to light propagation in optics. The method became known as Carathéodory's method of equivalent variational problems or the royal road to the calculus of variations.H. Boerner, Carathéodory und die Variationsrechnung, in A Panayotopolos (ed.), Proceedings of C. Carathéodory International Symposium, September 1973, Athens (Athens, 1974), 80–90. A key advantage of Carathéodory's work on this topic is that it illuminates the relation between the calculus of variations and partial differential equations.
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In this approach, the domain is discretized into smaller elements, often triangles or tetrahedra, but other elements such as rectangles or cuboids are possible. The solution space is then approximated using so called form-functions of a pre-defined degree. The differential equation containing the Laplace operator is then transformed into a variational formulation, and a system of equations is constructed (linear or eigenvalue problems). The resulting matrices are usually very sparse and can be solved with iterative methods.
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The resultant variational conditions on the orbitals lead to a new one-electron operator, the Fock operator. At the minimum, the occupied orbitals are eigensolutions to the Fock operator via a unitary transformation between themselves. The Fock operator is an effective one-electron Hamiltonian operator being the sum of two terms. The first is a sum of kinetic-energy operators for each electron, the internuclear repulsion energy, and a sum of nuclear–electronic Coulombic attraction terms.
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167–187, April 1987. #Lindeberg, Tony "Edge detection and ridge detection with automatic scale selection", International Journal of Computer Vision, 30, 2, pp 117—154, 1998. (Includes the differential approach to non-maximum suppression.) #Kimmel, Ron and Bruckstein, Alfred M. "On regularized Laplacian zero crossings and other optimal edge integrators", International Journal of Computer Vision, 53(3):225–243, 2003. (Includes the geometric variational interpretation for the Haralick–Canny edge detector.) #Moeslund, T. (2009, March 23).
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Chaotic motion in three-body problem (computer simulation). Poincaré published two now classical monographs, "New Methods of Celestial Mechanics" (1892–1899) and "Lectures on Celestial Mechanics" (1905–1910). In them, he successfully applied the results of their research to the problem of the motion of three bodies and studied in detail the behavior of solutions (frequency, stability, asymptotic, and so on). They introduced the small parameter method, fixed points, integral invariants, variational equations, the convergence of the asymptotic expansions.
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The main advantages of the hybrid Trefftz method over the conventional method are: # the formulation calls for integration along the element boundaries only which allows for curve-sided or polynomial shapes to be used for the element boundary, # presents expansion bases for elements that do not satisfy inter- element continuity through the variational functional, and # this method allows for the development of crack singular or perforated elements through the use of localized solution functions as the trial functions.
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In 1967 he became the finalist of the 18th Mathematical Olympiad. In 1972, he graduated in mathematics at Adam Mickiewicz University in Poznań. He obtained his doctorate in Institute of Mathematics of the Polish Academy of Sciences in 1977, based on the work Lefschetz Numbers of Maps Commuting with an Action of a Group written under the direction Kazimierz Gęba. He got habilitation there in 1991, based on the work Invariant topology methods used in variational problems.
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He taught mathematics at the University of Heidelberg for the rest of his life. He did research on differential equations, the calculus of variations and mechanics. His research on the integration of partial differential equations and a search to determine maxima and minima using variational methods brought him close to the investigations that Sophus Lie was carrying out around the same time. Several letters were exchanged between Mayer and mathematician Felix Klein from 1871 to 1907.
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In the 1980s Jordan started developing recurrent neural networks as a cognitive model. In recent years, his work is less driven from a cognitive perspective and more from the background of traditional statistics. Jordan popularised Bayesian networks in the machine learning community and is known for pointing out links between machine learning and statistics. He was also prominent in the formalisation of variational methods for approximate inference and the popularisation of the expectation-maximization algorithm in machine learning.
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He is a theoretical physicist, professionally associated with the Centre for Theoretical Physics in Warsaw. He is also a professor of the Faculty of Mathematics and Natural Sciences Cardinal Stefan Wyszynski University in Warsaw and a visiting professor at universities in Cologne, Leipzig, Turin, Milan, Rome (Sapienza University of Rome), Marseille, Tours, Louvain-la-Neuve and polytechnics in Aachen (RWTH, Rheinisch-Westfälische Technische Hochschule Aachen) and Claushal (Technische Universität Clausthal). From 1980–1985 he was a deputy director of the CFT PAN, in 1991–1992, he was the director of this center and, in 1996–1999, he held the position of the Head of the Department of Mathematical Methods in Physics, University of Warsaw. Scientific achievements of Professor Kijowski include the introduction of operator 'transition time' in Quantum mechanics, new interpretation of uncertainty principle for the time and energy, providing a new, completely original variational principle for Einstein equations and the discovery of the affine, variational principle for General Relativity Theory, as well as providing new, original positivity proof for gravitational energy.
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The connection to the Hamilton–Jacobi equation from classical physics was first drawn by Rudolf Kálmán. In discrete-time problems, the corresponding difference equation is usually referred to as the Bellman equation. While classical variational problems, such as the brachistochrone problem, can be solved using the Hamilton–Jacobi–Bellman equation, the method can be applied to a broader spectrum of problems. Further it can be generalized to stochastic systems, in which case the HJB equation is a second-order elliptic partial differential equation.
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The Hamilton–Jacobi–Bellman equation (HJB) is a partial differential equation which is central to optimal control theory. The solution of the HJB equation is the 'value function', which gives the optimal cost-to-go for a given dynamical system with an associated cost function. Classical variational problems, for example, the brachistochrone problem can be solved using this method as well. The equation is a result of the theory of dynamic programming which was pioneered in the 1950s by Richard Bellman and coworkers.
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Due to all of the mathematical manipulations involved, it is easy to lose track of the big picture. The important things are: #The idea of variational Bayes is to construct an analytical approximation to the posterior probability of the set of unobserved variables (parameters and latent variables), given the data. This means that the form of the solution is similar to other Bayesian inference methods, such as Gibbs sampling — i.e. a distribution that seeks to describe everything that is known about the variables.
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As a young professor in 1972, Kleinert visited Caltech and was impressed by noted US physicist Richard Feynman. Later, Kleinert was to collaborate with Feynman in some of the latter's last work. This collaboration led to a mathematical method for converting divergent weak-coupling power series into convergent strong- coupling ones. This so-called variational perturbation theory yields at present the most accurate theory of critical exponents Kleinert, H., "Critical exponents from seven-loop strong-coupling φ4 theory in three dimensions".
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Frieden is best known for his extensive work on Fisher information as a grounding principle for deriving and elaborating physical theory. (Examples are the Schrödinger wave equation of quantum mechanics, and the Maxwell–Boltzmann distribution of statistical mechanics.) Such theories take the form of differential equations or probability distribution functions. Central to Frieden's derivations is the mathematical variational principle of extreme physical information (EPI). This principle builds on the well-known idea that the observation of a "source" phenomenon is never completely accurate.
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In 1842, Carl Gustav Jacobi tackled the problem of whether the variational principle found minima or other extrema (e.g. a saddle point); most of his work focused on geodesics on two-dimensional surfaces. The first clear general statements were given by Marston Morse in the 1920s and 1930s, leading to what is now known as Morse theory. For example, Morse showed that the number of conjugate points in a trajectory equaled the number of negative eigenvalues in the second variation of the Lagrangian.
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Although some authors speak of general method of solving "isoperimetric problems", the eighteenth century meaning of this expression amounts to "problems in variational calculus", reserving the adjective "relative" for problems with isoperimetric-type constraints. The celebrated method of Lagrange multipliers, which applies to optimization of functions of several variables subject to constraints, did not appear until much later. See Lagrange also applied his ideas to problems of classical mechanics, generalising the results of Euler and Maupertuis. Euler was very impressed with Lagrange's results.
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Max Shiffman (30 October 1914, New York City – 2 July 2000, Hayward, California) was an American mathematician, specializing in the calculus of variations, partial differential equations, (According to this article by Lax, "He gave an invited address ... at the International Congress of Mathematicians in Cambridge in 1950." This is wrong. Shiffman gave a Section Address entitled On variational analysis in the large. Shiffman's talk was officially approved but he was not an Invited Speaker of the ICM in 1950 in Cambridge, Massachusetts.
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This often leads to simpler formulas by avoiding the need for the square-root. Thus, for example, the geodesic equations may be obtained by applying variational principles to either the length or the energy. In the latter case, the geodesic equations are seen to arise from the principle of least action: they describe the motion of a "free particle" (a particle feeling no forces) that is confined to move on the manifold, but otherwise moves freely, with constant momentum, within the manifold.
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Thereafter he developed together with Guido Brunnett and Paolo Santarelli the Variational Design methodology and a solution to the twist input and compatibility twist problem of the Coons patches. Triangular patches did not have "curvature modeling facilities" for many years. Hans Hagen had a strong impact on the Geometric Modeling community. He started the world-class Dagstuhl Seminar Series on Geometric Modeling that regularly brings together the leading experts of the field in a relaxed unmatched format that stimulated a lot of ideas.
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At Cornell University he is since 2004 a full professor and since 2018 the Samuel B. Eckert Professor of Engineering in the School of Operations Research and Information Engineering. From 2010 to 2013 he served as the School's director. Lewis has held visiting appointments at academic institutions in France, Italy, New Zealand, the United States, and Spain. He is a co-editor for Mathematical Programming, Series A and an associate editor for Set-Valued and Variational Analysis and for Mathematika.
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The issue of deriving the equations of motion or the field equations in any physical theory is considered by many researchers to be appealing. A fairly universal way of performing these derivations is by using the techniques of variational calculus, the main objects used in this being Lagrangians. Many consider this approach to be an elegant way of constructing a theory, others as merely a formal way of expressing a theory (usually, the Lagrangian construction is performed after the theory has been developed).
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The distinguishing features of Galerkin's method were the following: he did not associate the method, developed by him, with any variational problem direct solution, but considered it to be common for solving differential equations. He interpreted it, using the probable displacements principle. These ideas showed to be very productive, not only in structural mechanics, but for mathematical physics at large. The Galerkin method (or Bubnov-Galerkin method) with Galerkin's (or "weak") differential equations problem statement form are known all over the world.
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The finite element method (FEM) is a numerical technique for finding approximate solutions to boundary value problems for differential equations. It uses variational methods (the calculus of variations) to minimize an error function and produce a stable solution. Analogous to the idea that connecting many tiny straight lines can approximate a larger circle, FEM encompasses all the methods for connecting many simple element equations over many small subdomains, named finite elements, to approximate a more complex equation over a larger domain.
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In mathematics and theoretical physics, resummation is a procedure to obtain a finite result from a divergent sum (series) of functions. Resummation involves a definition of another (convergent) function in which the individual terms defining the original function are re-scaled, and an integral transformation of this new function to obtain the original function. Borel resummation is probably the most well-known example. The simplest method is an extension of a variational approach to higher order based on a paper by R.P. Feynman and H. Kleinert.
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Most such micromechanical methods use periodic homogenization, which approximates composites by periodic phase arrangements. A single repeating volume element is studied, appropriate boundary conditions being applied to extract the composite's macroscopic properties or responses. The Method of Macroscopic Degrees of Freedom can be used with commercial FE codes, whereas analysis based on asymptotic homogenization typically requires special-purpose codes. The Variational Asymptotic Method for Unit Cell Homogenization (VAMUCH) and its development, Mechanics of Structural Genome (see below), are recent Finite Element based approaches for periodic homogenization.
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In statistics, measures of central tendency and statistical dispersion, such as the mean, median, and standard deviation, are defined in terms of metrics, and measures of central tendency can be characterized as solutions to variational problems. In penalized regression, "L1 penalty" and "L2 penalty" refer to penalizing either the norm of a solution's vector of parameter values (i.e. the sum of its absolute values), or its norm (its Euclidean length). Techniques which use an L1 penalty, like LASSO, encourage solutions where many parameters are zero.
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Improvements in the performance of belief propagation algorithms are also achievable by breaking the replicas symmetry in the distributions of the fields (messages). This generalization leads to a new kind of algorithm called survey propagation (SP), which have proved to be very efficient in NP-complete problems like satisfiability and graph coloring. The cluster variational method and the survey propagation algorithms are two different improvements to belief propagation. The name generalized survey propagation (GSP) is waiting to be assigned to the algorithm that merges both generalizations.
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Since 2009 he is full member of Serbian Academy of Sciences and Arts and in 2014 he was elected Professor emeritus of the University of Novi Sad. At present he is Secretary of the Branch in Novi Sad of Serbian Academy of Sciences and Arts.Novi Sad Branch of the Serbian Academy of Sciences and Arts His research interests include Continuum mechanics, Variational principles of mechanics, Shape memory materials, Visco-elasticity of fractional type, Biomechanics, Stability of elastic systems, and the problems of Shape optimization of elastic rods.
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In this paradigm, the distinction between the E and M steps disappears. If using the factorized Q approximation as described above (variational Bayes), solving can iterate over each latent variable (now including θ) and optimize them one at a time. Now, k steps per iteration are needed, where k is the number of latent variables. For graphical models this is easy to do as each variable's new Q depends only on its Markov blanket, so local message passing can be used for efficient inference.
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In a model where a Dirichlet prior distribution is placed over a set of categorical-valued observations, the marginal joint distribution of the observations (i.e. the joint distribution of the observations, with the prior parameter marginalized out) is a Dirichlet-multinomial distribution. This distribution plays an important role in hierarchical Bayesian models, because when doing inference over such models using methods such as Gibbs sampling or variational Bayes, Dirichlet prior distributions are often marginalized out. See the article on this distribution for more details.
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The eudemon of tree of thousands of years has no power to sustain the life force of the whole enchanted forest. So the green place turned into desolation and the local water became dead lake in this forest. There were variational creatures everywhere or residents whose minds have been affected. The reason why Lily controlled tree eudemon was that she wanted to get dragon crystals. She also imprisoned her own and Yu’ teacher Xian Yi and wanted to obtain skills to make Su Long evolve ultimately.
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Many of these originate from computer science, as well as other branches of engineering such as computer engineering, electrical engineering, and bioengineering. clustering in machine learning. The field of information engineering is based heavily on mathematics, particularly probability, statistics, calculus, linear algebra, optimization, differential equations, variational calculus, and complex analysis. Information engineers often hold a degree in information engineering or a related area, and are often part of a professional body such as the Institution of Engineering and Technology or Institute of Measurement and Control.
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Wagner's early work was focused on mathematical population genetics. Together with the mathematician Reinhard Bürger at the University of Vienna, he contributed to the theory of mutation–selection balance and the evolution of dominance modifiers. Later Wagner shifted his focus on issues of the evolution of variational properties like canalization and modularity. He introduced the seminal distinction between variation and variability, the former describing the actually existing differences among individuals while the latter measures the tendency to vary, as measured in mutation rate and mutational variance.
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Starting from his variational principle, Schwinger derived a set of equations for Green's functions non-perturbatively, which generalize Dyson's equations to the Schwinger–Dyson equations for the Green functions of quantum field theories. Today they provide a non-perturbative approach to quantum field theories and applications can be found in many fields of theoretical physics, such as solid-state physics and elementary particle physics. Schwinger also derived an equation for the two-particle irreducible Green functions, which is nowadays referred to as the inhomogeneous Bethe–Salpeter equation.
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Ambrosetti has studied at the University of Padua and he is professor of mathematics at the International School for Advanced Studies. He is known for his basic work on topological methods in the calculus of variations. These provide tools aimed at establishing the existence of solutions to variational problems when classical direct methods of the calculus of variations cannot be applied. In particular, the so-called mountain pass theorem he established with Paul Rabinowitz is nowadays a classical tool in the context of nonlinear analysis problems.
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Mark Krasnosel'skii's has authored or co-authored some three hundred papers and fourteen monographs. Nonlinear techniques are roughly classified into analytical, topological and variational methods. Mark Krasnosel'skii has contributed to all three aspects in a significant way, as well as to their application to many types of integral, differential and functional equations coming from mechanics, engineering, and control theory. Mark Krasnosel'skii was the first to investigate the functional analytical properties of fractional powers of operators, at first for self-adjoint operators and then for more general situations.
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The most famous example (which led to Luchins and Luchins' coining of the term) is the Luchins water jar experiment, in which subjects were asked to solve a series of water jar problems. After solving many problems which had the same solution, subjects applied the same solution to later problems even though a simpler solution existed (Luchins, 1942).. Other experiments on the Einstellung effect can be found in The Effect of Einstellung on Compositional Processes and Rigidity of Behavior, A Variational Approach to the Effect of Einstellung.
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For irreversible thermodynamics, Biot utilized the variational approach and was the first to introduce the dissipation function and the minimum dissipation principle to account for the dissipation phenomenon, which led to the development of thermoelasticity, heat transfer, viscoelasticity, and thermorheology. Biot’s interest in the non-linear effects of initial stress and the inelastic behavior of solids led to his mathematical theory of folding of stratified rocks. In the period between 1935 and 1962 Biot published a number of scientific papersScientific Articles of Maurice A. Biot.
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He has been a member of the editorial boards of Mathematics of Operations Research, the SIAM Journal on Optimization, the SIAM Journal on Matrix Analysis and Applications, the SIAM Journal on Control and Optimization, and the MPS/SIAM Series on Optimization. Much of his research deals with "semi-algebraic optimization and variational properties of eigenvalues." With Jonathan Borwein he co-authored the book Convex Analysis and Nonlinear Optimization (2000, 2nd edition 2006). Lewis holds British and Canadian citizenship and permanent residency in the USA.
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Quimby praised the quality of printing and binding which make the book attractive. In the Journal of the Franklin Institute, Rupen Eskergian noted that the first edition of Classical Mechanics offers a mature take on the subject using vector and tensor notations and with a welcome emphasis on variational methods. This book begins with a review of elementary concepts, then introduces the principle of virtual work, constraints, generalized coordinates, and Lagrangian mechanics. Scattering is treated in the same chapter as central forces and the two-body problem.
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Steven R. White (born December 26, 1959 in Lawton, Oklahoma) is a professor of physics at the University of California, Irvine. He graduated from the University of California, San Diego; he then received his Ph.D. at Cornell University, where he was a shared student with Kenneth Wilson and John Wilkins. He is most known for inventing the Density Matrix Renormalization Group (DMRG) in 1992. This is a numerical variational technique for high accuracy calculations of the low energy physics of quantum many-body systems.
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While the original proof of this result due to Richard Schoen and Shing-Tung Yau used variational methods, Witten's proof used ideas from supergravity theory to simplify the argument. A third area mentioned in Atiyah's address is Witten's work relating supersymmetry and Morse theory, a branch of mathematics that studies the topology of manifolds using the concept of a differentiable function. Witten's work gave a physical proof of a classical result, the Morse inequalities, by interpreting the theory in terms of supersymmetric quantum mechanics.
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As in general relativity, equations structurally identical to the Einstein field equations are derivable from a variational principle. A spin tensor can also be supported in a manner similar to Einstein–Cartan–Sciama–Kibble theory. GTG was first proposed by Lasenby, Doran, and Gull in 1998 as a fulfillment of partial results presented in 1993. The theory has not been widely adopted by the rest of the physics community, who have mostly opted for differential geometry approaches like that of the related gauge gravitation theory.
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Generally, the exact Occam factor is intractable, but approximations such as Akaike information criterion, Bayesian information criterion, Variational Bayesian methods, false discovery rate, and Laplace's method are used. Many artificial intelligence researchers are now employing such techniques, for instance through work on Occam Learning or more generally on the Free energy principle. Statistical versions of Occam's razor have a more rigorous formulation than what philosophical discussions produce. In particular, they must have a specific definition of the term simplicity, and that definition can vary.
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E. Mayer, "Review: David D. Bleecker, Gauge theory and variational principles", Bull. Amer. Math. Soc. (N.S.) 9 (1983), no. 1, 83--92Alexandre Guay, Geometrical aspects of local gauge symmetry (2004) The affine connection is interesting because it does not require any concept of a metric tensor to be defined; the curvature of an affine connection can be understood as the field strength of the gauge potential. When a metric is available, then one can go in a different direction, and define a connection on a frame bundle.
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A rectified Gaussian distribution is semi-conjugate to the Gaussian likelihood, and it has been recently applied to factor analysis, or particularly, (non- negative) rectified factor analysis. Harva proposed a variational learning algorithm for the rectified factor model, where the factors follow a mixture of rectified Gaussian; and later Meng proposed an infinite rectified factor model coupled with its Gibbs sampling solution, where the factors follow a Dirichlet process mixture of rectified Gaussian distribution, and applied it in computational biology for reconstruction of gene regulatory networks.
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In the field of time–frequency analysis, several signal formulations are used to represent the signal in a joint time–frequency domain.L. Cohen, "Time–Frequency Analysis," Prentice-Hall, New York, 1995. There are several methods and transforms called "time-frequency distributions" (TFDs), whose interconnections were organized by Leon Cohen.L. Cohen, "Generalized phase- space distribution functions," J. Math. Phys., 7 (1966) pp. 781–786, doi:10.1063/1.1931206 L. Cohen, "Quantization Problem and Variational Principle in the Phase Space Formulation of Quantum Mechanics," J. Math. Phys.
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In two dimensions, parabolic LCSs are also solutions of the global shearless variational principle described above for hyperbolic LCSs. As such, parabolic LCSs are composed of shrink lines and stretch lines that represent geodesics of the Lorentzian metric tensor D^{t_1}_{t_0}. In contrast to hyperbolic LCSs, however, parabolic LCSs satisfy more robust boundary conditions: they remain stationary curves of the material-line-averaged shear functional even under variations to their endpoints. This explains the high degree of robustness and observability that jet cores exhibit in mixing.
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Following his PhD, Ghahramani moved to the University of Toronto in 1995 as an ITRC Postdoctoral Fellow in the Artificial Intelligence Lab, working with Geoffrey Hinton. From 1998 to 2005, he was a member of the faculty at the Gatsby Computational Neuroscience Unit, University College London. Ghahramani has made significant contributions in the areas of Bayesian machine learning (particularly variational methods for approximate Bayesian inference), as well as graphical models and computational neuroscience. His current research focuses on nonparametric Bayesian modelling and statistical machine learning.
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Hamilton originally matured his ideas before putting pen to paper. The discoveries, papers, and treatises previously mentioned might well have formed the whole work of a long and laborious life. But not to speak of his enormous collection of books, full to overflowing with new and original matter, which have been handed over to Trinity College, Dublin, the previous mentioned works barely form the greater portion of what Hamilton has published. Hamilton developed the variational principle, which was reformulated later by Carl Gustav Jacob Jacobi.
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Since 1986, he has been working at ETH Zürich, initially as an assistant professor, becoming a full professor in 1993. His specialisms included nonlinear partial differential equations and calculus of variations. He is joint editor of the journals Calculus of Variations, Commentarii Mathematici Helvetici, International Mathematical Research Notices and Mathematische Zeitschrift. His publications include the book Variational methods (Applications to nonlinear PDE and Hamiltonian systems) (Springer-Verlag, 1990), which was praised by Jürgen Jost as "very useful" with an "impressive range of often difficult examples".
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He established the theorem of instability for the equations of a perturbed motion. Working on the perturbations of stable motions of Hamiltonian system he formulated and proved the theorem of the properties of the Poincaré variational equations that states: “If the unperturbed motion of a holonomic potential system is stable, then, first, the characteristic numbers of all solutions of the variational equations are equal to zero, second, these equations are regular in the sense of Lyapunov and are reduced to a system of equations with constant coefficients and have a quadratic integral of definite sign”. The Chetaev's theorem generalizes the Lagrange's theorem on an equilibrium and the Poincaré–Lyapunov theorem on a periodic motion. According to the theorem, for a stable unperturbed motion of a potential system, an infinitely near perturbed motion has an oscillatory, wave-like, character. # Chetaev’s method of constructing Lyapunov functions as a coupling (combination) of first integrals. The previous result gave rise to and substantiated the Chetaev’s concept of constructing Lyapunov functions using first integrals initially implemented in his famous book “Stability of Motion” as a coupling of first integrals in quadratic form .
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The jury appointed three prize winners and eight honourable mentions. Young-Gook Park, Kim Dea Hyun, Choi Jin Kyu and Kim Won Ill from Hanyang University, South Korea, won the first prize for their project Constellation of Light Field. Ma Xin, Wang Rui and Yang Meng from the Architecture School of Tianjin University, China, won the second prize for their project Condensation of Variational Sunlight Influences. Joe Wu from Delft University of Technology, TU Delft, the Netherlands, also won the second prize for his project Lightscape between gaps.
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By contrast, there are an infinite number of horizontal subspaces to choose from, in forming the direct sum. The horizontal bundle concept is one way to formulate the notion of an Ehresmann connection on a fiber bundle. Thus, for example, if E is a principal G-bundle, then the horizontal bundle is usually required to be G-invariant: such a choice then becomes equivalent to the definition of a connection on the principal bundle.David Bleeker, Gauge Theory and Variational Principles (1981) Addison- Wesely Publishing Company (See theorem 1.2.
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The range of temperatures and water vapour concentrations over which the optical depth computations are valid depends on the training datasets which were used. The spectral range of the RTTOV9.1 model is 3-20 micrometres (500 – 3000 cm-1) in the infrared. RTTOV contains forward, tangent linear, adjoint and K (full Jacobian matrices) versions of the model; the latter three modules for variational assimilation or retrieval applications. One of several applications of RTTOV are retrievals of brightness temperature and sea surface temperature from Advanced Very High Resolution Radiometer sensor.
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It also implied that the Hall conductance can be characterized in terms of a topological invariable called Chern number which was formulated by Thouless and collaborators. Shortly after, in 1982, Horst Störmer and Daniel Tsui observed the fractional quantum Hall effect where the conductance was now a rational multiple of a constant. Laughlin, in 1983, realized that this was a consequence of quasiparticle interaction in the Hall states and formulated a variational method solution, named the Laughlin wavefunction. The study of topological properties of the fractional Hall effect remains an active field of research.
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In the 1970s, many numerical models were devised. Particularly variational approaches, such as those relying on Duvaut and Lion’s existence and uniqueness theories. Over time, these grew into finite element approaches for contact problems with general material models and geometries, and into half-space based approaches for so-called smooth- edged contact problems for linearly elastic materials. Models of the first category were presented by LaursenLaursen, T.A., 2002, Computational Contact and Impact Mechanics, Fundamentals of Modeling Interfacial Phenomena in Nonlinear Finite Element Analysis, Springer, Berlin and by Wriggers.
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Consider some model with parameters \theta and a prior probability density on those parameters p(\theta). The posterior belief about \theta after seeing the data p(\theta\mid y) is given by Bayes rule: The second line of Equation 1 is the model evidence, which is the probability of observing the data given the model. In practice, the posterior cannot usually be computed analytically due to the difficulty in computing the integral over the parameters. Therefore, the posteriors are estimated using approaches such as MCMC sampling or variational Bayes.
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In 1967 Stampacchia was elected President of the Unione Matematica Italiana. It was about this time that his research efforts shifted toward the emerging field of variational inequalities, which he modeled after boundary value problems for partial differential equations.Guido Stampacchia on The MacTutor History of Mathematics archive He was also director of the Istituto per le Applicazioni del Calcolo of Consiglio Nazionale delle Ricerche from December 1968Silvia Mazzone, Guido Stampacchia to 1974. Stampacchia accepted the position of Professor Mathematical Analysis at the University of Rome in 1968 and returned to Pisa in 1970.
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It happens that minimizers of E(\gamma) also minimize L(\gamma), because they turn out to be affinely parameterized, and the inequality is an equality. The usefulness of this approach is that the problem of seeking minimizers of E is a more robust variational problem. Indeed, E is a "convex function" of \gamma, so that within each isotopy class of "reasonable functions", one ought to expect existence, uniqueness, and regularity of minimizers. In contrast, "minimizers" of the functional L(\gamma) are generally not very regular, because arbitrary reparameterizations are allowed.
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In a famous 1947 paper, Korringa showed how his multiple scattering theory (MST) could be used to find the energy as a function of wavevector for electrons in a periodic solid. In 1954, Nobel laureate Walter Kohn and Norman Rostoker, who went on to have a successful career in nuclear physics, derived the same equations using the Kohn variational method. Two of Korringa's students, Sam Faulkner. and Harold Davis, started a program at the Oak Ridge National Laboratory using the Korringa-Kohn-Rostoker (KKR) band-theory equations to calculate the properties of solids.
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It is a method to approximate the energy states of an electron in a crystal lattice. The basic approximation lies in the potential in which the potential is assumed to be spherically symmetric in the muffin-tin region and constant in the interstitial region. Wave functions (the augmented plane waves) are constructed by matching solutions of the Schrödinger equation within each sphere with plane-wave solutions in the interstitial region, and linear combinations of these wave functions are then determined by the variational method. Many modern electronic structure methods employ the approximation.
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The coherent potential approximation (or CPA) is a method, in physics, of finding the Green's function of an effective medium. It is a useful concept in understanding how sound waves scatter in a material which displays spatial inhomogeneity. One version of the CPA is an extension to random materials of the muffin-tin approximation, used to calculate electronic band structure in solids. A variational implementation of the muffin-tin approximation to crystalline solids using Green's functions was suggested by Korringa and by Kohn and Rostoker, and is often referred to as the KKR method.
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He employed granular material physics to describe fragment ejecta behaviour and to predict the impact depth of projectiles as a function of impact velocity. Noting an anomalous behaviour from the experimental observations, he applied the variational perturbation theory to reveal and explain the role played by the increase in mass density during the failure of brittle materials under dynamic compression. In 1999, he accepted a postdoctoral researcher position from the Massachusetts Institute of Technology to enhance his studies in applied mechanics. After his postdoctoral studies, he lectured in ocean engineering at MIT.
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His treatise Theorie des fonctions analytiques laid some of the foundations of group theory, anticipating Galois. In calculus, Lagrange developed a novel approach to interpolation and Taylor series. He studied the three-body problem for the Earth, Sun and Moon (1764) and the movement of Jupiter's satellites (1766), and in 1772 found the special-case solutions to this problem that yield what are now known as Lagrangian points. Lagrange is best known for transforming Newtonian mechanics into a branch of analysis, Lagrangian mechanics, and presented the mechanical "principles" as simple results of the variational calculus.
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His contributions to the calculus of variation are mainly devoted to the proof of existence and uniqueness theorems for maxima and minima of functionals of particular form, in conjunction with his studies on variational inequalities and linear elasticity in theoretical and applied problems: in the paper a semicontinuity theorem for a functional introduced in the same paper is proved in order to solve the Signorini problem, and this theorem was extended in to the case where the given functional has general linear operators as arguments, not necessarily partial differential operators.
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The main mathematical tools to study critical points are renormalization group, which takes advantage of the Russian dolls picture or the self-similarity to explain universality and predict numerically the critical exponents, and variational perturbation theory, which converts divergent perturbation expansions into convergent strong-coupling expansions relevant to critical phenomena. In two- dimensional systems, conformal field theory is a powerful tool which has discovered many new properties of 2D critical systems, employing the fact that scale invariance, along with a few other requisites, leads to an infinite symmetry group.
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In geometry, minimax eversions are a class of sphere eversions, constructed by using half-way models. It is a variational method, and consists of special homotopies (they are shortest paths with respect to Willmore energy); contrast with Thurston's corrugations, which are generic. The original method of half- way models was not optimal: the regular homotopies passed through the midway models, but the path from the round sphere to the midway model was constructed by hand, and was not gradient ascent/descent. Eversions via half-way models are called tobacco-pouch eversions by Francis and Morin.
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It is a special case of the configuration interaction method in which all Slater determinants (or configuration state functions, CSFs) of the proper symmetry are included in the variational procedure (i.e., all Slater determinants obtained by exciting all possible electrons to all possible virtual orbitals, orbitals which are unoccupied in the electronic ground state configuration). This method is equivalent to computing the eigenvalues of the electronic molecular Hamiltonian within the basis set of the above-mentioned configuration state functions. In a minimal basis set a full CI computation is very easy.
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Other formulations start by writing directly the phase-field equations, without referring to any thermodynamical functional (non-variational formulations). In this case the only reference is the sharp interface model, in the sense that it should be recovered when performing the small interface width limit of the phase-field model. Phase-field equations in principle reproduce the interfacial dynamics when the interface width is small compared with the smallest length scale in the problem. In solidification this scale is the capillary length d_o, which is a microscopic scale.
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Non-divergent wind fields are produced by a procedure based on the variational principle and a finite-element discretization. The dispersion model, LODI, solves the 3-D advection-diffusion equation using a Lagrangian stochastic, Monte Carlo method.Ermak, D.L., and J.S. Nasstrom (2000), A Lagrangian Stochastic Diffusion Method for Inhomogeneous Turbulence, Atmospheric Environment, 34, 7, 1059-1068. LODI includes methods for simulating the processes of mean wind advection, turbulent diffusion, radioactive decay and production, bio-agent degradation, first-order chemical reactions, wet deposition, gravitational settling, dry deposition, and buoyant/momentum plume rise.
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An important ingredient in the calculus on finite weighted graphs is the mimicking of standard differential operators from the continuum setting in the discrete setting of finite weighted graphs. This allows one to translate well-studied tools from mathematics, such as partial differential equations and variational methods, and make them usable in applications which can best be modeled by a graph. The fundamental concept which makes this translation possible is the graph gradient, a first-order difference operator on graphs. Based on this one can derive higher-order difference operators, e.g.
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Fox took an advanced course on electrodynamics in 1965 using the first edition of Jackson and taught graduate electrodynamics for the first time in 1978 using the second edition. Jagdish Mehra, a physicist and historian of science, wrote that Jackson's text is not as good as the book of the same name by Julian Schwinger et al. Whereas Jackson treats the subject as a branch of applied mathematics, Schwinger integrates the two, illuminating the properties of the mathematical objects used with physical phenomena. Unlike Jackson, Schwinger employs variational methods and Green's functions extensively.
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Two recipes from Our Savior's (Montevideo, Minnesota) Lutheran Church (1879-2004) 125 Years cookbook It is a popular dish in the Upper Midwest and other areas of the U.S. where potlucks are popular. Canned pineapple is usually used, but other canned fruit, e.g., fruit cocktail and/or mandarin oranges, can be substituted, and there are many slight variations that use additional ingredients. Watergate salad is similar to ambrosia salad, which includes pineapple and marshmallows as part of its base ingredients, and whipped topping and nuts as part of its variational repertoire.
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Federer's mathematical work separates thematically into the periods before and after his watershed 1960 paper Normal and integral currents, co- authored with Fleming. That paper provided the first satisfactory general solution to Plateau's problem — the problem of finding a (k+1)-dimensional least-area surface spanning a given k-dimensional boundary cycle in n-dimensional Euclidean space. Their solution inaugurated a new and fruitful period of research on a large class of geometric variational problems — especially minimal surfaces — via what came to be known as Geometric Measure Theory.
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Projected dynamical systems have evolved out of the desire to dynamically model the behaviour of nonstatic solutions in equilibrium problems over some parameter, typically take to be time. This dynamics differs from that of ordinary differential equations in that solutions are still restricted to whatever constraint set the underlying equilibrium problem was working on, e.g. nonnegativity of investments in financial modeling, convex polyhedral sets in operations research, etc. One particularly important class of equilibrium problems which has aided in the rise of projected dynamical systems has been that of variational inequalities.
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1361 Elected a corresponding member of the Romanian Academy in 1936,Otlăcan, p. 126, 127 he was stripped of his membership by the new communist regime in 1948, Păun Otiman, "1948–Anul imensei jertfe a Academiei Române", in Academica, Nr. 4 (31), December 2013, p. 123 but made a titular member in 1965. Membrii Academiei Române din 1866 până în prezent, at the Romanian Academy site His numerous articles on theoretical and applied mechanics covered topics such as the principles of variational mechanics, the mechanics of ideal fluid flow, the theory of elasticity and astronomy.
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Leonhard Euler gave a formulation of the action principle in 1744, in very recognizable terms, in the Additamentum 2 to his Methodus Inveniendi Lineas Curvas Maximi Minive Proprietate Gaudentes. Beginning with the second paragraph: As Euler states, ∫Mvds is the integral of the momentum over distance travelled, which, in modern notation, equals the abbreviated or reduced action Thus, Euler made an equivalent and (apparently) independent statement of the variational principle in the same year as Maupertuis, albeit slightly later. Curiously, Euler did not claim any priority, as the following episode shows.
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All the constructions in classical differential calculus have an analog in secondary calculus. For instance, higher symmetries of a system of partial differential equations are the analog of vector fields on differentiable manifolds. The Euler operator, which associates to each variational problem the corresponding Euler–Lagrange equation, is the analog of the classical differential associating to a function on a variety its differential. The Euler operator is a secondary differential operator of first order, even if, according to its expression in local coordinates, it looks like one of infinite order.
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This is to be contrasted with the highly sensitive and fading footprint of hyperbolic LCSs away from strongly hyperbolic regions in diffusive tracer patterns. Under variable endpoint boundary conditions, initial positions of parabolic LCSs turn out to be alternating chains of shrink lines and stretch lines that connect singularities of these line fields. These singularities occur at points where \lambda_1(x_0)=\lambda_2(x_0), and hence no infinitesimal deformation takes place between the two time instances t_0 and t_1. Fig. 14b shows an example of parabolic LCSs in Jupiter's atmosphere, located using this variational theory.
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Genetic architecture is the underlying genetic basis of a phenotypic trait and its variational properties. Phenotypic variation for quantitative traits is, at the most basic level, the result of the segregation of alleles at quantitative trait loci (QTL). Environmental factors and other external influences can also play a role in phenotypic variation. Genetic architecture is a broad term that can be described for any given individual based on information regarding gene and allele number, the distribution of allelic and mutational effects, and patterns of pleiotropy, dominance, and epistasis.
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A fundamental flaw of transition state theory is that it counts any crossing of the transition state as a reaction from reactants to products or vice versa. In reality, a molecule may cross this "dividing surface" and turn around, or cross multiple times and only truly react once. As such, unadjusted TST is said to provide an upper bound for the rate coefficients. To correct for this, variational transition state theory varies the location of the dividing surface that defines a successful reaction in order to minimize the rate for each fixed energy.
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The explanation of this small but positive value is an outstanding theoretical challenge, the so-called cosmological constant problem. Some early generalizations of Einstein's gravitational theory, known as classical unified field theories, either introduced a cosmological constant on theoretical grounds or found that it arose naturally from the mathematics. For example, Sir Arthur Stanley Eddington claimed that the cosmological constant version of the vacuum field equation expressed the "epistemological" property that the universe is "self-gauging", and Erwin Schrödinger's pure-affine theory using a simple variational principle produced the field equation with a cosmological term.
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The structure of PsH is as a diatomic molecule, with a chemical bond between the two positively charged centres. The electrons are more concentrated around the proton. Predicting the properties of PsH is a four body Coulomb problem. Calculated using the stochastic variational method, the size of the molecule is larger than dihydrogen, which has a bond length of 0.7413 Å. In PsH the positron and proton are separated on average by 3.66 a0 (1.94 Å). The positronium in the molecule is swollen compared to the positronium atom, increasing to 3.48 a0 compared to 3 a0.
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The second research area is specialized in the study of partial differential equations from a theoretical viewpoint, but also in their applications. Members of this team thus address issues appearing in different branches of mechanics (fluid mechanics, celestial mechanics, quantum mechanics), physics and chemistry (atomic and molecular physics, quantum chemistry) and optimization. Assorted tools and equations are used and studied: variational methods, methods of optimal transportation, transport equations (kinetic equations), classical methods of analysis of partial differential equations (functional analysis, asymptotic methods, a priori estimates, etc.). Some researchers in this research axis conduct research in very sophisticated image and general signal processing.
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Sepp Hochreiter developed "Factor Analysis for Bicluster Acquisition" (FABIA) for biclustering that is simultaneously clustering rows and columns of a matrix. A bicluster in transcriptomic data is a pair of a gene set and a sample set for which the genes are similar to each other on the samples and vice versa. In drug design, for example, the effects of compounds may be similar only on a subgroup of genes. FABIA is a multiplicative model that assumes realistic non-Gaussian signal distributions with heavy tails and utilizes well understood model selection techniques like a variational approach in the Bayesian framework.
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Lazar Aronovich Lyusternik (also Lusternik, Lusternick, Ljusternik; ; 31 December 1899, Zduńska Wola, Congress Poland, Russian Empire (present-day Republic of Poland) – 23 July 1981, Moscow, Russia, Soviet Union) was a Soviet mathematician. He is famous for his work in topology and differential geometry, to which he applied the variational principle. Using the theory he introduced, together with Lev Schnirelmann, he proved the theorem of the three geodesics, a conjecture by Henri Poincaré that every convex body in 3-dimensions has at least three simple closed geodesics. The ellipsoid with distinct but nearly equal axis is the critical case with exactly three closed geodesics.
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An ideal diode is a diode that conducts electricity in the forward direction with no resistance if a forward voltage is applied, but allows no current to flow in the reverse direction. Then if the reverse voltage is v(t), and the forward current is i(t), then there is a complementarity relationship between the two: : 0\leq v(t)\quad\perp\quad i(t)\geq 0 for all t. If the diode is in a circuit containing a memory element, such as a capacitor or inductor, then the circuit can be represented as a differential variational inequality.
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Similarly, Hamilton's equations of motion are another system of 2N first-order equations for the time evolution of the generalized coordinates and their conjugate momenta p_1,\, p_2, ... , p_N . Since the HJE is an equivalent expression of an integral minimization problem such as Hamilton's principle, the HJE can be useful in other problems of the calculus of variations and, more generally, in other branches of mathematics and physics, such as dynamical systems, symplectic geometry and quantum chaos. For example, the Hamilton–Jacobi equations can be used to determine the geodesics on a Riemannian manifold, an important variational problem in Riemannian geometry.
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Thus the (central) elliptical distortion of the Moon's orbit caused by the variation should not be confused with an undisturbed eccentric elliptical motion of an orbiting body. The variational effects due to the Sun would still occur even if the hypothetical undisturbed motion of the Moon had an eccentricity of zero (i.e. even if the orbit would be circular in the absence of the Sun). Newton expressed an approximate recognition that the real orbit of the Moon is not exactly an eccentric Keplerian ellipse, nor exactly a central ellipse due to the variation, but "an oval of another kind".
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The GLM method does not suffer from the strong drawback of the Lagrangian specification of the flow field – following individual fluid parcels – that Lagrangian positions which are initially close gradually drift far apart. In the Lagrangian frame of reference, it therefore becomes often difficult to attribute Lagrangian-mean values to some location in space. The specification of mean properties for the oscillatory part of the flow, like: Stokes drift, wave action, pseudomomentum and pseudoenergy – and the associated conservation laws – arise naturally when using the GLM method. The GLM concept can also be incorporated into variational principles of fluid flow.
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Algorithmic flowchart illustrating the Hartree–Fock method The variational theorem states that for a time-independent Hamiltonian operator, any trial wave function will have an energy expectation value that is greater than or equal to the true ground-state wave function corresponding to the given Hamiltonian. Because of this, the Hartree–Fock energy is an upper bound to the true ground-state energy of a given molecule. In the context of the Hartree–Fock method, the best possible solution is at the Hartree–Fock limit; i.e., the limit of the Hartree–Fock energy as the basis set approaches completeness.
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In mathematics, any Lagrangian system generally admits gauge symmetries, though it may happen that they are trivial. In theoretical physics, the notion of gauge symmetries depending on parameter functions is a cornerstone of contemporary field theory. A gauge symmetry of a Lagrangian L is defined as a differential operator on some vector bundle E taking its values in the linear space of (variational or exact) symmetries of L. Therefore, a gauge symmetry of L depends on sections of E and their partial derivatives.Giachetta (2008) For instance, this is the case of gauge symmetries in classical field theory.
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In mathematics, specifically in the calculus of variations, a variation of a function can be concentrated on an arbitrarily small interval, but not a single point. Accordingly, the necessary condition of extremum (functional derivative equal zero) appears in a weak formulation (variational form) integrated with an arbitrary function . The fundamental lemma of the calculus of variations is typically used to transform this weak formulation into the strong formulation (differential equation), free of the integration with arbitrary function. The proof usually exploits the possibility to choose concentrated on an interval on which keeps sign (positive or negative).
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A number of formulations of the phase-field model are based on a free energy function depending on an order parameter (the phase field) and a diffusive field (variational formulations). Equations of the model are then obtained by using general relations of statistical physics. Such a function is constructed from physical considerations, but contains a parameter or combination of parameters related to the interface width. Parameters of the model are then chosen by studying the limit of the model with this width going to zero, in such a way that one can identify this limit with the intended sharp interface model.
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Nutation refers to the bending movements of stems, roots, leaves and other plant organs caused by differences in growth in different parts of the organ. Circumnutation refers specifically to the circular movements often exhibited by the tips of growing plant stems, caused by repeating cycles of differences in growth around the sides of the elongating stem. Nutational movements are usually distinguished from 'variational' movements caused by temporary differences in the water pressure inside plant cells (turgor). Simple nutation occurs in flat leaves and flower petals, caused by unequal growth of the two sides of the surface.
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Research interests of Professor Hasanoğlu include nonlinear differential equations, variational methods, inverse problems, mathematical and computational modeling in engineering sciences. He is an author of more than 100 scientific papers in international journals and conference proceedings, author of 4 books and co-author of 7 books, and conference proceedings, co-editor of special issues. His research projects has been supported by U.S.S.R Academy of Sciences institutions (1982–1989), Kocaeli Governorship, Arcelik A. S., Istanbul Municipality, TUBİTAK, Turkey (1994–2009), Office of Naval Research, USA (2002), INTAS, Brussels (2007–2009), Science for Peace and Security Section, NATO, Brussels (2008–2010).
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Weierstrass also made advances in the field of calculus of variations. Using the apparatus of analysis that he helped to develop, Weierstrass was able to give a complete reformulation of the theory that paved the way for the modern study of the calculus of variations. Among several axioms, Weierstrass established a necessary condition for the existence of strong extrema of variational problems. He also helped devise the Weierstrass–Erdmann condition, which gives sufficient conditions for an extremal to have a corner along a given extremum and allows one to find a minimizing curve for a given integral.
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In 1993, Pierre SuquetSuquet P., « Overall potentials and flow stresses of ideally plastic or power law materials », J. Mech. Phys. Solids, 41, 1993, pp. 981–1002 proposed a series of bollards for non-linear phase composites, using a method different from those available at the time (Willis, 1988, Ponte Castañeda, 1991), then showed in 1995Suquet P., « Overall properties of nonlinear composites: a modified secant moduli approach and its link with Ponte Casta\~neda's nonlinear variational procedure », C. R. Acad. Sc. Paris, IIb, 320, 1995, pp. 563–571Ponte Castaneda P., Suquet P., « Nonlinear composites », Advances in Applied Mechanics, 34, 1998, pp.
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Hu mainly generalized some versatile variational principles especially in elastic mechanics and promoted their corresponding applications such as in the spacecraft system design. Hu worked for China's spacecraft system design as early as 1966. Hu was in charge of the general system and structure design both for the Dong Fang Hong I and Dong Fang Hong II satellites in their early development phases. In 1993, Hu became a senior advisor and member of the Science and Technology Commission of the China Aerospace Science and Technology Corporation, a technical consultant of the China Academy of Space Technology.
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His theory, and Joseph-Louis Lagrange's improvement on the calculation (applying the variational principle), do not take into account relativistic effects, which were unknown at that time. Even so, Newton's theory is thought to be exceptionally accurate in the limit of weak gravitational fields and low speeds. Newton's theory enjoyed its greatest success when it was used to predict the existence of Neptune based on motions of Uranus that could not be accounted by the actions of the other planets. Calculations by John Couch Adams and Urbain Le Verrier both predicted the general position of the planet.
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Federer is perhaps best known for his treatise Geometric Measure Theory, published in 1969. Intended as both a text and a reference work, the book is unusually complete, general and authoritative: its nearly 600 pages cover a substantial amount of linear and multilinear algebra, give a profound treatment of measure theory, integration and differentiation, and then move on to rectifiability, theory of currents, and finally, variational applications. Nevertheless, the book's unique style exhibits a rare and artistic economy that still inspires admiration, respect—and exasperation. A more accessible introduction may be found in F. Morgan's book listed below.
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In mathematical physics, covariant classical field theory represents classical fields by sections of fiber bundles, and their dynamics is phrased in the context of a finite-dimensional space of fields. Nowadays, it is well known that jet bundles and the variational bicomplex are the correct domain for such a description. The Hamiltonian variant of covariant classical field theory is the covariant Hamiltonian field theory where momenta correspond to derivatives of field variables with respect to all world coordinates. Non-autonomous mechanics is formulated as covariant classical field theory on fiber bundles over the time axis ℝ.
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In quantum mechanical and quantum chemical computations matrix diagonalization is one of the most frequently applied numerical processes. The basic reason is that the time-independent Schrödinger equation is an eigenvalue equation, albeit in most of the physical situations on an infinite dimensional space (a Hilbert space). A very common approximation is to truncate Hilbert space to finite dimension, after which the Schrödinger equation can be formulated as an eigenvalue problem of a real symmetric, or complex Hermitian matrix. Formally this approximation is founded on the variational principle, valid for Hamiltonians that are bounded from below.
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The hook length formula also has important applications to the analysis of longest increasing subsequences in random permutations. If \sigma_n denotes a uniformly random permutation of order n, L(\sigma_n) denotes the maximal length of an increasing subsequence of \sigma_n, and \ell_n denotes the expected (average) value of L(\sigma_n), Anatoly Vershik and Sergei Kerov Vershik, A. M.; Kerov, C. V. (1977), "Asymptotics of the Plancheral measure of the symmetric group and a limiting form for Young tableaux", Dokl. Akad. Nauk SSSR 233: 1024–1027 and independently Benjamin F. Logan and Lawrence A. SheppB. F. Logan and L. A. Shepp, A variational problem for random Young tableaux, Advances in Math.
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The principle of Galilean invariance/relativity was merely implicit in Newton's theory of motion. Having ostensibly reduced the Keplerian celestial laws of motion as well as Galilean terrestrial laws of motion to a unifying force, Newton achieved great mathematical rigor, but with theoretical laxity.Imre Lakatos, auth, Worrall J & Currie G, eds, The Methodology of Scientific Research Programmes: Volume 1: Philosophical Papers (Cambridge: Cambridge University Press, 1980), pp 213–214, 220 In the 18th century, the Swiss Daniel Bernoulli (1700–1782) made contributions to fluid dynamics, and vibrating strings. The Swiss Leonhard Euler (1707–1783) did special work in variational calculus, dynamics, fluid dynamics, and other areas.
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Also notable was the Italian-born Frenchman, Joseph-Louis Lagrange (1736–1813) for work in analytical mechanics: he formulated Lagrangian mechanics) and variational methods. A major contribution to the formulation of Analytical Dynamics called Hamiltonian dynamics was also made by the Irish physicist, astronomer and mathematician, William Rowan Hamilton (1805-1865). Hamiltonian dynamics had played an important role in the formulation of modern theories in physics, including field theory and quantum mechanics. The French mathematical physicist Joseph Fourier (1768 – 1830) introduced the notion of Fourier series to solve the heat equation, giving rise to a new approach to solving partial differential equations by means of integral transforms.
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Fundamental works of Nikolay Bogoliubov were devoted to asymptotic methods of nonlinear mechanics, quantum field theory, statistical field theory, variational calculus, approximation methods in mathematical analysis, equations of mathematical physics, theory of stability, theory of dynamical systems, and to many other areas. He built a new theory of scattering matrices, formulated the concept of microscopical causality, obtained important results in quantum electrodynamics, and investigated on the basis of the edge-of-the-wedge theorem the dispersion relations in elementary particle physics. He suggested a new synthesis of the Bohr theory of quasiperiodic functions and developed methods for asymptotic integration of nonlinear differential equations which describe oscillating processes.
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He is remembered for his achievements on the Plateau's problem, on the theory of parametric minimal surfaces, on Lebesgue measure of continuous and related other variational problems: he also worked in the field of optimal control and studied periodic solutions of systems of nonlinear ordinary differential equations by using methods of nonlinear functional analysis. In the paper he introduced a generalization of functions of bounded variation to the multi-dimensional setting, now acknowledged as the most versatile of such generalizations. He wrote about 250 scientific works on topics such as non linear functional analysis, measure theory, optimal control: his published works include the fundamental monographs , and .
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A solution to the lack of anti-symmetry in the Hartree method came when it was shown that a Slater determinant, a determinant of one- particle orbitals first used by Heisenberg and Dirac in 1926, trivially satisfies the antisymmetric property of the exact solution and hence is a suitable ansatz for applying the variational principle. The original Hartree method can then be viewed as an approximation to the Hartree–Fock method by neglecting exchange. Fock's original method relied heavily on group theory and was too abstract for contemporary physicists to understand and implement. In 1935, Hartree reformulated the method to be more suitable for the purposes of calculation.
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Ralph Tyrrell Rockafellar (born February 10, 1935) is an American mathematician and one of the leading scholars in optimization theory and related fields of analysis and combinatorics. He is the author of four major books including the landmark text “Convex Analysis” (1970), which has been cited more than 27000 times according to Google Scholar and remains the standard reference on the subject, and "Variational Analysis" (1998, with Roger J-B Wets) for which the authors received the Frederick W. Lanchester Prize from the Institute for Operations Research and the Management Sciences (INFORMS). He is professor emeritus at the departments of mathematics and applied mathematics at the University of Washington, Seattle.
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In these conditions of small sinusoidal perturbations with wavelength bigger than its perimeter, the cylinder surface area becomes larger than the one of unperturbed cylinder with the same volume and thus it becomes unstable. Later, Hove Hove, W., Ph.D. Dissertation, Friendlich-Wilhelms, Universitat zu Berlin (1887) formulated the variational requirements for the stability of axisymmetric capillary surfaces (unbounded) in absence of gravity and with disturbances constrained to constant volume. He first solved Young-Laplace equation for equilibrium shapes and showed that the Legendre condition for the second variation is always satisfied. Therefore, the stability is determined by the absence of negative eigenvalue of the linearized Young-Laplace equation.
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Variational Multiscale Method (VMS) was introduced by Hughes in 1995. Broadly speaking, VMS is a technique used to get mathematical models and numerical methods which are able to catch multiscale phenomena; in fact, it is usually adopted for problems with huge scale ranges, which are separated into a number of scale groups. The main idea of the method is to design a sum decomposition of the solution as u = \bar u + u' , where \bar u is denoted as coarse-scale solution and it is solved numerically, whereas u' represents the fine scale solution and is determined analytically eliminating it from the problem of the coarse scale equation.
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His work in this field can be divided into several branches:He is, according to , one of the pioneers of modern numerical analysis together with Boris Galerkin, Alexander Ostrowski, John von Neumann, Walter Ritz and Mauro Picone. in the following text, four main branches are described, and a sketch of his last researches is also given. The papers within the first branch are summarized in the monograph , which contain the study of convergence of variational methods for problems connected with positive operators, in particular, for some problems of mathematical physics. Both "a priori" and "a posteriori" estimates of the errors concerning the approximation given by these methods are proved.
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Canny also introduced the notion of non-maximum suppression, which means that given the presmoothing filters, edge points are defined as points where the gradient magnitude assumes a local maximum in the gradient direction. Looking for the zero crossing of the 2nd derivative along the gradient direction was first proposed by Haralick. R. Haralick, (1984) "Digital step edges from zero crossing of second directional derivatives", IEEE Transactions on Pattern Analysis and Machine Intelligence, 6(1):58–68. It took less than two decades to find a modern geometric variational meaning for that operator that links it to the Marr–Hildreth (zero crossing of the Laplacian) edge detector.
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Ekeland has written influential monographs and textbooks on nonlinear functional analysis, the calculus of variations, and mathematical economics, as well as popular books on mathematics, which have been published in French, English, and other languages. Ekeland is known as the author of Ekeland's variational principle and for his use of the Shapley–Folkman lemma in optimization theory. He has contributed to the periodic solutions of Hamiltonian systems and particularly to the theory of Kreĭn indices for linear systems (Floquet theory).According to D. Pascali, writing for Mathematical Reviews () Ekeland helped to inspire the discussion of chaos theory in Michael Crichton's 1990 novel Jurassic Park.
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Mabuchi is well-known for his introduction, in 1986, of the Mabuchi energy, which gives a variational interpretation to the problem of Kähler metrics of constant scalar curvature. In particular, the Mabuchi energy is a real-valued function on a Kähler class whose Euler-Lagrange equation is the constant scalar curvature equation. In the case that the Kähler class represents the first Chern class of the complex manifold, one has a relation to the Kähler-Einstein problem, due to the fact that constant scalar curvature metrics in such a Kähler class must be Kähler-Einstein. Owing to the second variation formulas for the Mabuchi energy, every critical point is stable.
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The simplest type of ab initio electronic structure calculation is the Hartree-Fock (HF) scheme, in which the instantaneous Coulombic electron- electron repulsion is not specifically taken into account. Only its average effect (mean field) is included in the calculation. This is a variational procedure; therefore, the obtained approximate energies, expressed in terms of the system's wave function, are always equal to or greater than the exact energy, and tend to a limiting value called the Hartree-Fock limit as the size of the basis is increased. Many types of calculations begin with a Hartree- Fock calculation and subsequently correct for electron-electron repulsion, referred to also as electronic correlation.
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Given a set of experimental data that looks to be clustered about a line, a linear ansatz could be made to find the parameters of the line by a least squares curve fit. Variational approximation methods use ansätze and then fit the parameters. Another example could be the mass, energy, and entropy balance equations that, considered simultaneous for purposes of the elementary operations of linear algebra, are the ansatz to most basic problems of thermodynamics. Another example of an ansatz is to suppose the solution of a homogeneous linear differential equation to take an exponential form, or a power form in the case of a difference equation.
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In 1912, The structure of crystalline solids was studied by Max von Laue and Paul Knipping, when they observed the X-ray diffraction pattern of crystals, and concluded that crystals get their structure from periodic lattices of atoms. In 1928, Swiss physicist Felix Bloch provided a wave function solution to the Schrödinger equation with a periodic potential, known as Bloch's theorem. Calculating electronic properties of metals by solving the many-body wavefunction is often computationally hard, and hence, approximation methods are needed to obtain meaningful predictions. The Thomas–Fermi theory, developed in the 1920s, was used to estimate system energy and electronic density by treating the local electron density as a variational parameter.
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Some systems and processes are, however, in a useful sense, near enough to thermodynamic equilibrium to allow description with useful accuracy by currently known non-equilibrium thermodynamics. Nevertheless, many natural systems and processes will always remain far beyond the scope of non-equilibrium thermodynamic methods due to the existence of non variational dynamics, where the concept of free energy is lost. The thermodynamic study of non-equilibrium systems requires more general concepts than are dealt with by equilibrium thermodynamics. One fundamental difference between equilibrium thermodynamics and non-equilibrium thermodynamics lies in the behaviour of inhomogeneous systems, which require for their study knowledge of rates of reaction which are not considered in equilibrium thermodynamics of homogeneous systems.
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In computational physics and chemistry, the Hartree–Fock (HF) method is a method of approximation for the determination of the wave function and the energy of a quantum many-body system in a stationary state. The Hartree–Fock method often assumes that the exact N-body wave function of the system can be approximated by a single Slater determinant (in the case where the particles are fermions) or by a single permanent (in the case of bosons) of N spin- orbitals. By invoking the variational method, one can derive a set of N-coupled equations for the N spin orbitals. A solution of these equations yields the Hartree–Fock wave function and energy of the system.
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The balance between the self-focusing refraction and self-attenuating diffraction by ionization and rarefaction of a laser beam of terawatt intensities, created by chirped pulse amplification, in the atmosphere creates "filaments" which act as waveguides for the beam thus preventing divergence. Competing theories, that the observed filament was actually an illusion created by an axiconic (bessel) or moving focus instead of a "waveguided" concentration of the optical energy, were put to rest by workers at Los Alamos National Laboratory in 1997. Though sophisticated models have been developed to describe the filamentation process, a model proposed by Akozbek et al.N Aközbek, CM Bowden, A Talebpour, SL Chin, Femtosecond pulse propagation in air: Variational analysis, Phys. Rev.
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In general it is not easy to compute this invariant, which was initially introduced by Lazar Lyusternik and Lev Schnirelmann in connection with variational problems. It has a close connection with algebraic topology, in particular cup-length. In the modern normalization, the cup-length is a lower bound for the LS-category. It was, as originally defined for the case of X a manifold, the lower bound for the number of critical points that a real-valued function on X could possess (this should be compared with the result in Morse theory that shows that the sum of the Betti numbers is a lower bound for the number of critical points of a Morse function).
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In the calculus of variations, a field of mathematical analysis, the functional derivative (or variational derivative) relates a change in a functional to a change in a function on which the functional depends. In the calculus of variations, functionals are usually expressed in terms of an integral of functions, their arguments, and their derivatives. In an integral of a functional, if a function is varied by adding to it another function that is arbitrarily small, and the resulting integrand is expanded in powers of , the coefficient of in the first order term is called the functional derivative. For example, consider the functional : J[f] = \int_a^b L( \, x, f(x), f \, '(x) \, ) \, dx \ , where .
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Reddy's teaching and research activities reflect his multidisciplinary perspectives, which he pursues largely through the Centre for Research in Computational and Applied Mechanics (CERECAM) at the University of Cape Town, a centre comprising academic staff and their students in five different departments, straddling the engineering disciplines, mathematics, and biomedical sciences. He has made major contributions to the analysis of problems in solid mechanics, most notably plasticity. He has developed and analysed new variational formulations, as well as associated solution algorithms, which have been implemented computationally, for both classical and gradient theories. The second area in which he has a substantial international reputation is in the development and analysis of mixed and related finite element methods.
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He carried out the first calculations of the F-H-H system with a study of the energy requirements for the reaction H + HF → H2 \+ F and followed this work with calculations for F + H2 → HF + H, a reaction basic to the understanding of molecular dynamics. Trajectory calculations for the HI + HI reaction, a rare event, led to his work on predicting rare events in molecular dynamics by sampling trajectories crossing a surface in phase space. Initially called “variational theory of reaction rate” by James C. Keck (1960), it has since 1973 often been called “the reactive flux method.” Anderson extended Keck’s original method and defended it against a number of critics.
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Mingione received his Ph.D. in mathematics from the University of Naples Federico II in 1999 having Nicola Fusco as advisor; he is professor of mathematics at the University of Parma. He has mainly worked on regularity aspects of the Calculus of Variations, solving a few longstanding questions about the Hausdorff dimension of the singular sets of minimisers of vectorial integral functionals and the boundary singularities of solutions to nonlinear elliptic systems. This connects to the work of authors as Almgren, De Giorgi, Morrey, Giusti, who proved theorems asserting regularity of solutions outside a singular set (i.e. a closed subset of null measure) both in geometric measure theory and for variational systems of partial differential equations.
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Robert Ronald Jensen (born 6 April 1949) is an American mathematician, specializing in nonlinear partial differential equations with applications to physics, engineering, game theory, and finance. Jensen graduated in 1971 with B.S. in mathematics from Illinois Institute of Technology. He received in 1975 his Ph.D. from Northwestern University with thesis Finite difference approximation to the free boundary of a parabolic variational inequality under the supervision of Avner Friedman. Jensen was from 1975 to 1977 an assistant professor at the University of California, Los Angeles and from 1977 to 1980 a visiting assistant professor at the University of Wisconsin's Mathematics Research Center. At the University of Kentucky he was from 1977 to 1980 an assistant professor and from 1980 to 1987 an associate professor.
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These attempts initially concentrated on additional geometric notions such as vierbeins and "distant parallelism", but eventually centered around treating both the metric tensor and the affine connection as fundamental fields. (Because they are not independent, the metric-affine theory was somewhat complicated.) In general relativity, these fields are symmetric (in the matrix sense), but since antisymmetry seemed essential for electromagnetism, the symmetry requirement was relaxed for one or both fields. Einstein's proposed unified-field equations (fundamental laws of physics) were generally derived from a variational principle expressed in terms of the Riemann curvature tensor for the presumed space-time manifold. In field theories of this kind, particles appear as limited regions in space-time in which the field strength or the energy density are particularly high.
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Inspired by Einstein's approach to a unified field theory and Eddington's idea of the affine connection as the sole basis for differential geometric structure for space-time, Erwin Schrödinger from 1940 to 1951 thoroughly investigated pure-affine formulations of generalized gravitational theory. Although he initially assumed a symmetric affine connection, like Einstein he later considered the nonsymmetric field. Schrödinger's most striking discovery during this work was that the metric tensor was induced upon the manifold via a simple construction from the Riemann curvature tensor, which was in turn formed entirely from the affine connection. Further, taking this approach with the simplest feasible basis for the variational principle resulted in a field equation having the form of Einstein's general- relativistic field equation with a cosmological term arising automatically.
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ONETEP (Order-N Electronic Total Energy Package) is a linear-scaling density functional theory software package able to run on parallel computers. It uses a basis of non-orthogonal generalized Wannier functions (NGWFs) expressed in terms of periodic cardinal sine (psinc) functions, which are in turn equivalent to a basis of plane-waves. ONETEP therefore combines the advantages of the plane-wave approach (controllable accuracy and variational convergence of the total energy with respect to the size of the basis) with computational effort that scales linearly with the size of the system. The ONETEP approach involves simultaneous optimization of the density kernel (a generalization of occupation numbers to non-orthogonal basis, which represents the density matrix in the basis of NGWFs) and the NGWFs themselves.
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In mathematics, differential forms on a Riemann surface are an important special case of the general theory of differential forms on smooth manifolds, distinguished by the fact that the conformal structure on the Riemann surface intrinsically defines a Hodge star operator on 1-forms (or differentials) without specifying a Riemannian metric. This allows the use of Hilbert space techniques for studying function theory on the Riemann surface and in particular for the construction of harmonic and holomorphic differentials with prescribed singularities. These methods were first used by in his variational approach to the Dirichlet principle, making rigorous the arguments proposed by Riemann. Later found a direct approach using his method of orthogonal projection, a precursor of the modern theory of elliptic differential operators and Sobolev spaces.
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Energy minimization may ultimately prove more effective, however, as different authors recently showed that the energy optimization is more effective than the variance one. There are different motivations for this: first, usually one is interested in the lowest energy rather than in the lowest variance in both variational and diffusion Monte Carlo; second, variance optimization takes many iterations to optimize determinant parameters and often the optimization can get stuck in multiple local minimum and it suffers of the "false convergence" problem; third energy-minimized wave functions on average yield more accurate values of other expectation values than variance minimized wave functions do. The optimization strategies can be divided into three categories. The first strategy is based on correlated sampling together with deterministic optimization methods.
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The book is divided into three parts: of these, the first treats of the general theory of functions, and gives an algebraic proof of Taylor's theorem, the validity of which is, however, open to question; the second deals with applications to geometry; and the third with applications to mechanics. Another treatise on the same lines was his Leçons sur le calcul des fonctions, issued in 1804, with the second edition in 1806. It is in this book that Lagrange formulated his celebrated method of Lagrange multipliers, in the context of problems of variational calculus with integral constraints. These works devoted to differential calculus and calculus of variations may be considered as the starting point for the researches of Cauchy, Jacobi, and Weierstrass.
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Others expand the true multi-electron wave function in terms of a linear combination of Slater determinants—such as multi-configurational self-consistent field, configuration interaction, quadratic configuration interaction, and complete active space SCF (CASSCF). Still others (such as variational quantum Monte Carlo) modify the Hartree–Fock wave function by multiplying it by a correlation function ("Jastrow" factor), a term which is explicitly a function of multiple electrons that cannot be decomposed into independent single- particle functions. An alternative to Hartree–Fock calculations used in some cases is density functional theory, which treats both exchange and correlation energies, albeit approximately. Indeed, it is common to use calculations that are a hybrid of the two methods—the popular B3LYP scheme is one such hybrid functional method.
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A numerical solution to the heat equation on a pump casing model using the finite element method. Historically, applied mathematics consisted principally of applied analysis, most notably differential equations; approximation theory (broadly construed, to include representations, asymptotic methods, variational methods, and numerical analysis); and applied probability. These areas of mathematics related directly to the development of Newtonian physics, and in fact, the distinction between mathematicians and physicists was not sharply drawn before the mid-19th century. This history left a pedagogical legacy in the United States: until the early 20th century, subjects such as classical mechanics were often taught in applied mathematics departments at American universities rather than in physics departments, and fluid mechanics may still be taught in applied mathematics departments.
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The hybrid Trefftz finite-element method has been considerably advanced since its introduction about 30 years ago. The conventional method of finite element analysis involves converting the differential equation that governs the problem into a variational functional from which element nodal properties – known as field variables – can be found. This can be solved by substituting in approximate solutions to the differential equation and generating the finite element stiffness matrix which is combined with all the elements in the continuum to obtain the global stiffness matrix. Application of the relevant boundary conditions to this global matrix, and the subsequent solution of the field variables rounds off the mathematical process, following which numerical computations can be used to solve real life engineering problems.
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Note that Lie groups do not come equipped with a metric. A yet more complicated, yet more accurate and geometrically enlightening, approach is to understand that the gauge covariant derivative is (exactly) the same thing as the exterior covariant derivative on a section of an associated bundle for the principal fiber bundle of the gauge theory;David Bleecker, "Gauge Theory and Variational Principles" (1982) D. Reidel Publishing (See chapter 3) and, for the case of spinors, the associated bundle would be a spin bundle of the spin structure.David Bleecker, op. cit. (See Chapter 6.) Although conceptually the same, this approach uses a very different set of notation, and requires a far more advanced background in multiple areas of differential geometry.
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Hans F. Weinberger (September 27, 1928 in Vienna - September 15, 2017 in Durham, North Carolina)homepage was an Austrian-American mathematician, known for his contributions to variational methods for eigenvalue problems, partial differential equations, and fluid dynamics. He obtained an M.S. in physics from Carnegie Institute of Technology (1948) where he also got his Sc.D. on the thesis Fourier Transforms of Moebius Series advised by Richard Duffin (1950). He then worked at the institute for Fluid Dynamics at University of Maryland, College Park (1950-60), and as professor at University of Minnesota (1961-98) where he was department head (1967-69) and now is Professor Emeritus (1998-). Weinberger was the first director of Institute for Mathematics and its Applications (1981-87).
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A Bayesian network can thus be considered a mechanism for automatically applying Bayes' theorem to complex problems. The most common exact inference methods are: variable elimination, which eliminates (by integration or summation) the non-observed non-query variables one by one by distributing the sum over the product; clique tree propagation, which caches the computation so that many variables can be queried at one time and new evidence can be propagated quickly; and recursive conditioning and AND/OR search, which allow for a space–time tradeoff and match the efficiency of variable elimination when enough space is used. All of these methods have complexity that is exponential in the network's treewidth. The most common approximate inference algorithms are importance sampling, stochastic MCMC simulation, mini-bucket elimination, loopy belief propagation, generalized belief propagation and variational methods.
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Dal Maso studied at Scuola Normale Superiore under the guidance of Ennio De Giorgi and is professor of mathematics at the International School for Advanced Studies at Trieste, where he also serves as deputy director. Dal Maso has dealt with a number of questions related to partial differential equations and calculus of variations, covering a range of topics going from lower semicontinuity problems for multiple integrals to existence theorem for so called free discontinuity problems, from the study of asymptotic behaviour of variational problems via so called Γ-convergence methods to fine properties of solutions to obstacle problems. In the last years he has been considerably involved in the study of problems arising from applied mathematics, developing methods aimed at describing the evolution of fractures in plasticity problems.
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It establishes that systems minimise a free energy function of their internal states, which entail beliefs about hidden states in their environment. The implicit minimisation of free energy is formally related to variational Bayesian methods and was originally introduced by Karl Friston as an explanation for embodied perception in neuroscience, where it is also known as active inference. The free energy principle explains the existence of a given system by modeling it through a Markov blanket that tries to minimize the difference between their model of the world and their sense and associated perception. This difference can be described as "surprise" and is minimized by continuous correction of the world model of the system. As such, the principle is based on the Bayesian idea of the brain as an “inference engine”.
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However, in stronger bias regimes a more sophisticated treatment is required, as there is no longer a variational principle. In the elastic tunneling case (where the passing electron does not exchange energy with the system), the formalism of Rolf Landauer can be used to calculate the transmission through the system as a function of bias voltage, and hence the current. In inelastic tunneling, an elegant formalism based on the non-equilibrium Green's functions of Leo Kadanoff and Gordon Baym, and independently by Leonid Keldysh was advanced by Ned Wingreen and Yigal Meir. This Meir-Wingreen formulation has been used to great success in the molecular electronics community to examine the more difficult and interesting cases where the transient electron exchanges energy with the molecular system (for example through electron-phonon coupling or electronic excitations).
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Recall that the torsion of a connection \omega can be expressed as :\Theta_\omega = D\theta = d\theta + \omega \wedge \theta where \theta is the solder form (tautological one-form). The subscript \omega serves only as a reminder that this torsion tensor was obtained from the connection. By analogy to the lowering of the index on torsion tensor on the section above, one can perform a similar operation with the solder form, and construct a tensor :\Sigma_\omega(X,Y,Z)=\langle\theta(Z), \Theta_\omega(X,Y)\rangle + \langle\theta(Y), \Theta_\omega(Z,X)\rangle \- \langle\theta(X), \Theta_\omega(Y,Z)\rangle Here \langle,\rangle is the scalar product. This tensor can be expressed asDavid Bleecker, "Gauge Theory and Variational Principles" (1982) D. Reidel Publishing (See theorem 6.2.
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Thesis, "Ceramic Production in Middle Woodland Communities of Practice: A Cordage Twist Analysis in Tidewater Virginia." The College of William and Mary, 2009 Algonquian speakers from the Great Lakes region likely began migrating into the Middle Atlantic region around 100 or 200 CE.Potter 1993; Hayden 2009:8 Their dominant pottery preference was decorated with a S-twist cordage technique.Peterson 1996:95; Potter 1993; Hayden 2009:8 As many as six peoples shared a short period of transitioning in West Virginia. The earliest hamlet village farmers of Fort Ancient and Monongahela were concurrent with the latest Wood, Parkline, Montane and Buck Garden peoples for relatively short passage of time to a new way of living in the state using shell tempered pottery with variational pottery decorations and bow with arrows.
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The eccentric Keplerian ellipse is another and separate approximation for the Moon's orbit, different from the approximation represented by the (central) variational ellipse. The Moon's line of apses, i.e. the long axis of the Moon's orbit when approximated as an eccentric ellipse, rotates once in about nine years, so that it can be oriented at any angle whatever relative to the direction of the Sun at any season. (The angular difference between these two directions used to be referred to, in much older literature, as the "annual argument of the Moon's apogee".) Twice in every period of just over a year, the direction of the Sun coincides with the direction of the long axis of the eccentric elliptical approximation of the Moon's orbit (as projected on to the ecliptic).
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However, many of Hartree's contemporaries did not understand the physical reasoning behind the Hartree method: it appeared to many people to contain empirical elements, and its connection to the solution of the many- body Schrödinger equation was unclear. However, in 1928 J. C. Slater and J. A. Gaunt independently showed that the Hartree method could be couched on a sounder theoretical basis by applying the variational principle to an ansatz (trial wave function) as a product of single-particle functions. In 1930, Slater and V. A. Fock independently pointed out that the Hartree method did not respect the principle of antisymmetry of the wave function. The Hartree method used the Pauli exclusion principle in its older formulation, forbidding the presence of two electrons in the same quantum state.
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The approach of extending the real line with the values infinity and negative infinity and then allowing (convex) functions to take these values can be traced back to Rockafellar’s dissertation and, independently, the work by Jean-Jacques Moreau around the same time. The central role of set-valued mappings (also called multivalued functions) was also recognized in Rockafellar’s dissertation and, in fact, the standard notation ∂f(x) for the set of subgradients of a function f at x originated there. Rockafellar contributed to nonsmooth analysis by extending the rule of Fermat, which characterizes solutions of optimization problems, to composite problems using subgradient calculus and variational geometry and thereby bypassing the implicit function theorem. The approach broadens the notion of Lagrange multipliers to settings beyond smooth equality and inequality systems.
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For first published works, see references in The result was derived using ideas from the classical calculus of variations. After a slight perturbation of the optimal control, one considers the first- order term of a Taylor expansion with respect to the perturbation; sending the perturbation to zero leads to a variational inequality from which the maximum principle follows. Widely regarded as a milestone in optimal control theory, the significance of the maximum principle lies in the fact that maximizing the Hamiltonian is much easier than the original infinite-dimensional control problem; rather than maximizing over a function space, the problem is converted to a pointwise optimization. A similar logic leads to Bellman's principle of optimality, a related approach to optimal control problems which states that the optimal trajectory remains optimal at intermediate points in time.
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Far removed from the variational sociolinguistics as from ideological interpretations, he attempts to answer with new perspectives to the problem of why and how languages live and die. He's been a proponent of ecolinguistics since the mid-eighties of the 20th century.SÁNCHEZ CARRIÓN, José María (1985): La nueva sociolingüistica y la ecología de las lenguas, Donostia-San Sebastián: Eusko Ikaskuntza. However, his most notorious work is his doctoral thesis Un futuro para nuestro pasado [A future for our past] (1987), as it has offered to many Basque language loyalists a theoretical framework for their activity.URLA, Jacqueline (2008): "Kafe Antzokia: The Global meets the Local in Basque Cultural Politics", in ROSEMAN, Sharon R. & PARKHURST, Shawn S. (2008): Recasting Culture and Space in Iberian Contexts, New York: State University of New York, pp. 259–261.
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The most advanced quantum Monte Carlo approaches provide an exact solution to the many-body problem for non- frustrated interacting boson systems, while providing an approximate, yet typically very accurate, description of interacting fermion systems. Most methods aim at computing the ground state wavefunction of the system, with the exception of path integral Monte Carlo and finite-temperature auxiliary-field Monte Carlo, which calculate the density matrix. In addition to static properties, the time-dependent Schrödinger equation can also be solved, albeit only approximately, restricting the functional form of the time-evolved wave function, as done in the time-dependent variational Monte Carlo. From the probabilistic point of view, the computation of the top eigenvalues and the corresponding ground states eigenfunctions associated with the Schrödinger equation relies on the numerical solving of Feynman–Kac path integration problems.
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Variational principles are found among earlier ideas in surveying and optics. The rope stretchers of ancient Egypt stretched corded ropes between two points to measure the path which minimized the distance of separation, and Claudius Ptolemy, in his Geographia (Bk 1, Ch 2), emphasized that one must correct for "deviations from a straight course"; in ancient Greece Euclid states in his Catoptrica that, for the path of light reflecting from a mirror, the angle of incidence equals the angle of reflection; and Hero of Alexandria later showed that this path was the shortest length and least time. This was generalized to refraction by Pierre de Fermat, who, in the 17th century, refined the principle to "light travels between two given points along the path of shortest time"; now known as the principle of least time or Fermat's principle.
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The Hartree–Fock method, despite its physically more accurate picture, was little used until the advent of electronic computers in the 1950s due to the much greater computational demands over the early Hartree method and empirical models. Initially, both the Hartree method and the Hartree–Fock method were applied exclusively to atoms, where the spherical symmetry of the system allowed one to greatly simplify the problem. These approximate methods were (and are) often used together with the central field approximation, to impose the condition that electrons in the same shell have the same radial part, and to restrict the variational solution to be a spin eigenfunction. Even so, calculating a solution by hand using the Hartree–Fock equations for a medium-sized atom was laborious; small molecules required computational resources far beyond what was available before 1950.
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One-shot learning is an object categorization problem, found mostly in computer vision. Whereas most machine learning based object categorization algorithms require training on hundreds or thousands of samples/images and very large datasets, one-shot learning aims to learn information about object categories from one, or only a few, training samples/images. The primary focus of this article will be on the solution to this problem presented by Fei-Fei Li, R. Fergus and P. Perona in IEEE Transactions on Pattern Analysis and Machine Intelligence, Vol 28(4), 2006, which uses a generative object category model and variational Bayesian framework for representation and learning of visual object categories from a handful of training examples. Another paper, presented at the International Conference on Computer Vision and Pattern Recognition (CVPR) 2000 by Erik Miller, Nicholas Matsakis, and Paul Viola will also be discussed.
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In the "Story Notes" section of Stories of Your Life and Others, Chiang writes that inspiration for "Story of Your Life" came from his fascination in the variational principle in physics. When he saw American actor Paul Linke's performance in his play Time Flies When You’re Alive, about his wife's struggle with breast cancer, Chiang realized he could use this principle to show how someone deals with the inevitable. Regarding the theme of the story, Chiang said that Kurt Vonnegut summed it up in his introduction in the 25th anniversary edition of his novel Slaughterhouse-Five: In a 2010 interview Chiang said that "Story of Your Life" addresses the subject of free will. The philosophical debates about whether or not we have free will are all abstract, but knowing the future makes the question very real.
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The variational multiscale method (VMS) is a technique used for deriving models and numerical methods for multiscale phenomena. The VMS framework has been mainly applied to design stabilized finite element methods in which stability of the standard Galerkin method is not ensured both in terms of singular perturbation and of compatibility conditions with the finite element spaces. Stabilized methods are getting increasing attention in computational fluid dynamics because they are designed to solve drawbacks typical of the standard Galerkin method: advection-dominated flows problems and problems in which an arbitrary combination of interpolation functions may yield to unstable discretized formulations. The milestone of stabilized methods for this class of problems can be considered the Streamline Upwind Petrov-Galerkin method (SUPG), designed during 80s for convection dominated-flows for the incompressible Navier–Stokes equations by Brooks and Hughes.
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At the same time, he also taught at the new Novosibirsk University. In November 1967, he defended a doctorate at Moscow University named "A Periodic Problem of Variational Calculus", focused around Fet's theorem about two closed geodesic arcs, which became classical. In 1968, Fet signed the "Letter of 46" in defense of imprisoned dissidents, which became the reason for his dismissal both from the research institute and from the university. The real reason, though, was not the very fact of signing the letter but his independent character and straightforwardness with which he spoke about the professional and human qualities of his co-workers, about the intrigues of functionaries in science and about the privileges in Academgorodok (a limited- access grocery shop for residents, a special medical center and other privileges for the science town management and for doctors of sciences and their families).
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Variational forms are those in which variation is an important formative element. Theme and Variations: a theme, which in itself can be of any shorter form (binary, ternary, etc.), forms the only "section" and is repeated indefinitely (as in strophic form) but is varied each time (A,B,A,F,Z,A), so as to make a sort of sectional chain form. An important variant of this, much used in 17th-century British music and in the Passacaglia and Chaconne, was that of the ground bass—a repeating bass theme or basso ostinato over and around which the rest of the structure unfolds, often, but not always, spinning polyphonic or contrapuntal threads, or improvising divisions and descants. This is said by Scholes (1977) to be the form par excellence of unaccompanied or accompanied solo instrumental music.
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Junuthula N. Reddy (born 12 August 1945) is a Distinguished Professor, Regents' Professor and inaugural holder of the Oscar S. Wyatt Endowed Chair in Mechanical Engineering at Texas A&M; University, College Station, Texas, USA. at Texas A&M; University He is one of the researchers responsible for the development of the Finite Element Method (FEM). He has made seminal contributions in the areas of finite element method, plate theory, solid mechanics, variational methods, mechanics of composites, functionally graded materials, fracture mechanics, plasticity, biomechanics, classical and non- Newtonian fluid mechanics, and applied functional analysis. Reddy has over 620 journal papers, 20 books (with several second and third editions), and has given numerous (over 150) national and international talks. He has served as a member of International Advisory Committee at ICTACEM, 2001 and keynote addressing in 2014.
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Variational orbit: nearly an ellipse, with the Earth at the center. The diagram illustrates the perturbing effect of the Sun on the Moon's orbit, using some simplifying approximations, e.g. that in the absence of the Sun, the Moon's orbit would be circular with the Earth at its center In 1687 Newton published, in the 'Principia', his first steps in the gravitational analysis of the motion of three mutually-attracting bodies. This included a proof that the Variation is one of the results of the perturbation of the motion of the Moon caused by the action of the Sun, and that one of the effects is to distort the Moon's orbit in a practically elliptical manner (ignoring at this point the eccentricity of the Moon's orbit), with the centre of the ellipse occupied by the Earth, and the major axis perpendicular to a line drawn between the Earth and Sun.
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D T Whiteside (ed.) (1973), The Mathematical papers of Isaac Newton, Volume VI: 1684-1691, Cambridge University Press, at page 533. Newton did not give an explicit expression for the form of this "oval of another kind"; to an approximation, it combines the two effects of the central-elliptical variational orbit and the Keplerian eccentric ellipse. Their combination also continually changes its shape as the annual argument changes, and also as the evection shows itself in libratory changes in the eccentricity, and in the direction, of the long axis of the eccentric ellipse. The Variation is the second-largest solar perturbation of the Moon's orbit after the Evection, and the third-largest inequality in the motion of the Moon altogether; (the first and largest of the lunar inequalities is the equation of the centre, a result of the eccentricity - which is not an effect of solar perturbation).
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His work in this area has involved using a combination of both computational/theoretical models coupled with biochemical experiments, which are designed to test and refine these models. Most notably, Stultz’ research group has developed methods for analyzing and modeling intrinsically disordered proteins (IDPs) that are involved in neurodegenerative disorders. In the mid 2010s, he and his coworkers developed a novel method for modeling IDPs that uses Bayesian statistics to quantify the uncertainty in the underlying structural ensemble. Stultz and his lab have also developed a variational Bayes’ method that enables them to apply these methods to larger systems in a fraction of the CPU time that would be required using a standard Bayes’ formalism. In recent years, work in Stultz’s group has shifted to the application of signal processing and machine learning tools to help identify patients at elevated risk of cardiovascular death after an acute coronary syndrome.
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Fomenko is a full member (Academician) of the Russian Academy of Sciences (1994), the International Higher Education Academy of Sciences (1993) and Russian Academy of Technological Sciences (2009), as well as a doctor of physics and mathematics (1972), a professor (1980), and head of the Differential Geometry and Applications Department of the Faculty of Mathematics and Mechanics in Moscow State University (1992). Fomenko is the author of the theory of topological invariants of an integrable Hamiltonian system. He is the author of 180 scientific publications, 26 monographs and textbooks on mathematics, a specialist in geometry and topology, variational calculus, symplectic topology, Hamiltonian geometry and mechanics, and computational geometry. Fomenko is also the author of a number of books on the development of new empirico-statistical methods and their application to the analysis of historical chronicles as well as the chronology of antiquity and the Middle Ages.
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The concept of bi-directional associativity between components, views, and annotations was a distinguishing feature of Revit for many releases. The ease of making changes inspired the name Revit, a contraction of Revise-Instantly. At the heart of Revit is a parametric change propagation engine that relied on a new technology, context-driven parametrics, that was more scalable than the variational and history-driven parametrics used in mechanical CAD software. The term Parametric Building Model was adopted to reflect the fact that changes to parameters drove the whole building model and associated documentation, not just individual components. The company was renamed Revit Technology Corporation in January 2000. Revit version 1.0 was released on April 5, 2000. The software progressed rapidly, with version 2.0, 3.0, 3.1, 4.0, and 4.1 released in August 2000; October 2000; February 2001; June 2001; November 2001; and January 2002, respectively. The software was initially offered only as a monthly rental, with no option to purchase.
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Historically, engineering mathematics consisted mostly of applied analysis, most notably: differential equations; real and complex analysis (including vector and tensor analysis); approximation theory (broadly construed, to include asymptotic, variational, and perturbative methods, representations, numerical analysis); Fourier analysis; potential theory; as well as linear algebra and applied probability, outside of analysis. These areas of mathematics were intimately tied to the development of Newtonian physics, and the mathematical physics of that period. This history also left a legacy: until the early 20th century subjects such as classical mechanics were often taught in applied mathematics departments at American universities, and fluid mechanics may still be taught in (applied) mathematics as well as engineering departments. The success of modern numerical computer methods and software has led to the emergence of computational mathematics, computational science, and computational engineering (the last two are sometimes lumped together and abbreviated as CS&E;), which occasionally use high-performance computing for the simulation of phenomena and the solution of problems in the sciences and engineering.
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In 1924, at the age of 15, Nikolay Bogolyubov wrote his first published scientific paper On the behavior of solutions of linear differential equations at infinity. In 1925 he entered Ph.D. program at the Academy of Sciences of the Ukrainian SSR and obtained the degree of Kandidat Nauk (Candidate of Sciences, equivalent to a Ph.D.) in 1928, at the age of 19, with the doctoral thesis titled On direct methods of variational calculus. In 1930, at the age of 21, he obtained the degree of Doktor nauk (Doctor of Sciences, equivalent to Habilitation), the highest degree in the Soviet Union, which requires the recipient to have made a significant independent contribution to his or her scientific field. This early period of Bogolyubov's work in science was concerned with such mathematical problems as direct methods of the calculus of variations, the theory of almost periodic functions, methods of approximate solution of differential equations, and dynamical systems.
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Ceperley has pioneered novel methods for stochastic computation of quantum systems: variational Monte Carlo techniques for fermions, the fixed-node approximation and nodal release methods, the use of Metropolis steps to enforce reversibility in approximate Green's functions, the development of importance-sampled Diffusion Monte Carlo (DMC) method that has largely superseded other methods, the use of twist-averaged boundary conditions to reduce systematic size errors, the extension of DMC to systems having broken time-reversal symmetry, the fixed phase method. These are essential ingredients to make the methods quantitative and accurate. Ceperley has also introduced and developed the Coupled Electron-Ion Monte Carlo, a first- principles simulation method to perform statistical calculations of finite temperature quantum nuclei using electronic energies and has established a first-order phase transition in the metal-insulator transition of liquid hydrogen. Richard Martin and Ceperley started the annual workshop series, Recent Developments in Electronic Structure Methods in 1989.
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Uhlenbeck is one of the founders of the field of geometric analysis, a discipline that uses differential geometry to study the solutions to differential equations and vice versa. She has also contributed to topological quantum field theory and integrable systems.. Together with Jonathan Sacks in the early 1980s, Uhlenbeck established regularity estimates that have found applications to studies of the singularities of harmonic maps and the existence of smooth local solutions to the Yang–Mills–Higgs equations in gauge theory. In particular, Donaldson describes their joint 1981 paper The existence of minimal immersions of 2-spheres as a "landmark paper... which showed that, with a deeper analysis, variational arguments can still be used to give general existence results" for harmonic map equations. Building on these ideas, Uhlenbeck initiated a systematic study of the moduli theory of minimal surfaces in hyperbolic 3-manifolds (also called minimal submanifold theory) in her 1983 paper, Closed minimal surfaces in hyperbolic 3-manifolds.
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A wide variety of machine learning techniques have been used in IoT domain ranging from traditional methods such as regression, support vector machine, and random forest to advanced ones such as convolutional neural networks, LSTM, and variational autoencoder. In the future, the Internet of Things may be a non- deterministic and open network in which auto-organized or intelligent entities (web services, SOA components) and virtual objects (avatars) will be interoperable and able to act independently (pursuing their own objectives or shared ones) depending on the context, circumstances or environments. Autonomous behavior through the collection and reasoning of context information as well as the object's ability to detect changes in the environment (faults affecting sensors) and introduce suitable mitigation measures constitutes a major research trend, clearly needed to provide credibility to the IoT technology. Modern IoT products and solutions in the marketplace use a variety of different technologies to support such context- aware automation, but more sophisticated forms of intelligence are requested to permit sensor units and intelligent cyber-physical systems to be deployed in real environments.
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In terms of an electron density distribution's gradient vector field, this corresponds to a complete, non-overlapping partitioning of a molecule into three-dimensional basins (atoms) that are linked together by shared two-dimensional separatrices (interatomic surfaces). Within each interatomic surface, the electron density is a maximum at the corresponding internuclear saddle point, which also lies at the minimum of the ridge between corresponding pair of nuclei, the ridge being defined by the pair of gradient trajectories (bond path) originating at the saddle point and terminating at the nuclei. Because QTAIM atoms are always bounded by surfaces having zero flux in the gradient vector field of the electron density, they have some unique quantum mechanical properties compared to other subsystem definitions, including unique electronic kinetic energy the satisfaction of an electronic virial theorem analogous to the molecular electronic virial theorem and some interesting variational properties. QTAIM has gradually become a method for addressing possible questions regarding chemical systems, in a variety of situations hardly handled before by any other model or theory in chemistry.
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Concerning the Saint- Venant's principle, he was able to prove it using a variational approach and a slight variation of a technique employed by Richard Toupin to study the same problem: in the paper The same paper was previously published in Russian in a volume in honour of Ilia Vekua: see for the exact reference. there is a complete proof of the principle under the hypothesis that the base of the cylinder is a set with piecewise smooth boundary. Also he is known for his researches in the theory of hereditary elasticity: the paper emphasizes the necessity of analyzing very well the constitutive equations of materials with memory in order to introduce models where an existence and uniqueness theorems can be proved in such a way that the proof does not rely on an implicit choice of the topology of the function space where the problem is studied. At last, it is worth to mention that Clifford Truesdell invited him to write the contributions and for Siegfried Flügge's Handbuch der Physik.
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An important aspect of solving the functional requires us to find solutions that satisfy the given boundary conditions and satisfy inter-element continuity since we define independently the properties over each element domain. The hybrid Trefftz method differs from the conventional finite element method in the assumed displacement fields and the formulation of the variational functional. In contrast to the conventional method (based on the Rayleigh-Ritz mathematical technique) the Trefftz method (based on the Trefftz mathematical technique) assumes the displacement field is composed of two independent components; the intra- element displacement field which satisfies the governing differential equation and is used to approximate the variation of potential within the element domain, and the conforming frame field which specifically satisfies the inter- element continuity condition, defined on the boundary of the element. The frame field here is the same as that used in the conventional finite element method but defined strictly on the boundary of the element – hence the use of the term "hybrid" in the method's nomenclature.
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This is perhaps best seen in his work on topological methods in nonlinear analysis which he developed into a universal method for finding answers to such qualitative problems such as evaluating the number of solutions, describing the structure of a solution set and conditions for the connectedness of this set, convergence of Galerkin type approximations, the bifurcation of solutions in nonlinear systems, and so on. Krasnosel'skii also presented many new general principles on solvability of a large variety of nonlinear equations, including one-sided estimates, cone stretching and contractions, fixed-point theorems for monotone operators and a combination of the Schauder fixed point and contraction mapping theorems that was the genesis of condensing operators. He suggested a new general method for investigating degenerate extremals in variational problems and developed qualitative methods for studying critical and bifurcation parameter values based on restricted information of nonlinear equations. such as the properties of equations linearized at zero or at infinity, which have been very useful in determining the existence of bounded or periodic solutions.
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Professor Reissner is perhaps best known for the Reissner shear deformation plate theory, which resolved the classical boundary-condition paradox of Kirchhoff, and for establishment of the Reissner variational principle in solid mechanics, for which he received an award from the American Institute of Aeronautics and Astronautics. Professor Reissner also has been honored by the American Society of Civil Engineers with the Theodore von Kármán Medal, by the American Society of Mechanical Engineers with the Timoshenko Medal, and by the University of Hanover, Germany, with an honorary doctorate. He was elected a fellow of the American Academy of Arts and Sciences and the American Institute of Aeronautics and Astronautics, a member of the National Academy of Engineering and the International Academy of Astronautics, and an honorary member of the American Society of Mechanical Engineers and the German Society for Applied Mathematics and Mechanics (Gesellschaft für Angewandte Mathematik und Mechanik). He wrote nearly 300 articles published in scientific and technical journals and continued these contributions to the advancement of knowledge until the last few months of his illness.
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For example, he highlights findings from the Novara Expedition of 1861–67 where "a vast number of measurements of various parts of the body in different races were made, and the men were found in almost every case to present a greater range of variation than the women" (p. 275). To Darwin, the evidence from the medical community at the time, which suggested a greater prevalence of physical abnormalities among men than women, was also indicative of man's greater physical variability. Although Darwin was curious about sex differences in variability throughout the animal kingdom, variability in humans was not a chief concern of his research. The first scholar to carry out a detailed empirical investigation on the question of human sex differences in variability in both physical and mental faculties, was the sexologist Havelock Ellis. In his 1894 publication Man and Woman: A Study of Human Secondary Sexual Characters, Ellis dedicated an entire chapter to the subject, entitled “The Variational Tendency of Men”. In this chapter he posits that “both the physical and mental characters of men show wider limits of variation than do the physical and mental characters of women” (p. 358).
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His Sixth Quartet constitutes one of Tsintsadze's finest creations: on the one hand, it represents the culmination in the development and maturation of the composer's individual style; on the other, it reflects his continued search for new means of expression. A composition consisting of one movement yet divided into five structurally open sections, with its development based on monothematic techniques that serve to integrate the parts into a whole, this quartet is written in a form close to that of rondo-sonata, with a prominent role being given to variational continuation. The first section, marked Andante sostenuto, in which the theme is expounded, is wrought with emotion; in the ensuing Allegro assai the musical development is of a dramatic intensity that finds its culmination in the fugato; in the third section, also marked Andante sostenuto, the theme, filled with concealed sorrow, moves from sighs of lament to a rhythmic acceleration; in the Allegro scherzando, which sounds not unlike a grotesque and fantastic dance, the theme is subjected to a number of contrapuntal devices; finally, in the Andante molto sostenuto, the theme returns in its tragic colouring, as if posing a question to which there comes no reply.
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