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Автор: Grafiati
Опубліковано: 6 вересня 2023

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Статті в журналах з теми "Variational theory"

1

SISSAKIAN, ALEXEY, IGOR SOLOVTSOV, and OLEG SHEVCHENKO. "VARIATIONAL PERTURBATION THEORY." International Journal of Modern Physics A 09, no. 12 (May 10, 1994): 1929–99. http://dx.doi.org/10.1142/s0217751x94000832.

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A nonperturbative method — variational perturbation theory (VPT) — is discussed. A quantity we are interested in is represented by a series, a finite number of terms of which not only describe the region of small coupling constant but reproduce well the strong coupling limit. The method is formulated only in terms of the Gaussian quadratures, and diagrams of the conventional perturbation theory are used. Its efficiency is demonstrated for the quantum-mechanical anharmonic oscillator. The properties of convergence are studied for series in VPT for the [Formula: see text] model. It is shown that it is possible to choose variational additions such that they lead to convergent series for any values of the coupling constant. Upper and lower estimates for the quantities under investigation are considered. The nonperturbative Gaussian effective potential is derived from a more general approach, VPT. Various versions of the variational procedure are explored and the preference for the anharmonic variational procedure in view of convergence of the obtained series is argued. We investigate the renormalization procedure in the φ4 model in VPT. The nonperturbative β function is derived in the framework of the proposed approach. The obtained result is in agreement with four-loop approximation and has the asymptotic behavior as g3/2 for a large coupling constant. We construct the VPT series for Yang-Mills theory and study its convergence properties. We introduce coupling to spinor fields and demonstrate that they do not influence the VPT series convergence properties.
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2

Urban, Zbyněk, and Demeter Krupka. "Foundations of higher-order variational theory on Grassmann fibrations." International Journal of Geometric Methods in Modern Physics 11, no. 07 (August 2014): 1460023. http://dx.doi.org/10.1142/s0219887814600238.

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A setting for higher-order global variational analysis on Grassmann fibrations is presented. The integral variational principles for one-dimensional immersed submanifolds are introduced by means of differential 1-forms with specific properties, similar to the Lepage forms from the variational calculus on fibred manifolds. Prolongations of immersions and vector fields to the Grassmann fibrations are defined as a geometric tool for the variations of immersions, and the first variation formula in the infinitesimal form is derived. Its consequences, the Euler–Lagrange equations for submanifolds and the Noether theorem on invariant variational functionals are proved. Examples clarifying the meaning of the Noether theorem in the context of variational principles for submanifolds are given.
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3

Jackson, A. D., A. Lande, and R. A. Smith. "Planar Theory Made Variational." Physical Review Letters 54, no. 14 (April 8, 1985): 1469–71. http://dx.doi.org/10.1103/physrevlett.54.1469.

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4

Prestipino, Santi, and Erio Tosatti. "Variational theory of preroughening." Physical Review B 59, no. 4 (January 15, 1999): 3108–24. http://dx.doi.org/10.1103/physrevb.59.3108.

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5

Lekner, John. "Variational Theory of Reflection." Australian Journal of Physics 38, no. 2 (1985): 113. http://dx.doi.org/10.1071/ph850113.

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Schwinger's variational method for the scattering phase shift produced by a central potential is adapted to reflection by a planar potential barrier (or well). The formulation is general, for an arbitrary transition between any two media, but the application here is limited to reflection at a barrier between media of equal potential energy. The simplest variational estimate for the reflection amplitude correctly tends to -1 at grazing incidence, as it must for any finite barrier. This is in contrast to the first order perturbation reflection amplitude, which diverges at grazing incidence. The same variational estimate is also correct to second order in the ratio of the interface thickness to the wavelength of the incident wave. The theory applies also to the reflection of the electromagnetic s (or transverse electric) wave at an interface between two media.
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6

Hamad, Esam Z., and G. Ali Mansoori. "Variational theory of mixtures." Fluid Phase Equilibria 37 (January 1987): 255–85. http://dx.doi.org/10.1016/0378-3812(87)80055-9.

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7

Sasaki, Yoshi K. "Entropic Balance Theory and Variational Field Lagrangian Formalism: Tornadogenesis." Journal of the Atmospheric Sciences 71, no. 6 (May 30, 2014): 2104–13. http://dx.doi.org/10.1175/jas-d-13-0211.1.

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Abstract The entropic balance theory has been applied with outstanding results to explain many important aspects of tornadic phenomena. The theory was originally developed in variational (probabilistic) field Lagrangian formalism, or in short, variational formalism, with Lagrangian density and action appropriate for supercell-storm and tornadic phenomena. The variational formalism is broadly used in in modern physics, not only in classical mechanics, with Lagrangian density and action designed for each physical problem properly. The Clebsch transformation (equation) was derived in the classical variational formalism but has not been used because of the unobservable and nonmeteorological Lagrange multiplier. The entropic balance condition is thus developed from the Clebsch transformation, changing the unobservable nonmeteorological Lagrange multiplier to observable meteorological rotational flow velocity with entropy and making it applicable to tornadic phenomena. Theoretical details of the entropic balance are presented such as the entropic right-hand rule, entropic dipole, source and sink, overshooting mechanism of hydrometeors against westerlies and the existence of single and multiple vortices and their relation to tornadogenesis. These results are in reasonable agreement with the many observations and data analysis publications. The Clebsch transformation and entropic balance are the new balance conditions, different from the known other balance conditions such as hydrostatic, (quasi-)geostrophic, cyclostrophic, Boussinesq, and anelastic balance. The variations in calculus of variations and in the classical variational formalism are hypothetical. However, this article suggests that the hypothetical variations could be physical, relating to quantum variations and their interaction with the classical systems.
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8

Yuan, Xiao, Suguru Endo, Qi Zhao, Ying Li, and Simon C. Benjamin. "Theory of variational quantum simulation." Quantum 3 (October 7, 2019): 191. http://dx.doi.org/10.22331/q-2019-10-07-191.

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The variational method is a versatile tool for classical simulation of a variety of quantum systems. Great efforts have recently been devoted to its extension to quantum computing for efficiently solving static many-body problems and simulating real and imaginary time dynamics. In this work, we first review the conventional variational principles, including the Rayleigh-Ritz method for solving static problems, and the Dirac and Frenkel variational principle, the McLachlan's variational principle, and the time-dependent variational principle, for simulating real time dynamics. We focus on the simulation of dynamics and discuss the connections of the three variational principles. Previous works mainly focus on the unitary evolution of pure states. In this work, we introduce variational quantum simulation of mixed states under general stochastic evolution. We show how the results can be reduced to the pure state case with a correction term that takes accounts of global phase alignment. For variational simulation of imaginary time evolution, we also extend it to the mixed state scenario and discuss variational Gibbs state preparation. We further elaborate on the design of ansatz that is compatible with post-selection measurement and the implementation of the generalised variational algorithms with quantum circuits. Our work completes the theory of variational quantum simulation of general real and imaginary time evolution and it is applicable to near-term quantum hardware.
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9

Tessarotto, Massimo, and Claudio Cremaschini. "The Principle of Covariance and the Hamiltonian Formulation of General Relativity." Entropy 23, no. 2 (February 10, 2021): 215. http://dx.doi.org/10.3390/e23020215.

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The implications of the general covariance principle for the establishment of a Hamiltonian variational formulation of classical General Relativity are addressed. The analysis is performed in the framework of the Einstein-Hilbert variational theory. Preliminarily, customary Lagrangian variational principles are reviewed, pointing out the existence of a novel variational formulation in which the class of variations remains unconstrained. As a second step, the conditions of validity of the non-manifestly covariant ADM variational theory are questioned. The main result concerns the proof of its intrinsic non-Hamiltonian character and the failure of this approach in providing a symplectic structure of space-time. In contrast, it is demonstrated that a solution reconciling the physical requirements of covariance and manifest covariance of variational theory with the existence of a classical Hamiltonian structure for the gravitational field can be reached in the framework of synchronous variational principles. Both path-integral and volume-integral realizations of the Hamilton variational principle are explicitly determined and the corresponding physical interpretations are pointed out.
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10

Laurin-Kovitz, Kirsten F., and E. E. Lewis. "Variational Nodal Transport Perturbation Theory." Nuclear Science and Engineering 123, no. 3 (July 1996): 369–80. http://dx.doi.org/10.13182/nse96-a24200.

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Дисертації з теми "Variational theory"

1

Aghassi, Michele Leslie. "Robust optimization, game theory, and variational inequalities." Thesis, Massachusetts Institute of Technology, 2005. http://hdl.handle.net/1721.1/33670.

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Анотація:
Thesis (Ph. D.)--Massachusetts Institute of Technology, Sloan School of Management, Operations Research Center, 2005.
Includes bibliographical references (p. 193-109).
We propose a robust optimization approach to analyzing three distinct classes of problems related to the notion of equilibrium: the nominal variational inequality (VI) problem over a polyhedron, the finite game under payoff uncertainty, and the network design problem under demand uncertainty. In the first part of the thesis, we demonstrate that the nominal VI problem is in fact a special instance of a robust constraint. Using this insight and duality-based proof techniques from robust optimization, we reformulate the VI problem over a polyhedron as a single- level (and many-times continuously differentiable) optimization problem. This reformulation applies even if the associated cost function has an asymmetric Jacobian matrix. We give sufficient conditions for the convexity of this reformulation and thereby identify a class of VIs, of which monotone affine (and possibly asymmetric) VIs are a special case, which may be solved using widely-available and commercial-grade convex optimization software. In the second part of the thesis, we propose a distribution-free model of incomplete- information games, in which the players use a robust optimization approach to contend with payoff uncertainty.
(cont.) Our "robust game" model relaxes the assumptions of Harsanyi's Bayesian game model, and provides an alternative, distribution-free equilibrium concept, for which, in contrast to ex post equilibria, existence is guaranteed. We show that computation of "robust-optimization equilibria" is analogous to that of Nash equilibria of complete- information games. Our results cover incomplete-information games either involving or not involving private information. In the third part of the thesis, we consider uncertainty on the part of a mechanism designer. Specifically, we present a novel, robust optimization model of the network design problem (NDP) under demand uncertainty and congestion effects, and under either system- optimal or user-optimal routing. We propose a corresponding branch and bound algorithm which comprises the first constructive use of the price of anarchy concept. In addition, we characterize conditions under which the robust NDP reduces to a less computationally demanding problem, either a nominal counterpart or a single-level quadratic optimization problem. Finally, we present a novel traffic "paradox," illustrating counterintuitive behavior of changes in cost relative to changes in demand.
by Michele Leslie Aghassi.
Ph.D.
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2

Worthing, Rodney A. (Rodney Alan). "Contributions to the variational theory of convection." Thesis, Massachusetts Institute of Technology, 1996. http://hdl.handle.net/1721.1/10577.

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3

Gmeineder, Franz Xaver. "Regularity theory for variational problems on BD." Thesis, University of Oxford, 2017. https://ora.ox.ac.uk/objects/uuid:1f412087-de70-44a8-a045-8923f1e29611.

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In this thesis we provide regularity results for convex and semiconvex variational problems which are of linear growth and depend on the symmetric rather than the full gradient. By the non-availability of Korn's Inequality (known as Ornstein's Non-Inequality), usual approaches need to be modified in order to obtain higher regularity of generalised minima.
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4

Scott, Matthew. "Theory of electrode polarization, application of variational methods." Thesis, National Library of Canada = Bibliothèque nationale du Canada, 2000. http://www.collectionscanada.ca/obj/s4/f2/dsk1/tape3/PQDD_0015/MQ55238.pdf.

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5

Türköz, Ş. (Şemsettin). "Variational procedure for [phi]4-scalar field theory." Thesis, Massachusetts Institute of Technology, 1989. http://hdl.handle.net/1721.1/52913.

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Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Physics, 1990.
On t.p. "[phi]" is the original Greek letter.
Includes bibliographical references (leaves 81-83).
by Ş. Türköz.
Ph.D.
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6

Santambrogio, Filippo. "Variational problems in transport theory with mass concentration." Doctoral thesis, Scuola Normale Superiore, 2006. http://hdl.handle.net/11384/85701.

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7

Buquicchio, Luke J. "Variational Open Set Recognition." Digital WPI, 2020. https://digitalcommons.wpi.edu/etd-theses/1377.

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In traditional classification problems, all classes in the test set are assumed to also occur in the training set, also referred to as the closed-set assumption. However, in practice, new classes may occur in the test set, which reduces the performance of machine learning models trained under the closed-set assumption. Machine learning models should be able to accurately classify instances of classes known during training while concurrently recognizing instances of previously unseen classes (also called the open set assumption). This open set assumption is motivated by real world applications of classifiers wherein its improbable that sufficient data can be collected a priori on all possible classes to reliably train for them. For example, motivated by the DARPA WASH project at WPI, a disease classifier trained on data collected prior to the outbreak of COVID-19 might erroneously diagnose patients with the flu rather than the novel coronavirus. State-of-the-art open set methods based on the Extreme Value Theory (EVT) fail to adequately model class distributions with unequal variances. We propose the Variational Open-Set Recognition (VOSR) model that leverages all class-belongingness probabilities to reject unknown instances. To realize the VOSR model, we design a novel Multi-Modal Variational Autoencoder (MMVAE) that learns well-separated Gaussian Mixture distributions with equal variances in its latent representation. During training, VOSR maps instances of known classes to high-probability regions of class-specific components. By enforcing a large distance between these latent components during training, VOSR then assumes unknown data lies in the low-probability space between components and uses a multivariate form of Extreme Value Theory to reject unknown instances. Our VOSR framework outperforms state-of-the-art open set classification methods with a 15% F1 score increase on a variety of benchmark datasets.
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8

Black, Joshua. "Development and applications of Quasi-Variational Coupled-Cluster theory." Thesis, Cardiff University, 2017. http://orca.cf.ac.uk/105353/.

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The Quasi-Variational (QV) family of methods are a set of single-reference algorithms that can be used to investigate multireference systems with large nondynamic correlation effects. Within this current work, the Quasi-Variational Coupled Cluster Doubles (QVCCD) equations are derived and implemented into Molpro’s Integrated Tensor Framework (ITF), to produce fast and efficient code. This code, coupled with a new orbital optimisation implementation, is used to calculate potential energy curves for third-row diatomic molecules. In contrast to Traditional Coupled-Cluster methods, the QV methods are able to correctly describe the dissociation of these molecules. QV and several other single-reference methods are also applied to 5 chemical databases comprising of 88 unique reactions. From this, the activation and reaction energies are determined and contrasted. The QV methods produce larger activation energies that may correct the shortcomings of the perturbative triples correction. These results also include a new QV method with n ‘asymmetricrenormalised’ triples correction. The numerical results show there is little difference between this procedure and ‘symmetric-renormalised’ triples. Currently, only closed-shell QVCCD programs exist. Unrestricted QVCCD equations are derived and presented in the hope that this will facilitate the realisation of an open-shell QVCCD program. Finally, calculating the rate of a chemical reaction is of fundamental importance to chemistry. Knowledge of how quickly a reaction proceeds allows for an understanding of macroscopic chemical change. Rate constants are calculated with the on-the-fly Instanton method. In contrast to semi-classical Transition State Theory, the Instanton method incorporates quantum effects like atomic tunnelling into its rate constants. The effects of hydrogen tunnelling are examined for a reaction involving a Criegee intermediate. It is discovered that tunnelling does play a role in the reaction rate and may increase it by a factor of 1000. Combination of the Instanton calculations with the QV methods are discussed.
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9

Brown, Bruce J. L. "A variational approach to local optimality in control theory." Doctoral thesis, University of Cape Town, 2001. http://hdl.handle.net/11427/4869.

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Анотація:
Bibliography: leaves 203-212.
A new approach to control theory is investigated in this thesis. The approach is based on a locally specified state space model of the control dynamics; together with a goal function, which defines a generalized distance from each state position to the desired equilibrium point or trajectory. A feedback control function is sought, which will result in a system response which approximates the gradient descent trajectories of the specified goal function. The approximation is chosen so that the resulting trajectories satisfy a certain local optimality criterion, involving the averaged second derivative of the goal function along the trajectories.
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10

Laatz, C. D. "Cosmological perturbation theory and the variational principle in gravitation." Master's thesis, University of Cape Town, 2000. http://hdl.handle.net/11427/6671.

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Summary in English.
Include bibliographical references.
In this thesis firstly the theory of relativistic cosmological perturbations is studies, in the process being reviewed over the period 1960-1993. Secondly the variational principle, apropos of gravitation, is formulated and discussed. These two fields are then synthesised via a variational formulation of general relativity and cosmological perturbation theory. In the process new light is shed on Covariant Perturbation Theory via the development of generalised alternative variables, culminating in a unique variational formulation.
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Книги з теми "Variational theory"

1

Huang, Zhonglian, and Yongzhong Zhang. Variational Translation Theory. Singapore: Springer Singapore, 2020. http://dx.doi.org/10.1007/978-981-15-9271-3.

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2

Bezhaev, Anatoly Yu, and Vladimir A. Vasilenko. Variational Theory of Splines. Boston, MA: Springer US, 2001. http://dx.doi.org/10.1007/978-1-4757-3428-7.

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3

A, Vasilenko V., ed. Variational theory of splines. New York: Kluwer Academic/Plenum Publishers, 2001.

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4

Cheng, Zhengqian. Variational Discrete Action Theory. [New York, N.Y.?]: [publisher not identified], 2021.

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5

Bezhaev, Anatoly Yu. Variational Theory of Splines. Boston, MA: Springer US, 2001.

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6

Postnikov, M. M. The variational theory of geodesics. Mineola, N.Y: Dover Publications, 2003.

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7

Bleecker, David. Gauge theory and variational principles. Mineola, N.Y: Dover Publications, 2005.

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8

Libai, Avinoam. Variational principles in nonlinear shell theory. Haifa: Technion Israel Institute of Technology, 1987.

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9

Masiello, A. Variational methods in Lorentzian geometry. Harlow, Essex, England: Longman Scientific & Technical, 1994.

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10

Fixed point theory, variational analysis, and optimization. Boca Raton: CRC Press, Taylor & Francis Group, 2014.

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Частини книг з теми "Variational theory"

1

Lekner, John. "Variational theory." In Theory of Reflection of Electromagnetic and Particle Waves, 77–92. Dordrecht: Springer Netherlands, 1987. http://dx.doi.org/10.1007/978-94-015-7748-9_4.

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2

Lekner, John. "Variational Theory." In Theory of Reflection, 95–114. Cham: Springer International Publishing, 2016. http://dx.doi.org/10.1007/978-3-319-23627-8_4.

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3

Huang, Zhonglian, and Yongzhong Zhang. "Variational Translation Theory: An Emerging Translation Theory." In Variational Translation Theory, 19–45. Singapore: Springer Singapore, 2020. http://dx.doi.org/10.1007/978-981-15-9271-3_2.

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4

Chang, Kung-Ching. "Morse Theory for Harmonic Maps." In Variational Methods, 431–46. Boston, MA: Birkhäuser Boston, 1990. http://dx.doi.org/10.1007/978-1-4757-1080-9_30.

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5

Leipholz, Horst. "Linear Variational Equations." In Stability Theory, 24–60. Wiesbaden: Vieweg+Teubner Verlag, 1987. http://dx.doi.org/10.1007/978-3-663-10648-7_3.

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6

Ahlbrandt, Calvin D., and Allan C. Peterson. "Discrete Variational Theory." In Discrete Hamiltonian Systems, 153–97. Boston, MA: Springer US, 1996. http://dx.doi.org/10.1007/978-1-4757-2467-7_4.

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7

Nagurney, Anna. "Variational Inequality Theory." In Advances in Computational Economics, 3–37. Dordrecht: Springer Netherlands, 1993. http://dx.doi.org/10.1007/978-94-011-2178-1_1.

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8

Nagurney, Anna. "Variational Inequality Theory." In Advances in Computational Economics, 3–48. Boston, MA: Springer US, 1999. http://dx.doi.org/10.1007/978-1-4757-3005-0_1.

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9

Huang, Zhonglian, and Yongzhong Zhang. "Variational Translation System." In Variational Translation Theory, 81–88. Singapore: Springer Singapore, 2020. http://dx.doi.org/10.1007/978-981-15-9271-3_5.

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10

Huang, Zhonglian, and Yongzhong Zhang. "Complete Translation and Variational Translation." In Variational Translation Theory, 1–17. Singapore: Springer Singapore, 2020. http://dx.doi.org/10.1007/978-981-15-9271-3_1.

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Тези доповідей конференцій з теми "Variational theory"

1

Ding, Rui. "Lipschitz Variational Approximation of Total Variation Distance." In 5th International Conference on Statistics: Theory and Applications (ICSTA 2023). Avestia Publishing, 2023. http://dx.doi.org/10.11159/icsta23.138.

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2

Tatchyn, Roman. "Variational Theory of Insertion Devices." In International Conference on Insertion Devices for Synchrotron Sources, edited by Ingolf E. Lindau and Roman O. Tatchyn. SPIE, 1986. http://dx.doi.org/10.1117/12.950945.

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3

Preston, Serge. "Variational theory of balance systems." In Proceedings of the 10th International Conference on DGA2007. WORLD SCIENTIFIC, 2008. http://dx.doi.org/10.1142/9789812790613_0057.

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4

Panta Pazos, Rube´n, and Marco Tu´llio de Vilhena. "Variational Approach in Transport Theory." In 12th International Conference on Nuclear Engineering. ASMEDC, 2004. http://dx.doi.org/10.1115/icone12-49233.

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In this work we present a variational approach to some methods to solve transport problems of neutral particles. We consider a convex domain X (for example the geometry of slab, or a convex set in the plane, or a convex bounded set in the space) and we use discrete ordinates quadrature to get a system of differential equations derived from the neutron transport equation. The boundary conditions are vacuum for a subset of the boundary, and of specular reflection for the complementary subset of the boundary. Recently some different approximation methods have been presented to solve these transport problems. We introduce in this work the adjoint equations and the conjugate functions obtained by means of the variational approach. First we consider the general formulation, and then some numerical methods such as spherical harmonics and spectral collocation method.
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5

XIAN, Y. "A VARIATIONAL COUPLED-CLUSTER THEORY." In A Festschrift in Honour of the 65th Birthdays of John W Clark, Alpo J Kallio, Manfred L Ristig and Sergio Rosati. WORLD SCIENTIFIC, 2001. http://dx.doi.org/10.1142/9789812799760_0007.

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6

"VARIATIONAL REGION GROWING." In International Conference on Computer Vision Theory and Applications. SciTePress - Science and and Technology Publications, 2009. http://dx.doi.org/10.5220/0001790001660171.

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7

Hershey, John R., Peder A. Olsen, and Ramesh A. Gopinath. "Variational sampling approaches to word confusability." In 2007 Information Theory and Applications Workshop. IEEE, 2007. http://dx.doi.org/10.1109/ita.2007.4357616.

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8

Khosravifard, M., D. Fooladivanda, and T. A. Gulliver. "Exceptionality of the Variational Distance." In 2006 IEEE Information Theory Workshop - ITW '06 Chengdu. IEEE, 2006. http://dx.doi.org/10.1109/itw2.2006.323802.

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9

Wang, Hongwei, Hang Yu, Michael Hoy, Justin Dauwels, and Heping Wang. "Variational Bayesian dynamic compressive sensing." In 2016 IEEE International Symposium on Information Theory (ISIT). IEEE, 2016. http://dx.doi.org/10.1109/isit.2016.7541533.

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Rodriguez-Galvez, Borja, Ragnar Thobaben, and Mikael Skoglund. "A Variational Approach to Privacy and Fairness." In 2021 IEEE Information Theory Workshop (ITW). IEEE, 2021. http://dx.doi.org/10.1109/itw48936.2021.9611429.

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Звіти організацій з теми "Variational theory"

1

Truhlar, Donald G. Variational Transition State Theory. Office of Scientific and Technical Information (OSTI), September 2016. http://dx.doi.org/10.2172/1324939.

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2

Truhlar, D. G. Variational transition state theory. Office of Scientific and Technical Information (OSTI), January 1990. http://dx.doi.org/10.2172/6453957.

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3

Iarve, E. Spline Variational Theory for Composite Bolted Joints. Fort Belvoir, VA: Defense Technical Information Center, January 1997. http://dx.doi.org/10.21236/ada328258.

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4

Iarve, E., and R. Y. Kim. Spline Variational Theory for Composite Bolted Joints. Fort Belvoir, VA: Defense Technical Information Center, January 1998. http://dx.doi.org/10.21236/ada351476.

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5

Iarve, E. V., and R. Y. Kim. Spline Variational Theory for Composite Bolted Joints. Fort Belvoir, VA: Defense Technical Information Center, April 2000. http://dx.doi.org/10.21236/ada387153.

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6

Zako, R. L. Hamiltonian lattice field theory: Computer calculations using variational methods. Office of Scientific and Technical Information (OSTI), December 1991. http://dx.doi.org/10.2172/5736347.

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7

Zako, Robert L. Hamiltonian lattice field theory: Computer calculations using variational methods. Office of Scientific and Technical Information (OSTI), December 1991. http://dx.doi.org/10.2172/10132471.

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8

Zoltani, C. K., S. Kovesi-Domokos, and G. Domokos. Variational Method in the Statistical Theory of Turbulent Two-Phase Flows. Fort Belvoir, VA: Defense Technical Information Center, June 1992. http://dx.doi.org/10.21236/ada252263.

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9

Tadjbakhsh, Iradj G., and Dimitris C. Lagoudas. Variational Theory of Deformations of Curved, Twisted and Extensible Elastic Rods. Fort Belvoir, VA: Defense Technical Information Center, January 1993. http://dx.doi.org/10.21236/ada260331.

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10

Tadjbakhsh, Iradj, and Dimitris C. Lagoudas. Variational Theory of Motion of Curved, Twisted and Extensible Elastic Rods. Fort Belvoir, VA: Defense Technical Information Center, January 1993. http://dx.doi.org/10.21236/ada261028.

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