Relevant bibliographies by topics / Lie Symmetry group of SDE

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  2. Dissertations / Theses
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Academic literature on the topic 'Lie Symmetry group of SDE'

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Published: 1 February 2025

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Journal articles on the topic "Lie Symmetry group of SDE"

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Muniz, Michelle, Matthias Ehrhardt, and Michael Günther. "Approximating Correlation Matrices Using Stochastic Lie Group Methods." Mathematics 9, no. 1 (January 4, 2021): 94. http://dx.doi.org/10.3390/math9010094.

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Specifying time-dependent correlation matrices is a problem that occurs in several important areas of finance and risk management. The goal of this work is to tackle this problem by applying techniques of geometric integration in financial mathematics, i.e., to combine two fields of numerical mathematics that have not been studied yet jointly. Based on isospectral flows we create valid time-dependent correlation matrices, so called correlation flows, by solving a stochastic differential equation (SDE) that evolves in the special orthogonal group. Since the geometric structure of the special orthogonal group needs to be preserved we use stochastic Lie group integrators to solve this SDE. An application example is presented to illustrate this novel methodology.
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YU, JUN, and HANWEI HU. "FINITE SYMMETRY GROUP AND COHERENT SOLITON SOLUTIONS FOR THE BROER–KAUP–KUPERSHMIDT SYSTEM." International Journal of Bifurcation and Chaos 23, no. 09 (September 2013): 1350156. http://dx.doi.org/10.1142/s0218127413501563.

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A modified CK direct method is generalized to find finite symmetry groups of nonlinear mathematical physics systems. For the (2 + 1)-dimensional Broer–Kaup–Kupershmidt (BKK) system, both the Lie point symmetry and the non-Lie symmetry groups are obtained by this method. While using the traditional Lie approach, one can only find the Lie symmetry groups. Furthermore, abundant localized structures of the BKK equation are also obtained from the non-Lie symmetry group.
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Chen, Yong, and Xiaorui Hu. "Lie Symmetry Group of the Nonisospectral Kadomtsev-Petviashvili Equation." Zeitschrift für Naturforschung A 64, no. 1-2 (February 1, 2009): 8–14. http://dx.doi.org/10.1515/zna-2009-1-202.

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The classical symmetry method and the modified Clarkson and Kruskal (C-K) method are used to obtain the Lie symmetry group of a nonisospectral Kadomtsev-Petviashvili (KP) equation. It is shown that the Lie symmetry group obtained via the traditional Lie approach is only a special case of the symmetry groups obtained by the modified C-K method. The discrete group analysis is given to show the relations between the discrete group and parameters in the ansatz. Furthermore, the expressions of the exact finite transformation of the Lie groups via the modified C-K method are much simpler than those obtained via the standard approach.
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Kötz, H. "A Technique to Classify the Similarity Solutions of Nonlinear Partial (Integro-)Differential Equations. I. Optimal Systems of Solvable Lie Subalgebras." Zeitschrift für Naturforschung A 47, no. 11 (November 1, 1992): 1161–74. http://dx.doi.org/10.1515/zna-1992-1114.

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Abstract Lie group analysis is a powerful tool for obtaining exact similarity solutions of nonlinear (integro-) differential equations. In order to calculate the group-invariant solutions one first has to find the full Lie point symmetry group admitted by the given (integro-)differential equations and to determine all the subgroups of this Lie group. An effective, systematic means to classify the similarity solutions afterwards is an "optimal system", i.e. a list of group-invariant solutions from which every other such solution can be derived. The problem to find optimal systems of similarity solutions leads to that to "construct" the optimal systems of subalgebras for the Lie algebra of the known Lie point symmetry group. Our aim is to demonstrate a practicable technique for determining these optimal subalgebraic systems using the invariants relative to the group of the inner automorphisms of the Lie algebra in case of a finite-dimensional Lie point symmetry group. Here, we restrict our attention to optimal subsystems of solvable Lie subalgebras. This technique is applied to the nine-dimensional real Lie point symmetry group admitted by the two-dimensional non-stationary ideal magnetohydrodynamic equations
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SHIRKOV, DMITRIJ V. "RENORMALIZATION GROUP SYMMETRY AND SOPHUS LIE GROUP ANALYSIS." International Journal of Modern Physics C 06, no. 04 (August 1995): 503–12. http://dx.doi.org/10.1142/s0129183195000356.

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We start with a short discussion of the content of a term Renormalisation Group in modern use. By treating the underlying solution property as a reparametrisation symmetry, we relate it with the self-similarity symmetry well-known in mathematical physics and explain the notion of Functional Self-similarity. Then we formulate a program of constructing a regular approach for discovering RG-type symmetries in different problems of mathematical physics. This approach based upon S. Lie group analysis allows one to analyse a wide class of boundary problems for different type of equations. Several examples are mentioned.
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Nadjafikhah, Mehdi, and Seyed-Reza Hejazi. "SYMMETRY ANALYSIS OF TELEGRAPH EQUATION." Asian-European Journal of Mathematics 04, no. 01 (March 2011): 117–26. http://dx.doi.org/10.1142/s1793557111000101.

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Lie symmetry group method is applied to study the telegraph equation. The symmetry group and one-parameter group associated to the symmetries with the structure of the Lie algebra symmetries are determined. The reduced version of equation and its one-dimensional optimal system are given.
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Nadjafikhah, Mehdi, and Mehdi Jafari. "Some General New Einstein Walker Manifolds." Advances in Mathematical Physics 2013 (2013): 1–8. http://dx.doi.org/10.1155/2013/591852.

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Lie symmetry group method is applied to find the Lie point symmetry group of a system of partial differential equations that determines general form of four-dimensional Einstein Walker manifold. Also we will construct the optimal system of one-dimensional Lie subalgebras and investigate some of its group invariant solutions.
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Almutiben, Nouf, Ryad Ghanam, G. Thompson, and Edward L. Boone. "Symmetry analysis of the canonical connection on Lie groups: six-dimensional case with abelian nilradical and one-dimensional center." AIMS Mathematics 9, no. 6 (2024): 14504–24. http://dx.doi.org/10.3934/math.2024705.

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<abstract><p>In this article, the investigation into the Lie symmetry algebra of the geodesic equations of the canonical connection on a Lie group was continued. The key ideas of Lie group, Lie algebra, linear connection, and symmetry were quickly reviewed. The focus was on those Lie groups whose Lie algebra was six-dimensional solvable and indecomposable and for which the nilradical was abelian and had a one-dimensional center. Based on the list of Lie algebras compiled by Turkowski, there were eight algebras to consider that were denoted by $ A_{6, 20} $–$ A_{6, 27} $. For each Lie algebra, a comprehensive symmetry analysis of the system of geodesic equations was carried out. For each symmetry Lie algebra, the nilradical and a complement to the nilradical inside the radical, as well as a semi-simple factor, were identified.</p></abstract>
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Johnpillai, Andrew G., Abdul H. Kara, and Anjan Biswas. "Exact Group Invariant Solutions and Conservation Laws of the Complex Modified Korteweg–de Vries Equation." Zeitschrift für Naturforschung A 68, no. 8-9 (September 1, 2013): 510–14. http://dx.doi.org/10.5560/zna.2013-0027.

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We study the scalar complex modified Korteweg-de Vries (cmKdV) equation by analyzing a system of partial differential equations (PDEs) from the Lie symmetry point of view. These systems of PDEs are obtained by decomposing the underlying cmKdV equation into real and imaginary components. We derive the Lie point symmetry generators of the system of PDEs and classify them to get the optimal system of one-dimensional subalgebras of the Lie symmetry algebra of the system of PDEs. These subalgebras are then used to construct a number of symmetry reductions and exact group invariant solutions to the system of PDEs. Finally, using the Lie symmetry approach, a couple of new conservation laws are constructed. Subsequently, respective conserved quantities from their respective conserved densities are computed.
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Mehdi Nadjafikhah and Omid Chekini. "Invariant solutions of Barlett and Whitaker’s equations." Malaya Journal of Matematik 2, no. 02 (April 1, 2014): 103–7. http://dx.doi.org/10.26637/mjm202/002.

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Lie symmetry group method is applied to study the Barlett and Whitaker’s equations. The symmetry group and its optimal system are given,and group invariant solutions associated to the symmetries are obtained. Finally the structure of the Lie algebra symmetries is determined.
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Dissertations / Theses on the topic "Lie Symmetry group of SDE"

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Ouknine, Anas. "Μοdèles affines généralisées et symétries d'équatiοns aux dérivés partielles." Electronic Thesis or Diss., Normandie, 2024. http://www.theses.fr/2024NORMR085.

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Cette thèse se consacre à étudier les symétries de Lie d'une classe particulière d'équations différentielles partielles (EDP), désignée sous le nom d'équation de Kolmogorov rétrograde. Cette équation joue un rôle essentiel dans le cadre des modèles financiers, notamment en lien avec le modèle de Longstaff-Schwartz, qui est largement utilisé pour la valorisation des options et des produits dérivés.Dans un contexte plus générale, notre étude s'oriente vers l'analyse des symétries de Lie de l'équation de Kolmogorov rétrograde, en introduisant un terme non linéaire. Cette généralisation est significative, car l'équation ainsi modifiée est liée à une équation différentielle stochastique rétrograde et progressive (EDSRP) via la formule de Feynman-Kac généralisée (non linéaire). Nous nous intéressons également à l'exploration des symétries de cette équation stochastique, ainsi qu'à la manière dont les symétries de l'EDP sont connectées à celles de l'EDSRP.Enfin, nous proposons un recalcul des symétries de l'équation différentielle stochastique rétrograde (EDSR) et de l'EDSRP, en adoptant une nouvelle approche. Cette approche se distingue par le fait que le groupe de symétries qui opère sur le temps dépend lui-même du processus $Y$, qui constitue la solution de l'EDSR. Cette dépendance ouvre de nouvelles perspectives sur l'interaction entre les symétries temporelles et les solutions des équations
This thesis is dedicated to studying the Lie symmetries of a particular class of partialdifferential equations (PDEs), known as the backward Kolmogorov equation. This equa-tion plays a crucial role in financial modeling, particularly in relation to the Longstaff-Schwartz model, which is widely used for pricing options and derivatives.In a broader context, our study focuses on analyzing the Lie symmetries of thebackward Kolmogorov equation by introducing a nonlinear term. This generalization issignificant, as the modified equation is linked to a forward backward stochastic differ-ential equation (FBSDE) through the generalized (nonlinear) Feynman-Kac formula.We also examine the symmetries of this stochastic equation and how the symmetriesof the PDE are connected to those of the BSDE.Finally, we propose a recalculation of the symmetries of the BSDE and FBSDE,adopting a new approach. This approach is distinguished by the fact that the symme-try group acting on time itself depends also on the process Y , which is the solutionof the BSDE. This dependence opens up new perspectives on the interaction betweentemporal symmetries and the solutions of the equations
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Nikolaishvili, George. "Investigation of the Equations Modelling Chemical Waves Using Lie Group Analysis." Thesis, Blekinge Tekniska Högskola, Sektionen för ingenjörsvetenskap, 2012. http://urn.kb.se/resolve?urn=urn:nbn:se:bth-3996.

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A system of nonlinear di fferential equations, namely, the Belousov-Zhabotinskii reaction model has been investigated for nonlinear self-adjointness using the recent work of Professor N.H.Ibragimov. It is shown that the model is not nonlinearly self-adjoint. The symmetries of the system and nonlinear conservation laws are calculated. The modi fied system, which is nonlinearly self-adjoint, is also analysed. Its symmetries and conservation laws are presented.
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Wiseman, Robin D. "The Jahn-Teller effect in icosahedral symmetry : unexpected lie group symmetries and their exploitation." Thesis, University of Oxford, 1998. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.299385.

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Lindman, Hornlund Josef. "Sigma-models and Lie group symmetries in theories of gravity." Doctoral thesis, Universite Libre de Bruxelles, 2011. http://hdl.handle.net/2013/ULB-DIPOT:oai:dipot.ulb.ac.be:2013/209911.

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En utilisant des modèles sigma non-linéaires de fonctions d'un espace-temps D-dimensionnel à un espace symétrique G/H, nous discutons de solutions de type trou noir et membrane noire dans diverses théories de gravité supersymétriques. Un espace symétrique est une variété, riemannienne ou pseudo-riemannienne, pour laquelle le tenseur de Riemann est covariantement constant. L'utilisation du dictionnaire Kac-Moody/supergravité et les techniques de réduction dimensionnelles nous permettent de décrire des trous noirs de cohomogénéité un comme des géodésiques sur G/H. Un espace-temps M, potentiellement agrémenté d'un trou noir, est de cohomogénéité un s'il existe un groupe d'isométries Iso qui agit sur M et dont le quotient M/Iso est uni-dimensionnel. L'utilisation d'algèbres de Kac-Moody dans les théories de gravité a été développé dans l'espoir de décourvrir la symétrie sous-jacente de la théorie des cordes, aussi appelée théorie M. Les techniques de réduction dimensionnelle ont depuis longtemps été utilisées pour dévoiler les symétries cachées des théories de gravité. Dans la description du modèle sigma, les trous noirs extrémaux ou branes noires sont des géodésiques nulles et correspondent à un élément nilpotent de l'algèbre de Lie g de G. Un élément X nilpotent est caractérisé par la propriété X^n = 0. En utilisant le formalisme mathématique decrivant les orbites nilpotentes, nous classifions tous les trous noirs extrémaux dans la supergravité N=2 minimale à quatre dimensions, N=2 S^3 supergravité en quatre dimensions et la supergravité minimale en cinq dimensions. De la même manière, quand G est un sous-groupe d'un groupe Kac-Moody, très-étendu ou sur-étendu, on envoie l'orbite nilpotente minimale, en utilisant le plus haut poids de g, sur des solutions supersymétriques et non-supersymétriques de type brane dans les théories de supergravité à dix et onze dimensions. Nos résultats montrent que les symétries du groupe de Lie sont très utiles de ces solutions pour classer et trouver de nouvelles solutions de type trou noir. Afin de prouver l'unicité et plusieurs autres résultats formels, nous avons développé des méthodes préliminaires dans l'espoir qu'elles puissent être utilisées à l'avenir pour l'étude des trous noirs.
Doctorat en Sciences
info:eu-repo/semantics/nonPublished
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Tempesta, Patricia. "Simmetries in binary differential equations." Universidade de São Paulo, 2017. http://www.teses.usp.br/teses/disponiveis/55/55135/tde-11072017-170308/.

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The purpose of this thesis in to introduce the systematic study of symmetries in binary differential equations (BDEs). We formalize the concept of a symmetric BDE, under the linear action of a compact Lie group. One of the main results establishes a formula that relates the algebraic and geometric effects of the occurrence of the symmetry in the problem. Using tools from invariant theory and representation theory for compact Lie groups we deduce the general forms of equivariant binary differential equations under compact subgroups of O(2). A study about the behavior of the invariant straight lines on the configuration of homogeneous BDEs of degree n is done with emphasis on cases in which n = 0 and n = 1. Also for the linear case (n = 1) the equivariant normal forms are presented. Symmetries of linear 1-forms are also studied and related with symmetries of tangent orthogonal vectors fields associated with it.
O objetivo desta tese é introduzir o estudo sistemático de simetrias em equações diferenciais binárias (EDBs). Neste trabalho formalizamos o conceito de EDB simétrica sobre a ação de um grupo de Lie compacto. Um dos principais resultados é uma fórmula que relaciona o efeito geométrico e algébrico das simetrias presentes no problema. Utilizando ferramentas da teoria invariante e de representação para grupos compactos deduzimos as formas gerais para EDBs equivariantes. Um estudo sobre o comportamento das retas invariantes na configuração de EDBs com coeficientes homogêneos de grau n é feito com ênfase nos casos de grau 0 e 1, ainda no caso de grau 1 são apresentadas suas formas normais. Simetrias de 1-formas lineares são também estudadas e relacionadas com as simetrias dos seus campos tangente e ortogonal.
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Correa, Diego Paolo Ferruzzo. "Symmetric bifurcation analysis of synchronous states of time-delay oscillators networks." Universidade de São Paulo, 2014. http://www.teses.usp.br/teses/disponiveis/3/3139/tde-29122014-180651/.

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In recent years, there has been increasing interest in studying time-delayed coupled networks of oscillators since these occur in many real life applications. In many cases symmetry, patterns can emerge in these networks; as a consequence, a part of the system might repeat itself, and properties of this symmetric subsystem represent the whole dynamics. In this thesis, an analysis of a second order N-node time-delay fully connected network is made. This study is carried out using symmetry groups. The existence of multiple eigenvalues forced by symmetry is shown, as well as the possibility of uncoupling the linearization at equilibria, into irreducible representations due to the symmetry. The existence of steady-state and Hopf bifurcations in each irreducible representation is also proved. Three different models are used to analyze the network dynamics, namely, the full-phase, the phase, and the phase-difference model. A finite set of frequencies ω is also determined, which might correspond to Hopf bifurcations in each case for critical values of the delay. Although we restrict our attention to second order nodes, the results could be extended to higher order networks provided the time-delay in the connections between nodes remains equal.
Nos últimos anos, tem havido um crescente interesse em estudar redes de osciladores acopladas com retardo de tempo uma vez que estes ocorrem em muitas aplicações da vida real. Em muitos casos, simetria e padrões podem surgir nessas redes; em consequência, uma parte do sistema pode repetir-se, e as propriedades deste subsistema simétrico representam a dinâmica da rede toda. Nesta tese é feita uma análise de uma rede de N nós de segunda ordem totalmente conectada com atraso de tempo. Este estudo é realizado utilizando grupos de simetria. É mostrada a existência de múltiplos valores próprios forçados por simetria, bem como a possibilidade de desacoplamento da linearização no equilíbrio, em representações irredutíveis. É também provada a existência de bifurcações de estado estacionário e Hopf em cada representação irredutível. São usados três modelos diferentes para analisar a dinâmica da rede: o modelo de fase completa, o modelo de fase, e o modelo de diferença de fase. É também determinado um conjunto finito de frequências ω, que pode corresponder a bifurcações de Hopf em cada caso, para valores críticos do atraso. Apesar de restringir a nossa atenção para nós de segunda ordem, os resultados podem ser estendido para redes de ordem superior, desde que o tempo de atraso nas conexões entre nós permanece igual.
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Al, Sayed Nazir. "Modèles LES invariants par groupes de symétries en écoulements turbulents anisothermes." Phd thesis, Université de La Rochelle, 2011. http://tel.archives-ouvertes.fr/tel-00605655.

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Comme le groupe de symétries de Lie des équations aux dérivées partielles représentent les propriétés physiques intrinsèques contenues dans les équations, il offre un outil efficace pour étudier et modéliser les phénomènes physiques. Ainsi, dans cette thèse, on se propose d'appliquer la théorie du groupe de symétries de Lie à la modélisation des écoulements anisothermes.On calcule alors des lois de paroi, et, plus généralement des lois d'échelle, pour la vitesse et la température dans le cas d'un écoulement parallèle. En fait, ces lois d'échelle se révèlent être simplement des solutions auto-similaires des équations de Navier-Stokes moyennées par rapport aux symétries des équations.Ensuite, par l'approche de la théorie des groupes de Lie, on construit une classe de modèles de sous-maille qui sont invariants par les symétries des équations de Navier-Stokes anisothermes.Ces modèles ont l'avantage de respecter les propriétés physiques des équations qui sont contenues dans les symétries. De plus, par cette approche, le modèle de flux de chaleur apparaît naturellement,sans qu'on ait besoin de faire appel à la notion de nombre de Prandtl de sous-maille,ce qui augmente la portée de ces modèles par rapport à la plupart des modèles existants. Par ailleurs, le comportement proche de la paroi de certains des modèles proposés est étudié. Enfin,des tests numériques en convection naturelle et en convection mixtes sont effectués.
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Altafini, Claudio. "Geometric control methods for nonlinear systems and robotic applications." Doctoral thesis, Stockholm : Tekniska högsk, 2001. http://urn.kb.se/resolve?urn=urn:nbn:se:kth:diva-3151.

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John, Tyson. "Set Stabilization for Systems with Lie Group Symmetry." Thesis, 2010. http://hdl.handle.net/1807/25642.

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This thesis investigates the set stabilization problem for systems with Lie group symmetry. Initially, we examine left-invariant systems on Lie groups where the target set is a left or right coset of a closed subgroup. We broaden the scope to systems defined on smooth manifolds that are invariant under a Lie group action. Inspired by the solution of this problem for linear time-invariant systems, we show its equivalence to an equilibrium stabilization problem for a suitable quotient control system. We provide necessary and sufficient conditions for the existence of the quotient control system and analyze various properties of such a system. This theory is applied to the formation stabilization of three kinematic unicycles, the path stabilization of a particle in a gravitational field, and the conversion and temperature control of a continuously stirred tank reactor.
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Mamboundou, Hermane Mambili. "Lie group analysis of equations arising in non-Newtonian fluids." Thesis, 2009. http://hdl.handle.net/10539/6879.

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It is known now that the Navier-Stokes equations cannot describe the behaviour of fluids having high molecular weights. Due to the variety of such fluids it is very difficult to suggest a single constitutive equation which can describe the properties of all non-Newtonian fluids. Therefore many models of non-Newtonian fluids have been proposed. The flow of non-Newtonian fluids offer special challenges to the engineers, modellers, mathematicians, numerical simulists, computer scientists and physicists alike. In general the equations of non-Newtonian fluids are of higher order and much more complicated than the Newtonian fluids. The adherence boundary conditions are insufficient and one requires additional conditions for a unique solution. Also the flow characteristics of non-Newtonian fluids are quite different from those of the Newtonian fluids. Therefore, in practical applications, one cannot replace the behaviour of non-Newtonian fluids with Newtonian fluids and it is necessary to examine the flow behaviour of non-Newtonian fluids in order to obtain a thorough understanding and improve the utilization in various manufactures. Although the non-Newtonian behaviour of many fluids has been recognized for a long time, the science of rheology is, in many respects, still in its infancy, and new phenomena are constantly being discovered and new theories proposed. Analysis of fluid flow operations is typically performed by examining local conservation relations, conservation of mass, momentum and energy. This analysis gives rise to highly non-linear relationships given in terms of differential equations, which are solved using special non-linear techniques. Advancements in computational techniques are making easier the derivation of solutions to linear problems. However, it is still difficult to solve non-linear problems analytically. Engineers, chemists, physicists, and mathematicians are actively developing non-linear analytical techniques, and one such method which is known for systematically searching for exact solutions of differential equations is the Lie symmetry approach for differential equations. Lie theory of differential equations originated in the 1870s and was introduced by the Norwegian mathematician Marius Sophus Lie (1842 - 1899). However it was the Russian scientist Ovsyannikov by his work of 1958 who awakened interest in modern group analysis. Today, the Lie group approach to differential equations is widely applied in various fields of mathematics, mechanics, and theoretical physics and many results published in these area demonstrates that Lie’s theory is an efficient tool for solving intricate problems formulated in terms of differential equations. The conditional symmetry approach or what is called the non-classical symmetry approach is an extension of the Lie approach. It was proposed by Bluman and Cole 1969. Many equations arising in applications have a paucity of Lie symmetries but have conditional symmetries. Thus this method is powerful in obtaining exact solutions of such equations. Numerical methods for the solutions of non-linear differential equations are important and nowadays there several software packages to obtain such solutions. Some of the common ones are included in Maple, Mathematica and Matlab. This thesis is divided into six chapters and an introduction and conclusion. The first chapter deals with basic concepts of fluids dynamics and an introduction to symmetry approaches to differential equations. In Chapter 2 we investigate the influence of a time-dependentmagnetic field on the flow of an incompressible third grade fluid bounded by a rigid plate. Chapter 3 describes the modelling of a fourth grade flow caused by a rigid plate moving in its own plane. The resulting fifth order partial differential equation is reduced using symmetries and conditional symmetries. In Chapter 4 we present a Lie group analysis of the third oder PDE obtained by investigating the unsteady flow of third grade fluid using the modified Darcy’s law. Chapter 5 looks at the magnetohydrodynamic (MHD) flow of a Sisko fluid over a moving plate. The flow of a fourth grade fluid in a porous medium is analyzed in Chapter 6. The flow is induced by a moving plate. Several graphs are included in the ensuing discussions. Chapters 2 to 6 have been published or submitted for publication. Details are given in the references at the end of the thesis.
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Books on the topic "Lie Symmetry group of SDE"

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Robinson, Matthew B. Symmetry and the standard model: Mathematics and particle physics. New York: Springer, 2011.

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Ortaçgil, Ercüment H. The Symmetry Group. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198821656.003.0016.

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This chapter, which ends Part II on some consequences of the new approach, introduces an alternative prolongation theory of Klein geometries that is more geometric and intuitive than the well-known prolongation theory of a linear Lie algebra developed by Guillemin, Singer, and Sternberg.
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Vergados, J. D. Group and Representation Theory. World Scientific Publishing Co Pte Ltd, 2016.

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O'Raifeartaigh, L. Group Structure of Gauge Theories. Cambridge University Press, 2012.

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O'Raifeartaigh, L. Group Structure of Gauge Theories. Cambridge University Press, 2011.

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Wallach, Nolan R., and Roe Goodman. Symmetry, Representations, and Invariants. Springer, 2010.

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Sato, Ryuzo, and Rama V. Ramachandran. Symmetry and Economic Invariance. Springer London, Limited, 2013.

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Sato, Ryuzo, and Rama V. Ramachandran. Symmetry and Economic Invariance. Springer Japan, 2016.

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Sato, Ryuzo, and Rama V. Ramachandran. Symmetry and Economic Invariance. Springer, 2013.

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Sato, Ryuzo, and Rama V. Ramachandran. Symmetry and Economic Invariance. T Kobayashi, 2013.

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Book chapters on the topic "Lie Symmetry group of SDE"

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Baldeaux, Jan, and Eckhard Platen. "Lie Symmetry Group Methods." In Functionals of Multidimensional Diffusions with Applications to Finance, 101–40. Cham: Springer International Publishing, 2013. http://dx.doi.org/10.1007/978-3-319-00747-2_4.

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Singer, Stephanie Frank. "Symmetries are Lie Group Actions." In Symmetry in Mechanics, 83–100. Boston, MA: Birkhäuser Boston, 2004. http://dx.doi.org/10.1007/978-1-4612-0189-2_6.

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Ibragimov, N. H. "Symbolic Software for Lie Symmetry Analysis." In CRC Handbook of Lie Group Analysis of Differential Equations, Volume III, 367–414. Boca Raton: CRC Press, 2024. http://dx.doi.org/10.1201/9781003575221-16.

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Yahalom, Asher. "A New Diffeomorphism Symmetry Group of Magnetohydrodynamics." In Lie Theory and Its Applications in Physics, 461–68. Tokyo: Springer Japan, 2013. http://dx.doi.org/10.1007/978-4-431-54270-4_33.

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Vassilev, Vassil M., Petar A. Djondjorov, and Ivaïlo M. Mladenov. "Lie Group Analysis of the Willmore and Membrane Shape Equations." In Similarity and Symmetry Methods, 365–76. Cham: Springer International Publishing, 2014. http://dx.doi.org/10.1007/978-3-319-08296-7_7.

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Ibragimov, N. H., W. F. Ames, R. L. Anderson, V. A. Dorodnitsyn, E. V. Ferapontov, R. K. Gazizov, N. H. Ibragimov, and S. R. Svirshchevskii. "Symmetry of Finite-Difference Equations." In CRC Handbook of Lie Group Analysis of Differential Equations, Volume I, 365–403. Boca Raton: CRC Press, 2023. http://dx.doi.org/10.1201/9781003419808-22.

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Ibragimov, N. H., W. F. Ames, R. L. Anderson, V. A. Dorodnitsyn, E. V. Ferapontov, R. K. Gazizov, N. H. Ibragimov, and S. R. Svirshchevskii. "Nonlocal Symmetry Generators via Bäcklund Transformations." In CRC Handbook of Lie Group Analysis of Differential Equations, Volume I, 68–73. Boca Raton: CRC Press, 2023. http://dx.doi.org/10.1201/9781003419808-10.

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Ibragimov, N. H. "Approximate Transformation Groups and Deformations of Symmetry Lie Algebras." In CRC Handbook of Lie Group Analysis of Differential Equations, Volume III, 31–68. Boca Raton: CRC Press, 2024. http://dx.doi.org/10.1201/9781003575221-3.

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Ibragimov, N. H. "Calculation of Symmetry Groups for Integro-Differential Equations." In CRC Handbook of Lie Group Analysis of Differential Equations, Volume III, 139–46. Boca Raton: CRC Press, 2024. http://dx.doi.org/10.1201/9781003575221-6.

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Mitropolsky, Yu A., and A. K. Lopatin. "Asymptotic Decomposition of Differential Systems with Small Parameter in the Representation Space of Finite-dimensional Lie Group." In Nonlinear Mechanics, Groups and Symmetry, 219–58. Dordrecht: Springer Netherlands, 1995. http://dx.doi.org/10.1007/978-94-015-8535-4_6.

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Conference papers on the topic "Lie Symmetry group of SDE"

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Pulov, Vladimir I., Ivan M. Uzunov, Edy J. Chacarov, and Valentin L. Lyutskanov. "Lie group symmetry classification of solutions to coupled nonlinear Schrodinger equations." In SPIE Proceedings, edited by Peter A. Atanasov, Tanja N. Dreischuh, Sanka V. Gateva, and Lubomir M. Kovachev. SPIE, 2007. http://dx.doi.org/10.1117/12.726994.

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Lindgren, B., J. Osterlund, and A. Johansson. "Evaluation of scaling laws derived from lie group symmetry methods in turbulent boundary layers." In 40th AIAA Aerospace Sciences Meeting & Exhibit. Reston, Virigina: American Institute of Aeronautics and Astronautics, 2002. http://dx.doi.org/10.2514/6.2002-1103.

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Yu, Jingjun, Shouzhong Li, Shusheng Bi, and Guanghua Zong. "Symmetry Design in Flexure Systems Using Kinematic Principles." In ASME 2013 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. American Society of Mechanical Engineers, 2013. http://dx.doi.org/10.1115/detc2013-12385.

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Abstract:
Inspiration for the creation of mechanical devices often comes from observing the natural structures and movements of living organisms. Understanding the wide use of modularity and compliance in nature may lead to the design of high-performance flexure systems or compliant devices. One of the most important nature-inspired paradigms for constructing flexure systems is based on the effective use of symmetry. With a rigid mathematical foundation called screw theory and Lie group. The research of this paper mainly focuses on: (i) Mathematical explanation or treatment of symmetry design wildly used in flexure systems, concerning with a series of topics such as the relationship between degree of freedom (DOF), constraint, overconstraint, decouple motion and symmetrical geometry, and How to guarantee the mobility unchanged when using symmetry design? (ii) A compliance-based analytical verification for demonstrating that the symmetry design can effectively improve accuracy and dynamic performances. (iii) The feasibility of improving accuracy performance by connecting symmetry design with the principle of elastic averaging. The whole content is organized around a case study, i.e. symmetrical design of 1-DOF translational flexure mechanisms. The results are intent to provide a rigid theoretical foundation and significant instruction for the symmetry design philosophy in flexure systems using kinematic principles.
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Changizi, M. Amin, Ali Abolfathi, and Ion Stiharu. "MEMS Wind Speed Sensor: Large Deflection of Curved Micro-Cantilever Beam Under Uniform Horizontal Force." In ASME 2015 International Mechanical Engineering Congress and Exposition. American Society of Mechanical Engineers, 2015. http://dx.doi.org/10.1115/imece2015-50560.

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Micro-cantilever beams are currently employed as sensor in various fields. Of main applications, is using such beams in wind speed sensors. For this purpose, curved out of plane micro-cantilever beams are used. Uniform pressure on such beams causes a large deflection of beam. General mechanics of material theory deals with small deflection and thus cannot be used for explaining this deflection. Although there are a body of works on analysing of large deflection [1], nonlinear deflection, of curved beams [2], yet there is no research on large deflection of curved beam under horizontal uniform distributed force. Theoretically, the wind force is applying horizontally on curved micro-cantilever beam. Here, we neglect the effect of moving weather from beam sides. We first aim how to drive the governed equation. A curved beam does not have a calculable centroid. Also large deflection of beam changes its curvature which would change the centroid of beam consciously. The variation of centroid makes very though calculating the bending moment of each cross section in the beam. To address this issue, an integral equation will be used. The total force will be considered as a single force applied at the centroid. The second challenge is solving the governed nonlinear ordinary differential equation (ODE). Although there are several methods to solve analytically nonlinear ODE, Lie symmetry method, with all its complication, is a general method for this kind of equations. This approach covers all current methods in analytical solving nonlinear ODEs. In this method, an infinitesimal transformation should be calculated. All transformations under one parameter creates a group that called Lie group. A value of parameter which transfers the equation onto itself is called invariant of ODE. One can calculate canonical coordinates ODEs by the invariant. Solving the canonical coordinates ODEs yields to calculating the canonical coordinates. Canonical coordinate are used to reduce the order of nonlinear ODE [3]. By repeating this method one can solve high order ODEs. Our last question is how to do numerical solution of ODE. The possible answer will help to explain the phenomena of deflection clearly and compare the analytical solution with numerical results. Small dimensions of beam, small values of applied force from one side and Young modules value from the other side, will create a stiff ODE. Authors experience in this area shows that the best method to sole these kind of equations is LSODE. This method can be used in Maple. Here, primary calculations show that the governed equation is second order nonlinear ODE and we propose two possible invariants to solve ODE. Overall, the primary numerical solution has shown perfect match with the exact solution.
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