Academic literature on the topic 'Lie Algebras Expansion'
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Journal articles on the topic "Lie Algebras Expansion"
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Rowe, D. J., and J. Carvalho. "Boson expansion of lie algebras: The fermion pair algebra." Physics Letters B 175, no. 3 (August 1986): 243–48. http://dx.doi.org/10.1016/0370-2693(86)90848-8.
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Caroca, R., N. Merino, and P. Salgado. "S expansion of higher-order Lie algebras." Journal of Mathematical Physics 50, no. 1 (January 2009): 013503. http://dx.doi.org/10.1063/1.3036177.
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Curry, Charles, Kurusch Ebrahimi-Fard, and Brynjulf Owren. "The Magnus expansion and post-Lie algebras." Mathematics of Computation 89, no. 326 (May 26, 2020): 2785–99. http://dx.doi.org/10.1090/mcom/3541.
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Nieto, L. M., J. Negro, and M. Santander. "Two Dimensional Cayley-Klein Algebras Generated by Expansions." International Journal of Modern Physics A 12, no. 01 (January 10, 1997): 259–64. http://dx.doi.org/10.1142/s0217751x97000384.
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Expansion methods are applied in order to get extended two dimensional Cayley-Klein Lie algebras. The keystone of the construction will be the two dimensional extended Galilei algebra whose generators are used to define, by means of formal functions, generators closing any of the nine two-dimensional extended Cayley-Klein Lie algebras.
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Yan, Wang, and Yufeng Zhang. "Expansion of the Lie algebras and integrable couplings." Chaos, Solitons & Fractals 38, no. 2 (October 2008): 541–47. http://dx.doi.org/10.1016/j.chaos.2006.12.002.
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Andrianopoli, L., N. Merino, F. Nadal, and M. Trigiante. "General properties of the expansion methods of Lie algebras." Journal of Physics A: Mathematical and Theoretical 46, no. 36 (August 21, 2013): 365204. http://dx.doi.org/10.1088/1751-8113/46/36/365204.
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Ebrahimi-Fard, Kurusch, and Dominique Manchon. "Twisted dendriform algebras and the pre-Lie Magnus expansion." Journal of Pure and Applied Algebra 215, no. 11 (November 2011): 2615–27. http://dx.doi.org/10.1016/j.jpaa.2011.03.004.
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Mencattini, Igor, and Alexandre Quesney. "Crossed morphisms, integration of post-Lie algebras and the post-Lie Magnus expansion." Communications in Algebra 49, no. 8 (March 28, 2021): 3507–33. http://dx.doi.org/10.1080/00927872.2021.1900212.
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Pei, Yufeng, and Jinwei Yang. "Strongly graded vertex algebras generated by vertex Lie algebras." Communications in Contemporary Mathematics 21, no. 08 (October 20, 2019): 1850069. http://dx.doi.org/10.1142/s0219199718500694.
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We construct three families of vertex algebras along with their modules from appropriate vertex Lie algebras, using the constructions in [Vertex Lie algebra, vertex Poisson algebras and vertex algebras, in Recent Developments in Infinite-Dimensional Lie Algebras and Conformal Field Theory[Formula: see text] Proceedings of an International Conference at University of Virginia[Formula: see text] May 2000, in Contemporary Mathematics, Vol. 297 (American Mathematical Society, 2002), pp. 69–96] by Dong, Li and Mason. These vertex algebras are strongly graded vertex algebras introduced in [Logarithmic tensor category theory for generalized modules for a conformal vertex algebra, I: Introduction and strongly graded algebras and their generalized modules, in Conformal Field Theories and Tensor Categories[Formula: see text] Proceedings of a Workshop Held at Beijing International Center for Mathematics Research, eds. C. Bai, J. Fuchs, Y.-Z. Huang, L. Kong, I. Runkel and C. Schweigert, Mathematical Lectures from Beijing University, Vol. 2 (Springer, New York, 2014), pp. 169–248] by Huang, Lepowsky and Zhang in their logarithmic tensor category theory and can also be realized as vertex algebras associated to certain well-known infinite dimensional Lie algebras. We classify irreducible [Formula: see text]-gradable weak modules for these vertex algebras by determining their Zhu’s algebras. We find examples of strongly graded generalized modules for these vertex algebras that satisfy the [Formula: see text]-cofiniteness condition introduced in [Differential equations and logarithmic intertwining operators for strongly graded vertex algebra, Comm. Contemp. Math. 19(2) (2017) 1650009] by the second author. In particular, by a result of the second author [Differential equations and logarithmic intertwining operators for strongly graded vertex algebra, Comm. Contemp. Math. 19(2) (2017) 1650009, 26 pp.], the convergence and extension property for products and iterates of logarithmic intertwining operators in [Y.-Z. Huang, J. Lepowsky and L. Zhang, Logarithmic tensor category theory for generalized modules for a conformal vertex algebra, VII: Convergence and extension properties and applications to expansion for intertwining maps, preprint (2011); arXiv:1110.1929 ] among such strongly graded generalized modules is verified.
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Quesne, C. "Boson realisations of Lie algebras and expansion of shift operators." Journal of Physics A: Mathematical and General 20, no. 12 (August 21, 1987): L753—L758. http://dx.doi.org/10.1088/0305-4470/20/12/001.
Full textDissertations / Theses on the topic "Lie Algebras Expansion"
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Rocha, Eugénio Alexandre Miguel. "Uma Abordagem Algébrica à Teoria de Controlo Não Linear." Doctoral thesis, Universidade de Aveiro, 2003. http://hdl.handle.net/10773/21444.
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Doutoramento em MatemáticaNesta tese de Doutoramento desenvolve-se principalmente uma abordagem algébrica à teoria de sistemas de controlo não lineares. No entanto, outros tópicos são também estudados. Os tópicos tratados são os seguidamente enunciados: fórmulas para sistemas de controlo sobre álgebras de Lie livres, estabilidade de um sistema de corpos rolantes, algoritmos para aritmética digital, e equações integrais de Fredholm não lineares. No primeiro e principal tópico estudam-se representações para as soluções de sistemas de controlo lineares no controlo. As suas trajetórias são representadas pelas chamadas séries de Chen. Estuda-se a representação formal destas séries através da introdução de várias álgebras não associativas e técnicas específicas de álgebras de Lie livres. Sistemas de coordenadas para estes sistemas são estudados, nomeadamente, coordenadas de primeiro tipo e de segundo tipo. Apresenta-se uma demonstração alternativa para as coordenadas de segundo tipo e obtêm-se expressões explícitas para as coordenadas de primeiro tipo. Estas últimas estão intimamente ligadas ao logaritmo da série de Chen que, por sua vez, tem fortes relações com uma fórmula designada na literatura por “continuous Baker-Campbell- Hausdorff formula”. São ainda apresentadas aplicações à teoria de funções simétricas não comutativas. É, por fim, caracterizado o mapa de monodromia de um campo de vectores não linear e periódico no tempo em relação a uma truncatura do logaritmo de Chen. No segundo tópico é estudada a estabilizabilidade de um sistema de quaisquer dois corpos que rolem um sobre o outro sem deslizar ou torcer. Constroem-se controlos fechados e dependentes do tempo que tornam a origem do sistema de dois corpos num sistema localmente assimptoticamente estável. Vários exemplos e algumas implementações em Maple°c são discutidos. No terceiro tópico, em apêndice, constroem-se algoritmos para calcular o valor de várias funções fundamentais na aritmética digital, sendo possível a sua implementação em microprocessadores. São também obtidos os seus domínios de convergência. No último tópico, também em apêndice, demonstra-se a existência e unicidade de solução para uma classe de equações integrais não lineares com atraso. O atraso tem um carácter funcional, mostrando-se ainda a diferenciabilidade no sentido de Fréchet da solução em relação à função de atraso.
In this PhD thesis several subjects are studied regarding the following topics: formulas for nonlinear control systems on free Lie algebras, stabilizability of nonlinear control systems, digital arithmetic algorithms, and nonlinear Fredholm integral equations with delay. The first and principal topic is mainly related with a problem known as the continuous Baker-Campbell-Hausdorff exponents. We propose a calculus to deal with formal nonautonomous ordinary differential equations evolving on the algebra of formal series defined on an alphabet. We introduce and connect several (non)associative algebras as Lie, shuffle, zinbiel, pre-zinbiel, chronological (pre-Lie), pre-chronological, dendriform, D-I, and I-D. Most of those notions were also introduced into the universal enveloping algebra of a free Lie algebra. We study Chen series and iterated integrals by relating them with nonlinear control systems linear in control. At the heart of all the theory of Chen series resides a zinbiel and shuffle homomorphism that allows us to construct a purely formal representation of Chen series on algebras of words. It is also given a pre-zinbiel representation of the chronological exponential, introduced by A.Agrachev and R.Gamkrelidze on the context of a tool to deal with nonlinear nonautonomous ordinary differential equations over a manifold, the so-called chronological calculus. An extensive description of that calculus is made, collecting some fragmented results on several publications. It is a fundamental tool of study along the thesis. We also present an alternative demonstration of the result of H.Sussmann about coordinates of second kind using the mentioned tools. This simple and comprehensive proof shows that coordinates of second kind are exactly the image of elements of the dual basis of a Hall basis, under the above discussed homomorphism. We obtain explicit expressions for the logarithm of Chen series and the respective coordinates of first kind, by defining several operations on a forest of leaf-labelled trees. It is the same as saying that we have an explicit formula for the functional coefficients of the Lie brackets on a continuous Baker-Campbell-Hausdorff-Dynkin formula when a Hall basis is used. We apply those formulas to relate some noncommutative symmetric functions, and we also connect the monodromy map of a time-periodic nonlinear vector field with a truncation of the Chen logarithm. On the second topic, we study any system of two bodies rolling one over the other without twisting or slipping. By using the Chen logarithm expressions, the monodromy map of a flow and Lyapunov functions, we construct time-variant controls that turn the origin of a control system linear in control into a locally asymptotically stable equilibrium point. Stabilizers for control systems whose vector fields generate a nilpotent Lie algebra with degree of nilpotency · 3 are also given. Some examples are presented and Maple°c were implemented. The third topic, on appendix, concerns the construction of efficient algorithms for Digital Arithmetic, potentially for the implementation in microprocessors. The algorithms are intended for the computation of several functions as the division, square root, sines, cosines, exponential, logarithm, etc. By using redundant number representations and methods of Lyapunov stability for discrete dynamical systems, we obtain several algorithms (that can be glued together into an algorithm for parallel execution) having the same core and selection scheme in each iteration. We also prove their domains of convergence and discuss possible extensions. The last topic, also on appendix, studies the set of solutions of a class of nonlinear Fredholm integral equations with general delay. The delay is of functional character modelled by a continuous lag function. We ensure existence and uniqueness of a continuous (positive) solution of such equation. Moreover, under additional conditions, it is obtained the Fr´echet differentiability of the solution with respect to the lag function.
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Al-Kaabi, Mahdi Jasim Hasan. "Bases de monômes dans les algèbres pré-Lie libres et applications." Thesis, Clermont-Ferrand 2, 2015. http://www.theses.fr/2015CLF22599/document.
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Dans cette thèse, nous étudions le concept d’algèbre pré-Lie libre engendrée par un ensemble (non-vide). Nous rappelons la construction par A. Agrachev et R. Gamkrelidze des bases de monômes dans les algèbres pré-Lie libres. Nous décrivons la matrice des vecteurs d’une base de monômes en termes de la base d’arbres enracinés exposée par F. Chapoton et M. Livernet. Nous montrons que cette matrice est unipotente et trouvons une expression explicite pour les coefficients de cette matrice, en adaptant une procédure suggérée par K. Ebrahimi-Fard et D. Manchon pour l’algèbre magmatique libre. Nous construisons une structure d’algèbre pré-Lie sur l’algèbre de Lie libre $\mathcal{L}$(E) engendrée par un ensemble E, donnant une présentation explicite de $\mathcal{L}$(E) comme quotient de l’algèbre pré-Lie libre $\mathcal{T}$^E, engendrée par les arbres enracinés (non-planaires) E-décorés, par un certain idéal I. Nous étudions les bases de Gröbner pour les algèbres de Lie libres dans une présentation à l’aide d’arbres. Nous décomposons la base d’arbres enracinés planaires E-décorés en deux parties O(J) et $\mathcal{T}$(J), où J est l’idéal définissant $\mathcal{L}$(E) comme quotient de l’algèbre magmatique libre engendrée par E. Ici, $\mathcal{T}$(J) est l’ensemble des termes maximaux des éléments de J, et son complément O(J) définit alors une base de $\mathcal{L}$(E). Nous obtenons un des résultats importants de cette thèse (Théorème 3.12) sur la description de l’ensemble O(J) en termes d’arbres. Nous décrivons des bases de monômes pour l’algèbre pré-Lie (respectivement l’algèbre de Lie libre) $\mathcal{L}$(E), en utilisant les procédures de bases de Gröbner et la base de monômes pour l’algèbre pré-Lie libre obtenue dans le Chapitre 2. Enfin, nous étudions les développements de Magnus classique et pré-Lie, discutant comment nous pouvons trouver une formule de récurrence pour le cas pré-Lie qui intègre déjà l’identité pré-Lie. Nous donnons une vision combinatoire d’une méthode numérique proposée par S. Blanes, F. Casas, et J. Ros, sur une écriture du développement de Magnus classique, utilisant la structure pré-Lie de $\mathcal{L}$(E)In this thesis, we study the concept of free pre-Lie algebra generated by a (non-empty) set. We review the construction by A. Agrachev and R. Gamkrelidze of monomial bases in free pre-Lie algebras. We describe the matrix of the monomial basis vectors in terms of the rooted trees basis exhibited by F. Chapoton and M. Livernet. Also, we show that this matrix is unipotent and we find an explicit expression for its coefficients, adapting a procedure implemented for the free magmatic algebra by K. Ebrahimi-Fard and D. Manchon. We construct a pre-Lie structure on the free Lie algebra $\mathcal{L}$(E) generated by a set E, giving an explicit presentation of $\mathcal{L}$(E) as the quotient of the free pre-Lie algebra $\mathcal{T}$^E, generated by the (non-planar) E-decorated rooted trees, by some ideal I. We study the Gröbner bases for free Lie algebras in tree version. We split the basis of E- decorated planar rooted trees into two parts O(J) and $\mathcal{T}$(J), where J is the ideal defining $\mathcal{L}$(E) as a quotient of the free magmatic algebra generated by E. Here $\mathcal{T}$(J) is the set of maximal terms of elements of J, and its complement O(J) then defines a basis of $\mathcal{L}$(E). We get one of the important results in this thesis (Theorem 3.12), on the description of the set O(J) in terms of trees. We describe monomial bases for the pre-Lie (respectively free Lie) algebra $\mathcal{L}$(E), using the procedure of Gröbner bases and the monomial basis for the free pre-Lie algebra obtained in Chapter 2. Finally, we study the so-called classical and pre-Lie Magnus expansions, discussing how we can find a recursion for the pre-Lie case which already incorporates the pre-Lie identity. We give a combinatorial vision of a numerical method proposed by S. Blanes, F. Casas, and J. Ros, on a writing of the classical Magnus expansion in $\mathcal{L}$(E), using the pre-Lie structure
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Kaur, Amandeep. "Analytic and numerical aspects of isospectral flows." Thesis, University of Cambridge, 2018. https://www.repository.cam.ac.uk/handle/1810/270631.
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In this thesis we address the analytic and numerical aspects of isospectral flows. Such flows occur in mathematical physics and numerical linear algebra. Their main structural feature is to retain the eigenvalues in the solution space. We explore the solution of Isospectral flows and their stochastic counterpart using explicit generalisation of Magnus expansion. \par In the first part of the thesis we expand the solution of Bloch--Iserles equations, the matrix ordinary differential system of the form $ X'=[N,X^{2}],\ \ t\geq0, \ \ X(0)=X_0\in \textrm{Sym}(n),\ N\in \mathfrak{so}(n), $ where $\textrm{Sym}(n)$ denotes the space of real $n\times n$ symmetric matrices and $\mathfrak{so}(n)$ denotes the Lie algebra of real $n\times n$ skew-symmetric matrices. This system is endowed with Poisson structure and is integrable. Various important properties of the flow are discussed. The flow is solved using explicit Magnus expansion and the terms of expansion are represented as binary rooted trees deducing an explicit formalism to construct the trees recursively. Unlike classical numerical methods, e.g.\ Runge--Kutta and multistep methods, Magnus expansion respects the isospectrality of the system, and the shorthand of binary rooted trees reduces the computational cost of the exponentially growing terms. The desired structure of the solution (also with large time steps) has been displayed. \par Having seen the promising results in the first part of the thesis, the technique has been extended to the generalised double bracket flow $ X^{'}=[[N,X]+M,X], \ \ t\geq0, \ \ X(0)=X_0\in \textrm{Sym}(n),$ where $N\in \textrm{diag}(n)$ and $M\in \mathfrak{so}(n)$, which is also a form of an Isospectral flow. In the second part of the thesis we define the generalised double bracket flow and discuss its dynamics. It is noted that $N=0$ reduces it to an integrable flow, while for $M=0$ it results in a gradient flow. We analyse the flow for various non-zero values of $N$ and $M$ by assigning different weights and observe Hopf bifurcation in the system. The discretisation is done using Magnus series and the expansion terms have been portrayed using binary rooted trees. Although this matrix system appears more complex and leads to the tri-colour leaves; it has been possible to formulate the explicit recursive rule. The desired structure of the solution is obtained that leaves the eigenvalues invariant in the solution space.
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Singh, Pranav. "High accuracy computational methods for the semiclassical Schrödinger equation." Thesis, University of Cambridge, 2018. https://www.repository.cam.ac.uk/handle/1810/274913.
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The computation of Schrödinger equations in the semiclassical regime presents several enduring challenges due to the presence of the small semiclassical parameter. Standard approaches for solving these equations commence with spatial discretisation followed by exponentiation of the discretised Hamiltonian via exponential splittings. In this thesis we follow an alternative strategy${-}$we develop a new technique, called the symmetric Zassenhaus splitting procedure, which involves directly splitting the exponential of the undiscretised Hamiltonian. This technique allows us to design methods that are highly efficient in the semiclassical regime. Our analysis takes place in the Lie algebra generated by multiplicative operators and polynomials of the differential operator. This Lie algebra is completely characterised by Jordan polynomials in the differential operator, which constitute naturally symmetrised differential operators. Combined with the $\mathbb{Z}_2$-graded structure of this Lie algebra, the symmetry results in skew-Hermiticity of the exponents for Zassenhaus-style splittings, resulting in unitary evolution and numerical stability. The properties of commutator simplification and height reduction in these Lie algebras result in a highly effective form of $\textit{asymptotic splitting:} $exponential splittings where consecutive terms are scaled by increasing powers of the small semiclassical parameter. This leads to high accuracy methods whose costs grow quadratically with higher orders of accuracy. Time-dependent potentials are tackled by developing commutator-free Magnus expansions in our Lie algebra, which are subsequently split using the Zassenhaus algorithm. We present two approaches for developing arbitrarily high-order Magnus--Zassenhaus schemes${-}$one where the integrals are discretised using Gauss--Legendre quadrature at the outset and another where integrals are preserved throughout. These schemes feature high accuracy, allow large time steps, and the quadratic growth of their costs is found to be superior to traditional approaches such as Magnus--Lanczos methods and Yoshida splittings based on traditional Magnus expansions that feature nested commutators of matrices. An analysis of these operatorial splittings and expansions is carried out by characterising the highly oscillatory behaviour of the solution.
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Ramos, Alberto Gil Couto Pimentel. "Numerical solution of Sturm–Liouville problems via Fer streamers." Thesis, University of Cambridge, 2016. https://www.repository.cam.ac.uk/handle/1810/256997.
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The subject matter of this dissertation is the design, analysis and practical implementation of a new numerical method to approximate the eigenvalues and eigenfunctions of regular Sturm–Liouville problems, given in Liouville’s normal form, defined on compact intervals, with self-adjoint separated boundary conditions. These are classical problems in computational mathematics which lie on the interface between numerical analysis and spectral theory, with important applications in physics and chemistry, not least in the approximation of energy levels and wave functions of quantum systems. Because of their great importance, many numerical algorithms have been proposed over the years which span a vast and diverse repertoire of techniques. When compared with previous approaches, the principal advantage of the numerical method proposed in this dissertation is that it is accompanied by error bounds which: (i) hold uniformly over the entire eigenvalue range, and, (ii) can attain arbitrary high-order. This dissertation is composed of two parts, aggregated according to the regularity of the potential function. First, in the main part of this thesis, this work considers the truncation, discretization, practical implementation and MATLAB software, of the new approach for the classical setting with continuous and piecewise analytic potentials (Ramos and Iserles, 2015; Ramos, 2015a,b,c). Later, towards the end, this work touches upon an extension of the new ideas that enabled the truncation of the new approach, but instead for the general setting with absolutely integrable potentials (Ramos, 2014).
Books on the topic "Lie Algebras Expansion"
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Tao, Terence. Expansion in finite simple groups of Lie type. Providence, Rhode Island: American Mathematical Society, 2015.
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Christensen, Jens Gerlach. Trends in harmonic analysis and its applications: AMS special session on harmonic analysis and its applications : March 29-30, 2014, University of Maryland, Baltimore County, Baltimore, MD. Providence, Rhode Island: American Mathematical Society, 2015.
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Dzhamay, Anton, Ken'ichi Maruno, and Christopher M. Ormerod. Algebraic and analytic aspects of integrable systems and painleve equations: AMS special session on algebraic and analytic aspects of integrable systems and painleve equations : January 18, 2014, Baltimore, MD. Providence, Rhode Island: American Mathematical Society, 2015.
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Alladi, Krishnaswami, Frank Garvan, and Ae Ja Yee. Ramanujan 125: International conference to commemorate the 125th anniversary of Ramanujan's birth, Ramanujan 125, November 5--7, 2012, University of Florida, Gainesville, Florida. Providence, Rhode Island: American Mathematical Society, 2014.
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Eynard, Bertrand. Random matrices and loop equations. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198797319.003.0007.
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This chapter is an introduction to algebraic methods in random matrix theory (RMT). In the first section, the random matrix ensembles are introduced and it is shown that going beyond the usual Wigner ensembles can be very useful, in particular by allowing eigenvalues to lie on some paths in the complex plane rather than on the real axis. As a detailed example, the Plancherel model is considered from the point of RMT. The second section is devoted to the saddle-point approximation, also called the Coulomb gas method. This leads to a system of algebraic equations, the solution of which leads to an algebraic curve called the ‘spectral curve’ which determines the large N expansion of all observables in a geometric way. Finally, the third section introduces the ‘loop equations’ (i.e., Schwinger–Dyson equations associated with matrix models), which can be solved recursively (i.e., order by order in a semi-classical expansion) by a universal recursion: the ‘topological recursion’.
Book chapters on the topic "Lie Algebras Expansion"
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Watanabe, Yu. "Expansion of Linear Operators by Generators of Lie Algebra $$\mathfrak {su}(d)$$ su ( d )." In Formulation of Uncertainty Relation Between Error and Disturbance in Quantum Measurement by Using Quantum Estimation Theory, 45–70. Tokyo: Springer Japan, 2013. http://dx.doi.org/10.1007/978-4-431-54493-7_5.
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Pierrus, J. "Static electric fields in vacuum." In Solved Problems in Classical Electromagnetism. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198821915.003.0002.
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This chapter begins using Coulomb’s law to derive Maxwell’s electrostatic equations for a vacuum. In doing this, the integral forms of the electrostatic potential Ф and field E are obtained. These results are then used to determine Ф and E for various charge distributions possessing some symmetry: either via Gauss’s law or by directly integrating a known charge density over a line, surface or volume. Applications which require the use of computer algebra software (Mathematica) are included. A multipole expansion of the potential Ф leads to the various multipolemoments of a static charge distribution. Examples which deal with important properties like origin independence are presented. A range of questions and their solutions, not usually encountered in standard textbooks, appear in this chapter.
Conference papers on the topic "Lie Algebras Expansion"
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CAROCA, RICARDO, NELSON MERINO, and PATRICIO SALGADO. "DUAL FORMULATION OF THE S-EXPANSION PROCEDURE FOR HIGHER ORDER LIE ALGEBRAS." In Proceedings of the MG12 Meeting on General Relativity. WORLD SCIENTIFIC, 2012. http://dx.doi.org/10.1142/9789814374552_0367.
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Di Matteo, A. "Line element-less method (LEM) for arbitrarily shaped nonlocal nanoplates: exact and approximate analytical solutions." In AIMETA 2022. Materials Research Forum LLC, 2023. http://dx.doi.org/10.21741/9781644902431-99.
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Abstract. This paper presents an innovative procedure for the analysis of nonlocal plates with arbitrary shape and various boundary conditions. In this regard, the Eringen’s nonlocal model is used to capture small length scale effects. The proposed procedure, referred to as Line Element-Less Method (LEM), is a completely meshfree approach requiring the evaluations of simple line integrals along the plate boundary parametric equation. Further, the deflection function is represented by a series expansion is terms of harmonic polynomials whose coefficients are found by performing variations of appropriately introduced functionals, leading to a linear system of algebraic. Notably, the proposed procedure yields approximate analytical solutions for general shapes and boundary conditions, and even exact solutions for some plate geometries.
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Kornhauser, Alan A. "Dynamics and Thermodynamics of a Free-Piston Expander-Compressor." In ASME 2011 International Mechanical Engineering Congress and Exposition. ASMEDC, 2011. http://dx.doi.org/10.1115/imece2011-63517.
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In a free-piston expander-compressor (FPEC) the work of an expanding gas or vapor is used to compress another gas or vapor through a direct connection of one piston-cylinder assembly to another, without rotary motion. Each section of the FPEC operates like the piston-cylinder of a shaft-connected reciprocating machine. The expander section operates like that of a steam engine: high-pressure gas or vapor is freely admitted through part of the expansion process, but stopped at the “cutoff” part way through the expansion. The compressor section is like that of a shaft-driven reciprocating compressor: low-pressure gas is admitted during the intake stroke, compressed during the compression stroke, and then discharged when it has reached the downstream high pressure. The design of an FPEC is complicated by the fact that the expander and compressor have different relationships between force and position. The force on the expander piston decreases as the stroke progresses, while that on the compressor piston increases. The force difference must be made up by momentum change of the pistons and connecting rod, which accelerate in the early part of the stroke and decelerate in the latter part. The balance between the expanding fluid, the moving mass, and the compressed fluid can be described either dynamically (force and momentum) or thermodynamically (work and energy). It is shown that the mechanical design (piston areas, stroke, oscillating mass, and frequency) of an idealized FPEC is highly constrained by the thermodynamics of the high-pressure stream expansion and the low-pressure stream compression. For simple cases, dynamic models in differential equation form can be compared to thermodynamic analyses in algebraic form. The thermodynamic models provide a baseline design point for an ideal FPSE. The dynamic models can then be used to study non-ideal cases.
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Wang, Chung-Hao. "Cylindrically Anisotropic Tubes Containing a Line Dislocation." In ASME 2006 Pressure Vessels and Piping/ICPVT-11 Conference. ASMEDC, 2006. http://dx.doi.org/10.1115/pvp2006-icpvt-11-93188.
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An analytical solution of the problem of a cylindrically anisotropic tube which contains a line dislocation is presented in this study. The state space formulation in conjunction with the eigenstrain theory is proved to be a feasible and systematic methodology to analyze a tube with the existence of dislocations. The state space formulation which expediently groups the displacements and the cylindrical surface traction can construct a governing differential matrix equation. By using Fourier series expansion and the well developed theory of matrix algebra, the asymmetrical solutions are not only explicit but also compact in form. The dislocation considered in this study is a kind of mixed dislocation which is the combination of edge dislocations and a screw dislocation and the dislocation line is parallel to the longitudinal axis of the tube. The degeneracy of the eigen relation and the technique to determine the inverse of a singular matrix are thoroughly discussed, so that the general solutions can be applied to the case of isotropic tubes, which is one of the novel features of this research. The results of isotropic problems, which are belong to the general solutions, are compared with the well-established expressions in the literature. The satisfied correspondences of these comparisons indicate the validness of this study. A cylindrically orthotropic tube is also investigated as an example and the numerical results for the displacements and tangential stress on the outer surface are displayed. The effects on surface stresses due to the existence of a dislocation appear to have a characteristic of localized phenomenon.
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Sekrani, Ghofrane, Jean-Sebastien Dick, Sébastien Poncet, and Sravankumar Nallamothu. "Numerical Investigation of Air-Oil Two-Phase Flow Pattern Transition in the Scavenge Line of an Aeroengine." In ASME Turbo Expo 2021: Turbomachinery Technical Conference and Exposition. American Society of Mechanical Engineers, 2021. http://dx.doi.org/10.1115/gt2021-58988.
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Abstract Since most research investments in aeroengines have been targeted at the hot and cold sections, the oil system has remained an area poorly understood. Optimum sizing of the oil system can directly reduce the engine’s weight and specific fuel consumption while maximizing service life. The understanding of air/oil interaction in scavenge lines is required to influence the design of the oil systems and achieve those objectives. The challenge is in the existence of numerous possible flow regimes and phase interactions. In scavenge lines, a complex two-phase flow results from the interaction of sealing airflow and lubrication oil. Scavenge lines can have bends, junctions and sudden area changes which complicates their modeling by amplifying pressure gradients and turbulence generation, and causing the flow to change morphology (annular, slug, stratified, bubbly, mist, etc.). Several multiphase flow approaches have been developed to model two-phase flow in straight scavenge lines. However, up until now, no methodology is preferred by the community for simulating two-phase flow in such application. There are still many unknowns regarding the modeling of turbulence, phase interaction and the compressibility of immiscible mixtures such as air and oil. The present study compares the performance of two numerical models: Volume of Fluid (VOF) and Algebraic Interfacial Area Density (AIAD), for simulating the air/oil flow in a suddenly expanding scavenge line against the experimental data of Ahmed et al. [1–2]. The AIAD model is a two-fluid Eulerian approach newly implemented on Ansys Fluent. Discrepancies between the two models for predicting pressure loss and void fraction are evaluated and discussed into details. The flow regime before and after the sudden expansion is identified using iso-surfaces of the void-fraction and compared against visual data. Based on the results presented, recommendations are formulated for further work regarding the calibration of AIAD modeling parameters.
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Akyuzlu, K. M., and K. Hallenbeck. "A Study of Unsteady Mixed Convection in a Lid Driven Flow Inside a Rectangular Cavity." In ASME 2010 International Mechanical Engineering Congress and Exposition. ASMEDC, 2010. http://dx.doi.org/10.1115/imece2010-37888.
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A numerical study is conducted to identify the unsteady characteristics of momentum and heat transfer in lid-driven cavity flows. The cavity under study is filled with a compressible fluid and is of rectangular shape. The bottom of the cavity is insulated and stationary where as the top of the cavity (the lid) is pulled at constant speed. The vertical walls of the cavity are kept at constant but unequal temperatures. A two-dimensional, mathematical model is adopted to investigate the shear and buoyancy driven circulation patterns inside this rectangular cavity. This physics based mathematical model consists of conservation of mass, momentum (two-dimensional, unsteady Navier-Stokes equations for compressible flows) and energy equations for the enclosed fluid subjected to appropriate boundary and initial conditions. The compressibility of the working fluid is represented by an ideal gas relation and its thermodynamic and transport properties are assumed to be function of temperature. The governing equations are discretized using second order accurate central differencing for spatial derivatives and second order finite differencing (based on Taylor expansion) for the time derivatives. The resulting nonlinear equations are then linearized using Newton’s linearization method. The set of algebraic equations that result from this process are then put into a matrix form and solved using a Coupled Modified Strongly Implicit Procedure (CMSIP) for the unknowns of the problem. Grid independence and time convergence studies were carried out to determine the accuracy of the square mesh adopted for the present study. Two benchmark cases (driven cavity and rectangular channel flows) were studied to verify the accuracy of the CMSIP. Numerical experiments were then carried out to simulate the unsteady development of the shear and buoyancy driven circulation patterns for different Richardson numbers in the range of 0.036<Ri<100 where the Re number is kept less than 2000 to assure laminar flow conditions inside the cavity. Simulations start with a stagnant fluid subjected to a sudden increase in one of the walls temperature. At the same time the upper lid of the cavity is accelerated, instantaneously, to a constant speed. The circulation patterns, temperature contours, vertical and horizontal velocity profiles were generated at different times of the simulation, and wall heat fluxes and Nusselt numbers were calculated for the steady state conditions. Only the results for a square cavity are presented in this paper. These results indicate that the heat transfer rates at the vertical walls of the cavity are enhanced with the decrease in Richardson number.
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El Dine Matbouli, Houssam, and Iyad Fayssal. "Performance Assessment of a NACA-2412 Surrogate Model Using Non-Intrusive Polynomial Chaos Method." In ASME 2020 Fluids Engineering Division Summer Meeting collocated with the ASME 2020 Heat Transfer Summer Conference and the ASME 2020 18th International Conference on Nanochannels, Microchannels, and Minichannels. American Society of Mechanical Engineers, 2020. http://dx.doi.org/10.1115/fedsm2020-20212.
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Abstract Computational Fluid Dynamics (CFD) models have been regarded as inevitable tools in the design of various aerodynamics applications like formula 1 cars, wind turbines, airfoil design, and many others. Such models rely on applying advanced numerical methods to solve the set of Navier-Stokes equations and other auxiliary models to quantify the aerodynamics performance, such as lift and drag in airfoil applications. However, currently adopted advanced mathematical models that are used to predict airfoil performance, though are essential tools in the advancement of aviation industry, incur large computational costs. As a primary step, an approximate model of an airfoil is built for the sake of reducing computational cost while preserving accuracy. The approximate model is based on creating a simplified algebraic model which is suitable then to perform aerodynamic design optimization rather than solving the highly non-linear set of continuity and momentum equations. In this work, the non-intrusive polynomial chaos expansion (NIPC) technique is employed on a NACA-2412 airfoil. The main objective of this research is to investigate the performance of the NIPC model with different orders on solution accuracy. A methodology is designed by first developing a full CFD model of the NACA-2412 airfoil. An experimental setup of the airfoil was then prepared with the variations of lift and drag coefficients under different operative conditions were reported. The experimental data were used to validate the numerical results generated by the CFD simulations. The surrogate model was then constructed via the NIPC technique. The performance of the surrogate model was then benchmarked with CFD results under different flow and operating conditions. Analysis was also performed to investigate the effect of polynomial order on the solution accuracy. Results showed that for a given data set size, increasing the order of the polynomial would deteriorate the solution accuracy. At a certain order k, the analytical solution starts diverging especially for values of angle of attack close to −60 and 60°.
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Hallenbeck, K., and K. M. Akyuzlu. "A Parametric Study of Laminar Mixed Convection in a Square Cavity Using Numerical Simulation Techniques." In ASME 2012 International Mechanical Engineering Congress and Exposition. American Society of Mechanical Engineers, 2012. http://dx.doi.org/10.1115/imece2012-86903.
Full textAbstract:
A parametric study is conducted using numerical experimentation to construct an empirical Nusselt number correlation in terms of Richardson and Prandtl numbers for laminar mixed convection in a square cavity. The square cavity under study is assumed to be filled with a compressible fluid. The bottom of the cavity is insulated and stationary where as the top of the cavity (the lid) is pulled at constant speed. The vertical walls of the cavity are kept at constant but unequal temperatures. A two-dimensional, mathematical model is adopted to predict the momentum and heat transfer inside this rectangular cavity. This physics based mathematical model consists of conservation of mass, momentum (two-dimensional, unsteady Navier-Stokes equations for compressible flows) and energy equations for the enclosed fluid subjected to appropriate boundary and initial conditions. The compressibility of the working fluid is represented by an ideal gas relation. The thermodynamic and transport properties of the working fluid are assumed to be constant. The governing equations are discretized using second order accurate central differencing for spatial derivatives and second order finite differencing (based on Taylor expansion) for the time derivatives. The resulting nonlinear equations are then linearized using Newton’s linearization method. The set of algebraic equations that result from this process are then put into a matrix form and solved using a Coupled Modified Strongly Implicit Procedure (CMSIP) for the unknowns of the problem. Grid independence and time convergence studies were carried to determine the accuracy of the square mesh adopted for the present study. Two benchmark cases (driven cavity and rectangular channel flows) were studied to verify the accuracy of the CMSIP. Numerical experiments were then carried out to simulate the heat transfer characteristics of mixed convection flow for different Richardson numbers in the range of 0.036<Ri<1.00 where the Reynolds number is kept less than 2000 to ensure laminar flow conditions inside the cavity. The velocity vector field maps (circulation patterns) and temperature contours, and temperature profiles along the horizontal axes were generated for different Prandtl numbers ranging from 0.3 to 1. Wall heat fluxes and Nusselt numbers were determined for each parametric study. The collected data from the numerical experiments were then used to construct an empirical Nusselt number correlation in terms of Richardson and Prandtl numbers.
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Surana, K. S., and H. Vijayendra Nayak. "Computations of the Numerical Solutions of Higher Class of Navier-Stokes Equations: 2D Newtonian Fluid Flow." In ASME 2001 Engineering Technology Conference on Energy. American Society of Mechanical Engineers, 2001. http://dx.doi.org/10.1115/etce2001-17143.
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Abstract This paper presents formulations, computations, investigations and consequences of the various aspects of the numerical solutions of classes C00 and C11 of the two dimensional Navier-Stokes equations in primitive variables u, v, p, τxx, τxy and τyy for incompressible, isothermal and laminar Newtonian fluid flows using p-version Least Squares Finite Element Formulations (LSFEF). The stick-slip problem is used as a model problem in all investigations since this model problem is typical of many other flow situations like contraction, expansion etc. The major thrust of the work presented is to attempt to resolve the local behavior of the solutions in the immediate vicinity of the stick-slip point. The investigations reveal the following: a) The manner in which the stresses are non-dimensionalized in the governing differential equations (GDEs) influences the performance of the iterative procedure of solving non-linear algebraic equations and thus, computational efficiency. b) Solutions of the class C00 are always the wrong class of solutions and thus are always spurious. c) In the flow domains, containing sharp gradients of dependent variables, conservation of mass is difficult to achieve specially at lower p-levels. d) C11 solutions of the Navier-Stokes equations are in conformity with the continuity considerations in the GDEs. e) An augmented form of the Navier-Stokes equations is proposed that always ensures conservation of mass regardless of mesh, p-levels and the nature of the solution gradients. This approach yields the most desired class of C11 solutions. f) It is mathematically established and numerically demonstrated using stick-slip problem that τij are in fact zero at the stick-slip point and the peak values of τxx and τyy must occur, and in fact do, past the stick-slip point in the free field and that peak values of τxy must occur before the stick-slip point on the no-slip boundary. Thus, there is no singularity of τij in the stick-slip problem at the stick-slip point. A significant finding is that imposition of symmetry boundary condition (necessary based on physics) at the stick-slip point even in C11 interpolations is not possible without deteriorating τij behavior in the vicinity of the stick-slip point. However, with the boundary condition, the peak of τxy does occur before the stick-slip point, while the locations of τxx and τyy remain past the stick-slip point in the free field. h) A significant feature of our research work is that we utilize straightforward p-version LSFEF with C00 and C11 type interpolation without linearizing GDEs and that SUPG, SUPG/DC, SUPG/DC/LS operators are neither needed nor used. All numerical studies are conducted and presented using three different meshes (progressively refined and graded) for two different velocities (0.01 and 100 m/s).