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Relevant bibliographies by topics / Differential graded Lie algebras

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Academic literature on the topic 'Differential graded Lie algebras'

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Published: 10 March 2023

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Journal articles on the topic "Differential graded Lie algebras"

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Piontkovskii, D. I. "On differential graded Lie algebras." Russian Mathematical Surveys 58, no. 1 (February 28, 2003): 189–90. http://dx.doi.org/10.1070/rm2003v058n01abeh000604.

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Bonezzi, Roberto, and Olaf Hohm. "Duality Hierarchies and Differential Graded Lie Algebras." Communications in Mathematical Physics 382, no. 1 (February 2021): 277–315. http://dx.doi.org/10.1007/s00220-021-03973-8.

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AbstractThe gauge theories underlying gauged supergravity and exceptional field theory are based on tensor hierarchies: generalizations of Yang-Mills theory utilizing algebraic structures that generalize Lie algebras and, as a consequence, require higher-form gauge fields. Recently, we proposed that the algebraic structure allowing for consistent tensor hierarchies is axiomatized by ‘infinity-enhanced Leibniz algebras’ defined on graded vector spaces generalizing Leibniz algebras. It was subsequently shown that, upon appending additional vector spaces, this structure can be reinterpreted as a differential graded Lie algebra. We use this observation to streamline the construction of general tensor hierarchies, and we formulate dynamics in terms of a hierarchy of first-order duality relations, including scalar fields with a potential.

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Kaneyuki, Soji, and Hiroshi Asano. "Graded Lie algebras and generalized Jordan triple systems." Nagoya Mathematical Journal 112 (December 1988): 81–115. http://dx.doi.org/10.1017/s002776300000115x.

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One frequently encounters (real) semisimple graded Lie algebras in various branches of differential geometry (e.g. [16], [9], [14], [18]). It is therefore desirable to study semisimple graded Lie algebras, including those which have been studied individually, in a unified way. One of our concerns is to classify (finite-dimensional) semisimple graded Lie algebras in a way that enables us to construct them.

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Wulkenhaar, Raimar. "Noncommutative geometry with graded differential Lie algebras." Journal of Mathematical Physics 38, no. 6 (June 1997): 3358–90. http://dx.doi.org/10.1063/1.532048.

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Wulkenhaar, Raimar. "Gauge theories with graded differential Lie algebras." Journal of Mathematical Physics 40, no. 2 (February 1999): 787–94. http://dx.doi.org/10.1063/1.532685.

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Wulkenhaar, Raimar. "Graded differential lie algebras and model building." Journal of Geometry and Physics 25, no. 3-4 (May 1998): 305–25. http://dx.doi.org/10.1016/s0393-0440(97)00029-6.

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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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Budur, Nero, and Botong Wang. "Cohomology jump loci of differential graded Lie algebras." Compositio Mathematica 151, no. 8 (March 6, 2015): 1499–528. http://dx.doi.org/10.1112/s0010437x14007970.

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To study infinitesimal deformation problems with cohomology constraints, we introduce and study cohomology jump functors for differential graded Lie algebra (DGLA) pairs. We apply this to local systems, vector bundles, Higgs bundles, and representations of fundamental groups. The results obtained describe the analytic germs of the cohomology jump loci inside the corresponding moduli space, extending previous results of Goldman–Millson, Green–Lazarsfeld, Nadel, Simpson, Dimca–Papadima, and of the second author.

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BENKHALIFA, MAHMOUD. "WHITEHEAD EXACT SEQUENCE AND DIFFERENTIAL GRADED FREE LIE ALGEBRA." International Journal of Mathematics 15, no. 10 (December 2004): 987–1005. http://dx.doi.org/10.1142/s0129167x04002673.

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Let R be a principal and integral domain. We say that two differential graded free Lie algebras over R (free dgl for short) are weakly equivalent if and only if the homologies of their corresponding enveloping universal algebras are isomophic. This paper is devoted to the problem of how we can characterize the weakly equivalent class of a free dgl. Our tool to address this question is the Whitehead exact sequence. We show, under a certain condition, that two R-free dgls are weakly equivalent if and only if their Whitehead sequences are isomorphic.

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Yang, Jinwei. "Vertex algebras associated to the affine Lie algebras of abelian polynomial current algebras." International Journal of Mathematics 27, no. 05 (May 2016): 1650046. http://dx.doi.org/10.1142/s0129167x16500464.

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We construct a family of vertex algebras associated to the affine Lie algebra of polynomial current algebras of finite-dimensional abelian Lie algebras, along with their modules and logarithmic modules. These vertex algebras and their (logarithmic) modules are strongly [Formula: see text]-graded and quasi-conformal. We then show that matrix elements of products and iterates of logarithmic intertwining operators among these logarithmic modules satisfy certain systems of differential equations. Using these systems of differential equations, we verify the convergence and extension property needed in the logarithmic tensor category theory developed by Huang, Lepowsky and Zhang.

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Dissertations / Theses on the topic "Differential graded Lie algebras"

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Fialowski, Alice, Michael Penkava, and fialowsk@cs elte hu. "Deformation Theory of Infinity Algebras." ESI preprints, 2000. ftp://ftp.esi.ac.at/pub/Preprints/esi906.ps.

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Yaseen, Hogar M. "Generalized root graded Lie algebras." Thesis, University of Leicester, 2018. http://hdl.handle.net/2381/42765.

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Let g be a non-zero finite-dimensional split semisimple Lie algebra with root system Δ. Let Γ be a finite set of integral weights of g containing Δ and {0}. We say that a Lie algebra L over F is generalized root graded, or more exactly (Γ,g)-graded, if L contains a semisimple subalgebra isomorphic to g, the g-module L is the direct sum of its weight subspaces Lα (α ∈ Γ) and L is generated by all Lα with α ̸= 0 as a Lie algebra. If g is the split simple Lie algebra and Γ = Δ∪{0} then L is said to be root-graded. Let g∼= sln and Θn = {0,±εi±ε j,±εi,±2εi | 1 ≤ i ̸= j ≤ n} where {ε1, . . . , εn} is the set of weights of the natural sln-module. Then a Lie algebra L is (Θn,g)-graded if and only if L is generated by g as an ideal and the g-module L decomposes into copies of the adjoint module, the natural module V, its symmetric and exterior squares S2V and ∧2V, their duals and the one dimensional trivial g-module. In this thesis we study properties of generalized root graded Lie algebras and focus our attention on (Θn, sln)-graded Lie algebras. We describe the multiplicative structures and the coordinate algebras of (Θn, sln)-graded Lie algebras, classify these Lie algebras and determine their central extensions.

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at, Andreas Cap@esi ac. "Graded Lie Algebras and Dynamical Systems." ESI preprints, 2001. ftp://ftp.esi.ac.at/pub/Preprints/esi1086.ps.

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Yang, Qunfeng. "Some graded Lie algebra structures associated with Lie algebras and Lie algebroids." Thesis, National Library of Canada = Bibliothèque nationale du Canada, 1999. http://www.collectionscanada.ca/obj/s4/f2/dsk1/tape9/PQDD_0007/NQ41350.pdf.

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Bagnoli, Lucia. "Z-graded Lie superalgebras." Master's thesis, Alma Mater Studiorum - Università di Bologna, 2017. http://amslaurea.unibo.it/14118/.

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This thesis investigates the role of filtrations and gradings in the study of Lie superalgebras. The main connections between the Z-grading of a Lie superalgebra and its structure are explained. As an example, the simplicity of the Lie superalgebras W(m,n) and S(m,n) is proved. Finally, the strongly symmetric gradings of length three and five of the Lie superalgebras W(m,n) and S(m,n) are classified.

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Pauksztello, David. "Homological properties of differential graded algebras." Thesis, University of Leeds, 2008. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.493288.

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In this thesis we consider various homological properties of differential graded algebras, and more generally, properties of arbitrary triangulated categories which have set indexed coproducts. A major example of such a triangulated category is the derived category of a differential graded algebra. We present the background material to the theory of triangulated categories, derived categories and differential graded algebras as well as a brief resume of classical homological algebra in Chapters 2 and 3.

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Shklyarov, Dmytro. "Hirzebruch-Riemann-Roch theorem for differential graded algebras." Diss., Manhattan, Kan. : Kansas State University, 2009. http://hdl.handle.net/2097/1381.

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Maycock, Daniel. "Properties of triangular matrix and Gorenstein differential graded algebras." Thesis, University of Newcastle upon Tyne, 2011. http://hdl.handle.net/10443/1359.

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The main goal of this thesis is to investigate properties of two types of Differential Graded Algebras (or DGAs), namely upper triangular matrix DGAs and Gorenstein DGAs. In doing so we extend a number of corresponding ring theory results to the more general setting of DGAs and DG modules. Chapters 2 and 3 contain background material. In chapter 2 we give a brief summary of some important aspects of homological algebra. Starting with the definition of an abelian category we give the construction of the derived category and the definition of derived functors. In chapter 3 we present the basics about Differential Graded Algebras and Differential Graded Modules in particular extending the definitions of the derived category and derived functors to the Differential Graded case before providing some results on Recollement of DGAs, Dualising DG-modules and Gorenstein DGAs. Chapters 4 and 5 contain the bulk of the work for the Thesis. In chapter 4 we look at upper triangular matrix DGAs and in particular we generalise a result for upper triangular matrix rings to the situation of upper triangular matrix differential graded algebras. An upper triangular matrix DGA has the form [R M / 0 S] where R and S are DGAs and M is a DG R-Sop-bimodule. We show that under certain conditions on the DG-module M, and given the existence of a DG R-module X from which we can build the derived category D(R), that there exists a derived equivalence between the upper triangular matrix DGAs [R M / 0 S] and [ S M’ / 0 R’], where the DG-bimodule M0 is obtained from M and X, and R0 is the endomorphism differential graded algebra of a K-projective resolution of X. In chapter 5 we turn our attention to Gorenstein DGAs and generalise some results from Gorenstein rings to Gorenstein DGAs. We present a number of Gorenstein Theorems which state, for certain types of DGAs, that being Gorenstein is equivalent to the bounded and finite versions of the Auslander and Bass classes being maximal. We also provide a new definition of a Gorenstein morphism for DGAs by considering a DG bimodule as a generalised morphism of DGAs. We then show that some existing results for Gorenstein morphism extend to these "Generalised Gorenstein Morphisms". We finally conclude with some examples of generalised Gorenstein morphisms for some well known DGAs.

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Chung, Myungsuk. "Lie derivations on rings of differential operators." Diss., Virginia Tech, 1995. http://hdl.handle.net/10919/37457.

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Martini, Alessio. "Algebras of differential operators on Lie groups and spectral multipliers." Doctoral thesis, Scuola Normale Superiore, 2010. http://hdl.handle.net/11384/85663.

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Let (X, μ) be a measure space, and let L1, . . . ,Ln be (possibly unbounded) selfadjoint operators on L2(X, μ), which commute strongly pairwise, i.e., which admit a joint spectral resolution E on Rn. A joint functional calculus is then defined via spectral integration: for every Borel function m : Rn → C, m(L) = m(L1, . . . ,Ln) = ∫ Rn m(λ) dE(λ) is a normal operator on L2(X, μ), which is bounded if and only if m - called the joint spectral multiplier associated to m(L) - is (E-essentially) bounded. However, the abstract theory of spectral integrals does not tackle the following problem: to find conditions on the multiplier m ensuring the boundedness of m(L) on Lp(X, μ) for some p ≠ 2. We are interested in this problem when the measure space is a connected Lie group G with a right Haar measure, and L1, . . . ,Ln are left-invariant differential operators on G. In fact, the question has been studied quite extensively in the case of a single operator, namely, a sublaplacian or a higher-order analogue. On the other hand, for multiple operators, only specific classes of groups and specific choices of operators have been considered in the literature. Suppose that L1, . . . ,Ln are formally self-adjoint, left-invariant differential operators on a connected Lie group G, which commute pairwise (as operators on smooth functions). Under the assumption that the algebra generated by L1, . . . ,Ln contains a weighted subcoercive operator --- a notion due to [ER98], including positive elliptic operators, sublaplacians and Rockland operators---we prove that L1, . . . ,Ln are (essentially) self-adjoint and strongly commuting on L2(G). Moreover, we perform an abstract study of such a system of operators, in connection with the algebraic structure and the representation theory of G, similarly as what is done in the literature for the algebras of differential operators associated with Gelfand pairs. Under the additional assumption that G has polynomial volume growth, weighted L1 estimates are obtained for the convolution kernel of the operator m(L) corresponding to a compactly supported multiplier m satisfying some smoothness condition. The order of smoothness which we require on m is related to the degree of polynomial growth of G. Some techniques are presented, which allow, for some specific groups and operators, to lower the smoothness requirement on the multiplier. In the case G is a homogeneous Lie group and L1, . . . ,Ln are homogeneous operators, a multiplier theorem of Mihlin-H\"ormander type is proved, extending the result for a single operator of [Chr91] and [MM90]. Further, a product theory is developed, by considering several homogeneous groups Gj , each of which with its own system of operators; a non-conventional use of transference techniques then yields a multiplier theorem of Marcinkiewicz type, not only on the direct product of the Gj , but also on other (possibly non-homogeneous) groups, containing homomorphic images of the Gj . Consequently, for certain non-nilpotent groups of polynomial growth and for some distinguished sublaplacians, we are able to improve the general result of [Ale94].

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Books on the topic "Differential graded Lie algebras"

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1944-, Gregory Thomas Bradford, and Premet Alexander 1955-, eds. The recognition theorem for graded lie algebras in prime characteristic. Providence, R.I: American Mathematical Society, 2009.

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Euler, Norbert. Continuous symmetries, Lie algebras, and differential equations. Mannheim, [Germany]: BI Wissenschaftsverlag, 1992.

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V, Savelʹev M., ed. Lie algebras, geometry and Toda-type systems. New York: Cambridge University Press, 1997.

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Allison, Bruce N. Lie algebras graded by the root systems BCr, r[greater than or equal to] 2. Providence, RI: American Mathematical Society, 2002.

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Xu, Xiaoping. Representations of Lie Algebras and Partial Differential Equations. Singapore: Springer Singapore, 2017. http://dx.doi.org/10.1007/978-981-10-6391-6.

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Meinrenken, Eckhard. Clifford Algebras and Lie Theory. Berlin, Heidelberg: Springer Berlin Heidelberg, 2013.

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Steeb, W. H. Continuous symmetries, Lie algebras, differential equations, and computer algebra. 2nd ed. Hackensack, N.J: World Scientific, 2007.

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Steeb, W. H. Continuous symmetries, Lie algebras, differential equations, and computer algebra. Singapore: World Scientific, 1996.

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Mackenzie, K. Lie groupoids and Lie algebroids in differential geometry. Cambridge [Cambridgeshire]: Cambridge University Press, 1987.

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Sabinin, Lev V. Mirror geometry of lie algebras, lie groups, and homogeneous spaces. New York: Kluwer Academic Publishers, 2004.

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Book chapters on the topic "Differential graded Lie algebras"

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Manetti, Marco. "Differential Graded Lie Algebras." In Springer Monographs in Mathematics, 127–58. Singapore: Springer Nature Singapore, 2022. http://dx.doi.org/10.1007/978-981-19-1185-9_5.

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Buijs, Urtzi, Yves Félix, Aniceto Murillo, and Daniel Tanré. "Complete Differential Graded Lie Algebras." In Lie Models in Topology, 71–91. Cham: Springer International Publishing, 2020. http://dx.doi.org/10.1007/978-3-030-54430-0_3.

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Félix, Yves, Stephen Halperin, and Jean-Claude Thomas. "Graded (differential) Lie algebras and Hopf algebras." In Graduate Texts in Mathematics, 283–98. New York, NY: Springer New York, 2001. http://dx.doi.org/10.1007/978-1-4613-0105-9_22.

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Stasheff, Jim. "Differential graded Lie algebras, quasi-hopf algebras and higher homotopy algebras." In Lecture Notes in Mathematics, 120–37. Berlin, Heidelberg: Springer Berlin Heidelberg, 1992. http://dx.doi.org/10.1007/bfb0101184.

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Sitaram, B. R. "Graded Lie Algebras." In Gravitation, Gauge Theories and the Early Universe, 481–85. Dordrecht: Springer Netherlands, 1989. http://dx.doi.org/10.1007/978-94-009-2577-9_24.

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Griffiths, Phillip, and John Morgan. "Differential Graded Algebras." In Rational Homotopy Theory and Differential Forms, 95–102. New York, NY: Springer New York, 2013. http://dx.doi.org/10.1007/978-1-4614-8468-4_10.

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Keller, Corina. "Differential Graded Algebras." In Chern-Simons Theory and Equivariant Factorization Algebras, 41–61. Wiesbaden: Springer Fachmedien Wiesbaden, 2019. http://dx.doi.org/10.1007/978-3-658-25338-7_3.

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Kaneyuki, Soji. "Semisimple Graded Lie Algebras." In Analysis and Geometry on Complex Homogeneous Domains, 107–26. Boston, MA: Birkhäuser Boston, 2000. http://dx.doi.org/10.1007/978-1-4612-1366-6_9.

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Connett, William C., and Alan L. Schwartz. "Hypergroups and Differential Equations." In Lie Groups and Lie Algebras, 109–15. Dordrecht: Springer Netherlands, 1998. http://dx.doi.org/10.1007/978-94-011-5258-7_7.

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Iachello, Francesco. "Differential Realizations." In Lie Algebras and Applications, 193–99. Berlin, Heidelberg: Springer Berlin Heidelberg, 2014. http://dx.doi.org/10.1007/978-3-662-44494-8_11.

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Conference papers on the topic "Differential graded Lie algebras"

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Iachello, F. "Graded Lie algebras and applications." In LATIN-AMERICAN SCHOOL OF PHYSICS XXXV ELAF; Supersymmetries in Physics and Its Applications. AIP, 2004. http://dx.doi.org/10.1063/1.1853199.

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Patera, J. "Graded contractions of Lie algebras, representations and tensor products." In Group Theory in Physics: Proceedings of the international symposium held in honor of Professor Marcos Moshinsky. AIP, 1992. http://dx.doi.org/10.1063/1.42858.

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He, J. W., and Q. S. Wu. "Koszul differential graded algebras and modules." In 5th China–Japan–Korea International Ring Theory Conference. WORLD SCIENTIFIC, 2008. http://dx.doi.org/10.1142/9789812818331_0007.

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IZAURIETA, FERNANDO, EDUARDO RODRÍGUEZ, ALFREDO PÉREZ, and PATRICIO SALGADO. "EXPANDING LIE AND GAUGE FREE DIFFERENTIAL ALGEBRAS THROUGH ABELIAN SEMIGROUPS." In Proceedings of the MG12 Meeting on General Relativity. WORLD SCIENTIFIC, 2012. http://dx.doi.org/10.1142/9789814374552_0442.

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WANG, HONG-YU. "NONLINEAR SCHRÖDINGER SYSTEMS ASSOCIATED WITH HERMITIAN SYMMETRIC LIE ALGEBRAS." In Proceedings of the International Conference on Modern Mathematics and the International Symposium on Differential Geometry. WORLD SCIENTIFIC, 2002. http://dx.doi.org/10.1142/9789812776419_0017.

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Kime, Katherine A. "Effect of the Spatial Extent of the Control in a Bilinear Control Problem for the Schroedinger Equation." In ASME 2009 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. ASMEDC, 2009. http://dx.doi.org/10.1115/detc2009-86440.

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We consider control of the one-dimensional Schroedinger equation through a time-varying potential. Using a finite difference semi-discretization, we consider increasing the extent of the potential from a single central grid-point in space to two or more gridpoints. With the differential geometry package in Maple 8, we compute and compare the corresponding Control Lie Algebras, identifying a trend in the number of elements which span the Control Lie Algebras.

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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).

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Reports on the topic "Differential graded Lie algebras"

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Hrivnak, Jiri Hrivnak. Associated Lie Algebras and Graded Contractions of the Pauli Graded sl(3,C). Journal of Geometry and Symmetry in Physics, 2012. http://dx.doi.org/10.7546/jgsp-6-2006-47-54.

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