Relevant bibliographies by topics / Jacobi manifolds

Academic literature on the topic 'Jacobi manifolds'

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Published: 10 March 2023

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Journal articles on the topic "Jacobi manifolds"

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GATSE, Servais Cyr. "Jacobi Manifolds, Contact Manifolds and Contactomorphism." Journal of Mathematics Research 13, no. 4 (July 29, 2021): 85. http://dx.doi.org/10.5539/jmr.v13n4p85.

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Let M be a smooth manifold and let D(M) be the module of first order differential operators on M. In this work, we give a link between Jacobi manifolds and Contact manifolds. We also generalize the notion of contactomorphism on M and thus, we characterize the Contact diffeomorphisms.
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Chen, Bang-Yen, Sharief Deshmukh, and Amira A. Ishan. "On Jacobi-Type Vector Fields on Riemannian Manifolds." Mathematics 7, no. 12 (November 21, 2019): 1139. http://dx.doi.org/10.3390/math7121139.

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In this article, we study Jacobi-type vector fields on Riemannian manifolds. A Killing vector field is a Jacobi-type vector field while the converse is not true, leading to a natural question of finding conditions under which a Jacobi-type vector field is Killing. In this article, we first prove that every Jacobi-type vector field on a compact Riemannian manifold is Killing. Then, we find several necessary and sufficient conditions for a Jacobi-type vector field to be a Killing vector field on non-compact Riemannian manifolds. Further, we derive some characterizations of Euclidean spaces in terms of Jacobi-type vector fields. Finally, we provide examples of Jacobi-type vector fields on non-compact Riemannian manifolds, which are non-Killing.
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Okassa, Eugène. "Symplectic Lie–Rinehart–Jacobi Algebras and Contact Manifolds." Canadian Mathematical Bulletin 54, no. 4 (December 1, 2011): 716–25. http://dx.doi.org/10.4153/cmb-2011-033-6.

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Blažić, Novica, Neda Bokan, and Zoran Rakić. "Osserman pseudo-Riemannian manifolds of signature (2,2)." Journal of the Australian Mathematical Society 71, no. 3 (December 2001): 367–96. http://dx.doi.org/10.1017/s1446788700003001.

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AbstractA pseudo-Riemannian manifold is said to be timelike (spacelike) Osserman if the Jordan form of the Jacobi operator Kx is independent of the particular unit timelike (spacelike) tangent vector X. The first main result is that timelike (spacelike) Osserman manifold (M, g) of signature (2, 2) with the diagonalizable Jacobi operator is either locally rank-one symmetric or flat. In the nondiagonalizable case the characteristic polynomial of Kx has to have a triple zero, which is the other main result. An important step in the proof is based on Walker's study of pseudo-Riemannian manifolds admitting parallel totally isotropic distributions. Also some interesting additional geometric properties of Osserman type manifolds are established. For the nondiagonalizable Jacobi operators some of the examples show a nature of the Osserman condition for Riemannian manifolds different from that of pseudo-Riemannian manifolds.
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Vitagliano, Luca, and Aïssa Wade. "Holomorphic Jacobi manifolds." International Journal of Mathematics 31, no. 03 (February 14, 2020): 2050024. http://dx.doi.org/10.1142/s0129167x2050024x.

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In this paper, we develop holomorphic Jacobi structures. Holomorphic Jacobi manifolds are in one-to-one correspondence with certain homogeneous holomorphic Poisson manifolds. Furthermore, holomorphic Poisson manifolds can be looked at as special cases of holomorphic Jacobi manifolds. We show that holomorphic Jacobi structures yield a much richer framework than that of holomorphic Poisson structures. We also discuss the relationship between holomorphic Jacobi structures, generalized contact bundles and Jacobi–Nijenhuis structures.
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Haji-Badali, Ali, and Amirhesam Zaeim. "Commutative curvature operators over four-dimensional homogeneous manifolds." International Journal of Geometric Methods in Modern Physics 12, no. 10 (October 25, 2015): 1550123. http://dx.doi.org/10.1142/s0219887815501236.

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Four-dimensional pseudo-Riemannian homogeneous spaces whose isotropy is non-trivial with commuting curvature operators have been studied. The only example of homogeneous Einstein four-manifold which is curvature-Ricci commuting but not semi-symmetric has been presented. Non-trivial examples of semi-symmetric homogeneous four-manifolds which are not locally symmetric, also Jacobi–Jacobi commuting manifolds which are not flat have been presented.
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Bascone, Francesco, Franco Pezzella, and Patrizia Vitale. "Topological and Dynamical Aspects of Jacobi Sigma Models." Symmetry 13, no. 7 (July 5, 2021): 1205. http://dx.doi.org/10.3390/sym13071205.

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The geometric properties of sigma models with target space a Jacobi manifold are investigated. In their basic formulation, these are topological field theories—recently introduced by the authors—which share and generalise relevant features of Poisson sigma models, such as gauge invariance under diffeomorphisms and finite dimension of the reduced phase space. After reviewing the main novelties and peculiarities of these models, we perform a detailed analysis of constraints and ensuing gauge symmetries in the Hamiltonian approach. Contact manifolds as well as locally conformal symplectic manifolds are discussed, as main instances of Jacobi manifolds.
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Ibáñez, Raul, Manuel de León, Juan C. Marrero, and Edith Padrón. "Nambu-Jacobi and generalized Jacobi manifolds." Journal of Physics A: Mathematical and General 31, no. 4 (January 30, 1998): 1267–86. http://dx.doi.org/10.1088/0305-4470/31/4/015.

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Gilkey, Peter, and Veselin Videv. "Jacobi–Jacobi Commuting Models and Manifolds." Journal of Geometry 92, no. 1-2 (January 24, 2009): 60–68. http://dx.doi.org/10.1007/s00022-008-2061-9.

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Cariñena, J. F., X. Gràcia, G. Marmo, E. Martínez, M. C. Muñoz-Lecanda, and N. Román-Roy. "Structural aspects of Hamilton–Jacobi theory." International Journal of Geometric Methods in Modern Physics 13, no. 02 (January 26, 2016): 1650017. http://dx.doi.org/10.1142/s0219887816500171.

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In our previous papers [J. F. Cariñena, X. Gràcia, G. Marmo, E. Martínez, M. C. Muñoz-Lecanda and N. Román-Roy, Geometric Hamilton–Jacobi theory, Int. J. Geom. Meth. Mod. Phys. 3 (2006) 1417–1458; Geometric Hamilton–Jacobi theory for nonholonomic dynamical systems, Int. J. Geom. Meth. Mod. Phys. 7 (2010) 431–454] we showed that the Hamilton–Jacobi problem can be regarded as a way to describe a given dynamics on a phase space manifold in terms of a family of dynamics on a lower-dimensional manifold. We also showed how constants of the motion help to solve the Hamilton–Jacobi equation. Here we want to delve into this interpretation by considering the most general case: a dynamical system on a manifold that is described in terms of a family of dynamics (slicing vector fields) on lower-dimensional manifolds. We identify the relevant geometric structures that lead from this decomposition of the dynamics to the classical Hamilton–Jacobi theory, by considering special cases like fibered manifolds and Hamiltonian dynamics, in the symplectic framework and the Poisson one. We also show how a set of functions on a tangent bundle can determine a second-order dynamics for which they are constants of the motion.
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Dissertations / Theses on the topic "Jacobi manifolds"

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Arias, Marco Teresa. "Study of homogeneous DÀtri spaces, of the Jacobi operator on g.o. spaces and the locally homogeneous connections on 2-dimensional manifolds with the help of Mathematica." Doctoral thesis, Universitat de València, 2007. http://hdl.handle.net/10803/9954.

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Nowadays, the concept of homogeneity is one of the fundamental notions in geometry although its meaning must be always specified for the concrete situations. In this thesis, we consider the homogeneity of Riemannian manifolds and the homogeneity of manifolds equipped with affine connections. The first kind of homogeneity means that, for every smooth Riemannian manifold (M, g), its group of isometries I(M) is acting transitively on M. Part I of this thesis fits into this philosophy. Afterwards in Part II, we treat the homogeneity concept of affine connections. This homogeneity means that, for every two points of a manifold, there is an affine diffeomorphism which sends one point into another. In particular, we consider a local version of the homogeneity, that is, we accept that the affine diffeomorphisms are given only locally, i.e., from a neighborhood onto a neighborhood. More specifically, we devote the first Chapter of Part I to make a brief overview of some special kinds of homogeneous Riemannian manifolds which will be of special relevance in Part I and to show how the software MATHEMATICA© becomes useful. For that, we prove that "the three-parameter families of flag manifolds constructed by N. R. Wallach in "Compact homogeneous Riemannian manifols with strictly positive curvature, Ann. of Math. 96 (1972), p. 276-293" are D'Atri spaces if and only if they are naturally reductive spaces. Thus, we improve the previous results given by D'Atri, Nickerson and by Arias-Marco, Naveira.Moreover, in Chapter 2 we obtain the complete 4-dimensional classification of homogeneous spaces of type A. This allows us to prove correctly that every 4-dimensional homogeneous D'Atri space is naturally reductive. Therefore, we correct, complete and improve the results presented by Podestà, Spiro, Bueken and Vanhecke. Chapter 3 is devoted to prove that the curvature operator has constant osculating rank over g.o. spaces. It is mean that a real number 'r' exists such that under some assumptions, the higher order derivatives of the curvature operator from 1 to r are linear independent and from 1 to r + 1 are linear dependent. As a consequence, we also present a method valid on every g.o. space to solve the Jacobi equation. This method extends the method given by Naveira and Tarrío for naturally reductive spaces. In particular, we prove that the Jacobi operator on Kaplan's example (the first known g.o. space that it is not naturally reductive) has constant osculating rank 4. Moreover, we solve the Jacobi equation along a geodesic on Kaplan's example using the new method and the well-known method used by Chavel, Ziller and Berndt,Tricerri, Vanhecke. Therefore, we are able to present the main differences between both methods.In Part II, we classify (locally) all locally homogeneous affine connections with arbitrary torsion on two-dimensional manifolds. Therefore, we generalize the result given by Opozda for torsion-less case. Moreover, from our computations we obtain interesting consequences as the relation between the classifications given for the torsion less-case by Kowalski, Opozda and Vlá ek. In addition, we obtain interesting consequences about flat connections with torsion.In general, the study of these problems sometimes requires a big number of straightforward symbolic computations. In such cases, it is a quite difficult task and a lot of time consuming, try to make correctly this kind of computations by hand. Thus, we try to organize our computations in (possibly) most systematic way so that the whole procedure is not excessively long. Also, because these topics are an ideal subject for a computer-aided research, we are using the software MATHEMATICA©, throughout this work. But we put stress on the full transparency of this procedure.
En esta tesis, se consideran dos tipos bien diferenciados de homogeneidad: la de las variedades riemannianas y la de las variedades afines. El primer tipo de homogeneidad se define como aquel que tiene la propiedad de que el grupo de isometrías actúa transitivamente sobre la variedad. La Parte I, recoge todos los resultados que hemos obtenido en esta dirección. Sin embargo, en la Parte II se presentan los resultados obtenidos sobre conexiones afines homogéneas. Una conexión afín se dice homogénea si para cada par de puntos de la variedad existe un difeomorfismo afín que envía un punto en otro. En este caso, se considera una versión local de homogeneidad. Más específicamente, la Parte I de esta tesis está dedicada a probar que "las familias 3-paramétricas de variedades bandera construidas por Wallach son espacios de D'Atri si y sólo si son espacios naturalmente reductivos". Más aún, en el segundo Capítulo, se obtiene la clasificación completa de los espacios homogéneos de tipo A cuatro dimensionales que permite probar correctamente que todo espacio de D'Atri homogéneo de dimensión cuatro es naturalmente reductivo.Finalmente, en el tercer Capítulo se prueba que en cualquier g.o. espacio el operador curvatura tiene rango osculador constante y, como consecuencia, se presenta un método para resolver la ecuación de Jacobi sobre cualquier g.o. espacio. La Parte II se destina a clasificar (localmente) todas las conexiones afines localmente homogéneas con torsión arbitraria sobre variedades 2-dimensionales. Para finalizar el cuarto Capítulo, se prueban algunos resultados interesantes sobre conexiones llanas con torsión.En general, el estudio de estos problemas requiere a veces, un gran número de cálculos simbólicos aunque sencillos. En dichas ocasiones, realizarlos correctamente a mano es una tarea ardua que requiere mucho tiempo. Por ello, se intenta organizar todos estos cálculos de la manera más sistemática posible de forma que el procedimiento no resulte excesivamente largo. Este tipo de investigación es ideal para utilizar la ayuda del ordenador; así, cuando resulta conveniente, utilizamos la ayuda del software MATHEMATICA para desarrollar con total transparencia el método de resolución que más se adecua a cada uno de los problemas a resolver.
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Figalli, Alessio. "Optimal transportation and action-minimizing measures." Doctoral thesis, Scuola Normale Superiore, 2007. http://hdl.handle.net/11384/85683.

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Figalli, Alessio. "Optimal transportation and action-minimizing measures." Doctoral thesis, Lyon, École normale supérieure (sciences), 2007. http://www.theses.fr/2007ENSL0422.

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Larsson, Agnes. "Automatic Mesh Repair." Thesis, Linköpings universitet, Informationskodning, 2013. http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-98734.

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To handle broken 3D models can be a very time consuming problem. Several methods aiming for automatic mesh repair have been presented in the recent years. This thesis gives an extensive evaluation of automatic mesh repair algorithms, presents a mesh repair pipeline and describes an implemented automatic mesh repair algorithm. The presented pipeline for automatic mesh repair includes three main steps: octree generation, surface reconstruction and ray casting. Ray casting is for removal of hidden objects. The pipeline also includes a pre processing step for removal of intersecting triangles and a post processing step for error detection. The implemented algorithm presented in this thesis is a volumetric method for mesh repair. It generates an octree in which data from the input model is saved. Before creation of the output, the octree data will be patched to remove inconsistencies. The surface reconstruction is done with a method called Manifold Dual Contouring. First new vertices are created from the information saved in the octree. Then there is a possibility to cluster vertices together for decimation of the output. Thanks to a special Manifold criterion, the output is guaranteedto be manifold. Furthermore the output will have sharp and clear edges and corners thanks to the use of Singular Value Decomposition during determination of the positions of the new vertices.
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Sakamoto, Noboru, and der Schaft Arjan J. van. "An analytical approximation method for the stabilizing solution of the Hamilton-Jacobi equation based on stable manifold theory." IEEE, 2007. http://hdl.handle.net/2237/9430.

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Prieto, Martínez Pere Daniel. "Geometrical structures of higher-order dynamical systems and field theories." Doctoral thesis, Universitat Politècnica de Catalunya, 2014. http://hdl.handle.net/10803/284215.

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Geometrical physics is a relatively young branch of applied mathematics that was initiated by the 60's and the 70's when A. Lichnerowicz, W.M. Tulczyjew and J.M. Souriau, among many others, began to study various topics in physics using methods of differential geometry. This "geometrization" provides a way to analyze the features of the physical systems from a global viewpoint, thus obtaining qualitative properties that help us in the integration of the equations that describe them. Since then, there has been a strong development in the intrinsic treatment of a variety of topics in theoretical physics, applied mathematics and control theory using methods of differential geometry. Most of the work done in geometrical physics since its first days has been devoted to study first-order theories, that is, those theories whose physical information depends on (at most) first-order derivatives of the generalized coordinates of position (velocities). However, there are theories in physics in which the physical information depends explicitly on accelerations or higher-order derivatives of the generalized coordinates of position, and thus more sophisticated geometrical tools are needed to model them acurately. In this Ph.D. Thesis we pretend to give a geometrical description of some of these higher-order theories. In particular, we focus on dynamical systems and field theories whose dynamical information can be given in terms of a Lagrangian function, or a Hamiltonian that admits Lagrangian counterpart. More precisely, we will use the Lagrangian-Hamiltonian unified approach in order to develop a geometric framework for autonomous and non-autonomous higher-order dynamical system, and for second-order field theories. This geometric framework will be used to study several relevant physical examples and applications, such as the Hamilton-Jacobi theory for higher-order mechanical systems, relativistic spin particles and deformation problems in mechanics, and the Korteweg-de Vries equation and other systems in field theory.
La física geomètrica és una branca relativament jove de la matemàtica aplicada que es va iniciar als anys 60 i 70 qua A. Lichnerowicz, W.M. Tulczyjew and J.M. Souriau, entre molts altres, van començar a estudiar diversos problemes en física usant mètodes de geometria diferencial. Aquesta "geometrització" proporciona una manera d'analitzar les característiques dels sistemes físics des d'una perspectiva global, obtenint així propietats qualitatives que faciliten la integració de les equacions que els descriuen. D'ençà s'ha produït un fort desenvolupamewnt en el tractament intrínsic d'una gran varietat de problemes en física teòrica, matemàtica aplicada i teoria de control usant mètodes de geometria diferencial. Gran part del treball realitzat en la física geomètrica des dels seus primers dies s'ha dedicat a l'estudi de teories de primer ordre, és a dir, teories tals que la informació física depèn en, com a molt, derivades de primer ordre de les coordenades de posició generalitzades (velocitats). Tanmateix, hi ha teories en física en les que la informació física depèn de manera explícita en acceleracions o derivades d'ordre superior de les coordenades de posició generalitzades, requerint, per tant, d'eines geomètriques més sofisticades per a modelar-les de manera acurada. En aquesta Tesi Doctoral ens proposem donar una descripció geomètrica d'algunes d'aquestes teories. En particular, estudiarem sistemes dinàmics i teories de camps tals que la seva informació dinàmica ve donada en termes d'una funció lagrangiana, o d'un hamiltonià que prové d'un sitema lagrangià. Per a ser més precisos emprarem la formulació unificada Lagrangiana-Hamiltoniana per tal de desenvolupar marcs geomètrics per a sistemes dinàmics d'ordre superior autònoms i no autònoms, i per a teories de camps de segon ordre. Amb aquest marc geomètric estudiarem alguns exemples físics rellevants i algunes aplicacions, com la teoria de Hamilton-Jacobi per a sistemes mecànics d'ordre superior, partícules relativístiques amb spin i problemes de deformació en mecànica, i l'equació de Korteweg-de Vries i altres sistemes en teories de camps.
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Rizziolli, Elíris Cristina. ""Variedades de Thom-Boardman, ideais Jacobianos e singularidades de aplicações diferenciáveis"." Universidade de São Paulo, 2001. http://www.teses.usp.br/teses/disponiveis/55/55135/tde-27082003-095607/.

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Neste trabalho é desenvolvido um estudo sobre a relação entre as variedades de Thom-Boardman e os ideais jacobianos iterados associados a estas variedades. Inicialmente são estudadas as singularidades de Thom-Boardman associadas a germes de aplicações analíticas com a finalidade de introduzir as varidades de Thom-Boardman no espaço dos jatos. Posteriormente são estudados os ideais jacobianos extendidos, seguindo a construção de Morin. Finalmente é definida a multiplicidade c_i(f) associada a um símbolo de Boardman i=(i_1,...,i_k) e ao extrato (Sigma)^1(f).
In this work we study the relation between the Thom-Boardman manifolds and the iterated jacobian ideals associate to these manifolds. First, we study the Thom-Boardman singularities associate to analitic map germs with the objective to introduce Thom-Boardman manifolds in the jet space. After, we study the extended jacobians ideals, following Morin's construction. We give the definition of the mulitiplicity c_i(f) associate to a Boadman symbol i=(i_1,...,i_k) and the stratum (Sigma)^i(f).
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TORTORELLA, ALFONSO GIUSEPPE. "Deformations of coisotropic submanifolds in Jacobi manifolds." Doctoral thesis, 2017. http://hdl.handle.net/2158/1077777.

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In this thesis, we investigate deformation theory and moduli theory of coisotropic submanifolds in Jacobi manifolds. Originally introduced by Kirillov as local Lie algebras with one dimensional fibers, Jacobi manifolds encompass, unifying and generalizing, locally conformal symplectic manifolds, locally conformal Poisson manifolds, and non-necessarily coorientable contact manifolds. We attach an L-infinity-algebra to any coisotropic submanifold in a Jacobi manifold. Our construction generalizes and unifies analogous constructions by Oh-Park (symplectic case), Cattaneo-Felder (Poisson case), and Le-Oh (locally conformal symplectic case). As a completely new case we also associate an L-infinity-algebra with any coisotropic submanifold in a contact manifold. The L-infinity-algebra of a coisotropic submanifold S controls the formal coisotropic deformation problem of S, even under Hamiltonian equivalence, and provides criteria both for the obstructedness and for the unobstructedness at the formal level. Additionally we prove that if a certain condition ("fiberwise entireness") is satisfied then the L-infinity-algebra controls the non-formal coisotropic deformation problem, even under Hamiltonian equivalence. We associate a BFV-complex with any coisotropic submanifold in a Jacobi manifold. Our construction extends an analogous construction by Schatz in the Poisson setting, and in particular it also applies in the locally conformal symplectic/Poisson setting and the contact setting. Unlike the L-infinity-algebra, we prove that, with no need of any restrictive hypothesis, the BFV-complex of a coisotropic submanifold S controls the non-formal coisotropic deformation problem of S, even under both Hamiltonian equivalence and Jacobi equivalence. Notwithstanding the differences there is a close relation between the approaches to the coisotropic deformation problem via L-infinity-algebra and via BFV-complex. Indeed both the L-infinity-algebra and the BFV-complex of a coisotropic submanifold S provide a cohomological reduction of S. Moreover they are L-infinity quasi-isomorphic and so they encode equally well the moduli space of formal coisotropic deformations of S under Hamiltonian equivalence. In addition we exhibit two examples of coisotropic submanifolds in the contact setting whose coisotropic deformation problem is obstructed at the formal level. Further we provide a conceptual explanation of this phenomenon both in terms of the L-infinity-algebra and in terms of the BFV-complex.
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Books on the topic "Jacobi manifolds"

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Southeast Geometry Seminar (15th 2009 University of Alabama at Birmingham). Geometric analysis, mathematical relativity, and nonlinear partial differential equations: Southeast Geometry Seminars Emory University, Georgia Institute of Technology, University of Alabama, Birmingham, and the University of Tennessee, 2009-2011. Edited by Ghomi Mohammad 1969-. Providence, Rhode Island: American Mathematical Society, 2013.

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Topology and geometry in dimension three: Triangulations, invariants, and geometric structures : conference in honor of William Jaco's 70th birthday, June 4-6, 2010, Oklahoma State University, Stillwater, OK. Providence, R.I: American Mathematical Society, 2011.

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Carlson, James. Period Domains and Period Mappings. Edited by Eduardo Cattani, Fouad El Zein, Phillip A. Griffiths, and Lê Dũng Tráng. Princeton University Press, 2017. http://dx.doi.org/10.23943/princeton/9780691161341.003.0004.

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This chapter seeks to develop a working understanding of the notions of period domain and period mapping, as well as familiarity with basic examples thereof. It first reviews the notion of a polarized Hodge structure H of weight n over the integers, for which the motivating example is the primitive cohomology in dimension n of a projective algebraic manifold of the same dimension. Next, the chapter presents lectures on period domains and monodromy, as well as elliptic curves. Hereafter, the chapter provides an example of period mappings, before considering Hodge structures of weight. After expounding on Poincaré residues, this chapter establishes some properties of the period mapping for hypersurfaces and the Jacobian ideal and the local Torelli theorem. Finally, the chapter studies the distance-decreasing properties and integral manifolds of the horizontal distribution.
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Mann, Peter. The Hamiltonian & Phase Space. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198822370.003.0014.

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This chapter discusses the Hamiltonian and phase space. Hamilton’s equations can be derived in several ways; this chapter follows two pathways to arrive at the same result, thus giving insight into the motivation for forming these equations. The importance of deriving the same result in several ways is that it shows that, in physics, there are often several mathematical avenues to go down and that approaching a problem with, say, the calculus of variations can be entirely as valid as using a differential equation approach. The chapter extends the arenas of classical mechanics to include the cotangent bundle momentum phase space in addition to the tangent bundle and configuration manifold, and discusses conjugate momentum. It also introduces the Hamiltonian as the Legendre transform of the Lagrangian and compares it to the Jacobi energy function.
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Book chapters on the topic "Jacobi manifolds"

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Marle, Charles-Michel. "On Jacobi Manifolds and Jacobi Bundles." In Mathematical Sciences Research Institute Publications, 227–46. New York, NY: Springer US, 1991. http://dx.doi.org/10.1007/978-1-4613-9719-9_16.

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Yang, Jae-Hyun. "Geometry and Arithmetic on the Siegel–Jacobi Space." In Geometry and Analysis on Manifolds, 275–325. Cham: Springer International Publishing, 2015. http://dx.doi.org/10.1007/978-3-319-11523-8_10.

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Berceanu, Stefan. "A Useful Parametrization of Siegel–Jacobi Manifolds." In Geometric Methods in Physics, 99–108. Basel: Springer Basel, 2013. http://dx.doi.org/10.1007/978-3-0348-0645-9_8.

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Berceanu, Stefan. "Classical and Quantum Evolution on the Siegel-Jacobi Manifolds." In Geometric Methods in Physics, 43–52. Basel: Springer Basel, 2012. http://dx.doi.org/10.1007/978-3-0348-0448-6_3.

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Jackson, David M., and Iain Moffatt. "Jacobi Diagrams on a 1-Manifold." In CMS Books in Mathematics, 293–308. Cham: Springer International Publishing, 2019. http://dx.doi.org/10.1007/978-3-030-05213-3_16.

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Caseiro, R., and J. M. Nunes da Costa. "Integrable Systems and Recursion Operators on Symplectic and Jacobi Manifolds." In Encyclopedia of Mathematical Physics, 87–93. Elsevier, 2006. http://dx.doi.org/10.1016/b0-12-512666-2/00466-1.

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Beris, Antony N., and Brian J. Edwards. "Introduction." In Thermodynamics of Flowing Systems: with Internal Microstructure. Oxford University Press, 1994. http://dx.doi.org/10.1093/oso/9780195076943.003.0005.

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The investigation of dynamical phenomena in gases, liquids, and solids has attracted the interest of physicists, chemists, and engineers from the very beginning of the modern science. The early work on transport phenomena focussed on the description of ideal flow behavior as a natural extension to the dynamical behavior of a collection of discrete particles, which dominated so much of the classical mechanics of the last century. As far back as 1809, the mathematical techniques which later came to be known as Hamiltonian mechanics began to emerge, as well as an appreciation of the inherent symmetry and structure of the mathematical forms embodied by the Poisson bracket. It was in this year that S. D. Poisson introduced this celebrated bracket [Poisson, 1809, p. 281], and in succeeding years that such famous scholars as Hamilton, Jacobi, and Poincaré laid the foundation for classical mechanics upon the earlier bedrock of Euler, Lagrange, and d'Alembert. This surge of interest in Hamiltonian mechanics continues well into the waning years of the twentieth century, where scholars are just beginning to realize the wealth of information to be gained through the use of such powerful analytic tools as the Hamiltonian/Poisson formalism and the development of symplectic methods on differential manifolds. Specifically, the study of the dynamics of ideal continua, which is analogous to the discrete particle dynamics studied by Hamilton, Jacobi, and Poisson, has recently benefited significantly by the adaptation of the equations of motion into Hamiltonian form. The inherent structure and symmetry of this form of the equations is particularly well suited for many mathematical analyses which are extremely difficult when conducted in terms of the standard forms of the dynamical equations, for instance, stability and perturbation analyses of ideal fluid flows. Thus, classical mechanics and its outgrowth, continuum mechanics, seem to be on the verge of some major developments. Yet, further progress in this area was hindered by the fact that the traditional form of the Hamiltonian structure can only describe conservative systems, thus placing a severe constraint on the applicability of these mathematically elegant and computationally powerful techniques to real systems.
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Ehrenschwendtner, Marie-Luise. "Devoted Episcopalians, Reluctant Jacobites? George and James Garden and their Spiritual Environment." In Scottish Liturgical Traditions and Religious Politics, 138–53. Edinburgh University Press, 2021. http://dx.doi.org/10.3366/edinburgh/9781474483056.003.0010.

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This chapter examines the spirituality and religious environment of the brothers James and George Garden as theologians from Aberdeen; explores whether their religious convictions tie in with their political persuasions; and whether the religious or the political components decided their actions and loyalties. Both brothers had manifold links to Jacobite circles with interests in mystical literature, and the younger one, George, was openly spreading the ideas of the Flemish mystic Antoinette Bourignon. Episcopacy did not automatically equate to Jacobitism and some non-jurors in their emphatic rejection of Calvinism were given more to pietism than political action. The brothers moved in the direction of Flemish mysticism. Others flirted with Coptic Christianity or Greek and Russian Orthodoxy.
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"The Patriarch and His Manifold Descendants: Jacob as Visionary between Jews and Christians in the Apocryphal Ladder of Jacob." In The Embroidered Bible: Studies in Biblical Apocrypha and Pseudepigrapha in Honour of Michael E. Stone, 237–49. BRILL, 2017. http://dx.doi.org/10.1163/9789004357211_016.

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"Construction of Jacobian functions of a given type. Theta functions and Abelian functions. Abelian and Picard manifolds." In Translations of Mathematical Monographs, 115–52. Providence, Rhode Island: American Mathematical Society, 2006. http://dx.doi.org/10.1090/mmono/096/04.

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Conference papers on the topic "Jacobi manifolds"

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Xu, Xiaoqiang, Shikui Chen, Xianfeng David Gu, and Michael Yu Wang. "Conformal Topology Optimization of Heat Conduction Problems on Manifolds Using an Extended Level Set Method (X-LSM)." In ASME 2021 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. American Society of Mechanical Engineers, 2021. http://dx.doi.org/10.1115/detc2021-67819.

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Abstract In this paper, the authors propose a new dimension reduction method for level-set-based topology optimization of conforming thermal structures on free-form surfaces. Both the Hamilton-Jacobi equation and the Laplace equation, which are the two governing PDEs for boundary evolution and thermal conduction, are transformed from the 3D manifold to the 2D rectangular domain using conformal parameterization. The new method can significantly simplify the computation of topology optimization on a manifold without loss of accuracy. This is achieved due to the fact that the covariant derivatives on the manifold can be represented by the Euclidean gradient operators multiplied by a scalar with the conformal mapping. The original governing equations defined on the 3D manifold can now be properly modified and solved on a 2D domain. The objective function, constraint, and velocity field are also equivalently computed with the FEA on the 2D parameter domain with the properly modified form. In this sense, we are solving a 3D topology optimization problem equivalently on the 2D parameter domain. This reduction in dimension can greatly reduce the computing cost and complexity of the algorithm. The proposed concept is proved through two examples of heat conduction on manifolds.
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VIDEV, VESELIN, and MARIA VANOVA. "CHARACTERIZATION OF A FOUR-DIMENSIONAL RIEMANNIAN MANIFOLDS WITH COMMUTING STANILOV CURVATURE OPERATOR WITH RESPECT TO ORTHOGONAL PLANE." In INTERNATIONAL SCIENTIFIC CONFERENCE MATHTECH 2022. Konstantin Preslavsky University Press, 2022. http://dx.doi.org/10.46687/lqcr1576.

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In the present paper using commuting conditions of the skew-symmetric Stanilov curvature operator and the generalized Jacobi operator of order 2 defined with respect to orthogonal plane we characterize a four-dimensional Riemannian manifold of constant sectional curvature
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Wang, Deshi, Renbin Xiao, and Ming Yang. "The Attitude Stability for Longitudinal Motion of Underwater Vehicle." In ASME 2001 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. American Society of Mechanical Engineers, 2001. http://dx.doi.org/10.1115/detc2001/vib-21607.

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Abstract Although the equations describing the longitudinal motions of underwater vehicles are typically nonlinear, the linearized equations are still employed to design the depth controller by the traditional analysis methods in engineering for the sake of simplicity. The reduction of the nonlinearity loses the dynamics near the singular points, which may be responsible for the sudden climb or dive. The nonlinear systems limited in the longitudinal plane of the underwater vehicles are analyzed on center manifold through the bifurcation theory. It focuses on the case that single zero root in Jacobi matrix occurs at equilibrium points corresponding to nominal trajectory with varied angles of the elevator or the direction change of the flows. The center manifolds are calculated and one-dimensional bifurcation equations on the center manifolds are obtained and analyzed. Based on the transcritical bifurcation diagram, we have found the mechanism of the attitude stability loss as well as the abnormal trajectory of autonomous underwater vehicles. It gives good explainations to the practical climbing jump and diving fall and delivers the theoretical tools to design the controller and to design dynamics. Numerical simulation verifies the results.
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Nusawardhana, Antonius, and Stanislaw H. Zak. "Optimality of Synergetic Controllers." In ASME 2006 International Mechanical Engineering Congress and Exposition. ASMEDC, 2006. http://dx.doi.org/10.1115/imece2006-14839.

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Optimality properties of synergetic controllers are analyzed using the Euler-Lagrange conditions and the Hamilton-Jacobi-Bellman equation. First, a synergetic control strategy is compared with the variable structure sliding mode control. The synergetic control design methodology turns out to be closely related to the methods of variable structure sliding mode control. In fact, the method of sliding surface design from the sliding mode control are essential for designing similar manifolds in the synergetic control approach. It is shown that the synergetic control strategy can be derived using tools from the calculus of variations. The synergetic control laws have simple structure because they are derived from the associated first-order differential equation. It is also shown that the synergetic controller for a certain class of linear quadratic optimal control problems has the same structure as the one generated using the linear quadratic regulator (LQR) approach by solving the associated Riccati equation.
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Ye, Qian, Yang Guo, Shikui Chen, Xianfeng David Gu, and Na Lei. "Topology Optimization of Conformal Structures Using Extended Level Set Methods and Conformal Geometry Theory." In ASME 2018 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. American Society of Mechanical Engineers, 2018. http://dx.doi.org/10.1115/detc2018-85655.

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In this paper, we propose a new method to approach the problem of structural shape and topology optimization on manifold (or free-form surfaces). A manifold is conformally mapped onto a 2D rectangle domain, where the level set functions are defined. With conformal mapping, the corresponding covariant derivatives on a manifold can be represented by the Euclidean differential operators multiplied by a scalar. Therefore, the topology optimization problem on a free-form surface can be formulated as a 2D problem in the Euclidean space. To evolve the boundaries on a free-form surface, we propose a modified Hamilton-Jacobi equation and solve it on a 2D plane following the conformal geometry theory. In this way, we can fully utilize the conventional level-set-based computational framework. Compared with other established approaches which need to project the Euclidean differential operators to the manifold, the computational difficulty of our method is highly reduced while all the advantages of conventional level set methods are well preserved. We hope the proposed computational framework can provide a timely solution to increasing applications involving innovative structural designs on free-form surfaces in different engineering fields.
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Baron, Luc, and Ghislain Bernier. "The Design of Parallel Manipulators of Star Topology Under Isotropic Constraint." In ASME 2001 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. American Society of Mechanical Engineers, 2001. http://dx.doi.org/10.1115/detc2001/dac-21025.

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Abstract This paper proposes a necessary and sufficient number of 18 geometrical parameters allowing to describe the design manifold of the Star topological class, i.e., all geometries of manipulators having the same topology and mobility as the Y-Star parallel manipulator. The isotropic constraints are then applied on this manifold in order to define the constraint manifold of isotropic designs, i.e., those having isotropic Jacobian matrices at their home position. This constraint manifold is of dimension 11 and greatly facilitates the design of isotropic manipulators in this topological class.
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Abdel-Malek, K., Walter Seaman, and Harn-Jou Yeh. "An Exact Method for NC Verification of up to 5-Axis Machining." In ASME 1999 Design Engineering Technical Conferences. American Society of Mechanical Engineers, 1999. http://dx.doi.org/10.1115/detc99/dac-8560.

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Abstract The motion of a cutter tool is modeled as a surface undergoing a sweep operation along another geometric entity. A numerically controlled machining verification method is developed based on a formulation for delineating the volume generated by the motion of a cutting tool on the workpiece (stock). Varieties and subvarieties that are subsets of some Eucledian space defined by the zeros of a finite number of analytic functions are computed and are characterized as closed form equations of surface patches of this volume. A topological space describing the swept volume will be built as a stratified manifold with corners. Singularities of the variety are loci of points where the Jacobian of the manifold has lower rank than maximal. It is shown that varieties appearing inside the manifold representing the removed material are due to a lower degree strata of the Jacobian. Some of the varieties are complicated (so as not to confuse with varieties in complex Cn) and will be shown to be reducible because of their parametrization and are addressed. Benefits of this method are evident in its ability to depict the manifold and to compute a value for the volume.
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Li, Ju, and J. Michael McCarthy. "Singularity Variety of a 3SPS-1S Spherical Parallel Manipulator." In ASME 2016 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. American Society of Mechanical Engineers, 2016. http://dx.doi.org/10.1115/detc2016-60416.

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In this paper, we study the manifold of configurations of a 3SPS-1S spherical parallel manipulator. This manifold is obtained as the intersection of quadrics in the hypersphere defined by quaternion coordinates and is called its constraint manifold. We then formulate Jacobian for this manipulator and consider its singular. This is a quartic algebraic manifold called the singularity variety of the parallel manipulator. A survey of the architectures that can be defined for the 3SPS-1S spherical parallel manipulators yield a number of special cases, in particular the architectures with coincident base or moving pivots yields singularity varieties that factor into two quadric surfaces.
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Sakamoto, Noboru, and Arjan J. van der Schaft. "An analytical approximation method for the stabilizing solution of the Hamilton-Jacobi equation based on stable manifold theory." In 2007 American Control Conference. IEEE, 2007. http://dx.doi.org/10.1109/acc.2007.4282581.

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Butuk, Nelson, and JeanPaul Pemba. "Computing CHEMKIN Sensitivities Using Complex Variables." In ASME 2001 Engineering Technology Conference on Energy. American Society of Mechanical Engineers, 2001. http://dx.doi.org/10.1115/etce2001-17013.

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Abstract This paper discusses an accurate numerical approach of computing the Jacobian Matrix for the calculation of low dimensional manifolds for kinetic chemical mechanism reduction. The approach is suitable for numerical computations of large scale problems and is more accurate than the finite difference approach of computing Jacobians. The method is demonstrated via a highly stiff reaction mechanism for the synthesis of Bromide acid and a H2/Air mechanism using a modified CHEMKIN package. The Bromide mechanism consisted of five species participating in six elementary chemical reactions and the H2/Air mechanism consisted of 11 species and 23 reactions. In both cases it is shown that the method is superior to the finite difference approach of computing derivatives with an arbitrary computational step size, h.
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