Relevant bibliographies by topics / Transverse invariants

Contents

  1. Journal articles
  2. Dissertations / Theses
  3. Book chapters
  4. Conference papers

Academic literature on the topic 'Transverse invariants'

Author: Grafiati
Published: 10 March 2023

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Journal articles on the topic "Transverse invariants"

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Collari, Carlo. "Transverse invariants from Khovanov-type homologies." Journal of Knot Theory and Its Ramifications 28, no. 01 (January 2019): 1950012. http://dx.doi.org/10.1142/s0218216519500123.

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In this paper, we introduce a family of transverse invariants arising from the deformations of Khovanov homology. This family includes the invariants introduced by Plamenevskaya and by Lipshitz, Ng, and Sarkar. Then, we investigate the invariants arising from Bar-Natan’s deformation. These invariants, called [Formula: see text]-invariants, are essentially equivalent to Lipshitz, Ng, and Sarkar’s invariants [Formula: see text]. From the [Formula: see text]-invariants, we extract two non-negative integers which are transverse invariants (the [Formula: see text]-invariants). Finally, we give several conditions which imply the non-effectiveness of the [Formula: see text]-invariants, and use them to prove several vanishing criteria for the Plamenevskaya invariant [Formula: see text], and the non-effectiveness of the vanishing of [Formula: see text], for all prime knots with less than 12 crossings.
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López Machí, Rafael, and José Martínez Alfaro. "Invariants of transverse foliations." Topology and its Applications 159, no. 2 (February 2012): 519–25. http://dx.doi.org/10.1016/j.topol.2011.09.027.

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Lisca, Paolo, and András I. Stipsicz. "Contact surgery and transverse invariants." Journal of Topology 4, no. 4 (October 25, 2011): 817–34. http://dx.doi.org/10.1112/jtopol/jtr022.

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GRANT, MARK. "ON SELF-INTERSECTION INVARIANTS." Glasgow Mathematical Journal 55, no. 2 (August 2, 2012): 259–73. http://dx.doi.org/10.1017/s0017089512000481.

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AbstractWe prove that the Hatcher–Quinn and Wall invariants of a self-transverse immersion f: Nn ↬ M2n coincide. That is, we construct an isomorphism between their target groups, which carries one onto the other. We also employ methods of normal bordism theory to investigate the Hatcher–Quinn invariant of an immersion f: Nn ↬ M2n−1.
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DING, FAN, and HANSJÖRG GEIGES. "LEGENDRIAN KNOTS AND LINKS CLASSIFIED BY CLASSICAL INVARIANTS." Communications in Contemporary Mathematics 09, no. 02 (April 2007): 135–62. http://dx.doi.org/10.1142/s0219199707002381.

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It is shown that Legendrian (respectively transverse) cable links in S3 with its standard tight contact structure, i.e. links consisting of an unknot and a cable of that unknot, are classified by their oriented link type and the classical invariants (Thurston–Bennequin invariant and rotation number in the Legendrian case, self-linking number in the transverse case). The analogous result is proved for torus knots in the 1-jet space J1(S1) with its standard tight contact structure.
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Plamenevskaya, Olga. "Braid monodromy, orderings and transverse invariants." Algebraic & Geometric Topology 18, no. 6 (October 18, 2018): 3691–718. http://dx.doi.org/10.2140/agt.2018.18.3691.

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Lipshitz, Robert, Lenhard Ng, and Sucharit Sarkar. "On transverse invariants from Khovanov homology." Quantum Topology 6, no. 3 (2015): 475–513. http://dx.doi.org/10.4171/qt/69.

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Ito, Tetsuya. "Braids, chain of Yang–Baxter like operations, and (transverse) knot invariants." Journal of Knot Theory and Its Ramifications 27, no. 11 (October 2018): 1843009. http://dx.doi.org/10.1142/s0218216518430095.

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We introduce a notion of a chain of Yang–Baxter like operations. This is a sequence of solutions of an asymmetric variant of the Yang–Baxter equation and is a multi-operator generalization of (bi)rack/quandles. We discuss knot and link invariants coming from a chain of Yang–Baxter like operations, and give potential applications. Among them, we define a cocycle invariant for transverse links.
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CALSAMIGLIA, GABRIEL, and YOHANN GENZMER. "Classification of regular dicritical foliations." Ergodic Theory and Dynamical Systems 37, no. 5 (March 23, 2016): 1443–79. http://dx.doi.org/10.1017/etds.2015.123.

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In this paper we give complete analytic invariants for the set of germs of holomorphic foliations in $(\mathbb{C}^{2},0)$ that become regular after a single blow-up. Some of the invariants describe the holonomy pseudogroup of the germ and are called transverse invariants. The other invariants lie in a finite dimensional complex vector space. Such singularities admit separatrices tangentially to any direction at the origin. When enough separatrices are leaves of a radial foliaton (a condition that can always be attained if the multiplicity of the germ at the origin is at most four) we are able to describe and realize all the analytical invariants geometrically and provide analytic normal forms. As a consequence, we prove that any two such germs sharing the same transverse invariants are conjugated by a very particular type of birational transformation. We also provide explicit examples of universal equisingular unfoldings of foliations that cannot be produced by unfolding functions. With these at hand we are able to explicitly parametrize families of analytically distinct foliations that share the same transverse invariants.
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Freed, Alan D. "Transverse-Isotropic Elastic and Viscoelastic Solids." Journal of Engineering Materials and Technology 126, no. 1 (January 1, 2004): 38–44. http://dx.doi.org/10.1115/1.1631030.

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A set of invariants are presented for transverse-isotropic materials whose gradients produce strain fields, instead of deformation fields as is typically the case. Finite-strain theories for elastic and K-BKZ-type viscoelastic solids are derived. Shear-free and simple shearing deformations are employed to illustrate the constitutive theory.
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More sources

Dissertations / Theses on the topic "Transverse invariants"

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Tosun, Bulent. "Legendrian and transverse knots and their invariants." Diss., Georgia Institute of Technology, 2012. http://hdl.handle.net/1853/44880.

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In this thesis, we study Legendrian and transverse isotopy problem for cabled knot types. We give two structural theorems to describe when the (r,s)- cable of a Legendrian simple knot type K is also Legendrian simple. We then study the same problem for cables of the positive trefoil knot. We give a complete classification of Legendrian and transverse cables of the positive trefoil. Our results exhibit many new phenomena in the structural understanding of Legendrian and transverse knots. we then extend these results to the other positive torus knots. The key ingredient in these results is to find necessary and sufficient conditions on maximally thickened contact neighborhoods of the positive torus knots in three sphere.
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Hill, Jonathan William. "Invariants of legendrian curves and transverse knots." Thesis, University of Liverpool, 1999. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.367639.

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Song, Min Jae. "Direct tensor expression by Eulerian approach for constitutive relations based on strain invariants in transversely isotropic green elasticity - finite extension and torsion." [College Station, Tex. : Texas A&M University, 2006. http://hdl.handle.net/1969.1/ETD-TAMU-1667.

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Parejas, Jorge Luis Crisostomo. "Medidas transversas, correntes e sistemas dinâmicos." Universidade de São Paulo, 2013. http://www.teses.usp.br/teses/disponiveis/55/55135/tde-20032013-160120/.

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Neste trabalho, fazemos um estudo das correntes e das medidas transversas invariantes por holonomia, e mostraremos o resultado de D. Sullivan [23] sobre a correspondência biunívoca entre estes dois objetos. Em particular mostraremos um resultado conhecido de J. Plante [17] sobre a existência de medidas transversas invariantes sob a hipótese de crescimento sub-exponencial. Apresentamos também, o resultado devido a Ruelle-Sullivan [19] de que a medida de máxima entropia de um difeomorfismo topologicamente mixing pode-se expressar como o produto de duas medidas transversas invariantes para as folheações estáveis e instáveis. Por último, mostramos que os difeomorfismos de Anosov topologicamente mixing, que preservam a orientação das folhas estáveis e folhas instáveis induzem elementos da cohomologia de DeRham
In this work, we make a study of currents and holonomy invariant transverse measure, and we will show the result of D. Sullivan [23] about the biunivocal correspondence between these two objects. In particular we show a known result of J. Plante [17] about the existence of invariant transverse measures under the hypothesis of sub-exponential growth. Also we will present, the result due to Ruelle-Sullivan [19] that the maximum entropy measure of a diffeomorphism topologically mixing can be expressed as the product of two invariant transverse measures for stable and unstable foliations. Finally, we show that the Anosov diffeomorphisms topologically mixing, which preserve the orientation of the leaves stable and unstable induce elements DeRham cohomology
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Blyuss, Kostyantyn B. "Perturbed multi-symplectic systems : intersections of invariant manifolds and transverse instability." Thesis, University of Surrey, 2004. http://epubs.surrey.ac.uk/843502/.

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This thesis deals with two aspects of dynamics in systems described by multi-symplectic partial differential equations. The first part is devoted to the study of heteroclinic intersections in systems which govern the dynamics of travelling waves in multi-symplectic partial differential equations with perturbations. In this study a version of the Melnikov method is developed which takes into account the symmetry of the systems under consideration. The presence of the symmetry leads to various interesting differences between the method we develop and the standard approach. In particular, a result about persistence of the fixed point of the Poincare map under perturbations has to be amended since the unperturbed fixed point is non-hyperbolic. The symmetry also results in the necessity to consider separately two cases: when the perturbation has no component in the group direction at all, or when on average it has no component in the group direction when evaluated on the unperturbed solutions. For each of those cases we discuss persistence of the fixed points of the Poincare map and persistence of invariant manifolds, where the knowledge of the symmetry in incorporated in the geometrical constructions. Finally, we derive Melnikov-type conditions in both aforementioned cases which guarantee the existence of transverse intersections of the stable and unstable manifolds. We discuss some possible areas of applications of the Melnikov-type method derived and illustrate the method on the examples of a perturbed Korteweg-de Vries equation and a perturbed nonlinear Schrodinger equation. Implications of the transverse or topological intersections of the manifolds for possible chaotic behaviour in the systems are discussed together with directions of further investigation. The second part of this thesis considers the stability of solitary waves with respect to perturbations which are transverse to the basic direction of propagation of these waves. Using various analytical and numerical techniques, we study this problem for the solitary waves of the (2+l)-dimensional Boussinesq equation and the generalised fifth-order Kadomtsev-Petviashvili equation. For both equations we use a geometric condition for transverse instability based on the multi-symplectic formulation of the equation to derive a condition for transverse instability in the long-wavelength regime. Then an Evans function approach is employed to determine the dependence of the instability growth rate on the transverse wavenumber for all possible wavenumbers. In the case of the (2+l)-dimensional Boussinesq equation this is done analytically, while for the generalised fifth-order Kadomtsev-Petviashvili equation we have to resort to numerical simulations. Finally, for the (2+l)-dimensional Boussinesq equation we perform direct numerical simulations of the full equation to investigate the nonlinear stage of the evolution of the transversely unstable solitary waves, and the result is that the instability leads to the collapse of the solitary wave. The thesis is concluded by a discussion of some open problems.
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Pereira, Rodrigo Frehse. "Perturbações em sistemas com variabilidade da dimensão instável transversal." UNIVERSIDADE ESTADUAL DE PONTA GROSSA, 2013. http://tede2.uepg.br/jspui/handle/prefix/902.

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Made available in DSpace on 2017-07-21T19:26:04Z (GMT). No. of bitstreams: 1 Rodrigo Frehse Pereira.pdf: 4666622 bytes, checksum: b2dcf2959eef9f7fd82301c2e45ac87f (MD5) Previous issue date: 2013-03-01
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Unstable dimension variability (UDV) is an extreme form of nonhyperbolicity. It is a structurally stable phenomenon, typical for high dimensional chaotic systems, which implies severe restrictions to shadowing of perturbed solutions. Perturbations are unavoidable in modelling Physical phenomena, since no system can be made completely isolated, states and parameters cannot be determined without uncertainties and any numeric approach to such models is affected by truncation and/or roundoff errors. Thus, the lack of shadowability in systems exhibiting UDV presents a challenge for modelling. Aiming to unveil the effect of perturbations a class of nonhyperbolic systems is studied. These systems present transversal unstable dimension variability (TUDV), which means the dynamics can be split in a skew direct product form, i. e. the phase space is decomposed in two components: a hyperbolic chaotic one, called longitudinal, and a nonhyperbolic transversal one. Moreover, in the absence of perturbations, the longitudinal component is a global attractor of the system. A prototype composed of two coupled piecewise-linear chaotic maps is presented in order to study the TUDV effects. This system has an invariant subspace S which characterizes the complete chaos synchronization and UDV, when present, is transversal to it. Taking advantage of (piecewise) linearity of the equations, an analytical method for unstable periodic orbits’ computation is presented. The set of all unstable periodic orbits (UPOs) is one of the building block of chaotic dynamics and its properties provide valuable informations about the asymptotic behaviour of the system as, for instance, the invariant natural measure. Therefore, the TUDV’s intensity is analytically studied by computing the contrast measure, which quantifies the difference between the statistical weights associated to UPOs with different unstable dimension. The effect of perturbations is modelled by the introduction of a small parameter mismatch, instead of noise addition, in order to keep the model’s determinism. Consequently, the characterization of dynamics by means of UPOs is still possible. It is shown the existence of a dense set G of UPOs outside the invariant subspace consistent with a chaotic repeller. When perturbation takes place, G merges with the set H of UPOs previously in S, given rise to a new nonhyperbolic stationary state. The analysis of G ∪H provides a topological explanation to the behaviour of systems with TUDV under perturbations. Moreover, the relation between the set of UPOs embedded in a chaotic attractor and its natural measure, proven only for hyperbolic systems, is successfully applied to this system: the error between the natural measure estimated both numerically and by means of UPOs is shown to be decreasing with p, the considered UPOs’ period. It is conjectured the coincidence between both in limit. Hence, a positive answer to reliability of numerical estimation to natural measure in nonhyperbolic systems via unstable dimension variability is presented.
A variabilidade da dimensão instável (VDI) é uma forma extrema de não-hiperbolicidade. É um fenômeno estruturalmente estável, típico para sistemas caóticos de alta dimensionalidade, que implica restrições severas ao sombreamento de soluções perturbadas. As perturbações¸ s são inevitáveis na modelagem de fenômenos fíısicos, uma vez que nenhum sistema pode ser isolado completamente, os estados e os parâmetros não podem ser determinados sem incertezas e qualquer abordagem numérica dos modelos é afetada por erros de arredondamento e/ou truncamento. Portanto, a falta da sombreabilidade em sistemas exibindo VDI apresenta um desafio à modelagem. Visando revelar os efeitos das perturbações, uma classe desses sistemas não hiperbó licos é estudada. Esses sistemas apresentam variabilidade da dimensão instável transversal (VDIT), significando que a dinâmica pode ser decomposta na forma de um produto direto assimétrico, i. e. o espação de fase é dividido em dois componentes: um hiperbólico e caótico, dito longitudinal, e um transversal e não-hiperbólico. Mais ainda, na ausência de perturbações, o componente longitudinal é um atrator global do sistema. Um protótipo composto de dois mapas ca´oticos lineares por partes acoplados é apresentado para o estudo dos efeitos da VDIT. Esse sistema possui um subespaço invariante S que caracteriza a sincronização completa de caos e a VDI, quando presente, é transversal a esse subespaço. Valendo-se da linearidade (por partes) das equações, um método analítico para o cálculo das órbitas periódicas instáveis é apresentado. O conjunto de todas as órbitas periódicas instáveis (OPIs) é um dos fundamentos da dinâmica caótica e suas propriedades fornecem informaões, valiosas sobre o comportamento assintótico do sistema como, por exemplo, a medida natural invariante. Assim, a intensidade da VDIT é estudada analiticamente pelo cálculo da medida de contraste, que quantifica a diferença entre o peso estatístico associado às OPIs com dimensão instável distintas. O efeito das perturbações é modelado pela introdução de um pequeno desvio nos parâmetros, ao invés da adição de ruído, a fim de manter o determinismo do modelo. Consequentemente, a caracterização da dinâmica em termos das OPIs ainda é possível. Demonstra-se a existência de um conjunto denso G de OPIs fora do subespaço invariante consistente com um repulsor caótico. Na presença de perturbações, G se funde com o conjunto H das OPIs previamente em S, dando origem a um novo estado estacionario não-hiperbólico. A análise de G ∪H fornece uma explicação topológica ao comportamento de sistemas com variabilidade da dimensão instável sob a açãoo de perturbações. Mais ainda, a relação entre o conjunto de OPIs imersas em um atrator caótico e sua medida natural, provada apenas para sistemas hiperbólicos, é aplicada com sucesso nesse sistema: mostra-se que o erro entre as medidas naturais estimadas numericamente e pelas OPIs é decrescente com p, o período das OPIs consideradas. Conjectura-se, portanto, a coincidência entre ambas no limite . Logo, apresenta-se uma resposta positiva à estimativa numérica da medida natural em sistemas não-hiperbólicos via variabilidade da dimensão instável.
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Moussa, Miled Hassan Youssef. "Eestudo do crossover no modelo XY com campo transverso." Universidade de São Paulo, 1990. http://www.teses.usp.br/teses/disponiveis/54/54131/tde-06092007-093907/.

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Em decorrência do grande avanço alcançado pela mecânica estatística devido a introdução das idéias de invariância conforme às teorias de escala para sistemas finitos, retomamos, neste trabalho, o estudo do modelo XY em campo transverso. A princípio, apresentamos uma análise detalhada do comportamento \"crossover\" característico do modelo, onde incluímos cálculos melhorados dos expoentes da susceptibilidade e do gap de energia anteriormente apresentados por dos Santos e Stinch-combe. Em seguida, uma análise numérica do espectro foi desenvolvida, considerando-se condições livres de contorno, e comparada com as previsões da invariância conforme. Finalmente, as correções à energia do estado fundamental de cadeias finitas foram utilizadas para obter o parâmetro que caracteriza as classes de universalidade (a carga central c).
In view of the great advance attached from statistical mechanics due to the conformal invariance ideas introduced to the scale theories, we take over at this work, the study of the XY model in a transverse field. At first, we present a detailed analysis on the sample\'s typical crossover behavior. An improved calculation of the susceptibility and gap exponents early presented by dos Santos and Stinchcombe is included. Nest, a numerical analysis of the spectrum, regarding free boundary condi¬tions was developed and compared with conformal invariance predictions. Finally, the fundamental state energy corrections of finite chains were used to obtain the parameter which ,distingoishes the universality classes (the central charge c).
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Batog, Guillaume. "Problèmes classiques en vision par ordinateur et en géométrie algorithmique revisités via la géométrie des droites." Phd thesis, Université Nancy II, 2011. http://tel.archives-ouvertes.fr/tel-00653043.

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Systématiser: tel est le leitmotiv des résultats de cette thèse portant sur trois domaines d'étude en vision et en géométrie algorithmique. Dans le premier, nous étendons toute la machinerie du modèle sténopé des appareils photos classiques à un ensemble d'appareils photo (deux fentes, à balayage, oblique, une fente) jusqu'à présent étudiés séparément suivant différentes approches. Dans le deuxième, nous généralisons avec peu d'effort aux convexes de $\R^3$ l'étude des épinglages de droites ou de boules, menée différemment selon la nature des objets considérés. Dans le troisième, nous tentons de dégager une approche systématique pour élaborer des stratégies d'évaluation polynomiale de prédicats géométriques, les méthodes actuelles étant bien souvent spécifiques à chaque prédicat étudié. De tels objectifs ne peuvent être atteints sans un certain investissement mathématique dans l'étude des congruences linéaires de droites, des propriétés différentielles des ensembles de tangentes à des convexes et de la théorie des invariants algébriques, respectivement. Ces outils ou leurs utilisations reposent sur la géométrie des droites de $\p^3(\R)$, construite dans la seconde moitié du XIX\ieme{} siècle mais pas complètement assimilée en géométrie algorithmique et dont nous proposons une synthèse adaptée aux besoins de la communauté.
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Canales, Gonzalez Carolina. "Hypersurfaces Levi-plates et leur complément dans les surfaces complexes." Thesis, Université Paris-Saclay (ComUE), 2015. http://www.theses.fr/2015SACLS249/document.

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Dans ce mémoire nous étudions les hypersurfaces Levi-plates analytiques dans les surfaces algébriques complexes. Il s'agit des hypersurfaces réelles qui admettent un feuilletage par des courbes holomorphes, appelé le feuilletage de Cauchy Riemann (CR). Dans un premier temps nous montrons que si ce dernier admet une dynamique chaotique (i.e. s'il n'admet pas de mesure transverse invariante) alors les composantes connexes de l'extérieur de l'hypersurface sont des modifications de domaines de Stein. Ceci permet d'étendre le feuilletage CR en un feuilletage algébrique singulier sur la surface complexe ambiante. Nous appliquons ce résultat pour montrer, par l'absurde, qu'une hypersurface Levi-plate analytique qui admet une structure affine transverse dans une surface algébrique complexe possède une mesure transverse invariante. Ceci nous amène à conjecturer que les hypersurfaces Levi-plates dans les surfaces algébriques complexes qui sont difféomorphes à un fibré hyperbolique en tores sur le cercle sont des fibrations par courbes algébriques
In this work we study analytic Levi-flat hypersurfaces in complex algebraic surfaces. These are real hypersurfaces that admit a foliation by holomorphic curves, called Cauchy Riemann foliation (CR). First, we show that if this foliation admits chaotic dynamics (i.e. if it doesn't admit an invariant transverse measure), then the connected components of the complement of the hypersurface are Stein. This allows us to extend the CR foliation to a singular algebraic foliation on the ambient complex surface. We apply this result to prove, by contradiction, that analytic Levi-flat hypersurfaces admitting a transverse affine structure in a complex algebraic surface have a transverse invariant measure. This leads us to conjecture that Levi-flat hypersurfaces in complex algebraic surfaces that are diffeomorphic to a hyperbolic tori bundle over the circle are fibrations by algebraic curves
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Schapira, Barbara. "Propriétés ergodiques du feuilletage horosphérique d'une variété à courbure négative." Phd thesis, Université d'Orléans, 2003. http://tel.archives-ouvertes.fr/tel-00163420.

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Cette thèse est consacrée à l'étude des propriétés ergodiques du feuilletage horosphérique d'une variété géométriquement finie à courbure négative $M$. Un de nos principaux résultats est la classification des mesures transverses quasi-invariantes dont la dérivée de Radon-Nikodym est un cocycle höldérien fixé, associé à une mesure de Gibbs. À un tel cocycle, nous associons certaines moyennes sur les horosphères et montrons qu'elles s'équidistribuent vers la mesure de Gibbs correspondante lorsque $M$ est compacte ou convexe-cocompacte. Lorsqu'elle n'est ni compacte ni convexe-cocompacte, nous limitons l'étude aux moyennes associées à la mesure d'entropie maximale. Nous montrons qu'elles forment une suite tendue, ce qui, dans le cas des surfaces, nous permet d'obtenir leur équidistribution vers cette mesure d'entropie maximale. En corollaire, nous obtenons l'équidistribution des orbites du flot horocyclique d'une surface hyperbolique géométriquement finie mais de volume infini.
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Book chapters on the topic "Transverse invariants"

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"Invariants of Legendrian and transverse knots." In Mathematical Surveys and Monographs, 215–46. Providence, Rhode Island: American Mathematical Society, 2015. http://dx.doi.org/10.1090/surv/208/12.

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"Appendix E. The Transverse Invariant Wn2/B." In Conversations on Electric and Magnetic Fields in the Cosmos, 167–68. Princeton University Press, 2007. http://dx.doi.org/10.1515/9781400847433-018.

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Cotón, Carlos Meniño. "Transverse invariant measures extend to the ambient space." In Foliations 2012, 103–13. WORLD SCIENTIFIC, 2013. http://dx.doi.org/10.1142/9789814556866_0006.

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Camanho, P. P., A. Arteiro, G. Catalanotti, A. R. Melro, and M. Vogler. "Three-dimensional invariant-based failure criteria for transversely isotropic fibre-reinforced composites." In Numerical Modelling of Failure in Advanced Composite Materials, 111–50. Elsevier, 2015. http://dx.doi.org/10.1016/b978-0-08-100332-9.00005-0.

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Conference papers on the topic "Transverse invariants"

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Okamoto, Ruth J., Yuan Feng, Guy M. Genin, and Philip V. Bayly. "Anisotropic Behavior of White Matter in Shear and Implications for Transversely Isotropic Models." In ASME 2013 Summer Bioengineering Conference. American Society of Mechanical Engineers, 2013. http://dx.doi.org/10.1115/sbc2013-14039.

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Experimental studies [1] have shown that white matter (WM) in the brain is mechanically anisotropic. Based on its fibrous structure, transversely isotropic (TI) material models have been suggested to capture WM behavior. TI hyperelastic material models involve strain energy density functions that depend on the I4 and I5 pseudo-invariants of the Cauchy-Green strain tensor to account for the effects of stiff fibers. The pseudo-invariant I4 is the square of the stretch ratio in the fiber direction; I5 contains contributions of shear strain in planes parallel to the fiber axis. Most, if not all, published models of WM depend on I4 but not on I5.
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Romeo, Francesco, and Achille Paolone. "Propagation Properties of Three-Coupled Periodic Mechanical Systems." In ASME 2005 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. ASMEDC, 2005. http://dx.doi.org/10.1115/detc2005-85617.

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General three-coupled periodic systems are dealt with by means of transfer matrices of single units. The solutions of the associated characteristic equation are discussed in terms of invariant quantities by exploiting the well-known reversibility of its coefficients. An exhaustive description of the free wave propagation patterns is given on the invariants’ space where propagation domains with qualitatively different character are identified. Afterwards, two three-coupled periodic mechanical models are considered: pipes and truss beams. A nonlinear mapping from the invariants’ space to the physical parameters plane provides with a concise representation of the pattern of the propagation domains. A mechanical interpretation associated with the boundaries of these regions is given. The analyzed models give rise to equations of motion where the three-coupled nature stems from the coupling between longitudinal (mono-coupled) and transversal (bi-coupled) dynamics. The evolution of the propagation properties when the coupling parameters tend to vanish is eventually discussed.
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Vairo, Antonio. "Transverse momentum broadening and gauge invariance." In QCD@WORK 2012: International Workshop on Quantum Chromodynamics: Theory and Experiment. AIP, 2012. http://dx.doi.org/10.1063/1.4763541.

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Kappos, Efthimios. "Bifurcations on Control-Transverse Dynamics." In ASME 2010 Dynamic Systems and Control Conference. ASMEDC, 2010. http://dx.doi.org/10.1115/dscc2010-4141.

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To every submanifold transverse to the control distribution of a given control system is associated in a canonical way a dynamical behavior. By varying this transverse manifold, we can effect bifurcations in these dynamics. In this paper we explain how this is done and draw the analogies with the more familiar aspects of bifurcation theory. The second crucial point is that the manifold can be made invariant for a control flow and the dynamics in the directions transverse to it can be assigned arbitrarily — in particular, the manifold can be made asymptotically stable.
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Georgiou, Ioannis T., and Ira B. Schwartz. "Decoupling the Free Axial-Transverse Motions of a Nonlinear Plate: An Invariant Manifold Approach." In ASME 1995 Design Engineering Technical Conferences collocated with the ASME 1995 15th International Computers in Engineering Conference and the ASME 1995 9th Annual Engineering Database Symposium. American Society of Mechanical Engineers, 1995. http://dx.doi.org/10.1115/detc1995-0320.

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Abstract We approximate the nonlinearly coupled transverse-axial motions of an isotropic elastic plate with three nonlinearly coupled fundamental oscillators, and show that transverse motions can be decoupled from in-plane motions. We demonstrate this decoupling by showing analytically and numerically the existence of a global two-dimensional nonlinear invariant manifold. The invariant manifold carries a continuum of slow, periodic motions. In particular, for any motion on the slow invariant manifold, the transverse oscillator executes a periodic motion and it slaves the in-plane oscillators into periodic motions of half its period. The spectrum of the in-plane slaved motions consists of two distinct harmonics with frequencies twice and quadruple than that of the dominant harmonic of the transverse motion. Furthermore, as the energy level of motion on the slow manifold increases the frequency of the largest harmonic of the in-plane motions approaches the in-plane natural frequencies. This causes the in-plane oscillators to oscillate in pure compression.
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Cheng, Yu Chieh, Pei Yu Wang, Ramon Herrero, Muriel Botey, and Kestutis Staliunas. "Meta-mirrors with transverse invariance for beam shaping." In Metamaterials, Metadevices, and Metasystems 2019, edited by Nader Engheta, Mikhail A. Noginov, and Nikolay I. Zheludev. SPIE, 2019. http://dx.doi.org/10.1117/12.2530176.

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Rahn, Christopher D., and C. D. Mote. "Axial Force Stabilization of Transverse Beam Vibration." In ASME 1993 Design Technical Conferences. American Society of Mechanical Engineers, 1993. http://dx.doi.org/10.1115/detc1993-0217.

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Abstract An axial force stabilizes the transverse vibration of a beam with translational and rotational boundary springs and arbitrary geometry. The nonlinearly coupled, longitudinal and transverse equations of motion of the beam with axial force control are derived and simplified using a quasistatic assumption. Lyapunov’s direct method and an invariance principle for distributed systems show that an axial damper can asymptotically and simultaneously stabilize all transverse vibration modes. Asymptotic stability is guaranteed if the eigenvalues are simple and nonzero and if there are no internal resonances between coupled modes.
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Tsai, Juei-Hung, Bo-Zhi Huang, Zheng-Jia Zeng, Tzu-Yi Chuang, and Yu-Chieh Cheng. "Near-field flat focusing mirrors with an transverse invariance." In 2018 7th International Symposium on Next Generation Electronics (ISNE). IEEE, 2018. http://dx.doi.org/10.1109/isne.2018.8394653.

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Mehtar-Tani, Yacine, and Renaud Boussarie. "On gauge invariance of transverse momentum dependent distributions at small x." In 10th International Conference on Hard and Electromagnetic Probes of High-Energy Nuclear Collisions. Trieste, Italy: Sissa Medialab, 2021. http://dx.doi.org/10.22323/1.387.0182.

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Einstein, D. R., A. D. Freed, and I. Vesley. "Invariant Theory for Dispersed Transverse Isotropy: An Efficient Means for Modeling Fiber Splay." In ASME 2004 International Mechanical Engineering Congress and Exposition. ASMEDC, 2004. http://dx.doi.org/10.1115/imece2004-61236.

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Microstructual studies suggest that in some tissues, collagen fibers are approximately normally distributed about a mean preferred fiber direction. Structural constitutive equations that account for this dispersion of fibers have been shown to capture the mechanical complexity of these tissues quite well. However, such descriptions are computationally cumbersome for two-dimensional fiber distributions, let alone for fully three dimensional fiber populations. We have developed a new constitutive law for such tissues, based on a novel invariant theory for dispersed transverse isotropy. The model is polyconvex and fits biaxial data for aortic valve tissue as accurately as the standard structural model. Modification of the fiber stress-strain law requires no re-formulation of the constitutive tangent matrix, making the model flexible for different types of soft-tissues. Most importantly, the model is computationally expedient in a finite element analysis.
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