Journal articles: 'Real tori'

1

Yang, Jae-Hyun. "POLARIZED REAL TORI." Journal of the Korean Mathematical Society 52, no. 2 (March 1, 2015): 269–331. http://dx.doi.org/10.4134/jkms.2015.52.2.269.

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2

Ma, Hui, and Yujie Ma. "Totally real minimal tori in." Mathematische Zeitschrift 249, no. 2 (July 6, 2004): 241–67. http://dx.doi.org/10.1007/s00209-004-0693-5.

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3

Duval, Julien, and Damien Gayet. "Riemann surfaces and totally real tori." Commentarii Mathematici Helvetici 89, no. 2 (2014): 299–312. http://dx.doi.org/10.4171/cmh/320.

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4

Kaletha, Tasho. "Decomposition of Splitting Invariants in Split Real Groups." Canadian Journal of Mathematics 63, no. 5 (October 18, 2011): 1083–106. http://dx.doi.org/10.4153/cjm-2011-024-5.

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Abstract For a maximal torus in a quasi-split semi-simple simply-connected group over a local field of characteristic 0, Langlands and Shelstad constructed a cohomological invariant called the splitting invariant, which is an important component of their endoscopic transfer factors. We study this invari- ant in the case of a split real group and prove a decomposition theorem which expresses this invariant for a general torus as a product of the corresponding invariants for simple tori. We also show how this reduction formula allows for the comparison of splitting invariants between different tori in the given real group.

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Marcolli, Matilde. "Solvmanifolds and noncommutative tori with real multiplication." Communications in Number Theory and Physics 2, no. 2 (2008): 421–76. http://dx.doi.org/10.4310/cntp.2008.v2.n2.a4.

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6

ABE, Ryuji. "On Correspondences between Once Punctured Tori and Closed Tori: Fricke Groups and Real Lattices." Tokyo Journal of Mathematics 23, no. 2 (December 2000): 269–93. http://dx.doi.org/10.3836/tjm/1255958671.

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7

Gong, Xianghong. "Unimodular invariants of totally real tori in Cn." American Journal of Mathematics 119, no. 1 (1997): 19–54. http://dx.doi.org/10.1353/ajm.1997.0001.

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8

Catanese, Fabrizio, Keiji Oguiso, and Thomas Peternell. "On volume-preserving complex structures on real tori." Kyoto Journal of Mathematics 50, no. 4 (2010): 753–75. http://dx.doi.org/10.1215/0023608x-2010-013.

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9

Černe, Miran. "Analytic varieties with boundaries in totally real tori." Michigan Mathematical Journal 45, no. 2 (September 1998): 243–56. http://dx.doi.org/10.1307/mmj/1030132181.

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10

Manoliu, Mihaela. "Quantization of symplectic tori in a real polarization." Journal of Mathematical Physics 38, no. 5 (May 1997): 2219–54. http://dx.doi.org/10.1063/1.531970.

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11

Bruzzo, U., G. Marelli, and F. Pioli. "A Fourier transform for sheaves on real tori." Journal of Geometry and Physics 39, no. 2 (August 2001): 174–82. http://dx.doi.org/10.1016/s0393-0440(01)00009-2.

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Bruzzo, U., G. Marelli, and F. Pioli. "A Fourier transform for sheaves on real tori." Journal of Geometry and Physics 41, no. 4 (April 2002): 312–29. http://dx.doi.org/10.1016/s0393-0440(01)00065-1.

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13

Burelle, Jean-Philippe, Virginie Charette, Dominik Francoeur, and William M. Goldman. "Einstein tori and crooked surfaces." Advances in Geometry 21, no. 2 (March 11, 2021): 237–50. http://dx.doi.org/10.1515/advgeom-2020-0023.

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Abstract In hyperbolic space, the angle of intersection and distance classify pairs of totally geodesic hyperplanes. A similar algebraic invariant classifies pairs of hyperplanes in the Einstein universe. In dimension 3, symplectic splittings of a 4-dimensional real symplectic vector space model Einstein hyperplanes and the invariant is a determinant. The classification contributes to a complete disjointness criterion for crooked surfaces in the 3-dimensional Einstein universe.

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14

Chinta, Gautam, Jay Jorgenson, and Anders Karlsson. "Zeta functions, heat kernels, and spectral asymptotics on degenerating families of discrete tori." Nagoya Mathematical Journal 198 (June 2010): 121–72. http://dx.doi.org/10.1215/00277630-2009-009.

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AbstractBy a discrete torus we mean the Cayley graph associated to a finite product of finite cycle groups with the generating set given by choosing a generator for each cyclic factor. In this article we examine the spectral theory of the combinatorial Laplacian for sequences of discrete tori when the orders of the cyclic factors tend to infinity at comparable rates. First, we show that the sequence of heat kernels corresponding to the degenerating family converges, after rescaling, to the heat kernel on an associated real torus. We then establish an asymptotic expansion, in the degeneration parameter, of the determinant of the combinatorial Laplacian. The zeta-regularized determinant of the Laplacian of the limiting real torus appears as the constant term in this expansion. On the other hand, using a classical theorem by Kirchhoff, the determinant of the combinatorial Laplacian of a finite graph divided by the number of vertices equals the number of spanning trees, called the complexity, of the graph. As a result, we establish a precise connection between the complexity of the Cayley graphs of finite abelian groups and heights of real tori. It is also known that spectral determinants on discrete tori can be expressed using trigonometric functions and that spectral determinants on real tori can be expressed using modular forms on general linear groups. Another interpretation of our analysis is thus to establish a link between limiting values of certain products of trigonometric functions and modular forms. The heat kernel analysis which we employ uses a careful study of I-Bessel functions. Our methods extend to prove the asymptotic behavior of other spectral invariants through degeneration, such as special values of spectral zeta functions and Epstein-Hurwitz–type zeta functions.

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15

Chinta, Gautam, Jay Jorgenson, and Anders Karlsson. "Zeta functions, heat kernels, and spectral asymptotics on degenerating families of discrete tori." Nagoya Mathematical Journal 198 (June 2010): 121–72. http://dx.doi.org/10.1017/s0027763000009958.

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AbstractBy a discrete torus we mean the Cayley graph associated to a finite product of finite cycle groups with the generating set given by choosing a generator for each cyclic factor. In this article we examine the spectral theory of the combinatorial Laplacian for sequences of discrete tori when the orders of the cyclic factors tend to infinity at comparable rates. First, we show that the sequence of heat kernels corresponding to the degenerating family converges, after rescaling, to the heat kernel on an associated real torus. We then establish an asymptotic expansion, in the degeneration parameter, of the determinant of the combinatorial Laplacian. The zeta-regularized determinant of the Laplacian of the limiting real torus appears as the constant term in this expansion. On the other hand, using a classical theorem by Kirchhoff, the determinant of the combinatorial Laplacian of a finite graph divided by the number of vertices equals the number of spanning trees, called thecomplexity, of the graph. As a result, we establish a precise connection between the complexity of the Cayley graphs of finite abelian groups and heights of real tori. It is also known that spectral determinants on discrete tori can be expressed using trigonometric functions and that spectral determinants on real tori can be expressed using modular forms on general linear groups. Another interpretation of our analysis is thus to establish a link between limiting values of certain products of trigonometric functions and modular forms. The heat kernel analysis which we employ uses a careful study ofI-Bessel functions. Our methods extend to prove the asymptotic behavior of other spectral invariants through degeneration, such as special values of spectral zeta functions and Epstein-Hurwitz–type zeta functions.

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16

Doković, Dragomir Ž., and Nguyêñ Quôć Thăńg. "Conjugacy Classes of Maximal Tori in Simple Real Algebraic Groups and Applications." Canadian Journal of Mathematics 46, no. 4 (August 1, 1994): 699–717. http://dx.doi.org/10.4153/cjm-1994-039-5.

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AbstractLet G be an almost simple complex algebraic group defined over R, and let G(R) be the group of real points of G. We enumerate the G(R)-conjugacy classes of maximal R-tori of G. Each of these conjugacy classes is also a single G(R)˚-conjugacy class, where G(R)˚ is the identity component of G(R), viewed as a real Lie group. As a consequence we also obtain a new and short proof of the Kostant-Sugiura's theorem on conjugacy classes of Cartan subalgebras in simple real Lie algebras.A connected real Lie group P is said to be weakly exponential (w.e.) if the image of its exponential map is dense in P. This concept was introduced in [HM] where also the question of identifying all w.e. almost simple real Lie groups was raised. By using a theorem of A. Borel and our classification of maximal R-tori we answer the above question when P is of the form G(R)˚.

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17

Peterzil, Ya'acov, and Sergei Starchenko. "Algebraic and o-minimal flows on complex and real tori." Advances in Mathematics 333 (July 2018): 539–69. http://dx.doi.org/10.1016/j.aim.2018.05.040.

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18

Zhong, Xu, Jun Wang, and XiaoXiang Jiao. "Totally real conformal minimal tori in the hyperquadric Q 2." Science China Mathematics 56, no. 10 (March 4, 2013): 2015–23. http://dx.doi.org/10.1007/s11425-013-4600-6.

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19

Polishchuk, A. "Noncommutative two-tori with real multiplication as noncommutative projective varieties." Journal of Geometry and Physics 50, no. 1-4 (April 2004): 162–87. http://dx.doi.org/10.1016/j.geomphys.2003.11.007.

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20

Kruzhilin, N. G. "Holomorphic disks with boundaries in totally real tori in ℂ2." Mathematical Notes 56, no. 6 (December 1994): 1244–48. http://dx.doi.org/10.1007/bf02266692.

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21

Keynes, Harvey B., Nelson G. Markley, and Michael Sears. "The structure of automorphisms of real suspension flows." Ergodic Theory and Dynamical Systems 11, no. 2 (June 1991): 349–64. http://dx.doi.org/10.1017/s0143385700006180.

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AbstractThis paper is motivated by the connections between automorphisms of real suspension flows and ℝ2 suspension actions. Automorphisms which naturally lead to ℤ2-cocyles are examined from the viewpoint of covering theory in terms of an associated cylinder flow. A natural type of automorphisms (called simple) is analyzed via ergodic methods. It is shown that all automorphisms of suspensions built over minimal rotations on tori satisfy this condition. A more general approach using eigenfunctions extends this result to minimal affines, Furstenberg-type distal flows, certain nilmanifolds and a class of non-distal flows on the 2-torus.

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22

Niu, Yanmin, and Xiong Li. "The Existence of Invariant Tori and Quasiperiodic Solutions of the Nosé–Hoover Oscillator." Complexity 2020 (November 11, 2020): 1–9. http://dx.doi.org/10.1155/2020/6864573.

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In this paper, we consider an equivalent form of the Nosé–Hoover oscillator, x ′ = y , y ′ = − x − y z , and z ′ = y 2 − a , where a is a positive real parameter. Under a series of transformations, it is transformed into a 2-dimensional reversible system about action-angle variables. By applying a version of twist theorem established by Liu and Song in 2004 for reversible mappings, we find infinitely many invariant tori whenever a is sufficiently small, which eventually turns out that the solutions starting on the invariant tori are quasiperiodic. The discussion about quasiperiodic solutions of such 3-dimensional system is relatively new.

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23

Momot, Aleksander. "On the classification of complex tori arising from real Abelian surfaces." Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg 79, no. 2 (June 9, 2009): 283–98. http://dx.doi.org/10.1007/s12188-009-0024-1.

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24

Kim, Joontae. "Unknottedness of real Lagrangian tori in $$S^2\times S^2$$." Mathematische Annalen 378, no. 3-4 (July 27, 2020): 891–905. http://dx.doi.org/10.1007/s00208-020-02049-7.

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25

LIANG, ZHENGUO, JUN YAN, and YINGFEI YI. "Viscous stability of quasi-periodic tori." Ergodic Theory and Dynamical Systems 34, no. 1 (November 27, 2012): 185–210. http://dx.doi.org/10.1017/etds.2012.120.

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AbstractThis paper is devoted to the study of $P$-regularity of viscosity solutions $u(x,P)$, $P\in {\Bbb R}^n$, of a smooth Tonelli Lagrangian $L:T {\Bbb T}^n \rightarrow {\Bbb R}$ characterized by the cell equation $H(x,P+D_xu(x,P))=\overline {H}(P)$, where $H: T^* {\Bbb T}^n\rightarrow {\Bbb R}$ denotes the Hamiltonian associated with $L$ and $\overline {H}$ is the effective Hamiltonian. We show that if $P_0$ corresponds to a quasi-periodic invariant torus with a non-resonant frequency, then $D_xu(x,P)$ is uniformly Hölder continuous in $P$ at $P_0$ with Hölder exponent arbitrarily close to $1$, and if both $H$ and the torus are real analytic and the frequency vector of the torus is Diophantine, then $D_xu(x,P)$ is uniformly Lipschitz continuous in $P$ at $P_0$, i.e., there is a constant $C\gt 0$ such that $\|D_xu(\cdot ,P)-D_xu(\cdot ,P_0)\|_{\infty }\le C\|P-P_0\|$ for $\|P-P_0\|\ll 1$. Similar P-regularity of the Peierls barriers associated with $L(x,v)- \langle P,v \rangle $is also obtained.

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26

JOHNSON, M. A., and F. C. MOON. "EXPERIMENTAL CHARACTERIZATION OF QUASIPERIODICITY AND CHAOS IN A MECHANICAL SYSTEM WITH DELAY." International Journal of Bifurcation and Chaos 09, no. 01 (January 1999): 49–65. http://dx.doi.org/10.1142/s0218127499000031.

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We present an electro-mechanical system with finite delay whose construction was motivated by delay differential equations used to describe machine tool vibrations [Johnson, 1996; Moon & Johnson, 1998]. We show that the electro-mechanical system is capable of exhibiting periodic, quasiperiodic and chaotic vibrations. We provide a novel experimental technique for creating real-time Poincaré sections for systems with delay. This experimental technique was also applied to machine tool vibrations [Johnson, 1996]. Experimental Poincaré sections clearly show the existence of tori, and reveal the tori bifurcation sequence which leads to chaotic vibrations. The electro-mechanical system can be modeled by a single second-order differential equation with delay and a cubic nonlinearity. We show that the simple mathematical model fully replicates the bifurcation sequence seen in the electro-mechanical system.

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27

Lehrer, G. I., and J. van Hamel. "Euler characteristics of the real points of certain varieties of algebraic tori." Proceedings of the London Mathematical Society 94, no. 3 (March 21, 2007): 715–48. http://dx.doi.org/10.1112/plms/pdm002.

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28

Hultgren, Jakob. "Permanental Point Processes on Real Tori, Theta Functions and Monge–Ampère Equations." Annales de la faculté des sciences de Toulouse Mathématiques 28, no. 1 (2019): 11–65. http://dx.doi.org/10.5802/afst.1592.

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29

Liebendorfer, Adam. "Machine learning mitigates tearing modes in plasmas tori using real-time feedback." Scilight 2020, no. 6 (February 7, 2020): 061104. http://dx.doi.org/10.1063/10.0000729.

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30

CHIERCHIA, LUIGI, and FABIO PUSATERI. "Analytic Lagrangian tori for the planetary many-body problem." Ergodic Theory and Dynamical Systems 29, no. 3 (June 2009): 849–73. http://dx.doi.org/10.1017/s0143385708000503.

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AbstractIn 2004, Féjoz [Démonstration du ‘théoréme d’Arnold’ sur la stabilité du système planétaire (d’après M. Herman). Ergod. Th. & Dynam. Sys.24(5) (2004), 1521–1582], completing investigations of Herman’s [Démonstration d’un théoréme de V.I. Arnold. Séminaire de Systémes Dynamiques et manuscripts, 1998], gave a complete proof of ‘Arnold’s Theorem’ [V. I. Arnol’d. Small denominators and problems of stability of motion in classical and celestial mechanics. Uspekhi Mat. Nauk. 18(6(114)) (1963), 91–192] on the planetary many-body problem, establishing, in particular, the existence of a positive measure set of smooth (C∞) Lagrangian invariant tori for the planetary many-body problem. Here, using Rüßmann’s 2001 KAM theory [H. Rüßmann. Invariant tori in non-degenerate nearly integrable Hamiltonian systems. R. & C. Dynamics2(6) (2001), 119–203], we prove the above result in the real-analytic class.

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31

Okonek, Christian, and Andrei Teleman. "Abelian Yang-Mills Theory on Real Tori and Theta Divisors of Klein Surfaces." Communications in Mathematical Physics 323, no. 3 (September 17, 2013): 813–58. http://dx.doi.org/10.1007/s00220-013-1793-z.

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32

Matsumoto, Kazuko, and Takeo Ohsawa. "On the real analytic Levi flat hypersurfaces in complex tori of dimension two." Annales de l’institut Fourier 52, no. 5 (2002): 1525–32. http://dx.doi.org/10.5802/aif.1923.

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33

Doković, D. Ž., and Nguyên Q. Thăńg. "Correction To: Conjugacy Classes of Maximal Tori in Simple Real Algebraic Groups and Applications." Canadian Journal of Mathematics 46, no. 06 (December 1994): 1208–10. http://dx.doi.org/10.4153/cjm-1994-068-7.

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34

Xu, Junxiang. "Persistence of lower dimensional degenerate invariant tori with prescribed frequencies in Hamiltonian systems with small parameter." Nonlinearity 34, no. 12 (November 10, 2021): 8192–247. http://dx.doi.org/10.1088/1361-6544/ac2c91.

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Abstract In this paper we develop some KAM techniques to prove the persistence of lower dimensional elliptic-type degenerate invariant tori with prescribed frequencies in Hamiltonian systems. The proof is based on a formal KAM theorem, which allows us to solve the equation of equilibrium points and choose the parameter of small divisors after the KAM iteration, instead of in each KAM step. The proof is also based on the Leray–Schauder continuation theorem, which insures the existence of a path of real roots of an approximating odd-order real polynomial which depends continuously on parameters. This result is very important for us to tackle the Melnikov condition in the elliptic-type degenerate case.

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35

Overkamp, Otto. "Jumps and Motivic Invariants of Semiabelian Jacobians." International Mathematics Research Notices 2019, no. 20 (January 29, 2018): 6437–79. http://dx.doi.org/10.1093/imrn/rnx303.

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Abstract We investigate Néron models of Jacobians of singular curves over strictly Henselian discretely valued fields and their behavior under tame base change. For a semiabelian variety, this behavior is governed by a finite sequence of (a priori) real numbers between 0 and 1, called jumps. The jumps are conjectured to be rational, which is known in some cases. The purpose of this paper is to prove this conjecture in the case where the semiabelian variety is the Jacobian of a geometrically integral curve with a push-out singularity. Along the way, we prove the conjecture for algebraic tori which are induced along finite separable extensions and generalize Raynaud’s description of the identity component of the Néron model of the Jacobian of a smooth curve (in terms of the Picard functor of a proper, flat, and regular model) to our situation. The main technical result of this paper is that the exact sequence that decomposes the Jacobian of one of our singular curves into its toric and Abelian parts extends to an exact sequence of Néron models. Previously, only split semiabelian varieties were known to have this property.

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36

CARBERRY, EMMA, and IAN MCINTOSH. "MINIMAL LAGRANGIAN 2-TORI IN $\mathbb{CP}^2$ COME IN REAL FAMILIES OF EVERY DIMENSION." Journal of the London Mathematical Society 69, no. 02 (March 29, 2004): 531–44. http://dx.doi.org/10.1112/s0024610703005039.

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37

Losev, Ivan. "On categories for quantized symplectic resolutions." Compositio Mathematica 153, no. 12 (September 7, 2017): 2445–81. http://dx.doi.org/10.1112/s0010437x17007382.

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In this paper we study categories ${\mathcal{O}}$ over quantizations of symplectic resolutions admitting Hamiltonian tori actions with finitely many fixed points. In this generality, these categories were introduced by Braden et al. We establish a family of standardly stratified structures (in the sense of the author and Webster) on these categories ${\mathcal{O}}$. We use these structures to study shuffling functors of Braden et al. (called cross-walling functors in this paper). Most importantly, we prove that all cross-walling functors are derived equivalences that define an action of the Deligne groupoid of a suitable real hyperplane arrangement.

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38

MASSETTI, JESSICA ELISA. "Normal forms for perturbations of systems possessing a Diophantine invariant torus." Ergodic Theory and Dynamical Systems 39, no. 8 (December 12, 2017): 2176–222. http://dx.doi.org/10.1017/etds.2017.116.

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We give a new proof of Moser’s 1967 normal-form theorem for real analytic perturbations of vector fields possessing a reducible Diophantine invariant quasi-periodic torus. The proposed approach, based on an inverse function theorem in analytic class, is flexible and can be adapted to several contexts. This allows us to prove in a unified framework the persistence, up to finitely many parameters, of Diophantine quasi-periodic normally hyperbolic reducible invariant tori for vector fields originating from dissipative generalizations of Hamiltonian mechanics. As a byproduct, generalizations of Herman’s twist theorem and Rüssmann’s translated curve theorem are proved.

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39

Lin, Huaxin. "Homomorphisms From C(X) Into C*-Algebras." Canadian Journal of Mathematics 49, no. 5 (October 1, 1997): 963–1009. http://dx.doi.org/10.4153/cjm-1997-050-9.

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AbstractLet A be a simple C*-algebra with real rank zero, stable rank one and weakly unperforated K0(A) of countable rank. We show that a monomorphism Φ: C(S2) → A can be approximated pointwise by homomorphisms from C(S2) into A with finite dimensional range if and only if certain index vanishes. In particular,we show that every homomorphism ϕ from C(S2) into a UHF-algebra can be approximated pointwise by homomorphisms from C(S2) into the UHF-algebra with finite dimensional range.As an application, we show that if A is a simple C*-algebra of real rank zero and is an inductive limit of matrices over C(S2) then A is an AF-algebra. Similar results for tori are also obtained. Classification of Hom (C(X), A) for lower dimensional spaces is also studied.

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40

Trapanese, Gemma. "Luoghi, forme e tempi dell'inconscio nella clinica psicoanalitica di famiglia." INTERAZIONI, no. 2 (November 2021): 33–50. http://dx.doi.org/10.3280/int2021-002003.

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L'interesse che la psicoanalisi da tempo ha sviluppato per il genealogico, per il pro¬blema della trasmissione e dell'eredità psichica, ha consentito di mettere a fuoco quelle deli¬cate vicen-de familiari che accompagnano il processo di soggettivazione individuale e che vedono nell'evoluzione dei percorsi identificatori un preciso snodo. L'autrice intende met¬tere a fuoco, con l'aiuto di una vicenda clinica, l'ascolto psicoanalitico orientato al ricono¬scimento della real-tà psichica familiare che consente alla famiglia intera, a partire dai geni¬tori, di assumere la re-sponsabilità delle correnti emotive e delle rappresentazioni inconsce che circolano nel suo in-terno. In modo particolare nei casi di adozione.

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41

Corner, Alexander S., and Claire Cornock. "Applications and props: the impact on engagement and understanding." MSOR Connections 17, no. 1 (October 18, 2018): 3. http://dx.doi.org/10.21100/msor.v17i1.908.

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Problems based on applications or objects were added into a first year pure module in gaps where real-life problems were missing. Physical props were incorporated within the teaching sessions where it was possible. The additions to the module were the utilities problem whilst studying planar graphs, data storage when looking at number bases, RSA encryption after modular arithmetic and the Euclidean algorithm, as well as molecules and the mattress problem when looking at group theory. The physical objects used were tori, molecule models and mini mattresses. Evaluation was carried out through a questionnaire to gain the students' opinions of these additions and their general views of applications. Particular attention was paid to the effect on engagement and understanding.

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42

KHARBACH, J., S. DEKKAKI, A. T. H. OUAZZANI, and M. OUAZZANI-JAMIL. "BIFURCATIONS OF THE COMMON LEVEL SETS OF ATOMIC HYDROGEN IN VAN DER WAALS POTENTIAL." International Journal of Bifurcation and Chaos 13, no. 01 (January 2003): 107–14. http://dx.doi.org/10.1142/s0218127403006364.

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The classical dynamics of a hydrogen atom in a generalized van der Waals potential is investigated. In order to carry out the analytical and numerical investigations for a range of parametric values, we removed the singularity of the problem using Levi–Civita regularization and converted the problem into that of two coupled sextic anharmonic oscillators. We give a complete description of the real phase space structure of the converted system and give also an explicit periodic solution for singular common-level sets of the first integrals. All generic bifurcations of Liouville tori were determined theoretically. Numerical investigations are carried out for all generic bifurcations and we observe chaos-order-chaos transition when one of the system parameters is varied.

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43

Odaka, Yuji. "Tropical Geometric Compactification of Moduli, II: A g Case and Holomorphic Limits." International Mathematics Research Notices 2019, no. 21 (January 31, 2018): 6614–60. http://dx.doi.org/10.1093/imrn/rnx293.

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Abstract We compactify the classical moduli variety Ag of principally polarized abelian varieties of complex dimension g, by attaching the moduli of flat tori of real dimensions at most g in an explicit manner. Equivalently, we explicitly determine the Gromov–Hausdorff limits of principally polarized abelian varieties. This work is analogous to [50], where we compactified the moduli of curves by attaching the moduli of metrized graphs. Then, we also explicitly specify the Gromov–Hausdorff limits along holomorphic families of abelian varieties and show that these form special nontrivial subsets of the whole boundary. We also do the same for algebraic curves case and observe a crucial difference with the case of abelian varieties.

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44

YANG, YUXING, and LINGLING ZHANG. "Prescribed Hamiltonian Connectedness of 2D Torus." Journal of Interconnection Networks 20, no. 01 (March 2020): 2050001. http://dx.doi.org/10.1142/s0219265920500012.

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Tori are important fundamental interconnection networks for multiprocessor systems. Hamiltonian paths are important in information communication of multiprocessor systems, and Hamiltonian path embedding capability is an important aspect to determine if a network topology is suitable for a real application. In real systems, some links may have better performance. Therefore, when embedding Hamiltonian path into interconnection networks, it is desirable that these Hamiltonian paths would pass through the links with better performance. Given a two two-dimensional torus T (m, n) with m, n ≥ 5 odd, let L be a linear forest with at most two edges in T (m, n) and let u and v be two distinct vertices in T (m, n) such that none of the paths in L has u or v as internal node or both of them as end nodes. In this paper, we construct a hamiltonian path of T (m, n) between u and v passing through L.

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45

Fortunati, Alessandro, and Stephen Wiggins. "Transient Invariant and Quasi-Invariant Structures in an Example of an Aperiodically Time Dependent Fluid Flow." International Journal of Bifurcation and Chaos 28, no. 05 (May 2018): 1830015. http://dx.doi.org/10.1142/s021812741830015x.

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Starting from the concept of invariant KAM tori for nearly-integrable Hamiltonian systems with periodic or quasi-periodic nonautonomous perturbation, the paper analyzes the “analogue” of this class of invariant objects when the dependence on time is aperiodic. The investigation is carried out in a model motivated by the problem of a traveling wave in a channel over a smooth, quasi- and asymptotically flat (from which the “transient” feature) bathymetry, representing a case in which the described structures play the role of barriers to fluid transport in phase space. The paper provides computational evidence for the existence of transient structures also for “large” values of the perturbation size, as a complement to the rigorous results already proven by the first author for real-analytic bathymetry functions.

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Bertola, Marco, Gennady A. El, and Alexander Tovbis. "Rogue waves in multiphase solutions of the focusing nonlinear Schrödinger equation." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 472, no. 2194 (October 2016): 20160340. http://dx.doi.org/10.1098/rspa.2016.0340.

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Rogue waves appearing on deep water or in optical fibres are often modelled by certain breather solutions of the focusing nonlinear Schrödinger (fNLS) equation which are referred to as solitons on finite background (SFBs). A more general modelling of rogue waves can be achieved via the consideration of multiphase, or finite-band, fNLS solutions of whom the standard SFBs and the structures forming due to their collisions represent particular, degenerate, cases. A generalized rogue wave notion then naturally enters as a large-amplitude localized coherent structure occurring within a finite-band fNLS solution. In this paper, we use the winding of real tori to show the mechanism of the appearance of such generalized rogue waves and derive an analytical criterion distinguishing finite-band potentials of the fNLS equation that exhibit generalized rogue waves.

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Zhang, Min, Zhe Hu, and Yonggang Chen. "Invariant Tori for a Two-Dimensional Completely Resonant Beam Equation with a Quintic Nonlinear Term." Journal of Function Spaces 2022 (October 5, 2022): 1–15. http://dx.doi.org/10.1155/2022/7106366.

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This paper focuses on a two-dimensional completely resonant beam equation with a quintic nonlinear term. This means studying u t t + Δ 2 u + ε f u = 0 , x ∈ T 2 , t ∈ ℝ , under periodic boundary conditions, where ε is a small positive parameter and f u is a real analytic odd function of the form f u = f 5 u 5 + ∑ i ^ ≥ 3 f 2 i ^ + 1 u 2 i ∧ + 1 , f 5 ≠ 0 . It is proved that the equation admits small-amplitude, Whitney smooth, linearly stable quasiperiodic solutions on the phase-flow invariant subspace ℤ † 2 = r = r 1 , r 2 , r 1 ∈ 4 ℤ − 1 , r 2 ∈ 4 ℤ . Firstly, the corresponding Hamiltonian system of the equation is transformed into an angle-dependent block-diagonal normal form by using symplectic transformation, which can be achieved by selecting the appropriate tangential position. Finally, the existence of a class of invariant tori is proved, which implies the existence of quasiperiodic solutions for most values of frequency vector by an abstract KAM (Kolmogorov-Arnold-Moser) theorem for infinite dimensional Hamiltonian systems.

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Jánosi, Dániel, György Károlyi, and Tamás Tél. "Climate change in mechanical systems: the snapshot view of parallel dynamical evolutions." Nonlinear Dynamics 106, no. 4 (November 1, 2021): 2781–805. http://dx.doi.org/10.1007/s11071-021-06929-8.

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AbstractWe argue that typical mechanical systems subjected to a monotonous parameter drift whose timescale is comparable to that of the internal dynamics can be considered to undergo their own climate change. Because of their chaotic dynamics, there are many permitted states at any instant, and their time dependence can be followed—in analogy with the real climate—by monitoring parallel dynamical evolutions originating from different initial conditions. To this end an ensemble view is needed, enabling one to compute ensemble averages characterizing the instantaneous state of the system. We illustrate this on the examples of (i) driven dissipative and (ii) Hamiltonian systems and of (iii) non-driven dissipative ones. We show that in order to find the most transparent view, attention should be paid to the choice of the initial ensemble. While the choice of this ensemble is arbitrary in the case of driven dissipative systems (i), in the Hamiltonian case (ii) either KAM tori or chaotic seas should be taken, and in the third class (iii) the best choice is the KAM tori of the dissipation-free limit. In all cases, the time evolution of the chosen ensemble on snapshots illustrates nicely the geometrical changes occurring in the phase space, including the strengthening, weakening or disappearance of chaos. Furthermore, we show that a Smale horseshoe (a chaotic saddle) that is changing in time is present in all cases. Its disappearance is a geometrical sign of the vanishing of chaos. The so-called ensemble-averaged pairwise distance is found to provide an easily accessible quantitative measure for the strength of chaos in the ensemble. Its slope can be considered as an instantaneous Lyapunov exponent whose zero value signals the vanishing of chaos. Paradigmatic low-dimensional bistable systems are used as illustrative examples whose driving in (i, ii) is chosen to decay in time in order to maintain an analogy with case (iii) where the total energy decreases all the time.

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Huang, Wentao, Chengcheng Cao, and Dongping He. "Quasi-Periodic Motion and Hopf Bifurcation of a Two-Dimensional Aeroelastic Airfoil System in Supersonic Flow." International Journal of Bifurcation and Chaos 31, no. 02 (February 2021): 2150018. http://dx.doi.org/10.1142/s0218127421500188.

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In this article, the complex dynamic behavior of a nonlinear aeroelastic airfoil model with cubic nonlinear pitching stiffness is investigated by applying a theoretical method and numerical simulation method. First, through calculating the Jacobian of the nonlinear system at equilibrium, we obtain necessary and sufficient conditions when this system has two classes of degenerated equilibria. They are described as: (1) one pair of purely imaginary roots and one pair of conjugate complex roots with negative real parts; (2) two pairs of purely imaginary roots under nonresonant conditions. Then, with the aid of center manifold and normal form theories, we not only derive the stability conditions of the initial and nonzero equilibria, but also get the explicit expressions of the critical bifurcation lines resulting in static bifurcation and Hopf bifurcation. Specifically, quasi-periodic motions on 2D and 3D tori are found in the neighborhoods of the initial and nonzero equilibria under certain parameter conditions. Finally, the numerical simulations performed by the fourth-order Runge–Kutta method provide a good agreement with the results of theoretical analysis.

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LÜ, JINHU, and GUANRONG CHEN. "GENERATING MULTISCROLL CHAOTIC ATTRACTORS: THEORIES, METHODS AND APPLICATIONS." International Journal of Bifurcation and Chaos 16, no. 04 (April 2006): 775–858. http://dx.doi.org/10.1142/s0218127406015179.

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Over the last two decades, theoretical design and circuit implementation of various chaos generators have been a focal subject of increasing interest due to their promising applications in various real-world chaos-based technologies and information systems. In particular, generating complex multiscroll chaotic attractors via simple electronic circuits has seen rapid development. This article offers an overview of the subject on multiscroll chaotic attractors generation, including some fundamental theories, design methodologies, circuit implementations and practical applications. More precisely, the article first describes some effective design methods using piecewise-linear functions, cellular neural networks, nonlinear modulating functions, circuit component design, switching manifolds, multifolded tori formation, and so on. Based on different approaches, computer simulation and circuit implementation of various multiscroll chaotic attractors are then discussed in detail, with some theoretical proofs and laboratory experiments presented for verification and demonstration. It is then followed by some discussion on potential applications of multiscroll chaotic attractors, including secure and digital communications, synchronous prediction, random bit generation, and so on. The article is finally concluded with some future research outlooks, putting the important subject into engineering perspective.

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Journal articles on the topic 'Real tori'

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