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Journal articles on the topic 'Poincaré-Birkhoff theorem'

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Published: 14 March 2023

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1

Bonfiglioli, Andrea, and Roberta Fulci. "A New Proof of the Existence of Free Lie Algebras and an Application." ISRN Algebra 2011 (March 7, 2011): 1–11. http://dx.doi.org/10.5402/2011/247403.

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The existence of free Lie algebras is usually derived as a consequence of the Poincaré-Birkhoff-Witt theorem. Moreover, in order to prove that (given a set and a field of characteristic zero) the Lie algebra of the Lie polynomials in the letters of (over the field ) is a free Lie algebra generated by , all available proofs use the embedding of a Lie algebra into its enveloping algebra . The aim of this paper is to give a much simpler proof of the latter fact without the aid of the cited embedding nor of the Poincaré-Birkhoff-Witt theorem. As an application of our result and of a theorem due to Cartier (1956), we show the relationships existing between the theorem of Poincaré-Birkhoff-Witt, the theorem of Campbell-Baker-Hausdorff, and the existence of free Lie algebras.

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2

Michaelis, Walter. "The Dual Poincaré-Birkhoff-Witt Theorem." Advances in Mathematics 57, no. 2 (August 1985): 93–162. http://dx.doi.org/10.1016/0001-8708(85)90051-9.

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3

Berger, Roland. "The quantum Poincaré-Birkhoff-Witt theorem." Communications in Mathematical Physics 143, no. 2 (January 1992): 215–34. http://dx.doi.org/10.1007/bf02099007.

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4

Le Calvez, Patrice, and Jian Wang. "Some remarks on the Poincaré-Birkhoff theorem." Proceedings of the American Mathematical Society 138, no. 02 (October 7, 2009): 703–15. http://dx.doi.org/10.1090/s0002-9939-09-10105-3.

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5

Winkelnkemper, H. E. "A generalization of the Poincaré-Birkhoff theorem." Proceedings of the American Mathematical Society 102, no. 4 (April 1, 1988): 1028. http://dx.doi.org/10.1090/s0002-9939-1988-0934887-5.

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6

Kirillov, Alexander, and Victor Starkov. "Some extensions of the Poincaré–Birkhoff theorem." Journal of Fixed Point Theory and Applications 13, no. 2 (June 2013): 611–25. http://dx.doi.org/10.1007/s11784-013-0127-2.

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7

Margheri, Alessandro, Carlota Rebelo, and Fabio Zanolin. "Fixed points for planar maps with multiple twists, with application to nonlinear equations with indefinite weight." Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 379, no. 2191 (January 4, 2021): 20190385. http://dx.doi.org/10.1098/rsta.2019.0385.

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In this paper, we investigate the dynamical properties associated with planar maps which can be represented as a composition of twist maps together with expansive–contractive homeomorphisms. The class of maps we consider present some common features both with those arising in the context of the Poincaré–Birkhoff theorem and those studied in the theory of topological horseshoes. In our main theorems, we show that the multiplicity results of fixed points and periodic points typical of the Poincaré–Birkhoff theorem can be recovered and improved in our setting. In particular, we can avoid assuming area-preserving conditions and we also obtain higher multiplicity results in the case of multiple twists. Applications are given to periodic solutions for planar systems of non-autonomous ODEs with sign-indefinite weights, including the non-Hamiltonian case. The presence of complex dynamics is also discussed. This article is part of the theme issue ‘Topological degree and fixed point theories in differential and difference equations’.

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8

Franks, John. "Erratum to “Generalizations of the Poincaré–Birkhoff theorem”." Annals of Mathematics 164, no. 3 (November 1, 2006): 1097–98. http://dx.doi.org/10.4007/annals.2006.164.1097.

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9

Makar-Limanov, L. "A Version of the Poincaré-Birkhoff-Witt Theorem." Bulletin of the London Mathematical Society 26, no. 3 (May 1994): 273–76. http://dx.doi.org/10.1112/blms/26.3.273.

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10

Li, Yong, and Zheng Hua Lin. "A constructive proof of the Poincaré-Birkhoff theorem." Transactions of the American Mathematical Society 347, no. 6 (June 1, 1995): 2111–26. http://dx.doi.org/10.1090/s0002-9947-1995-1290734-4.

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11

Casas, José Manuel, Manuel A. Insua, and Manuel Ladra. "Poincaré–Birkhoff–Witt theorem for Leibniz n-algebras." Journal of Symbolic Computation 42, no. 11-12 (November 2007): 1052–65. http://dx.doi.org/10.1016/j.jsc.2007.05.003.

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12

Zanini, Chiara. "Rotation numbers, eigenvalues, and the Poincaré–Birkhoff theorem." Journal of Mathematical Analysis and Applications 279, no. 1 (March 2003): 290–307. http://dx.doi.org/10.1016/s0022-247x(03)00012-x.

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13

BARGE, MARCY, and THOR MATISON. "A Poincaré–Birkhoff theorem on invariant plane continua." Ergodic Theory and Dynamical Systems 18, no. 1 (February 1998): 41–52. http://dx.doi.org/10.1017/s0143385798097569.

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14

Zhang, Zerui, Yuqun Chen, and Leonid A. Bokut. "Free Gelfand–Dorfman–Novikov superalgebras and a Poincaré–Birkhoff–Witt type theorem." International Journal of Algebra and Computation 29, no. 03 (May 2019): 481–505. http://dx.doi.org/10.1142/s0218196719500115.

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We construct linear bases of free Gelfand–Dorfman–Novikov (GDN) superalgebras. As applications, we prove a Poincaré–Birkhoff–Witt (PBW) type theorem, that is, every GDN superalgebra can be embedded into its universal enveloping associative differential supercommuative algebra. An Engel theorem is given.

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15

Oh, Sei-Qwon, Chun-Gil Park, and Yong-Yeon Shin. "A POINCARÉ-BIRKHOFF-WITT THEOREM FOR POISSON ENVELOPING ALGEBRAS." Communications in Algebra 30, no. 10 (January 12, 2002): 4867–87. http://dx.doi.org/10.1081/agb-120014673.

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16

Lambre, Thierry, Cyrille Ospel, and Pol Vanhaecke. "Poisson enveloping algebras and the Poincaré–Birkhoff–Witt theorem." Journal of Algebra 485 (September 2017): 166–98. http://dx.doi.org/10.1016/j.jalgebra.2017.05.001.

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17

Fonda, Alessandro, and Antonio J. Ureña. "A higher dimensional Poincaré–Birkhoff theorem for Hamiltonian flows." Annales de l'Institut Henri Poincaré C, Analyse non linéaire 34, no. 3 (May 2017): 679–98. http://dx.doi.org/10.1016/j.anihpc.2016.04.002.

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18

Eastwood, Michael. "A geometric proof of the Poincaré-Birkhoff-Witt Theorem." São Paulo Journal of Mathematical Sciences 12, no. 2 (August 1, 2018): 246–51. http://dx.doi.org/10.1007/s40863-018-0095-y.

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19

Fonda, Alessandro, and Antonio J. Ureña. "A higher-dimensional Poincaré–Birkhoff theorem without monotone twist." Comptes Rendus Mathematique 354, no. 5 (May 2016): 475–79. http://dx.doi.org/10.1016/j.crma.2016.01.023.

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20

Fonda, Alessandro, and Paolo Gidoni. "An avoiding cones condition for the Poincaré–Birkhoff Theorem." Journal of Differential Equations 262, no. 2 (January 2017): 1064–84. http://dx.doi.org/10.1016/j.jde.2016.10.002.

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21

Qian, Dingbian, and Pedro J. Torres. "Bouncing solutions of an equation with attractive singularity." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 134, no. 1 (February 2004): 201–13. http://dx.doi.org/10.1017/s0308210500003164.

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For any n, m ∈ N, we prove the existence of 2mπ-periodic solutions, with n bouncings in each period, for a second-order forced equation with attractive singularity by using the approach of successor map and Poincaré-Birkhoff twist theorem.

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22

ANDRADE, L. C. GARCIA DE. "ON TIME DISLOCATION SOLUTION OF EINSTEIN FIELD EQUATIONS." Modern Physics Letters A 14, no. 02 (January 20, 1999): 93–97. http://dx.doi.org/10.1142/s0217732399000122.

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The theory considered here is not Einstein general relativity, but is a Poincaré type gauge theory of gravity, therefore the Birkhoff theorem is not applied and the external solution is not vacuum spherically symmetric and tachyons may exist outside the core defect.

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23

Ma, Tiantian, and Zaihong Wang. "Infinitely Many Periodic Solutions of Duffing Equations with Singularities via Time Map." Abstract and Applied Analysis 2014 (2014): 1–8. http://dx.doi.org/10.1155/2014/398512.

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We study the periodic solutions of Duffing equations with singularitiesx′′+g(x)=p(t). By using Poincaré-Birkhoff twist theorem, we prove that the given equation possesses infinitely many positive periodic solutions provided thatgsatisfies the singular condition and the time map related to autonomous systemx′′+g(x)=0tends to zero.

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24

Bautista, César. "A Poincaré–Birkhoff–Witt theorem for generalized Lie color algebras." Journal of Mathematical Physics 39, no. 7 (July 1998): 3828–43. http://dx.doi.org/10.1063/1.532471.

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25

Fonda, Alessandro, and Antonio J. Ureña. "A Poincaré–Birkhoff theorem for Hamiltonian flows on nonconvex domains." Journal de Mathématiques Pures et Appliquées 129 (September 2019): 131–52. http://dx.doi.org/10.1016/j.matpur.2018.12.007.

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26

OMORI, Hideki, Yoshiaki MAEDA, and Akira YOSHIOKA. "A Poincaré-Birkhoff-Witt theorem for infinite dimensional Lie algebras." Journal of the Mathematical Society of Japan 46, no. 1 (January 1994): 25–50. http://dx.doi.org/10.2969/jmsj/04610025.

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27

Braverman, Alexander, and Dennis Gaitsgory. "Poincaré–Birkhoff–Witt Theorem for Quadratic Algebras of Koszul Type." Journal of Algebra 181, no. 2 (April 1996): 315–28. http://dx.doi.org/10.1006/jabr.1996.0122.

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28

Song, Chunmei, Qihuai Liu, and Guirong Jiang. "Harmonic and subharmonic solutions of quadratic Liénard type systems with sublinearity." AIMS Mathematics 6, no. 11 (2021): 12913–28. http://dx.doi.org/10.3934/math.2021747.

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<abstract><p>In this paper, we prove the existence of harmonic solutions and infinitely many subharmonic solutions of dissipative second order sublinear differential equations named quadratic Liénard type systems. The method of the proof is based on the Poincaré-Birkhoff twist theorem.</p></abstract>

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29

Yang, Xuxin, Weibing Wang, and Jianhua Shen. "Existence of Periodic Solutions for the Duffing Equation with Impulses." Mathematical Problems in Engineering 2012 (2012): 1–13. http://dx.doi.org/10.1155/2012/903653.

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We study the existence of solutions to the Duffing equation with impulses. By means of the Poincaré-Birkhoff fixed point theorem under given conditions, we obtain the sufficient condition of existence of infinitely many solutions. Our results generalize those of T. R. Ding. An example is presented to demonstrate applications of our main result.

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30

Pascoletti, Anna, and Fabio Zanolin. "A Topological Approach to Bend-Twist Maps with Applications." International Journal of Differential Equations 2011 (2011): 1–20. http://dx.doi.org/10.1155/2011/612041.

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In this paper we reconsider, in a purely topological framework, the concept of bend-twist map previously studied in the analytic setting by Tongren Ding in (2007). We obtain some results about the existence and multiplicity of fixed points which are related to the classical Poincaré-Birkhoff twist theorem for area-preserving maps of the annulus; however, in our approach, like in Ding (2007), we do not require measure-preserving conditions. This makes our theorems in principle applicable to nonconservative planar systems. Some of our results are also stable for small perturbations. Possible applications of the fixed point theorems for topological bend-twist maps are outlined in the last section.

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31

Reyes, Armando, and Jason Hernández-Mogollón. "A Survey on Some Algebraic Characterizations of Hilbert’s Nullstellensatz for Non-commutative Rings of Polynomial Type." Ingeniería y Ciencia 16, no. 31 (June 19, 2020): 27–52. http://dx.doi.org/10.17230/ingciencia.16.31.2.

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In this paper we present a survey of some algebraic characterizations of Hilbert’s Nullstellensatz for non-commutative rings of polynomial type. Using several results established in the literature, we obtain a version of this theorem for the skew Poincaré-Birkhoff-Witt extensions. Once this is done, we illustrate the Nullstellensatz with examples appearing in noncommutative ring theory and non-commutative algebraic geometry.

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32

Shepler, Anne V., and Sarah Witherspoon. "A Poincaré-Birkhoff-Witt theorem for quadratic algebras with group actions." Transactions of the American Mathematical Society 366, no. 12 (July 21, 2014): 6483–506. http://dx.doi.org/10.1090/s0002-9947-2014-06118-7.

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33

Campos, J., A. Margheri, R. Martins, and C. Rebelo. "A note on a modified version of the Poincaré–Birkhoff theorem." Journal of Differential Equations 203, no. 1 (August 2004): 55–63. http://dx.doi.org/10.1016/j.jde.2004.03.022.

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34

Kolesnikov, P. S. "Gröbner–Shirshov Bases for Replicated Algebras." Algebra Colloquium 24, no. 04 (November 15, 2017): 563–76. http://dx.doi.org/10.1142/s1005386717000372.

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We establish a universal approach to solutions of the word problem in the varieties of di- and tri-algebras. This approach, for example, allows us to apply Gröbner–Shirshov bases method for Lie algebras to solve the ideal membership problem in free Leibniz algebras (Lie di-algebras). As another application, we prove an analogue of the Poincaré–Birkhoff–Witt Theorem for universal enveloping associative tri-algebra of a Lie tri-algebra (CTD!-algebra).

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35

Kirillov, Alexander. "An extension of the Poincaré-Birkhoff fixed point theorem to noninvariant annuli." Fixed Point Theory 22, no. 1 (February 1, 2021): 251–62. http://dx.doi.org/10.24193/fpt-ro.2021.1.18.

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36

López-Gómez, Julián, Eduardo Muñoz-Hernández, and Fabio Zanolin. "The Poincaré–Birkhoff Theorem for a Class of Degenerate Planar Hamiltonian Systems." Advanced Nonlinear Studies 21, no. 3 (July 17, 2021): 489–99. http://dx.doi.org/10.1515/ans-2021-2137.

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Abstract In this paper, we investigate the problem of the existence and multiplicity of periodic solutions to the planar Hamiltonian system x ′ = - λ ⁢ α ⁢ ( t ) ⁢ f ⁢ ( y ) x^{\prime}=-\lambda\alpha(t)f(y) , y ′ = λ ⁢ β ⁢ ( t ) ⁢ g ⁢ ( x ) y^{\prime}=\lambda\beta(t)g(x) , where α , β \alpha,\beta are non-negative 𝑇-periodic coefficients and λ > 0 \lambda>0 . We focus our study to the so-called “degenerate” situation, namely when the set Z := supp ⁡ α ∩ supp ⁡ β Z:=\operatorname{supp}\alpha\cap\operatorname{supp}\beta has Lebesgue measure zero. It is known that, in this case, for some choices of 𝛼 and 𝛽, no nontrivial 𝑇-periodic solution exists. On the opposite, we show that, depending of some geometric configurations of 𝛼 and 𝛽, the existence of a large number of 𝑇-periodic solutions (as well as subharmonic solutions) is guaranteed (for λ > 0 \lambda>0 and large). Our proof is based on the Poincaré–Birkhoff twist theorem. Applications are given to Volterra’s predator-prey model with seasonal effects.

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37

Yamane, Hiroyuki. "A Poincaré-Birkhoff-Witt theorem for the quantum group of type $A_N$." Proceedings of the Japan Academy, Series A, Mathematical Sciences 64, no. 10 (1988): 385–86. http://dx.doi.org/10.3792/pjaa.64.385.

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38

Martins, Rogério, and Antonio J. Ureña. "The star-shaped condition on Ding's version of the Poincaré-Birkhoff theorem." Bulletin of the London Mathematical Society 39, no. 5 (August 23, 2007): 803–10. http://dx.doi.org/10.1112/blms/bdm064.

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39

Jiang, Fangfang, Jianhua Shen, and Yanting Zeng. "Applications of the Poincaré–Birkhoff theorem to impulsive Duffing equations at resonance." Nonlinear Analysis: Real World Applications 13, no. 3 (June 2012): 1292–305. http://dx.doi.org/10.1016/j.nonrwa.2011.10.006.

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40

Fel’shtyn, Alexander. "Nielsen theory, Floer homology and a generalisation of the Poincaré-Birkhoff theorem." Journal of Fixed Point Theory and Applications 3, no. 2 (August 14, 2008): 191–214. http://dx.doi.org/10.1007/s11784-008-0085-2.

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41

Bonino, Marc. "A topological version of the Poincaré-Birkhoff theorem with two fixed points." Mathematische Annalen 352, no. 4 (April 22, 2011): 1013–28. http://dx.doi.org/10.1007/s00208-011-0669-9.

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42

Fonda, Alessandro, and Rodica Toader. "Subharmonic solutions of Hamiltonian systems displaying some kind of sublinear growth." Advances in Nonlinear Analysis 8, no. 1 (July 28, 2017): 583–602. http://dx.doi.org/10.1515/anona-2017-0040.

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Abstract We prove the existence and multiplicity of subharmonic solutions for Hamiltonian systems obtained as perturbations of N planar uncoupled systems which, e.g., model some type of asymmetric oscillators. The nonlinearities are assumed to satisfy Landesman–Lazer conditions at the zero eigenvalue, and to have some kind of sublinear behavior at infinity. The proof is carried out by the use of a generalized version of the Poincaré–Birkhoff Theorem. Different situations, including Lotka–Volterra systems, or systems with singularities, are also illustrated.

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43

CHAPOTON, FRÉDÉRIC, and FRÉDÉRIC PATRAS. "ENVELOPING ALGEBRAS OF PRELIE ALGEBRAS, SOLOMON IDEMPOTENTS AND THE MAGNUS FORMULA." International Journal of Algebra and Computation 23, no. 04 (June 2013): 853–61. http://dx.doi.org/10.1142/s0218196713400134.

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We study the internal structure of enveloping algebras of preLie algebras. We show in particular that the canonical projections arising from the Poincaré–Birkhoff–Witt theorem can be computed explicitly. They happen to be closely related to the Magnus formula for matrix differential equations. Indeed, we show that the Magnus formula provides a way to compute the canonical projection on the preLie algebra. Conversely, our results provide new insights on classical problems in the theory of differential equations and on recent advances in their combinatorial understanding.

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44

YAN, PING, and MEIRONG ZHANG. "ROTATION NUMBER, PERIODIC FUČIK SPECTRUM AND MULTIPLE PERIODIC SOLUTIONS." Communications in Contemporary Mathematics 12, no. 03 (June 2010): 437–55. http://dx.doi.org/10.1142/s0219199710003877.

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In this paper, we will introduce the rotation number for the one-dimensional asymmetric p-Laplacian with a pair of periodic potentials. Two applications of this notion will be given. One is a clear characterization of two unbounded sequences of Fučik curves of the periodic Fučik spectrum of the p-Laplacian with potentials. With the help of the Poincaré–Birkhoff fixed point theorem, the other application is some existence result of multiple periodic solutions of nonlinear ordinary differential equations concerning with the p-Laplacian.

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45

Hryniewicz, Umberto, Al Momin, and Pedro A. S. Salomão. "A Poincaré–Birkhoff theorem for tight Reeb flows on $$S^3$$ S 3." Inventiones mathematicae 199, no. 2 (April 1, 2014): 333–422. http://dx.doi.org/10.1007/s00222-014-0515-2.

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46

Segal, D. "Free Left-Symmetrical Algebras and an Analogue of the Poincaré-Birkhoff-Witt Theorem." Journal of Algebra 164, no. 3 (March 1994): 750–72. http://dx.doi.org/10.1006/jabr.1994.1088.

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47

Lezama, Oswaldo, and Edward Latorre. "Non-commutative algebraic geometry of semi-graded rings." International Journal of Algebra and Computation 27, no. 04 (May 26, 2017): 361–89. http://dx.doi.org/10.1142/s0218196717500199.

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In this paper, we introduce the semi-graded rings, which extend graded rings and skew Poincaré–Birkhoff–Witt (PBW) extensions. For this new type of non-commutative rings, we will discuss some basic problems of non-commutative algebraic geometry. In particular, we will prove some elementary properties of the generalized Hilbert series, Hilbert polynomial and Gelfand–Kirillov dimension. We will extend the notion of non-commutative projective scheme to the case of semi-graded rings and we generalize the Serre–Artin–Zhang–Verevkin theorem. Some examples are included at the end of the paper.

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48

ARDIZZONI, ALESSANDRO. "UNIVERSAL ENVELOPING ALGEBRAS OF PBW TYPE." Glasgow Mathematical Journal 54, no. 1 (August 2, 2011): 9–26. http://dx.doi.org/10.1017/s0017089511000310.

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AbstractWe continue our investigation of the general notion of universal enveloping algebra introduced in [A. Ardizzoni, A Milnor–Moore type theorem for primitively generated braided Bialgebras, J. Algebra 327(1) (2011), 337–365]. Namely, we study a universal enveloping algebra when it is of Poincaré–Birkhoff–Witt (PBW) type, meaning that a suitable PBW-type theorem holds. We discuss the problem of finding a basis for a universal enveloping algebra of PBW type: as an application, we recover the PBW basis both of an ordinary universal enveloping algebra and of a restricted enveloping algebra. We prove that a universal enveloping algebra is of PBW type if and only if it is cosymmetric. We characterise braided bialgebra liftings of Nichols algebras as universal enveloping algebras of PBW type.

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49

Hausrath, Alan R., and Raúl F. Manasevich. "Periodic solutions of a periodically perturbed Lotka-Volterra equation using the Poincaré-Birkhoff Theorem." Journal of Mathematical Analysis and Applications 157, no. 1 (May 1991): 1–9. http://dx.doi.org/10.1016/0022-247x(91)90132-j.

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50

Margheri, A., C. Rebelo, and F. Zanolin. "Maslov index, Poincaré–Birkhoff Theorem and Periodic Solutions of Asymptotically Linear Planar Hamiltonian Systems." Journal of Differential Equations 183, no. 2 (August 2002): 342–67. http://dx.doi.org/10.1006/jdeq.2001.4122.

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