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Добірка наукової літератури з теми "Billaud Conjecture"

Автор: Grafiati
Опубліковано: 10 грудня 2022
Оновлено: 28 січня 2023

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Статті в журналах з теми "Billaud Conjecture"

1

Balitskiy, Alexey. "Equality Cases in Viterbo’s Conjecture and Isoperimetric Billiard Inequalities." International Mathematics Research Notices 2020, no. 7 (April 19, 2018): 1957–78. http://dx.doi.org/10.1093/imrn/rny076.

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Abstract We apply the billiard technique to deduce some results on Viterbo’s conjectured inequality between the volume of a convex body and its symplectic capacity. We show that the product of a permutohedron and a simplex (properly related to each other) delivers equality in Viterbo’s conjecture. Using this result as well as previously known equality cases, we prove some special cases of Viterbo’s conjecture and interpret them as isoperimetric-like inequalities for billiard trajectories.
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2

Church, Kevin E. M., and Clément Fortin. "Computer-Assisted Methods for Analyzing Periodic Orbits in Vibrating Gravitational Billiards." International Journal of Bifurcation and Chaos 31, no. 08 (June 26, 2021): 2130021. http://dx.doi.org/10.1142/s0218127421300214.

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Using rigorous numerical methods, we prove the existence of 608 isolated periodic orbits in a gravitational billiard in a vibrating unbounded parabolic domain. We then perform pseudo-arclength continuation in the amplitude of the parabolic surface’s oscillation to compute large, global branches of periodic orbits. These branches are themselves proven rigorously using computer-assisted methods. Our numerical investigations strongly suggest the existence of multiple pitchfork bifurcations in the billiard model. Based on the numerics, physical intuition and existing results for a simplified model, we conjecture that for any pair [Formula: see text], there is a constant [Formula: see text] for which periodic orbits consisting of [Formula: see text] impacts per period [Formula: see text] cannot be sustained for amplitudes of oscillation below [Formula: see text]. We compute a verified upper bound for the conjectured critical amplitude for [Formula: see text] using our rigorous pseudo-arclength continuation.
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3

Bialy, Misha, and Andrey E. Mironov. "Angular billiard and algebraic Birkhoff conjecture." Advances in Mathematics 313 (June 2017): 102–26. http://dx.doi.org/10.1016/j.aim.2017.04.001.

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4

Gomes, Sean P. "Percival’s conjecture for the Bunimovich mushroom billiard." Nonlinearity 31, no. 9 (July 27, 2018): 4108–36. http://dx.doi.org/10.1088/1361-6544/aa776f.

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5

SIMÁNYI, NÁNDOR, and DOMOKOS SZÁSZ. "Non-integrability of cylindric billiards and transitive Lie group actions." Ergodic Theory and Dynamical Systems 20, no. 2 (April 2000): 593–610. http://dx.doi.org/10.1017/s0143385700000304.

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A conjecture is formulated and discussed which provides a necessary and sufficient condition for the ergodicity of cylindric billiards (this conjecture improves a previous one of the second author). This condition requires that the action of a Lie-subgroup ${\cal G}$ of the orthogonal group $SO(d)$ ($d$ being the dimension of the billiard in question) be transitive on the unit sphere $S^{d-1}$. If $C_1, \dots, C_k$ are the cylindric scatterers of the billiard, then ${\cal G}$ is generated by the embedded Lie subgroups ${\cal G}_i$ of $SO(d)$, where ${\cal G}_i$ consists of all transformations $g\in SO(d)$ of ${\Bbb R}^d$ that leave the points of the generator subspace of $C_i$ fixed ($1 \le i \le k$). In this paper we can prove the necessity of our conjecture and we also formulate some notions related to transitivity. For hard ball systems, we can also show that the transitivity holds in general: for an arbitrary number $N\ge 2$ of balls, arbitrary masses $m_1, \dots, m_N$ and in arbitrary dimension $\nu \ge 2$. This result implies that our conjecture is stronger than the Boltzmann–Sinai ergodic hypothesis for hard ball systems. We also note a somewhat surprising characterization of the positive subspace of the second fundamental form for the evolution of a special orthogonal manifold (wavefront), namely for the parallel beam of light. Thus we obtain a new characterization of sufficiency of an orbit segment.
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6

Carlo, Gabriel, Eduardo Vergini, and Alejandro J. Fendrik. "Numerical verification of Percival’s conjecture in a quantum billiard." Physical Review E 57, no. 5 (May 1, 1998): 5397–403. http://dx.doi.org/10.1103/physreve.57.5397.

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7

Liboff, Richard L. "Nodal‐surface conjectures for the convex quantum billiard." Journal of Mathematical Physics 35, no. 8 (August 1994): 3881–88. http://dx.doi.org/10.1063/1.530453.

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8

Bialy, Misha, and Andrey E. Mironov. "A survey on polynomial in momenta integrals for billiard problems." Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 376, no. 2131 (September 17, 2018): 20170418. http://dx.doi.org/10.1098/rsta.2017.0418.

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Анотація:
In this paper, we give a short survey of recent results on the algebraic version of the Birkhoff conjecture for integrable billiards on surfaces of constant curvature. We also discuss integrable magnetic billiards. As a new application of the algebraic technique, we study the existence of polynomial integrals for the two-sided magnetic billiards introduced by Kozlov and Polikarpov. This article is part of the theme issue ‘Finite dimensional integrable systems: new trends and methods’.
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9

Kibkalo, V., A. Fomenko, and I. Kharcheva. "Realizing integrable Hamiltonian systems by means of billiard books." Transactions of the Moscow Mathematical Society 82 (March 15, 2022): 37–64. http://dx.doi.org/10.1090/mosc/324.

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Fomenko’s conjecture that the topology of the Liouville foliations associated with integrable smooth or analytic Hamiltonian systems can be realized by means of integrable billiard systems is discussed. An algorithm of Vedyushkina and Kharcheva’s realizing 3-atoms by billiard books, which has been simplified significantly by formulating it in terms of f f -graphs, is presented. Note that, using another algorithm, Vedyushkina and Kharcheva have also realized an arbitrary type of the base of the Liouville foliation on the whole 3-dimensional isoenergy surface. This algorithm is illustrated graphically by an example where the invariant of the well-known Joukowsky system (the Euler case with a gyrostat) is realized for a certain energy range. It turns out that the entire Liouville foliation, rather than just the class of its base, is realized there; that is, the billiard and mechanical systems turn out to be Liouville equivalent. Results due to Vedyushkina and Kibkalo on constructing billiards with arbitrary values of numerical invariants are also presented. For billiard books without potential that possess a certain property, the existence of a Fomenko–Zieschang invariant is shown; it is also proved that they belong to the class of topologically stable systems. Finally, an example is presented when the addition of a Hooke potential to a planar billiard produces a splitting nondegenerate 4-singularity of rank 1.
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10

Marco, Jean-Pierre. "Entropy of billiard maps and a dynamical version of the Birkhoff conjecture." Journal of Geometry and Physics 124 (January 2018): 413–20. http://dx.doi.org/10.1016/j.geomphys.2017.11.012.

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Дисертації з теми "Billaud Conjecture"

1

Nevisi, Hossein. "Conditions on the existence of unambiguous morphisms." Thesis, Loughborough University, 2012. https://dspace.lboro.ac.uk/2134/10282.

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A morphism $\sigma$ is \emph{(strongly) unambiguous} with respect to a word $\alpha$ if there is no other morphism $\tau$ that maps $\alpha$ to the same image as $\sigma$. Moreover, $\sigma$ is said to be \emph{weakly unambiguous} with respect to a word $\alpha$ if $\sigma$ is the only \emph{nonerasing} morphism that can map $\alpha$ to $\sigma(\alpha)$, i.\,e., there does not exist any other nonerasing morphism $\tau$ satisfying $\tau(\alpha) = \sigma(\alpha)$. In the first main part of the present thesis, we wish to characterise those words with respect to which there exists a weakly unambiguous \emph{length-increasing} morphism that maps a word to an image that is strictly longer than the word. Our main result is a compact characterisation that holds for all morphisms with ternary or larger target alphabets. We also comprehensively describe those words that have a weakly unambiguous length-increasing morphism with a unary target alphabet, but we have to leave the problem open for binary alphabets, where we can merely give some non-characteristic conditions. \par The second main part of the present thesis studies the question of whether, for any given word, there exists a strongly unambiguous \emph{1-uniform} morphism, i.\,e., a morphism that maps every letter in the word to an image of length $1$. This problem shows some connections to previous research on \emph{fixed points} of nontrivial morphisms, i.\,e., those words $\alpha$ for which there is a morphism $\phi$ satisfying $\phi(\alpha) = \alpha$ and, for a symbol $x$ in $\alpha$, $\phi(x) \neq x$. Therefore, we can expand our examination of the existence of unambiguous morphisms to a discussion of the question of whether we can reduce the number of different symbols in a word that is not a fixed point such that the resulting word is again not a fixed point. This problem is quite similar to the setting of Billaud's Conjecture, the correctness of which we prove for a special case.
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Частини книг з теми "Billaud Conjecture"

1

Łopaciuk, Szymon, and Daniel Reidenbach. "The Billaud Conjecture for $${|{\varSigma } |} = 4$$, and Beyond." In Developments in Language Theory, 213–25. Cham: Springer International Publishing, 2022. http://dx.doi.org/10.1007/978-3-031-05578-2_17.

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