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Journal articles on the topic 'Billaud Conjecture'

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Published: 10 December 2022

Last updated: 28 January 2023

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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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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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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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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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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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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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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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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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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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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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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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Balitskiy, Alexey. "Shortest closed billiard trajectories in the plane and equality cases in Mahler’s conjecture." Geometriae Dedicata 184, no. 1 (April 6, 2016): 121–34. http://dx.doi.org/10.1007/s10711-016-0160-6.

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POHL, ANKE D. "Odd and even Maass cusp forms for Hecke triangle groups, and the billiard flow." Ergodic Theory and Dynamical Systems 36, no. 1 (August 4, 2014): 142–72. http://dx.doi.org/10.1017/etds.2014.64.

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By a transfer operator approach to Maass cusp forms and the Selberg zeta function for cofinite Hecke triangle groups, Möller and the present author found a factorization of the Selberg zeta function into a product of Fredholm determinants of transfer-operator-like families:$$\begin{eqnarray}Z(s)=\det (1-{\mathcal{L}}_{s}^{+})\det (1-{\mathcal{L}}_{s}^{-}).\end{eqnarray}$$In this article we show that the operator families${\mathcal{L}}_{s}^{\pm }$arise as families of transfer operators for the triangle groups underlying the Hecke triangle groups, and that for$s\in \mathbb{C}$,$\text{Re}s={\textstyle \frac{1}{2}}$, the operator${\mathcal{L}}_{s}^{+}$(respectively${\mathcal{L}}_{s}^{-}$) has a 1-eigenfunction if and only if there exists an even (respectively odd) Maass cusp form with eigenvalue$s(1-s)$. For non-arithmetic Hecke triangle groups, this result provides a new formulation of the Phillips–Sarnak conjecture on non-existence of even Maass cusp forms.

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Gensane, Th. "Dense Packings of Equal Spheres in a Cube." Electronic Journal of Combinatorics 11, no. 1 (May 27, 2004). http://dx.doi.org/10.37236/1786.

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We describe an adaptation of the billiard algorithm for finding dense packings of equal spheres inside a domain of the euclidean space. In order to improve the convergence of this stochastic algorithm, we introduce systematic perturbations in it. We apply this perturbed billiard algorithm in the case of $n$ spheres in a cube and display all the optimal and best known packings up to $n=32$. We improve the previous record packings for all $11\leq n\leq 26$ except $n=13,14,18$. We prove the existence of the displayed packings for $n=11,12,15,17,20,21,22,26,32$, by constructing them explicitly. For example, the graph of the conjectured optimal packing of twenty-two spheres is composed of five octahedrons and four isolated points. We also conjecture that the minimum distance $d_n$ between spheres centers of the optimal packings is constant in the range $29\leq n \leq 32$.

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Rudolf, Daniel. "Viterbo’s conjecture as a worm problem." Monatshefte für Mathematik, December 18, 2022. http://dx.doi.org/10.1007/s00605-022-01806-x.

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AbstractIn this paper, we relate Viterbo’s conjecture from symplectic geometry to Minkowski versions of worm problems which are inspired by the well-known Moser worm problem from geometry. For the special case of Lagrangian products this relation provides a connection to systolic Minkowski billiard inequalities and Mahler’s conjecture from convex geometry. Moreover, we use the above relation in order to transfer Viterbo’s conjecture to a conjecture for the longstanding open Wetzel problem which also can be expressed as a systolic Euclidean billiard inequality and for which we discuss an algorithmic approach in order to find a new lower bound. Finally, we point out that the above mentioned relation between Viterbo’s conjecture and Minkowski worm problems has a structural similarity to the known relationship between Bellmann’s lost-in-a-forest problem and the original Moser worm problem.

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Bialy, Misha, and Andrey E. Mironov. "The Birkhoff-Poritsky conjecture for centrally-symmetric billiard tables." Annals of Mathematics 196, no. 1 (July 1, 2022). http://dx.doi.org/10.4007/annals.2022.196.1.2.

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"High orders of the Weyl expansion for quantum billiards: resurgence of periodic orbits, and the Stokes phenomenon." Proceedings of the Royal Society of London. Series A: Mathematical and Physical Sciences 447, no. 1931 (December 8, 1994): 527–55. http://dx.doi.org/10.1098/rspa.1994.0154.

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A formalism is developed for calculating high coefficients c r of the Weyl (high energy) expansion for the trace of the resolvent of the Laplace operator in a domain B with smooth boundary ∂ B The c r are used to test the following conjectures. ( a ) The sequence of c r diverges factorially, controlled by the shortest accessible real or complex periodic geodesic. ( b ) If this is a 2-bounce orbit, it corresponds to the saddle of the chord length function whose contour is first crossed when climbing from the diagonal of the Möbius strip which is the space of chords of B . ( c ) This orbit gives an exponential contribution to the remainder when the Weyl series, truncated at its least term, is subtracted from the resolvent; the exponential switches on smoothly (according to an error function) where it is smallest, that is across the negative energy axis (Stokes line). These conjectures are motivated by recent results in asymptotics. They survive tests for the circle billiard, and for a family of curves with 2 and 3 bulges, where the dominant orbit is not always the shortest and is sometimes complex. For some systems which are not smooth billiards (e. g. a particle on a ring, or in a billiard where ∂ B is a polygon), the Weyl series terminates and so no geodesics are accessible; for a particle on a compact surface of constant negative curvature, only the complex geodesics are accessible from the Weyl series.

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"Calculation of spectral determinants." Proceedings of the Royal Society of London. Series A: Mathematical and Physical Sciences 447, no. 1930 (November 8, 1994): 413–37. http://dx.doi.org/10.1098/rspa.1994.0148.

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A method for regularizing spectral determinants is developed which facilitates their computation from a finite number of eigenvalues. This is used to calculate the determinant ∆ for the hyperbola billiard over a range which includes 46 quantum energy levels. The result is compared with semiclassical periodic orbit evaluations of ∆ using the Dirichlet series, Euler product, and a Riemann-Siegel-type formula. It is found that the Riemann-Siegel-type expansion, which uses the least number of orbits, gives the closest approximation. This provides explicit numerical support for recent conjectures concerning the analytic properties of semiclassical formulae, and in particular for the existence of resummation relations connecting long and short pseudo-orbits.

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Aistleitner, Christoph, Daniel El-Baz, and Marc Munsch. "Difference Sets and the Metric Theory of Small Gaps." International Mathematics Research Notices, December 21, 2021. http://dx.doi.org/10.1093/imrn/rnab354.

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Abstract Let $(a_n)_{n \geq 1}$ be a sequence of distinct positive integers. In a recent paper, Rudnick established asymptotic upper bounds for the minimal gaps of $\{a_n \alpha \mod 1, ~1 \leq n \leq N\}$ as $N \to \infty $, valid for Lebesgue-almost all $\alpha $ and formulated in terms of the additive energy of $\{a_1, \dots , a_N\}$. In the present paper, we argue that the metric theory of minimal gaps of such sequences is not controlled by the additive energy, but rather by the cardinality of the difference set of $\{a_1, \dots , a_N\}$. We establish a (complicated) sharp convergence/divergence test for the typical asymptotic order of the minimal gap and prove (slightly weaker) general upper and lower bounds that allow for a direct application. A major input for these results comes from the recent proof of the Duffin–Schaeffer conjecture by Koukoulopoulos and Maynard. We show that our methods give very precise results for slowly growing sequences whose difference set has relatively high density, such as the primes or the squares. Furthermore, we improve a metric result of Blomer, Bourgain, Rudnick, and Radziwiłł on the order of the minimal gap in the eigenvalue spectrum of a rectangular billiard.

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Yu, Pei, Weihua Zhang, Barbara Dietz, and Liang Huang. "Quantum signatures of chaos in relativistic quantumbilliards with shapes of circle- and ellipse-sectors." Journal of Physics A: Mathematical and Theoretical, April 19, 2022. http://dx.doi.org/10.1088/1751-8121/ac6840.

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Abstract According to the Berry-Tabor conjecture, the spectral properties of typical nonrelativistic quantum systems with an integrable classical counterpart agree with those of Poissonian random numbers. We investigate to what extend it applies to relativistic neutrino billiards (NBs) consisting of a spin-1/2 particle confined to a bounded planar domain by imposing suitable boundary conditions (BCs). In distinction to nonrelativistic quantum billiards (QBs), NBs do not have a well-defined classical counterpart. However, the peaks in the length spectra, that is, the modulus of the Fourier transform of the spectral density from wave number to length, of NBs are just like for QBs at the lengths of periodic orbits of the classical billiard (CB). This implies that there must be a connection between NBs and the dynamic of the CB. We demonstrate that NBs with shapes of circle- and ellipse-sectors with an integrable classical dynamic, obtained by cutting the circle and ellipse NB along symmetry lines, have no common eigenstates with the latter and that, indeed, their spectral properties can be similar to those of classically chaotic QBs. These features orginate from the intermingling of symmetries of the spinor components and the discontinuity in the BCs leading to contradictory conditional equations at corners connecting curved and straight boundary parts. To corroborate the necessity of the curved boundary part in order to generate GOE- like behavior, we furthermore consider the right-angled triangle NB constructed by halving the equilateral-triangle NB along a symmetry axis. For an understanding of these findings in terms of purely classical quantities we use the semiclassical approach recently developed for massive NBs, and Poincaré-Husimi distributions of the eigenstates in classical phase space. The results indicate, that in the ultrarelativistic limit these NBs do not show the behavior expected for classically chaotic QBs.

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