Billiards Optimization Algorithm:A New Game-Based Metaheuristic Approach

Metaheuristic algorithms are one of themost widely used stochastic approaches in solving optimization problems.In this paper,a new metaheuristic algorithm entitled Billiards Optimization Algorithm(BOA)is proposed and designed to be used in optimization applications.The fundamental inspiration in BOA design is the behavior of the players and the rules of the billiards game.Various steps of BOA are described and then its mathematical model is thoroughly explained.The efficiency of BOA in dealing with optimization problems is evaluated through optimizing twenty-three standard benchmark functions of different types including unimodal,high-dimensional multimodal,and fixed-dimensionalmultimodal functions.In order to analyze the quality of the results obtained by BOA,the performance of the proposed approach is compared with ten well-known algorithms.The simulation results show that BOA,with its high exploration and exploitation abilities,achieves an impressive performance in providing solutions to objective functions and is superior and far more competitive compared to the ten competitor algorithms.

Level spacing statistics for two-dimensional massless Dirac billiards

Classical-quantum correspondence has been an intriguing issue ever since quantum theory was proposed. The search- ing for signatures of classically nonintegrable dynamics in quantum systems comprises the interesting field of quantum chaos. In this short review, we shall go over recent efforts of extending the understanding of quantum chaos to relativistic cases. We shall focus on the level spacing statistics for two-dimensional massless Dirac billiards, i.e., particles confined in a closed region. We shall discuss the works for both the particle described by the massless Dirac equation （or Weyl equation） and the quasiparticle from graphene. Although the equations are the same, the boundary conditions are typically different, rendering distinct level spacing statistics.

一组Billiard系统的衰减率(英文)

对具有慢混合率的奇异动力系统的研究方法是近几年才发展起来的,但应用到具体模型上还比较困难.研究了一组新型的半发散billiards系统,证明该系统碰撞映射的相关函数的收敛速率的上界为(lnn)3/n2,其中n表示离散时间.

具有零曲率边界Billiard系统的混合率(英文)

对具有慢混合率的奇异动力系统的研究方法是近年才发展起来的,但应用到具体模型上还很困难.研究了一类新型的具有0曲率边界的半发散billiards系统,证明了这个系统碰撞映射的相关函数的收敛速率的上界是(lnn)8/3/n5/3,其中n表示离散的时间.

Semiclassical Analysis of Quarter Stadium Billiards

An expansion method for stationary states is applied to obtain the eigenfunctions and the eigenenergies of the quarter stadium billiard, and its nearest energy-level spacing distribution is obtained. The histogram is consistent with the standard Wigner distribution, which indicates that the stadium billiard system is chaotic. Particular attention is paid to pursuing the quantum manifestations of such classical chaos. The correspondences between the Fourier transformation of quantum spectra and classical orbits are investigated by using the closed-orbit theory. The analytical and numerical results are in agreement with the required resolution, which corroborates that the semiclassical method provides a physically meaningful image to understand such chaotic systems.

The quantum spectra analysis of the circular billiards in wells

We use a recently defined quantum spectral function and apply the method of closed-orbit theory to the 2D circular billiard system. The quantum spectra contain rich information of all classical orbits connecting two arbitrary points in the well. We study the correspondence between quantum spectra and classical orbits in the circular, 1/2 circular and 1/4 circular wells using the analytic and numerical methods. We find that the peak positions in the Fourier- transformed quantum spectra match accurately with the lengths of the classical orbits. These examples show evidently that semi-classlcal method provides a bridge between quantum and classical mechanics.

EQUIDISTRIBUTION OF CLOSED ORBITS OF BILLIARD FLOWS ON THE PLANE

The authors consider the billiard system with finitely many convex scatters with smooth boundary satisfying the visibility assumption on the plane and prove that the closed orbits for the billiard flow is uniformly distributed.

Experimental investigation of the fluctuations in nonchaotic scattering in microwave billiards

We report on the experimental investigation of the properties of the eigenvalues and wavefunctions and the fluctuation properties of the scattering matrix of closed and open billiards, respectively, of which the classical dynamics undergoes a transition from integrable via almost integrable to fully chaotic. To realize such a system, we chose a billiard with a 60° sector shape of which the classical dynamics is integrable, and introduced circular scatterers of varying number, size,and position. The spectral properties of generic quantum systems of which the classical counterpart is either integrable or chaotic are universal and well understood. If, however, the classical dynamics is pseudo-integrable or almost-integrable,they exhibit a non-universal intermediate statistics, for which analytical results are known only in a few cases, e.g., if it corresponds to semi-Poisson statistics. Since the latter is, above all, clearly distinguishable from those of integrable and chaotic systems, our aim was to design a billiard with these features which indeed is achievable by adding just one scatterer of appropriate size and position to the sector billiard. We demonstrated that, while the spectral properties of almostintegrable billiards are sensitive to the classical dynamics, this is not the case for the distribution of the wavefunction components, which was analyzed in terms of the strength distribution, and the fluctuation properties of the scattering matrix which coincide with those of typical, fully chaotic systems.

Semiclassical approximation for transmission through open Sinai billiards

We use a semiclassical approximation to study the transport through the weakly open chaotic Sinai quantum billiards which can be considered as the schematic of a Sinai mesoscopic device,with the diffractive scatterings at the lead openings taken into account.The conductance of the ballistic microstructure which displays universal fluctuations due to quantum interference of electrons can be calculated by Landauer formula as a function of the electron Fermi wave number,and the transmission amplitude can be expressed as the sum over all classical paths connecting the entrance and the exit leads.For the Sinai billiards,the path sum leads to an excellent numerical agreement between the peak positions of power spectrum of the transmission amplitude and the corresponding lengths of the classical trajectories,which demonstrates a good agreement between the quantum theory and the semiclassical theory.

The quantum spectral analysis of the two-dimensional annular billiard system

Based on the extended closed-orbit theory together with spectral analysis, this paper studies the correspondence between quantum mechanics and the classical counterpart in a two-dimensional annular billiard. The results demonstrate that the Fourier-transformed quantum spectra are in very good accordance with the lengths of the classical ballistic trajectories, whereas spectral strength is intimately associated with the shapes of possible open orbits connecting arbitrary two points in the annular cavity. This approach facilitates an intuitive understanding of basic quantum features such as quantum interference, locations of the wavefunctions, and allows quantitative calculations in the range of high energies, where full quantum calculations may become impractical in general. This treatment provides a thread to explore the properties of microjunction transport and even quantum chaos under the much more general system.

Unpredictable Property of a Kicked Billiard Ball

A simultaneous change in the systemic property of a kicked billiard ball is observed from an entirely smooth and conservative state to a piecewise smooth and quasi-dissipative state when a single controlling parameter has been adjusted. The transition induces a sudden change of a typical conservative stochastic web into a transient web. The iterations on the transient web eventually escape to some elliptic islands. In the meantime, a fat fractal forbidden web, which appears also at the threshold, grows up and cuts away increasingly more parts from the original conservative stochastic web. We numerically show that the initial conditions that generated different attractors are mixed in a random manner and the pattern remains unchanged even when smaller and smaller scales are used for examination, indicating a riddle-like basin structure that practically rules out the possibility of predicting the attractors from a given initial condition.

Negative Magneto—Resistance Beyond Weak Localization in Three—Dimensional Billiards：Effect of Arnold Diffusion

We investigate a semiclassical conductance for ballistic open three-dimensional (3-d) billiards. For partially or completely broken-ergodic 3-d billiards such as SO(2) symmetric billiards, the dependence of the conductance on the Fermi wavenumber is dramatically changed by the lead orientation. Application of a symmetry-breaking weak magnetic field brings about mixed phase-space structures of 3-d billiards which ensures a novel Arnold diffusion that cannot be seen in 2-d billiards. In contrast to the 2-d case, the anomalous increment of the conductance should inevitably include a contribution arising from Arnold diffusion as well as a weak localization correction. Discussions are devoted to the physical condition for observing this phenomenon.

Universality in Statistical Measures of Trajectories in Classical Billiard Systems

For classical billiards, we suggest that a matrix of action or length of trajectories in conjunction with statistical measures, level spacing distribution and spectral rigidity, can be used to distinguish chaotic from integrable systems. As examples of 2D chaotic billiards, we considered the Bunimovich stadium billiard and the Sinai billiard. In the level spacing distribution and spectral rigidity, we found GOE behaviour consistent with predictions from random matrix theory. We studied transport properties and computed a diffusion coefficient. For the Sinai billiard, we found normal diffusion, while the stadium billiard showed anomalous diffusion behaviour. As example of a 2D integrable billiard, we considered the rectangular billiard. We found very rigid behaviour with strongly correlated spectra similar to a Dirac comb. These findings present numerical evidence for universality in level spacing fluctuations to hold in classically integrable systems and in classically fully chaotic systems.

On Differential Equations of Integrable Billiard Tables

We introduce a method to find differential equations for functions which define tables,such that associated billiard systems admit a local first integral.We illustrate this method in three situations:the case of(locally)integrable wire billiards,for finding surfaces in R^(3)with a first integral of degree one in velocities,and for finding a piece-wise smooth surface in R^(3)homeomorphic to a torus,being a table of a billiard admitting two additional first integrals.

基于有限元分析的一体式台球杆力学性能评价

为探究台球杆杆体材料的选择依据,采用有限元分析法评价不同树种木材制作的台球杆杆体的力学性能。杆体材料选取白蜡木(Fraxinus chinensis)、榉木(Zelkova schneideriana)、槲栎(Quercus aliena)、槭木(Acer mono)、降香黄檀(Dalbergia odorifera)和核桃木(Juglans regia)等6种木材,借助ANSYS Workbench仿真呈现台球杆击球时碰撞情况,分析不同材料台球杆的力学性能:通过对比不同台球杆击球后台球获得的速度值大小,判断杆体传力性能优劣,并根据台球杆在击球过程中承受的最大应力值判断其强度性能。结果表明:6种材料的台球杆中,白蜡木台球杆的传力性能最佳,而降香黄檀台球杆的传力性能最差,台球杆的传力性能与木材密度、ER、ET、μRT、μLT呈现负相关,与GLT、GLR和μLR均呈正相关,6种台球杆都具有足够的强度保证在击球时台球杆不发生破坏。

中式台球赛事发展研究

采用文献资料法,对近年来中式台球运动项目的发展演变和赛事发展进行研究,目的是发现中式台球发展过程中存在的问题,讨论并提出中式台球赛事良性发展的措施,弘扬中式台球文化。通过研究发现,中式台球赛事仍处于萌芽发展时期,赛事发展面临巨大挑战的同时,也有良好的发展前景;品牌赛事相对较少,普及程度不高;职业化发展体系未能更好地被运用。建议中式台球赛事发展规模扩大,满足群众、学生、职业选手的竞赛需求;扩大广告和筹资渠道,增加赛事影响力,吸引人们的关注,推动地方品牌台球赛事;促进台球文化传播,营造良好的台球文化环境。

Surface Billiards and Nonvanishing Lyapunov Exponents

We study in this paper the billiards on surfaces with mix-valued Gaussian curvature and the condition which gives nonvanishing Lyapunov exponents of the system.We introduce a criterion upon which a small perturbation of the surface will also produce a system with positive Lyapunov exponents.Some examples of such surfaces are given in this article.

Estimates for Correlations in Billiards with Large Arcs

Bunimovich billiards are ergodic and mixing. However, if the billiard table contains very large arcs on its boundary then if there exist trajectories experience infinitely many collisions in the vicinity of periodic trajectories on the large arc. The hyperbolicity is nonuniform and the mixing rate is very slow. The corresponding dynamics are intermittent between regular and chaotic, which makes them particularly interesting in physical studies. The study of mixing rates in intermittent chaotic systems is more difficult than that of truly chaotic ones, and the resulting estimates may depend on delicate details of the dynamics in the traps. We present a rigorous analysis of the corresponding singularities and correlations to certain class of billiards and show the mixing rate is of order 1/n.

On the Integrability of Strictly Convex Billiard Tables with Boundaries Close to Ellipses with Small Eccentricities

In this paper,we introduce a new notion of integrability for billiard tables,namely,integrability away from the boundary.One key feature of our notion is that the integrable region could be disjoint from the boundary with a positive distance.We prove that if a strictly convex billiard table,whose boundary is a small perturbation of an ellipse with small eccentricity,is integrable in this sense,then its boundary must be itself an ellipse.

带有辅助瞄准功能的中式台球母球定位器

该文阐述一种带有辅助瞄准功能,并且可以确定母球运动方向的中式台球母球定位器,可以在母球产生静电时固定母球的位置,帮助初学者和想要提高中式台球水平的玩家掌握一定的技巧,培养自己的打球习惯,提高进球准确率,改掉错误击球法。该装置由圆弧卡球器、三角形指针、轻质钉状物、增稳部和底盘构成。此定位器可帮助爱好者准确掌握袋口、目标球和直射点三者空间与位置关系,可以培养根据袋口和目标球位置,对直射点的空间进行定位判断的能力;还可以帮助爱好者掌握分离角指示原理,从而提高进球的准确率。