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.

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.

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.

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.

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.

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.

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.

Semiclassical Analysis of Quarter Stadium Billiards

为静止状态的一个扩大方法被使用获得特徵函数和 1/4 个体育场台球的 eigenenergies，并且它的最近的 energy-levelspacing 分发被获得。直方图与标准 Wigner 分发一致，它显示体育场台球系统是混乱的。特别注意对追求如此的古典混乱的量表明被给予。在量系列和古典轨道的 Fouriertransformation 之间的通讯被使用 closed-orbittheory 调查。分析、数字的结果与要求的分辨率一致，它支持半古典作品方法提供一幅身体上有意义的图象理解如此的混乱系统。

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.

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.

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。

台球运动中主球运动轨迹的仿真分析

在台球完全弹性碰撞的假设下,运用基本的理论力学原理对主球在碰撞目标球后、碰桌边前的运动规律进行了分析,给出了这段时间内主球的运动控制方程。用Matlab仿真分析了主球的质心速度和旋转角速度对其运动轨迹的影响,并提出了一些针对球手训练的建议。

正方体量子台球的半经典分析(英文)

近几年,量子台球问题引起人们的广泛兴趣.以前有很多对二维量子台球做过研究,相对于二维台球来说,三维台球更接近实际体系.本文以三维正方体量子台球为例,利用半经典闭合轨道理论计算了正方体量子台球中的经典开轨道,并研究量子谱函数与经典轨道长度之间的对应关系,发现他们之间对应的很好.这将有助于我们分析开放型量子台球中输运性质问题.利用这种方法物理图像清晰,计算量小并且可以帮助理解一些混沌体系的性质.这是半经典理论为联系量子力学与经典力学起桥梁作用的又一证明。