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We develop an analytical framework and numerical approach to obtain the coefficient of self-diffusivity for the transport of a rarefied gas in channels in the limit of large Knudsen number. This framework provides a method for determining the influence of channel surface microstructure on the value of diffusivity that is particularly effective when the microstructure exhibits relatively low roughness. This method is based on the observation that the Markov transition (scattering) operator determined by the microstructure, under the condition of weak surface scattering, has a universal form given, up to a multiplicative constant, by the classical Legendre differential operator. We also show how characteristic numbers of the system--namely, geometric parameters of the microstructure, the spectral gap of a Markov operator, and the tangential momentum accommodation coefficient of a commonly used model of surface scattering--are all related. Examples of microstructures are investigated to illustrate the relation of these quantities numerically and analytically. [ABSTRACT FROM AUTHOR]

Discovered by William Chapple in 1746, the Poristic family is a set of variable-perimeter triangles with common Incircle and Circumcircle. By definition, the family has constant Inradius-to-Circumradius ratio. Interestingly, this invariance also holds for the family of 3-periodics in the Elliptic Billiard, though here Inradius and Circumradius are variable and perimeters are constant. Indeed, we show one family is mapped onto the other via a varying similarity transform. This implies that any scale-free quantities and invariants observed in one family must hold on the other. [ABSTRACT FROM AUTHOR]

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 (k , p) , there is a constant ξ for which periodic orbits consisting of k impacts per period p cannot be sustained for amplitudes of oscillation below ξ. We compute a verified upper bound for the conjectured critical amplitude for (k , p) = (2 , 2) using our rigorous pseudo-arclength continuation. [ABSTRACT FROM AUTHOR]

• Identifies issue of "body coherence" in curved-space VR games. • Presents solution to body coherence problem in curved-space billiards. • Discusses "native-inhabitant view" vs. "tourist view" in curved-space VR. • Simplifies understanding of projection transformations in curved-space VR. [Display omitted] Virtual-reality simulations of curved space are most effective and most fun when presented as a game (for example, curved-space billiards), so the user not only has something to see in the curved space, but also has something fun to do there. However, such simulations encounter a geometrical problem: they must track the player's hands as well as her head, and in curved space the effects of holonomy would quickly lead to violations of body coherence. That is, what the player sees with her eyes would disagree with what she feels with her hands. This article presents a solution to the body coherence problem, as well as several other questions that arise in interactive VR simulations in curved space (radians vs. meters, visualization of the projection transformation, native-inhabitant view vs. tourist view, and mental models of curved space). [ABSTRACT FROM AUTHOR]

We study the dynamical billiards on a symmetric lemon table Q (b) , where Q (b) is the intersection of two unit disks with center distance b. We show that there exists δ 0 > 0 such that for all b ∈ (1.5 , 1.5 + δ 0) (except possibly a discrete subset), the billiard map F b on the lemon table Q (b) admits crossing homoclinic and heteroclinic intersections. In particular, such lemon billiards have positive topological entropy. [ABSTRACT FROM AUTHOR]

We show that there exists a C 2 open dense set of convex bodies with smooth boundary whose billiard map exhibits a non-trivial hyperbolic basic set. As a consequence billiards in generic convex bodies have positive topological entropy and exponential growth of the number of periodic orbits. [ABSTRACT FROM AUTHOR]

Copyright of Cuadernos del Centro de Estudios de Diseño y Comunicación is the property of Cuadernos del Centro de Estudios de Diseno y Comunicacion and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Billiards is a sport that not only requires numerous skills, but also emphasizes the strategies of planning the path of the contact and shot. Traditional billiards training mainly focuses on skills practicing, while the strategies for facing different kinds of challenges in billiards competitions are usually ignored. Therefore, referring to social constructivism, the current study proposed an ASQI (annotation, summarizing, questioning and interflow) mobile flipped learning approach to strengthen students’ pre-class learning effectiveness via engaging them in social interaction contexts to share experience and strategies of dealing with different billiards competition cases. An experiment was conducted to evaluate the effectiveness of the proposed approach. The participants were 80 university students from two classes; one class learned with the ASQI mobile flipped learning approach while the other adopted the traditional flipped learning approach. The results showed that ASQI could significantly enhance students’ billiards skills performance and online learning engagement with no influence on task load. In addition, according to the students’ feedback, ASQI could help students “make effective use of strategies,” “improve learning performance” and “promote reflection.”

We show that certain billiard flows on planar billiard tables with horns can be modeled as suspension flows over Young towers (Ann. Math. 147:585–650, 1998) with exponential tails. This implies exponential decay of correlations for the billiard map. Because the height function of the suspension flow itself is polynomial when the horns are Torricelli-like trumpets, one can derive Limit Laws for the billiard flow, including Stable Limits if the parameter of the Torricelli trumpet is chosen in (1, 2). [ABSTRACT FROM AUTHOR]

We present the probability distribution function of the nodal domain number for the right-isosceles triangle billiard. This is the first explicit result for a nonseparable plane polygonal billiard. The distribution function has a peak at (the same as for integrable billiards) and also at . The only exact results previously known in this direction correspond to integrable billiards that are separable. [ABSTRACT FROM AUTHOR]

In this article, we study the decay rates of the correlation functions for a hyperbolic system T : M → M with singularities that preserves a unique mixing SRB measure μ. We prove that, under some general assumptions, the correlations decay exponentially as n → ∞ for each pair of piecewise Hölder observables f, g ∈ Lp(μ) and for each p > 1. As an application, we prove that the autocorrelations of the first return time functions decay exponentially for the induced maps of various billiard systems, which include the semi-dispersing billiards on a rectangle, billiards with cusps, and Bunimovich stadia. These estimates of the decay rates of autocorrelations of the first return time functions for the induced maps have an essential importance in the study of the statistical properties of nonuniformly hyperbolic systems (with singularities). [ABSTRACT FROM AUTHOR]

We obtain sharp error rates in the local limit theorem for the Sinai billiard map (one and two dimensional) with infinite horizon. This result allows us to further obtain higher order terms and thus, sharp mixing rates in the speed of mixing of dynamically Hölder observables for the planar and tubular infinite horizon Lorentz gases in the map (discrete time) case. We also obtain an asymptotic estimate for the tail probability of the first return time to the initial cell. In the process, we study families of transfer operators for infinite horizon Sinai billiards perturbed with the free flight function and obtain higher order expansions for the associated families of eigenvalues and eigenprojectors. [ABSTRACT FROM AUTHOR]

Consider a strictly convex set Ω in the plane, and a homogeneous, stationary magnetic field orthogonal to the plane whose strength is B on the complement of Ω and 0 inside Ω. The trajectories of a charged particle in this setting are straight lines concatenated with circular arcs of Larmor radius μ. We examine the dynamics of such a particle and call this inverse magnetic billiards. Comparisons are made to standard Birkhoff billiards and magnetic billiards, as some theorems regarding inverse magnetic billiards are consistent with each of these billiard variants while others are not. [ABSTRACT FROM AUTHOR]

A system of ordinary differential equations that describes the motion of a self-propelling rigid disk is studied. In this system, the disk moves along a straight-line and reflects from a boundary. Interestingly, numerical simulation shows that the angle of reflection is greater than that of incidence. The purpose of this study is to present a mathematical proof for this attractive phenomenon. Moreover, the reflection law is numerically investigated. Finally, existence and asymptotic stability of a square-shaped closed orbit for billiards in square table with inelastic reflection law are discussed. [ABSTRACT FROM AUTHOR]

We consider a class of random mechanical systems called random billiards to study the problem of quantifying the irreversibility of nonequilibrium macroscopic systems. In a random billiard model, a point particle evolves by free motion through the interior of a spatial domain, and reflects according to a reflection operator, specified in the model by a Markov transition kernel, upon collision with the boundary of the domain. We derive a formula for entropy production rate that applies to a general class of random billiard systems. This formula establishes a relation between the purely mathematical concept of entropy production rate and textbook thermodynamic entropy, recovering in particular Clausius' formulation of the second law of thermodynamics. We also study an explicit class of examples whose reflection operator, referred to as the Maxwell-Smoluchowski thermostat, models systems with boundary thermostats kept at possibly different temperatures. We prove that, under certain mild regularity conditions, the class of models are uniformly ergodic Markov chains and derive formulas for the stationary distribution and entropy production rate in terms of geometric and thermodynamic parameters. [ABSTRACT FROM AUTHOR]

We introduce the asymmetric wedge billiard as a generalization of the wedge billiard first studied by Lehtihet and Miller in 1986. This is a billiard system in which the billiard ball moves under the influence of a constant gravitational field, colliding elastically with two wedge walls with the collisions obeying the reflection law. Collision maps are given from which derivatives and area-preservation (or lack thereof) were determined. Expressions for the fixed points of the collision maps were also calculated and discussed. Long-term dynamics were determined computationally from which we observed integrable, quasi-periodic and chaotic behavior which were all dependent on the wedge angles. [ABSTRACT FROM AUTHOR]

In this paper, a novel symmetric image encryption algorithm is proposed on the basis of the chaotic billiard (Sinai billiard). These chaotic systems, characterized by the highest degree of chaos, have been recently introduced in cryptography. The design of the algorithm relies on the random walk of three-point particles that move freely in the billiard. It includes several rounds of the confusion and diffusion processes. In each ciphering round, the permutation of the image pixels is performed according to the random numbers obtained from the output of a particle system, by using a new calculation technique. Thus, the pixel values are hidden by the pseudo-random sequences generated by the systems of the other two particles. The simulation results demonstrate that the algorithm is simple to implement and is highly secure. It has successfully passed all security tests and has shown the desirable properties in a good secure cryptosystem. [ABSTRACT FROM AUTHOR]

Background: This study aimed to develop new test protocols for evaluating 9-ball expertise levels in cue sports players. Methods: Thirty-one male 9-ball players at different playing levels were recruited (recreational group, n = 8; university team, n = 15; national team, n = 8). A 15-ball test was administered to indicate overall performance by counting the number of balls potted. Five skill tests (power control, cue alignment, angle, back spin, and top spin) were conducted to evaluate specific techniques by calculating error distances from pre-set targets using 2D video analysis. Results: Intra-class correlation analyses revealed excellent intra-rater and inter-rater reliability in four out of five skill tests (ICC > 0.95). Significant between-group differences were found in 15-ball test performance (p < 0.001) and absolute error distances in the angle (p < 0.001), back spin (p = 0.006), and top spin tests (p = 0.045), with the recreational group performing worst while the national team performing best. Greater inter-trial variability was observed in recreational players than the more skilled players (p < 0.005). Conclusions: In conclusion, the 9-ball test protocols were reliable and could successfully discriminate between different playing levels. Coaches and researchers may employ these protocols to identify errors, monitor training, and rank players. [ABSTRACT FROM AUTHOR]

Motor-learning literature focuses on simple laboratory-tasks due to their controlled manner and the ease to apply manipulations to induce learning and adaptation. Recently, we introduced a billiards paradigm and demonstrated the feasibility of real-world-neuroscience using wearables for naturalistic full-body motion-tracking and mobile-brain-imaging. Here we developed an embodied virtual-reality (VR) environment to our real-world billiards paradigm, which allows to control the visual feedback for this complex real-world task, while maintaining sense of embodiment. The setup was validated by comparing real-world ball trajectories with the trajectories of the virtual balls, calculated by the physics engine. We then ran our short-term motor learning protocol in the embodied VR. Subjects played billiard shots when they held the physical cue and hit a physical ball on the table while seeing it all in VR. We found comparable short-term motor learning trends in the embodied VR to those we previously reported in the physical real-world task. Embodied VR can be used for learning real-world tasks in a highly controlled environment which enables applying visual manipulations, common in laboratory-tasks and rehabilitation, to a real-world full-body task. Embodied VR enables to manipulate feedback and apply perturbations to isolate and assess interactions between specific motor-learning components, thus enabling addressing the current questions of motor-learning in real-world tasks. Such a setup can potentially be used for rehabilitation, where VR is gaining popularity but the transfer to the real-world is currently limited, presumably, due to the lack of embodiment. [ABSTRACT FROM AUTHOR]

Recently, physical billiards were introduced where a moving particle is a hard sphere rather than a point as in standard mathematical billiards. It has been shown that in the same billiard tables, the physical billiards may have totally different dynamics than mathematical billiards. This difference appears if the boundary of a billiard table has visible singularities (internal corners if the billiard table is two-dimensional); i.e., the particle may collide with these singular points. Here, we consider the collision of a hard ball with a visible singular point and demonstrate that the motion of the smooth ball after collision with a visible singular point is indeed the one that was used in the studies of physical billiards. Therefore, such collision is equivalent to the elastic reflection of hard ball's center off a sphere with the center at the singular point and the same radius as the radius of the moving particle. However, a ball could be rough, not smooth. In difference with a smooth ball, a rough ball also acquires rotation after reflection off a point of the boundary, which leads to more complicated dynamics. [ABSTRACT FROM AUTHOR]

Asymmetric lemon billiards was introduced in Chen et al (2013 Chaos 23 043137), where the billiard table Q(r, b, R) is the intersection of two round disks with radii r ⩽ R, respectively, and b measures the distance between the two centres. It is conjectured Bunimovich et al (2016 Commun. Math. Phys. 341 781–803) that the asymmetric lemon billiards is hyperbolic when the arc Γr is a major arc and R is large. In this paper we prove this conjecture for sufficiently large R. [ABSTRACT FROM AUTHOR]

A class of force evolutionary billiards is discovered that realizes important integrable Hamiltonian systems on all regular isoenergy 3-surfaces simultaneously, i.e., on the phase 4-space. It is proved that the well-known Euler and Lagrange integrable systems are billiard equivalent, although the degrees of their integrals are different (two and one). [ABSTRACT FROM AUTHOR]

We consider polygonal billiards with collisions contracting the reflection angle towards the normal to the boundary of the table. In previous work, we proved that such billiards have a finite number of ergodic SRB measures supported on hyperbolic generalized attractors. Here we study the relation of these measures with the ergodic absolutely continuous invariant probabilities (acips) of the slap map, the 1-dimensional map obtained from the billiard map when the angle of reflection is set equal to zero. We prove that if a convex polygon satisfies a generic condition called (*), and the reflection law has a Lipschitz constant sufficiently small, then there exists a one-to-one correspondence between the ergodic SRB measures of the billiard map and the ergodic acips of the corresponding slap map, and moreover that the number of Bernoulli components of each ergodic SRB measure equals the number of the exact components of the corresponding ergodic acip. The case of billiards in regular polygons and triangles is studied in detail. [ABSTRACT FROM AUTHOR]

A Circumconic passes through a triangle's vertices; an Inconic is tangent to the sidelines. We study the variable geometry of certain conics derived from the 1d family of 3-periodics in the Elliptic Billiard. Some display intriguing invariances such as aspect ratio and pairwise ratio of focal lengths. We also define the Circumbilliard, a circumellipse to a generic triangle for which the latter is a 3-periodic. [ABSTRACT FROM AUTHOR]

Copyright of Revista Soletras is the property of Editora da Universidade do Estado do Rio de Janeiro (EdUERJ) and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

We introduce the Iris billiard that consists of a point particle enclosed by a unit circle around a central scattering ellipse of fixed elongation (defined as the ratio of the semi-major to the semi-minor axes). When the ellipse degenerates to a circle, the system is integrable; otherwise, it displays mixed dynamics. Poincaré sections are presented for different elongations. Recurrence plots are then applied to the long-term chaotic dynamics of trajectories launched from the unstable period-2 orbit along the semi-major axis, i.e., one that initially alternately collides with the ellipse and the circle. We obtain numerical evidence of a set of critical elongations at which the system undergoes a transition to global chaos. The transition is characterized by an endogenous escape event, E , which is the first time a trajectory launched from the unstable period-2 orbit misses the ellipse. The angle of escape, θ e s c , and the distance of the closest approach, d m i n , of the escape event are studied and are shown to be exquisitely sensitive to the elongation. The survival probability that E has not occurred after n collisions is shown to follow an exponential distribution. [ABSTRACT FROM AUTHOR]

We construct families of two-dimensional Sinai billiards whose transfer operators have Ruelle resonances arbitrarily close to 1. Our method involves taking a large enough cover of an initial billiard table, and relating the transfer operator of the covering table to twisted transfer operators of the initial table. We also study the distribution of these resonances which are close to 1. [ABSTRACT FROM AUTHOR]

The article is devoted to the study of billiard sport as a specific type of sports activity. The history, its types and features of the rules of the game of Billiards are presented.

In a previous study of Dettmann and Georgiou, it was considered such a situation that point-wise particles repeatedly and elastically collided with the wall of a container without any particle-particle collision and eventually escaped through a small window from the container, which can be modeled by an open billiard. In the present study, point-wise particles were replaced with finite-size disks and disk-disk collisions were also considered, and inverse power laws of distributions of dwell time in the container were obtained from numerical simulations. [ABSTRACT FROM AUTHOR]