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We rigorously state the connection between the EHZ-capacity of convex Lagrangian products K × T ⊂ R n × R n and the minimal length of closed (K, T)-Minkowski billiard trajectories. This connection was made explicit for the first time by Artstein–Avidan and Ostrover under the assumption of smoothness and strict convexity of both K and T. We prove this connection in its full generality, i.e., without requiring any conditions on the convex bodies K and T. This prepares the computation of the EHZ-capacity of convex Lagrangian products of two convex polytopes by using discrete computational methods. [ABSTRACT FROM AUTHOR]

Given a random map (T1,T2,T3,T4,p1,p2,p3,p4), we define a random billiard map on a surface of constant curvature (Euclidean plane, hyperbolic plane, or the sphere). The Liouville measure is invariant for this billiard map. Finally, we show some dynamical properties such as ergodicity in the case of random circular billiards. [ABSTRACT FROM AUTHOR]

We construct surface codes corresponding to genus greater than one in the context of quantum error correction. The architecture is inspired by the topology of invariant integral surfaces of certain non-integrable classical billiards. Corresponding to the fundamental domains of rhombus and square torus billiard, surface codes of genus two and five are presented here. There is significant improvement in encoding rates and code distance, in addition to immunity against noise. [ABSTRACT FROM AUTHOR]

This study aimed to investigate individual trial-to-trial performance in three tests to define adaptive regulation as a key feature of expertise in nine-ball. Thirty-one male players were assigned into the low-skilled (n = 11), intermediate (n = 10), or high-skilled groups (n = 10). The power control, cue alignment, and angle tests were selected to assess participants' ability to control the power applied in shots, strike the ball straight, and understand the ball paths, respectively. Error distance and correction of error distance were identified for each shot using 2D video analysis. Results of one-way analysis of variance showed that the high-skilled group performed better in two out of the three tests than the other two groups (p = .010 for the cue alignment test; p = .002 for the angle test). However, the adaptation effect represented by the decreased error distances across trials was not observed. Pearson correlation revealed only a few significant correlations between the error distance and its correction within each participant in all tests (p < .05), and hence, the hypothesis that "low correction happened after small error and vice versa" is not supported. [ABSTRACT FROM AUTHOR]

In this paper we consider rectangular billiard tables having vertices with integer coordinates, and side lengths equal to integer multiples of the norms of the side directions. We also assume that all the bouncing points of a billiard ball are constrained to belong to the integer lattice Z 2. We address several questions concerning combinatorial and geometric properties of the allowed orbits, that, due to the integer constraint, are called integer orbits. We give a complete classification of integer orbits, and the parameters contributing to their structure are precisely determined. This leads to understand how the orbit fills the lattice billiard before it really propagates. In particular, one can characterize the trajectories that reach a billiard pocket, as well as all the closed orbits, by the simple knowledge of the size of the billiard table, and of the starting moving direction. The characterization bases on the explicit determination of the numerical sequences corresponding to clockwise, and counterclockwise, bouncing. We also investigate the geometrical structure of an allowed orbit in terms of special sub-patterns, called Z -paths, pointing out the allowed lengths of different Z -paths in a same orbit. This is of independent interest, and is related to the configurations known as switching components, that play a crucial role in discrete tomography, and in problems concerning image reconstruction. [ABSTRACT FROM AUTHOR]

In this paper, we consider covering graphs obtained by lifting a tree with a loop at each vertex as a voltage graph over a cyclic group. We generalize a tool of Hell et al. (2020), known as the billiard strategy, for constructing Hamiltonian cycles in the covering graphs of paths. We show that our extended tool can be used to provide new sufficient conditions for the Hamiltonicity of covering graphs of trees that are similar to those of Batagelj and Pisanski (1982) and of Hell et al. (2020). Next, we focus specifically on covering graphs obtained from trees lifted as voltage graphs over cyclic groups Z p of large prime order p. We prove that for a given reflexive tree T whose edge labels are assigned uniformly at random from a finite set, the corresponding lift is almost surely Hamiltonian for a large enough prime-ordered cyclic group Z p. Finally, we show that if a reflexive tree T is lifted over a group Z p of a large prime order, then for any assignment of nonzero elements of Z p to the edges of T , the corresponding cover of T has a large circumference. [ABSTRACT FROM AUTHOR]

Keywords: Billiards; systems with impacts; projective billiards; periodic orbits; open sets of periodic orbits; periodic points; Ivrii's conjecture; Weyl's law.GraphBy Corentin FierobeReported by Author [Extracted from the article]

Starting from billiard partitions which arose recently in the description of periodic trajectories of ellipsoidal billiards in d-dimensional Euclidean space, we introduce a new type of separable integer partition classes, called type B. We study the numbers of basis partitions with d parts and relate them to the Fibonacci sequence and its natural generalizations. Remarkably, the generating series of basis partitions can be related to the quiver generating series of symmetric quivers corresponding to the framed unknot via knots-quivers correspondence, and to the count of Schröder paths. [ABSTRACT FROM AUTHOR]

Abstract This research paper presents a comprehensive investigation into the utilization of color image processing technologies and deep learning algorithms in the development of a robot vision system specifically designed for 8-ball billiards. The sport of billiards, with its various games and ball arrangements, presents unique challenges for robotic vision systems. The proposed methodology addresses these challenges through two main components: object detection and ball pattern recognition. Initially, a robust algorithm is employed to detect the billiard balls using color space transformation and thresholding techniques. This is followed by determining the position of the billiard table through strategic cropping and isolation of the primary table area. The crucial phase involves the intricate task of recognizing ball patterns to differentiate between solid and striped balls. To achieve this, a modified convolutional neural network is utilized, leveraging the Xception network optimized by an innovative algorithm known as the Improved Chaos African Vulture Optimization (ICAVO) algorithm. The ICAVO algorithm enhances the Xception network's performance by efficiently exploring the solution space and avoiding local optima. The results of this study demonstrate a significant enhancement in recognition accuracy, with the Xception/ICAVO model achieving remarkable recognition rates for both solid and striped balls. This paves the way for the development of more sophisticated and efficient billiards robots. The implications of this research extend beyond 8-ball billiards, highlighting the potential for advanced robotic vision systems in various applications. The successful integration of color image processing, deep learning, and optimization algorithms shows the effectiveness of the proposed methodology. This research has far-reaching implications that go beyond just billiards. The cutting-edge robotic vision technology can be utilized for detecting and tracking objects in different sectors, transforming industrial automation and surveillance setups. By combining color image processing, deep learning, and optimization algorithms, the system proves its effectiveness and flexibility. The innovative approach sets the stage for creating advanced and productive robotic vision systems in various industries.

This research paper presents a comprehensive investigation into the utilization of color image processing technologies and deep learning algorithms in the development of a robot vision system specifically designed for 8-ball billiards. The sport of billiards, with its various games and ball arrangements, presents unique challenges for robotic vision systems. The proposed methodology addresses these challenges through two main components: object detection and ball pattern recognition. Initially, a robust algorithm is employed to detect the billiard balls using color space transformation and thresholding techniques. This is followed by determining the position of the billiard table through strategic cropping and isolation of the primary table area. The crucial phase involves the intricate task of recognizing ball patterns to differentiate between solid and striped balls. To achieve this, a modified convolutional neural network is utilized, leveraging the Xception network optimized by an innovative algorithm known as the Improved Chaos African Vulture Optimization (ICAVO) algorithm. The ICAVO algorithm enhances the Xception network's performance by efficiently exploring the solution space and avoiding local optima. The results of this study demonstrate a significant enhancement in recognition accuracy, with the Xception/ICAVO model achieving remarkable recognition rates for both solid and striped balls. This paves the way for the development of more sophisticated and efficient billiards robots. The implications of this research extend beyond 8-ball billiards, highlighting the potential for advanced robotic vision systems in various applications. The successful integration of color image processing, deep learning, and optimization algorithms shows the effectiveness of the proposed methodology. This research has far-reaching implications that go beyond just billiards. The cutting-edge robotic vision technology can be utilized for detecting and tracking objects in different sectors, transforming industrial automation and surveillance setups. By combining color image processing, deep learning, and optimization algorithms, the system proves its effectiveness and flexibility. The innovative approach sets the stage for creating advanced and productive robotic vision systems in various industries. [ABSTRACT FROM AUTHOR]

We show that smooth and strongly convex bodies in the symplectic |$\mathbb R^{2n}$| for |$n>1$| with all characteristics planar, or all outer billiard trajectories planar, are affine symplectic images of balls. [ABSTRACT FROM AUTHOR]

In this paper, two models of interest for Celestial Mechanics are presented and analysed, using both analytic and numerical techniques, from the point of view of the possible presence of regular and/or chaotic motion, as well as the stability of the considered orbits. The first model, presented in a Hamiltonian formalism, can be used to describe the motion of a satellite around Earth, taking into account both the non-spherical shape of our planet and the third-body gravitational influence of Sun and Moon. Using semi-analytical techniques coming from Normal Form and Nekhoroshev theories it is possible to provide stability estimates for the orbital elements of its geocentric motion. The second dynamical system presented can be used as a simplified model to describe the motion of a particle in an elliptic galaxy having a central massive core; it is constructed as a refraction billiard where an inner dynamics, induced by a Keplerian potential, is coupled with an external one, where a harmonic oscillator-type potential is considered. The investigation of the dynamics is carried on by using results of ODEs' theory and is focused on studying the trajectories' properties in terms of periodicity, stability and, possibly, chaoticity. [ABSTRACT FROM AUTHOR]

Given a polygon \(P\) in the triangular grid, we obtain a permutation \(\pi_P\) via a natural billiards system in which beams of light bounce around inside of \(P\). The different cycles in \(\pi_P\) correspond to the different trajectories of light beams. We prove that \[\operatorname{area}(P)\geq 6\operatorname{cyc}(P)-6\quad\text{and}\quad\operatorname{perim}(P)\geq\frac{7}{2}\operatorname{cyc}(P)-\frac{3}{2},\] where \(\operatorname{area}(P)\) and \(\operatorname{perim}(P)\) are the (appropriately normalized) area and perimeter of \(P\), respectively, and \(\operatorname{cyc}(P)\) is the number of cycles in \(\pi_P\). The inequality concerning \(\operatorname{area}(P)\) is tight, and we characterize the polygons \(P\) satisfying \(\operatorname{area}(P)=6\operatorname{cyc}(P)-6\). These results can be reformulated in the language of Postnikov's plabic graphs as follows. Let \(G\) be a connected reduced plabic graph with essential dimension \(2\). Suppose \(G\) has \(n\) marked boundary points and \(v\) (internal) vertices, and let \(c\) be the number of cycles in the trip permutation of \(G\). Then we have \[v\geq 6c-6\quad\text{and}\quad n\geq\frac{7}{2}c-\frac{3}{2}.\]Mathematics Subject Classifications: 05D99, 51M04Keywords: Triangular grid, billiards, plabic graph, membrane

Orbital invariants of integrable billiards on two-dimensional book tables are studied at constant energy values. These invariants are calculated from rotation functions defined on one-parameter families of Liouville 2-tori. For two-dimensional billiard books, a complete analogue of Liouville's theorem is proved, action–angle variables are introduced, and rotation functions are defined. A general formula for the rotation functions of such systems is obtained. For a number of examples, the monotonicity of these functions is studied, and edge orbital invariants (rotation vectors) are calculated. It turned out that not all billiards have monotonic rotation functions, as was originally assumed by A. Fomenko's hypothesis. However, for some series of billiards, this hypothesis is true. [ABSTRACT FROM AUTHOR]

We construct convex bodies that can be “captured by nets.” More precisely, for each dimension n ≥ 2 , we construct a family of Riemannian n-spheres, each with a stable geodesic net, which is a stable 1-dimensional integral varifold. Small perturbations of a stable geodesic net must lengthen it. These stable geodesic nets are composed of multiple geodesic loops based at the same point, and also do not contain any closed geodesic. All of these Riemannian n-spheres are isometric to convex hypersurfaces of R n + 1 with positive sectional curvature. [ABSTRACT FROM AUTHOR]

In this text we study billiards on symmetric ovals and investigate some consequences of the symmetry of the boundary on the dynamics. As it simplifies some calculations, the symmetry helps to obtain the results. We focus on periodic orbits with the same symmetry of the boundary which always exist and prove that typically half of them are elliptic and Moser stable and the other half are hyperbolic with homo(hetero)clinic intersections. [ABSTRACT FROM AUTHOR]

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 ℝ3 with a first integral of degree one in velocities, and for finding a piece-wise smooth surface in ℝ3 homeomorphic to a torus, being a table of a billiard admitting two additional first integrals. [ABSTRACT FROM AUTHOR]

Exact expressions for the relationship between angles of incidence and reflection from a boundary wall in nonlinear dissipative particle models are extremely rare. Here, we study a particle model of a camphor disk floating on water in a low speed limit, for which the model was derived. We begin with a very rough and inaccurate approximation, based in part on a Hamiltonian limit of the model, and then, using symmetry arguments supported by a series of experiments, show that this rough approximation can likely be repaired by introducing a factor dependent upon model parameters, then extract the full parameter dependence of this factor, and finally conjecture a simple and exact asymptotic relationship between the angles of incidence and reflection. There are reasons to believe that this asymptotic relationship may be universal for such models in their corresponding limits. [ABSTRACT FROM AUTHOR]

In a recent work, Baladi and Demers constructed a measure of maximal entropy for finite horizon dispersing billiard maps and proved that it is unique, mixing and moreover Bernoulli. We show that this measure enjoys natural probabilistic properties for Hölder continuous observables, such as at least polynomial decay of correlations and the Central Limit Theorem. The results of Baladi and Demers are subject to a condition of sparse recurrence to singularities. We use a similar and slightly stronger condition, and it has a direct effect on our rate of decay of correlations. For billiard tables with bounded complexity (a property conjectured to be generic), we show that the sparse recurrence condition is always satisfied and the correlations decay at a super‐polynomial rate. [ABSTRACT FROM AUTHOR]

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In this paper, we estimate the largest Lyapunov exponent for open billiards in the plane. We show that the largest Lyapunov exponent is differentiable with respect to a billiard deformation. [ABSTRACT FROM AUTHOR]

While quantum multifractality has been widely studied in the physics literature and is by now well understood from the point of view of physics, there is little work on this subject in the mathematical literature. I will report on the proof of multifractal scaling laws for arithmetic Seba billiards. I will explain the mathematical approach to defining the Rényi entropy associated with a sequence of eigenfunctions and sketch how arithmetic methods permit us to obtain a precise asymptotic in the semiclassical regime and how this allows us to compute the fractal exponents explicitly. Moreover, I will discuss how the symmetry relation for the fractal exponent is related to the functional equation of certain zeta functions. [ABSTRACT FROM AUTHOR]

We demonstrate the ability to control the scattering properties of a two-dimensional wave-chaotic microwave billiard through the use of tunable metasurfaces located on the interior walls of the billiard. The complex reflection coefficient of the metasurfaces can be varied by applying a DC voltage bias to varactor diodes on the mushroom-shaped resonant patches, and this proves to be very effective at perturbing the eigenmodes of the cavity. Placing multiple metasurfaces inside the cavity allows us to engineer desired scattering conditions, such as coherent perfect absorption, by actively manipulating the poles and zeros of the scattering matrix through the application of multiple voltage biases. We demonstrate the ability to create on-demand coherent perfect absorption conditions at a specific frequency, and document the near-null of output power as a function of four independent parameters tuned through the coherent perfect absorption point. A remarkably low output-to-input power ratio Pout=Pin = 3:71 x 10-8 is achieved near the coherent perfect absorption point at 8.54 GHz. [ABSTRACT FROM AUTHOR]

In the paper, we establish Squash Rigidity Theorem—the dynamical spectral rigidity for piecewise analytic Bunimovich squash-type stadia whose convex arcs are homothetic. We also establish Stadium Rigidity Theorem—the dynamical spectral rigidity for piecewise analytic Bunimovich stadia whose flat boundaries are a priori fixed. In addition, for smooth Bunimovich squash-type stadia we compute the Lyapunov exponents along the maximal period two orbit, as well as the value of the Peierls' Barrier function from the maximal marked length spectrum associated to the rotation number 2 n 4 n + 1 . [ABSTRACT FROM AUTHOR]

This paper investigates the behaviour of open billiard systems in high-dimensional spaces. Specifically, we estimate the largest Lyapunov exponent, which quantifies the rate of divergence between nearby trajectories in a dynamical system. This exponent is shown to be continuous and differentiable with respect to a small perturbation parameter. A theoretical analysis forms the basis of the investigation. Our findings contribute to the field of dynamical systems theory and have significant implications for the stability of open billiard systems, which are used to model physical phenomena. The results provide a deeper comprehension of the behaviour of open billiard systems in high-dimensional spaces and emphasise the importance of taking small perturbations into consideration when analysing these systems. [ABSTRACT FROM AUTHOR]

Oblique angle collisions between two balls or two disks have been addressed by many authors. This paper describes oblique angle collisions between three or four billiard balls. Measurements and calculations are presented for cases where one ball is incident obliquely on two or three identical balls, which are in contact and initially at rest. Conservation laws alone are not sufficient to predict the outcome. Adding information about the impact forces on each ball (and ignoring friction forces) provides good theoretical agreement with experimental results. [ABSTRACT FROM AUTHOR]

We offer some theorems, mainly finiteness results, for certain patterns in elliptical billiards, related to periodic trajectories; these seem to be the first finiteness results in this context. For instance, if two players hit a ball at a given position and with directions forming a fixed angle in (0,π)$(0,\pi)$, there are only finitely many directions for both trajectories being periodic. Another instance is the finiteness of the billiard shots which send a given ball into another one so that this falls eventually in a hole. These results (which are shown not to hold for general billiards) have their origin in 'relative' cases of the Manin–Mumford conjecture and constitute instances of how arithmetical content may affect chaotic behaviour (in billiards). We shall also interpret the statements through a variant of the dynamical Mordell–Lang conjecture. In turn, this variant embraces cases, which, somewhat surprisingly, sometimes can be treated (only) by completely different methods compared to the former ones; here we shall offer an explicit example related to diophantine equations in algebraic tori. [ABSTRACT FROM AUTHOR]

A quadrilateral contains a closed billiard circuit that bounces consecutively on the interiors of its sides each exactly once per cycle, if and only if it is cyclic and its interior contains the center of its circumscribed circle. [ABSTRACT FROM AUTHOR]

Two-dimensional quantum billiards are one of the most important paradigms for exploring the connection between quantum and classical worlds. Researchers are mainly focused on nonintegrable and irregular shapes to understand the quantum characteristics of chaotic billiards. The emergence of the scarred modes relevant to unstable periodic orbits (POs) is one intriguing finding in nonintegrable quantum billiards. On the other hand, stable POs are abundant in integrable billiards. The quantum wavefunctions associated with stable POs have been shown to play a key role in ballistic transport. A variety of physical systems, such as microwave cavities, optical fibers, optical resonators, vibrating plates, acoustic waves, and liquid surface waves, are used to analogously simulate the wave properties of quantum billiards. This article gives a comprehensive review for the subtle connection between the quantum level clustering and the classical POs for three integrable billiards including square, equilateral triangle, and circular billiards. [ABSTRACT FROM AUTHOR]

The main objective of this research is to study the properties of a billiard system in an unbounded domain with moving boundary. We consider a system consisting of an infinite rod (a straight line) and a ball (a massless point) on the plane. The rod rotates uniformly around one of its points and experiences elastic collisions with the ball. We define a mathematical model for the dynamics of such a system and write down asymptotic formulae for its motions. In particular, we determine existence and uniqueness of solutions. We find all possible grazing impacts of the ball. Besides, we demonstrate that for almost every initial condition, the ball goes to infinity exponentially fast, with the time intervals between neighboring collisions tending to zero. The approach developed in this paper is an original combination of methods of Billiards and Vibro-Impact Dynamics. It could be a base for studying more complicated systems of similar types. [ABSTRACT FROM AUTHOR]

Background. Simulation models have been applied to analyze daily living activities and some sports movements. However, it is unknown whether the current upper extremity musculoskeletal models can be utilized for investigating cue sports movements to generate corresponding kinematic and muscle activation profiles. This study aimed to test the feasibility of applying simulation models to investigate cue sports players' cueing movements with OpenSim. Preliminary muscle forces would be calculated once the model is validated. Methods. A previously customized and validated unimanual upper extremity mus- culoskeletal model with six degrees of freedom at the scapula, shoulder, elbow, and wrist, as well as muscles was used in this study. Two types of cueing movements were simulated: (1) the back spin shot, and (2) 9-ball break shot. Firstly, kinematic data of the upper extremity joints were collected with a 3D motion capture system. Using the experimental marker trajectories of the back spin shot on 10 male cue sports players, the simulation on the cueing movements was executed. The model was then validated by comparing the model-generated joint angles against the experimental results using statistical parametric mapping (SPM1D) to examine the entire angle-time waveform as well as t -tests to compare the discrete variables (e.g., joint range of motion). Secondly, simulation of the break shot was run with the experimental marker trajectories and electromyographic (EMG) data of two male cue sports players as the model inputs. A model-estimated muscle activation calculation was performed accordingly for the upper extremity muscles. Results. The OpenSim-generated joint angles for the back spin shot corresponded well with the experimental results for the elbow, while the model outputs of the shoulder deviated from the experimental data. The discrepancy in shoulder joint angles could be due to the insufficient kinematic inputs for the shoulder joint. In the break shot simulation, the preliminary findings suggested that great shoulder muscle forces could primarily contribute to the forward swing in a break shot. This suggests that strengthening the shoulder muscles may be a viable strategy to improve the break shot performance. Conclusion. It is feasible to cater simulation modeling in OpenSim for biomechanical investigations of the upper extremity movements in cue sports. Model outputs can help better understand the contributions of individual muscle forces when performing cueing movements. [ABSTRACT FROM AUTHOR]

The use of virtual reality or augmented reality systems in billiards sports are useful tools for pure entertainment or improving the player's skills. Depending on the purpose of these systems, tracking algorithms based on computer vision must be used. These algorithms are especially useful in systems aiming to reconstruct the trajectories followed by the balls after a strike. However, depending on the billiard modality, the problem of tracking multiple small identical objects, such as balls, is a complex task. In addition, when an amateur or nontop professional player uses low-frame-rate and low-resolution devices, problems such as blurred balls, blurred contours, or fuzzy edges, among others, arise. These effects have a negative impact on ball-tracking accuracy and reconstruction quality. Thus, this work proposes two contributions. The first contribution is a new tracking algorithm called "multiobject local tracking (MOLT)". This algorithm can track balls with high precision and accuracy even with motion blur caused by low-resolution and low-frame-rate devices. Moreover, the proposed MOLT algorithm is compared with nine tracking methods and four different metrics, outperforming the rest of the methods in the majority of the cases and providing a robust solution. The second contribution is a whole system to track (using the MOLT algorithm) and reconstruct the movements of the balls on a billiard table in a 3D virtual world using computer vision. The proposed system covers all steps from image capture to 3D reconstruction. The 3D reconstruction results have been qualitatively evaluated by different users through a series of questionnaires, obtaining an overall score of 7.6 (out of 10), which indicates that the system is a promising and useful tool for training. Finally, both the MOLT algorithm and the reconstruction system are tested in three billiard modalities: blackball, carom billiards, and snooker. [ABSTRACT FROM AUTHOR]

We investigate a rotated, orthogonal gravitational wedge billiard — a special case of the asymmetric gravitational wedge billiard — in which the dynamics are integrable. We derive equations and conditions under which periodic orbits may be constructed for this model, and show that any other trajectory will be dense in the configuration space. [ABSTRACT FROM AUTHOR]

Galperin introduced an interesting method to learn the digits of π by counting the collisions of two billiard balls and a hard wall. This paper studies two quantum versions of the Galperin billiards. It is shown that the digits of π can be observed in the phase shifts of the quantum models. [ABSTRACT FROM AUTHOR]

We give closed-form expressions for recently discovered invariants of Poncelet N-periodic trajectories in the elliptic billiard with spatial integrals evaluated over the boundary of the elliptic billiard. The integrand is weighed by a measure equal to the density of rays hitting a given boundary point. We find that aperiodic averages are smooth and monotonic on caustic eccentricity, and are consistent with the invariant values predicted at the discrete caustic parameters which admit a given N-periodic family. Our results are verified by numerical simulation. [ABSTRACT FROM AUTHOR]

The propagation of electromagnetic waves in a closed domain with a reflecting boundary amounts, in the eikonal approximation, to the propagation of rays in a billiard. If the inner medium is uniform, then the symplectic reflection map provides the polygonal rays' paths. The linear response theory is used to analyze the stability of any trajectory. The Lyapunov and reversibility error invariant indicators provide an estimate of the sensitivity to a small initial random deviation and to a small random deviation at any reflection, respectively. A family of chaotic billiards is considered to test the chaos detection effectiveness of the above indicators. [ABSTRACT FROM AUTHOR]

Experimental data from Dunfield et al. using random grid diagrams suggest that the genus of a knot grows linearly with respect to the crossing number. Using billiard table diagrams of Chebyshev knots developed by Koseleff and Pecker and a random model of 2-bridge knots via these diagrams developed by the author with Krishnan and then with Even-Zohar and Krishnan, we introduce a further-truncated model of all 2-bridge knots of a given crossing number, almost all counted twice. We present a convenient way to count Seifert circles in this model and use this to compute a lower bound for the average Seifert genus of a 2-bridge knot of a given crossing number. [ABSTRACT FROM AUTHOR]

We show that in a typical polygon the billiard map as well as its associated subshift obtained by coding orbits by the sequence of sides they visit are topologically weakly mixing. [ABSTRACT FROM AUTHOR]

We consider billiards obtained by removing from the plane finitely many strictly convex analytic obstacles satisfying the non-eclipse condition. The restriction of the dynamics to the set of non-escaping orbits is conjugated to a subshift, which provides a natural labeling of periodic orbits. We show that under suitable symmetry and genericity assumptions, the Marked Length Spectrum determines the geometry of the billiard table. [ABSTRACT FROM AUTHOR]

We construct Birkhoff cones for dispersing billiards, which are contracted by the action of the transfer operator. This construction permits the study of statistical properties not only of regular dispersing billiards but also of sequential billiards (the billiard changes at each collision in a prescribed manner), open billiards (the dynamics exits some region or dies when hitting some obstacle) and many other examples. In particular, we include applications to chaotic scattering and the random Lorentz gas. [ABSTRACT FROM AUTHOR]

The article titled "Somewhere in This Town" is a baseball-themed poem written by Rick Burton. The poem reflects on memories of going to baseball games with the narrator's father and the nostalgia associated with those experiences. The narrator reminisces about specific players and moments from the past, highlighting the bond between the narrator and their father. The poem evokes a sense of sentimentality and appreciation for the game of baseball and its ability to create lasting memories. [Extracted from the article]

The coefficient of sliding friction is usually regarded as a constant for any given surfaces although it is sometimes known to vary with sliding speed. An experiment is described where the coefficient of friction for a billiard ball decreases to a very small value as the sliding speed decreases to zero, after which the ball starts rolling without sliding. A simple theoretical explanation is provided. [ABSTRACT FROM AUTHOR]