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We present a hierarchical model for ball action anticipation in football broadcast video. Given a 30-second observation window, the system predicts actions occurring in the subsequent 5-second window across 10 classes. A shared local Transformer encodes clip-level features within each 5-second sub-window; a GRU then aggregates temporal context across all sub-windows; finally, a Transformer decoder with K input-conditioned event slots decodes the anticipation target via three decoupled heads (objectness, class, temporal offset). We introduce frequency-reweighted Hungarian matching that systematically favours rare action classes, and Gaussian soft targets for temporal bin supervision. On the SoccerNet Ball Action Anticipation benchmark, our method achieves 17.91% mAP on the test server.

Balls-and-bins games have been a wildly successful tool for modeling load balancing problems. In this paper, we study a new scenario, which we call the ball recycling game, defined as follows: Throw m balls into n bins i.i.d. according to a given probability distribution p. Then, at each time step, pick a non-empty bin and recycle its balls: take the balls from the selected bin and re-throw them according to p. This balls-and-bins game closely models memory-access heuristics in databases. The goal is to have a bin-picking method that maximizes the recycling rate, defined to be the expected number of balls recycled per step in the stationary distribution. We study two natural strategies for ball recycling: Fullest Bin, which greedily picks the bin with the maximum number of balls, and Random Ball, which picks a ball at random and recycles its bin. We show that for general p, Random Ball is constant-optimal, whereas Fullest Bin can be pessimal. However, when p = u, the uniform distribution, Fullest Bin is optimal to within an additive constant.

We describe our system for the SoccerNet 2026 Player-Centric Ball-Action Spotting Challenge, which requires predicting who performs which action and when, across eight classes in broadcast soccer. Building on the three FOOTPASS baselines [1] (TAAD, TAAD+GNN, and TAAD+DST), we contribute four extensions: (1) gradient check pointing to enable full-backbone fine-tuning on a single GPU; (2) fusion of GNN logits into the DST encoder, combining graph-based tactical context with per-player visual features; (3) square-root frequency class weighting to address the 213:1 pass-to-tackle imbalance in the training data; and (4) a post processing pipeline comprising per-class logit gating, temporal frame refinement, jersey re-assignment, and a two-model ensemble. Our system achieves 0.548 Macro F1 on the test set and 0.446 on the challenge set (server evaluation).

The gauge-mediated SUSY-breaking (GMSB) model needs entropy production at a relatively low temperature in the thermal history of the Universe for the unwanted relics to be diluted. This requires a mechanism for the baryogenesis after the entropy production, and the Affleck and Dine (AD) mechanism is a promising candidate for it. The AD baryogenesis in the GMSB model predicts the existence of the baryonic Q ball, that is the B ball, and this may work as the dark matter in the Universe. In this article, we discuss the stability of the B ball in th presence of baryon-number violating interactions. We find that the evaporation rate increases monotonically with the B-ball charge because the large field value inside the B ball enhances the effect of the baryon-number-violating operators. While there are some difficulties to evaluate the evaporation rate of the B ball, we derive the evaporation time (lifetime) of the B ball for the mass-to-charge ratio $ω_0\gsim 100 \MEV$. The lifetime of the B ball and the distortion of the cosmic ray positron flux and the cosmic background radiation from the B ball evaporation give constraints on the baryon number of the B ball and the interaction, if t

We consider free boundary minimal submanifolds in geodesic balls in the hyperbolic space $\mathbb H^n$ and in the round upper hemisphere $\mathbb S^n_+$. Recently, Lima and Menezes have found a connection between free boundary minimal surfaces in geodesic balls in $\mathbb S^n_+$ and maximal metrics for a functional, defined on the set of Riemannian metrics on a given compact surface with boundary. This connection is similar to the connection between free boundary minimal submanifolds in Euclidean balls and the critical metrics of the functional "the $k$-th normalized Steklov eigenvalue", introduced by Faser and Schoen. We define two natural functionals on the set of Riemannian metrics on a compact surface with boundary. One of these functionals is the high order generalization of the functional, introduced by Lima and Menezes. We prove that the critical metrics for these functionals arise as metrics induced by free boundary minimal immersions in geodesic balls in $\mathbb H^n$ and in $\mathbb S^n_+$, respectively. We also prove a converse statement. Besides that, we discuss the (Morse) index of free boundary minimal submanifolds in geodesic balls in $\mathbb H^n$ or $\mathbb S^n_+

Visualization of data is an important step in the understanding of data and the evaluation of statistical models. Topological Data Analysis Ball Mapper (TDABM) after Dlotko (2019), provides a model free means to visualize multivariate datasets without information loss. To permit the construction of a TDABM graph, each variable must be ordinal and have sufficiently many values to make a scatterplot of interest. Where a scatterplot works with two, or three, axes, the TDABM graph can handle any number of axes simultaneously. The result is a visualization of the structure of data. The TDABM graph also permits coloration by additional variables, enabling the mapping of outcomes across the joint distribution of axes. The strengths of TDABM for understanding data, and evaluating models, lie behind a rapidly expanding literature. This guide provides an introduction to TDABM with code in Python.

Directional wavelet dictionaries are hierarchical representations which efficiently capture and segment information across scale, location and orientation. Such representations demonstrate a particular affinity to physical signals, which often exhibit highly anisotropic, localised multiscale structure. Many physically important signals are observed over spherical domains, such as the celestial sky in cosmology. Leveraging recent advances in computational harmonic analysis, we design new highly distributable and automatically differentiable directional wavelet transforms on the $2$-dimensional sphere $\mathbb{S}^2$ and $3$-dimensional ball $\mathbb{B}^3 = \mathbb{R}^+ \times \mathbb{S}^2$ (the space formed by augmenting the sphere with the radial half-line). We observe up to a $300$-fold and $21800$-fold acceleration for signals on the sphere and ball, respectively, compared to existing software, whilst maintaining 64-bit machine precision. Not only do these algorithms dramatically accelerate existing spherical wavelet transforms, the gradient information afforded by automatic differentiation unlocks many data-driven analysis techniques previously not possible for these spaces. We p

We study Nevanlinna theory of meromorphic mappings from a geodesic ball of a general complete Kähler manifold with non-negative Ricci curvature into a complex projective manifold by introducing a heat kernel method. When dimension of a target manifold is not greater than one of a source manifold, we establish a second main theorem which is a generalization of the classical second main theorem for a ball of $\mathbb C^m.$ If a source manifold is non-compact and it carries a positive global Green function, then we establish a global second main theorem for the source manifold. As a result, we obtain a Picard's theorem for complete Kähler manifolds with non-negative Ricci curvature.

We first introduce an appropriate family of conformally covariant boundary operators associated to the Siegel domain ${\mathcal U}^{n+1}$ with the Heisenberg group $\mathbb{H}^{n}$ as its boundary and the complex ball $\mathbb{B}_{\mathbb{C}}^{n+1}$ with the complex sphere $\mathbb{S}^{2n+1}$ as its boundary. We provide the explicit formulas of these conformally covariant boundary operators. Second, we establish all higher order extension theorems of Caffarelli-Silvestre type for the Siegel domain and complex ball. Third, we prove all higher order CR Sobolev trace inequalities for the Siegel domain ${\mathcal U}^{n+1}$ and the complex ball $\mathbb{B}_{\mathbb{C}}^{n+1}$.In particular, we generalize the Sobolev trace inequalityfor $γ\in (0, 1)$ in the CR setting by Frank-González-Monticelli-Tan to the case for all $γ\in (0, n+1)\backslash \mathbb{N}$. The family of higher order conformally covariant boundary operators we define are naturally intrinsic to the higher order Sobolev trace inequalities on both the Siegel domain ${\mathcal U}^{n+1}$ and complex ball $\mathbb{B}_{\mathbb{C}^{n+1}}$. Finally, we give an explicit solution to the scattering problem on the complex hyperbolic

For any $g_1, g_2 \ge 0$, this paper shows that there is a cocompact lattice $Γ< \mathrm{PU}(2,1)$ such that the ball quotient $Γ\backslash \mathbb{B}^2$ is birational to a product $C_1 \times C_2$ of smooth projective curves $C_j$ of genus $g_j$. The only prior examples were $\mathbb{P}^1 \times \mathbb{P}^1$, due to Deligne--Mostow and rediscovered by many others, and a lesser-known product of elliptic curves whose existence follows from work of Hirzebruch. Combined with related new examples, this answers the rational variant of a question of Gromov in the positive for surfaces of Kodaira dimension $κ\le 0$, namely that they admit deformations $V^\prime$ such that there is a compact ball quotient $Γ\backslash \mathbb{B}^2$ with a rational map $Γ\backslash \mathbb{B}^2 \dashrightarrow V^\prime$. Often the proof gives the stronger conclusion that $V^\prime$ is birational to a ball quotient orbifold. It also follows that every simply connected $4$-manifold is dominated by a complex hyperbolic manifold. All examples considered in this paper are shown to be arithmetic, and even arithmeticity of Hirzebruch's example appears to be new.

We present a full geometric characterization of the $1$-dimensional (semialgebraic) images $S$ of either $n$-dimensional closed balls $\overline{\mathcal B}_n\subset{\mathbb R}^n$ or $n$-dimensional spheres ${\mathbb S}^n\subset{\mathbb R}^{n+1}$ under polynomial, regular and regulous maps for some $n\geq1$. In all the previous cases one can find an alternative polynomial, regular or regulous map on either $\overline{\mathcal B}_1:=[-1,1]$ or ${\mathbb S}^1$ such that $S$ is the image under such map of either $\overline{\mathcal B}_1:=[-1,1]$ or ${\mathbb S}^1$. As a byproduct, we provide a full characterization of the images of ${\mathbb S}^1\subset{\mathbb C}\equiv{\mathbb R}^2$ under Laurent polynomials $f\in{\mathbb C}[{\tt z},{\tt z}^{-1}]$, taking advantage of some previous works of Kobalev-Yang and Wilmshurst. We also alternatively prove that all polynomial maps ${\mathbb S}^k\to{\mathbb S}^1$ are constant if $k\geq2$.

In this paper, the dynamics of a bouncing ball is described for several common ball types having different bounce characteristics. Results are presented for a tennis ball, a baseball, a golf ball, a superball, a steel ball bearing, a plasticene ball, and a silly putty ball. The plasticene ball was studied as an extreme case of a ball with a low coefficient of restitution (in fact zero, since the collision is totally inelastic) and the silly putty ball was studied because it has unusual elastic properties. The first three balls were studied because of their significance in the physics of sports. For each ball, a dynamic hysteresis curve is presented to show how energy is lost during and after the collision. The measurement technique is quite simple, it is suited for undergraduate laboratory experiments, and it may provide a useful method to test and approve balls for major sporting events.

A simple model is applied to study a high temperature rather dense plasma ball. It is assumed that the ions and delocalized electrons are distributed uniformly throughout the ball, and extra/missing charge is found in a thin layer on the surface of a ball. It is shown in the framework of this model that regarded plasma ball can be relatively stable as a metastable state. Calculations show that electrostatic forces, repulsive forces between the ions, and atmospheric pressure can provide stability of a plasma ball. Presented model could be useful to understand the nature of a ball lightning.

A detailed simple model is applied to study a high temperature plasma ball. It is assumed that the ions and delocalized electrons are distributed randomly throughout the charged plasma ball (extra/missing charge is assumed to be found in a thin layer on the surface of a ball). The energy of the microscopic electrostatic field around the ions is taken into account and calculated. It is shown in the framework of this model that regarded charged plasma ball can be stable as a metastable state, when subjected to an external (atmospheric) pressure. Equilibrium radius of a ball is calculated.

The author (physicist)has observed the very strange,beautiful and frightening Lightning Ball (LB). He has never forgotten this phenomenon. During his working life he could not devote himself to the problem of LB-formation.Only two years ago as he has been reading different unbelievable models of LB-formation, he decided to work on this problem. By studying the literature and the crucial points of his observation the author succeeded in creating a completely new model of Lightning Ball(LB) and Ball Lightning(BL)-formation based on the symmetry breaking of the hydrodynamic vortex ring.This agrees fully with the observation and overcomes the shortcomings of current models of LB formation. This model provides answers to the questions: Why are LBs so rarely observed,why do BLs in rare cases have such a high energy and how can we generate LB in the laboratory? Moreover the author differentiates between LB and BL, the latter having a high energy and occuring in 5 % of the observations. Keywords: ball lightning, hydrodynamic vortex ring, symmetry breaking, electroluminescence, triboelectrification.

We propose a hybrid control algorithm that guarantees fast convergence and uniform global asymptotic stability of the unique minimizer of a continuously differentiable, convex objective function. The algorithm, developed using hybrid system tools, employs a uniting control strategy, in which Nesterov's accelerated gradient descent is used "globally" and the heavy ball method is used "locally," relative to the minimizer. Without knowledge of its location, the proposed hybrid control strategy switches between these accelerated methods to ensure convergence to the minimizer without oscillations, with a (hybrid) convergence rate that preserves the convergence rates of the individual optimization algorithms. We analyze key properties of the resulting closed-loop system including existence of solutions, uniform global asymptotic stability, and convergence rate. Additionally, stability properties of Nesterov's method are analyzed, and extensions on convergence rate results in the existing literature are presented. Numerical results validate the findings and demonstrate the robustness of the uniting algorithm.

In this paper, the author brings further details regarding his Lightning Ball observation that were not mentioned in the first one (Ref.1-2). Additionally, he goes more into detail as the three forces that are necessary to allow the residual crescent form the hydrodynamic vortex ring to shrink into a sphere.Further topics are the similarities and analogies between the Lightning Ball formation's theory and the presently undertaken Tokamak-Stellarator-Spheromak fusion reactor experiments. A new theory and its experimental realisation are proposed as to make the shrinking of the hot plasma of reactors into a ball possible by means of the so called long range electromagnetic forces. In this way,the fusion ignition temperature could possibly atteined.

The Casimir energy of a solid ball placed in an infinite medium is calculated by a direct frequency summation using the contour integration. It is assumed that the permittivity and permeability of the ball and medium satisfy the condition $ε_1 μ_1=ε_2μ_2$. Upon deriving the general expression for the Casimir energy, a dilute compact ball is considered $(ε_1 -ε_2)^2/(ε_1+ε_2)^2\ll 1$. In this case the calculations are carried out which are of the first order in $ξ^2$ and take account of the five terms in the Debye expansion of the Bessel functions involved. The implication of the obtained results to the attempts of explaining the sonoluminescence via the Casimir effect is shortly discussed.

The kinetic theory of gases provides methods for calculating Lyapunov exponents and other quantities, such as Kolmogorov-Sinai entropies, that characterize the chaotic behavior of hard-ball gases. Here we illustrate the use of these methods for calculating the Kolmogorov-Sinai entropy, and the largest positive Lyapunov exponent, for dilute hard-ball gases in equilibrium. The calculation of the largest Lyapunov exponent makes interesting connections with the theory of propagation of hydrodynamic fronts. Calculations are also presented for the Lyapunov spectrum of dilute, random Lorentz gases in two and three dimensions, which are considerably simpler than the corresponding calculations for hard-ball gases. The article concludes with a brief discussion of some interesting open problems.