arXiv research

In this paper the problem of estimating the number of periodical billiard trajectories is considered. The main result is the theorem on Morse theory for periodical billiard trajectories.

Lower bounds on periodic Finsler billiard trajectories in convex hypersurfaces.

Study on billiard trajectories with fixed bounces.

We consider billiard trajectories in a smooth convex body in $\mathbb R^d$ and estimate the number of distinct periodic trajectories that make exactly $p$ reflections per period at the boundary of the body. In the case of prime $p$ we obtain the lower bound $(d-2)(p-1)+2$, which is much better than the previous estimat…

We give lower bound on the number of periodic billiard trajectories inside a generic smooth strictly convex closed surface in 3-space: for odd n, there are at least 2(n-1) such trajectories. We apply a topological approach based on the calculation of cohomology of certain configuration spaces.

A lower bound for the number of 3-periodical billiard trajectories in a manifold embedded in Euclidean space is obtained.

Given a domain or, more generally, a Riemannian manifold with boundary, a billiard is the motion of a particle when the field of force is absent. Trajectories of such a motion are geodesics inside the domain; and the particle reflects from the boundary making the angle of incidence equal the angle of reflection. The bi…

Generative model captures repetitive industrial processes with varying durations and dynamics.

For a closed symplectic manifold $(M,ω)$, a compatible almost complex structure $J$, a 1-periodic time dependent symplectic vector field $Z$ and a homotopy class of closed curves $γ$ we define a Floer complex based on 1-periodic trajectories of $Z$ in the homotopy class $γ$. We suppose that the closed 1-form $i_{Z_t}ω$…

Analytic Euler fields on non-torus bundles have periodic orbits.

Study of closed trajectories in hyperbolic plane with specific curvature constraints.

Study magnetic trajectories on 2-step nilpotent Lie groups.

In this paper we use the Ekeland-Hofer-Zehnder symplectic capacity to provide several bounds and inequalities for the length of the shortest periodic billiard trajectory in a smooth convex body in ${\mathbb R}^{n}$. Our results hold both for classical billiards, as well as for the more general case of Minkowski billiar…

We give lowed bounds on the number of periodic trajectories in strictly convex smooth billiards in $\R^{m+1}$ for $m\ge 3$. For plane billiards (when m=1) such bounds were obtained by G. Birkhoff in the 1920's. Our proof is based on topological methods of calculus of variations - equivariant Morse and Lusternik - Schir…

Dan Reznik discovered conserved quantities for ellipses using billiard maps.

Study trajectories on homothety surfaces, linking to square torus and revealing mixed behaviors.

We study periodic wind-tree models, billiards in the plane endowed with $\mathbb{Z}^2$-periodically located identical connected symmetric right-angled obstacles. We show asymptotic formulas for the number of (isotopy classes of) closed billiard trajectories (up to $\mathbb{Z}^2$-translations) on the wind-tree billiard.…

New periodic solutions found in 2n-body problem, braids of pseudo-Anosov type with stretch factors as metallic ratios.

Proposes a deep learning method for modeling dynamic individual-level latent trajectories with changing parameters.

A hierarchical model learns multi-agent trajectories from weak labels.

Study reveals 2020 stock crashes were mostly endogenous, not exogenous.

New findings on magnetic geodesic flows and periodic motions.

We give topological lower bounds on the number of periodic and closed trajectories in strictly convex smooth billiards. We use variational reduction admitting a finite group of symmetries and apply topological approach based on equivariant Morse and Lusternik - Schnirelman theories. The paper continues results publishe…

KOMET identifies Koopman operators from model parameter trajectories to adapt to evolving data distributions.

When the Poincaré map associated with a periodic orbit of a hybrid dynamical system has constant-rank iterates, we demonstrate the existence of a constant-dimensional invariant subsystem near the orbit which attracts all nearby trajectories in finite time. This result shows that the long-term behavior of a hybrid model…

Study orbits in right triangles, deducing periodic billiard paths and classifying orbit closures.

Poisson plane and sphere --- homogeneous spaces of Poisson groups E(2) and SU(2) (resp.) --- have phase spaces (corresponding symplectic groupoids), in which a free Hamiltonian is naturally defined. We solve the equations of motion and point out some unexpected features: free motion on the plane is bounded (periodic) a…

In this paper we describe a 1-dimensional family of initial conditions Σthat provides reduced periodic solution of the three body problem. This family Σcontains a bifurcation point and extend the periodic solution described in (Perdomo, http://arxiv.org/pdf/1507.01100.pdf). This 1-dimensional family is the union of two…

Floer homology proves noncontractible orbits in Finsler manifolds.

Study of intersections in Hamiltonian orbits on cotangent bundles.

The paper uses persistent homology to estimate recurrence times in multi-variate time series.

Proposes a novel anomaly detection method for echocardiogram videos.

Based on the assumption that economic complexity is characterised by the interactions of economic agents (who) constantly change their actions and strategies in response to the outcome they mutually create, this paper presents how network models can be used a proxies for the mapping, quantification and analysis of Roma…

New method subtypes irregular patient data for disease progression.

We describe the curves of constant (geodesic) curvature and torsion in the three-dimensional round sphere. These curves are the trajectory of a point whose motion is the superposition of two circular motions in orthogonal planes. The global behavior may be periodic or the curve may be dense in a Clifford torus embedded…

Neural network approach simplifies multiscale problem homogenization.

We solve a multi-period portfolio optimization problem using D-Wave Systems' quantum annealer. We derive a formulation of the problem, discuss several possible integer encoding schemes, and present numerical examples that show high success rates. The formulation incorporates transaction costs (including permanent and t…

Proposes SSC for estimating counterfactual survival trajectories from observational data.

Gradient flow with weight decay shows grokking effect in deep learning.

Optimal algorithms for non-linear ridge bandits reduce burn-in cost.

Extends tracking guarantees for time-varying variational inequalities.

Study of straight-line flows on a unique infinite surface.

Study of magnetic curves in SL(2,R) with quantization and horocycle projections.

Reinforcement learning is explored as a candidate machine learning technique to enhance existing analytical solutions for optimal trade execution with elements from the market microstructure. Given a volume-to-trade, fixed time horizon and discrete trading periods, the aim is to adapt a given volume trajectory such tha…

We apply two non-parametric methods to test further the hypothesis that log-periodicity characterizes the detrended price trajectory of large financial indices prior to financial crashes or strong corrections. The analysis using the so-called (H,q)-derivative is applied to seven time series ending with the October 1987…

Neural network outperforms traditional methods in chaotic dynamics classification.

UK's rapid vaccine rollout linked to reduced COVID-19 mortality.

DHLNN improves deep hedging for financial derivatives with faster convergence and better stability.