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Milena Radnovic

Publications and source records attributed to Milena Radnovic.

At least 19 recordsLinked to original sources
arXiv2026

The Weighted Walks in Quadrant with Finite Groups: an Algebro-Geometric Approach

We classify weighted small-step lattice walks in the quadrant whose associated birational group is finite. Using an algebro-geometric description of the kernel curves and Cayley-type finite-order conditions, we relate the group $G_W$ of a walk to the family of groups $\Gamma_t$ acting on its kernel curves. Together with the uniform upper bound on the order of $G_W$, this allows us to analyse all possible finite orders. We obtain explicit necessary and sufficient conditions for the group to have order 4, 6, 8, or 10, and prove that no weighted walk in the quadrant has a group of order 12. This yields a complete classification of weighted quadrant walks with finite groups.

math.CO↗
arXiv2025

Poncelet porism in singular cases

The celebrated Poncelet porism is usually studied for a pair of smooth conics that are in a general position. Here we discuss Poncelet porism in the real plane - affine or projective, when that is not the case, i.e. the conics have at least one point of tangency or at least one of the conics is not smooth. In all such cases, we find necessary and sufficient conditions for the existence of an n-gon inscribed in one of the conics and circumscribed about the other.

math.AG↗
arXiv2022

Resonance of ellipsoidal billiard trajectories and extremal rational functions

We study resonant billiard trajectories within quadrics in the $d$-dimensional Euclidean space. We relate them to the theory of approximation, in particular the extremal rational functions on the systems of $d$ intervals on the real line. This fruitful link enables us to prove fundamental properties of the billiard dynamics and to provide a comprehensive study of a large class of non-periodic trajectories of integrable billiards. A key ingredient is a functional-polynomial relation of a generalized Pell type. Applying further these ideas and techniques to $s$-weak billiard trajectories, we come to a functional-polynomial relation of the same generalized Pell type.

math.DS↗
arXiv2022

Billiard Ordered Games and Books

The aim of this work is to put together two novel concepts from the theory of integrable billiards: billiard ordered games and confocal billiard books. Billiard books appeared recently in the work of Fomenko's school, in particular of V. Vedyushkina. These more complex billiard domains are obtained by gluing planar sets bounded by arcs of confocal conics along common edges. Such domains are used in this paper to construct the configuration space for billiard ordered games. We analyse dynamical and topological properties of the systems obtained in that way.

math.DS↗
arXiv2021

Integrable Billiards on a Minkowski Hyperboloid: Extremal Polynomials and Topology

We consider billiard systems within compact domains bounded by confocal conics on a hyperboloid of one sheet in the Minkowski space. We derive conditions for elliptic periodicity for such billiards. We describe the topology of those billiard systems in terms of Fomenko invariants. We provide then periodicity conditions in terms of functional Pell equations and related extremal polynomials. Several examples are computed in terms of elliptic functions and classical Chebyshev and Zolotarev polynomials, as extremal polynomials over one or two intervals. These results are contrasted with the cases of billiards in the Minkowski and the Euclidean planes.

math.DS↗
arXiv2020

Pseudo-Euclidean Billiards within Confocal Curves on the Hyperboloid of One Sheet

We consider a billiard problem for compact domains bounded by confocal conics on a hyperboloid of one sheet in the Minkowski space. We show that there are two types of confocal families in such setting. Using an algebro-geometric integration technique, we prove that the billiard within generalized ellipses of each type is integrable in the sense of Liouville. Further, we prove a generalization of the Poncelet theorem and derive Cayley-type conditions for periodic trajectories and explore geometric consequences.

math.DS↗
arXiv2019

Periodic billiards within conics in the Minkowski plane and Akhiezer polynomials

We derive necessary and sufficient conditions for periodic and for elliptic periodic trajectories of billiards within an ellipse in the Minkowski plane in terms of an underlining elliptic curve. We provide several examples of periodic and elliptic periodic trajectories with small periods. We observe relationship between Cayley-type conditions and discriminantly separable and factorizable polynomials. Equivalent conditions for periodicity and elliptic periodicity are derived in terms of polynomial-functional equations as well. The corresponding polynomials are related to the classical extremal polynomials. In particular, the light-like periodic trajectories are related to the classical Chebyshev polynomials. The similarities and differences with respect to previously studied Euclidean case are indicated.

math.DS↗
arXiv2019

Combinatorics of periodic ellipsoidal billiards

We study combinatorics of billiard partitions which arose recently in the description of periodic trajectories of ellipsoidal billiards in d-dimensional Euclidean and pseudo-Euclidean spaces. Such partitions uniquely codify the sets of caustics, up to their types, which generate periodic trajectories. The period of a periodic trajectory is the largest part while the winding numbers are the remaining summands of the corresponding partition. In order to take into account the types of caustics as well, we introduce weighted partitions. We provide closed forms for the generating functions of these partitions.

math.CO↗
arXiv2019

Periodic ellipsoidal billiard trajectories and extremal polynomials

A comprehensive study of periodic trajectories of billiards within ellipsoids in $d$-dimensional Euclidean space is presented. The novelty of the approach is based on a relationship established between periodic billiard trajectories and extremal polynomials on the systems of $d$ intervals on the real line. By leveraging deep, but yet not widely known results of the Krein-Levin-Nudelman theory of generalized Chebyshev polynomials, fundamental properties of billiard dynamics are proven for any $d$, viz., that the sequences of winding numbers are monotonic. By employing the potential theory we prove the injectivity of the frequency map. As a byproduct, for $d=2$ a new proof of the monotonicity of the rotation number is obtained. The case study of trajectories of small periods $T$, $d\le T\le 2d$ is given. In particular, it is proven that all $d$-periodic trajectories are contained in a coordinate-hyperplane and that for a given ellipsoid, there is a unique set of caustics which generates $d+1$-periodic trajectories. A complete catalog of billiard trajectories with small periods is provided for $d=3$.

math.DS↗
arXiv2018

Caustics of Poncelet polygons and classical extremal polynomials

A comprehensive analysis of periodic trajectories of billiards within ellipses in the Euclidean plane is presented. The novelty of the approach is based on a relationship recently established by the authors between periodic billiard trajectories and extremal polynomials on the systems of $d$ intervals on the real line and ellipsoidal billiards in $d$-dimensional space. Even in the planar case, systematically studied in the present paper it leads to new results in characterizing $n$ periodic trajectories vs. so-called $n$ elliptic periodic trajectories, which are $n$-periodic in elliptical coordinates. The characterizations are done both in terms of the underlying elliptic curve and divisors on it and in terms of polynomial functional equations, like Pell's equation. This new approach also sheds light on some classical results. In particular we connect search for caustics which generate periodic trajectories with three classical classes of extremal polynomials on two intervals, introduced by Zolotarev and Akhiezer. The main classifying tool are winding numbers, for which we provide several interpretations, including one in terms of numbers of points of alternance of extremal polynomials. The latter implies important inequality between the winding numbers, which as a consequence, provides another proof of monotonicity of rotation numbers. A complete catalog of billiard trajectories with small periods is provided for $n=3, 4, 5, 6$ along with an effective search for caustics. As a byproduct, an intriguing connection between Cayle type conditions and discriminantly separable polynomials has been observed for all those small periods.

math.DS↗
arXiv2018

Asymptotic behaviour of the third Painlevé transcendents in the space of initial values

We study the asymptotic behaviour of the solutions of the generic ($D_6^{(1)}$-type) third Painlevé equation in the space of initial values as the independent variable approaches infinity (or zero) and show that the limit set of each solution is compact and connected. Moreover, we prove that any solution with essential singularity at infinity has an infinite number of poles and zeroes, and similarly at the origin.

nlin.SI↗
arXiv2015

Integrable lattices of hyperplanes related to billiards within confocal quadrics

We introduce a new discrete system that arises from ellipsoidal billiards and is closely related to the double reflection nets. The system is defined on the lattice of a uniform honeycomb consisting of rectified hypercubes and cross polytopes. In the $2$-dimensional case, the lattice is regular and it incorporates dynamics both in the original space and its dual. In the $3$-dimensional case, the lattice consists of tetrahedra and cuboctahedra.

math.DG↗
arXiv2013

Bicentennial of the Great Poncelet Theorem (1813-2013): Current Advances

The paper gives a review of very recent results related to the Poncelet Theorem, on the occasion of its bicentennial. We are telling the story of one of the most beautiful theorems of Geometry, recalling for the general mathematical audience the dramatic historic circumstances which led to its discovery, a glimpse of its intrinsic appeal, and importance of its relationship to the dynamics of billiards within confocal conics. We focus on the three main issues: A) The case of Pseudo-Euclidean spaces, presenting a recent notion of relativistic quadrics, and applying it to the description of periodic trajectories of billiards within quadrics. B) The relationship between so-called billiard algebra and foundations of modern discrete differential geometry which leads to the Double-reflection nets. C) We introduce a new class of dynamical systems -- pseudo-integrable billiards generated by the boundary composed of several arcs of confocal conics having nonconvex angles. The dynamics of such billiards has several extraordinary properties. They are related to the interval exchange transformations and generate families of flows which are minimal but not uniquely ergodic. This type of dynamics provides a novel type of the Poncelet porisms -- the local ones.

nlin.SI↗
arXiv2012

Ellipsoidal billiards in pseudo-Euclidean spaces and relativistic quadrics

We study geometry of confocal quadrics in pseudo-Euclidean spaces of an arbitrary dimension $d$ and any signature, and related billiard dynamics. The goal is to give a complete description of periodic billiard trajectories within ellipsoids. The novelty of our approach is based on introduction of a new discrete combinatorial-geometric structure associated to a confocal pencil of quadrics, a colouring in $d$ colours, by which we decompose quadrics of $d+1$ geometric types of a pencil into new relativistic quadrics of $d$ relativistic types. Deep insight of related geometry and combinatorics comes from our study of what we call discriminat sets of tropical lines $Σ^+$ and $Σ^-$ and their singularities. All of that enable usto get an analytic criterion describing all periodic billiard trajectories, including the light-like ones as those of a special interest.

math.AG↗
arXiv2012

Billiard algebra, integrable line congruences, and double reflection nets

The billiard systems within quadrics, playing the role of discrete analogues of geodesics on ellipsoids, are incorporated into the theory of integrable quad-graphs. An initial observation is that the Six-pointed star theorem, as the operational consistency for the billiard algebra, is equivalent to an integrabilty condition of a line congruence. A new notion of the double-reflection nets as a subclass of dual Darboux nets associated with pencils of quadrics is introduced, basic properies and several examples are presented. Corresponding Yang-Baxter maps, associated with pencils of quadrics are defined and discussed.

nlin.SI↗