相关论文: Complexity and growth for polygonal billiards
E. Gutkin found a remarkable class of convex billiard tables in the plane which have a constant angle invariant curve. In this paper we prove that in dimension 3 only round sphere has such a property. For dimension greater than 3 it must be…
In this letter we give fourth-order autonomous recurrence relations with two invariants, whose degree growth is cubic or exponential. These examples contradict the common belief that maps with sufficiently many invariants can have at most…
We present analytical and numerical solutions of the Lippmann-Schwinger equation for the scattered wavefunctions generated by confocal parabolic billiards and parabolic segments with various $\delta$-type potential-strength functions. The…
This paper presents an analysis of complexity of double dummy bridge. Values of both, a state-space (search-space) complexity and a game tree complexity have been estimated. ----- Oszacowanie z{\l}o\.zono\'sci problemu rozgrywki w otwarte…
This survey is based on a series of talks I gave at the conference "Dynamical systems and diophantine approximation" at l'Instut Henri Poincar\'e in June 2003. I will present asymptotic results (transitivity, ergodicity, weak-mixing) for…
The aim of the present paper is to establish a Bialy-Mironov type rigidity for centrally symmetric symplectic billiards. For a centrally symmetric $C^2$ strongly-convex domain $D$ with boundary $\partial D$, assume that the symplectic…
The notion of a rough two-dimensional (convex) body is introduced, and to each rough body there is assigned a measure on $\TTT^3$ describing billiard scattering on the body. The main result is characterization of the set of measures…
We define the topological complexity sequence of a group as the sequence of topological complexities of its Milnor constructions. This sequence may be regarded as an intrinsic refinement of the topological complexity of a group and, unlike…
We consider algebraic geometrical properties of the integrable billiard on a quadric Q with elastic impacts along another quadric confocal to Q. These properties are in sharp contrast with those of the ellipsoidal Birkhoff billiards.…
Probabilistic justification logic is a modal logic with two kind of modalities: probability measures and explicit justification terms. We present a tableau procedure that can be used to decide the satisfiability problem for this logic in…
It has been previously shown by two of the authors that some polynomial Julia sets are algorithmically impossible to draw with arbitrary magnification. On the other hand, for a large class of examples the problem of drawing a picture has…
The conjugation problem for billiard maps conjectures that if two strictly convex billiards have conjugated billiard maps, the billiard tables must be homothetic to each other. We show that if two billiard maps are conjugated, the…
We define billiards in the context of sub-Finsler Geometry. We provide symplectic and variational (or rather, control theoretical) descriptions of the problem and show that they coincide. We then discuss several phenomena in this setting,…
It is an open question whether the linear extension complexity of the Cartesian product of two polytopes P, Q is the sum of the extension complexities of P and Q. We give an affirmative answer to this question for the case that one of the…
We examine the complexity of the ``Texas Hold'em'' variant of poker from a topological perspective. We show that there exists a natural simplicial complex governing the multi-way winning probabilities between various hands, and that this…
We study growth and complexity of \'etale groupoids in relation to growth of their convolution algebras. As an application, we construct simple finitely generated algebras of arbitrary Gelfand-Kirillov dimension $\ge 2$ and simple finitely…
This is a survey of the geometry of complex cubic fourfolds with a view toward rationality questions. Topics include classical constructions of rational examples, Hodge structures and special cubic fourfolds, associated K3 surfaces and…
We investigate the rotation sets of billiards on the $m$-dimensional torus with one small convex obstacle and in the square with one small convex obstacle. In the first case the displacement function, whose averages we consider, measures…
A systematic study of non-trivial cubic extensions of the four-dimensional Poincar\'e algebra is undertaken. Explicit examples are given with various techniques (Young tableau, characters etc).
We construct the quaternion algebra [10] "geometrically" by a three dimensional analogue of the classic two dimensional geometric description of the complex field. The algebraic description of the multiplication operation in three…