Related papers: Limiting behavior in growth of Bulgarian Solitaire…
Bulgarian Solitaire is an interesting self-map on the set of integer partitions of a fixed number $n$. As a finite dynamical system, its long-term behavior is well-understood, having recurrent orbits parametrized by necklaces of beads with…
We consider a stochastic version of Bulgarian solitaire: A number of cards are distributed in piles; in every round a new pile is formed by cards from the old piles, and each card is picked independently with a fixed probability. This game…
Bulgarian solitaire is played on $n$ cards divided into several piles; a move consists of picking one card from each pile to form a new pile. In a recent generalization, $\sigma$-Bulgarian solitaire, the number of cards you pick from a pile…
The Bulgarian solitaire is a mathematical card game played by one person. A pack of n cards is divided into several decks (or "piles"). Each move consists of the removing of one card from each deck and collecting the removed cards to form a…
We consider a stochastic variant of the game of Bulgarian solitaire [M. Gardner (1983), Sci. Amer. 249, 12-21]. For the stationary measure of the random Bulgarian solitaire, we prove that most of its mass is concentrated on (roughly)…
We introduce \emph{$p_n$-random $q_n$-proportion Bulgarian solitaire} ($0<p_n,q_n\le 1$), played on $n$ cards distributed in piles. In each pile, a number of cards equal to the proportion $q_n$ of the pile size rounded upward to the nearest…
It is a well known that, for odd $n$, the number of subsets of $\{1,2,\dots,n\}$ the sum of whose elements is divisible by $n$ equals the number of binary necklaces of length $n$. In this paper generalize this result in two directions. On…
Austrian Solitaire is a variation of Bulgarian Solitaire. It may be described as a card game, a method of asset inventory management, or a discrete dynamical system on integer partitions. We prove that the limit cycles in Austrian Solitaire…
We study a lattice regularization of the BFKL evolution, showing its bulk dynamics is governed by an abelian Knizhnik--Zamolodchikov equation. The Hamiltonian combines long-range hopping with virtual corrections encoded by harmonic numbers.…
Let $B$ be the group of invertible upper-triangular complex $n\times n$ matrices, $\mathfrak{u}$ the space of upper-triangular complex matrices with zeroes on the diagonal and $\mathfrak{u}^*$ its dual space. The group $B$ acts on…
In this paper, we show that the protocol complex of a Byzantine synchronous system can remain $(k - 1)$-connected for up to $\lceil t/k \rceil$ rounds, where $t$ is the maximum number of Byzantine processes, and $t \ge k \ge 1$. This…
Whether or not system is unitary can be seen from the way it, if perturbed, relaxes back to equilibrium. The relaxation of semiclassical black hole can be described in terms of correlation function which exponentially decays with time. In…
We report on results of fully consistent N-body simulations of globular cluster models with N = 100 000 members containing neutron stars and black holes. Using the improved `algorithmic regularization' method of Hellstrom and Mikkola for…
We show that the complexity of the billiard in a typical polygon grows cubically and the number of saddle connections grows quadratically along certain subsequences. It is known that the set of points whose first n-bounces hits the same…
The conjugacy growth function counts the number of distinct conjugacy classes in a ball of radius $n$. We give a lower bound for the conjugacy growth of certain branch groups, among them the Grigorchuk group. This bound is a function of…
A study of the quantum triangular billiard requires consideration of a boundary value problem for the Green's function of the Laplacian on a trianglar domain. Our main result is a reformulation of this problem in terms of coupled…
The finite orbits of the braid group action on Stokes matrices are studied and are shown to be the orbits on ordered sets of reflections, generating finite groups. All invariants of a reflection arrangement are determined. Determination of…
To describe a general bound binary black hole system, we need to consider orbital eccentricity and the misalignment of black holes' spin vectors with respect to the orbital angular momentum. While binary black holes produced through many…
Cosmological billiards arise as a map of the solution to the Einstein equations, when the most general symmetry of the metric tensor is implemented, under the BKL (named after Belinskii, Khalatnikov and Lifshitz) paradigm, for which points…
Let $G$ be a quasi-simple algebraic group defined over an algebraically closed field $k$ and $B$ a Borel subgroup of $G$ acting on the nilradical $\mathfrak{n}$ of its Lie algebra $\mathfrak{b}$ via the Adjoint representation. It is known…