Related papers: On regulated partitions
Using the notion of proper Cantor colorings we prove the following theorem. For any countably infinite group $\Gamma$, there exists a free continuous action $\zeta: \Gamma \curvearrowright C$ on the Cantor set, which is universal in the…
This paper concerns two-player alternating play combinatorial games (Conway 1976) in the normal-play convention, i.e. last move wins. Specifically, we study impartial vector subtraction games on tuples of nonnegative integers (Golomb 1966),…
Let $S$ be a Polish space and $(X_n:n\geq1)$ an exchangeable sequence of $S$-valued random variables. Let $\alpha_n(\cdot)=P(X_{n+1}\in \cdot\mid X_1,\...,X_n)$ be the predictive measure and $\alpha$ a random probability measure on $S$ such…
We study the descriptive complexity of sets of points defined by placing restrictions on statistical behaviour of their orbits in dynamical systems on Polish spaces. A particular examples of such sets are the set of generic points of a…
Numerous congruences for partitions with designated summands have been proven since first being introduced and studied by Andrews, Lewis, and Lovejoy. This paper explicitly characterizes the number of partitions with designated summands…
The moduli space of planar polygons with generic side lengths is a closed, smooth manifold. Mapping a polygon to its reflected image across the $X$-axis defines a fixed-point-free involution on these moduli spaces, making them into free…
We study an elementary problem of the topological robotics: collective motion of a set of $n$ distinct particles which one has to move from an initial configuration to a final configuration, with the requirement that no collisions occur in…
In a number of recent papers, the idea of generalized boundaries has found use in fractal and in multiresolution analysis; many of the papers having a focus on specific examples. Parallel with this new insight, and motivated by quantum…
The Satisfactory Partition problem consists in deciding if the set of vertices of a given undirected graph can be partitioned into two nonempty parts such that each vertex has at least as many neighbours in its part as in the other part.…
Tarski's Circle Squaring Problem from 1925 asks whether it is possible to partition a disk in the plane into finitely many pieces and reassemble them via isometries to yield a partition of a square of the same area. It was finally resolved…
We study Polish spaces for which a set of possible distances $A \subseteq \mathbb{R}^+$ is fixed in advance. We determine, depending on the properties of $A$, the complexity of the collection of all Polish metric spaces with distances in…
In this paper we study partitions whose successive ranks belong to a given set. We enumerate such partitions while keeping track of the number of parts, the largest part, the side of the Durfee square, and the height of the Durfee…
In this paper, we study two kinds of combinatorial objects, generalized integer partitions and tilings of two dimensional zonotopes, using dynamical systems and order theory. We show that the sets of partitions ordered with a simple…
Let $w$ be a word in alphabet $\{x,D\}$ with $m$ $x$'s and $n$ $D$'s. Interpreting "$x$" as multiplication by $x$, and "$D$" as differentiation with respect to $x$, the identity $wf(x) = x^{m-n}\sum_k S_w(k) x^k D^k f(x)$, valid for any…
Thus far in the search for, and classification of, `physical' modular invariant partition functions $\sum N_{LR}\,\c_L\,\C_R$ the attention has been focused on the {\it symmetric} case where the holomorphic and anti-holomorphic sectors, and…
A function on the real line is called regulated if it has a left limit and a right limit at each point. If $f$ is a Schwartz distribution on the real line such that $f=F'$ (distributional or weak derivative) for a regulated function $F$…
The lattice of partitions of a set and its d-divisible generalization have been much studied for their combinatorial, topological, and representation-theoretic properties. An ordered set partition is a set partition where the subsets are…
This paper deals with the problem of when, given a collection $\mathcal C$ of weakly compact operators between separable Banach spaces, there exists a separable reflexive Banach space $Z$ with a Schauder basis so that every element in…
We provide polynomial upper bounds for the minimal sizes of distal cell decompositions in several kinds of distal structures, particularly weakly $o$-minimal and $P$-minimal structures. The bound in general weakly $o$-minimal structures…
We study algorithmic problems on subsets of Euclidean space of low fractal dimension. These spaces are the subject of intensive study in various branches of mathematics, including geometry, topology, and measure theory. There are several…