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A new approach to the results of Kovari, Sos, and Turan concerning rectangle-free subsets of the grid
Turan concerning rectangle-free subsets of the grid Combinatorics
2012/6/21
For positive integers $m$ and $n$, define $f(m,n)$ to be the smallest integer such that any subset $A$ of the $m \times n$ integer grid with $|A| \geq f(m,n)$ contains a rectangle; that is, there are ...
Lowering topological entropy over subsets revisited
entropy principal extension lowerable hereditarily lowerable
2012/6/19
Let $(X, T)$ be a topological dynamical system. Denote by $h (T, K)$ and $h^B (T, K)$ the covering entropy and dimensional entropy of $K\subseteq X$, respectively. $(X, T)$ is called D-{\it lowerable}...
Affine open subsets in A^3 without the cancellation property
Cancellation problem Koras-Russell threefolds Algebraic Geometry
2011/9/1
Abstract: We give families of examples of principal open subsets of the affine space \mathbb{A}^{3} which do not have the cancellation property. We show as a by-product that the cylinders over Koras-R...
A Kronecker-Weyl theorem for subsets of abelian groups
uniform distribution Weyl criterion Zariski closure Zariski topology
2011/2/22
Let N be the set of non-negative integer numbers, T the circle group and c the cardinality
of the continuum. Given an abelian group G of size at most 2c and a countable family E of
infinite subsets ...
Variational principles for topological entropies of subsets
Topological entropies Measure-theoretical entropies
2011/1/18
Let (X, T ) be a topological dynamical system. We define the measure-theoretical lower and upper entropies hμ(T ), hμ(T ) for any μ ∈ M(X), where M(X) denotes the collection of all Borel probability m...
Rational subsets of groups
Free groups inverse automata Stallings automata rational subsets
2011/3/2
Over the years, finite automata have been used effectively in the theory of infinite
groups to represent rational subsets.
Ordering subsets of (partially) ordered sets: Representation theorems
partial orders ordering subsets of a poset
2010/9/20
In many practical situations, we have a (partially) ordered set V of different values. For example, we may have the set of all possible values of temperature, or the set of all possible degrees of con...
Counting packings of generic subsets in finite groups
Counting generic subsets finite groups
2010/11/9
A packing of subsets $\mathcal S_1,..., \mathcal S_n$ in a group $G$ is a sequence $(g_1,...,g_n)$ such that $g_1\mathcal S_1,...,g_n\mathcal S_n$ are disjoint subsets of $G$. We give a formula for t...
We prove a Marstrand type theorem for a class of subsets of the integers. More specifically, after defining the counting dimension D(E) of subsets of Z and the concepts of regularity and compatibilit...
Given a family $F$ of subsets of a group $G$ we describe the structure of its thin-completion $\tau^*(F)$, which is the smallest thin-complete family that contains $I$. A family $F$ of subsets of $G$...
In this exposition the space of at most 3-element subsets of the circle, first identified by Borsuk and Bott, is used as a motivation to introduce the readers to a variety of methods in (algebraic) to...
Vertex subsets with minimal width and dual width in $Q$-polynomial distance-regular graphs
Vertex subsets $Q$-polynomial distance-regular graphs
2010/11/15
We study $Q$-polynomial distance-regular graphs from the point of view of what we call descendents, that is to say, those vertex subsets with the property that the width $w$ and dual width $w^*$ sati...
G-subsets and G-orbits of Q(sqrt n) under action of the modular group
Real quadratic irrational number Congruences Quadratic residues Linear-fractional transformations
2010/12/10
It is well known that G = hx, y : x2 = y3 = 1i represents the modular group PSL(2,Z), where x : z → −1 z , y : z → z−1 z are linear fractional transformations. Let n = k2m, where k is any ...
On sets of directions and angles determined by subsets of ${\Bbb R}^d$
directions and angles determined ${\Bbb R}^d$
2010/12/9
Given E Rd, d 2, de ne D(E) n xy jxyj : x; y 2 E o Sd1; the set of directions
determined by E. We prove that if the Hausdor dimension of E is greater than d
...
On triangles determined by subsets of the Euclidean plane, the associated bilinear operators and applications to discrete geometry
triangles determined Euclidean plane associated bilinear operators discrete geometry
2010/12/6
We prove that if the Hausdorff dimension of the set E R2 is greater than 74 , then the three-dimensional Lebesgue measure of the set of triangles determined by E is positive. We
establish this fact...