A fractional Helly theorem for convex lattice sets

A set of the form , where is convex and denotes the integer lattice, is called a convex lattice set. It is known that the Helly number of d-dimensional convex lattice sets is 2^d. We prove that the fractional Helly number is only d+1: For every d and every α∈(0,1] there exists β>0 such that whenever F₁,…,F_n are convex lattice sets in such that for at least index sets I⊆{1,2,…,n} of size d+1, then there exists a (lattice) point common to at least βn of the F_i. This implies a (p,d+1)-theorem for every p?d+1; that is, if is a finite family of convex lattice sets in such that among every p sets of , some d+1 intersect, then has a transversal of size bounded by a function of d and p.     

Forwarding and optical indices of a graph

Motivated by wavelength-assignment problems for all-to-all traffic in optical networks, we study graph parameters related to sets of paths connecting all pairs of vertices. We consider sets of both undirected and directed paths, under minimisation criteria known as edge congestion and wavelength count; this gives rise to four parameters of a graph G: its edge forwarding index π(G), arc forwarding index , undirected optical index , and directed optical index .In the paper we address two long-standing open problems: whether the equality holds for all graphs, and whether indices π(G) and are hard to compute. For the first problem, we give an example of a family of planar graphs {G_k} such that . For the second problem, we show that determining either π(G) or is NP-hard.     

Majorization of regular measures and weights with finite and positive critical exponent

For the sets , 1?p<∞, of positive finite Borel measures μ on the real axis with the set of algebraic polynomials P dense in Lp(R,dμ), we establish a majorization principle of their “boundaries,” i.e. for every there exists such that dμ/dν?1. A corresponding principle holds for the sets , p>0, of non-negative upper semi-continuous on R functions (weights) w such that P is dense in the space : For every there exists such that w?ω.     

Sequences of integers with missing quotients

Let S be any set of natural numbers, and A be a given set of rational numbers. We say that S is an A-quotient-free set if x,y∈S implies y/x∉A. Let and , where the supremum is taken over all A-quotient-free sets S, and are the upper and lower asymptotic densities of S respectively. Let ρ(A)=sup_Sδ(S), where the supremum is taken over all A-quotient-free sets S such that δ(S) exists. In this paper we study the properties of , and ρ(A).     

Pointlike sets with respect to R and J

We present an algorithm to compute the pointlike subsets of a finite semigroup with respect to the pseudovariety of all finite R-trivial semigroups. The algorithm is inspired by Henckell’s algorithm for computing the pointlike subsets with respect to the pseudovariety of all finite aperiodic semigroups. We also give an algorithm to compute -pointlike sets, where denotes the pseudovariety of all finite J-trivial semigroups. We finally show that, in contrast with the situation for , the natural adaptation of Henckell’s algorithm to computes pointlike sets, but not all of them.     

A one-to-one correspondence between colorings and stable sets

Given a graph G, we construct an auxiliary graph with vertices such that the set of all stable sets of is in one-to-one correspondence with the set of all colorings of G. Then, we show that the Max-Coloring problem in G reduces to the Maximum Weighted Stable set problem in .     

Mass transport generated by a flow of Gauss maps

Let A⊂Rd, d?2, be a compact convex set and let be a probability measure on A equivalent to the restriction of Lebesgue measure. Let be a probability measure on equivalent to the restriction of Lebesgue measure. We prove that there exists a mapping T such that ν=μ○T⁻¹ and T=φ⋅n, where is a continuous potential with convex sub-level sets and n is the Gauss map of the corresponding level sets of φ. Moreover, T is invertible and essentially unique. Our proof employs the optimal transportation techniques. We show that in the case of smooth φ the level sets of φ are governed by the Gauss curvature flow , where K is the Gauss curvature. As a by-product one can reprove the existence of weak solutions to the classical Gauss curvature flow starting from a convex hypersurface.     

Product of diagonal entries of the unitary orbit of a 3-by-3 normal matrix

Let N be a 3×3 normal matrix. We investigate the sets where U(3) is the group of 3×3 unitary matrices and 1?k?3. Geometric properties of these sets are studied, namely, star-shapedness and simple connectedness are investigated. A method for the numerical estimation of is also provided for normal matrices of size 3.     

Fractal analysis for sets of non-differentiability of Minkowski's question mark function

In this paper we study various fractal geometric aspects of the Minkowski question mark function Q. We show that the unit interval can be written as the union of the three sets , , and . The main result is that the Hausdorff dimensions of these sets are related in the following way:

dimH(νF)<dimH(Λ_∼)=dimH(Λ_∞)=dimH(L(htop))<dimH(Λ₀)=1.     

Elementary differences between the degrees of unsolvability and degrees of compressibility

Given two infinite binary sequences A,B we say that B can compress at least as well as A if the prefix-free Kolmogorov complexity relative to B of any binary string is at most as much as the prefix-free Kolmogorov complexity relative to A, modulo a constant. This relation, introduced in Nies (2005) [14] and denoted by A≤_LKB, is a measure of relative compressing power of oracles, in the same way that Turing reducibility is a measure of relative information. The equivalence classes induced by ≤_LK are called LK degrees (or degrees of compressibility) and there is a least degree containing the oracles which can only compress as much as a computable oracle, also called the ‘low for K’ sets. A well-known result from Nies (2005) [14] states that these coincide with the K-trivial sets, which are the ones whose initial segments have minimal prefix-free Kolmogorov complexity.We show that with respect to ≤_LK, given any non-trivial sets X,Y there is a computably enumerable set A which is not K-trivial and it is below X,Y. This shows that the local structures of and Turing degrees are not elementarily equivalent to the corresponding local structures in the LK degrees. It also shows that there is no pair of sets computable from the halting problem which forms a minimal pair in the LK degrees; this is sharp in terms of the jump, as it is known that there are sets computable from which form a minimal pair in the LK degrees. We also show that the structure of LK degrees below the LK degree of the halting problem is not elementarily equivalent to the or structures of LK degrees. The proofs introduce a new technique of permitting below a set that is not K-trivial, which is likely to have wider applications.     

An abstract form of a theorem of Helson and applications to sets of synthesis and sets of uniqueness

Let E be a compact perfect subset of the real line R such that the restriction of the Fourier transform from L¹(R) into C(E) is onto. Helson proved that then, for μ∈M(E), is possible only if μ=0. In this paper we present an abstract version of this theorem of Helson and provide some applications of it to the study of sets of spectral synthesis and sets of uniqueness.     

Sampling sets and sufficient sets for A

We give new characterizations of the subsets S of the unit disc of the complex plane such that the topology of the space A^−∞ of holomorphic functions of polynomial growth on coincides with the topology of space of the restrictions of the functions to the set S. These sets are called weakly sufficient sets for A^−∞. Our approach is based on a study of the so-called (p,q)-sampling sets which generalize the A^−p-sampling sets of Seip. A characterization of (p,q)-sampling and weakly sufficient rotation invariant sets is included. It permits us to obtain new examples and to solve an open question of Khôi and Thomas.     

Unique expansions of real numbers

It was discovered some years ago that there exist non-integer real numbers q>1 for which only one sequence (ci) of integers ci∈[0,q) satisfies the equality . The set of such “univoque numbers” has a rich topological structure, and its study revealed a number of unexpected connections with measure theory, fractals, ergodic theory and Diophantine approximation.In this paper we consider for each fixed q>1 the set Uq of real numbers x having a unique representation of the form with integers ci belonging to [0,q). We carry out a detailed topological study of these sets. For instance, we characterize their closures, and we determine those bases q for which Uq is closed or even a Cantor set. We also study the set consisting of all sequences (ci) of integers ci∈[0,q) such that . We determine the numbers r>1 for which the map (defined on (1,∞)) is constant in a neighborhood of r and the numbers q>1 for which is a subshift or a subshift of finite type.     

Transitive resolvable idempotent quasigroups and large sets of resolvable Mendelsohn triple systems

We first define a transitive resolvable idempotent quasigroup (TRIQ), and show that a TRIQ of order v exists if and only if 3∣v and . Then we use TRIQ to present a tripling construction for large sets of resolvable Mendelsohn triple systems s, which improves an earlier version of tripling construction by Kang. As an application we obtain an for any integer n≥1, which provides an infinite family of even orders.