A version of the precise definition of Euler–Venn diagram for a given family of subsets of a universal set is presented. Certain geometrical properties of such diagrams are discussed and close connections with purely combinatorial problems and with the theory of convex sets are indicated. In particular, some geometrical realizations of uncountable independent families of sets are considered. 20...

In An asymptotic result on compressed sensing matrices [4], a new construction for compressed sensing matrices using combinatorial design theory was introduced. In this paper, we analyse the performance of these matrices using deterministic and probabilistic methods. We provide a new recovery algorithm and detailed simulations. These simulations suggest that the construction is competitive with...

For a free ultrafilter U on ω we study several cardinal characteristics which describe part of the combinatorial structure of U . We provide various consistency results; e.g. we show how to force simultaneously many characters and many π–characters. We also investigate two ideals on the Baire space ω naturally related to U and calculate cardinal coefficients of these ideals in terms of cardinal...

Improving a result of Dyer, Frieze and Greenhill [Journal of Combinatorial Theory, Series B, 2015], we determine the q-colorability threshold in random k-uniform hypergraphs up to an additive error of ln2+εq , where limq→∞ εq = 0. The new lower bound on the threshold matches the “condensation phase transition” predicted by statistical physics considerations [Krzakala et al., PNAS 2007]. Mathema...

Previous work has studied the pattern count on singly restricted permutations. In this work, we focus on patterns of length 3 in multiply restricted permutations, especially for double and triple pattern-avoiding permutations. We derive explicit formulae or generating functions for various occurrences of length 3 patterns on multiply restricted permutations, as well as some combinatorial interp...

A normal surface singularity is rational if and only if the dual intersection graph of a desingularization satisfies some combinatorial properties. In fact, the graphs defined in this way are trees. In this paper we give geometric features of these trees. In particular, we prove that the number of vertices of valency ≥ 3 in the dual intersection tree of the minimal desingularization of a ration...

We study the combinatorial and structural properties of the circle map sequences. We introduce an embedding procedure which gives a map Φ : Ω → W := {R,L} from the hull(closure of the set of translates) to the sequence of embedding operations through which we study the structure of Ω. We also study the set of admissible words and classify them in terms of their appearance. Mathematics Subject C...

We consider a strongly regular graph, G, with adjacency matrix A, and associate a three dimensional Euclidean Jordan algebra to A. Then, by considering convergent series of Hadamard powers of the idempotents of the unique complete system of orthogonal idempotents of the Euclidean Jordan algebra associated to A, we establish new admissibility conditions for the existence of strongly regular grap...

The Shortest lattice Vector Problem is central in lattice-based cryptography, as well as in many areas of computational mathematics and computer science, such as computational number theory and combinatorial optimisation. We present an algorithm for solving it in time 2 and space 2, where n is the lattice dimension. This improves the best previously known algorithm, by Micciancio and Voulgaris ...

The Determinant vs. Permanent problem has been of particular interest as it gives an example of an “easy” problem, which under some seemingly insignificant alteration becomes vastly more difficult to solve. In sections 2 and 3 I will discuss and prove theorems regarding the complexity of each problem. In section 4 I explore some important combinatorial interpretations of determinant and permane...