Related papers: A combinatorial study on product of filter large s…
Hindman and Leader first introduced the notion of semigroup of ultrafilters converging to zero for a dense subsemigroups of $((0,\infty),+)$. Using the algebraic structure of the Stone-$\breve{C}$ech compactification, Tootkabani and Vahed…
The questions of the measure and finding open intervals in certain sets of sums and products of elements of the middle third Cantor set (or a variant of it), have generated considerable interest recently. A broad general framework that…
In literature, many important combinatorial properties of subsets of N have been studied both with nonstandard techniques and from the point of view of N. In this thesis we mix these two different approaches in a technique that, at the same…
Given a finite set of roots of unity, we show that all power sums are non-negative integers iff the set forms a group under multiplication. The main argument is purely combinatorial and states that for an arbitrary finite set system the…
We introduce a new combinatorial structure: the superselector. We show that superselectors subsume several important combinatorial structures used in the past few years to solve problems in group testing, compressed sensing, multi-channel…
The paper introduces the notion of the size of countable sets that preserves the Part-Whole Principle and generalizes the notion of the cardinality of finite sets. The sizes of natural numbers, integers, rational numbers, and all their…
In this paper we want to apply the notion of product between ultrafilters to answer several questions which arise around the Connes' embedding problem. For instance, we will give a simplification and generalization of a theorem by…
We give a notably simpler and shorter proof of H. B. Neumann's result which is stated, cursorly, like this. For any well-ordered subset, A, of a totally ordered semigroup, the set of products of any finite number of elements of A is itself…
Hypercomplex Fourier transforms are increasingly used in signal processing for the analysis of higher-dimensional signals such as color images. A main stumbling block for further applications, in particular concerning filter design in the…
H. Furstenberg introduced the notion of central sets in terms of topological dynamics and established the famous Central Sets Theorem. Later in [A new and stronger Central Sets Theorem, Fund. Math. 199 (2008), 155-175], D. De, N. Hindman,…
We study the growth of product sets in some finite three-dimensional matrix groups. In particular, we prove two results about the group of $2\times 2$ upper triangular matrices over arbitrary finite fields: a product set estimate using…
The study of the size of subsets in a semigroup have shown that many of these subsets have strong combinatorial properties and contribute richly to the algebraic structure of the Stone-Cech compactification of a discrete semigroup. N.…
Suppose $K_{_Y}$ and $K_{_X}$ are large sets of observed and reference signals, respectively, each containing $N$ signals. Is it possible to construct a filter $F$ that requires a priori information only on few signals, $p\ll N$, from…
We investigate higher dimensional chain conditions, where the largeness notion is given by Fubini products of a given ideal. From strong saturation properties of an ideal, we derive abstractly versions of higher dimensional $\Delta$-system…
Let $\mathbf{k}$ denote the totally ordered set (or chain) on $k$ elements. The product $\mathbf{k}^t=\mathbf{k}\times\cdots\times\mathbf{k}$ is a poset called a grid. This paper discusses several loosely related results on the Ramsey…
We develop large sample theory for merged data from multiple sources. Main statistical issues treated in this paper are (1) the same unit potentially appears in multiple datasets from overlapping data sources, (2) duplicated items are not…
Let $G$ be a multiplicative group, let $A,B \subseteq G$ be finite and nonempty, and define the product set $AB = {ab \mid $a \in A$ and $b \in B$}$. Two fundamental problems in combinatorial number theory are to find lower bounds on…
We study the combinatorics of two classes of basic hypergeometric series. We first show that these series are the generating functions for certain overpartition pairs defined by frequency conditions on the parts. We then show that when…
Starting from filters over the set of indices, we introduce structures in a product of sets where the coordinate sets have the given structures.
We show that any $2$-coloring of $\mathbb{N}$ contains infinitely many monochromatic sets of the form $\{x,y,xy,x+y\},$ and more generally monochromatic sets of the form $\{x_i,\prod x_i,\sum x_i: i\leq k\}$ for any $k\in\mathbb{N}.$ Along…