Partitions and sums of integers with repetition

A partition of N is called “admissible” provided some cell has arbitrarily long arithmetic progressions of even integers in a fixed increment. The principal result is that the statement “Whenever {A_i}_{i < r} is an admissible partition of N, there are some i < r and some sequence 〈x_n〉_{n < ω} of distinct members of N such that x_n + x_m?A_i whenever {m, n} ? ω″ is true when r = 2 and false when r ? 3.     

Largeness of the set of finite products in a semigroup

We investigate when the set of finite products of distinct terms of a sequence 〈x _n 〉 _n=1^∞ in a semigroup (S,⋅) is large in any of several standard notions of largeness. These include piecewise syndetic, central, syndetic, central*, and IP*. In the case of a “nice” sequence in (S,⋅)=(ℕ,+) one has that FS(〈x _n 〉 _n=1^∞) has any or all of the first three properties if and only if {x _n+1−∑ _t=1 ⁿ x _t :n∈ℕ} is bounded from above. N. Hindman acknowledges support received from the National Science Foundation via Grant DMS-0554803.     

The research of thirteen students at Howard University

Results from, and problems related to, the dissertations of my Ph.D. students are discussed.     

Nearly prime subsemigroups ofβℕ

We show that there exist relatively small subsemigroupsM ofβℕ with the property that ifp+q andq+p are inM then bothp andq are inM + ℤ. We also show that it is consistent with the usual axioms of set theory that there is some idempotent e inβℕ such that ifp+q=e, then bothp andq are ine + ℤ. The first author acknowledges support received from the National Science Foundation (USA) via grant DMS90-25025.     

Partition theorems for subspaces of vector spaces

The principal result of this paper provides a nearly complete answer to the following question. For which cardinal numbers $\text{t}$, $\text{m}$, $\text{n}$, $\text{q}$ and $\text{r}$ is it true that whenever the $\text{t}$-dimensional subspaces of an $\text{n}$-dimensional vector space V over a field of $\text{q}$ elements are partitioned into $\text{r}$ classes, there must be some $\text{m}$-dimensional subspace of V, all of whose $\text{t}$-dimensional subspaces lie in the same class? This question is answered completely if $\text{r}$ ≤ $\text{N}$₀. The contributions of this paper are in the form of negative answers, since it turns out that all affirmative answers (which we have) were already known or easily deducible from known results.     

Refining families for ultrafilters

Ohne Zusammenfassung     

Notions of size in a semigroup: an update from a historical perspective

Semigroup Forum - Previous papers have investigated relationships among several notions of largeness in a semigroup, some of which have their origins in topological dynamics, others with pure...     

Some new results in multiplicative and additive Ramsey theory

There are several notions of largeness that make sense in any semigroup, and others such as the various kinds of density that make sense in sufficiently well-behaved semigroups including and . It was recently shown that sets in which are multiplicatively large must contain arbitrarily large geoarithmetic progressions, that is, sets of the form , as well as sets of the form . Consequently, given a finite partition of , one cell must contain such configurations. In the partition case we show that we can get substantially stronger conclusions. We establish some combined additive and multiplicative Ramsey theoretic consequences of known algebraic results in the semigroups and , derive some new algebraic results, and derive consequences of them involving geoarithmetic progressions. For example, we show that given any finite partition of there must be, for each , sets of the form together with , the arithmetic progression , and the geometric progression in one cell of the partition. More generally, we show that, if is a commutative semigroup and a partition regular family of finite subsets of , then for any finite partition of and any , there exist and such that is contained in a cell of the partition. Also, we show that for certain partition regular families and of subsets of , given any finite partition of some cell contains structures of the form for some .

     

Nanocomposite-based lignocellulosic fibers 1. Thermal stability of modified fibers with clay-polyelectrolyte multilayers

The layer-by-layer (LbL) assembly process of creating highly structured thin films derived from layers of polyelectrolytes and nanoparticles was adopted in this study to modify the surface of lignocellulosic fibers. Aqueous dispersions of clay nanoplatelets were created with ultrasonication and characterized with dynamic light scattering and atomic force microscopy in which confirmed the presence of individual clay nanoplatelets. Film thickness of never-dried clay and poly(diallyldimethylammonium chloride) (PDDA) multilayers was studied with a quartz crystal microbalance with dissipation monitoring (QCM-D). Using identical LbL deposition parameters, a slurry of steam-exploded wood fibers was modified by alternate adsorption of PDDA and clay with multiple rinsing steps after each adsorption cycle. Zeta potential measurements were used to characterize the fiber surface charges after each adsorption step while SEM images revealed that the LbL film masked the cellulose microfibril structure. Using a thermogravimetric analyzer, LbL modified steam-exploded wood fibers were observed to attain increased thermal stability relative to the unmodified material tested in both air and nitrogen atmospheres. Significant char for the LbL clay coated steam-exploded wood suggests the multilayer film serves as a barrier creating an insulating layer to prevent further decomposition of the material. This nanotechnology may have a positive impact on the processing of lignocellulosic fibers in thermoplastic matrices, designing of paper-based overlays for building products, and modification of cellulosic fibers for textiles.