Sums of sets of lattice points and unimodular coverings of polytopes

If P is a lattice polytope (that is, the convex hull of a finite set of lattice points in \({\mathbf{R}^n}\)), then every sum of h lattice points in P is a lattice point in the h-fold sumset hP. However, a lattice point in the h-fold sumset hP is not necessarily the sum of h lattice points in P. It is proved that if the polytope P is a union of unimodular simplices, then every lattice point in the h-fold sumset hP is the sum of h lattice points in P.     

N-graphs,Modular Sidon and Sum-Free Sets,and Partition Identities

Using a new graphical representation for partitions, the author obtains a family of partition identities associated with partitions into distinct parts of an arithmetic progression, or, more generally, with partitions into distinct parts of a set that is a finite union of arithmetic progressions associated with a modular sum-free Sidon set. Partition identities are also constructed for sets associated with modular sum-free sets.     

On sums and products of integers

Erdos and Szemerédi conjectured that if is a set of positive integers, then there must be at least integers that can be written as the sum or product of two elements of . Erdos and Szemerédi proved that this number must be at least for some and . In this paper it is proved that the result holds for .

     

Collisions of DCl with pure and salty glycerol: enhancement of interfacial D --> H exchange by dissolved NaI

The fate of DCl molecules striking pure glycerol and a 2.6 M NaI-glycerol solution is investigated using scattering, uptake, and residence time measurements. We find that dissolved Na+ and I- ions alter every gas-liquid pathway from the moment of contact of DCl with the surface to its eventual emergence as HCl. In particular, the salt enhances both trapping-desorption of DCl and interfacial DCl --> HCl exchange at the expense of DCl entry into the bulk solution. The reduced entry and enhanced desorption of thermalized DCl molecules are interpreted by assuming that Na+ and I- ions bind to interfacial OH groups and tie up surface sites that would otherwise capture incoming DCl molecules. These ion-glycerol interactions may also be responsible for enhancing interfacial D --> H exchange by disrupting the interfacial hydrogen bond network that carries the newly formed H+ ion away from its Cl- pair. This disruption may increase the fraction of interfacial Cl- and H+ that recombine and desorb immediately as HCl before the ions separate and diffuse deeply into the bulk.     

Asymptotic Density and the Asymptotics of Partition Functions

Let A be a set of positive integers with gcd (A) = 1, and let p _A (n) be the partition function of A. Let c ₀ = 2/3. If A has lower asymptotic density and upper asymptotic density , then lim inf log p _A (n)/c ₀ n and lim sup log p _A (n)/c ₀ n . In particular, if A has asymptotic density > 0, then log p _A (n) c₀n. Conversely, if > 0 and log p _A (n) c ₀ n, then the set A has asymptotic density .     

Growth Polynomials for Additive Quadruples and (h,k)-Tuples

Consider the interval of integers I _m,n = {m, m + 1, m + 2, . . . , m + n ? 1}. For fixed integers h, k, m, and c, let \({\Phi^{(c)}_{h,k,m}(n)}\) denote the number of solutions of the equation (a ₁ +  . . . +  a _h ) ? (a _h+1 +  . . . +  a _h+k ) =  c with \({a_i \in I_{m,n}}\) for all i =  1, . . . , h +  k. This is a polynomial in n for all sufficiently large n, and the growth polynomial is constructed explicitly.     

Linear Forms in Finite Sets of Integers

Let A_1, ..., A_r be finite, nonempty sets of integers, and let h_1,..., h_r be positive integers. The linear formh_1A_1 + ··· + h_rA_r is the set of all integers of the form b_1 + ··· + b_r, where b_i is an integer that can be represented as the sum ofh_i elements of the set A_i. In this paper, the structure of the linear form h_1A_1 + ··· + h_rA_r is completely determined for all sufficiently large integersh_i .