We present structural results on solutions to the Diophantine system Ay = b, y ∈ Z ≥0 with the smallest number of non-zero entries. Our tools are algebraic and number theoretic in nature and include Siegel’s Lemma, generating functions, and commutative algebra. These results have some interesting consequences in discrete optimization.

For positive integers m and n, define f(m,n) to be the smallest integer such that any subset A of the m × n integer grid with |A| ≥ f(m,n) contains a rectangle; that is, there are x ∈ [m] and y ∈ [n] and d1, d2 ∈ Z+ such that all four points (x, y), (x + d1, y), (x, y + d2), and (x + d1, y + d2) are contained in A. In 1954, Kövari, Sós, and Turán showed that lim k→∞ f(k, k) k3/2 = 1. They also ...

A subgroup H H of a group G"> G encoding="application/x-tex">G is confined if the -o...

By gluing together copies of an all-right angled Coxeter polytope a number of open hyperbolic 6-manifolds with Euler characteristic −1 are constructed. They are the first known examples of hyperbolic 6-manifolds having the smallest possible volume.

We have designed, implemented and tested (by simulation on a serial computer) the new dynamic ordering for the parallel one-sided block-Jacobi SVD algorithm. Our idea is based on the estimation of the cosines of principal angles between two block columns X and Y of the same width without explicitly forming the matrix product XY (or Y X) and computing its SVD. Instead, we propose to use a fixed ...

We show that the arithmetic hyperbolic 3-manifold of smallest volume is the Weeks manifold. The next smallest one is the Meyerhoff manifold.

in the exterior domain Ω(1) := {x ∈ RN : ‖x‖ > 1}, where p > 1, x = (x1, . . . ,xN ) ∈ RN , N ≥ 2, Du= (∂u/∂x1, . . . ,∂u/∂xN ), ‖x‖ is usual Euclidean norm in RN . Throughout this paper we will assume that (A1) p ∈ C loc(Ω(1)), 0 < μ < 1, and p > 1 constant, (A2) A = (Aij(x))N×N is a real symmetric positive define matrix function with Aij ∈ C 1+μ loc (Ω(1)), i, j = 1, . . . ,N , and 0 < μ < 1....

We report the draft genome sequence of Commensalibacter papalotli strain MX01, isolated from the intestines of an overwintering monarch butterfly. The 2,332,652-bp AT-biased genome of C. papalotli MX01 is the smallest genome for a member of the Acetobacteraceae family and provides the first evidence of plasmids in Commensalibacter.