The Frobenius problem, sums of powers of integers, and recurrences for the Bernoulli numbers

In the Frobenius problem with two variables, one is given two positive integers a and b that are relative prime, and is concerned with the set of positive numbers NR that have no representation by the linear form ax+by in nonnegative integers x and y. We give a complete characterization of the set NR, and use it to establish a relation between the power sums over its elements and the power sums over the natural numbers. This relation is used to derive new recurrences for the Bernoulli numbers.     

Minimum L1-Distance Projection onto the Boundary of a Convex Set: Simple Characterization

We show that the minimum distance projection in the L ₁-norm from an interior point onto the boundary of a convex set is achieved by a single, unidimensional projection. Application of this characterization when the convex set is a polyhedron leads to either an elementary minmax problem or a set of easily solved linear programs, depending upon whether the polyhedron is given as the intersection of a set of half spaces or as the convex hull of a set of extreme points. The outcome is an easier and more straightforward derivation of the special case results given in a recent paper by Briec (Ref. 1).