Arbitrarily large <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:mi mathvariant="script">O</mml:mi></mml:math>-Morita Frobenius numbers

Limiting behaviour of large Frobenius numbers

Sequences of Pseudorandom Numbers with Arbitrarily Large Periods

A b st ract. We show t hat t he rest riction of th e Berno ulli shift map x <-1' X (mod L) to t he set of rati onals between 0 and 1 can generate sequences of uniformly dist ributed pseudorandom numbers whose periods exceed any given k > 0, however large. Here r > 1 is an integer.

Bounded-Degree Graphs can have Arbitrarily Large Slope Numbers

We construct graphs with n vertices of maximum degree 5 whose every straight-line drawing in the plane uses edges of at least n1/6−o(1) distinct slopes. A straight-line drawing of a graph G = (V (G), E(G)) is a layout of G in the plane such that the vertices are represented by distinct points, the edges are represented by (possibly crossing) line segments connecting the corresponding point pair...

Euler–frobenius Numbers and Rounding

We study the Euler–Frobenius numbers, a generalization of the Eulerian numbers, and the probability distribution obtained by normalizing them. This distribution can be obtained by rounding a sum of independent uniform random variables; this is more or less implicit in various results and we try to explain this and various connections to other areas of mathematics, such as spline theory. The mea...

Bounds on generalized Frobenius numbers

Let N ≥ 2 and let 1 < a1 < · · · < aN be relatively prime integers. The Frobenius number of this N -tuple is defined to be the largest positive integer that has no representation as PN i=1 aixi where x1, ..., xN are nonnegative integers. More generally, the s-Frobenius number is defined to be the largest positive integer that has precisely s distinct representations like this. We use techniques...