A universal cycle for permutations is a word of length n! such that each of the n! possible relative orders of n distinct integers occurs as a cyclic interval of the word. We show how to construct such a universal cycle in which only n + 1 distinct integers are used. This is best possible and proves a conjecture of Chung, Diaconis and Graham.

A conjecture of D. Peterson, proved by W. Graham [Gr], states that the structure constants of the (T−)equivariant cohomology of a homogeneous space G/P satisfy a certain positivity property. In this paper we show that this positivity property holds in the more general situation of equivariant quantum cohomology.

The estimate of the Phase Change Rate of the local clock makes more efficient a clock synchronization protocol, like IEEE1588 Precision Time Protocol. We show how to obtain such parameter with a lightweight protocol that uses one-way messages from a Master to a Slave

For permutations π and τ of lengths |π| ≤ |τ |, let t(π, τ) be the probability that the restriction of τ to a random |π|-point set is (order) isomorphic to π. We show that every sequence {τj} of permutations such that |τj| → ∞ and t(π, τj) → 1/4! for every 4-point permutation π is quasirandom (that is, t(π, τj) → 1/|π|! for every π). This answers a question posed by Graham.

Two simple proofs are given to an earlier partial result about an extremal set theoretic conjecture of Chung, Frank!, Graham, Shearer and Faudree, Schelp, S6s, respectively. The statement is slightly strengthened within a matroid theoretic framework. The first proof re lies on results from matroid theory, while the second is based on an explicit constJuction providing an elementary proof. @ 199...

We investigate generalizations of pebbling numbers and of Graham’s pebbling conjecture that π(G × H) ≤ π(G)π(H), where π(G) is the pebbling number of the graph G. We develop new machinery to attack the conjecture, which is now twenty years old. We show that certain conjectures imply others that initially appear stronger. We also find counterexamples that show that Sjöstrand’s theorem on cover p...

We solve a problem posed by Ronald Graham about the minimum number, over all 2-colorings of [1, n], of monochromatic (x, y, x + ay) triples, a ≥ 2. We show that the minimum number of such triples is n 2 2a(a+2a+3) + O(n). We also find a new upper bound for the minimum number, over all r-colorings of [1, n], of monochromatic Schur triples, for r ≥ 3.

R. L. Graham and H. O. Pollak observed that the sequence u1 = 1, un+1 = ⌊√ 2 ( un + 1 2 )⌋ , n ≥ 1, has the curious property that the sequence of numbers (u2n+1 − 2u2n−1)n≥1 gives the binary digits of √ 2. We present an extension of the Graham–Pollak sequence which allows to get – in a fancy way – the binary digits of 759250125 √ 2 and other numbers.

This paper presents a fast implementation of the Graham scan on the GPU. The proposed algorithm is composed of two stages: (1) two rounds of preprocessing performed on the GPU and (2) the finalization of finding the convex hull on the CPU. We first discard the interior points that locate inside a quadrilateral formed by four extreme points, sort the remaining points according to the angles, and...

The goal of this paper is to prove the equivariant version of Bloch’s Riemann-Roch isomorphism between the higher algebraic K-theory and the higher Chow groups of smooth varieties. We show that for a linear algebraic group G acting on a smooth variety X , although there is no Chern character map from the equivariant K-groups to equivariant higher Chow groups, there is indeed such a map K i (X)⊗...