The Hales–Jewett theorem is one of the pillars of Ramsey theory, from which many other results follow. A celebrated theorem of Shelah says that Hales–Jewett numbers are primitive recursive. A key tool used in his proof, now known as the cube lemma, has become famous in its own right. In its simplest form, this lemma says that if we color the edges of the Cartesian product Kn × Kn in r colors th...

We numerically study in the one-dimensional case the validity of the functional calculated by Graham and coworkers (R. Graham and T. Tel, Phys. Rev. A 42, 4661 (1990), O. Descalzi and R. Graham, Z. Phys. B 93, 509 (1994)) as a Lyapunov potential for the Complex Ginzburg-Landau equation. In non-chaotic regions of parameter space the functional decreases monotonically in time towards the plane wa...

In this paper we prove a conjecture of Bulter and Graham [2] on the existence of a certain way of marking the lines in [k] for any prime k. The conjecture states that there exists a way of marking each line of [k] one point so that every point in [k] is marked exactly a or b times as long as the parameters (a, b, n, k) satisfy the condition that there are integers s, t such that s + t = k and a...

We introduce a family of rings of symmetric functions depending on an infinite sequence of parameters. A distinguished basis of such a ring is comprised by analogues of the Schur functions. The corresponding structure coefficients are polynomials in the parameters which we call the Littlewood–Richardson polynomials. We give a combinatorial rule for their calculation by modifying an earlier resu...

We explore two of Ron Graham’s passions—juggling patterns and perfect card shuffling— through one of our passions, mathematical fonts. First, for each letter of the English alphabet, we design a one-person three-ball juggling pattern where the balls trace out the letter (possibly rotated 90◦). Second, using a deck of 26 cards labeled A through Z, we show how to perform a sequence of in/out perf...

Many applications in wireless sensor networks require sensor nodes to obtain their absolute geographical positions. There are various localization algorithms have been recently proposed. Localization – finding the position of individual sensor nodes-remains one of the most difficult research challenges. Many applications based on networked sensors require location awareness; a node must be able...

We show that a number of conditions on oriented graphs, all of which are satisfied with high probability by randomly oriented graphs, are equivalent. These equivalences are similar to those given by Chung, Graham and Wilson [5] in the case of unoriented graphs, and by Chung and Graham [3] in the case of tournaments. Indeed, our main theorem extends to the case of a general underlying graph G th...

IAl > ((l/k) + E)n then there is a subset B L A such that 0 < 1 BI 0, let snd(nt) denote the smallest integer that does not divide PI. We prove that for every I-: > 0 there is a constant c = ~(8:) z I, such that for every n > 0 and every rn, n ' +' 6 WI < n'llog'n...

Graham deduces this from a more general result about varieties with finitely many unipotent orbits, which is proved using induction and a calculation in the rank-one case. (In fact, A. Knutson points out that Graham’s proof yields a stronger result: as a polynomial in all the negative roots, cuv is a nonnegative combination of squarefree monomials.) The goal of this note is to give a short, geo...

The Radon transform on a group A is a linear operator on the space of functions /: A-+ C. It is shown that if A = Z;: then the Radon transform with respect to a subset B c .4 is not invertible if and only if B has the same number of elements in every coset of some maximal subgroup of A. The same does not hold in general for arbitrary finite abelian groups. ' IW7 ACxhlK I%\\. 1°C Let A be a fini...