Chris Smyth

Nonblocking photonic switch networks

A practical algorithm is described, with examples, for deciding whether or not a given network of 2*2 photonic switches is truly unblocking. Nonblocking photonic networks with specific additional properties, e.g. planarity, short path length, are also discussed, and specific networks, which have advantages over the square networks so far fabricated, are described. It is also shown that a correspondence can be set up between classical switch networks and networks of 2*2 switches, so that the classical results on blocking can be carried over to these networks. >

Salem Numbers, Pisot Numbers, Mahler Measure, and Graphs

We use graphs to define sets of Salem and Pisot numbers and prove that the union of these sets is closed, supporting a conjecture of Boyd that the set of all Salem and Pisot numbers is closed. We find all trees that define Salem numbers. We show that for all integers n the smallest known element of the nth derived set of the set of Pisot numbers comes from a graph. We define the Mahler measure of a graph and find all graphs of Mahler measure less than ½ (1+√5). Finally, we list all small Salem numbers known to be definable using a graph.

There are Salem Numbers of Every Trace

We show that there are Salem numbers of every trace. The nontrivial part of this result is for Salem numbers of negative trace. The proof has two main ingredients. The first is a novel construction, using pairs of polynomials whose zeros interlace on the unit circle, of polynomials of specified negative trace having one factor a Salem polynomial, with any other factors being cyclotomic. The second is an upper bound for the exponent of a maximal torsion coset of an algebraic torus in a variety defined over the rationals. This second result, which may be of independent interest, enables us to refine our construction to avoid getting cyclotomic factors, giving a Salem polynomial of any specified trace, with a trace-dependent bound for its degree. We show also how our interlacing construction can be easily adapted to produce Pisot polynomials, giving a simpler, and more explicit, construction for Pisot numbers of arbi- trary trace than previously known.

Salem Numbers of Trace -2 and Traces of Totally Positive Algebraic Integers

Until recently, no Salem numbers were known of trace below -1. In this paper we provide several examples of trace -2, including an explicit infinite family. We establish that the minimal degree for a Salem number of trace -2 is 20, and exhibit all Salem numbers of degree 20 and trace -2. Indeed there are just two examples.

Salem numbers of negative trace

We prove that, for all d > 4, there are Salem numbers of degree 2d and trace - 1, and that the number of such Salem numbers is » d/ (log log d) 2 . As a consequence, it follows that the number of totally positive algebraic integers of degree d and trace 2d - 1 is also » d/ (log log d) 2 .

On the absolute Mahler measure of polynomials having all zeros in a sector

Let α be an algebraic integer of degree d, not 0 or a root of unity, all of whose conjugates α i are confined to a sector |arg z| ≤ θ. We compute the greatest lower bound c(θ) of the absolute Mahler measure (Π i=1 d max(1, |α i |)) 1/d of α, for θ belonging to nine subintervals of [0, 2π/3]. In particular, we show that c(π/2) = 1.12933793, from which it follows that any integer α ¬= 1 and α ¬= e ±iπ/3 all of whose conjugates have positive real part has absolute Mahler measure at least c(π/2). This value is achieved for α satisfying α + 1/α = β 0 2 , where β 0 = 1.3247... is the smallest Pisot number (the real root of β 0 3 = β 0 + 1)

Birch reduction of C60—a new appraisal

Contrary to a previous report that Birch reduction of C60 affords C60H36 as the principal product, laser desorption-laser photoionisation time-of-flight (L2TOF), laser desorption Fourier transform ion cyclotron resonance (FTICR), and liquid secondary ion mass spectrometry (LSIMS) show collectively that a mixture of polyhydrofullerenes, containing C60H18 through to C60H36 with a skewed distribution centred on C60H32 is formed, the discrepancy in results arising from the thermal lability of this mixture of polyhydrofullerenes when subjected to the elevated temperatures (>250 °C) required for mass spectroscopic studies using direct-insertion heated probes.

Seventy years of Salem numbers

I survey results about, and recent applications of, Salem numbers.

VARIATIONS ON THE THEME OF HILBERT'S THEOREM 90

Given an algebraic number field K, we find two separate necessary and sufficient conditions on a given algebraic number for it to be expressible as a quotient, or as a difference, of two algebraic numbers which are conjugate over K.

Solving algebraic equations in roots of unity

Abstract. This paper is devoted to finding solutions of polynomial equations in roots of unity. It was conjectured by S. Lang and proved by M. Laurent that all such solutions can be described in terms of a finite number of parametric families called maximal torsion cosets. We obtain new explicit upper bounds for the number of maximal torsion cosets on an algebraic subvariety of the complex algebraic -torus . In contrast to earlier work that gives the bounds of polynomial growth in the maximum total degree of defining polynomials, the proofs of our results are constructive. This allows us to obtain a new algorithm for determining maximal torsion cosets on an algebraic subvariety of .