Dr Judy-Anne Osborn

Let D(n) be the maximal determinant for n × n {±1}-matrices, and R(n) = D(n)/nn/2 be the ratio of D(n) to the Hadamard upper bound. Using the probabilistic method, we prove new lower bounds on D(n) and R(n) in terms of d = n - h, where h is the order of a Hadamard matrix and h is maximal subject to h = n. For example, (Formula Presented) By a recent result of Livinskyi, d2/h1/2 ¿ 0 as n ¿ 8, so the second bound is close to (pe/2)-d/2 for large n. Previous lower bounds tended to zero as n¿8with d fixed, except in the cases d ¿ {0, 1}. For d = 2, our bounds are better for all sufficiently large n. If the Hadamard conjecture is true, then d = 3, so the first bound above shows that R(n) is bounded below by a positive constant (pe/2)-3/2 > 0.1133.

We consider several families of binomial sum identities whose definition involves the absolute value function. In particular, we consider centered double sums of the form (formula presented) obtaining new results in the cases a = 1, 2. We show that there is a close connection between these double sums in the case a = 1 and the single centered binomial sums considered by Tuenter.

Abstract We prove an upper bound on sums of squares of minors of {+1, -1\}-matrices. The bound is sharp for Hadamard matrices, a result due to de Launey and Levin [' (1,-1)-matrices with near-extremal properties', SIAM J. Discrete Math. 23(2009), 1422-1440], but our proof is simpler. We give several corollaries relevant to minors of Hadamard matrices. Copyright © 2012 Australian Mathematical Publishing Association Inc.

We present an analysis of a partially directed walk model of a polymer which at one end is tethered to a sticky surface and at the other end is subjected to a pulling force at fixed angle away from the point of tethering. Using the kernel method, we derive the full generating function for this model in two and three dimensions and obtain the respective phase diagrams. We observe adsorbed and desorbed phases with a thermodynamic phase transition in between. In the absence of a pulling force this model has a secondorder thermal desorption transition which merely gets shifted by the presence of a lateral pulling force. On the other hand, if the pulling force contains a non-zero vertical component this transition becomes first order. Strikingly, we find that, if the angle between the pulling force and the surface is below a critical value, a sufficiently strong force will induce polymer adsorption, no matter how large the temperature of the system. Our findings are similar in two and three dimensions, an additional feature in three dimensions being the occurrence of a re-entrance transition at constant pulling force for low temperature, which has been observed previously for this model in the presence of pure vertical pulling. Interestingly, the re-entrance phenomenon vanishes under certain pulling angles, with details depending on how the three-dimensional polymer is modeled. © 2010 IOP Publishing Ltd and SISSA.

We prove a constant term theorem which is useful for finding weight polynomials for Ballot/Motzkin paths in a strip with a fixed number of arbitrary 'decorated' weights as well as an arbitrary 'background' weight. Our CT theorem, like Viennot's lattice path theorem from which it is derived primarily by a change of variable lemma, is expressed in terms of orthogonal polynomials which in our applications of interest often turn out to be non-classical. Hence, we also present an efficient method for finding explicit closed-form polynomial expressions for these non-classical orthogonal polynomials. Our method for finding the closed-form polynomial expressions relies on simple combinatorial manipulations of Viennot's diagrammatic representation for orthogonal polynomials. In the course of the paper we also provide a new proof of Viennot's original orthogonal polynomial lattice path theorem. The new proof is of interest because it uses diagonalization of the transfer matrix, but gets around difficulties that have arisen in past attempts to use this approach. In particular we show how to sum over a set of implicitly defined zeros of a given orthogonal polynomial, either by using properties of residues or by using partial fractions. We conclude by applying the method to two lattice path problems important in the study of polymer physics as the models of steric stabilization and sensitized flocculation. © 2009 IOP Publishing Ltd.