Judy-anne Heather Osborn - Publications

1. J. H. Osborn, The Hadamard Maximal Determinant Problem, Honours Thesis, University of Melbourne 2002, 144 pp.
Pages 1-25, 36-70, 71-105, 106-144.

Summary. The Hadamard maximal determinant problem (maxdet problem for short) is: "given n, what is the largest possible determinant of an n times n matrix R with elements +-1?". We understand this to include related problems such as the construction of maxdet matrices, the number of equivalence classes of such matrices, and connections with other branches of mathematics/statistics such as coding theory and the design of experiments (saturated D-optimal designs). Hadamard's problem has many applications to signal processing, error-correcting codes, experimental design theory, and cryptography. This thesis is a survey of the Hadamard maximal determinant problem.

2. J. H. Osborn, Combinatorics of Pavings and Paths, PhD thesis, University of Melbourne, April 2007, xxvi+292 pp.

Abstract. The principle theme of this thesis is the enumeration of directed lattice paths. An ancillary theme is the development of a notion of pavings due to Viennot. We then utilize pavings in the solution of path problems.

Pavings:

We build on Viennot's conception of "pavings" on a path graph, along with associated "paving polynomials"; and Viennot's bijection between these pavings and cycles on the digraphs associated with Ballot and Motzkin paths.

• We show that a very natural generalization extends Viennot's pavings and cycle bijection to d-up and Lukasiewicz paths; but not to Jumpstep paths.
• We further generalize the mapping between cycles and pavings to enable the calculation of Jump-step paving polynomials.
• We define a new kind of path graph paving called a "Laurent paving", which has some complexity/combinatorial advances over the original pavings; and prove the combinatorial relationship between the two.
• We use the combinatorics of pavings and Laurent pavings to enable closed-form expressions for paving polynomials and Laurent paving polynomials associated with weighted Ballot, Motzkin and other more general paths.
Paths:
• We derive the generating function for n-banded directed ballot paths in the half plane explicitly for all n, thus showing that it is always a quadratic function. In the case of bi-banded directed ballot paths in the half plane, we also find the weight function explicitly, and in so doing proved a new combinatorial interpretation of the Narayana numbers.
• We give a new combinatorial derivation of the known single variable constant coecient recurrence relations on path weight polynomials. This denes a new combinatorial derivation of generating functions for directed paths in a strip in terms of pavings.
• We define a new form of `Constant Term' method applicable to weighted Ballot-like and Motzkin-like paths with nite numbers of decorations. We use the method to solve an open enumeration problem from the 1970's and generalizations thereof.

3. R. Brak, J. Essam, J. H. Osborn, A. L. Owczarek and A. Rechnitzer, Lattice paths and the constant term, J. Phys.: Conf. Ser.. 42 (2006), 47-58.

Abstract. We firstly review the constant term method (CTM), illustrating its combinatorial connections and show how it can be used to solve a certain class of lattice path problems. We show the connection between the CTM, the transfer matrix method (eigenvectors and eigenvalues), partial difference equations, the Bethe Ansatz and orthogonal polynomials. Secondly, we solve a lattice path problem first posed in 1971. The model stated in 1971 was only solved for a special case. We solve the full model.

4. R. Brak and J. H. Osborn, Chebyshev type lattice path weight polynomials by a constant term method, J. Phys. A: Math. Theor. 42, 44 (2009), 22 pp.

Abstract. 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 nonclassical. Hence, we also present an efficient method for finding explicit closedform 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.

Accepted

5. Jennifer Badham and Judy-anne Osborn, "Zombies in the City: a Netlogo Model".

Book chapter in refereed book: `Academics on Zombies'. Editor Robert Smith? (The question mark is part of the name and not a typo.)

Abstract. We present a NetLogo model of an infestation of `Zombies' in a city environment. Our model offers hope to those who may have been dejected by previous mathematical modelling of zombie attack, which painted a bleak picture for the survival of humanity under most scenarios. We show that factors alluded to in a genre defining paper \cite{MHIS}, but not incorporated into their mathematical model, are of essential importance in survival scenarios for humans in the face of zombie infiltration - in particular humans' higher speed and capacity to gain in skill through experience. Thus in our model we observe the full triad of possibilities: zombie-win, human-win and stalemate. This demonstrates the usefulness of Agent Based Models to complement analytical approaches.

6. J. H. Osborn, Bi-banded Dyck paths, a bijection and the Narayana numbers, Accepted for publication in the Australasian Journal of Combinatorics 30 July 2010. arXiv:1005.1521v1

Abstract. We find a new interpretation of Narayana numbers as coefficients of weight polynomials enumerating bi-banded Dyck paths, which class of paths has arisen naturally in a solution of the stationary state of a stochastic process for traffic modeling called TASEP. Our proof of the enumerative result is by means of a bijection between bi-banded Dyck paths and corner-counting Dyck paths; thus adding one more to the list of combinatorial families known to be counted by Narayana numbers.

Submitted

7. J. Osborn and T. Prellberg, Forcing Adsorption of a Tethered Polymer by Pulling, Submitted for publication 20 July 2010. arXiv:1007.3160

Abstract. 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 second-order 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 beneath 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 reentrance transition at constant pulling force for small temperature, which has been observed previously for this model in the presence of pure vertical pulling. Interestingly, the reentrance phenomenon vanishes under certain pulling angles, with details depending on how the three-dimensional polymer is modeled.

In Preparation

8. R. P. Brent and J. H. Osborn, Minors of maximal determinant matrices, in preparation.

9. R. P. Brent, W. H. Orrick, J. H. Osborn and P. Zimmermann, The maximal {-1,1}-determinant for matrices of orders 19 and 37, in preparation. Abstract