2020 
O Toole JM, McKoy K, Freestone M, Osborn JA, ' Scientific Literacy : An Exercise in Model Building', Education Sciences, 10 116 (2020) [C1]



2020 
Osborn JA, Larkins JA, McBain B, Ellerton P, Black J, Borwein N, et al., 'Foundations of the DEFT Project: tertiary educators Developing Expertise Fostering Thinking', International Journal of Innovation in Science and Mathematics Education, 28 215 (2020)
We describe the rationale, creation, and activity of a longterm coconstructed voluntary professional development initiative for tertiary educators. This is a Community of Practi... [more]
We describe the rationale, creation, and activity of a longterm coconstructed voluntary professional development initiative for tertiary educators. This is a Community of Practice (CoP) formed to investigate ¿thinking¿ as a topic which may be explicitly taught. The aim of this paper is to share the value of this CoP in one context and insights into how similar approaches may be useful to other tertiary educators. The project has run for a year to date, involving a small butgrowing collective of tertiary educators, withmembers from one Canadian and several Australian Universities. Our methodology is participatory: we regularly meet,reflect,and record our reflections. Our records contain data relatingto our motivation, our insights, and the impact of these upon our choices in our teaching practices. In particular,our rationaleincludes the mutual desire to invest in developing understanding of our teaching challenges, to enable us to create thoughtful teaching approaches fit for our purposes and contexts. Hence, the central focus of our CoP isthe Development of our Expertise in Fostering Thinking (DEFT). This focus hasilluminatedgaps in existing scholarly literaturepertainingto communal development of theory, personal development of schemata, capacity for reflexivity, and instantiation in our disciplines. Opportunities and risks associated with our other sources of professional learningare identified and discussed. We elaborate on adoublelayered approach, in which we explore the constructionofour ownschemataas a precursor to helpingstudents buildtheirschemataas a foundation for their own understanding, and the role offlexible, critical,and creativethinking on our part. We utilise the scholarship of expertise, frequently returning to such questionsas ¿How do we know what our students are thinking?¿Insights gleaned from our reflections are shared, and recommendations are presented on the formation of similar projects.



2019 
Butler E, PrietoRodriguez E, Osborn JA, Howley P, Lloyd A, Kepert A, Roberts M, 'Learning Across Discipline Boundaries Through Narrative Inquiry: A Study of Collaboration to Improve Mathematics Teacher Education', Mathematics Teacher Education and Development, 21 87105 (2019) [C1]



2017 
Osborn JH, Hussain M, Trudgian T, 'Number Theory Down Under 2016. Dedicated to Professor Richard Brent on his 70th birthday.', Gazette of the Australian Mathematical Society, 44 8288 (2017)



2017 
Osborn JH, 'Guttmann 2015: 70 and counting', Gazette of the Australian Mathematical Society, 44 1825 (2017)



2016 
Brent RP, Osborn JAH, Smith WD, 'Probabilistic lower bounds on maximal determinants of binary matrices', Australasian Journal of Combinatorics, 66 350364 (2016) [C1]
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 lo... [more]
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.



2016 
Brent RP, Ohtsuka H, Osborn JAH, Prodinger H, 'Some binomial sums involving absolute values', Journal of Integer Sequences, 19 (2016) [C1]
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... [more]
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.



2015 
Brent RP, Osborn JAH, Smith WD, 'Note on best possible bounds for determinants of matrices close to the identity matrix', LINEAR ALGEBRA AND ITS APPLICATIONS, 466 2126 (2015) [C1]



2015 
PrietoRodriguez E, Howley P, Holmes K, Osborn J, Roberts M, Kepert A, 'Quality Teaching Rounds in Mathematics Teacher Education', Mathematics Teacher Education and Development (MTED), 17 98110 (2015) [C1]



2013 
Brent RP, Osborn JAH, 'On minors of maximal determinant matrices', Journal of Integer Sequences, 16 (2013) [C1]



2013 
Brent RP, Osborn JAH, 'Bounds on minors of binary matrices', Bulletin of the Australian Mathematical Society, 88 280285 (2013) [C1]
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)m... [more]
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 nearextremal properties', SIAM J. Discrete Math. 23(2009), 14221440], but our proof is simpler. We give several corollaries relevant to minors of Hadamard matrices. Copyright © 2012 Australian Mathematical Publishing Association Inc.



2013 
Brent RP, Osborn JAH, 'General lower bounds on maximal determinants of binary matrices', ELECTRONIC JOURNAL OF COMBINATORICS, 20 (2013) [C1]



2012 
Borwein JM, Osborn JAH, 'Response to 'Experimental Approaches to Theoretical Thinking: Artefacts and Proofs Proof and Proving in Mathematics Education'', Proof and Proving in Mathematics Education: The 19th ICMI Study, 15 138143 (2012) [C3]



2011 
Borwein JM, Osborn JAH, 'Loving and Hating Mathematics by Reuben Hersh and Vera JohnSteiner [Book Review]', The Mathematical Intelligencer, 33 6369 (2011) [C3]



2010 
Osborn J, Prellberg T, 'Forcing adsorption of a tethered polymer by pulling', Journal of Statistical Mechanics: Theory and Experiment, 2010 (2010)
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 fix... [more]
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 nonzero 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 reentrance 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 reentrance phenomenon vanishes under certain pulling angles, with details depending on how the threedimensional polymer is modeled. © 2010 IOP Publishing Ltd and SISSA.



2010 
Osborn JAH, 'Bibanded paths, a bijection and the Narayana numbers', Australasian Journal of Combinatorics, 48 243252 (2010) [C1]



2009 
Brak R, Osborn J, 'Chebyshev type lattice path weight polynomials by a constant term method', Journal of Physics A: Mathematical and Theoretical, 42 (2009)
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 ... [more]
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 nonclassical orthogonal polynomials. Our method for finding the closedform 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.



2006 
Brak R, Essam J, Osborn J, Owczarek AL, Rechnitzer A, 'Lattice Paths and the Constant Term', J Phys Conf Ser, 4758 (2006)


