Breadth-first search and the Andrews-Curtis conjecture

Andrews and Curtis conjectured in 1965 that every balanced presentation of the trivial group can be transformed into a standard presentation by a finite sequence of elementary transformations. Recent computational work by Miasnikov and Myasnikov on this problem has been based on genetic algorithms. We show that a computational attack based on a breadth-first search of the tree of equivalent presentations is also viable, and seems to outperform that based on genetic algorithms. It allows us to extract shorter proofs (in some cases, provably shortest) and to consider the length thirteen case for two generators. We prove that, up to equivalence, there is a unique minimum potential counterexample.

Irreducible cyclic presentations of the trivial group

We produce families of irreducible cyclic presentations of the trivial group. These families comprehensively answer questions about such presentations asked by Dunwoody and by Edjvet, Hammond, and Thomas. Our theorems are purely theoretical, but their derivation is based on practical computations. We explain how we chose the computations and how we deduced the theorems.

The epidemiology of invasive group A streptococcal disease in Victoria, Australia

Objective To estimate the incidence and severity of invasive group A streptococcal infection in Victoria, Australia. Design Prospective active surveillance study. Setting Public and private laboratories, hospitals and general practitioners throughout Victoria. Patients eople in Victoria diagnosed with group A streptococcal disease notified to the surveillance system between 1 March 2002 and 31 August 2004. Main outcome measure Confirmed invasive group A streptococcal disease. Results We identified 333 confirmed cases: an average annualised incidence rate of 2.7 (95% CI, 2.3-3.2) per 100000 population per year. Rates were highest in people aged 65 years and older and those younger than 5 years. The case-fatality rate was 7.8%. Streptococcal toxic shock syndrome occurred in 48 patients (14.4%), with a case-fatality rate of 23%. Thirty cases of necrotising fasciitis were reported; five (17%) of these patients died. Type 1 (23%) was the most frequently identified emm sequence type in all, age groups. All tested isolates were susceptible to penicillin and clindamycin. Two isolates (4%) were resistant to erythromycin. Conclusion The incidence of invasive group A streptococcal disease in temperate Australia is greater than previously appreciated and warrants greater public health attention, including its designation as a notifiable disease.

Making quality teachers : the role of an explicit values-based pedagogy

There is little conjecture that quality teaching is essential to student achievement and well-being. Whilst much has been written about the importance of quality teaching, including the link to pre-service teacher education, to date there has been little investigation into specific pedagogical practices that can enhance quality teaching dimensions within a pre-service teacher education programme. This paper reports on a small-scale qualitative research study, undertaken in an Australian university, which linked the fields of quality teaching, pre-service teacher education and values education. The study followed the journey of five pre-service teacher education students as they undertook their second field experience unit where the focus was centred on the values-based pedagogy of Philosophy in the Classroom. The research findings, collected via interviews, demonstrated that an explicit values-based pedagogy can have a positive impact on the development of quality teaching dimensions. This new knowledge has potential for further research into examining the ways quality teaching dimensions are gained and practised by pre-service teacher education students and these findings and recommendations are discussed in this paper.

John Andrews : Architecture a Performing Art

The distinguished Australian architect surveys his career and examines how his architectural theories are expressed in his designs.

Equivariant Birch-Swinnerton-Dyer Conjecture for the Base Change of Elliptic Curves: An Example

Let E be an elliptic curve defined over Q and let K/Q be a finite Galois extension with Galois group G. The equivariant Birch-Swinnerton-Dyer conjecture for h(1)(E x(Q) K)(1) viewed as amotive over Q with coefficients in Q[G] relates the twisted L-values associated with E with the arithmetic invariants of the same. In this paper I prescribe an approach to verify this conjecture for a given data. Using this approach, we verify the conjecture for an elliptic curve of conductor 11 and an S-3-extension of Q.

A Conjecture Concerning Transformation Of A Supercooled Hard-Sphere Liquid To A Metastable Disordered Solid

It is conjectured that the hard sphere system has several distinct solid phases, all but one of which are metastable. The bifurcation theory analysis of freezing is extended to the description of the transition between a supercooled liquid and a disordered solid by defining a restricted phase space for the disordered solid. This approach leads to the prediction of a first order transition between a supercooled hard sphere fluid and a disordered metastable hard sphere solid. The results of the calculation are in qualitative agreement with the results of Woodcock's molecular dynamics computer simulations. The Journal of Chemical Physics is copyrighted by The American Institute of Physics.

Albert Einstein with Robert Andrews Millikan, Albert Michelson and unidentified men; Pasadena, California Portraits Groups; Men

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Curves on threefolds and a conjecture of Griffiths-Harris

We prove that any arithmetically Gorenstein curve on a smooth, general hypersurface of degree at least 6, is a complete intersection. This gives a characterisation of complete intersection curves on general type hypersurfaces in . We also verify that certain 1-cycles on a general quintic hypersurface are non-trivial elements of the Griffiths group.

Proof of a conjecture of Frankl and Furedi

We give a simple linear algebraic proof of the following conjecture of Frankl and Furedi [7, 9, 13]. (Frankl-Furedi Conjecture) if F is a hypergraph on X = {1, 2, 3,..., n} such that 1 less than or equal to /E boolean AND F/ less than or equal to k For All E, F is an element of F, E not equal F, then /F/ less than or equal to (i=0)Sigma(k) ((i) (n-1)). We generalise a method of Palisse and our proof-technique can be viewed as a variant of the technique used by Tverberg to prove a result of Graham and Pollak [10, 11, 14]. Our proof-technique is easily described. First, we derive an identity satisfied by a hypergraph F using its intersection properties. From this identity, we obtain a set of homogeneous linear equations. We then show that this defines the zero subspace of R-/F/. Finally, the desired bound on /F/ is obtained from the bound on the number of linearly independent equations. This proof-technique can also be used to prove a more general theorem (Theorem 2). We conclude by indicating how this technique can be generalised to uniform hypergraphs by proving the uniform Ray-Chaudhuri-Wilson theorem. (C) 1997 Academic Press.

The Herzog-Vasconcelos conjecture for affine semigroup rings

Let S be a simplicial affine semigroup such that its semigroup ring A = k[S] is Buchsbaum. We prove for such A the Herzog-Vasconcelos conjecture: If the A-module Der(k)A of k-linear derivations of A has finite projective dimension then it is free and hence A is a polynomial ring by the well known graded case of the Zariski-Lipman conjecture.

OOCs, Partial Relative Difference Families and a Conjecture of Golomb

The cyclic difference sets constructed by Singer are also examples of perfect distinct difference sets (DDS). The Bose construction of distinct difference sets, leads to a relative difference set. In this paper we introduce the concept of partial relative DDS and prove that an optical orthogonal code (OOC) construction due to Moreno et. al., is a partial relative DDS. We generalize the concept of ideal matrices previously introduced by Kumar and relate it to the concepts of this paper. Another variation of ideal matrices is introduced in this paper: Welch ideal matrices of dimension n by (n - 1). We prove that Welch ideal matrices exist only for n prime. Finally, we recast an old conjecture of Golomb on the Welch construction of Costas arrays using the concepts of this paper. This connection suggests that our construction of partial relative difference sets is in a sense, unique