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.

Balanced critical sets in Latin squares

In this note we first introduce balanced critical sets and near balanced critical sets in Latin squares. Then we prove that there exist balanced critical sets in the back circulant Latin squares of order 3n for n even. Using this result we decompose the back circulant Latin squares of order 3n, n even, into three isotopic and disjoint balanced critical sets each of size 3n. We also find near balanced critical sets in the back circulant Latin squares of order 3n for n odd. Finally, we examine representatives of each main class of Latin squares of order up to six in order to determine which main classes contain balanced or near balanced critical sets.

Variance-balanced designs under interference for dependent observations

It is shown that variance-balanced designs can be obtained from Type I orthogonal arrays for many general models with two kinds of treatment effects, including ones for interference, with general dependence structures. These designs can be used to obtain optimal and efficient designs. Some examples and design comparisons are given. (C) 2002 Elsevier B.V. All rights reserved.

A report of a national mutation testing service for the MEN1 gene: clinical presentations and implications for mutation testing

Introduction: Mutation testing for the MEN1 gene is a useful method to diagnose and predict individuals who either have or will develop multiple endocrine neoplasia type 1 ( MEN 1). Clinical selection criteria to identify patients who should be tested are needed, as mutation analysis is costly and time consuming. This study is a report of an Australian national mutation testing service for the MEN1 gene from referred patients with classical MEN 1 and various MEN 1- like conditions. Results: All 55 MEN1 mutation positive patients had a family history of hyperparathyroidism, had hyperparathyroidism with one other MEN1 related tumour, or had hyperparathyroidism with multiglandular hyperplasia at a young age. We found 42 separate mutations and six recurring mutations from unrelated families, and evidence for a founder effect in five families with the same mutation. Discussion: Our results indicate that mutations in genes other than MEN1 may cause familial isolated hyperparathyroidism and familial isolated pituitary tumours. Conclusions: We therefore suggest that routine germline MEN1 mutation testing of all cases of classical'' MEN1, familial hyperparathyroidism, and sporadic hyperparathyroidism with one other MEN1 related condition is justified by national testing services. We do not recommend routine sequencing of the promoter region between nucleotides 1234 and 1758 ( Genbank accession no. U93237) as we could not detect any sequence variations within this region in any familial or sporadic cases of MEN1 related conditions lacking a MEN1 mutation. We also suggest that testing be considered for patients < 30 years old with sporadic hyperparathyroidism and multigland hyperplasia

Existence of complementary pairs of proportionally balanced designs

Proportionally balanced designs (pi BDs) were introduced by Gray and Matters in response to a need for the allocation of markers of the Queensland Core Skills Test to have a certain property. Subsequent papers extended the theoretical results relating to such designs and provided further instances and general constructions. This work focused on designs comprising blocks of precisely two sizes, and when each variety occurs with one of precisely two possible frequencies. Two designs based on the set V of varieties are complementary if, whenever B is a block of one, then its complement with regard to the set V is a block of the other. Here we present necessary conditions for the existence of complementary pairs of such pi BDs and provide lists of some restricted parameter sets satisfying these necessary conditions. The lists are arranged according to the number of blocks. We demonstrate that not all of these parameter sets give rise to designs. However we establish by construction of the sets of blocks that, for every feasible number of blocks less than or equal to 100, with the possible exception of 63, there exists at least one pair of complementary pi BDs. We also investigate the conditions under which the complementary design can be isomorphic to the original design, and again provide a list of feasible parameters for pairs of such designs with at most 400 blocks.

Review of Beginning Reading: A balanced approach to teaching literacy during the first three years at school by Yola Centre

Beginning Reading: A Balanced Approach to Literacy Instruction during the First Three Years at School by Yola Center is intended for those involved in teaching and supporting literacy practices in regular classrooms and addresses literacy practices for learners with disabilities and those experiencing difficulties in literacy learning.

A note on the Balanced method

Recently the Balanced method was introduced as a class of quasi-implicit methods for solving stiff stochastic differential equations. We examine asymptotic and mean-square stability for several implementations of the Balanced method and give a generalized result for the mean-square stability region of any Balanced method. We also investigate the optimal implementation of the Balanced method with respect to strong convergence.