The Fa,b,c conjecture is true, II

In 1977 a five-part conjecture was made about a family of groups related to trivalent graphs and subsequently two parts of the conjecture were proved. The conjecture completely determines all finite members of the family. Here we complete the proof of the conjecture by giving proofs for the remaining three parts. (c) 2006 Elsevier Inc. All rights reserved.

Varieties of algebras arising from K-perfect m-cycle systems

A class of algebras forms a variety if it is characterised by a collection of identities. There is a well-known method, often called the standard construction, which gives rise to algebras from m-cycle systems. It is known that the algebras arising from {1}-perfect m-cycle systems form a variety for m is an element of {3, 5} only, and that the algebras arising from {1, 2}-perfect m-cycle systems form a variety for m is an element of {3, 5, 7} only. Here we give, for any set K of positive integers, necessary and sufficient conditions under which the algebras arising from K-perfect m-cycle systems form a variety. (c) 2006 Elsevier B.V. All rights reserved.

A flaw in the use of minimal defining sets for secret sharing schemes

It is shown that in some cases it is possible to reconstruct a block design D uniquely from incomplete knowledge of a minimal defining set for D. This surprising result has implications for the use of minimal defining sets in secret sharing schemes.

Maximal triangle trades with foundation 5 (mod 6) - the final case

The maximum possible volume of a simple, non-Steiner (v, 3, 2) trade was determined for all v by Xhosrovshahi and Torabi (Ars Combinatoria 51 (1999), 211-223), except that in the-case v equivalent to 5 (mod 6), v >= 23, they were only able to provide an upper, bound on the volume. In this paper we construct trades with volume equal to that bound for all v equivalent to 5 (mod 6), thus completing the problem.

Minimal Homogeneous Latin Trades

A latin trade is a subset of a latin square which may be replaced with a disjoint mate to obtain a new latin square. A d-homogeneous latin trade is one which intersects each row, each column and each entry of the latin square either 0 or d times. In this paper we give a construction for minimal d-homogeneous latin trades of size dm, for every integer d >= 3, and m >= 1.75d(2) + 3. We also improve this bound for small values of d. Our proof relies on the construction of cyclic sequences whose adjacent sums are distinct. (c) 2006 Elsevier B.V. All rights reserved.

I-Spi: innovative shared practical ideas: a support guide for literacy and numeracy in the early years of schooling, for home tutor/supervisors in distance education

Innovative Shared Practical Ideas (I-Spi) is a guide to help you and your children learn together. It is designed to affirm, support and strengthen your role as home tutor/supervisors in your daily learning sessions with your children. In this guide particular emphasis is given to the value of talk, formal and informal early literacy and numeracy practices (including ideas from distance school lessons, from home tutor/supervisors, research, and beyond), assessment of these practices together with informal assessment ideas for gauging your children’s literacy and numeracy progress, and stepping in and building on strategies

Effects of inducer continuity on auditory stream segregation:comparison of physical and perceived continuity in different contexts

The factors influencing the stream segregation of discrete tones and the perceived continuity of discrete tones as continuing through an interrupting masker are well understood as separate phenomena. Two experiments tested whether perceived continuity can influence the build-up of stream segregation by manipulating the perception of continuity during an induction sequence and measuring streaming in a subsequent test sequence comprising three triplets of low and high frequency tones (LHL-…). For experiment 1, a 1.2-s standard induction sequence comprising six 100-ms L-tones strongly promoted segregation, whereas a single extended L-inducer (1.1 s plus 100-ms silence) did not. Segregation was similar to that following the single extended inducer when perceived continuity was evoked by inserting noise bursts between the individual tones. Reported segregation increased when the noise level was reduced such that perceived continuity no longer occurred. Experiment 2 presented a 1.3-s continuous inducer created by bridging the 100-ms silence between an extended L-inducer and the first test-sequence tone. This configuration strongly promoted segregation. Segregation was also increased by filling the silence after the extended inducer with noise, such that it was perceived like a bridging inducer. Like physical continuity, perceived continuity can promote or reduce test-sequence streaming, depending on stimulus context.

A variational method and approximations of a Cauchy problem for elliptic equations

A Cauchy problem for general elliptic second-order linear partial differential equations in which the Dirichlet data in H½(?1 ? ?3) is assumed available on a larger part of the boundary ? of the bounded domain O than the boundary portion ?1 on which the Neumann data is prescribed, is investigated using a conjugate gradient method. We obtain an approximation to the solution of the Cauchy problem by minimizing a certain discrete functional and interpolating using the finite diference or boundary element method. The minimization involves solving equations obtained by discretising mixed boundary value problems for the same operator and its adjoint. It is proved that the solution of the discretised optimization problem converges to the continuous one, as the mesh size tends to zero. Numerical results are presented and discussed.