The intersection problem for K_4-e designs

A G-design of order n is a pair (P,B) where P is the vertex set of the complete graph K-n and B is an edge-disjoint decomposition of K-n into copies of the simple graph G. Following design terminology, we call these copies ''blocks''. Here K-4 - e denotes the complete graph K-4 with one edge removed. It is well-known that a K-4 - e design of order n exists if and only if n = 0 or 1 (mod 5), n greater than or equal to 6. The intersection problem here asks for which k is it possible to find two K-4 - e designs (P,B-1) and (P,B-2) of order n, with \B-1 boolean AND B-2\ = k, that is, with precisely k common blocks. Here we completely solve this intersection problem for K-4 - e designs.

3-Homogeneous latin trades

Let T be a partial latin square and L be a latin square with T subset of L. We say that T is a latin trade if there exists a partial latin square T' with T' boolean AND T = theta such that (LT) U T' is a latin square. A k-homogeneous latin trade is one which intersects each row, each column and each entry either 0 or k times. In this paper, we construct 3-homogeneous latin trades from hexagonal packings of the plane with circles. We show that 3-homogeneous latin trades of size 3 m exist for each m >= 3. This paper discusses existence results for latin trades and provides a glueing construction which is subsequently used to construct all latin trades of finite order greater than three. Crown Copyright (c) 2005 Published by Elsevier B.V. All rights reserved.

Lithium for schizophrenia revisited: A systematic review and meta-analysis of randomized controlled trials

Background: Clinicians frequently use lithium to augment antipsychotic medication in schizophrenia. Therefore, we undertook a systematic review and meta-analysis of the use of lithium in the treatment of schizophrenia. Data sources and study selection: Randomized controlled trials examining lithium (as a sole or an adjunctive compound) in participants with schizophrenia or related disorders were searched in the register of the Cochrane Schizophrenia Group. No language restrictions were applied. The Boolean phrase [lithium* or lithicarb or eskalith or lithobid or lithane or cibalith-s or quilonum or hypnorex] was used to locate articles. The search strategy initially identified 90 references. The authors of the included studies were contacted to obtain original patient data. The data were combined in a meta-analysis. The main outcome parameters were the number of patients with a clinically significant response and the number of patients leaving the studies early. Results: The meta-analysis includes 20 studies (N = 611). The evidence shows that lithium as a sole agent is ineffective in the treatment of schizophrenia. Eleven trials examined the augmentation of antipsychotics with lithium. More patients who received lithium augmentation than those who received antipsychotics alone were classified as responders. However, the superiority was not consistent across different response thresholds, and when patients with prominent affective symptoms were excluded from the analysis, the advantage of lithium augmentation was not significant (p = .07). Significantly more patients taking lithium left the trials early, suggesting a lower acceptability of lithium augmentation compared with that of taking antipsychotics alone. Conclusion: Despite some evidence in favor of lithium augmentation, the overall results are inconclusive. A large trial of lithium augmentation of antipsychotic medications will be required in order to detect a benefit of small effect size in patients with schizophrenia who lack affective symptoms.