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.

A census of critical sets in the Latin squares of order at most six

A critical set in a Latin square of order n is a set of entries from the square which can be embedded in precisely one Latin square of order n, Such that if any element of the critical set. is deleted, the remaining set can be embedded, in more than one Latin square of order n.. In this paper we find all the critical sets of different sizes in the Latin squares of order at most six. We count the number of main and isotopy classes of these critical sets and classify critical sets from the main classes into various strengths. Some observations are made about the relationship between the numbers of classes, particularly in the 6 x 6 case. Finally some examples are given of each type of critical set.

An investigation of 2-critical sets in latin squares

In this paper we focus on the existence of 2-critical sets in the latin square corresponding to the elementary abelian 2-group of order 2(n). It has been shown by Stinson and van Rees that this latin square contains a 2-critical set of volume 4(n) - 3(n). We provide constructions for 2-critical sets containing 4(n) - 3(n) + 1 - (2(k-1) + 2(m-1) + 2(n-(k+m+1))) entries, where 1 less than or equal to k less than or equal to n and 1 less than or equal to m less than or equal to n - k. That is, we construct 2-critical sets for certain values less than 4(n) - 3(n) + 1 - 3 (.) 2([n /3]-1). The results raise the interesting question of whether, for the given latin square, it is possible to construct 2-critical sets of volume m, where 4(n) - 3(n) + 1 - 3 (.) 2([n/3]-1) < m < 4(n) - 3(n).

Mixed partitions of sets of triples into small planes

We continue our study of partitions of the set of all ((v)(3)) triples chosen from a v-set into pairwise disjoint planes with three points per line. We develop further necessary conditions for the existence of partitions of such sets into copies of PG(2, 2) and copies of AG(2, 3), and deal with the cases v = 13, 14, 15 and 17. These partitions, together with those already known for v = 12, 16 and 18, then become starters for recursive constructions of further infinite families of partitions. (C) 2004 Elsevier B.V. All rights reserved.

Searching oligo sets of human chromosome 12 using evolutionary strategies

DNA Microarray is a powerful tool to measure the level of a mixed population of nucleic acids at one time, which has great impact in many aspects of life sciences research. In order to distinguish nucleic acids with very similar composition by hybridization, it is necessary to design microarray probes with high specificities and sensitivities. Highly specific probes correspond to probes having unique DNA sequences; whereas highly sensitive probes correspond to those with melting temperature within a desired range and having no secondary structure. The selection of these probes from a set of functional DNA sequences (exons) constitutes a computationally expensive discrete non-linear search problem. We delegate the search task to a simple yet effective Evolution Strategy algorithm. The computational efficiency is also greatly improved by making use of an available bioinformatics tool.

Construction of representative transcript and protein sets of human, mouse, and rat as a platform for their transcriptome and proteome analysis

The number of mammalian transcripts identified by full-length cDNA projects and genome sequencing projects is increasing remarkably. Clustering them into a strictly nonredundant and comprehensive set provides a platform for functional analysis of the transcriptome and proteome, but the quality of the clustering and predictive usefulness have previously required manual curation to identify truncated transcripts and inappropriate clustering of closely related sequences. A Representative Transcript and Protein Sets (RTPS) pipeline was previously designed to identify the nonredundant and comprehensive set of mouse transcripts based on clustering of a large mouse full-length cDNA set (FANTOM2). Here we propose an alternative method that is more robust, requires less manual curation, and is applicable to other organisms in addition to mouse. RTPSs of human, mouse, and rat have been produced by this method and used for validation. Their comprehensiveness and quality are discussed by comparison with other clustering approaches. The RTPSs are available at ftp://fantom2.gsc.riken.go.jp/RTPS/. (C). 2004 Elsevier Inc. All rights reserved.

The effectiveness of hylan G-F 20 in patients with knee osteoarthritis: an application of two sets of response criteria developed by the OARSI and one set developed by OMERACT-OARSI

Objective: Secondary analyses of a previously conducted 1-year randomized controlled trial were performed to assess the application of responder criteria in patients with knee osteoarthritis (OA) using different sets of responder criteria developed by the Osteoarthritis Research Society International (OARSI) (Propositions A and B) for intra-articular drugs and Outcome Measures in Arthritis Clinical Trials (OMERACT)-OARSI (Proposition D). Methods: Two hundred fifty-five patients with knee OA were randomized to appropriate care with hylan G-F 20 (AC + H) or appropriate care without hylan G-F 20 (AC). A patient was defined as a responder at month 12 based on change in Western Ontario and McMaster Universities Osteoarthritis Index pain and function (0-100 normalized scale) and patient global assessment of OA in the study knee (at least one-category improvement in very poor, poor, fair, good and very good). All propositions incorporate both minimum relative and absolute changes. Results: Results demonstrated that statistically significant differences in responders between treatment groups, in favor of hylan G-F 20, were detected for Proposition A (AC + H = 53.5%, AC = 25.2%), Proposition B (AC + H = 56.7%, AC = 32.3%) and Proposition D (AC + H = 66.9%, AC = 42.5%). The highest effectiveness in both treatment groups was observed with Proposition D, whereas Proposition A resulted in the lowest effectiveness in both treatment groups. The treatment group differences always exceeded the required 20% minimum clinically important difference between groups established a priori, and were 28.3%, 24.4% and 24.4% for Propositions A, B and D, respectively. Conclusion: This analysis provides evidence for the capacity of OARSI and OMERACT-OARSI responder criteria to detect clinically important statistically detectable differences between treatment groups. (C) 2004 OsteoArthritis Research Society International. Published by Elsevier Ltd. All rights reserved.

A new class of critical sets in Latin squares

Previously the process of finding critical sets in Latin squares has been inside cumbersome by the complexity and number of Latin trades that, must be constructed. In this paper we develop a theory of Latin trades that yields more transparent constructions. We use these Latin trades to find a new class of critical sets for Latin squares which are a product of the Latin square of order 2 with a. back circulant Latin square of odd order.

Toddler subtraction with large sets: further evidence for an analog-magnitude representation of number

Two experiments were conducted to test the hypothesis that toddlers have access to an analog-magnitude number representation that supports numerical reasoning about relatively large numbers. Three-year-olds were presented with subtraction problems in which initial set size and proportions subtracted were systematically varied. Two sets of cookies were presented and then covered The experimenter visibly subtracted cookies from the hidden sets, and the children were asked to choose which of the resulting sets had more. In Experiment 1, performance was above chance when high proportions of objects (3 versus 6) were subtracted from large sets (of 9) and for the subset of older participants (older than 3 years, 5 months; n = 15), performance was also above chance when high proportions (10 versus 20) were subtracted from the very large sets (of 30). In Experiment 2, which was conducted exclusively with older 3-year-olds and incorporated an important methodological control, the pattern of results for the subtraction tasks was replicated In both experiments, success on the tasks was not related to counting ability. The results of these experiments support the hypothesis that young children have access to an analog-magnitude system for representing large approximate quantities, as performance on these subtraction tasks showed a Webers Law signature, and was independent of conventional number knowledge.

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.