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.

On the completion of Latin rectangles to symmetric Latin squares

We find necessary and sufficient conditions for completing an arbitrary 2 by n latin rectangle to an n by n symmetric latin square, for completing an arbitrary 2 by n latin rectangle to an n by n unipotent symmetric latin square, and for completing an arbitrary 1 by n latin rectangle to an n by n idempotent symmetric latin square. Equivalently, we prove necessary and sufficient conditions for the existence of an (n - 1)-edge colouring of K-n (n even), and for an n-edge colouring of K-n (n odd) in which the colours assigned to the edges incident with two vertices are specified in advance.

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).

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.

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.

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.

Reliability of the Knee Examination in Osteoarthritis: Effect of Standardization

Objective. To assess the reliability of physical examination of the osteoarthritic (OA) knee by rheumatologists, and to evaluate the benefits of standardization. Methods. Forty-two physical signs and techniques were evaluated using a 6 X 6 Latin square design. Patients with mild to severe knee OA, based on physical and radiographic signs, were examined in random order prior to and following standardization of techniques. For those signs with dichotomous scales, agreement among the rheumatologists was calculated as the prevalence-adjusted bias-adjusted kappa (PABAK), while for the signs with continuous and ordinal scales, a reliability coefficient (R-c) was calculated using analysis of variance. A PABAK of >0.60 and an Re of >0.80 were considered to indicate adequate reliability. Results. Adequate poststandardization reliability was achieved for 30 of 42 physical signs/techniques (71%). The most highly reliable signs identified by physical examination of the OA knee included alignment by goniometer (R-c = 0.99), bony swelling (R-c = 0.97), general passive crepitus (R-c = 0.96), gait by inspection (PABAK = 0.78), effusion bulge sign (R-c = 0.97), quadriceps atrophy (R. = 0.97), medial tibiofemoral tenderness (R-c = 0.94), lateral tibiofemoral tenderness (R-c = 0.85), patellofemoral tenderness by grind test (R-c = 0.94), and flexion contracture (R-c = 0.95). The standardization process resulted in substantial improvements in reliability for evaluation of a number of physical signs, although for some signs, minimal or no effect of standardization was noted. After standardization, warmth (PABAK = 0.14), medial instability at 30degrees flexion (PABAK = 0.02), and lateral instability at 30degrees flexion (PABAK = 0.34) were the only 3 signs that were highly unreliable. Conclusion. With the exception of physical examinations for instability, a comprehensive knee examination can be performed with adequate reliability. Standardization further improves the reliability for some physical signs and techniques. The application of these findings to future OA studies will contribute to improved outcome assessments in OA.

Feeding strategies for grain-fed cattle in a hot environment

Six Bos taurus (Hereford) steers (body weight 324 22 kg) were used in a 45-day study with a replicated 3 x 3 Latin-square design. Three treatments [ad libitum feeding (ADLIB); limit feeding, 85% of ad libitum (LIMIT); bunk management feeding where steers were only given access to feed from 1600 to 0800 hours the following day (BUNK)] were imposed over 3 periods, with 2 steers assigned to each treatment in each period. Cattle were managed in a temperature-controlled metabolism unit and were exposed to both thermoneutral (17.7degreesC-26.1degreesC) and hot (16.7degreesC-32.9degreesC) environmental conditions. By design, during the thermoneutral period, the ADLIB cattle displayed greater intake (P < 0.05) than the LIMIT group, with the BUNK group being intermediate. However, during the hot period, both the LIMIT and BUNK treatment groups increased feed intake 4-5%, whereas feed intake of the ADLIB treatment group declined nearly 2%. During both periods respiration rate (RR, breath/min) followed the same pattern that was observed for feed intake, with the greatest (P < 0.05) RR found in the ADLIB treatment group (81.09 and 109.55, thermoneutral and hot, respectively) and lowest (P < 0.05) RR in the LIMIT treatment group (74.47 and 102.76, thermoneutral and hot, respectively). Rectal temperature (RT) did not differ among treatments during the thermoneutral period or the first hot day, although during the thermoneutral period the ADLIB treatment group did tend to display a lower RT, possibly as a result of other physiological processes (pulse rate and RR) aiding to keep RT lower. During the hot period, differences in RT were found on Day 5, with the LIMIT cattle having lower (P < 0.10) RT (38.92degreesC) than the ADLIB (39.18degreesC) cattle, with BUNK cattle RT (39.14degreesC) being intermediate. However, when hourly data were examined, the ADLIB cattle had greater(P < 0.05) RT than the BUNK and LIMIT at 1800 hours and greater RT (P < 0.05) than the LIMIT group at 1400, 1500, and 1600 hours. Clearly, a change in diurnal RT pattern was obtained by using the LIMIT and BUNK feeding regimen. Both of these groups displayed a peak RT during the hot conditions, between 2100 and 2200 hours, whereas the ADLIB group displayed a peak RT between 1400 and 1500 hours, a time very close to when peak climatic stress occurs. Based on these results it is apparent that feedlot managers could alleviate the effects of adverse hot weather on cattle by utilising either a limit-feeding regimen or altering bunk management practices to prevent feed from being consumed several hours prior to the hottest portion of the day.

Impact of acute weight loss and/or thermal stress on rowing ergometer performance

Purpose: The impact of acute weight loss on rowing performance was assessed when generous nutrient intake was provided in 2 h of recovery after making weight. Methods: Competitive rowers (N = 17) completed four ergometer trials, each separated by 48 h. Two trials were performed after a 4% body mass loss in the previous 24 h (WT) and two were performed after no weight restrictions, that is, unrestricted (UNR). In addition, two trials (I X WT, I X UNR) were in a thermoneutral environment (NEUTRAL, mean 21.1 +/- SD 0.7 degrees C, 29.0 +/- 4.5% RH) and two were in the heat (HOT 32.4, +/- 0.4 degrees C, 60.4 +/- 2.7% RH). Trials were performed in a counterbalanced fashion according to a Latin square design. Aggressive nutritional recovery strategies (WT 2.3 g(.)kg(-11) carbohydrate, 34 mg-kg(-1) Na, 28.4 mL(.)kg(-1) fluid; UNR ad libitum) were employed in the 2 h after weigh-in. Results: Both WT (mean 2.1, 95% CI 0.7-3.4 s; P = 0.003) and HOT (4.1, 2.7 - 5.4 s; P < 0.001) compromised 2000-m time-trial performance. Whereas WT resulted in hypohydration, the associated reduction in plasma volume explained only part of the performance compromise observed (0.2 s for every 1% decrement) Moreover, WT did not influence core temperature or indices of cardiovascular function. Conclusions: Acute weight loss compromised performance, despite generous nutrient intake in recovery, although the effect was small. Performance decrements were further exacerbated when exercise was performed in the heat.