Smallest weak and smallest totally weak critical sets in the latin squares of order at most seven

A critical set in a latin square of order n is a set of entries in a latin square which can be embedded in precisely one latin square of order n. Also, 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 smallest weak and smallest totally weak critical sets for all the latin squares of orders six and seven. Moreover, we computationally prove that there is no (totally) weak critical set in the back circulant latin square of order five and we find a totally weak critical set of size seven in the other main class of latin squares of order five.

On the direct product of two weak uniquely completable partial latin squares

In this note we show by counter-example that the direct product of two weak uniquely completable partial latin squares is not necessarily a uniquely completable partial latin square. This counter-example rejects a conjecture by Gower (see [3]) on the direct product of two uniquely completable partial latin squares.

On the intersection problem for 1-factorizations and near 1-factorizations of K-v

The number of 1-factors (near 1-factors) that mu 1-factorizations (near 1-factorizations) of the complete graph K-v, v even (v odd), can have in common, is studied. The problem is completely settled for mu = 2 and mu = 3.

An in vitro study of non-axial forces upon the retention of an O-ring attachment

Objective The purpose of this study was to evaluate the retention force of an O-ring attachment system in different inclinations to the ideal path of insertion, using devices to compensate angulations. Material and methods Two implants were inserted into an aluminum base, and ball attachments were screwed to implants. Cylinders with O-rings were placed on ball attachments and connected to the test device using positioners to compensate implant angulations (0 degrees, 7 degrees, and 14 degrees). Plexiglass bases were used to simulate implant angulations. The base and the test device were positioned in a testing apparatus, which simulated insertion/removal of an overdenture. A total of 2900 cycles, simulating 2 years of overdenture use, were performed and 36 O-rings were tested. The force required for each cycle was recorded with computer software. Longitudinal sections of ball attachment-positioner-cylinder with O-rings of each angulation were obtained to analyze the relationship among them, and O-ring sections tested in each angulation were compared with an unused counterpart. A mixed linear model was used to analyze the data, and the comparison was performed by orthogonal contrasts (alpha=0.05). Results At 0 degrees, the retention force decreased significantly over time, and the retention force was significantly different in all comparisons, except from 12 to 18 months. When the implants were positioned at 7 degrees, the retention force was statistically different at 0 and 24 months. At 14 degrees, significant differences were found from 6 and 12 to 24 months. Conclusions Within the limitations of this study, it was concluded that O-rings for implant/attachments perpendicular to the occlusal plane were adequately retentive over the first year and that the retentive capacity of O-ring was affected by implant inclinations despite the proposed positioners. To cite this article:Rodrigues RCS, Faria ACL, Macedo AP, Sartori IAM, de Mattos MGC, Ribeiro RF. An in vitro study of non-axial forces upon the retention of an O-ring attachment.Clin. Oral Impl. Res. 20, 2009; 1314-1319.doi: 10.1111/j.1600-0501.2009.01742.x.

Star factorizations of graph products

A k-star is the graph K-1,K-k. We prove a general theorem about k-star factorizations of Cayley graphs. This is used to give necessary and sufficient conditions for the existence of k-star factorizations of any power (K-q)(S) of a complete graph with prime power order q, products C-r1 x C-r2 x ... x C-rk of k cycles of arbitrary lengths, and any power (C-r)(S) of a cycle of arbitrary length. (C) 2001 John Wiley & Sons, Inc.

On the Oberwolfach problem with two similar length cycles

For all in greater than or equal to 3, the Oberwolfach problem is solved for the case where the 2-factors consist of two cycles of lengths in and m + 1, and for the case where the 2-factors consist of two cycles of lengths m and m + 2.

Dam worms

Human and animal infection rates with the Oriental schistosome have steadily declined in China over the last half-century, but the Three Gorges Dam may reverse this decline by creating new, or enlarging existing, ideal environments for the worm and its aquatic snail intermediate host.

The influence of genetic and environmental factors in estimations of current body size, desired body size, and body dissatisfaction

The objective was to investigate the genetic epidemiology of figural stimuli. Standard figural stimuli were available from 5,325 complete twin pairs: 1,751 (32.9%) were monozygotic females, 1,068 (20.1%) were dizygotic females, 752 (14.1%) were monozygotic males, 495 (9.3%) were dizygotic males, and 1,259 (23.6%) were dizygotic male-female pairs. Univariate twin analyses were used to examine the influences on the individual variation in current body size and ideal body size. These data were analysed separately for men and women in each of five age groups. A factorial analysis of variance, with polychoric correlations between twin pairs as the dependent variable, and age, sex, zygosity, and the three interaction terms (age x sex, age x zygosity, sex x zygosity) as independent variables, was used to examine trends across the whole data set. Results showed genetic influences had the largest impact on the individual variation in current body size measures, whereas non-shared environmental influences were associated with the majority of individual variation in ideal body size. There was a significant main effect of zygosity (heritability) in predicting polychoric correlations for current body size and body dissatisfaction. There was a significant main effect of gender and zygosity in predicting ideal body size, with a gender x zygosity interaction. In common with BMI, heritability is important in influencing the estimation of current body size. Selection of desired body size for both men and women is more strongly influenced by environmental factors.

Energy expenditure and the body cell mass in cystic fibrosis

Poor nutritional status in patients with cystic fibrosis (CF) is associated with severe lung disease, and possible causative factors include inadequate intake, malabsorption, and increased energy requirements. Body cell mass (which can be quantified by measurement of total body potassium) provides an ideal standard for measurements of energy expenditure. The aim of this study was to compare resting energy expenditure (REE) in patients with CF with both predicted values and age-matched healthy children and to determine whether REE was related to either nutritional status or pulmonary function. REE was measured by indirect calorimetry and body cell mass by scanning with total body potassium in 30 patients with CF(12 male, mean age = 13.07 +/- 0.55 y) and 18 healthy children (six male, mean age = 12.56 +/- 1.25 y). Nutritional status was expressed as a percentage of predicted total body potassium; Lung function was measured in the CF group by spirometry and expressed as the percentage of predicted forced expiratory volume in 1 s. Mean REE was significantly increased in the patients with CF compared with healthy children (119.3 +/- 3.1% predicted versus 103.6 +/- 5% predicted, P < 0.001) and, using multiple regression techniques, REE for total body potassium was significantly increased in patients with CF (P = 0.0001). There was no relation between REE and nutritional status or pulmonary disease status in the CF group. In conclusion, REE is increased in children and adolescents with CF but is not directly related to nutritional status or pulmonary disease. Nutrition 2001;17:22-25. (C)Elsevier Science Inc. 2001.

Ni(II)-doped CsCdBrCl2: Variation of spectral and structural properties via mixed-halide coordination

A new addition to the family of single-molecule magnets is reported: an Fete cage stabilized with benzoate and pyridonate ligands. Monte Carlo methods have been used to derive exchange parameters within the cage, and hence model susceptibility behavior.