Multi-latin squares

A multi-latin square of order $n$ and index $k$ is an $n\times n$ array of multisets, each of cardinality $k$, such that each symbol from a fixed set of size $n$ occurs $k$ times in each row and $k$ times in each column. A multi-latin square of index $k$ is also referred to as a $k$-latin square. A $1$-latin square is equivalent to a latin square, so a multi-latin square can be thought of as a generalization of a latin square. In this note we show that any partially filled-in $k$-latin square of order $m$ embeds in a $k$-latin square of order $n$, for each $n\geq 2m$, thus generalizing Evans' Theorem. Exploiting this result, we show that there exist non-separable $k$-latin squares of order $n$ for each $n\geq k+2$. We also show that for each $n\geq 1$, there exists some finite value $g(n)$ such that for all $k\geq g(n)$, every $k$-latin square of order $n$ is separable. We discuss the connection between $k$-latin squares and related combinatorial objects such as orthogonal arrays, latin parallelepipeds, semi-latin squares and $k$-latin trades. We also enumerate and classify $k$-latin squares of small orders.Comment: Final version as sent to journa

Investigation of macrocyclisation routes to 1,4,7-triazacyclononanes : efficient syntheses from 1,2-ditosylamides

Two routes to the synthesis of a cyclohexyl-fused 1,4,7-triazacyclononane involving macrocyclisations of tosamides have been investigated. In the first approach, using a classic Richman-Atkins-type cyclisation of a cyclohexyl-substituted 1,4,7-tritosamide with ethylene glycol ditosylate, afforded the cyclohexyl-fused 1,4,7-triazacyclononane in 5.86% overall yield in four steps. The second, more concise, approach involving the macrocyclisation of trans-cyclohexane-1,2-ditosamide with the tritosyl derivative of diethanolamine initially gave poor yields (< 25%). The well-documented problems with efficiencies in macrocyclisations using 1,2-ditosamides led to the use of a wider range of 1,2-ditosamides including ethane-1,2-ditosamide and propane-1,2-ditosamide. These extended studies led to the development of an efficient macrocyclisation protocol using lithium hydride. This new method afforded 1,4,7-tritosyl-1,4,7-triazacyclononanes in good yield (57-90%) from 1,2-ditosamides in a single step. These efficient methods were then applied to the preparation of a chiral cyclohexyl-fused 1,4,7-tritosyl-1,4,7-triazacyclononane (65-70%). This key chiral intermediate was then converted into a copper(II) complex following detosylation and N-methylation. The resulting chiral copper(II) complex catalysed the aziridination of styrene but it did so in a racemic fashion

Control for Uterine Fibroid Embolisation- An Initial Experience in East Africa

Uterine fibroid embolisation (UFE) generates moderate to severe post-procedural pain. We present a case series of 24 patients who underwent UFE during our first experience in managing the sometimes excruciating pain that accompanies embolisation of the uterine arteries. We also show the evolution of our protocol for post-procedural pain management from a first to second round of procedures

Model-free measurement of the pair potential in colloidal fluids using optical microscopy

We report a straightforward, model-free approach for measuring pair potentials from particle-coordinate data, based on enforcing consistency between the pair distribution function measured separately by the distance-histogram and test-particle insertion routes. We demonstrate the method's accuracy and versatility in simulations of simple fluids, before applying it to an experimental system composed of superparamagnetic colloidal particles. The method will enable experimental investigations into many-body interactions and allow for effective coarse-graining of interactions from simulations.Comment: 8 pages, 3 figures including supplemental materia

Highly efficient aqueous phase reduction of nitroarenes catalyzed by phosphine-decorated polymer immobilized ionic liquid stabilized PdNPs

Palladium nanoparticles stabilized by lightly cross-linked phosphine-decorated polymer immobilized ionic liquids (PIIL) and their PEGylated counterparts (PEGPIIL) are highly effective catalysts for the aqueous phase hydrogenation and sodium borohydride-based reduction of a wide range of nitroaromatic and heteroaromatic compounds under mild conditions with low catalyst loadings. Introduction of extensive cross-linking with tris(4-vinylphenyl)phosphine to isolate the phosphine-based heteroatom and limit the number of surface Pd⋯P interactions did not have a significant influence on catalyst performance. Comparative testing revealed PdNPs immobilized on lightly cross-linked phsophine-decoarted PEGylated polymer to be a highly efficient catalyst for the aqueous phase reduction of nitroarenes with a TON of 36 000 (TOF = 2580 h−1) for hydrogenation and a TON of 274 000 (TOF = 17 125 h−1) for transfer hydrogenation. Even though these reactions occur under diffusion control due the poor solubility of the substrate these values are the highest to be reported for the room temperature aqueous phase reduction of nitroarenes catalyzed by a nanoparticle-based system. A continuous flow reduction of nitrobenzene in a packed bed reactor operated over a period of 250 min with no sign of catalyst deactivation and the corresponding space-time-yield of 0.738 g L−1 min−1 is a marked improvement on that of 0.384 g L−1 min−1 obtained in batch. The same system also catalyzes a tandem Suzuki–Miyaura cross coupling-nitroarene reduction sequence to afford high yields of biaryl amine in an operationally straightforward single-pot procedure. This is a highly versatile protocol which will enable the aromatic nitro fragment to be introduced as a nitro-substituted aryl or heteroaryl halide and as such will lend itself to rapid diversification for the synthesis of a wide range of amines

The Impact of Ethnicity on Wilms Tumor: Characteristics and Outcome of a South African Cohort

Background. Nephroblastoma is the commonest renal tumour seen in children. It has a good prognosis in developed countries with survival rates estimated to be between 80% and 90%, while in Africa it remains low. Method. Retrospective study of patients diagnosed with nephroblastoma who are seen at 4 paediatric oncology units, representing 58.5% of all South African children with nephroblastoma and treated following SIOP protocol between January 2000 and December 2010. Results. A total of 416 patients were seen at the 4 units. Over 80% of our patients were African and almost 10% of mixed ethnicity. The most common stage was stage 4. The median survival was 28 months after diagnosis with the mixed ethnicity patients recording the longest duration (39 months) and the white patients had the shortest median survival. The overall 5-year survival rate was estimated to be 66%. Stage 2 patients did significantly better (85%). Conclusions. Our patients are similar with regard to gender ratio, median age, and age distribution as described in the literature, but in South Africa the more advanced stage disease seen than in other developed countries is translated into low overall survival rate

The lived experience of discrimination by white women in committed interracial relationships with black men

Abstract: This study explores the experiences of discrimination by white women in committed interracial relationships with black men within the South African context from a descriptive phenomenological perspective. Three white females in committed interracial relationships with black males were recruited and interviewed. Open-ended interviews were conducted in order to elicit rich and in-depth first-person descriptions of the participants’ lived experiences of discrimination as a result of being in committed interracial relationships. The data analysis entailed a descriptive phenomenological content analysis and description. The results of this study suggest that white women in committed interracial relationships with black men experienced discrimination in various contexts where discrimination manifests as either a negative or a positive encounter; in addition, discrimination evokes various emotional responses and is coped with in either maladaptive or adaptive ways. Finally the experience of discrimination, although personal, necessarily impacts on the interracial relationship. Discrimination experienced by white women in committed interracial relationships with black men is thus multi-layered and both an intra-personal and inter-personal phenomenon