The Lovász Local Lemma is a powerful probabilistic technique for proving the existence of combinatorial objects. It is especially useful for colouring graphs and hypergraphs with bounded maximum ...degree. This paper presents a general theorem for colouring hypergraphs that in many instances matches or slightly improves upon the bounds obtained using the Lovász Local Lemma. Moreover, the theorem directly shows that there are exponentially many colourings. The elementary and self-contained proof is inspired by a recent result for nonrepetitive colourings by Rosenfeld 2020. We apply our general theorem in the setting of proper hypergraph colouring, proper graph colouring, independent transversals, star colouring, nonrepetitive colouring, frugal colouring, Ramsey number lower bounds, and for \(k\)-SAT.
Hepatic vitamin A (VA) is concentrated in lipocytes (fat-storing cells) but the actual concentration in various cell types has not been measured. In this study hepatocytes from normal and ...VA-pre-treated rats were isolated and the VA content was measured. Because most lipocytes were lysed in the procedure, VA content within lipocytes was estimated indirectly by subtracting total hepatic VA from hepatocellular VA. Hepatocytes contained 10.9% of the total hepatic VA (7.2% in VA-treated rats). The volume density of lipocytes in whole liver was 0.23% as determined morphometrically (1.45% in VA-treated rats). From these results it is estimated that the VA concentration in lipocytes is 39 mg/ml (3015 times that in hepatocytes) in normal rats and 69 mg/ml in VA-treated rats. Lipocyte lipid droplets are estimated to be 26% retinyl palmitate (by volume) in normal rats and 31% in VA-treated rats.
There are finitely many graphs with diameter \(2\) and girth 5. What if the girth 5 assumption is relaxed? Apart from stars, are there finitely many triangle-free graphs with diameter \(2\) and no ...\(K_{2,3}\) subgraph? This question is related to the existence of triangle-free strongly regular graphs, but allowing for a range of co-degrees gives the question a more extremal flavour. More generally, for fixed \(s\) and \(t\), are there infinitely many twin-free triangle-free \(K_{s,t}\)-free graphs with diameter 2? This paper presents partial results regarding these questions, including computational results, potential Cayley-graph and probabilistic constructions.
Nodular regenerative hyperplasia (NRH) is an uncommon hepatic lesion often associated with noncirrhotic portal hypertension (PHT). We have noted that NRH and PHT are frequent occurrences in a colony ...of dogs with the genetic storage disease, mucopolysaccharidosis I (MPS‐I). This observation provides the opportunity to study the histology and pathogenesis of NRH and noncirrhotic PHT in a new animal model. Thirteen of 32 dogs (41%) with MPS‐I developed multiple portocaval shunts between 4 and 48 months of age that were grossly visible at necropsy. Seven of the 13 developed marked ascites, whereas all those without shunts and littermates (n = 24) heterozygous for the mutated α‐l‐iduronidase allele (carriers unaffected by the storage disease) did not. The large and medium‐sized portal veins were widely patent without thrombosis or vascular malformations. Hepatic parenchymal fibrosis was absent or mild and did not correlate with shunt formation. All 32 livers had varying degrees of diffuse periportal hepatocellular hyperplasia with multifocal atrophy and compression of centrolobular cords (NRH) most prominent in dogs with shunts. Many small portal veins were reduced in diameter or absent, especially in animals with shunts. Noncirrhotic PHT and NRH appear to be related to the obliteration of small portal veins in these dogs, but the pathogenesis of this vascular change remains unknown.
Liver resection or transplantation offers the best opportunity for cure of hepatocellular carcinoma (HCC). To determine the relative roles for resection and transplantation and to evaluate the ...patient and tumor characteristics that might predict survival, the records of 125 patients treated for nonfibrolamellar HCC at The Toronto Hospital between 1981 and 1996 were reviewed. No adjuvant chemotherapy or antiviral protocols were used. Resection was the first operation in 67 patients; one underwent re-resection. Sixty patients underwent transplantation including two who had previously had a resection; 40 had known or suspected HCC and 20 had incidental tumors identified in the explanted liver. The incidence of cirrhosis was 49% for resection and 88% for transplantation. The incidence of hepatitis B virus (HBV) was 58% and 33%, respectively. The operative mortality rate for resection was 4.4% (9.4% in cirrhotic and 0 in noncirrhotic patients) and 13.3% for transplantation. The 5-year cumulative recurrence rate was 55% following resection and 20% following transplantation (
P <0.001). The 5-year Kaplan-Meier survival rates were 38% for resection and 45% for transplantation—60% for transplanted HBV-negative and 17% for HBV-positive patients (
P <0.001). After resection, recurrent HCC accounted for 86% of deaths, whereas recurrent HBV was responsible for 42% of deaths after transplantation. By univariate analysis, following resection, vascular invasion, advanced stage, multiple tumors, and lack of a capsule were predictive of survival; cirrhosis, HBV, age, tumor size, number, and grade were not. By multivariate analysis, only vascular invasion was predictive for resection and HBV for transplantation. Resection and transplantation are complementary methods of treating HCC. With the current organ shortage, resection should be considered first-line treatment. HBV-positive patients with HCC should only undergo transplantation in combination with effective antiviral therapy.
An array is row-Latin if no symbol is repeated within any row. An array is Latin if it and its transpose are both row-Latin. A transversal in an \(n\times n\) array is a selection of \(n\) different ...symbols from different rows and different columns. We prove that every \(n \times n\) Latin array containing at least \((2-\sqrt{2}) n^2\) distinct symbols has a transversal. Also, every \(n \times n\) row-Latin array containing at least \(\frac14(5-\sqrt{5})n^2\) distinct symbols has a transversal. Finally, we show by computation that every Latin array of order \(7\) has a transversal, and we describe all smaller Latin arrays that have no transversal.
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