The hexagonal versus the square lattice

We establish Schmutz Schaller's conjecture that the hexagonal lattice is `better' than the square lattice. Schmutz Schaller (Bulletin of the AMS 35 (1998), p. 201), motivated by considerations from hyperbolic geometry, conjectured that in dimensions 2 to 8 the best known lattice sphere packings have `maximal lengths' and goes on to write: "In dimension 2 the conjecture means in particular that the hexagonal lattice is `better' than the square lattice. More precisely, let 0<h_1<h_2<... be the positive integers, listed in ascending order, which can be written as h_i=x^2+3y^2 for integers x and y. Let 0<q_1<q_2<... be the positive integers, listed in ascending order, which can be written as q_i=x^2+y^2 for integers x and y. Then the conjecture is that q_i<=h_i for i=1,2,3,..." Our proof requires computational prime number theory in combination with methods from a preprint of the first author (to appear in Math. Comp.), arXiv:math.NT/0112100.Comment: 24 pages, 6 figures, 2 table

Liver transplantation

Human liver transplantation, which was first performed in man in Denver 15 years ago, has finally come of age in the past 2 years. The 1 year survival has improved from 28 per cent to 50 per cent in the recent Denver Second Series. Past experience has shown that long-term prognosis can usually be determined based on the 1 year assessment. Patients who are fit with a well functioning liver are likely to remain well. This applied to the 45 per cent of the 1 year survivors in the First Denver Series, who are still alive today at between 2 5/6 and 8 5/6 years. It has however, been a much more frequent finding in the Second Series, which suggests that a significant number of patients should be long-term survivors in the future. Improved survival has been attributed to a number of factors including a better understanding of the rejection and infection problems in immunosuppressed liver recipients. Postoperative hepatic dysfunction is no longer as easily ascribed to rejection, and an aggressive diagnostic approach has helped to prevent over-immunosuppression. Furthermore, new approaches to the biliary anastomosis, and a better understanding of the blood supply of the human bile duct, is currently preventing many of the earlier catastrophes related to this, the Achilles heel of liver tranplantation

Pilot trial of fk 506 in the management of steroid-resistant nephrotic syndrome

Seven patients with steroid-resistant nephrotic syndrome were treated with FK 506 monotherapy. Four patients were children with focal sclerosing glomerulonephritis (FSGS). Three of these had evldence for chronic progressive renal disease consisting of interstitial fibrosis and tubular atrophy on pretreatment renal biopsies. Two patients had also failed cyclosporin A (CsA), two cyclophosphamide, and one chlorambucil prior to treatment with FK 506. Three patients were adults wlth mesangial proliferative. membranoproliferative, and membranous glomerulonephritis. Three patterns of response were noted: (1) a reduction in proteinuria to normal levels, (2) partial response (50% reduction) or; (3) no improvement. All patients except one experienced at least a 50% reduction in protein excretion at some time during FK 506 therapy. Two of the children and one adult reduced protein excretion to essentially normal values. One patient had no sustained reduction In Droteln excretion and is considered to be a treatment fallure, although her protein excretion was approximately 50% of pretreatment values intermittently. The drug was generally well tolerated. The most common side-effect was nephrotoxlclty, whlch was reversible. These encouraging results suggest that FK 506 monotherapy may be effective in controlling the proteinuria of somc patlents with steroid-resistant nephrotic syndrome The use of this drug may extend our understanding of the role of T lymphocytes and cytokines in the pathogenesls of glomerulonephritis. Further study of this agent In a larger population of patlents is warranted. © 1993 European Dialysis and Transplant Assoiation-European Renal Association