Partitioning 3-homogeneous latin bitrades

A latin bitrade $(T^{\diamond}, T^{\otimes})$ is a pair of partial latin squares which defines the difference between two arbitrary latin squares $L^{\diamond} \supseteq T^{\diamond}$ and $L^{\diamond} \supseteq T^{\otimes}$ of the same order. A 3-homogeneous bitrade $(T^{\diamond}, T^{\otimes})$ has three entries in each row, three entries in each column, and each symbol appears three times in $T^{\diamond}$. Cavenagh (2006) showed that any 3-homogeneous bitrade may be partitioned into three transversals. In this paper we provide an independent proof of Cavenagh's result using geometric methods. In doing so we provide a framework for studying bitrades as tessellations of spherical, euclidean or hyperbolic space.Comment: 13 pages, 11 figures, fixed the figures. Geometriae Dedicata, Accepted: 13 February 2008, Published online: 5 March 200

Decomposing complete equipartite graphs into closed trails of length k

Necessary conditions for a simple connected graph G to admit a decomposition into closed trails of length k a parts per thousand yen 3 are that G is even and its total number of edges is a multiple of k. In this paper we show that these conditions are sufficient in the case when G is the complete equipartite graph having at least three parts, each of the same size

Final analysis of survival outcomes in the phase 3 FIRST trial of up-front treatment for multiple myeloma

This FIRST trial final analysis examined survival outcomes in patients with transplant-ineligible newly diagnosed multiple myeloma (NDMM) treated with lenalidomide and low-dose dexamethasone until disease progression (Rd continuous), Rd for 72 weeks (18 cycles; Rd18), or melphalan, prednisone, and thalidomide (MPT; 72 weeks). The primary endpoint was progression-free survival (PFS; primary comparison: Rd continuous vs MPT). Overall survival (OS) was a key secondary endpoint (final analysis prespecified ‡60 months’ follow-up). Patients were randomized to Rd continuous (n 5 535), Rd18 (n 5 541), or MPT (n 5 547). At a median follow-up of 67 months, PFS was significantly longer with Rd continuous vs MPT (hazard ratio [HR], 0.69; 95% confidence interval [CI], 0.59-0.79; P < .00001) and was similarly extended vs Rd18. Median OS was 10 months longer with Rd continuous vs MPT (59.1 vs 49.1 months; HR, 0.78; 95% CI, 0.67-0.92; P 5 .0023), and similar with Rd18 (62.3 months). In patients achieving complete or very good partial responses, Rd continuous had an 30-month longer median time to next treatment vs Rd18 (69.5 vs 39.9 months). Over half of all patients who received second-line treatment were given a bortezomib-based therapy. Second-line outcomes were improved in patients receiving bortezomib after Rd continuous and Rd18 vs after MPT. No new safety concerns, including risk for secondary malignancies, were observed. Treatment with Rd continuous significantly improved survival outcomes vs MPT, supporting Rd continuous as a standard of care for patients with transplant-ineligible NDMM. This trial was registered at www.clinicaltrials.gov as #NCT00689936 and EudraCT as 2007-004823-39. © 2018 by The American Society of Hematology