Small Minimal Blocking Sets inPG(2, q3)

AbstractWe extend the results of Polverino (1999, Discrete Math., 208/209, 469–476; 2000, Des. Codes Cryptogr., 20, 319–324) on small minimal blocking sets in PG(2,p3 ), p prime, p≥ 7, to small minimal blocking sets inPG (2, q3), q=ph, p prime, p≥ 7, with exponent e≥h. We characterize these blocking sets completely as being blocking sets of Rédei-type

Concomitant and alternating radiation therapy (RT) and chemotherapy (CT) for inoperable, M0, non-small cell lung cancers (NSCLC): a consensus report

Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/31248/1/0000154.pd

Recent advances in understanding the roles of whole genome duplications in evolution

Ancient whole-genome duplications (WGDs)—paleopolyploidy events—are key to solving Darwin’s ‘abominable mystery’ of how flowering plants evolved and radiated into a rich variety of species. The vertebrates also emerged from their invertebrate ancestors via two WGDs, and genomes of diverse gymnosperm trees, unicellular eukaryotes, invertebrates, fishes, amphibians and even a rodent carry evidence of lineage-specific WGDs. Modern polyploidy is common in eukaryotes, and it can be induced, enabling mechanisms and short-term cost-benefit assessments of polyploidy to be studied experimentally. However, the ancient WGDs can be reconstructed only by comparative genomics: these studies are difficult because the DNA duplicates have been through tens or hundreds of millions of years of gene losses, mutations, and chromosomal rearrangements that culminate in resolution of the polyploid genomes back into diploid ones (rediploidisation). Intriguing asymmetries in patterns of post-WGD gene loss and retention between duplicated sets of chromosomes have been discovered recently, and elaborations of signal transduction systems are lasting legacies from several WGDs. The data imply that simpler signalling pathways in the pre-WGD ancestors were converted via WGDs into multi-stranded parallelised networks. Genetic and biochemical studies in plants, yeasts and vertebrates suggest a paradigm in which different combinations of sister paralogues in the post-WGD regulatory networks are co-regulated under different conditions. In principle, such networks can respond to a wide array of environmental, sensory and hormonal stimuli and integrate them to generate phenotypic variety in cell types and behaviours. Patterns are also being discerned in how the post-WGD signalling networks are reconfigured in human cancers and neurological conditions. It is fascinating to unpick how ancient genomic events impact on complexity, variety and disease in modern life

Blocking sets in PG(r, q(n))

... and B ̄ be, respectively, an (n − 2)-dimensional subspace of an element of S and a minimal blocking set of an ((r − 1)n + 1)-dimensional subspace of PG(rn, q) skew to . Denote by K the cone with vertex and base B̄, and consider the point set B defined by B = (K \ ) ∪ {X ∈ S: X ∩ K = ∅} in the Barlotti–Cofman representation of PG(r, qn) in PG(rn, q) associated to the (n − 1)-spread S. Generalizing the constructions of Mazzocca and Polverino (J Algebraic Combin, 24(1):61–81, 2006), under suitable assumptions on B̄, we prove that B is a minimal block-ing set in PG(r, qn). In this way, we achieve new classes of minimal blocking sets and we find new sizes of minimal blocking sets in finite projective spaces of non-prime order. In particular, for q a power of 3, we exhibit examples of r-dimensional minimal blocking set