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Let G be an abstract Kac-Moody group over a finite field and G the closure of the image of G in the automorphism group of its positive building. We show that if the Dynkin diagram associated to G is irreducible and neither of spherical nor of affine type, then the contraction groups of elements in G which are not topologically periodic are not closed. (In those groups there always exist elements which are not topologically periodic.)
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Complete achromatopsia is a rare autosomal recessive disease associated with CNGA3, CNGB3, GNAT2 and PDE6C mutations. This retinal disorder is characterized by complete loss of color discrimination due to the absence or alteration of the cones function. The purpose of the present study was the clinical and the genetic characterization of achromatopsia in a large consanguineous Tunisian family. Ophthalmic evaluation included a full clinical examination, color vision testing and electroretinography. Linkage analysis using microsatellite markers flanking CNGA3, CNGB3, GNAT2 and PDE6C genes was performed. Mutations were screened by direct sequencing. A total of 12 individuals were diagnosed with congenital complete achromatopsia. They are members of six nuclear consanguineous families belonging to the same large consanguineous family. Linkage analysis revealed linkage to GNAT2. Mutational screening of GNAT2 revealed three intronic variations c.119-69G>C, c.161+66A>T and c.875-31G>C that co-segregated with a novel mutation p.R313X. An identical GNAT2 haplotype segregating with this mutation was identified, indicating a founder mutation. All patients were homozygous for the p.R313X mutation. This is the first report of the clinical and genetic investigation of complete achromatopsia in North Africa and the largest family with recessive achromatopsia involving GNAT2; thus, providing a unique opportunity for genotype-phenotype correlation for this extremely rare condition.
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Abstract Despite the popularity of auction theoretical thinking, it appears that no one has presented an elementary equilibrium analysis of the first-price sealed-bid auction mechanism under complete information. This paper aims to remedy that omission. We show that the existence of pure strategy undominated Nash equilibria requires that the bidding space is not "too divisible" (that is, a continuum). In fact, when bids must form part of a finite grid there always exists a "high price equilibrium". However, there might also be "low price equilibria" and when the bidding space is very restrictive the revenue obtained in these "low price equilibria" might be very low. We discuss the properties of the equilibria and an application of auction theoretical thinking in which "low price equilibria" may be relevant. Keywords: First-price auctions, undominated Nash equilibria. JEL Classification Numbers: C72 (Noncooperative Games), D44 (Auctions).