On well-rounded ideal lattices - II

We study well-rounded lattices which come from ideals in quadratic number fields, generalizing some recent results of the first author with K. Petersen. In particular, we give a characterization of ideal well-rounded lattices in the plane and show that a positive proportion of real and imaginary quadratic number fields contains ideals giving rise to well-rounded lattices.Comment: 13 pages; to appear in the International Journal of Number Theor

A solid-phase enzyme-linked assay for vitamin B 12

A new solid-phase enzyme-linked competitive binding assay for vitamin B 12 (cyanocobalamin) is described. The assay is based on the competition between analyte B 12 molecules and a glucose-6-phosphate dehydrogenase-vitamin B 12 conjugate for a limited number of R-protein binding sites immobilized on sepharose particles. After appropriate incubation and washing steps, the enzyme activity bound to the solid-phase is inversely related to the concentration of B 12 in the sample. Under optimized conditions, the method can detect B 12 in the range of 3×10 −10 −1×10 −8 M (using 100 μ l sample) with high selectivity over other biological molecules.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/41626/1/604_2005_Article_BF01197285.pd

Clinical, functional and genetic characterization of 16 patients suffering from chronic granulomatous disease variants - identification of 11 novel mutations in CYBB

Chronic granulomatous disease (CGD) is a rare inherited disorder in which phagocytes lack nicotinamide adenine dinucleotide phosphate (NADPH) oxidase activity. The most common form is the X-linked CGD (X91-CGD), caused by mutations in the CYBB gene. Clinical, functional and genetic characterizations of 16 CGD cases of male patients and their relatives were performed. We classified them as suffering from different variants of CGD (X910, X91- or X91+), according to NADPH oxidase 2 (NOX2) expression and NADPH oxidase activity in neutrophils. Eleven mutations were novel (nine X910-CGD and two X91--CGD). One X910-CGD was due to a new and extremely rare double missense mutation Thr208Arg-Thr503Ile. We investigated the pathological impact of each single mutation using stable transfection of each mutated cDNA in the NOX2 knock-out PLB-985 cell line. Both mutations leading to X91--CGD were also novel; one deletion, c.-67delT, was localized in the promoter region of CYBB; the second c.253-1879A>G mutation activates a splicing donor site, which unveils a cryptic acceptor site leading to the inclusion of a 124-nucleotide pseudo-exon between exons 3 and 4 and responsible for the partial loss of NOX2 expression. Both X91--CGD mutations were characterized by a low cytochrome b558 expression and a faint NADPH oxidase activity. The functional impact of new missense mutations is discussed in the context of a new three-dimensional model of the dehydrogenase domain of NOX2. Our study demonstrates that low NADPH oxidase activity found in both X91--CGD patients correlates with mild clinical forms of CGD, whereas X910-CGD and X91+-CGD cases remain the most clinically severe forms.</p

On powerful numbers

A powerful number is a positive integer n satisfying the property that p2 divides n whenever the prime p divides n; i.e., in the canonical prime decomposition of n, no prime appears with exponent 1. In [1], S.W. Golomb introduced and studied such numbers. In particular, he asked whether (25,27) is the only pair of consecutive odd powerful numbers. This question was settled in [2] by W.A. Sentance who gave necessary and sufficient conditions for the existence of such pairs. The first result of this paper is to provide a generalization of Sentance's result by giving necessary and sufficient conditions for the existence of pairs of powerful numbers spaced evenly apart. This result leads us naturally to consider integers which are representable as a proper difference of two powerful numbers, i.e. n=p1−p2 where p1 and p2 are powerful numbers with g.c.d. (p1,p2)=1. Golomb (op.cit.) conjectured that 6 is not a proper difference of two powerful numbers, and that there are infinitely many numbers which cannot be represented as a proper difference of two powerful numbers. The antithesis of this conjecture was proved by W.L. McDaniel [3] who verified that every non-zero integer is in fact a proper difference of two powerful numbers in infinitely many ways. McDaniel's proof is essentially an existence proof. The second result of this paper is a simpler proof of McDaniel's result as well as an effective algorithm (in the proof) for explicitly determining infinitely many such representations. However, in both our proof and McDaniel's proof one of the powerful numbers is almost always a perfect square (namely one is always a perfect square when n≢2(mod4)). We provide in §2 a proof that all even integers are representable in infinitely many ways as a proper nonsquare difference; i.e., proper difference of two powerful numbers neither of which is a perfect square. This, in conjunction with the odd case in [4], shows that every integer is representable in infinitely many ways as a proper nonsquare difference. Moreover, in §2 we present some miscellaneous results and conclude with a discussion of some open questions.Peer Reviewe

On powerful numbers

A powerful number is a positive integer n satisfying the property that p2 divides n whenever the prime p divides n; i.e., in the canonical prime decomposition of n, no prime appears with exponent 1. In [1], S.W. Golomb introduced and studied such numbers. In particular, he asked whether (25,27) is the only pair of consecutive odd powerful numbers. This question was settled in [2] by W.A. Sentance who gave necessary and sufficient conditions for the existence of such pairs. The first result of this paper is to provide a generalization of Sentance's result by giving necessary and sufficient conditions for the existence of pairs of powerful numbers spaced evenly apart. This result leads us naturally to consider integers which are representable as a proper difference of two powerful numbers, i.e. n=p1−p2 where p1 and p2 are powerful numbers with g.c.d. (p1,p2)=1. Golomb (op.cit.) conjectured that 6 is not a proper difference of two powerful numbers, and that there are infinitely many numbers which cannot be represented as a proper difference of two powerful numbers. The antithesis of this conjecture was proved by W.L. McDaniel [3] who verified that every non-zero integer is in fact a proper difference of two powerful numbers in infinitely many ways. McDaniel's proof is essentially an existence proof. The second result of this paper is a simpler proof of McDaniel's result as well as an effective algorithm (in the proof) for explicitly determining infinitely many such representations. However, in both our proof and McDaniel's proof one of the powerful numbers is almost always a perfect square (namely one is always a perfect square when n≢2(mod4)). We provide in §2 a proof that all even integers are representable in infinitely many ways as a proper nonsquare difference; i.e., proper difference of two powerful numbers neither of which is a perfect square. This, in conjunction with the odd case in [4], shows that every integer is representable in infinitely many ways as a proper nonsquare difference. Moreover, in §2 we present some miscellaneous results and conclude with a discussion of some open questions