The Cardinality of Sets of <Emphasis Type="Italic">k</Emphasis>-Independent Vectors over Finite Fields

A set of vectors is k-independent if all its subsets with no more than k elements are linearly independent. We obtain a result concerning the maximal possible cardinality Ind _q (n, k) of a k-independent set of vectors in the n-dimensional vector space F _q ⁿ over the finite field F _q of order q. Namely, we give a necessary and sufficient condition for Ind _q (n, k) = n + 1. We conclude with some pertinent remarks re applications of our results to codes, graphs and hypercubes. Supported, in part by grants EP/C000285, NSF-DMS-0439734 and NSF-DMS-0555839. S. B. Damelin thanks the Institute for Mathematics and Applications for their hospitality.     

The geometry of sets of orthogonal frequency hypercubes

We extend the notion of a framed net, introduced by D. Jungnickel, V. C. Mavron, and T. P. McDonough, J Combinatorial Theory A, 96 (2001), 376–387, to that of a d‐framed net of type ?, where d ≥ 2 and 1 ≤ ? ≤ d‐1, and we establish a correspondence between d‐framed nets of type ? and sets of mutually orthogonal frequency hypercubes of dimension d. We provide a new proof of the maximal size of a set of mutually orthogonal frequency hypercubes of type ? and dimension d, originally established by C. F. Laywine, G. L. Mullen, and G. Whittle, Monatsh Math 119 (1995), 223–238, and we obtain a geometric characterization of the framed net when this bound is satisfied as a PBIBD based on a d‐class association Hamming scheme H(d,n). © 2006 Wiley Periodicals, Inc. J Combin Designs 15: 449–459, 2007     

N-Boc 4-nitropiperidine: preparation and conversion into a spiropiperidine analogue of the eastern part of maraviroc

Previously unreported N-Boc 4-nitropiperidine was prepared in two steps from N-Boc-piperidone. The synthetic utility of this new intermediate was demonstrated by the development of a new and simple route to spirolactam piperidines. Further synthetic work involving a challenging triazole cyclisation allowed the preparation of a spiropiperidine analogue of the eastern part of maraviroc.     

The Cardinality of Sets of k -Independent Vectors over Finite Fields

A set of vectors is k-independent if all its subsets with no more than k elements are linearly independent. We obtain a result concerning the maximal possible cardinality Ind _q (n, k) of a k-independent set of vectors in the n-dimensional vector space F _q ⁿ over the finite field F _q of order q. Namely, we give a necessary and sufficient condition for Ind _q (n, k) = n + 1. We conclude with some pertinent remarks re applications of our results to codes, graphs and hypercubes.     

Time dependence of the source recoilless fraction for a cobalt-57 in rhodium source

The effect of source resonance self absorption (SRSA) has been measured for⁵⁷Co in a rhodium matrix by determining the ratio of the intensity of the 14.4 keV gamma transition to that of the associated 22 keV rhodium x-ray, directly counting both intensities from a stationary source. A ratio, ρ, is defined asf _s/f _so, wheref _s is the recoil-free fraction with the SRSA effect present, andf _so is in the absence of SRSA effects [1]. The value of ρ is to within an additive constant, proportional to the gamma over the X-ray intensities. In the absence of SRSA it is expected that both the 14.4 keV and the 22 keV decay rate would be identical to that of the⁵⁷Co parent, and ρ would be unity and time independent. But because of SRSA a fraction of the 14.4 keV gammas are reabsorbed by⁵⁷Fe, which is increasing as a result of⁵⁷Co decay, and hence, ρ decreases with time. By direct measurement our results confirm that the effect of SRSA can be appreciable even over short time intervals, and we find that in many cases SRSA may have a pronounced effect on Mössbauer experiments, affecting both the line width andf-value, and contrary to general practice in the field, needs to be evaluated in high intensity source experiments.     

Diastereoselective 1,3-dipolar cycloaddition of pyrylium ylides with chiral enamides

Chiral enamides 5f-i were found to react with pyrylium ylides to give cycloadducts 6d-i in good yields with an excellent level of stereoselectivity. The chiral auxiliary was successfully removed on hydrogenolysis of compound 6f in continuous flow (H-Cube) resulting in the first asymmetric synthesis of complex amine 8.     

Proton‐Detected Solid‐State NMR Spectroscopy of Natural‐Abundance Peptide and Protein Pharmaceuticals

The natural way : A sensitive NMR spectroscopic method is developed to obtain well‐resolved two‐dimensional spectra (¹⁵N–¹H and ¹³C–¹H) for natural‐abundance (that is, without the need for isotopic enrichment) large‐molecule samples, such as biopharmaceuticals. This method gives structural insights on two lyophilized aprotinin samples and three insulin samples in lyophilized, microcrystalline suspension formulation (red; see picture) and fibril (green) forms.