21,655 research outputs found

    An analytic solution to the Busemann-Petty problem on sections of convex bodies

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    We derive a formula connecting the derivatives of parallel section functions of an origin-symmetric star body in R^n with the Fourier transform of powers of the radial function of the body. A parallel section function (or (n-1)-dimensional X-ray) gives the ((n-1)-dimensional) volumes of all hyperplane sections of the body orthogonal to a given direction. This formula provides a new characterization of intersection bodies in R^n and leads to a unified analytic solution to the Busemann-Petty problem: Suppose that K and L are two origin-symmetric convex bodies in R^n such that the ((n-1)-dimensional) volume of each central hyperplane section of K is smaller than the volume of the corresponding section of L; is the (n-dimensional) volume of K smaller than the volume of L? In conjunction with earlier established connections between the Busemann-Petty problem, intersection bodies, and positive definite distributions, our formula shows that the answer to the problem depends on the behavior of the (n-2)-nd derivative of the parallel section functions. The affirmative answer to the Busemann-Petty problem for n\le 4 and the negative answer for n\ge 5 now follow from the fact that convexity controls the second derivatives, but does not control the derivatives of higher orders.Comment: 13 pages, published versio

    New Records of Michigan Cicadidae (Homoptera), With Notes on the Use of Songs to Monitor Range Changes

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    We present records of Diceroprocta vitripennis, Tibicen chloromera, and Tibicen pruinosa (new state record) in Michigan. Monitoring geographic range changes and population size differences by song suggests several population situations for cicadas: (1) sizable populations in most areas of apparently good habitat; (2) widely separated single individuals or small populations on the edges of populated regions, representing range extensions that may be of limited duration; (3) one or a few individuals present only once, probably transferred in soil on roots, and ultimately unsuccessful. Species- specific calling songs allow sensitive measurement of species\u27 range changes

    Self-assembly of polyphosphazene immunoadjuvant with poly(ethylene oxide) enables advanced nanoscale delivery modalities and regulated pH-dependent cellular membrane activity

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    AbstractWater-soluble polyphosphazene polyacids, such as poly[di(carboxylatophenoxy)phosphazene] (PCPP), have been of significant interest due to their unique immunoadjuvant and vaccine delivery properties. We report that PCPP can spontaneously self-assemble into intermolecular complexes with common formulation excipients − polyethers in aqueous solutions at neutral pH through the establishment of hydrogen bonds. The resulting advanced PCPP delivery modalities can range from macromolecular assemblies at the nanoscale level to physically cross-linked hydrogels and the physical state can be modulated through varying polymer ratios and molecular weight of polyether. It has been demonstrated that such macromolecular complexes maintain protein-binding ability − a key characteristics of the delivery system. Importantly, the non-covalent modification of PCPP immunoadjuvant with polyethers introduces pH dependent membrane disruptive activity, which is not characteristic for PCPP itself, and is typically correlated to the ability of macromolecular carrier to facilitate endosomal escape. This can potentially affect the mechanism of immunoadjuvant action displayed by PCPP, afford means for its fine-tuning, as well as provide important insights for understanding the relationship between fundamental physico-chemical characteristics of polyphosphazene immunoadjuvants and their activity in vivo

    A Two-Coordinate Nickel Imido Complex That Effects C−H Amination

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    An exceptionally low coordinate nickel imido complex, (IPr*)Ni═N(dmp) (2) (dmp = 2,6-dimesitylphenyl), has been prepared by the elimination of N_2 from a bulky aryl azide in its reaction with (IPr*)Ni(η^6-C_7H_8) (1). The solid-state structure of 2 features two-coordinate nickel with a linear C−Ni−N core and a short Ni−N distance, both indicative of multiple-bond character. Computational studies using density functional theory showed a Ni═N bond dominated by Ni(dπ)−N(pπ) interactions, resulting in two nearly degenerate singly occupied molecular orbitals (SOMOs) that are Ni−N π* in character. Reaction of 2 with CO resulted in nitrene-group transfer to form (dmp)NCO and (IPr*)Ni(CO)_3 (3). Net C−H insertion was observed in the reaction of 2 with ethene, forming the vinylamine (dmp)NH(CH═CH_2) (5) via an azanickelacyclobutane intermediate, (IPr*)Ni{N,C:κ^2-N(dmp)CH_2CH_2} (4)
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