A width-diameter inequality for convex bodies

AbstractA special case of the Blaschke-Santaló inequality regarding the product of the volumes of polar reciprocal convex bodies is shown to be equivalent to a power-mean inequality involving the diameters and widths of a convex body. This power-mean inequality leads to strengthened versions of various known inequalities

Identification, cloning and nucleotide sequencing of the granulin genes of Trichoplusia ni and Pieris brassicae granulosis viruses

Certain insect baculoviruses are occluded in a proteinaceous, crystalline structure which serves to protect the virus outside its insect host. Two groups of occluded viruses have been defined: the nuclear polyhedrosis viruses (NPVs) (Baculovirus Subgroup A), having many virions occluded per crystal, and the granulosis viruses (GVs) (Baculovirus Subgroup B), having only one virion per crystal. Using the cloned polyhedrin gene from the Orgyia pseudotsugata MNPV (OpMNPV) as a hybridization probe, a 2 kilobase Sal I fragment from the Trichoplusia ni granulosis virus (TnGV) was identified as the locus of the granulin gene. This fragment was cloned into pUC8 and mapped with restriction enzymes. Three fragments of about 400 base-pairs each were subcloned into reciprocal M13 vectors and their nucleotide sequences were determined by the dideoxy method. A 1.3 kilobase Eco RI clone containing most of the granulin gene from TnGV was used as a hybridization probe to identify the granulin gene from Pieris brassicae granulosis virus (PbGV). This gene was also cloned, mapped, subcloned and dideoxy sequenced. The amino acid sequences derived from both the PbGV and TnGV granulin gene sequences are about 70% conserved with respect to each other and about 50% conserved related to the Lepidopteran NPV polyhedrins. This suggests that the GVs form a distinct branch of Baculoviruses which evolved before the extensive divergence of the Lepidopteran NPVs. Highly conserved sequences within these genes suggest regions of granulin and polyhedrin which might be determinants of higher-order structure. Comparison of the 5' flanking regions of both granulins, several NPVs and another hyperexpressed late gene, the "10k" protein of Autographa dalifornica MNPV, reveals a highly conserved sequence which may be a regulatory element involved in governing the expression of these genes

Carbon Monoxide in type II supernovae

Infrared spectra of two type II supernovae 6 months after explosion are presented. The spectra exhibit a strong similarity to the observations of SN 1987A and other type II SNe at comparable epochs. The continuum can be fitted with a cool black body and the hydrogen lines have emissivities that are approximately those of a Case B recombination spectrum. The data extend far enough into the thermal region to detect emission by the first overtone of carbon monoxide. The molecular emission is modeled and compared with that in the spectra of SN 1987A. It is found that the flux in the CO first overtone is comparable to that found in SN 1987A. We argue that Carbon Monoxide forms in the ejecta of all type II SNe during the first year after explosion.Comment: 6 pages, 6 figures, accepted for publications in A&

Subdiffusion and weak ergodicity breaking in the presence of a reactive boundary

We derive the boundary condition for a subdiffusive particle interacting with a reactive boundary with finite reaction rate. Molecular crowding conditions, that are found to cause subdiffusion of larger molecules in biological cells, are shown to effect long-tailed distributions with identical exponent for both the unbinding times from the boundary to the bulk and the rebinding times from the bulk. This causes a weak ergodicity breaking: typically, an individual particle either stays bound or remains in the bulk for very long times. We discuss why this may be beneficial for in vivo gene regulation by DNA-binding proteins, whose typical concentrations are nanomolarComment: 4 pages, 1 figure, REVTeX4, accepted to Phys Rev Lett, some typos correcte

On the geometric dilation of closed curves, graphs, and point sets

The detour between two points u and v (on edges or vertices) of an embedded planar graph whose edges are curves is the ratio between the shortest path in in the graph between u and v and their Euclidean distance. The maximum detour over all pairs of points is called the geometric dilation. Ebbers-Baumann, Gruene and Klein have shown that every finite point set is contained in a planar graph whose geometric dilation is at most 1.678, and some point sets require graphs with dilation at least pi/2 = 1.57... We prove a stronger lower bound of 1.00000000001*pi/2 by relating graphs with small dilation to a problem of packing and covering the plane by circular disks. The proof relies on halving pairs, pairs of points dividing a given closed curve C in two parts of equal length, and their minimum and maximum distances h and H. Additionally, we analyze curves of constant halving distance (h=H), examine the relation of h to other geometric quantities and prove some new dilation bounds.Comment: 31 pages, 16 figures. The new version is the extended journal submission; it includes additional material from a conference submission (ref. [6] in the paper

Coloring translates and homothets of a convex body

We obtain improved upper bounds and new lower bounds on the chromatic number as a linear function of the clique number, for the intersection graphs (and their complements) of finite families of translates and homothets of a convex body in \RR^n.Comment: 11 pages, 2 figure

A Generalization of the Convex Kakeya Problem

Given a set of line segments in the plane, not necessarily finite, what is a convex region of smallest area that contains a translate of each input segment? This question can be seen as a generalization of Kakeya's problem of finding a convex region of smallest area such that a needle can be rotated through 360 degrees within this region. We show that there is always an optimal region that is a triangle, and we give an optimal \Theta(n log n)-time algorithm to compute such a triangle for a given set of n segments. We also show that, if the goal is to minimize the perimeter of the region instead of its area, then placing the segments with their midpoint at the origin and taking their convex hull results in an optimal solution. Finally, we show that for any compact convex figure G, the smallest enclosing disk of G is a smallest-perimeter region containing a translate of every rotated copy of G.Comment: 14 pages, 9 figure

The Fermat-Torricelli problem in normed planes and spaces

We investigate the Fermat-Torricelli problem in d-dimensional real normed spaces or Minkowski spaces, mainly for d=2. Our approach is to study the Fermat-Torricelli locus in a geometric way. We present many new results, as well as give an exposition of known results that are scattered in various sources, with proofs for some of them. Together, these results can be considered to be a minitheory of the Fermat-Torricelli problem in Minkowski spaces and especially in Minkowski planes. This demonstrates that substantial results about locational problems valid for all norms can be found using a geometric approach

Covering convex bodies by cylinders and lattice points by flats

In connection with an unsolved problem of Bang (1951) we give a lower bound for the sum of the base volumes of cylinders covering a d-dimensional convex body in terms of the relevant basic measures of the given convex body. As an application we establish lower bounds on the number of k-dimensional flats (i.e. translates of k-dimensional linear subspaces) needed to cover all the integer points of a given convex body in d-dimensional Euclidean space for 0<k<d