Pointwise tube formulas for fractal sprays and self-similar tilings with arbitrary generators

In a previous paper by the first two authors, a tube formula for fractal sprays was obtained which also applies to a certain class of self-similar fractals. The proof of this formula uses distributional techniques and requires fairly strong conditions on the geometry of the tiling (specifically, the inner tube formula for each generator of the fractal spray is required to be polynomial). Now we extend and strengthen the tube formula by removing the conditions on the geometry of the generators, and also by giving a proof which holds pointwise, rather than distributionally. Hence, our results for fractal sprays extend to higher dimensions the pointwise tube formula for (1-dimensional) fractal strings obtained earlier by Lapidus and van Frankenhuijsen. Our pointwise tube formulas are expressed as a sum of the residues of the "tubular zeta function" of the fractal spray in $\mathbb{R}^d$. This sum ranges over the complex dimensions of the spray, that is, over the poles of the geometric zeta function of the underlying fractal string and the integers $0,1,...,d$. The resulting "fractal tube formulas" are applied to the important special case of self-similar tilings, but are also illustrated in other geometrically natural situations. Our tube formulas may also be seen as fractal analogues of the classical Steiner formula.Comment: 43 pages, 13 figures. To appear: Advances in Mathematic

Minkowski measurability results for self-similar tilings and fractals with monophase generators

In a previous paper [arXiv:1006.3807], the authors obtained tube formulas for certain fractals under rather general conditions. Based on these formulas, we give here a characterization of Minkowski measurability of a certain class of self-similar tilings and self-similar sets. Under appropriate hypotheses, self-similar tilings with simple generators (more precisely, monophase generators) are shown to be Minkowski measurable if and only if the associated scaling zeta function is of nonlattice type. Under a natural geometric condition on the tiling, the result is transferred to the associated self-similar set (i.e., the fractal itself). Also, the latter is shown to be Minkowski measurable if and only if the associated scaling zeta function is of nonlattice type.Comment: 18 pages, 1 figur

The StarScan plate measuring machine: overview and calibrations

The StarScan machine at the U.S. Naval Observatory (USNO) completed measuring photographic astrograph plates to allow determination of proper motions for the USNO CCD Astrograph Catalog (UCAC) program. All applicable 1940 AGK2 plates, about 2200 Hamburg Zone Astrograph plates, 900 Black Birch (USNO Twin Astrograph) plates, and 300 Lick Astrograph plates have been measured. StarScan comprises of a CCD camera, telecentric lens, air-bearing granite table, stepper motor screws, and Heidenhain scales to operate in a step-stare mode. The repeatability of StarScan measures is about 0.2 micrometer. The CCD mapping as well as the global table coordinate system has been calibrated using a special dot calibration plate and the overall accuracy of StarScan x,y data is derived to be 0.5 micrometer. Application to real photographic plate data shows that position information of at least 0.65 micrometer accuracy can be extracted from course grain 103a-type emulsion astrometric plates. Transformations between "direct" and "reverse" measures of fine grain emulsion plate measures are obtained on the 0.3 micrometer level per well exposed stellar image and coordinate, which is at the limit of the StarScan machine.Comment: 24 pages, 8 figures, accepted for PAS

A Radio--Optical Reference Frame VIII. CCD observations from KPNO and CTIO: internal calibration and first results

In this pilot investigation, precise optical positions in the FK5 system are presented for a set of 16 compact extragalactic radio sources, which will be part of the future radio--optical reference frame. The 0.9 m KPNO and CTIO telescopes equipped with 2K CCD's have been used for this project. The astrometric properties of these instruments are investigated in detail. New techniques of using wide field CCD observations for astrometry in general are developed. An internal precision of 5 to 31 mas in position per single exposure is found, depending on the brightness of the object. The tie to the primary optical reference system is established by photographic astrometry using dedicated astrographs on both hemispheres. An accuracy of $\approx 30$ mas per source is estimated for the multi--step reduction procedure when based on the future Hipparcos catalog, while the FK5--based positions suffer from system errors of 100 to 200 mas as compared to the radio positions. This work provides a contribution to the international effort to link the Hipparcos instrumental coordinate system to the quasi--inertial VLBI radio reference frame. Precise radio and optical astrometry of a large sample of compact extragalactic sources will also contribute to the astrophysics of these objects by comparing the respective centers of emission at the optical and radio wavelengths.Comment: AAS v.4 LaTeX, 2 parts on 1 file (main text + deluxetable), accepted by AJ, Dec.95, fig. with reprint

Correlations of the IR Luminosity and Eddington Ratio with a Hard X-ray Selected Sample of AGN

We use the SWIFT Burst Alert Telescope (BAT) sample of hard x-ray selected active galactic nuclei (AGN) with a median redshift of 0.03 and the 2MASS J and K band photometry to examine the correlation of hard x-ray emission to Eddington ratio as well as the relationship of the J and K band nuclear luminosity to the hard x-ray luminosity. The BAT sample is almost unbiased by the effects of obscuration and thus offers the first large unbiased sample for the examination of correlations between different wavelength bands. We find that the near-IR nuclear J and K band luminosity is related to the BAT (14 - 195 keV) luminosity over a factor of $10^3$ in luminosity ($L_{IR} \approx L_{BAT}^{1.25}$)and thus is unlikely to be due to dust. We also find that the Eddington ratio is proportional to the x-ray luminosity. This new result should be a strong constraint on models of the formation of the broad band continuum.Comment: accepted to ApJ

Optical measurement of heteronuclear cross-relaxation interactions in Tm:YAG

We investigate cross-relaxation interactions between Tm and Al in Tm:YAG using two optical methods: spectral holeburning and stimulated echoes. These interactions lead to a reduction in the hyperfine lifetime at magnetic fields that bring the Tm hyperfine transition into resonance with an Al transition. We develop models for measured echo decay curves and holeburning spectra near a resonance, which are used to show that the Tm-Al interaction has a resonance width of 10~kHz and reduces the hyperfine lifetime to 0.5 ms. The antihole structure is consistent with an interaction dominated by the Al nearest neighbors at 3.0 Angstroms, with some contribution from the next nearest neighbors at 3.6 Angstroms.Comment: 13 pages, 9 figure

A rapid change in virulence gene expression during the transition from the intestinal lumen into tissue promotes systemic dissemination of Salmonella.

Bacterial pathogens causing systemic disease commonly evolve from organisms associated with localized infections but differ from their close relatives in their ability to overcome mucosal barriers by mechanisms that remain incompletely understood. Here we investigated whether acquisition of a regulatory gene, tviA, contributed to the ability of Salmonella enterica serotype Typhi to disseminate from the intestine to systemic sites of infection during typhoid fever. To study the consequences of acquiring a new regulator by horizontal gene transfer, tviA was introduced into the chromosome of S. enterica serotype Typhimurium, a closely related pathogen causing a localized gastrointestinal infection in immunocompetent individuals. TviA repressed expression of flagellin, a pathogen associated molecular pattern (PAMP), when bacteria were grown at osmotic conditions encountered in tissue, but not at higher osmolarity present in the intestinal lumen. TviA-mediated flagellin repression enabled bacteria to evade sentinel functions of human model epithelia and resulted in increased bacterial dissemination to the spleen in a chicken model. Collectively, our data point to PAMP repression as a novel pathogenic mechanism to overcome the mucosal barrier through innate immune evasion