Slicing Sets and Measures, and the Dimension of Exceptional Parameters

We consider the problem of slicing a compact metric space \Omega with sets of the form \pi_{\lambda}^{-1}\{t\}, where the mappings \pi_{\lambda} \colon \Omega \to \R, \lambda \in \R, are \emph{generalized projections}, introduced by Yuval Peres and Wilhelm Schlag in 2000. The basic question is: assuming that \Omega has Hausdorff dimension strictly greater than one, what is the dimension of the 'typical' slice \pi_{\lambda}^{-1}{t}, as the parameters \lambda and t vary. In the special case of the mappings \pi_{\lambda} being orthogonal projections restricted to a compact set \Omega \subset \R^{2}, the problem dates back to a 1954 paper by Marstrand: he proved that for almost every \lambda there exist positively many $t \in \R$ such that \dim \pi_{\lambda}^{-1}{t} = \dim \Omega - 1. For generalized projections, the same result was obtained 50 years later by J\"arvenp\"a\"a, J\"arvenp\"a\"a and Niemel\"a. In this paper, we improve the previously existing estimates by replacing the phrase 'almost all \lambda' with a sharp bound for the dimension of the exceptional parameters.Comment: 31 pages, three figures; several typos corrected and large parts of the third section rewritten in v3; to appear in J. Geom. Ana

Porosities and dimensions of measures

We introduce a concept of porosity for measures and study relations between dimensions and porosities for two classes of measures: measures on $R^n$ which satisfy the doubling condition and strongly porous measures on $R$.Comment: Jarvenpaa = J\"arvenp\"a\"

Super star clusters and Supernovae in interacting LIRGs unmasked by NIR adaptive optics

We report on an on-going near-IR adaptive optics survey targeting interacting luminous IR galaxies. High-spatial resolution NIR data are crucial to enable interpretation of kinematic, dynamical and star formation (SF) properties of these very dusty objects. Whole progenitor nuclei in the interactions can be missed if only optical HST imaging is used. Here we specifically present the latest results regarding core-collapse supernovae found within the highly extincted nuclear regions of these galaxies. Direct detection and study of such highly obscured CCSNe is crucial for revising the optically-derived SN rates used for providing an independent measurement of the SF history of the Universe. We also present thus-far the first NIR luminosity functions of super star cluster (SSC) candidates. The LFs can then be used to constrain the formation and evolution of SSCs via constraints based on initial mass functions and cluster disruption models.Comment: 6 pages. To appear in proceedings of 'Galaxies and their Masks' (Namibia, April 2010), published by Springer, New York, eds. D.L. Block, K.C. Freeman, I. Puerar

Existence of principal values of some singular integrals on Cantor sets, and Hausdorff dimension

Consider a standard Cantor set in the plane of Hausdorff dimension 1. If the linear density of the associated measure $\mu$ vanishes, then the set of points where the principal value of the Cauchy singular integral of $\mu$ exists has Hausdorff dimension 1. The result is extended to Cantor sets in $\mathbb{R}^d$ of Hausdorff dimension $\alpha$ and Riesz singular integrals of homogeneity $-\alpha$, 0 < $\alpha$ < d : the set of points where the principal value of the Riesz singular integral of $\mu$ exists has Hausdorff dimension $\alpha$. A martingale associated with the singular integral is introduced to support the proof.Comment: 14 pages, minor revision after the referee's report, to appear in Pacific J. of Mat

Infrared Photometry of Starless Dense Cores

Deep JHKs photometry was obtained towards eight dense molecular cores and J-H vs. H-Ks color-color plots are presented. Our photometry, sensitive to the detection of a 1 solar mass, 1 X 10^6 year old star through approx. 35 - 50 magnitudes of visual extinction, shows no indication of the presence of star/disk systems based on J-H vs. H-Ks colors of detected objects. The stars detected towards the cores are generally spatially anti-correlated with core centers suggesting a background origin, although we cannot preclude the possibility that some stars detected at H and Ks alone, or Ks alone, are not low mass stars or brown dwarfs (< 0.3 Solar Masses) behind substantial amounts of visual extinction (e.g. 53 magnitudes for L183B). Lower limits to optical extinctions are estimated for the detected background stars, with high extinctions being encountered, in the extreme case ranging up to at least Av = 46, and probably higher. The extinction data are used to estimate cloud masses and densities which are comparable to those determined from molecular line studies. Variations in cloud extinctions are consistent with a systematic nature to cloud density distributions and column density variations and extinctions are found to be consistent with submillimeter wave continuum studies of similar regions. The results suggest that some cores have achieved significant column density contrasts (approx. 30) on sub-core scales (approx. 0.05 pc) without having formed known stars.Comment: 44 pages including tables and figures, accepted ApJ, March 24, 200

A Phase Transition for Circle Maps and Cherry Flows

We study $C^{2}$ weakly order preserving circle maps with a flat interval. The main result of the paper is about a sharp transition from degenerate geometry to bounded geometry depending on the degree of the singularities at the boundary of the flat interval. We prove that the non-wandering set has zero Hausdorff dimension in the case of degenerate geometry and it has Hausdorff dimension strictly greater than zero in the case of bounded geometry. Our results about circle maps allow to establish a sharp phase transition in the dynamics of Cherry flows

Differentiability of fractal curves

While self-similar sets have no tangents at any single point, self-affine curves can be smooth. We consider plane self-affine curves without double points and with two pieces. There is an open subset of parameter space for which the curve is differentiable at all points except for a countable set. For a parameter set of codimension one, the curve is continuously differentiable. However, there are no twice differentiable self-affine curves in the plane, except for parabolic arcs

Early-time Spitzer observations of the type II-Plateau supernova, 2004dj

We present mid-infrared observations with the Spitzer Space Telescope of the nearby type II-P supernova, SN 2004dj, at epochs of 89 to 129 days. We have obtained the first mid-IR spectra of any supernova apart from SN 1987A. A prominent [NiII] 6.64 micron line is observed, from which we deduce that the mass of stable nickel must be at least 2.2e10(-4) Msun. We also observe the red wing of the CO-fundamental band. We relate our findings to possible progenitors and favour an evolved star, most likely a red supergiant, with a probable initial mass between ~10 and 15 Msun.Comment: ApJ Letters (accepted