Reflectionless measures and the Mattila-Melnikov-Verdera uniform rectifiability theorem

A new proof is given of the Mattila-Melnikov-Verdera theorem on the uniform rectifiability of an Ahlfors-David regular measure whose associated Cauchy transform operator is bounded.Comment: 31 pages, 5 figure

Supernovae in the nuclear regions of starburst galaxies

The feasibility of using near-infrared observations to discover supernovae in the nuclear and circumnuclear regions of nearby starburst galaxies is investigated. We provide updated estimates of the intrinsic core-collapse supernova rates in these regions. We discuss the problem of extinction, and present new estimates of the extinction towards 33 supernova remnants in the starburst galaxy M 82. This is done using H I and H_2 column density measurements. We estimate the molecular to atomic hydrogen mass ratio to be 7.4 +- 1.0 in M 82. We have assembled near-infrared photometric data for a total of 13 core-collapse supernovae, some unpublished hitherto. This constitutes the largest database of IR light curves for such events. We show that the IR light curves fall into two classes, ``ordinary'' and ``slow-declining''. Template JHKL light curves are derived for both classes. For ordinary core-collapse supernovae, the average peak JHKL absolute magnitudes are -18.4, -18.6, -18.6, and -19.0 respectively. The slow-declining core-collapse SNe are found to be significantly more luminous than the ordinary events, even at early times, having average peak JHKL absolute magnitudes of -19.9, -20.0, -20.0, and -20.4 respectively. We investigate the efficiency of a computerised image subtraction method in supernova detection. We then carry out a Monte Carlo simulation of a supernova search using K-band images of NGC 5962. The effects of extinction and observing strategy are discussed. We conclude that a modest observational programme will be able to discover a number of nuclear supernovae.Comment: 19 pages, 7 figures; accepted in MNRA

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

Singular integrals and rectifiability

We shall discuss singular integrals on lower dimensional subsets of $\mathbf{R}^n$. A survey of this topic was given in [M4]. The first part of this paper gives a quick review of some results discussed in [M4] and a survey on some newer results and open problems. In the second part we prove some results on the Riesz kernels in $\mathbf{R}^n$. As far I know, they have not been explicitly stated and proved, but they are very closely related to some earlier results and methods

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

Filopodia: molecular architecture and cellular functions

Filopodia are thin, actin-rich plasma-membrane protrusions that function as antennae for cells to probe their environment. Consequently, filopodia have an important role in cell migration, neurite outgrowth and wound healing and serve as precursors for dendritic spines in neurons. The initiation and elongation of filopodia depend on the precisely regulated polymerization, convergence and crosslinking of actin filaments. The increased understanding of the functions of various actin-associated proteins during the initiation and elongation of filopodia has provided new information on the mechanisms of filopodia formation in distinct cell types

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\"

Child perspectives on income and time use in Finnish families in the 1990s

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