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

Progenitor's signatures in Type Ia supernova remnants

The remnants of Type Ia supernovae can provide important clues about their progenitor-histories. We discuss two well-observed supernova remnants (SNRs) that are believed to result from a Type Ia SN and use various tools to shed light on the possible progenitor history. We find that Kepler's SNR is consistent with a symbiotic binary progenitor consisted of a white dwarf and an AGB star. Our hydrosimulations can reproduce the observed kinematic and morphological properties. For Tycho's remnant we use the characteristics of the X-ray spectrum and the kinematics to show that the ejecta has likely interacted with dense circumstellar gas.Comment: 4 pages, 9 figures, proceedings for IAU Symposium 281, Padova, July 201

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\"

The Core-Collapse Supernova Rate in Arp299 Revisited

We present a study of the CCSN rate in nuclei A and B1 of the luminous infrared galaxy Arp299, based on 11 years of Very Large Array monitoring of their radio emission at 8.4 GHz. Significant variations in the nuclear radio flux density can be used to identify the CCSN activity in the absence of high-resolution very long baseline interferometry observations. In the case of the B1-nucleus, the small variations in its measured diffuse radio emission are below the fluxes expected from radio supernovae, thus making it well-suited to detect RSNe through flux density variability. In fact, we find strong evidence for at least three RSNe this way, which results in a lower limit for the CCSN rate of 0.28 +/- 0.16 per year. In the A-nucleus, we did not detect any significant variability and found a SN detection threshold luminosity which allows only the detection of the most luminous RSNe known. Our method is basically blind to normal CCSN explosions occurring within the A-nucleus, which result in too small variations in the nuclear flux density, remaining diluted by the strong diffuse emission of the nucleus itself. Additionally, we have attempted to find near-infrared counterparts for the earlier reported RSNe in the Arp299 nucleus A, by comparing NIR adaptive optics images from the Gemini-N telescope with contemporaneous observations from the European VLBI Network. However, we were not able to detect NIR counterparts for the reported radio SNe within the innermost regions of nucleus A. While our NIR observations were sensitive to typical CCSNe at 300 mas from the centre of the nucleus A, suffering from extinction up to A_v~15 mag, they were not sensitive to such highly obscured SNe within the innermost nuclear regions where most of the EVN sources were detected. (abridged)Comment: 12 pages, 4 figures and 7 tables. Accepted for publication in MNRA