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

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

How large dimension guarantees a given angle?

We study the following two problems: (1) Given $n\ge 2$ and \al, how large Hausdorff dimension can a compact set A\su\Rn have if $A$ does not contain three points that form an angle \al? (2) Given \al and \de, how large Hausdorff dimension can a %compact subset $A$ of a Euclidean space have if $A$ does not contain three points that form an angle in the \de-neighborhood of \al? An interesting phenomenon is that different angles show different behaviour in the above problems. Apart from the clearly special extreme angles 0 and $180^\circ$, the angles $60^\circ,90^\circ$ and $120^\circ$ also play special role in problem (2): the maximal dimension is smaller for these special angles than for the other angles. In problem (1) the angle $90^\circ$ seems to behave differently from other angles

Higher-dimensional multifractal value sets for conformal infinite graph directed Markov systems

We give a description of the level sets in the higher dimensional multifractal formalism for infinite conformal graph directed Markov systems. If these systems possess a certain degree of regularity this description is complete in the sense that we identify all values with non-empty level sets and determine their Hausdorff dimension. This result is also partially new for the finite alphabet case.Comment: 20 pages, 1 figur

Sixty Years of Fractal Projections

Sixty years ago, John Marstrand published a paper which, among other things, relates the Hausdorff dimension of a plane set to the dimensions of its orthogonal projections onto lines. For many years, the paper attracted very little attention. However, over the past 30 years, Marstrand's projection theorems have become the prototype for many results in fractal geometry with numerous variants and applications and they continue to motivate leading research.Comment: Submitted to proceedings of Fractals and Stochastics

Metal-ligand interplay in strongly-correlated oxides: a parametrized phase diagram for pressure induced spin transitions

We investigate the magnetic properties of archetypal transition-metal oxides MnO, FeO, CoO and NiO under very high pressure by x-ray emission spectroscopy at the K\beta line. We observe a strong modification of the magnetism in the megabar range in all the samples except NiO. The results are analyzed within a multiplet approach including charge-transfer effects. The pressure dependence of the emission line is well accounted for by changes of the ligand field acting on the d electrons and allows us to extract parameters like local d-hybridization strength, O-2p bandwidth and ionic crystal field across the magnetic transition. This approach allows a first-hand insight into the mechanism of the pressure induced spin transition.Comment: 5 pages, 3 figure

Temperature and pressure-induced spin-state transitions in LaCoO3

We report the continuous variation of the spin moment of cobalt in LaCoO3 across its temperature and pressure-induced spin transitions evidenced with K\beta emission spectra. The first thermal transition is best described by a transition to an orbitally nondegenerate intermediate spin (S=1) state. In parallel, continuous redistribution of the 3d electrons is also indicated by partial fluorescence yield X-ray absorption spectra. At high pressure, our study confirms that the material becomes low spin between 40 and 70 kbar at room temperature

Supernovae and radio transients in M 82

We present optical and near-infrared (IR) photometry and near-IR spectroscopy of SN 2004am, the only optically detected supernova (SN) in M 82. These demonstrate that SN 2004am was a highly reddened type II-P SN similar to the low luminosity type II-P events such as SNe 1997D and 2005cs. We show that SN 2004am was located coincident with the obscured super star cluster M 82-L, and from the cluster age infer a progenitor mass of 12 +7/-3 Msun. In addition to this, we present a high spatial resolution Gemini-N K-band adaptive optics image of the site of SN 2008iz and a second transient of uncertain nature, both detected so far only at radio wavelengths. Using image subtraction techniques together with archival data from the Hubble Space Telescope, we are able to recover a near-IR transient source co-incident with both objects. We find the likely extinction towards SN 2008iz to be not more than Av ~ 10. The nature of the second transient remains elusive and we regard an extremely bright microquasar in M 82 as the most plausible scenario.Comment: 14 pages, 8 figures, accepted for publication in MNRA