31 research outputs found
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 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
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
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The utility of Dynamic Contrast Enhanced Intranodal MR Lymphangiography in the investigation of primary lymphatic anomalies.
Background
Primary lymphatic anomalies are often complicated by central lymphatic abnormalities. Intranodal dynamic contrast enhanced MR lymphangiography (DCMRL) is a relatively new technique for imaging the central lymphatic system. Historically, the only method of imaging the central lymphatics well was with pedal lymphangiography, which is technically challenging, time consuming, involves the use of ionising radiation and has risks associated with the use of lipiodol. The treatment options for primary lymphatic disorders have also been limited to symptomatic management.
Purpose
To describe the technique of DCMRL to identify central lymphatic abnormalities in patients with primary lymphatic anomalies and discuss utility of the findings.
Materials and Methods
28 patients with primary lymphatic abnormalities underwent dynamic MR imaging following injection of gadolinium directly into inguinal lymph nodes at a tertiary lymphovascular referral centre.
Results
Technical success was achieved in 23 patients (82.1%). Pathological imaging findings included obstructed, hypoplastic or absent lymphatic channels with collateralisation/rerouting or reflux of flow, lymphangiectasia, lymphatic pseudoaneurysms and lymph leaks. Protocol modifications for improved imaging are highlighted including technical aspects of lymph node injection, image acquisition and MRI parameters. In two patients, imaging findings warranted embolization of the abnormal lymphatic channels with subsequent symptomatic improvement.
Conclusion
DCMRL has been shown to be a safe, reproducible technique in patients with primary lymphatic anomalies enabling imaging of the central lymphatic system