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

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

Evaluating the Accuracy and Reliability of Splenic Artery Aneurysm Assessment on Computed Tomography Imaging

BACKGROUND: Splenic artery aneurysms (SAAs) are the third most common abdominal arterial aneurysm. Several radiological features have been associated with their growth rate and rupture risk. We aim to evaluate the accuracy and reliability in assessing these features on computed tomography scans. METHODS: Radiological reports were searched for scans positive for SAA between 2012 and 2021 inclusive. These scans were assessed with our novel radiological proforma by 2 radiologists to determine those criteria which were reliably reported. A nonexpert was then compared against this baseline to assess the reliability of SAA assessment by nonradiologists. Pearson and Spearman rank correlation coefficients and Cohen Kappa statistics was used to assess agreement. RESULTS: A cohort of 170 scans was assessed. A high degree of agreement (correlation coefficient r = 0.89-0.91) was achieved by radiologists for SAA diameter measurement. A high level of agreement was also achieved for calcification (Kappa = 0.827) and previous intervention (1.0), moderate agreement for presence of SAA (0.563), calcification percentage (0.563), morphology (0.446), and presence of thrombus (0.516). Rupture and pseudoaneurysm morphology demonstrated poor agreement (Kappa <0.01) but were rare events limiting interpretation. A nonexpert could achieve comparable diameter measurements (P < 0.05) but had consistently lower kappa agreements for all aneurysm characteristics compared to radiologists. CONCLUSION: Aneurysm diameter, calcification, and previous intervention were the only features reported with high agreement between radiologists. Other SAA characteristics only achieved moderate or poor agreement and so should be used with caution in research and clinical settings

The Application of Constrained Maxima and Minima to some Actuarial Problems

A tool that is missing from the standard equipment of the actuarial student is the technique of maximizing, or minimizing, a function of several variables which are subject to certain constraints. The conception of an extreme value of a function, subject to constraints, is, we feel, as important as the mathematical technique of evaluating such a value in a particular instance.We shall first deal with the technique and its mathematical development and then demonstrate some applications. It is as if we were demonstrating a new car, how it works and how to drive it. The first example is simple and is in the nature of a trial run to show how the car handles. Succeeding examples go further and we hope that the actuarial countryside into which we go will prove as interesting as the mechanical means of getting there. To many the vehicle itself will be familiar, but perhaps some of our byroads may lead to unfamiliar places or to familiar places by unfamiliar routes.</jats:p

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