Shape selection in non-Euclidean plates

We investigate isometric immersions of disks with constant negative curvature into $\mathbb{R}^3$, and the minimizers for the bending energy, i.e. the $L^2$ norm of the principal curvatures over the class of $W^{2,2}$ isometric immersions. We show the existence of smooth immersions of arbitrarily large geodesic balls in $\mathbb{H}^2$ into $\mathbb{R}^3$. In elucidating the connection between these immersions and the non-existence/singularity results of Hilbert and Amsler, we obtain a lower bound for the $L^\infty$ norm of the principal curvatures for such smooth isometric immersions. We also construct piecewise smooth isometric immersions that have a periodic profile, are globally $W^{2,2}$, and have a lower bending energy than their smooth counterparts. The number of periods in these configurations is set by the condition that the principal curvatures of the surface remain finite and grows approximately exponentially with the radius of the disc. We discuss the implications of our results on recent experiments on the mechanics of non-Euclidean plates

Linear algebra and multidimensional geometry

495 p.; 20 cm

A review of magnetic resonance enterography-based indices for quantification of Crohn’s disease inflammation

Magnetic resonance enterography (MRE) is a leading radiological modality in Crohn’s disease (CD) and is used together with laboratory findings and endoscopic examinations for the evaluation of patients during initial diagnosis and follow up. Over the years, there has been great progress in the understanding of CD and there is a continuous strive to achieve better monitoring of patients and to develop new modalities which will predict disease course and thus help in clinical decisions making. An objective evaluation of CD using a quantification score is not a new concept and there are different clinical, endoscopies, radiological and combined indices which are used in clinical practice. Such scores are a necessity in clinical trials on CD for evaluation of disease response, however, there is no consensus of the preferred MRE score and they are not routinely used. This review presents MRE-based indices in use in the last decade: the Magnetic Resonance Index of Activity (MaRIA), the Clermont score, the Crohn’s Disease Magnetic Resonance Imaging (MRI) Index (CDMI), the Magnetic Resonance Enterography Global Score (MEGS) and the Lemann index. We compare the different indices and evaluate the clinical research that utilized them. The aim of this review is to provide a reference guide for researchers and clinicians who incorporate MRE indices in their work. When devising future indices, accumulated data of the existing indices must be taken into account, as each of the current indices has its own strengths and weakness