The algebro-geometric study of range maps

Localizing a radiant source is a widespread problem to many scientific and technological research areas. E.g. localization based on range measurements stays at the core of technologies like radar, sonar and wireless sensors networks. In this manuscript we study in depth the model for source localization based on range measurements obtained from the source signal, from the point of view of algebraic geometry. In the case of three receivers, we find unexpected connections between this problem and the geometry of Kummer's and Cayley's surfaces. Our work gives new insights also on the localization based on range differences.Comment: 38 pages, 18 figure

Curvature based triangulation of metric measure spaces

We prove that a Ricci curvature based method of triangulation of compact Riemannian manifolds, due to Grove and Petersen, extends to the context of weighted Riemannian manifolds and more general metric measure spaces. In both cases the role of the lower bound on Ricci curvature is replaced by the curvature-dimension condition ${\rm CD}(K,N)$. We show also that for weighted Riemannian manifolds the triangulation can be improved to become a thick one and that, in consequence, such manifolds admit weight-sensitive quasimeromorphic mappings. An application of this last result to information manifolds is considered. Further more, we extend to weak ${\rm CD}(K,N)$ spaces the results of Kanai regarding the discretization of manifolds, and show that the volume growth of such a space is the same as that of any of its discretizations.Comment: 24 pages, submitted for publicatio

Slow sedimentation and deformability of charged lipid vesicles

The study of vesicles in suspension is important to understand the complicated dynamics exhibited by cells in vivo and in vitro. We developed a computer simulation based on the boundary-integral method to model the three dimensional gravity-driven sedimentation of charged vesicles towards a flat surface. The membrane mechanical behavior was modeled using the Helfrich Hamiltonian and near incompressibility of the membrane was enforced via a model which accounts for the thermal fluctuations of the membrane. The simulations were verified and compared to experimental data obtained using suspended vesicles labelled with a fluorescent probe, which allows visualization using fluorescence microscopy and confers the membrane with a negative surface charge. The electrostatic interaction between the vesicle and the surface was modeled using the linear Derjaguin approximation for a low ionic concentration solution. The sedimentation rate as a function of the distance of the vesicle to the surface was determined both experimentally and from the computer simulations. The gap between the vesicle and the surface, as well as the shape of the vesicle at equilibrium were also studied. It was determined that inclusion of the electrostatic interaction is fundamental to accurately predict the sedimentation rate as the vesicle approaches the surface and the size of the gap at equilibrium, we also observed that the presence of charge in the membrane increases its rigidity

New Directions in Geometric and Applied Knot Theory

The aim of this book is to present recent results in both theoretical and applied knot theory—which are at the same time stimulating for leading researchers in the field as well as accessible to non-experts. The book comprises recent research results while covering a wide range of different sub-disciplines, such as the young field of geometric knot theory, combinatorial knot theory, as well as applications in microbiology and theoretical physics

Longitudinale Evaluierung der Aortenhämodynamik bei Patienten mit Aortenklappenpathologie mittels vierdimensionaler kardiovaskulärer Magnetresonanztomographie

Background Four-Dimensional Flow Magnetic Resonance Imaging (4D flow MRI) provides a non-invasive assessment of aortic hemodynamics and insight into blood flow patterns and wall shear stress (WSS). Aortic valve stenosis (AS) and bicuspid aortic valves (BAV) cause substantial changes in blood flow patterns and elevated and asymmetrically distributed WSS in the ascending aorta (1,2,3,4). The AS and BAV patients exhibited more helical, vortical, and eccentric flow (1,2). This study aimed to understand better the evolution of aortic hemodynamics in aortic valve pathologies. Methods After receiving ethical approval and informed consent, 20 patients from the primary studies (1,3) were reevaluated using cardiovascular MRI. There were 14 patients with BAV and 6 with stenotic tricuspid aortic valves (TAV). Between primary (P) and follow-up (FU) examination, 1 TAV and 6 BAV underwent aortic valve replacement (AVR). The patients were divided into 2 groups: non-operated (NOP) (n=13) and operated (OP) (n=7). The average duration from P to FU was 4 years. The mean age at FU was 73.3±4.4 years for OP and 57.5±15.9 for NOP. MRI studies were performed on 1.5 T (n=4) and 3.0 T (n=16) systems. Aortic valve morphology, LV function, and mass were assessed using ECG-gated breath-hold SSFP cine imaging. Aortic dimensions were measured using non-contrast magnetic resonance angiography in FU and axial SSFP imaging of the thorax in P. Standard cardiac MRI analysis was performed using cvi42. Thoracic aorta 4D flow MRI using ECG gating and respiratory navigators was acquired. Hemodynamic parameters were evaluated, including net flow, peak systolic velocity, WSS, and helical and vortical flow patterns. Results NOP and OP groups significantly differed in age and BMI. After surgery, LV mass and WSS magnitude were significantly decreased in OP in FU. WSS was significantly correlated with peak systolic velocity, helical flow, and LV mass. Peak velocity in the ascending aorta and aortic arch of OP declined significantly in FU but remained constant in NOP. There was a significant increase in net flow in certain aortic planes in OP in FU. In some cases, the helical and vortical flow patterns changed.Hintergrund Die vierdimensionale Fluss-Magnetresonanztomographie (4D-Flow-MRT) ermöglicht eine nicht-invasive Beurteilung der Aortenhämodynamik und bietet Einblick in die Blutflussmuster und die Wandschubspannung (WSS). Aortenklappenstenose (AS) und bikuspide Aortenklappen (BAV) verursachen erhebliche Veränderungen der Blutflussmuster und erhöhte und asymmetrisch verteilte WSS in der Aorta ascendens (1,2,3,4). Ziel der Studie war es, die Entwicklung der Aortenhämodynamik bei Aortenklappenpathologien besser zu verstehen. Methodik Nach Erhalt der ethischen Genehmigung und informierten Zustimmung wurden 20 Patienten aus den Primärstudien (1,3) mittels CMR erneut untersucht. Es handelte sich um 14 Patienten mit BAV und 6 mit stenotischen trikuspidalen Aortenklappen (TAV). Zwischen der Erst- (P) und Nachuntersuchung (FU) unterzogen sich 7 Patienten einem Aortenklappenersatz (AVR). Die Patienten wurden in 2 Gruppen eingeteilt: nicht operiert (NOP) und operiert (OP). Die Dauer von P bis FU betrug circa 4 Jahre. Die MRT-Untersuchungen wurden mit 1,5 T und 3,0 T Systemen durchgeführt. Die Aortenklappenmorphologie, LV-Funktion und Masse wurden mittels SSFP-Cine-Bildgebung beurteilt. Die Aortenabmessungen wurden mittels kontrastfreier Magnetresonanzangiographie bei FU und axialer SSFP-Bildgebung des Thorax bei P gemessen. Die Standardanalyse der kardialen MRT wurde mit cvi42 durchgeführt. Die 4D-Flow-MRT der Aorta wurde mit EKG-Gating und Atemnavigator durchgeführt. Es wurden hämodynamische Parameter ausgewertet, darunter Nettofluss, systolische Spitzengeschwindigkeit, WSS sowie helikale und vortikale Flussmuster. Ergebnisse Die NOP- und OP unterschieden sich signifikant in Alter und BMI. Nach der AVR waren die LV-Masse und die WSS-Größe bei OP in FU signifikant verringert. Die WSS korrelierte signifikant mit der systolischen Spitzengeschwindigkeit, dem helikalen Fluss und der LV-Masse. Die Spitzengeschwindigkeit in der Aorta ascendens und im Aortenbogen der OP nahm in FU signifikant ab. Der Nettofluss in bestimmten Aortenebenen war bei OP in FU signifikant erhöht. In einigen Fällen veränderten sich die helikalen und vortikalen Flussmuster. Fazit Aortenklappenpathologien können die Hämodynamik des linken Ventrikels und der Aorta beeinträchtigen. Die AVR verbessert die Parameter. Die hämodynamischen Veränderungen können zur Aortendilatation und Dissektion beitragen. In dieser Studie wurde festgestellt, dass das WSS der Aorta in Gebieten mit der höchsten Dissektionshäufigkeit höher war. Das Management von Aortenerkrankungen sucht nach neuen Parametern, die über die Aortengröße hinausgehen, um chirurgische Entscheidungen zu verbessern und Dissektionen vorherzusagen. Wir schlagen vor, Kriterien, die nicht die Größe betreffen, wie das WSS, als potenziellen Parameter für die individuelle Behandlung von Aortenerkrankungen zu entwickeln. Letztlich haben wir Hypothesen für weitere, umfassendere Studien aufgestellt, um unsere Ergebnisse zu validiere

Multi-Scale Local Shape Analysis and Feature Selection in Machine Learning Applications

We introduce a method called multi-scale local shape analysis, or MLSA, for extracting features that describe the local structure of points within a dataset. The method uses both geometric and topological features at multiple levels of granularity to capture diverse types of local information for subsequent machine learning algorithms operating on the dataset. Using synthetic and real dataset examples, we demonstrate significant performance improvement of classification algorithms constructed for these datasets with correspondingly augmented features.Comment: 15 pages, 6 figures, 8 table