Concept# Leonhard Euler

Summary

Leonhard Euler (ˈɔɪlər , ˈleːɔnhaʁt ˈɔʏlɐ; 15 April 1707 18 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in many other branches of mathematics such as analytic number theory, complex analysis, and infinitesimal calculus. He introduced much of modern mathematical terminology and notation, including the notion of a mathematical function. He is also known for his work in mechanics, fluid dynamics, optics, astronomy and music theory.
Euler is held to be one of the greatest mathematicians in history and the greatest of the 18th century. Several great mathematicians who produced their work after Euler's death have recognised his importance in the field as shown by quotes attributed to many of them: Pierre-Simon Laplace expressed Euler's influence on mathematics by stating, "Read Euler, read Euler, he is the master o

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In this project, we deepen the analysis of a tumour growth model, recently proposed by Garcke et al. in [1]. This model describes tumour and healthy cells evolution as well as tumour cells’ nutrients, mixture velocity and pressure in the domain. Furthermore, it takes into account chemotaxis and apoptosis death of tumour cells, through a system of parabolic nonlinear PDE, that is a Cahn-Hilliard Darcy model, together with an advection-diffusion-reaction equation describing the evolution of nutrients. We perform a dimensional analysis and we build a numerical solver by use of the finite element method in space, a Backward Euler in time and a Newton method to tackle the nonlinearity. We perform several numerical simulations in order to recover results obtained in the article and to catch a general growth of the tumour depending on parameters of interest. Finally, a PDE-constrained optimization problem is formulated and solved, aiming at determining the shape of the tumur after a fixed time from an initial guess of its location. From the numerical simulations we obtained for the nutrients, we notice that the concentration of nutrients in an observable zone around the tumor region could possibly bring enough information to achieve this goal. Therefore, a previous numerical simulation of nutrients will be taken as a target, in order to recover the controlled tumor function, previously simulated numerically. In this respect, preliminary numerical results show that, to some extents, it is possible to identify the general shape of the tumor, even if the exact result of the numerical simulation could not be recovered.

2016The Hubble constant H0 is one of the most important parameters in cosmology, as it encodes the age of the Universe and is necessary for any distance determination at a cosmological scale. It is, however, only poorly constrained by traditional methods. The current favored value, H0 = 72±8 km s-1 Mpc-1, is provided by the HST Hubble constant Key Project (Freedman et al. 2001), which combines several Cepheid-calibrated distance indicators. This roughly 10% error nevertheless denotes only the statistical uncertainty in the determination of H0, while the possible systematical errors in the first step of the distance ladder (the distance to the Large Magellanic Cloud) may be of the same order of magnitude. Time delays between gravitationally lensed images of distant quasars can yield a more precise measurement of the Hubble constant, on a truly cosmic scale, and independently of any local distance calibrator. At the beginning of this thesis, time delays had been measured in only ten lensed systems, nine of which gave H0 estimates. However before 2004, no concerted and long term action has succeeded to apply the time delay method at a level of precision really competitive with other techniques. The major difficulties arise from the modeling of the lens mass distribution, and from the uncertainty on the time delay measurement itself, which was typically of about 10% in past monitoring programs. COSMOGRAIL (COSmological MOnitoring of GRAvItational Lenses) is an international collaboration initiated in April 2004 at the Laboratory of Astrophysics of EPFL, and which aims at measuring precise time delays for most known lensed quasars, in order to determine the Hubble constant down to an uncertainty of a few percent. This thesis took place at the beginning of COSMOGRAIL and consisted in setting up this large photometric monitoring. It addressed both issues of carrying out accurate photometry of faint blended sources and of obtaining well sampled light curves, in order to measure precise time delays. As part of the COSMOGRAIL project, I have been managing the monitoring of over twenty gravitationally lensed quasars with the 1-2m telescopes involved both in the Northern and Southern hemispheres, and organizing the data. The first crucial work of this thesis was then to develop an automated reduction pipeline able to produce an homogeneous data set from images acquired with very different telescopes. This pipeline was also needed to perform aperture photometry of all lensed quasars, in order to study their variability and define the monitoring priorities. The powerful MCS deconvolution algorithm (Magain, Courbin, & Sohy 1998) was greatly used in this work and allowed to highly improve the image resolution, with the aim of obtaining accurate photometric measurements of the individual quasar lensed images. I have finally tested and improved three different numerical techniques to determine time delays between the quasar components from their light curves. In this thesis, time delays have been determined in four systems. The first one was measured in the doubly imaged quasar SDSS J1650+4251, after two years of monitoring with the 1.5m telescope of Maidanak Observatory, in Uzbekistan. The quadruply lensed system RXS J1131–1231 was then studied and three time delays determined from 3-year observations with the Swiss Euler 1.2m telescope located at La Silla, in Chile. The photometric monitoring of the quadruple WFI J2033–4723 was also carried out with the Euler telescope, and data were then merged with those obtained by a second monitoring group, with the SMARTS 1.3m telescope at the Cerro Tololo Interamerican Observatory (CTIO), also located in Chile. Two time delays were measured in this system, after three years of observations, the close pair A1 – A2 remaining unresolved. Three time delays were determined in the quadruply imaged quasar HE0435–1223, after four years of optical monitoring with Euler, Mercator and Maidanak telescopes, to which photometric measurements by SMARTS 1.3m telescope were added. Euler and SMARTS merged data for the doubly imaged quasar QJ0158–4325 were also analysed, the size of the source accretion disk was measured, but we failed to determine a time delay due to the high amplitude of the microlensing variability in this system. The accuracies on time delay measurements reached in this thesis are of the order of 3-4% and show a clear improvement from the typical 10% uncertainties of past monitoring programs. These results were finally converted into estimates of the Hubble constant following different models of the lensing mass potential. The H0 mean value obtained when considering the individual determinations from twelve gravitationally lensed quasars with known time delays is H0 = 60 ± 7 km s-1 Mpc-1. This result is consistent with the current favored value, and above all promising, as including additional systems to this ensemble will surely provide tighter bounds on H0. In conclusion, the increasing number of time delay measurements and improvements in lens modeling should reduce the errors on the Hubble constant estimate provided by gravitational lensing. Conversely, the determination of more time delays should put further constraints on lens galaxy density profiles when using a prior on H0 from other studies.

This work is devoted to the study of the main models which describe the motion of incompressible fluids, namely the Navier-Stokes, together with their hypodissipative version, and the Euler equations. We will mainly focus on the analysis of non-smooth weak solutions to those equations. Most of the results have been obtained by using the convex integration techniques introduced by Camillo De Lellis and László Székelyhidi in the context of the Euler equations, which recently led to the proof of the Onsager's conjecture on the anomalous dissipation of the kinetic energy. With various refinements of those iterative schemes we prove ill-posedness of Leray-Hopf weak solutions of the hypodissipative Navier-Stokes equations, sharpness of the kinetic energy regularity for Euler, typicality results in the sense of Baire's category for both Euler and Navier-Stokes, estimate on the dimension of the singular set in time of non-conservative Hölder weak solutions of the Euler equations. Moreover, building on different techniques, we also address some regularizing effects of those equations in various classes of weak solutions with some fractional differentiability in terms of Hölder, Sobolev and Besov regularity. The latter make use of new abstract interpolation results for multilinear operators which we developed for our specific context but which may also have independent interests.