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Concept# Rigour

Résumé

Rigour (British English) or rigor (American English; see spelling differences) describes a condition of stiffness or strictness. These constraints may be environmentally imposed, such as "the rigours of famine"; logically imposed, such as mathematical proofs which must maintain consistent answers; or socially imposed, such as the process of defining ethics and law.
Etymology
"Rigour" comes to English through old French (13th c., Modern French rigueur) meaning "stiffness", which itself is based on the Latin rigorem (nominative rigor) "numbness, stiffness, hardness, firmness; roughness, rudeness", from the verb rigere "to be stiff". The noun was frequently used to describe a condition of strictness or stiffness, which arises from a situation or constraint either chosen or experienced passively. For example, the title of the book Theologia Moralis Inter Rigorem et Laxitatem Medi roughly translates as "mediating theological morality between rigour and laxness". The book details

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Mathématiques

thumb|upright|Raisonnement mathématique sur un tableau.
Les mathématiques (ou la mathématique) sont un ensemble de connaissances abstraites résultant de raisonnements logiques appliqués à des objets

Géométrie

La géométrie est à l'origine la branche des mathématiques étudiant les figures du plan et de l'espace (géométrie euclidienne). Depuis la fin du , la géométrie étudie également les figures appartenan

Physique mathématique

La physique mathématique est un domaine de recherche commun à la physique et aux mathématiques s'intéressant au développement des méthodes mathématiques spécifiques aux problèmes physiques ou plus g

It is a fundamental question in disease modeling how the initial seeding of an epidemic, spreading over a network, determines its final outcome. One important goal has been to find the seed configuration, which infects the most individuals. Although the identified optimal configurations give insight into how the initial state affects the outcome of an epidemic, they are unlikely to occur in real life. In this paper we identify two important seeding scenarios, both motivated by historical data, that reveal a complex phenomenon. In one scenario, the seeds are concentrated on the central nodes of a network, while in the second one, they are spread uniformly in the population. Comparing the final size of the epidemic started from these two initial conditions through data-driven and synthetic simulations on real and modeled geometric metapopulation networks, we find evidence for a switchover phenomenon: When the basic reproduction number R0 is close to its critical value, more individuals become infected in the first seeding scenario, but for larger values of R0, the second scenario is more dangerous. We find that the switchover phenomenon is amplified by the geometric nature of the underlying network and confirm our results via mathematically rigorous proofs, by mapping the network epidemic processes to bond percolation. Our results expand on the previous finding that, in the case of a single seed, the first scenario is always more dangerous and further our understanding of why the sizes of consecutive waves of a pandemic can differ even if their epidemic characters are similar.

2021Snow microstructure and its evolution play an important role for various applications of snow physics in cryospheric sciences. The main modes of microstructure evolution in snow are referred to as isothermal and temperature gradient metamorphism. The former describes the coarsening driven by interfacial energy while the latter is dominated by recrystallization processes induced by temperature gradients. An accurate description of these processes in snowpack models is of key importance. However common snowpack models are still based on traditional grain metrics, originally tailored to field observations, and empirical evolution laws. This treatment of snow microstructure is essentially unrelated to recent advances of snow observations by microcomputed tomography (ÎŒCT). The present thesis contributes to the solution of this problem by i) identifying suitable microstructure parameters ii) deriving evolution equations for these from first principles and iii) developing methods that allow to utilize 4D ÎŒCT measurements of snow as a link between local ice crystal growth and upscaled microstructure as relevant on the scales of interest for common snowpack models. To this end, three studies have been conducted. The first study focuses on estimating local ice-crystal growth rates from interface tracking by analyzing 4D ÎŒCT data of in-situ snow metamorphism experiments under isothermal and temperature gradient conditions. For temperature gradient metamorphism diffusion-limited growth is considered, while for isothermal metamorphism the data is compared to kinetics and diffusion limited growth. Despite considerable scatter, in both cases the significance of underlying growth laws could be statistically confirmed. The second study uses ÎŒCT images from a variety of snow samples to investigate the role of grain shape in the context of microwave and optical properties of snow. Grain shape can be objectively defined via size-dispersity of structure from the second moment of either the mean curvature distribution or the chord-length distribution. In addition, a quantitative link between these quantities and the exponential correlation length is shown. The latter is relevant for parameterizing macroscopic properties such as microwave scattering coefficients, dielectric permittivity and thermal conductivity. Finally, a rigorous, upscaled microstructure scheme is developed by deriving mathematically exact evolution equations for the density, specific surface area, the mean and Gaussian curvature and the second moment of mean curvature. The microstructural evolution is driven by local ice crystal growth. All parameters are upscaled by volume averaging and the correctness of the model is confirmed for the time evolution of idealized grains. The model can be compared to 4D ÎŒCT data without any a-priori assumptions. This benchmarking reveals the uncertainties of the interface tracking method which are largely caused by limited temporal and spatial resolution. The model allows to statistically assess the validity of ice crystal growth laws during snow metamorphism. For a temperature gradient experiment it is shown that a diffusion limited growth law is not consistent with the observed decay of the specific surface area. The developed model is a powerful and rigorous tool that is tailored to 4D ÎŒCT data. It connects microscale ice-crystal growth thermodynamics with the macroscale snowpack modeling.

Matthieu Martin Jean-André Simeoni

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