Euclidean plane | EPFL Graph Search

In mathematics, a Euclidean plane is a Euclidean space of dimension two, denoted E2. It is a geometric space in which two real numbers are required to determine the position of each point. It is an affine space, which includes in particular the concept of parallel lines. It has also metrical properties induced by a distance, which allows to define circles, and angle measurement. A Euclidean plane with a chosen Cartesian coordinate system is called a Cartesian plane. The set \mathbb{R}^2 of the pairs of real numbers (the real coordinate plane), equipped with the dot product, is often called the Euclidean plane, since every Euclidean plane is isomorphic to it. History Euclidean geometry#History Books I through IV and VI of Euclid's Elements dealt with two-dimensional geometry, developing such notions as similarity of shapes, the Pythagorean theorem (Proposition 47), equality of angles and areas, parallelism, the sum of the angles in a triangle, and the three c

ANALYSE PARASISMIQUE D’UN BÂTIMENT EN VALAIS

Camille Spinelli

La Suisse est un pays à l’aléa sismique non négligeable dans certaines régions, comme par exemple en Valais. Les constructions dans ces zones doivent satisfaire des exigences en matière de sécurité parasismique. La structure (porteuse et non porteuse) des bâtiments doit être dimensionnée de sorte à résister à un séisme. Ce projet de master se concentre sur l’étude d’une construction neuve en béton armé soumis à un risque sismique élevé. Le bâtiment se situe à Crans-Montana (zone 3b) et se compose d’un sous-sol commun de 6 étages ainsi que deux bâtiments émergents de chacun de 6 étages également. Différents outils sont utilisés pour comprendre le comportement dynamique et dimensionner la structure. Un pré-dimensionnement à l’aide de la méthode des forces de remplacement est effectué sur la base d’un modèle simplifié. Divers modèles aux éléments finis sont réalisés, partant d’un modèle général où seuls les éléments horssol sont considérés pour arriver à un modèle final qui prend en compte l’ensemble des conditions d’appuis de la structure. Les modèles numériques sont analysés sur la base de deux méthodes distinctes : la méthode des forces de remplacement et la méthode du spectre de réponse. Les résultats de cette étude ont permis de mettre en évidence que pour une analyse sismique : - la méthode du spectre de réponse est à privilégier car elle donne des résultats non-surestimés (comparé à la méthode des forces) - les modèles numériques tenant compte des conditions d’appuis réelles de la structure permettent de déterminer les redistributions les plus justes des efforts internes

2018

Thermal scanning probe lithography for 2D material-based gas sensing

Berke Erbas

Recently, two-dimensional (2D) material based gas sensing, especially transition metal dichalcogenide-based sensing, has been widely investigated thanks to its room temperature sensing ability. Unlike metal oxide based sensors, 2D material-based sensing can be carried out without a heater, therefore it reduces the power consumption and makes them promising candidates for smartphones and wearable electronic devices. Besides gas sensing ability, these 2D materials are interesting and promising candidates for electronics and optoelectronics applications. For example, monolayer molybdenum disulfide (MoS2), which is a 2D semiconductor with direct bandgap, is one of the promising candidates for transistor but it has low mobility compared to silicon. To replace silicon-based devices, strain engineering of MoS2 is widely investigated. It is observed that tensile strain tunes the bandgap of MoS2, enhances carrier mobility and improves NO2 gas sensing by piezotronic effect. Different strategies have been used to strain MoS2 such as flexible substrates to stretch the exfoliated material on top of them, silicon/ dielectric substrates with surface roughness or silicon/dielectric substrates patterned with nanocone structure to locally strain transferred MoS2. Herein, field effect transistor with three-dimensionally patterned gate oxide is designed, fabricated and electrically characterized for sensing application. Thermal scanning probe lithography, which is a direct three-dimensional nanofabrication technique, is used for the fabrication of sinusoidal surfaces. By transferring the exfoliated MoS2 flakes on top of these sinusoidal gate oxide, biaxial strain is induced. Unlike counterparts, thermal scanning probe lithography-based patterning provides us design flexibility, which controls the direction of strain and the strain rate. As a result of Raman spectroscopy, the shifts in out-of-plane vibration (A1g) and in-plane vibration (E1 2g) modes are up to 0.87 cm−1 and 2.60 cm−1, respectively. According to the theoretical Raman spectra calculations of strained MoS2 [1], these shifts correspond to an average strain of 0.58% in a circular area with 1 μm diameter. The strain of 0.58% is obtained for a sinusoidal pattern having a 23 nm amplitude and a 440 nm period. Besides, as a result of photoluminescence characterization, it is shown that an average strain of 0.39% leads to a 35 meV redshift for A excitation peak position. Consequently, the strain of MoS2 on the surface patterned by thermal scanning probe lithography tunes the bandgap of MoS2 by 90.11 meV/%.

2021

Covariance Estimation for Random Surfaces beyond Separability

Tomas Masák

This thesis focuses on non-parametric covariance estimation for random surfaces, i.e.~functional data on a two-dimensional domain. Non-parametric covariance estimation lies at the heart of functional data analysis, andconsiderations of statistical and computational efficiency often compel the use of separability of the covariance, when working with random surfaces. We seek to provide efficient alternatives to this ambivalent assumption.In Chapter 2, we study a setting where the covariance structure may fail to be separable locally -- either due to noise contamination or due to the presence of a non-separable short-range dependent signal component. That is, the covariance is an additive perturbation of a separable component by a non-separable but banded component. We introduce non-parametric estimators hinging on shifted partial tracing -- a novel concept enjoying strong denoising properties. We illustrate the usefulness of the proposed methodology on a data set of mortality surfaces.In Chapter 3, we propose a distinctive decomposition of the covariance, which allows us to understand separability as an unconventional form of low-rankness. From this perspective, a separable covariance has rank one. Allowing for a higher rank suggests a structured class in which any covariance can be approximated up to an arbitrary precision. The key notion of the partial inner product allows us to generalize the power iteration method to general Hilbert spaces and estimate the aforementioned decomposition from data. Truncation and retention of the leading terms automatically induces a non-parametric estimator of the covariance, whose parsimony is dictated by the truncation level. Advantages of this approach, allowing for estimation beyond separability, are demonstrated on the task of classification of EEG signals.While Chapters 2 and 3 propose several generalizations of separability in the densely sampled regime, Chapter 4 deals with the sparse regime, where the latent surfaces are observed only at few irregular locations. Here, a separable covariance estimator based on local linear smoothers is proposed, which is the first non-parametric utilization of separability in the sparse regime. The assumption of separability reduces the intrinsically four-dimensional smoothing problem into several two-dimensional smoothers and allows the proposed estimator to retain the classical minimax-optimal convergence rate for two-dimensional smoothers. The proposed methodology is used for a qualitative analysis of implied volatility surfaces corresponding to call options, and for prediction of the latent surfaces based on information from the entire data set, allowing for uncertainty quantification. Our quantitative results show that the proposed methodology outperforms the common approach of pre-smoothing every implied volatility surface separately.Throughout the thesis, we put emphasis on computational aspects, since those are the main reason behind the immense popularity of separability. We show that the covariance structures of Chapters 2 and 3 come with no (asymptotic) computational overhead relative to assuming separability. In fact, the proposed covariance structures can be estimated and manipulated with the same asymptotic costs as the separable model. In particular, we develop numerical algorithms that can be used for efficient inversion, as required e.g.~for prediction. All the methods are implemented in R and available on~GitHub.

EPFL2022