Bornes sur l'élongation du plus court chemin dans la triangulation de Delaunay d'un processus de Poisson de grande densité

Let $X:=X_n\cup\{(0,0),(1,0)\}$, where $X_n$ is a planar Poisson point process of intensity $n$. We prove that the distance between the expected length of the shortest path between $(0,0)$ and $(1,0)$ in the Delaunay triangulation associated with $X$ belongs to $[1+2.47\cdot 10^{-11}, 1.182]$ when the intensity of $X_n$ goes to infinity. Experimental values indicate that the correct value is about 1.04. We also prove that the expected number of Delaunay edges crossed by the line segment $[(0,0),(1,0)]$ is $\frac{64}{3\pi^2}\sqrt{n}+O(1)\simeq2.16\sqrt{n}$.Simulation code and Maple sheet are available with the research report.Soit $X:=X_n\cup\{(0,0),(1,0)\}$, où $X_n$ est un processus ponctuel de Poisson planaire d'intensité $n$. Nous montrons que la longueur du plus court chemin entre $(0,0)$ et $(1,0)$ dans la triangulation de Delaunay de $X$ est dans l'intervalle $[1+2.47\cdot 10^{-11}, 1.182]$ quand $n$ tends vers l'infini.Les résultats expérimentaux indiquent que la valeur correcte est 1.04.Nous montrons aussi que l'espérance du nombre d'arêtes de Delaunay coupées par le segment $[(0,0),(1,0)]$ est $\frac{64}{3\pi^2}\sqrt{n}+O(1)\simeq2.16\sqrt{n}$.Le code pour les simulations et la feuille de calcul Maple sont disponibles avce ce rapport de recherche

Fast and Accurate Visibility Preprocessing

Visibility culling is a means of accelerating the graphical rendering of geometric models. Invisible objects are efficiently culled to prevent their submission to the standard graphics pipeline. It is advantageous to preprocess scenes in order to determine invisible objects from all possible camera views. This information is typically saved to disk and may then be reused until the model geometry changes. Such preprocessing algorithms are therefore used for scenes that are primarily static. Currently, the standard approach to visibility preprocessing algorithms is to use a form of approximate solution, known as conservative culling. Such algorithms over-estimate the set of visible polygons. This compromise has been considered necessary in order to perform visibility preprocessing quickly. These algorithms attempt to satisfy the goals of both rapid preprocessing and rapid run-time rendering. We observe, however, that there is a need for algorithms with superior performance in preprocessing, as well as for algorithms that are more accurate. For most applications these features are not required simultaneously. In this thesis we present two novel visibility preprocessing algorithms, each of which is strongly biased toward one of these requirements. The first algorithm has the advantage of performance. It executes quickly by exploiting graphics hardware. The algorithm also has the features of output sensitivity (to what is visible), and a logarithmic dependency in the size of the camera space partition. These advantages come at the cost of image error. We present a heuristic guided adaptive sampling methodology that minimises this error. We further show how this algorithm may be parallelised and also present a natural extension of the algorithm to five dimensions for accelerating generalised ray shooting. The second algorithm has the advantage of accuracy. No over-estimation is performed, nor are any sacrifices made in terms of image quality. The cost is primarily that of time. Despite the relatively long computation, the algorithm is still tractable and on average scales slightly superlinearly with the input size. This algorithm also has the advantage of output sensitivity. This is the first known tractable exact solution to the general 3D from-region visibility problem. In order to solve the exact from-region visibility problem, we had to first solve a more general form of the standard stabbing problem. An efficient solution to this problem is presented independently

Applications of Molecular Dynamics simulations for biomolecular systems and improvements to density-based clustering in the analysis

Molecular Dynamics simulations provide a powerful tool to study biomolecular systems with atomistic detail. The key to better understand the function and behaviour of these molecules can often be found in their structural variability. Simulations can help to expose this information that is otherwise experimentally hard or impossible to attain. This work covers two application examples for which a sampling and a characterisation of the conformational ensemble could reveal the structural basis to answer a topical research question. For the fungal toxin phalloidin—a small bicyclic peptide—observed product ratios in different cyclisation reactions could be rationalised by assessing the conformational pre-organisation of precursor fragments. For the C-type lectin receptor langerin, conformational changes induced by different side-chain protonations could deliver an explanation of the pH-dependency in the protein’s calcium-binding. The investigations were accompanied by the continued development of a density-based clustering protocol into a respective software package, which is generally well applicable for the use case of extracting conformational states from Molecular Dynamics data