Confidence Bands for Convex Median Curves using Sign-Tests

Suppose that one observes pairs (x1,Y1), (x2,Y2), ..., (xn,Yn), where x1 < x2 < ... < xn are fixed numbers while Y1, Y2, ..., Yn are independent random variables with unknown distributions. The only assumption is that Median(Yi) = f(xi) for some unknown convex or concave function f. We present a confidence band for this regression function f using suitable multiscale sign tests. While the exact computation of this band seems to require O(n4) steps, good approximations can be obtained in O(n2) steps. In addition the confidence band is shown to have desirable asymptotic properties as the sample size n tends to infinity.

HIV infection disrupts the sympatric host-pathogen relationship in human tuberculosis

The phylogeographic population structure of Mycobacterium tuberculosis suggests local adaptation to sympatric human populations. We hypothesized that HIV infection, which induces immunodeficiency, will alter the sympatric relationship between M. tuberculosis and its human host. To test this hypothesis, we performed a nine-year nation-wide molecular-epidemiological study of HIV-infected and HIV-negative patients with tuberculosis (TB) between 2000 and 2008 in Switzerland. We analyzed 518 TB patients of whom 112 (21.6%) were HIV-infected and 233 (45.0%) were born in Europe. We found that among European-born TB patients, recent transmission was more likely to occur in sympatric compared to allopatric host-pathogen combinations (adjusted odds ratio [OR] 7.5, 95% confidence interval [95% CI] 1.21-infinity, p = 0.03). HIV infection was significantly associated with TB caused by an allopatric (as opposed to sympatric) M. tuberculosis lineage (OR 7.0, 95% CI 2.5-19.1, p<0.0001). This association remained when adjusting for frequent travelling, contact with foreigners, age, sex, and country of birth (adjusted OR 5.6, 95% CI 1.5-20.8, p = 0.01). Moreover, it became stronger with greater immunosuppression as defined by CD4 T-cell depletion and was not the result of increased social mixing in HIV-infected patients. Our observation was replicated in a second independent panel of 440 M. tuberculosis strains collected during a population-based study in the Canton of Bern between 1991 and 2011. In summary, these findings support a model for TB in which the stable relationship between the human host and its locally adapted M. tuberculosis is disrupted by HIV infection.

Image Space Adaptive Rendering

In this thesis, we develop an adaptive framework for Monte Carlo rendering, and more specifically for Monte Carlo Path Tracing (MCPT) and its derivatives. MCPT is attractive because it can handle a wide variety of light transport effects, such as depth of field, motion blur, indirect illumination, participating media, and others, in an elegant and unified framework. However, MCPT is a sampling-based approach, and is only guaranteed to converge in the limit, as the sampling rate grows to infinity. At finite sampling rates, MCPT renderings are often plagued by noise artifacts that can be visually distracting. The adaptive framework developed in this thesis leverages two core strategies to address noise artifacts in renderings: adaptive sampling and adaptive reconstruction. Adaptive sampling consists in increasing the sampling rate on a per pixel basis, to ensure that each pixel value is below a predefined error threshold. Adaptive reconstruction leverages the available samples on a per pixel basis, in an attempt to have an optimal trade-off between minimizing the residual noise artifacts and preserving the edges in the image. In our framework, we greedily minimize the relative Mean Squared Error (rMSE) of the rendering by iterating over sampling and reconstruction steps. Given an initial set of samples, the reconstruction step aims at producing the rendering with the lowest rMSE on a per pixel basis, and the next sampling step then further reduces the rMSE by distributing additional samples according to the magnitude of the residual rMSE of the reconstruction. This iterative approach tightly couples the adaptive sampling and adaptive reconstruction strategies, by ensuring that we only sample densely regions of the image where adaptive reconstruction cannot properly resolve the noise. In a first implementation of our framework, we demonstrate the usefulness of our greedy error minimization using a simple reconstruction scheme leveraging a filterbank of isotropic Gaussian filters. In a second implementation, we integrate a powerful edge aware filter that can adapt to the anisotropy of the image. Finally, in a third implementation, we leverage auxiliary feature buffers that encode scene information (such as surface normals, position, or texture), to improve the robustness of the reconstruction in the presence of strong noise.

A new model construction by making a detour via intuitionistic theories I: Operational set theory without choice is Π1-equivalent to KP

We introduce a version of operational set theory, OST−, without a choice operation, which has a machinery for Δ0Δ0 separation based on truth functions and the separation operator, and a new kind of applicative set theory, so-called weak explicit set theory WEST, based on Gödel operations. We show that both the theories and Kripke–Platek set theory KPKP with infinity are pairwise Π1Π1 equivalent. We also show analogous assertions for subtheories with ∈-induction restricted in various ways and for supertheories extended by powerset, beta, limit and Mahlo operations. Whereas the upper bound is given by a refinement of inductive definition in KPKP, the lower bound is by a combination, in a specific way, of realisability, (intuitionistic) forcing and negative interpretations. Thus, despite interpretability between classical theories, we make “a detour via intuitionistic theories”. The combined interpretation, seen as a model construction in the sense of Visser's miniature model theory, is a new way of construction for classical theories and could be said the third kind of model construction ever used which is non-trivial on the logical connective level, after generic extension à la Cohen and Krivine's classical realisability model.

Ambient cosmology and spacetime singularities

We present a new approach to the issues of spacetime singularities and cosmic censorship in general relativity. This is based on the idea that standard 4-dimensional spacetime is the conformal infinity of an ambient metric for the 5-dimensional Einstein equations with fluid sources. We then find that the existence of spacetime singularities in four dimensions is constrained by asymptotic properties of the ambient 5-metric, while the non-degeneracy of the latter crucially depends on cosmic censorship holding on the boundary.

Eigenvalue estimates for non-selfadjoint Dirac operators on the real line

We show that the non-embedded eigenvalues of the Dirac operator on the real line with complex mass and non-Hermitian potential V lie in the disjoint union of two disks, provided that the L1-norm of V is bounded from above by the speed of light times the reduced Planck constant. The result is sharp; moreover, the analogous sharp result for the Schrödinger operator, originally proved by Abramov, Aslanyan and Davies, emerges in the nonrelativistic limit. For massless Dirac operators, the condition on V implies the absence of non-real eigenvalues. Our results are further generalized to potentials with slower decay at infinity. As an application, we determine bounds on resonances and embedded eigenvalues of Dirac operators with Hermitian dilation-analytic potentials.

Topology of the ambient boundary and the convergence of causal curves

We discuss the topological nature of the boundary spacetime, the conformal infinity of the ambient cosmological metric. Due to the existence of a homothetic group, the bounding spacetime must be equipped not with the usual Euclidean metric topology but with the Zeeman fine topology. This then places severe constraints to the convergence of a sequence of causal curves and the extraction of a limit curve, and also to our ability to formulate conditions for singularity formation.