An Empirical Process Central Limit Theorem for Multidimensional Dependent Data

Let $(U_n(t))_{t\in\R^d}$ be the empirical process associated to an $\R^d$-valued stationary process $(X_i)_{i\ge 0}$. We give general conditions, which only involve processes $(f(X_i))_{i\ge 0}$ for a restricted class of functions $f$, under which weak convergence of $(U_n(t))_{t\in\R^d}$ can be proved. This is particularly useful when dealing with data arising from dynamical systems or functional of Markov chains. This result improves those of [DDV09] and [DD11], where the technique was first introduced, and provides new applications.Comment: to appear in Journal of Theoretical Probabilit

An extended quantitative model for super-resolution optical fluctuation imaging (SOFI)

Super-resolution optical fluctuation imaging (SOFI) provides super-resolution (SR) fluorescence imaging by analyzing fluctuations in the fluorophore emission. The technique has been used both to acquire quantitative SR images and to provide SR biosensing by monitoring changes in fluorophore blinking dynamics. Proper analysis of such data relies on a fully quantitative model of the imaging. However, previous SOFI imaging models made several assumptions that can not be realized in practice. In this work we address these limitations by developing and verifying a fully quantitative model that better approximates real-world imaging conditions. Our model shows that (i) SOFI images are free of bias, or can be made so, if the signal is stationary and fluorophores blink independently, (ii) allows a fully quantitative description of the link between SOFI imaging and probe dynamics, and (iii) paves the way for more advanced SOFI image reconstruction by offering a computationally fast way to calculate SOFI images for arbitrary probe, sample and instrumental properties

On symmetries of Chern-Simons and BF topological theories

We describe constructing solutions of the field equations of Chern-Simons and topological BF theories in terms of deformation theory of locally constant (flat) bundles. Maps of flat connections into one another (dressing transformations) are considered. A method of calculating (nonlocal) dressing symmetries in Chern-Simons and topological BF theories is formulated

Homogeneous variational problems: a minicourse

A Finsler geometry may be understood as a homogeneous variational problem, where the Finsler function is the Lagrangian. The extremals in Finsler geometry are curves, but in more general variational problems we might consider extremal submanifolds of dimension $m$. In this minicourse we discuss these problems from a geometric point of view.Comment: This paper is a written-up version of the major part of a minicourse given at the sixth Bilateral Workshop on Differential Geometry and its Applications, held in Ostrava in May 201

Time series prediction via aggregation : an oracle bound including numerical cost

We address the problem of forecasting a time series meeting the Causal Bernoulli Shift model, using a parametric set of predictors. The aggregation technique provides a predictor with well established and quite satisfying theoretical properties expressed by an oracle inequality for the prediction risk. The numerical computation of the aggregated predictor usually relies on a Markov chain Monte Carlo method whose convergence should be evaluated. In particular, it is crucial to bound the number of simulations needed to achieve a numerical precision of the same order as the prediction risk. In this direction we present a fairly general result which can be seen as an oracle inequality including the numerical cost of the predictor computation. The numerical cost appears by letting the oracle inequality depend on the number of simulations required in the Monte Carlo approximation. Some numerical experiments are then carried out to support our findings

Symmetries of Helmholtz forms and globally variational dynamical forms

Invariance properties of classes in the variational sequence suggested to Krupka et al. the idea that there should exist a close correspondence between the notions of variationality of a differential form and invariance of its exterior derivative. It was shown by them that the invariance of a closed Helmholtz form of a dynamical form is equivalent with local variationality of the Lie derivative of the dynamical form, so that the latter is locally the Euler--Lagrange form of a Lagrangian. We show that the corresponding local system of Euler--Lagrange forms is variationally equivalent to a global Euler--Lagrange form.Comment: Presented at QTS7 - Quantum Theory and Symmetries VII, Prague 7-13/08/201

Adaptive density estimation for stationary processes

We propose an algorithm to estimate the common density $s$ of a stationary process $X_1,...,X_n$. We suppose that the process is either $\beta$ or $\tau$-mixing. We provide a model selection procedure based on a generalization of Mallows' $C_p$ and we prove oracle inequalities for the selected estimator under a few prior assumptions on the collection of models and on the mixing coefficients. We prove that our estimator is adaptive over a class of Besov spaces, namely, we prove that it achieves the same rates of convergence as in the i.i.d framework

A Study of Concurrency Bugs and Advanced Development Support for Actor-based Programs

The actor model is an attractive foundation for developing concurrent applications because actors are isolated concurrent entities that communicate through asynchronous messages and do not share state. Thereby, they avoid concurrency bugs such as data races, but are not immune to concurrency bugs in general. This study taxonomizes concurrency bugs in actor-based programs reported in literature. Furthermore, it analyzes the bugs to identify the patterns causing them as well as their observable behavior. Based on this taxonomy, we further analyze the literature and find that current approaches to static analysis and testing focus on communication deadlocks and message protocol violations. However, they do not provide solutions to identify livelocks and behavioral deadlocks. The insights obtained in this study can be used to improve debugging support for actor-based programs with new debugging techniques to identify the root cause of complex concurrency bugs.Comment: - Submitted for review - Removed section 6 "Research Roadmap for Debuggers", its content was summarized in the Future Work section - Added references for section 1, section 3, section 4.3 and section 5.1 - Updated citation

Constraint algorithm for k-presymplectic Hamiltonian systems. Application to singular field theories

The k-symplectic formulation of field theories is especially simple, since only tangent and cotangent bundles are needed in its description. Its defining elements show a close relationship with those in the symplectic formulation of mechanics. It will be shown that this relationship also stands in the presymplectic case. In a natural way, one can mimick the presymplectic constraint algorithm to obtain a constraint algorithm that can be applied to $k$-presymplectic field theory, and more particularly to the Lagrangian and Hamiltonian formulations of field theories defined by a singular Lagrangian, as well as to the unified Lagrangian-Hamiltonian formalism (Skinner--Rusk formalism) for k-presymplectic field theory. Two examples of application of the algorithm are also analyzed.Comment: 22 p