A Robust Iterative Unfolding Method for Signal Processing

There is a well-known series expansion (Neumann series) in functional analysis for perturbative inversion of specific operators on Banach spaces. However, operators that appear in signal processing (e.g. folding and convolution of probability density functions), in general, do not satisfy the usual convergence condition of that series expansion. This article provides some theorems on the convergence criteria of a similar series expansion for this more general case, which is not covered yet by the literature. The main result is that a series expansion provides a robust unbiased unfolding and deconvolution method. For the case of the deconvolution, such a series expansion can always be applied, and the method always recovers the maximum possible information about the initial probability density function, thus the method is optimal in this sense. A very significant advantage of the presented method is that one does not have to introduce ad hoc frequency regulations etc., as in the case of usual naive deconvolution methods. For the case of general unfolding problems, we present a computer-testable sufficient condition for the convergence of the series expansion in question. Some test examples and physics applications are also given. The most important physics example shall be (which originally motivated our survey on this topic) the case of pi^0 --> gamma+gamma particle decay: we show that one can recover the initial pi^0 momentum density function form the measured single gamma momentum density function by our series expansion.Comment: 23 pages, 9 figure

A mathematical model for induction hardening including mechanical effects

In most structural components in mechanical engineering, there are surface parts, which are particularly stressed. The aim of surface hardening is to increase the hardness of the corresponding boundary layers by rapid heating and subsequent quenching. This heat treatment leads to a change in the microstructure, which produces the desired hardening effect. The mathematical model accounts for electromagnetic effects that lead to the heating of the workpiece as well as thermomechanical effects that cause the hardening of the workpiece. The new contribution of this paper is that we put a special emphasis on the thermomechanical effects caused by the phase transitions. We formulate a consistent model which takes care of effects like transformation strain and transformation plasticity induced by the phase transitions and allows for physical parameters depending on the respective phase volume fractions. The coupling between the electromagnetic and the thermomechanical part of the model is given through the temperature-dependent electric conductivity on the one hand and through the Joule heating term on the other hand, which appears in the energy balance and leads to the rise in temperature. Owing to the quadratic Joule heat term and a quadratic mechanical dissipation term in the energy balance, we obtain a parabolic equation with L1 data. We prove existence of a weak solution to the complete system using a truncation argument

The statistics of spikes trains: a stochastic calculus approach

We discuss the statistics of spikes trains for different types of integrate-and-fire neurons and different types of synaptic noise models. In cotnrast with the usual approaches in neuroscience, mainly based on statistical physics methods such as the Fokker-Planck equation or the mean-field theory, we chose the point of the view of the stochastic calculus theory to characterize neurons in noisy environments. We present four stochastic calculus techniques that can be used to find the probability distributions attached to the spikes trains. We illustrate the power of these techniques for four types of widely used neuron models. Despite the fact that these techniques are mathematically intricate we believe that they can be useful for answering questions in neuroscience that naturally arise from the variability of neuronal activity. For each technique we indicate its range of application and its limitation

Bounded Variation in Time

International audienceDescribing a motion consists in defining the state or position $q$ of the investigated system as a function of the real variable $t$, the time. Commonly, $q$ takes its values in some set $Q$, suitably structured for the velocity $u$ to be introduced as the derivative of $t\to q$, when it exists. This, in fact, makes sense if $Q$ is a topological linear space or, more generally, a differential manifold modelled on such a space.For smooth situations, classical dynamics rests, in turn, on the consideration of the acceleration. This is the derivative of $t \to u$, if it exists in the sense of the topological linear structure of $Q$, or, when $Q$ is a manifold, in the sense of some connection. But, from its early stages, classical dynamics has also had to face shocks, i.e. velocity jumps. For isolated shocks, one traditionally resorts to the equations of the dynamics of percussions. Even in the absence of impact, it has been known for a long time that systems submitted to such nonsmooth effects as dry friction may exhibit time discontinuity of the velocity. Furthermore, nonsmooth mechanical constraints may also prevent $t\to u$ from admitting a derivative. In all these cases, the laws governing the motion can no longer be formulated in terms of acceleration

Stochastic Processes and Hitting Times in Mathematical Neurosciences

Memoire de Master 2: Processus Stochastiques.In this research report we define a new event-based mathematical framework for studying the dynamics of networks of integrate-and-fire neuron driven by external noise. Such networks are classically studied using the Fokker-Planck equation (Brunel, Hakim). In this study, we use the powerful tools developed for communication networks theory and define a formalism for the study of spiking neuron networks driven by an external noise. With this formalism, we address biological questions to characterize the different network regimes. In this framework, the probability distribution of the interspike interval is a fundamental parameter. We developed and apply several tools for defining and computing the probability density function (pdf) of the time of the first spike, using stochastic analysis. This point of view gives us an event-driven strategy for simulating this type of random networks. This strategy has been implemented in an extension of the event-driven simulator Mvaspike

First hitting time of Double Integral Processes to curved boundaries

The problem of finding the first hitting time of a Double Integral Process (DIP) such as the Integrated Wiener Proces (IWP) has been a central and difficult endeavor in stochastic calculus and has applications in many fields of physics (first exit time of a particle in a noisy force field) or in biology and neuroscience (spike time of an integrate-and-fire neuron with exponentially decaying synaptic current). The only results available so far were an approximation of the stationnary mean crossing time and the distribution of the first hitting time of the IWP to a constant boundary. In this paper, we generalize those results and find an analytical formula for the first hitting time of the IWP to piecewise cubic boundaries. We use this formula to approximate the law of the first hitting time of a general DIP to a smooth curved boundary, and we provide an estimation of the convergence of this method. This approximation formula is the first analytical description of the hitting time of a DIP to a curved boundary, and allows us to infer properties of this random variable and provides a way for computing accurately its law. The accuracy of the approximation is computed in the general case for the IWP and the calculation of crossing probability can be carried out through a Monte-Carlo method

Merging of Opinions under Uncertainty

We consider long-run behavior of agents assessing risk in terms of dynamic convex risk measures or, equivalently, utility in terms of dynamic variational preferences in an uncertain setting. By virtue of a robust representation, we show that all uncertainty is revealed in the limit and agents behave as expected utility maximizer under the true underlying distribution regardless of their initial risk anticipation. In particular, risk assessments of distinct agents converge. This result is a generalization of the fundamental Blackwell-Dubins Theorem, cp. [Blackwell & Dubins, 62], to convex risk. We furthermore show the result to hold in a non-time-consistent environment.Dynamic Convex Risk Measures, Multiple Priors, Uncertainty, Robust Representation, Time-Consistency, Blackwell-Dubins

Constrained extremal problems in the Hardy space H2 and Carleman's formulas

We study some approximation problems on a strict subset of the circle by analytic functions of the Hardy space H2 of the unit disk (in C), whose modulus satisfy a pointwise constraint on the complentary part of the circle. Existence and uniqueness results, as well as pointwise saturation of the constraint, are established. We also derive a critical point equation which gives rise to a dual formulation of the problem. We further compute directional derivatives for this functional as a computational means to approach the issue. We then consider a finite-dimensional polynomial version of the bounded extremal problem