arXiv subjects
Publications and source records attributed to Claire David.
In the sequel, we recall and comment some classical results on the non-increasing rearrangement and Lorentz spaces. There are papers in the existing literature that seemed to have been bypassed as regards its contractive property in~$L^p$ spaces. Also, we provide detailed proofs and some properties that does not seem to arise in the existing literature.
In the following, bypassing dynamical systems tools, we propose a simple means of computing the box dimension of the graph of the classical Weierstrass function defined, for any real number~$x$, by~$ {\cal W}(x)=\displaystyle \sum_{n=0}^{+\infty} λ^n\,\cos \left ( 2\, π\,N_b^n\,x \right) $, where~$λ$ and~$N_b$ are two real numbers such that~\mbox{$0 <λ<1$},~\mbox{$ N_b\,\in\,\N$} and~$ λ\,N_b > 1 $, using a sequence a graphs that approximate the studied one.
In this work, we describe how to approximate solutions of some partial differential equations using the finite difference method defined on the Minkowski self-similar curve.
In the sequel, we question the validity of multiple choice questionnaires for undergraduate level math courses. Our study is based on courses given in major French universities, to numerous audiences.
Following our work on the graph of the Weierstrass function, in the spirit of those of J. Kigami and R. S. Strichartz, which enabled us to build a Laplacian on the aforementioned graph, it was natural to go further and give the related explicit resistance metric. The aim of this work is twofold. We had a special interest in the study of the spectral properties of the Laplacian. In our previous work, we have given the explicit the spectrum on the graph of the Weierstrass function. In the case of Laplacians on post-critically finite fractals, existing results of J. Kigami and M. Lapidus, R. S. Strichartz, make the link between resistance metric, and asymptotic properties of the spectrum of the Laplacian, by means of an analoguous of Weyl's formula. So we asked ourselves wether those results were still valid, for the graph of the Weierstrass function.
J. Kigami has laid the foundations of what is now known as analysis on fractals, by allowing the construction of an operator of the same nature of the Laplacian, defined locally, on graphs having a fractal character. The Sierpinski gasket stands out of the best known example. It has, since then, been taken up, developed and popularized by R. S. Strichartz. The Laplacian is obtained through a weak formulation, obtained by means of Dirichlet forms, built by induction on a sequence of graphs that converges towards the considered domain. It is these Dirichlet forms that enable one to obtain energy forms on this domain. Yet, things are not that simple. If, for domains like the Sierpinski gasket, the Laplacian is obtained in a quite natural way, one must bear in mind that Dirichlet forms solely depend on the topology of the domain, and not of its geometry. Which means that, if one aims at building a Laplacian on a fractal domain, the topology of which is the same as, for instance, a line segment, one has to find a way of taking account a very specific geometry. We came across that problem in our work on the graph of the Weierstrass function The solution was thus to consider energy forms more sophisticated than classical ones, by means of normalization constants that could, not only bear the topology, but, also, the very specific geometry of, from now on, we will call W-curves. We presently aim at investigating the links between energy forms and geometry. We have chosen to consider fractal curves, specifically, the Sierpiński arrowhead curve, the limit of which is the Sierpiński gasket. Does one obtain the same Laplacian as for the triangle ? The question appears as worth to be investigated.
There are hundreds of papers dealing with the Black-Scholes equation. Yet, no one seems to have ever used the scaling invariance coming from the heat equation. And there appears to be very few studies on the "control aspect" of the equation. In previous works, J.Y. Chemin and Cl. David obtained new results for the mass critical nonlinear Schrödinger equation, with a simple principle: one uses the fact that for this nonlinear equation, solutions of scales which are different enough almost do not interact. The nonlinearity of the original equation just required to determine a condition about the size of the scale which depends continuously on the data. The idea is to apply the building technique of the map described above, starting from the point that the Black-Scholes equation can be transformed into a linear heat equation, which has a natural scaling invariance, and that the linearity of the transformed equation enables one to obtain exact solutions, and, thus, yield interesting results. The aim of this paper is to present a new way to change either the Delta, or other greeks, by a given amount ; this technique appears thus as a new way to control the equation, without resorting to control functions. Indeed, one can change the greeks directly using the general solution of the Black-Scholes equation, nevertheless, our technique appears as an alternative and simple one to implement.
Numerous work revolve around the Sierpinski gasket. Its three-dimensional analogue, the Sierpinski tetrahedron, obtained by means of an iterative process which consists in repeatedly contracting a regular 3-simplex to one half of its original height, put together four copies, the frontier corners of which coincide with the initial simplex, appears as a natural extension. Yet, very few works concern the Sierpinski tetrahedron in the existing literature. We go further and, after a detailed study, we give the explicit spectrum of the Laplacian, with a specific presentation of the first eigenvalues. This enables us to obtain an estimate of the spectral counting function (analogous of Weyl's law)
In April 2016, a community of researchers working in the area of Principles of Data Management (PDM) joined in a workshop at the Dagstuhl Castle in Germany. The workshop was organized jointly by the Executive Committee of the ACM Symposium on Principles of Database Systems (PODS) and the Council of the International Conference on Database Theory (ICDT). The mission of this workshop was to identify and explore some of the most important research directions that have high relevance to society and to Computer Science today, and where the PDM community has the potential to make significant contributions. This report describes the family of research directions that the workshop focused on from three perspectives: potential practical relevance, results already obtained, and research questions that appear surmountable in the short and medium term.
Graph games of infinite length are a natural model for open reactive processes: one player represents the controller, trying to ensure a given specification, and the other represents a hostile environment. The evolution of the system depends on the decisions of both players, supplemented by chance. In this work, we focus on the notion of randomised strategy. More specifically, we show that three natural definitions may lead to very different results: in the most general cases, an almost-surely winning situation may become almost-surely losing if the player is only allowed to use a weaker notion of strategy. In more reasonable settings, translations exist, but they require infinite memory, even in simple cases. Finally, some traditional problems becomes undecidable for the strongest type of strategies.
A linear dispersive mechanism leading to a burst in the $L_\infty$ norm of the error in numerical simulation of polychromatic solutions is identified. This local error pile-up corresponds to the existence of spurious caustics, which are allowed by the dispersive nature of the numerical error. From the mathematical point of view, spurious caustics are related to extrema of the numerical group velocity and are physically associated to interactions between rays defined by the characteristic lines of the discrete system. This paper extends our previous work about classical schemes to dispersion-relation preserving schemes.
The goal of this work is to determine whole classes of solitary wave solutions general for wave equations.
The goal of this work is to determine classes of travelling solitary wave solutions for a differential approximation of a finite difference scheme by means of a hyperbolic ansatz. It is shown that spurious solitary waves can occur in finite-difference solutions of nonlinear wave equation. The occurance of such a spurious solitary wave, which exhibits a very long life time, results in a non-vanishing numerical error for arbitrary time in unbounded numerical domain. Such a behavior is referred here to has a structural instability of the scheme, since the space of solutions spanned by the numerical scheme encompasses types of solutions (solitary waves in the present case) that are not solution of the original continuous equations. This paper extends our previous work about classical schemes to dispersion-relation preserving schemes.
The aim of this paper is to propose methods that enable us to build new numerical schemes, which preserve the Lie symmetries of the original differential equations. To this purpose, the compound Burgers-Korteweg-de Vries (\textit{CBKDV}) equation is considered. The particular case of the Burgers equation is taken as a numerical example, and the resulting semi-invariant scheme is exposed.
In this paper, we study a compound Korteweg-de Vries-Burgers equation with a higher-order nonlinearity. A class of solitary wave solutions is obtained by means of a series expansion.
Finite difference schemes are here solved by means of a linear matrix equation. The theoretical study of the related algebraic system is exposed, and enables us to minimize the error due to a finite difference approximation, while building a new DRP scheme in the same time.
Differential equations arising in fluid mechanics are usually derived from the intrinsic properties of mechanical systems, in the form of conservation laws, and bear symmetries, which are not generally preserved by a finite difference approximation, and leading to inaccurate numerical results. This paper proposes a method that enables us to build a scheme that preserves some of those symmetries.
A Mathematica based program has been elaborated in order to determine the symmetry group of a finite difference equation, by means of its differential representation. The package provides functions which enable us to solve the determining equations of the related Lie group