Linear Perturbations for the Vacuum Axisymmetric Einstein Equations

In axial symmetry, there is a gauge for Einstein equations such that the total mass of the spacetime can be written as a conserved, positive definite, integral on the spacelike slices. This property is expected to play an important role in the global evolution. In this gauge the equations reduce to a coupled hyperbolic–elliptic system which is formally singular at the axis. Due to the rather peculiar properties of the system, the local in time existence has proved to resist analysis by standard methods. To analyze the principal part of the equations, which may represent the main source of the difficulties, we study linear perturbation around the flat Minkowski solution in this gauge. In this article we solve this linearized system explicitly in terms of integral transformations in a remarkable simple form. This representation is well suited to obtain useful estimates to apply in the non-linear case.     

Mathematical modeling of industrial aluminum electrolysis

The problem of aluminum electrolysis is discussed. The mathematical model of an industrial electrolyzer presented in the paper is written under the assumption that the electrolyte and metal media are immiscible. At the basis of the mathematical statement is a three-dimensional, nonstationary, and nonlinear system of magnetic hydrodynamics equations which is written separately in the aluminum medium and in the electrolyte medium with a geometric account for wall accretion, skull, and arrangement of anodes. The proposed system allows one to model various forms of anodes, the number of anodes in a bath, and their sizes. Interfaces of media are connected by a viscous friction. Initial values of speeds and electromagnetic fields in the media and the medium interface are considered as set. On the skull, bottom, and anodes the attachment conditions are set. The speed of change of a magnetic field in the metal and electrolyte on the interface is considered zero. On the basis of a numerical method of solution of the system of equations, there is a well-proved method of division over physical processes. The analysis of results of the numerical experiment has shown that it is actually possible to allocate a ??middle?? layer for modeling of the electrolysis process. The proposed model allows one to investigate media behavior upon the occurrence of a long anode effect due to sharp reduction of the electric conductivity of the electrolyte and subsequent sharp growth of the electric-field strength.     

A Trotter product formula for gradient flows in metric spaces

We prove a Trotter product formula for gradient flows in metric spaces. This result is applied to establish convergence in the L ²-Wasserstein metric of the splitting method for some Fokker-Planck equations and porous medium type equations perturbed by a potential.     

From Constructive Field Theory to Fractional Stochastic Calculus. (I) An introduction: Rough Path Theory and Perturbative Heuristics

Let B = (B ₁(t), . . . , B _d (t)) be a d-dimensional fractional Brownian motion with Hurst index α ≤ 1/4, or more generally a Gaussian process whose paths have the same local regularity. Defining properly iterated integrals of B is a difficult task because of the low H?lder regularity index of its paths. Yet rough path theory shows it is the key to the construction of a stochastic calculus with respect to B, or to solving differential equations driven by B. We intend to show in a forthcoming series of papers how to desingularize iterated integrals by a weak singular non-Gaussian perturbation of the Gaussian measure defined by a limit in law procedure. Convergence is proved by using “standard” tools of constructive field theory, in particular cluster expansions and renormalization. These powerful tools allow optimal estimates of the moments and call for an extension of the Gaussian tools such as for instance the Malliavin calculus. This first paper aims to be both a presentation of the basics of rough path theory to physicists, and of perturbative field theory to probabilists; it is only heuristic, in particular because the desingularization of iterated integrals is really a non-perturbative effect. It is also meant to be a general motivating introduction to the subject, with some insights into quantum field theory and stochastic calculus. The interested reader should read for a second time the companion article (Magnen and Unterberger in From constructive theory to fractional stochastic calculus. (II) The rough path for \frac16 < a < \frac14{\frac{1}{6} < \alpha < \frac{1}{4}}: constructive proof of convergence, 2011, preprint) for the constructive proofs.     

Intermediate long-wave equation on a half-line

We consider the initial-boundary value problem for intermediate long-wave equation on a half-line. We study traditionally important problems of the theory of nonlinear partial differential equations, such as global in time existence of solutions to the initial-boundary value problem and the asymptotic behavior of solutions for large time.     

Radon Transform on the Torus

We consider the Radon transform on the (flat) torus \mathbbTⁿ = \mathbbRⁿ/\mathbbZⁿ{\mathbb{T}^{n} = \mathbb{R}^{n}/\mathbb{Z}^n} defined by integrating a function over all closed geodesics. We prove an inversion formula for this transform and we give a characterization of the image of the space of smooth functions on \mathbbTⁿ{\mathbb{T}^{n}} .     

Sharply k-Transitive Permutation Groups Viewed as Galois Groups

The classification of finite sharply k-transitive groups was achieved by the efforts of Jordan (1873), Dickson (1905), and Zassenhaus (1936). Likewise for other families of finite groups, one expects that they are realizable as Galois groups over the field of rational numbers \mathbbQ{\mathbb{Q}}. In this article, we study some properties of the polynomials f ? \mathbbQ[x]{f \in \mathbb{Q}[x]} such that the Galois group Gal(f) acts sharply k-transitively on its roots.     

Logic of Determination of Objects (LDO): How to Articulate ��Extension�� with ��Intension�� and ��Objects�� with ��Concepts��

From a logical viewpoint, object is never defined, even by a negative definition. This paper is a theoretical contribution about object using a new constructivist logical approach called Logic of Determination of Objects founded on a basic operation, called determination. This new logic takes into account cognitive problems such as the inheritance of properties by non typical occurrences or by indeterminate atypical objects in opposition to prototypes that are typical completely determinate objects. We show how extensional classes, intensions, more and less determined objects, more or less typical representatives of a concept and prototypes are defined and organized, using a determination operation that constructs a class of indeterminate objects from an object representation of a concept called typical object.     

Consensus of partitions : a constructive approach

Given a profile (family) ?? of partitions of a set of objects or items X, we try to establish a consensus partition containing a maximum number of joined or separated pairs in X that are also joined or separated in the profile. To do so, we define a score function, S _?? associated to any partition on X. Consensus partitions for ?? are those maximizing this function. Therefore, these consensus partitions have the median property for the profile and the symmetric difference distance. This optimization problem can be solved, in certain cases, by integer linear programming. We define a polynomial heuristic which can be applied to partitions on a large set of items. In cases where an optimal solution can be computed, we show that the partitions built by this algorithm are very close to the optimum which is reached in practically all the cases, except for some sets of bipartitions.     

Decorrelation Estimates for the Eigenlevels of the Discrete Anderson Model in the Localized Regime

The purpose of the present work is to establish decorrelation estimates for the eigenvalues of the discrete Anderson model localized near two distinct energies inside the localization region. In dimension one, we prove these estimates at all energies. In higher dimensions, the energies are required to be sufficiently far apart from each other. As a consequence of these decorrelation estimates, we obtain the independence of the limits of the local level statistics at two distinct energies.