Showing papers by "Paris Dauphine University published in 1995"

Stochastic Navier-Stokes Equations

Abstract: The purpose of this article is to survey some results related to the theory of stochastic Navier-Stokes equations (SNSE). The interest of SNSE arises from modelling turbulence. We begin to show how SNSE can be introduced intuitively from the random motion of particles. We then review briefly the deterministic theory and present the main core of existence theory for NSE. We also discuss uniqueness issues. We end up by showing how the splitting-up method provides a useful constructive approach to existence, and by presenting some extensions, like weakening assumptions or considering the special case of small initial data.

The economic cost of obesity: the French situation.

Abstract: Objective To estimate the economic burden of obesity in France. Design A prevalence-based approach identifying the costs incurred during a given year (1992) by obese subjects. Measurements Direct costs (personal health care, hospital care, physician services, drugs) and indirect costs (lost output as a result of cessation or reduction of productivity caused by morbidity and mortality); economic benefits due to the reduced incidence of hip fractures. Results The direct costs of obesity (BMI > or = 27) were 11.89 billion French Francs (FF), which corresponded to about 2% of the expenses of the French care system. Hypertension represented 33% of the total amount and cancer 2.5% of the direct cost of obesity. Indirect costs represented FF 0.6 billion. These are conservative estimates as far as all obesity-related diseases and all health care and indirect costs were not included due to missing information. Conclusion These results were remarkably similar to previous reports on the economic costs of obesity in other western countries (USA, Sweden, Netherlands, Australia) which concluded that the cost of obesity amounted to around 2% to 5% of the total cost of health care in industrialized societies.

Stationary states of the nonlinear Dirac equation: a variational approach

Abstract: In this paper we prove the existence of stationary solutions of some nonlinear Dirac equations. We do it by using a general variational technique. This enables us to consider nonlinearities which are not necessarily compatible with symmetry reductions.

Quasicrystals, Diophantine approximation and algebraic numbers

Abstract: Quasicrystals can be characterized by a remarkable Diophantine approximation property. This permits to define the dual quasicrystal Λ* as the collection of y in ℝn such that |e iy·x −1| ≤ 1 for each x in the given quasicrystal Λ. In many cases one obtains Λ** = Λ and this duality is nicely related to the spectral properties of quasicrystals.

Eliciting Information Concerning the Relative Importance of Criteria

Abstract: The notion of Relative Importance of Criteria (RIC) is central in the domain of Multiple Criteria Decision Aid (MCDA). It aims at differentiating the role of each criterion in the construction of comprehensive preferences, thus allowing to discriminate among pareto-optimal alternatives. In most aggregation procedures, this notion takes the form of importance parameters.

Convergence of MUSCL and filtered schemes for scalar conservation laws and Hamilton-Jacobi equations

Abstract: This paper considers the questions of convergence of: (i) MUSCL type (i.e. second-order, TVD) finite-difference approximations towards the entropic weak solution of scalar, one-dimensional conservation laws with strictly convex flux and (ii) higher-order schemes (filtered to ``preserve'' an upper-bound on some weak second-order finite differences) towards the viscosity solution of scalar, multi-dimensional Hamilton-Jacobi equations with convex Hamiltonians.

Remarks on eigenvalues and eigenvectors of Hermitian matrices, berry phase, adiabatic connections and quantum Hall effect

Abstract: The singularities of the stratification of the space of the Hermitian matrices according to the multiplicities of the eigenvalues are described as an informal complexification of the previous study of the space of the real symmetric matrices. The degeneration of the spectral sequence associated to this stratification provides some strange combinatorial identities. The eigenvector bundles over the manifold of the Hermitian matrices with simple spectra are equiped with the natural connections, describing also the adiabatic approximation to the oscillations of the linear systems defined by the slowly varying skew Hermitian matrices. The curvature of this connection is singular at the codimension 3 variety of the Hermitian matrices having multiple eigenvalues. The resulting jumps of the integrals of the curvature form at the crossings of this variety by the moving surface of integration are responsible for the quantum Hall effect.

Stochastic inertial manifold

Abstract: A nonlinear stochastic evolution equation in Hilbert space with generalized additive white noise is considered. A concept of stochastic mertial manifold is introduced, defined as a random manifold depending on time, which is finite dimensional, invariant for the dynamic, and attracts exponentially fast all the trajectories as t → ∞. Under the classical spectral gap condition of the deterministic theory, the existence of a stochastic inertial manifold is proved. It is obtained as the solution of a stochastic partial differential equation of degenerate parabolic type, studied by a variant of Bernstein method. A result of existence and uniqueness of a stationary inertial manifold is also proved; the stationary inertial manifold contains the random attractor, introduced in previous works.

Legendrian distributions with applications to relative Poincaré series

Abstract: Let $X$ be a compact Kahler manifold and $L\\to X$ a quantizing holomorphic Hermitian line bundle. To immersed Lagrangian submanifolds $\\Lambda$ of $X$ satisfying a Bohr-Sommerfeld condition we associate sequences $\\{ |\\Lambda, k\\rangle \\}_{k=1}^\\infty$, where $\\forall k$ $|\\Lambda, k\\rangle$ is a holomorphic section of $L^{\\otimes k}$. The terms in each sequence concentrate on $\\Lambda$, and a sequence itself has a symbol which is a half-form, $\\sigma$, on $\\Lambda$. We prove estimates, as $k\\to\\infty$, of the norm squares $\\langle \\Lambda, k|\\Lambda, k\\rangle$ in terms of $\\int_\\Lambda \\sigma\\overline{\\sigma}$. More generally, we show that if $\\Lambda_1$ and $\\Lambda_2$ are two Bohr-Sommerfeld Lagrangian submanifolds intersecting cleanly, the inner products $\\langle\\Lambda_1, k|\\Lambda_2, k\\rangle$ have an asymptotic expansion as $k\\to\\infty$, the leading coefficient being an integral over the intersection $\\Lambda_1\\cap\\Lambda_2$. Our construction is a quantization scheme of Bohr-Sommerfeld Lagrangian submanifolds of $X$. We prove that the Poincar\\'e series on hyperbolic surfaces are a particular case, and therefore obtain estimates of their norms and inner products.

Some results on risk-sensitive control with full observation

Abstract: We consider in this presentation the risk-sensitive control problem under fairly general assumptions, in particular as far growth conditions are concerned. This is a much more delicate issue than in the classical stochastic control problem. We then consider the case of small variance of the noises and study the limit, which is related to robust control. The limit problem corresponds to a differential game. In this paper, the complete observation case is considered.

Mutational equations of shapes and vision-based control

Abstract: Basic idea of vision-based control in robotics is to include the vision system directly in the control servo loop of the robot. When images are binary, this problem corresponds to the control of the evolution of a geometric domain. The present paper proposes mathematical tools derived from shape analysis and optimization to study this problem in a quite general way, i.e., without any regularity assumptions or modelsa priori on the domains that we deal with. Indeed, despite the lackness of a vectorial structure, one can develop a differential calculus in the metric space of all non-empty compact subsets of a given domain ofR n , and adapt ideas and results of classical differential systems to study and control the evolution of geometric domains. For instance, a shape Lyapunov characterization allows to investigate the asymptotic behavior of these geometric domains using the notion of directional shape derivative. We apply this inR2 to the visual servoing problem using the optical flow equations and some experimental simulations illustrate this approach.

Stochastic equity volatility related to the leverage effect II: valuation of European equity options and warrants

Abstract: We propose a general framework to assess the value of the financial claims issued by the firm, European equity options and warrantsin terms of the stock price. In our framework, the firm's asset is assumed to follow a standard stationary lognormal process with constant volatility. However, it is not the case for equity volatility. The stochastic nature of equity volatility is endogenous, and comes from the impact of a change in the value of the firm's assets on the financial leverage. In a previous paper we studied the stochastic process for equity volatility, and proposed analytic approximations for different capital structures. In this companion paper we derive analytic approximations for the value of European equity options and warrants for a firm financed by equity, debt and warrants. We first present the basic model, which is an extension of the Black-Scholes model, to value corporate securities either as a function of the stock price, or as a function of the firm's total assets. Since stock prices a...

Symbolic Reachability Graph and Partial Symmetries

Abstract: The construction of symbolic reachability graphs is a useful technique for reducing state explosion in High-level Petri nets. Such a reduction is obtained by exploiting the symmetries of the whole net [1]. In this paper, we extend this method to deal with partial symmetries. In a first time, we introduce an example which shows the interest and the principles of our method. Then we develop the general algorithm. Lastly we enumerate the properties of this Extended Symbolic Reachability Graph, including the reachability equivalence.

Ergodic Bellman systems for stochastic games in arbitrary dimension

Abstract: Stochastic games with cost functionals J ( i ) ρ, x ( v ) = E ∫ ∞ 0 e – ρ t l i ( y, v ) d t , i = 1, 2 with controls v = v 1 , v 2 and state y ( t ) with y (0) = x are considered. Each player wants to minimize his (her) cost functional. E denotes the expected value and the state variables y are coupled with the controls v via a stochastic differential equation with initial value x . The corresponding Bellman system, which is used for the calculation of feedback controls v = v ( y ) and the solvability of the game, leads to a class of diagonal second-order nonlinear elliptic systems, which also occur in other branches of analysis. Their behaviour concerning existence and regularity of solutions is, despite many positive results, not yet well understood, even in the case where the l i , are simple quadratic functions. The objective of this paper is to give new insight to these questions for fixed ρ > 0, and, primarily, to analyse the limiting behaviour as the discount ρ → 0. We find that the modified solutions of the stochastic games converge, for subsequences, to the solution of the so-called ergodic Bellman equation and that the average cost converges. A former restriction of the space dimension has been removed. A reasonable class of quadratic integrands may be treated. More specifically, we consider the Bellman systems of equations – ∆ z + λ = H ( x, Dz ), where the space variable x belongs to a periodic cube (for the sake of simplifying the presentation). They are shown to have smooth solutions. If u ρ is the solution of – ∆ u ρ + ρ u ρ = H ( x, Du ρ ) then the convergence of u ρ — ῡ ρ to z , as ρ tends to 0, is established. The conditions on H are such that some quadratic growth in Du is allowed.

Evaluation of high level Petri nets by means of aggregation and decomposition

Abstract: We introduce a new method combining the powerful stochastic well formed Petri net (SWN) model with the decomposition approach initiated by B. Plateau (1985). We derive necessary conditions on the modeled systems that allow for the two methods to be combined. For parallel systems satisfying these necessary conditions we develop two models with corresponding algorithms. The first model, which forbids color synchronization between components of the system yields to a strict generalization of each of the two methods. The second model which covers a large range of real life systems needs theoretical study which we have undertaken. An example shows the intuitive ideas behind these developments.

Homoclinic orbits on compact hypersufaces in 293-1293-1293-1, of restricted contact type

Abstract: Consider a smooth Hamiltonian system in ℝ2N,\(\dot x = JH'(x)\), the energy surface Σ={x/H(x)=H(0)} being compact, and 0 being a hyperbolic equilibrium. We assume, moreover, that Σ∖{0} is of restricted contact type. These conditions are symplectically invariant. By a variational method, we prove the existence of an orbit homoclinic, i.e. non-constant and doubly asymptotic, to 0.

Periodic solutions of elliptic type for strongly nonlinear Hamiltonian systems

Abstract: It is shown that, for autonomous Hamiltonian systems, every convex energy level which is symmetric with respect to the origin carries a periodic solution of elliptic type. An extension is given to nonautonomous systems.

Economic sociology: a historical perspective

Abstract: In this paper we briefly recall what is meant by economic sociology, basing our definition principally on the works of R. Swedberg and M. Cranovetter. We then focus on the questions and problematics which are more particularly relevant to the history of economic and sociological thought in such a way as to make explicit the kind of past in relation to which economic sociology is most pertinent. We shall thus be induced to correct certain explanations and to propose a slightly different perspective on the origin of economic sociology. This historical work enables us to formulate a definition of economic sociology based on two key concepts - economic institution and economic action - and allows us to articulate what economic sociology actually is, rather than merely describing what it is not. Finally, we show that these propositions are not without interest in relation to the present-day economic sociology that tends to go by the name - made fasionable by Granovetter and Swedberg - of ‘New Economic Sociology.’

A new efficient heuristic for the minimum set covering problem

Abstract: We solve approximately the minimum set covering problem by developing a new heuristic, which is essentially based on the flow algorithm originally developed by Ford and Fulkerson. We perform a comparative study of the performances (concerning solution qualities and execution times) of the flow algorithm as well as of the natural greedy heuristic for set covering originally studied by Johnson and Lovasz.

Auxiliary variables for deformable models

Abstract: We present a mathematical formulation for curve and surface reconstruction algorithms by introduction of auxiliary variables. For deformable models and templates, two step iterative algorithms have been often used where, at each iteration, the model is first locally deformed according to the potential data attraction and then globally smoothed. We show how these approaches can be interpreted as the introduction of auxiliary variables and the minimization of a two variables energy. This permits us to transform an implicit data constraint defined by a non convex potential into an explicit convex reconstruction problem. We show some mathematical properties and results on this new auxiliary problem, in particular when the potential is a function of the distance to the closest feature point. We then illustrate our approach for some deformable models and templates and image restoration. >

Image restoration by constrained total variation minimization and variants

Abstract: We study a signal or image restoration method proposed by Rudin and Osher, namely, the constrained total variation (TV) minimization. This very powerful method gives excellent results on nearly piecewise constant signals, but fails on more complicated data. We propose a very general way to combine convex potentials like the TV, and in this setting we introduce a variant that performs better on piecewise regular--but non constant-- signals.

The myth of the income effect

Abstract: Microeconomics traditionally splits the consequences of price changes into two different effects: the substitution effect and the income effect.' The effects are supposed to work in diverging directions. The precise reaction of demand or supply to a relative price change can operate either way. The supply curve is not necessarily upward-sloping and the demand curve is not necessarily downward-sloping. The income effect plays a crucial role in mainstream microeconomics, as well as in applied economics, because i t induces economists to think, for instance, that there are backward bending curves. Such curves are particularly assumed to exist for the supply of labor. Within a certain range, a decrease in the real wage would not bring about a decrease in the supply of labor, since people want to maintain their income and, therefore, increase their supply of labor. If this is t rue in tax theory, the \"Laffer-effect\"' would not exist, a t least in certain conditions in which the supply curve for labor is supposed to be atypical. With the income effect, an increase in the tax rate on labor income-i.e., a decrease in the real after-tax wage-would be compensated for by working more. The \"Laffer-effect\" assumes that there is always an inverse relation between the tax rate and productive efforts. In a similar way, the income effect would

Newton and other continuation methods for multivalued inclusions

Abstract: Viability theory provides an efficient framework for looking for zeros of multivalued equations: 0 ∈F(x). The main idea is to consider solutions of a suitable differential inclusion, viable in graph ofF. The choice of the differential inclusion is guided necessarily by the fact that any solution should converge or go through a zero of the multivalued equation. We investigate here a new understanding of the well-known Newton's method, generalizing it to set-valued equations and set up a class of algorithms which involve generalization of some homotopic path algorithms.

On Some Recent Methods for Nonlinear Partial Differential Equations

Abstract: We wish to present here some aspects of a few general methods that have been introduced recently in order to solve nonlinear partial differential equations and related problems in nonlinear analysis.

Generalized sentinels defined via least squares

Abstract: We address the problem of monitoring a linear functional (c, x)Eof an unknown vectorx of a Hilbert spaceE, the available data being the observationz, in a Hilbert spaceF, of a vectorAx depending linearly onx through some known operatorAeℒ(E; F) WhenE=E 1×E 2,c=(c 1 0), andA is injective and defined through the solution of a partial differential equation, Lions ([6]–[8]) introduced sentinelsseF such that (s, Ax)Fis sensitive to x1 eE 1 but insensitive to x2 e E2 In this paper we prove the existence, in the general case, of (i) a generalized sentinel (s, σ) e ℱ ×E, where ℱ ⊃F withF dense in 80, such that for anya priori guess x0 ofx, we have 〈s, Ax〉ℱℱ + (σ, x0)E=(c, x)E, where x is the least-squares estimate ofx closest tox 0, and (ii) a family of regularized sentinels (s n , σ n ) e F×E which converge to (s, σ) Generalized sentinels unify the least-squares approach (by construction !) and the sentinel approach (whenA is injective), and provide a general framework for the construction of “sentinels with special sensitivity” in the sense of Lions [8])

Perspective invariant movie analysis for depth recovery

Abstract: Processing entire movies and taking advantage of the interframe redundancy is the key of shape-from-motion analysis. Thus, recovering the depth of a fixed scene from an image sequence can be viewed as a movie processing problem: how to focus the redundant depth information of a noisy image sequence into a perfect depth-coherent movie? We present a natural set of axioms in agreement with the depth recovery, in the simple case of a straight movement of the camera parallel to the focal plane. According to these axioms, we show that there is a unique depth-coherent way of processing movies, described by a nonlinear partial differential equation. The corresponding multiscale analysis has the property of smoothing the motion field of a movie, leading naturally to a perfect motion field compatible with a depth interpretation. Moreover, in the case of an ideal movie, i.e. coherent with the observation of a fixed 3D scene, this analysis can be viewed as a simple filtering of the camera movement preserving the depth interpretation given by the movie, and is thereby perspective invariant. Last, we study a numerical scheme, compatible with the theoretical axioms, and produce some experiments on synthetic noisy movies.