An inverse problem in population dynamics

The evolution of population densities of two interacting species in presence of diffusion phenomena is governed by a system of semilinear Volterra integrodifferential parabolic equations. In this system there are time convolution integrals, accounting for past history effects, which are essentially characterized by kernels depending on time only. These delay kernels can be viewed as entries of a 2x2 matrix K. The inverse problem of determining K via suitable population measurements is analyzed.     

Quantum Lefschetz hyperplane theorem

The mirror theorem is generalized to any smooth projective variety X. That is, a fundamental relation between the Gromov–Witten invariants of X and Gromov–Witten invariants of complete intersections Y in X is established. Oblatum 21-IV-2000 & 11-I-2001?Published online: 2 April 2001     

Stochastic cascades and 3-dimensional Navier–Stokes equations

Summary. In this article, we study the incompressible Navier–Stokes equations in ℝ³. The non linear integral equation satisfied by the Fourier transform of the Laplacian of the velocity field can be interpreted in terms of a branching process and a composition rule along the associated tree. We derive from this representation new classes where global existence and uniqueness can be proven. Received: 27 November 1996 / In revised form: 30 May 1997     

Uniqueness of Dirichlet forms associated with systems of infinitely many Brownian balls in ℝ d

Summary. Dirichlet forms associated with systems of infinitely many Brownian balls in ℝ ^d are studied. Introducing a linear operator L ₀ defined on a space of smooth local functions, we show the uniqueness of Dirichlet forms associated with self adjoint Markovian extensions of L ₀. We also discuss the ergodicity of the reversible process associated with the Dirichlet form. Received: 18 July 1996/In revised form: 13 February 1997     

Interpolatory quadrature formulas on the unit circle for Chebyshev weight functions

Summary. In this paper, interpolatory quadrature formulas based upon the roots of unity are studied for certain weight functions. Positivity of the coefficients in these formulas is deduced along with computable error estimations for analytic integrands. A comparison is made with Szeg? quadrature formulas. Finally, an application to the interval [-1,1] is also carried out. Received February 29, 2000 / Published online August 17, 2001     

A new proof of a theorem of Magidor

We give a new proof using iterated Prikry forcing of Magidor's theorem that it is consistent to assume that the least strongly compact cardinal is the least supercompact cardinal. Received: 8 December 1997 / Revised version: 12 November 1998     

Continuous version of Filippov’s theorem for fractional differential inclusions

Using Bressan-Colombo results, concerning the existence of continuous selections of lower semicontinuous multifunctions with decomposable values, we prove a continuous version of Filippov’s theorem for a fractional differential inclusion.     

An implementation of Vincent's theorem

Summary A new method is presented for the isolation of the real roots of a polynomial equation; it is based on Vincent's forgotten theorem of 1836 and has been implemented using exact (infinite precision) integer arithmetic algorithms. A theoretical analysis of the computing time of this method is given along with some empirical results.     

A Sard theorem for the distance function

In this article, we prove that the set of all critical values of the distance function from a submanifold of a complete Riemannian manifold whose dimension is less than 5 is of Lebesgue measure zero. Received November 26, 1999 / Published online April 12, 2001     

A new inductive approach to the lace expansion for self-avoiding walks

Summary. We introduce a new inductive approach to the lace expansion, and apply it to prove Gaussian behaviour for the weakly self-avoiding walk on ℤ ^d where loops of length m are penalised by a factor e ^{−β/m p} (0<β≪1) when: (1) d>4, p≥0; (2) d≤4, . In particular, we derive results first obtained by Brydges and Spencer (and revisited by other authors) for the case d>4, p=0. In addition, we prove a local central limit theorem, with the exception of the case d>4, p=0. Received: 29 October 1997 / In revised form: 15 January 1998     

Pointwise Remez- and Nikolskii-type inequalities for exponential sums

Let So is the collection of all n + 1 term exponential sums with constant first term. We prove the following two theorems. Theorem 1 (Remez-type inequality for $E_n$ at 0). Let $s \in \left( 0, \frac 12 \right]\,.$ There are absolute constants $c_1 > 0$ and $c_2 > 0$ such that where the supremum is taken for all $f \in E_n$ satisfying Theorem 2 (Nikolskii-type inequality for $E_n$ ). There are absolute constants $c_1 > 0$ and $c_2 > 0$ such that for every $a < y < b$ and $q > 0\,.$ It is quite remarkable that, in the above Remez- and Nikolskii-type inequalities, behaves like , where denotes the collection of all algebraic polynomials of degree at most n with real coefficients. Received: 4 November 1998 / in final form: 2 March 1999     

Sur l'erreur d'approximation par éléments finis en utilisant la méthode des plaquettes splines

Résumé. On établit des majorations de l'erreur d'approximation par éléments finis à partir de données de Lagrange pour des fonctions appartenant à un espace de Sobolev d'ordre convenable, lorsque les degrés de liberté sont approchés à l'aide de la méthode des plaquettes splines introduite par A. Le Méhauté (cf. [13], [14], [15]). Les résultats obtenus s'appliquent notamment à la construction de surfaces de classe . Received May 29, 1995 / Revised version received August 20, 1995     

Boolean Functions With Low Average Sensitivity Depend On Few Coordinates

. The sensitivity of a point is dist, i.e. the number of neighbors of the point in the discrete cube on which the value of differs. The average sensitivity of is the average of the sensitivity of all points in . (This can also be interpreted as the sum of the influences of the variables on , or as a measure of the edge boundary of the set which is the characteristic function of.) We show here that if the average sensitivity of is then can be approximated by a function depending on coordinates where is a constant depending only on the accuracy of the approximation but not on . We also present a more general version of this theorem, where the sensitivity is measured with respect to a product measure which is not the uniform measure on the cube. Received: November 12, 1996