Poincaré series

Let N_α denote the number of solutions to the congruence F(x_i,..., x_m) ≡ 0 (mod p^α) for a polynomial F(x_i,..., x_m) with integral p-adic coefficients. We examine the series \(\varphi (t) = \sum\nolimits_{\alpha = 0}^\infty {N_{\alpha ^{t^\alpha } } } \) . called the Poincaré series for the polynomial F. In this work we prove the rationality of the series ?(t) for a class of isometrically equivalent polynomials of m variables, m ≥ 2, containing the sum of two forms ?_n(x, y) + ?_n+1(x, y) respectively of degrees n and n+1, n ≥ 2. In particular the Poincaré series for any third degree polynomial F₃(x, y) (over the set of unknowns) with integral p-adic coefficients is a rational function of t.     

A Poincaré Cone Condition in the Poincaré Group

Ben Arous and Gradinaru (Potential Anal 8(3):217–258, 1998) described the singularity of the Green function of a general sub-elliptic diffusion. In this article we first adapt their proof to the more general context of a hypoelliptic diffusion. In a second time, we deduce a Wiener criterion and a Poincaré cone condition for a relativistic diffusion with values in the Poincaré group (i.e the group of affine direct isometries of the Minkowski space-time).     

Compatibilité d'un fibré de Dirac avec une connexion

We extend to an almost Dirac structure and affine connections the notion of compatibility of a bivectors field or a 2-differential form with a pseudo-metric. Compatibility with a symmetric connection implies integrability. We shall be interested especially by such structures on Lie groups or Riemannian homogeneous spaces.     

Partial Padé prediction

When the first terms of a sequence (called sequence to predict) are known, a prediction method is a method which gives us an approximation of the following terms (the so-constructed sequence is called the predicted sequence). In this paper, we state two prediction methods, respectively called _p-prediction and partial Padé prediction, which are generalizations of Aitken's ²-prediction of Brezinski and Redivo-Zaglia [6] and Padé prediction of Gilewicz [8] which are very simple to use. In order to choose among the different partial Padé predictions, we study some of their properties. The most important points of this paper are: the use of an extrapolation algorithm (the -algorithm), to obtain a prediction algorithm for each partial Padé prediction (which avoids solving a system); the results about consistency obtained for the partial Padé prediction (i.e., under certain conditions, each term of the predicted sequence converges to the analogous term of the sequence to predict).     

Convexité holomorphe intermediaire

The purpose of the present article is to give a generalisation of the notion of holomorphic convexity. We begin with an analogue of Remmert's reduction Theorem by showing that aq-holomorphically convex complex space is naturally endowed with a proper morphism on aq-complete space. We show after that theq-complete spaces arising in this situation are special: they enjoy in particular the property that each relatively compact open subset carries a Kähler form.

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Une grande partie de cet article a été conçue lors d'un sejour du premier auteur à l'Université de Rome La Sapienza; celui-ci tient à remercier le CNR et le Départment G. Castelnuovo     

p-Poincaré inequality versus ∞-Poincaré inequality: some counterexamples

We point out some of the differences between the consequences of p-Poincaré inequality and that of ∞-Poincaré inequality in the setting of doubling metric measure spaces. Based on the geometric characterization of ∞-Poincaré inequality given in Durand-Cartagena et al. (Mich Math J 60, 2011), we obtain a geometric property implied by the support of a p-Poincaré inequality, and demonstrate by examples that an analogous geometric characterization for finite p is not possible. The examples we give are metric measure spaces which are doubling and support an ∞-Poincaré inequality, but support no finite p-Poincaré inequality. In particular, these examples show that one cannot expect a self-improving property for ∞-Poincaré inequality in the spirit of Keith–Zhong (Ann Math 167(2):575–599, 2008). We also show that the persistence of Poincaré inequality under measured Gromov–Hausdorff limits fails for ∞-Poincaré inequality.     

Perversité et variation

Let S be the spectrum of a strictly henselian discrete valuation ring with residue characteristic p and =/, where is a prime number p and is an integer 1. For a scheme X of finite type over S and smooth over S along the special fiber X _s outside a closed point x, we study the vanishing cycles complex R() and the tame variation , for in the tame inertia group I _t . In particular, we show that if X is regular, flat over S of relative dimension n1, and is a topological generator of I _t , then R ^q () _x =0 for qn and is an isomorphism. Mathematics Subject Classification (2000):14F20, 14D05, 14D06     

Distributions of Poles to Painlevé Transcendents via Padé Approximations

A version of the Fair–Luke algorithm has been used to find the Padé approximate solutions to the Painlevé I, II, and IV equations. The distributions of poles in the complex plane are studied to check the dynamics of movable poles and the emergence of rational and truncated solutions, as well as various patterns formed by the poles. The high-order approximations allow us to check asymptotic expansions at infinity and estimate the range of asymptotic domains. The Coulomb gas interpretation of the pole ensembles is discussed in view of the patterns arising in Painlevé IV transcendents.     

The convergence of diagonal Padé approximants and the Padé Conjecture

Questions related to the convergence problem of diagonal Padé approximants are discussed. A central place is taken by the Padé Conjecture (also known as the Baker-Gammel-Wills Conjecture). Partial results concerning this conjecture are reviewed and weaker and more special versions of the conjecture are formulated and their plausibility is investigated. Great emphasis is given to the role of spurious poles of the approximants. A conjecture by Nuttall (1970) about the number and distribution of such poles is stated and its importance for the Padé Conjecture is analyzed.     

On the Neumann-Poincaré operator

Let be a rectifiable Jordan curve in the finite complex plane which is regular in the sense of Ahlfors and David. Denote by L _C ² () the space of all complex-valued functions on which are square integrable w.r. to the arc-length on . Let L ²() stand for the space of all real-valued functions in L _C ² () and put Since the Cauchy singular operator is bounded on L _C ² (), the Neumann-Poincaré operator C ₁ sending each h L ²() into , is bounded on L ²(). We show that the inclusion characterizes the circle in the class of all AD-regular Jordan curves .     

Matrix Painlevé II equations

Theoretical and Mathematical Physics - We use the Painlevé–Kovalevskaya test to find three matrix versions of the Painlevé II equation. We interpret all these equations as...     

Simultaneous partial Padé approximants

In this paper the concept of partial Padé approximation, introduced by Claude Brezinski, is generalised to the case of simultaneous rational approximation with common denominator. The use of information about known poles and/or zeros, can lead to approximants with a better numerical behaviour than in the case of ordinary simultaneous Padé approximation.     

The A-Stability of Methods with Padé and Generalized Padé Stability Functions

This paper surveys some stability results and suggests the use of order arrows as an alternative to order stars in studying questions about the possible A-stability of a numerical method. A discussion of the so-called Butcher–Chipman conjecture includes a proof of a partial result.     

Construction d'une cavité supraconductrice

Sans résumé     

Klystron stabilisé en phase

Summary In an earlier communication on High Resolution Microwave Spectroscopy²⁷) we described a spectrograph using a phase-stabilized klystron oscillator as a local oscillator in the detection system. The stabilizing system has a holding range of several megacycles which permits the sweeping of the reference crystal oscillator.     

Le degré de similarité d'une algèbre d'opérateurs

Let H be a Hilbert space and let A ⊂B(H) be a unital closed subalgebra (not necessarily self adjoint). We introduce the “degree of similarity” of A, denoted by d(A), which is a non-negative integer or possibly ∞, and we study its significance in connection with the notions of amenability for groups and operator algebras.