On the Positivity of Scattering Operators for Poincaré-Einstein Manifolds

In this paper, we mainly study the scattering operators for a Poincaré-Einstein manifold $(X^{n+1}, g_+)$, which define the fractional GJMS operators $P_{2\gamma}$ of order $2\gamma$ for $0<\gamma<\frac{n}{2}$ for the conformal infinity $(M, [g])$. We generalise Guillarmou-Qing's positivity results in [8] to the higher order case. Namely, if $(X^{n+1}, g_+)$ $(n\geq 5)$ is a hyperbolic Poincaré-Einstein manifold and there exists a smooth representative $g$ for the conformal infinity such that the scalar curvature $R_g$ is a positive constant and $Q_4$ is semi-positive on $(M, g)$, then $P_{2\gamma}$ is positive for $\gamma\in [1,2]$ and the first real scattering pole is less than $\frac{n}{2}-2$.     

On the interlacing of the zeros of Poincaré series

The Ramanujan Journal - Rankin proved that the Poincaré series for $$mathbf{SL}(2,{{mathbb {Z}}})$$ that are not cusp forms have all their zeros on the unit circle in the standard...     

On the homotopy type of Poincaré spaces

We study the homotopy type of finite-oriented Poincaré spaces (and, in particular, of closed topological manifolds) in even dimension. Our results relate polarized homotopy types over a stage of the Postnikov tower with the concept of CW-tower of categories due to Baues. This fact allows us to obtain a new formula for the top-dimensional obstruction for extending maps to homotopy equivalences. Then we complete the paper with an algebraic characterization of high-dimensional handlebodies. Received: April 14, 1999?Published online: October 2, 2001     

Prix Bolyai. avec un rapport par Henri Poincaré

Conclusions Après cet exposé, un long commentaire serait inutile. On voit quelle a été la variété des recherches deM. Hilbert, l'importance des problèmes auxquelles il s'est attaqué. Nous signalerons l'élégance et la simplicité des méthodes, la clarté de l'exposition, le souci de l'absolue riguer. En cherchant à être parfaitement rigoureux, on risque parfois d'être long, et ce n'est pas là acheter trop cher une correction sans laquelle les mathématiques ne seraient rien. MaisM. Hilbert a su éviter ce que ces longueurs auraient pu avoir d'un peu pénible pour ses lecteurs, en ne leur laissant jamais perdre de vue le fil conducteur qui lui a servi à s'orienter. On voit toujours aisément par quel encha?nement d'idées il a été amené à se poser un problème et à en trouver la solution. On sent que, plus analyste que géomètre au sensordinaire du mot, il a néanmoins aper?u l'ensemble de son travail d'un coup d'œil, avant d'en distinguer les détails, et il sait faire profiter le lecteur de cette vue d'ensemble. M. Hilbert a exercé une influence considérable sur les progrès récents des sciences mathématiques, non seulement par ses travaux personnels, mais par son enseignement, par les conseils qu'il donait à ses élèves et qui leur permettaient de contribuer à leur tour à ce développement de nos connaissances en se servant des méthodes créées par leur ma?tre. Il n'est pas besion, ce semble, d'en dire davantage pour justifier le choix de la Commission qui a été unanime à attribuer àM. Hilbert le prix Bolyai pour la période 1905–1909.     

On extension and refinement of the Poincaré inequality

The aim of this paper is to analyze the heat semigroup ${(\mathcal{N}_{t})_{t >0 } = \{e^{t \Delta}\}_{t >0 }}$ generated by the usual Laplacian operator Δ on ${\mathbb{R}^{d}}$ equipped with the d-dimensional Lebesgue measure. We obtain and study, via a method involving some semigroup techniques, a large family of functional inequalities that does not exist in the literature and with the local Poincaré and reverse local Poincaré inequalities as particular cases. As a consequence, we establish in parallel a new functional and integral inequality related to the Ornstein–Uhlenbeck semigroup.     

On the Uniform Poincaré Inequality

We give a proof of the Poincaré inequality in W ^{1, p} (Ω) with a constant that is independent of Ω ? , where  is a set of uniformly bounded and uniformly Lipschitz domains in ? ⁿ . As a byproduct, we obtain the following: The first non vanishing eigenvalues λ₂(Ω) of the standard Neumann (variational) boundary value problem on Ω for the Laplace operator are bounded below by a positive constant if the domains Ω vary and remain uniformly bounded and uniformly Lipschitz regular.     

On the Denominator of the Poincaré Series for Monomial Quotient Rings

Let S = 𝕜 [x ₁,…, x _n ] be a polynomial ring over a field 𝕜 and I a monomial ideal of S. It is well known that the Poincaré series of 𝕜 over S/I is rational. We describe the coefficients of the denominator of the series and study the multigraded homotopy Lie algebra of S/I.     

On the direct proof of the Poincaré theorem on invariant tori

We present the direct proof of the Poincaré theorem on invariant tori.     

Poincaré's recurrence theorem and the unitarity of the S-matrix

A scattering process can be described by suitably closing the system and considering the first return map from the entrance onto itself. This scattering map may be singular and discontinuous, but it will be measured preserving as a consequence of the recurrence theorem applied to any region of a simpler map. In the case of a billiard this is the Birkhoff map. The semiclassical quantization of the Birkhoff map can be subdivided into an entrance and a repeller. The construction of a scattering operator then follows in exact analogy to the classical process. Generically, the approximate unitarity of the semiclassical Birkhoff map is inherited by the S-matrix, even for highly resonant scattering where direct quantization of the scattering map breaks down.     

On Approximate Summation of Poincaré Series in the Schottky Model

Doklady Mathematics - For approximate summation of Poincaré theta series in the Schottky model of real hyperelliptic curves, modifications of the Bogatyrev and Schmies algorithms are proposed...     

On the Existence of Solutions of Poisson Equation and Poincaré–Lelong Equation

One of the main goals of this paper is to solve the Poincaré–Lelong equation on a class of Kähler manifolds with nonnegative holomorphic bisectional curvature, $\mathrm{Ric}(x)\geq \left(a\ln\ln\left(10+r(x)\right)\right)\Big/\big.\left(\left(1+r^2(x)\right)\ln(10+r(x))\right)One of the main goals of this paper is to solve the Poincaré–Lelong equation on a class of K?hler manifolds with nonnegative holomorphic bisectional curvature, for some a > 67(n + 4)². We will also study the Poisson equation on complete noncompact manifolds which satisfy volume doubling and Poincaré inequality.     

On the Structure of Bézier Nets

The distance between two neighbouring multivariate Bézier nets is proved to be $O(m^{-2})$ in this paper. As a consequence, the sequence of Bézier nets is uniformly convergent with the optimal approximation order $O(m^{-1})$. Furthermore, the structures of Bézier nets are explored by investigating how the piecewise linear surface tends to the Bézier surface of $C^{\infty}$.