Poincaré polynomial of the moduli spaces of parabolic bundles

In this paper we use Weil conjectures (Deligne’s theorem) to calculate the Betti numbers of the moduli spaces of semi-stable parabolic bundles on a curve. The quasi parabolic analogue of the Siegel formula, together with the method of HarderNarasimhan filtration gives us a recursive formula for the Poincaré polynomials of the moduli. We solve the recursive formula by the method of Zagier, to give the Poincaré polynomial in a closed form. We also give explicit tables of Betti numbers in small rank, and genera.     

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 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 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.     

Poincaré series of Drinfeld type

Let K be the rational function field $\mathbb{F}_q (t)$ . We construct Poincaré series on the Bruhat-Tits tree of GL₂ over K _∞ and show that they generate the space of automorphic cusp forms of Drinfeld type.     

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.     

On action-angle coordinates and the Poincaré coordinates

This article is a review of two related classical topics of Hamiltonian systems and celestial mechanics. The first section deals with the existence and construction of action-angle coordinates, which we describe emphasizing the role of the natural adiabatic invariants “∮_γ p dq”. The second section is the construction and properties of the Poincaré coordinates in the Kepler problem, adapting the principles of the former section, in an attempt to use known first integrals more directly than Poincaré did.     

The radius of convergence of Poincaré series of loop spaces

LetR _S (resp.R _A) be the radius of convergence of the Poincaré series of a loop space (S) (resp. of the Betti-Poincaré series of a noetherian connected graded commutative algebraA over a field of characteristic zero).IfS is a finite 1-connected CW-complex, the rational homotopy Lie algebra ofS is finite dimensional if and only ifR _S-1. OtherwiseR _S<1.There is an easily computable upper bound (usually less than 1) forR _S ifS is formal or coformal.On the other handR _A=+ if and only ifA is a polynomial algebra andR _A=1 if and only ifA is a complete intersection (Golod and Gulliksen conjecture). OtherwiseR _A<1 and the sequence dim Tor _p ^H grows exponentially withp.     

On the Poincaré inequality for vector fields

We prove the Poincaré inequality for vector fields on the balls of the control distance by integrating along subunit paths. Our method requires that the balls are representable by means of suitable “controllable almost exponential maps”. Both authors were partially supported by the University of Bologna, funds for selected research topics.     

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 fractional Poincaré inequalities

We show that fractional (p, p)-Poincaré inequalities and even fractional Sobolev-Poincaré inequalities hold for bounded John domains, and especially for bounded Lipschitz domains. We also prove sharp fractional (1,p)-Poincaré inequalities for s-John domains.     

Modulus and the Poincaré inequality on metric measure spaces

The purpose of this paper is to develop the understanding of modulus and the Poincaré inequality, as defined on metric measure spaces. Various definitions for modulus and capacity are shown to coincide for general collections of metric measure spaces. Consequently, modulus is shown to be upper semi-continuous with respect to the limit of a sequence of curve families contained in a converging sequence of metric measure spaces. Moreover, several competing definitions for the Poincaré inequality are shown to coincide, if the underlying measure is doubling. One such characterization considers only continuous functions and their continuous upper gradients, and extends work of Heinonen and Koskela. Applications include showing that the p-Poincaré inequality (with a doubling measure), for p1, persists through to the limit of a sequence of converging pointed metric measure spaces — this extends results of Cheeger. A further application is the construction of new doubling measures in Euclidean space which admit a 1-Poincaré inequality. Mathematics Subject Classification (2000):31C15, 46E35.     

Poincaré polynomials of moduli spaces of stable bundles over curves

Given a curve over a finite field, we compute the number of stable bundles of not necessarily coprime rank and degree over it. We apply this result to compute the virtual Poincaré polynomials of the moduli spaces of stable bundles over a curve. A similar formula for the virtual Hodge polynomials and motives is conjectured.     

An extension of the Beckner’s type Poincaré inequality to convolution measures on abstract Wiener spaces

We generalize the Beckner’s type Poincaré inequality (Beckner, W. Proc. Amer. Math. Soc. (1989) 105:397–400) to a large class of probability measures on an abstract Wiener space of the form μ?ν, where μ is the reference Gaussian measure and ν is a probability measure satisfying a certain integrability condition. As the Beckner inequality interpolates between the Poincaré and logarithmic Sobolev inequalities, we utilize a family of products for functions which interpolates between the usual point-wise multiplication and the Wick product. Our approach is based on the positivity of a quadratic form involving Wick powers and integration with respect to those convolution measures. In addition, we prove that in the finite-dimensional case the class of densities of convolutions measures satisfies a point-wise covariance inequality.     

On the Poincaré Inequality for Infinitely Divisible Measures

We completely characterize the Poincaré inequality for bilinear forms of gradient type defined on L²-spaces w.r.t. infinitely divisible measures m in terms of the canonical measure associated with m. The characterization is based on an elementary algebraic observation concerning certain quadratic forms associated with m and , which is of its own interest (see Lemma 3.4). Examples include canonical Dirichlet forms on configuration spaces and Dirichlet forms associated to continuous state branching processes. As an application, a strong law of large numbers for time-inhomogeneous one-dimensional subordinators is obtained.Mathematics Subject Classifications (2000) 31C25 (60E07, 60G57, 60H07, 60J80).     

On a Poincaré type formula for solutions of singular and degenerate elliptic equations

We provide a geometric Poincaré type formula for stable solutions of ?Δ _p (u) = f(u). From this, we derive a symmetry result in the plane. This work is a refinement of previous results obtained by the authors under further integrability and regularity assumptions.     

On Poincaré bundles of vector bundles on curves

An arbitrary cubic function field can have 0, 1, or 2 for its unit rank. This paper presents the complete classification of unit rank of an arbitrary cubic function field by its discriminant and the polynomial discriminant of its generating polynomial. The notions of Kummer Theory and Cardanos formula are used.Mathematics Subject Classification (2000): 11R27, 11R16Acknowledgement The author expresses her gratitude to the referee for very helpful comments.