Poincaré Duality for Logarithmic Crystalline Cohomology

We prove Poincaré duality for logarithmic crystalline cohomology of log smooth schemes whose underlying schemes are reduced. This is a generalization of the result of P. Berthelot for usual smooth schemes and that of O. Hyodo for the special fibers of semi-stable families and trivial coefficients.     

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.     

Extending Persistence Using Poincaré and Lefschetz Duality

Persistent homology has proven to be a useful tool in a variety of contexts, including the recognition and measurement of shape characteristics of surfaces in ℝ³. Persistence pairs homology classes that are born and die in a filtration of a topological space, but does not pair its actual homology classes. For the sublevelset filtration of a surface in ℝ³, persistence has been extended to a pairing of essential classes using Reeb graphs. In this paper, we give an algebraic formulation that extends persistence to essential homology for any filtered space, present an algorithm to calculate it, and describe how it aids our ability to recognize shape features for codimension 1 submanifolds of Euclidean space. The extension derives from Poincaré duality but generalizes to nonmanifold spaces. We prove stability for general triangulated spaces and duality as well as symmetry for triangulated manifolds. An erratum to this article can be found at     

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.     

The Auslander–Reiten Quiver of a Poincaré Duality Space

In a previous paper, Auslander–Reiten triangles and quivers were introduced into algebraic topology. This paper shows that over a Poincaré duality space, each component of the Auslander–Reiten quiver is isomorphic to . Presented by Yuri Drozd     

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     

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.     

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.     

Higher-Order Multilinear Poincaré and Sobolev Inequalities in Carnot Groups

The notions of higher-order weighted multilinear Poincaré and Sobolev inequalities in Carnot groups are introduced. As an application, weighted Leibniz-type rules in Campanato-Morrey spaces are established.     

Differentiable Connections with Poincaré Groups of Local Transformations. I

In the canonical smooth fiber bundles , we study generalized differentiable connections constructed by the author in his previous works. Special emphasis is laid on the investigation of the behavior of these connections under local transformations of the classical Poincaré groups and extented Poincaré groups canonically acting in the given connections. We found all firstorder nonholonomic affine, connections with the groups and of local transformations and also constructed classes of the corresponding invariant secondorder connections.     

Differentiable Connections with Poincaré Groups of Local Transformations. II

In the canonical smooth fiber bundles :ⁿ⁺¹ⁿ, we study generalized differentiable connections constructed by the author in his previous works. Special emphasis is laid on the investigation of the behavior of these connections under local transformations of the classical Poicaré (1,n) and extended Poincaré groups canonically acting in the given connections. We found all the firstorder nonholonomic affine, ₁, ₂, and _1,2connections with the groups (1,n) and of local transformations and also constructed classes of the corresponding invariant secondorder connections.     

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 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.     

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.     

The Poincaré Series of Relative Invariants of Finite Pseudo-Reflection Groups

Let F be a field with characteristic 0,V=F~n the n-dimensional vector space over F and let G be a finite pseudo-reflection group which acts on V.Let χ:G→F~* be a 1-dimensional representation of G.In this article we show that X(g)=(detg)~α(0≤α≤r-1),where g∈G and r is the order of g.In addition,we characterize the relation between the relative invariants and the invariants of the group G,and then we use Molien's Theorem of invariants to compute the Poincaré series of relative invariants.