A Remark on the Steklov–Poincaré Inequality

Mathematical Notes - In an $$n$$ -dimensional bounded domain $$\Omega_n$$ , $$n\ge 2$$ , we prove the Steklov–Poincaré inequality with the best constant in the case where $$\Omega_n$$ is...     

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 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).     

Integration over the Space of Functions and Poincaré Series Revisited

Earlier (2000) the authors introduced the notion of the integral with respect to the Euler characteristic over the space of germs of functions on a variety and over its projectivization. This notion allowed the authors to rewrite known definitions and statements in new terms and also turned out to be an effective tool for computing the Poincar´e series of multi-index filtrations in some situations. However, the “classical” (initial) notion can be applied only to multi-index filtrations defined by so-called finitely determined valuations (or order functions). Here we introduce a modified version of the notion of the integral with respect to the Euler characteristic over the projectivization of the space of function germs. This version can be applied in a number of settings where the “classical approach” does not work. We give examples of the application of this concept to definitions and computations of the Poincar´e series (including equivariant ones) of collections of plane valuations which contain valuations not centred at the origin.     

An Analog of the Poincaré Model for a Quaternion Hyperbolic Space

We construct an analog of the Poincaré model for a quaternion hyperbolic space.     

On the Estimation of Parameters of Poisson Processes

Let(N(t),t≥0)be a poisson process on probability spaces,i.e.(N(t),t≥0)is stochastic process with independent increments under P_θ and satisfies (a)N(0)=0 a.s.P_θ, (b)For all.     

Ruelle''''s Inequality for the Entropy of Diffusion Processes

In this paper we prove the Ruelle's inequality for the entropy and Lyapunovexponents of diffusion processes, that is, suppose h_μ,λ(i)(x),m_i (x) are respectivelythe entropy, Lyapunov exponents and its multiplicity for the random diffeomorphismsarising from SDE:     

On the direct proof of the Poincaré theorem on invariant tori

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

Spectral structure of the Neumann–Poincaré operator on tori

We address the question whether there is a three-dimensional bounded domain such that the Neumann–Poincaré operator defined on its boundary has infinitely many negative eigenvalues. It is proved in this paper that tori have such a property. It is done by decomposing the Neumann–Poincaré operator on tori into infinitely many self-adjoint compact operators on a Hilbert space defined on the circle using the toroidal coordinate system and the Fourier basis, and then by proving that the numerical range of infinitely many operators in the decomposition has both positive and negative values.     

Multiple periodic points of the Poincaré map of Lagrangian systems on tori

In this paper, suppose , A is positive definite and symmetric, and both A and V are and 1-periodic in all of their variables. We prove that the Poincaré map (i.e. the time-1-solution map) of the Lagrangian system possesses infinitely many periodic points on produced by contractible integer periodic solutions. Received July 23, 1997; in final form December 17, 1998     

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...     

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     

A problem of Kreck on Poincaré manifolds

An 1-connected closed manifold M is called a Poincaré manifold if any other 1-connected closed manifold with the same homology as that of M is homeomorphic to M. In the metastable range 3pq<2p–3 we answer the question raised by Kreck : for which p and q is the product of two spheres S ^p ×S ^q a Poincaré manifold ?. The second named author is partially supported by the 973 program of China.     

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).     

The Pathwise Connected Components of Path Space on Manifold M_n

Let Ω_(A_1)~(A_2) be the path space consists of all piecewise C~∞-path joining point pair A_1≠A_2 on a n-dimensional connected differential manifold M_n. We know that the connectivity numbers of Ω_(A_1)~(A_2) are invariants of M_n. In other words,these quantities are independent of the point pair A_1≠A_2 on M_n. Ω_(A_1)~(A_2) is not pathwise connected in general. So we can divide Ω_(A_1)~(A_2) to several pathwise connected components. Let C_(A_1)~(A_2)(h) be the pathwise connected component involving the piecewise C~∞-path h.     

Uniformization and the Poincaré metric on the leaves of a foliation by curves

In this paper we prove that a holomorphic foliation by curves, on a complex compact manifoldM, whose singularities are non degenerated and whose tangent line bundle admits a metric of negative curvature, satisfies the following properties:(a): All leaves are hyperbolic.(b): The Poincaré metric on the leaves is continuous.(c): The set of uniformizations of the leaves by the Poincaré disc D is normal. Moreover, if ( _n ) _n 1 is a sequence of uniformizations which converges to a map : D, then either is a constant map (a singularity), or is an uniformization of some leaf. This result generalizes Theorem B of [LN], in which we prove the same facts for foliations of degree 2 on projective spaces.This research was partially supported by Pronex-Dynamical Systems, FINEP-CNPq.     

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 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.     

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.     

Global bifurcation of fixed points and the Poincaré translation operator on manifolds

Let M be a differentiable manifold and [0, )×MM be a C¹ map satisfying the condition (0, p)=p for all pM. Among other results, we prove that when the degree (also called Hopf index or Euler characteristic) of the tangent vector field wMTM, given by w(p)=(/)(0, p), is well defined and nonzero, then the set (of nontrivial pairs) admits a connected subset whose closure is not compact and meets the slice {0}×M of [0, )×M. This extends known results regarding the existence of harmonic solutions of periodic ordinary differential equations on manifolds.