Plane poisson manifolds and finite-dimensional pseudogroups

Translated from Matematicheskie Zametki, Vol. 45, No. 3, pp. 53–65, March, 1989.     

The bitangential inverse spectral problem for canonical systems

The theory of the direct and bitangential inverse input impedance problem is used to solve the direct and bitangential inverse spectral problem. The analysis of the direct spectral problem uses and extends a number of results that appear in the literature. Special attention is paid to the class of canonical integral systems with matrizants that are strongly regular J-inner matrix valued functions in the sense introduced in [7]. The bitangential inverse spectral problem is solved in this class. In our considerations, the data for this inverse problem is a given nondecreasing p×p matrix valued function σ(μ) on and a normalized monotonic continuous chain of pairs , of entire inner p×p matrix valued functions. Each such chain defines a class of canonical integral systems in which we find a solution of the inverse problem for the given spectral function σ(μ). A detailed comparison of our investigations of inverse problems with those of Sakhnovich is presented.     

Holonomy on poisson manifolds and the modular class

(1) It is shown that ifc is real-valued measurable then the Maharam type of (c, P(c),σ) is 2 ^c . This answers a question of D. Fremlin [Fr, (P2f)]. (2) A different construction of a model with a real-valued measurable cardinal is given from that of R. Solovay [So]. This answers a question of D. Fremlin [Fr, (P1)]. (3) The forcing with aκ-complete ideal over a setX, |X| ≥κ cannot be isomorphic to Random × Cohen or Cohen × Random. The result forX=κ was proved in [Gi-Sh1] but, as was pointed out to us by M. Burke, the application of it in [Gi-Sh2] requires dealing with anyX. The application is: ifA _n is a set of reals forn<ω then for some pairwise disjointB _n (forn<ω) we haveB _n ⊆A _n but they have the same outer Lebesgue measure. Partially supported by the Israeli Basic Research Fund. Publ. Number 582.     

Analytic invariant manifolds for sequences of diffeomorphisms

We obtain real analytic invariant manifolds for trajectories of maps assuming only the existence of a nonuniform exponential behavior. We also consider the more general case of sequences of maps, which corresponds to a nonautonomous dynamics with discrete time. We emphasize that the maps that we consider are defined in a real Euclidean space, and thus, one is not able to obtain the invariant manifolds from a corresponding procedure to that in the nonuniform hyperbolicity theory in the context of holomorphic dynamics. We establish the existence both of stable (and unstable) manifolds and of center manifolds. As a byproduct of our approach we obtain an exponential control not only for the trajectories on the invariant manifolds, but also for all their derivatives.     

Exceptional sequences of eigenfunctions for hyperbolic manifolds

Examples are given of hyperbolic manifolds in every dimension at least five which support sequences of eigenfunctions for the Laplacian whose -norms grow as a power of the eigenvalue while their -norms are one.

     

Periodic manifolds with spectral gaps

We investigate spectral properties of the Laplace operator on a class of non-compact Riemannian manifolds. For a given number N we construct periodic manifolds such that the essential spectrum of the corresponding Laplacian has at least N open gaps. We use two different methods. First, we construct a periodic manifold starting from an infinite number of copies of a compact manifold, connected by small cylinders. In the second construction we begin with a periodic manifold which will be conformally deformed. In both constructions, a decoupling of the different period cells is responsible for the gaps.     

The Cartan model of the canonical vector bundles over the Grassmannian manifolds

We give a representation of the canonical vector bundles

Open image in new window

over the Grassmannian manifolds G(n, p) as noncompact symmetric affine spaces together with their Cartan model in the group of the Euclidean motions SE(n).

     

The singularity of the canonical model of compact Kähler manifolds

Partially supported by the Fûjukai foundation     

The spectral sequence of the canonical foliation of a Vaisman manifold

In this paper we investigate the spectral sequence associated with a Riemannian foliation which arises naturally on a Vaisman manifold. Using the Betti numbers of the underlying manifold we establish a lower bound for the dimension of some terms of this cohomological object. This way we obtain cohomological obstructions for two-dimensional foliations to be induced from a Vaisman structure. We show that if the foliation is quasi-regular the lower bound is realized. In the final part of the paper we discuss two examples.     

On the spectral flow in Lorentzian manifolds

In this paper we use functional analytical techniques to determine the differential equation satisfied by the eigenvalues of a smooth family of Fredholm operators, obtained from the index form along a Lorentzian geodesic. The formula is then applied to the study of the evolution of the index function, and, using a perturbation argument, we prove a version of the classical Morse index theorem for stationary Lorentzian manifolds. Received: January 31, 2000; in final form: March 13, 2002?Published online: February 20, 2003 The second author is partially sponsored by CNPq (Brazil), Grant 200615/01-7.     

Extreme value theory for dependent sequences via the stein-chen method of poisson approximation

In 1970 Stein introduced a new method for bounding the approximation error in central limit theory for dependent variables. This was subsequently developed by Chen for Poisson approximation and has proved very successful in the areas to which it has been applied. Here we show how the method can be applied to extreme value theory for dependent sequences, focussing particularly on the nonstationary case. The method gives new and shorter proofs of some known results, with explicit bounds for the approximation error.     

Piecewise linear spectral sequences

We study a class of orthonormal exponential bases for the space and introduce the concept of spectral sequences. We characterize piecewise linear spectral sequences with the knot at and investigate the non-continuity of the piecewise linear spectral sequences. From a special construction of a piecewise constant spectral sequence, the classical Walsh system is recovered.

     

The spectral property of dual matrix strings as a consequence of the existence of spectral matrix functions for canonical systems

The term dual string for scalar strings was introduced in [KK1], where some connections between the spectra of a string and its dual were studied. In [KK2] it was shown that if () is a spectral function of a scalar stringS ₁ with nonnegative spectrum (in the sense of [KK2]), then the function

Structure of exchangeable infinitely divisible sequences of poisson random vectors

De Finetti's Theorem reveals a simple explicit structure for an infinite exchangeable sequence of zero-one random variables. Although more general results are known, simple explicit results might be expected in particular settings. In this paper such results are obtained for exchangeable sequences of infinitely divisible Poisson random variables and random vectors. The methods employed are elementary, except in that they involve appeal to moment theorems.     

Subordinate solutions and spectral measures of canonical systems

The theory of 2×2 trace-normed canonical systems of differential equations on ?⁺ can be put in the framework of abstract extension theory, cf. [9]. This includes the theory of strings as developed by I.S. Kac and M.G. Kre?n. In the present paper the spectral properties of such canonical systems are characterized by means of subordinate solutions. The theory of subordinacy for Schrödinger operators on the halfline ?⁺, was originally developed D.J. Gilbert and D.B. Pearson. Its extension to the framework of canonical systems makes it possible to describe the spectral measure of any Nevanlinna function in terms of subordinate solutions of the corresponding trace-normed canonical system, which is uniquely determined by a result of L. de Branges.