Some topological properties of ω-covering sets

We prove the following theorems:1. There exists an -covering with the property s ₀.2. Under cov there exists X such that is not an -covering orX \ B is not an -covering].3. Also we characterize the property of being an -covering.     

Two consistency results concerning thin-tall Boolean algebras

In this paper we show that the following is relatively consistent withZFC +CH: There is no superatomic Boolean algebra of height ₂+1 and width, and there is no superatomic Boolean algebraA with for 0<<₁ and Presented by J. Mycielski.     

An application of Shoenfield's absoluteness theorem to the theory of uniform distribution

IfC is a Polish probability space, a Borel set whose sectionsW _x ( have measure one and are decreasing , then we show that the set _x W _x has measure one. We give two proofs of this theorem—one in the language of set theory, the other in the language of probability theory, and we apply the theorem to a question on completely uniformly distributed sequences.Supported by DFG grant Ko 490/7-1.     

Über die Diskrepanz von Folgen in bewerteten Körpern

Let k be a field which is locally compact with respect to a valuation ||, let X denote the subset X={ k, ||1}. This paper is concerned with sequences of elements of X with respect to the theory of uniform distribution. Let D_N() denote the discrepancy of the sequence =(x_n). It is proved that for archimedian valuations with a certain constant for every sequence . In the non-archimedian case there exist sequences with .     

Equivalence Bimodule Between Non-Commutative Tori

The non-commutative torus C ^*(ⁿ,) is realized as the C*-algebra of sections of a locally trivial C*-algebra bundle over S with fibres isomorphic to C ^*n/S, ₁) for a totally skew multiplier ₁ on ⁿ/S. D. Poguntke [9] proved that A is stably isomorphic to C(S) C(^*( Zⁿ/S, ₁) C(S) A M_kl( C) for a simple non-commutative torus A and an integer kl. It is well-known that a stable isomorphism of two separable C*-algebras is equivalent to the existence of equivalence bimodule between them. We construct an A-C(S) A-equivalence bimodule.     

On local automorphisms of certain quadrics of codimension 2

The article considers nondegenerate quadrics in Cⁿ⁺¹ with codimension 2 that are of the form M={zCⁿ, C²: Im_j=z, z_j; j=1, 2}, where are Hermitian forms, and thje stability groups Aut_x M that preserve the point x. It is proved that if the matrix ¹ is stable and the matrix (¹)^–12 has more than two different eigenvalues, all automorphisms of Aut_x M are linear transformations.Translated from Matematicheskie Zametki, Vol. 52, No. 1, pp. 9–14, July, 1992.     

Pinning effect in peierls impurity systems with deviation from half-filling of the energy band

A study is made of the effect of deviation from half-filling of the energy band (0) on the Fröhlich collective mode in onedimensional impurity systems. A low impurity concentration is considered, and the infinite series of impurity scattering is taken into account self-consistently in the determination of the collective mode Green's function. The conductivity () is found in terms of this Green's function, and an analytic expression is obtained for () at _T ( _T is the pinning frequency). It is shown that for the ratio Re(()/_max) a universal formula arises. It differs from the results of Kurihara in the expression for _T , which contains an essential dependence on in the incommensurate state of the charge density wave. It is also shown that the width of the peak in the dependence () and its position increase with increasing .Institute of Applied Physics, Moldovan Academy of Sciences. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 101, No. 1, pp. 110–122, October, 1994.     

Wiener-Hopf operators on L_{\omega}^{2}(\mathbb{R}^{+})

Let be a weighted space with weight . In this paper we show that for every Wiener-Hopf operator T on and for every a I, there exists a function such that

The differences between Kurepa trees and Jech-Kunen trees

Summary By an ₁ we mean a tree of power ₁ and height ₁. An ₁-tree is called a Kurepa tree if all its levels are countable and it has more than ₁ branches. An ₁-tree is called a Jech-Kunen tree if it has branches for some strictly between ₁ and . In Sect. 1, we construct a model ofCH plus , in which there exists a Kurepa tree with not Jech-Kunen subtrees and there exists a Jech-Kunen tree with no Kurepa subtrees. This improves two results in [Ji1] by not only eliminating the large cardinal assumption for [Ji1, Theorem 2] but also handling two consistency proofs of [Ji1, Theorem 2 and Theorem 3] simultaneously. In Sect. 2, we first prove a lemma saying that anAxiom A focing of size ₁ over Silver's model will not produce a Kurepa tree in the extension, and then we apply this lemma to prove that, in the model constructed for Theorem 2 in [Ji1], there exists a Jech-Kunen tree and there are no Kurepa trees.     

Parametrized Partitions of Products of Finite Sets*

For every infinite sequence of positive integers and every Borel partition c : ×[]{0, 1} there is H[] and a sequence of subsets of , with |H_i|=m_i for every i, such that c is constant on .* Research partially supported by CNRS-FONACIT Project PI 2000001471. This author thanks the University of Paris VII for hospitality.     

Universal property of chern character forms of the canonical connection

Let _q,n denote the complex Stiefel bundle over the complex Grassmannian and let ₀ be the universal connection on this bundle. Consider the Chern character form of ₀ defined by the formula where ₀ is the curvature form of the connection ₀. Let M be a manifold of dimension m and a closed 2k-form on M. Suppose, there exists a continuous map which pulls back the cohomology class of ch_k(₀) onto the cohomology class of . We prove that if q and n are greater than certain numbers (which we determine in this paper) then there exists a smooth map such that f*ch_k(₀) = .     

The conformal structure of Riemann surfaces with boundary parametrizing minimal surfaces

Let F denote a surface with boundary F, being contained in a Riemann surface R, such that R\F is somedisk. If we vary the boiundary curve _o parametrizing F, we will get a manifold of real dimension 6g–3, such that any bounds some F and any local deformation of F is conformally equivalent to just one F for .This result also implies that none of the conformal invariants of R will be an invariant of this F, since its neighbors {F|} cover all possible deformations of F at all.     

On the best approximation in the metric of L to certain classes of functions by Haar-system polynomials

Let H, H ^L be classes of functionsf(x) whose modulus of continuity (f; t) and, respectively, integral modulus of continuity(f; t)_L do not exceed a given modulus of continuity(t), while H_v is a class of functionsf(x) whose variation fdoes not exceed a given number V > 0. Bounds are obtained for the upper limit of the best approximations in the metric of L by Haar-system polynomials on the classes just introduced (on the class H ^L only when (t)=Kt). These bounds are exact for class H_V and, in case(t) is convex, also for the classes H and H ^L .Translated from Matematicheskie Zametki, Vol. 6, No. 1, pp. 47–54, July, 1969.The author wishes to thank N. P. Korneichuk for having posed the problem and for his constant attention to this work.     

Sur les Solutions D'un Opérateur Différentiel Singulier Semi-Linéaire

This paper deals with the following Dirichlet problem Lu = 1A ( Au – qu = – f ( , u ) on ] 0, [ , u, ( 0 ) = 0, u ( ) = 0, where ] 0, + ], q 0 is continuous on [ 0, [ × ] 0, + [ ] 0, + [ is continuous and A satisfies some appropriate conditions. The main result is the existence and the uniqueness of a strictly positive regular solution of the problem ( ). Moreover, we study the behaviour of this solution in a neighbourhood of . Our approach is based on the use of the Green's function of the homogeneous equation and Schauder's fixed point theorem.     

Estimating non-linear functions of the spectral density,using a data-taper

Letf() be the spectral density of a Gaussian stationary process. Consider periodogram-based estimators of integrals of certain non-linear functions off(), like , where () is a bounded function of bounded variation, possibly depending on the sample sizeT. Then it is known that, under mild conditions on , a central limit theorem holds for these statisticsH _T if the non-tapered periodogramI _T() is used. In particular, Taniguchi (1980,J. Appl. Probab.,17, 73–83) gave a consistent and asymptotic normal estimator of , choosing to be a suitable transform of a given function . In this work we shall generalize this result to statisticsH _T where a taper-modified periodogram is used. We apply our result to the use of data-tapers in nonparametric peak-insensitive spectrum estimation. This was introduced in von Sachs (1994,J. Time Ser. Anal.,15, 429–452) where the performance of this estimator was shown to be substantially improved by using a taper.This work has been supported by the Deutsche Forschungsgemeinschaft.     

On Periodic Solutions of Nonlinear Functional Differential Equations

Sufficient conditions are established for the existence and uniqueness of an -periodic solution of the functional differential equation where f is a continuous operator acting from the space of n-dimensional -periodic continuous vector functions into the space of n-dimensional -periodic and summable on [0,] vector functions.     

Linear response of a viscous liquid sheet to external pressure perturbation. Long wave approximations

This paper is concerned with the linear signal response analysis of a thin viscous liquid sheet which is at rest in an appropriate frame of reference and in contact with passive external media, and on which localized external pressures act from the passive media as sources of perturbation.The frame of the analysis is provided by general formulae for the response signals of the sheet in the two excitation modes, sinuous and varicose, which result as the solution of the appropriate fluid dynamic initial-boundary value problems by the Fourier-Laplace transform technique. These formulae display how the signals depend on the nature of the perturbation and on the spectrum of the (linear) eigenmodes of the sheet.The signals can be evaluated either numerically or, as initiated in this paper, analytically, in long wave approximations. The long wave approximation will be seen in the sequel to concentrate on particular eigenmodes of the sheet spectrum, with small values k of the wave vector along the sheet. The present paper is devoted mainly to a detailed analysis of the spectrum of eigenmodes of the sheet, and to the formulation of long wave approximations of (linear) response signals of the sheet in the context of this analysis.It turns out that the sheet spectrum of eigenmodes depends on only one characteristic number , which depends on the (positive) fluid parameters: the surface tension at (both) the interfaces of the sheet, the density of the fluid, its kinematic viscosity , and on the thickness h of the sheet: This number is closely related, by to the Ohnesorge number where is the dynamic viscosity of the liquid.It will be shown in the sequel that of the infinitely many branches of the sheet spectrum only two (pairs of) soft branches, one sinuous and one varicose, will be relevant for a long wave approximation. For these branches asymptotic expansions of the dispersion relations (k) between the (complex) mode frequencies and for long waves (i.e. for k  0), which obey (k)  0 with k  0, will be derived.The analytic long wave dispersion relations for the (soft) sinuous and varicose spectral branches allow a very favourable insight into the qualitative mode behaviour, including analogies beyond liquid sheets. They show e.g. that to their lowest orders in k the soft sinuous modes are nondispersive, i.e. and only weakly absorptive, i.e.   k² for low viscosity and   k⁴ for high viscosity ( < 0). Their signals are therefore expected to have some resemblance to signals of a flexible membrane on the one side and those of a (2-dimensional) diffusion process, or a hyperdiffusion process where is replaced by The behaviour of the soft varicose modes will be seen to depend on the value of : For ² < 4 the asymptotic expansion for k  0 gives

Differentiability of the minimal average action as a function of the rotation number

Let be a finite composition of exact twist diffeomorphisms. For any real number , letA() denote the minimal average action of -invariant measures with angular rotation number . We prove thatA() is differentiable at every irrational number and that for generic it is not differentiable at rational , thus verifying conjectures of S. Aubry. Moreover, we show that these results are valid for a variational principleh which satisfies the condition which we have called elsewhere (H). As a consequence, we generalize a result due to Bangert concerning geodesics on a two dimensional torus with an arbitrary, but sufficiently smooth metric.supported by NSF grant no. DMS-8806067.01 and a Guggenheim Fellowship.     

Divisor Functions in the Ring of Gaussian Integers Weighted by the Generalized Klosterman Sum

Let _a,b,c () be the number of factorizations of a Gaussian number in the form = ₁ ^a ₂ ^b ₃ ^c , where a, b, and c are natural numbers. In the ring of Gaussian numbers, we construct an asymptotic formula for a summatory function of _a,b,c () weighted by the generalized Klosterman function.     

Random fixed point theorems for a certain class of mappings in banach spaces

Let (, ) be a measurable space and C a nonempty bounded closed convex separable subset of p-uniformly convex Banach space E for some p > 1. We prove random fixed point theorems for a class of mappings T: × C C satisfying: for each x, y C, and integer n 1,