Outer functions from the analytic O. V. Besov classes

For the case , we completely describe the set of outer functions, in the sense of the Nevanlinna factorization, for the analytic O. V. Besov classes AB _pq ^r+α . Corollaries that are of the type of are obtained. Bibliography:14 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 217, 1994, pp. 172–217.     

Geodesic inversion and Sobolev spaces on Heisenberg type groups

Let σ be the geodesic inversion on a Heisenberg type groupN with homogeneous dimensionQ, and denote byS the jacobian of σ. We prove that, for , the operators are bounded on certain homogeneous Sobolev spaces if and only ifN is an IwasawaN-group. The research for this paper was carried out at the University of New South Wales. The authors thank the Italian G.N.A.M.P.A., the Progetto Cofinanziato M.U.R.S.T. “Analisi Armonica”, and the School of Mathematics of the University of New South Wales for their support.     

Precise asymptotics in the Baum-Katz and davis law of large numbers for positively associated sequences

§ 1　 Introduction and resultsL et { X,Xi;i≥ 1} be a sequence of i.i.d.random variables,and set Sn= ni=1 Xi,n≥1.Hsu and Robbins[1 ] introduced the conceptof complete convergence.They together withErdos[2 ] proved n≥ 1 P(|Sn|≥εn) <∞ ,ε>0 (1)if and only if EX=0 and EX2 <∞ .L ater,Spitzer[3] proved n≥ 11n P(|Sn|≥εn) <∞ ,ε>0if and only if EX =0 and E|X|<∞ .More generally,it was shown by Baum and Katz[4 ]that,for 0 0 (…     

On representations of Ψ* andC *-algebras ofb-pseudo-differential operators on manifolds with corners

The author computes the Jacobson topology and the corresponding Borel structure on the spectrum of . Here denotes the C^*-algebra generated by the algebra of b-pseudo-differential operators of orderO on a compact manifold with corners (in the sense of Melrose). The author also reduces the problem of classification of representations to the problem of classification of positive Borel measures on certain well-known Borel spaces. Bibliography:29 titles. Translated fromProblemy Matematicheskogo Analiza, No. 18, 1998, pp. 85–117.     

The solutions of the<Emphasis Type="Italic">n</Emphasis>-dimensional Bessel diamond operator and the Fourier-Bessel transform of their convolution

In this article, the operator is introduced and named as the Bessel diamond operator iteratedk times and is defined by

Toeplitz operators related to strongly singular Calderón-Zygmund operators

In this paper, the boundedness of Toeplitz operator T _b(f) related to strongly singular Calderón-Zygmund operators and Lipschitz function b ε (ℝⁿ) is discussed from L ^p(ℝⁿ) to L ^q(ℝⁿ), , and from L ^p(ℝⁿ) to Triebel-Lizorkin space . We also obtain the boundedness of generalized Toeplitz operator Θ _α0 ^b from L ^p(ℝⁿ) to L ^q(ℝⁿ), . All the above results include the corresponding boundedness of commutators. Moreover, the boundedness of Toeplitz operator T _b(f) related to strongly singular Calderón-Zygmund operators and BMO function b is discussed on L ^p(ℝⁿ), 1 < p < ∞.     

Equivariant D-modules attached to nilpotent orbits in a semisimple Lie algebra

Let _u be a compact Lie algebra and let _u be its complexification. Let ζ^−1/2 be the inverse on the set of regular elements of _u of a square root of the discriminant of . Generalizing a result of W. Lichtenstein in the case _u = (n, ℂ) or (nℝ), we prove that ∂(q).ζ^1/2 is non zero for all harmonic polynomialsq ∈S( ) \ {0}. This fact is deduced from results about equivariantD-modules supported on the nilpotent cone of .     

Integrals involving products of bessel functions

Summary The integrals and , where n is any positive integer, are evaluated in terms ofMacRobert E-functions and generalized hypergeometric functions.     

On the existence of components of the Noether-Lefschetz locus with given codimension

It is known that the codimension c of a component of the Noether-Lefschetz locus NL(d) satisfies . We prove that ford≥47 and for every integer there exists a component of NL(d) with codimension c. This is done with families of surfaces of degree d in ℙ³ containing a curve lying on a cubic or on a quartic surface or a curve with general moduli. Moreover we produce an explicit example, for everyd≥4, of components of maximum codimension , thus giving a new proof of the fact that these components are dense in the locus of smooth surfaces (density theorem).     

Diffusion times and stability exponents for nearly integrable analytic systems

For a positive integer n and R>0, we set . Given R>1 and n≥4 we construct a sequence of analytic perturbations (H _j ) of the completely integrable Hamiltonian on , with unstable orbits for which we can estimate the time of drift in the action space. These functions H _j are analytic on a fixed complex neighborhood V of , and setting the time of drift of these orbits is smaller than (C(1/ɛ _j )^1/2(n-3)) for a fixed constant c>0. Our unstable orbits stay close to a doubly resonant surface, the result is therefore almost optimal since the stability exponent for such orbits is 1/2(n−2). An analogous result for Hamiltonian diffeomorphisms is also proved. Two main ingredients are used in order to deal with the analytic setting: a version of Sternberg's conjugacy theorem in a neighborhood of a normally hyperbolic manifold in a symplectic system, for which we give a complete (and seemingly new) proof; and Easton windowing method that allow us to approximately localize the wandering orbits and estimate their speed of drift.     

The true order of magnitude of Lamé rounding error sums

Let (z ∈ ℝ). Further let λ denote a large real parameter. We show that for arbitrary real numbersk and α withk>=2.7013 and 0<α≦1,

Asymptotic properties of the maximum likelihood estimate in the first order autoregressive process

Summary In this paper we obtain an asymptotic expansion of the distribution of the maximum likelihood estimate (MLE) based onT observations from the first order Gaussian process up to the term of orderT ⁻¹. The expansion is used to compare with a generalized estimate including the least square estimate (LSE) , based on the asymptotic probabilities around the true value of the estimates up to the terms of orderT ⁻¹. It is shown that (or the modified MLE ) is better than (or the modified estimate ). Further, we note that does not attain the bound for third order asymptotic median unbiased estimates.     

Continuity and differentiability of Nemytskii operators on the Hardy space\mathcal{H}^{1,1} (T^1 )

Let denote the Hardy space of real-valued functions on the unit circle with weak derivatives in the usual real Hardy space . It is shown that when the weak derivative of a locally Lipschitz continuous functionf has bounded variation on compact sets the Nemytskii operatorF, defined byF(u)=f·u, maps continuously into itself. A further condition sufficient for the continuous Fréchet differentiability ofF is then added.     

多线性分数次积分算子交换子的有界性

Ibαf ( x) =∫R ∏mj=1( bj( x) - bj( y) ) 1| x - y| n-αf ( y) dyare considered.The following priori estimates are proved.For 1 01Φ1t| {y∈Rn:| Ibαf( y) | >t}| 1q ≤csupt>01Φ1t| {y∈Rn:ML( log L) 1r ,α(‖b‖f ) ( y) >t}| 1q,where‖b‖=∏mj=1‖bj‖Oscexp Lrj,Φ( t) =t( 1 + log+t) 1r,1r =1r1+ ...+ 1rm,ML(…     

How big are the increments of lp-valued Gaussian processes?

Let be a sequence of independent Gaussian processes with (h) Put . The large increments forY(·) with bounded σ(p, h) are investigated. As an example the large increments of infinite-dimensional fractional Ornstein-Uhlenbeck process in I^P are given. The method can also be applied to certain processes with stationary increments. Project supported by the National Natural Science Foundation of China and the Natural Science Foundation of Zhejiang Province.     

The closure diagram for nilpotent orbits of the split real form of E8

Let and be adjoint nilpotent orbits in a real semisimple Lie algebra. Write ≥ if is contained in the closure of . This defines a partial order on the set of such orbits, known as the closure ordering. We determine this order for the split real form of the simple complex Lie algebra, E ₈. The proof is based on the fact that the Kostant-Sekiguchi correspondence preserves the closure ordering. We also present a comprehensive list of simple representatives of these orbits, and list the irreeducible components of the boundaries and of the intersections .     

Inequalities for a distribution with monotone hazard rate

Summary LetX be a positive random variable with the survival function and the densityf. LetX have the moments μ=E(X) and μ₂=E(X ²) and put ε=|1-μ₂/2μ²|. Put and . It is proved that the following inequalities hold: , for allx>0, ifq(x) is monotone and that , ifq ₁ (x) is monotone. It is also shown that Brown's inequality which holds wheneverq ₁ (x) is increasing is not valid in general whenq ₁ is decreasing. The Institute of Statistical Mathematics     

Asymptotic behaviour of the phase in non-smooth obstacle scattering

In this paper, we study the asymptotic behaviour of the scattering phases(λ) of the Dirichlet Laplacian associated with obstacle , where Ω is a bounded open subset of ℝ ⁿ (n≥2) with non-smooth boundary ∂Ω and connected complement Ω _e =ℝ ⁿ . We can prove that if Ω satisfies a certain geometrical condition, then

On the word problem and the conjugacy problem for groups of the formF/V(R)

LetF be a free group with at most countable system of free generators, letR be its normal subgroup recursively enumerable with respect to , and let be a variety of groups that differs from and for which the corresponding verbal subgroupV of the free group of countable rank is recursive. It is proved that the word problem inF/V(R) is solvable if and only if this problem is solvable inF/R, and if , then there exists anR such, that the conjugacy problem inF/R is solvable, but this problem is unsolvable inF/V(R) for any Abelian variety (all algorithmic problems are regarded with respect to the images of under the corresponding natural epimorphisms). Translated fromMatematicheskie Zametki, Vol. 61, No. 1, pp. 3–9, January, 1997. Translated by M. I. Anokhin