A note on a theorem of Daboussi

The following assertion is proved. Let be the set of integers the number of the prime power of which is . Let be the size of . Then for each irrational , uniformly in , \begin{equation*} \frac{1}{\pi_k(x)} \bigg|\sum _{\alul{n\le x}{n\in\cN_k}} f(n) e^{2\pi in\alpha}\bigg|\to 0, \end{equation*} where is an arbitrary multiplicative function with , is a positive constant. This revised version was published online in June 2006 with corrections to the Cover Date.     

Stability results for scattered‐data interpolation on Euclidean spheres

Let denote the unit sphere in and the geodesic distance in . A spherical‐basis function approximant is a function of the form , where are real constants, is a fixed function, and is a set of distinct points in . It is known that if is a strictly positive definite function in , then the interpolation matrix is positive definite, hence invertible, for every choice of distinct points and every positive integer M. The paper studies a salient subclass of such functions , and provides stability estimates for the associated interpolation matrices. This revised version was published online in June 2006 with corrections to the Cover Date.     

Finite rank operators in Jacobson radical \mathcal{R}_{\mathcal{N} \otimes \mathcal{M}}

In this paper we investigate finite rank operators in the Jacobson radical of Alg( ), where are nests. Based on the concrete characterizations of rank one operators in Alg( ) and , we obtain that each finite rank operator in can be written as a finite sum of rank one operators in and the weak closure of equals Alg( ) if and only if at least one of is continuous.     

Some properties of the absolute galois group of a hilbertian field

LetK be a hilbertian field,G(K) its absolute Galois group. IfK is countable, then for a.a. inG(K) ^e , and there is no intermediate field with . Let ∈G(K) ^e . Then for a.a. in .     

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.     

Marcinkiewicz-\theta-Summability of Fourier Transforms

A general summability method of two-dimensional Fourier transforms is given with the help of an integrable function . Under some conditions on we show that the maximal operator of the Marcinkiewicz- -means of a tempered distribution is bounded from to for all and, consequently, is of weak type , where depends only on . As a consequence we obtain a generalization for Fourier transforms of a summability result due to Marcinkievicz and Zhizhiashvili, more exactly, the Marcinkiewicz- -means of a function converge a.e. to the function in question. Moreover, we prove that the Marcinkiewicz- -means are uniformly bounded on the spaces and so they converge in norm . Some special cases of the Marcinkievicz- -summation are considered, such as the Weierstrass, Picar, Bessel, Fejér, de la Vallée-Poussin, Rogosinski and Riesz summations. This revised version was published online in June 2006 with corrections to the Cover Date.     

Closed polynomials and saturated subalgebras of polynomial algebras

The behavior of closed polynomials, i.e., polynomials such that the subalgebra is integrally closed in , is studied under extensions of the ground field. Using some properties of closed polynomials, we prove that, after shifting by constants, every polynomial can be factorized into a product of irreducible polynomials of the same degree. We consider some types of saturated subalgebras , i.e., subalgebras such that, for any , a generative polynomial of f is contained in A. Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 12, pp. 1587–1593, December, 2007.     

Three-dimensional sets with small sumset

We describe the structure of three dimensional sets of lattice points, having a small doubling property. Let be a finite subset of ℤ³ such that dim = 3. If and , then lies on three parallel lines. Moreover, for every three dimensional finite set that lies on three parallel lines, if , then is contained in three arithmetic progressions with the same common difference, having together no more than terms. These best possible results confirm a recent conjecture of Freiman and cannot be sharpened by reducing the quantity υ or by increasing the upper bounds for .     

Large deviations of the steady-state distribution of reflected processes with applications to queueing systems

We consider a Skorohod map which takes paths in to paths which stay in the positive orthant . We let be the domain of definition of . A convex and lower semi-continuous function and a set are given. We are concerned with the calculation of the infimum of the value for t ⩾ 0 and absolutely continuous subject to the conditions and . We show that such minimization problems characterize large deviation asymptotics of tail probabilities of the steady-state distribution of certain reflected processes. We approximate the infimum by a sequence of finite-dimensional minimization problems. This approximation allows to formulate an algorithm for the calculation of the infimum and to derive analytical bounds for its value. Several applications are discussed including large deviations of generalized processor sharing and large deviations of heavily loaded queueing networks. This revised version was published online in June 2006 with corrections to the Cover Date.     

Decomposition and embedment of trajectories after explosion for a birth and death process

Let X = {X(t),t ＜σ} (σ is lifespan) be a birth and death process with explosion whose characteristic triple (Mα,MC,MD) of MX in terms of (α, C, D) and M. This means that a lot of given birth and death processes can be embedded in one and the same birth and death process. If κ∈ E and M = {κ},we decompose X into κX, κ∈ E.     

Measure-valued almost periodic functions

We consider Stepanov almost periodic functions μ ∈ ranging in the metric space of Borel probability measures on a complete separable metric space is equipped with the Prokhorov metric). The main result is as follows: a function , belongs to if and only if for each bounded continuous function , the function is Stepanov almost periodic (of order 1) and

Baer cotorsion Pairs

LetR be a unital associative ring and two classes of leftR-modules. In [St3] the notion of a ( ) pair was introduced. In analogy to classical cotorsion pairs, a pair (V,W) of subclasses is called a ( ) pair if it is maximal with respect to the classes and the condition Ext _R ¹ (V, W)=0 for all . In this paper we study pairs whereR = ℤ and is the class of all torsion-free abelian groups andT is the class of all torsion abelian groups. A complete characterization is obtained assumingV=L. For example, it is shown that every pair is singly cognerated underV=L. The author was supported by a DFG grant.     

Supersaturation For Ramsey-Turán Problems

For an l-graph , the Turán number is the maximum number of edges in an n-vertex l-graph containing no copy of . The limit is known to exist [8]. The Ramsey–Turán density is defined similarly to except that we restrict to only those with independence number o(n). A result of Erdős and Sós [3] states that as long as for every edge E of there is another edge E′of for which |E∩E′|≥2. Therefore a natural question is whether there exists for which . Another variant proposed in [3] requires the stronger condition that every set of vertices of of size at least εn (0<ε<1) has density bounded below by some threshold. By definition, for every . However, even is not known for very many l-graphs when l>2. We prove the existence of a phenomenon similar to supersaturation for Turán problems for hypergraphs. As a consequence, we construct, for each l≥3, infinitely many l-graphs for which . We also prove that the 3-graph with triples 12a, 12b, 12c, 13a, 13b, 13c, 23a, 23b, 23c, abc, satisfies . The existence of a hypergraph satisfying was conjectured by Erdős and Sós [3], proved by Frankl and R?dl [6], and later by Sidorenko [14]. Our short proof is based on different ideas and is simpler than these earlier proofs. * Research supported in part by the National Science Foundation under grants DMS-9970325 and DMS-0400812, and an Alfred P. Sloan Research Fellowship. † Research supported in part by the National Science Foundation under grants DMS-0071261 and DMS-0300529.     

Martin boundary of unlimited covering surfaces

LetW be an open Riemann surface and ap-sheeted (1<p<∞) unlimited covering surface ofW. Denote by Δ¹ (resp., ) the minimal Martin boundary ofW (resp., ). For ζ ∈ Δ, let ζ be the (cardinal) number of the set of pionts which lie over ζ and the class of open connected subsetsM ofW such thatM∪{ζ} is a minimal fine neighborhood of ζ. Our main result is the following: , where is the number of components of π^-1 M and π is the projection of ontoW. Moreover, some applications of the above results are discussed whenW is the unit disc.     

A new class of contact riemannian manifolds

N. Tanaka ([10]) defined the canonical affine connection on a nondegenerate integrable CR manifold. In the present paper, we introduce a new class of contact Riemannian manifolds satisfying (C) ( for any unit -geodesic ( , where is the generalized Tanaka connection. In particular, when the associated CR structure of a given contact Riemannian manifold is integrable we have a structure theorem and find examples which are neither Sasakian nor locally symmetric but satisfy the condition (C). This work was supported in part by BSRI 98-1425.     

The normal subgroup structure of the extended Hecke groups

We consider the extended Hecke groups generated by T(z) = −1/z, S(z) = −1/(z + λ) and R(z) = 1/z with λ ≥ 2. In this paper, firstly, we study the fundamental region of the extended Hecke groups . Then, we determine the abstract group structure of the commutator subgroups , the even subgroup , and the power subgroups of the extended Hecke groups . Also, finally, we give some relations between them.     

The Reduced Minimum Modulus in <Emphasis>C</Emphasis><Superscript>*</Superscript>-Algebras

We define the reduced minimum modulus of a nonzero element a in a unital C ^*-algebra by . We prove that . Applying this result to and its closed two side ideal , we get that dist , and for any if RR = 0, where and is the quotient homomorphism and . These results generalize corresponding results in Hilbert spaces.     

On the representation of differentiations by Hamiltonians,operating from an algebra of local observable spin systems

Let and be algebras of local and quasilocal observable spin systems corresponding to the group Zr, be a differentiation invariant with respect to displacements. The question of representation of D in the form of formal Hamiltonian formed by the displacements of an elementx ε is considered. It is shown that such a representation exists if the condition holds, where means an element obtained from the elements [T_kX,a] by some r-multiple process of summation. Translated from Matematicheskii Zametki, Vol. 21, No. 1, pp. 93–98, January, 1977.     

On the range of a generalized derivation

A generalized derivation , is defined by the formula , where and is the Banach algebra of bounded linear operators in a Hilbert space . Sufficient conditions under which and are given. Bibliography: 8 titles. Translated fromProblemy Matematicheskogo Analiza, No. 20, 2000, pp. 111–119.     

On the Order of Magnitude of the Divisor Function

Let D be an increasing sequence of positive integers, and consider the divisor functions： d（n, D） =∑d｜n,d∈D,d≤√n1, d2（n,D）=∑[d,δ]｜n,d,δ∈D,[d,δ]≤√n1, where [d,δ]=1.c.m.（d,δ）. A probabilistic argument is introduced to evaluate the series ∑n=1^∞and（n,D） and ∑n=1^∞and2（n,D）.