Let h: [0,1] → R be such that and define .In 1966, Erdős [8] proved that holds for almost all n, which by using a simple argument implies that in the case h(u) = u, for almost all n, He further obtained that, for every z > 0 and almost all n, and that where ϕ, ψ, are continuous distribution functions. Several other results concerning the normal growth of prime factors of integers were obtained by Galambos [10], [11] and by De Koninck and Galambos [6].

Let χ = ﹛xm : w ∈ N﹜ be a sequence of real numbers such that limm→∞ xm = +∞. For each x ∈ χ let be a set of primes p ≤x. Denote by p(n) the smallest prime factor of n. In this paper, we investigate the number of prime divisors p of n, belonging to for which Th(n,p) > z. Given Δ < 1, we study the behaviour of the function We also investigate the two functions , where, in each case, h belongs to a large class of functions.

We give a lower estimate for the central value μ*n(e) of the nth convolution power μ*···*μ of a symmetric probability measure μ on a polycyclic group G of exponential growth whose support is finite and generates G. We also give a similar large time diagonal estimate for the fundamendal solution of the equation (∂/∂t + L)u = 0, where L is a left invariant sub-Laplacian on a unimodular amenable Lie group G of exponential growth.

In this paper it is shown that the construction of measures on standard spaces via Loeb measures and the standard part map does not depend on the full structure of the internal algebra being used. A characterization of universal Loeb measurability is given for completely regular Hausdorff spaces, and the behavior of this property under various topological operations is investigated.

Let {Pi} be a sequence of real (Laurent) polynomials each of which has no negative coefficients, and suppose that f is a real polynomial. Consider the problem of deciding whether

for all integers k, there exists Nsuch that the product of polynomials

(*) Pk+1. Pk+2.....Pk+N·ƒ has no negative coefficients.

We extend the famous Kodaira-Spencer's completeness theorem for a family of deformations of complex structures (see [12]). As an application, we show that the canonical family constructed in [9] is versai.

The following paper defines a new type of ordering of higher level on a commutative ring. This definition allows the set of all orderings of level n to be given a topology which we show is consistent with the topology of the real spectrum.

Let G be a complex reductive group, and G^ its set of irreducible admissible representations. The Bruhat order on G^ is defined in a natural way. We prove that this Bruhat order is preserved by transfer. This gives new proofs of some results by the author on L-functions.

We prove a homogenization formula for a sub-Laplacian are left invariant Hörmander vector fields) on a connected Lie group Gof polynomial growth. Then using a rescaling argument inspired from M. Avellanedaand F. H. Lin [2], we prove Harnack inequalities for the positive solutions of the equation (∂/∂t+ L)u= 0. Using these inequalities and further exploiting the algebraic structure of Gwe prove that the Riesz transforms , are bounded on Lq,1 < q <+∞ and from L1to weak-L1.

Given a bivariate function defined on some subset of the Cartesian product of two sets, it is natural to ask when that function can be decomposed as the sum of two univariate functions. In particular, is a pointwise limit of such functions itself decomposable? At first glance this might seem obviously true but, as we show, the possibilities are quite subtle. We consider the question of existence and uniqueness of such decompositions for this case and for many generalizations to multivariate functions and to cases where the sets and functions have topological or measure theoretic structure.

Eigenvalue problems for selfadjoint quadratic operator polynomials L(λ) = Iλ2 + Bλ+ C on a Hilbert space H are considered where B, C∈ℒ(H), C >0, and |B| ≥ kI + k-l C for some k >0. It is shown that the spectrum of L(λ) is real. The distribution of eigenvalues on the real line and other spectral properties are also discussed. The arguments rely on the well-known theory of (weakly) hyperbolic operator polynomials.

This paper defines three simplicial approximation properties for maps of 2-cells and 2-spheres into spaces, each providing homotopical tameness conditions on the approximating images. These are the general position properties used in the two main results. The first shows that a resolvable generalized 3-manifold is a genuine 3- manifold if and only if it has the weakest of these approximation properties as well as a mild 3-dimensional disjoint disks condition known as the Light Map Separation Property. The second shows a resolvable generalized 3-manifold to be a 3-manifold if and only if it satisfies the strongest of these approximation properties.

An interesting class of matrices is shown to have the property that the spectrum of each of its elements is invariant under multiplication by p-th. roots of unity. For this class and for a class of Hamiltonian matrices improved spectral variation bounds are obtained.

This paper gives another construction of (0, p)-forms on a complex analytic space and of the operator. This construction is independent of the one in [1] and apart from the general result of Section 1 of [1], it can be read independently. As in [1], the hypotheses on S are the following: S has normal singularities, its singular locus X is smooth, the exceptional divisor in a desingularization of S is irreducible.

A pointwise estimate for the rate of approximation by polynomials , For 0 ≤ ƛ ≤ 1, integer r, and δn(x) = n-1 + φ(x), is achieved here. This formula bridges the gap between the classical estimate mentioned in most texts on approximation and obtained by Timan and others (ƛ = 0) and the recently developed estimate by Totik and first author (ƛ = 1 ). Furthermore, a matching converse result and estimates on derivatives of the approximating polynomials and their rate of approximation are derived. These results also cover the range between the classical pointwise results and the modern norm estimates for C[— 1,1].

Our main result is the following monotonicity property for moment sequences μ. Let p be fixed, 1 ≤ p < ∞: then

is an increasing function of r(r = 1,2,…). From this we derive a sharp lower bound for an arbitrary Hausdorff matrix acting on ℓp.The corresponding upper bound problem was solved by Hardy.

In the past few years, much progress have been made on several open problems in infinite dimensional Banach space theory. Here are some of the most recent results:

1. 1) The existence of boundedly complete basic sequences in a large class of Banach spaces including the ones with the so-called Radon-Nikodym property ([G-M2], [G-M4]).

2. 2) The embedding of separable reflexive Banach spaces into reflexive spaces with basis (fZl).

3. 3) The existence of long sequences of projections and hence of locally uniformly convex norms in the duals of Asplund spaces. ([F-G])

In this paper we study the fixed point algebra of the automorphism of the rotation algebra , θ = p/q with p, q coprime positive integers, given by u → v-1, v → u. We give a general characterization of the fixed point algebra, determine its K-theory and consider the related crossed-product algebra ⋊Ƭ Z4.

This paper examines the simultaneous doubling of multiple intervals of a lattice in great detail. In the case of a finite set of W-failure intervals, it is shown that there in a unique smallest lattice mapping homomorphically onto the original lattice, in which the set of W-failures is removed. A nice description of this new lattice is given. This technique is used to show that every lattice that is a bounded homomorphic image of a free lattice has a projective cover. It is also used to give a sufficient condition for a fintely presented lattice to be weakly atomic and shows that the problem of which finitely presented lattices are finite is closely related to the problem of characterizing those finite lattices with a finite W-cover.

Extreme points of the unit sphere in any Orlicz space over a measure space that contains no atoms of infinite measure are characterized. In the case of a finite-valued Orlicz function and a nonatomic measure space, exposed points of the unit sphere in these spaces are characterized too. Some corollaries and examples are also given.

Let u1 and u2 be subharmonic (plurisubharmonic) on overlapping sets K1 and K2 respectively. We seek to approximate u1 and u2 simultaneously by a single subharmonic (plurisubharmonic) function u.