Classes of Operator-Smooth Functions - II. Operator-Differentiable Functions

This paper studies the spaces of Gateaux and Frechet Operator Differentiable functions of a real variable and their link with the space of Operator Lipschitz functions. Apart from the standard operator norm on B(H), we consider a rich variety of spaces of Operator Differentiable and Operator Lipschitz functions with respect to symmetric operator norms. Our approach is aimed at the investigation of the interrelation and hierarchy of these spaces and of the intrinsic properties of Operator Differentiable functions. We apply the obtained results to the study of the functions acting on the domains of closed *-derivations of C*-algebras and prove that Operator Differentiable functions act on all such domains.We also obtain the following modification of this result: any continuously differentiable, Operator Lipschitz function acts on the domains of all weakly closed *-derivations of C*-algebras.     

Generalized Moments of Additive Functions. II

We obtain upper and lower bounds for the generalized moments of additive arithmetical functions. The results extend relevant theorems proved by I. Z. Ruzsa and A. Hildebrand.     

Normal Families of Zero-free Meromorphic Functions Ⅱ

Let k, m be two positive integers with m ≤ k and let F be a family of zero-free meromorphic functions in a domain D, let h(z) ≡ 0 be a meromorphic function in D with all poles of h has multiplicity at most m. If, for each f ∈ F, f(k)(z) = h(z) has at most k- m distinct roots(ignoring multiplicity) in D, then F is normal in D. This extends the results due to Chang[1], Gu[3], Yang[11]and Deng[1]etc.     

线性同胚于星象函数的一族解析函数(II)

本文继续研究[1]中引进的线性同胚于星象函数的一族解析函数A(λ,α),给出了A(λ,α)族的卷积定理以及系数和的一个不等式.     

线性同胚于星象函数的一族解析函数（Ⅱ）

本文继续研究了「１」中引进的线性同胚于星象函数的一族解析函数Ａ（λ,α）,给出了Ａ（λ,α）族的卷积定理以及系数和的一个不等式。     

Uniform Asymptotic Expansions for Ellipsoidal Wave Functions II. Ellipsoidal Wave Functions

This paper is concerned with the determination of uniform asymptoticexpansion for ellipsoidal wave functions. An alternative normalizationconvention for ellipsoidal wave functions is suggested. An illustrativetable of the leading terms of one type of ellipsoidal wave functionis supplied.     

Functions of the Lipschitz class and summability of Fourier series

In this paper we investigate summability of the general orthonormal systems (GOS) of the functions f(x) ∈ Lip 1.     

Topologically Inseparable Functions II: Infinitary Case

   Abstract. Given a set A and a function

Sieve Functions in Arithmetic Bands,II

An arithmetic function f is called a sieve function of range Q if its Eratosthenes transform g = f * μ is supported in [1,Q] ∩ N, where g(q) ? _ε q ^ε (?ε > 0). We continue our study of the distribution of f(n) over short arithmetic bands, n ≡ ar + b (mod q), with n ∈ (N,2N] ∩ N, 1 ≤ a ≤ H = o(N) and r,b ∈ Z such that g:c:d:(r,q) = 1. In particular, the optimality of some results is discussed.     

Planar Functions and Planes of Lenz-Barlotti Class II

Planar functions were introduced by Dembowski and Ostrom [4] to describe projective planes possessing a collineation group with particular properties. Several classes of planar functions over a finite field are described, including a class whose associated affine planes are not translation planes or dual translation planes. This resolves in the negative a question posed in [4]. These planar functions define at least one such affine plane of order 3^e for every e 4 and their projective closures are of Lenz-Barlotti type II. All previously known planes of type II are obtained by derivation or lifting. At least when e is odd, the planes described here cannot be obtained in this manner.     

On BMO-Regular Lattices of Measurable Functions. II

The main result of a previous author’s paper is refined slightly and extended to certain quasi-Banach lattices. Bibliogrhaphy: 3 titles.__________Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 303, 2003, pp. 161–168.     

A class of vertex-transitive digraphs,II

(1). We determine the number of non-isomorphic classes of self-complementary circulant digraphs with pq vertices, where p and q are distinct primes. The non-isomorphic classes of these circulant digraphs with pq vertices are enumerated. (2). We also determine the number of non-isomorphic classes of self-complementary, vertex-transitive digraphs with a prime number p vertices, and the number of self-complementary strongly vertex-transitive digraphs with p vertices. The non-isomorphic classes of strongly vertex-transitive digraphs with p vertices are also enumerated.     

The Schur Algorithm for Generalized Schur Functions II: Jordan Chains and Transformations of Characteristic Functions

 In the first paper of this series (Daniel Alpay, Tomas Azizov, Aad Dijksma, and Heinz Langer: The Schur algorithm for generalized Schur functions I: coisometric realizations, Operator Theory: Advances and Applications 129 (2001), pp. 1–36) it was shown that for a generalized Schur function s(z), which is the characteristic function of a coisometric colligation V with state space being a Pontryagin space, the Schur transformation corresponds to a finite-dimensional reduction of the state space, and a finite-dimensional perturbation and compression of its main operator. In the present paper we show that these formulas can be explained using simple relations between V and the colligation of the reciprocal s(z)⁻¹ of the characteristic function s(z) and general factorization results for characteristic functions. Received October 31, 2001; in revised form August 21, 2002 RID="a" ID="a" Dedicated to Professor Edmund Hlawka on the occasion of his 85th birthday     

The Hausdorff Dimension of Julia Sets of Meromorphic Functions II

A family of transcendental meromorphic functions, f_p(z), p N is considered. It is shown that, if p 6, then the Hausdorffdimension of the Julia set of f_p satisfies dim J(f_p) 1/p, for0 < < 1/6^p, and dim J(f_p) 1–(30 ln ln p/ln p),for p^4p–1/10⁵ ln p < < p^4p–1/10⁴ ln p. Theseresults are used elsewhere to show that, for each d (0, 1),there exists a transcendental meromorphic function for whichdim J(f) = d.     

On the Asymptotic Behaviour of General Partition Functions,II

Let = {a ₁, a ₂,...} be a set of positive integers and let p (n) and q (n) denote the number of partitions of n into a's, resp. distinct a's. In an earlier paper the authors studied large values of log(max (2,p (n)))/log(max(2,q (n))). In this paper the small values of the same quotient are studied.