Let A be a uniform algebra and M(A) the maximal ideal space of A. A sequence {an} in M(A) is called l1-interpolating if for every sequence (αn) in l1 there exists a function f in A such that f (an) = αn for all n. In this paper, an l1-interpolating sequence is studied for an arbitrary uniform algebra. For some special uniform algebras, an l1-interpolating sequence is equivalent to a familiar l-interpolating sequence. However, in general these two interpolating sequences may be different from each other.

In this note the authors give the L2(n) boundedness of a class of parametric Marcinkiewicz integral with kernel function Ω in L log+L(Sn−1) and radial function h(|x|) ∈ l ∞ l(Lq)(+) for 1 < q ≦.

As its corollary, the Lp (n)(2 < p < ∞) boundedness of and and with Ω in L log+L (Sn-1) and h(|x|) ∈ l∞ (Lq)(+) are also obtained. Here and are parametric Marcinkiewicz functions corresponding to the Littlewood-Paley g*λ-function and the Lusin area function S, respectively.

We prove that polynomial rings in one indeterminate over nil rings are antiregular radical and uniformly strongly prime radical. These give some approximations of Köthe's problem. We also study the uniformly strongly prime and superprime radicals of polynomial rings in non-commuting indeterminates. Moreover, we show that the semi-uniformly strongly prime radical coincides with the uniformly strongly prime radical and that the class of semi-superprime rings is closed under taking finite subdirect sums.

Calderón type reproducing formulae with applications have been studied on one- and higher-dimensional Lipschitz graphs. In this note we study higher order Calderón reproducing formulae on star-shaped and non-star-shaped closed Lipschitz curves and surfaces.

An algebra A is homogeneous if the automorphism group of A acts transitively on the one dimensional subspaces of A. Suppose A is a homogeneous algebra over an infinite field k. Let La denote left multiplication by any nonzero element a ∈ A. Several results are proved concerning the structure of A in terms of La. In particular, it is shown that A decomposes as the direct sum A = ker La Im La. These results are then successfully applied to the problem of classifying the infinite homogeneous algebras of small dimension.

Westwick's convexity theorem on the numerical range is generalized in the context of compact connected Lie groups.

Suppose that a graph process begins with n isolated vertices, to which edges are added randomly one by one so that the maximum degree of the induced graph is always at most d. In a previous article, the authors showed that as n → ∞, with probability tending to 1, the result of this process is a d-regular graph. This graph is shown to be connected with probability asymptotic to 1.

A point of departure for this paper is the famous theorem of Hermite and Biehler: If f (z) is a polynomial with complex coefficients ak and its zeros zk satisfy Im Zk < 0, then the polynomials with coefficients Re ak, and Im ak have only real zeros.

We generalize this theorem for some entire functions. The entire functions in Theorem 2 and Theorem 3 are of first and second genus respectively.

In this paper we give the definition of a meromorphic function which is geometrically finite and investigate some properties of geometrically finite meromorphic functions and the Lebesgue measure of their Julia sets.

We investigate the location and separation of zeros of certain three-term linear combination of translates of polynomials. In particular, we find an interval of the form I = [−1, 1 + h], h > 0 such that for a polynomial f, all of whose zeros are real, and λ ∈ I, all zeros of f (x + 2ic) + 2λf (x) + f (x – 2ic) are also real.

Here we study the dimension δ(m, X) of the general fibers of the m-Gaussian map of a singular n-dimensional variety X ⊂ Pn. We show that for all integers a, b, c, d with n ≦ a < b ≦ c < d ≦ N − 1 and a + d = b + c we have δ (a, X) + δ(d, X) > δ(b, X) + δ(c, X). If δ(X, N − 1) is very large we give some classification results which extend to the singular case some results of Ein.

Suppose that f is meromorphic in the plane, and that there is a sequence Zn → ∞ and a sequence of positive numbers ∈n → 0, such that ∈n|zn|f#(zn)/log|zn| → ∞. It is shown that if f is analytic and non-zero in the closed discs Δn = {z: |z – zn| ≦∈n|zn|}, n = 1, 2, 3 …, then, given any positive integer K, there are arbitrarily large values of n and there is a point z in Δn such that │f (z)| 〉 |Z│k. Examples are given to show that the hypotheses cannot be relaxed.

In this paper we define and study a generalized Drazin inverse xD for ring elements x, and give a characterization of elements a, b for which aaD = bbD. We apply our results to the study of EP elements in a ring with involution.