The classical theorem of Müntz and Szász says that the span of

is dense in C[0,1] in the uniform norm if and only if . We prove that, if {λi} is lacunary, we can replace the underlying interval [0,1] by any set of positive measure. The key to the proof is the establishment of a bounded Remez-type inequality for lacunary Müntz systems. Namely if A ⊆ [0,1] and its Lebesgue measure µ(A) is at least ε > 0 then

where c depends only on ε and Λ (not on n and A) and where Λ:=infiλi+1/λi>1.

For a measurable function f on the unit ball B in ℂn we define (M1f)(w), |w|<1, to be the mean modulus of f over a hyperbolic ball with center at w and of a fixed radius. The space , 0<p<∞, is defined by the requirement that M1f belongs to the Lebesgue space Lp. It is shown that the subspace of Lp spanned by holomorphic functions coincides with the corresponding subspace of . It is proved that if s>(n+1)(p−1−1), 0<p<1, then this subspace is complemented in by the projection whose reproducing kernel is . As corollaries we get an extension of the Forelli–Rudin projection theorem and we show that a holomorphic function f is Lp-integrable, 0<p<∞, over the unit ball B iff u = Ref is Lp-integrable over B. Finally, we sketch an alternative proof of the main result of this paper in the case 0<p<1.

Precise asymptotic estimates for the eigenvalues of a uniformly right definite two parameter system of Sturm–Liouville problems are developed. The work extends recent results of B. P. Rynne.

A correspondence is established between a class of coverings of an inverse semigroup S and a class of embeddings of S, generalising results of McAlister and Reilly on E-unitary covers of inverse semigroups.

The relationship between the prime ideals and the primal ideals of a Banach algebra is investigated. It is shown that the closure of the prime radical of a Banach algebra may be properly contained in the intersection of the closed primal ideals of the algebra.

Lazarsfeld proved a bound for the excess dimension of an intersection of irreducible and reduced schemes. Flenner and Vogel gave another approach for reduced, non-degenerate schemes which are connected in codimension one, using the intersection algorithm of Stückrad and Vogel and defining a new multiplicity k. Renschuch and Vogel considered a condition to ensure that there is no degeneration for more than two schemes. We define an integer which enables us to unify these methods. This allows us to generalize the result of Flenner and Vogel to non-reduced schemes by comparing the multiplicities j and k. Using this point of view we give applications to converses of Bézout's theorem; in particular we investigate the Cohen-Macaulay case.

In this paper, we investigate the complex oscillation theory of

where A, F≢0 are entire functions, and obtain general estimates of the exponent of convergence of the zero-sequence and of the order of growth of solutions for the above equation.

It is proved that if G is any compact connected Hausdorff group with weight w(G)≦c, ℝ is the topological group of all real numbers and n is a positive integer, then the topological group G × ℝn can be topologically generated by n + 1 elements, and no fewer elements will suffice.

Suppose that K is a function field of a conic over a subfield K0. Let v0 be a valuation of K0 with residue field k0 of characteristic ≠2. Let v be an extension of v0 to K having residue field k. It has been proved that either k is an algebraic extension of k0 or k is a regular function field of a conic over a finite extension of k0. This result can also be deduced from the genus inequality of Matignon (cf. [On valued function fields I, Manuscripta Math. 65 (1989), 357–376]) which has been proved using results about vector space defect and methods of rigid analytic geometry. The proof given here is more or less self-contained requiring only elementary valuation theory.

In this paper we consider the questions of existence and uniqueness of solutions to a singular, nonlinear boundary value problem arising from a model problem in isothermal autocatalytical chemical kinetics. The boundary value problem occurs in the construction of a small time asymptotic solution to an initial-boundary value problem (King and Needham [14]), and existence and uniqueness for the boundary value problem are required for consistency of this formal asymptotic solution.

A knot K is said to have tunnel number 1 if there is an embedded arc A in S3, with endpoints on K, whose interior is disjoint from K and such that the complement of a regular neighbourhood of K ∪ A is a genus 2 handlebody. In particular the fundamental group of the complement of a tunnel number one knot is 2-generator. There has been some interest in the question as to whether there exists a hyperbolic tunnel number one knot whose complement contains a closed essential surface. The aim of this paper is to prove the existence of infinitely many 2-generator hyperbolic 3-manifolds with a single cusp which contain a closed essential surface. One such example is a knot complement in RP3. The methods used are of interest as they include the possibility that one of our examples is a knot complement in S3.

The idea of applying isoperimetric functions to group theory is due to M. Gromov [8]. We introduce the concept of a “bicombing of narrow shape” which generalizes the usual notion of bicombing as defined for example in [5], [2], and [10]. Our bicombing is related to but different from the combings defined by M. Bridson [4]. If they Cayley graph of a group with respect to a given set of generators admits a bicombing of narrow shape then the group is finitely presented and satisfies a sub-exponential isoperimetric inequality, as well as a polynomial isodiametric inequality. We give an infinite class of examples which are not bicombable in the usual sense but admit bicombings of narrow shape.

It is shown that an extended Schlömilch formula for Stirling-type pairs of numbers and the inversion formula of Lagrange are implied by each other. Also proved are some congruence relations modulo a prime number p(>2) associated with generalized Stirling numbers. The third result is concerned with the asymptotic expansions of Stirling-type pairs involving large parameters.

In this work we prove the existence and uniqueness of positive solutions of the nonlinear singular boundary value problem

where 0<σ<1.

Extensions of the above results to the case of Δ2u−f(x, u) = 0 with appropriate singularity built into f are also given.