We consider the error term P3(R) = #{n ∈ ℤ3 :|n| [les ] R} − 4|3πR3 which occurs in the counting of lattice points in a sphere of radius R. By considering second and third power moments, we prove that P3(R) = Ω±(R√log R). An upper bound for the gap between the sign changes of P3(R) is also proved.

In his book Abelian groups, L. Fuchs raised the question as to whether, in general, in the factorization of a finite (cyclic) abelian group one factor may always be replaced by some subgroup. The answer turned out to be negative in general, but positive in certain cases. In this paper the complete answer for cyclic groups is given. In all previously unsolved cases, the answer turns out to be positive. It is shown that a cyclic group has the property that in every factorization, one factor may be replaced by a subgroup if and only if the group has order equal to the product of a prime and a square-free integer. Certain results are also given in non-cyclic cases.

Let Tt be the semigroup of linear operators generated by a Schrödinger operator − A = Δ − V, where V is a non-negative polynomial, and let ∫∞0λdEA(λ) be the spectral resolution of A. We say that f is an element of HpA if the maximal function [Mscr ]f(x) = supt>0[mid ]Ttf(x)[mid ] belongs to Lp. We prove a criterion of Mihlin type on functions F which implies boundedness of the operators F(A) = ∫∞0F(λ)dEA(λ) on HpA, 0 < p [les ] 1.

It is proved that the infinitesimal generator A of a strongly continuous semigroup of linear operators on a Hilbert space also generates a strongly continuous group if and only if the resolvent of −A, (λ+A)−1, is also a bounded function on some right-hand-side half plane of complex numbers, and converges strongly to zero as the real part of λ tends to infinity. An application to a partial differential equation is given.

Analogues of the Funk–Hecke formula for spherical harmonics are proved for Dunkl's h-harmonics associated to the reflection groups, and for orthogonal polynomials related to h-harmonics on the unit ball. In particular, an analogue and its application are discussed for the weight function (1−[mid ]x[mid ]2)μ−1/2 on the unit ball in ℝd.

Let μ be a real number. The Möbius group Gμ is the matrix group generated by

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It is known that Gμ is free if [mid ]μ[mid ] [ges ] 2 (see [1]) or if μ is transcendental (see [3, 8]). Moreover, there is a set of irrational algebraic numbers μ which is dense in (−2,2) and for which Gμ is non-free [2, p. 528]. We may assume that μ > 0, and in this paper we consider rational μ in (0, 2). The following problem is difficult.

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Let [Gscr ] denote the set of all rational numbers μ in (0, 2) for which Gμ is non-free. In 1969 Lyndon and Ullman [8] proved that [Gscr ]nf contains the elements of the forms p/(p2 + 1) and 1/(p + 1), where p = 1, 2, …, and that if μ0 ∈ [Gscr ]nf then μ0/p ∈ [Gscr ]nf for p = 1, 2, …. In 1993 Beardon [2] studied problem (P) by means of the words of the form ArBsAt and ArBsAtBuAv, and he obtained a sufficient condition for solvability of (P), included implicitly in [2, pp. 530–531], by means of the following Diophantine equations:

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In the analysis of hyperbolic boundary value problems, the construction of Kreiss' symmetrizers relies on a suitable block structure decomposition of the symbol of the system. In this paper, we show that this block structure condition is satisfied by all symmetrizable hyperbolic systems of constant multiplicity.

We study deformations of zero-dimensional complete intersections in the plane, and prove the following results. (1) Two complex non-singular curves intersecting at r points with multiplicities d1, …, dr can be deformed into curves intersecting (at some points) with multiplicities d′1, …, d′s which are arbitrary prescribed partitions of the numbers d1, …, dr. (2) Two real curves intersecting with multiplicity at most 2 at each of their real common points can be deformed so that all real multiple intersection points split into real simple intersection points.

Using properties of the Steinberg character, we obtain a congruence modulo p for the number of ways in which a p-regular element may be expressed as a commutator in a finite simple group G of Lie type of characteristic p. This congruence shows that such an element is a commutator in G. We also show that if K and L are conjugacy classes of G consisting of elements whose centralizers have order relatively prime to p, then any p-regular element of G is expressible as the product of an element of K and an element of L.

We examine the range of universal Taylor series. We prove that every universal Taylor series on the unit disc assumes every complex number, with one possible exception, infinitely often. On the other hand, we prove that on any simply connected domain there exist universal functions that omit one value.

We prove that if A is a prime non-commutative JB*-algebra which is neither quadratic nor commutative, then there exist a prime C*-algebra B and a real number λ with ½ < λ [les ] 1 such that A = B as involutive Banach spaces, and the product of A is related to that of B (denoted by ∘, say) by means of the equality xy = λx ∘ y + (1 − λ)y ∘ x.

A potential for the one-dimensional Dirac operator is constructed such that its essential spectrum does not cover the whole real line, whereas the potential q(x) tends to ∞ as [mid ]x[mid ] → ∞. Furthermore, a criterion by Hartman and Wintner for points of the essential spectrum of Sturm–Liouville operators is generalised to a purely operator-theoretical setting, and a simplified proof is given.

In this paper, we define a Hardy–Littlewood maximal function operator M associated with the Jacobi transform, and prove that M is of weak (1, 1) and strong (p, p) (p > 1) type.

The aim of this paper is to prove that for certain generalized Weyl algebras A (including the first Weyl algebra A1 over a field of characteristic zero) and for every simple left (right) A-module M, there are infinitely many non-isomorphic simple left (right) A-modules {Ni} such that Ext1A (M, Ni) ≠ 0 (respectively Ext1A(Ni, M) ≠ 0.

Using basic harmonic analysis on the n-torus, it is shown that the non-commutative n-tori are the only primitive C*-algebras (up to isomorphism) on which the n-torus [ ]n acts freely and ergodically. An explicit construction recovers the generating unitaries in a natural way from the action.

Functions with bounded variation and with a (total) variation are examined within Bishop's constructive mathematics. It is shown that the property of having a variation is hereditary downward on compact intervals, and hence that a real-valued function f with a variation on a compact interval can be expressed as a difference of two increasing functions. Moreover, if f is sequentially continuous, then the corresponding variation function, and hence f itself, is uniformly continuous.

In this paper we give answers to some open questions concerning generation and enumeration of finite transitive permutation groups. In [1], Bryant, Kovács and Robinson proved that there is a number c′such that each soluble transitive permutation group of degree n [ges ] 2 can be generated by [c′ n/ √log n] elements, and later A. Lucchini [5] extended this result (with a different constant c′) to finite permutation groups containing a soluble transitive subgroup. We are now able to prove this theorem in full generality, and this solves the question of bounding the number of generators of a finite transitive permutation group in terms of its degree. The result obtained is the following.

Let Ω⊆ℝd be an open set of measure 1. An open set D⊆ℝd is called a ‘tight orthogonal packing region’ for Ω if D−D does not intersect the zeros of the Fourier transform of the indicator function of Ω, and D has measure 1. Suppose that Λ is a discrete subset of ℝd. The main contribution of this paper is a new way of proving the following result: D tiles ℝd when translated at the locations Λ if and only if the set of exponentials EΛ = {exp 2πi〈λ, x〉 [ratio ] λ ∈ Λ} is an orthonormal basis for L2(Ω). (This result has been proved by different methods by Lagarias, Reeds and Wang [9] and, in the case of Ω being the cube, by Iosevich and Pedersen [3]. When Ω is the unit cube in ℝd, it is a tight orthogonal packing region of itself.) In our approach, orthogonality of EΛ is viewed as a statement about ‘packing’ ℝd with translates of a certain non-negative function and, additionally, we have completeness of EΛ in L2(Ω) if and only if the above-mentioned packing is in fact a tiling. We then formulate the tiling condition in Fourier analytic language, and use this to prove our result.

Let 1 [les ] p [les ] ∞. For each n-dimensional Banach space E = (E, ∥ · ∥), we define a norm ∥ · ∥p on E × ℝ as follows:

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It is shown that the correspondence (E, ∥ · ∥) [map ] (E × ℝ, ∥ · ∥p) defines a topological embedding of one Banach–Mazur compactum, BM(n), into another, BM(n + 1), and hence we obtain a tower of Banach–Mazur compacta: BM(1) ⊂ BM(2) ⊂ BM(3) ⊂ ···. Let BMp be the direct limit of this tower. We prove that BMp is homeomorphic to Q∞ = dir lim Qn, where Q = [0, 1]ω is the Hilbert cube.

Let [Mscr ]S be the universal maximal operator over unit vectors of arbitrary directions. This operator is not bounded in L2(R2). We consider a sequence of operators over sets of finite equidistributed directions converging to [Mscr ]S. We provide a new proof of N. Katz's bound for such operators. As a corollary, we deduce that [Mscr ]S is bounded from some subsets of L2 to L2. These subsets are composed of positive functions whose Fourier transforms have a logarithmic decay or which are supported on a disc.