We consider the quadratic polynomial m2 + D and study the asymptotic formula for the number of integers m, 1≤ m ≤ M, for which the values of the polynomial are square-free. We are interested in particular in the question of how small we can take m in relation to D and still have the asymptotic hold.

Given an embedded hypersurface $M$ in $\mathbb{R}^4$ we consider families of projections $H$ of $M$ to lines and families of projections $P$ of $M$ to 3-spaces. We characterize generically the singularities of these projections. We also show that there is a duality relation between some strata of the bifurcation sets of $H$ and $P$, and deduce geometric properties about these sets.

AMS 2000 Mathematics subject classification: Primary 58K05. Secondary 58K40

The nonlinear eigenvalue problem

for 0 ≤ x < ∞, fixed p ∈ (1, ∞), and with y′(0)/y(0) specified is studied under various conditions on the coefficients s and q, leading to either oscillatory or non-oscillatory situations.

Nearfields and near-rings are related to many other structures and needed for several representation theorems. Therefore it is important to gain knowledge about the structure of near-rings and nearfields and to find construction methods. The first examples of proper nearfields were constructed by L.E. Dickson 1905, they were finite. 30 years later H. Zassenhaus completely determined the finite nearfields his attention having been attracted to them by the study of certain permutation groups. By the axiomatization of Dickson's methods, H. Karzel succeeded in giving new examples of infinite nearfields. In the extensive generalization by J. Timm the so-called “Dicksonian processes” are the most important tool for constructions of nearfields and near-rings. The report (44) by H. Wähling is a summary of the results on nearfields obtained so far.

In the 1970s, Feldman and Moore classified separably acting von Neumann algebras containing Cartan maximal abelian self-adjoint subalgebras (MASAs) using measured equivalence relations and 2-cocycles on such equivalence relations. In this paper we give a new classification in terms of extensions of inverse semigroups. Our approach is more algebraic in character and less point-based than that of Feldman and Moore. As an application, we give a restatement of the spectral theorem for bimodules in terms of subsets of inverse semigroups. We also show how our viewpoint leads naturally to a description of maximal subdiagonal algebras.

It is known that ℚ-derived univariate polynomials (polynomials defined over ℚ, with the property that they and all their derivatives have all their roots in ℚ) can be completely classified subject to two conjectures: that no quartic with four distinct roots is ℚ-derived, and that no quintic with a triple root and two other distinct roots is ℚ-derived. We prove the second of these conjectures.

AMS 2000 Mathematics subject classification: Primary 11G30

In this paper we investigate the weighted convolution algebras lp(ωn), where 1≦p<∞ and {ωn} is a sequence of positive weights satisfying the following conditions. If p = 1 we require ω0 = 1, and ωt+s≦ωtωs.Then

We introduce the notion of functionally compact sets into the theory of nonlinear generalized functions in the sense of Colombeau. The motivation behind our construction is to transfer, as far as possible, properties enjoyed by standard smooth functions on compact sets into the framework of generalized functions. Based on this concept, we introduce spaces of compactly supported generalized smooth functions that are close analogues to the test function spaces of distribution theory. We then develop the topological and functional–analytic foundations of these spaces.

We introduce a general definition for coloured cyclic operads over a symmetric monoidal ground category, which has several appealing features. The forgetful functor from coloured cyclic operads to coloured operads has both adjoints, each of which is relatively simple. Explicit formulae for these adjoints allow us to lift the Cisinski–Moerdijk model structure on the category of coloured operads enriched in simplicial sets to the category of coloured cyclic operads enriched in simplicial sets.

We give a new very concrete description of the C*-envelope of the tensor algebra associated to a multivariable dynamical system. In the surjective case, this C*-envelope is described as a crossed product by an endomorphism and as a groupoid C*-algebra. In the non-surjective case, it is a full corner of such an algebra. We also show that when the space is compact the C*-envelope is simple if and only if the system is minimal.

We investigate topological realizations of higher-rank graphs. We show that the fundamental group of a higher-rank graph coincides with the fundamental group of its topological realization. We also show that topological realization of higher-rank graphs is a functor and that for each higher-rank graph Λ, this functor determines a category equivalence between the category of coverings of Λ and the category of coverings of its topological realization. We discuss how topological realization relates to two standard constructions for k-graphs: projective limits and crossed products by finitely generated free abelian groups.

Several papers on the subject of spatial distance in General Relativity appeared a few years ago, and a simple extension of this idea to any pair of points in any Riemannian space was given by me in a thesis. A distance invariant was defined, and this was found to depend upon a certain two-point invariant which was first introduced by H. S. Ruse in a study of Laplace's Equation. This invariant, now written ρ and defined in (3), has lately re-appeared, and it may now be of interest to publish the results found earlier. These include a geometrical interpretation of ρ, a simple method of calculation, and an expansion as a power series in the geodesic arc. The dependence of ρ upon the geodesic arc is also considered.

Let R be a ring with identity, let Ω be an infinite set and let M be the free R-module R(Ω). In [1] we investigated the problem of locating and classifying the normal subgroups of GL(Ω, R), the group of units of the endomorphism ring EndRM, where R was an arbitrary ring with identity. (This extended the work of [3] and [8] where it was necessary for R to satisfy certain finiteness conditions.) When R is a division ring, the complete classification of the normal subgroups of GL(Ω, R) is given in [9] and the corresponding results for a Hilbert space are given in [6] and [7]. The object of this paper is to extend the methods of [1] to yield a classification of the subnormal subgroups of GL(Ω, R) along the lines of that given by Wilson in [10] in the finite dimensional case.

Let r be an integer with 2 ≤ r ≤ 24 and let pr(n) be defined by $\sum _{n=0}^\infty p_r(n) q^n = \prod _{k=1}^\infty (1-q^k)^r$. In this paper, we provide uniform methods for discovering infinite families of congruences and strange congruences for pr(n) by using some identities on pr(n) due to Newman. As applications, we establish many infinite families of congruences and strange congruences for certain partition functions, such as Andrews's smallest parts function, the coefficients of Ramanujan's ϕ function and p-regular partition functions. For example, we prove that for n ≥ 0,

\[ \textrm{spt}\bigg( \frac{1991n(3n+1) }{2} +83\bigg) \equiv \textrm{spt}\bigg(\frac{1991n(3n+5)}{2} +2074\bigg) \equiv 0\ (\textrm{mod} \ 11), \]

\[ \textrm{spt}\bigg( \frac{143\times 5^{6k} +1 }{24}\bigg)\equiv 2^{k+2} \ (\textrm{mod}\ 11), \]

An algebraic equation

determines, in general, s as a many valued function of z. If s and z can be expressed as one valued functions of a third variable t, then t is called the uniformising variable. As Poincaré showed, s and z are automorphic functions of t.

Methods for solving boundary value problems in linear, second order, partial differential equations in two variables tend to be somewhat rigidly partitioned in some of the standard text-books. Problems for elliptic equations are sometimes solved by finding the fundamental solution which is defined as a solution with a given singularity at a certain point. Another approach is by way of Green's functions which are usually defined as solutions of the original homogeneous equations now made inhomogeneous by the introduction of adelta function on the right hand side. The Green's function coincides with the fundamental solution for elliptic equations but exhibits a totally different type of singularity for parabolic or hyperbolic equations. Boundary value problems for hyperbolic equations can often by solved by Riemann's method which depends on the existence of an auxiliary function called the Riemann or sometimes the Riemann-Green function. The main object of this paper is to show the close relationship between Riemann's method and the method of Green's functions. This not only serves to unify different methods of solution of boundary value problems but also provides an additional method of determining Riemann functions for given hyperbolic equations. Before establishing these relationships we shall survey the general approach to boundary value problems through the use of the Green's function.

We prove that for F: [0,∞) × ℝn → K (ℝn) a Lipschitzian multifunction with compact values, the set of derivatives of solutions of the Cauchy problem

is a retract of

Basic properties of symplectic reflection algebras over an algebraically closed field k of positive characteristic are laid out. These algebras are always finite modules over their centres, in contrast to the situation in characteristic 0. For the subclass of rational Cherednik algebras, we determine the PI-degree and the Goldie rank, and show that the Azumaya and smooth loci of the centre coincide.

1. The chief purpose of this paper is to demonstrate the existence of a plane quartic curve with eight undulations, an “undulation” being a point at which the tangent has four-point contact. It is shown that the curve

where (x, y, z) are homogeneous point-coordinates and f a constant, has undulations at the eight points

The curve has, in addition to these undulations, eight inflections which are; in general, distinct. But there are two geometrically different possibilities of their not being distinct, and in either instance they coincide in pairs at four further undulations. Thus two types of curve arise without any ordinary inflections at all, their 24 inflections coinciding in pairs at 12 undulations.

It is proved in this paper that for any non-elementary subgroup $G$ of $\mathrm{PSL}(2,\sGa_n)$, which has no elliptic element, to be not strict, there is a minimal generating system of $G$ consisting of loxodromic elements, and that if $G$ is a non-elementary subgroup of $\mathrm{PSL}(2,\sGa_n)$ of which each loxodromic element is hyperbolic, then $G$ is conjugate to a subgroup of $\mathrm{PSL}(2,\mathbb{R})$.

AMS 2000 Mathematics subject classification: Primary 30F40. Secondary 20H10