Let Fn, Mn and Un denote the free group of rank n, the free metabelian group of rank n, and the free abelian-by-(nilpotent of class 2) group of rank n, respectively. Thus Mn≅ Fn/F″″n and Un≅Fn/ [γ3(Fn), γ3(Fn)], where γ3(Fn) =[F′n, Fn], the third term of the lower central series of Fn.

Consider an arbitrary epimorphism θ[ratio ]Fn[Rarr ]Fk, where k<n. A routine argument (see [10, Theorem 3.3]) using the Nielsen reduction process shows that there exists a free basis {y1, y2, …, yn} of Fn such that kerθ is the normal closure in Fn of {y1, y2, …, yn−k}. A similar result has been obtained for free metabelian groups by C. K. Gupta, N. D. Gupta and G. A. Noskov [5]. Indeed, it follows immediately from [5, Theorem 3.1] that if k<n and θ[ratio ]Mn[Rarr ]Mk is an epimorphism, then there exists a free basis {m1, m2, …, mn} of Mn such that kerθ is the normal closure in Mn of {m1, m2, …, mn−k}.

In their seminal work [1] on the fields of fractions of the enveloping algebra of an algebraic Lie algebra, Gel'fand and Kirillov formulate the following conjecture. Assume that [gfr ] is a finite-dimensional algebraic Lie algebra over a field of characteristic zero. Then D([gfr ]) is a Weyl skew-field over a purely transcendental extension of the base field.

They showed that neither the conjecture nor its negation holds for all non-algebraic algebras. In [2], A. Joseph gave a particularly easy non-algebraic counterexample devised by L. Makar-Limanov: this is a non-algebraic 5-dimensional solvable Lie algebra, providing a counterexample despite the fact that the centre is one-dimensional. Besides, he raised a question of generalization of this method for any completely solvable Lie algebra.

On the other hand, consider [Ascr ](V, δ, Γ), the McConnell algebra for the triple (V, δ, Γ) as defined in [4, 14.8.4] and below. McConnell in [3] described the completely prime quotients of the enveloping algebra of a solvable Lie algebra in terms of [Ascr ](V, δ, Γ), and found a complete set of invariants to separate them. In [2], A. Joseph raised the question whether the fields of fractions of these McConnell algebras remain non-isomorphic. The purpose of this note is to extend the work of L. Makar-Limanov reported in [2, Section 6], and so provide an integer-valued invariant which, for McConnell algebras defined over ℤ, says precisely when this skew-field is isomorphic to a Weyl skew-field: this number has simply to be positive. This result therefore gives a large supply of skew-fields which ‘resemble’ a Weyl skew-field very nearly, but nevertheless are not isomorphic to it.

In order to present the results of this note, we begin with some definitions. Consider a differential system

formula here

where I⊆ℝ is an open interval, and f(t, x), (t, x)∈I×ℝn, is a continuous vector function with continuous first derivatives δfr/δxs, r, s=1, 2, …, n.

Let Dxf(t, x), (t, x)∈I×ℝn, denote the Jacobi matrix of f(t, x), with respect to the variables x1, …, xn. Let x(t, t0, x0), t∈I(t0, x0) denote the maximal solution of the system (1) through the point (t0, x0)∈I×ℝn.

For two vectors x, y∈ℝn, we use the notations x>y and x[Gt ]y according to the following definitions:

formula here

An n×n matrix A=(ars) is called reducible if n[ges ]2 and there exists a partition

formula here

(p[ges ]1, q[ges ]1, p+q=n) such that

formula here

The matrix A is called irreducible if n=1, or if n[ges ]2 and A is not reducible.

The system (1) is called strongly monotone if for any t0∈I, x1, x2∈ℝn

formula here

holds for all t>t0 as long as both solutions x(t, t0, xi), i=1, 2, are defined. The system is called cooperative if for all (t, x)∈I×ℝn the off-diagonal elements of the n×n matrix Dxf(t, x) are nonnegative.

A dinilpotent group is a group that can be written as the product of two nilpotent subgroups. There is an extensive literature dealing with such groups (see, for example, the recent book of Amberg, Franciosi and de Giovanni [1]). In 1955, Itô proved in [6] that the product of two abelian groups is always metabelian, and in the following year, Hall and Higman proved, as a special case of [3, Theorem 1.2.4], that if G=AB is a finite soluble group with A and B nilpotent of coprime orders and classes c and d respectively, then G has derived length at most c+d. Wielandt [9] showed that if the finite group G=AB has A and B nilpotent and of coprime orders, then G is necessarily soluble. Kegel [7] then showed that the condition that the orders be coprime was unnecessary. A natural next question to ask is if the derived length of a finite dinilpotent group is bounded by some function of the classes of the factors, and, in the light of the Hall–Higman result and the result of Itô, the following conjecture seems natural. (The first explicit reference to this conjecture that we know of is in Kegel [8]; see also, for example, Problem 5 on page 36 of [1].)

In this paper we show that, given 0[les ]m, n[les ]∞, any monoid can be embedded into a monoid with left cohomological dimension m and right cohomological dimension n. The embedding can be chosen to preserve various monoid-theoretical properties.

The structural stability of constrained polynomial differential systems of the form a(x, y)x′+b(x, y)y′=f(x, y), c(x, y)x′+d(x, y)y′=g(x, y), under small perturbations of the coefficients of the polynomial functions a, b, c, d, f and g is studied. These systems differ from ordinary differential equations at ‘impasse points’ defined by ad−bc=0. Extensions to this case of results for smooth constrained differential systems [7] and for ordinary polynomial differential systems [5] are achieved here.

Let Γ be a Fuchsian group. We denote by ax(Γ) the set of axes of hyperbolic elements of Γ. Define Fuchsian groups Γ1 and Γ2 to be isoaxial if ax(Γ1)=ax(Γ2). The main result in this note is to show (see Section 2 for definitions) the following.

Theorem 1.1. Let Γ1and Γ2be isoaxial arithmetic Fuchsian groups. Then Γ1and Γ2are commensurable.

This result was motivated by the results in [6], where it is shown that if Γ1 and Γ2 are finitely generated non-elementary Fuchsian groups having the same non-empty set of simple axes, then Γ1 and Γ2 are commensurable. The general question of isoaxial was left open. Theorem 1.1 is therefore a partial answer. Arithmetic Fuchsian groups are a very special subclass of Fuchsian groups (for example, there are at most finitely many conjugacy classes of such groups of a fixed signature), and the general case at present seems much harder to resolve.

The technical result of this paper which implies Theorem 1.1 is Theorem 2.4 (below), proved in Section 4. Denoting the commensurability subgroup by Comm(Γ) and the subgroup of PGL(2, R) which preserves the set ax(Γ) by Σ(Γ) (careful definitions are given below), we have the following theorem which describes the commensurability subgroup of an arithmetic Fuchsian group geometrically.

Theorem 2.4. Let Γ be an arithmetic Fuchsian group. Then Comm(Γ)=Σ(Γ).

The problem of ‘isoaxial implies commensurable’ is like a dual problem to ‘isospectral implies commensurable’. It was shown in [8] that this latter statement holds for arithmetic Fuchsian groups. As in the case considered here, the general statement about isospectrality implying commensurability is still open.

In this paper we are concerned with the asymptotic description of the sum of digits function of canonical number systems in number fields. We derive an asymptotic formula for its summatory function whose second term contains a periodic fluctuation. In imaginary quadratic fields this function is uniformly distributed with respect to the argument.

Using the BMO-H1 duality (among other things), D. R. Adams proved in [1] the strong type inequality

∫Mf(x)dHα(x) [les ]C∫[mid ]f(x)[mid ]dHα(x), 0<α<n, (1)

where C is some positive constant independent of f. Here M is the Hardy–Littlewood maximal operator in ℝn, Hα is the α-dimensional Hausdorff content, and the integrals are taken in the Choquet sense. The Choquet integral of ϕ[ges ]0 with respect to a set function C is defined by

formula here

Precise definitions of M and Hα will be given below. For an application of (1) to the Sobolev space W1, 1 (ℝn), see [1, p. 114].

The purpose of this note is to provide a self-contained, direct proof of a result more general than (1).

Let A be an order integral over a valuation ring V in a central simple F-algebra, where F is the fraction field of V. We show that (a) if (Vh, Fh) is the Henselization of (V, F), then A is a semihereditary maximal order if and only if A[otimes ]VVh is a semihereditary maximal order, generalizing the result by Haile, Morandi and Wadsworth, and (b) if J(V) is a principal ideal of V, then a semihereditary V-order is an intersection of finitely many conjugate semihereditary maximal orders; if not, then there is only one maximal order containing the V-order.

Extending earlier work, we establish the finiteness of the number of two-generator arithmetic Kleinian groups with one generator parabolic and the other either parabolic or elliptic. We also identify all the arithmetic Kleinian groups generated by two parabolic elements. Surprisingly, there are exactly 4 of these, up to conjugacy, and they are all torsion free.

Let T be the Volterra operator on L2[0, 1]

formula here

where f∈L2[0, 1], 0[les ]x[les ]1. It is well known that ∥n!Tn∥=O(1/n!). In a recent paper [1], D. Kershaw has proved that

formula here

a result which was first conjectured by Lao and Whitley in [2]. It is easy to prove that

formula here

For completeness, we give the proof using the familiar Schmidt norm estimate for the norm of an integral operator (see Section 2 below). Kershaw proves that

formula here

by analysing the special positivity preserving properties of T*T. He uses one of the many abstract theorems on eigenvalues and eigenfunctions of compact operators which preserve a cone. In this paper we shall reprove (1), giving a short and direct proof of (2).

Using techniques from geometric measure theory and descriptive set theory, we prove a general result concerning sets in the plane which meet each straight line in exactly two points. As an application, we show that no such ‘two-point’ set can be expressed as the union of countably many rectifiable sets together with a set with Hausdorff 1-measure zero. Also, as a corollary, we show that no analytic set can be a two-point set provided that every purely unrectifiable set meets some line in at least three points. Some generalizations are given to ‘n-point’ sets and some other geometric constructions.

Two-sided bounds for the heat kernel of the Schrödinger operator with a singular potential are obtained with explicit constants.

Some years ago, Blatter [1] gave a result of the form

formula here

for any function f regular and univalent in D[ratio ][mid ]z[mid ]<1, where ρ is the hyperbolic distance between z1 and z2. Kim and Minda [5] pointed out that the multiplier on the right is incorrect. They say that Blatter's proof gives the correct multiplier, but Blatter's formula for ρ in terms of z1, z2 is wrong. Kim and Minda proved the generalized formula

formula here

where D1(f)=f′(z) (1−[mid ]z[mid ]2), valid for p[ges ]P with some P, 1<P[les ]3/2. In each case there was an appropriate equality statement. Kim and Minda made the important and easily verified remark that these problems are linearly invariant in the sense that if the result is proved for f, then it follows for f˜=UfT, where U is a linear transformation of the plane onto itself and T is a linear transformation of D onto itself. This means that we need to prove such results only in an appropriately normalized context.

The theory of isometric group actions on R-trees is extended to actions by homeomorphisms with the following non-nesting property: no group element maps an arc properly into itself. A finitely presented group acting freely by homeomorphisms on an R-tree is free abelian or splits over a (possibly trivial) cyclic group.

In this paper we find the norm of powers of the indefinite integral operator V, acting on L2(0, 1). This answers a question raised by Halmos, and supplements some recent results of Manakov in [9]. Using results of Stepanov in [13], we show that the operator norm of Vn is asymptotically equal to the Hilbert–Schmidt norm as n→∞.

We study the conditions for truncated symmetric products of manifolds to be manifolds. In particular, we show that suitably defined spaces of systems of real roots of real polynomials are homeomorphic to real projective spaces.

We study zeros of elliptic integrals I(h)=∫∫H[les ]hR(x, y)dx dy, where H(x, y) is a real cubic polynomial with a symmetry of order three, and R(x, y) is a real polynomial of degree at most n. It turns out that the vector space [Ascr ]n formed by such integrals is a Chebishev system: the number of zeros of each elliptic integral I(h)∈[Ascr ]n is less than the dimension of the vector space [Ascr ]n.

Let S be an elementary operator on a C*-algebra with minimal length [lscr ](S)[ges ]1. Then S is completely positive if and only if S is n-positive for some natural number n[ges ]([lscr ](S)−1)/2.