Let r1(n) denote the number of representations of the positive integer n as the sum of a square of a positive integer and the square of a positive prime number. We prove an asymptotic evaluation for [sum ]n[les ]xr1(n)2, as x → ∞, thereby improving upon a O-result of Rieger [7]. We further prove an asymptotic formula for the number of positive integers n [les ] x with r1(n) [ges ] 1, which answers a question stated at the end of [7]. Our result in particular shows that for almost all integers represented as the sum of a square and a square of a prime, the representation is unique.

Let I = [α, β] be a subinterval of [0, 1]. For each positive integer Q, we denote by [Fscr ]I(Q) the set of Farey fractions of order Q from I, that is

and order increasingly its elements γj = aj/qj as α [les ] γ1 < γ2 < … < γNI(Q) [les ] β. The number of elements of [Fscr ]I(Q) is

We simply let [Fscr ](Q) = [Fscr ][0,1](Q), N(Q) = N[0,1](Q).

Farey sequences have been studied for a long time, mainly because of their role in problems related to diophantine approximation. There is also a connection with the Riemann zeta function which has motivated their study. Farey sequences seem to be distributed as uniformly as possible along [0, 1]; a way to prove it is to show that

for all ε > 0, as Q → ∞. Yet this is a very strong statement, as Franel and Landau [3, 4] have shown that (1·3) is equivalent to the Riemann Hypothesis.

Our object here is to investigate the distribution of spacings between Farey points in subintervals of [0, 1]. Various results related to this problem have been obtained by [2, 3, 5–8, 10–13].

1·1. Background. Throughout this note Q stands for a finitely generated multiplicative Abelian group of torsion-free rank n, R for a commutative ring with 1, and M for a finitely generated RQ-module. The geometric invariant ΣM of M was introduced in [BS1, BS2]. It can be viewed as a subset of the ℝ-vector space of all (additive) characters of Q, Q* = Hom(Q, ℝ) ≅ ℝn, as follows: for every character χ: Q → ℝ one considers the submonoid Qχ = {q ∈ Q [mid ] χ(q) [ges ] 0} of Q and puts

Note that 0 ∈ ΣM. It is often convenient to work with the complement ΣcM of ΣM in Q*.

The geometric invariant ΣM has been investigated for two reasons. Firstly, if R is a Dedekind domain, then ΣM turns out to be a polyhedral (i.e. a finite union of finite intersection of (open) vector half spaces) subset of Q*. This rather subtle fact was conjectured, for R a field, by Bergman [B] and established by Bieri and Groves in [BG2]; it opens the possibility for computations and imposes arithmetic restrictions on automorphisms of M. Secondly, for R = ℤ, ΣM contains interesting information on the (metabelian) groups G which are extensions of M by Q (i.e. G fits into a short exact sequence M [rarrtl ] G [Rarr ] Q). In [BS1] it is proved that G has a finite presentation if and only if ΣM ∪ −ΣM = Q*. A number of attempts have been undertaken to extend this result to a characterization of the higher dimensional finiteness property that G is of typeA group of G is of type FPm if the trivial G-module ℤ admits a free resolution F [Rarr ] ℤ with finitely generated m-skeleton. For metabelian groups G it is known, by [BS1], that FP2 is equivalent to finite presentability.FPm for m > 2, and they all revolve around the following:

FPm-Conjecture: G is of type FPmif and only if 0 ∈ Q* is not in the convex hull of m points of ΣcM.

The conjecture appeared in [BG1].

In this paper, we will study the local cohomology modules HiI(S) of a polynomial ring S = k[x1, …, xn] with supports in a (radical) monomial ideal I. When S/I is a Cohen–Macaulay ring of dimension d (more generally, if Extn−d(S/I, [wfr ]S) is Cohen–Macaulay), we can ‘visualize’ a ℤn-graded minimal injective resolution of Hn−dI(S) using Stanley–Reisner's simplicial complex of I.

Let I be an ideal of a commutative ring R and M an R-module. It is well known that the I-adic completion functor ΛI defined by ΛI(M) = lim←tM/ItM is an additive exact covariant functor on the category of finitely generated R-modules, provided R is Noetherian. Unfortunately, even if R is Noetherian, ΛI is neither left nor right exact on the category of all R-modules. Nevertheless, we can consider the sequence of left derived functors {LIi} of ΛI, in which LI0 is right exact, but in general LI0 ≠ ΛI. Therefore the computation of these functors is in general very difficult. For the case that R is a local Noetherian ring with the maximal ideal [mfr ] and I is generated by a R-regular sequence, Matlis proved in [9, 10] that

where D(−) = HomR(−; E(R/[mfr ])) is the Matlis dual functor, and that

In [18, 5] A.-M. Simon shows that LIo(M) = M and LIi(M) = 0 for i > 0, provided that M is complete with respect to the I-adic topology.

Later, Greenlees and May [3] using the homotopy colimit, or telescope, of the cochain of Koszul complexes to define so-called local homology groups of a module M (over a commutative ring R) by

where x is a finitely generated system of I. Then they showed, under some conditions on x which are satisfied when R is Noetherian, that HI[bull ](M) ≅ LI[bull ](M). Recently, Tarrío, López and Lipman [1] have presented a sheafified derived-category generalization of Greenlees–May results for a quasi-compact separated scheme. The purpose of this paper is to study, with elementary methods of homological and commutative algebra, local homology modules for the category of Artinian modules over Noetherian rings.

According to the result by Dyer and Formanek [4], the automorphism group of a finitely generated free two-step nilpotent group is complete except in the case when this group is a one- or three-generator (the three-generator groups have automorphism tower of height 2). The purpose of this paper is to prove that the automorphism group of an infinitely generated free two-step nilpotent group is also complete. (Recall that a group G is said to be complete if G is centreless and every automorphism is inner.)

The paper may be considered as a contribution to the study of automorphism towers of relatively free groups.

Let G be a nilpotent, connected and simply connected real Lie group and let Γ be a lattice in G. A necessary and sufficient condition is given for a Lie subgroup of G to act ergodically on Γ[setmn ]G. As an application, we characterize the irreducible unitary representations of G for which the restriction to Γ remains irreducible.

Suppose M is an oriented 3-manifold. A Dehn surgery on M (defined below) is a process by which M is altered by deleting a tubular neighbourhood of an embedded circle and replacing it again via some diffeomorphism of the boundary torus. It was shown by Lickorish [Li] and Wallace [Wa] that any closed oriented connected 3-manifold can be obtained from any other such manifold by a finite sequence of Dehn surgeries. Thus under this equivalence relation all closed oriented 3-manifolds are equivalent. We shall investigate this same question for more restricted classes of surgeries. In particular we shall insist that our Dehn surgeries preserve the integral (or rational) homology groups. Specifically, if M0 and M1 have isomorphic integral (respectively rational) homology groups, is there a sequence of Dehn surgeries, each of which preserves integral (respectively rational) homology, that transforms M0 to M1? What is the situation if we further restrict the Dehn surgeries to preserve more of the fundamental group? Is there a difference if we require ‘integral’ surgeries? We also show that these Dehn surgery relations are strongly connected to the following questions concerning another point of view towards understanding 3-manifolds. Is there a Heegard splitting of M0, M0 = H1 ∪fH2 (Hi are handlebodies of genus g and f is a homeomorphism of their common boundary surface), and a homeomorphism g of ∂H1 such that M1 has a Heegard splitting using g ∘ f as the identification? Since there are many natural subgroups of the mapping class group, such as the Torelli subgroup and the ‘Johnson subgroup’, one can ask the same question where g is restricted to lie in one of these subgroups. This is related to work of Morita on Casson's invariant for homology 3-spheres [Mo1]. Even under these restrictions it has been known for some time that any homology 3-sphere is related to S3. This fact has been used to define, calculate and understand invariants of homology 3-spheres (such as Casson's invariant) by choosing such a ‘path to S3’ in the ‘space’ of 3-manifolds.

In 1922–23, Julia and Fatou proved that any 2 rational functions f and g of degree at least 2 such that f(g(z)) = g(f(z)), have the same Julia set. Baker then asked whether the result remains true for nonlinear entire functions. In this paper, we shall show that the answer to Baker's question is true for almost all nonlinear entire functions. The method we use is useful for solving functional equations. It actually allows us to find out all the entire functions g which permute with a given f which belongs to a very large class of entire functions.

For a general family of weight functions, a relation between orthogonal polynomials on the unit sphere in ℝd+m and orthogonal polynomials on the unit ball in ℝd is discussed and used to show that the summability of the Fourier orthogonal series on the unit ball follows from the summability on the unit sphere. Convergence of the Cesàro summability is established for a number of weight functions on the sphere and on the ball.

In this paper we apply pointwise subsequence ergodic theory to show that certain sequences of integers constructed randomly are multiply intersective. Analogous results for sequences constructed from Hardy fields are also indicated.

Let [Efr ] be a Banach space with a normalized, 1-unconditional basis. Each operator on the [Efr ]-direct sum of a sequence ([Xfr ]i)i∈ℕ of Banach spaces corresponds to an infinite matrix. We study whether this correspondence is multiplicative, in which case we say that matrix multiplication works. We prove that matrix multiplication works if at least one of the following two conditions is satisfied:

(i) for each i ∈ ℕ, each operator from [Xfr ]i to [Efr ] is compact;

(ii) the basis of [Efr ] is shrinking and, for each i ∈ ℕ, each operator from [Efr ] to [Xfr ]i is compact.

In the case where [Efr ] is either c0 or [lscr ]p, where 1 [les ] p < ∞, the converse also holds.

We show that, in some cases, the projective and the injective tensor products of two Banach spaces do not have the Dunford–Pettis property (DPP). As a consequence, we obtain that (c0 &[otimes ]circ;πc0)** fails the DPP. Since (c0 &[otimes ]circ;πc0)* does enjoy it, this provides a new space with the DPP whose dual fails to have it. We also prove that, if E and F are [Lscr ]1-spaces, then E &[otimes ]circ;ε has the DPP if and only if both E and F have the Schur property. Other results and examples are given.