We are going to investigate translation invariant derivations on Lp spaces of locally compact abelian groups, 1[les ]p<∞. By these we mean densely defined closed linear operators which commute with translations and obey a Leibniz rule, that is, T(fg) =(Tf)·g+f·(Tg); see Definition 1 for details.

The original motivation for studying these operators was to find an abstract description of constant coefficient partial differential operators as a link to perturbation theory which is usually formulated in terms of abstract operator theoretic notions. This fits, for example, Schrödinger operator theory (as in [1]). Here, in view of the applications to perturbation theory, the point is that this identification is not just a formal one but includes assertions about domains (Theorem 1).

In this paper, however, we shall concentrate on groups other than ℝn.

The number (up to isomorphism) of positive-definite, even, unimodular lattices of rank 8r grows rapidly with r. However, Bannai [1] has shown that, when counted according to weight, those with non-trivial automorphisms make up a fraction of the whole, which goes rapidly to zero as r→∞. Therefore it is of some interest to produce families of positive-definite, even, unimodular lattices with large automorphism groups and unbounded ranks.

Suppose that G is a finite group and V is an irreducible ℚ[G]-module such that V[otimes ]ℝ is still irreducible. Then, as observed by Gross [8], the space of G-invariant symmetric bilinear forms on V is one-dimensional and is necessarily generated by a positive-definite form, unique up to scaling by non-zero positive rationals. Thompson [23] showed that, if V is also irreducible modp for all primes p, then it contains an invariant lattice (unique up to scaling) which is even and unimodular with appropriate scaling of the quadratic form. Examples arising in this manner are the E8-lattice of rank 8, the Leech lattice of rank 24 and the Thompson–Smith lattice of rank 248. Gow [6] has also constructed some examples associated with the basic spin representations of 2An and 2Sn.

We use generalized power series to construct algebraically a nonstandard model of the theory of the real field with exponentiation. This model enables us to show the undefinability of the zeta function and certain non-elementary and improper integrals. We also use this model to answer a question of Hardy by showing that the compositional inverse to the function (log x) (log log x) is not asymptotic as x→+∞ to a composition of semialgebraic functions, log and exp.

This paper is devoted to the study of different types of twisting points of conformal maps. We define the sets of gyration, spiral and oscillation points and we prove, in the case that f is conformal almost nowhere, that the above sets have Hausdorff dimension one. Also we define points of bounded radial oscillation. It is proved that there are always points of π-bounded radial oscillation but there exists a conformal map without points of small bounded radial oscillation.

Suppose that [Pscr ] is a distribution of N points in the unit square U=[0, 1]2. For every x=(x1, x2)∈U, let B(x)=[0, x1]×[0, x2] denote the aligned rectangle containing all points y=(y1, y2)∈U satisfying 0[les ]y1[les ]x1 and 0[les ]y2[les ]x2. Denote by Z[[Pscr ]; B(x)] the number of points of [Pscr ] that lie in B(x), and consider the discrepancy function

D[[Pscr ]; B(x)]=Z[[Pscr ]; B(x)]−Nμ(B(x)),

where μ denotes the usual area measure.

In this paper we study the value distribution of the least primitive root to a prime modulus, as the modulus varies. For each odd prime number p, we shall denote by g(p) and G(p) the least primitive root and the least prime primitive root (mod p), respectively. Numerical examples show that, in most cases, g(p) and G(p) are very small (cf. §4). We can support this observation by a probabilistic argument [14, §1]. In fact, on the assumption of a good distribution of the primitive residue classes modulo p, we can surmise that

formula here

where π(x) denotes the number of primes not exceeding x.

A univalent harmonic map of the unit disk Δ[ratio ]={z∈[Copf ][ratio ][mid ]z[mid ]<1} is a complex-valued function f(z) on Δ that satisfies Laplace's equation fzz[bar]=0 and is injective. The Jacobian J[ratio ]=[mid ]fz[mid ]2 −[mid ]fz[bar][mid ]2 of a univalent harmonic map can never vanish [18], and so we might as well assume that J>0 throughout Δ. Then [mid ]fz[mid ]>0 and a short computation verifies that the analytic dilatation ω[ratio ] =f[bar]z[bar]/fz is indeed an analytic function, with [mid ]ω[mid ]<1 since J>0. Clearly ω≡0 when f is a conformal map, and in general the dilatation ω measures how far f is from being conformal. Also, if ω happens to be the square of an analytic function, then f ‘lifts’ to give an isothermal coordinate map for a minimal surface, and in that case i/√ω equals the stereographic projection of the Gauss map of the surface.

A method is given to solve any Thue–Mahler equation when the coefficients and variables come from the ring of integers of a number field. The method involves using an algorithm for solving an S-unit equation and a method of reducing the number of exponential variables. The method is used to solve the equation

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One of the principal impulses for this paper was the consideration of the question, first raised in 1973 by G. Baumslag [2], of whether every finitely generated Abelian by polycyclic group can be embedded in a finitely presented Abelian by polycyclic group. That this cannot be done follows at once from

THEOREM A. Every finitely presented Abelian by polycyclic group has a metanilpotent normal subgroup of finite index.

In this paper, we consider a basic question in commutative algebra: if I and J are ideals of a commutative ring S, when does IJ=I∩J? More precisely, our setting will be in a polynomial ring k[x0, …, xn], and the ideals I and J will define subschemes of the projective space ℙnk over k. In this situation, we are able to relate the equality of product and intersection to the behavior of the cohomology modules of the subschemes defined by I and J. By doing this, we are able to prove several general algebraic results about the defining ideals of certain subschemes of projective space.

Our main technique in this paper is a study of the deficiency modules of a subscheme V of ℙn. These modules are important algebraic invariants of V, and reflect many of the properties of V, both geometric and algebraic. For instance, when V is equidimensional and dim V[ges ]1, the deficiency modules of V are invariant (up to a shift in grading) along the even liaison class of V [14, 11, 15, 7], although they do not in general completely determine the even liaison class, except in the case of curves in ℙ3 [14]. On the algebraic side, at least for curves in ℙ3, the deficiency modules have been shown to have connections to the number and degrees of generators of the saturated ideal defining V [12]. One of our main goals in this paper is to extend these results to subschemes of arbitrary codimension in any projective space ℙn.

We now describe the contents of this paper more precisely. In the first section, we set up our notation and give the basic definitions which we will use. Then we prove our main technical result: if I and J define subschemes V and Y, respectively, of ℙn, we relate the quotient module (I∩J)/IJ to the cohomology of V, at least when V and Y meet properly. We are then able to give a different proof of a general statement due to Serre about when there is an equality of intersection and product.

In the second section, we give an extension of Dubreil's Theorem on the number of generators of ideals in a polynomial ring. Specifically, our generalization works for an ideal I defining a locally Cohen–Macaulay, equidimensional subscheme V of any codimension in ℙn, and relates the number of generators of the defining ideal to the length of certain Koszul homologies of the cohomology of V. The results in this section depend crucially on the identification done in Section 1 of the intersection modulo the product.

Finally, in Section 3, we give an extension of a surprising result of Amasaki [1] showing a lower bound for the least degree of a minimal generator of the ideal of a Buchsbaum subscheme. Originally, Amasaki gave a bound in the case of Buchsbaum curves in ℙ3 (and later gave a natural extension to Buchsbaum codimension 2 subschemes of ℙn [2]). Easier proofs were subsequently given by Geramita and Migliore in [6], based on a determination of the free resolution of the ideal from a resolution of its deficiency module. For Buchsbaum codimension 2 subschemes of ℙn whose intermediate cohomology vanishes, we are able to extend these considerations.

In this paper we continue our study of the Canonical Element Conjecture (henceforth C.E.C.) via the dualizing complex. Throughout the work (A, m, k) will denote a noetherian complete local ring A of dimension n, m its maximal ideal and k=A/m. Since A is complete, we can find a complete local Gorenstein ring (R, mR, k) (complete intersection) such that dim R=dim A and A=R/I. Let Ω denote the canonical module of A, that is, Ω=HomR (A, R), which may be identified with the annihilator of I in R, an ideal of R. When A is a domain, we change notation and denote I by P; in this case P is a height 0 prime ideal of R.

All rings in this paper are commutative, with identity. If the ring R is either finitely generated, or semi-local (by which I mean Noetherian and possessing only finitely many maximal ideals), then R has only a finite number of ideals of each finite index. Thus it makes sense to study the function n[map ]an(R), where an(R) denotes the number of ideals of index n in R. In an earlier note [10], I determined an(R) for certain 2-dimensional domains R. The results to be discussed here are less precise, but more general; in particular, they deal with submodules of a finitely generated module. This makes it possible to translate them into results about the growth of normal subgroups in certain metabelian groups, the original motivation for this work: for example, we determine the finitely generated metabelian groups with ‘polynomial normal subgroup growth’ (Theorem 6.1).

This work was motivated in part by the following general question: given an ideal I in a Cohen–Macaulay (abbreviated to CM) local ring R such that dim R/I=0, what information about I and its associated graded ring can be obtained from the Hilbert function and Hilbert polynomial of I? By the Hilbert (or Hilbert–Samuel) function of I, we mean the function HI(n) =λ(R/In) for all n[ges ]1, where λ denotes length. Samuel [23] showed that for large values of n, the function HI(n) coincides with a polynomial PI(n) of degree d=dim R. This polynomial is referred to as the Hilbert, or Hilbert–Samuel, polynomial of I. The Hilbert polynomial is often written in the form

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where e0(I), [ctdot ], ed(I) are integers uniquely determined by I. These integers are known as the Hilbert coefficients of I.

This work was started as an attempt to apply theory from noncommutative graded algebra to questions about the holonomy algebra of a hyperplane arrangement. We soon realized that these algebras and their deformations, which form a class of quadratic graded algebras, have not been studied much and yet are interesting to algebra, arrangement theory and combinatorics.

Let G be a group, and let Fn[G] be the free G-group of rank n. Then Fn[G] is just the natural non-abelian analogue of the free ℤG-module of rank n, and correspondingly the group Φn(G) of equivariant automorphisms of Fn[G] is a natural analogue of the general linear group GLn(ℤG). The groups Φn(G) have been studied recently in [4, 8, 5]. In particular, in [5] it was shown that if G is not finitely presentable (f.p.) then neither is Φn(G), and conversely, that Φn(G) is f.p. if G is f.p. and n≠2. It is a common phenomenon that for small ranks the automorphism groups of free objects may behave unstably (see the survey article [2]), and the main aim of the present paper is to show that this turns out to be the case for the groups Φ2(G).

Nash-Williams [6] formulated a condition that is necessary and sufficient for a countable family [Ascr ]=(Ai)i∈I of sets to have a transversal. In [7] he proved that his criterion applies also when we allow the set I to be arbitrary and require only that [xcap ]i∈JAi=[emptyv ] for any uncountable J⊆I. In this paper, we formulate another criterion of a similar nature, and prove that it is equivalent to the criterion of Nash-Williams for any family [ufr ]. We also present a self-contained proof that if [xcap ]i∈JAi=[emptyv ] for any uncountable J⊆I, then our condition is necessary and sufficient for the family [ufr ] to have a transversal.

We develop a theory of harmonic analysis and duality for finite commutative hypergroups by considering somewhat more general objects called signed hypergroups. A notion of entropy is defined, and a Second Law of Thermodynamics is established. Applications to group theory and to the fusion rule algebras of conformal field theory are given.

This paper brings together two apparently unrelated results about locally compact groups G by giving them a common proof. The first concerns the number of topologically left invariant means on L∞(G), while the second states that the topological centre of the largest semigroup compactification of G is simply G itself. On the way, we introduce as vital tools some new compactifications of the half line ([0, ∞), +), we produce a right invariant pseudometric on a compactly generated G for which the bounded sets are precisely the relatively compact sets, and we receive striking confirmation that some algebraic properties of semigroups can be transferred by maps which are quite far from being homomorphisms.

We describe a general technique for embedding certain amalgamated products into direct products. This technique provides us with a way of constructing a host of finitely presented subgroups of automatic groups which are not even asynchronously automatic. We can also arrange that such subgroups satisfy, at best, an exponential isoperimetric inequality.

Large classes of integrable Hamiltonian systems can be expressed as systems over coadjoint orbits in a loop algebra defined over a semi-simple Lie algebra [gfr ]. These systems can then be integrated via the classical, symplectic Liouville–Arnold method. On the other hand, the existence of spectral curves as constants of motion allows one to integrate these systems in terms of flows of line bundles on the curves. This note links the symplectic geometry of the coadjoint orbits with the algebraic geometry of these curves for arbitrary semi-simple [gfr ], which then allows us to reconcile the two integration methods.