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Let φ be ab ounded linear functional on A ,w hereA is a commutative Banach algebra, then the bilinear functional ˇ φ is defined as ˇ φ(a, b )= φ(ab) − φ(a)φ(b )f or eacha and b in A .I f thenorm of ˇ φ is small then φ is approximately multiplicative, and it is of interest whether or not � ˇ φ� being small implies that φ is near to a multiplicative functional. If this property holds for a commutative Banach algebra then A is an AMNM algebra (approximately multiplicative functionals are near multiplicative functionals). The main result of the paper shows that C N [0, 1] M (the complex valued functions defined on [0, 1] M with all Nth order partial derivatives continuous) is AMNM. It is also shown that a similar proof can be applied to certain Lipschitz algebras.

In this paper we prove an almost-orthogonality principle for maximal operators over arbitrary sets of directions in R 2 . Namely, we obtain L p -bounds for an operator of this type from the corresponding L p -bounds of the maximal functions associated to a certain partition of the set of directions, and from the particular structure of this partition. We give applications to several types of maximal operators.

Removable singularities for Hardy spaces Hp(Ω) = {f ∈ Hol(Ω): |f|p ⩽ u in Ω for some harmonic u}, 0 < p < ∞ are studied. A set E = Ω is a weakly removable singularity for Hp(Ω\E) if Hp(Ω\E) ⊂ Hol(Ω), and a strongly removable singularity for Hp(Ω\E) if Hp(Ω\E) = Hp(Ω). The two types of singularities coincide for compact E, and weak removability is independent of the domain Ω. The paper looks at differences between weak and strong removability, the domain dependence of strong removability, and when removability is preserved under unions. In particular, a domain Ω and a set E ⊂ Ω that is weakly removable for all Hp, but not strongly removable for any Hp(Ω\E), 0 < p < ∞, are found. It is easy to show that if E is weakly removable for Hp(Ω\E) and q > p, then E is also weakly removable for Hq(Ω\E). It is shown that the corresponding implication for strong removability holds if and only if q/p is an integer. Finally, the theory of Hardy space capacities is extended, and a comparison is made with the similar situation for weighted Bergman spaces.

LetG be a compact group andC(G )b e theC-algebra of continuous complex-valued functions onG. The paper constructs an imbedding of the Fourier algebra A(G )o fG into the algebra V(G )= C(G) h C(G) (Haagerup tensor product) and deduces results about parallel spectral synthesis, generalizing a result of Varopoulos. It then characterizes which diagonal sets in GG are sets of operator synthesis with respect to the Haar measure, using the denition of operator synthesis due to Arveson. This result is applied to obtain an analogue of a result of Froelich: a tensor formula for the algebras associated with the pre-orders dened by closed unital subsemigroups of G.

Throughout this paper, we shall suppose that all algebraic number fields, namely, all algebraic extensions over the rational field , are contained in the complex field . Let be the set of all prime numbers. For any algebraic number field , let denote the ideal class group of and, writing for the maximal real subfield of , let denote the kernel of the norm map from to the ideal class group of ; for each , let denote the -class group of , that is, the -primary component of , and let denote the -primary component of . Furthermore, for each , we denote by the ring of -adic integers.

A transitive nite permutation group is called elusive if it contains no nontrivial semiregular subgroup. The purpose of the paper is to collect known information about elusive groups. The main results are recursive constructions of elusive permutation groups, using various product operations and ane group constructions. A brief historical introduction and a survey of known elusive groups are also included. In a sequel, Giudici has determined all the quasiprimitive elusive groups. Part of the motivation for studying this class of groups was a conjecture due to Maru si c, Jordan and Klin asserting that there is no elusive 2-closed permutation group. It is shown that the constructions given will not build counterexamples to this conjecture.

In this paper we shall prove that the measure algebra M(G) of a locally compact group G is amenable as a Banach algebra if and only if G is discrete and amenable as a group. Our contribution is to resolve a conjecture by proving that M(G) is not amenable in the case where the group G is not discrete. Indeed, we shall prove a much stronger result: the measure algebra of a non-discrete, locally compact group has a non-zero, continuous point derivation at a certain character on the algebra.

The paper computes the Hausdorff dimension of sets of real numbers which are close to infinitely many real algebraic integers of bounded degree. It also investigates the distribution of real algebraic integers of bounded degree, which are proved to be evenly spaced.

Suppose that A is a pointed CW-complex. The paper looks at how difficult it is to construct an A-cellular space B from copies of A by repeatedly taking homotopy colimits, this is determined by an ordinal number called the complexity of B. Studying the complexity leads to an iterative technique, based on resolutions, for constructing the A-cellular approximation CWA(X) of an arbitrary space X.

Much of the recent progress in the representation theory of infinitesimal group schemes rests on the application of algebro-geometric techniques related to the notion of cohomological support varieties (cf. [6, 8–10]). The noncohomological characterization of these varieties via the so-called rank varieties (see [21, 22]) involves schemes of additive subgroups that are the infinitesimal counterparts of the elementary abelian groups. In this note we introduce another geometric tool by considering schemes of tori of restricted Lie algebras. Our interest in these derives from the study of infinitesimal groups of tame representation type, whose determination [12] necessitates the results to be presented in §4 and §5 as well as techniques from abstract representation theory.In contrast to the classical case of complex Lie algebras, the information on the structure of a restricted Lie algebra that can be extracted from its root systems is highly sensitive to the choice of the underlying maximal torus. Schemes of tori obviate this defect by allowing us to study algebraic families of root spaces. Accordingly, these schemes may also shed new light on various aspects of the structure theory of restricted Lie algebras. We intend to pursue these questions in a forthcoming paper [13], and focus here on first applications within representation theory.

The paper gives examples of knots with equal knot polynomials, but distinct signatures, 4-genera, double branched cover homology groups and unknotting numbers. This generalizes some examples given by Lickorish and Millett. It is also shown that there are knots with minimal (crossing number) diagrams of different unknotting number (thus answering a question of Bleiler), and an alternative proof is given of Rudolph's result that there are knots of [les ] 15n crossings with unit Alexander polynomial and 4-genus or unknotting number n.

The paper considers the iterated function systems of similitudes which satisfy a separation condition weaker than the open set condition, in that it allows overlaps in the iteration. Such systems include the well-known Bernoulli convolutions associated with the PV numbers, and the contractive similitudes associated with integral matrices. The latter appears frequently in wavelet analysis and the theory of tilings. One of the basic questions is studied: the absolute continuity and singularity of the self-similar measures generated by such systems. Various conditions to determine the dichotomy are given.

Let k[x]((x)) be the polynomial ring k[x] localized in the maximal ideal (x) subset of k[x]. The paper studies the Hilbert functor parametrizing ideals of colength n in this ring having support at ...

In this paper we establish some subnormal embeddings of groups into groups with additional properties; in particular embeddings of countable groups into 2-generated groups with some extra properties. The results obtained are generalizations of theorems of P. Hall, R. Dark, B. Neumann, Hanna Neumann, G. Higman on embeddings of that type. Considering subnormal embeddings of finite groups into finite groups with additional properties we bring some illustration to a result of H. Heineken and J. Lennox., 10 pages

We examine the commutative rings for which every proper ideal has a proper prime factor. Among this class of rings, we classify the rings for which every proper ideal is a product of invertible and prime ideals. This continues and corrects a classification begun in an earlier paper

In this paper we provide a minimal constructive integration process of Riemann type which includes the Lebesgue integral and also integrates the derivatives of differentiable functions. We provide a new solution to the classical problem of recovering a function from its derivative by integration, which, unlike the solution provided by Denjoy, Perron and many others, does not possess the generality which is not needed for this purpose.The descriptive version of the problem was treated by A. M. Bruckner, R. J. Fleissner and J. Foran in [2]. Their approach was based on the trivial observation that for the required minimal integral, a function F is the indefinite integral of f if and only if F' = f almost everywhere and there exists a differentiable function H such that F – H is absolutely continuous. They strengthen this definition by proving that F – H can have arbitrary small variation. Nevertheless, their definition still needs a choice of a differentiable function.

Recently Bergelson and Leibman proved an extension to van der Waerden's theorem involving polynomials. They also generalised the Hales–Jewett theorem in a similar way. The paper presents short and purely combinatorial proofs of those results.

in integers x1, . . . , xs has been considered by a number of authors over the last quartercentury, starting with the work of Cook [9] and Pitman [13] on the case t = 2. More recently, Brudern and Cook [8] have shown that the above system is soluble provided that s is sufficiently large in terms of k and t and that the forms F1, . . . , Ft satisfy certain additional conditions. What has not yet been considered is the possibility of allowing the forms F1, . . . , Ft to have different degrees. However, with the recent work of Wooley [18], [20] on the corresponding problem for equations, the study of such systems has become a feasible prospect. In this paper we take a first step in that direction by studying the analogue of the system considered in [18] and [20]. Let λ1, . . . , λs and μ1, . . . , μs be real numbers such that for each i either λi or μi is nonzero. We define the forms F (x) = λ1x 3 1 + · · ·+ λsxs G(x) = μ1x 2 1 + · · ·+ μsxs

If r is an automorphism and d is a r-derivation of a ring R, then the subring of invariants is the set R(d) fl† r ‘ R r d(r) fl 0·. The main result of this paper is ‘let R be a semiprime ring with an algebraic r-derivation d such that R(d) is central; then R is commutative’. This theorem generalizes results on the invariants of automorphisms and derivations and is proved by reducing down to the special cases of automorphisms and derivations.

Nash-Williams [ 6 ] formulated a condition that is necessary and sufficient for a countable family [Ascr ]=( A i ) 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 ∈ J A i =[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 ∈ J A i =[emptyv ] for any uncountable J ⊆ I , then our condition is necessary and sufficient for the family [ufr ] to have a transversal.

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 show that given an affine algebraic group G over a field K and a finite subgroup scheme H of G there exists a finite dimensional G -module V such that V [mid ] H is free. The problem is raised in the recent paper by Kuzucuo˘glu and Zalesskiiˇ [ 15 ] which contains a treatment of the special case in which K is the algebraic closure of a finite field and H is reduced. Our treatment is divided into two parts, according to whether K has zero or positive characteristic. The essence of the characteristic 0 case is a proof that, for given n , there exists a polynomial GL n (ℚ)-module V of dimension 2 n Π p p ( n 2 ) , where the product is over all primes less than or equal to n +1, such that V is free as a ℚ H -module for every finite subgroup H of GL n (ℚ). The module V is the tensor product of the exterior algebra Λ*( E ), on the natural GL n (ℚ)-module E , and Steinberg modules St p , one for each prime not exceeding n +1. The Steinberg modules also play the major role in the case in which K has characteristic p >0 and the key point in our treatment is to show that for a finite subgroup scheme H of a general linear group scheme (or universal Chevalley group scheme) G over K , the Steinberg module St p n for G is injective (and projective) on restriction to H for n [Gt ]0. A curious consequence of this is that, despite the wild behaviour of the modular representation theory of finite groups, one has the following. Let H be a finite group and V a finite dimensional vector space. Then there exists a (well-understood) faithful rational representation π: GL( V )→GL( W ) such that, for each faithful representation ρ: H →GL( V ), the composite πορ: H →GL( W ) is free, in particular all representations πορ are equivalent.

In this paper, we continue our investigation of the distribution of the values of 1x 3 1 + 2x 3 2+sx 3 s at integer points when s = 7 and 8. Under suitable conditions on the real algebraic numbers s it is shown that for any real the inequality j 1x 3 1 +sx 3 s - j

An asymptotic field is a special type of Hardy field in which, module an oracle for constants, one can determine asymptotic behaviour of elements. In a previous paper, it was shown in particular that limits of real Liouvillian functions can thereby be computed. Let F denote an asymptotic field and let f is an element of F. We prove here that if G is meromorphic at the limit of f (which may be infinite) and satisfies an algebraic differential equation over R(x), then F(Gof) is an asymptotic field. Hence it is possible (module an oracle for constants) to compute asymptotic forms for elements of F(Gof). An example is given to show that the result may fail if G has an essential singularity at limf.

Let pc(d) be the critical probability for percolation in Z . In this paper it is shown that limd→∞ 2d pc(d) = 1. The proof uses the properties of a random subgraph of an m-ary d-dimensional cube. If each edge in this cube is included with probability > 1 2d(1−3/m) , then for large d, the cube will have a connected component of size cm for some c > 0. This generalizes a result of Ajtai, Komlos and Szemeredi.

The concepts of a fractional Cartesian product and combinatorial dimension, cast in discrete frameworks of multidimensional lattices, were bases on a measurement of interdependencies between coordinates of points within a given lattice set. In the present paper, we continue the study of this measurement in the framework of [0, 1] J , where J is a positive integer

Let $R$ be an associative ring with centre $Z$. The aim of this paper is to study how the ideal structure of the Lie ring of derivations of $R$, denoted $D(R)$, is determined by the ideal structure of $R$. If $R$ is a simple (respectively semisimple) finite-dimensional $Z$-algebra and δ$(z)$ = 0 for all δ ∈ $D(R)$, then every derivation of $R$ is inner and $D(R)$ is known to be a simple (respectively semisimple) Lie algebra (see [7, 5]). Here we are interested in extending these results to the case where $R$ is a prime or semi-prime ring.

The purpose of this paper is to characterize the complemented translation-invariant subspaces of H (R) in terms of the zero set of the Fourier transform. It is shown that if X is such a subspace then X = I(A) where A is in the ring generated by arithmetic progressions and lacunary sequences and A is e-separated for some E > 0. This proves a conjecture of the author and D. Ullrich.

P. J. CAMERO ANN MD. DEZA1. IntroductionThe lattic oef flats of a matroid or combinatorial geometr cay n be regarde d as asublattice (with rank function) of the lattice of subsets of a set, having the property that,given an element of rank r of the lattice and a point outside it, a unique element of rankr+1 covers both. This paper develop a theorsy of permutation geometries, whic arhesimilarly defined object ins the semilattice of subpefmiitations (partia 1-l 1 mappings)on a set.As well as providing this basic analogy, raatroid theor is relatey d in other ways toour theor oyf permutation geometries In. the semilattice of all subpermutations th, einterval consistin ogf all elements below a given bound is a lattice isomorphi tco thelattice of all subsets of a set. In a permutation geometry, any such lower interva ils thelattice of flats of a matroid.The set of subpermutations, as well as being a semilattice, has additional structure:it is an inverse semigroup, where maximal element (thse permutations) form a group.The most interesting permutatio annd thos geometriese to whic wh e, give mostattention, are the ones whose maximal elements a for subgroum p of the symmetricgroup. (We call a group which arises in this way a geometric group.) Our results includethe determination of doubly transitive geometric groups, and some examples which arenot doubly transitive.Connections also exist with the theories of permutation cliques, sharp permutationgroups, and Jordan groups.The author ars e gratefu tl o C Landaue. r for helpfu l remarks.2. SubpermutationsLet N = {1, 2,...,«}. A subpermutation c of N is a bijection between subset osf N;its height \\c\\ is the cardinalit oy f it domais n c (or of it range)s I. f c and d aresubpermutations, we write c ^ d, and say c is covered by d, if c is the restriction of d to asubset of d. The meet c A d of two subpermutations c and d is the subpermutation edefined on the set of points{iecn d\c{i) = d(i)}and equal to each of c and d on its domain. The inverse c~

In this paper, we continue our work in [9] on the univalence of derivatives of functions defined by gap power series. We consider regular functions, f, defined in some region containing 0, and denote by pn and pJc> the radius of univalence and the radius of convexity, respectively, off@). For ease of notation, we shall sometimes write pn = p(n). Earlier, we considered functions defined by power series with gaps of at least a certain length. By that we meant the following: Suppose F(z) = Cj”=, Aizj. Then F is defined by a power series with gaps at least of length K provided that K is a nonnegative integer such that, if A, # 0 for some n, then Ancj = 0 for 1 < j < K. We proved the following [9].

where the integral is in the appropriate sense; let CL(a, b), D*(a,b) and D(a, b) be the associated separated spaces. Miss W. L. C. Sargent [3] has established several theorems of Banach-Steinhaus type for these spaces; from a modern point of view it is evident that these results may be interpreted as proving that each of the above spaces is barrelled. In this paper we present a direct proof of this observation. In particular these spaces provide the earliest examples of barrelled spaces which are meager and normable; the standard example of such a space [1; p. 157, Ex. 10] is perhaps less natural and less accessible as it is obtained essentially from the fact that the tensor product of two Banach spaces equipped with its projective tensor product topology is barrelled and normable but not, in general, a Baire space. 2. The main theorems