In [9], Shelah introduced a class of first order theories, which he called simple, properly containing the class of stable theories. Here we prove for simple theories, (i) the equivalence of forking and dividing, (ii) the symmetry and transivity of forking.

In this paper, we prove the following theorems. (i) Let G be a graph of minimum degree δ[ges ]5. If G is embeddable in a surface σ and satisfies (δ−5)[mid ]V(G)[mid ]+6χ(σ)[ges ]0, then G is edge reconstructible. (ii) Any graph of minimum degree 4 that triangulates a surface is edge reconstructible. (iii) Any graph which triangulates a surface of characteristic χ[ges ]0 is edge reconstructible.

In this paper we study the L1 mean

formula here

of the exponential sum M(α)= [sum ]n[les ]Xμ(n)e(nα), where μ(n) is the Möbius function and e(x)=e2πix. From the Cauchy–Schwarz inequality and Parseval's identity, we have

formula here

and it is an interesting problem to investigate whether (2) reflects the true order of magnitude of (1).

The Schur algebra S(n, r) has a basis (described in [6, §2.3]) consisting of certain elements ζi,j, where i, j∈I(n, r), the set of all ordered r-tuples of elements from the set n={1, 2, …, n}. The multiplication of two such basis elements is given by a formula known as Schur's product rule. In recent years, a q-analogue Sq(n, r) of the Schur algebra has been investigated by a number of authors, particularly Dipper and James [3, 4]. The main result of the present paper, Theorem 3.6, shows how to embed the q-Schur algebra in the rth tensor power Tr(Mn) of the n×n matrix ring. This embedding allows products in the q-Schur algebra to be computed in a straightforward manner, and gives a method for generalising results on S(n, r) to Sq(n, r). In particular we shall make use of this embedding in subsequent work to prove a straightening formula in Sq(n, r) which generalises the straightening formula for codeterminants due to Woodcock [12].

We shall be working mainly with three types of algebra: the quantized enveloping algebra U(gln) corresponding to the Lie algebra gln, the q-Schur algebra Sq(n, r), and the Hecke algebra, [Hscr ](Ar−1). It is often convenient, in the case of the q-Schur algebra and the Hecke algebra, to introduce a square root of the usual parameter q which will be denoted by v, as in [5]. This corresponds to the parameter v in U(gln). We shall denote this ‘extended’ version of the q-Schur algebra by Sv(n, r), and we shall usually refer to it as the v-Schur algebra. All three algebras are associative and have multiplicative identities, and the base field will be the field of rational functions, ℚ(v), unless otherwise stated. The symbols n and r shall be reserved for the integers given in the definitions of these three algebras.

Hammocks have been considered by Brenner [1], who gave a numerical criterion for a finite translation quiver to be the Auslander–Reiten quiver of some representation-finite algebra. Ringel and Vossieck [11] gave a combinatorial definition of left hammocks which generalised the concept of hammocks in the sense of Brenner, as a translation quiver H and an additive function h on H (called the hammock function) satisfying some conditions. They showed that a thin left hammock with finitely many projective vertices is just the preprojective component of the Auslander–Reiten quiver of the category of [Sscr ]-spaces, where [Sscr ] is a finite partially ordered set (abbreviated as ‘poset’). An important role in the representation theory of posets is played by two differentiation algorithms. One of the algorithms was developed by Nazarova and Roiter [8], and it reduces a poset [Sscr ] with a maximal element a to a new poset [Sscr ]′=a∂[Sscr ]. The second algorithm was developed by Zavadskij [13], and it reduces a poset [Sscr ] with a suitable pair (a, b) of elements a, b to a new poset [Sscr ]′=∂(a,b)[Sscr ]. The main purpose of this paper is to construct new left hammocks from a given one, and to show the relationship between these new left hammocks and the Nazarova–Roiter algorithm. In a later paper [5], we discuss the relationship between hammocks and the Zavadskij algorithm.

Recently J. Coates and R. Greenberg introduced a new important class of extensions of local number fields with finite residue field which they call deeply ramified fields. These extensions play an essential role in their study of the arithmetics of abelian varieties over local fields with finite residue fields [1].

The first aim of this paper is to provide an ‘elementary’ treatment of deeply ramified extensions of local fields with arbitrary perfect residue fields using a method different from the original approach of Coates and Greenberg. Equivalent properties (1), (3)–(8) of deeply ramified extensions in Section 1 are due to them, and for their proofs a presentation of the different as an integral was involved. We translate the most important constructions into the language of the Hasse–Herbrand function (Section 1) and then apply the methods of [3, Chapter 3], where the Hasse–Herbrand function is defined in terms of the norm map. Some of the properties of deeply ramified extensions (two implications in the language of this paper) have been already studied by M. Matignon [7] and J. Fresnel, M. Matignon [5] for different purposes (see Remark 1.6 and the beginning of Section 2).

The second aim of this text is to expose relations among classes of deeply ramified extensions, that of arithmetically profinite extensions (Subsections 2.1–2.4) and that of p-adic Lie extensions (Subsections 2.5–2.7). For local fields with finite residue field we give in Subsection 2.2 an example of a Galois deeply ramified extension with infinite residue extension in which every Galois deeply ramified subextension is not arithmetically profinite; and in Subsection 2.4 we give an example of a Galois deeply ramified extension with finite residue field extension and a nondiscrete set of breaks (that means that this extension is not arithmetically profinite). The main result is that for local fields with finite residue field the class of Galois deeply ramified extensions with finite residue extension and a discrete set of breaks coincides with the class of infinite Galois arithmetically profinite extensions (Proposition 2.3). However, in the case of the fields with infinite residue fields Proposition 2.3 does not hold (Subsection 2.1). In Subsection 2.5 we construct an example that shows that the class of infinite Galois totally ramified arithmetically profinite extensions is strictly larger than the class of the most natural arithmetic origin – the class of totally ramified p-adic Lie extensions. Subsection 2.6 contains an example of a Galois deeply and totally ramified extension L of a local field F with finite residue field such that the norm group of L/E is of finite index in E*.

Two results, the Auslander–Buchsbaum and Bass Theorems from the homological theory of commutative noetherian rings, are generalized to important classes of non-commutative ℕ-graded algebras over fields. As a corollary to the generalized Auslander–Buchsbaum Theorem, it is found that under weak (so-called χ−) conditions, a non-commutative ℕ-graded connected noetherian algebra of finite global dimension is in fact Artin–Schelter regular.

Throughout this paper, D denotes a division ring (possibly commutative) and V a left vector space over D, usually, but not exclusively, infinite-dimensional. We consider irreducible subgroups G of GL(V) and are particularly interested in such G that contain an element g the fixed-point set CV(g) of which is non-zero but finite-dimensional (over D). We then use this to derive conclusions about cofinitary groups, an element g of GL(V) being cofinitary if dimDCV(g) is finite, and a subgroup of GL(V) being cofinitary if all its non-identity elements are cofinitary.

Suppose that G is a cofinitary subgroup of GL(V). There are two extreme cases. If dimDV is finite the cofinitary condition is vacuous. At the other extreme, if G acts fixed-point freely on V then the fixed-point sets CV(g) for g∈G[setmn ]〈1〉 are as small as possible, namely {0}. Work of Blichfeldt and his successors shows that certain irreducible linear groups G of dimension at least 2 over, for example, the complexes are always imprimitive. This is the case if G is nilpotent, or supersoluble, or metabelian. Apart from the two extreme cases, the same is frequently true for irreducible cofinitary subgroups G of GL(V). For example, this is the case if G is finitely generated nilpotent [9, 1.2] or more generally if G is supersoluble [10, 1.1], but not in general if G is metabelian [10, 7.1] or parasoluble (a group G is parasoluble if it has a normal series of finite length such that every subgroup of each of its factors is Abelian and normalised by G) (see [10, 7.2]). Further, it is also the case if G is Abelian-by-finite [10, 3.4], and every supersoluble group is finitely generated and nilpotent-by-finite. Collectively, these results suggest that one should consider nilpotent-by-finite groups.

The purpose of this paper is to give characterizations of weights for which reverse inequalities of Hölder type for quasi-monotone functions are satisfied. Our inequalities with general weights and with sharp constants complement the results of [2, 8, 9] and [16, 17] for the values of parameters p and q with 1[les ]p[les ]q<∞.

Lower bounds are established for the number of zeros of the logarithmic derivative of a meromorphic function of small growth with few poles.

A theorem of Hall extends the sylow theory to π-subgroups of a finite solvable group for any set of primes π. This theorem has been generalized to solvable periodic linear groups [21] and to other similar settings [19]. We consider the same problem in a model-theoretic context.

In model theory, the study of ω-stable groups generalizes the classical theory of algebraic groups in the case of a algebraically closed base field, and is closely connected with certain aspects of the general structure theory developed by Shelah, particularly in connection with the more geometric theory of Zilber, Buechler, Hrushovski, Pillay, and others. In this connection, the primary concern is with the structure of simple ω-stable groups of finite rank. One way to approach the study of ω-stable groups is by generalizing some of the methods of finite group theory to this setting. One anticipates that character theory and counting arguments will not be available, though one would hope to recover much of the ‘local analysis’. Unfortunately there is as yet no very satisfactory sylow theory for ω-stable groups, except for the prime 2 [4]. One of us (Nesin) noticed initially that there is a sylow theorem for connected solvable ω-stable groups of finite Morley rank; this prompted us to look for a generalization along the following lines.

Throughout this paper k denotes a fixed commutative ground ring. A Cohen–Macaulay complex is a finite simplicial complex satisfying a certain homological vanishing condition. These complexes have been the subject of much research; introductions can be found in, for example, Björner, Garsia and Stanley [6] or Budach, Graw, Meinel and Waack [7]. It is known (see, for example, Cibils [8], Gerstenhaber and Schack [10]) that there is a strong connection between the (co)homology of an arbitrary simplicial complex and that of its incidence algebra. We show how the Cohen–Macaulay property fits into this picture, establishing the following characterization.

A pure finite simplicial complex is Cohen–Macaulay over k if and only if the incidence algebra over k of its augmented face poset, graded in the obvious way by chain lengths, is a Koszul ring.

The main result in this paper, Theorem 1.2, generalizes a theorem of Zerner [26] concerning sufficient conditions for the holomorphic continuability of a solution of a linear holomorphic partial differential equation across a point of a hypersurface, on one side of which it is holomorphic. The point of the new theorem is, roughly speaking, that it applies also to regular solutions of partial differential equations whose coefficients may have certain kinds of singularities. This enables us to deduce some new results (see §2) on elliptic partial differential equations in ℝ2[ratio ]Theorem 2.1 extends a result of Vekua on the size of the domain of holomorphy of solutions to elliptic equations, in the case where singularities are permitted in the coefficients; Theorem 2.2 is of an apparently novel type, showing (roughly) that under certain conditions the solution to Cauchy's problem is real-analytic in a domain whose size depends only on the principal part of the operator, which is assumed to be the Laplacian, and the Cauchy data on the real axis. (Results of this kind are very delicate, as we shall illustrate in §4 with a simple counterexample.) Theorem 2.2 is new and non-trivial even for equations with analytic coefficients, in which case though, Theorem 1.2 is not needed for the proof.

In [17, 18, 19], we began to investigate the continuity properties of homomorphisms from (non-abelian) group algebras. Already in [19], we worked with general intertwining maps [3, 12]. These maps not only provide a unified approach to both homomorphisms and derivations, but also have some significance in their own right in connection with the cohomology comparison problem [4].

The present paper is a continuation of [17, 18, 19]; this time we focus on groups which are connected or factorizable in the sense of [26]. In [26], G. A. Willis showed that if G is a connected or factorizable, locally compact group, then every derivation from L1(G) into a Banach L1(G)-module is automatically continuous. For general intertwining maps from L1(G), this conclusion is false: if G is connected and, for some n∈ℕ, has an infinite number of inequivalent, n-dimensional, irreducible unitary representations, then there is a discontinuous homomorphism from L1(G into a Banach algebra by [18, Theorem 2.2] (provided that the continuum hypothesis is assumed). Hence, for an arbitrary intertwining map θ from L1(G), the best we can reasonably hope for is a result asserting the continuity of θ on a ‘large’, preferably dense subspace of L1(G). Even if the target space of θ is a Banach module (which implies that the continuity ideal [Iscr ](θ) of θ is closed), it is not a priori evident that θ is automatically continuous: the proofs of the automatic continuity theorems in [26] rely on the fact that we can always confine ourselves to restrictions to L1(G) of derivations from M(G) [25, Lemmas 3.1 and 3.4]. It is not clear if this strategy still works for an arbitrary intertwining map from L1(G) into a Banach LL1(G)-module.

A simple geometrical argument shows that every pair of projections on a finite-dimensional complex vector space has a common invariant subspace of dimension 1 or 2. The idea extends to certain pairs of projections on an infinite-dimensional Hilbert space H. In particular every projection on H has a reducing subspace, although a finite-dimensional one need not exist. In a final section, the results are extended to the existence of hyperinvariant subspaces for pairs of projections.

We investigate the closability of those derivations D defined on a (non necessarily closed) subalgebra B of a complex Banach algebra A for which the conditions

BAB⊂B and dim[Bk∩Rad(A)]<∞

hold for some k∈ℕ, where Rad(A) stands for the Jacobson radical of A. In this situation we show that the separating subspace [Sscr ](D) for D satisfies the property

B[B∩[Sscr ](D)]B⊂Rad(A).

Furthermore, we demonstrate several specially relevant situations in which we get a ‘closability property’ which is more precise than the former one.

Let Y be a topological space; and suppose that the circle group, S1, which will be denoted by G throughout this paper, is acting on Y. Then the equivariant cohomology H*G(Y) of Y is defined to be H*(YG), where YG=(EG×Y)/G, the Borel construction. Because of the bundle map YG→BG, the cohomology H*G(Y) is an H*(BG)-module. With rational coefficients H*(BG; ℚ)=ℚ[u], where deg(u)=2. And, when Y is a reasonably nice finite-dimensional space, for example, a finite-dimensional G-CW-complex, then the localized module H*G(Y; ℚ)[u−1] is isomorphic to H*G (YG; ℚ)[u−1] via the restriction homomorphism, where YG⊆Y is the fixed point set. (See, for example, [2].) That this very useful Localization Theorem does not generalize to infinite dimensional spaces can be seen by taking Y=EG, for example.

A noninjective bounded-to-one factor map from an irreducible shift of finite type onto a sofic system can be factored as a composition of other such maps in only finitely many ways (up to isomorphism). This generalizes to factor maps from systems with canonical coordinates to finitely presented dynamical systems.

Let Xn for n[ges ]1 be independent random variables with EXn=0 and EX2n=1. Set Sk,n= [sum ]i1<…<ik[les ]nXi1…Xik. Define Tk,c,m= inf{n[ges ]m[ratio ][mid ]k!Sk,n[mid ] >cnk/2}. We study critical values ck,p for k[ges ]2 and p>0, such that ETpk,c,m<∞ for c<ck,p and all m, and ETpk,c,m=∞ for c>ck,p and all sufficiently large m. In particular, c1,1=c2,1=1, c3,1=2 and c4,1=3 under certain moment conditions on X1, when Xn are identically distributed. We also investigate perturbed stopping rules of the form Th,m=inf{n[ges ]m[ratio ] h(S1,n/n1/2) <ζn or >ζn} for continuous functions h and random variables ζn∼a and ζn∼b with a<b. Related stopping rules of the Wiener process are also considered via the Uhlenbeck process.