This paper introduces a stratification of the variety of modules of a fixed dimension over a finite-dimensional algebra. For a large and interesting class of algebras, called subfinite, this stratification is used to compute the dimension of this variety and give bounds on the number of its irreducible components.

Suppose that K is a linear space of functions analytic in some domain D in the complex plane. A sequence Λ = (λk) of distinct points from D is said to be a set of uniqueness for K if ƒ∈K and ƒ(λk) = 0 for all k imply ƒ≡0. Depending on the dispersion and the density of Λ on the one hand, and the growth of the functions in K on the other, one may often require only |ƒ(λk)| [les ]ak for some sequence of positive numbers ak, and still conclude that ƒ≡0 for ƒ∈K. Of particular interest are sharp conditions on the decay of ak, which reflect the interplay between growth and decay of analytic functions.

We prove that if all the rank-one bounded operators on a Banach space X attain their numerical radii, then X must be reflexive, but the converse does not hold. In fact, every reflexive space with basis can be renormed in such a way that there is a rank-one operator not attaining the numerical radius.

functional identities for the classical rogers dilogarithm associated to simply laced dynkin diagrams are proved uniformly in the setting of $y$-systems related to clusters.

If G is a finitely presented group and [Kscr ] is any (G,2)-complex (that is, a finite 2-complex with fundamental group G), then it is well known that χ([Kscr ]) [ges ] ν(G), where ν(G) = 1−rk H1G+dH2G. We define χ(G) to be min{χ([Kscr ]): [Kscr ] a (G, 2)-complex}, and we say that G is efficient if χ(G) = ν(G). In this paper we give sufficient conditions for a Coxeter group to be efficient (Theorem 4.2). We also give examples of inefficient Coxeter groups (Theorem 5.1). In fact, we give an infinite family Gn(n = 2, 3, 4, . . . ) of Coxeter groups such that χ(Gn)−ν(Gn) [xrarr ] ∞ as n [xrarr ] ∞.

For n a positive integer, a group G is called core-n if H/HG has order at most n for every subgroup H of G (where HG is the normal core of H, the largest normal subgroup of G contained in H). It is proved that a locally finite core-n group G has an abelian subgroup whose index in G is bounded in terms of n.

an index of a collection of 1-forms on a complex isolated complete intersection singularity corresponding to a chern number is defined and – in the case when the 1-forms are complex analytic – expressed as the dimension of a certain algebra.

We classify all finite 2-groups G for which the maximum proportion of elements inverted by an automorphism of G is a half. These groups constitute 10 isoclinism families.

Let $S$ be a compact Riemann surface of genus ${\rm g}\,{>}\,1$. By using Teichmüller theory, a new approach is obtained to the existence of a Jenkins–Strebel differential with prescribed type and heights on $S$. Moreover, the geometric structures of horizontal trajectories of certain classes of quadratic differentials are discussed.

A set A of real numbers is called universal (in measure) if every measurable set of positive measure necessarily contains an affine copy of A. All finite sets are universal, but no infinite universal sets are known. Here we prove some results related to a conjecture of Erdős that there is no infinite universal set. For every infinite set A, there is a set E of positive measure such that (x+tA)⊆E fails for almost all (Lebesgue) pairs (x, t). Also, the exceptional set of pairs (x, t) (for which (x+tA)⊆E) can be taken to project to a null set on the t-axis. Finally, if the set A contains large subsets whose minimum gap is large (in a scale-invariant way), then there is E⊆R of positive measure which contains no affine copy of A.

We give geometric proofs of some of the basic structure theorems for compact Lie groups. The goal is to take a fresh look at these theorems, prove some that are difficult to find in the literature, and illustrate an approach to the theorems that can be imitated in the homotopy theoretic setting of p-compact groups.

We examine the range of universal Taylor series. We prove that every universal Taylor series on the unit disc assumes every complex number, with one possible exception, infinitely often. On the other hand, we prove that on any simply connected domain there exist universal functions that omit one value.

The relationship between two natural definitions of the relative pressure function, for a locally constant potential function and a factor map from a shift of finite type, is clarified by showing that they coincide almost everywhere with respect to every invariant measure. With a suitable extension of one of the definitions, the same holds true for any continuous potential function.

In this note we discuss the radial positive solutions of

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where D is the annulus {s∈Rn [ratio ]b<|s|<1}. Here 0<b<1, n[ges ]2, and g[ratio ]R→R is a suitable C1 function with g(0)[ges ]0 but with g changing sign. We answer a question on page 223 of the original Gidas–Ni–Nirenberg paper [8], by showing that the maximum of these solutions occurs at a point sε, where |sε|→b as ε→0. This also shows that the non-negativity condition in the result on page 223 of [8] is necessary.

We provide a new lower bound for the greatest prime factor of the norm of algebraic numbers of the form axm+byn. The improvement concerns the dependence on the number field containing a, b, x and y.

Since the very beginning of the multidimensional theory of quasiregular mappings, it has been widely believed that the class of K-quasiregular mappings in ℝn is closed with respect to uniform convergence, where K stands for the linear dilatation. In this note we give a striking example which refutes this belief. The key element of our construction is that the linear dilatation function fails to be rank-one convex in dimensions higher than 2.

We prove upper and lower heat kernel bounds for the Laplacian on weighted graphs which include the case that the weights have no strictly positive lower bound. Our estimates give rise to a very explicit probabilistic interpretation, and can be formulated in terms of a weighted metric. Interestingly, this metric is not equivalent to the intrinsic metric.

Investigations concerning the generating function associated with the kth powers,

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originate with Hardy and Littlewood in their famous series of papers in the 1920s, ‘On some problems of “Partitio Numerorum”’ (see [7, Chapters 2 and 4]). Classical analyses of this and similar functions show that when P is large the function approaches P in size only for α in a subset of (0, 1) having small measure. Moreover, although it has never been proven, there is some expectation that for ‘most’ α, the generating function is about √P in magnitude. The main evidence in favour of this expectation comes from mean value estimates of the form

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An asymptotic formula of the shape (1.2), with strong error term, is immediate from Parseval's identity when s=2, and follows easily when s=4 and k>2 from the work of Hooley [2, 3, 4], Greaves [1], Skinner and Wooley [5] and Wooley [9]. On the other hand, (1.2) is false when s>2k (see [7, Exercise 2.4]), and when s=4 and k=2. However, it is believed that when t<k, the total number of solutions of the diophantine equation

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with 1[les ]xj, yj[les ]P (1[les ]j[les ]t), is dominated by the number of solutions in which the xi are merely a permutation of the yj, and the truth of such a belief would imply that (1.2) holds for even integers s with 0[les ]s<2k.

The purpose of this paper is to investigate the extent to which knowledge of the kind (1.2) for an initial segment of even integer exponents s can be used to establish information concerning the general distribution of fP(α), and the behaviour of the moments in (1.2) for general real s.

Given an affine domain of Gelfand–Kirillov dimension 2 over an algebraically closed field, it is shown that the centralizer of any non-scalar element of this domain is a commutative domain of Gelfand–Kirillov dimension 1 whenever the domain is not polynomial identity. It is shown that the maximal subfields of the quotient division ring of a finitely graded Goldie algebra of Gelfand–Kirillov dimension 2 over a field $F$ all have transcendence degree 1 over $F$. Finally, centralizers of elements in a finitely graded Goldie domain of Gelfand–Kirillov dimension 2 over an algebraically closed field are considered. In this case, it is shown that the centralizer of a non-scalar element is an affine commutative domain of Gelfand–Kirillov dimension 1.

For every infinite cardinal κ, we construct a primitive permutation group which has a finite suborbit paired with a suborbit of size κ. This answers a question of Peter M. Neumann.