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.

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 ] ∞.

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.

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.

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.

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.

If $\Gamma$ is a discrete group, conditions are given that ensure that the full group C*-algebra $C^*(\Gamma)$ is primitive, thereby partially answering a question posed by P. de la Harpe.

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.

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.

The exact best possible range of p is determined such that any dyadic $A_{1}$ weight w on $\mathbb{R}^{n}$ satisfies a reverse Hölder inequality for p, which depends on the dimension n and the corresponding $A_{1}$ constant of w. The proof is based on an effective linearization of the dyadic maximal operator applied to dyadic step functions.

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.

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 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.

Let Pf(x) =−if′(x) and Qf(x) = xf(x) be the canonical operators acting on an appropriate common dense domain in L2(ℝ). The derivations DP(A) = i(PA−AP) and DQ(A) = i(QA−AQ) act on the *-algebra [Ascr ] of all integral operators having smooth kernels of compact support, for example, and one may consider the noncommutative ‘Laplacian’, L = D2P+D2Q, as a linear mapping of [Ascr ] into itself.

L generates a semigroup of normal completely positive linear maps on [Bscr ](L2(ℝ)), and this paper establishes some basic properties of this semigroup and its minimal dilation to an E0-semigroup. In particular, the author shows that its minimal dilation is pure and has no normal invariant states, and he discusses the significance of those facts for the interaction theory introduced in a previous paper.

There are similar results for the canonical commutation relations with n degrees of freedom, where 1 [les ] n < 1.

The minimum value of the Mahler measure of a nonreciprocal polynomial whose coefficients are all odd integers is proved here to be the golden ratio. The smallest measures of reciprocal polynomials with $\pm1$ coefficients and degree at most 72 are also determined.

In terms of the upper bounds of a second-order elliptic operator acting on specific Lyapunov-type functions with compact level sets, sufficient conditions are presented for the corresponding Dirichlet form to satisfy the Poincaré and the super-Poincaré inequalities. Here, the elliptic operator is assumed to be symmetric on $L^2(\mu)$ with some probability measure $\mu$. As applications, proofs are given for a class of (non-symmetric) diffusion operators generating $C_0$-semigroups on $L^1(\mu)$: that their $L^p(\mu)$-essential spectrum is empty for $p>1$. This follows since it is proved that their $C_0$-semigroups are compact.Supported in part by the NNSFC for Distinguished Young Scholars (10025105), the 973-Project and TRAPOYT in China.

Mather has shown that almost all linear projections of a smooth submanifold of a vector space to a subspace yield a generic mapping (so in the nice dimensions, most linear projections give rise to ‘stable’ mappings). In this paper we show that the same is true if we replace the submanifold by the image of a stable mapping.

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.