Two classes of vector-valued BMOA spaces are defined, in the complex ball and on the complex sphere, respectively. In the case of the complex sphere, vector measures are involved, since the argument in the scalar setting is not appropriate. Several properties (the Lp-equivalent norm theorem, exponential decay, the Baernstein theorem, and so on) of BMOA in the complex ball are extended to the Banach space setting. The two classes of BMOA spaces are proved to be isomorphic; in particular, the corresponding John–Nirenberg exponential decay is shown. Finally, the vector-valued H1-BMOA duality theorem is proved.

As a generalization of the dynamics of a single polynomial, Hinkkanen and Martin studied the dynamics of polynomial semigroups of finite type. Among the issues addressed were possible generalizations of the filled-in Julia set of a single polynomial and the basin of attraction for infinity. In the paper, these concepts are further refined, and the connections between the dynamics of a single function and the dynamics of semigroups are strengthened.

Given two self-adjoint Hilbert space operators A, B, and a continuous function f, we prove several inequalities of the form

formula here

involving the Lp-norm of the derivative f′ and the Besov Br∞,1-norm of f.

The ideas of Friedman and Qin are extended to find the wall-crossing formulae for the Donaldson invariants of algebraic surfaces with pg = 0, q > 0 and anticanonical divisor −K effective, for any wall ζ with lζ = ¼(ζ2 − p1) being 0 or 1.

Let $N$ be a point of an orbit closure $\overline{{\mathcal O}}_M$ in a module variety such that its orbit ${\mathcal O}_N$ has codimension 2 in $\overline{{\mathcal O}}_M$. We show that under some additional conditions the pointed variety $(\overline{{\mathcal O}}_M,N)$ is smoothly equivalent to a cone over a rational normal curve.

The purpose of this note is to generalise, and give a more illuminating proof, of a theorem of [13] (Theorem 1.1 below). Before stating it, we provide some introductory information. Consider the following two sequences of pictures: in each we see a 1-parameter family Xℝ,t of real algebraic hypersurfaces, which undergoes a bifurcation when the parameter t is equal to 0. Note that in Figure 1, both (i) (a) and (i) (b), and in (ii) (b), the surface Xℝ,t has a purely 1-dimensional part, which we have indicated with a dotted line, and that in (i) (b) we have drawn a curve vertically along the middle of the surface to make clearer the way it passes through itself. The reader will observe that in (a) the surface Xℝ,t is homotopically a 2-sphere when t>0 and a 0-sphere when t<0, while in (b) Xℝ,t is a homotopy 1-sphere both for t<0 and t>0.

Such sequences are typical in singularity theory; each is in fact the family of algebraic closures of images of a versal deformation of a codimension 1 singularity of mapping.

Now suppose that the complexification X[Copf ],t is a homotopy n-sphere. In [13] the second author pointed out that it follows that Xℝ,t is a homotopy sphere for t≠0 (allowing the empty set as a −1-sphere). Indeed, in the local situation, or globally in the weighted homogeneous case, there are well-defined integers k+ and k− between −1 and n such that Xℝ,t≃Sk+ for t>0 and Xℝ,t≃Sk− for t<0.

We describe Xℝ,t for t∈ℝ−0 as ‘good’ if the homotopy dimension of Xℝ,t is equal to n. In this case the inclusion Xℝ,t[rarrhk ]Xt is a homotopy equivalence [13, 1.1].

For a prime p, a homology decomposition of the classifying space BG of a finite group G consist of a functor F : D → spaces from a small category into the category of spaces and a map hocolim F → BG from the homotopy colimit to BG that induces an isomorphism in mod-p homology. Associated to a modular representation G → Gl(n; [ ]p), a family of subgroups is constructed that is closed under conjugation, which gives rise to three different homology decompositions, the so-called subgroup, centralizer and normalizer decompositions. For an action of G on an [ ]p-vector space V, this collection consists of all subgroups of G with nontrivial p-Sylow subgroup which fix nontrivial (proper) subspaces of V pointwise. These decomposition formulas connect the modular representation theory of G with the homotopy theory of BG.

It is shown that the splitting modulo a prime p of a given monic, integral, irreducible cubic with non-square discriminant is equivalent to p being represented by forms in a certain subgroup of index 3 in the form class group of discriminant equal to the discriminant of the field defined by the cubic.

Let $\textbf{F}\,{=}\,(F_1, \ldots, F_m)$ be an $m$-tuple of primitive positive binary quadratic forms and let $U_{\textbf{F}}(x)$ be the number of integers not exceeding $x$ that can be represented simultaneously by all the forms $F_j$, $j = 1, \ldots, m$. Sharp upper and lower bounds for $U_{\textbf{F}}(x)$ are given uniformly in the discriminants of the quadratic forms.

As an application, a problem of Erdős is considered. Let $V(x)$ be the number of integers not exceeding $x$ that are representable as a sum of two squareful numbers. Then $V(x) = x(\log x)^{-\alpha+o(1)}$ with $\alpha\,{=}\,1-2^{-1/3}\,{=}\,0.206\ldots$.

Let $p$ be a prime number, $F$ a number field, and ${\cal H}_F^n$ the set of all unramified cyclic extensions over $F$ of degree $p$ having a relative normal integral basis. When $\z_p \in F^{\times}$, Childs determined the set ${\cal H}_F^n$ in terms of Kummer generators. When $p=3$ and $F$ is an imaginary quadratic field, Brinkhuis determined this set in a form which is, in a sense, analogous to Childs's result. The paper determines this set for all $p \ge 3$ and $F$ with $\z_p \not\in F^{\times}$ (and satisfying an additional condition), using the result of Childs and a technique developed by Brinkhuis. Two applications are also given.

For positive integers n and c, with n [ges ] 2, let Gn, c be a relatively free group of finite rank n in the variety N2A ∧ AN2 ∧ Nc. It is shown that the subgroup of the automorphism group Aut(Gn, c) of Gn, c generated by the tame automorphisms and an explicitly described finite set of IA-automorphisms of Gn, c has finite index in Aut(Gn, c). Furthermore, it is proved that there are no non-trivial elements of Gn, c fixed by every tame automorphism of Gn, c.

It is known from Vaughan and Wooley's work on Waring's problem that every sufficiently large natural number is the sum of at most 17 fifth powers [13]. It is also known that at least six fifth powers are required to be able to express every sufficiently large natural number as a sum of fifth powers (see, for instance, [5, Theorem 394]). The techniques of [13] allow one to show that almost all natural numbers are the sum of nine fifth powers. A problem of related interest is to obtain an upper bound for the number of representations of a number as a sum of a fixed number of powers. Let R(n) denote the number of representations of the natural number n as a sum of four fifth powers. In this paper, we establish a non-trivial upper bound for R(n), which is expressed in the following theorem.

The purpose of the paper is to illustrate how vanishing theorems can be used to give effective criteria for a generically finite morphism $f\,{:}\,X\,{\longrightarrow}\,Y$ of smooth complex projective algebraic varieties to be birational. In particular, as a consequence of a non-vanishing theorem of Kollár, it is shown that if $Y$ is of general type and has generically large algebraic fundamental group, then $f$ is birational if and only if $P_2(X)\,{=}\,P_2(Y)$.

My paper [2], which appeared in the special issue dedicated to the late Professor Brian Hartley, contained an error in Lemma 2.2 as a result of overlooking the case that the exponents ki on p. 359 may be divisible by arbitrary powers of p, which was pointed out by Professor H. Smith. In order to overcome this situation all of Section 2, except Lemma 2.1, has been rewritten. However this does not cause any change in the rest of the paper. In particular the original assertions remain correct.

Graded versions of the principal series modules of the category $\cO$ of a semisimple complex Lie algebra $\mg$ are defined. Their combinatorial descriptions are given by some Kazhdan–Lusztig polynomials. A graded version of the Duflo–Zhelobenko four-term exact sequence is proved. This gives results about composition factors of quotients of the universal enveloping algebra of $\mg$ by primitive ideals; in particular an upper bound is obtained for the multiplicities of such composition factors. Explicit descriptions are given of principal series modules for Lie algebras of rank $2$. It can be seen that these graded versions of principal series representations are neither rigid nor Koszul modules.

Consider a discrete-time regenerative phenomenon with associated renewal sequence un. General results for the supremum of um+1, …, um+n are developed for those renewal sequences {un} for which the first m + 1 elements match those of a fixed renewal sequence {vn}, that is, u0 = v0, …, um = vm. A series of associated lemmas are developed in the process.

It is shown that, for finitely generated residually soluble groups, a condition weaker than polynomial growth guarantees virtual nilpotence. Let $G$ be a residually soluble group having a finite generating set $X$, and suppose that the number $\ga_X(n)$ of elements of $G$ that are products of at most $n$ elements of $X\cup X^{-1}$ satisfies $\ga_X(n)\leqslant e^{\alpha(n)}$ for each $n$, where $\alpha(n)/e^{(1/2)(\ln n)^{1/2}}\to\infty$ as $n\to\infty$; then $G$ is virtually nilpotent.

Given a smooth, compactly supported hypersurface S in ℝn that does not pass through the origin, and denoting by tS the surface dilated by a factor t>0, we can consider the averaging operator defined for functions f∈[Sscr ], the Schwartz class of functions, by

formula here

where dσ is Lebesgue measure on S. We can now define a maximal average operator,

formula here

In the case where S is Sn−1, the sphere of unit radius in ℝn, we are looking at Stein's spherical maximal function, as treated in his paper [10]. Stein proved that M is a bounded operator on the Lp spaces if and only if p>n/(n−1) when n>3. Subsequently, Bourgain [2] showed that if S is any compactly supported smooth curve with non-vanishing Gaussian curvature, then M will be bounded on Lp, if and only if p>2, thus dealing with the case of the circular maximal function in the plane. (For related results, see also [6] and [7]). In the case where S is a curve whose curvature vanishes to order at most m−2 at a single point, Iosevich [4] showed that M is bounded on Lp for p>m, and unbounded if p = m. If we study curves given by Γ(s) = (s, γ(s)+1), s∈[0, 1], for some suitably smooth γ, where γ(0) = γ′(0) = … = γ(m−1)(0) ≠ γ(m)(0)>0, then we can reinterpret his results as follows. Define

formula here

for Schwartz functions f. Iosevich proved that

formula here

for p>2. If we note that κ(s), the curvature of the curve Γ(s) is approximately 2−k(m−2) whenever s∈[2−k, 21−k], then we have that the operator

formula here

is bounded on Lp for some p>2, if σ is sufficiently large, since

formula here

which is finite so long as σ>(m/p−1) (m−2)−1. If we want to choose σ independent of m>2, the type of the curve, such that Mσ is bounded on Lp for some fixed p>2, then clearly we can take σ = 1/p. In this paper we show that Mσ will be bounded on Lp for p>max{σ−1, s} for a class of infinitely flat, convex curves in the plane. Counterexamples will show that this is the best possible result, in that there exist flat curves for which Mσ is unbounded for 2<p[les ]σ−1.

It is shown that the compact matrix quantum groups SUq(2) are non-isomorphic to each other for q∈[−1, 1][setmn ]{0}, and that the compact matrix quantum groups SUq(n) are non-isomorphic to each other for q∈(0, 1]. Some invariants for compact quantum groups are also discussed.

A classical result of M. Gerstenhaber [5] states that finite-dimensional semisimple complex Lie algebras are rigid. One interpretation of this fact is as follows. Let [Lfr ] be a semisimple complex Lie algebra of dimension d and let [Bfr ] be a [Copf ]-basis of [Lfr ]. Let L(d) ⊂ [Copf ]d·(d2) denote the affine variety of all structure constants of [Copf ]-Lie algebras of dimension d and let c ∈ L(d) denote the point corresponding to the structure constants of [Lfr ] with respect to [Bfr ]. Then there exists an open neighbourhood in the metric topology U ⊂ L(d) of c ∈ L(d) such that [Lfr ]c* is isomorphic to [Lfr ] for all c* ∈ U, where [Lfr ]c* denotes the Lie algebra defined by the structure constants c* ∈ L(d). Our aim is to generalize this result to semisimple Lie algebras over discrete valuation domains and to apply these results to powerful pro-p-groups.