One of the principal impulses for this paper was the consideration of the question, first raised in 1973 by G. Baumslag [2], of whether every finitely generated Abelian by polycyclic group can be embedded in a finitely presented Abelian by polycyclic group. That this cannot be done follows at once from

THEOREM A. Every finitely presented Abelian by polycyclic group has a metanilpotent normal subgroup of finite index.

A relation is described between Arnold's strange duality and a polar duality between the Newton polytopes which is mostly due to M. Kobayashi. It is shown that this relation continues to hold for the extension of Arnold's strange duality found by C. T. C. Wall and the author. By a method of Ehlers–Varchenko, the characteristic polynomial of the monodromy of a hypersurface singularity can be computed from the Newton diagram. This method is generalized to the isolated complete intersection singularities embraced in the extended duality. This is used to explain the duality of characteristic polynomials of the monodromy discovered by K. Saito for Arnold's original strange duality and extended by the author to the other cases.

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.

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.

This paper deals with a formula satisfied by ‘r-reduced’ Schur functions. Schur functions originally appear as irreducible characters of general linear group over the complex number field. In this paper they are considered as weighted homogeneous polynomials with respect to the power sum symmetric functions. More precisely, for a Young diagram λ of size n, the Schur function indexed by λ reads

formula here

where χλ(v) is the character value of the irreducible representation Sλ of the group algebra ℚ[Sfr ]n, evaluated at the conjugacy class of the cycle type v = (1v12v2 … nvn). Setting tjr = 0 for j = 1, 2, … in Sλ(t), we have the r-reduced Schur function S(r)λ(t). The set of all r-reduced Schur functions spans the polynomial ring P(r) = ℚ[tj; j[nequiv ]0 (mod r)]. We show that a good choice of basis elements leads to an explicit description of all other r-reduced Schur functions involving the Littlewood–Richardson coefficients.

The formula has not only a purely combinatorial meaning, but also nice implications in two different fields. One is about the basic representation of the affine Lie algebra A(1)r−1. We show that the basis in the main theorem gives in turn a weight basis of the basic A(1)r−1-module realised in P(r). The other implication is about modular representations of the symmetric group. Our explicit formula implies that the determination of the decomposition matrices reduces to that for the basic set we give in this paper.

The paper is organised as follows. In Section 1 we introduce generalised Maya diagrams and associated r-reduced Schur functions. In Section 2 we discuss combinatorics of Young diagrams. Section 3 is devoted to the main theorem. In Section 4 we describe weight vectors of the basic A(1)r−1-module. In Section 5 the formula is translated into that in the modular representation theory.

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 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.

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.

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.

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.

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.

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.

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.

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 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.

For every Hausdorff function h, the paper gives a construction of a compact metric space of infinite diameter-based h-packing measure which has no subsets of positive finite measure. It then indicates how such a construction can be modified to provide the same result for a radius-based packing measure in the case of certain Hausdorff functions which do not satisfy a doubling condition. This is in surprising contrast to another result which shows that for a particular radius-based definition of packing measure no such example can be found.