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

formula here

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

formula here

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.

Extending a theorem of S. Mori [6], we characterize two-dimensional factorial algebras which are graded by a finitely generated abelian group of rank one and which are affine over an algebraically closed field. The result is related to the classification of weighted projective lines [2, 5] and to the topic of canonical algebras [7].

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 Fn, Mn and Un denote the free group of rank n, the free metabelian group of rank n, and the free abelian-by-(nilpotent of class 2) group of rank n, respectively. Thus Mn≅ Fn/F″″n and Un≅Fn/ [γ3(Fn), γ3(Fn)], where γ3(Fn) =[F′n, Fn], the third term of the lower central series of Fn.

Consider an arbitrary epimorphism θ[ratio ]Fn[Rarr ]Fk, where k<n. A routine argument (see [10, Theorem 3.3]) using the Nielsen reduction process shows that there exists a free basis {y1, y2, …, yn} of Fn such that kerθ is the normal closure in Fn of {y1, y2, …, yn−k}. A similar result has been obtained for free metabelian groups by C. K. Gupta, N. D. Gupta and G. A. Noskov [5]. Indeed, it follows immediately from [5, Theorem 3.1] that if k<n and θ[ratio ]Mn[Rarr ]Mk is an epimorphism, then there exists a free basis {m1, m2, …, mn} of Mn such that kerθ is the normal closure in Mn of {m1, m2, …, mn−k}.

Using the BMO-H1 duality (among other things), D. R. Adams proved in [1] the strong type inequality

∫Mf(x)dHα(x) [les ]C∫[mid ]f(x)[mid ]dHα(x), 0<α<n, (1)

where C is some positive constant independent of f. Here M is the Hardy–Littlewood maximal operator in ℝn, Hα is the α-dimensional Hausdorff content, and the integrals are taken in the Choquet sense. The Choquet integral of ϕ[ges ]0 with respect to a set function C is defined by

formula here

Precise definitions of M and Hα will be given below. For an application of (1) to the Sobolev space W1, 1 (ℝn), see [1, p. 114].

The purpose of this note is to provide a self-contained, direct proof of a result more general than (1).

Some years ago, Blatter [1] gave a result of the form

formula here

for any function f regular and univalent in D[ratio ][mid ]z[mid ]<1, where ρ is the hyperbolic distance between z1 and z2. Kim and Minda [5] pointed out that the multiplier on the right is incorrect. They say that Blatter's proof gives the correct multiplier, but Blatter's formula for ρ in terms of z1, z2 is wrong. Kim and Minda proved the generalized formula

formula here

where D1(f)=f′(z) (1−[mid ]z[mid ]2), valid for p[ges ]P with some P, 1<P[les ]3/2. In each case there was an appropriate equality statement. Kim and Minda made the important and easily verified remark that these problems are linearly invariant in the sense that if the result is proved for f, then it follows for f˜=UfT, where U is a linear transformation of the plane onto itself and T is a linear transformation of D onto itself. This means that we need to prove such results only in an appropriately normalized context.

We study the conditions for truncated symmetric products of manifolds to be manifolds. In particular, we show that suitably defined spaces of systems of real roots of real polynomials are homeomorphic to real projective spaces.

Two lines of gunmen face each other, there being initially m on one side, n on the other. Each person involved is a hopeless shot, but keeps firing at the enemy until either he himself is killed or there is no one left on the other side. Let μ(m, n) be the expected number of survivors. Clearly, we have boundary conditions:

μ(m, 0)=m, μ(0, n)=n. (1.1)

We also have the equation

formula here

This is because the probability that the first successful shot is made by the side with m gunmen is m/(m+n). On using the recurrence relation (1.2) together with the boundary condition (1.1), the computer produces Table 1 below, in which

m=8192+k, n=8192−k, d(m, n) =√(m2−n2 =128√(2k).

Let B=(Bt)t[ges ]0 be standard Brownian motion started at zero. We prove

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for all c>1 and all stopping times τ for B satisfying E(τr)<∞ for some r>1/2. This inequality is sharp, and equality is attained at the stopping time

τ*=inf{t>0[mid ]St [ges ]u*, Xt =1∨αSt},

where u*=1+1/ec(c−1) and α=(c−1)/c for c>1, with Xt=[mid ]Bt[mid ] and St= max0[les ]r[les ]t[mid ]Br[mid ]. Likewise, we prove

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for all c>1 and all stopping times τ for B satisfying E(τr<∞ for some r>1/2. This inequality is sharp, and equality is attained at the stopping time

σ*=inf{t>0[mid ]St [ges ]v*, Xt =αSt},

where v*=c/e(c−1) and α=(c−1)/c for c>1. These results contain and refine the results on the Llog L-inequality of Gilat [6] which are obtained by analytic methods. The method of proof used here is probabilistic and is based upon solving the optimal stopping problem with the payoff

formula here

where F(x) equals either xlog+x or xlog x. This optimal stopping problem has some new interesting features, but in essence is solved by applying the principle of smooth fit and the maximality principle. The results extend to the case when B starts at any given point (as well as to all non-negative submartingales).

For all smooth nondecreasing functions f on [0, 1], the local time Lf spent by standard Brownian motion on the graph of f satisfies P(Lf<x) [ges ]P(2L0<x), for x>0. The constant 2 in this inequality may not be replaced with a smaller one.

Richard Rado was born in Berlin; he was the second son of Leopold Rado, from Budapest. At one stage of his education he had to decide whether to become a concert pianist or a mathematician. He chose the latter in the belief that he could continue with music as a hobby, but that he could never treat mathematics in that way. He studied at the University of Berlin, but also spent some time in Göttingen. He took a DPh at Berlin with his thesis ‘Studien zur Kombinatorik’ [3] under Issai Schur in 1933. During this period he was also influenced by Erhard Schmidt.

On 16 March 1933, he married Luise Zadek, the elder daughter of Hermann Zadek, whom he had earlier come to know when he needed a partner to play piano duets. It was indeed a remarkable partnership.

As Hitler came to power in 1933, the Rados, being Jewish, made their way to England, Richard having obtained a scholarship of £300 p.a. from Sir Robert Mond through the recommendation of Professor Lindemann (later Lord Cherwell), who had interviewed him in Berlin, to enable him to study at Cambridge.