Frank Smithies was born in Edinburgh, on 10 March 1912. His father, also Frank Smithies, and his mother, Mary (née Blakemore) had met through their involvement in the socialist movement. They had two children, Frank and his sister Violet, some four years younger.

Let G be the group given by the following presentation:

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The subgroup generated by ab is infinite-cyclic and normal, with quotient the dihedral group of order 6, so G is cyclic-by-finite. The subgroups H = 〈a, c〉 and K = 〈b, c〉 are both dihedral of order 6, and G is isomorphic to the free product of H and K amalgamating L = H∩K. We study K0(kG), the Grothendieck group of isomorphism classes of finitely generated projective kG-modules, and, in particular, the dependence of K0(kG) on the choice of field k. As usual, let ℚ, ℝ and [Copf ] stand for the rationals, reals and complex numbers, respectively. We prove the following.

It is shown that if a C*-algebra A contains a semi-scattered C*-algebra B such that the pure states of B and the zero functional extend uniquely to A, and the canonical mapping Bˆ → Â is injective, then there exists a (unique) projection of norm one R : A → B. In certain circumstances, the conditional expectation R can be effected by a unitary averaging process using unitary elements in the centre of the multiplier algebra M(B).

Let $\mathfrak{g}=\mathfrak{g}_0\oplus\mathfrak{g}_1$ be a $\mathbb Z_2$-graded simple Lie algebra. Fix a Borel subalgebra $\mathfrak b_0\subset\mathfrak{g}_0$. Let $\mathfrak a\subset\mathfrak{g}_1$ be a $\mathfrak b_0$-stable subalgebra. Then $\mathfrak a$ is automatically commutative. It is known that if $\overline{G{\cdot}\mathfrak a}$ is a spherical $G$-variety, then $G_0{\cdot}\mathfrak a$ is a spherical $G_0$-variety. We describe all $\mathbb Z_2$-gradings having the property that $\overline{G{\cdot}\mathfrak a}$ is a spherical $G$-variety for any $\mathfrak a$.

A proof is given to show that a positive function on the unit disk admits a harmonic majorant if and only if it has a certain explicit upper envelope that admits a superharmonic majorant. The (logarithmic) Lipschitz regularity of this superharmonic majorant is discussed.

Let G be a finite group, let V be an [ ]G-module of finite dimension d, and denote by β(V, G) the minimal number m such that the invariant ring S(V)G is generated by finitely many elements of degree at most m. A classical result of E. Noether says that β(V, G)[les ][mid ]G[mid ] provided that char [ ] is coprime to [mid ]G[mid ]!. If char [ ] divides [mid ]G[mid ], then no bounds for β(V, G) are known except for very special choices of G. In this paper we present a constructive proof of the following. If H[les ]G with [G[ratio ]H]∈[ ]*, and if the restriction V[mid ]H is a permutation module (for example, if V is a projective [ ]G-module and H∈Sylp(G)), then β(V, G)[les ]max{[mid ]G[mid ], d([mid ]G[mid ]−1)} regardless of char [ ].

Let q and N be integers, let a be an integer coprime to q, and let zN be defined implicitly by q = (log N)log22−ZN √(log2N). We show that for large N, an integer n has at least one divisor d with q [les ] d [les ] N and d ≡ a(mod q) with probability approximately Φ(zN), where Φ denotes the distribution function of the Gaussian Law. This solves a conjecture of Hall.

In this paper, a proof is given that certain boundedness properties of operators yield distributional estimates that have exponential decay at infinity. Such distributional estimates imply local exponential integrability, and apply to many operators such as $m$-linear Calderón–Zygmund operators and their maximal counterparts.

A new property of a complete intersection in a graded Noetherian ring is proved; this result, in conjunction with the Jacobian criterion, is then applied to the description of the loci of singularities in a family of affine varieties introduced in a well-known paper on forms of many variables.

A deterministic self-similar Cantor set F defines a natural probability space, the elements of which are neighbourhoods of various sizes of all points in F. Considering random variables on this probability space, some porosity and density parameters are explicitly calculated.

Tao has shown that hard summation (summing Fourier-type series by using terms in order of decreasing size of the Fourier coefficients) works for wavelets, but the present author has shown that it fails for classical Fourier series. This paper, which is intended for a general audience, exhibits the underlying ideas in the context of Haar and Walsh series, where many of the proofs simplify.

Let Ω be the set {1, 2, [ctdot ], n}, and let [emptyv ] be the empty set. Let [Gscr ] be the family of all non-empty sets of subsets of Ω. For [Ascr ]∈[Gscr ] and X⊆Ω, put

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An important discovery is the following.

Let G be either a free product with amalgamation A[midast ]CB or an HNN group A[midast ]C, where all normal subgroups of C are finitely generated. Suppose that both A and B have no non-trivial finitely generated normal subgroups of infinite indices. We show that if G contains a finitely generated normal subgroup N which intersects A or B non-trivially but is not contained in C, then the index of N in G is finite.

Cet article présente des minorations universelles de la première valeur propre non nulle du laplacien des formes différentielles sur les domaines euclidiens ou hémisphériques convexes ou les hypersurfaces euclidiennes convexes.

The semilinear elliptic eigenvalue problem with superlinear pure power nonlinearity is considered. This problem is treated from the standpoint of $L^2$-theory and the precise asymptotic formula for the eigenvalue parameter $\lambda \,{=}\, \lambda(\alpha)$ as $\alpha \,{\to}\, \infty$ is established, where $\alpha$ is the $L^2$-norm of the solution $u$ associated with $\lambda$.

Let $\sigma$ be a non-degenerate positive $M_n$-valued measure on a locally compact group $G$ with $\|\sigma\|\,{=}\,1$. An $M_n$-valued Borel function $f$ on $G$ is called $\sigma$-harmonic if $f(x) = \int_G f(xy^{-1})\,d\sigma(y)$ for all $x\,{\in}\,G$. Given such a function $f$ which is bounded and left uniformly continuous on $G$, it is shown that every central element in $G$ is a period of $f$. Further, it is shown that $f$ is constant if $G$ is nilpotent or central.

In the early 1930s, Wiener proved that if f(x) is a strictly positive periodic function whose Fourier series is absolutely convergent, then the Fourier series of g(x)=1/f(x) is also absolutely convergent [8, pp. 10–14]. This phenomenon can be easily understood nowadays using Banach algebra techniques (see, for example, [4, pp. 202–203]). In fact, these techniques allow us to study the absolute convergence of g(x)=F(f(x)), where F is holomorphic in an open subset of [Copf ] that contains the range of f(x) (for x∈ℝ). In this context, Wiener's original problem corresponds to the choice F(z)=1/z.

In this work we want to analyse the constraints on the simultaneous rate of vanishing of the Fourier coefficients fˆ(n) and ĝ(n) as n→∞. We shall focus on g=1/f, but we shall also study the general case g=F(f). In either case, there are obviously no constraints when f is a constant function.

Although this problem does not seem to be directly related to uncertainty inequalities for the Fourier Transform, we observe that there are some analogies, both in the nature of the results and in the proof techniques. The general fact with which we are dealing is that fˆ(n) and ĝ(n) cannot vanish too quickly at the same time as n→∞, unless f(x) is constant. The general fact that underlies uncertainty inequalities is that a non-periodic function ϕ(x) and its Fourier Transform ϕcirc;(u) cannot vanish too quickly at the same time as x→∞ and u→∞, unless ϕ(x) is zero (almost everywhere). For a simple introduction to some aspects of uncertainty inequalities, see [5]; for a thorough and recent introduction to this vast subject, see [3].

It is known that, if f is a hyperbolic rational function, then the Hausdorff, packing and box dimensions of the Julia set, J(f), are equal. In this paper it is shown that, for a hyperbolic transcendental meromorphic function f, the packing and upper box dimensions of J(f) are equal, but can be strictly greater than the Hausdorff dimension of J(f).