The papers [2] and [3] study the function g(k, n), defined for integers k > 1 and n > 1 as the smallest r with the property that every integer is a sum of r kth powers mod n. This note identifies g′(k), defined as the maximum over all n of g(k, n), with the function Γ(k) studied by Hardy and Littlewood [1] fifty years ago in connection with Waring's problem.

It is shown, using the Borwein–Preiss variational principle that for every continuous convex function $f$ on a weakly compactly generated space $X$, every ${{x}_{0}}\in X$ and every weakly compact convex symmetric set $K$ such that $\overline{\text{span}}K=X$, there is a point of Gâteaux differentiability of $f$ in ${{x}_{0}}+K$. This extends a Klee's result for separable spaces.

An arithmetic function f is said to be multiplicative if f(mn) = f(m)f(n), whenever (m, n) = 1 and f(1) = 1. The Dirichlet convolution of two arithmetic functions f and g, denoted by f • g, is defined by f • g(n) = Σd|nf(d)g(n/d). Let w(n) denote the product of the distinct prime factors of n, with w(l) = 1. R.

Let L1, L2 ⊂ Cn be two totally real subspaces of real dimension n, and such that L1 ∩ L2 = {0}. We show that continuous functions on L1 ∪L2 allow Carleman approximation by entire functions if and only if L1 ∪L2 is polynomially convex. If the latter condition is satisfied, then a function f:L1 ∪L2 —> C such that f|LiCk(Li), i = 1,2, allows Carleman approximation of order k by entire functions if and only if f satisfies the Cauchy-Riemann equations up to order k at the origin.

It is shown that simple rings over which proper cyclic right modules are continuous coincide with simple right PCI-rings, introduced by Faith.

Let B be a Boolean algebra and let ℳ and n be two systems of subsets of B, both containing all finite subsets of B. Let us assume further that the join ∨M of every set M∊ℳ and the meet ∧N of every set N∊n exist. Several authors have treated the question under which conditions there exists an isomorphism φ between B and a field δ of sets, satisfying the conditions:

The number of cyclic cubic fields with a given conductor and a given index is determined.

A simpler proof of the theorem on the spinor structure of space-time is given. Some geometrical insights are provided.

The investigations reported in this paper were prompted by a remark by A. M. Krall in [2] that certain functional which appear in the boundary conditions of the system adjoint to a given linear differential boundary value problem seem artificial in that setting.

Consider the extended hull of a weak model set together with its natural shift action. Equip the extended hull with the Mirsky measure, which is a certain natural pattern frequency measure. It is known that the extended hull is a measure-theoretic factor of some group rotation, which is called the underlying torus. Among other results, in the article Periods and factors of weak model sets, we showed that the extended hull is isomorphic to a factor group of the torus, where certain periods of the window of the weak model set have been factored out. This was proved for weak model sets having a compact window. In this note, we argue that the same results hold for arbitrary measurable and relatively compact windows. Our arguments crucially rely on Moody’s work on uniform distribution in model sets. We also discuss implications for the diffraction of such weak model sets and discuss a new class of examples which are generic for the Mirsky measure.

In a variety generated by a finite non-Abelian group, the commutator subgroup is not definable by a first-order formula.

compact Riemannian 4-symmetric space M can be regarded as a fibre bundle over a Riemannian 2-symmetric space with totally geodesic fibres isometric to a 2-symmetric space. Here the result of R. Crittenden for conjugate and cut points in a 2-symmetric space is extended to the focal points of the fibres of M. Also the restriction of the exponential map of M up to the first focal locus in the normal bundle of a fibre is proved to yield a covering map onto its image. It is shown that for the noncompact dual M*, the fibres have no focal points and hence the exponential map of M* restricted to the normal bundle of a fibre is a covering map. The classification of the compact simply connected 4-symmetric spaces G/L with G classical simple provides a large class of examples of these fibrations.

Mutants of boundary links may not be boundary links, not even homology boundary links. Hence mutants of homology boundary links may not be homology boundary links.

The surjectivity of epimorphisms in the category of planar graphs and edge-preserving maps follows from and is implied by the Four Colour Theorem. The argument that establishes the equivalence is not combinatorially complex.

Let $T$ be a contraction on a complex, separable, infinite dimensional Hilbert space and let $\sigma (T)\,(\text{resp}\text{.}\,{{\sigma }_{e}}(T))$ be its spectrum (resp. essential spectrum). We assume that $T$ is an essentially isometric operator; that is, ${{I}_{H}}\,-\,T*T$ is compact. We show that if $D\backslash \sigma (T)\,\ne \,\varnothing $, then for every $f$ from the disc-algebra

$${{\sigma }_{e}}\left( f\left( T \right) \right)\,=\,f\left( {{\sigma }_{e}}\left( T \right) \right),$$

where $D$ is the open unit disc. In addition, if $T$ lies in the class ${{C}_{0}}.\,\bigcup \,C{{.}_{0}}$, then

$${{\sigma }_{e}}\left( f\left( T \right) \right)\,=\,f\left( \sigma \left( T \right)\,\bigcap \,\Gamma \right),$$

where $\Gamma $ is the unit circle. Some related problems are also discussed.

In this note we study dependence in matroids as an exercise in combinational algebra. Because the work seems to have little connection with graph theory we will not use Tutte′s approach (1) which uses dual concepts. To define a matroid we use Edmunds (2).

A wavefront set is defined to be an image of Legendrian mapping. In this note we prove that generic wavefront sets have stable local topological structures by using Mather's stratification theory.

Let f(t) be integrable L(-π, π) and periodic with period 2π, and let

1.1

be its Fourier series. The series

1.2

obtained by the term by term differentiation of the series (1. 1), is called the Derived Fourier series of (1. 1). The series conjugate to (1. 2),

1.3

is called the Derived conjugate series of the Fourier series of f.

The paper proves that the group of infinite bounded Nielsen transformations is generated by elementary simultaneous Nielsen transformations modulo the subgroup of those transformations which are equivalent to the identical transformation while acting in a free abelian group. This can be formulated somewhat differently: the group of bounded automorphisms of a free abelian group of countably infinite rank is generated by the elementary simultaneous automorphisms. This proves D. Solitar's conjecture for the abelian case.