If light of a frequency which corresponds to an energy greater than the ionisation potential falls on an atom, an electron may be ejected and energy absorbed. To calculate the absorption coefficient, or the rate of absorption of energy per unit intensity of incident radiation for a given frequency, one must first choose a model for the atom. If we confine ourselves to the inner K electrons there will be two electrons in this shell for the heavier atoms, and a fairly good model of the atom is obtained by considering each electron to be moving independently in a central field of force due to the charged nucleus: i.e. we neglect electronic interaction and assume that the wave functions for the system are hydrogenic. Some writers make a partial correction for this neglect of interaction by modifying the central charge through the introduction of a screening factor which is so chosen that the minimum calculated energy required to remove one of the K electrons will agree with the experimental value provided by the K absorption edge. In general, however, the approximation is fairly good, and this is particularly so in the interior of a star where the atoms are highly ionised. It is not so good when the atom is bound as in a metal, and, of course, most of the laboratory work has been carried out on atoms in this bound state.

Additive representations of natural numbers by mixtures of squares, cubes and biquadrates belong to the class of more interesting special cases which form the object of attention for testing the general expectation that any sufficiently large natural number n is representable in the form

formula here

as soon as the reciprocal sum [sum ]sj=1k−1j is reasonably large. With the exception of a handful of very special problems, in the current state of knowledge the latter reciprocal sum must exceed 2, at the very least, in order that it be feasible to successfully apply the Hardy–Littlewood method to treat the corresponding additive problem. Here we remove a case from the list of those combinations of exponents which have defied treatment thus far.

Let Γ be a Fuchsian group of the first kind acting on the upper half plane H+. Let be a Ford fundamental region for Γ in H+. Let ξ be a real number (a limit point) and let L( = Lξ) = {ξ + iy|0 ≤ y < 1}. L can be broken into successive intervals each one of which can be mapped by an element of Γ into . Since L is a hyperbolic line (h-line), this gives us a set of h-arcs in which we will call the image.

The celebrated Kerékjártó theorem asserts that planar continuous periodic maps can be continuously linearised. We prove that for each k ∈ {1, 2,. . ., ∞}, ${\Mathcal {C}}$k-planar periodic maps can be ${\Mathcal {C}}$k-linearised.

It is a matter of some interest to determine which closed manifolds have the same homology as a closed aspherical manifold (7) (9). This paper is concerned with the more restrictive question of which spheres have the same homology as some closed aspherical manifold.

Recently, Lutz and Stull used methods from algorithmic information theory to prove two new Marstrand-type projection theorems, concerning subsets of Euclidean space which are not assumed to be Borel, or even analytic. One of the theorems states that if \[K \subset {\mathbb{R}^n}\] is any set with equal Hausdorff and packing dimensions, then

\begin{equation} \[{\dim _{\text{H}}}{\pi _e}(K) = \min \{ {\dim _{\text{H}}}{\text{ }}K{\text{, 1}}\} \] \end{equation}

for almost every \[e \in {S^{n - 1}}\]. Here \[{\pi _e}\] stands for orthogonal projection to span (\[e\]). The primary purpose of this paper is to present proofs for Lutz and Stull’s projection theorems which do not refer to information theoretic concepts. Instead, they will rely on combinatorial-geometric arguments, such as discretised versions of Kaufman’s “potential theoretic” method, the pigeonhole principle, and a lemma of Katz and Tao. A secondary purpose is to generalise Lutz and Stull’s theorems: the versions in this paper apply to orthogonal projections to m-planes in \[{\mathbb{R}^n}\], for all \[0 < m < n\].

We show that projective unitary groups, or more generally, central quotients of products of unitary groups are totally $N$-determined when considered as $p$-compact groups at odd primes $p$.

We study the set ${\mathcal{S}}$ of labelled seeds of a cluster algebra of rank n inside a field ${\mathcal{F}}$ as a homogeneous space for the group Mn of (globally defined) mutations and relabellings. Regular equivalence relations on ${\mathcal{S}}$ are associated to subgroups W of AutMn(${\mathcal{S}}$), and we thus obtain groupoids W\${\mathcal{S}}$. We show that for two natural choices of equivalence relation, the corresponding groups Wc and W+ act on ${\mathcal{F}}$, and the groupoids Wc\${\mathcal{S}}$ and W+\${\mathcal{S}}$ act on the model field ${\mathcal{K}}$=ℚ(x1,. . .,xn). The groupoid W+\${\mathcal{S}}$ is equivalent to Fock–Goncharov's cluster modular groupoid. Moreover, Wc is isomorphic to the group of cluster automorphisms, and W+ to the subgroup of direct cluster automorphisms, in the sense of Assem–Schiffler–Shramchenko.

We also prove that, for mutation classes whose seeds have mutation finite quivers, the stabiliser of a labelled seed under Mn determines the quiver of the seed up to ‘similarity’, meaning up to taking opposites of some of the connected components. Consequently, the subgroup Wc is the entire automorphism group of ${\mathcal{S}}$ in these cases.

An indefinite binary quadratic form ƒ gives rise to a certain function M on the torus. The properties of M, especially those related to its maximum – the so-called inhomogeneous minimum of ƒ – are the subject of numerous papers. Here we continue this study, putting more emphasis on the general behaviour of M.

In this paper we study the asymptotic behaviour of large axisymmetric deformations of closed axisymmetric nonlinearly elastic shells under internal hydrostatic pressure. These shells can suffer flexure, extension, and shear. Since there are spherical shells that can enclose an arbitrarily large volume at a finite pressure (cf. [1]), we take the volume rather than the pressure as the large parameter.

In this note we provide proofs for two facts related to connected topological K-theory, expected by Ib Madsen.

Many models of the universe have been proposed, by de Sitter, Milne, Bondi and Gold, Hoyle and others. The observed data being insufficient, the models are usually based on some simple hypothesis. The simplest is the cosmological principle, namely, that apart from local irregularities the universe presents the same general aspect at every point. Milne (5) has used a restricted form of the principle, namely, that the aspect is independent of spatial position but is dependent on the observed time from some fixed epoch in the past. Bondi and Gold(i) have proposed the ‘perfect cosmological principle’ that the aspect is completely independent of space and time.

In (1) Akemann and Rosenfeld introduced a property for Banach algebras which they called (*). A Banach algebra satisfies (*) if every maximal one-sided ideal in is closed. They proved that certain classes of Banach algebra with satisfy (*), and they mentioned a conjecture that if is a Banach algebra with , then satisfies (*). In this paper we show that, if is a Banach algebra with a bounded right (left) approximate identity, then maximal left (right) ideals in are closed, and we give a counter-example to the above conjecture. We also give an independent proof that C*-algebras satisfy (*).

Green's function for SH-line source inside a wedge shaped medium with its apex removed by a circular cylindrical surface has been obtained. Exact location of the image sources facilitates the determination of the reflexion and diffraction at the circular and also at the plane boundary of the multiple phases clearly. The scattered field from the corner is indicated. The connexion of this solution with that of complete wedge has been shown with suitable contour integrations.

A classical theorem of Lindelöf asserts that if ƒ is a function analytic and bounded in the unit disc δ which has the asymptotic value L at a point ξ ε ∂ δ then it has the non-tangential limit L at ξ. This result does not remain true for functions f analytic in δ whose maximum modulus grows to infinity arbitrarily slowly. However, the second author has recently obtained some results of Lindelöf type valid for these functions. In this paper we obtain new results of this kind. We prove that if f is an analytic function of slow growth in δ and ξ ε ∂ δ, then certain restrictions on the growth of ƒ′ along a segment which ends at ξ do imply that ƒ has a non-tangential limit at ξ.

A necessary and sufficient condition is given for a unital C*-algebra A to admit a discontinuous homomorphism into a Banach algebra which is continuous on its centre. The condition is that A must have a Glimm ideal G such that the C*-algebra A/G admits a discontinuous homomorphism into a Banach algebra.

Let X be a complex normed space and let T be a Hermitian linear operator on X (i.e. the numerical range of T is real). It was proved by Bonsall (see (4); Theorem 10.13) that then

for every u ∈ X. This inequality was improved by the author (3) to

Inequality (1) is closely related (see (3)) to some inequalities of Hadamard (6) and Kolmogorov (7) about the successive derivatives of functions in L∞(−∞, ∞). It was also shown in (3) (and was, in fact, shown already in (7)) that the constant 2 is best possible in (1). However, as we shall see, inequality (1) can be sharpened considerably if T attains its norm on u, i.e. if ‖Tu‖ = ‖T‖‖u‖.

Limiting solutions are derived for the flexure of simply supported many-sided regular polygons, as the number of sides is increased indefinitely. It is shown that these solutions are different from those for simply supported circular plates. For axisymmetric loading, circular plate solutions overestimate the deflexions and the moments by significant factors.