Suppose that (ni) = (n1, n2,…) and (mi) are infinite sequences of positive integers with ni < mi < ni+1. It is well-known and easily proved that, if a series σak is (C, 1) summable to s and has lacunae (ni, mi) such that ak = 0 (ni < k < mi) with

then where

The idea of “free” electrons is purely classical, but it is convenient to introduce it also in the quantum theory of metals. In order to calculate the number of free electrons N F from a quantum mechanical model it is necessary to give an appropriate definition.

This paper investigates the qualitative behaviour of solutions of the second order equation x¨+f(x, x¯) x¯+g(x)=0. The sufficient conditions for all solutions to be oscillatory are established. The theorems on the existence of a nontrivial periodic solution are proved and the criteria for the origin to be a global centre and globally asymptotically stable are given.

The present paper is a sequel to a previous paper (1) in which I proved a conjecture made by R. M. Robinson concerning sums of three roots of unity. For any algebraic integer α, we denote by the maximum absolute value of α and all its algebraic conjugates. We say that α and β are equivalent if α = εβ′, where ε is some root of unity and β′ is some conjugate of β; when this happens we obviously have

The particular form of Taylor's theorem considered states that: If ƒ(x)is n times differentiable at a thenwhere.

Consider the pencil of [4]'s through the [3] which contains l15, l25, l34, any [4] of the pencil meets the V25 again in a conic, q.

A non-linear Klein–Gordon equation is used to discuss the theory of slowly varying, weakly non-linear wave trains. An averaged variational principle is used to obtain transport equations for the slow variations which incorporate the leading order modulation and non-linear terms. Linearized transport equations are used to discuss instabilities.

The Mellin transforms of generalized hypergeometric functions are discussed in this paper, and it is shown how some of the most general integrals of the Mellin type can be deduced from them. Four general theorems are considered and a number of special cases are given in detail.

We consider a 2-dimensional square lattice which is partitioned into a periodic array of rectangular cells, on which a nearest neighbour random walk with symmetric increments is defined whose transition probabilities only depend on the relative position within a cell. On the basis of a determinantal identity proved in this paper, we obtain a result for finite Markov chains which shows that the diffusion constants for the random walk are monotonic functions of the individual transition probabilities. We point out the similarity of this monotonicity property to Rayleigh's Monotonicity Law for electric networks or, equivalently, reversible random walks.

The properties of centre-of-mass systems at high energies are discussed, and the single time form of field theory is used to set up state vectors that describe the simpler centre-of-mass states of fields. These states, which are labelled by the eigenvalues of total angular momentum, are expanded in terms of the spherical harmonic resolution of the fields about a fixed point. A discussion of the application of the method to pion-nucleon scattering is given.

The usual formula for the rate of loss of energy by a fast particle contains a factor z2, the square of the charge, which might suggest that He+ should lose energy at a quarter of the rate that He++ does. It is shown that when Born's approximation is applicable the energy-loss is reduced instead by a factor of about ⅝.

Let G be a connected perfect real Lie group. We show that there exists α < dim G and p ∈ $\mathbb{N}$* such that if μ is a compactly supported α-Frostman Borel measure on G, then the pth convolution power μ*p is absolutely continuous with respect to the Haar measure on G, with arbitrarily smooth density. As an application, we obtain that if A ⊂ G is a Borel set with Hausdorff dimension at least α, then the p-fold product set Ap contains a non-empty open set.

Let χ(t) = a0tn – a1tn−1 + ⋯ + (−1)rartn−r be the chromatic polynomial of a graph, the characteristic polynomial of a matroid, or the characteristic polynomial of an arrangement of hyperplanes. For any integer k = 0, 1, …, r and real number x ⩾ k − r − 1, we obtain a linear bound of the coefficient sequence, that is

\begin{align*} {r+x\choose k}\leqslant \sum_{i=0}^{k}a_{i}{x\choose k-i}\leqslant {m+x\choose k}, \end{align*}

We characterise the topological spaces which arise as the primitive ideal spaces of the Cuntz–Krieger algebras of graphs satisfying condition (K): directed graphs in which every vertex lying on a loop lies on at least two loops. We deduce that the spaces which arise as ${\rm Prim}\;C^*(E)$ are precisely the spaces which arise as the primitive ideal spaces of AF-algebras. Finally, we construct a graph $\wt{E}$ from E such that $C^*(\wt{E})$ is an AF-algebra and ${\rm Prim}\;C^*(E)$ and ${\rm Prim}\;C^*(\wt{E})$ are homeomorphic.

In several recent papers G. L. Lunc (7, 8, 9, 10, 11) has examined the series

where {mn} is a sequence of positive integers or zeros, and μn > 0, μn ↑ ∞. The discussion has been aimed towards extending results which are well-known for Dirichlet series to series (1) which Lunc has very reasonably called a Taylor–Dirichlet series.

We give necessary and sufficient conditions for a conformal foliation locally generated by conformal vector fields to produce harmonic morphisms. Natural constructions of harmonic maps and morphisms are thus obtained. Also we obtain reducibility results for harmonic morphisms induced by (infinitesimal) conformal actions on Einstein manifolds.

Let (Ω, ,μ) be a probability space, and let be a sub-sigma-algebra of . Let X be a uniformly convex Banach space. Let A =L∞(Ω, , μ X) denote the Banach space of (equivalence classes of) essentially bounded μ-Bochner integrable functions g: Ω.→ X, normed by the function ∥.∥∞ defined for g ∈ A by

(cf. [6] for a discussion of this space). Let B = L∞(Ω, , μ X), and let f ε A. A sufficient condition for g ε B to be a best L∞-approximation to f by elements of B is established herein.

Let R be a Dedekind domain with field of quotients K. Let A be a finite-dimensional K-algebra. We consider isomorphism classes and genera in a category whose objects are indexed sets of full R-lattices in some ambient A-module and whose morphisms are the A-homomorphisms of the ambient A-modules which map each lattice into its corresponding lattice. We find conditions under which the stable A-isomorphism class of one particular lattice in an indexed set will determine the stable class of the indexed set within its genus. We apply our methods to show that if L/K is a tame Galois extension of algebraic number fields then the stable isomorphism class of the set of ambiguous ideals in L considered as Galois modules over K is determined by the class of the ring of integers in L together with the inertia subgroups and their standard representations over the respective residue fields of R.

The Morrison–Kawamata Cone Conjecture predicts that the action of the automorphism group on the effective nef cone and the action of the pseudo-automorphism group on the effective movable cone of a klt Calabi–Yau pair have rational, polyhedral fundamental domains. In [CPS], we proved the conjecture for certain blowups of Fano manifolds of index n - 1. In this paper, we consider the Morrison–Kawamata conjecture for blowups of Fano manifolds of index n - 2.