This paper is a sequel to [2]. By a semigroup of high rank we mean a semigroup such that for s1≠s2, 〈S\{s1,s2}〉⊂S (properly). Semigroups of high rank such that 〈S\{s}〉⊂S(royal semigroups) were classified in [2], where it was also shown that for a noble semigroup (i.e. a semigroup of high rank such that there exists a superfluous element z in S for which 〈S\{z}〉S) there exists eithere exactly one superfluous element or exactly two superfluous elements [2, Theorem 3.7].

The following formulae may be added to the list given in the Proceedings of the Edinburgh Mathematical Society, Vol. XII., pp. 86-102 (1894).

and also on.

1. Many investigations have been concerned with a squaro matrix P with non-negative coefficients (elements). It is remarkable that many interesting properties of P are determined by the set Σ of index pairs of positive (i.e. non-zero) coefficients of P, the actual values of these coefficients being irrelevant. Thus, for example, the number of characteristic roots equal in absolute value to the largest non-negative characteristic root p depends on Σ alone, if P is irreducible. If P is reducible, then Σ determines the standard forms of P (cf. § 3). The multiplicity of p depends on Σ, and on the set S of indices of those submatrices in the diagonal in a standard form of P which have p as a characteristic root. It has apparently not been considered before whether Σ and S also determine the elementary divisors associated with p. We shall show that, in general, the elementary divisors do not depend on these sets alone, but that necessary and sufficient conditions may be found in terms of Σ and S (a) for the elementary divisors associated with p to be simple, and (b) that there is only one elementary divisor associated with p.

Since the work of R. A. Fisher [2] and Kolmogorov, Petrovskij and Piskunov [7] (see [6] for further references) the problem of travelling fronts in reaction–diffusion equations has been extensively studied. For the equation

with F(0) = F(1) = 0 a travelling front is a solution

where the function of one variable φ is decreasing and satisfies φ(−∞) = 1, φ(+∞) = 0. The function φ describes the shape of the front and the constant c is the speed of propagation. There are two main types of the problem. In the all-positive case, where the function F satisfies

there is a half-line [c0, ∞), c0>0, of speeds. For each c∈[c0, ∞) there is, up to translation, a unique travelling front. Fronts for different c can be distinguished by the rate of decay towards +∞. In the threshold case, where F has the property, for some λ∈(0, 1),

there is a unique speed c0 with a travelling front, which is unique up to translation. In this case the sign of c0 is determined by the sign of the integral

Let G be a finite group, p be a prime and K be a field of characteristic p. Let

be a decomposition of the group ring of G over K as a sum of indecomposable two-sided ideals. An irreducible K(G)-module is said to be in the block Bi if it occurs as a composition factor of Bi. The block containing the trivial K(G)-module is called the principal block of G.

Let $K$ be an algebraically closed field of characteristic zero, complete for an ultrametric absolute value. We show that the $p$-adic main Nevanlinna Theorem holds for meromorphic functions inside an ‘open’ disc in $K$. Let $P_{n,c}$ be the Frank–Reinders’s polynomial

$$ (n-1)(n-2)X^n-2n(n-2)X^{n-1}+ n(n-1)X^{n-2}-c\qq (c\neq0,\ c\neq1,\ c\neq2) $$

and let $S_{n,c}$ be the set of its $n$ distinct zeros. For every $n\geq 7$, we show that $S_{n,c}$ is an $n$-points unique range set (counting multiplicities) for unbounded analytic functions inside an ‘open disc’, and for every $n\geq10$, we show that $S_{n,c}$ is an $n$-points unique range set ignoring multiplicities for the same set of functions. Similar results are obtained for meromorphic functions whose characteristic function is unbounded: we obtain unique range sets ignoring multiplicities of $17$ points. A better result is obtained for an analytic or a meromorphic function $f$ when its derivative is ‘small’ comparatively to $f$. In particular, for every $n\geq5$ we show that $S_{n,c}$ is an $n$-points unique range set ignoring multiplicities for unbounded analytic functions with small derivative. Actually, in each case, results also apply to pairs of analytic functions when just one of them is supposed unbounded. The method we use is based upon the $p$-adic Nevanlinna Theory, and Frank–Reinders’s and Fujimoto’s methods used for meromorphic functions in $\mathbb{C}$. Among other results, we show that the set of functions having a bounded characteristic function is just the field of fractions of the ring of bounded analytic functions in the disc.

AMS 2000 Mathematics subject classification: Primary 12H25. Secondary 12J25; 46S10

The following is a direct proof of a theorem by Zia-ud-Din.

Let {ν} ≡ {ν1, ν2, …, νp)} be any S-function of weight r + s such that

in the alternant denoted in the theorem by A(αβγ….). Let {μ} be an S-function of weight s, equal to and let {λ} be an S-function of weight r, such that {λ}δ(x1, …,xp) is obtained with coefficient gλμν by diminishing the indices in the alternant {ν}δ(x1, …,xp) according to the theorem.

1. The Gontcharoff interpolation series

where

has been studied in various special cases. For example, if an = a0 (all n), (1.0) reduces to the Taylor expansion of F(z). If an = (−1)n, J. M. Whittaker showed that the series (1.0) converges to F(z) provided F(z) is an integral function whose maximum modulus satisfies

the constant ¼π being the “best possible”. In the case |an| ≤ 1, I have shown that the series converges to F(z) provided F(z) is an integral function whose maximum modulus satisfies

and that while ·7259 is not the “best possible” constant here, it cannot be replaced by a number as great as ·7378.

This paper generalizes, in two senses, work of Petzl and Sharp, who showed that, for a $\mathbb{Z}$-graded module $M$ over a $\mathbb{Z}$-graded commutative Noetherian ring $R$, the graded Cousin complex for $M$ introduced by Goto and Watanabe can be regarded as a subcomplex of the ordinary Cousin complex studied by Sharp, and that the resulting quotient complex is always exact. The generalizations considered in this paper are, firstly, to multigraded situations and, secondly, to Cousin complexes with respect to more general filtrations than the basic ones considered by Petzl and Sharp. New arguments are presented to provide a sufficient condition for the exactness of the quotient complex in this generality, as the arguments of Petzl and Sharp will not work for this situation without additional input.

AMS 2000 Mathematics subject classification: Primary 13A02; 13E05; 13D25; 13D45

Bonsall and Tomiuk have shown, in (3), the connection between the local compactness of a monothetic semi-algebra and the spectral properties of a generating element. This theme was developed, in (4), to give a complete characterisation of prime, strict locally compact monothetic semi-algebras in terms of the spectrum of a generator (Theorem A). Here we extend this result to the case of a semi-simple locally compact monothetic semi-algebra (Theorem B).

The object of this note is to derive a form of Poisson's equation from general relativistic mechanics, without assuming the field to be either static or “weak”. The problem is essentially a “local” problem, all observations being made by one observer; this observer determines the apparent gravitational field in his vicinity by observing the motions of free (isolated) particles. Defining gravitational mass by means of Poisson's equation, we find the relation between the densities of gravitational and inertial mass relative to any observer. We also find what may be called the non-rotating frame of reference belonging to any observer.

Let G be a group and K a field. We shall denote by U(KG) the group of units of the group ring of G over K. Also, if X is a group, T(X) will denote the torsion subset of X, i.e., the set of all elements of finite order in X.

Group theoretical properties of U(KG) have been studied intensively in recent years and it has been found that some conditions about U(KG) imply that T = T(G) must be a subgroup of G and that every idempotent of KT must be central in KG.

Let X be a real or complex Banach space with norm ∥·∥· Let G denote the set of all isometric automorphisms on X. Then G is a bounded subgroup of the group of all invertible operators GL(X) in B(X). We shall call G the group of isometries with respect to the norm ∥·∥· A bounded subgroup of GL(X) is said to be maximal if it is not contained in any larger bounded subgroup. The Banach space X has maximal norm if G is maximal. Hilbert spaces have maximal norm. For the (real or complex) spaces c0, lp (1≦p<∞), Lp[0,1] (1≦p<∞), Pelczynski and Rolewicz have shown that the standard norms are maximal ([3], pp. 252–265). In finite dimensional spaces the only maximal groups of isometries are the groups of orthogonal transformations. Given any bounded group H in B(X), X can be renormed equivalently so that each T∈H is an isometry, by ‖x‖1=sup{|Tx‖; T∈H}. Therefore corresponding to every maximal subgroup G there is at least one maximal norm for which G is the group of isometries. In this paper we shall investigate those maximal groups G for which there is only one maximal norm with G as its group of isometries.