About forty years ago, when Spencer was rising into philosophic fame, it used often to be said by his admirers that he was an accomplished mathematician. This statement was accepted without demur, though it was known that he had not measured himself against rivals of his own age, or, what is more important, had not produced anything new in this old science.

Let G be a permutation group of degree m. Let x be an irreducible complex character of G. If A = (aij) is an m-square matrix, the generalised matrix function of A based on G and x is defined by

For example if G = Sm, the full symmetric group, and x is the alternating character, then d = determinant. If G = Sm and x is identically 1, then d = permanent.

1. The following note examines, more closely than the text books usually do, some results of reciprocation, and has special reference to properties of a certain infinite set of triangles and connected conies.

Let Ω be a finite set and let G be a permutation group acting on it. A subset H of G is called t-intersecting if any two elements in H agree on at least t points. Let SDn and SBn be the classical Coxeter group of type Dn and type Bn, respectively. We show that the maximum-sized (2t)-intersecting families in SDn and SBn are precisely cosets of stabilizers of t points in [n] ≔ {1, 2, …, n}, provided n is sufficiently large depending on t.

Let S(X) denote the semigroup of all continuous selfmaps of the topological space X. Let ℒ(S(X)) and ℛ(S(X)) denote the partially ordered families of all ℒ-classes and ℛ-classes, respectively, of S(X) where the partial orders are the usual ones [3, p. 29]. In [6] we made the following

Conjecture. The following statements are equivalent about any two compact 0-dimensional metric spaces X and Y:

(1) ℒ(S(X)) and ℒ(S(Y)) are order isomorphic.

(2) ℛ(S(X)) and (S{Y)) are order isomorphic.

(3) The semigroupsS(X) and S(Y) are isomorphic.

(4) The spaces X and Y are homeomorphic.

An asymptotic tract of a real function u harmonic and non-constant in ℂ is a component of the set {z:u(z)≠c}, for some real number c; a quasi-tractT(≠ℂ) is an unbounded simply-connected domain in ℂ such that there exists a function u that is positive, unbounded and harmonic in T such that, for each point ζ∈∂T∩ℂ,

and a ℱ-tract is an unbounded simply-connected domain T in ℂ whose every prime end that contains ∞ in its impression is of the first kind.

The authors study the growth of a harmonic function in one of its asymptotic tracts, and the question of whether a quasi-tract is an asymptotic tract. The branching of either type of tract is also taken into consideration.

Using a combined dominant condition, we obtain general results concerning the complex oscillation for a class of homogeneous linear differential equations w(k) + + … + A1w′ + (A0 + A)w = 0 with k ≥ 2, which has been investigated by many authors. In particular, we discover that there exists a unique case that possesses k linearly independent zero-free solutions for these equations, and we resolve an open problem and simultaneously answer a question of Bank.

Let be a linear differential expression involving n independent variables xi the coefficients Aik Bi, and C being functions of the independent variables but not involving the dependent variable u. Associated with F(u) is the adjoint expression

In (2) Bruhat has developed a theory of differentiable functions and distributions on a locally compact group in order to apply it to the study of the irreducible representations of the p-adic groups. Later, Whyburn (8) defined differentiable forms on a locally compact group and proved an analog of the de Rham theorem concerningthe relationship between the Čech cohomology and the De Rham cohomology. In (4) Ihave introduced the notions of “generalised manifold” (roughly speaking a projective limit of smooth manifolds) and of “differentiable forms” on it, extending some of the results due to Bruhat and Whyburn.

We define a new functional which is gauge invariant on the space of all smooth connections of a vector bundle over a compact Riemannian manifold. This functional is a generalization of the classical Yang-Mills functional. We derive its first variation formula and prove the existence of critical points. We also obtain the second variation formula.

In this paper we continue our study of the tensor product of distributive lattices which was begun in (2). We obtain a representation of the tensor product as a ring of sets and we describe a simple way to construct the tensor product of finite distributive lattices.

A sequence x = {si} has been defined to be summable-(Z, p) to s if

Where p is a positive integer and

When the Edinburgh Mathematical Society was founded, it was deemed impracticable, from the expense that would be involved, to print the papers read at its meetings. It was, however, resolved that copies of them should be deposited with the Secretary, and that these copies should as far as possible be made accessible to the members of the Society. Want of leisure during the first session prevented me from doing my part in carrying out this intention of the Society, and committing to writing the paper which was delivered at the second meeting. During its second session, the Society resolved to print its proceedings in whole or in abstract; and a beginning was made with the second volume, the first being left over for future consideration, as the cost of printing absorbed nearly the entire income of the Society.

It is shown that a condition on the order of a meromorphic function in a result of A. A. Gol’dberg cannot be relaxed.

AMS 2000 Mathematics subject classification: Primary 30D35

Let

be an entire function, where (αn) is a strictly increasing sequence of non-negativeintegers. The maximum modulus, M(r), the minimum modulus, m(r), and the maxi-mum term, μ(r), of f defined by

A characterization is given of those unital, 2-subhomogeneous, Fell C*-algebras which have only inner derivations. This proves Sproston and Strauss's conjecture from 1992. Various examples are given of phenomena which cannot occur for separable C*-algebras. In particular, an example is given of a C*-algebra with only inner derivations which has a quotient algebra admitting outer derivations. This answers a question of Akemann, Elliott, Pedersen and Tomiyama from 1976.

The explicit defining equations of a new family of curves whose members have a trivial automorphism group are given. Each member is defined for characteristic zero and all but a finite number of characteristics greater than zero. This family, in conjunction with a previously appearing family of the author’s, provides explicit examples of algebraic curves which possess only the trivial automorphism for each genus greater than three. The family is then used to construct Riemann surfaces without anticonformal automorphisms and Klein surfaces with no non-trivial automorphisms.

AMS 2000 Mathematics subject classification: Primary 14H37; 30F50; 30F99

The general heat equation is defined as

where v is a fixed positive number and α is a fixed number. If v = α = 0, then (1.1) reduces to the ordinary heat equation

where u(x,t) is regarded as the temperature at a point x at time t, in an infinite insulated rod extended along the x-axis in the xt-plane. If we set , then (1.1) becomes

In [3] Fuller introduced an index (now called the Fuller index) in order to study periodic solutions of ordinary differential equations. The objective of this paper is to give a simple generalisation of the Fuller index which can be used to study periodic points of flows in Banach spaces. We do not claim any significant breakthrough but merely suggest that the simplistic approach, presented here, might prove useful for the study of non-linear differential equations. We show our results can be used to study functional differential equations.

The transpositions that generate a symmetric group can be represented as real reflections: symmetry operations of a regular simplex. Analogous unitary reflections serve to generate other factor groups of the braid group; they are symmetry operations of regular complex polytopes. Certain relationships among these groups have, as geometric counterparts, unexpected plane sections of the polytopes, beginning with the square sections of the regular tetrahedron. In Section 6, 5-dimensional coordinates will be used to exhibit pentagonal sections of the 4-dimensional regular simplex. The most spectacular instance of such “equatorial” sections occurs in the case of the Witting polytope in complex 4-space, so exquisitely drawn by Peter McMullen for the frontispiece of Regular Complex Polytopes [6]. This has a plane section 3{<5}3 which appears thare as Fig. 4.8B on page 48. Shephard [9, p. 92] called it 3(360)3. Its 120 vertices will be seen to be situated “inside” 120 of the 2160 faces 3{3}3 of the Witting polytope. These “faces” are self-inscribed octagons [7, p. 290].