A new, simplified proof is given of a theorem of J.L. Hickman (to be published; see also J. London Math. Soc. (2) 9 (1974), 239–244), giving an upper bound for the sums of a well-ordered series of ordinals under permutation.

For a constrained minimization problem with cone constraints, lagrangean necessary conditions for a minimum are well known, but are subject to certain hypotheses concerning cones. These hypotheses are now substantially weakened, but a counter example shows that they cannot be omitted altogether. The theorem extends to minimization in a partially ordered vector space, and to a weaker kind of critical point (a quasimin) than a local minimum. Such critical points are related to Kuhn-Tucker conditions, assuming a constraint qualification; in certain circumstances, relevant to optimal control, such a critical point must be a minimum. Using these generalized critical points, a theorem analogous to duality is proved, but neither assuming convexity, nor implying weak duality.

High level programming languages can generally be compiled in many different ways. Thus it is natural to ask if there is a best way, for example in the sense of achieving complete execution of the program in a minimum number of steps; such a complete, minimal execution procedure will be called optimal.

In recent years this question has been studied and answered for several simple models of programming languages, and a technique for proving procedures to be optimal has gradually emerged. Even for model languages the proofs of optimality become intricate; thus it is natural to emphasize the simplicity of the underlying technique by generalising it to an abstract system. That is the purpose of this paper. The general method to be given applies to prove all theorems (on optimal executions for model languages) which are known to the author. None of the previously known proofs has explicitly used the method, but in no case is it particularly difficult to modify the known proof so as to conform with the method.

Minimal congruences on a Rees matrix semigroup S having at least one proper congruence are described. Necessary and sufficient conditions for S to te subdirectly irreducible are given in two cases according to whether the structure group of S is trivial.

Necessary and sufficient conditions are given for a square matrix to te the matrix of distances of a circulant code. These conditions are used to obtain some inequalities for cyclic difference sets, and a necessary condition for the existence of circulant weighing matrices.

Let p be a rational prime and Qp be the field of p–adic numbers. Jean-Pierre Serre [Lecture Notes in Mathematics, 350, 191–268 (1973)] had defined p–adic modular forms as the limits of sequences of modular forms over the modular group SL2(Z). He proved that with each non-zero p–adic modular form there is associated a unique element called its weight k. The p–adic modular forms having the same weight form a Qp–vector space.

The object of this paper is to obtain a basis of p–adic modular forms and thus to know precisely all p–adic modular forms of a given weight k. The dimension of such modular forms as a Qp–vector space is countably infinite.

A review of the development of the theory of existence and uniqueness of solutions to initial-value problems for mostly reduced versions of the nonlinear Maxwell-Boltzmann equation with a cut-off of intermolecular interaction, precedes the formulation and discussion of a somewhat generalized initial-value problem for the full nonlinear Maxwell-Boltzmann equation, with or without a cut-off. This is followed by a derivation of a new existence-uniqueness result for a particular Cauchy problem for the full nonlinear Maxwell-Boltzmann equation with a cut-off, under the assumption that the monatomic Boltzmann gas in the unbounded physical space X is acted upon by a member of a broad class of external conservative forces with sufficiently well-behaved potentials, defined on X and bounded from below. The result represents a significant improvement of an earlier theorem by this author which was until now the strongest obtained for Cauchy problems for the full Maxwell-Boltzmann equation. The improvement is basically due to the introduction of equivalent norms in a Banach space, the definition of which is connected with an exponential function of the total energy of a free-streaming molecule.

Let (α β) denote the class of functions

analytic in the unit disc Δ ≡{z: |z| < 1} and satisfying

for some α, β (0 ≤ α < 1, 0 < β ≤ 1) and for all z ∈ Δ. In the present paper, sharp coefficient estimates for functions in (α, β) have been obtained. The results thus obtained not only generalize the corresponding results of Thomas H. MacGregor (Michigan Math. J. 10 (1963), 277–281), A.V. Boyd (Proc. Amer. Math. Soc. 17 (1966), 1016–1018) and others, but also give rise to analogous results for various other subclasses of starlike functions.

The article contains the basic theory resulting from the presentation of a dense functor N: A → C by means of an expansion K(k, C) ० NJk ≅ C the term dense functor being used instead of the equivalent term left-adequate functor. Results by various authors on the density type of a functor are formulated in the V-context for V symmetric monoidal closed, and elementary proofs are given. In particular a characterisation theorem containing the well-known results of Beck and Ulmer is established.

This paper presents three arguments, one involving orientability, and the others Milnor duality and, respectively, the injectivity of cup product into H2 for an abelian group and free finite group actions on homotopy 3-spheres to show that there are high dimensional knot groups which are not the groups of knotted 2-spheres in S4, thus answering a question of Fox (“Some problems in knot theory”, Topology of 3-manifolds and related topics”, 168–176 (Proceedings of the University of Georgia Institute, 1961. Prentice-Hall, Englewood Cliffs, New Jersey, 1962).

The classical Wiener-Pitt phenomenon for measures may be formulated as an existence theorem for Fourier multipliers with irregular, spectral properties and the result has been refined in various ways over the years. The most recent development is due to Zafran, who exhibits abnormal spectral behaviour in multipliers whose transforms vanish at infinity and which operate on Lp where 1 < p < ∞, p ≠ 2. Zafran's methods use Littlewood-Paley theory, interpolation, and the construction of a certain class of measures. We show here how the constructive element of his proof may be considerably simplified and sharpened.