In this paper we investigate the structure of a set of n reals that contains a maximal number of l-term arithmetic progressions. This problem has been indicated by J. Riddell. Let l and n be positive integers with 2 ≦ l ≦ n. By F1(n) we denote the maximal number of l-term arithmetic progressions that a set of n reals can contain. A set of n reals containing F1(n)l-progressions will be called an Fl,(n)-set.

In the usual definition of an inductive limit of locally convex spaces, one is given a linear space E, a family (Eα) of locally convex spaces and a set (iα) of linear maps from Eα into E. Garling in [2] studies an extension of this, looking at absolutely convex subsets Sα of Eα and restrictions jα of iα to such sets. If, in the definition of Garling [2, p. 3], each Sα is instead a balanced semiconvex set, then the finest linear (not necessarily locally convex) topology on E for which the maps ja are continuous, will be referred to as the generalized *-inductive limit topology of the semiconvex sets. This topology is our object of study in the present paper; we find applications in the closed graph theorem.

A systematic and easily automated least squares procedure, not using integral equations or special functions, is presented for approximating the solutions of general dual trigonometric equations. This is desirable, since current analytic methods apply only to special equations, require the use of integral equation and special function theory, and do not lend themselves easily to numerical work; see, e.g. [1, 2, 6, 8, 9,10, 11, 12, 13, 14, 15, 16, 17].

It is proved that a pure-injective module over a commutative ring with unity is a summand of a product of duals of finitely presented modules, where duals are to be understood with reference to the circle group T, with induced module structures. Using similar techniques, it is also shown that an R-module has its underlying group pure-injective precisely when it is a submodule of a product of duals of cyclic modules and also a summand as abelian group of the same product.

All rings considered are commutative with unity and all modules are unitary. Let Mod-R be the category of modules over a ring R. An exact sequence 0 → A → B → C → 0 in Mod-R is pure-exact if, for any N in Mod-R, 0 → A⊗N → B⊗N → C⊗N → 0 is exact. A module M is pure-injective if it has the injective property relative to the class of pure-exact sequences in Mod-R. A module P is FP (finitely presented) if it is the image of a finitely generated free module with a finitely generated kernel. A module M is compact if it carries a Hausdorff compact topology so that M is a topological R-module. Let T denote the circle group—the group of real numbers modulo the integers—and let X* denote the dual module Homℤ(X, T) of the module X.

Let {an} be a sequence of non-negative real numbers. Suppose that

Then M1,n is the arithmetic mean, MO,n the geometric mean, and Mr,n the generalized mean of order r, of a1, a2, …, an. By a result of Everitt [1] and McLaughlin and Metcalf [5], {n(Mr,n–Ms,n)}, where r ≧ l ≧ s, is a monotonic increasing sequence. It follows that this sequence tends to a finite or an infinite limit as n → ∞. Everitt [2, 3] found a necessary and sufficient condition for the finiteness of this limit in the cases r, s = 1, 0 and r ≧ 1 > s > 0. His results are included in the following theorem.

Let A be a complex Banach algebra with unit e of norm one. We show that A can be represented on a compact Hausdorff space ω which arises entirely out of the algebraic and norm structures of A. This space induces an order structure on A that is preserved by the representation. In the commutative case, ω is the spectrum of A, and we have a generalization of Gelfand's representation theorem for commutative complex Banach algebras with unit. Various aspects of this representation are illustrated by considering algebras of n × n complex matrices.

This paper establishes a relationship (Theorem 4.1) between the approaches of A. C. Thompson [8, 9] and E. G. Effros [2] to the representation of simplex algebras, that is, real unital Banach algebras that are simplex spaces with the unit for order identity. It proves that the (nonempty) interior of the associated cone is contained in the principal component of the set of all regular elements of the algebra. It also conjectures that each maximal ideal (in the order sense—see below) of a simplex algebra contains a maximal left ideal of the algebra. This conjecture and other aspects of the relationship are illustrated by considering algebras of n × n real matrices.

A tight Riesz group is a partially ordered group which satisfies a strengthened form of the Riesz interpolation property. The term “tight” was introduced by Miller in [8], and the tight interpolation property has been considered in papers by Fuchs [3], Miller [8, 9], Loy and Miller [7] and Wirth [12]. If the closure of the cone P, in the interval topology, of such a partially ordered group G contains no pseudozeros, then is itself the cone of a partial order on G. Loy and Miller found of particular interest the case in which this associated partial orderis a lattice order. This situation was then considered in reverse by A. Wirth [12] who investigated under what circumstances a lattice ordered group would permit the existence of a tight Riesz order (called a compatible tight Riesz order) for which the initial lattice order is the order defined by the closure of the cone of the tight Riesz order.Wirth gave two fundamental anduseful characterizations of those subsets of the cone of a lattice ordered group that canbe the strict cone of a compatible tight Riesz order; one is in terms of archimedean classes and the other is an elementwise characterization. Although Loy, Miller and Wirth restricted their attention to abelian groups, much of what they do carries over verbatim to nonabelian groups. In the main result of this paper (Theorem 2.6) a description of the strict cone of a compatible tight Riesz order on a lattice ordered group Gis given in terms of the prime subgroups of G.This is particularly useful when one is attempting to identify the compatible tight Riesz orders on some particular lattice ordered group or class of lattice ordered groups, since it narrows down to a convenient family of subsets the possible candidates for strict cones of compatible tight Riesz orders. These can then be tested under Wirth's criteria. This technique is illustrated in § 5, where the compatible tight Riesz orders are determined o a lattice ordered group of the type V(Γ, Gγ), where Γ is of finite width, and in § 6, where two examples are considered.

Let f(z) be an entire function (of several variables). We define the function

which is increasing. The orderof f(z) is the constant (perhaps infinite)

The problem of Josephus and the forty Jewsis well known [1, 3]. In its most general form, this problem is equivalent to the problem of m-enumeration of a set, as described below.

Define the ordered set

We choose and remove cyclically, from left to right, each with element of Zn until the set is exhausted. The chosen elements are ordered into a new ordered set

which is therefore a permutation of Zn, obtained by what we call m-enumeration of theset Zn.

An exact solution of triple trigonometrical equations is obtained by using the finiteHilbert transform. The solution of these equations is used to solve a two-dimensional electrostatic problem. The problem of determining the electrostatic potential due to two parallel coplanar strips of equal length, charged to equal and opposite potentials, each parallel to and equidistant from an earthed strip, is considered. Both the charged strips lie along the x-axis and they are equally spaced with respect to the y-axis. Finally the expression for the surface charge density (per unit depth) of the strip is derived

The object of this paper is to find the solutions of some dual equations involving the inverses of certain Mellin type transforms that were first introduced by D. Naylor in his paper [1]. Because these transforms are relatively unknown we shall begin by defining them and making a note of some of their properties. The main result is contained in the following theorem.

Let A be a Banach *-algebra. We assume neither that A has an identity element, nor that the involution on A is continuous.

It is implicit in a result of Kapp and Schneider [3] that, if Sisa completely simple semigroup, then the lattice Λ(S) of congruences on S can be embedded in the product of certain sublattices. In this paper we consider the problem of embedding Λ(S) in a product of sublattices, when S is an arbitrary band of groups. The principal tool is the θ-relation of Reilly and Scheiblich [7]. The class of θ-modular bands of groups is definedby means of a type of modularity condition on Λ(S). It is shown that the θ-modular bands of groups are precisely those for which a certain function is an embedding of Λ(S) into a product of sublattices. The problem of embedding the inverse semigroup congruences into a certain product lattice is also considered.

The Jacobi expansion

is well known and easily obtained from the generating function of the Besselcoefficients. The sum of the series on the right of equation (1) when sin (n+½)x is replaced by cos (n+½)x cannot be found in this way but it can be expressed in terms of a definite integral as shown below. The result so obtained is useful in reducing certain triple cosine series to dual series and so simplifying the solution given by one of us for such series in an earlier paper [1].

Let Sbe a compact Riemann surface of genus g ≥ 2 and σ an automorphism (conformal self-homeomorphism) of S of order n. Let S* = S/ « σ« have genus g*. In [5], Schoeneberg gave a sufficient condition that a fixed point P ∈ S of σ should be a Weierstrass point of S, i.e., that Sshould support a function that has a pole of order less than or equal to g at P and is elsewhere regular.

The object of this paper is to derive, using a version of the large sieve for function fields due to J. Johnsen [6], explicit lower boundsfor the average number of distinct values taken by a polynomial over a finite field.