In 1851 Prouhet (2) stated that any bk+1 consecutive positive integers (b a positive integer ⩾ 2) can be separated into b sets o, …, b-1 each with bk members in such a way that where designates the sum of the tth powers of the numbers in j. In 1947 Lehmer (1) generalized this result.

A few years ago Bailey (1) gave certain transformations of both terminating and non-terminating nearly-poised hypergeometric series of the ordinary type and later on he also deduced basic analogues of some of his transformations. Recently, (3) I gave certain transformations of both ordinary and basic terminating nearly-poised bilateral hypergeometric series which generalized Bailey's results.

Let n be a fixed positive integer, and consider an alphabet consisting of the numbers 1, 2, … , n. With this alphabet form all possible k-letter words (a1a2 … ak), where k is also fixed. There are evidently nk such words in all.

More than sixty years ago, Sylvester (13) proposed the following problem: Let n given points have the property that the straight line joining any two of them passes through a third point of the set. Must the n points all lie on one line?

An alleged solution (not by Sylvester) advanced at the time proved to be fallacious and the problem remained unsolved until about 1933 when it was revived by Erdös (7) and others.

Let {p, q, r} be the regular 4-dimensional poly tope for which each face is a {p, q} and each vertex figure is a {q, r}, where {p, q}, for example, is the regular polyhedron with p-gonal faces, q at each vertex. A Petrie polygon of {p, q} is a skew polygon made up of edges of {p, q} such that every two consecutive sides belong to the same face, but no three consecutive sides do. Then a Petrie polygon of {p, g, r} is defined by the property that every three consecutive sides belong to a Petrie polygon of a bounding {p, q}, but no four do. Let hPqr be the number of sides of such a polygon, and gp,q,r the order of the group of symmetries of {p, g, r}.

Let {Xn} (n = 1, 2 , …) be a stochastic process. The random variables comprising it or the process itself will be said to be interchangeable if, for any choice of distinct positive integers i1, i2, H3 … , ik, the joint distribution of

depends merely on k and is independent of the integers i1, i2, … , ik. It was shown by De Finetti (3) that the probability measure for any interchangeable process is a mixture of probability measures of processes each consisting of independent and identically distributed random variables.

If S1, S2, S3, … , Sn are subsets of a set M then it is known that a necessary and sufficient condition that it is possible to choose representatives at such that ai is in Si for (i = 1, 2, 3, … , n) and such that ai ≠ aj for i ≠ j , is that for k = 1, 2, 3, … , n, the union of any k of the sets S1, S2, … , Sn, contains at least k elements. The theorem has a number of consequences amongst which we list the following.

In his paper on the algebraic approximation of differentiable manifolds Nash (1) introduced the concept of a sheet of a real algebraic variety (see the definition in §16 below) and raised certain questions of a general nature. In attempting to answer these questions it has been necessary to evolve some sort of technique for manipulating curves on algebraic varieties, and, in particular, to set up a criterion for the possibility of approximating a sequence of analytic arcs (definition in §1) joined end to end by a single analytic arc.

Denote by Fα,β the multiplicative group generated by the two matrices where α and β are complex numbers. Sanov (3) proved that Fα,β is free, and Brenner (1) showed that Fm,m is free if m ⩾ 2.

An integral domain E is said to be Euclidean if there exists a non-negative, integer-valued function g denned on the non-zero elements of E such that for every non-zero x and y in E,

(1) g(xy) ⩾ g(x);

(2) (division algorithm) if x does not divide y then there exists an element q in E, depending on x and y, with .

A well-known theorem asserts that if K is a field, a prime ideal in the polynomial ring S = K[X1, … Xn] and d the transcendence degree of S / over K

n = rank + d.

In the first half of this paper we extend this result to the case of arbitrary commutative noetherian K, as well as giving a purely homological proof of the classical theorem. In the second half we use our first result to compute the analogue of the dimension of the product and intersection of two affine varieties when K is a Dedekind ring.

L'ensemble des nombres ∈ Z/n, premiers avec l'entier n, forme un groupe (le groupe géométrique) G, par rapport à la multiplication. Etant donné un ensemble de nombres réels, M, un groupoïde Q, formé d'éléments quelconques, x, est automorphe par le groupe géométrique si (i) pour tout élément x ∈ Q et tout nombre m ∈ M, la multiplication xm est définie; (ii) l'application (x→xm) est un automorphisme de Q.

Dans un article antérieur (3), j'ai donné quelques théorèmes généraux sur les représentations unitaires des groupes de Lie nilpotents simplement connexes. On va, dans le présent article, construire explicitement les représentations unitaires irréductibles et la formule de Plancherel des groupes de Lie nilpotents simplement connexes de dimension ⩽ 5.

The purpose of the present paper1 is to simplify the calculation of the Betti numbers of the simple compact Lie groups.

For the unimodular group and the orthogonal group on a space of odd dimension the form of the Poincaré polynomial was correctly guessed by E. Cartan in 1929 (5, p. 183). The proof of his conjecture and its extension to the four classes of classical groups was given by L. Pontrjagin (13) using topological arguments and then by R. Brauer (2) using algebraic methods.

Let us compare two properties of sets of non-negative integers: (1) the set a has property Γ, if there exists an effective procedure which when applied to any element of a different from its maximum (which α does not necessarily possess) yields the next larger element of α; (2) the set α has property Δ, if there exists an effective procedure which when applied to any element of α different from its minimum yields the next smaller element of α. It is readily seen that every recursive set has both properties.

Let D be a non-commutative division ring with centre C, and let Δ be a proper division subring not contained in C. In (4) Cartan raised the question: is it ever possible for each inner automorphism of D to induce an automorphism of Δ? As is well-known, Cartan (4, Théorème 4) with the aid of his Galois Theory answered this negatively in case D is a finite dimensional division algebra. Later Brauer (3), and Hua (8), using elegant, elementary methods, extended Cartan's theorem to arbitrary division rings.

Marston Morse and William Transue (9, 10), motivated by their theory of bilinear functions, introduced and studied vector function spaces called MT-spaces for which each element of the dual is represented by an integral with respect to a suitable (C) measure. In this paper the definition of real MT-spaces is generalized to give spaces, called MT*-spaces, for which part but not all of the dual is of integral type and this part is called the MT*-conjugate of the space.

Our aim in this paper is to extend a known theorem about the convergence of subsequences of the partial sums of the Fourier series in one variable of class L2 to Fourier series in two variables of the same class, (1, p. 396). The theorem asserts that for each function ƒ in L2, there is a sequence {mV} of positive integers of upper density one such that

Smv(X;ƒ)

converges to ƒ almost everywhere where sm(x;f) denotes the mth partial sum of the Fourier series of ƒ.

Among the variety of integrals which have been devised for integrating vector-valued functions the most widely used is that of Bochner (2), perhaps because of the simplicity of its formulation. Other approaches, including one by Birkhoff (1), have yielded more general integrals yet none of them seems to have supplanted the Bochner integral to a significant extent.

Letƒ(x) be a measurable function denned in the interval (a, b), and let If the limit of exists and is finite at the point x, as h → 0, it is called the wth generalized Riemann derivative of ƒ(x) at the point x, Dnƒ(x).