A (group) divisible design is a tactical configuration for which the v points are split into m classes of n each, such that points have joining number λ (resp. λ2) if and only if they are in the same (resp. in different) classes. We are interested in such designs with a nice automorphism group. We first investigate divisible designs with equally many points and blocks admitting an automorphism group acting regularly on all points and on all blocks, i.e., with a Singer group (Singer [50] obtained the first result in this direction for the finite projective spaces).

As in the case of block designs, one may expect a divisible design with a Singer group to be equivalent to some sort of difference set; as it turns out, one here obtains a generalisation of the relative difference sets of Butson and Elliott [11] and [20].

Let X be an arbitrary compact subset of the real line R which has at least two points. For each finite complex valued function f on X we denote by V(f; X) (and call it the weak variation of f on X) the least upper bound of the numbers ∑i|f(bi ) – f(ai )| where {[ai , bi ]} is any sequence of non-overlapping intervals whose end points belong to X. A function f is said to be of bounded variation (BV) on X if V(f; X) < ∞. A function f is said to be absolutely continuous (AC) on X, if given any ∈ > 0 there exists an n > 0 such that for every sequence of non-overlapping intervals {[au bi]} whose end points belong to X, the inequality

implies that

([7], p. 221, 223).

A useful general strategy for the construction of interesting families of vertex-transitive graphs is to begin with some family of transitive permutation groups and to construct for each group Γ in the family all graphs G whose vertex–set is the orbit V of Γ and for which Γ ≦ Aut (G), where Aut (G) denotes the automorphism group of G. For example, if we consider the family of cyclic groups 〈(0 1 … n – 1)〉 generated by cycles (0, 1 … n – 1) of length n, then the corresponding graphs are the n-vertex circulant graphs.

In this paper we consider transitive permutation groups of degree mn generated by a “rotation” ρ which is a product of m disjoint cycles of length n and by a “twisted translation” t; such that τρτ–l = ρα for some α.

Each Hausdorff compactification of a given Tychonoff space is the Smirnov compactification associated with a compatible proximity on the space. Also each realcompactification of a given Tychonoff space is the underlying topological space of the completion of a compatible uniformity on the space. But if T is a realcompactification of a Tychonoff space X which is contained in a particular compactification Z of X, then it is not always possible to find a compatible uniformity on X such that T is the underlying topological space of the completion of (X, ) and induces the proximity on X associated with Z. We shall call a Hausdorff compactification Z of a Tychonoff space X a rich compactification of X (and the associated proximity on X a rich proximity) if every realcompactification of X contained in Z can be obtained as the underlying topological space of the completion of a compatible uniformity on X which induces the proximity on X associated with Z.

The fact that simple links in locally compact connected metric spaces are nondegenerate was probably first established by C. Kuratowski and G. T. Whyburn in [2], where it is proved for Peano continua. J. L. Kelley in [3] established it for arbitrary metric continua, and A. D. Wallace extended the theorem to Hausdorff continua in [4]. In [1], B. Lehman proved this theorem for locally compact, locally connected Hausdorff spaces. We will show that the locally connected property is not necessary.

A continuum is a compact connected Hausdorff space. For any two points a and b of a connected space M, E(a, b) denotes the set of all points of M which separate a from b in M. The interval ab of M is the set E(a, b) ∪ {a, b}.

In a recent note [2] we showed that if R is a prime ring and d ≠ 0 a derivation of R such that d(x)d(y) = d(y)d(x) for all x, y ∈ R then, if R is not a characteristic 2, R must be commutative. (If char R = 2 we showed that R must be an order in a 4-dimensional simple algebra.)

In this paper we shall consider a similar problem, namely, that of a prime ring R with involution * where d(x)d(y) = d(y)d(x) not for all x, y ∈ R but merely for symmetric elements x* = x and y* = y. Although it is clear that some results can be obtained if R is of characteristic 2, we shall only be concerned with the case char R ≠ 2. Even in this case one cannot hope to extend the result cited in the first paragraph, that is, to show that R is commutative.

In the theory of Jordan algebras one encounters several definitions of the trace, and it is sometimes unclear whether the different notions are equivalent or not. If we restrict attention to the so–called JB–algebras studied in [2] and their weakly closed analogues JBW–algebras [8], we shall in the present note show that the different concepts are all equivalent for JBW–algebras, and that the conditions not involving projections are equivalent for JB–algebras. Among the seven equivalent conditions we shall consider, the second (ii) was used by Alfsen and Shultz [1] to show that if the JBW–algebra, is the self–adjoint part of a von Neumann algebra, then the condition characterizes traces on the enveloping von Neumann algebra. Condition (iii) appears in Robertson's paper [7] together with the implication (ii) ⇒ (iii). The inequality (iv) is a Jordan analogue of Gardner's inequality | ϕ(x) ≦ ϕ(|x|)|, [3], characterizing traces on C ast;–algebras.

A symmetric game consists of a set of pure strategies indexed by {0, …, n} and a real payoff matrix (aij ). When two players choose strategies i and j the payoffs are aij and aji to the i-player and j-player respectively. In classical game theory of Von Neumann and Morgenstern [16] the payoffs are measured in units of utility, i.e., desirability, or in units of some desirable good, e.g. money. The problem of game theory is that of a rational player who seeks to choose a strategy or mixture of strategies which will maximize his return. In evolutionary game theory of Maynard Smith and Price [13] we look at large populations of game players. Each player's opponents are selected randomly from the population, and no information about the opponent is available to the player. For each one the choice of strategy is a fixed inherited characteristic.

Introduction. Let f be a modulus, ei = (δij) and E = {ei , i = 1, 2, …}. The L(f) spaces were created (to the best of our knowledge) by W. Ruckle in [2] in order to construct an example to answer a question of A. Wilansky. It turned out that these spaces are interesting spaces. For example lp , 0 < p ≦ 1 is an L(f) space with f(x) = xp , and every FK space contains an L(f) space [2]. A natural question is: For which f is L(f) a locally convex space? It is known that L(f) ⊆ l 1, for all f modulus (see [2]), and l 1 is the smallest locally convex FK space in which E is bounded (see [1]). Thus the question becomes: For which f does L(f) equal l 1? In this paper we characterize such f. (An FK space need not be locally convex here.) We also characterize those f for which L(f) contains a convex ball. The final result of this paper is to show that if f satisfies f(x · y) ≦ f(x) · f(y) and L(f) ≠ l 1 then L(f) contains no infinite dimensional subspace isomorphic to a Banach space.

Dans [11] (page 32 formule (5)), Tricomi donne la formule suivante:

0.1

Dans [8] (page 704, formule 36) Meixner donne une formule analogue.

La formule (0.1) peut donc être considérée comme la formule de Meixner-Tricomi, la formule de Meixner citée étant antérieure, semblet-il, à celle de Tricomi.

L'objet de ce travail est de généraliser le résultat (0.1) qui sera considéré comme la formule de Meixner-Tricomi.

Pour cela, on va considérer l'équation différentielle ordinaire suivante:

0.2

Let L be a semilattice and let a ∊ L. We refer the reader to Definitions 2.2, 2.4, 2.5 and 2.12 below for the terminology. If L is a-implicative, let Ca be the set of a-closed elements of L, and let Da be the filter of a-dense elements of L. Then Ca is a Boolean algebra. If a = 0, then C 0 and D 0 are the usual closed algebra and dense filter of L. If L is a-admissible and f : Ca × Da → Da is the corresponding admissible map, we can form a quotient semilattice Ca × D 0 f. In case a = 0, Murty and Rao [4] have shown that C 0 × D 0/f is isomorphic to L, and hence that C 0 × D 0 is 0-admissible. In case L is in fact implicative, Nemitz [5] has shown that C 0 × D 0/f is isomorphic to L, and that C 0 × D 0/f is also implicative.

We wish to consider the class of (0,1)-matrices with prescribed row and column sums. Let R = (r 1, r 2, …, rm ) and S= (s 1, s 2, …, sn ) be vectors with nonnegative integral entries and r 1 + r 2 + … + rm = s 1 + s 2 + … + sn . We define the class to be the set of m × n (0, 1)-matrices with i th row sum ri and j th column sum sj for 1 ≦ i ≦ m and 1 ≦ j ≦ n.

Gale and Ryser independently found simple necessary and sufficient conditions for to be nonempty [9 , 14]. From R, we form an m × n (0, 1)-matrix Ā as follows. The ith row sum of Ā is ri and the 1‘s are as far to the left as possible. Let be the j th column sum of A. We define the sequence to be the conjugate of the sequence (ri).

Let H be a graded commutative algebra with a nice set of algebra generators. Let H also be a comodule over a Hopf algebra A. In Section 2 we give conditions under which certain of these generators of H can be rechosen to be primitive. In addition we give explicit formulas expressing these primitive generators in terms of the original set of generators.

In Section 3 we apply the theory of Section 2 to the mod p homology of the Thorn spectra MO, MU and MSp. In particular we give two explicit descriptions of the image of the Hurewicz homomorphism for MO. One of these makes explicit the recursive computation of E. Brown and F. Peterson [1].

In Section 4 we give a variation of the theory of Section 2 which computes primitive generators of certain Hopf algebras. This theory is applied to study the primitive elements of H*(BU) and H*(SO; Z2).

Let G be a bounded, open, connected, non-empty subset of the complex plane C. We put the usual two dimensional (Lebesgue) area measure on G and consider the Hilbert space L 2(G) that consists of the complex-valued, measurable functions defined on G that are square integrable. The inner product on L 2(G) is given by the norm ‖h‖2 of a function h in L 2(G) is given by ‖h‖2 = (∫G |h|2)1/2.

The Bergman space of G, denoted La 2(G), is the set of functions in L 2(G) that are analytic on G. The Bergman space La 2(G) is actually a closed subspace of L 2(G) (see [12 , Section 1.4]) and thus it is a Hilbert space.

Let G denote the closure of G and let C(G) denote the set of continuous, complex-valued functions defined on G.

For G a group, S a subset of G which generates G, the length problem in G with respect to S is to find, for g ∈ G, the least integer r such that g can be written as the product of r elements of S. For G an orthogonal group Of(F) (here F is a field, and the elements of Of(F) preserve the quadratic form f) and S the set of reflections in Of(F) the length problem has been studied by E. Cartan [2], J. Dieudonné [4, 5], E. Ellers [7], P. Scherk [8], and others. In all of these investigations, however, the problem posed by requiring that S be a single conjugacy class of reflections in Of(F) has been ignored. And it is generally the case that the reflections in Of(F) fall into several conjugacy classes.

In this paper we will give brief proofs of two results on the undecidability of a first-order theory using a construction which we call a modified Boolean power. Modified Boolean powers were introduced by Burris in late 1978, and the first results were announced in [2]. Subsequently we succeeded in using this construction to prove the results in this paper, namely Ershov's theorem that every variety of groups containing a finite non-abelian group has an undecidable theory, and Zamjatin's theorem that a variety of rings with unity which is not generated by finitely many finite fields has an undecidable theory. Later McKenzie further modified the construction mentioned above, and combined it with a variant of one of Zamjatin's constructions to prove the sweeping main result of [3]. The proofs given here have the advantage (over the original proofs) that they use a single construction.

The problem of additivity of the Pn -integral on abutting intervals was considered in [2] and in [5]. It was noted in [2] that the necessary and sufficient conditions for additivity for the P 2-integral obtained by Skvorcov in [5] could be completely generalized to the Pn -integral, n> 2, if a key lemma (corresponding to Skvorcov's Lemma 3 [6]) could be proved. We provide a proof of that lemma in this paper and hence obtain the general additivity result.

The definitions and notation of [2] are used in the following, except that we shall take the following as the definition of Pn -major and minor functions:

Definition 1.1. Let f(x) be a function defined in [a, b] and let a 1, i = 1, 2, …, n, be fixed points such that a = a 1 < a 2 < … < an = b.