A combinatorial interpretation of Schur functions in terms of Young tableaux is well-known. (For example, see Littlewood [1] or Thomas [4]). The purpose of this paper is to present a combinatorial interpretation of the wreath product (or plethysm) of two Schur functions.

The central theorem of this note was proposed to me by I. Juhâsz on a napkin in Canter's restaurant in Pittsburgh, Pennsylvania. We agreed that if the proposition could be proven it would have some elegant applications and although it required a slight change before it could be proven, hopefully those expectations will be realized in this note.

If A is a (linear) transformation acting on a (finitedimensional, non-zero, complex) Hilbert space H the family of (linear) subspaces of H which are invariant under A is denoted by Lat A. The family of subspaces of H which are invariant under every transformation commuting with A is denoted by Hyperlat A. Since A commutes with itself we have Hyperlat A ⊆ Lat A. Set-theoretic inclusion is an obvious partial order on both these families of subspaces. With this partial order each is a complete lattice; joins being (linear) spans and meets being set-theoretic intersections. Also, each has H as greatest element and the zero subspace (0) as least element. With this lattice structure being understood, Lat A (respectively Hyperlat A) is called the lattice of invariant (respectively, hyper invariant) subspaces of A.

In this paper we consider representations of weakly regular algebras with permutable congruences and a Boolean orthogonality as direct products of orthoprime algebras. Our theorems generalize results of Abian [2; 3] and Speed [24] which characterize direct products of integral domains, and results of Abian [1] and Chacron [7] which characterize direct products of division rings.

Let A = ﹛a0, a1, … , a3﹜ be a set of relatively prime integers such that 0 < a0 < a1 < … < as = n. Let ϕ (A) denote the smallest integer such that, for N ≧ ϕ (A), the equation

should always have a solution in nonnegative integers.

Let denote the space of functions holomorphic on an open Riemann surface R, where has the topology of uniform convergence on compact sets. In this note, we characterize the dual space *. The result is not new, for it is implicitly contained in the more general results of Tillmann [5] and, in case R is planar, in those of Köthe [4]. However, the paper of Gunning and Narasimhan [3], which appeared subsequently, allows us to give a short proof of this important result. Actually, our characterization is a natural one in terms of differentials, while the Köthe-Tillmann characterization is in terms of functions, but we show that these two characterizations are isomorphic. We end our paper by using our characterization to prove an interpolation result. The second author gratefully acknowledges a helpful discussion with Professor George Szekeres.

Nous allons montrer que tout système faible de Tchebycheff est voisin d'un système de Tchebycheff. Ce fait permet de prolonger aux systèmes faibles de Tchebycheff plusieurs résultats valides pour les systèmes de Tchebycheff, tels le théorème d'alternance et le théorème de de la Vallée- Poussin qui sont reliés au problème de la meilleure approximation uniforme par une fonction d'un système de Tchebycheff. Il arrive que certaines classes de fonctions splines forment un système faible de Tchebycheff. Pour ces classes, on pourra donc caractériser les meilleures approximations par des propriétés d'alternance de l'écart à la fonction à approcher.

Consider a system of N linear transformations A1, … , AN: V → W, where F and IF are complex vector spaces. Denote it for short by (F, W). A pair of subspaces X ⊂ V, Y ⊂ W such that determines a subsystem (X, Y) and a quotient system (V/X, W/Y) (with the induced transformations). The subsystem (X, Y) is of finite codimension in (V, W) if and only if V/X and W / Y are finite-dimensional. It is a direct summand of (V, W) in case there exist supplementary subspaces P of X in F and Q of F in IF such that (P, Q) is a subsystem.

Let C denote an irreducible, algebraic curve defined over an algebraically closed field k. Let ? be a singular point of C. We shall employ the following notation throughout the rest of this paper: R will denote the local ring at P, K the quotient field of the integral closure of R in K, A the completion of R with respect to its radical topology, and Ā the integral closure of A in its total quotient ring.

Let G be a n affine algebraic group over the algebraically closed field k, and let V be an affine, normal algebraic variety over k on which G acts. Suppose that the ring of invariants k [F]G is finitely generated over k, and let W be the affine variety with k[W] = k[V]G. The purpose of this paper is to show that the induced homomorphism from the étale fundamental group of V to that of W is surjective, and to examine the consequences of this observation in terms of the relations between the Picard and divisor class groups of k[V] and k[W],

A remarkable theorem of Herstein [1, Theorem 2] of which we have made several uses states: If R is a semiprime ring of characteristic different from 2 and if U is both a Lie ideal and a subring of R then either U ⊂ Z (the centre of R) or U contains a nonzero ideal of R. In a recent paper [3] Herstein extends the above mentioned result significantly and has proved that if R is a semiprime ring of characteristic different from 2 and V is an additive subgroup of R such that [V, U] ⊂ V, where U is a Lie ideal of R, then either [V, U] = 0 or V ⊃ [M, R] ≠ 0 where M is an ideal of R. In this paper our object is to prove the following.

In this paper, an induction theorem for rearrangements involving w-tuples in Rn is proved, showing that a certain proposition regarding a pair of w-tuples related by the weak spectral order ≪ is true for any integer n ≧ 2 if and only if it is true for n = 2. This theorem contains as particular cases a well-known theorem of Hardy-Littlewood-Pólya [4, Lemma 2, p. 47], a theorem of Pólya [8], a theorem of Rado [9, pp. 1-2], two theorems of Mirsky [6, Theorem 2, p. 232; 7, Theorem 4, p. 90], a result given in [1, Corollary 2.4, p. 1333] and also [2, Proposition 2.1, p. 439].

In [1], Bachmuth and Mochizuki conjecture, by analogy with a celebrated result of Tits on linear groups [8], that a finitely generated group of automorphisms of a finitely generated soluble group either contains a soluble subgroup of finite index (which may of course be taken to be normal) or contains a non-abelian free subgroup. They point out that their conjecture holds for nilpotent-by-abelian groups and in some other cases.

Let denote the 2-fold cyclic covering space branched over a link L in S3. We wish to describe an infinite family of prime knots and links in which each member L exhibits two minimal 6-plat representations, where the associated Heegaard splittings of are minimal and inequivalent. Thus each knot or link of that family admits at least two equivalence classes of 6-plat representations which are minimal.

Let X be a σ-finite measure space and let Tk, k any integer, be a group of positive linear transformations in Lp(X) such that

with C independent of / and k. From now on / will be a positive function in Lp(X) and we will use the following notation:

A method for constructing the product of two Schur functions was stated, but not proved in the most general case, by Littlewood and Richardson [1] in 1934. This method, which came to be known as the Littlewood-Richardson rule, was later proved completely by Robinson [2] in 1938. In this proof, Robinson describes an operation on a finite sequence of positive integers. It is this operation, set in a more general context, that is the subject of this paper.

Davis and Bolz (1974) considered, and to some extent classified, compatible tight Riesz order on the group of all order-preserving permutations of a totally ordered field. Glass (1976) carried out a more general study of compatible tight Riesz orders on ordered permutation groups and, in particular, showed the importance of determining compatible tight Riesz orders on O-primitive ordered permutation groups. However, the general problems of existence and classification of compatible tight Riesz orders on O-primitive ordered permutation groups remained open.

A formula is derived for the dimension of a symmetry class of tensors (over a finite dimensional complex vector space) associated with an arbitrary finite permutation group G and a linear character of x of G. This generalizes a result of the first author [3] which solved the problem in case G is a cyclic group.

Let Γ be a Jordan curve in the extended complex plane C. Γ is called a quasiconformal circle if it is the image of a circle by a homeomorphism ƒ which is quasiconformal in a neighborhood of that circle. If q(zi, z2) is the chordal distance from z1 to z2, the chordal cross ratio of a quadruple z1, z2, z3, z4 in C is

Heilbronn [6] proved that for any ϵ > 0 there exists C(ϵ) such that for any real θ and N ≧ 1 there is an integer x satisfying

where ||α|| denotes the difference between α and the nearest integer, taken positively. Danicic [2] obtained an analogous result for the fractional parts of θxk and in 1967 Davenport [4] generalized Heilbronn's result to polynomials of degree with no constant term. The last condition is essential, for if there is a constant term then no analogous result can hold (see Koksma [7, Kap. 6 SatzlO]).