In a recent paper Robinson has obtained an explicit formula for the expression of an invariant matrix of an invariant matrix as a direct sum of invariant matrices. The object of the present note is to show that this formula may be deduced from known properties of Schur functions, with the aid of a result which the author has proved elsewhere.

Let k and A denote commutative rings with identity and assume that A is a k-algebra. A qth order k-derivation δ of A into an A -module V is an element of Homk(A, V) such that for any q + 1 elements a0, … , aq of A, the following identity holds:

Thus, a 1st-order derivation is just an ordinary derivation of A into V.

From the time that the basic existence and regularity problems for partial differential equations have been solved many interesting new variational and control problems could be studied. In general a differential equation or boundary value problem is used to define a class of admissible functions, and then the problem is that of finding the extrema of a given functional defined on that class of functions.

A triangulation of a hyperbolic 3-manifold is $L$-thick if each tetrahedron having all vertices in the thick part of $M$ is $L$-bilipschitz diffeomorphic to the standard Euclidean tetrahedron. We show that there exists a fixed constant $L$ such that every complete hyperbolic 3-manifold has an $L$-thick geodesic triangulation. We use this to prove the existence of universal bounds on the principal curvatures of ${{\pi }_{1}}$-injective surfaces and strongly irreducible Heegaard surfaces in hyperbolic 3-manifolds.

Soient K un corps local, sa clôture algébrique, R, L, M trois corps tels K ⊂ R ⊂ L ⊂ et K ⊂ R ⊂ M ⊂ . Supposons que L et M soient linéairement disjoints sur R. Quel est le complété du corps LM (compositum de L et M) pour une norme de R-algèbre qui coïncide avec la valeur absolue sur L ∪ M? On sait que LM est isomorphe en tant que R-algèbre au produit tensoriel L ⨂RM et il est facile de montrer que la valeur absolue est la plus petite norme du type considéré ci-dessus et que la norme tensorielle en est la plus grande. C'est pourquoi nous étudions d'abord le complété de L ⊗RM pour la norme tensorielle. Citons dans ce cas les résultats essentiels lorsque la caractéristique de K est nulle.

Let U be a k-dimensional vector space over the complex numbers. Let ⊗m U denote the mth tensor power of U where m ≧ 2. For each permutation σ in the symmetric group Sm, there exists a linear mapping P(σ) on ⊗mU such that

for all x1, …, xm in U.

Let G be a subgroup of Sm and λ an irreducible (complex) character on G. The symmetrizer

is a projection of ⊗ mU. Its range is denoted by Uλm(G) or simply Uλ(G) and is called the symmetry class of tensors corresponding to G and λ.

Given a suitable regular ring R, we construct a sheaf-like representation for R as a ring of continuous sections from a completely regular space to an appropriately toplogized disjoint union of factor rings corresponding to ‘'extremal“ pseudo-rank functions. Applied to rings which are complete with respect to a rank function this representation is an isomorphism, the completely regular space is extremally disconnected and compact, and the * ‘stalks” are the simple factor rings.

Let T be a Grothendieck topology, Ab the category of abelian groups, and . the category of Ab-valued sheaves on T. It is known that is an abelian A B5 category with a set of generators [2, Theorem 1.6(i), p. 30] and, hence, has injective envelopes [10, Theorem 3.2, p. 89]. Consider an object F of .

In this paper we study the q-Carleson measures for a space of M-harmonic potentials in the unit ball of Cn, when q < p. We obtain some computable sufficient conditions, and study the relations among them.

In this work we deal with algebraic properties of some fields of functions meromorphic in the complex plane with a view to determining the possible isomorphisms between two such fields. Interest in problems of this type began with a paper by Bers (2), in which it was shown that the algebraic structure of the ring of functions analytic on a plane region determines the conformai structure of the region to within conformai or anti-conformal equivalence, and this result was later extended to arbitrary non-compact Riemann surfaces by Nakai (7).

Definition 1.1. Let be analytic for |z| < 1. If ƒ is univalent, we say that ƒ belongs to the class S.

Definition 1.2. Let ƒ ∈ S, 0 ≦ α < 1. Then ƒ belongs to the class of convex functions of order α, denoted by Kα, provided

(1)

and if > 0 is given, there exists Z0, |Z0| < 1, such that

Let ƒ ∈ Kα and consider the Jordan curve ϒτ = ƒ(|z| = r), 0 < r < 1. Let s(r, θ) measure the arc length along ϒτ; and let ϕ(r, θ) measure the angle (in the anti-clockwise sense) that the tangent line to ϒτ at ƒ(reiθ) makes with the positive real axis.

In this paper we prove a complete convergence theorem for attractive, reversible, super-critical nearest particle systems satisfying a natural regularity condition. In particular this implies that under these conditions there exist precisely two extremal invariant measures. The result we prove is relevant to question seven of Liggett (1985), Chapter VII.

Throughout this article c0, c1, c2, … will denote effectively computable positive absolute constants. Denote the cardinality of a set X by |X|. Let N be a positive integer and let A and B be non-empty subsets of {1, …,N}. Put

In [3], Balog and Sá;rközy proved that if N > c0 and

(1)

then there exist a0 and b0 with a0 ∊ A0 and b0 ∊ B0 and a prime number p such that

and

(2)

If follows from this result that if |A| ≫ N and |B| ≫ N then there exist a in A and b in B and a prime p such that p2|(a + b) with

In an earlier paper (1), published in this journal, a necessary condition was given which the reciprocal of a polynomial without multiple roots must satisfy in order to be a characteristic function. This condition is, however, valid for a wider class of functions since it can be shown (2, theorem 2 and corollary to theorem 3) that it holds for all analytic characteristic functions. The proof given in (1) is elementary and has some methodological interest since it avoids the use of theorems on singularities of Laplace transforms. Moreover the method used in (1) yields some additional necessary conditions which were not given in (1) and which do not seem to follow easily from the properties of analytic characteristic functions.

In [11] it is shown that the deficiency of a translation experiment with respect to another on a σ-finite, amenable, locally compact group can be calculated in terms of probability measures on the group. This interesting result, brought to the writer's notice by [1], does not seem to be as wellknown in the theory of amenable groups as it should be. The present note presents a simple proof of the result, removing the σ-finiteness condition and repairing a gap in Torgersen's argument. The main novelty is the use of Wendel's multiplier theorem to replace Torgersen's approach which is based on disintegration of a bounded linear operator from L1(G) into C(G)* for G σ-finite (cf. [5], VI.8.6). The writer claims no particular competence in mathematical statistics, but hopes that the discussion of the above result from the “harmonic analysis” perspective may prove illuminating.

We then investigate a similar issue for discrete semigroups. A set of transition operators, which reduce to multipliers in the group case, is introduced, and a semigroup version of Torgersen's theorem is established.

Several authors (1; 2; 3; 4; 5; 6) have recently studied the existence and non-existence of certain types of extensions of a given field. In this note we prove a theorem closely related to these results which, in a sense, contains essential portions of each of these.

We prove that the maximum number of rational points on a smooth, geometrically irreducible genus 4 curve over the field of 8 elements is 25. The body of the paper shows that 27 points is not possible by combining techniques from algebraic geometry with a computer verification. The appendix shows that 26 points is not possible by examining the zeta functions.

Commutator theory has its origins in constructive quantum field theory. It was initially developed by Glirnm and Jaffe [7] as a method to establish self-adjointness of quantum fields and model Hamiltonians. But it has subsequently proved useful for a variety of other problems in field theory [17] [15] [8] [3], quantum mechanics [5], and Lie group theory [6]. Despite all these detailed applications no attempt appears to have been made to systematically develop the theory although reviews have been given in [22] and [9]. The primary aim of the present paper is to partially correct this situation. The secondary aim is to apply the theory to the analysis of first and second order partial differential operators associated with a Lie group.

Except for using “algebra” rather than “ring” and “compact” rather than “bicompact”, we adopt the terminology used in (6). Every symmetric algebra, Ai, will have an identity, ei All representations will be cyclic and symmetric. Sets of functionals will carry the relative weak* topology.

A subset ${\mathcal{X}}$ of a C*-algebra ${\mathcal{A}}$ is called irredundant if no $A\in {\mathcal{X}}$ belongs to the C*-subalgebra of ${\mathcal{A}}$ generated by ${\mathcal{X}}\setminus \{A\}$. Separable C*-algebras cannot have uncountable irredundant sets and all members of many classes of nonseparable C*-algebras, e.g., infinite dimensional von Neumann algebras have irredundant sets of cardinality continuum.

There exists a considerable literature showing that the question whether every AF commutative nonseparable C*-algebra has an uncountable irredundant set is sensitive to additional set-theoretic axioms, and we investigate here the noncommutative case.

Assuming $\diamondsuit$ (an additional axiom stronger than the continuum hypothesis), we prove that there is an AF C*-subalgebra of ${\mathcal{B}}(\ell _{2})$ of density $2^{\unicode[STIX]{x1D714}}=\unicode[STIX]{x1D714}_{1}$ with no nonseparable commutative C*-subalgebra and with no uncountable irredundant set. On the other hand we also prove that it is consistent that every discrete collection of operators in ${\mathcal{B}}(\ell _{2})$ of cardinality continuum contains an irredundant subcollection of cardinality continuum.

Other partial results and more open problems are presented.