Dans tout ce qui suit, H désigne un espace de Hilbert séparable, A un opérateur fermé de domaine D(A) dans H, on note B(H) l'ensemble des opérateurs bornés de H dans lui-même et N(A), R(A) respectivement le noyau de A, l'image de A.

En 1958, T. Kato a démontré dans [7] le théorème suivant.

Let Gbe a finite group, and P:G → GL(n, ) be such that for all x, y ∈ G

(i) P(x)P(y) = α(x, y)P(xy), and

(ii)P(l) = In,

where α(x, y) ∈ *; then P is a projective representation of G with cocycle α and degree n. For other basic definitions concerning projective representations see [4].

Let R be a ring, see below for other notation. The functor categories (mod-R, Ab) and ((R-mod)op, Ab) have received considerable attention since the 1960s. The first of these has achieved prominence in the model theory of modules and most particularly in the investigation of the representation theory of Artinian algebras. Both [11, Chapter 12] and [8] contain accounts of the use (mod-R, Ab) may be put to in the model theoretic setting, and Auslander's review, [1], details the application of (mod-R, Ab) to the study of Artinian algebras. The category ((R-mod)op, Ab) has been less fully exploited. Much work, however, has been devoted to the study of the transpose functor between R-mod and mod-R. Warfield's paper, [13], describes this for semiperfect rings, and this duality is an essential component in the construction of almost split sequences over Artinian algebras, see [4]. In comparison, the general case has been neglected. This paper seeks to remedy this situation, giving a concrete description of the resulting equivalence between (mod-R, Ab) and ((R-mod)op, Ab) for an arbitrary ring R.

The braid group on n strings Bn has a presentation as a group with generators σ1, …, σn−1 and relations

In this paper we prove an identity between sums of reciprocals of Fibonacci and Lucas numbers. The Fibonacci numbers are defined for all n ≥ 0 by the recurrence relation Fn + 1 = Fn + Fn-1 for n ≥ 1, where F0 = 0 and F1 = 0. The Lucas numbers Ln are defined for all n ≥ 0 by the same recurrence relation, where L0 = 2 and L1 = 1 We prove the following identify.

A semilattice of cancellative semigroups S is a p.o. semigroup with the order relation a ≤ b iff ab = a2. If S is a strong semilattice of cancellative semigroups (i.e., multiplication in S is given by structure maps ϕe,f (f ≤ e in E)), for each supremum-preserving completion Ē of the semilattice E there is a strong semilattice of cancellative semigroups T over Ē which is a supremum-preserving completion of S in ≤. Given Ē, T is constructed directly. In this paper it is shown that multiplication by an element of S distributes over suprema in ≤ if E has this property (called strong distributivity). Next it is shown that the completion construction also applies to a semilattice of cancellative semigroups which is not strong if S is commutative and Ē is strongly distributive. Finally, it is shown that for semilattices of cancellative monoids a completion is completely determined, up to isomorphism over S, by completions of E.

Let

be a positive definite quadratic form with determinant αβ−X2 = 1. A special form of this kind is

We consider the Epstein zeta-function

the series converging for s > 1. For s ≥ 1·035 Rankin [1] proved the following

STatement R.

The sign of equality is needed only when h is equivalent to Q.

The problem of determining, within the limits of the classical theory of elasticity, the displacements and stresses in the interior of a semi-infinite solid (z ≧ 0) when a part of the boundary surface (z = 0) is forced to rotate through a given angle ω about an axis which is normal to the undeformed plane surface of the solid, has been discussed by several authors [7, 8, 9, 1, 11, and others]. All of this work is concerned with rotating a circular area of the boundary surface and the field equation to be solved is, essentially, J. H. Mitchell's equation for the torsion of bars of varying circular cross-sections.

1. Introduction: Suppose the group G is generated by subnormal subgroups H and K, and that A, B are normal subgroups of finite index in H, Krespectively. The following question has been asked by J. C. Lennox: Under what circumstances is the subgroupJ = (A, B) subnormal in G? In particular, it is of interest to know when J has finite index in G, for, if this is the case, we may factor out by the normal core of J in G and apply Wielandt's theorem on joins of subnormal subgroups of finite groups [11] to deduce that J is subnormal in G. Here we prove the following result.

In 1956, Jacobson asked whether the intersection of the powers of the Jacobson radical, J(R), of a right Noetherian ring R, must always be zero [4, p. 200]. His question was answered in the negative by I. N. Herstein [3], who noted that , where Z(2) denotes the ring of rational numbers with denominator prime to 2, affords a counterexample. In contrast, the ring , though similar in appearance to R1, satisfies . (Here, k denotes a field.)

In this paper we presuppose a familiarity with the terminology and the basic results of radical theory, all of which can be found in Divinsky [2].

Let G by any given group. A homomorphic mapping μ of a subgroup A of G onto a second subgroup B of G, where A and B need not be distinct, is called a partial endomorphism of G. When μ is defined on the whole of G, that is when A = G, we call μ a total endomorphism of G; or simply an endomorphism of G.

A partial (or total) endomorphism μ* of a supergroup G* of G is said to extend (or continue) μ if μ* is defined on a supergroup A* of A, that is, μ* is defined for at least the elements for which μ. is defined, and moreover μ* coincides with μ on A.

Remarks 1, 3 and 5 are incorrect as stated. They should be supplemented by the following observations:

(i) In case the perturbing term is linear in y, i.e. f(t, y) = B(t)y, the conclusion of Theorem 1 will follow from Lemma 1 when applied to equation (15) if we assume, instead of (6),

The hypothesis given in Trench's theorem is sufficient to imply (*) but not (6). A similar comment applies to Remark 5.

Let A be a commutative Noetherian ring (with non-zero identity). The Cousin complex C(A) for A is described in [19, Section 2]: it is a complex of A-modules and A-homomorphisms

with the property that, for each n ∈ N0 (we use N0 to denote the set of non-negative integers),

Cohen–Macaulay rings can be characterized in terms of the Cousin complex: A is a Cohen–Macaulay ring if and only if C(A) is exact [19, (4.7)]. Also, the Cousin complex provides a natural minimal injective resolution for a Gorenstein ring (see [19,(5.4)]).

Let H be a Hilbert space and let B denote the Banach algebra of all bounded linear operators on H with K denoting the closed ideal of compact operators in B. If T ∈ B, σ(T) and r(T) will denote the spectrum and spectral radius of T, respectively, and π the canonical mapping of B onto the Calkin algebra B/K.

Let F = F(g, n) be an oriented surface of genus g≥1 with n<2 boundary components and let M(F) be its mapping class group. Then M(F) is generated by Dehn twists about a finite number of non-bounding simple closed curves in F([6, 5]). See [1] for the definition of a Dehn twist. Let e be a non-bounding simple closed curve in F and let E denote the isotopy class of the Dehn twist about e. Let Nbe the normal closure of E2in M(F). In this paper we answer a question of Birman [1, Qu 28 page 219]:

Theorem 1. The subgroup N is of finite index in M(F).

A number of integrals of E-functions with respect to their parameters have been given by Ragab [1, 5]. Two further integrals of this type are given in § 2 and § 3.

In § 4 it is shown that these can be employed to sum series of products of E-functions.