Let SO*(2n) be the group of quaternionic n × n matrices A satisfying A*JA = J, where J is a fixed skew-hermitian invertible matrix. An element R ∊ SO*(2n) is called a reflection if R2 = In and R — In has rank one. We assume that n ≧ 2, in which case S*(2n) is generated by reflections. The length of A ∊ SO*(2n) is the smallest integer k(≧0) such that A can be written as A = R1R2 … Rk where R1, …, Rk are reflections. In this paper, for each A ∊ SO*(2n), we compute its length l(A). Set r(A) = rank (A — In). Already in Section 3 we are able to show that the difference δ = l(A) – r(A) can take only three values 0, 1, or 2. The remainder of the paper deals with the problem of separating these three possibilities. The main results are stated in Section 4 and proved in Section 6. The intermediate Section 5 consists of a sequence of lemmas which are needed for the proof. Clearly l(A) depends only on the conjugacy class of A and the main results in Section 4 are stated in terms of conjugacy classes. For the description of conjugacy classes in SO*(2n) we refer the reader to [1]. The present paper relies heavily on our previous paper [5] where the analogous problem was solved for the groups U(p, q). It is worth remarking that only the various lemmas from that paper were used but not the main theorem.

Nous commençons par caractériser les algèbres infraséquentielles (cf [11]) à l'aide du rayon de régularité. Nous montrons qu'une algèbre topologique (E, τ) est infraséquentielle si, et seulement si, le rayon de régularité β est borné (i.e., l'image, par β, de tout borné de (E, τ) est un borné de R+). Nous dirons qu'une telle algèbre est β-régulière. Nous montrons qu'une algèbre β-régulière pseudo-complète (cf [3]) a tous ses caractères bornés. En fait nous obtenons que l'ensemble des caractères (algébriques) M*, de E, est équiborné. Ce résultat recouvre et généralise ceux de T. Husaïn et S.B. Ng ([12]), de G.A. Joseph ([13]) et de T. Husaïn. ([11]). Dans le cas des algèbres séquentielles pseudo-complètes, nous montrons que tout caractère est séquentiellement continu. Nous obtenons des résultats analogues sur les formes faiblement positives.

Studies of optimization problems have led in recent years to definitions of several types of generalized directional derivatives. Those derivatives of primary interest in this paper were introduced and investigated by F. M. Clarke ([5], [6], [7], [8]), J. B. Hiriart-Urruty ([12]), Lebourg ([16], [17]), R. T. Rockafellar ([23], [24], [26], [27]), Penot ([21], [22]) among others.

In an attempt to explore in more detail relationships between various types of generalized directional derivatives we discovered some unexpected results which were not observed by the above mentioned authors. We are able to give simple conditions which characterize directionally Lipschitzian functions defined on a Baire metrizable locally convex topological vector space.

Dans cet article, tous les semigroupes sont supposés finis, sauf dans le cas d'un semigroupe ou d'un monoïde libre. Les références de base sont [1, 2].

L'objet de cet article est de poursuivre l'étude du semigroupe des parties d'un semigroupe, étude qui a connu récemment des développements inattendus [3, 4, 5, 6, 8, 10, 11, 12]. Putcha [10] avait caractérisé les semigroupes S tels que le semigroupe des parties soit apériodique, autrement dit -trivial. Je donne ici une nouvelle démonstration du théorème de Putcha basée sur le théorème de Ramsey et je caractérise les semigroupes S tels que soit -trivial (resp. ℒ-trivial, ℐ-trivial): ce sont les semigroupes apériodiques tels que es = ese (se = ese, es = se) pour tout s ∊ S et pour tout idempotent e ∊ S. Il en résulte que la variété des semigroupes V2 (resp V3, V1) définie par ces conditions est la variété maximale V telle que la variété PV engendrée par les , S ∊ V, soit contenue dans la variété des monoïdes -triviaux (ℒ-trivial, ℐ-trivial).

Let R denote the set of real numbers. For f:R → R and h ∈ R, the difference function Δhf is defined by

The function H:R → R is called additive if it satisfies Cauchy's equation

Let ℱ be a class of real valued functions defined on R. ℱ is said to have the difference property if, for every function f:R → R satisfying Δhf ∈ ℱ for every h ∈ R, there exists an additive function H such that f — H ∈ ℱ

It was conjectured by P. Erdos that the class of continuous functions has the difference property. This conjecture was proved by N. G. de Bruijn in [1], where the difference property of several other classes was verified as well. (For other references, see [6].)

Let X be a compactum in En of dimension at most n – 2. In [9, Theorem 4.1] it was shown that there is an arbitrarily small homeomorphism h of En fixed outside any given neighborhood of X, so that h(X) has vertical order n – 1 provided n ≠ 3. If X is a 0-dimensional set or a tame 1-dimensional set in E3 then the result is still true. However, the examples of tangled continua of Bothe [2] and McMillan and Row [7] are not amenable to the techniques used in dimensions other than three. This prompted Wright [9] to make the following conjecture.

Let k be a commutative field, and A a finite-dimensional k-algebra. By a module will always be meant a finitely generated right module. Following [8], we shall call a module TA a tilting module if (1) pdTA ≦ 1, (2) Ext1A(T, T) = 0 and (3) there is a short exact sequence

with T’ and T” direct sums of direct summands of T. Given a tilting module TA, the full subcategories

and

of the category modA of A -modules are respectively the torsion-free class and the torsion class of a torsion theory on modA[8]. The aim of the present paper is to find conditions on a torsion theory in order that it be induced by a tilting module.

This paper deals with two conditions which, when stated, appear similar, but when applied to finitely generated solvable groups have very different effect. We first establish the notation before stating these conditions and their implications. If H is a subgroup of a group G, let denote the set

We say G has the isolator property if is a subgroup for all H ≦ G. Groups possessing the isolator property were discussed in [2]. If we define the relation ∼ on the set of subgroups of a given group G by the rule H ∼ K if and only if , then ∼ is an equivalence relation and every equivalence class has a maximal element which may not be unique. If , we call H an isolated subgroup of G.

The “sets” in this paper are actually multi-sets. That is, we allow an element to occur several times in a set and distinguish between the number of elements in a set and the number of distinct elements in the set. On the few occasions when we need to avoid repetition we will use the term “ordinary set.“

Definition. Let G be a group and let S a set of elements of G. An r-sum in S is an ordered subset of S of cardinality r; the result of that r-sum is the product of its elements in the designated order.

Definition. If S is a set, r(x, S) denotes the number of times x appears in S and [x, S] is a set consisting of r(x, S) copies of x. An n-set or n-subset is a set consisting of n elements. Hence [x, S] is an r(x, S)-subset of S.

Similarity invariants of certain families of Volterra operators acting in Lp[0, 1] were first determined by Kalisch in [4] and [5]. Subsequently, Kantorovitz made an extensive study of perturbations of the form Tα = M + αN, where M is the operation of multiplication by x, and N is the Volterra operator

both acting in Lp[0, 1], and determined that Re α is a similarity invariant for the class of Tα's (cf. [7], and earlier work cited there). This result was eventually generalized to unbounded M (cf. [6]), and one direction was proved in [2] for both M and N unbounded: if Re α = Re β, then Tα is similar to Tβ. Properties of the Riemann-Liouville semigroup of fractional integration play a key role, for in all cases the ensuing similarity turns out to be implemented by N(i(lm(α — β))), where {N(iγ)} is the boundary group of the semigroup of fractional integration.

In 1930, M. H. A. Newman proved a rather remarkable theorem which has become one of the classical theorems in topology. It has many important applications. A special case of Newman's Theorem is that a periodic homeomorphism of period n > 1 of a sphere S onto itself must have some orbit which is not contained in a “cap” smaller than a hemisphere. The general theorem is as follows:

THEOREM ([15]). Suppose that Mn is a connected (metric) n-manifold, U is a domain in Mn, andp is an integer greater than 1. Then there is a positive number d such that no uniformly continuous homeomorphism h of Mn onto itself of period p moves every point of U a distance < d. That is, there is x ∊ U so that the orbit of x under h has diameter ≦ d.

A rational normal curve C of order n in [n] has, at each point P, a nest of osculating spaces

as P moves on C the [n — 2] generates a primal Dn-1 of order 2n — 2.

Hilbert [3] found the multiplicities on Dn-1 not only of the vμ+1 generated by Dμ for each lesser value of μ but also those of all submanifolds common to these various Dμ.

A surface Φ in higher space has, as explained [4] by del Pezzo, a nest of tangent spaces

of respective dimensions

they raise the problem of finding the orders of manifolds generated by them and the multiplicity of each on the higher manifolds to which it belongs: the task does not seem to have been attempted, but it may well be eased if Φ is rational and normal.

The theory of associated prime ideals is one of the most basic notions in the study of modules over commutative Noetherian rings. For modules over non-Noetherian rings however, the classical associated primes are not so useful and in fact do not exist for some modules M. In [4] [22] a prime ideal P of a ring R is said to be attached to an R-module M if for each finite subset I of P there exists m ∊ M such that I ⊆ annR(m) ⊆ P. In [4] the attached primes were compared to the associated primes and the results of [4], [22], [23], [24] show that the attached primes are a useful alternative in non-Noetherian rings to associated primes. Several other methods of associating a set of prime ideals to a module M over a non-Noetherian ring have proven very useful in the past. The most common of these is the set Assf(M) of weak Bourbaki primes of M [2, pp. 289-290].

Let Φ be a regular closed C2 curve on a sphere S in Euclidean three-space. Let H(S)[H(Φ) ] denote the convex hull of S[Φ]. For any point p ∈ H(S), let O(p) be the set of points of Φ whose osculating plane at each of these points passes through p.

1. THEOREM ([8]). If Φ has no multiple points and p ∈ H(Φ), then |0(p) | ≧ 3[4] when p is [is not] a vertex of Φ.

2. THEOREM ( [9]). a) If the only self intersection point of Φ is a doublepoint and p ∈ H(Φ) is not a vertex of Φ, then |O(p)| ≧ 2.

b) Let Φ possess exactly n vertices. Then

• (1) |O(p)| ≦ nforp ∈ H(S) and

• (2) if the osculating plane at each vertex of Φ meets Φ at exactly one point, |O(p)| = n if and only if p ∈ H(Φ) is not vertex.

It is an old conjecture by P. Bankston that the category CompHaus of compact Hausdorff spaces and their continuous maps is not dually equivalent to any elementary P-class of finitary algebras (taken as a category with all homomorphisms between its members as maps), where elementary means defined by first order axioms, and a P-class is one closed under arbitrary (cartesian) products. One motivation for this conjecture is the fact that such a dual equivalence would make ultracopowers of compact Hausdorff spaces correspond to ultrapowers of finitary algebras, and one might expect this to have contradictory consequences.

As a possible step towards proving his conjecture, Bankston [2] showed that no elementary SP-class of finitary algebras can be dually equivalent to CompHaus. However, it was subsequently proved in [1] that the same holds for any SP-class of finitary algebras, using an argument independent of ultrapowers.

For a functor G:𝒜→𝔛 and a class 𝔇 of small categories containing the terminal category 1 we form the extension

and call G right 𝔇-pro-adjoint if and only if Pro (𝔇, G) is right adjoint. Here Pro (𝔇, 𝒜) is the completion of 𝒜with respect to 𝔇; it coincides with the usual pro-category of 𝒜 in case 𝔇 = directed sets. For this 𝔇 a full embedding Gis dense in the sense of Mardešić [11] if and only if it is right 𝔇-pro-adjoint in the above sense; this has been proved recently by Stramaccia [15]. The most important example is the embedding of the homotopy category of pointed CW-complexes into the homotopy category of pointed topological spaces (cf. [2]).

A large number of important integral transforms, such as Laplace, Fourier sine and cosine, Hankel, Stieltjes, and Riemann- Liouville fractional integral transforms, can be put in the form

1.1

where f(t) and the kernel, h(t), are locally integrable functions on (0,∞), and x is a positive parameter. Recently, two important techniques have been developed to give asymptotic expansions of I(x) as x → + ∞ or x → 0+. One method relies heavily on the theory of Mellin transforms [8] and the other is based on the use of distributions [24]. Here, of course, the integral I(x) is assumed to exist in some ordinary sense.

If the above integral does not exist in any ordinary sense, then it may be regarded as an integral transform of a distribution (generalized function). There are mainly two approaches to extend the classical integral transforms to distributions.

Let E be a vector space of dimension n ∊ N over a field Γ, of characteristic 0. Choose E*, dual to E, and form

the space of exterior powers of E, as well as ΛE*. Finally, let

In [3], C. Davis showed that if a convex polyhedral cone C (the positive span of a finite set of vectors in Euclidean space) contains no nonzero linear subspace, then C is linearly isomorphic to the set V+ of nonnegative points in a linear subspace V of Rn. Moreover n can be taken to be the number of facets (maximal proper faces) of C.

The purpose of this paper is to continue the investigation into Massey products defined on two dimensional polyhedra, initiated in [13]. It will be shown that for many such spaces there is a hyperbolic model which can be used to study Massey products. More precisely, Massey products may be interpreted as intersections of geodesies in the Poincaré model. These elements are called minimal Massey products and are the analogue of Massey products over a system considered in Porter's paper. They enjoy the property of being uniquely defined (without indeterminacy) and of being multilinear and natural. Minimal products also satisfy symmetry properties generalising the symmetry properties enjoyed by cup products.

A device which will be useful in the proof of the main theorem, 7.4, is the introduction of a class of complexes called basic complexes. These generalise the notion of a surface and each one houses a standard copy of a Massey product.