We prove that, for non purely atomic measures, Lp (μ, X) is a Grothendieck space if and only if X is reflexive.

In any field of nonlinear analysis Nemytskij operators, the superposition operators generated by appropriate functions, play a crucial part. Their analytic properties depend on the postulated properties of the defining function and on the function space in which they are considered. A rich source for related questions is the monograph by J. Appell and P. P. Zabrejko [2] and the survey paper by J. Appell [1].

We use the principle of symmetric criticality to connect the Willmore variational problem for surfaces in a warped product space with base a circle, and the free elastica variational problem for curves on its fiber. In addition we obtain a rational oneparameter family of closed helices in the anti De Sitter 3-space which are critical points of the total squared curvature functional. This means they are free elasticae. Also they are spacelike; this allows us to construct a corresponding family of spacelike Willmore tori in a certain kind of spacetime close to the Robertson-Walker spaces.

Let X be a complex normed space, with dual space X′, and T a bounded linear operator on X. The numerical range V(T) of T is defined as {f(Tx): x∊X, f∊ X′, ∥x∥ = ∥f∥ = f(x) = 1}. Let ⃒V(T)⃒ denote sup {⃒λ⃒: λ∊ V(T)}. Our purpose is to prove the following theorem.

Let F be an orientable surface with or without boundary and let M(F) be the mapping class group of F, i.e. the group of isotopy classes of orientation preserving diffeomorphisms of F. To each essential simple closed curve c on F we can associate an element C of M(F) called the Dehn twist about c. We refer the reader to [1] for definitions. It is well known (see [1]) that, at least in the case where F has no more than one boundary component, M(F) is generated by Dehn twists. Further, there are important subgroups of M(F) which are also generated by Dehn twists or simple products of Dehn twists; for example the Torelli group, the kernel of the homology action map M(F)→ Aut(H1(F;Z)) = Sp(H1(F;Z)), where Sp(H1(F;Z)) denotes the symplectic group, is known to be generated by Dehn twists about bounding curves and by “bounding pairs”. See [8] for proofs and definitions. Also Dehn twists crop up as geometric monodromy maps associated to Picard–Lefschetz vanishing cycles for plane curve singularities (see [5]).

In 1927 Schreier [8] proved the Nielsen-Schreier Theorem that a subgroup H of a free group F is a free group by selecting a left transversal for H in F possessing a certain cancellation property. Hall and Rado [5] call a subset T of a free group F a Schreier system in F if it possesses this cancellation property, and consider the existence of a subgroup H of F such that a given Schreier system T is a left transversal for H in F.

In [12], Loy and Miller proved that a locally compact, eudoxian TR group is algebraically and order-theoretically (and hence, topologically) isomorphic to a finite product of copies of the real numbers. In [18], Wirth used their result to describe the subgroup of a locally compact TR group generated by the compact neighbourhoods of zero. The proof of Loy and Miller relied heavily on a result of Mackey (cf. [10], p. 390) and either the finite-dimensional case of the Choquet-Kendall Theorem (cf. [15], pp. 9–10) or the representation theory of Kakutani (cf. [11], Appendix). Below we use only elementary topological results and order-theoretic arguments and a theorem of Conrad [4] to characterize all non-secular, locally compact TRL groups (Theorem 3). Our proof of Theorem 3 allows us to deduce algebraically the theorems both of Loy and Miller and of Wirth, in both cases without appealing to the theorem of Conrad.

A locally convex space (E, ) has the Mazur Property if and only if every linear -sequential continuous functional is -continuous (see [11]).

In the Banach space setting, a Banach space X is a Mazur space if and only if the dual space X* endowed with the w*-topology has the Mazur property. The Mazur property was introduced by S. Mazur, and, for Banach spaces, it is investigated in detail in [4], where relations with other properties and applications to measure theory are listed. T. Kappeler obtained (see [8]) certain results for the injective tensor product and showed that L1(μ, X), the space of Bochner-integrable functions over a finite and positive measure space (S, σ, μ), is a Mazur space provided X is also, and ℓ1 does not embed in X.

In [18] Shioda proved that the space of holomorphic 2-forms on a certain type of elliptic surface is canonically isomorphic to the space of modular forms of weight three for the associated Fuchsian group. Later, Hunt and Meyer [6] made an observation that the holomorphic 2-forms on a more general elliptic surface should in fact be identified with mixed automorphic forms associated to an automorphy factor of the form

for z in the Poincaré upper half plane ℋ, g = and χ(g) = , where g is an element of the fundamental group Γ⊂PSL(2, R) of the base space of the elliptic fibration, χ-Γ→SL(2, R) the monodromy representation, and w: ℋ→ℋ the lifting of the period map of the elliptic surface.

Theorem 3 of the paper, published in the Glasgow Mathematical Journal 35 (1993), 95–98, is stated incorrectly.

The subgroup G′ ∩ L(G) is nilpotent but not, in general, finitely generated (a suitable counterexample being provided by the group whose presentation is given n the Introduction). In groups with the property σ the Fitting radical is itself finitely generated and so the conclusion of Theorem 3 holds in this special case. The words “finitely generated” should be deleted from the paragraph preceding the statement of Theorem 4, but both this theorem and its proof are correct.

Let Sn denote the “surface” of an n-dimensional unit sphere in Euclidean space of n dimensions. We may suppose that the sphere is centred at the origin of coordinates O, so that the points P(x1, x2, …, xn) of Sn satisfy

We suppose that n≥2.

Given a group G and a positive integer k, let vk(G) denote the number of conjugacy classes of subgroups of G which are not subnormal of defect at most k. Groups G such that vkG) < ∝ for some k are considered in Section 2 of [1], and Theorem 2.4 of that paper states that an infinite group G for which vk(G) < ∝ (for some k) is nilpotent provided only that all chief factors of G are locally (soluble or finite). Now it is easy to see that a group G whose chief factors are of this type is locally graded, that is, every nontrivial, finitely generated subgroup F of G has a nontrivial finite image (since there is a chief factor H/K of G such that F is contained in H but not in K). On the other hand, every (locally) free group is locally graded and so there is in general no restriction on the chief factors of such groups. The class of locally graded groups is a suitable class to consider if one wishes to do no more than exclude the occurrence of finitely generated, infinite simple groups and, in particular, Tarski p-groups. As pointed out in [1], Ivanov and Ol'shanskiĭ have constructed (finitely generated) infinite simple groups all of whose proper nontrivial subgroups are conjugate; clearly a group G with this property satisfies v1(G) = l. The purpose of this note is to provide the following generalization of the above-mentioned theorem from [1].

If E is a Hausdorff locally convex space and M is an -dimensional subspace of the algebraic dual E* that is transverse to the continuous dual E′, then, according to [7], the Mackey topology τ(E, E′ + M) is a countable enlargement (CE) of τ(E, E′) [or of E]. Much is still unknown as to when CEs preserve barrelledness (cf. [14]). E is quasidistinguished (QD) if each bounded subset of the completion Ê is contained in the completion of a bounded subset of E [12]. Clearly, each normed space is QD, and Tsirulnikov [12] asked if each CE of a normed space must be a QDCE, i.e., must preserve the QD property. Since CEs preserve metrizability (but not normability), her question was whether metrizable spaces so obtained must be QD, and was moderated by Amemiya's negative answer (cf. [5, p. 404]) to Grothendieck's query, who had asked if all metrizable spaces are QD, having proved the separable ones are [4].

An almost Hermitian manifold (, J, g) with Riemannian connection is called nearly Kaehlerian if (xJ)X = 0 for any . The typical example is the sphere S6. The nearly Kaehlerian structure J for S6 is constructed in a natural way by making use of Cayley division algebra [3]. It is because of this nearly Kaehler, non-Kaehler, structure that S6 has attracted attention. Different classes of submanifolds of S6 have been considered by A. Gray [4], K. Sekigawa [5] and N. Ejiri [2]. In this paper we study 2-dimensional totally real submanifolds of S6. These are submanifolds with the property that for every x є M, J (Tx M) belongs to the normal bundle v. For this class we have obtained the following result.

Let K be an arbitrary field with characteristic p > 0, G a finite group of order pag′ with (p, g′) = 1, P a p-Sylow subgroup of G and G′ the commutator subgroup of G. For a ring R denote by J(R) the Jacobson radical of R and by Z(R) the centre of R. We write KG for the group algebra of G over K.

In [1], J. M. Howie considered the semigroup of transformations of sets and proved (Theorem 1) that every transformation of a finite set which is not a permutation can be written as a product of idempotents. In view of the analogy between the theories of transformations of finite sets and linear transformations of finite dimensional vector spaces, Howie's theorem suggests a corresponding result for matrices. The purpose of this note is to prove such a result.

Over a field of characteristic p the group algebra of a finite group has a non-trivial radical if and only if the order of the group is divisible by the prime p. It would be of interest to determine the powers of the radical in the non-semi-simple case [2, p. 61]. In the particular case of p-groups the solution to the problem is known through the work of Jennings [6]. We here consider the special case of group algebras whose radicals have square zero and we relate this condition to the structure of the group itself.