Showing papers in "Archiv der Mathematik in 1993"
TL;DR: In this paper, the authors define an immersion from an n-dimensional (n > 0) mani- fold into a Euclidean re, space and define the Laplacian operator of M with respect to the induced metric.
Abstract: By BANG-YEN CHEN 1. Introduction. Let x: M -* E" be an immersion from an n-dimensional (n > 0) mani- fold into a Euclidean re, space. Denote by A the Laplacian operator of M with respect to the induced metric. Then we have the following formula of Beltrami: (1.1)
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TL;DR: In this article, the aim of the present paper is to establish some new topological mini-max theorems as applications of an alternative principle for multifunctions (several further consequences of which will be presented in successive papers).
Abstract: does hold. The relevant literature is by now really impressive. For a first approach to it, we refer the reader to the references quoted in [4], [5], [7], [9]. However, as well stressed by the lucid Introduction of [9], it is rather easy to recognize a few research currents within which practically each existing contribution can be located. In particular, one of these currents is that concerning the so called topological mini-max theorems, the ancestors of which are the results of [13] and, more properly, of [16] (see also [2]). For a sharp discussion on such kind 0f theorems, we refer to a very recent paper by H. K6nig [8] which also contains the best results in this area, up to now. Since the aim of the present paper is to establish some new topological mini-max theorems as applications of an alternative principle for multifunctions (several further consequences of which will be presented in successive papers), we consider just [8] as a starting point to introduce the main novelties of our results. So, we state now a particular case of the main theorem of [8].
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TL;DR: In this article, it was shown that an indecomposable projective A-module P is directed if and only if the radical of P is directing as a module over the trace ideal of P.
Abstract: Let A be an Artin algebra. The A-modules which we consider are always left modules of finite length. If X, Y, Z are A-modules, the composition of maps f : X ~ Y and g : Y ~ Z is denoted by f 9 : X ~ Z. The category of (finite length) A-modules is denoted by A-mod. If X, Y are indecomposable A-modules, we denote by rad (X, Y) the set of non-invertible maps from X to Y. A path in A-mod is a sequence (X o . . . . . Xs) of (isomorphism classes of) indecomposable A-modules X i, 0 1, and X o = Xs, then the path (Xo, . . . , X~) is called a cycle. A indecomposable A-module is called directing if X does not occur in a cycle. Our first aim will be to extend the definition of a directing module to decomposable modules. We show that an indecomposable projective A-module P is directing if and only if the radical of P is directing. In case the top of P is injective it follows that P is directing if and only if the radical of P is directing as a module over the factor algebra of A by the trace ideal of P.
TL;DR: A submeasure 4) is called pathological if whenever 2 is a measure satisfying 0 0 there exists N ~ N such that whenever {A 1..,... AN} are disjoint sets in ~ then as discussed by the authors.
Abstract: haram problem ([1], [5], [6] and [15]) asks whether every exhaustive submeasure is equivalent to a measure. A submeasure 4) is called pathological if whenever 2 is a measure satisfying 0 0 there exists N ~ N such that whenever {A 1 . . . . . AN} are disjoint sets in ~ then
TL;DR: In this paper, it was shown that the topology of finitely open sets is not a vector space topology except when E is countable-dimensional, which coincides with the finest locally convex topology.
Abstract: Introduced by Hille (t948), the topology of f initely open sets (called the ~ topology by some) on a real vector space E is defined by the condition that a set G is open if and only if for every finite-dimensional subspace F of E the set G c~ F is open in the usual topology on F. By a strikingly simple example, Harremo~s (1987) proved that the topology of finitely open sets is not locally convex unless E is countable-dimensional. The aim of the present note is to show that the topology of finitely open sets is not a vector space topology except when E is countable-dimensional. In the latter case, the topology of finitely open sets coincides with the finest locally convex topology, by definition the finest of all locally convex vector space topologies on E cf. Schaefer (1971) or Berg, Christensen, and Ressel (1984), Ch. 1. Among the results preparing for the main theorem are a formula for the closure of a set, a characterization of neighbourhoods of 0, and a strange lemma. Everywhere, " E " denotes a real vector space, considered with the topology of finitely open sets except when other topologies are specifically introduced.
TL;DR: In this article, it was shown that a group of units in a group basis with linear independence is a group-basis, provided no rational prime divisor of the elements of the subgroup is a unit in R.
Abstract: By V(RG) we denote the units in RG, which have augmentation 1. The group of units in RG is then the product of the units in R and V(RG). A subgroup H of V(RG) with \\H\\ = |G| is called a group basis, provided the elements of H are linearly independent. This latter condition is automatic, provided no rational prime divisor of \\H\\ is a unit in R [1]. If H is a group basis, then RG = RH as augmented algebras and conversely. The object of this note is to prove the following
TL;DR: Theorem 1.1 as discussed by the authors assumes that the Sylow p-subgroups of the centralizer CG(K) are not conjugate to a subgroup of any of the groups in H.
Abstract: Theorem 1.1 Suppose that G is a finite group, and k is a field of characteristic p. Let H be a collection of subgroups of G. Denote by K the collection of subgroups K of G with the property that the Sylow p-subgroups of the centralizer CG(K) are not conjugate to a subgroup of any of the groups in H. Let J be the ideal in H∗(G, k) given by the sum of the images of transfer from subgroups in H, and J ′ be the ideal in H∗(G, k) given by the intersection of the kernels of restriction to subgroups in K. Then the ideals J and J ′ have the same radical, √ J = √ J ′.