Isometrische Immersionen der Kodimension 2 von Raumformen

Let M, resp. , denote Riemannian manifolds of dimensions m>4, resp. =m+2, and of constant sectional curvatures C, resp. , with 0 and is a standard space form, then the foliation L is a (globally) trivial fibre bundle with fibre S^m–1.     

Complétude des métriques lorentziennes de\mathbb{T}^2 et difféomorphismes du cercle

Let and the foliations by the null geodesics of some lorentzian metricg on the torus . We analyse how geodesic completeness properties ofg are related to the dynamics of and .     

Der Satz von v. Staudt-Schleiermacher in Minkowski-Ebenen

Let be a Minkowski-(incidence-)plane and let be the group of free projectivities of , i. e. the subgroup generated by pairs of proper perspectivities with identical centers. Our theorem then asserts that is miquelian if satisfies condition (P ₅), i. e. every free projectivity with 5 fixed points is the identity. But first a lemma is shown, which holds in Möbius- and Laguerre-(incidence-)planes too: if fulfills (P ₅), then every affine derivation of is pappian.     

Théorème de division avec des conditions au bord

Let be an open subset of ⁿ and be a subalgebra of the algebra of analytic functions on . We suppose that satisfies some weak conditions of noetherianity such that we can construct a finite stratification for each ideal of . We also suppose that satifies global £ojasiewicz's inequalities. We prove the following: Let andf C on flat on ; if for eacha the Taylor's serie off ata, T _a f, is in the ideal generated byT _a f ₁,...,T _a f _p in the ring of formal power series, then there exist ₁,..., _p ,C on flat on such that . This result extends the classic Hormander's theorem of division (for a polynomial) or the £ojasiewicz-Malgrange theorem in the local analytic case.Reherches menées dans le cadre du Programme d'Appui à la Recherche Scientifique (PARS MI 33)     

Operator inequalities and construction of Krein spaces

Let be a Hilbert space. A continuous positive operatorT on uniquely determines a Hilbert space which is continuously imbedded in and for which with the canonical imbedding . A Kreîn space version of this result, however, is not valid in general. This paper provides a necessary and sufficient condition for that a continuous selfadjoint operatorT uniquely determines a Kreîn space ( ) which is continuously imbedded in and for which with the canonical imbedding .     

Homogeneous algebraic varieties defined by Jordan pairs

Every Jordan pair defines an algebraic varietyX containing as a dense open subset.X is projective (affine) if and only if is separable (radical). The Picard group ofX is generated by the irreducible factors of the generic norm of . If is separable then the automorphism group ofX is the projective group of .     

Universality of the automorphism group of the real line

We discuss the question, whether each automorphism group (of cardinality at most ) of a linear order is embeddable into the automorphism group of the real line. We show that the answer to this question is independent of the axioms of ZFC: the answer is positive, if we assume < and Souslin's hypothesis; the answer is negative, if we assume orV=L.     

Nonconvergence results for the application of least-squares estimation to Ill-posed problems

One standard approach to solvingf(x)=b is the minimization of f(x)–b² overx in , where corresponds to a parametric representation providing sufficiently good approximation to the true solutionx*. Call the minimizerx=A( ). Take = _N for a sequence { _N } of subspaces becoming dense, and so determine an approximating sequences {x _N A ( _N )}. It is shown, withf linear and one-to-one, that one need not havex _Nx* iff ^–1 is not continuous.This work was supported by the US Army Research Office under Grant No. DAAG-29-77-G-0061. The author is indebted to the late W. C. Chewning for suggesting the topic in connection with computing optimal boundary controls for the heat equation (Ref. 2).     

A Uniqueness Theorem for the Dual Problem Associated to a Variational Problem with Linear Growth

Uniqueness is proved for solutions of the dual problem that is associated with the minimum problem among the mappings with prescribed Dirichlet boundary data and for smooth strictly convex integrands f of linear growth. No further assumptions on f or its conjugate function are imposed, in particular, is not assumed to be strictly convex. A special solution of the dual problem is seen to be a mapping into the image of , which immediately implies uniqueness. Bibliography: 13 titles.     

Topological and bornological characterisations of ideals in von Neumann algebras: I

Suppose is a von Neumann algebra on a Hilbert space and is any ideal in . We determine a topology on , for which the members of that are to norm continuous are exactly those in ; and a bornology on such that the elements of which map the unit ball to an element of , equivalently those members of that are norm to bounded, are exactly those in . This is achieved via analogues of the notions of injectivity and surjectivity in the theory of operator ideals on Banach spaces.     

Rings of regular functions on nilpotent orbits and their covers

Summary LetG be a complex semisimple algebraic group with Lie algebra . Let be a nilpotentG-orbit, its ring of regular functions. We derive a formula for as aG-module and prove some partial results on a cover of . We then relate this formula to various existing multiplicity formulas forK-types in Harish-Chandra bimodules ofG.Supported by National Science Foundation Grant DMS-8505550     

Functions of translation type and functorial properties of Segal algebras II

In this paper we investigate functorial properties of the Segal algebra which consists of all functionsf in Wiener's algebra onG with Fourier transform in Wiener's algebra on the dual group . Especially may serve as a very large and natural domain for Poisson's formula. Moreover, there is introduced a Segal algebraE ₀(G) containing as a subspace, but still eachfE ₀(S) satisfies Poisson's formula.     

L 2 harmonic forms and stability of hypersurfaces with constant mean curvature

We prove that a complete noncompact oriented strongly stable hypersurfaceM ⁿ with cmc (constant mean curvature)H in a complete oriented manifoldN ⁿ⁺¹ with bi-Ricci curvature, satisfying alongM, admits no nontrivialL ² harmonic 1-forms. This implies ifM ⁿ (2n4) is a complete noncompact strongly stable hypersurface in hyperbolic spaceH ⁿ⁺¹(–1) with cmc , there exist no nontrivialL ² harmonic 1-forms onM. We also classify complete oriented strongly stable surfaces with cmcH in a complete oriented manifoldN ³ with scalar curvature satisfying .     

Differentiability of the minimal average action as a function of the rotation number

Let be a finite composition of exact twist diffeomorphisms. For any real number , letA() denote the minimal average action of -invariant measures with angular rotation number . We prove thatA() is differentiable at every irrational number and that for generic it is not differentiable at rational , thus verifying conjectures of S. Aubry. Moreover, we show that these results are valid for a variational principleh which satisfies the condition which we have called elsewhere (H). As a consequence, we generalize a result due to Bangert concerning geodesics on a two dimensional torus with an arbitrary, but sufficiently smooth metric.supported by NSF grant no. DMS-8806067.01 and a Guggenheim Fellowship.     

Completion of an additive measure to a σ-additive measure by an extension of the space

For an algebra of subsets of a set X there is constructed a set and an algebra of its subsets so that the mapping is a one-to-one correspondence between and and for each additive measure on the measure on defined by the equation is countably additive.Translated from Matematicheskie Zametki, Vol. 3, No. 1, pp. 71–76, January, 1968.The author wishes to express his deep appreciation to S. V. Fomin, under whose guidance this paper was written.     

Commutator-finite orthomodular lattices

The class of orthomodular lattices which have only finitely many commutators is investigated. The following theorems are proved: contains the block-finite orthomodular lattices. Every irreducible element of is simple. Every element of is a direct product of a Boolean algebra and finitely many simple orthomodular lattices. The irreducible elements of which are modular, or are M-symmetric with at least one atom, have height two or less.     

Common invariant subspaces for collections of operators

Let be a collection of bounded operators on a Banach spaceX of dimension at least two. We say that is finitely quasinilpotent at a vectorx ₀X whenever for any finite subset of the joint spectral radius of atx ₀ is equal 0. If such collection contains a non-zero compact operator, then and its commutant have a common non-trivial invariant, subspace. If in addition, is a collection of positive operators on a Banach lattice, then has a common non-trivial closed ideal. This result and a recent remarkable theorem of Turovskii imply the following extension of the famous result of de Pagter to semigroups. Let be a multiplicative semigroup of quasinilpotent compact positive operators on a Banach lattice of dimension at least two. Then has a common non-trivial invariant closed ideal.This work was supported by the Research Ministry of Slovenia.     

Lower bounds for faithful,preinjective modules

Let be the path algebra for some representation-infinite quiver over some field k. There exists a bound such that ^mI is faithful for all indecomposable injective -modules I and all , and such that there exists an indecomposable injective -module J satisfying that J is not faithful, denotes the Auslander-Reiten-translation. Let m() be the maximum of the taken over all possible orientations of the underlying graph . In this article we determine the bounds m() for representation-infinite quivers for which is a tree.     

Robust tests in group sequential analysis: One- and two-sided hypotheses in the linear model

Consider the linear modelY=X+E in the usual matrix notation where the errors are independent and identically distributed. We develop robust tests for a large class of one- and two-sided hypotheses about when the data are obtained and tests are carried out according to a group sequential design. To illustrate the nature of the main results, let and be anM- and the least squares estimator of respectively which are asymptotically normal about with covariance matrices ²(X ^t X)^–1 and ²(X ^t X)^–1 respectively. Let the Wald-type statistics based on and be denoted byRW andW respectively. It is shown thatRW andW have the same asymptotic null distributions; here the limit is taken with the number of groups fixed but the numbers of observations in the groups increase proportionately. Our main result is that the asymptotic Pitman efficiency ofRW relative toW is (²/²). Thus, the asymptotic efficiency-robustness properties of relative to translate to asymptotic power-robustness ofRW relative toW. Clearly, this is an attractive result since we already have a large literature which shows that is efficiency-robust compared to . The results of a simulation study show that with realistic sample sizes,RW is likely to have almost as much power asW for normal errors, and substantially more power if the errors have long tails. The simulation results also illustrate the advantages of group sequential designs compared to a fixed sample design, in terms of sample size requirements to achieve a specified power.     

The algebra of implicit operations

For an ordinal and a class of topological algebras of a given type (which may be infinite and may contain inflnitary operations), an-aryimplicit operation on is any new-ary operation whose introduction does not eliminate any continuous homomorphisms between members of . The set of all-ary implicit operations on is denoted by and forms an algebra of the given type which is endowed with the least topology making continuous all homomorphisms into members of . With this topology, is a topological algebra in which the subalgebra of all-ary operations on which are induced by terms is dense, provided that is closed under the formation of closed subalgebras and finitary direct products. This is obtained by realizing as an inverse limit of-generated members of . These results are applied to pseudovarieties of topological and finite algebras.This work was supported, in part, by INIC grant 85/CEX/4. This paper was written while the author was a faculty member at the Universidade do Minho.Presented by Ralph McKenzie.