Sul problema di Dirichlet in un cono diR n inL p

Summary This work is concerned with Dirichlet boundary-value problem for the equation (1) in a cone G di Rⁿ. This problem is investigated in Sobolev weight spaces W _r,p ^s (G): existence and uniqueness theorem si given.

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Entrata in Redazione il 5 dicembre 1977.

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Lavoro eseguito nell'ambito del G.N.A.F.A. del C.N.R.     

Equazioni quasi ellittiche e spaziL p, θ (ω, δ) (I)

Summary Regularity theorems inL ^{2, θ} (ω, δ) spaces are proved for weak solutions of quasielliptic differential equations. In particular, regularization results are obtained in the class of holder continuous functions (with respect to a suitable metric related to the operator). As a consequence, we obtain results and estimates in L^p andL ^{p, θ} spaces for the solution of the Dirichlet problem.

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Lavoro eseguito nell’ambito del Gruppo di Ricerca n^o 46 del Comitato per la Matematica del C N.R.     

Proprietà spettrali per una classe di operatori ellittici

Summary In this paper we study, in the framework of Sobolev weight spaces, differential operators of elliptic type, with singular coefficients, perturbed by a potential. Conditions for compactness of embedding W _s,σ ^r,p (Ω)↪W _s,0 ^0,p (Ω)=L _s ^p (Ω) for density ofD(Ω) on W _s,σ ^r,p (Ω) and discretness conditions for the spectrum of A_σ in the Banach spaces L _s ^p (Ω), have been established.

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Entrata in Redazione il 20 dicembre 1977.

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Lavoro eseguito, nell'ambito del Gruppo Nazionale per l'Analisi Funzionale e le sue Applicazioni del C.N.R.     

Problemi discontinui e di trasmissione per due equazioni ellittiche di ordine diverso a coefficienti variabili relativi ad angoli consecutivi

Summary In this paper we continue our investigations (see[5]) devoted to transmission problems in consecutive angles, for elliptic operators. In[5] the above mentioned boundary value problems in the case of homogeneous constant coefficient operators and within the Sobolev weight spaces W _s1,s2 ^r were investigated. This work is concerned with the problem for general operators, within the Sobolev weight spaces W _s1,s2 ^r and W _s1,s2 ^*r . Existence theorems and stability theorems for nullity, deficiency, index have been proved.

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Entrata in Redazione il 28 marzo 1975.     

Projektive Abschliessbarkeit von Kinematischen Räumen

Kinematic spaces of an order greater than 3 with only affine or projective lines are slit spaces i.e. trace spaces of projective spaces where a subspace is omitted.

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Herrn Professor Dr. Helmut Karzel zum 50. Geburtstag     

Ein einfacher Beweis des Satzes von Alexandroff-Lester

By presenting counterexamples and a new proof, we determine all metric vector spaces, for which the theorem of ALEXANDROFF-LESTER holds. In this context, the theorem will be extended to certain nonregular metric vector spaces of characteristic 2.

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Dedicated to Professor Giuseppe Tallini on the occasion of his 60th birthday

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Herrn H. Lenz danke ich für mehrere Vorschläge zur durchsichtigeren Formulierung der Beweise.     

Hyperbolische Räume

Hyperbolic spaces are characterized as ordered spaces with a congruence relation on the set of pairs of points such that any plane is a hyperbolic plane.

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Helmut Karzel zum 70. Geburtstag     

Kennzeichnung von Berührstrukturen, die Kettengeometrien sind

Chain geometries already were characterized by means of chain spaces possessing a distinguished automorphism group. Here instead of this the weaker concept of contact spaces is used.

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Herrn Prof.Dr.Helmut Karzel zum 70.Geburtstag gewidmet     

Zur Erzeugung von kinematischen RÄumen aus einbettbaren Berührstrukturen

In [4] it was shown, that exactly those algebras (A,K) belong to chain geometries embeddable in projektive spaces which are kinematic, that means x² Kx + K for all x A. Those algebras were studied by KARZEL for describing certain incidence groups namely the kinematic spaces. It will be shown how to construct the kinematic space from the embedded chain geometry.

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Herrn Professor Dr. W. Burau zum 70. Geburtstag gewidmet     

Die richtigen Räume für Analysis im Unendlich-Dimensionalen

The aim of this paper is to characterize those locally convex spaces, which have the following properties. 1. Any curve, which is differentiable if composed with continuous linear forms, is differentiable for its own. 2. Any differentiable curve is Riemann integrable. 3. The topology is determined by the differentiable curves. 4. Linear mappings are continuous iff they are differentiable. This category of thec -complete bornological spaces is symetrically monoidal closed and includes the LF-spaces.

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Unterstützt durch das Forschungsstipendium GZ 61 622/134-14/80 des Bundesministeriums für Wissenschaft und Forschung.     

Spazi di Sobolev con peso e problemi ellittici in un angolo. (I)

Summary This work, which consists of three parts, is concerned with boundary value problems for linear partial differential equations of elliptic type in a plane sector. They have been investigated within suitable Sobolev weight spaces. In this first part of the work Sobolev weight spaces are studied. Duality, embedding and trace theorems have been proved.

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Lavoro eseguito con contributo del C.N.R. nell'ambito del Gruppo Nazionale per l'Analisi Funzionale e le sue Applicazioni.

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Entrato in Redazione il 28 novembre 1972.     

Problemi al contorno con condizioni omogenee per le equazioni quasi-ellittiche

Summary We are concerned with non-variational boundary value problems, with omogeneus boundary conditions, for linear partial differential equations of quasi-elliptic type in a bounded domain Θ in Rⁿ. It is well known that some of difficulties which arise in treating such problems, in comparison with ? regular ? elliptic problems, are connected with the presence of angular points in Θ: let us point out withB. Pini [32] that ? a bounded domain for which it is possible to assign a correct boundary value problem for a quasi-elliptic but not elliptic equation always has angular points ?. We suppose Θ is a cartesian product of a finite number of open sets and, in order to overcome the difficulties attached to the presence of angular points in Θ, taking as a model the two previous papers[33], [34] devoted to elliptic problems with singular data, we investigate the problem within suitable Sobolev weight spaces, connected with the angular points of Θ and included in the ones we have studied in[35]. Within such spaces we get existence and uniqueness theorems.

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Lavoro eseguito con contributo del C. N. R.

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Entrata in Redazione il 30 ottobre 1971.     

Lineare Optimierung in halbgeordneten Vektorräumen

Summary The problem of linear programming in partially ordered vector spaces is formulated as an immediate generalization of the same problem in Euclidean spaces. Sufficient conditions for the existence of solutions of the problem and its dual are obtained. In the special case of function spaces the sufficient conditions for the solvability of the dual problem are satisfied if a certain regularity condition is assumed.

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This research was supported by the Air Force Office of Scientific Research under grant AF-AFO SR-93 7-6 7     

Varietà parakähleriane

Sunto Vengono qui considerate varietà di tipo intermedio tra le varietà k?hleriane e le varietà quasi hermitiane (varietà parak?leriane). Esse costituiscono una classe che presenta notevoli proprietà di curvatura, dà luogo ad interessanti relazioni con alcune note classi di varietà e può venire caratterizzata utilizzando il tensore di Riemann ovvero la curvatura mista di Bompiani.

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Summary A class of almost Hermite spaces, generalising K?hler spaces is considered (parakàhler spaces) and some curvature theorems are proven. This class appears to be in connection with known classes of spaces and can be characterized by means of Riemann tensor as well of the bi-sectional curvature.

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Entrata in Redazione il 6 novembre 1972.

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Lavoro eseguito con contributo del C.N.R. nell'ambito del Gruppo Nazionale Strutture Algebriche, Geometriche e loro Applicazioni. La maggior parte dei risultati è stata annunciata in una riunione di Gruppo a Milano (marzo 1971) ed in una conferenza tenuta all'Istituto Matematico Università Perugia (aprile 1971).     

Spazi di Sobolev con peso e problemi ellittici in un angolo. - II

Summary In part I of this work (see [3]) Sobolev weight spaces related to boundary value problems for linear partial differential equations of elliptic type in a plane sector were discussed. This second part is concerned with such boundary value problems for homogeneous constant coefficient operators. They have been investigated within Sobolev weight spaces included in those previously discussed (see [3]). Existence and uniqueness theorems have been proved

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Entrata in Redazione il 28 dicembre 1972.

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Lavoro eseguito con contributo del C.N.R. nell'ambito del Gruppo Nazionale per l'Analisi Funzionale e le sue Applicazioni.     

Pseudo-Euklidische Ebenen und euklidische Räume

Euclidean planes and spaces as well as pseudo-euclidean planes are characterized solely by their incidence structure and a congruence relation on the set of the pairs of points.

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Herrn Professor Dr. WERNER BURAU zum 70. Geburtstag     

Die SchmiegrÄume an die Segre-Mannigfaltigkeiten

In this paper will be given a formula of the dimensions of the osculating spaces on a Segre-manifold in a point of it, or along a sub-Segre-manifold.

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Herrn Prof. Dr. Werner Burau zum 70. Geburtstag gewidmet     

I piani di inversione come modelli di una classe di spazi metrici

Summary The purpose of this paper is to represent a class of metric spaces on the elliptic and hyperbolic inversive planes by inversions and their transformations.

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Lavoro eseguito nell'ambito del Gruppo Nazionale per le Strutture Algebriche e Geometriche (sez. n. 4) del C.N.R., presso l'Istituto di Matematica Applicata della Facoltà di Ingegneria, Roma.

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Entrata in Redazione il 24 maggio 1972.     

Über einige Klassen von Iterationsverfahren in Banach-Räumen

Summary Local existence and uniqueness theorems for a solution of an operator equation and convergence theorems for certain classes of higher order iteration methods, namely the well-known methods of Tchebysheff, Halley, Ehrmann and a method here called minimal linearization method, in Banach spaces are given. The simple and weak conditions under which the theorems are proved are jointly valid for all classes of methods considered. The error bounds obtained are simpler and much sharper then the known ones, especially in function spaces.

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Herrn Professor Dr. C. Schmieden zum 65. Geburtstag gewidmet

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Auszug aus der von der Fakultät für Mathematik und Physik der Technischen Hochschule Darmstadt (Referenten: Prof. Dr. C. Schmieden, Prof. Dr. E. Martensen) anerkannten Habilitations-schrift des Verfassers.     

Automorphiesummanden und Cousin-I-Probleme auf Faserräumen

This article deals with the classification of summands of automorphy on locally trivial fibre spaces E(X, Y), the base space X having the property H¹ (X,O)=0 and the fibre Y being a compact Kähler manifold. Further we assume that the structure group is connected and that the transgression H¹(Y)H²(X) vanishes. Then every summand of automorphy on E is equivalent to a homomorphism on B₁(E) with values in the ring of holomorphic functions on X (theorem I). As a corollary we prove a theorem on the solvability of Cousin-I-Problems on E by additive functions (theorem II).

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Meinem verehrten Lehrer KARL STEIN in Dankbarkeit zum 60. Geburtstag gewidmet