Sul teorema di Riesz-Thorin

Summary We give an extension of the Riesz-Thorin theorem to linear operators which map Morrey spaces in spaces of the same or similar type.

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A Bruno Finzi nel suo 70^mo compleann

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Entrata in Redazione il 10 dicembre 1969.     

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 Bewegungen in der absoluten Geometrie

In this paper we improve a theorem on the representation of isometries in (generalized) euclidean and non-euclidean geometries of arbitrary dimension presented in a previous paper [3]. As an application we give a characterization of isometries in spaces of three dimensions.

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W. BURAU zum 70. Geburtstag gewidmet     

On applications of the Taylor formula in some quasispaces

We consider some metric spaces with quasimetric (quasispaces) comprising uniformly regular (equiregular) Carnot — Carathéodory quasispaces whose quasimetric is induced by C ^ϒ−1-smooth vector fields of formal degree not higher than ϒ. For these spaces, some analogues of the Campbell — Hausdorff formula are derived, which allows us to prove a theorem on a nilpotent tangent cone, a theorem on isomorphism of various nilpotent tangent cones defined at a common point, and a local approximation theorem.     

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.     

Lineare Komplexbündel im einfach isotropen Raum

According to K. Strubecker ([5]–[8]) a three dimensional real affine space with the metric ds²=dx²+dy² is called a simply iso-tropic space J₃(1). In J₃(1) exist 41 types of bundles of linear line complexes. In this paper we study the bundles of type 1, 2-called cyclic bundle and 38-called Strubecker-bundle. For a cyclic bundle and a Strubecker-bundle we give only the fundamental theorem and one metric result, however for a cyclic bundle the metric theory is completely developed.

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Herrn o. Prof. Dr. K. Strubecker zum 80. Geburtstag gewidmet     

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     

Alcune osservazioni riguardanti i gruppi di LieG 2 e Spin(7), candidati a gruppi di olonomia

Summary The Lie groups G₂ and Spin(7) can be considered as automorphisms groups of euclidean vector spaces (of dimension 7, 8 resp.) endowed with a suitable vector product (cfr. [12]). Here one put in evidence several geometric properties of certain special subspaces of such euclidean spaces and the manifolds of special subspaces are determined as well known homogeneous spaces. One considers also riemannian manifolds with holonomy group G₂ or Spin(7) establishing that in the analytic case the existence of a totally geodesic submanifold of codimension 1 imply local reducibility.

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Lavoro svolto nell'ambito del «Gruppo Nazionale Structure Algebriche, Geometriche e Applicazioni (Consiglio Nazionale delle Ricerche)».     

The octahedron theorem in Minkowski three-space a metric vector space proof of Miquel's theorem in the Laguerre plane

As is well known [1, p.480], the cycles and spears of the real Laguerre plane can be represented by the points and null planes of three-dimensional Minkowski space. Miquel's theorem in the Laguerre plane can then be expressed as an intrinsically interesting Minkowski space theorem the octahedron theorem. We outline the correspondence between the two theorems, and then give a metric vector space proof of the octahedron theorem, thereby providing an alternate proof of Miquel's theorem. We then discuss the generalization of both theorems to more general spaces.     

Factorization of involutions in characteristic two orthogonal groups: An application of the Jordan form to group theory

This paper uses the theory of the Jordan canonical form for a matrix and the theory of orthogonal sums of isometries in metric vector spaces (quadratic spaces) in order to prove a theorem on the factorization of involutions in the orthogonal groups of metric vector spaces over fields of characteristic two. Using this theorem, a classification scheme for such involutions is devised. This scheme is similar to the scheme for involutions when the field is of characteristic not equal of two.     

Was Sie schon immer über affine Punkträume wissen wollten oder Keine Angst vor Translationen

The concept of an Affine Point Space to be found in many textbooks is complicated and easily misunderstood. Looking at the system of all cosets of subspaces of a vector space yields a more lucid and mathematically clearer definition of affine spaces.

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Herrn Professor Dr. Walter Benz zum 60. Geburtstag gewidmet     

On an equivalence of topological vector space valued cone metric spaces and metric spaces

Scalarization method is an important tool in the study of vector optimization as corresponding solutions of vector optimization problems can be found by solving scalar optimization problems. Recently this has been applied by Du (2010) [14] to investigate the equivalence of vectorial versions of fixed point theorems of contractive mappings in generalized cone metric spaces and scalar versions of fixed point theorems in general metric spaces in usual sense. In this paper, we find out that the topology induced by topological vector space valued cone metric coincides with the topology induced by the metric obtained via a nonlinear scalarization function, i.e any topological vector space valued cone metric space is metrizable, prove a completion theorem, and also obtain some more results in topological vector space valued cone normed spaces.     

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.     

New existence theorems for quasi-equilibrium problems and a minimax theorem on complete metric spaces

Motivated by the Suzuki’s type fixed point theorems, we give several new existence theorems for scalar quasi-equilibrium problems, and vector quasi-equilibrium problem on complete metric spaces. We give important examples for our results. Note that the solution of quasi-equilibrium problem (resp. vector quasi-equilibrium problem) is unique under suitable conditions, and we can find the unique solution by the Picard iteration. Besides, we also give a new coincidence theorem on complete metric spaces. Finally, we give a new minimax theorem on complete metric spaces. Note that the solution of minimax theorem is unique under suitable conditions, and we can find the unique solution by the Picard iteration.     

On the global sections of a vector bundle over a projective variety

Here we study a few general formal properties related to the dimensions of the spaces of global sections of subsheaves of a vector bundle.

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Sunto SiaE un fibrato vettoriale algebrico su una varietà proiettiva. In questo lavoro studiamo alcune nozioni correlate alle dimensioni degli spazi di sezioni globali dei sottofasci diE.

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The author was partially supported by MURST and GNSAGA of CNR (Italy).     

Some applications of the KKM-mapping principle in hyperconvex metric spaces

In 1996 Khamsi proved a result in hyperconvex metric spaces which is an analogue of the famous KKM-maps principle proved by Ky Fan in 1972 for topological vector spaces.In this note, using the above result due to Khamsi, we establish two results in hyperconvex metric spaces which are analogues of the famous Ky Fan inequality and Sion minimax theorem proved for topological vector spaces in 1972 and 1958, respectively.     

A coincidence point result in Menger spaces using a control function

In the present work we prove a coincidence point theorem in Menger spaces with a t-norm T which satisfies the condition sup{T(t,t):t<1}=1. As a corollary of our theorem we obtain some existing results in metric spaces and probabilistic metric spaces. Particularly our result implies a probabilistic generalization of Banach contraction mapping theorem. We also support our result by an example.     

Das Pinchingproblem in Fastriemannschen Finslerschen Mannigfaltigkeiten

In a preceeding paper ([7]), there was defined, what are almost Riemannian metrics. Here this concept is used to deal with the pinching problem, which characterizes the type of a manifold, when the sectional curvature of a metric is sufficiently pinched. Main purpose is a generalisation of the GROMOLL-CALABI theorem, which is done by constructing an osculating Riemannian metric with a sectional curvature that behaves well, when it is derived from a closely pinched almost Riemannian metric.

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Diese Arbeit beruht auf meiner Dissertation (Mainz 1970) und wurde unterstützt von der Deutschen Forschungsgemeinschaft.     

An Extension of Set-Valued Contraction Principle for Mappings on a Metric Space with a Graph and Application

In this paper, we obtain an existence theorem for fixed points of contractive set-valued mappings on a metric space endowed with a graph. This theorem unifies and extends several fixed point theorems for mappings on metric spaces and for mappings on metric spaces endowed with a graph. As an application, we obtain a theorem on the convergence of successive approximations for some linear operators on an arbitrary Banach space. This result yields the well-known Kelisky–Rivlin theorem on iterates of the Bernstein operators on C[0,1].     

Compactness for manifolds and integral currents with bounded diameter and volume

By Gromov??s compactness theorem for metric spaces, every uniformly compact sequence of metric spaces admits an isometric embedding into a common compact metric space in which a subsequence converges with respect to the Hausdorff distance. Working in the class of oriented k-dimensional Riemannian manifolds (with boundary) and, more generally, integral currents in metric spaces in the sense of Ambrosio?CKirchheim and replacing the Hausdorff distance with the filling volume or flat distance, we prove an analogous compactness theorem in which however we only assume uniform bounds on volume and diameter.