Die Flächen mit geraden Linien als Affinen Zentraflächen

If the (euclidean) Gauss curvature of a surface in 3-space is nowhere vanishing, we have a uniqueaffine normal in every point of the surface and the set of affine normals forms a line congruence. According to the euclidean situation one can discuss existence and degeneration ofaffine focal surfaces. A cylinder of revolution in euclidean space can be characterized by the property, that all normals intersecttwo straight lines. The corresponding property in affine geometry leads to certainaffine surfaces of revolution, whose meridian curves (plane sections through a fixed axis) can be determined. If we assume the affine focal surfaces to coincide (the affine Weingarten endomorphism has double eigenvalues in every point) and degenerate intoone straight line, then the surface is aruled surface the generators of which areparallel to a plane.

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Herrn Professor Dr. Gerhard Geise zum 70. Geburtstag gewidmet     

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     

Scheitelsätze für Regelflächen

In this present paper there are given some extensions of the four-vertex theorem for closed regular plane curves on regular ruled surfaces with closed striction lines (closed ruled surfaces).

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Herrn Professor Dr. Hans Robert Müller zum 60. Geburtstag gewidmet     

Continuous Linear Right Inverses for Partial Differential Operators on Non - Quasianalytic Classes and on Ultradistributions

Characterizations are given of those linear partial differential operators with constant coefficients which admit a continuous linear right inverse on ε_(ω()ω) )resp. ε_[ω](Ω) and/or D′_(ω)(Ω) (resp. D′_[ω](Ω), where Ω is an open set in ?ⁿ. The characterisations are in the same spirit as in the previous results of the authors on the existence of right inverses on C^∞ and/or D′(Ω).     

Sopra alcune proprietà dell’entropia dipendente dall’errore

Riassunto Si introducono alcune operazioni sullo spazio delle funzioni di ripartizione e si studiano, in corrispondenza, le proprietà dell’entropia con errore .

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Summary Some operations on the space of distribution functions are introduced. The corresponding invariance properties of the entropy with error , are then studied.

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Lavoro eseguito nell’ambito del G.N.f.M. del C.N.R.     

Zur Abhängigkeit von Anordnungsaxiomen

A halfordered or ordered plane can be described in terms of a betweenness function a on each of its lines, which fulfills some different axioms. In this paper we will discuss which of these axioms are equivalent to each other.

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Herrn Professor Dr. Helmut Karzel zum 60. Geburtstag     

Die windschiefen Flächen mit einer stetigen Schar ebener Schattengrenzen II

In part I of this subject it has been shown that each ruled surface of the projective 3-spaceII with a continuous set of plane shade lines (ES-Regelfächen) is a ruled surface ofBlank with two conjugate families of such lines.—In this paper ES-Regelfächen will be constructed by using the specific projective motion of a plane, defined by each continuous set of plane shade lines on a ruled surface (central motion of a plane inII). We show that each such central motion of a shade-plane is the restriction of a one-parametric continuous group of projective collineations of the 3-space to a plane (theorem 5). Using this it is possible to characterize ES-Regelflächen as special surfaces with two conjugate families of plane shade lines (theorems 6 and 7). Finally moulding ruled surfaces in projective, affine, euclidian and non-euclidian 3-spaces are interpreted as ES-Regelflächen, and all those surfaces are listed completely.

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Teil I zu diesem Thema ist in Mh. Math.91, 39–71 (1981) erschienen. Die Numerierung der Abschnitte, Sätze und Fußnoten von Teil II schließt an Teil I an. Die Bezeichnung der auftretenden geometrischen Objekte stimmt mit jener in Teil I überein.     

Die windschiefen Flächen mit einer stetigen Schar ebener Schattengrenzen I

Already in 1933J. Blank found all ruled surfaces with two conjugated families of plane shade lines. In the late sixtiesH. Brauner solved a similar problem by determining all algebraic surfaces with only one family of plane shade lines. — We consider here only differentiable ruled surfaces withBrauner's condition, that carry at least one continuous family of plane shade lines. By using methods of projective differential geometry, algebra and synthetic projective geometry, it is possible to find all projective representatives of those surfaces. Although the assumption ofBrauner is much weaker thanBlank's, each of these surfaces is an analytic, ruled surface ofBlank.

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Mit 2 Abbildungen     

Sulla rappresentazione gruppale dei piani metrici sui piani di inversione

Summary The purpose of this paper is to give a group representation of the metric planes over the real numbers (exactly the elliptic plane, the hyperbolic plane, the DeSitter plane, the euclidean plane and the Minkowski plane) on the elliptic and hyperbolic inversive planes by inversions and their transformations.

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Dedicato al Prof. BeniaminoSegre per il 70^mo compleanno.

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

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Entrata in Redazione il 12 febbraio 1973.     

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     

Eine gruppentheoretische Kennzeichnung ebener kommutativer punktiert-affiner Inzidenzgruppen

In this paper we give a group-theoretical characterization of plane commutative punctured affine incidence groups as introduced by G.P. Kist in [1].

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Herrn Professor Dr. WERNER BURAU zum 70. Geburtstag     

Sistemi ellittici di tipo A.D.N. a coefficienti variabili e condizioni di radiazione

Summary I study an elliptic system, in the sense of Agmon-Douglis-Nirenberg, of partial differential equations with variable coefficients. The matrix operator is of type P(D) + + λR(x, D) where λ εC, P(D) has constant coefficients, is elliptic, and his determinant admits a special elementary solution. On the coefficients in R(x, D), sufficiently smooth, a certain behaviour at the infinity is assumed. For suitable known vectors f, the problem P(D)u - λR(x, D)u=f is shown to be equivalent to a system of singular integral equations in special subspaces of [W^l,p]^N, if N is the rank of the system, as is studied in [5]. This is possible when the unknown vector u belongs to a class that, generally, is stricter than the one of existence and uniquencess for P(D) [4]. Then results on the solvability of the system follow when λ is such that P(D+λR(x, D) is elliptic.

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Entrata in Redazione l'8 giugno 1977.

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Lavoro svolto nell'ambito del ? Laboratorio per la matematica applicata ? del C.N.R. presso l'Università di Genova.     

Una proprietà gruppale delle involuzioni planari che mutano in sé un'ovale di un piano proiettivo finito

Summary The purpose of the present paper is to prove the following theorem: Let Ω be an oval in the projective plane P of odd order n. If P admits a collineation group G wich maps Ω onto itself and is doubly transitive on Ω, then P is desarguesian, Ω is a conic and G contains all collineations in the little projective group PSL(2, n) of P wich leaves Ω invariant.

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Entrata in Redazione il 5 april 1977.     

Eléments analytiques et filtres percés sur un ensemble infraconnexe

Summary Let K be a complete ultrametric algebraically closed field, let D be an infraconnected bounded closed set of K and let H(D) be the Banach algebra of the analytical elements on D. The properties of the elements f of H(D) are learnt introducing a function v(f, μ) continuous and affine on pieces in the intervals where it is bounded. We learn the elements f ε H(D) which are not a product of a polynomial with an invertible element. We introduce the notion of monotonous filter, related with the continuous multiplicative semi-norms of H(D) and we prove these such elements are annulated by a monotonous filter and more precisely, a pierced monotonous filter.

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Entrata in Redazione il 13 settembre 1975.     

Winkelmessung und Orientierung

In an affine plane oriented distance d and oriented angle w (between non-oriented lines) are introduced as mapping into , fulfilling certain axioms. Examples show, that the use of w is of advantage in elementary geometry. Axiomatic consequences of generalizing d, w to mappings into ordered abelian groups are considered.

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Herrn Helmut Salzmann zum 60. Geburtstag     

Almansi decomposition for Dunkl operators

Let Ω be a G-invariant convex domain in ℝ^N including 0, where G is a Coxeter group associated with reduced root system R. We consider functions f defined in Ω which are Dunkl polyharmonic, i.e. (Δh)ⁿf = 0 for some integer n. Here_333-01is the Dunkl Laplacian, and D_j is the Dunkl operator attached to the Coxeter group G,

$$\mathcal{D}_j f(x) = \frac{\partial }{{\partial x_j }}f(x) + \sum\limits_{v \in R_ + } {\kappa _v \frac{{f(x) - f(\sigma _v x)}}{{\left\langle {x,v} \right\rangle }}} v_j ,$$

where K_v is a multiplicity function on R and σ_v is the reflection with respect to the root v. We prove that any Dunkl polyharmonic function f has a decomposition of the form

$$f(x) = f_0 (x) + \left| x \right|^2 f_1 (x) + \cdots + \left| x \right|^{2(n - 1)} f_{n - 1} (x), \forall x \in \Omega ,$$

where f_j are Dunkl harmonic functions, i.e. Δ_hf_j = 0. This generalizes the classical Almansi theorem for polyharmonic functions as well as the Fischer decomposition.

     

SOME PROPERTIES OF m-ISOMETRIES AND m-INVERTIBLE OPERATORS ON BANACH SPACES

This article has two purposes:the first is to give some structure results for the class of m-isometries,and the second purpose is to extend the notions of left and right inverses to m-left and m-right ...     

A proof of a relation between the numbers of singularities of a closed polygon

Let (P) denote a closed plane polygon, t_e and t_i the numbers of exterior and interior double supporting lines, respectively, d the number of double points and i the number of inflectional edges of (P). T.F. Banchoff has shown that these numbers satisfy the relation t_e–t_i=d+1/2i.A new and simple proof of this equation is presented.     

Strukturen geometrischer Ornamente

We consider double-periodic tilings of the whole plane from the view of graph theory, not with respect to symmetry groups. We suppose that the graph is planar and connected and that the fundamental domain contains a finite number of vertices and edges. We assign to every tiling a tableau. There exists a fundamental formula connecting the number of all numbers of the tableau with the sum of the reciprocals of all these numbers and with the number p of lines in the tableau; the formula is proved even if multiple edges or loops occur. By this way we get a graph-theoretic classification of the tilings. We introduce families F of tilings and their ranks. The family F={k₁,k₂...,k_s} (with k₁>k₂>...>k_s>O) is the set of all tilings, the tableau of which contains all the numbers k_j and no others. The smallest number p of lines (which occur for the tilings of the family F) is the rank of F and has special geometric interest. Some open questions are mentioned at the end.

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Herrn Helmut Karzel zum 60. Geburtstag gewidmet     

Parallelität der Erzeugenden und Fasern Verallgemeinerter Regelflächen

In euclidean 3-space it is well known the statement of Darboux and Myller: On a ruled surface the generators are parallel along the striction line. For generalised ruled (k+1)-surfaces with ruled central-surfaces we proof all generalisations of this statement and their inverses.

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Herrn Prof. Drs. H. Brauner zum 50. Geburtstag