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     

Parallelodromie-Ebenen

The André-representation of affine translation planes by congruence partitions of abelian groups ([1]) is generalized to a representation of those affine Barbilian structures, which satisfy the little axiom of parallelodromy.

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Herrn WERNER BURAU zum 70. Geburtstag gewidmet.     

Eine Bemerkung über absolute Ebenen

By separating the compatibility conditions for incidence, order and congruence the axioms for absolute planes in [3] are put in a somewhat more lucid form.

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Die absoluten Ebenen aus [3] entsprechen den Ebenen, die die ebenen Axiome der Verknüpfung, die Axiome der Anordung und die Axiome der Kongruenz aus D.HilbertsGrundlagen der Geometrie erfüllen.

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Werner Leiner zum 60. Geburtstag     

Triangle congruence and the Moulton plane

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Triangle congruence and the Moulton plane

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Equilibrium and congruence on a rotation surface

Summary Equilibrium and congruence are analyzed for surfaces applicable on a rotation surface: congruence conditions are shown to be in an essential number of two, linked by a differential identity, and the displacement field induced by a congruent strain is obtained. Then the solution of the homogeneous equilibrium equation is determined, when the equilibrium problem is the adjoint problem of the congruence.

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Sunto Si analizzano i problemi dell'equilibrio e della congruenza per superfici applicabili su superfici di rotazione: si mostra che le condizioni di congruenza essenziali sono due, legate da una identità differenziale, e si determina lo spostamento indotto da una deformazione congruente. A partire dalle condizioni di congruenza si determina poi la soluzione dell'equazione di equilibrio omogenea, nella ipotesi che il problema dell'equilibrio sia il problema aggiunto della congruenza.

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This work was done in the sphere of activity of the C.N.R. — Group for mathematical research.

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Istituto di Matematica del Politecnico — Via Bonardi, 9 — 20133 Milano.

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Entrata in Redazione il 3 Marzo 1973.     

Affine Ebenen,in denen der Dreispiegelungssatz für Punktbüschel gilt

Affine planes with reflections on every line such that the 3-reflection theorem is valid for any three lines through one point are either euclidean planes or elliptic planes of characteristic 2. A slight weakening of the conditions yields to pseudo euclidean planes and hyperbolic planes of characteristic 2.

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Herrn Professor Dr. Hanfried Lenz zum 70. Geburtstag     

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     

Einige Resultate über die III-Hauptflächen eines Strahlensystems

Stephanidis has introduced the third quadratic differential form III of a rectilinear congruence S in the real 3-dimensional Euclidean space [6]. In the present paper we prove that the form III can be expressed in terms of the first and the second quadratic form of the congruence S and of the second fundamental form of the middle enveloppe M(u,v) of S. Further, we study the congruences, for which the III-principle surfaces coincide with some well-known ruled surfaces of the congruence S.

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Herrn Professor Dr. Walter Benz zum 60. Geburtstag gewidmet     

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.     

Une Generalisation des Ensembles de Dyck

A certain class of Thue congruence relations are studied, which abstract some of combinatorial properties of the special congruence relation defining Dyck sets. It is proved that the congruence classes of these “Dyck congruences” as well as their complements are, unambiguous context-free languages.

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Cet auteur a preparé ce travail lors d’un sejour de quelques semaines à l’Université de Jerusalem en Avril, 1969. Que le department de mathématiques de cette université et tout particulièrement le professeur E. Shamir soit ici chaleureusement remerciés.     

Eine Variante des Bilinski-Modells Der Ebenen Hyperbolischen Geometrie

A new model of the 2-dimensional hyperbolic geometry is constructed in the congruence of bisecants of a twisted cubic. This model is a variant of the Bilinski model, constructed in the torus-plane (a plane with the topological connectivity of a torus).

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Herrn Prof.Dr. Helmut Karzel zum 60. Geburtstag gwidmet     

Ein Beweis von H. Karzel und I. Pieper

An incidence space of dimension at least 2 and order at least 4 in which all planes are affine planes is an affine space ([1 ] and [2]). Karzel's and Pieper's proof of this theorem is represented here lightened from some material relevant to the report [2] but not to this theorem.

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Herrn Professor Dr. Adriano Barlotti zum 65. Geburtstag     

Zur algebraischen Kennzeichnung mehrwertiger Halbordnungen

The radical R(T) of a planar ternary ring (PTR) T, which plays an important role in the theory of halforderings and orderings of PTRs, allows also to give an algebraic characterization ofJunkers' multiple-valued halforderings of projective planes. As an application we obtain some information on the group of projectivities in projective planes.

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Herrn Professor G. Tallini zum 60. Geburtstag gewidmet     

über Kurven vom TYP (m;n) und Beispiele total m-regulärer (k,n)-Kurven

In this paper we are concerned with a special kind of subsets of finite projective planes and give some new examples of totally m-regular (k,n)-arcs.

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Herrn Professor Dr. WERNER BURAU zum 70. Geburtstag     

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 die affin vollständigen,endlich erzeugbaren Moduln

A universal algebraA is calledk-affine complete, if any function of the Cartesian powerA ^k intoA, which is compatible with all congruence relations ofA, is a polynomial function.A is called affine complete, if it isk-affine complete for every integerk. In this paper, all affine complete finitely generated modules are characterized. Moreover, the paper contains some results on functions compatible with all congruence relations of an algebra, and on affine complete algebras in general.

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Herrn Prof. Dr. E. Hlawka zum 60. Geburtstag gewidmet     

Eine Verallgemeinerung des Satzes von Holditch

The Holditch Theorems for the euclidian plane and the euclidian sphere are generalized for the euclidian space. If we move a hyperbolic linear congruence of linesN along a closed ruled surfaceX, each line ofN generates a closed ruled surfaceY. Some global invariants ofY are linear functions of some global invariants ofX.

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Herrn Prof. Dr. K. Strubecker zum 70. Geburtstag gewidmet     

Equi-isoclinic planes of Euclidean spaces

In a Euclidean space, a p-set of equi-isoclinic planes is a set of p isoclinic planes of which each pair has the same non-zero angle.The Euclidean 4-space E⁴ contains a unique congruence class of quadruples of equi-isoclinic planes, whereas quintuples of equi-isoclinic planes do not exist in E⁴.In the following a method is given to derive sets of equi-isoclinic planes in Euclidean spaces. We find again the well-known sets of equi-isoclinic planes of E⁴. The quadruples of equi-isoclinic planes in E⁵ are derived. It turns out that E⁵ contains one congruence class of such quadruples which are not flat quadruples and one congruence class of quintuples of equi-isoclinic planes, whereas sextuples of equi-isoclinic planes do not exist in E⁵.It appears that the symmetry group of that quintuple is isomorphic to the symmetric group S₅.     

Halbordnungstopologien in Projektiven Ebenen

The order topology of an ordered projective plane leads to a topological projective plane in which intersection and joining are continuous operations (c.f. O.WYLER [5]).Here continuity properties in projective planes with half order topology derived from a SPERNER'sche Ordnungsfunktion are analysed. Central projections are continuous and planes which are connected with respect to the half order topology are (fully) ordered and therefore topological planes.

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Herrn Prof. Dr. WERNER BURAU zum 70. Geburtstag     

Su un problema unidimensionale di diffusione in un mezzo a contorno mobile con condizioni ai limiti non lineari

Summary It is considered a one dimensional heat diffusion probelm in a medium bounded by two parallel planes, x=0 and x=s(t), with a temperature specification on the moving face and a nonlinear relationship between the heat flux and the temperature on the other one. Existence, uniquenesses and computability of the solution are investigated. Continuous and monotone dependence on the data and related properties are also studied. The case s(0)=0 is then widely discussed. The class of boundary problems for which these results hold seems to be very wide.

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Entrata in Redazione il 9 ottobre 1971.

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Lavoro eseguito nell'ambito dell'attività del Gruppo Nazionale per la Fisica Matematica del C.N.R.