Zum Drehvorgang der Darboux-Achse einer Raumkurve

In one of his papers J. Hartl says that he decomposes the Darboux rotation for the Frenet frame of a space curve into two simultaneous rotations. What he does, however, is to decompose the rotation for the Darboux axis of the space curve into two simultaneous rotations. In the following approach this rotation for the Darboux axis is being described from a more general aspect.

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Herrn N.K. Stephanidis zum 65. Geburtstag am 27. November 1993 gewidmet     

On Darboux helices in Euclidean 4‐space

The notion of Darboux helix in Euclidean 3‐space was introduced and studied by Yayl? et al. 2012. They show that the class of Darboux helices coincide with the class of slant helices. In a special case, if the curvature functions satisfy the equality κ² + τ² = constant, then these curves are curve of the constant precession. In this paper, we study Darboux helices in Euclidean 4‐space, and we give a characterization for a curve to be a Darboux helix. We also prove that Darboux helices coincide with the general helices. In a special case, if the first and third curvatures of the curve are equal, then Darboux helix, general helix, and V₄‐slant helix are the same concepts.     

Zur Geometrie höherer Planetenumschwungbewegungen

The product ofn rotations with constant velocities around parallel axes in the Euclidean 3-space is called a planetary motion of degreen. The paper discusses spatial motions (P-motions) defined as the result of a planetary motion and a special translation in the direction of the axes; this translation is the sum of a finite number of harmonic translations with different frequencies.

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Mit 2 Abbildungen     

Invariants of Multidimensional Time Series Based on Their Iterated-Integral Signature

We introduce a novel class of features for multidimensional time series that are invariant with respect to transformations of the ambient space. The general linear group, the group of rotations and the group of permutations of the axes are considered. The starting point for their construction is Chen’s iterated-integral signature.

     

Pseudo-spherical Darboux images and lightcone images of principal-directional curves of nonlightlike curves in Minkowski 3-space

Choosing an alternative frame, which is the Frenet frame of the principal-directional curve along a nonlightlike Frenet curve γ , we define de Sitter Darboux images, hyperbolic Darboux images, and lightcone images generated by the principal directional curves of nonlightlike Frenet curves and investigate geometric properties of these associated curves under considerations of singularity theory, contact, and Legendrian duality. It is shown that pseudo-spherical Darboux images and lightcone images can occur singularities (ordinary cusp) characterized by some important invariants. More interestingly, the cusp is closely related to the contact between nonlightlike Frenet curve γ and a slant helix, the principal-directional curve ψ of γ and a helix or the principal-directional curve ψ and a slant helix. In addition, some relations of Legendrian dualities between C-curves and pseudo-spherical Darboux images or lightcone images are shown. Some concrete examples are provided to illustrate our results.     

Some remarks on quasi-complete intersection space curves

Summary This paper is devoted to the study of quasi-complete intersection space curves. First, we give a Castelnuovo bound on the index of regularity fork-Buchsbaum, quasi-complete intersection space curves. Then, we prove that, smooth, arithmetically Buchsbaum, quasi-complete intersection space curves of maximal rank are unobstructed. We conclude by studying some examples and adding some remarks.

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Riassunto Questo articolo è dedicato allo studio delle curve spaziali quasi-complete intersezioni. Dapprima, noi diamo un limite di Castelnuovo per l'indice di regolarità delle curvek-Buchsbaum, quasi-complete intersezioni. Inoltre, dimostriamo che le curve liscie, aritmeticamente di Buchsbaum, di rango massimo quasi complete intersezioni sono non ostruite. Si conclude studiando alcuni esempi e aggiungendo alcune osservazioni.

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To Joan     

Moltiplicatori diH 1,2(Ω)

Riassunto In questo lavoro viene studiato l'insieme dei moltiplicatori diH ^1,2(Ω) con Ω aperto diR ⁿ. Nella prima parte vengono analizzate alcune proprietà di struttura dello spazio dei moltiplicatori, nella seconda viene data una caratterizzazione degli elementi di tale spazio.

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Summary In this paper we study the space of multipliers ofH ^1,2(Ω). We give first some general properties of such space, then we give a characterization of its elements.

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Lavoro eseguito nell'ambito dell'attività dei Gruppi di Ricerca del C.N.R. nell'anno 1970.

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L'autore è borsista del C.N.R., contratto No. 115.3059.05177.     

The conjugating map for commutative groups of circle diffeomorphisms

For a single aperiodic, orientation preserving diffeomorphism on the circle, all known local results on the differentiability of the conjugating map are also known to be global results. We show that this does not hold for commutative groups of diffeomorphisms. Given a set of rotation numbers, we construct commuting diffeomorphisms inC ^2-ε for all ε>0 with these rotation numbers that are not conjugate to rotations. On the other hand, we prove that for a commutative subgroupF ⊂C ^1+β, 0<β<1, containing diffeomorphisms that are perturbations of rotations, a conjugating maph exists as long as the rotation numbers of this subset jointly satisfy a Diophantine condition.     

Uniqueness of bounded solutions for some abstract differential equations

Summary For some differential equations in Hilbert space we prove uniqueness of bounded solutions in the Stepanoff norm.

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Riassunto Si dimostra l’unicità delle soluzioniS ²-limitate per alcune equazioni differenziali astratte.

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This research is supported by a grant of the N.R.C. of Canada.     

Ensembles invariants non enlacés des difféomorphismes du tore et de l’anneau

We introduce the notion of unlinked invariant set for a diffeomorphism of the 2-torus or the closed annulus and we look at the properties of these sets which generalize the Aubry-Mather sets. We prove that for any irrational number in the rotation set of an area-preserving diffeomorphism of the annulus, there exists an unliked invariant set whose rotation set is reduced to . In the same way we prove that any minimal diffeomorphism of the 2-torus homotopic to the identity may be approximated in the C⁰-topology by a periodic diffeomorphism.

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Mathematics Subject Classification (2000) 37E30, 37E45, 37J10     

Extended Darboux function in a Weyl hypersurface

In this paper, by means of the prolonged differentiation, we introduce a pair of operators D andD ^* which enable us to extend the Darboux functions associated with a surface to the case of hypersurface in a Weyl space and discuss a few properties of these functions.     

Curve aritmeticamente Buchsbaum su superfici di grado 3 e 4 dello spazio proiettivo

Riassunto Si studiano curve aritmeticamente Buchsbaum nello spazio proiettivoP ³, tali che l’ordine minimo di una superficie che le contiene è 3 o 4. Per tali curve si determinano l’ordine, il genere aritmetico, il carattere numerico connesso, il modulo di Hartshorne-Rao e la curva legata di ordine minimo. Nel caso di curve situate su superfici cubiche lisce si determinano anche i multigradi corrispondenti.

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Summary In this paper we study arithmetically Buchsbaum curves in the projective spaceP ³, such that the minimal degree of a surface containing them is 3 or 4. For such curves we determine the degree, the aritmethic genus, the connected numerical character, the Hartshorne-Rao module, and the linked curves having minimal degree. For curves lying on smooth cubic surfaces ofP ³ we determine also the associated multidegrees.

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Lavoro eseguito sotto gli auspici del G.N.S.A.G.A. del C.N.R.     

Groupe de Chow de codimension deux des variétés définies sur un corps de nombres: un théorème de finitude pour la torsion

Summary LetX be a smooth, projective variety defined over a number field and let CH² (X) denote the Chow group of codimension two cycles modulo rational equivalence. We show that if the cohomology groupH ²(X,O_x) vanishes then the torsion subgroup of CH² (X) is a finite group. This result covers all previous results in this direction. The hypothesisH ²(X,O_x)=0 is used to lift line bundles.

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Oblatum 17-IX-1990     

Die metrische Theorie der linearen Komplexbündel von Typ 1 des einfach isotropen RaumesJ 3 (1)

According toK. Strubecker ([18]–[21]) a three dimensional real affine space with the metricds ²=dx ²+dy ² is called a simply isotropic spaceJ ₃ ⁽¹⁾ . InJ ₃ ⁽¹⁾ exist 41 types of bundles of linear line complexes. In this paper we study the metric theory of a bundle of type 1. Especially we investigate the congruence of axis, we give some isotropic and affine results and we study the complementary bundle.

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Herrn em. o. Prof. Dr. W. Wunderlich zum 75. Geburtstag gewidmet

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Im folgenden beziehen wir uns stets auf die Bezeichnungen und Resultate aus [15].     

Infinitesimal Rotary Deformations of Surfaces and Their Application to the Theory of Elastic Shells

We present a variational generalization of the problem of infinitesimal geodesic deformations of surfaces in the Euclidean space E ³. By virtue of rotary deformation, the image of every geodesic curve is an isoperimetric extremal of rotation (in the principal approximation). The results are associated in detail with rotary-conformal deformations. The application of these results to the mechanics of elastic shells is given.     

Problemi al contorno per equazioni a derivate parziali ellittiche a coefficienti costanti ed in due variabili,relativi ad un angolo

Summary We pose the problems, that are in the title, for functions u(x) from Λ=R ₊ ² \{(0, 0)} into a Banach space B: such functions are required to satisfy asintotical conditions for |x| →0 and |x| → + ∞. Existence and uniqueness theorems are proved (no.5) and the regularity of C^n,λ type up to vertex of Λ is discussed (no.6). The main treatment is the study about an integral function g joined with operators which define the differential conditions (no.3). This function arises from the transformation of those problems by using a rapresentation's formula stated in the previous paper [1].

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Entrata in Redazione il 22 dicembre 1971.

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Lavoro eseguito con finanziamento del C.N.R. nell'ambito della attività del gruppo di Ricerca di Analisi Funzionale.     

New constructions for balanced Howell rotations

Howell rotations have been used in bridge tournaments for a long time. But it was not until 1955 that Parker and Mood first gave a rigorous definition of a balanced Howell rotation and began a systematic study of its mathematical properties. Later, Berlekamp and Hwang extended this work to the study of complete balanced Howell rotations (which are special cases of balanced Howell rotations). Surprisingly, even though the concept of balanced Howell rotations precedes that of complete balanced Howell rotations, systematic construction methods have been studied only for the latter. Most of these construction methods use the properties of a Galois field GF(p^γ) where p^γ is a prime power. In this paper, we use the properties of a Galois domain GD(p^γq^s) to construct balanced Howell rotations for n partnerships where n ? 1 is the product of two prime powers satisfying certain conditions. In particular, we construct a balanced Howell rotation for 36 partnerships, this being the smallest number for which the existence of a balanced Howell rotation was not previously known. We also give two composition methods for the constructions of balanced Howell rotations.     

A note on the zero sets ofA(B) andA ∞ B) in transverse curves

Summary In this note we give two constructive proofs concerning the zero sets ofA(B) andA ^∞(B) of transverse curves contained in the unit sphere of C ⁿ .

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Riassunto Si forniscono due prove costruttive riguardanti l’insieme degli zeri diA(B) eA ^∞(B) delle curve traverse contenute nella sfera di C ⁿ .

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This work has been supported in part by the grant PB85-0374 from the C.I.C.Y.T.     

On the locus of curves with automorphisms

Sunto Si descrivono le componenti del luogo delle curve con automorfismi non banali all'interno dello spazio dei moduli delle curve algebriche di genere maggiore o uguale a tre. Si descrivono poi le componenti del luogo singolare dello spazio dei moduli delle curve di genere maggiore o uguale a due.

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Durante questa ricerca l'autore ha usufruito di un finanziamento del Ministero della Pubblica Istruzione.

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An erratum to this article is available at .     

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.