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  • 1
    Electronic Resource
    Electronic Resource
    On the total absolute curvature of closed curves in spheres (1986)
    Springer
    Manuscripta mathematica 57 (1986), S. 101-108 
    Details
    ISSN: 1432-1785
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract The total absolute curvature of a closed curve in a Euclidean space is always greater or equal to 2 (Fenchel's inequality,1929, [3], [1], [13]); especially for a knotted curve it is always greater than 4 (Fary-Milnor's inequality, 1949, [2], [7], [5], [4]). For the total absolute curvature of closed curves in spheres no such lower bounds exist because there are closed geodesies. Here we derive similar bounds which depend on the length of the curve resp.the area of surfaces of disk-type bounded by the curve. In order to prove these inequalities we start from the computation of the total absolute curvature as mean value of the number of critical points of certain level functions ([11],[12]); we use some topological considerations and Poincaré's integralgeometric formula for the computation of length resp. area.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01172493
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  • 2
    Electronic Resource
    Electronic Resource
    Eine differentialtopologische Berechnung der Totalen Krümmung und Totalen Absolutkrümmung in der sphärischen Differentialgeometrie (1980)
    Springer
    Manuscripta mathematica 31 (1980), S. 119-147 
    Details
    ISSN: 1432-1785
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract The aim of this article is to compute the total (absolute) curvature, i.e. the mean value of the (absolute) Lipschitz-Killing-curvature, of an immersion f: M→Sn of a compact manifold into the unit sphere in a differential topological manner. Through a generalization of KUIPERs treatment of immersions in Euclidean spaces it can be computed as the mean value of the number of critical points—weighted by (−1)k (k=Index) resp.not weighted—of certain functions. These functions are the pullback via f of “level-functions”, which are defined almost everywhere on Sn. Such a “level-function” is constructed by taking any oriented great circle as a “leveling-scale” and the orthogonal great (n−1)-spheres as “level-surfaces”.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01303271
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  • 3
    Electronic Resource
    Electronic Resource
    Anwendungen der differentialtopologischen Berechnung der Totalen Krümmung und Totalen Absolutkrümmung in der sphärischen Differentialgeometrie (1980)
    Springer
    Manuscripta mathematica 32 (1980), S. 239-262 
    Details
    ISSN: 1432-1785
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract In a previous article [16] we have shown how the total absolute (Lipschitz-Killing) curvature of the immersion f:M→Sn of a compact manifold into a sphere can be computed in a differential topological manner as the mean-value of the number of critical points of certain level-functions. (And similar the total curvature.) (comp. [12], prob. 15) Now we consider the gradient vector field of the level-functions and achieve a relation between the total curvature and the Euler characteristic, of the manifold, which can be sharpened in some cases to inequalities. Moreover it leads to the formula of Allendoerfer-Weil for compact n-dim. submanifolds of the sphere Sn.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01299604
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  • 4
    Electronic Resource
    Electronic Resource
    Differential topology and the computation of total absolute curvature (1982)
    Springer
    Mathematische Annalen 258 (1982), S. 471-480 
    Details
    ISSN: 1432-1807
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01453980
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  • 5
    Electronic Resource
    Electronic Resource
    A generalization of the isoperimetric inequality in the hyperbolic plane (1991)
    Springer
    Archiv der Mathematik 57 (1991), S. 508-513 
    Details
    ISSN: 1420-8938
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01246751
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  • 6
    Electronic Resource
    Electronic Resource
    Kinematische Berührung im Äquiaffinen (1990)
    Springer
    Geometriae dedicata 33 (1990), S. 317-323 
    Details
    ISSN: 1572-9168
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract Given two curves in the real affine plane, one is fixed and the other undergoes volume-preserving affinities. Through transversal affinities we define a contact measure on the subset consisting of those affinities, which cause third-order contact between the fixed and the transformed curve. A kinematic formula expresses this contact measure in terms of affine lengths and affine curvatures of the given curves. In a similar way, parallel supporting planes of closed convex surfaces in affine space are treated.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00181336
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  • 7
    Electronic Resource
    Electronic Resource
    Cyclovinyloge Additionen von 1,3-Diphenylbenzo[c]furan an 1,3,5-Cycloheptatrien (1983)
    Weinheim : Wiley-Blackwell
    Berichte der deutschen chemischen Gesellschaft 116 (1983), S. 618-629 
    Details
    ISSN: 0009-2940
    Keywords: Chemistry ; Inorganic Chemistry
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Chemistry and Pharmacology
    Description / Table of Contents: Cyclovinylogous Additions of 1,3-Diphenylbenzo[c]furan to 1,3,5-Cycloheptatriene1,3-Diphenylbenzo[c]furan (1) reacts thermally with 3,4-dimethylenecyclobutene (2) and 1,3,5-cycloheptatriene (5) to give exo- and endo-Diels-Alder adducts. The reaction of 1 with potassium cycloheptatrienide leads to substitutive adducts (13, 14) at low temperature, but to [4 + 2]-ring closure (15) at higher temperatures, via the anionic intermediate 12. The photoaddition of 1 to 5 gives the exo- and endo-[4 + 2]-, endo-[4 + 6]-, exo-[4 + 4]- und cis-1,4(1,3)-substitutive adducts 7, 6, 16, 17 und 14, in divergence to recent literature data. The structures of the products are elucidated by 1H NMR spectra and in part by chemical transformations, their formation can be interpreted in terms of the approved diradical mechanism. The endo-/exo-isomerization of 16 into 19 proceeds solvolytically. The products 7 and 16 are formed out of 17 by suprafacial thermal 1,3-alkyl shifts. Further exo- und endo-[4 + 2]- as well as [4 + 4]-additions of 1 are achieved with dimethyl maleate (23, 24) and anthracene (26), resp.
    Notes: 1,3-Diphenylbenzo[c]furan (1) reagiert thermisch mit 3,4-Dimethylencyclobuten (2) und 1,3,5-Cycloheptatrien (5) zu exo- und endo-Diels-Alder-Addukten. Die Reaktion von 1 mit Kaliumcycloheptatrienid führt über das Anionzwischenprodukt 12 bei tiefer Temperatur zu Substitutionsaddukten (13, 14), bei höherer Temperatur dagegen zum [4 + 2]-Ringschluß (15). Die Photoaddition von 1 an 5 führt abweichend von Literaturangaben zu den exo- und endo-[4 + 2]-, endo-[4 + 6]-, exo-[4 + 4]- und cis-1,4(1,3)-Substitutions-Addukten 7, 6, 16, 17 und 14, deren Konstitution 1H-NMR-spektroskopisch sowie z. T. durch chemische Umwandlungen belegt wird und deren Bildung auf der Grundlage des bewährten Diradikalmechanismus gedeutet werden kann. Die endo-/exo-Isomerisierung von 16 zu 19 gelingt solvolytisch. Durch suprafaciale thermische 1,3-Alkylverschiebungen entstehen aus 17 die Produkte 7 und 16. Weitere exo- und endo-[4 + 2]- sowie [4 + 4]-Additionen von 1 werden mit Maleinsäure-dimethylester (23, 24) und Anthracen (26) erreicht.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/cber.19831160221
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  • 8
    Electronic Resource
    Electronic Resource
    Photoadditionen von Anthracenen an 2,4-Hexadien und 1,3,5-Cycloheptatrien (1983)
    Weinheim : Wiley-Blackwell
    Berichte der deutschen chemischen Gesellschaft 116 (1983), S. 630-644 
    Details
    ISSN: 0009-2940
    Keywords: Chemistry ; Inorganic Chemistry
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Chemistry and Pharmacology
    Description / Table of Contents: Photoadditions of Anthracenes to 2,4-Hexadiene and 1,3,5-CycloheptatrieneElectronically excited 9-anthracenecarbonitrile (1*), in contrast to theoretical predictions, is trapped by 2,4-hexadiene to give the [4 + 4]-adduct 3 with a trans-double bond, which forms the [4 + 2]-adduct 4 above 0°C via suprafacial 1,3-shift. Also 1,3,5-cycloheptatriene (6) quenches 1* with the result of a thermolabile [4 + 4]-addukt 7 (→ 8). In addition to 8, further adducts (9, 10, 11, 12) are formed via isomeric diradicals. Anthracene (19*) and 6 generate the [4 + 6]-, [4 + 4]-, [4 + 2]-, substitutive, and peripheral adducts 20, 21, 22, 23, 24, and 24. 9-Phenylanthracene (25*) and 6 form the corresponding, partly isomeric, adducts (26; 29; 27, 28; 30, 31; 32). The reactions are interpreted in terms of diradical mechanisms. Numerous literature reports, which havealready been classified theoretically, are amended or corrected, the reasons for erroneous theoretical predictions are discussed. The diversity of products can be understood in detail by empirical extrapolations on the basis of the experimentally secured cyclovinylogy principle.
    Notes: Elektronisch angeregtes 9-Anthracencarbonitril (1*) wird von 2,4-Hexadien (2) entgegen theoretischen Erwartungen zum [4 + 4]-Addukt 3 mit trans-Doppelbindung abgefangen, aus dem oberhalb 0°C durch suprafaciale 1,3-Verschiebung das [4 + 2]-Addukt 4 entsteht. Auch 1,3,5-Cycloheptatrien (6) reagiert mit 1* zu einem thermolabilen [4 + 4]-Addukt 7 (→ 8). Daneben entstehen weitere Addukte (9, 10, 11, 12) über isomere Diradikale. Anthracen (19*) bildet mit 6 die [4 + 6]-, [4 + 4]-, [4 + 2]-, Substitutions- und Peripher-Addukte 20, 21, 22, 23, 24. Mit 9-Phenylanthracen (25*) und 6 gibt es die entsprechenden, z. T. stellungsisomeren, Addukte (26; 29; 27, 28; 30, 31; 32). Die Reaktionen werden an Hand von Diradikalmechanismen gedeutet. Zahlreiche bereits theoretisch klassifizierte Literaturangaben werden ergänzt oder berichtigt, die Gründe für die fehlerhaften theoretischen Vorhersagen diskutiert. Die Produktevielfalt kann durch empirische Extrapolation im Rahmen des experimentell gesicherten Cyclovinylogieprinzips detailliert verstanden werden.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/cber.19831160222
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  • 9
    Electronic Resource
    Electronic Resource
    Dien-katalysierte Photo-Dimerisierung über Excimere (Exciplexe) oder Dienaddition an Anthracen und 9-Phenylanthracen? (1980)
    Weinheim : Wiley-Blackwell
    Berichte der deutschen chemischen Gesellschaft 113 (1980), S. 3669-3674 
    Details
    ISSN: 0009-2940
    Keywords: Chemistry ; Inorganic Chemistry
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Chemistry and Pharmacology
    Notes: Diene-Catalyzed Photodimerization via Excimers (Exciplexes) or Diene Addition to Anthracene and 9-PhenylanthraceneThe [4 + 4]-photodimerization of anthracene (1) is not enhanced by 2,4-hexadiene (3) but impeded due to formation of [4 + 2]- (4) and [4 + 4]-adducts (5, 6). 6 and 1 combine to give the bisadduct 7 at room temperature. The selective photolysis of 9-phenylanthracene (8) leads to the anti-[4 + 4]-dimer (9). This can be compared with the [4 + 4]-photodimerization of 9-methylanthracene which affords the anti- and the thermolabile syn-product. 1,3-Dienes such as 1,3-pentadiene (10) or 1,3-cyclohexadiene (13) are not necessary for the formation of 9 but undergo cycloadditions with 8 to yield [4 + 2]- (11, 12, 14), [4 + 4]- (15), and [2 + 2]-adducts (16, 17). The structures of the products are established primarily by high field 1H-NMR spectroscopy.
    Additional Material: 1 Tab.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/cber.19801131126
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  • 10
    Electronic Resource
    Electronic Resource
    1,4-Addition von Kalium-cycloheptatrienid an (9-Phenyl-)Anthracen (1982)
    Weinheim : Wiley-Blackwell
    Berichte der deutschen chemischen Gesellschaft 115 (1982), S. 3208-3211 
    Details
    ISSN: 0009-2940
    Keywords: Chemistry ; Inorganic Chemistry
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Chemistry and Pharmacology
    Notes: 1,4-Addition of Potassium Cycloheptatrienide to (9-Phenyl-)AnthraceneIn liquid ammonia the cycloheptatriene anion adds to anthracene (3) and 9-phenylanthracene (9) to give the anions 4 and 10, resp., which are quenched at -35°C to give the substitutive 1,4-adducts 5 and 11 + 12, resp. Upon heating up to 20°C the intermediate anion 4 cyclizes to the Diels-Alder product 6, which is quenched to give the asymmetric and symmetric [4 + 2]-adducts 7 and 8. The structures of the products have been established primarily by high-field 1H-NMR spectroscopy.
    Additional Material: 1 Tab.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/cber.19821150924
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