Flächenprobleme der Variationsrechnung bei fester und variabler Begrenzung und für geschlossene Flächen

Ohne Zusammenfassung Dem Andenken von Leo Lichtenstein gewidmet.     

Universal ZN-graded differential calculus

We investigate the propeties of differential algebras generated by an operator d satisfying the property d^N = 0 instead of d² = 0 as in the usual case. Several examples of realizations of such differential algebras are given, either in the context of Z_N-graded N × N matrix algebras, or as a generalized differential calculus on manifolds.     

Z 3-Graded exterior differential calculus and gauge theories of higher order

We present a possible generalization of the exterior differential calculus, based on the operator d such that d³=0, but d²0. The entities dx ⁱ and d² x ^k generate an associative algebra; we shall suppose that the products dx ⁱ dx ^k are independent of dx ^k dx ⁱ , while theternary products will satisfy the relation: dx ⁱ dx ^k dx ^m =jdx ^k dx ^m dx ⁱ =j ²dx ^m dx ^m dx ⁱ dx ^k , complemented by the relation dx ⁱ d² x ^k =jd² x ^k dx ⁱ , withj:=e^2i/3.We shall attribute grade 1 to the differentials dx ⁱ and grade 2 to the second differentials d² x ^k ; under the associative multiplication law the grades add up modulo 3.We show how the notion ofcovariant derivation can be generalized with a 1-formA so thatD:=d+A, and we give the expression in local coordinates of thecurvature 3-form defined as :=d² A+d(A ²)+AdA+A ³.Finally, the introduction of notions of a scalar product and integration of theZ ₃-graded exterior forms enables us to define the variational principle and to derive the differential equations satisfied by the 3-form . The Lagrangian obtained in this way contains the invariants of the ordinary gauge field tensorF _ik and its covariant derivativesD _i F _km .     

Synthesis and properties of macrocyclic diazene switch with binaphthalene unit attached via acrylamide linkers

2,2′-Diiodo-1,1′-binaphthalene undergoes a tandem Heck reaction with methyl acrylate to afford methyl 2-(7H-dibenzo[c,g]fluoren-7-ylidene)acetate. As a consequence, the target macrocyclic diazene with binaphthalene unit attached via acrylamide linker was prepared by the stepwise building of acrylamide at a binaphthalene moiety, including the Doebner modification of the Knoevenagel condensation, and completed by oxidative macrocyclisation of aniline end-groups. Despite being an equimolar mixture of monomer and dimer, it exhibited remarkable changes in CD spectra due to reversible (E/Z) isomerisation of N=N diazene bonds upon irradiation at 365/465 nm. Although the dimer isomerises from (E) to (Z) isomer 7.4 times faster than the monomer, the latter’s contribution to the change in ellipticity at 307 nm in the photostationary state is 2.4 times greater.     

Quantitative gas-chromatographische Bestimmung des Wirkstoffrückstandes auf Weintrauben,in Traubenmost und in Wein nach Anwendung von Ortho-Phaltan

A quantitative gas-chromatographic method for the determination of the fungicide Ortho-Phaltan (N-trichloromethylthiophthalimide) on grapes, in must and wine is described. The residues are extracted with chloroform and after purification on a silica-gel column analysed by electron-capture gas-chromatography. The method is sensitive to 0,1 ppm. The average recovery of residues was 94% on grapes, 84% in must and 85% in wine.     

Approximate radiative solutions of the Yang-Mills field equations

In this paper we look for the asymptotic radiative solutions of the Yang-Mills field equations. Considering the potential of the Yang-Mills field as a connection in a principal fibre bundle gives us a fully covariant formalism similar to the formalism of the General Relativity. Then we apply directly the results obtained by Mme Choquet-Bruhat for the gravitational field by means of the W.K.B. method. After deriving the equations for the asymptotic waves and interpreting the zero-order conditions as the initial conditions, we consider some known trivial solutions of the Yang-Mills field equations as the background field and construct the asymptotic waves explicitly. All the solutions considered turn out to be of the electromagnetic type, with some extra restrictions of the algebraic type.     

Geschlossene geodätische Linien auf einem Kreistorus

Ohne Zusammenfassung     

Geometrical construction of a generalized Skyrme Lagrangian

A geometrical scheme is proposed that leads to a straightforward generalization of a Skyrme Lagrangian, comprising higher-order terms up to the eighth power in pion fields.Boursière DGRST.