A finiteness result for Heegaard splittings

In this paper we show that for a given 3-manifold and a given Heegaard splitting there are finitely many preferred decomposing systems of 3g−3 disjoint essential disks. These are characterized by a combinatorial criterion which is a slight strengthening of Casson-Gordon's rectangle condition. This is in contrast to fact that in general there can exist infinitely many such systems of disks which satisfy just the Casson-Gordon rectangle condition.     

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.     

Universal WCG banach spaces and universal Eberlein Compacts

We prove that for cardinalsτ satisfying τ^ω=τ and forτ=ω ₁, there do not exist universal Eberlein Compacts of weightτ, or universal WCG spaces of density characterτ. Ifτ is a strong limit cardinal of countable cofinality such universal spaces do exist. Thus under GCH universal spaces exist forτ iff cof(τ)=ω. The research of the second author was supported by a grant from the United States-Israel Binational Science Foundation and the Fund for the Promotion of Research at the Technion.     

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.     

Finite quotients of the multiplicative group of a finite dimensional division algebra are solvable

We prove that finite quotients of the multiplicative group of a finite dimensional division algebra are solvable. Let be a finite dimensional division algebra having center , and let be a normal subgroup of finite index. Suppose is not solvable. Then we may assume that is a minimal nonsolvable group (MNS group for short), i.e. a nonsolvable group all of whose proper quotients are solvable. Our proof now has two main ingredients. One ingredient is to show that the commuting graph of a finite MNS group satisfies a certain property which we denote Property . This property includes the requirement that the diameter of the commuting graph should be , but is, in fact, stronger. Another ingredient is to show that if the commuting graph of has Property , then is open with respect to a nontrivial height one valuation of (assuming without loss of generality, as we may, that is finitely generated). After establishing the openness of (when is an MNS group) we apply the Nonexistence Theorem whose proof uses induction on the transcendence degree of over its prime subfield to eliminate as a possible quotient of , thereby obtaining a contradiction and proving our main result.