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∗-Algebras in Several Complex Variables
Published in Kehe Zhu, Handbook of Analytic Operator Theory, 2019
Operators on Hilbert spaces of holomorphic functions are also of interest when a domain in Cn is replaced by a complex subvariety. This is related to the so-called Hilbert module program initiated by R. Douglas. For a polynomial ideal ℐ consider the algebraic variety I°:={z∈Cn:f(z)=0∀f∈I}.
The index theorem
Published in Peter B. Gilkey, Invariance Theory, the Heat Equation, and the Atiyah-Singer Index Theorem, 2018
Remark: We say M is Kaehler if it is a holomorphic manifold and if it admits a Kaehler metric. The Fubini-Study metric on CPk discussed in §2.3 is Kaehler. Any submanifold of a Kaehler manifold is again Kaehler so any smooth algebraic variety is Kaehler. On the other hand, there are Kaehler manifolds which are not smooth algebraic varieties; the Riemann period relations are obstructions to tori being algebraic varieties for example. There are holomorphic manifolds which do not admit Kaehler structures; S1 × S2k–1 being examples of such for k ≥ 2.
Nanomagnetic NH2·MIL-101(Fe)/ED as a new highly efficient catalyst for the synthesis of thiopyran and oxospiro-indolinethiopyran derivatives
Published in Journal of Sulfur Chemistry, 2023
F. Matloubi Moghaddam, Bagher Aghamiri, Aida Yazdani Motlagh, Atefeh Jarahiyan
As shown in Table 2, not using a catalyst resulted in a low yield of the final product (entries 1-5). Also, the use of catalyst components separately provided a moderate yield of the desired product (entries 13-18). In contrast, it is noteworthy that a considerable amount of 5a was formed in the presence of magnetic NH2.MIL-101(Fe)/ED after 30 min in a smooth, meaningful manner (entry 11). Hence, magnetic NH2.MIL-101(Fe)/ED was chosen as the optimized catalyst for carrying out this reaction. As another attempt for increasing the yield of the final product, we changed the reaction temperature from ambient to reflux condition. It was observed that increasing the reaction temperature has no beneficial effect on reaction time and the final product yield (entries 8 and 12). In the next step, different solvents were screened. According to the achieved results, DCM was selected as an efficient solvent in this synthetic procedure. All in all, the best results were obtained with magnetic NH2.MIL-101(Fe)/ED as the catalyst and DCM as solvent at 25 °C and 30 min of stirring. Encouraged by these promising results, the substrate scope, generality, and versatility of the optimized conditions were extended to other series of substituted arylaldehyde derivatives (Table 3). The reaction proceeded well with a variety of substituents on the aromatic ring. To our delight, the desired heterocyclic products were obtained in high yield.
Atom-canonicity in varieties of cylindric algebras with applications to omitting types in multi-modal logic
Published in Journal of Applied Non-Classical Logics, 2020
Let . For a variety between and , a set algebra is a subalgebra of an algebra, having the same signature as , of the form , say, where is identical to the interpretation of in the class of quasipolyadic equality set algebras. Let be an algebra having the same signature of ; then is a representable algebra, or simply representable ⇔ is isomorphic to a subdirect product of set algebras. We write for the class of representable algebras
Monotone nonlinear realisations of response maps
Published in International Journal of Control, 2019
In general the map ξ* is not an isomorphism of the observation universes and . This may be interpreted as that the realisation is not minimal (in some sense) and one may look for another realisation for which this map is an isomorphism. In Bartosiewicz (1988), such a realisation was constructed for polynomial systems (the canonical realisation on some algebraic variety), so the results of this section may be the starting point for a similar research for systems of other classes, in particular analytic or smooth.