Numerical methods for finding multiple eigenvalues of matrices depending on parameters

Summary. Let be a square matrix dependent on parameters and , of which we choose as the eigenvalue parameter. Many computational problems are equivalent to finding a point such that has a multiple eigenvalue at . An incomplete decomposition of a matrix dependent on several parameters is proposed. Based on the developed theory two new algorithms are presented for computing multiple eigenvalues of with geometric multiplicity . A third algorithm is designed for the computation of multiple eigenvalues with geometric multiplicity but which also appears to have local quadratic convergence to semi-simple eigenvalues. Convergence analyses of these methods are given. Several numerical examples are presented which illustrate the behaviour and applications of our methods. Received December 19, 1994 / Revised version received January 18, 1996     

Nitrostyrene Derivatives of Adenosine 5′-Glutarates Modified with an Alkyl Spacer and their inhibitory activity on epidermal growth factor receptor protein tyrosine kinase

β-Nitrostyrene derivatives of adenosine 5′-glutarates are potent and selective bisubstrate-type inhibitors of the epidermal growth factor receptor protein tyrosine kinase (EGF-R PTK). In an attempt to improve the inhibitory activity, this type of compounds was modified with alkyl spacers of varying length between the nitrostyrene and the glutaryl units. The spacers consisted of 1, 3, 4, and 5 atoms to give compounds of the benzyl, oxyethyl, oxypropyl, and oxybutyl series, respectively (Schemes 1 and 2). Adenosine 5′-esters were prepared in the benzyl and oxypropyl series only. Compared to the compounds in the parent series without spacer (IC₅₀ = 0.7–12 μM ), most of the modified compounds inhibited the EGF-R PTK only marginally or were inactive (IC₅₀ ≥ 100 μM ). The only exceptions were the free acids 19 and 20 with IC₅₀ values of ca. 5 μM . It is noteworthy that esterification of these two hydrogen glutarates with either MeOH or adenosine yielded inactive compounds, which is in contrast to the corresponding substances without spacers.     

Derivatives of (2R)-N-Glyoxylbornane-10,2-sultam and Their Use as auxiliaries in the diastereoselective spirocyclization of 2-substituted tryptamines

A crude hydrate 6 and a crystalline hemiacetal 7 of glyoxylamide 4 were prepared from crotonamide 5 (Scheme 2). Particularly hemiacetal 7 , but also 6 and the ‘dimer’ 8 (obtained from 7 ) may serve as homochiral auxiliaries. The structure of 8 was determined by X-ray analysis. By arenesulfonyl halides, tryptimines 12–14 of 4 were diastereoselectively transformed into spirotricycles 15–17 and 19 .     

Synthesis of Benzazepine Analogues of Noscapine

The synthesis of benzazepine analogues of the opium alkaloid noscapine ( 1 ) is described. The benzazepines 2 and 3 were prepared starting from nornarceine ethyl ester ( 4 ; readily available from 1 ) in several steps. X-Ray analysis of compound 2 revealed that it is not a diastereosisomer mixture but a racemate of the threo-form and thus has the same configuration as 1 .     

An approach to triquinane synthesis using the Pauson-Khand reaction

Tricyclic skeletons have been generated from acyclic enyne precursors by using an intramolecular Pauson-Khand reaction in combination with aldol, Michael and alkylation reactions.     

Cleaving functors and controlled wild algebras

We show that the cleaving functors introduced in [Bautista et al., Invent. Math. 81 (1985) 217] as a tool for proving infinite representation type of finite-dimensional algebras can also be used to establish controlled wildness. The main application is that an algebra is controlled wild if there is an indecomposable projective module with a Loewy factor having a homogeneous direct summand which is of length at least 3. As a second application we derive Han's covering criterion.