Inversion of circulant matrices over

In this paper we consider the problem of inverting an circulant matrix with entries over . We show that the algorithm for inverting circulants, based on the reduction to diagonal form by means of FFT, has some drawbacks when working over . We present three different algorithms which do not use this approach. Our algorithms require different degrees of knowledge of and , and their costs range, roughly, from to operations over . Moreover, for each algorithm we give the cost in terms of bit operations. We also present an algorithm for the inversion of finitely generated bi-infinite Toeplitz matrices. The problems considered in this paper have applications to the theory of linear cellular automata.

     

Synthesis of new enantiomerically enriched beta-Hydroxy-gamma-amino phosphines by selective transformation of naturally occurring amino acids

Ring opening of amino epoxides derived from naturally occurring amino acids with lithium diphenylphosphido borane is reported as an efficient approach to a new family of enantiomerically enriched multifunctional phosphines.     

Formation of a triply bridged mu-oxo diiron(III) core stabilized by two deprotonated carboxamide groups upon photorelease of NO from a [Fe-NO]6 iron nitrosyl

The iron nitrosyl [(PaPy2Q)Fe(NO)](ClO4)2 (2), derived from the quinoline-based ligand PaPy2QH (N,N-bis(2-pyridylmethyl)amine-N-ethyl-2-quinoline-2-carboxamide, where H is dissociable proton) has been characterized by spectroscopy and X-ray diffraction techniques. The 1H NMR spectrum (S = 0 ground state) and v(NO) value of 1885 cm(-1) indicate that 2 is a [Fe-NO]6 nitrosyl. Although 2 is stable in the dark, exposure of an acetonitrile solution of 2 (lambdamax = 510 nm) to light in the visible range causes rapid release of NO and formation of the solvato species [(PaPy2Q)Fe(MeCN)](ClO4)2 (6). Quantum yield (Phi) measurements indicate that 2 is a more efficient NO donor (Phi = 0.258) than [(PaPy3)Fe(NO)](ClO4)2 (1, Phi = 0.185), a complex derived from a similar but pyridine-based ligand. Interestingly, when the photoproduct 6 is exposed to water or a small amount of base, the triply bridged diiron(III) species [(PaPy2Q)FeOFe(PaPy2Q)](ClO4)2 (3) forms in good yield. This species can be independently synthesized from aerobic oxidation of the Fe(II) species [(PaPy2Q)Fe(MeCN)](ClO4) in acetonitrile. The structure of 3 reveals a unique Fe(III)-O-Fe(III) link supported by two (eta2,mu2)mu-NCO bridges derived from the carboxamido groups of the two (PaPy2Q)Fe(III) moieties.     

Block tridiagonal reduction of perturbed normal and rank structured matrices

It is well known that if a matrix A∈C^n×n$A\in {\mathbb{C}}^{n\times n}$ solves the matrix equation f(A,A^H)=0$f(A,A^H)=0$, where f(x,y)$f(x,y)$ is a linear bivariate polynomial, then A is normal; A   and A^H$A^H$ can be simultaneously reduced in a finite number of operations to tridiagonal form by a unitary congruence and, moreover, the spectrum of A is located on a straight line in the complex plane. In this paper we present some generalizations of these properties for almost normal matrices which satisfy certain quadratic matrix equations arising in the study of structured eigenvalue problems for perturbed Hermitian and unitary matrices.     

Synthesis of new polysubstituted piperazines and dihydro-2H-pyrazines by selective reduction of 2-oxo-piperazines

New enantiomerically enriched 1,4,5-piperazines and 1,4,5-dihydro-2H-pyrazines have been prepared by reduction of the corresponding 2-oxo-piperazines. Selective reduction can be achieved by careful control of the reaction conditions using LiAlH₄. Notably the two nitrogen atoms of the final compounds are orthogonally protected.     

Photokinetic methods: A mathematical analysis of the rate equations in photochromic systems

In this article the analytical integration of kinetic equations describing the dynamic behavior of reversible photoreactions has been undertaken. Photochemical systems giving rise to competing thermal and photochemical steps are kinetically analyzed; the rate law is set up and analytical solutions are obtained under precisely defined boundary conditions. More complicated kinetic systems, where several species interact both thermally and photochemically, are also investigated. The kinetic treatment leads to a system of coupled differential equations which are amenable to analytical solutions, under the appropriate experimental and boundary conditions. The usefulness of the equations developed is illustrated by their application to some spirooxazine and chromene photochromic systems: two examples are described in detail. © 1999 John Wiley & Sons, Inc., Int J Chem Kinet 31: 303–313, 1999     

Small ring constrained peptidomimetics. Synthesis of epoxy peptidomimetics, inhibitors of cysteine proteases.

Different dipeptide analogues containing an oxirane ring in the place of the peptidic bond were prepared starting from naturally occurring amino acids. N-Fmoc-amino aldehydes were transformed into the corresponding methoxyvinyl derivatives through a Wittig reaction, and the addition of PhSeCl gave a series of different alpha-phenylselenyl aldehydes. Mukajiama reaction with silylketene acetals gave an intermediate product that was finally transformed into the desired oxiranyl peptidomimetics. Following this strategy we were able to control three new contiguous stereocenters starting from the enantiomerically pure amino acid. The dipeptide analogues could be used in SPPS on a SASRIN resin as the final epoxides were relatively unstable under acidic conditions. Moreover the synthesis of the single dipeptide mimetics was carried out on solid phase to generate a small library of epoxy peptidomimetics. Some of the products prepared in this work resulted as time-dependent reversible inhibitors of cysteine protease.