Rank-Augmented LU-Algorithm for Computing Generalized Matrix Inverses

A rank-augmnented LU-algorithm is suggested for computing a generalized inverse of a matrix. Initially suitable diagonal corrections are introduced in (the symmetrized form of) the given matrix to facilitate decomposition; a backward-correction scheme then yields a desired generalized inverse.

Construction of inverses with prescribed zero minors and applications to decentralized stabilization

We examine the following question: Suppose R is a principal ideal domain, and that F is an n × m matrix with elements in R, with n>m. When does there exist an m × n matrix G such that GF = Im, and such that certain prescribed minors of G equal zero? We show that there is a simple necessary condition for the existence of such a G, but that this condition is not sufficient in general. However, if the set of minors of G that are required to be zero has a certain pattern, then the condition is necessary as well as sufficient. We then show that the pattern mentioned above arises naturally in connection with the question of the existence of decentralized stabilizing controllers for a given plant. Hence our result allows us to derive an extremely simple proof of the fact that a necessary and sufficient condition for the existence of decentralized stabilizing controllers is the absence of unstable decentralized fixed modes, as well as to derive a very clean expression for these fixed modes. In addition to the application to decentralized stabilization, we believe that the result is of independent interest.

Conformation of the LL and LD hairpin bends with internal hydrogen bonds in proteins and peptides

The conformation of three linked peptide units having an internal 4 → 1 type of hydrogen bond has been studied in detail, and the low energy conformations are listed. These conformations all lead to the reversal of the chain direction, and may therefore be called as “hairpin bends” or “U-bends”. Since this bend can occur at the end of two chains hydrogen-bonded in the antiparallel β-conformation, it is also known as the “β-bend”. Two types of conformation are possible when the residues at the second and third Cα atoms are both of type L (the LL bend), while only one type is possible for the LD and the DL bend. The LL bend can also accommodate the sequences LG, GL, GG (G = glycine), while the LD bend can accommodate the sequences LG, GD and GG. The conformations for the sequences DD and DL are exact inverses (or mirror images) of those for the sequences LL and LD, respectively, and have dihedral angles (phi2, ψ2), (phi3, ψ3) of the same magnitudes, but of opposite signs as those for the former types, which are listed, along with the characteristics (length, angle and energy) of the hydrogen bonds. A comparison of the theoretical predictions with experimental data (from X-ray diffraction and NMR studies) on proteins and peptides, show reasonably good agreement. However, a systematic trend is observable in the experimental data, slightly deviating from theory, which indicates that some deformations occur in the shapes of the peptide units forming the bend, differing from that of the standard planar peptide unit.

Free vibration of rectangular beams of arbitrary depth

The state space approach is extended to the two dimensional elastodynamic problems. The formulation is in a form particularly amenable to consistent reduction to obtain approximate theories of any desired order. Free vibration of rectangular beams of arbitrary depth is investigated using this approach. The method does not involve the concept of the shear coefficientk. It takes into account the vertical normal stress and the transverse shear stress. The frequency values are calculated using the Timoshenko beam theory and the present analysis for different values of Poisson's ratio and they are in good agreement. Four cases of beams with different end conditions are considered.Die Zustandsraum-Technik wird auf zweidimensionale elastodynamische Probleme ausgedehnt. Die Formulierung ist besonders geeignet für die Aufstellung von Näherungstheorien beliebigen Grades. Freie Schwingungen von Rechteckbalken beliebiger Höhe wurden mit Hilfe dieser Technik untersucht. Das Verfahren umgeht den Begriff des Schubbeiwertsk. Es berücksichtigt die senkrechte Normalbeanspruchung und die Querkraft. Die Frequenzwerte werden mit Hilfe der Balkentheorie von Timoshenko und der vorliegenden Analyse berechnet, und zwar für verschiedene Werte der Querdehnzahl. Die berechneten Werte befinden sich in guter Übereinstimmung. Vier Fälle von Balken mit verschiedenen Endbedingungen werden untersucht.

On the explicit determination of the polar decomposition in n-dimensional vector spaces

A method for the explicit determination of the polar decomposition (and the related problem of finding tensor square roots) when the underlying vector space dimension n is arbitrary (but finite), is proposed. The method uses the spectral resolution, and avoids the determination of eigenvectors when the tensor is invertible. For any given dimension n, an appropriately constructed van der Monde matrix is shown to play a key role in the construction of each of the component matrices (and their inverses) in the polar decomposition.

Physical chemistry of mixed conducting zirconia ceramics

Suivant la pression partielle d'oxygène, la zircone peut être conducteur ionique ou électronique. Mise au point de méthodes de mesures de f.é.m. permettant de s'affranchir des sources d'erreur introduites par ces propriétés.