Least Squares Properties of Generalized Inverses

The aim of this paper is to systematize solutions of some systems of linear equations in terms of generalized inverses.As a significant application of the Moore-Penrose inverse,the best approximation solution to linear matrix equations (i.e.both least squares and the minimal norm) is considered.Also,characterizations of least squares solution and solution of minimum norm are given.Basic properties of the Drazin-inverse solution and the outer-inverse so-lution are present.Motivated by recent research,important least square prop-erties of composite outer inverses are collected.     

Synthesis,Structure and Characterization of Dinuclear Pentacoordinate Molybdenum(V) Complexes with Thiosemicarbazone Ligands

New dinuclear pentacoordinate molybdenum(V) complexes, [Mo₂^VO₃L₂] [L = thiosemicarbazonato ligand: C₆H₄(O)CH:NN:C(S)NHR′ and C₁₀H₆(O)CH:NN:C(S)NHR′; R′ = H, CH₃, C₆H₅) were obtained either by oxygen atom abstraction from Mo^VIO₂L with triphenylphosphine or by using [Mo₂O₃(acac)₄] in the reaction with the corresponding ligands H₂L. Crystal and molecular structure of [Mo₂O₃{C₆H₄(O)CH:NN:C(S)NHC₆H₅}₂] · CH₃CN has been determined by the single‐crystal X‐ray diffraction method.     

Catalan matrix and related combinatorial identities

We introduce the notion of the Catalan matrix whose non-zero elements are expressions which contain the Catalan numbers arranged into a lower triangular Toeplitz matrix. Inverse of the Catalan matrix is derived. Correlations between the matrix and the generalized Pascal matrix are considered. Some combinatorial identities involving Catalan numbers, binomial coefficients and the generalized hypergeometric function are derived using these correlations. Moreover, an additional explicit representation of the Catalan number, as well as an explicit representation of the sum of the first m Catalan numbers are given.     

Extensions of Faddeev’s algorithms to polynomial matrices

Starting from algorithms introduced in [Ky M. Vu, An extension of the Faddeev’s algorithms, in: Proceedings of the IEEE Multi-conference on Systems and Control on September 3-5th, 2008, San Antonio, TX] which are applicable to one-variable regular polynomial matrices, we introduce two dual extensions of the Faddeev’s algorithm to one-variable rectangular or singular matrices. Corresponding algorithms for symbolic computing the Drazin and the Moore-Penrose inverse are introduced. These algorithms are alternative with respect to previous representations of the Moore-Penrose and the Drazin inverse of one-variable polynomial matrices based on the Leverrier-Faddeev’s algorithm. Complexity analysis is performed. Algorithms are implemented in the symbolic computational package MATHEMATICA and illustrative test examples are presented.     

Unstable Manifolds of Relative Periodic Orbits in the Symmetry-Reduced State Space of the Kuramoto–Sivashinsky System

Systems such as fluid flows in channels and pipes or the complex Ginzburg–Landau system, defined over periodic domains, exhibit both continuous symmetries, translational and rotational, as well as discrete symmetries under spatial reflections or complex conjugation. The simplest, and very common symmetry of this type is the equivariance of the defining equations under the orthogonal group O(2). We formulate a novel symmetry reduction scheme for such systems by combining the method of slices with invariant polynomial methods, and show how it works by applying it to the Kuramoto–Sivashinsky system in one spatial dimension. As an example, we track a relative periodic orbit through a sequence of bifurcations to the onset of chaos. Within the symmetry-reduced state space we are able to compute and visualize the unstable manifolds of relative periodic orbits, their torus bifurcations, a transition to chaos via torus breakdown, and heteroclinic connections between various relative periodic orbits. It would be very hard to carry through such analysis in the full state space, without a symmetry reduction such as the one we present here.     

Theories with constants and three countable models

We prove that a countable, complete, first-order theory with infinite dcl() and precisely three non-isomorphic countable models interprets a variant of Ehrenfeucht’s or Peretyatkin’s example.     

Stochastic stability of a rotating shaft

The stochastic stability problem of an elastic, balanced rotating shaft subjected to action of axial forces at the ends is studied. The shaft is of circular cross-section, it rotates at a constant rate about its longitudinal axis of symmetry. The effect of rotatory inertia of the shaft cross-section is included in the present formulation. Each force consists of a constant part and a time-dependent stochastic function. Closed form analytical solutions are obtained for simply supported boundary conditions. By using the direct Liapunov method almost sure asymptotic stability conditions are obtained as the function of stochastic process variance, damping coefficient, damping ratio, angular velocity, mode number and geometric and physical parameters of the shaft. Numerical calculations are performed for the Gaussian process with a zero mean and as well as an harmonic process with random phase.     

Conformational analysis of oleandomycin and its 8-methylene-9-oxime derivative by NMR and molecular modelling

Conformations of the 14-membered macrolide antibiotic oleandomycin and its 8-methylene-9-oxime derivative were determined in various solvents. The experimental NMR data--coupling constants and NOE contacts--were compared with the results of molecular modelling--molecular mechanics calculations and molecular dynamics simulations. The conformational changes, on the right-hand side of the 14-membered ring, affected mostly the 3JH2,H3 values and NOE crosspeaks H3 or H4 to H11. Oleandomycin was found to be present predominantly in the C3-C5 folded-in conformations in DMSO-d6 solution, whereas in buffered D2O, acetone-d6 and CDCl3, there was a mixture of folded-in and folded-out conformational families. The predominant conformation of the 8-methylene-oleandomycin-9-oxime derivative in solution was a folded-out one although different amounts of folded-in conformation were also present depending on the solvent. Oleandrose and desosamine sugar moieties adopted the usual and expected chair conformation. The conformation around the glycosidic bonds, governing the relative orientation of sugars vs. the lactone ring, showed a certain flexibility within two conformationally close families. We believe that by combining the experimental NMR data and the molecular modelling techniques, as reported in this paper, we have made significant progress in understanding the conformational behaviour and properties of macrolides. Our belief is based on our own current studies on oleandomycins as well as on the previously reported results and best practices concerning other macrolides. A rational for macrolide conformational studies and advances in methodology has been suggested accordingly.     

Potentiometric and Multinuclear Magnetic Resonance Study of the Solution Equilibria Between Aluminium(III) Ion and L-Aspartic Acid

Solution equilibria between aluminium(III) ion and L-aspartic acid were studied by potentiometric, ²⁷Al, ¹³C, and ¹H NMR measurements. Glass electrode equilibrium potentiometric studies were performed on solutions with ligand to metal concentration ratios 1:1, 3:1, and 5:1 with the total metal concentration ranging from 0.5 to 5.0 mmol/dm³ in 0.1 mol/dm³ LiCl ionic medium, at 298 K. The pH of the solutions was varied from ca. 2.0 to 5.0. The non-linear least squares treatment of the data performed with the aid of the Hyperquad program, indicated the formation of the following complexes with the respective stability constants log β_p,q,r given in parenthesis (p, q, r are stoichiometric indices for metal, ligand, and proton, respectively): Al(HAsp)²⁺ (log β_1,1,1 = 11.90 ± 0.02); Al(Asp)⁺ (log β_1,1,0 = 7.90 ± 0.03); Al(OH)Asp⁰ (log β_1,1,−1 = 3.32 ± 0.04); Al(OH)₂Asp⁻ (log β_1,1−2 = −1.74 ± 0.08), and Al₂(OH) Asp³⁺ (log β_2,1,−1 = 6.30 ± 0.04). ²⁷Al NMR spectra of Al³⁺ + aspartic acid solutions (pH 3.85) indicate that sharp symmetric resonance at δ∼10 ppm can be assigned to (1, 1, 0) complex. This resonance increases in intensity and slightly broadens upon further increasing the pH. In Al(Asp)⁺ complex the aspartate is bound tridentately to aluminum. The ¹H and ¹³C NMR spectra of aluminium + aspartic acid solutions at pH 2.5 and 3.0 indicate that β-methylene group undergoes the most pronounced changes upon coordination of aluminum as well as α-carboxylate group in ¹³C NMR spectrum. Thus, in Al(HAsp)²⁺ which is the main complex in this pH interval the aspartic acid acts as a bidentate ligand with –COO⁻ and –NH₂ donors closing a five-membered ring.     

Study of Equilibria in the Aluminium(III)-Glycine and -Alanine Systems

The formation of various hydrolytic and mixed hydrolytic complexes of the aluminium(III) ion in the presence of glycine and L-alanine, has been studied in 0.5 mol dm^?3 (Na)NO₃ medium at 25deg;C, by emf method. The concentration ratios of amine acids to aluminium(III) were varied from 1 : 1 to 10 : 1. The least-squares treatment of the data obtained, in the absence of the amino acids, indicates the formation of the dimer, [Al₂(OH)₂]⁴⁺, and monomer, [AlOH]²⁺, with the stability constants log β₂₂ = ?7.03 ± 0.03 and log β₁₁ = ?5.65 ± 0.09, respectively. At pH values higher then ～4.0 formation of the trimer [Al₃(OH)₄]⁵⁺ (log β₃₄ = ?12.60 ± 0.08) becomes significant. In the presence of amino acids the evidence has been found for the formation of [Al₂(OH)₄]²⁺ (log β₂₄ = ?15.65 ± 0.09). Besides the formation of the pure hydrolytic complexes, equilibria in the title systems can be explained by assuming the main reaction products to have the compositions [Al(OH)₃Gly] (log β₁₃₁ = ?7.53 ± 0.04), [Al₂(OH)₂(Gly)₂] (log β₂₂₂ = 6.56 ± 0.09) and [Al(OH)₃Ala] (log β₁₃₁ = ?7.70 ± 0.03), [Al₂(OH)₂Ala₂] (log β₂₂₂ = 7.23 ± 0.07).