Various solution methods,accompanied by dynamic investigation,for the same problem as a means for enriching the mathematical toolbox

A geometrical task is presented with multiple solutions using different methods, in order to show the connection between various branches of mathematics and to highlight the importance of providing the students with an extensive ‘mathematical toolbox’. Investigation of the property that appears in the task was carried out using a computerized tool.     

Percentile residual life orders

In this paper we study a family of stochastic orders of random variables defined via the comparison of their percentile residual life functions. Some interpretations of these stochastic orders are given, and various properties of them are derived. The relationships to other stochastic orders are also studied. Finally, some applications in reliability theory and finance are described. Copyright © 2010 John Wiley & Sons, Ltd.     

Preconditioning spectral element schemes for definite and indefinite problems

Spectral element schemes for the solution of elliptic boundary value problems are considered. Preconditioning methods based on finite difference and finite element schemes are implemented. Numerical experiments show that inverting the preconditioner by a single multigrid iteration is most efficient and that the finite difference preconditioner is superior to the finite element one for both definite and indefinite problems. A multigrid preconditioner is also derived from the finite difference preconditioner and is found suitable for the CGS acceleration method. It is pointed out that, for the finite difference and finite element preconditioners, CGS does not always converge to the accurate algebraic solution. © 1999 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 15: 535–543, 1999     

Diastereoisomerically selective enantiomerically pure titanium complexes of Salan ligands: synthesis, structure, and preliminary activity studies

The degree of diastereoselectivity in the wrapping of four new chiral Salan ligands to form chiral-at-metal titanium complexes ranged from mild to perfect as a function of the ligands' N substituents; the enantiomerically pure complexes catalyzed the addition of diethyl zinc to benzaldehyde in 73-76% enantiomeric excess.     

Mechanistic Insight into the Stereochemical Control of Lactide Polymerization by Salan–Aluminum Catalysts

Alkyl aluminum complexes of chiral salan ligands assembled around the 2,2′‐bipyrrolidine core form as single diastereomers that have identical configurations of the N donors. Active catalysts for the polymerization of lactide were formed upon the addition of benzyl alcohol. Polymeryl exchange between enantiomorphous aluminum species had a dramatic effect on the tacticity of the poly(lactic acid) (PLA) in the polymerization of racemic lactide (rac‐LA): The enantiomerically pure catalyst of the nonsubstituted salan ligand led to isotactic PLA, and the racemic catalyst exhibited lower stereocontrol. The enantiomerically pure catalyst of the chloro‐substituted salan ligand led to PLA with a slight tendency toward heterotacticity, whereas the racemic catalyst led to PLA of almost perfect heterotacticity following an insertion/auto‐inhibition/exchange mechanism.     

The boundary trace and generalized boundary value problem for semilinear elliptic equations with coercive absorption

In the first part of the paper we establish the existence of a boundary trace for positive solutions of the equation ?Δu + g(x, u) = 0 in a smooth domain Ω ? ?^N, for a general class of positive nonlinearities. This class includes every space independent, monotone increasing g which satisfies the Keller‐Osserman condition as well as degenerate nonlinearities g_α,q of the form g_α,q (x, u) = d(x, ?Ω)^α |u|^q?1 u, with α > ?2 and q > 1. The boundary trace is given by a positive regular Borel measure which may blow up on compact sets. In the second part we concentrate on the family of nonlinearities {g_α,q}, determine the critical value of the exponent q (for fixed α > ?2) and discuss (a) positive solutions with an isolated singularity, for subcritical nonlinearities and (b) the boundary value problem for ?Δu + g_α,q (x, u) = 0 with boundary data given by a positive regular Borel measure (possibly unbounded). We show that, in the subcritical case, the problem possesses a unique solution for every such measure. © 2003 Wiley Periodicals, Inc.     

Analysis and design of nonlinear fiber Bragg gratings and their application for optical compression of reflected pulses

We demonstrate a novel split-step solution for analyzing nonlinear fiber Bragg gratings. The solution is used for designing nonlinear fiber Bragg gratings with a low reflectivity. The structure of the grating is designed according to the profiles of the incident and reflected pulses. We demonstrate our method for nonlinear compression of a pulse reflected from a fiber Bragg grating. The method allows us to obtain compressed pulses with a very low wing intensity.     

Eilatin complexes of ruthenium and osmium. synthesis, electrochemical behavior, and near-IR luminescence

The synthesis and characterization of new Ru(II) and Os(II) complexes of the ligand eilatin (1) are described. The new complexes [Ru(bpy)(eil)(2)](2+) (2), [Ru(eil)(3)](2+) (3), and [Os(eil)(3)](2+) (4) (bpy = 2,2'-bipyridine; eil = eilatin) were synthesized and characterized by NMR, fast atom bombardment mass spectrometry, and elemental analysis. In the series of complexes [Ru(bpy)(x)(eil)(y)()](2+) (x + y = 3), the effect of sequential substitution of eil for bpy on the electrochemical and photophysical properties was examined. The absorption spectra of the complexes exhibit several bpy- and eil-associated pi-pi and metal-to-ligand charge-transfer (MLCT) transitions in the visible region (400-600 nm), whose energy and relative intensity depend on the number of ligands bound to the metal center (x and y). On going from [Ru(bpy)(2)(eil)](2+) (5) to 2 to 3, the d(pi)(Ru) --> pi(eil) MLCT transition undergoes a red shift from 583 to 591 to 599 nm, respectively. Electrochemical measurements performed in dimethyl sulfoxide reveal several ligand-based reduction processes, where each eil ligand can accept up to two electrons at potentials that are significantly anodically shifted (by ca. 1 V) with respect to the bpy ligands. The complexes exhibit near-IR emission (900-1100 nm) of typical (3)MLCT character, both at room temperature and at 77 K. Along the series 5, 2, and 3, upon substitution of eil for bpy, the emission maxima undergo a blue shift and the quantum yields and lifetimes increase. The radiative and nonradiative processes that contribute to deactivation of the excited level are discussed in detail.     

Vanadium(III) and vanadium(V) amine tris(phenolate) complexes

The coordination chemistry of amine tris(phenolate) ligands around V(III) and V(V) is described for the first time. Three amine tris(phenolate) ligands were employed featuring different steric and electronic influence exerted by the phenolate substituents in the ortho and para positions being either t-Bu, Me, or Cl. V(III) complexes of all ligands (1-3) were readily obtained by reaction between the ligand precursors and VCl3(THF)3 in the presence of triethylamine. The complexes obtained were pentacoordinate, a THF ligand completing the coordination sphere of the metal, which was found to be of almost perfect TBP geometry, as revealed by crystallography. V(V) oxo complexes of all the ligands (4-6) were readily obtained by a reaction between the ligand precursors and VO(OPr)3. The oxo complexes of the alkyl-bearing ligands (4 and 5) could also be synthesized by the air oxidation of the corresponding V(III) complexes (1 and 2); however, the attempted air oxidation of the V(III) complex bound to the electron-poor ligand (3) did not yield the corresponding oxo complex 6. 1H NMR and crystallographic analysis of complexes 4 and 5 supported their TBP structures. Complex 6, on the other hand, was found to be composed of a TBP complex (6a) and an octahedral complex (6b) in equilibrium, the octahedral complex being more stable at lower temperatures. An X-ray structure of 6b revealed a mononuclear oxo complex, the sixth coordination site being occupied by an aqua ligand to which two THF molecules are H-bonded. Complexes 4-6 catalyze the epoxidation of olefins by t-BuOOH, albeit slowly. These complexes may thus be considered as structural and functional models of vanadium-dependent haloperoxidase enzymes.