Continuation of eigenvalues and invariant pairs for parameterized nonlinear eigenvalue problems

Invariant pairs have been proposed as a numerically robust means to represent and compute several eigenvalues along with the corresponding (generalized) eigenvectors for matrix eigenvalue problems that are nonlinear in the eigenvalue parameter. In this work, we consider nonlinear eigenvalue problems that depend on an additional parameter and our interest is to track several eigenvalues as this parameter varies. Based on the concept of invariant pairs, a theoretically sound and reliable numerical continuation procedure is developed. Particular attention is paid to the situation when the procedure approaches a singularity, that is, when eigenvalues included in the invariant pair collide with other eigenvalues. For the real generic case, it is proven that such a singularity only occurs when two eigenvalues collide on the real axis. It is shown how this situation can be handled numerically by an appropriate expansion of the invariant pair. The viability of our continuation procedure is illustrated by a numerical example.     

Evaluating American put options on zero-coupon bonds by a penalty method

In this paper, American put options on zero-coupon bonds are priced under a single factor model of short-term rate. The linear complementarity problem of the option value is solved numerically by a penalty method, by which the problem is transformed into a nonlinear PDE by adding a power penalty term. The solution of the penalized problem converges to that of the original problem. A numerical scheme is established by using the finite volume method and the corresponding stability and convergence are discussed. Numerical results are presented to show the usefulness of the method.     

Tunable magnetic properties of nanoparticle two-dimensional assemblies addressed by mixed self-assembled monolayers

Assemblies of magnetic nanoparticles (NPs) are intensively studied due to their high potential applications in spintronic, magnetic and magneto-electronic. The fine control over NP density, interdistance, and spatial arrangement onto substrates is of key importance to govern the magnetic properties through dipolar interactions. In this study, magnetic iron oxide NPs have been assembled on surfaces patterned with self-assembled monolayers (SAMs) of mixed organic molecules. The modification of the molar ratio between coadsorbed 11-mercaptoundecanoic acid (MUA) and mercaptododecane (MDD) on gold substrates is shown to control the size of NPs domains and thus to modulate the characteristic magnetic properties of the assemblies. Moreover, NPs can be used to indirectly probe the structure of SAMs in domains at the nanometer scale.     

New exact penalty function for solving constrained finite min-max problems

This paper introduces a new exact and smooth penalty function to tackleconstrained min-max problems.By using this new penalty function and adding justone extra variable,a constrained min-max problem is transformed into an unconstrainedoptimization one.It is proved that,under certain reasonable assumptions and when thepenalty parameter is sufficiently large,the minimizer of this unconstrained optimizationproblem is equivalent to the minimizer of the original constrained one.Numerical resultsdemonstrate that this penalty function method is an effective and promising approach forsolving constrained finite min-max problems.     

Synthesis and characterization of palladium(II) and platinum(II) complexes with ferrocenylimidazole

The synthesis and characterization of ferrocenylimidazole complexes of platinum(II) and palladium(II) are described. Reaction of ferrocenylimidazoles with K₂MCl₄ (M = Pd, Pt) using a biphasic system of dichloromethane and ethanol/water provided the corresponding complexes 2a–2j in good yields. New synthetic routes for the synthesis of ferrocenylbenzylethers 2k–2o, bis(4-ferrocenylbenzyl)carbonate [2p] and 4-ferrocenylbenzylacetate [2q] are also described. These products were obtained by the reaction of 4-ferrocenylbenzyl-1H-imidazole-carboxylate and K₂PtCl₄ under various conditions. Compounds 2k–2o were also obtained by alternative routes which do not involve the use of a platinum salt. The crystal structures of 2b, 2q and plausible mechanisms for the formation of 2k, 2p and 2q are reported.     

Mackey Formula in Type A

The statement (and the proof) of Theorem 4.1.1 of the paper‘Mackey formula in type A’ [Proc. London. Math.Soc. (3) 80 (2000) 545–574] is false. In this corrigendumwe provide a correct statement (and a correct proof). As a consequence,we see that Corollary 4.1.2 of the paper holds. So all the otherstatements in the paper are correct (up to minor misprints...).2000 Mathematical Subject Classification: 20G05, 20G40.     

Inherent Reactivity of Spiro-Activated Electrophilic Cyclopropanes

The kinetics of the ring-opening reactions of thiophenolates with geminal bis(acceptor)-substituted cyclopropanes in DMSO at 20 °C was monitored by photometric methods. The determined second-order rate constants of the S_N2 reactions followed linear relationships with Mayr nucleophilicity parameters (N/s_N) and Brønsted basicities (pK_aH) of the thiophenolates as well as with Hammett substituent parameters (σ) for groups attached to the thiophenolates. Phenyl-substituted cyclopropanes reacted by up to a factor of 15 faster than their unsubstituted analogues, in accord with the known activating effect of adjacent π-systems in S_N2 reactions. Variation of the electronic properties of substituents at the phenyl groups of the cyclopropanes gave rise to parabolic Hammett relationships. Thus, the inherent S_N2 reactivity of electrophilic cyclopropanes is activated by electron-rich π-systems because of the more advanced C1−C2 bond polarization in the transition state. On the other hand, electron-poor π-systems also lower the energetic barriers for the attack of anionic nucleophiles owing to attractive electrostatic interactions.     

Mesoscale Characterization of Nanoparticles Distribution Using X‐ray Scattering

The properties of many functional materials depend critically on the spatial distribution of an active phase within a support. In the case of solid catalysts, controlling the spatial distribution of metal (oxide) nanoparticles at the mesoscopic scale offers new strategies to tune their performance and enhance their lifetimes. However, such advanced control requires suitable characterization methods, which are currently scarce. Here, we show how the background in small‐angle X‐ray scattering patterns can be analyzed to quantitatively access the mesoscale distribution of nanoparticles within supports displaying hierarchical porosity. This is illustrated for copper catalysts supported on meso‐ and microporous silica displaying distinctly different metal distributions. Results derived from X‐ray scattering are in excellent agreement with electron tomography. Our strategy opens unprecedented prospects for understanding the properties and to guide the synthesis of a wide array of functional nanomaterials.