Spectral asymmetry of the massless Dirac operator on a 3-torus

Consider the massless Dirac operator on a 3-torus equipped with Euclidean metric and standard spin structure. It is known that the eigenvalues can be calculated explicitly: the spectrum is symmetric about zero and zero itself is a double eigenvalue. The aim of the paper is to develop a perturbation theory for the eigenvalue with smallest modulus with respect to perturbations of the metric. Here the application of perturbation techniques is hindered by the fact that eigenvalues of the massless Dirac operator have even multiplicity, which is a consequence of this operator commuting with the antilinear operator of charge conjugation (a peculiar feature of dimension 3). We derive an asymptotic formula for the eigenvalue with smallest modulus for arbitrary perturbations of the metric and present two particular families of Riemannian metrics for which the eigenvalue with smallest modulus can be evaluated explicitly. We also establish a relation between our asymptotic formula and the eta invariant.

Scale-invariant moving finite elements for nonlinear partial differential equations in two dimensions

A scale-invariant moving finite element method is proposed for the adaptive solution of nonlinear partial differential equations. The mesh movement is based on a finite element discretisation of a scale-invariant conservation principle incorporating a monitor function, while the time discretisation of the resulting system of ordinary differential equations is carried out using a scale-invariant time-stepping which yields uniform local accuracy in time. The accuracy and reliability of the algorithm are successfully tested against exact self-similar solutions where available, and otherwise against a state-of-the-art h-refinement scheme for solutions of a two-dimensional porous medium equation problem with a moving boundary. The monitor functions used are the dependent variable and a monitor related to the surface area of the solution manifold. (c) 2005 IMACS. Published by Elsevier B.V. All rights reserved.

Are the spatial patterns of weeds scale-invariant?

In previous empirical and modelling studies of rare species and weeds, evidence of fractal behaviour has been found. We propose that weeds in modern agricultural systems may be managed close to critical population dynamic thresholds, below which their rates of increase will be negative and where scale-invariance may be expected as a consequence. We collected detailed spatial data on five contrasting species over a period of three years in a primarily arable field. Counts in 20×20 cm contiguous quadrats, 225,000 in 1998 and 84,375 thereafter, could be re-structured into a wide range of larger quadrat sizes. These were analysed using three methods based on correlation sum, incidence and conditional incidence. We found non-trivial scale invariance for species occurring at low mean densities and where they were strongly aggregated. The fact that the scale-invariance was not found for widespread species occurring at higher densities suggests that the scaling in agricultural weed populations may, indeed, be related to critical phenomena.

Molybdenum eta(3)-allyl dicarbonyl complexes as a new class of precursors for highly reactive epoxidation catalysts with tert-butyl hydroperoxide

New Mo(II) diimine derivatives of [Mo(q (3)allyl)X(CO)(2)(CH3CN)(2)] (allyl = C3H5 and C5H5O; X = Cl, Br) were prepared, and [MO(eta(3)-C3H5)Cl(CO)(2)(BIAN)] (BIAN = 1,4-(4-chloro)phenyl-2,3-naphthalene-diazabutadiene) (7) was structurally characterized by single-crystal X-ray diffraction. This complex adopted an equatorial-axial arrangement of the bidentate ligand (axial isomer), in contrast with the precursors, found as the equatorial isomer in the solid and fluxional in solution. The new complexes of the type [Mo(eta(3)-allyl)X(CO)(2)(N-N)l (N-N is a bidentate chelating dinitrogen ligand) were tested for the catalytic epoxidation of cyclooctene using tert-butyl hydroperoxide as oxidant. All catalytic systems were 100% selective toward epoxide formation. While their turnover frequencies paralleled those of related Mo(eta) carbonyl compounds or Mo(VI) compounds bearing similar N-donor ligands, they exhibited similar olefin conversions in consecutive catalytic runs. The acetonitrile precursors were generally more active than the diimine complexes, and the chloro derivatives more active than the bromo ones. Combined vibrational and NMR spectroscopy and computational studies (DFT) were used to investigate the nature of the molybdenum species formed in the catalytic system with [Mo(eta(3)-C3H5)Cl(CO)(2){1,4-(2,6-dimethyl)phenyl-2.3-dimethyldiazabuta diene}] (4) and to propose that the resulting species may be dimeric bearing oxide bridges.

Immobilisation of eta(3)-allyidicarbonyl complexes of Mo-II with bidentate nitrogen ligands within aluminium-pillared clays

Molybdenum(II) complexes [MOX(CO)(2)(eta(3)-allyl)(CH3CN)(2)] (X = Cl or Br) were encapsulated in an aluminium-pillared natural clay or a porous clay heterostructure and allowed to react with bidentate diimine ligands. All the materials obtained were characterised by several solid-state techniques. Powder XRD, and Al-27 and Si-29 MAS NMR were used to investigate the integrity of the pillared clay during the modification treatments. C-13 CP MAS NMR, FTIR, elemental analyses and low-temperature nitrogen adsorption showed that the immobilisation of the precursor complexes was successful as well as the in situ ligand-substitution reaction. The new complex [MoBr(CO)(2)(eta(3)-allyl)(2-aminodipyridyl)] was characterised by single-crystal X-ray diffraction and spectroscopic techniques, and NMR studies were used to investigate its fluxional behaviour in solution. The prepared materials are active for the oxidation of cis-cyclooctene using tert-butyl hydroperoxide as oxidant, though the activity of the isolated complexes is higher. ((c) Wiley-VCH Verlag GmbH & Co. KGaA, 69451 Weinheim, Germany, 2008).

Mono- and binuclear bipyridyl derivatives of the Mo(eta(3)-C3H5)(CO)(2) fragment: structural studies and fluxionality in solution

New mono- and binuclear complexes of the Mo(eta(3)-C3H5)(CO)(2) fragment, containing bipyridyl ligands (2,2'-bpy, 4,4'-Me-2-2,2'-bpy) as chelates, and mono- (4-CNpy, 4-Mepy, NCMe, Br) or bidentate nitrogen ligands (4,4'-bpy, bipyridylethylene, pyrazine) as terminal or bridging ligands, respectively, were prepared. The binuclear complex [{Mo(eta(3)-C3H5)(CO)(2)(2,2'-bpy)}(2)(mu-4,4'-bpy)][PF6](2) (2) was shown by X-ray diffraction to assemble in the crystal forming large channels with a rectangular section. A longer bridge, such as bipyridylethylene, led to a different structure (3). 4-CNpy behaved as monodentate ligand (4), coordinating through the pyridine nitrogen as a terminal ligand. NMR spectroscopy studies showed that the complexes exhibited a fluxional behavior in solution, the endo and exo forms of the more symmetrical equatorial isomers being usually present and interconverting in solution. The solid state structures of the complexes revealed a preference for the more symmetrical equatorial isomer, with the two chelate ligands in trans positions in the binuclear species. The rings tended to become parallel in the organized crystal. (C) 2003 Elsevier B.V. All rights reserved.

Nitrogen donor ligands bearing N-H groups: Effect on catalytic and cytotoxic activity of molybdenum eta(3)-allyldicarbonyl complexes

Reactions of [Mo(eta(3)-C3H5)Br(CO)(2)(NCMe)(2)] with the bidentate nitrogen ligands 2-(2'-pyridyl)imidazole (L1), 2-(2'-pyridyl)benzimidazole (L2), N,N'-bis(2'-pyridinecarboxamido)-1,2-ethane (L3), and 2,2'-bisimidazole (L4) led to the new complexes [Mo(eta(3)-C3H5)Br(CO)(2)(L)] (L = L1, 1; L2, 2; L4, 4) and [{Mo(eta(3)-C3H5) Br(CO)(2)}(2)(mu-L-3)] (3). The reaction of complexes 2 and 3 with Tl[CF3SO3] afforded [Mo(eta(3)-C3H5)(CF3SO3)(CO)(2)(L2)] (2T) and [{Mo(eta(3)-C3H5)(CF3SO3)(CO)(2)}(2)(mu-L-3)] (3T). Complexes 3 and 2T were structurally characterized by single crystal X-ray diffraction, showing the facial allyl/carbonyls arrangement and the formation of the axial isomer. In 2T, two molecules are assembled in a hydrogen bond dimer. The four complexes 1-4 were tested as precursors in the catalytic epoxidation of cyclooctene and styrene, in the presence of t-butylhydroperoxide (TBHP), with moderate conversions and turnover frequencies for complexes 1-3 and very low ones for 4. The increasing number of N-H groups in the complexes seems to be responsible for the loss of catalytic activity, compared with other related systems. The cytotoxic activities of all the complexes were evaluated against HeLa cells. The results showed that compounds 1,2,4, and 2T exhibited significant activity, complexes 2 and 2T being particularly promising. (C) 2008 Elsevier B.V. All rights reserved.

A phosphorus analogue of Bis(eta(4)-cyclobutadiene)iron(0)

P makes it possible: The convenient oxidative synthesis of the 16-electron organophosphorus iron sandwich complex [Fe(4-P2C2tBu2)2] suggests that the elusive all-carbon complex [Fe(4-C4H4)2] is a viable synthetic target.

Spectrally-invariant behavior of zenith radiance around cloud edges simulated by radiative transfer

In a previous paper, we discovered a surprising spectrally-invariant relationship in shortwave spectrometer observations taken by the Atmospheric Radiation Measurement (ARM) program. The relationship suggests that the shortwave spectrum near cloud edges can be determined by a linear combination of zenith radiance spectra of the cloudy and clear regions. Here, using radiative transfer simulations, we study the sensitivity of this relationship to the properties of aerosols and clouds, to the underlying surface type, and to the finite field-of-view (FOV) of the spectrometer. Overall, the relationship is mostly sensitive to cloud properties and has little sensitivity to other factors. At visible wavelengths, the relationship primarily depends on cloud optical depth regardless of cloud phase function, thermodynamic phase and drop size. At water-absorbing wavelengths, the slope of the relationship depends primarily on cloud optical depth; the intercept, by contrast, depends primarily on cloud absorbing and scattering properties, suggesting a new retrieval method for cloud drop effective radius. These results suggest that the spectrally-invariant relationship can be used to infer cloud properties near cloud edges even with insufficient or no knowledge about spectral surface albedo and aerosol properties.

Spectral invariant behavior of zenith radiance around cloud edges observed by ARM SWS

The ARM Shortwave Spectrometer (SWS) measures zenith radiance at 418 wavelengths between 350 and 2170 nm. Because of its 1-sec sampling resolution, the SWS provides a unique capability to study the transition zone between cloudy and clear sky areas. A spectral invariant behavior is found between ratios of zenith radiance spectra during the transition from cloudy to cloud-free. This behavior suggests that the spectral signature of the transition zone is a linear mixture between the two extremes (definitely cloudy and definitely clear). The weighting function of the linear mixture is a wavelength-independent characteristic of the transition zone. It is shown that the transition zone spectrum is fully determined by this function and zenith radiance spectra of clear and cloudy regions. An important result of these discoveries is that high temporal resolution radiance measurements in the clear-to-cloud transition zone can be well approximated by lower temporal resolution measurements plus linear interpolation.

Power properties if invariant tests for spatial autocorrelation in linear regression

This paper derives some exact power properties of tests for spatial autocorrelation in the context of a linear regression model. In particular, we characterize the circumstances in which the power vanishes as the autocorrelation increases, thus extending the work of Krämer (2005). More generally, the analysis in the paper sheds new light on how the power of tests for spatial autocorrelation is affected by the matrix of regressors and by the spatial structure. We mainly focus on the problem of residual spatial autocorrelation, in which case it is appropriate to restrict attention to the class of invariant tests, but we also consider the case when the autocorrelation is due to the presence of a spatially lagged dependent variable among the regressors. A numerical study aimed at assessing the practical relevance of the theoretical results is included

A phosphorus analogue of Bis(eta(4)-cyclobutadiene)iron(0)

Iron is a pivotal element in organometallic chemistry, enabling fundamental insights with high-impact applications.[1] Ferrocene derivatives have countless uses,[2] and the recent advances in iron catalysis are equally impressive.[3]