Influence functions and efficiencies of the canonical correlation and vector estimates based on scatter and shape matrices

In this paper, the influence functions and limiting distributions of the canonical correlations and coefficients based on affine equivariant scatter matrices are developed for elliptically symmetric distributions. General formulas for limiting variances and covariances of the canonical correlations and canonical vectors based on scatter matrices are obtained. Also the use of the so-called shape matrices in canonical analysis is investigated. The scatter and shape matrices based on the affine equivariant Sign Covariance Matrix as well as the Tyler's shape matrix serve as examples. Their finite sample and limiting efficiencies are compared to those of the Minimum Covariance Determinant estimators and S-estimator through theoretical and simulation studies. The theory is illustrated by an example.     

Cyclohexanone Dimethyl Acetals: A 13C NMR, Thermodynamic, and Ab Initio Study

The spatial structures of a number of mono- and disubstituted 1,1-dimethoxycyclohexanes (cyclohexanone dimethyl acetals) were studied by ¹³C NMR spectroscopy. In the monosubstituted acetals, substituents (Me, Et, i-Pr, and MeO) on C-2 are axially oriented, contrary to their normal, equatorial orientation on C-3 and C-4. Besides the spectroscopic study, the relative thermodynamic stabilities of the cis-trans isomers of a few 2,X-dialkyl (X = 3, 4, 5, or 6) derivatives of the parent cyclohexanone dimethyl acetal were determined by acid-catalyzed chemical equilibration in MeOH solution. In the most stable isomeric form, the 2-substituent is axial and the other equatorial. In the less stable isomer, both substituents are equatorial, excluding the cis-2,6-dimethyl derivative, where the ¹³C NMR shift data point to a predominance of the diaxial form. In general, the enthalpy difference between the isomeric forms is ca. 9 kJ mol^–1, while the entropy term favors the less stable isomer by 4 to 16 J K^–1 mol^–1. In the 2,6-dimethyl derivatives, however, the trans form is favored by only 0.8 kJ mol^–1 in G _m at 298.15 K. The main findings of the experimental work are in good agreement with ab initio calculations.     

Relative thermodynamic stabilities of 2-substituted 4-methylene-1,3-dioxolanes and 4-methyl-1,3-dioxoles

The relative thermodynamic stabilities of 24 pairs of carbon-carbon double-bondexo-endo isomeric 2-substituted 4-methylene-1,3-dioxolanes (a) and 4-methyl-1,3-dioxoles (b) have been determined by base-catalyzed chemical equilibration in DMSO solution. In all cases, theendo isomer (b) is the favored species at thermodynamic equilibrium. A single alkyl substitutent on C-2 gives only a negligible contribution to the relative stability of the isomeric forms, but the presence of two alkyl groups on C-2 increases the relative stability of theendo isomer by 2–3 kL mol^–1. A still higher effect in favor of theendo isomer is produced by introduction of a single alkoxy group on C-2; this effect is further slightly accentuated by 2,2-dialkoxy substitution at C-2. The origin of the favorable effect of 2-alkoxy substitution on the relative stability of theendo isomer is not clear, but it seems to arise from an unexpected stability of theendo isomer rather than from an enhanced destabilization of theexo form.     

IR Study of p–π Conjugation in 2-Substituted 4-Methylene-1,3-dioxolanes and 4-Methyl-1,3-dioxoles

Selected group frequencies of the IR spectra of a number of 2-substituted 4-methylene-1,3-dioxolanes (a) and their endocyclic isomers, 2-substituted 4-methyl-1,3-dioxoles (b), have been studied to establish their usefulness as a measure of the strength of p– conjugation in the O—C=C moieties of the title compounds. In the exo compounds (a), the C=C stretching frequencies C=C and the in-phase, out-of-plane bending (wagging) frequencies CH₂ w of the =CH₂ group were found to be linearly related, besides to each other, also to the electron-donating character of the substituents and the ¹³C NMR chemical shift of the C atom of the exocyclic CH₂ group. This suggests that the two IR group frequencies of the exo isomers are linearly related to the strength of p– conjugation in the O—C=C moiety. The C=C stretching absorption of the exo compounds in the 1600–1700 cm^–1 region appeared as a doublet, shown to arise from Fermi resonance of the C=C fundamental with the CH₂ w overtone. The C=C frequencies of the endo compounds support some previous findings of an unexpected strength of p– conjugation in the 2-alkoxy derivatives. Finally, DFT calculations at the B3LYP/6-31G* level of theory are shown to give accurate predictions of the effect of 2-substituents on the group frequencies.     

Korn inequality for a thin rod with rounded ends

We consider an elastic rod with rounded ends and diameter proportional to a small parameter h > 0. The roundness of the ends is described by an exponent m ∈ (0,1). We derive for the rod an asymptotically sharp Korn inequality with a special weighted anisotropic norm. Weight factors with m‐dependent powers of h appear both in the anisotropic norm and the Korn inequality itself, and we discover three critical values m = 1 ∕ 4, m = 1 ∕ 2 and m = 3 ∕ 4 at which these exponents are crucially changed. Copyright © 2014 John Wiley & Sons, Ltd.     

Spectral gaps in the dirichlet and neumann problems on the plane perforated by a doubleperiodic family of circular holes

We show that the spectrum of the Dirichlet and Neumann problems for the Laplace operator in the plane perforated by a double–periodic family of circular holes contains gaps (even any a priori given number of gaps) of certain radii of holes. The result is obtained by asymptotic analysis of the cell spectral problem, interpreted as a problem in a domain with thin bridges. Some open questions are stated.     

Kinetics of bioconjugate nanoparticle label binding in a sandwich-type immunoassay

Nanoparticle labels have enhanced the performance of diagnostic, screening, and other measurement applications and hold further promise for more sensitive, precise, and cost-effective assay technologies. Nevertheless, a clear view of the biomolecular interactions on the molecular level is missing. Controlling the ratio of molecular recognition over undesired nonspecific adhesion is the key to improve biosensing with nanoparticles. To improve this ratio with an aim to disallow nonspecific binding, a more detailed perspective into the kinetic differences between the cases is needed. We present the application of two novel methods to determine complex binding kinetics of bioconjugate nanoparticles, interferometry, and force spectroscopy. Force spectroscopy is an atomic force microscopy technique and optical interferometry is a direct method to monitor reaction kinetics in second-hour timescale, both having steadily increasing importance in nanomedicine. The combination is perfectly suited for this purpose, due to the high sensitivity to detect binding events and the ability to investigate biological samples under physiological conditions. We have attached a single biofunctionalized nanoparticle to the outer tip apex and studied the binding behavior of the nanoparticle in a sandwich-type immunoassay using dynamic force spectroscopy in millisecond timescale. Utilization of the two novel methods allowed characterization of binding kinetics in a time range spanning from 50 ms to 4 h. These experiments allowed detection and demonstration of differences between specific and nonspecific binding. Most importantly, nonspecific binding of a nanoparticle was reduced at contact times below 100 ms with the solid-phase surface.

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Figure A single biofunctionalized nanoparticle was attached to the outer tip apex and the binding behavior of the nanoparticle in a sandwich-type immunoassay, A) without analyte, B) with analyte and C) saturating analyte concentration, was recorded using dynamic force spectroscopy in millisecond timescale. The setting allowed measurement of the association speed of nonspecific binding.

     

Relative Thermodynamic Stabilities of Isomeric Vinyl Ethers Derived from 2-Acetyland 2-Isobutyryltetrahydrofurans. An Experimental and DFT Study

The relative thermodynamic stabilities of four isomeric vinyl ethers obtained by acid-catalyzed cleavage of MeOH from the dimethyl acetals of both 2-acetyltetrahydrofuran and 2-isobutyryltetrahydrofuran have been determined by chemical equilibration in cyclohexane solution. In addition, the structures and relative energies of many of these compounds have also been studied theoretically by DFT calculations. In each series of compounds, the endocyclic isomers, 5-(1-methoxyethyl)-2,3-dihydrofuran and 5-(1-methoxy-2-methylpropyl)-2,3-dihydrofuran, respectively, show the highest thermodynamic stabilities. This is particularly prominent for the vinyl ethers derived from 2-isobutyryltetrahydrofuran in which the equilibrium mixture consists of 90% of the endocyclic isomer at room temperature. The geometrical (exocyclic) isomers of the latter series of compounds show unexpected relative thermodynamic stabilities, but on the basis of the geometry-optimized structures provided by the DFT calculations, a reasonable explanation can be found.     

On the spectrum of the Steklov problem in a domain with a peak

We consider the spectral Steklov problem in a domain with a peak on the boundary. It is shown that the spectrum on the real nonnegative semi-axis can be either discrete or continuous depending on the sharpness of the exponent.