Multivariate approximation in the worst case setting over reproducing kernel Hilbert spaces

We study the worst case setting for approximation of d variate functions from a general reproducing kernel Hilbert space with the error measured in the L_∞ norm. We mainly consider algorithms that use n arbitrary continuous linear functionals. We look for algorithms with the minimal worst case errors and for their rates of convergence as n goes to infinity. Algorithms using n function values will be analyzed in a forthcoming paper.We show that the L_∞ approximation problem in the worst case setting is related to the weighted L₂ approximation problem in the average case setting with respect to a zero-mean Gaussian stochastic process whose covariance function is the same as the reproducing kernel of the Hilbert space. This relation enables us to find optimal algorithms and their rates of convergence for the weighted Korobov space with an arbitrary smoothness parameter α>1, and for the weighted Sobolev space whose reproducing kernel corresponds to the Wiener sheet measure. The optimal convergence rates are n^-(α-1)/2 and n^-1/2, respectively.We also study tractability of L_∞ approximation for the absolute and normalized error criteria, i.e., how the minimal worst case errors depend on the number of variables, d, especially when d is arbitrarily large. We provide necessary and sufficient conditions on tractability of L_∞ approximation in terms of tractability conditions of the weighted L₂ approximation in the average case setting. In particular, tractability holds in weighted Korobov and Sobolev spaces only for weights tending sufficiently fast to zero and does not hold for the classical unweighted spaces.     

Inter-hydrogen bond dynamical coupling effects in polarized IR spectra of N-methylacetamide single crystals

This article presents the investigation results of polarized IR spectra of the hydrogen bond in N-methylacetamide (NMA) crystals measured in the frequency range of the proton and deuteron stretching vibration bands, ν_N–H and ν_N–D. A similar study was also performed for crystals of the deuterium isotopomers of the compound, D₇-NMA (CD₃CONDCD₃) and D₆-NMA (CD₃CONHCD₃). On the basis of the analysis of the linear dichroic and temperature effects, the two-branch structure of the ν_N–H bands in the spectra was ascribed to centrosymmetric hydrogen bond pairs in the lattice. Each hydrogen bond in such a dimeric system belonged to another chain of the associated molecules. The exciton interactions involving the dimer hydrogen bonds were considered to be responsible for the band shape generation. For the deuterium-bonded crystals the exciton interactions were found to be weaker since the ν_N–D bands were less split. Within an individual hydrogen or deuterium bond chain the in-chain exciton couplings involving hydrogen bonds were estimated as considerably weaker than the inter-chain ones. The exciton dilution retains the two-branch fine structure pattern of the “residual” ν_N–H and ν_N–D bands. This means that the inter-chain couplings involving hydrogen bonds do not change, when the in-chain couplings vanish. These results are the evidence of the influence of non-conventional co-operative interactions occurring in the hydrogen bond systems on the spectra. These co-operative interactions are responsible for the non-random distribution of the hydrogen isotope atoms in the hydrogen bridge lattices, namely for the grouping of identical hydrogen isotope atoms in the dimers. The proposed interpretation of the IR spectra of the hydrogen bond in N-methylacetamide (NMA) crystals casts light on the spectra generation mechanisms and gives a new meaning to the traditional nomenclature applied for describing the ν_N–H band structure pattern in IR spectra of amides.     

Effects of dynamical couplings in hydrogen bond systems in the polarized IR spectra of 3-hydroxybenzaldehyde and 4-hydroxybenzaldehyde crystals

This paper presents the investigation results of the polarized IR spectra of 3-hydroxybenzaldehyde and 4-hydroxybenzaldehyde crystals measured at 293 and 77 K. Analysis of the results concerned the linear dichroic, H/D isotopic and temperature effects observed in the spectra of the hydrogen and deuterium bond at the frequency ranges of the ν_O–H and the ν_O–D bands, respectively. The main spectral properties of the crystals were interpreted in terms of the “strong-coupling” theory on the basis of the hydrogen bond dimer model. The spectra revealed that the strongest vibrational exciton coupling involved the closely-spaced hydrogen bonds, each belonging to a different chain of associated molecules. The reason for two different crystalline systems, are characterized by almost identical ν_O–H and ν_O–D band shapes, is explained. It was proved that a random distribution of the protons and deuterons took place in the lattices of the isotopically diluted crystals.     

Calorimetric studies of interactions of some peptides with electrolytes,urea and ethanol in water at 298.15 K

The standard molar enthalpies of solution at infinite dilution \Updelta_\textsol H_\textm^￥ \Updelta_{\text{sol}} H_{\text{m}}^{\infty } of glycylglycine, dl-alanyl-dl-alanine and glycylglycylglycine in aqueous solutions of potassium chloride and ethanol as well as of glycylglycine and glycylglycylglycine in the solutions containing urea and water have been determined by calorimetry at the temperature 298.15 K. Changes of solution enthalpy, expressed in a form so-called heterotactic interaction coefficients, h_\textxy h_{\text{xy}} were used for analysis of interactions occurring between the investigated solutes in water. The group contributions illustrating the interactions of KCl, urea and ethanol with selected functional groups in the peptide molecules, namely CH₂, “pep,” and “ion” groups, were calculated and discussed.     

Synthesis and structural characterization of novel 2-benzimidazolylthioureas: adducts of natural isothiocyanates and 2-amino-1-methylbenzimidazole

Adducts of natural allyl, phenethyl, and benzyl isothiocyanates and 2-amino-1-methylbenzimidazole were synthesized. After optimization of the reaction conditions, the target 2-benzimidazolylthioureas were obtained in reasonable yields. The detailed molecular and crystal structures of these compounds were characterized by spectroscopic and X-ray methods. Spectral analysis demonstrated that N-(1-methylbenzimidazolyl)-N′-allylthiourea, N-(1-methylbenzimidazolyl)-N′-benzylthiourea, and N-(1-methylbenzimidazolyl)-N′-phenethylthiourea exist in solution in an unprecedented three tautomeric forms, whose structures were corroborated unambiguously.     

The phase diagram for coexisting d-wave superconductivity and charge-density waves: cuprates and beyond

Phase diagrams of d-wave superconductivity characterized by an order parameter Δ coexisting with charge-density waves (CDWs) characterized by an order parameter Σ were constructed for the two-dimensional Fermi surface (FS) appropriate to, e.g., cuprates. CDWs were considered as an origin of the pseudogap appearing at antinodal FS sections of the d(x2-y2) superconductor. Two types of the Σ-reentrance were found: with the temperature, T, and with the opening of the CDW sector, 2α. The angular plots in the momentum space for the resulting gap profile over the FS ('gap roses') were obtained. The gap patterns are rather involved, giving insight into the difficulties of the interpretation of photoemission spectra. It was shown that the Σ-Δ coexistence region exists even for the complete dielectric gapping due to the distinction between the superconducting and CDW order parameter symmetries. The checkerboard and unidirectional CDW configurations were examined, and both the phase diagrams and the behavior with T and α of the order parameters were found to differ. A more general case with a non-zero mismatch angle β between the superconducting lobes and the CDW sectors was analyzed, the case β = π/4 corresponding to the d(xy) symmetry of the superconducting order parameter. The phase diagrams were found to be sensitive to β-variations, showing that internal strains and external pressure can drastically affect the behavior of Σ(T) and Δ(T).     

X‐ray investigations of bicyclic α‐methyl­ene‐δ‐valerolactones. II. (4aS,8aS)‐4a‐Methyl‐3‐methyl­eneper­hydro­chromen‐2‐one

The title compound, C₁₁H₁₆O₂, adopts a semifolded conformation with the δ‐lactone and cyclo­hexane rings almost perpendicular to one another. The β‐methyl substituent occupies an axial position with respect to the cyclo­hexane ring. The δ‐lactone moiety adopts a slightly distorted half‐chair arrangement, while the cyclo­hexane ring exists in an almost ideal chair conformation.