Sub-Riemannian distance on the Lie group SL(2)

We find the distances between arbitrary elements of the Lie group SL(2) for the left invariant sub-Riemannian metric also invariant with respect to the right shifts by elements of the Lie subgroup SO(2) ? SL(2), in other words, the invariant sub-Riemannian metric on the weakly symmetric space (SL(2) × SO(2))/ SO(2) of Selberg.     

Calcium-mediated dearomatization, C-H bond activation, and allylation of alkylated and benzannulated pyridine derivatives

A facile and general synthetic pathway for the production of dearomatized, allylated, and C? H bond activated pyridine derivatives is presented. Reaction of the corresponding derivative with the previously reported reagent bis(allyl)calcium, [Ca(C₃H₅)₂] ( 1 ), cleanly affords the product in high yield. The range of N‐heterocyclic compounds studied comprised 2‐picoline ( 2 ), 4‐picoline ( 3 ), 2,6‐lutidine ( 4 ), 4‐tert‐butylpyridine ( 5 ), 2,2′‐bipyridine ( 6 ), acridine ( 7 ), quinoline ( 8 ), and isoquinoline ( 9 ). Depending on the substitution pattern of the pyridine derivative, either carbometalation or C? H bond activation products are obtained. In the absence of methyl groups ortho or para to the nitrogen atom, carbometalation leads to dearomatized products. C(sp³)? H bond activation occurs at ortho and para situated methyl groups. Steric shielding of the 4‐position in pyridine yields the ring‐metalated product through C(sp²)? H bond activation instead. The isolated compounds [Ca(2‐CH₂‐C₅H₄N)₂(THF)] ( 2 b ?(THF)), [Ca(4‐CH₂‐C₅H₄N)₂(THF)₂] ( 3 b ?(THF)₂), [Ca(2‐CH₂‐C₅H₃N‐6‐CH₃)₂(THF)_n] ( 4 b ?(THF)_n; n=0, 0.75), [Ca{2‐C₅H₃N‐4‐C(CH₃)₃}₂(THF)₂] ( 5 c ?(THF)₂), [Ca{4,4′‐(C₃H₅)₂‐(C₁₀H₈N₂)}(THF)] ( 6 a ?(THF)), [Ca(NC₁₃H₉‐9‐C₃H₅)₂(THF)] ( 7 a ?(THF)), [Ca(4‐C₃H₅‐C₉H₇N)₂(THF)] ( 8 b ?(THF)), and [Ca(1‐C₃H₅‐C₉H₇N)₂(THF)₃] ( 9 a ?(THF)₃) have been characterized by NMR spectroscopy and metal analysis. 9 a ?(THF)₄ and 4 b ?(THF)₃ were additionally characterized in the solid state by X‐ray diffraction experiments. 4 b ?(THF)₃ shows an aza‐allyl coordination mode in the solid state. Based on the results, mechanistic aspects are discussed in the context of previous findings.     

Electronic structure and stability of fullerene C82 isolated-pentagon-rule isomers

All nine isolated-pentagon-rule isomers of fullerene C(82) were investigated by the DFT method with the B3LYP functional at the 6-31G, 6-31G*, and 6-31+G* levels. The distribution of single, double, and delocalized π-bonds in the molecules of these isomers is shown for the first time. The obtained results are fully supported by DFT quantum-chemical calculations of electronic and geometrical structures of these isomers. The molecules of isomers 7 (C(3v)), 8 (C(3v)), and 9 (C(2v)) contain some radical substructures (such as the phenalenyl-radical substructure), which indicates that they are unstable and cannot be obtained as empty molecules. Thus, there is a possibility of obtaining them only as endohedral metallofullerenes or exohedral derivatives. Isomers 1 (C(2)), 2 (C(s)), 4 (C(s)), 5 (C(2)), and 6 (C(s)) with closed electronic shell are supposed to be stable, resembling isomer 3 (C(2)), which has just been extracted experimentally as an empty fullerene. We assume they can be produced as empty molecules.     

On certain classes of functional inclusions with causal operators in Banach spaces

We introduce the notion of a multivalued causal operator and consider an abstract Cauchy problem in a Banach space for various classes of functional inclusions with causal operators. The methods of the topological degree theory for condensing maps are applied to obtain local and global existence results for this problem and to study the continuous dependence of a solution set on initial data. As application we generalize some existence results for semilinear functional differential inclusions and Volterra integro-differential inclusions with delay.     

Correlation between the intensities of intermultiplet electric dipole transitions and the fine features of the stark structure of multiplets of Pr3+ ions in the LaCl3 compound

The Stark structure of the multiplets of Pr³⁺ ions in the LaCl₃ compound is described and the even and odd parameters of the crystal field, as well as the covalency parameters, are determined in the approximation of an intermediate interconfiguration interaction. The covalency parameters thus obtained agree well with the parameters calculated in the framework of microscopic models. The intensity parameters of intermultiplet electric dipole transitions are calculated using the odd parameters of the crystal field and the covalency parameters. The results of the calculations are in satisfactory agreement with the experimental data. This allows the conclusion that there is a correlation between the fine features of the Stark structure of the multiplets and the intensities of the intermultiplet electric dipole transitions.     

Anomalous proximity effect in superconducting oxide structures with an antiferromagnetic layer

The electrophysical parameters of superconductor/antiferromagnetic insulator structures based on the Nb/Au/Ca_1?x Sr _x CuO₂/YBa₂Cu₃O_7?δ hybrid heterostructure have been examined. YBa₂Cu₃O_7?δ and Ca_1?x Sr _x SuO₂ epitaxial films are grown by the laser ablation method on NdGaO₃ single crystal substrates, the thickness of the Ca_1?x Sr _x CuO₂ layer varies from 20 to 50 nm, and x = 0.15 and 0.5. The superconducting pair potential in the interface between the YBa₂Cu₃O_7?δ superconductor and Ca_1?x Sr _x CuO₂ antiferromagnet is found to penetrate into the antiferromagnet at distances much larger than the coherence length calculated for the ferromagnetic layer. The critical current of the superconducting transition manufactured at such an interface is highly sensitive to the magnetic field.     

Anomalies in the optical properties of nanocrystalline copper oxides CuO and Cu2O near the fundamental absorption edge

A 0.1–0.15-eV displacement of the fundamental absorption edge in the optical absorption spectra of nanocrystalline oxide n-CuO (relative to the position of the fundamental absorption edge in the spectra of CuO single crystals) towards lower energies (red shift) is observed against the background of strong blurring. Nanocrystalline n-Cu₂O exhibits a displacement of the fundamental absorption edge towards higher energies (blue shift) by approximately 0.35 eV. The size of crystallites in n-CuO and n-Cu₂O ranges from 10 to 90 nm. The blue shift of the fundamental absorption edge of n-Cu₂O is typical of classical wide-gap semiconductors and can be explained by size quantization upon a change in the particle size. The anomalous red shift of the fundamental absorption edge of the strongly correlated nanocrystalline oxide n-CuO can be attributed to the highly defective structure of n-CuO, anomalies in the electronic structure of strongly correlated compounds based on 3d metals, and their tendency to electronic phase separation with the formation of metal-like inclusions.     

Low-frequency noise in monodisperse platinum nanostructures near the percolation threshold

The percolation threshold p ₀ ≈ 0.6 is determined for monodisperse platinum nanostructures with 1.8-nm metallic particles deposited in a monolayer onto an insulating substrate through laser electrodispersion. It is shown that, in the “metallic” state (for p > p ₀), both the magnitude of the noise and its temperature dependence are close to those of pure metallic Pt layers. The frequency dependence of the normalized noise power spectral density is described by the relationship S _I /I ² ～ 1/f ^γ with the exponent γ close to unity. For current densities j ≥ 10⁷?10⁸ A/cm², the noise power spectral density S _I increases more rapidly with a further increase in the current as compared to I ² because of the current generating excess defects. For p < p ₀, the dependence of the conductivity σ on the temperature is adequately described by the standard relationship σ ～ exp[?(T ₀/T)^1/2]. The normalized noise power spectral density S _I /I ² exceeds the corresponding value for a quasi-metallic structure by many orders of magnitude. The noise power spectral density S _I is approximately proportional to the square of the current only for very low currents and increases steeply with a further increase in the current.     

Criterion for identification of ferroelectric domains in TGS crystals from AFM images

Criteria for identification of actual (dynamic) domains and morphologically similar domain “memory” regions are proposed based on the study of various types of contrast of topographic atomic-force microscopy images of lenslike regions on the polar surface of TGS crystals. Inaccuracy in identification may result in further errors in estimating the parameters of the domain structure. The images of ferroelectric domains in the spreading-resistance mode that indicate directly the presence of conductive properties of the domain walls have been obtained for the first time.     

The interference component of low-coherent radiation backscattering

An interference component of backscattering of low-coherent radiation is calculated for the first time based on the previously developed approach consisting of direct simulation of the Bethe-Salpeter equation in terms of the Monte Carlo method. It is shown that, upon an increase in the spatial coherence length, it plays the role of the characteristic scale of the backscattering instead of the transport length. The effect of localization of backscattering of an ultrashort pulse in relation to the penetration depth is revealed. Both the theory and the numerical simulation predict, in accordance with experiment, a considerable broadening of the peak of coherent backscattering with decreasing coherence length, which significantly extends the application field of the effect, especially in biomedical diagnostics.