Mean Value Integral Equations and the Helmholtz Equation

This paper considers the mean-value functional equation $$ f(x)={1 \over \mu({B_{R}(x))}} \int_{B_{R}(x)}{f(t)dt} $$ where B_r(x) is a ball of radius R centred at x ∈ R ⁿ, and μ represents Lebesgue measure. By using the spectral synthesis, we find that the space of continuous functions satisfying this equation (in the case n = 1,2) is characterized in terms of solutions of the Helmholtz equation.     

The characterization of Ba3Re2O9 and Sr3Re2O9

The compounds Ba₃Re₂O₉ and Sr₃Re₂O₉ were prepared by the solid state reaction of the corresponding alkaline-earth oxide with ReO₃ at 750 to 900°C in sealed, evacuated, fused silica tubes. The two compounds are isostructural, having the nine-layer ABO₃ structure with vacant central octahedra. The unit cell parameters are given. The magnetic susceptibility for Ba₃Re₂O₉ indicates Curie-Weiss behavior with a Re⁶⁺ moment having localized electrons. The magnetic data for Sr₃Re₂O₉ suggest delocalized electron behavior from its temperature-independent susceptibility. Both compounds appear to have semiconducting properties, but the strontium analog is a better conductor. Both compounds are unstable when heated in air above 400°C. They are readily decomposed by chemical oxidizing agents.     

A study on the PbCrO3 perovskite

The perovskite compound PbCrO₃ has been prepared from CrO₂ and PbO at high pressure. The product was formed as small black cubes suitable for electrical resistivity measurements. Magnetic susceptibility data were obtained in the region 77 to 550°K. The oxide phase was found to be a semiconductor with antiferromagnetic behavior below a transition temperature of about 160°K.     

Preparation of LiVO2 crystals

The synthesis of LiVO₂ crystals exhibiting two different morphologies is reported. Black hexagonal platelet-type and octahedral-like dendritic crystals were obtained using a fused-salt electrolytic method. DSC, TGA, magnetic susceptibility, and X-ray diffraction data are presented. Weak reflections observed in X-ray precession photographs of the platelets indicate a hexagonal superstructure with a unit cell a = 9.838(4) Å = 2√3a′, where a′ = 2.84Å, and c = 14.755(6) Å. Observed systematic absences are consistent with the space groups P6₂22 and P6₄22. The dendrites appear to be a pseudocubic phase with a face-centered unit cell, a = 8.31(1) Å.     

An investigation of the Cr1−xMnxO2 series and the characterization of orthorhombic CrMnO4

Several members of the Cr_1?xMn_xO₂ series were prepared in the tetrahedral anvil press by the reaction of CrO₂ with MnO₂. The tetragonal, rutile-type products were single-phase and have been characterized by crystallographic and magnetic measurements. The results are consistent with the formulations Cr⁴⁺_1?2xCr³⁺ Mn⁵⁺O₂ for 0 ? x ? 0.5. At low manganese concentration, x < 0.20, the magnetic moments are consistent with ferromagnetic contribution from Mn⁵⁺. A two-phase product was noted at the composition x = 0.90. The CrMnO₄ composition was found to have a powder pattern similar to that of orthorhombic PtO₂.     

μSR study of internal magnetic fields in superconducting La1.9Sr0.1CuO4 in the mK region

Muon-spin-rotation (μSR) experiments were performed on superconducting La_1.90Sr_0.10CuO₄ in zero external field. Below 2 K, a fast depolarization of the muon signal is observed, indicating that internal magnetic fields are present at the muon site. The average magnitude of the magnetic field in La_1.90Sr_0.10CuO₄ is in the order of 3 mT and thus approximately 14 times smaller than the corresponding field in antiferromagnetic La₂CuO₄. The present data support theoretical models of superconductivity in which magnetic correlations are responsible for pairing.     

A natural extension of the Pythagorean equation to higher dimensions

The Pythagorean equation is extended to higher dimensions via circulant matrices. This form allows for the set of solutions to be expressed in a clean yet non-trivial way. The cubic case, namely the equation x ³+y ³+z ³−3xyz=1, was studied by Ramanujan and displays many interesting properties. The general case highlights the use of circulant matrices and systems of differential equations. The structure of the solutions also allows parametrized solutions of the Fermat equation in degrees 3 and 4 to be given in terms of theta functions.      

Application of the Floquet theory to multiple quantum NMR of dipolar-coupled multi-spin systems under magic angle spinning

A new analytical Liouville-space representation of the time-propagator under magic angle spinning (MAS) is introduced using the formalized quantum Floquet theory. This approach has the advantage that it is applicable to the analysis of any type of NMR experiment where MAS is combined with multiple-pulse excitation. General relationships describing the spectral parameters in multiple-quantum (MQ) MAS spectra are derived in this representation. Their use is illustrated with an application to double-quantum (DQ) NMR spectra of dipolar-coupled multi-spin systems. Corresponding to the separation of the MAS time-propagator into a rotor modulated and a dephasing component, two distinct mechanisms for DQ excitation are identified. One of them exploits the rotor-modulated component to excite DQ coherences through dipolar-recoupling techniques, which are familiar for spin pairs. Analytical expressions of the integral intensities and linewidths in the resulting DQ sideband pattern are derived in the form of power series expansions of the inverse rotor frequency, of which coefficients depend on structural parameters. In a multi-spin system they can most reliably be extracted in the fast spinning regime. The other mechanism exploits the dephasing component, which is characteristic to multi-spin systems only. This is shown to give rise to DQ coherences by free evolution at full rotor periods. The possibility to exploit it for selective excitation of higher order MQ coherences is discussed. In either case, the dephasing component also leads to residual broadening. The main results of the theoretical developments are demonstrated experimentally on adamantane.