Isotropy Representations, Eigenvalues of a Casimir Element, and Commutative Lie Subalgebras

Let h be a reductive subalgebra of a semisimple Lie algebrag and C_h U(h) be the Casimir element determined by the restrictionof the Killing form on g to h. The paper studies eigenvaluesof C_h on the isotropy representation mg/h. Some general estimatesconnecting the eigenvalues and the Dynkin indices of m are given.If h is a symmetric subalgebra, it is shown that describingthe maximal eigenvalue of C_h on exterior powers of m is connectedwith possible dimensions of commutative Lie subalgebras in m,thereby extending a result of Kostant. In this situation, aformula is also given for the maximal eigenvalue of C_h on m.More generally, a similar picture arises if h = g, where isan automorphism of finite order m and m is replaced by the eigenspaceof corresponding to a primitive mth root of unity.     

An YBCO film as a Josephson medium near T c : Frequency and field dependences and scaling relationships

The diamagnetic response of a laser-deposited YBCO film was studied in weak exciting magnetic fields H _ac =H ₀sinωt (H ₀=2–500 mOe, ω/2π=1–30 kHz). An analysis of the experimentally measured diamagnetic response showed evidence of a ring character of the electric field flowing in the sample at small H ₀ values. The plots of T _m (the temperature of maximum dissipative losses χ″₁) versus H ₀ measured at various frequencies showed the existence of a certain critical field strength H*(ω) at which the (1?T _m /T _c ) versus H ₀ ^2/3 plots exhibit a break. A logarithmic dependence of T _m on the frequency ω of the exciting field H _ac is reliably valid only in the region of (ω/2π)>5 kHz. Verification of the scaling relationship showed that this relationship is valid (to within the experimental accuracy) for the frequencies ω above 5 kHz and the field amplitudes H ₀>H*(ω). However, the exponent n in the scaling relationship under consideration is not correlated with the exponent n in the power relationship describing the current-voltage characteristic.     

Effect of hydrostatic pressure and temperature on the EPR spectrum of the Mn2+ ion in Zn(BF4)2 · 6H2O

A study of the effect of hydrostatic pressure and temperature on the EPR spectrum of the Mn²⁺ ion in Zn(BF₄)₂ · 6H₂O is reported. The break in the temperature dependence of the b ₂ ⁰ parameter at 196 K is evidence of the existence of a phase transition accompanied by a change in the thermal expansion coefficient. It is shown that pressure considerably affects the spectral parameters by reducing the axial parameter b ₂ ⁰ and increasing the cubic one, b ₄ ⁰ . At 9 kbar, the b ₂ ⁰ parameter is temperature independent. A comparison of the pressure dependences of the spectra of Zn(BF₄)₂ · 6H₂O, ZnSiF₆ · 6H₂O, ZnTiF₆ · 6H₂O, and MgSiF₂ · 6H₂O crystals suggests equal hydrogen-bond lengths in these compounds. A ligand hyperfine structure has been detected, which originates from the Zeeman interaction with the proton nuclei surrounding Mn²⁺ and manifests itself in the formation of satellites at each EPR line, their separation being proportional to the magnetic field. The nonlinear pressure dependence of the linewidth is related to the structural features of the crystal under study.     

X-ray study of structural features of GaSb(Si) single crystals grown using a set of diffraction methods under various heat and mass transfer conditions

Structural features of GaSb(Si) single crystals grown under various heat and mass transfer conditions were studied by x-ray topography and by double- and triple-crystal diffractometry. It was shown that a decrease in the convective flow intensity during crystal growth by the vertical Bridgman method with axisymmetric upper heat supply in comparison with the Czochralski method eliminates growth striations (caused by microsegregation) and improves the uniformity of electrical parameters of materials. According to triple-crystal x-ray diffractometry data, the increased density of structural defects observed in some crystal regions causes significant lattice distortions, which makes an additional contribution to crystal inhomogeneity. Some specific features in x-ray topography images of growth striations, caused by a high silicon concentration and a dopant state deviation from an ideal substitutional solid solution, were revealed.     

A new a posteriori error estimate for convection–reaction–diffusion problems

A new a posteriori error estimate is derived for the stationary convection–reaction–diffusion equation. In order to estimate the approximation error in the usual energy norm, the underlying bilinear form is decomposed into a computable integral and two other terms which can be estimated from above using elementary tools of functional analysis. Two auxiliary parameter-functions are introduced to construct such a splitting and tune the resulting bound. If these functions are chosen in an optimal way, the exact energy norm of the error is recovered, which proves that the estimate is sharp. The presented methodology is completely independent of the numerical technique used to compute the approximate solution. In particular, it is applicable to approximations which fail to satisfy the Galerkin orthogonality, e.g. due to an inconsistent stabilization, flux limiting, low-order quadrature rules, round-off and iteration errors, etc. Moreover, the only constant that appears in the proposed error estimate is global and stems from the Friedrichs–Poincaré inequality. Numerical experiments illustrate the potential of the proposed error estimation technique.     

Characterization of single-crystal sapphire substrates by X-ray methods and atomic force microscopy

The possibility of characterizing a number of practically important parameters of sapphire substrates by X-ray methods is substantiated. These parameters include wafer bending, traces of an incompletely removed damaged layer that formed as a result of mechanical treatment (scratches and marks), surface roughness, damaged layer thickness, and the specific features of the substrate real structure. The features of the real structure of single-crystal sapphire substrates were investigated by nondestructive methods of double-crystal X-ray diffraction and plane-wave X-ray topography. The surface relief of the substrates was investigated by atomic force microscopy and X-ray scattering. The use of supplementing analytical methods yields the most complete information about the structural inhomogeneities and state of crystal surface, which is extremely important for optimizing the technology of substrate preparation for epitaxy.     

Nilpotent fusion categories

In this paper we extend categorically the notion of a finite nilpotent group to fusion categories. To this end, we first analyze the trivial component of the universal grading of a fusion category C, and then introduce the upper central series of C. For fusion categories with commutative Grothendieck rings (e.g., braided fusion categories) we also introduce the lower central series. We study arithmetic and structural properties of nilpotent fusion categories, and apply our theory to modular categories and to semisimple Hopf algebras. In particular, we show that in the modular case the two central series are centralizers of each other in the sense of M. Müger.