Bounds on changes in Ritz values for a perturbed invariant subspace of a Hermitian matrix

The Rayleigh-Ritz method is widely used for eigenvalue approximation. Given a matrix $X$ with columns that form an orthonormal basis for a subspace \X, and a Hermitian matrix $A$, the eigenvalues of $X^HAX$ are called Ritz values of $A$ with respect to \X. If the subspace \X is $A$-invariant then the Ritz values are some of the eigenvalues of $A$. If the $A$-invariant subspace \X is perturbed to give rise to another subspace \Y, then the vector of absolute values of changes in Ritz values of $A$ represents the absolute eigenvalue approximation error using \Y. We bound the error in terms of principal angles between \X and \Y. We capitalize on ideas from a recent paper [DOI: 10.1137/060649070] by A. Knyazev and M. Argentati, where the vector of absolute values of differences between Ritz values for subspaces \X and \Y was weakly (sub-)majorized by a constant times the sine of the vector of principal angles between \X and \Y, the constant being the spread of the spectrum of $A$. In that result no assumption was made on either subspace being $A$-invariant. It was conjectured there that if one of the trial subspaces is $A$-invariant then an analogous weak majorization bound should only involve terms of the order of sine squared. Here we confirm this conjecture. Specifically we prove that the absolute eigenvalue error is weakly majorized by a constant times the sine squared of the vector of principal angles between the subspaces \X and \Y, where the constant is proportional to the spread of the spectrum of $A$. For many practical cases we show that the proportionality factor is simply one, and that this bound is sharp. For the general case we can only prove the result with a slightly larger constant, which we believe is artificial.Comment: 12 pages. Accepted to SIAM Journal on Matrix Analysis and Applications (SIMAX

Renormalization of hole-hole interaction at decreasing Drude conductivity

The diffusion contribution of the hole-hole interaction to the conductivity is analyzed in gated GaAs/In$_x$Ga$_{1-x}$As/GaAs heterostructures. We show that the change of the interaction correction to the conductivity with the decreasing Drude conductivity results both from the compensation of the singlet and triplet channels and from the arising prefactor $\alpha_i<1$ in the conventional expression for the interaction correction.Comment: 6 pages, 5 figure

Metal-Insulator Transition in 2D: Experimental Test of the Two-Parameter Scaling

We report a detailed scaling analysis of resistivity \rho(T,n) measured for several high-mobility 2D electron systems in the vicinity of the 2D metal-insulator transition. We analyzed the data using the two parameter scaling approach and general scaling ideas. This enables us to determine the critical electron density, two critical indices, and temperature dependence for the separatrix in the self-consistent manner. In addition, we reconstruct the empirical scaling function describing a two-parameter surface which fits well the \rho(T,n) data.Comment: 4 pages, 4 figures, 1 tabl

Evaluation of strawberry cultivars as sources of high winter hardiness and productivity

Background. The realization of high productivity potential in strawberry plants significantly depends on their adaptability to soil and climate conditions of their cultivation. Winter hardiness is the most important trait in the central part of Russia. With this in view, the purpose of this study was to identify sources of high winter hardiness combined with high productivity and yield potential among the studied cultivars.Materials and methods. The research was carried out at the primary variety testing site of the All-Russian Research Institute of Fruit Crop Breeding (VNIISPK), Orel Province, in accordance with the published program and methodology of fruit, berry and nut variety studies. The testing covered 23 domestic and foreign strawberry cultivars. An Espec PSL-2KPH climate chamber (Japan) and a Polair refrigerator cabinet (Russia) were used for artificial freezing.Results. Winter hardiness of strawberry cultivars was assessed under artificial freezing conditions. Cvs. ‘Korona’, ‘Tsaritsa’, ‘Solovushka’ and ‘Sara’ were the most winter-hardy cultivars. The study of the cultivars in the field made it possible to identify the most productive and large-fruited ones: ‘Alba’, ‘Azia’, ‘Bereginya’ and ‘Tsaritsa’. Cv. ‘Tsaritsa’ combines high winter hardiness and high yield productivity and yield potential, being a valuable source of these traits in breeding practice. According to a set of indicators ‘Tsaritsa’ is promising for large-scale cultivation in the Central Black Earth Region of Russia