Symmetrizability of differential equations having orthogonal polynomial solutions

AbstractWe show that if a linear differential equation of spectral type with polynomial coefficients LN[y](x) = ∑i=0Nli(x) = λny(x) has an orthogonal polynomial system of solutions, then the differential operator LN[·] must be symmetrizable. We also give a few applications of this result

On mean convergence of Hermite–Fejér and Hermite interpolation for Erdős weights

AbstractWe investigate convergence of Hermite–Fejér and Hermite interpolation polynomials in Lp(0<p<∞) for Erdo&#x030B;s weights

Necessary conditions for weighted mean convergence of Lagrange interpolation for exponential weights

AbstractGiven a continuous real-valued function f which vanishes outside a fixed finite interval, we establish necessary conditions for weighted mean convergence of Lagrange interpolation for a general class of even weights w which are of exponential decay on the real line or at the endpoints of (−1,1)

Differential equations having orthogonal polynomial solutions

AbstractNecessary and sufficient conditions for an orthogonal polynomial system (OPS) to satisfy a differential equation with polynomial coefficients of the form (∗) LN[y] = ∑i=1Nli(x)y(i)(x) = λny(x) were found by H.L. Krall. Here, we find new necessary conditions for the equation (∗) to have an OPS of solutions as well as some other interesting applications. In particular, we obtain necessary and sufficient conditions for a distribution w(x) to be an orthogonalizing weight for such an OPS and investigate the structure of w(x). We also show that if the equation (∗) has an OPS of solutions, which is orthogonal relative to a distribution w(x), then the differential operator LN[·] in (∗) must be symmetrizable under certain conditions on w(x)

Orthogonal polynomials in two variables and second-order partial differential equations

AbstractWe study the second-order partial differential equations L[u] = Auxx +22Buxy + Cuyy + Dux + Euy = λnu, which have orthogonal polynomials in two variables as solutions. By using formal functional calculus on moment functionals, we first give new simpler proofs and improvements of the results by Krall and Sheffer and Littlejohn. We then give a two-variable version of Al-Salam and Chihara's characterization of classical orthogonal polynomials in one variable. We also study in detail the case when L[·] belongs to the basic class, that is, Ay = Cx = 0. In particular, we characterize all such differential equations which have a product of two classical orthogonal polynomials in one variable as solutions

The origin of the 90 degree magneto-optical Kerr rotation in CeSb

We calculate the linear magneto-optical Kerr rotation for CeSb in the near-infrared spectral range. Using an exact formula for large Kerr rotation angles and a simplified electronic structure of CeSb we find at \hbar \omega = 0.46 eV a Kerr rotation of 90 degree which then for decreasing \omega jumps to -90 degree as recently observed. We identify the general origin of possible 180 degree polarization rotations as resulting from mainly nonmagnetic optical properties, in particular from the ratio of the dominant interband resonance frequency to the plasma frequency. The dependence of the Kerr rotation on moments and magnetization is discussed.Comment: 6 pages, REVTEX, 5 eps figure

Temperature and Frequency Dependence of Complex Conductance of Ultrathin YBa2Cu3O7-x Films: A Study of Vortex-Antivortex Pair Unbinding

We have studied the temperature dependencies of the complex sheet conductance of 1-3 unit cell (UC) thick YBa2Cu3O7-x films sandwiched between semiconducting Pr0.6Y0.4Ba2Cu3O7-x layers at high frequencies. Experiments have been carried out in a frequency range between: 2 - 30 MHz with one-spiral coil technique, 100 MHz - 1 GHz frequency range with a new technique using the spiral coil cavity and at 30 GHz by aid of a resonant cavity technique. The real and imaginary parts of the mutual-inductance between a coil and a film were measured and converted to complex conductivity by aid of the inversion procedure. We have found a quadratic temperature dependence of the kinetic inductance, L_k^-1(T), at low temperatures independent of frequency, with a break in slope at T^dc_BKT, the maximum of real part of conductance and a large shift of the break temperature and the maximum position to higher temperatures with increasing frequency. We obtain from these data the universal ratio T^dc_BKT/L_k^-1(T^dc_BKT) = 25, 25, and 17 nHK for 1-, 2- and 3UC films, respectively in close agreement with theoretical prediction of 12 nHK for vortex-antivortex unbinding transition. The activated temperature dependence of the vortex diffusion constant was observed and discussed in the framework of vortex-antivortex pair pinning. PACS numbers: 74.80.Dm, 74.25.Nf, 74.72.Bk, 74.76.BzComment: PDF file, 10 pages, 6 figures, to be published in J. Low Temp. Phys.; Proc. of NATO ARW: VORTEX 200

A theory of L1L^1-dissipative solvers for scalar conservation laws with discontinuous flux

We propose a general framework for the study of $L^1$ contractive semigroups of solutions to conservation laws with discontinuous flux. Developing the ideas of a number of preceding works we claim that the whole admissibility issue is reduced to the selection of a family of "elementary solutions", which are certain piecewise constant stationary weak solutions. We refer to such a family as a "germ". It is well known that (CL) admits many different $L^1$ contractive semigroups, some of which reflects different physical applications. We revisit a number of the existing admissibility (or entropy) conditions and identify the germs that underly these conditions. We devote specific attention to the anishing viscosity" germ, which is a way to express the "$\Gamma$-condition" of Diehl. For any given germ, we formulate "germ-based" admissibility conditions in the form of a trace condition on the flux discontinuity line $x=0$ (in the spirit of Vol'pert) and in the form of a family of global entropy inequalities (following Kruzhkov and Carrillo). We characterize those germs that lead to the $L^1$-contraction property for the associated admissible solutions. Our approach offers a streamlined and unifying perspective on many of the known entropy conditions, making it possible to recover earlier uniqueness results under weaker conditions than before, and to provide new results for other less studied problems. Several strategies for proving the existence of admissible solutions are discussed, and existence results are given for fluxes satisfying some additional conditions. These are based on convergence results either for the vanishing viscosity method (with standard viscosity or with specific viscosities "adapted" to the choice of a germ), or for specific germ-adapted finite volume schemes

Golgi Outpost Synthesis Impaired by Toxic Polyglutamine Proteins Contributes to Dendritic Pathology in Neurons

Dendrite aberration is a common feature of neurodegenerative diseases caused by protein toxicity, but the underlying mechanisms remain largely elusive. Here, we show that nuclear polyglutamine (polyQ) toxicity resulted in defective terminal dendrite elongation accompanied by a loss of Golgi outposts (GOPs) and a decreased supply of plasma membrane (PM) in Drosophila class IV dendritic arborization (da) (C4 da) neurons. mRNA sequencing revealed that genes downregulated by polyQ proteins included many secretory pathway-related genes, including COPII genes regulating GOP synthesis. Transcription factor enrichment analysis identified CREB3L1/CrebA, which regulates COPII gene expression. CrebA overexpression in C4 da neurons restores the dysregulation of COPII genes, GOP synthesis, and PM supply. Chromatin immunoprecipitation (ChIP)-PCR revealed that CrebA expression is regulated by CREB-binding protein (CBP), which is sequestered by polyQ proteins. Furthermore, co-overexpression of CrebA and Rac1 synergistically restores the polyQ-induced dendrite pathology. Collectively, our results suggest that GOPs impaired by polyQ proteins contribute to dendrite pathology through the CBP-CrebA-COPII pathway. ? 2017 The Author(s)113Ysciescopu

Modelling of strain effects in manganite films

Thickness dependence and strain effects in films of $La_{1-x}A_xMnO_3$ perovskites are analyzed in the colossal magnetoresistance regime. The calculations are based on a generalization of a variational approach previously proposed for the study of manganite bulk. It is found that a reduction in the thickness of the film causes a decrease of critical temperature and magnetization, and an increase of resistivity at low temperatures. The strain is introduced through the modifications of in-plane and out-of-plane electron hopping amplitudes due to substrate-induced distortions of the film unit cell. The strain effects on the transition temperature and transport properties are in good agreement with experimental data only if the dependence of the hopping matrix elements on the $Mn-O-Mn$ bond angle is properly taken into account. Finally variations of the electron-phonon coupling linked to the presence of strain turn out important in influencing the balance of coexisting phases in the filmComment: 7 figures. To be published on Physical Review