Symbolic Computation of Newton Sum Rules for the Zeros of Polynomial Eigenfunctions of Linear Differential Operators

A symbolic algorithm based on the generalized Lucas polynomials of first kind is used in order to compute the Newton sum rules for the zeros of polynomial eigenfunctions of linear differential operators with polynomial coefficients.     

Influence of a Magnetic Field on the Nonlinear Optical Susceptibility of NiI2-Doped CdI2 Single Crystals

The influence of treatment in a magnetic field at low temperatures on the nonlinear optical properties of NiI₂-doped CdI₂ single crystals is investigated. The impurity ions in the interlayer space or more complex centers into which these ions enter can align on exposure to a magnetic field due to the interaction with the membrane vibrations of the lattice and create residual polarization in a specimen. This polarization causes redistribution of the electric charge on the bonds and improves the nonlinear optical properties of single crystals.     

On the Gibbsian Nature of the Random Field Kac Model Under Block-Averaging

We consider the Kac–Ising model in an arbitrary configuration of local magnetic fields = , in any dimension d, at any inverse temperature. We investigate the Gibbs properties of the 'renormalized' infinite volume measures obtained by block averaging any of the Gibbs-measures corresponding to fixed , with block-length small enough compared to the range of the Kac-interaction. We show that these measures are Gibbs measures for the same renormalized interaction potential. This potential depends locally on the field configuration and decays exponentially, uniformly in , for which we give explicit bounds. The construction of the potential is based on a high temperature-type cluster expansion.     

Mechanism Study for Thermal/Electric Field Poling of Fused Silica

On the basis of the experimental reports, the mechanism of the second-order susceptibility ⁽²⁾ for the thermal/electric field poling of fused silica is analyzed, and expressions for ⁽²⁾ are detailedly derived and numerically calculated for the first time. By comparison the theoretical value of ⁽²⁾ with the experiment results, we propose that the effective ⁽²⁾ is created via both the interaction of the intense electric field with the third-order susceptibility ⁽³⁾ and the dipole orientation. The theoretical results show that, in the differently applied voltage, the dipole orientation and ⁽³⁾ play different role in the formation of ⁽²⁾. This theory successfully explains some experiment results.     

Investigation of Hyperfine Interactions in CeIn3 byTDPAC

Magnetic hyperfine fields (mhf) at ¹¹¹Cd and ¹⁴⁰Ce nuclei, dilutely substituting the In and Ce sites, respectively, have been measured in the intermetallic compound CeIn₃ using perturbed angular correlation technique. A pure electric quadrupole interaction with an axially symmetric electric field gradient was observed at ¹¹¹In(EC)¹¹¹Cd probe nuclei at room temperature while a combined magnetic dipole and electric quadrupole interaction is observed below 10K. Below the ordering temperature, only a magnetic interaction is observed at ¹⁴⁰La(^–)¹⁴⁰Ce probe. The values of mhf measured experimentally as a function of temperature are discussed in terms of critical behavior.     

Steady Nonintegrable High-Dimensional Fluids

We consider the existence of steady incompressible fluids (solutions to the Euler equations) on Riemannian manifolds of dimensions three and higher. We demonstrate that, as in the case of the ABC fields in dimension three, there exist chaotic Beltrami fields – nonsingular eigenfields of the curl operator – in higher dimensions. We give an explicit set of analytic examples on a non-Euclidean five-torus T ⁵. We also detail a plug construction for inserting chaotic vortices into a Beltrami field. These constructions employ contact-topological techniques.     

Legendre Transformation for Regularizable Lagrangians in Field Theory

Hamilton equations based not only upon the Poincaré–Cartan equivalent of a first-order Lagrangian, but also upon its Lepagean equivalent are investigated. Lagrangians which are singular within the Hamilton–De Donder theory, but regularizable in this generalized sense are studied. Legendre transformation for regularizable Lagrangians is proposed and Hamilton equations, equivalent with the Euler–Lagrange equations, are found. It is shown that all Lagrangians affine or quadratic in the first derivatives of the field variables are regularizable. The Dirac field and the electromagnetic field are discussed in detail.     

High-field high-speed MAS resolution enhancement in 1H NMR spectroscopy of solids

Resolution in ¹H NMR spectra of solids can be significantly enhanced with fast magic-angle spinning and high magnetic fields. A variable field and spinning speed study up to 25 T and 40 kHz shows that the homogeneous line broadening is inversely proportional to the product of magnetic field strength and spinning speed. The combination of high field and fast speed yields a ¹H linewidth approaching the intrinsic limit determined by anisotropy of magnetic susceptibility. An analysis of the anisotropic magnetic susceptibility line broadening is presented.     

CLASSIFICATION OF THE PHASE PORTRAIT FOR PLANAR Z_8-EQUIVARIANT HAMILTONIAN SYSTEM OF DEGREE 7

The Hilbert's sixteenth problem, that is the problem on the rlurnber anddistribution of limit cycles of planar polynomial system, has not solved fOr acentury. Since the original problem is so difficuIt, in 1977, V.I.Arnold posed"weakened Hilbert's sixteenth problem"-- the possibility of the number anddistribution of limit cycles fOr polynomial HamiItonian system of degrees n -- lunder perturbation of the polynomial of degrees m 1. From l983, Prof.Li Jibin etc. began to study cubic vecto…     

Serre's generalization of Nagao's theorem: An elementary approach

Let be a smooth projective curve over a field . For each closed point of let be the coordinate ring of the affine curve obtained by removing from . Serre has proved that is isomorphic to the fundamental group, , of a graph of groups , where is a tree with at most one non-terminal vertex. Moreover the subgroups of attached to the terminal vertices of are in one-one correspondence with the elements of , the ideal class group of . This extends an earlier result of Nagao for the simplest case .

Serre's proof is based on applying the theory of groups acting on trees to the quotient graph , where is the associated Bruhat-Tits building. To determine he makes extensive use of the theory of vector bundles (of rank 2) over . In this paper we determine using a more elementary approach which involves substantially less algebraic geometry.

The subgroups attached to the edges of are determined (in part) by a set of positive integers , say. In this paper we prove that is bounded, even when Cl is infinite. This leads, for example, to new free product decomposition results for certain principal congruence subgroups of , involving unipotent and elementary matrices.