Coxeter group actions on Saalschützian F34(1) series and very-well-poised F67(1) series

AbstractIn this paper we consider a function L(x→)=L(a,b,c,d;e;f,g), which can be written as a linear combination of two Saalschützian F34(1) hypergeometric series or as a very-well-poised F67(1) hypergeometric series. We explore two-term and three-term relations satisfied by the L function and put them in the framework of group theory. We prove a fundamental two-term relation satisfied by the L function and show that this relation implies that the Coxeter group W(D5), which has 1920 elements, is an invariance group for L(x→). The invariance relations for L(x→) are given two classifications based on two double coset decompositions of the invariance group. The fundamental two-term relation is shown to generalize classical results about hypergeometric series. We derive Thomaeʼs identity for F23(1) series, Baileyʼs identity for terminating Saalschützian F34(1) series, and Barnesʼ second lemma as consequences. We further explore three-term relations satisfied by L(a,b,c,d;e;f,g). The group that governs the three-term relations is shown to be isomorphic to the Coxeter group W(D6), which has 23 040 elements. Based on the right cosets of W(D5) in W(D6), we demonstrate the existence of 220 three-term relations satisfied by the L function that fall into two families according to the notion of L-coherence. The complexity of the coefficients in front of the L functions in the three-term relations is studied and is shown to also depend on L-coherence

On the Zeros of the Solutions to Nonlinear Hyperbolic Equations with Delays

2002 Mathematics Subject Classification: Primary 35В05; Secondary 35L15We consider the nonlinear hyperbolic equation with delays uxy + λuxy(x - σ, у - τ) + c(x, у, u, ux, uy) = f(x, у) . We obtain sufficient conditions for oscillation of the solutions of problems of Goursat in the case, where λ>0.This work was partly supported by Grant MM-437 and Grant MM-516

Critical current anisotropy in Nd-1111 single crystals and the infuence of neutron irradiation

We report on angle-resolved magnetization measurements on NdFeAsO$_{0.65}$F$_{0.35}$ (Nd-1111) single crystals. The field dependence of the critical current density, $J_c$, is non-monotonous in these crystals at all orientations and temperatures due to the fishtail effect, which strongly influences the angular dependence of $J_c$. The currents decrease as the field is tilted from the crystallographic c-axis at low fields, but increase at high fields. A peak occurs in the angular dependence of $J_c$ at intermediate fields. The critical currents are significantly enhanced after irradiation with fast neutrons and the fishtail disappears. The different current anisotropies at low and high fields, however, persist. We discuss the data in the framework of the anisotropic scaling approach and propose a transition from dominant pinning by large defects of low density at low fields to pinning by small defects of high density at high fields in the pristine crystal. Strong pinning dominates at all fields after the irradiation, and the angular dependence of $J_c$ can be described by anisotropic scaling only after an appropriate extension to this pinning regime