Jones index theory for Hilbert C*-bimodules and its equivalence with conjugation theory

We introduce the notion of finite right (respectively left) numerical index on a bimodule $X$ over C*-algebras A and B with a bi-Hilbertian structure. This notion is based on a Pimsner-Popa type inequality. The right (respectively left) index element of X can be constructed in the centre of the enveloping von Neumann algebra of A (respectively B). X is called of finite right index if the right index element lies in the multiplier algebra of A. In this case we can perform the Jones basic construction. Furthermore the C*--algebra of bimodule mappings with a right adjoint is a continuous field of finite dimensional C*-algebras over the spectrum of Z(M(A)), whose fiber dimensions are bounded above by the index. We show that if A is unital, the right index element belongs to A if and only if X is finitely generated as a right module. We show that bi-Hilbertian, finite (right and left) index C*-bimodules are precisely those objects of the tensor 2-C*-category of right Hilbertian C*-bimodules with a conjugate object, in the sense of Longo and Roberts, in the same category.Comment: 59 pages, amste

On a q-difference Painlev\'e III equation: II. Rational solutions

Rational solutions for a $q$-difference analogue of the Painlev\'e III equation are considered. A Determinant formula of Jacobi-Trudi type for the solutions is constructed.Comment: Archive version is already official. Published by JNMP at http://www.sm.luth.se/math/JNMP

Hypergeometric solutions to the q-Painlev\'e equation of type A4(1)A_4^{(1)}

We consider the q-Painlev\'e equation of type $A_4^{(1)}$ (a version of q-Painlev\'e V equation) and construct a family of solutions expressible in terms of certain basic hypergeometric series. We also present the determinant formula for the solutions.Comment: 16 pages, IOP styl

Enhancement of the spin pumping efficiency by spin-wave mode selection

The spin pumping efficiency of lateral standing spin wave modes in a rectangular YIG/Pt sample has been investigated by means of the inverse spin-Hall effect (ISHE). The standing spin waves drive spin pumping, the generation of spin currents from magnetization precession, into the Pt layer which is converted into a detectable voltage due to the ISHE. We discovered that the spin pumping efficiency is significantly higher for lateral standing surface spin waves rather than for volume spin wave modes. The results suggest that the use of higher-mode surface spin waves allows for the fabrication of an efficient spin-current injector

On a q-difference Painlev\'e III equation: I. Derivation, symmetry and Riccati type solutions

A q-difference analogue of the Painlev\'e III equation is considered. Its derivations, affine Weyl group symmetry, and two kinds of special function type solutions are discussed.Comment: arxiv version is already officia

Rational solutions of the discrete time Toda lattice and the alternate discrete Painleve II equation

The Yablonskii-Vorob'ev polynomials $y_{n}(t)$, which are defined by a second order bilinear differential-difference equation, provide rational solutions of the Toda lattice. They are also polynomial tau-functions for the rational solutions of the second Painlev\'{e} equation ($P_{II}$). Here we define two-variable polynomials $Y_{n}(t,h)$ on a lattice with spacing $h$, by considering rational solutions of the discrete time Toda lattice as introduced by Suris. These polynomials are shown to have many properties that are analogous to those of the Yablonskii-Vorob'ev polynomials, to which they reduce when $h=0$. They also provide rational solutions for a particular discretisation of $P_{II}$, namely the so called {\it alternate discrete} $P_{II}$, and this connection leads to an expression in terms of the Umemura polynomials for the third Painlev\'{e} equation ($P_{III}$). It is shown that B\"{a}cklund transformation for the alternate discrete Painlev\'{e} equation is a symplectic map, and the shift in time is also symplectic. Finally we present a Lax pair for the alternate discrete $P_{II}$, which recovers Jimbo and Miwa's Lax pair for $P_{II}$ in the continuum limit $h\to 0$.Comment: 23 pages, IOP style. Title changed, and connection with Umemura polynomials adde