A system of difference equations with elliptic coefficients and Bethe vectors

An elliptic analogue of the $q$ deformed Knizhnik-Zamolodchikov equations is introduced. A solution is given in the form of a Jackson-type integral of Bethe vectors of the XYZ-type spin chains.Comment: 20 pages, AMS-LaTeX ver.1.1 (amssymb), 15 figures in LaTeX picture environment

Construction of some missing eigenvectors of the XYZ spin chain at the discrete coupling constants and the exponentially large spectral degeneracy of the transfer matrix

We discuss an algebraic method for constructing eigenvectors of the transfer matrix of the eight vertex model at the discrete coupling parameters. We consider the algebraic Bethe ansatz of the elliptic quantum group $E_{\tau, \eta}(sl_2)$ for the case where the parameter $\eta$ satisfies $2 N \eta = m_1 + m_2 \tau$ for arbitrary integers $N$, $m_1$ and $m_2$. When $m_1$ or $m_2$ is odd, the eigenvectors thus obtained have not been discussed previously. Furthermore, we construct a family of degenerate eigenvectors of the XYZ spin chain, some of which are shown to be related to the $sl_2$ loop algebra symmetry of the XXZ spin chain. We show that the dimension of some degenerate eigenspace of the XYZ spin chain on $L$ sites is given by $N 2^{L/N}$, if $L/N$ is an even integer. The construction of eigenvectors of the transfer matrices of some related IRF models is also discussed.Comment: 19 pages, no figure (revisd version with three appendices

Solvable vector nonlinear Riemann problems, exact implicit solutions of dispersionless PDEs and wave breaking

We have recently solved the inverse spectral problem for integrable PDEs in arbitrary dimensions arising as commutation of multidimensional vector fields depending on a spectral parameter $\lambda$. The associated inverse problem, in particular, can be formulated as a non linear Riemann Hilbert (NRH) problem on a given contour of the complex $\lambda$ plane. The most distinguished examples of integrable PDEs of this type, like the dispersionless Kadomtsev-Petviashivili (dKP), the heavenly and the 2 dimensional dispersionless Toda equations, are real PDEs associated with Hamiltonian vector fields. The corresponding NRH data satisfy suitable reality and symplectic constraints. In this paper, generalizing the examples of solvable NRH problems illustrated in \cite{MS4,MS5,MS6}, we present a general procedure to construct solvable NRH problems for integrable real PDEs associated with Hamiltonian vector fields, allowing one to construct implicit solutions of such PDEs parametrized by an arbitrary number of real functions of a single variable. Then we illustrate this theory on few distinguished examples for the dKP and heavenly equations. For the dKP case, we characterize a class of similarity solutions, a class of solutions constant on their parabolic wave front and breaking simultaneously on it, and a class of localized solutions breaking in a point of the $(x,y)$ plane. For the heavenly equation, we characterize two classes of symmetry reductions.Comment: 29 page

Eigenvalues of Ruijsenaars-Schneider models associated with An−1A_{n-1} root system in Bethe ansatz formalism

Ruijsenaars-Schneider models associated with $A_{n-1}$ root system with a discrete coupling constant are studied. The eigenvalues of the Hamiltonian are givein in terms of the Bethe ansatz formulas. Taking the "non-relativistic" limit, we obtain the spectrum of the corresponding Calogero-Moser systems in the third formulas of Felder et al [20].Comment: Latex file, 25 page

qq-analogue of modified KP hierarchy and its quasi-classical limit

A $q$-analogue of the tau function of the modified KP hierarchy is defined by a change of independent variables. This tau function satisfies a system of bilinear $q$-difference equations. These bilinear equations are translated to the language of wave functions, which turn out to satisfy a system of linear $q$-difference equations. These linear $q$-difference equations are used to formulate the Lax formalism and the description of quasi-classical limit. These results can be generalized to a $q$-analogue of the Toda hierarchy. The results on the $q$-analogue of the Toda hierarchy might have an application to the random partition calculus in gauge theories and topological strings.Comment: latex2e, a4 paper 15 pages, no figure; (v2) a few references are adde

Explorations of the Extended ncKP Hierarchy

A recently obtained extension (xncKP) of the Moyal-deformed KP hierarchy (ncKP hierarchy) by a set of evolution equations in the Moyal-deformation parameters is further explored. Formulae are derived to compute these equations efficiently. Reductions of the xncKP hierarchy are treated, in particular to the extended ncKdV and ncBoussinesq hierarchies. Furthermore, a good part of the Sato formalism for the KP hierarchy is carried over to the generalized framework. In particular, the well-known bilinear identity theorem for the KP hierarchy, expressed in terms of the (formal) Baker-Akhiezer function, extends to the xncKP hierarchy. Moreover, it is demonstrated that N-soliton solutions of the ncKP equation are also solutions of the first few deformation equations. This is shown to be related to the existence of certain families of algebraic identities.Comment: 34 pages, correction of typos in (7.2) and (7.5

WNT signalling in prostate cancer

Genome sequencing and gene expression analyses of prostate tumours have highlighted the potential importance of genetic and epigenetic changes observed in WNT signalling pathway components in prostate tumours-particularly in the development of castration-resistant prostate cancer. WNT signalling is also important in the prostate tumour microenvironment, in which WNT proteins secreted by the tumour stroma promote resistance to therapy, and in prostate cancer stem or progenitor cells, in which WNT-β-catenin signals promote self-renewal or expansion. Preclinical studies have demonstrated the potential of inhibitors that target WNT receptor complexes at the cell membrane or that block the interaction of β-catenin with lymphoid enhancer-binding factor 1 and the androgen receptor, in preventing prostate cancer progression. Some WNT signalling inhibitors are in phase I trials, but they have yet to be tested in patients with prostate cancer

Soft and non-soft structural transitions in disordered nematic networks

Properties of disordered nematic elastomers and gels are theoretically investigated with emphasis on the roles of non-local elastic interactions and crosslinking conditions. Networks originally crosslinked in the isotropic phase lose their long-range orientational order by the action of quenched random stresses, which we incorporate into the affine-deformation model of nematic rubber elasticity. We present a detailed picture of mechanical quasi-Goldstone modes, which accounts for an almost completely soft polydomain-monodomain (P-M) transition under strain as well as a ``four-leaf clover'' pattern in depolarized light scattering intensity. Dynamical relaxation of the domain structure is studied using a simple model. The peak wavenumber of the structure factor obeys a power-law-type slow kinetics and goes to zero in true mechanical equilibrium. The effect of quenched disorder on director fluctuation in the monodomain state is analyzed. The random frozen contribution to the fluctuation amplitude dominates the thermal one, at long wavelengths and near the P-M transition threshold. We also study networks obtained by crosslinking polydomain nematic polymer melts. The memory of initial director configuration acts as correlated and strong quenched disorder, which renders the P-M transition non-soft. The spatial distribution of the elastic free energy is strongly dehomogenized by external strain, in contrast to the case of isotropically crosslinked networks.Comment: 19 pages, 15 EPS figure