Darboux transformations for a twisted derivation and quasideterminant solutions to the super KdV equation

This paper is concerned with a generalized type of Darboux transformations defined in terms of a twisted derivation $D$ satisfying $D(AB)=D(A)+\sigma(A)B$ where $\sigma$ is a homomorphism. Such twisted derivations include regular derivations, difference and $q$-difference operators and superderivatives as special cases. Remarkably, the formulae for the iteration of Darboux transformations are identical with those in the standard case of a regular derivation and are expressed in terms of quasideterminants. As an example, we revisit the Darboux transformations for the Manin-Radul super KdV equation, studied in Q.P. Liu and M. Ma\~nas, Physics Letters B \textbf{396} 133--140, (1997). The new approach we take enables us to derive a unified expression for solution formulae in terms of quasideterminants, covering all cases at once, rather than using several subcases. Then, by using a known relationship between quasideterminants and superdeterminants, we obtain expressions for these solutions as ratios of superdeterminants. This coincides with the results of Liu and Ma\~nas in all the cases they considered but also deals with the one subcase in which they did not obtain such an expression. Finally, we obtain another type of quasideterminant solutions to the Main-Radul super KdV equation constructed from its binary Darboux transformations. These can also be expressed as ratios of superdeterminants and are a substantial generalization of the solutions constructed using binary Darboux transformations in earlier work on this topic

Nodeless superconductivity in Ca3Ir4Sn13: evidence from quasiparticle heat transport

We report resistivity $\rho$ and thermal conductivity $\kappa$ measurements on Ca$_3$Ir$_4$Sn$_{13}$ single crystals, in which superconductivity with $T_c \approx 7$ K was claimed to coexist with ferromagnetic spin-fluctuations. Among three crystals, only one crystal shows a small hump in resistivity near 20 K, which was previously attributed to the ferromagnetic spin-fluctuations. Other two crystals show the $\rho \sim T^2$ Fermi-liquid behavior at low temperature. For both single crystals with and without the resistivity anomaly, the residual linear term $\kappa_0/T$ is negligible in zero magnetic field. In low fields, $\kappa_0(H)/T$ shows a slow field dependence. These results demonstrate that the superconducting gap of Ca$_3$Ir$_4$Sn$_{13}$ is nodeless, thus rule out nodal gap caused by ferromagnetic spin-fluctuations.Comment: 5 pages, 4 figure

On solutions to the non-Abelian Hirota-Miwa equation and its continuum limits

In this paper, we construct grammian-like quasideterminant solutions of a non-Abelian Hirota-Miwa equation. Through continuum limits of this non-Abelian Hirota-Miwa equation and its quasideterminant solutions, we construct a cascade of noncommutative differential-difference equations ending with the noncommutative KP equation. For each of these systems the quasideterminant solutions are constructed as well.Comment: 9 pages, 1 figur