Carath\'{e}odory interpolation on the non-commutative polydisk

The Carath\'{e}odory problem in the $N$-variable non-commutative Herglotz--Agler class and the Carath\'{e}odory--Fej\'{e}r problem in the $N$-variable non-commutative Schur--Agler class are posed. It is shown that the Carath\'{e}odory (resp., Carath\'{e}odory--Fej\'{e}r) problem has a solution if and only if the non-commutative polynomial with given operator coefficients (the data of the problem indexed by an admissible set $\Lambda$) takes operator values with positive semidefinite real part (resp., contractive operator values) on $N$-tuples of $\Lambda$-jointly nilpotent contractive $n\times n$ matrices, for all $n\in\mathbb{N}$.Comment: to appear in J. Funct. Ana

Stable p-wave resonant two-dimensional Fermi-Bose dimers

We consider two-dimensional weakly-bound heterospecies molecules formed in a Fermi-Bose mixture with attractive Fermi-Bose and repulsive Bose-Bose interactions. Bosonic exchanges lead to an intermolecular attraction, which can be controlled and tuned to a p-wave resonance. Such attractive fermionic molecules can be realized in quasi-two-dimensional ultracold isotopic or heteronuclear mixtures. We show that they are stable with respect to the recombination to deeply-bound molecular states and with respect to the formation of higher-order clusters (trimers, tetramers, etc.

Metastable states and macroscopic quantum tunneling in a cold atom Josephson ring

We study macroscopic properties of a system of weakly interacting neutral bosons confined in a ring-shaped potential with a Josephson junction. We derive an effective low energy action for this system and evaluate its properties. In particular we find that the system possesses a set of metastable current-carrying states and evaluate the rates of transitions between these states due to macroscopic quantum tunneling. Finally we discuss signatures of different metastable states in the time-of-flight images and argue that the effect is observable within currently available experimental technique.Comment: 4 pages, 2 figure

Sensitivity analysis of utility-based prices and risk-tolerance wealth processes

In the general framework of a semimartingale financial model and a utility function $U$ defined on the positive real line, we compute the first-order expansion of marginal utility-based prices with respect to a ``small'' number of random endowments. We show that this linear approximation has some important qualitative properties if and only if there is a risk-tolerance wealth process. In particular, they hold true in the following polar cases: \begin{tabular}@p97mm@ for any utility function $U$, if and only if the set of state price densities has a greatest element from the point of view of second-order stochastic dominance;for any financial model, if and only if $U$ is a power utility function ($U$ is an exponential utility function if it is defined on the whole real line). \end{tabular}Comment: Published at http://dx.doi.org/10.1214/105051606000000529 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org