Relative unitary commutator calculus and applications

This note revisits localisation and patching method in the setting of generalised unitary groups. Introducing certain subgroups of relative elementary unitary groups, we develop relative versions of the conjugation calculus and the commutator calculus in unitary groups, which are both more general, and substantially easier than the ones available in the literature. For the general linear group such relative commutator calculus has been recently developed by the first and the third authors. As an application we prove the mixed commutator formula, for two form ideals of a form ring. This answers two problems posed in a paper by Alexei Stepanov and the second author.Comment: polishe

Exact solution of mean geodesic distance for Vicsek fractals

The Vicsek fractals are one of the most interesting classes of fractals and the study of their structural properties is important. In this paper, the exact formula for the mean geodesic distance of Vicsek fractals is found. The quantity is computed precisely through the recurrence relations derived from the self-similar structure of the fractals considered. The obtained exact solution exhibits that the mean geodesic distance approximately increases as an exponential function of the number of nodes, with the exponent equal to the reciprocal of the fractal dimension. The closed-form solution is confirmed by extensive numerical calculations.Comment: 4 pages, 3 figure

Study of the Wealth Inequality in the Minority Game

To demonstrate the usefulness of physical approaches for the study of realistic economic systems, we investigate the inequality of players' wealth in one of the most extensively studied econophysical models, namely, the minority game (MG). We gauge the wealth inequality of players in the MG by a well-known measure in economics known as the modified Gini index. From our numerical results, we conclude that the wealth inequality in the MG is very severe near the point of maximum cooperation among players, where the diversity of the strategy space is approximately equal to the number of strategies at play. In other words, the optimal cooperation between players comes hand in hand with severe wealth inequality. We also show that our numerical results in the asymmetric phase of the MG can be reproduced semi-analytically using a replica method.Comment: 9 pages in revtex 4 style with 3 figures; minor revision with a change of title; to appear in PR

Dynamical properties of a trapped dipolar Fermi gas at finite temperature

We investigate the dynamical properties of a trapped finite-temperature normal Fermi gas with dipole-dipole interaction. For the free expansion dynamics, we show that the expanded gas always becomes stretched along the direction of the dipole moment. In addition, we present the temperature and interaction dependences of the asymptotical aspect ratio. We further study the collapse dynamics of the system by suddenly increasing the dipolar interaction strength. We show that, in contrast to the anisotropic collapse of a dipolar Bose-Einstein condensate, a dipolar Fermi gas always collapses isotropically when the system becomes globally unstable. We also explore the interaction and temperature dependences for the frequencies of the low-lying collective excitations.Comment: 11 pages, 7 figure

From first-order magneto-elastic to magneto-structural transition in (Mn,Fe)1.95P0.50Si0.50 compounds

We report on structural, magnetic and magnetocaloric properties of MnxFe1.95-xP0.50Si0.50 (x > 1.10) compounds. With increasing the Mn:Fe ratio, a first-order magneto-elastic transition gradually changes into a first-order magneto-structural transition via a second-order magnetic transition. The study also shows that thermal hysteresis can be tuned by varying the Mn:Fe ratio. Small thermal hysteresis (less than 1 K) can be obtained while maintaining a giant magnetocaloric effect. This achievement paves the way for real refrigeration applications using magnetic refrigerants.Comment: 4 pages, 3 figures, Supplemental Materia

The yoga of commutators

In the present paper we discuss some recent versions of localisation methods for calculations in the groups of points of algebraic-like and classical-like groups. Namely, we describe relative localisation, universal localisation, and enhanced versions of localisation-completion. Apart from the general strategic description of these methods, we state some typical technical results of the conjugation calculus and the commutator calculus. Also, we state several recent results obtained therewith, such as relative standard commutator formulae, bounded width of commutators, with respect to the elementary generators, and nilpotent filtrations of congruence subgroups. Overall, this shows that localisation methods can be much more efficient, than expected