On critical cases of Sobolev's inequalities for Heisenberg groups

We prove some Trudinger-type inequalities and Brezis-Gallouet-Wainger inequality on the Heisenberg group, extending to this context the Euclidean results by T. Ozawa.     

Capable Lie algebras with the derived subalgebra of dimension 2 over an arbitrary field

In this paper, we classify all capable nilpotent Lie algebras with the derived subalgebra of dimension 2 over an arbitrary field. Moreover, the explicit structure of such Lie algebras of class 3 is given.     

磁性纳米颗粒系统偶极相互作用的Monte Carlo研究

采用局域Monte Carlo方法模拟不同易轴分布的简单立方排列单分散单畴Fe纳米颗粒系统的ZFC-FC曲线及磁滞回线.结果表明：随着偶极相互作用的增强，系统的阻塞温度T_B逐渐增大，且ZFC曲线的峰变宽.说明偶极相互作用使得系统的有效能垒提高，分布宽度增加.研究FC曲线磁化强度的倒数与温度关系，发现偶极相互作用系统中存在反铁磁有序.系统的阻塞态及超顺磁态的磁滞回线表明，极低低温下，随着偶极相互作用的增强，系统的矫顽力和剩磁减小，偶极相互作用阻碍系统的磁化；系统处于超顺磁态，各向异性作用及偶极相互作用使得系统的磁化曲线偏离Langevin曲线且偶极相互作用展现出退磁相互作用效应.偶极相互作用增强，系统磁化曲线与Langevin曲线偏差量的最大值向低场移动.在偶极相互作用下，易轴与外场夹角为45°的磁性纳米颗粒系统的平均有效能垒和有效能垒分布宽度较易轴随机分布系统的大.     

Weak attractors for the dissipative fractional Heisenberg equation

In this short paper, we consider the long time behaviors of the fractional Heisenberg equation and the existence of a global weak attractor is proved for the shift dynamics in the path space. The key ingredient is some new type of commutator structure introduced in this paper, which seems indispensable in proving the compactness of the dynamics. The technique introduced in this paper may also be useful to other fractional order partial differential equations.     

On a hidden symmetry of quantum harmonic oscillators

We consider a six-parameter family of the square integrable wave functions for the simple harmonic oscillator, which cannot be obtained by the standard separation of variables. They are given by the action of the corresponding maximal kinematical invariance group on the standard solutions. In addition, the phase space oscillations of the electron position and linear momentum probability distributions are computer animated and some possible applications are briefly discussed. A visualization of the Heisenberg uncertainty principle is presented.     

Magnetic behavior of a nano-disk constrained in an antiferromagnetic substrate

In this work we report a numerical Monte Carlo study of the behavior of a magnetic nano-disk put over an antiferromagnetic substrate. Three approaches were considered for describe the substrate: (1) A stacked antiferromagnetic configuration, (2) an Ising like arrangement and (3) Heisenberg like spins. For the Heisenberg case we still have considered an easy-plane and an easy-axis symmetry of the substrate. The hysteresis loop for the nano-disk is obtained by considering the three cases. The signature of the vortex in the nano-disk appears as small jumps in the hysteresis curve. Exchange bias effects are observed since the substrate has an easy axis symmetry.     

Generation of Bell states via Lyapunov control on a two-qubit system with an anisotropic XY Heisenberg interaction

We demonstrate that an arbitrary Bell state can be achieved in a two qubit anisotropic Heisenberg XY chain via Lyapunov control. During the whole process, we only need to apply an external field along a given direction to a single qubit. This control strategy is effective for all initial states in the four-dimensional Hilbert space where the target state is asymptotically stable. The effects of imperfections on the fidelity for the target state such as Gaussian leakage of local control and localized dephasi...     

Correlation functions from a unified variational principle: Trial Lie groups

Time-dependent expectation values and correlation functions for many-body quantum systems are evaluated by means of a unified variational principle. It optimizes a generating functional depending on sources associated with the observables of interest. It is built by imposing through Lagrange multipliers constraints that account for the initial state (at equilibrium or off equilibrium) and for the backward Heisenberg evolution of the observables. The trial objects are respectively akin to a density operator and to an operator involving the observables of interest and the sources. We work out here the case where trial spaces constitute Lie groups. This choice reduces the original degrees of freedom to those of the underlying Lie algebra, consisting of simple observables; the resulting objects are labeled by the indices of a basis of this algebra. Explicit results are obtained by expanding in powers of the sources. Zeroth and first orders provide thermodynamic quantities and expectation values in the form of mean-field approximations, with dynamical equations having a classical Lie–Poisson structure. At second order, the variational expression for two-time correlation functions separates–as does its exact counterpart–the approximate dynamics of the observables from the approximate correlations in the initial state. Two building blocks are involved: (i) a commutation matrix which stems from the structure constants of the Lie algebra; and (ii) the second-derivative matrix of a free-energy function. The diagonalization of both matrices, required for practical calculations, is worked out, in a way analogous to the standard RPA. The ensuing structure of the variational formulae is the same as for a system of non-interacting bosons (or of harmonic oscillators) plus, at non-zero temperature, classical Gaussian variables. This property is explained by mapping the original Lie algebra onto a simpler Lie algebra. The results, valid for any trial Lie group, fulfill consistency properties and encompass several special cases: linear responses, static and time-dependent fluctuations, zero- and high-temperature limits, static and dynamic stability of small deviations.     

Hausdorff operators on the Heisenberg group

This paper is devoted to the high-dimensional and multilinear Hausdorff operators on the Heisenberg group H n. The sharp bounds for the strong type(p, p)(1 ≤ p ≤∞) estimates of n-dimensional Hausdorff operators on H n are obtained. The sharp bounds for strong(p, p) estimates are further extended to multilinear cases. As an application, we derive the sharp constant for the multilinear Hardy operator on H n. The weak type(p, p)(1 ≤ p ≤∞) estimates are also obtained.     

The minimal dimensions of faithful representations for Heisenberg Lie superalgebras

This paper aims to determine the minimal dimensions and super-dimensions of faithful representations for Heisenberg Lie superalgebras over an algebraically closed field of characteristic zero.