Strong 1-boundedness of unimodular orthogonal free quantum groups

Recently, Brannan and Vergnioux showed that the orthogonal free quantum group factors [Formula: see text] have Jung’s strong [Formula: see text]-boundedness property, and hence are not isomorphic to free group factors. We prove an analogous result for the other unimodular case, where the parameter matrix is the standard symplectic matrix in [Formula: see text] dimensions [Formula: see text]. We compute free derivatives of the defining relations by introducing self-adjoint generators through a decomposition of the fundamental representation in terms of Pauli matrices, resulting in [Formula: see text]-boundedness of these generators. Moreover, we prove that under certain conditions, one can add elements to a [Formula: see text]-bounded set without losing [Formula: see text]-boundedness. In particular, this allows us to include the character of the fundamental representation, proving strong [Formula: see text]-boundedness.

Tame torsion and the tame inverse Galois problem

Fix a positive integer g and a squarefree integer m. We prove the existence of a genus g curve $$C/{\mathbb {Q}}$$ C / Q such that the mod m representation of its Jacobian is tame. The method is to analyse the period matrices of hyperelliptic Mumford curves, which could be of independent interest. As an application, we study the tame version of the inverse Galois problem for symplectic matrix groups over finite fields.

Enhanced Squeezing and Entanglement in Nondegenerate Three-Level Laser Coupled to Squeezed Vacuum Reservoir

Squeezing and entanglement of a two-mode cascade laser, produced by a three-level atom which is initially prepared by a coherent superposition of the top and bottom levels then injected into a cavity coupled to a two-mode squeezed vacuum reservoir is discussed. I obtain stochastic differential equations associated with the normal ordering using the pertinent master equation. Making use of the solutions of the resulting differential equations, we determined the mean photon number for the cavity mode and their correlation, EPR variables, smallest eigenvalue of the symplectic matrix, intensity difference fluctuation, and photon number correlation. It is found that the squeezed vacuum reservoir increases the degree of the statistical and nonclassical features of light produced by the system. Furthermore, using the criteria developed by logarithm negativity and Hillery-Zubairy criteria, the quantum entanglement of the cavity mode is quantified. It is found that the degree of the entanglement for the system under consideration increases with the squeezing parameter of the squeezed vacuum reservoir.

Variational principles for symplectic eigenvalues

If A is a real $2n \times 2n$ positive definite matrix, then there exists a symplectic matrix M such that $M^TAM=\text {diag}(D, D),$ where D is a positive diagonal matrix with diagonal entries $d_1(A)\leqslant \cdots \leqslant d_n(A).$ We prove a maxmin principle for $d_k(A)$ akin to the classical Courant–Fisher–Weyl principle for Hermitian eigenvalues and use it to derive an analogue of the Weyl inequality $d_{i+j-1}(A+B)\geqslant d_i(A)+d_j(B).$

Faddeev–Jackiw quantization of Christ–Lee model

We analyze the constraints of Christ–Lee model by means of modified Faddeev–Jackiw formalism in Cartesian as well as polar coordinates. Further, we accomplish quantization à la Faddeev–Jackiw by choosing appropriate gauge conditions in both the coordinate systems. Finally, we establish gauge symmetries of Christ–Lee model with the help of zero-modes of the symplectic matrix.

Decomposition of symplectic matrices into products of symplectic unipotent matrices of index 2

In this article, it is proved that every symplectic matrix can be decomposed into a product of three symplectic unipotent matrices of index 2, i.e., every complex matrix A satisfying ATJA = J with J = [0 -

Dynamic Analysis of Flexible Rotor Based on Transfer Symplectic Matrix

In view of the numerical instability and low accuracy of the traditional transfer matrix method in solving the high-order critical speed of the rotor system, a new idea of incorporating the finite element method into the transfer matrix is proposed. Based on the variational principle, the transfer symplectic matrix of gyro rotors suitable for all kinds of boundary conditions and supporting conditions under the Hamilton system is derived by introducing dual variables. To verify the proposed method in rotor critical speed, a numerical analysis is adopted. The simulation experiment results show that, in the calculation of high-order critical speed, especially when exceeding the sixth critical speed, the numerical accuracy of the transfer symplectic matrix method is obviously better than that of the reference method. The relative errors between the numerical solution and the exact solution are 0.0347% and 0.2228%, respectively, at the sixth critical speed. The numerical example indicates the feasibility and superiority of the method, which provides the basis for the optimal design of the rotor system.