A notion of a quantum automorphism group of a finite quantum group, generalising that of a classical automorphism group of a finite group, is proposed and a corresponding existence result proved.
Discrete choice models describe the choices made by decision makers among alternatives and play an important role in transportation planning, marketing research and other applications. The mixed multinomial logit (MMNL) model is a popular discrete ch
oice model that captures heterogeneity in the preferences of decision makers through random coefficients. While Markov chain Monte Carlo methods provide the Bayesian analogue to classical procedures for estimating MMNL models, computations can be prohibitively expensive for large datasets. Approximate inference can be obtained using variational methods at a lower computational cost with competitive accuracy. In this paper, we develop variational methods for estimating MMNL models that allow random coefficients to be correlated in the posterior and can be extended easily to large-scale datasets. We explore three alternatives: (1) Laplace variational inference, (2) nonconjugate variational message passing and (3) stochastic linear regression. Their performances are compared using real and simulated data. To accelerate convergence for large datasets, we develop stochastic variational inference for MMNL models using each of the above alternatives. Stochastic variational inference allows data to be processed in minibatches by optimizing global variational parameters using stochastic gradient approximation. A novel strategy for increasing minibatch sizes adaptively within stochastic variational inference is proposed.
The entanglement, purity and energy of two isolated two-level atoms which are initially prepared in Bell state and each interacts with a thermal cavity field are investigated by considering the atomic motion and the field-mode structure. We achieve t
he analytical solutions of the atomic qubits by using the algebraic dynamical approach and the influences of the field-mode structure parameter, the strength of the thermal field and the detuning on the entanglement, purity and energy are discussed. We also investigate the state evolution of the atomic qubits based on the entanglement-purity-energy diagrams. Our results show that the disentanglement process of the atomic qubits accompanies by excitations transferring from atomic subsystem to cavity field modes and atomic state from a pure state convert to the mixed states.
A graphene nanoribbon (GNR) with orientation along its principle axis was obtained through a mechanical tearing process, and a quantum dot device was fabricated from the GNR. We have studied the transport property of the GNR quantum dot device down t
o dilution refrigerator temperatures. Two-fold charging periodicity was observed in the Coulomb-blockade measurement, signaling a shell-filling process with broken valley degeneracy. In one of the smaller Coulomb diamonds, Kondo-like resonance were observed, with two conductance peaks displaced symmetrically from the zero bias voltage. The splitting of Kondo resonance at zero magnetic field suggests spin-polarization of the quantum dot, possibly due to the edge states of a zigzag GNR.
Using x-ray resonant magnetic scattering and x-ray magnetic circular dichroism, techniques that are element specific, we have elucidated the role of Ho3+ in multiferroic HoMnO3. In zero field, Ho3+ orders antiferromagnetically with moments aligned al
ong the hexagonal c direction below 40 K, and undergoes a transition to another magnetic structure below 4.5 K. In applied electric fields of up to 1x10^7 V/m, the magnetic structure of Ho3+ remains unchanged.
