NONCLASSICAL PLANE-CRYSTALLOGRAPHIC GROUPS AND THEIR APPLICATIONS Ⅰ

A new nonclassical crystallographic group (NCG) theoretical system is set up.This system can describe infinite kinds of nonclassical periodic structures,especially for those with locally n-fold rotational symmetries forbidden by the rules of the classical crystallography. The formal classification of NCGs is given.

Nonclassical Symmetries for Nonlinear Partial Differential Equations via Compatibility

The determining equations for the nonclassical symmetry reductions of nonlinear partial differential equations with arbitrary order can be obtained by requiring the compatibility between the original equations and the invariant surface conditions. The （2＋1）-dimensional shallow water wave equation, Boussinesq equation, and the dispersive wave equations in shallow water serve as examples i11ustrating how compatibility leads quickly and easily to the determining equations for their nonclassical symmetries.

Classical and nonclassical symmetry classifications of nonlinear wave equation with dissipation

A complete classical symmetry classification and a nonclassical symmetry classification of a class of nonlinear wave equations are given with three arbitrary parameter functions. The obtained results show that such nonlinear wave equations admit richer classical and nonclassical symmetries, leading to the conservation laws and the reduction of the wave equations. Some exact solutions of the considered wave equations for particular cases are derived.

Comparison of nonclassicality between photon-added and photon-subtracted squeezed vacuum states

In this paper, we introduce photon-added and photon-subtracted squeezed vacuum state （PASV and PSSV） and obtain their normalized factors, which have the similar forms involved in Lengendre polynomials. Moreover, we give the compact expressions of Wigner function, which are related to single-variable Hermite polynomials. Especially, we compare their nonclassicality in terms of Mandel Q-factor and the negativity of Wigner function.

A Note on Nonclassical Symmetries of a Class of Nonlinear Partial Differential Equations and Compatibility

The nonclassical symmetries of a class of nonlinear partial differential equations obtained by the compatibility method is investigated. We show the nonclassicaJ symmetries obtained in [J. Math. Anal. Appl. 289 （2004） 55, J. Math. Anal. Appl. 311 （2005） 479] are not all the nonclassical symmetries. Based on a new assume on the form of invariant surface condition, all the nonclassical symmetries for a class of nonlinear partial differential equations can be obtained through the compatibility method. The nonlinear Klein-Gordon equation and the Cahn-Hilliard equations all serve as examples showing the compatibility method leads quickly and easily to the determining equations for their all nonclassical symmetries for two equations.

Global Attractors for Nonclassical Diffusion Equations of Kirchhoff Type

In this article, the well-posedness and long-time behavior of a nonclassical diffusion equation of Kirchhoff type are considered. Using the method of Galerkin approximation, the existence and uniqueness of solutions are proved. At last, the existence of global attractors and its upper semicontinuous property are discussed.

Multi-crack imaging using nonclassical nonlinear acoustic method

Solid materials with cracks exhibit the nonclassical nonlinear acoustical behavior. The micro-defects in solid materials can be detected by nonlinear elastic wave spectroscopy （NEWS） method with a time-reversal （TR） mirror. While defects lie in viscoelastic solid material with different distances from one another, the nonlinear and hysteretic stress-strain relation is established with Preisach-Mayergoyz （PM） model in crack zone. Pulse inversion （PI） and TR methods are used in numerical simulation and defect locations can be determined from images obtained by the maximum value. Since false-positive defects might appear and degrade the imaging when the defects are located quite closely, the maximum value imaging with a time window is introduced to analyze how defects affect each other and how the fake one occurs. Furthermore, NEWS-TR- NEWS method is put forward to improve NEWS-TR scheme, with another forward propagation （NEWS） added to the existing phases （NEWS and TR）. In the added phase, scanner locations are determined by locations of all defects imaged in previous phases, so that whether an imaged defect is real can be deduced. NEWS-TR-NEWS method is proved to be effective to distinguish real defects from the false-positive ones. Moreover, it is also helpful to detect the crack that is weaker than others during imaging procedure.

Nonclassicality of a two-variable Hermite polynomial state

The nonclassicality of the two-variable Hermite polynomial state is investigated. It is found that the two-variable Hermite polynomial state can be considered as a two-mode photon subtracted squeezed vacuum state. A compact expression for the Wigner function is also derived analytically by using the Weyl-ordered operator invariance under similar transformations. Especially, the nonclassicality is discussed in terms of the negativity of the Wigner function. Then violations of Bell＇s inequality for the two-variable Hermite polynomial state are studied.

Exponential Attractors of the Nonclassical Diffusion Equations with Lower Regular Forcing Term

In this article, we prove the existence of exponential attractors of the nonclassical diffusion equation with critical nonlinearity and lower regular forcing term. As an additional product, we show that the fractal dimension of the global attractors of this problem is finite.

Realization of Nonclassical States and Two-Mode Coupling in the Motion of Two Trapped Ions

We propose a scheme to generate various nonclassical vibrational states in the collective motion of twotrapped ions, such as squeezed states, Schrodinger cat states, and SU(2) states. It is based on Raman-type excitations.Two-mode coupling between the center-of-mass and relative vibrational modes can also be realized.

Generation of Entanglement Between Two Noninteracting Qubits via Interaction with Nonclassical Radiation Field

We study the interaction between a single-mode quantized field and a quantum system composed of two qubits. We suppose that two qubits initially be prepared in the mixed and separable state, and study how entanglement between two qubits arises in the course of evolution according to the Jaynes-Cummings type interaction with nonclassical radiation field. We also investigate the relation between entanglement and purity of qubit subsystem. We show that different photon statistics have different effects on the dynamical behavior of the qubit subsystem. When the qubits are initially prepared in the maximally mixed and separable state, for coherent state field we cannot find entanglement between two qubits; for squeezed state field entanglement between two qubits exists in several short period of time; for even and odd coherent state fields of large photon number, the dynamical behavior of the entanglement between two qubits shows collapse and revival phenomenon. For odd coherent state field of small photon number, the entanglement with both qubits initially prepared in maximally mixed state can be stronger than that with both qubits initially prepared in pure states. For fields of small photon number, the entanglement strongly depends on the states they are initially prepared in. For coherent state field, and odd and even coherent state fields of large photon number, the entanglement only depends on the purity of the initial state of qubit subsystem. We also show that during the evolution the unentangled state may be purer than the entangled state, and the maximum degree of entanglement may not occur at the time when the qubit subsystem is in the purist state.

Barut-Girardello and Gilmore-Perelomov coherent states for pseudoharmonic oscillators and their nonclassical properties: Factorization method

In this paper we try to introduce the ladder operators associated with the pseudoharmonic oscillator, after solving the corresponding Schrrdinger equation by using the factorization method. The obtained generalized raising and lowering operators naturally lead us to the Dirac representation space of the system which is much easier to work with, in comparison to the functional Hilbert space. The SU（1, 1） dynamical symmetry group associated with the considered system is exactly established through investigating the fact that the deduced operators satisfy appropriate commutation relations. This result enables us to construct two important and distinct classes of Barut-Girardello and Gilmore-Perelomov coherent states associated with the system. Finally, their identities as the most important task are exactly resolved and some of their nonclassical properties are illustrated, numerically.

NONCLASSICAL POTENTIAL SYMMETRIES AND INVARIANT SOLUTIONS OF HEAT EQUATION

Some nonclassical potential symmetry generators and group-invariant solutions of heat equation and wave equation were determined. It is shown that some new explicit solutions of partial differential equations in conserved form can he constructed by using the nonclassical potential symmetry generators which are derived from their adjoint system. These explicit solutions cannot he obtained by using the Lie or Lie-Baeicklund symmetry group generators of differential equations.

Algebraic and group treatments to nonlinear displaced number states and their nonclassicality features:A new approach

Recently, nonlinear displaced number states （NDNSs） have been manually introduced, in which the deformation function f（n） has been artificially added to the previously well-known displaced number states （DNSs）. Indeed, just a simple comparison has been performed between the standard coherent state and nonlinear coherent state for the formation of NDNSs. In the present paper, after expressing enough physical motivation of our procedure, four distinct classes of NDNSs are presented by applying algebraic and group treatments. To achieve this purpose, by considering the DNSs and recalling the nonlinear coherent states formalism, the NDNSs are logically defined through an algebraic consideration. In addition, by using a particular class of Gilmore-Perelomov-type of SU（1,1） and a class of SU（2） coherent states, the NDNSs are introduced via group-theoretical approach. Then, in order to examine the nonclassical behavior of these states, sub-Poissonian statistics by evaluating Mandel parameter and Wigner quasi-probability distribution function associated with the obtained NDNSs are discussed, in detail.

Nonclassical Plane-crystallographic Groups and Their Applications Ⅲ

Six kinds of nonclassical periodic lattices with locally 10-fold rotational symmetries are proposed. They can be delineated via nonclassical plane-crystallographic groups. The projections on the planes of corresponding unit cells consisting of embedding polyhedra generate the periodic lattices, respectively. The Fourier-transform patterns of the periodiclattices have almost perfect 10-fold rotational symmetries, which are very similar to those displaying in the electron-diffraction patterns of so-called quasicrystals.

The transfer of the quantum correlation from two-mode nonclassical state field to the supercurrents in two distant SQUID rings

We have considered two distant mesoscopic superconducting quantum interference device （SQUID） rings A and B in the presence of two-mode nonclassical state fields and investigated the correlation of the supercurrents in the two rings using the normalized correlation function CAB. We show that when the parameter c~ is very small for the separable state with the density matrix ρ = {│α,-α） （α,-α│ ＋ │-α, α） （-α, α│}/2 and entangled coherent state {（ECS） [u） = N1（│α, -α） ＋ │-α, α）} fields, the dynamic behaviours of the normalized correlation function CAB are similar, but it is quite different for the entangled coherent state │u＇） = N2（│α,-α） - │-α, α）} field. When the parameter α is very large, the dynamic behaviours of CAB are almost the same for the separable state, entangled coherent state │u） and [u＇〉 fields. For the two-mode squeezed vacuum state field the maximum of CAB increases monotonically with the squeezing parameter γ, and as γ→ ∞ , CAB→ 1. This means that the supercurrents in the two rings A and B are quantum mechanically correlated perfectly. It is concluded that not all the quantum correlations in the two-mode nonclassical state field can be transferred to the supercurrents; and the transfer depends on the state of the two-mode nonclassical state field prepared.

Nonlinear Quantum Optical Springs and Their Nonclassical Properties

The original idea of quantum optical spring arises from the requirement of quantization of the frequency of oscillations in the Hamiltonian of harmonic oscillator. This purpose is achieved by considering a spring whose constant （and so its frequency） depends on the quantum states ofanother system. Recently, it is realized that by the assumption of frequency modulation of ω to ω √1＋ μα＋α the mentioned idea can be established. In the present paper, we generalize the approach of quantum optical spring with particular attention to the dependence or trequency to the intensity of radiation field that naturally observes in the nonlinear coherent states, from which we arrive at a physical system has been called by us as nonlinear quantum optical spring. Then, after the introduction of the generalized tlamiltonian of nonlinear quantum optical spring and it＇s solution, we will investigate the nonclassical properties of the obtained states. Specially, typical collapse and revival in the distribution functions and squeezing parameters, as particular quantum features, will be revealed.

Blow-up Results for the Nonclassical Diffusion Equations with Dynamical Boundary Condition

In this article,under suitable conditions on the internal term f(·)and the bound-ary nonlinear term g(·),we prove blow-up results for a class of nonclassical diffusion equations with nonlinear boundary condition.

Nonclassicality of photon-modulated atomic coherent states in the Schwinger bosonic realization

We theoretically introduce two new photon-modulated atomic coherent states(ACSs)via using the Schwinger bosonic representation of the angular momentum operators(the sequential operations J±n)on an ACS,and investigate their nonclassicality using the Wigner distribution,photon number distribution,and entanglement entropy.It is found that photonmodulated ACSs possess more stronger nonclassicality than the original ACS in certain regions ofτ,the nonclassicality enhances with increasing number n of the operations J±and the operation J+(-)n enhances the entanglement in the region of small(large)τ.

A NONCLASSICAL LAW OF ITERATED LOGARITHM FOR NEGATIVELY ASSOCIATED RANDOM VARIABLES

A nonclassical law of iterated logarithm that holds for a stationary negatively associated sequence of random variables with finite variance is proved in this paper. The proof is based on a Rosenthal type maximal inequality and the subsequence method.This result extends the work of Klesov,Rosalsky (2001) and Shao,Su (1999).