Vertex Operators of the q-Virasoro Algebra; Defining Relations,Adjoint Actions and Four Point Functions

Primary fields of the q-deformed Virasoro algebra are constructed. Commutation relations among the primary fields are studied. Adjoint actions of the deformed Virasoro current on the primary fields are represented by the shift operator f(x) = f(x). Four point functions of the primary fields enjoy the connection formula associated with the Boltzmann weights of the fusion Andrews–Baxter–Forrester model.     

Algebra of Screening Operators for the Deformed W n Algebra

We construct a family of intertwining operators (screening operators) between various Fock space modules over the deformed W _n algebra. They are given as integrals involving a product of screening currents and elliptic theta functions. We derive a set of quadratic relations among the screening operators, and use them to construct a Felder-type complex in the case of the deformed W ₃ algebra. Received: 3 March 1997 / Accepted: 20 May 1997     

Vertex Operators Arising from the Homogeneous Realization for

We use the underlying Fock space for the homogeneous vertex operator representation of the affine Lie algebra to construct a family of vertex operators. As an application, an irreducible module for an extended affine Lie algebra of type A _{N −1} coordinatized by a quantum torus ℂ _q of 2 variables (or 3 variables) is obtained. Moreover, this module turns out to be a highest weight module which is an analog of the basic module for affine Lie algebras. Received: 16 August 1999 / Accepted: 18 January 2000     

Interwining Operators for Vertex Representations of Toroidal Lie Algebras

The construction of the vertex representation of the toroidal Lie algebra ^[n] depends on the way of labelling the points in the dual ⁿ of the torus Tⁿ. Thus there is a built-in symmetry of the vertex representation with respect to the symmetry of Zⁿ. In conjunction with this, the energy operator L₀ gives rise to intertwining operators which reflect the symmetry of the vertex representation with respect to S¹ action on .     

Sharp Estimates in Ruelle Theorems for Matrix Transfer Operators

A matrix coefficient transfer operator , on the space of -sections of an m-dimensional vector bundle over n-dimensional compact manifold is considered. The spectral radius of is estimated by\para; and the essential spectral radius by

Matrix Representations for Vector Differential Operators in General Orthogonal Coordinates

The restrictions in arbitrary orthogonal coordinates to a unified matrix presentation of vector operations using generalized differential matrix operators [Y. Chen et al.: IEEE Trans. on Educ., Vol. 41, (1998), p. 61] are pointed out. The corrected matrix representation for the operations (A) and B(A) valid in arbitrary orthogonal curvilinear coordinates are obtained and shown to be consistent with the calculation using the dyadic technique. The presentation is accessible to undergraduate students.     

A Semiclassical Egorov Theorem and Quantum Ergodicity for Matrix Valued Operators

We study the semiclassical time evolution of observables given by matrix valued pseudodifferential operators and construct a decomposition of the Hilbert space L²(^d)ⁿ into a finite number of almost invariant subspaces. For a certain class of observables, that is preserved by the time evolution, we prove an Egorov theorem. We then associate with each almost invariant subspace of L²(^d)ⁿ a classical system on a product phase space T^*d×, where is a compact symplectic manifold on which the classical counterpart of the matrix degrees of freedom is represented. For the projections of eigenvectors of the quantum Hamiltonian to the almost invariant subspaces we finally prove quantum ergodicity to hold, if the associated classical systems are ergodic.     

Analytic Calculation of Arbitrary Matrix Elements for Boson Exponential Quadratic Operators

By means of the transformation relation between the ordinary form of boson exponential quadratic operators (BEQO) and its anti-normal product form, we present an effective method to conveniently calculate arbitrary matrix elements of BEQO. By this method, many important matrix elements can be calculated analytically. As a direct application, we obtain the exact solutions of the density matrix and partition function for general boson quadratic Hamiltonian without any information about the energy level.     

Analytic Expression of Arbitrary Matrix Elements for Boson Exponential Quadratic Polynomial Operators

Making use of the transformation relation among usual, normal, and antinormal ordering for the multimode boson exponential quadratic polynomial operators （BEQPO＇s）I we present the analytic expression of arbitrary matrix elements for BEQPO＇s. As a preliminary application, we obtain the exact expressions of partition function about the boson quadratic polynomial system, matrix elements in particle-number, coordinate, and momentum representation, and P representation for the BEQPO＇s.     

Trace Functionals for a Class of Pseudo-Differential Operators in R n

In this paper we define trace functionals on the algebra of pseudo-differential operators with cone-shaped exits to infinity. Furthermore, we improve the Weyl formula on the asymptotic distribution of eigenvalues and make use of it in order to establish inclusion relations between the interpolation normed ideals of compact operators in L ²(R ⁿ ) and the above operator classes.     

New Matrix Lie Algebra, a Powerful Tool for Constructing Multi-component C-KdV Equation Hierarchy

A set of new multi-component matrix Lie algebra is constructed, which is devoted to obtaining a new loop algebra A-2M. It follows that an isospectral problem is established. By making use of Tu scheme, a Liouville integrable multi-component hierarchy of soliton equations is generated, which possesses the multi-component Hamiltonian structures. As its reduction cases, the multi-component C-KdV hierarchy is given. Finally, the multi-component integrable coupling system of C-KdV hierarchy is presented through enlarging matrix spectral problem.     

KTiOPO4, KTiOAsO4, and KNbO3 crystals for mid-infrared femtosecond optical parametric amplifiers: analysis and comparison

We have designed a simple mid-IR femtosecond OPA seeded by a white-light continuum and delivering very stable, largely tunable and energetic output pulses. This OPA requires fewer than 200 7J of 120-fs Ti:sapphire pump pulses as energy. This system allowed us to experimentally compare the performances of potassium crystals (KTP, KTA, and KNbO₃) for the generation of up to 100-fs Fourier-transform-limited pulses tunable in 1-4.6 7m mid-IR wavelengths.     

Evaluation of Optical Properties of Self-Frequency-Doubling Crystal Yb：GdYAl3（BO3）4 for Laser Applications

Yb：GdYAl3 （B03）4 （Yb：GdYAB） is investigated as a new laser crystal for potential applications in self-frequency doubling. The emission and absorption properties of Yb：GdYAB crystal are studied, and the emission decay times of the upper laser level are measured. The emission cross sections are evaluated using the absorption cross section and principle of reciprocity. The other laser performance parameters, such as the minimum inversion fraction βmin, pump saturation intensity Isat and minimum pump intensity 1rain, are also calculated. The results are discussed in the framework of requirements for an effective diode-pumped Yb^3＋ laser system. Yb：GdYAB is expected to exhibit the most useful laser properties and to be superior to Yb：YAB crystal that has been excellent self-frequency-doubling crystal at present in many key spectroscopic parameter values.     

LaNiO3 under-electrode layers for the growth of textured (Pb0.4Zr0.6)TiO3 thin films using pulsed laser deposition

We investigated the structural properties of LaNiO₃ thin films of three different thicknesses deposited by pulsed laser deposition on Si(001) mainly by using a synchrotron X-ray scattering measurement. The LaNiO₃ thin films were grown with the (00l) preferred growth direction, showing completely random distribution in the in-plane direction. In the early stage of the growth, the film was almost unstrained. However, as the film grew further, tensile strain was markedly involved. Also its surface became rougher but its crystalline quality improved significantly with increasing film thickness. A completely (00l)-oriented (Pb_0.4Zr_0.6)TiO₃ thin film was successfully grown on such a LaNiO₃/Si(001) substrate at a substrate temperature of 350 °C by using the same pulsed laser deposition. Our results show that the LaNiO₃ film can serve effectively as a bottom electrode layer for the preparation of a well-oriented (Pb_0.4Zr_0.6)TiO₃ thin film on Si substrates.     

Waveform digitizing for LaBr3/NaI phoswich detector

The detection efficiency of phoswich detector starts to decrease when Compton scattering becomes significant. Events with energy deposit in both scintillators, if not rejected, are not useful for spectral analysis as the full energy of the incident photon cannot be reconstructed with conventional readout. We show that once the system response is carefully calibrated, the full energy of those double deposit events can be reconstructed using a waveform digitizer as the readout. Our experiment suggests that the efficiency of a photopeak at 662 keV can be increased by a factor of 2 using our LaBr₃/NaI phoswich detector.     

Drinfeld–Sokolov Reduction for Difference Operators and Deformations of -Algebras¶I. The Case of Virasoro Algebra

We propose a q-difference version of the Drinfeld-Sokolov reduction scheme, which gives us q-deformations of the classical -algebras by reduction from Poisson-Lie loop groups. We consider in detail the case of SL ₂ . The nontrivial consistency conditions fix the choice of the classical r-matrix defining the Poisson-Lie structure on the loop group LSL ₂ , and this leads to a new elliptic classical r-matrix. The reduced Poisson algebra coincides with the deformation of the classical Virasoro algebra previously defined in [19]. We also consider a discrete analogue of this Poisson algebra. In the second part [31] the construction is generalized to the case of an arbitrary semisimple Lie algebra. Received: 20 April 1997 / Accepted: 22 July 1997     

Energy Bounds for Static Spherically Symmetric Spacetime in f(R,G) Gravity

The main purpose of this paper is to investigate energy bounds in the context of f(R,G) gravity. To meet this aim, we choose static spherically symmetric spacetime in f(R,G) gravity to develop the field equations. We select three different models of f(R,G) gravity, which are thoroughly discussed in the literature. Firstly, the inequalities are formulated using energy bounds and then viability of the considered models are checked respectively. Graphical analysis show that specific f(R,G) gravity models are satisfied under suitable values of model parameters. It is shown that in a certain case energy bounds are satisfied expect SEC, which supports the late time acceleration expansion of unverse.