A Hierarchy of Integrable Nonlinear Lattice Equations and New Integrable Symplectic Map

A discrete spectral problem is discussed, and a hierarchy of integrable nonlinear lattice equations related to this spectral problem is devised. The new integrable symplectic map and finite-dimensional integrable systems are given by nonlinearization method. The binary Bargmann constraint gives rise to a B?cklund transformation for the resulting integrable lattice equations.

A New Integrable Lattice Hierarchy and Its Two Discrete Integrable Couplings

A new and efficient way is presented for discrete integrable couplings with the help of two semi-direct sum Lie algebras. As its applications, two discrete integrable couplings associated with the lattice equation are worked out. The approach can be used to study other discrete integrable couplings of the discrete hierarchies of solition equations.

AN ABLOWITZ-LADIK INTEGRABLE LATTICE HIERARCHY WITH MULTIPLE POTENTIALS

Within the zero curvature formulation,a hierarchy of integrable lattice equations is constructed from an arbitrary-order matrix discrete spectral problem of Ablowitz-Ladik type.The existence of infinitely many symmetries and conserved functionals is a consequence of the Lax operator algebra and the trace identity.When the involved two potential vectors are scalar,all the resulting integrable lattice equations are reduced to the standard Ablowitz-Ladik hierarchy.

Extended l＊-Module

Theory of uncertainty reasoning base on l-module of true value field,in this paper,an extended l-module was proposed,some properties and lattice measure were discussed,then the lattice integral on l-module was gained.

Influence of relative positions between RF coil and crucible on sapphire crystals by edge-defined film-fed growth(EFG) technique

To obtain the stable temperature field required for growing sapphire crystals, the influence of relative positions between RF coil and crucible on the performances of sapphires produced by edge-defined film-fed growth（EFG） technique was investigated. For comparison, the crucible was located at the top（case A） and the middle（case B） of the RF coil, respectively. Furthermore, the lattice integrities were studied by the double-crystal X-ray diffraction, and the dislocations were observed under the optical microscope and atomic force microscope after corroding in molten KOH at 390 ℃. The crystals in case B exhibit better lattice integrity with smaller full width at half maximum of 29.13 rad·s, while the value in case A is 45.17 rad·s. The morphologies of dislocation etch pits in both cases show typical triangular symmetry with smooth surfaces. However, the dislocation density of 2.8×104 cm-2 in case B is only half of that in case A, and the distribution is more uniform, compared to the U-shaper in case A.

A lattice measuring method based on integral imaging technology

Aiming at the problem that the lattice feature exceeds the view field of the scanning electron microscope(SEM)measuring system,a new lattice measuring method is proposed based on integral imaging technology.When the system works,the SEM measuring system is equivalent to an integral image acquisition system.Firstly,a lattice measuring method is researched based on integral imaging theory.Secondly,the system parameters are calibrated by the VLSI lattice standard.Finally,the value of the lattice standard to be tested is determined based on the calibration parameters and the lattice measuring algorithm.The experimental results show that,compared with the traditional electron microscope measurement method,the relative error of the measured value of the algorithm is maintained within 0.2%,with the same level of measurement accuracy,but it expands the field of view of the electron microscope measurement system,which is suitable for the measurement of samples under high magnification.

Electromagnetic Field Due to a Loop Antenna Buried in a Medium with Complex Boundaries

The radiation of a loop antenna embedded in a dissipative medium with complex boundaries isanalyzed by a perturbation method and an efficient fast multiple-integration technique. But theperturbation method can not be used directly because there is a finite-length metal cylinder in the vicinityof the loop antenna. The prolate ellipsoid equivalence of the metal cylinder is made, then the cylinder maybe removed and the perturbation method is valid. Numerical results indicate that the approach is accurateat low frequencies and stable.

Hamiltonian Forms for a Hierarchy of Discrete Integrable Coupling Systems

A semi-direct sum of two Lie algebras of four-by-four matrices is presented,and a discrete four-by-fourmatrix spectral problem is introduced.A hierarchy of discrete integrable coupling systems is derived.The obtainedintegrable coupling systems are all written in their Hamiltonian forms by the discrete variational identity.Finally,we prove that the lattice equations in the obtained integrable coupling systems are all Liouville integrable discreteHamiltonian systems.

Lax pair formulation for the open boundary Osp(1|2)spin chain

Based on the Lax pair formulation,we study the integrable conditions of the Osp(1|2)spin chain with open boundaries.We consider both the non-graded and graded versions of the Osp(1|2)chain.The Lax pair operators M_(±)for the boundaries can be induced by the Lax operator M_(j)for the bulk of the system.They correspond to the reflection equations(RE)and the Yang-Baxter equation,respectively.We further calculate the boundary K-matrices for both the non-graded and graded versions of the model with open boundaries.The double row monodromy matrix and transfer matrix of the spin chain have also been constructed.The K-matrices obtained from the Lax pair formulation are consistent with those from Sklyanin’s RE.This construction is another way to prove the quantum integrability of the Osp(1|2)chain.We find that the Lax pair formulation has advantages in dealing with the boundary terms of the supersymmetric model.