Discrete wavelet structure and discrete energy of classical plane light waves

We find by the wavelet transform that the classical plane light wave of linear polarization can be decomposed into a series of discrete Morlet wavelets.In the theoretical frame,the energy of the classical light wave becomes discrete;interestingly,the discretization is consistent with the energy division of P portions in Planck radiation theory,where P is an integer.It is shown that the changeable energy of a basic plane light wave packet or wave train is H_(0k)=nP0 kω(n=1,2,3,...;k=|k|),with discrete wavelet structure parameter n,wave vector k and idler frequency ω,and a constant p0 k.The wave-particle duality from the Mach-Zehnder interference of single photons is simulated by using random basic plane light wave packets.

Three New Hydrogenosuccinate and Succinate Adducts Complexes:Synthesis and Spectroscopic Study

One new hydrogenosuccinate and two succinate adduct and complex have been synthesized and studied by infrared,UV-Visible and NMR(Nuclear Magnetic Resonance Spectroscopy)^(119)Sn spectroscopies.The suggested structure is discrete,the hydrogenosuccinate behaving as a monodentate ligand or only involved in hydrogen bonding,the environment around the magnesium centre being triangular(compound 3).The succiate anion is a monochelating ligand(compound 1 and 2).In all the suggested structures,when extra hydrogen bonds are considered,supramolecular architectures are obtained(compound 2 and 3).

Some Salicylato New Organotin (IV) and Copper Chloride Adducts and Derivative: Synthesis and Spectroscopic Study

Three new salicylate complexes and derivatives have been synthesized and studied by infrared, 119Sn NMR and UV-visible spectroscopies. The suggested structures for the two compounds are discrete with NH-O and NH-Cl hydrogen bonds. The salicylate oxyanion is monochelating for the first salicylate compound with an octahedral tin (IV) centre and monodentate for the second salicylate compound, the environments around the tin centre being tetrahedral. For the cooper complex, the salicylate ligand is monochelating and the environments around the copper atom centre are tetrahedral.

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

Starting from a discrete spectral problem, a hierarchy of integrable lattice soliton equations is derived. It is shown that the hierarchy is completely integrable in the Liouville sense and possesses discrete bi-Hamiltonian structure. A new integrable symplectic map and finite-dimensional integrable systems are given by nonlinearization method. The binary Bargmann constraint gives rise to a Biicklund transformation for the resulting integrable lattice equations. At last, conservation laws of the hierarchy are presented.

A Hierarchy of Nonlinear Lattice Soliton Equations and Its Darboux Transformation

A hierarchy of nonlinear lattice soliton equations is derived from a new discrete spectral problem. The Hamiltonian structure of the resulting hierarchy is constructed by using a trace identity formula. Moreover, a Darboux transformation is established with the help of gauge transformations of Lax pairs for the typical lattice soliton equations. The exact solutions are given by applying the Darboux transformation.

New Positive and Negative Hierarchies of Integrable Differential-Difference Equations and Conservation Laws

Two hierarchies of nonlinear integrable positive and negative lattice equations are derived from a discrete spectrak problem. The two lattice hierarchies are proved to have discrete zero curvature representations associated with a discrete spectral problem, which also shows that the positive and negative hierarchies correspond to positive and negative power expansions of Lax operators with respect to the spectral parameter, respectively. Moreover, the integrable lattice models in the positive hierarchy are of polynomial type, and the integrable lattice models in the negative hierarchy are of rational type. Further, we construct infinite conservation laws about the positive hierarchy.

Integrable Properties Associated with a Discrete Three-by-Three Matrix Spectral Problem

Based on a new discrete three-by-three matrix spectral problem, a hierarchy of integrable lattice equations with three potentials is proposed through discrete zero-curvature representation, and the resulting integrable lattice equation reduces to the classical Toda lattice equation. It is shown that the hierarchy possesses a HamiItonian structure and a hereditary recursion operator. Finally, infinitely many conservation laws of corresponding lattice systems are obtained by a direct way.

Hamiltonian System of New Nonlinear Lattice Equations

A 3-dimensional Lie algebra sμ（3） is obtained with the help of the known Lie algebra. Based on the sμ（3）, a new discrete 3 × 3 matrix spectral problem with three potentials is constructed. In virtue of discrete zero curvature equations, a new matrix Lax representation for the hierarchy of the discrete lattice soliton equations is acquired. It is shown that the hierarchy possesses a Hamiltonian operator and a hereditary recursion operator, which implies that there exist infinitely many common commuting symmetries and infinitely many common commuting conserved functionals.

Algebraic Structure of Discrete Zero Curvature Equations and Master Symmetries of Discrete Evolution Equations

In this paper, based on a discrete spectral problem and the corresponding zero curvature representation,the isospectral and nonisospectral lattice hierarchies are proposed. An algebraic structure of discrete zero curvature equations is then established for such integrable systems. the commutation relations of Lax operators corresponding to the isospectral and non-isospectral lattice flows are worked out, the master symmetries of each lattice equation in the isospectral hierarchyand are generated, thus a τ-symmetry algebra for the lattice integrable systems is engendered from this theory.

A ubiquitous knowledgeable data representation model(UKRM) for three-dimensional geographic information systems(3D GIS)

In the face of complicated, diversified three-dimensional world, the existing 3D GIS data models suffer from certain issues such as data incompatibility, insufficiency in data representation and representation types, among others. It is often hard to meet the requirements of multiple application purposes(users) related to GIS spatial data management and data query and analysis, especially in the case of massive spatial objects. In this study, according to the habits of human thinking and recognition, discrete expressions(such as discrete curved surface(DCS), and discrete body(DB)) were integrated and two novel representation types(including function structure and mapping structure) were put forward. A flexible and extensible ubiquitous knowledgeable data representation model(UKRM) was then constructed, in which structurally heterogeneous multiple expressions(including boundary representation(B-rep), constructive solid geometry(CSG), functional/parameter representation, etc.) were normalized. GIS's ability in representing the massive, complicated and diversified 3D world was thus greatly enhanced. In addition, data reuse was realized, and the bridge linking static GIS to dynamic GIS was built up. Primary experimental results illustrated that UKRM was overwhelmingly superior to the current data models(e.g. IFC, City GML) in describing both regular and irregular spatial objects.

DISCRETE ENERGY ANALYSIS OF THE THIRD-ORDER VARIABLE-STEP BDF TIME-STEPPING FOR DIFFUSION EQUATIONS

This is one of our series works on discrete energy analysis of the variable-step BDF schemes.In this part,we present stability and convergence analysis of the third-order BDF(BDF3)schemes with variable steps for linear diffusion equations,see,e.g.,[SIAM J.Numer.Anal.,58:2294-2314]and[Math.Comp.,90:1207-1226]for our previous works on the BDF2 scheme.To this aim,we first build up a discrete gradient structure of the variable-step BDF3 formula under the condition that the adjacent step ratios are less than 1.4877,by which we can establish a discrete energy dissipation law.Mesh-robust stability and convergence analysis in the L^(2) norm are then obtained.Here the mesh robustness means that the solution errors are well controlled by the maximum time-step size but independent of the adjacent time-step ratios.We also present numerical tests to support our theoretical results.

Photo Switchable Two-step Photochromism in a Series of Ln-Phosphonate(Ln=Dy,Gd,Tb,Y)Dinuclear Complexes

Exploring the discrete complexes with multi-step coloration is still a challenge in the field of electron transfer photochromic materials. Herein, we synthesized a series of dinuclear Ln-diphosphonate compounds[Ln=Dy(1);Gd(2);Tb(3);Y(4)] with a remarkably and reversibly photoactive coloration phenomenon. These compounds showed two-step coloration behavior, which were the first discrete architectures in the reported electron transfer photochromic complexes. This two-step coloration phenomenon was originated from the large distortion of H3-TPT acceptors, which in turn reduced the π-conjugation of electron acceptors and slowed the decay process of electron transfer. The photogenerated stable doublet radicals originated from electron transfer from diphosphonate donor to polypyridine acceptor in these complexes were detected by UV-Vis and electron spin resonance(ESR) spectra. Furthermore, the photogenerated radicals were estimated by direct current magnetic susceptibilities and variable temperature ESR spectra, suggesting the doublet radicals in the dinuclear structure for all the compounds. This work revealed a series of discrete phosphonate-based systems with a multi-step coloration process, providing a new pathway for designing multicolor photochromic materials with potential photoswitching or other applications.

Adaptive Stabilization of Neural Control for Robot Trajectory Tracking

A neural network (NN) based adaptive control law is proposed for the tracking control of an n link robot manipulator with unknown dynamic nonlinearities. Basis function like nets are employed to approximate the plant nonlinearities, and the bound on the NN reconstruction error is assumed to be unknown. The proposed NN based adaptive control approach integrates an NN approach with an adaptive implementation of discrete variable structure control with a simple estimation law to estimate the upper bound on the NN reconstruction error and an additional control input to be updated as a function of the estimate. Lyapunov stability theory is used to prove the uniform ultimate boundedness of the tracking error.