Symmetry Groups and New Exact Solutions to （2＋1）-Dimensional Variable Coefficient Canonical Generalized KP Equation

In this paper, the modified CK＇s direct method to find symmetry groups of nonlinear partial differential equation is extended to （2＋1）-dimensional variable coeffficient canonical generalized KP （VCCGKP） equation. As a result, symmetry groups, Lie point symmetry group and Lie symmetry for the VCCGKP equation are obtained. In fact, the Lie point symmetry group coincides with that obtained by the standard Lie group approach. Applying the given Lie symmetry, we obtain five types of similarity reductions and a lot of new exact solutions, including hyperbolic function solutions, triangular periodic solutions, Jacobi elliptic function solutions and rational solutions, for the VCCGKP equation.

New Multiple Soliton-like and Periodic Solutions for （2＋l）-Dimensional Canonical Generalized KP Equation with Variable Coefficients

In this paper, the generalized ranch function method is extended to （2＋1）-dimensianal canonical generalized KP （CGKP） equation with variable coetfficients. Taking advantage of the Riccati equation, many explicit exact solutions, which contain multiple soliton-like and periodic solutions, are obtained for the （2＋1）-dimensional OGKP equation with variable coetffcients.

Variation principle of piezothermoelastic bodies,canonical equation and homogeneous equation

Combining the symplectic variations theory, the homogeneous control equation and isopaxametric element homogeneous formulations for piezothermoelastic hybrid laminates problems were deduced. Firstly, based on the generalized Hamilton variation principle, the non-homogeneous Hamilton canonical equation for piezothermoelastic bodies was derived. Then the symplectic relationship of variations in the thermal equilibrium formulations and gradient equations was considered, and the non-homogeneous canonical equation was transformed to homogeneous control equation for solving independently the coupling problem of piezothermoelastic bodies by the incensement of dimensions of the canonical equation. For the convenience of deriving Hamilton isopaxametric element formulations with four nodes, one can consider the temperature gradient equation as constitutive relation and reconstruct new variation principle. The homogeneous equation simplifies greatly the solution programs which axe often performed to solve nonhomogeneous equation and second order differential equation on the thermal equilibrium and gradient relationship.

The Unification of Classical and Quantum Gravity

The favored classical variables that are promoted to quantum operators are divided into three sets that feature constant positive curvatures, constant zero curvatures, as well as constant negative curvatures. This list covers the spin variables, the canonical variables, and the affine variables, and these three topics will be briefly reviewed. In this discussion, appropriate coherent states are introduced which are the principal items that are critical in the unification of relevant classical and quantum realms. This analysis can also serve to unify classical gravity and quantum gravity without any speculative aspects.

EXTREMES OF RATIOS OF DETERMINANTS AND CANONICAL CORRELATION VARIABLES

This article develops some extremes of the ratios of determinants. The results are themultivariate extensions of the extremes of quadratic forms, and can be applied to finding thecanonicai correlation variables of two random vectors. Hence a group of canonical correlationvariables is a solution of the extreme of the ratio of determinants.