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The symplectic method for the elastic problem of decagonal symmetric 2D quasicrystal plates with 2 opposite edges simply supported, was discussed. The basic equations of the elastic theory for decagonal symmetric 2D quasicrystals were transformed into the Hamilton dual equations. With the method of separation of variables, the symplectic eigenvalues of the corresponding Hamilton operator matrix and the symplectic eigenfunction system were obtained. The completeness of the symplectic eigenfunction system of the Hamilton operator matrix in the sense of the Cauchy principal value was proved. Based on the symplectic eigenfunction expansion of the Hamilton system, the analytical solution to the bending problem of the decagonal symmetric 2D quasicrystal plate was given. [ABSTRACT FROM AUTHOR]

In order to investigate the influence of small perturbations on the invariants of dynamical systems, we study the approximate Noether invariants of nonconservative systems under El-Nabulsi exponential model and El-Nabulsi power-law model. According to Hamilton’s principle with nonstandard Lagrangians as its action functional, the dynamics equations of El-Nabulsi type for nonconservative systems are established. On this basis, due to the invariance of the functional under infinitesimal transformation, the approximate Noether invariants of nonconservative systems under small disturbance are obtained. When not disturbed, the exact Noether invariants are given. The approximate Noether invariant theorems for nonconservative systems under El-Nabulsi exponential model and El-Nabulsi power‑law model are proved. This method provides a new idea for the study of nonconservative system dynamics, and two examples show the effectiveness of the results. [ABSTRACT FROM AUTHOR]

Noether’s symmetry and conserved quantity of singular systems under generalized operators were studied. Firstly, the Lagrangian equation of singular systems under generalized operators was established, and the primary constraints on the system were derived. Then the Lagrangian multiplier was introduced to establish the constrained Hamilton equation and the compatibility condition under generalized operators. Secondly, based on the invariance of the Hamilton action under the infinitesimal transformation, Noether’s theorem for constrained Hamiltonian systems under generalized operators was established, and the symmetry and corresponding conserved quantity of the system were given. Under certain conditions, Noether’s conservation of constrained Hamiltonian systems under generalized operators can be reduced to Noether’s conservation of integer-order constrained Hamiltonian systems. Finally, an example illustrates the application of the results. [ABSTRACT FROM AUTHOR]

The Harnack inequalities of a family of closed convex equi-affine curves based on equiaffine curve shortening flow were studied. Firstly, according to the geometric evolution property of the equi-affine curve, a new type of Harnack quantity on the closed convex equi-affine curve was defined, then the evolution equation about the Harnack quantity of the closed convex equiaffine curve was discovered. Secondly, by the maximum principle, the non-negativity of the Harnack quantity, i. e., the Harnack inequality of the closed convex equi-affine curve, was investigated. Moreover, the constraint conditions of parameters in the Harnack quantity were found. Then, the Hamilton' s Harnack inequality of the closed convex equi-affine curve was further explored using the newly defined Harnack quantity. Finally, the classical Harnack inequality was derived based on Harnack inequality of the closed convex equi-affine curve and Cauchy-Schwarz inequality. [ABSTRACT FROM AUTHOR]

Under generalized β plane approximation, based on the quasi-geostrophic potential vorticity equation, and by means of the Gardner-Morikawa transform and the weak nonlinear perturbation expansion method, a Boussinesq equation with external source and dissipation forcing was derived to describe the generation and evolution of the Rossby wave amplitude. The periodic wave solutions and solitary wave solutions for the Boussinesq equation were presented with the modified Jacobi elliptic function expansion method. The solution structure shows that, the generalized β effect, the shear basic flow, the external source and the dissipation are extremely important factors influencing the nonlinear Rossby wave. [ABSTRACT FROM AUTHOR]