In the present article, we investigate generalized quasi-Einstein spacetimes in the sense of Chaki with its Ricci curvature belonging to Gray's decomposition subspaces equipped with semisymmetric metric connections. Generalized Robertson-Walker spacetimes are also considered. It is established that a generalized quasi-Einstein spacetime with its Ricci curvature belonging to any Gray's decomposition subspace becomes perfect fluid spacetime under some restrictions. Further, it is established that a generalized quasi-Einstein generalized Robertson-Walker spacetime is also a perfect fluid spacetime.
We propose a new infinite-dimensional Lie superalgebra as an extension of the N = 2 Neveu-Schwarz algebra and study its related supersymmetric and local bi-superhamiltonian integrable systems.
In analytical mechanics of systems with ideal nonholonomic constraints, the question of the existence of constraints described by nonlinear in velocities equations remains unresolved. The few known examples of such constraints are usually constructed, as a rule, artificially, using auxiliary subsystems involving massless rigid rods, threads, and other devices whose mechanical motion is neglected. This paper analyses several published but incorrect examples of nonlinear nonholonomic constraints. Conversely, constraints of the second kind (servo constraints) are not ideal, may be described by rather complex nonlinear equations, and are realized using energy sources external to the system. Irrespective of the condition of ideality of constraints, there exists a most general definition of virtual displacements for constraints that are linear in accelerations. In certain cases, it is possible to derive a closed system of equations of motion without resorting to the expression for the virtual work of external forces. The article considers an example of such a system, related to the problem of stabilizing the upper equilibrium position of a movable rigid rod.
A framework is presented to solve partial differential equations (PDE) defined through a generalization of pseudo-differential operators, when the psi-DO symbol takes on coefficient functions that are singular distributions. Such treatments of PDE have not been treated systematically elsewhere. The result is a rigorous mathematical method, which does not rely upon renormalization or regularization, while producing bound state energy results in agreement with the literature. The procedure is the same in each dimension. So there is no ad hoc dimensional regularization. The method also does not rely upon self-adjoint extensions (SAEs) of the Laplace operator. This is important for non-Schro & uml;dinger systems, like the Dirac equation, where the kinetic energy is not prescribed by the Laplace operator. It is also a necessary property of any rigorous mathematical form of relativistic quantum field theories. SAEs only exist in dimensions less than four, whereas the method presented is unaffected by the number of dimensions. This new framework is applied to the Schro & uml;dinger equation in dimensions one, two, and three, for comparison with the method of SAEs. A precise definition of hyperlocal interaction is introduced, which provides a nonperturbative method for handling singular potentials. Lastly, a rigorous distributional version of the Hellmann-Feynman theorem is shown.
In this paper, the corresponding relationships of the integrability structures and the solution structures between a dispersionless Kadomtsev-Petviashvili (dKP) equation and a dispersionless modified Kadomtsev-Petviashvili (dmKP) equation are obtained. On the one hand, the equivalence of the conservation laws, potential systems, nonlocal conservation laws and extended potential systems for the two equations are explored. On the other hand, starting from the original equations and the extended potential systems, respectively, we establish the correspondences of the solution structures for these two equations. Furthermore, the discrete symmetries of the dmKP equation are presented. It is worth noting that we link the extended potential system with the Type-II hidden symmetry. This combination provides a new perspective for solving the extended potential system. In addition, this method of further studying equations through the Miura transformation can be applied to other equations as well, and it can help us better analyze these equations.
