Differential Invariants and Application to Riccati-Type Systems
Roman Popovych, Vyacheslav Boyko
备注：
12 pages
发表信息：
Proceedings of Fourth International Conference ``Symmetry in Nonlinear Mathematical Physics'' (9-15 July, 2001, Kyiv), Editors A.G. Nikitin, V.M. Boyko and R.O. Popovych, Proceedings of Institute of Mathematics, Kyiv, 2002, V.43, Part 1, 184-193
We generalize the classical Lie results on a basis of differential invariants for a one-parameter group of local transformations to the case of arbitrary number of independent and dependent variables. It is proved that if universal invariant of a one-parameter group is known then a complete set of functionally independent differential invariants can be constructed via one quadrature and differentiations. Some applications of first-order differential invariants to Riccati-type systems are also presented.
Shifted Riccati Procedure: Application to Conformal Barotropic FRW Cosmologies
在正压FRW宇宙学的情况下，共形时间的哈勃参数是一个简单的常系数Riccati方程的解。我们认为这些宇宙学的框架内的非相对论性超对称性，已在该地区的超对称量子力学的有效。回顾Faraoni [Amer. J. Phys. 67（1999），732-734]提出了如何将正压FRW微分方程组简化为简谐振子微分方程，我们将后者的方程组用超对称方法求解，并将其解分为两类“玻色子”（非奇异）和“费米子”（奇异）宇宙学零模解。费米子方程可以被认为是代表Stephani类型的宇宙学，即，在保形时间内，曲率变化。接下来，我们通过在这些超对称伙伴宇宙学的共同Riccati解中引入一个常数加性参数（用S表示）来应用所谓的移位Riccati过程。这导致正压Stephani宇宙学与周期性奇点在其空间曲率指数，我们称之为U和V宇宙学，第一个是玻色子型和费米子型后者。我们完全解决这些循环奇异宇宙学在其零模式的水平上，表明一个可接受的移位参数应该是纯粹的虚的，这反过来又引入了宇称时间（PT）属性的合作伙伴曲率指数。
