Abstract: In his book "Group Analysis of Differential Equations", Ovsiannikov poses a problem called the problem of reduction. Roughly speaking, it asks for necessary and sufficient conditions under which symmetry reduction, in a certain sense, is possible. When it *is*, the construction of non-invariant solutions -- as well as of invariant ones -- of partial differential equations becomes easier. Thus a solution of the reduction problem would be useful. A solution, under certain assumptions that appear to be natural, is presented in this paper. The method used indicates not only *when* the reduction is possible but also *how* it can be carried out.
2. Sanjay Rai and C.C.A. Sastri, "Group Classification and Invariant Boundary Conditions for Generalized Burgers Equations", Ind. J. Math., 43, 99-118 (2001)
Abstract: The Burgers equation u_t = u_xx - uu_x (or a related form of it) occurs in a variety of fields. Another widely studied equation is the reaction-diffusion equation u_t = u_xx + f(u). In this paper, the symmetry properties of the equation u_t = u_xx + f(u, u_x), which is a generalization of both the above equations, are studied, and a group classification of this equation considered as a system is carried out. The classification is complete when f is a function of u_x alone but incomplete when f depends on both u and u_x. Once the classification is completed, invariant boundary conditions can be found corresponding to the different forms of f. Examples are given to illustrate the point.
3. Terrance Quinn, Sanjay Rai, and C.C.A. Sastri, "Remarks on Partially Invariant Solutions of Differential Equations", Comm. Appl. Anal., 4, 475-480 (2000)
Abstract: A solution of a given system of PDEs is said to be partially invariant w.r.t a subgroup H of the group G of the Lie symmetries of the system if the solution manifold is not necessarily invariant under H but invariant under a subgroup H' of H. In this paper, a necessary and sufficient condition is given for a solution to be partially invariant.
4. A. Almudevar, R.N. Bhattacharya, and C.C.A. Sastri, "Estimating the Probability Mass of Unobserved Support in Random Sampling", J. Stat. Plann. Inference, 91, 91-105 (2000)
Abstract: The problem of estimating the probability mass of the support of a distribution not observed in random sampling is considered in the case where the distribution is discrete. An example of a situation in which the problem arises is that of species sampling: suppose that one wishes to determine the species of fish native to a body of water and that, after repeated sampling, one identifies a certain number of species. The problem is to estimate the proportion of the fish population belonging to the unobserved species. Since it is a rare event, ideas from large deviation theory play a role in answering the question. The result depends on the underlying distribution, which is unknown in general. Methods similar to nonparametric bootstrapping are therefore used to prove a limit theorem and obtain a confidence interval for the rate function.
5. Sanjay Rai and C.C.A. Sastri, "On the Simplification of Ovsiannikov's Method for the Construction of Partially Invariant Solutions", Ind. J. Math., 39, 65-74 (1997)
Abstract: The simplification of Ovsiannikov's method for the construction of partially invariant solutions (PIS) of a system of partial differential equations (PDEs) carried out earlier by Sastri, Dunn and Rao (1987) in the case of the one-dimensional heat equation u_t = u_xx is extended here to systems of nonlinear PDEs. Although a procedure that works for all systems of PDEs is not given here, the examples considered, namely the equations of the transonic flow of a gas, the Landau damping equation and the Burgers equation provide a basis for handling nonlinear equations in general. Invariant and partially invariant solutions are constructed for two different forms of the Burgers equation. Both types of solutions exhibit the blow-up property. The PIS, however, turn out to be reducible, i.e., invariant w.r.t a subgroup of the symmetry group. The problem of finding irreducible PIS for some form of the generalized Burgers equation will be addressed in a future publication.