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As the Black-Scholes equation can be transformed into the one-dimensional linear heat equation via two sets of transformations, an optimal system of one-dimensional subalgebras for the one-dimensional heat equation is exploited to obtain two classes of optimal systems of one-dimensional subalgebras for the well-known Black-Scholes equation of the mathematics of finance. Two methods for the derivation of the two classes of optimal systems of group-invariant solutions for this model are available. We present the simpler approach

Kinematic and dynamic expressions are derived for the Lie derivative of vorticity along a particle world line in a rigid motion. It is found that the evolution of vorticity in a rigid motion is governed by the electric part of the Weyl tensor. Necessary and sufficient kinematic and dynamic conditions are established for a rotating rigid motion to be isometric.

This study compared the in-sample forecasting accuracy of three forecasting nonlinear models namely: the Smooth Transition Regression (STR) model, the Threshold Autoregressive (TAR) model and the Markov-switching Autoregressive (MS-AR) model. Nonlinearity tests were used to confirm the validity of the assumptions of the study. The study used model selection criteria, SBC to select the optimal lag order and for the selection of appropriate models. The Mean Square Error (MSE), Mean Absolute Error (MAE) and Root Mean Square Error (RMSE) served as the error measures in evaluating the forecasting ability of the models. The MS-AR models proved to perform well with lower error measures as compared to LSTR and TAR models in most cases.

This paper uses symmetry methods to obtain nonequivalent similarity reductions of the steady two-dimensional thermal boundary layer equations of an incompressible laminar flow. New similarity reductions are uncovered and results obtained through the classical dimensional analysis are recovered.