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In this paper Lie symmetry analysis is used to find the solution to a third order first degree nonlinear ordinary differential equation (ODE) fourth degree in second derivative that arise in waves of systems like water in shallow oceans. This method gives exact solutions to problems and does not depend on initial boundary values.

In this paper, we present results on the necessary and sufficient conditions for positivity of operators in non-unital C*-algebras.Mathematics Subject Classification: 47B47, 47A30

In this note, we provide detailed characterization of operators in terms of norm-attainability and norm estimates in Banach algebras. In particular, we establish the necessary and sufficient conditions for norm-attainability of the derivations and also give their norm bounds in the norm-attainable classes.

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