## ABSTRACT

This chapter considers various problems from applied mathematics and engineering which employ the methods the authors have developed. It discusses enough probability to talk about quantum mechanics, and our discussion of quantum mechanics will be based on our work on Hilbert spaces. The chapter also considers some classes of special functions, in addition to complex exponentials, that form complete orthonormal systems. Legendre polynomials provide one of the most interesting examples of a complete orthonormal system. The Legendre polynomials have an elementary generating function: The function appears when one puts the Coulomb potential into spherical coordinates, and hence the Legendre polynomials are related to spherical harmonics. The chapter explains the linear time-invariant systems and illustrates the importance of convolution in applied mathematics. It talks about an elementary approach for finding rational roots for polynomials with rational coefficients.