## Math Tutorials

This section contains some tutorials that I have written which cover various aspects of applied mathematics.

• Harmonic Analysis (pdf)

This tutorial discusses some of the numerical aspects of practical harmonic analysis. Topics include historical background, Fourier Series and Integral approximations, convergence improvement, differentiation of Fourier Series and Sigma Factors, Chebyshev Polynomial approximations, the Tau method, Fast Fourier Transforms, and Fast Sine or Cosine transforms.

• Numerical Linear Algebra (PDF)

This tutorial describes many of the standard numerical methods used in Linear Algebra. Topics include Gaussian Elimination, LU and QR factorizations, The Singular Value Decomposition, eigenvalues and eigenvectors via the QR Method with shifts or the Divide-and-Conquer method, and the Conjugate Gradient and Lanczos iterative methods.

• Digital Encoding (PDF)

This tutorial describes methods for encoding digital information in an electric or optical signal. Topics include Manchester and 4B/5B encoding, Amplitude Modulation, Frequency Modulation, Phase Modulation, Minimum Shift Keying (MSK), Quadrature Amplitude Modulation (QAM), Orthogonal Frequency Division Multiplexing (OFDM), Frequency Hopping (FHSS), Direct Sequence Spread Spectrum (DSSS), and Code Division Multiple Access (CDMA).

• Modeling of Sonar Transducers and Arrays (PDF)

This tutorial describing some of the mathematical tools used in the modeling of Sonar transducers and arrays. Topics include the Acoustic Wave Equation, Acoustic array interactions, the Helmholtz and Kirchhoff integral equations, Infinite Element methods, The Wave Envelope method, Doubly Asymptotic methods, T-Matrix methods, The Finite Element method (structural, fluid, piezoelectric, and magnetostrictive ), simple variational approximations, and Structure-Acoustic Coupling. This tutorial contains links to many reference documents including the CHIEF 2004 User Manual. To download the pdf document together with many of the included reference documents click on Modeling.zip.

• Symmetry Reductions (PDF)

The numerical solution of many physical problems can be reduced to the solution of a system of linear algebraic equations. This paper shows how to use geometric symmetry in order to reduce the solution of the original system of equations to the solution of a number of smaller systems of equations. The reduction technique employs the spectral decomposition of an appropriate symmetry operator. The use of this technique can significantly reduce computation time.