## 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, as well as 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 document was originally distributed on a CD along with many of the referenced documents. The documents that were included on the CD are listed at the end of the document. You can view any of the documents by clicking on its reference link in the list. You can also download the entire contents of the CD by clicking 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.