 # MA304: Topics in Applied Mathematics

## Course Syllabus for "MA304: Topics in Applied Mathematics"

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### Learning Outcomes

Upon successful completion of this course, the student will be able to:

• Compute singular values of rectangular and singular square matrices.
• Perform singular value decomposition of rectangular matrices.
• Solve an arbitrary system of linear equations.
• Compute linear least-squares estimation.
• Compute principal components.
• Reduce data dimension.
• Compute DFT of vectors.
• Compute inverse DFT.
• Compute DCT of vectors.
• Compute inverse DCT.
• Compute two-dimensional DCT of matrix data array.
• Compute histogram of data sets.
• Formulate probability distribution based on histograms.
• Compute entropy from probability distribution.
• Construct Huffman trees and Huffman codes.
• Perform color transform.
• Outline the JPEG image compression scheme.
• Explain video compression.
• Compute Fourier series.
• Compute Fourier cosine series.
• Describe the importance of the Dirichlet and Fejer kernels.
• Apply the property of positive approximate identity to prove convergence theorems.
• Compute mean-square error of approximation by partial sums of Fourier series.
• Solve the Basel problem and its extension to higher even powers.
• Compute the Fourier transform of some simple functions.
• Compute the Fourier transform of the affine transformation of some simple functions.
• Compute the convolution of some simple functions with certain filters.
• Describe and apply the important Fourier transform property of mapping the convolution operation to product of the Fourier transform of the individual functions.
• Explain and apply the concept of localized Fourier and inverse Fourier transforms.
• Formulate the Fourier transform of a general Gaussian function.
• Explain the Uncertainty Principle.
• Compute the Gabor transform of some simple functions.
• Formulate local time-frequency basis functions from a given sliding time-window function.
• Formulate local cosine basis functions from a given sliding time-window function.
• Apply the Gaussian to solve the heat equation with the entire d-dimensional Euclidean space as the spatial domain, where d is any positive integer.
• Apply the method of separation of variables to separate a given linear PDE into a finite family of ODEs.
• Solve the corresponding eigenvalue problems for the spatial ODEs.
• Apply the Fourier series of the input function to formulate the superposition solution of boundary value problems.
• Give the relationship between scale and frequency for a given wavelet filter.
• Perform matrix extension to compute wavelet filters.
• Compute multi-scale data representation by applying the wavelet decomposition algorithm for the Haar wavelet.
• Identify the order of vanishing moments of a given wavelet.
• Apply the wavelet decomposition and reconstruction algorithms to multi-scale data analysis.
• Apply wavelets to digital image manipulation.

### Course Requirements

In order to take this course, you must:

√    Have continuous broadband Internet access.

√    Have the ability/permission to install plug-ins (e.g. Adobe Reader or Flash) and software.

√    Have the ability to download and save files and documents to a computer.

√    Have the ability to open Microsoft Office files and documents (.doc, .ppt, .xls, etc.).

√    Have competency in the English language.

√    Have read the Saylor Student Handbook.

√    Have completed the following courses from “The Core Program” of the mathematics major: MA101: Single-Variable Calculus I; MA102: Single-Variable Calculus II; MA103: Multivariable Calculus; MA211: Linear Algebra; MA221: Differential Equations; and MA241: Real Analysis I

√    Have completed the following courses from the “Advanced Mathematics” section of the mathematics major: MA212: Linear Algebra II; MA243: Complex Analysis; and MA222: Introduction to Partial Differential Equations.

### Course Information

Welcome to MA304: Topics in Applied Mathematics.  Below, please find some information on the course and its requirements.

Primary Resources: This course is comprised of a range of different free, online materials.  However, the course makes primary use of the following materials:

Requirements for Completion: In order to complete this course, you will need to work through each unit and all of its assigned materials.  You will also need to complete:

• The Final Exam

Note that you will only receive an official grade on your Final Exam.  However, in order to adequately prepare for this exam, you will need to work through the resources in each unit.

In order to “pass” this course, you will need to earn a 70% or higher on the Final Exam.  Your score on the exam will be tabulated as soon as you complete it.  If you do not pass the exam, you may take it again.

Time Commitment: Each unit includes a “time advisory” that lists the amount of time you should spend on each subunit.  These should help you plan your time accordingly.  It may be useful to take a look at these time advisories and to determine how much time you have over the next few weeks to complete each unit, and then to set goals for yourself.  For example, Unit 1 should take you 12.5 hours.  Perhaps you can sit down with your calendar and decide to complete subunit 1.1 (a total of 3.75 hours) on Monday night; subunit 1.2 (a total of 3.75 hours) on Tuesday night; etc.

Tips/Suggestions: As noted in the “Course Requirements,” there are several mathematics pre-requisites for this course.  If you are struggling with the mathematics as you progress through this course, consider taking a break and revisiting the applicable course listed as a pre-requisite.