01:640:357 - Topics in Applied Algebra

  • Course Code: 01:640:357
  • Semester(s) Offered: Spring
  • Credits: 3
  • Counts toward math major/minor?: Yes
  • Prerequisites: Math 250 and Calc III

Course Description:

This is a course aiming for undergraduate students majoring in math and engineering who are interested in understanding the importance of linear algebra and its application to problems within and outside mathematics. The basic concepts and results of linear algebra (vector spaces, linear transformations, matrices, determinants, eigenvalues and eigenvectors, orthogonality and diagonalization) will be reviewed. The course will cover the following topics: least square approximation, discrete Fourier transformation, numerical computations with matrices, graphs and networks, and image compression. Possible additional topics include: finite element method, systems of ordinary differential equations and linear programming.

The instructor may set MatLab assignments at their discretion.

Prerequisites:

Math 250 Introductory Linear Algebra and Math 251 Multivariable Calculus. 

Textbook:

For current textbook please refer to our Master Textbook List page

Published version of the textbook available in Spring 2016 from  World Scientific Publishing

Other Resources

Other Recommended Books (not required for course)

A. Jensen and A. la Cour-Harbo, Ripples in Mathematics: The Discrete Wavelet Transform
S. Allen Broughton and Kurt Bryan, Discrete Fourier Analysis and Wavelets
James S. Walker, A Primer on Wavelets and Their Scientific Applications (Second Edition)

 

Topics

Geometry of Linear Equations, Gaussian Elimination
Matrix Multiplication, LU Decomposition
Row Operations, Inverses and Transposes
Vector Spaces and Subspaces, Kernel and Range of Matrices
Linear Independence, Basis, Dimension
Solving Ax=b
Graphs and Networks
Linear Transformations
Orthogonal Vectors and Subspaces
Orthogonal Projection
Least Square Method
Gram-Schmidt Procedure, QR Decomposition
Fast Fourier Transform
Determinants and Their Properties
Applications of Determinants, Eigenvalues
Diagonalization of Matrices
Matrix Powers and Difference Equations
Matrix Exponentials and Differential Equations, Similarity Transformations
Minima, Maxima, and Saddle Points
Positive Definite Matrices
Singular Value Decomposition
Minimum Principles, Finite Element Method
Matrix Norms and Condition Number
Iterative Methods to Solve Ax=b

 

Sample Course Materials

Sample MATLAB Assignments

  • Project 1: Digital Signals and Vector Graphics  (pdf format)
  • Project 2: Convolution and Discrete Fourier Transform  (pdf format)
  • Project 3: Haar Wavelet Transform  (pdf format)
  • Project 4: Implementation of Wavelet Transforms (pdf format)
  • Project 5: Image Analysis by Wavelet Transforms (pdf format)

For Project 2 you will use the Finite Fourier transform graphic user interface.
Matlab m-file. Here is the link to download this m-file:  fftgui

For Projects 4 and 5 you will use the Uvi_Wave collection of Matlab m-files for wavelet transforms (developed at the University of Vigo, Spain). Here is the link to download these m-files:  Uvi_Wave zip file (unzip the file to use the package)

Using Matlab

Note: You can run Matlab on your own computer (without buying the program) by using the Rutgers X-application server.

  • Click on this apps server link.
  • Log in to the apps server using the connect button at the upper right-hand corner of the screen and your Rutgers NetID.
  • From the Main Menu at the lower left corner of the apps server toolbar, click on Education and then on Matlab
  • From the Main Menu click on Internet and then on Firefox Web Browser to access the Uvi_Wave files from the math 357 course web page.
  • Copy the fftgui.m file and the whole unzipped Uvi_Wave directory into a directory that your create on the X-apps server. Then set the Matlab path to this directory.

Course History

Taught by Prof. R. Goodman 2005-2008 and 2010-2014, Prof. V. Retakh 2009 and 2016, Dr. M. Thibault 2015.

 

Schedule of Sections:

01:640:357 Schedule of Sections

 

01:640:361 - Set Theory

  • Course Code: 01:640:361
  • Semester(s) Offered: Fall
  • Credits: 3
  • Counts toward math major/minor?: Yes
  • Prerequisites: Math 300 and either Math 250 or Calc 3

General Information

Catalogue listing for SET THEORY (3):
Introduction to set theory. The set-theoretic foundactions of mathematics, including the construction of the real number system, countable and uncountable sets, cardinal numers, and ordinals, the axiom of choice.
Prerequisite: 01:640:300 and either 01:640:250 or CALC3, or permission of department

Note: This course makes extensive use of the proof writing skills students develop in Math 300. Students who need further pratice with these skills are adviced to take Math 311 before taking Math 361, if possible

Textbook

Textbook:  For current textbook please refer to our Master Textbook List page

This course is taught every Fall term.

Schedule of Sections

Previous semesters:

  • Fall 2010, Prof. Cherlin
  • Fall 2009, Prof. Cherlin
  • Fall 2008. Prof. Deloro
  • Fall 2007. Prof. Weibel: MW 4th period (1:40-3 PM) in BECK 201 (Livingston campus)
  • Fall 2006: Prof. Schleimer

01:640:373 - Numerical Analysis I

  • Course Code: 01:640:373
  • Semester(s) Offered: Fall, Spring, Summer
  • Credits: 3
  • Counts toward math major/minor?: Yes
  • Prerequisites: Calc IV

Catalog Description

01:640:373-374. NUMERICAL ANALYSIS I,II (3,3)
Prerequisites: CALC4 and familiarity with a computer language. Credit not given for both these courses and 01:198:323,324.
Textbook: Numerical Analysis / Burden & Faires / Cengage / 978-1305253667 / 10th Edition / 2015

An analysis of numerical methods for the solution of linear and nonlinear equations, approximation of functions, numerical differentiation and integration, and the numerical solution of initial and boundary value problems for ordinary differential equations.

373 versus 374

The catalog description treats this a single two semester course with no fixed division of topics between the two parts. This allows some flexibility in organizing the course to follow the presentation in the textbook. One approach is to put things dealing with functions of one variable in the first semester, with multivariable methods in the second semester. In particular, techniques of numerical linear algebra are more likely to appear in the second semester, and the solution of differential equations in the first. The page for the current course should be consulted for a syllabus.

640:373 versus 198:323

The needs of the subject tends to blur the distinction between Mathematics and Computer Science. It is not unusual for the same textbook to be used in the two courses. Neither course is a collection of Numerical Recipes, although it is likely that programming considerations and questions of machine implementation would be more at home in a Computer Science course, while questions of the existence of solutions or the theoretical basis for error estimates are more suitable for a Mathematics course.

Prerequisites

Since the numerical solution of differential equations is a major topic in Math 373, prior exposure to the topic in a CALC4 course is essential. That course uses linear algebra, which is also used in other topics contained in Math 373 such as interpolation. The brief treatment of linear algebra in Math 244 will probably suffice for Math 373, but a course equivalent of Math 250 is strongly recommended for Math 374. Some prior programming experience is desirable, but not essential.

Programming

Part of the course involves computer implementation of the algorithms discussed, and therefore some prior programming experience is desirable, although not essential. The computer assignments will be fairly short, and although a computer language is not taught in the course, a description of Matlab commands that can be used to write the programs and examples of their use will be provided on the course webpage.

Schedule of Sections

Previous semester resources

01:640:325 - Foundations of Quantum Mechanics

  • Course Code: 01:640:325
  • Semester(s) Offered: Fall
  • Credits: 3
  • Counts toward math major/minor?: Yes
  • Prerequisites: Calc IV

General Information
Interdisciplinary course, intended primarily for juniors and seniors majoring in mathematics, physics, or philosophy, dealing with what can be concluded from quantum mechanics about the nature of reality.

Prerequisites: CALC4 or permission of instructor (As far as mathematics is concerned, students need a working knowledge of complex numbers, eigenvalues of matrices, and partial derivatives. Prior knowledge in physics is not required but helpful.)

Contents: It has been claimed that quantum mechanics entails the most radical consequences about the world and our knowledge of it, such as the existence of parallel universes, faster-than-light action-at-a-distance, limitations to what we can know, that reality itself can be paradoxical, or that electrons become real only when observed. On the other hand, it has been claimed that quantum mechanics, in its orthodox formulation, is "unprofessional" (J. Bell), "incoherent" (A. Einstein), "incomprehensible" (R. Feynman), and "insane" (E. Schrodinger). We will investigate these claims, their basis and merits. The course will involve advanced mathematics, as appropriate for a serious discussion of quantum mechanics, but will not focus on technical methods of problem-solving.

Topics will include most of the following: The Schrodinger equation, the Born rule, self-adjoint matrices, axioms of the quantum formalism, the double-slit experiment, non-locality, the paradox of Schrodinger's cat, the quantum measurement problem, Heisenberg's uncertainty relation, interpretations of quantum mechanics (Copenhagen, Bohm's trajectories, Everett's many worlds, spontaneous collapse theories, quantum logic, perhaps others), views of Bohr and Einstein, no-hidden-variables theorems, and identical particles.

Text: J. Bell: Speakable and unspeakable in quantum mechanics, Cambridge University Press.

Learning goals: To understand the rules of quantum mechanics; to understand several important views of how the quantum world works; to be familiar with the surprising phenomena and paradoxes of quantum mechanics.

This course is normally in Fall semesters.

 

Schedule of Sections:

 01:640:325 Schedule of Sections

01:640:403 - Introduction to Theory of Functions of a Complex Variable

  • Course Code: 01:640:403
  • Semester(s) Offered: Spring
  • Credits: 3
  • Counts toward math major/minor?: Yes
  • Prerequisites: Calc IV

Course description:

A first course in the theory of differentiable functions of one complex variable. Topics include line integrals, Cauchy's theorem and its applications, Taylor and Laurent expansions, singularities, and conformal mapping.
Prerequisite: CALC4

Textbook:  For current textbook please refer to our Master Textbook List page

Previous semesters:

01:640:411 - Mathematical Analysis I

  • Course Code: 01:640:411
  • Semester(s) Offered: Fall
  • Credits: 3
  • Counts toward math major/minor?: Yes
  • Prerequisites: Special permission only. Typically students have already taken Math 311.

General Information (Catalog Listing)

01:640:411-412 Mathematical Analysis I,II (3,3)
Rigorous analysis of the differential and integral calculus of one and several variables.
Prerequisites: Permission of department and instructor. For students preparing for graduate study in the mathematical sciences.

This course forms part of the Honors Track sequence. Special permission is required for admission to Math 411. Requests for admission are evaluated based on the student's prior achievements, level of interest, and potential for success in the course.

Textbook

Textbook:  For current textbook please refer to our Master Textbook List page

 

Math 411-412 is a year long sequence.

Schedule of Sections

 

Previous semesters:

01:640:412 - Mathematical Analysis II

  • Course Code: 01:640:412
  • Semester(s) Offered: Spring
  • Credits: 3
  • Counts toward math major/minor?: Yes
  • Prerequisites: Math 411

General Information (Catalog Listing)

01:640:411-412 Mathematical Analysis I,II (3,3)
Rigorous analysis of the differential and integral calculus of one and several variables.
Prerequisites: Permission of department and instructor. For students preparing for graduate study in the mathematical sciences.

Math 412 is part of the Honors Track sequence. It is intended to be taken immediately after Math 411. Students enrolled in Math 411 during the Fall semester should be able to preregister for Math 412 in the Spring without assistance. All other students should contact the Honors Advisor <>.

Textbook

Textbook:  For current textbook please refer to our Master Textbook List page

Math 411-412 is a year long sequence.

Schedule of Sections

 

Previous semesters:

  • Spring 2009: Prof. Speer
  • Spring 2008: Prof. E. Teixeira
  • Spring 2007: Prof. Bahri

01:640:421 - Advanced Calculus for Engineering

  • Course Code: 01:640:421
  • Semester(s) Offered: Fall, Spring, Summer
  • Credits: 3
  • Counts toward math major/minor?: Yes
  • Prerequisites: Calc IV. No credit for both Math 421 and Math 423. This version of the class is aimed at engineers and physics majors.

General Information (Catalog Listing)

01:640:421. Advanced Calculus for Engineering (3)
Primarily for mechanical engineering majors. Prerequisite: CALC4.
Credit not given for both this course and 01:640:423.
Covers Laplace transforms, numerical solution of ordinary differential equations, Fourier series, and separation of variables method applied to the linear partial differential equations of mathematical physics (heat, wave, and Laplace's equation).

Notes: CALC4 (Differential Equations) means Math 244, 252, or 292.
Math 423 is Elementary Partial Differential Equations. It covers similar material to Math 421, but is aimed at students majoring in Mathematics or Physics, rather than Engineering students.

Textbook

Textbook:  For current textbook please refer to our Master Textbook List page

Syllabus

Individual sections may vary, but chapters 4, 12 and 13 should be covered in detail, supplemented with a treatment of linearity including a review of Vector Calculus from Part 2. If time permits, Chapter 14 will introduce boundary value problems in non-rectangular coordinate systems.

Sample syllabus

Schedule of Sections

Archive:

Notes for Instructors

Comments and corrections by Peter Landweber for the 3rd edition of the text. (Similar to the 4th.)

A selection of recommended homework problems, from the 3rd edition.

 

01:640:423 - Elementary Partial Differential Equations

  • Course Code: 01:640:423
  • Semester(s) Offered: Fall, Spring
  • Credits: 3
  • Counts toward math major/minor?: Yes
  • Prerequisites: Calc IV. No credit for both Math 421 and Math 423. This version of the class is aimed at math majors.

Catalog Description

01:640:423. Elementary Partial Differerntial Equations (3)
Prerequisite: CALC4. Credit not given for both this course and 01:640:421.
Linear partial differential equations of mathematical physics (heat, wave, and Laplace's equation), separation of variables, Fourier series.

Textbook

Textbook:  For current textbook please refer to our Master Textbook List page

Math 421 vs Math 423: A comparison

These two courses are only superficially similar: 421 series to introduce techniques needed by the Mechanical Engineering program while 423 is an introduction to the mathematics of partial differential equations.

Prerequisities

Calculus through a course in Ordinary Differential Equations.

Schedule of Sections

Previous semester resources

Use the drop-down menu above to see who has taught the course since 2006. Some prior course pages are archived below.

01:640:428 - Graph Theory

  • Course Code: 01:640:428
  • Semester(s) Offered: Fall, Spring, Summer
  • Credits: 3
  • Counts toward math major/minor?: Yes
  • Prerequisites: Math 250 and Calc III

General Information (Catalog listing)

01:640:428 Graph Theory (3)
Colorability, connectedness, tournaments, eulerian and hamiltonian paths, orientability, and other topics from the theory of finite linear graphs, with an emphasis on applications chosen from social, biological, computer science, and physical problems.
Prerequisites: CALC3 and 01:640:250.

Textbook

Textbook:  For current textbook please refer to our Master Textbook List page

Syllabus

Syllabus may vary.

 

Schedule of Sections:

01:640:428 Schedule of Sections

 

Previous semesters:

  • Fall 2010 Prof. Weibel
  • Summer 2010: Wesley Pegden
  • Summer 2009: Prof. Beck
  • Fall 2008: Prof. Butler
  • Summer '08. A. Thanatipanonda
  • Fall 2007. Prof. Ocone
  • Summer 2007. Liviu Ilinca
  • Fall 2006. Prof. Beck. (Used a different text: Brualdi)
  • Fall 2005 (Schleimer)
  • Fall 2004 (Maclagan)
  • Fall 2003 (Zeilberger)
  • Fall 1997 (Weibel

 

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