Notes on Math 514: Numerical Analysis @ University of Wisconsin-Madison Your comments and criticism are greatly welcomed.

Textbook

An Introduction to Numerical Analysis - by Endre Süli

Course Notes (PDF version)

Ch 1: Simple Iteration Method

Ch 2: Solution of Systems of Linear Equations

LU Decomposition

QR Factorization

Norm & Condition Number

Ch 3: Special Matrices

Ch 4: Simultaneous Iteration Method

Ch 5: Eigen-Decomposition

Midterm Review

Ch 6-10: Approximation & Integration

Polynomial Approximation Theory

Polynomial Interpolation & Chebyshev Nodes

Polynomial Projection & Quadrature Method

Integration Rules & Undetermined Coefficients

Review for Approximation & Integration

Ch 12: Numerical ODE

Introduction to Numerical ODE

Euler & Trapezoidal & Runge-Kutta

Linear Multistep Methods & Stability

Review for Numerical ODE

Note: You can download a PDF version of the notes here

Notes on Math 632: Introduction to Stochastic Processes @ University of Wisconsin-Madison Your comments and criticism are greatly welcomed.

Resources

Past Exams

Textbook: Essentials of stochastic processes by Rick Durrett

Homeworks: HW01 HW02 HW03 HW04 HW05 HW06 HW07 HW08

Lecture Notes (PDF version)

Week 1
9/6	Review, Introduction to Stochastic Processes
Week 2
9/11	Introduction to Markov Chain
9/13	Simple Random Walk, P^n, Gambler's Ruin
Week 3
9/18	T, Strong Markov Property, T_y, ρ_yy, Recurrence
9/20	Recurrence, Closed, Irreducible, Communication
Week 4
9/24	Theorems Related to Recurrence
9/27	Stationary Distribution/Measure, Renewal Chain
Week 5
10/2	Positive/Null Recurrent, Limit Behavior
10/4	Periodicity, Limiting Behavior
Week 6
10/9	Convergence Theorem
10/11	Midterm 1
Week 7
10/16	Doubly Stochastic, Detailed Balance
10/18	Exit Distribution
Week 8
10/23	Exit Time
10/25	Probability Review for Poisson Process
Week 9
10/30	Introduction to Poisson Process
11/1	Compound Poisson Process
Week 10
11/6	Thinning, Superposition, and Conditioning
11/8	Poisson Process Comprehensive Problems
Week 11
11/13	More Exercises on Poisson Process
11/15	Midterm 2
Week 12
11/20	Introduction to Renewal Process
11/22	Thanksgiving
Week 13
11/27	Renewal Process, Age and Residual Life
11/29	Continuous Time Markov Processes
Week 14
12/4	M/M/s Queue, Kolmogorov Equations
12/6	Properties of CTMC
Week 15
12/11	CTMC Exercises

Note: You can download a PDF version of the notes here

Notes on Math 240: Introduction to Discrete Mathematics @University of Wisconsin-Madison Your comments and criticism are greatly welcomed.

Course Website

Homepage

Syllabus

Textbook

Kenneth H. Rosen, Discrete Mathematics and its Applications, seventh Edition

Lecture Notes

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0. Introductory Lecture

1. The Foundations: Logic and Proofs

1.1 Propositional Logic

1.2 Applications of Propositional Logic

1.3 Propositional Equivalences

1.4 Predicates and Quantifiers

1.5 Nested Quantifiers

1.6 Rules of Inference

1.7 Introduction to Proofs

1.8 Proof Methods and Strategy

2. Basic Structures: Sets, Functions, Sequences, Sums, and Matrices

2.1 Sets

2.2 Set Operations

2.3 Functions

2.4 Sequences and Summations

2.5 Cardinality of Sets

2.6 Matrices

3. Algorithms

3.1 Algorithms

3.2 The Growth of Functions

3.3 Complexity of Algorithms

4. Number Theory and Cryptography

4.1 Divisibility and Modular Arithmetic

4.2 Integer Representations and Algorithms

4.3 Primes and Greatest Common Divisors

4.4 Solving Congruences

5. Induction and Recursion

5.1 Mathematical Induction

5.2 Strong Induction and Well-Ordering

5.3 Recursive Definitions and Structural Induction

5.4 Recursive Algorithms

6. Counting

6.1 The Basics of Counting

6.2 The Pigeonhole Principle

6.3 Permutations and Combinations

6.4 Binomial Coefficients and Identities

6.5 Generalized Permutations and Combinations

7. Discrete Probability

7.1 An Introduction to Discrete Probability

9. Relations

9.1 Relations and Their Properties

9.3 Representing Relations

9.5 Equivalence Relations

9.6 Partial Orderings

10. Graphs

10.1 Graphs and Graph Models

10.2 Graph Terminology and Special Types of Graphs

10.3 Representing Graphs and Graph Isomorphism

10.4 Connectivity

11. Trees

11.1 Introduction to Trees

Lecture Slides

Past Exams

Spring 2002 - Exam 1

Spring 2002 - Exam 2

Spring 2002 - Final

Spring 2005 - Exam 1

Spring 2005 - Exam 2

Spring 2005 - Final

Spring 2008 - Exam 1

Spring 2008 - Exam 2

Spring 2008 - Exam 3

Spring 2008 - Final 

Spring 2011 - Exam 1 (Exam, Solution)

Spring 2011 - Exam 2 (Part I, Part II, Solution)

Spring 2011 - Final (Exam, Solution)

Spring 2015 - Exam 2

Spring 2016 - Exam 2

Notes on Math 521: Analysis I @ University of Wisconsin-Madison Your comments and criticism are greatly welcomed.

Course Website

Homepage

Syllabus

Textbook

Rudin, W. Principles of Mathematical Analysis. Third Edition

Lecture Notes

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Week 1
1/24	Number Systems, Irrationality of √2
1/26	Sets, Gaps in Q, Field
Week 2
1/29	Field, Order, Upper Bound and Lower Bound
1/31	Infimum and Supremum, Ordered Field
2/2	Ordered Field, Archimedean Property, Density of Q in R
Week 3
2/5	n-th Root of Real Number, Complex Numbers
2/7	Complex Numbers, Euclidean Spaces
2/9	Quiz
Week 4
2/12	Schwarz Inequality, Function, Cardinality
2/14	Finite and Infinite, Sequence
2/16	Set Operations, Countable and Uncountable
Week 5
2/19	Metric Space, Interval, Cell, Ball, Convex
2/21	Definitions in Metric Space
2/23	Neighborhood, Open and Closed, De Morgan's Law
Week 6
2/26	Open and Closed, Closure
2/28	Convergence and Divergence, Range, Bounded
3/2	Important Properties of Convergent Sequences
Week 7
3/5	Algebraic Limit Theorem
3/7	Convergence of Sequences in R^n, Compact Set
3/9	Exam 1
Week 8
3/12	Compact Subset, Cantor's Intersection Theorem
3/14	Nested Intervals Theorem, Compactness of k-cell
3/16	Heine-Borel, Weierstrass, Subsequence
Week 9
3/19	Cauchy Sequence, Diameter
3/21	Cauchy Sequence, Complete Metric Space, Monotonic
3/23	Upper and Lower Limits
Week 10
4/2	Some Special Sequences
4/4	Series, Cauchy Criterion for Series, Comparison Test
4/6	Convergence Tests for Series
Week 11
4/9	Power Series, Absolute Convergence, Rearrangement
4/11	Rearrangement, Limit of Functions
4/13	Exam 2
Week 12
4/16	Continuous Function and Open Set
4/18	Continuity and Compactness, Extreme Value Theorem
4/20	Uniform Continuity and Compactness
Week 13
4/23	Connected Set, Intermediate Value Theorem
4/25	Derivative, Chain Rule, Local Extrema
4/27	Mean Value Theorem, Monotonicity, Taylor's Theorem
Week 14
4/30	Riemann-Stieltjes Integral, Refinement
5/2	Fundamental Theorem of Calculus
5/4	Sequence of Functions, Uniform Convergence

Notes on Math 541: Modern Algebra @ University of Wisconsin-Madison Your comments and criticism are greatly welcomed.

Course Website

Homepage

Syllabus

Textbook

Abstract Algebra, by Dummit and Foote, Third Edition, 2004

Lecture Notes

Download PDF

Week 1
1/24	Divides, Equivalence Relations
1/26	Well-ordering of Z
Week 2
1/29	Division Algorithm, gcd
1/31	Euclidean Algorithm
2/2	Equivalence Class, Z/nZ, Group
Week 3
2/5	Group, Well-definedness, Z/nZ
2/7	(Z/nZ)*, Properties of Group
2/9	Order, Symmetric Group
Week 4
2/12	Symmetric Group, Cycle
2/14	Homomorphism, Isomorphism
2/16	Order, Homomorphism, Subgroup
Week 5
2/19	Dihedral Groups, Subgroup
2/21	Cyclic Group, lcm, Order of g^a
2/23	Cyclic Subgroup, Generating Set of a Group
Week 6
2/26	Finitely Generated Group
2/28	Coset, Normal Subgroup
3/2	Exam 1
Week 7
3/5	Quotient Group, Index, Lagrange's Theorem
3/7	Corollaries of Lagrange's Theorem
3/9	The First & Second Isomorphism Theorems
Week 8
3/12	The Third Isomorphism Theorem
3/14	Transposition, Sign of Permutation
3/16	Homework 6, The Correspondence Theorem
Week 9
3/19	Sign of Permutation, Alternating Group
3/21	Subgroups of A_4, Group Action, Orbit, Stabilizer
3/23	Orbit, Stabilizer, Cayley's Theorem
Week 10
4/2	Conjugacy Class, The Class Equation
4/4	Cauchy's Theorem, Recognizing Direct Products
4/6	Homework 8, Properties of Finite Abelian Group
Week 11
4/9	Fundamental Theorem of Finite Abelian Groups
4/11	Definition of Ring
4/13	Exam 2
Week 12
4/16	Properties of Ring, Zero-Divisor, Unit
4/18	Field, Product Ring, Integral Domain
4/20	Product Ring, Finite Domain and Field, Subring
Week 13
4/23	Polynomial Ring, Ideal, Principal Ideal
4/25	Examples of Ideals, Quotient Ring
4/27	Isomorphism Theorems for Rings
Week 14
4/30	Generating Ideal, Maximal Ideal, Prime Ideal
5/2	Prime Ideal, Euclidean Domain
5/4	Review, Galois Theory

Notes on Math 375: Topics in Multi-Variable Calculus and Linear Algebra Your comments and criticism are greatly welcomed.

Course Website

Syllabus

Lecture and Homework Schedule

Textbook

Calculus, Volume II, 2nd Edition, by Tom M. Apostol

Calendar

September

Mon (Dis)	Tue (Lec)	Wed (Dis)	Thu (Lec)	HW
-	-	-	9/7	HW1
9/11	9/12	9/13	9/14	HW2
9/18	9/19	9/20	9/21	HW3
9/25	9/26	9/27	9/28	HW4

All notes from September in PDF format

October

Mon (Dis)	Tue (Lec)	Wed (Dis)	Thu (Lec)	HW
10/2	Exam 1	10/4	10/5	HW5
10/9	10/10	10/11	10/12	HW6
10/16	10/17	10/18	10/19	HW7
10/23	10/24	10/25	10/26	-
10/30	10/31	11/1	11/2	HW8

Midterm 1 Practice

All notes from October in PDF format

November

Mon (Dis)	Tue (Lec)	Wed (Dis)	Thu (Lec)	HW
10/30	10/31	11/1	11/2	HW8
11/6	11/7	11/8	11/9	HW9
11/13	11/14	11/15	Exam 2	HW10
11/20	11/21	11/22	Holiday	-
11/27	11/28	11/29	11/30	HW11

Midterm 2 Practice 1

Midterm 2 Practice 2

All notes from November in PDF format

December

Mon (Dis)	Tue (Lec)	Wed (Dis)	Thu (Lec)	HW
12/4	12/5	12/6	12/7	HW12
12/11	12/12	12/13

Matrix Algebra Review

Lecture Outlines

Chapter 1: Linear Spaces

Week 01 - 9/7: Vector Space

Week 02 - 9/12: Proof Writing

Week 02 - 9/14: Subspace, Span of Vector Spaces, Linear Independence

Week 03 - 9/19: Linear Independence, Basis, Coordinates

Week 03 - 9/21: Theorems about Linear Dependence

Week 04 - 9/26: Inner Product, Length, Angle

Week 04 - 9/28: Distance, Triangle Inequality, Orthogonal, Gramm-Schmidt Process

Week 05 - 10/3: Midterm 1

Week 05 - 10/5: Best Approximation, Fourier Series

Chapter 2: Linear Transformations and Matrices

Week 06 - 10/10: Linear Transformations, Solving Systems of Equations

Week 06 - 10/12: Injective, Null Space, Range, Rank-Nullity Theorem

Week 07 - 10/17: Algebraic Operations on Linear Transformations, Injective, Inverse

Week 07 - 10/19: Matrix Representation of Linear Transformations, Matrix Multiplication

Week 08 - 10/24: Isomorphism Between Linear Transformations and Matrices, Solving Linear Equations using Matrix

Week 08 - 10/26: Solving Linear Equations

Chapter 3: Determinants

Week 09 - 10/31: Determinants

Week 09 - 11/2: Uniqueness Theorem, Properties of Determinants

Week 10 - 11/7: Determinant and Area, Inverse of Matrix, Minors and Cofactors

Week 10 - 11/9: Cofactor Expansion, Cramer's Rule, Linear Independence and Determinant

Chapter 4: Eigenvalues and Eigenvectors

Week 11 - 11/14: Eigenvalues, Eigenvectors

Week 11 - 11/16: Midterm 2

Week 12 - 11/21: Characteristic Polynomial, Trace, Diagonalization

Week 12 - 11/23: Thanksgiving Break

Chapter 8: Differential Calculus of Scalar and Vector Fields

Week 13 - 11/28: Open Balls, Limits, Continuity, Directional Derivative

Week 13 - 11/30: Partial Derivative, Total Derivative, Linear Approximation Formula

Week 14 - 12/5: Differentiable, Total Derivative, Continuity, Multivariable Chain Rule

Week 14 - 12/7: Differentiable, Chain Rule, Gradient, Level Sets

Week 15 - 12/12: Multivariable Chain Rule, Jacobian Matrix