## Lower Division Courses

## M301 College Algebra Syllabus

**Prerequisite and degree relevance:** A passing score on the mathematics section of the Texas Higher Education Assessment (THEA) test (or an appropriate assessment test). May not be counted toward a degree in mathematics. Credit for Mathematics 301 may not be earned after a student has received credit for any calculus course with a grade of C- or better.

**Course description:** Topics include a brief review of elementary algebra; linear, quadratic, exponential, and logarithmic functions; polynomials; systems of linear equations; applications. Usually offered only in the summer session.

**Text: **Durbin, **College Algebra, preliminary third edition, McGraw-Hill College Custom Series, 1993**

M 301 is the lowest-level "precalculus" course we offer. It should be an honest college algebra course, that is, not an intermediate algebra course (which is offered by community colleges and some four-year colleges and which is often equivalent to second-year high school algebra.) This syllabus is written for use in summer school (the only time we offer M 301). It assumes 26 lectures.

- Chapter 1 Five Fundamental Themes 5 sections 4 lectures
- Chapter 2 Algebraic Expressions 5 sections 4 lectures
- Chapter 3 Equations and Inequalities 5 sections 5 lectures
- Chapter 4 Graphs and Functions 4 sections 4 lectures
- Chapter 5 Polynomial and Rational Functions 4 sections 4 lectures
- Chapter 6 Exponential, Logarithmic Functions 4 sections 3 lectures
- Chapter 7 Systems of Equations, Inequalities 3 sections 2 lectures

## M302 Introduction to Mathematics Syllabus

**Prerequisite and degree relevance: **Texas Success Initiative (TSI) exemption or a TSI Mathematics Assessment score of 350 or higher. The placement test is not required. It may be used to satisfy Area C requirements for the Bachelor of Arts degree under Plan I or the mathematics requirement for the Bachelor of Arts degree under Plan II.

M302 is Intended primarily for general liberal arts students seeking knowledge of the nature of mathematics as well as training in mathematical thinking and problem-solving. Mathematics 302 and 303F may not both be counted. A student may not earn credit for Mathematics 302 after having received credit for any calculus course. May not be counted toward a degree in the College of Natural Sciences.

Responsible Parties: Jennifer Austin and Amanda Hager, June 2019

**Course Description: **Introduction to Mathematics is a terminal course satisfying the University's general education requirement in mathematics. Topics may may be chosen from: Fibonacci numbers, number theory (divisibility, prime numbers, the Fundamental Theorem of Arithmetic, gcd, Euclidean Algorithm, modular arithmetic, special divisibility tests), infinity, geometry (Pythagorean Theorem, Platonic Solids, the fourth dimension, rubber sheet geometry, the Moebius band), chaos and fractals, probability (definition, laws, permutations and combinations), network theory (Euler circuits, traveling salesman problem, bin packing), statistics, game theory, voting paradoxes. Some material is of the instructor's choosing.

**Texts:** **For all Practical Purpose** or **The Heart of Mathematics, Fourth Edition (preferred)**

There is a broad spectrum of students who take M302. Some are quite good at math and may even have had some calculus in high school. These, however, are greatly outnumbered by the students who have weak math skills and poor backgrounds. It is not at all uncommon for the students to exhibit a fear of and dislike for math and most have very low self-confidence about their ability to succeed in a math class. In answer to this, the goal of the course should be to demonstrate that math is not about memorizing formulas, but is rather a process of thinking which is relevant to them on a daily basis. The two recommended books, both are geared toward this type of course. **For All Practical Purpose** emphasizes applications of math in today's world such as scheduling problems and consumer finance models, for example. **The Heart of Mathematics**, while dealing with more theoretical topics such as number theory and topology, emphasizes that the problem-solving strategies used to solve mathematical problems are universal and can be applied to solving day-to-day problems. Both texts have proven to be successful at engaging this population of students and giving them a new appreciation of math as well as boosting their self-confidence.

The topics to be covered will depend on the choice of text. Both texts cover probability and statistics and at least 3 weeks of the course should be devoted to this topic. The coverage in **For all Practical Purpose **is more thorough, especially in the area of statistics. If this is the chosen text, then the syllabus should include chapter 5 and 7. Chapter 6 can be covered lightly, if at all, and chapter 8 should be considered optional. If **The Heart of Mathematics** is the chosen text, then all of chapter 7 should be covered.

**Sample Syllabus for "For All Practical Purpose": **

- Chapter 1 Street Networks All sections (3 days)
- Chapter 2Visiting Vertices(omit Minimum cost spanning trees) (4 days)
- Chapter 3 Planning and Scheduling(omit Bin Packing) (4 days)
- Chapter 5 Producing Data All sections(4 days)
- Chapter 6Exploring Data Cover lightly (2 days)
- Chapter 7 Probability All sections (5 days)
- Chapter 10 Transmitting Information (supplement the modular arithmetic and cover cryptography only) (4 days)
- Chapter 15 Game Theory All sections (5 days)
- Chapter 20 Consumer Finance Models All section (time permitting) (6 days)

Notes: **For all Practical Purposes **Chapter 1 is an introduction to graph theory and is a good chapter for establishing the course as one which is not "formula-based." Chapter 2 and 3 then follow up with some applications of graph theory.

As mentioned above, Chapters 5 and 7 should be covered thoroughly and Chapter 6 lightly. Chapters 9 and 10 introduce the concept of modular arithmetic with applications to error detecting codes cryptography. Students tend to find the arithmetic challenging, but in general, they enjoy the ideas in these chapters.

Chapter 13 on Fair Division is fun to do, however, it is difficult to get the ideas across. Students tend to get lost in the logic and may end up simply memorizing procedures.

Chapter 15, Game Theory, also gives the students a work-out in the area of following a logical argument and again they tend to memorize algorithms for finding good strategies. This chapter does give a chance to revisit expected value and they also appreciate the real-world applications of the "Prisoner's Dilemma" problems.

Chapter 20 deals with compound interest and annuities. The relevance of this material to their lives makes it one of the most widely-appreciated chapter on the part of the students.

** Sample syllabus for "The Heart of Mathematics":**

- Chapter 1: Fun and Games All sections (2 days)
- Chapter 2: Number Contemplation 2.1-2.3; 2.6-2.7 (7 days)
- Chapter 3: Infinity Sections 3.1-3.3 (4 days)
- Chapter 4: Geometric Gems 4.1, 4.3, 4.5, 4.7 (6 days)
- Chapter 5: Contortions of Space 5.1-5.3 (4-5 days)
- Chapter 6: Modeling Our World through Graphs 6.1-6.2 (2 days)
- Chapter 7: Chaos and Fractals 7.1-7.2, 7.6 (3 days)
- Chapter 8: Taming Uncertainty 8.1-8.4 (5 days)

**Notes: The Heart of Mathematics: **

Chapter 1 is excellent for setting the tone of the class and illustrating some problem-solving strategies. The puzzles also tie in with the material from later chapters.

Chapter 2 covers some topics from number theory and gives an appreciation of number theory as an ancient area of mathematics.

Chapter 3, on infinity, is guaranteed to provoke lively discussions as well as controversy.

Chapter 4 contains some nice sections on geometry. The section on the Pythagorean theorem gives the students several examples of geometric proofs. In the section on the Platonic solids, the students are encouraged to build the solids and explore the concept of duality. The section on the fourth dimension gives them the opportunity to experience an abstract idea through the process of generalization. The Moebius Band is a nice, concrete application.

Chapter 5 deals with some ideas from topology. The section on rubber sheet geometry has some fun and surprising results, but the students will probably need a model to convince them that the results are indeed true.

Chapter 6 deals with an introduction to graph theory. The section on the Euler characteristic ties in with chapter 4's section on Platonic Solids.

Chapter 7 deals with chaos and fractals. The key point here is for students to understand iterative processes and how they relate to fractals.

Chapter 8 deals with basic probability.

## M305G Preparation for Calculus Syllabus

**Texts: **Abramson, Algebra and Trigonometry, ISBN 978-1-947172-10-4 (Units 1-3) and Abramson, Precalculus, ISBN 978-1-947172-06-7 (Unit 4)

**Responsible party:** Amanda Hager, December 2017

**Prerequisite and degree relevance:** An appropriate score on the mathematics placement exam. Mathematics 305G and any college-level trigonometry course may not both be counted. A student may not earn credit for Mathematics 305G after having received credit for any calculus course with a grade of at least C-. Mathematics 301, 305G, and equivalent courses may not be counted toward a degree in mathematics.

**Course description:** The purpose of this course is to prepare students for calculus courses. Some students are taking this course as a review, many because they did not score high enough on the mathematics placement exam to enter calculus directly. The course emphasis is on techniques needed in calculus, with an emphasis on rigorous algebraic practice and on recognizing and interpreting graphs. It is assumed that the students have had at least three and a half years of high school mathematics.

**Timing and optional sections:** The following table contains suggestions for timing of topics and includes 36 class hours of content. Allowing for in-class exams, there remain 3-5 class hours for review or optional topics.

Topic |
Section |
No of class hours |

Unit 1: Algebra and function basics, 9 hours |
||

Exponents | 1.2 | 0.5 |

Quadratic Formula | 2.5 | 0.5 |

Absolute value equations | 2.6 | 0.5 |

Absolute value inequalities | 2.7 | 0.5 |

Quadratic, rational inequalities | 2.7 | 1 |

Functions, notation, domain/range | 3.1/3.2 | 0.5 |

Graph types (toolbox functions) | 3.1 | 0.5 |

Increasing/decreasing/pos/neg functions | 3.3 | 1 |

Function transformations | 3.5 | 2 |

Algebra and compositions of functions | 3.4 | 1 |

Domains of compositions, sums, etc. | 3.4 | 1 |

Unit 2: Exponential and logarithmic functions, 8 hours |
||

One-to-one, invertible functions | 3.7/5.7 | 1 |

Exponential functions | 6.1 | 0.5 |

Graphs of exponential functions | 6.2 | 0.5 |

Logarithmic functions | 6.3 | 1 |

Graphs of logarithmic functions | 6.4 | 1 |

Properties of logarithmic and exponential functions | 6.5 | 1 |

Solving exponential/logarithmic equations | 6.6 | 2 |

Modeling with exponential/logarithmic functions | 6.7 | 1 |

Unit 3: Trigonometry, 12 hours |
||

Angles/triangles/radians | 7.1 | 1 |

Unit circle trigonometry | 7.3/7.4 | 1 |

Graphing sine/cosine, computing amplitude/period | 8.1 | 1 |

Graphing tangent | 8.2 | 1 |

Transformations of trig graphs | 8.1/8.2 | 1 |

Solving trig equations | 9.5 | 2 |

Solving right triangles, angles of elevation | 7.2 | 2 |

Identities/trig identities | 9.1/9.2 | 2 |

Inverse trig functions | 8.3 | 1 |

Unit 4: Limits, 7 hours (uses Precalculus text) |
||

Piecewise defined functions | 1.2 | 1 |

Limits via graphs, tables | 12.1 | 2 |

Limits formally | 12.2 | 2 |

Limits at infinity, infinite limits, continuity | 12.2/12.3 | 2 |

## M408C Differential and Integral Calculus Syllabus

**Text: Stewart, Calculus, Early Transcendentals, Eighth Edition**

**Responsible Party**: Ray Heitmann June 2014

**Prerequisite and degree relevance**: An appropriate score on the mathematics placement exam or Mathematics 305G with a grade of at least B-.

Math majors are required to take both M408C and M408D (or either the equivalent sequence M408K, M408L, M408M; or the equivalent sequence M408N, M408S, M408M). Mathematics majors are required to make grades of C- or better in each of these courses.

408C may not be counted by students with credit for any of Mathematics 403K, 408K, 408N, or 408L.

**Course description:** M408C is the standard first-semester calculus course. It is directed at students in the natural sciences and engineering. The emphasis in this course is on problem-solving, not the theory of analysis. There should be some understanding of analysis, but the majority of the proofs in the text should not be covered in class.

The syllabus for M408C includes most of the basic topics in the theory of functions of a real variable: algebraic, trigonometric, logarithmic and exponential functions and their limits, continuity, derivatives, maxima and minima, integration, area under a curve, and volumes of revolution.

**Overview and Course Goals**

The following pages comprise the syllabus for M408C, and advice on teaching it. Calculus is a service course, and the material in it was chosen after interdepartmental discussions. Please do not make drastic changes. You will do your students a disservice and leave them ill-equipped for subsequent courses.

For those instructors who have taught M408C previously, some changes should be noted. Chapter 7 has been moved to M408D, allowing a slightly less hectic pace and more importantly the coverage of some topics which have been omitted or optional in the past. The formal definition of a limit should be covered, although you still shouldn’t expect delta-epsilon proofs. Sections 3.8, 3.9, 3.10, 4.7 are no longer optional. Sections 6.3, 6.4, 6.5 have been included as optional sections and some, but not all, of these topics, should be covered. These adjustments give more attention to applications of both differentiation and integration.

Remember that 408C/D is the fast sequence for students with good algebra skills; students who cannot maintain the pace are encouraged to take either the 408NSM or the 408KLM sequence.

**Resources for Students**

Many students find the study skills from high school are not sufficient for UT. Sanger Learning Center in Jester has a wide variety of material ( drills, videotaped lectures, computer programs, counseling, math anxiety workshops, algebra and trig review, calculus review) as well as tutoring options, all designed to help students through calculus. On request, (471-3614) they will come to your classroom and explain their services.

You can help your students by informing them of SLC services.

**Timing and Optional Sections**

A typical semester has 42-44 MWF days. The syllabus contains material for 37 days, allowing some time for testing and review. Those teaching on TTh should adjust the syllabus; a MWF lecture lasts 50 min; a TTh 75 min. The purpose of Chapter 6 is to provide applications showing students what integration really means. It does not matter which optional sections you cover, but it is crucial that you cover some of them or provide alternative examples.

37 Class Days As:

1 Functions and Models (Three Days )

- 1.4 Exponential Functions
- 1.5 Inverse Functions and Logarithms

2 Limits and Derivatives (Six Days)

- 2.1 The Tangent and Velocity Problems
- 2.2 The Limit of a Function
- 2.3 Calculating Limits Using the Limit Laws
- 2.4 The Precise Definition of a Limit
- 2.5 Continuity
- 2.6 Limits at Infinity; Horizontal Asymptotes
- 2.7 Derivatives and Rates of Change
- 2.8 The Derivative of a Function

3 Differentiation Rules (Eleven Days)

- 3.1 Derivatives of Polynomials and Exponential Functions
- 3.2 The Product and Quotient Rules
- 3.3 Derivatives of Trigonometric Functions
- 3.4 The Chain Rule
- 3.5 Implicit Differentiation
- 3.6 Derivatives of Logarithmic Functions
- 3.7 Rates of Change in the Natural and Social Sciences (
*optional*) - 3.8 Exponential Growth and Decay
- 3.9 Related Rates
- 3.10 Linear Approximations and Differentials
- 3.11 Hyperbolic Functions (
*quickly*)

4 Applications of Differentiation (Eight Days)

- 4.1 Maximum and Minimum Values
- 4.2 The Mean Value Theorem
- 4.3 How Derivatives Affect the Shape of a Graph
- 4.4 Indeterminate Forms and L'Hospital's Rule
- 4.5 Summary of Curve Sketching
- 4.7 Optimization Problems
- 4.9 Antiderivatives

5 Integrals (Five Days)

- 5.1 Areas and Distances
- 5.2 The Definite Integral
- 5.3 The Fundamental Theorem of Calculus
- 5.4 Indefinite Integrals and the Net Change Theorem
- 5.5 The Substitution Rule

6 Applications of Integration (Four Days)

- 6.1 Areas between Curves
- 6.2 Volume
- 6.3 Volumes by Cylindrical shells (optional)
- 6.4 Work (optional)
- 6.5 Average value of function (optional)

## M408D Sequences, Series, and Multivariable Calculus Syllabus

**Text: Stewart, Calculus, Early Transcendentals, 8th Edition**

Responsible Parties: Ray Heitmann and Lorenzo Sadun, June 2014

**Prerequisite and degree relevance:** A grade of C- or better in M 408C, M308L, M 408L, M 308S or M 408S. Only one of the following may be counted: Mathematics 403L, 408D, 408M (or 308M). Math majors are required to take both M408C and M408D (or either the equivalent sequence M408K, M408L, M408M; or the equivalent sequence M408N, M408S, M408M). Mathematics majors are required to make grades of C- or better in each of these courses.

Certain sections of this course are reserved as advanced placement or are honors sections; they are restricted to students who have scored well on the advanced placement AP exams or are honors students, or who have the approval of the faculty mathematics advisor. Such sections and their restrictions are listed in the Course Schedule each semester.

**Course description:** M 408C, M 408D is our standard first-year calculus sequence. It is designed for students in the natural and social sciences and engineering students. The emphasis in this course is on problem-solving, not on theory. While the course necessarily includes some discussion of theoretical notions, its primary objective is not the production of theorem-provers. M 408D contains a treatment of infinite series, and an introduction to vectors and vector calculus in 2-space and 3-space, including parametric equations, partial derivatives, gradients, and multiple integrals.

### Overview and Course Goals

The following pages comprise the syllabus for M 408D, and advice on teaching it. Calculus is a service course, and the material in it was chosen after interdepartmental discussions. Please do not make drastic changes (for example, skipping techniques of integration). You will do your students a disservice and leave them ill-equipped for subsequent courses.

This is not a course in the theory of calculus; the majority of the proofs in the text should not be covered in class. At the other extreme, some of our brightest math majors first found their passion for calculus; one ought not to bore them. In general it is fair to say that M 408D students will do better than M 408C students; on the other hand, M 408D is a more difficult course. Please keep in mind that students who pass this course meet the prerequisite for M 427K, where it assumed they have good calculus skills. The M 408C/D sequence is the fast sequence for students with good algebra skills; students who cannot maintain the pace are encouraged to take either the M 408N/S/M or the M 408K/L/M sequence.

**Resources for Students**

Some of our students have weak study skills. The Sanger Learning Center in Jester has a wide variety of material (drills, videotaped lectures, computer programs, counseling, math anxiety workshops, algebra and trig review, calculus review), as well as tutoring options, all designed to help students through calculus. On request, (471-3614) they'll come to your classroom and explain their services.

You can help your students by informing them of SLC services.

**Timing and Optional Sections**

A typical semester has 42-44 MWF days. The syllabus contains material for 38 days; this allows some time for testing, reviews, and optional material. In the spring semester, you will have more time to cover optional material. Those teaching on TTh should adjust the syllabus; a MWF lecture lasts 50 min; a TTh lasts 75 minutes.

### 38 Class Days As:

- 7 Techniques of Integration (eight days)
- Substitution Review
- 7.1 Integration by Parts
- 7.2 Trigonometric Integrals
- 7.3 Trigonometric Substitution
- 7.4 Integration of Rational Functions by Partial Fractions
- 7.5 Strategy for Integration (use as a reference with good problem set)
- 7.8 Improper Integrals

- 9 Differential Equations (six days)
- 9.1 Modeling with Differential Equations
- 9.2 Direction Fields and Euler’s Method
- 9.3 Separable Equations
- 9.4 Models for Population Growth
- 9.5 Linear Equations
- 9.6 Predator-prey Systems (optional)

- 10 Parametric Equations and Polar Coordinates (four days)
- 10.1 Curves Defined by Parametric Equations
- 10.2 Calculus with Parametric Curves
- 10.3 Polar Coordinates
- 10.4 Areas and Lengths in Polar Coordinates
- 10.5 Conic Sections (optional)
- 10.6 Conic Sections in Polar Coordinates (optional)

- 11 Infinite Sequences and Series (twelve days)
- 11.1 Sequences
- 11.2 Series
- 11.3 The Integral Test and Estimates of Sums
- 11.4 The Comparison Tests
- 11.5 Alternating Series
- 11.6 Absolute Convergence and the Ratio and Root Tests
- 11.7 Strategy for Testing Series
- 11.8 Power Series
- 11.9 Representations of Functions as Power Series
- 11.10 Taylor and Maclaurin Series
- 11.11 Applications of Taylor Polynomials

- 14 Partial Derivatives (three days)
- 14.1 Functions of Several Variables
- 14.2 Limits and Continuity
- 14.3 Partial Derivatives
- 14.5 The Chain Rule

- 15 Multiple Integrals (five days)
- 15.1 Double Integrals over Rectangles
- 15.2 Double Integrals over General Regions
- 15.3 Double Integrals in Polar Coordinates
- 15.4 Applications of Double Integrals (optional)
- 15.9 Change of Variables in Multiple Integrals (if time permits)

## M408K Differential Calculus Syllabus

**Text: Stewart, Calculus, Early Transcendentals, Eighth Edition**

** Responsible Parties:** Jane Arledge, Kathy Davis, Ray Heitmann, Diane Radin June 2011

**Core curriculum**

This course may be used to fulfill the mathematics component of the university core curriculum and addresses core objectives established by the Texas Higher Education Coordinating Board: communication skills, critical thinking skills, and empirical and quantitative skills.

Calculus is the theory of things that change, and so is essential for understanding a changing world. Students are expected to use calculus to compute optimal strategies in a variety of settings (Chapter 3, max/min), as well as to apply derivatives to understand changing quantities in physics, economics, and biology.

Students improve their number sense through qualitative reasoning and by comparing the results of formulas to those guiding principles.

Student activities include creating logically ordered, clearly written solutions to problems, and communicating with the instructor and their peers during a lecture by asking and responding to questions and discussion in lecture.

**Prerequisite and degree relevance: **An appropriate score on the mathematics placement exam or Mathematics 305G with a grade of at least B-. Only one of the following may be counted: M403K, M408C, M408K, M408N.

Calculus is offered in two equivalent sequences: a two-semester sequence, M 408C/408D, which is recommended only for students who score at least 600 on the mathematics Level I or IC Test, and a three-semester sequence, M 408K/408L/408M.

For some degrees, the two-semester sequence M 408K/408L satisfies the calculus requirement. This sequence is also a valid prerequisite for some upper-division mathematics courses, including M325K, 427K, 340L, and 362K.

Math majors are required to take both M408C and M408D (or either the equivalent sequence M408K, M408L, M408M; or the equivalent sequence M408N, M408S, M408M). Mathematics majors are required to make grades of C- or better in each of these courses.

**Course description:** M408K is one of two first-year calculus courses. It is directed at students in the natural and social sciences and at engineering students. In comparison with M408C, it covers fewer chapters of the text. However, some material is covered in greater depth, and extra time is devoted to the development of skills in algebra and problem-solving. This is not a course in the theory of calculus.

The syllabus for M 408K includes most of the basic topics in the theory of functions of a real variable: algebraic, trigonometric, logarithmic and exponential functions and their limits, continuity, derivatives, maxima, and minima, as well as definite integrals and the Fundamental Theorem of Calculus.

### Overview and Course Goals

The following pages comprise the syllabus for M 408K, and advice on teaching it. Calculus is a service course, and the material in it was chosen after interdepartmental discussions. Please do not make drastic changes (for example, skipping techniques of integration). You will do your students a disservice and leave them ill-equipped for subsequent courses.

This is not a course in the theory of calculus; the majority of the proofs in the text should not be covered in class. At the other extreme, some of our brightest math majors found their first passion in calculus; one ought not to bore them. Remember that 408K/L/M is the sequence designed for students who may not have taken calculus previously. Students who have seen calculus and have done well might be better placed in the faster M 408C/408D sequence.

**Resources for Students**

Many students find the study skills from high school are not sufficient for UT. The Sanger Learning Center (http://lifelearning.utexas.edu/) in Jester has a wide variety of material ( drills, videotaped lectures, computer programs, counseling, math anxiety workshops, algebra and trig review, calculus review) as well as tutoring options, all designed to help students through calculus. On request, they will come to your classroom and explain their services.

You can help your students by informing them of these services.

**Timing and Optional Sections**

A typical fall semester has 42 hours of lecture, 42 MWF and 28 TTh days, while the spring has 45 hours, 45 MWF and 30 TTh days (here, by one hour we mean 50 minutes -- thus in both cases there are three "hours" of lecture time per week). The syllabus contains suggestions as to timing and includes approximately 35 hours. Even after including time for exams, etc., there will be some time for the optional topics, reviews, and/or additional depth in some areas.

### Forty Class Days As:

- 1 Functions and Models (3 hours)
- 1.4 Exponential Functions
- 1.5 Inverse Functions and Logarithms

- 2 Limits and Derivatives (9 hours)
- 2.1 The Tangent and Velocity Problems
- 2.2 The Limit of a Function
- 2.3 Calculating Limits Using the Limit Laws
- 2.4 The Precise Definition of a Limit (optional)
- 2.5 Continuity
- 2.6 Limits at Infinity; Horizontal Asymptotes
- 2.7 Derivatives and Rates of Change
- 2.8 The Derivative of a Function

- 3 Differentiation Rules (10 hours)
- 3.1 Derivatives of Polynomials and Exponential Functions
- 3.2 The Product and Quotient Rules
- 3.3 Derivatives of Trigonometric Functions
- 3.4 The Chain Rule
- 3.5 Implicit Differentiation
- 3.6 Derivatives of Logarithmic Functions
- 3.7 Rates of Change in the Natural and Social Sciences
- 3.8 Exponential Growth and Decay (
*optional*) - 3.9 Related Rates
- 3.10 Linear Approximations and Differentials
- 3.11 Hyperbolic Functions (
*optional*)

- 4 Applications of Differentiation (9 hours)
- 4.1 Maximum and Minimum Values
- 4.2 The Mean Value Theorem
- 4.3 How Derivatives Affect the Shape of a Graph
- 4.4 Indeterminate Forms and L'Hospital's Rule
- 4.5 Summary of Curve Sketching
- 4.7 Optimization Problems
- 4.9 Antiderivatives

- 5 Integrals (4 hours)
- 5.1 Areas and Distances
- 5.2 The Definite Integral
- 5.3 The Fundamental Theorem of Calculus

## M408L Integral Calculus Syllabus

**Text: Stewart, Calculus, Early Transcendentals, 8th Edition**

**Responsible Parties**: Jane Arledge, Kathy Davis, Ray Heitmann, December 2011.

**Prerequisite and degree relevance: **One of M408C, M408N, or M408K, with a grade of at least C- or M408R with a grade of at least B. Only one of the following may be counted: Mathematics 403L, 408L (or 308L), 408S.

Calculus is offered in three equivalent sequences at UT: an accelerated two-semester sequence, M 408C/D, and two three-semester sequences, M 408K/L/M and M 408N/S/M. The latter is restricted to students in the College of Natural Sciences.

Completion (with grades of C- or better) of one of these calculus sequences is required for a mathematics major. For some degrees, the two-semesters M 408N/S satisfies the calculus requirement. These two courses are also a valid prerequisite for some upper-division mathematics courses, including M 325K, 427K, 340L, and 362K.

**Course description:** M 408L is the second-semester calculus course of the three-course calculus sequence. In comparison with M408D, it covers fewer chapters of the text. However, some material is covered in greater depth, and extra time is devoted to the development of skills in algebra and problem-solving. This is not a course in the theory of calculus.

Introduction to the theory and applications of integral calculus of functions of one variable. The syllabus for M 408L includes most of the basic topics of integration on functions of a single real variable: the fundamental theorem of calculus, applications of integrations, techniques of integration, sequences, and infinite series.

The emphasis in this course is on problem-solving, not on the presentation of theoretical considerations. While the course includes some discussion of theoretical notions, these are supporting rather than primary.

**Overview and Course Goals**

The following pages comprise the syllabus for M 408L, and advice on teaching it. Calculus is a service course, and the material in it was chosen after interdepartmental discussions. Please do not make drastic changes (for example, skipping techniques of integration). You will do your students a disservice and leave them ill-equipped for subsequent courses.

This is not a course in the theory of calculus; the majority of the proofs in the text should not be covered in class. At the other extreme, some of our brightest math majors found their first passion in calculus; one ought not to bore them. Remember that 408K/L/M is the sequence designed for students who may not have taken calculus previously. Students who have seen calculus and have done well might be better placed in the faster M 408C/408D sequence.

**Resources for Students**

Many students find the study skills from high school are not sufficient for UT. The Sanger Learning Center (http://lifelearning.utexas.edu/) in Jester has a wide variety of material ( drills, videotaped lectures, computer programs, counseling, math anxiety workshops, algebra and trig review, calculus review) as well as tutoring options, all designed to help students through calculus. On request, they will come to your classroom and explain their services.

You can help your students by informing them of these services.

**Timing and Optional Sections**

A typical fall semester has 42 hours of lecture, 42 MWF and 28 TTh days, while a typical spring has 44 MWF and 30 TTh days (here, by one hour we mean 50 minutes -- thus in both cases there are three "hours" of lecture time per week). The following syllabus contains suggestions as to timing and includes approximately 36 hours of required material. Even after including time for exams, etc., there will be some time for the optional topics, reviews, and/or additional depth in some areas.

** Syllabus**

- Ch. 5 Integrals (4 hours)
- 5.3 The Fundamental Theorem of Calculus (review)
- 5.4 Indefinite Integrals and the Net Change Theorem
- 5.5 The Substitution Rule

- Ch. 6 Applications of Integration (2 hours)
- 6.1 Areas between Curves
- 6.2 Volumes
- 6.3 Volumes by Cylindrical Shells (optional)

- Ch. 7 Techniques of Integration (9 hours)
- 7.1 Integration by Parts
- 7.2 Trigonometric Integrals (light)
- 7.3 Trigonometric Substitution
- 7.4 Integration of Rational Functions by Partial Fractions
- 7.5 Strategy for Integration
- 7.7 Approximate Integration (optional)
- 7.8 Improper Integrals

- Ch. 9 Differential Equations (optional -- not in special UT version of the book)
- 9.3 Separable Equations
- 9.4 Models for Population Growth

- Ch. 14 Partial Derivatives (1 hour)
- 14.3 Partial Derivatives

- Ch. 15 Multiple Integrals (4 hours)
- 15.1 Double Integrals over Rectangles
- 15.2 Double Integrals over General Regions

- Ch. 11 Infinite Sequences and Series (16 hours)
- 11.1 Sequences
- 11.2 Series
- 11.3 The Integral Test and Estimates of Sums
- 11.4 The Comparison Tests
- 11.5 Alternating Series
- 11.6 Absolute Convergence and the Ratio and Root Tests
- 11.7 Strategy for Testing Series
- 11.8 Power Series
- 11.9 Representations of Functions as Power Series
- 11.10 Taylor and Maclaurin Series
- 11.11 Applications of Taylor Polynomials (optional)

## M408M Multivariable Calculus Syllabus

**Text: Stewart, Calculus, Early Transcendentals, Eighth Edition**

**Responsible Parties:** Ray Heitmann and Jane Arledge, May 2012

**Prerequisite and degree relevance:** M408L or M408S with a grade of at least C-. Only one of the following may be counted: Mathematics 403L, 408D, 408M (or 308M).

Calculus is offered in two equivalent sequences: a two-semester sequence, M 408C/D, or either of two three-semester sequences, M 408N/S/M (for College of Natural Science Students) or M 408K/L/M. Completion of one of these sequences is required for mathematics majors, with a C- or better in each course.

For some degrees, M 408N/S or M 408K/L satisfy the calculus requirement. This sequence is also a valid prerequisite for some upper-division mathematics courses, including M 325K, M 427K, M 340L, and M 362K.

**Course description:** M 408M is directed at students in the natural and social sciences and at engineering students. In comparison with M408D, it covers fewer chapters of the text. However, some material is covered in greater depth. This is not a course in the theory of calculus.

The content includes an introduction to the theory and applications of differential and integral calculus of functions of several variables, including parametric equations, polar coordinates, vectors, vector calculus, functions of several variables, partial derivatives, gradients, and multiple integrals.

### Forty Class Days As:

- 10 Parametric Equations and Polar Coordinates (seven days)
- 10.1 Curves Defined by Parametric Equations
- 10.2 Calculus with Parametric Curves
- 10.3 Polar Coordinates
- 10.4 Areas and Lengths in Polar Coordinates
- 10.5 Conic Sections
- 10.6 Conic Sections in Polar Coordinates

- 12 Vectors and the Geometry of Space (eight days)
- 12.1 Three-Dimensional Coordinate Systems
- 12.2 Vectors
- 12.3 The Dot Product
- 12.4 The Cross Product
- 12.5 Equations of Lines and Planes
- 12.6 Cylinders and Quadric Surfaces

- 13 Vector Functions (five days)
- 13.1 Vector Functions and Space Curves
- 13.2 Derivatives and Integrals of Vector Functions
- 13.3 Arc Length and Curvature
- 13.4 Motion in Space: Velocity and Acceleration

- 14 Partial Derivatives (ten days)
- 14.1 Functions of Several Variables
- 14.2 Limits and Continuity
- 14.3 Partial Derivatives
- 14.4 Tangent Planes and Linear Approximations
- 14.5 The Chain Rule
- 14.6 Directional Derivatives and the Gradient Vector
- 14.7 Maximum and Minimum Values
- 14.8 Lagrange Multipliers

- 15 Multiple Integrals (ten days)(first three sections are review)

- 15.1 Double Integrals over Rectangles
- 15.2 Double Integrals over General Regions
- 15.3 Double Integrals in Polar Coordinates
- 15.4 Applications of Double Integrals (optional)
- 15.9 Change of Variables in Multiple Integrals (if time permits)

## M408N Differential Calculus for Science

**Text: Stewart, Calculus, Early Transcendentals, Eighth Edition**

** Responsible Parties:** Jane Arledge, Kathy Davis, Ray Heitmann, June 2011

**Prerequisite and degree relevance:** An appropriate score on the mathematics placement exam or Mathematics 305G with a grade of at least B-. Only one of the following may be counted: Mathematics 403K, 408C, 408K, 408N.

Calculus is offered in three equivalent sequences at UT: an accelerated two-semester sequence, M 408C/D, and two three-semester sequences, M 408K/L/M and M 408N/S/M. The latter is restricted to students in the College of Natural Sciences.

Completion (with grades of C- or better) of one of these calculus sequences is required for a mathematics major. For some degrees, the two-semester sequence M 408N/S satisfies the calculus requirement. These two courses are also a valid prerequisite for some upper-division mathematics courses, including M 325K, 427K, 340L, and 362K.

**Course description:** M 408N is the first-semester calculus course of the three-course calculus sequence. It is directed at students in the natural sciences and is restricted to College of Natural Science Students. The emphasis in this course is on problem-solving, not on the presentation of theoretical considerations. While the course includes some discussion of theoretical notions, these are supporting rather than primary.

The syllabus for M 408N includes most of the basic topics in the theory of differential calculus of functions of a real variable: algebraic, trigonometric, logarithmic and exponential functions and their limits, continuity, derivatives, maxima, and minima, as well as definite integrals and the Fundamental Theorem of Calculus.

**Overview and Course Goals**

The following pages comprise the syllabus for M 408N, and advice on teaching it. Calculus is a service course, and the material in it was chosen after interdepartmental discussions. Please cover the material that is not deemed "optional." You will do your students a disservice and leave them ill-equipped for subsequent courses.

This is not a course in the theory of calculus; the majority of the proofs in the text should not be covered in class. At the other extreme, some of our brightest math majors found their first passion in calculus; one ought not to bore them. Remember that 408N/S/M is the sequence designed for students who may not have taken calculus previously. Students who have seen calculus and have done well might be better placed in the faster M 408C/408D sequence.

**Resources for Students**

Many students find the study skills from high school are not sufficient for UT. The Sanger Learning Center (http://lifelearning.utexas.edu/) in Jester has a wide variety of material ( drills, videotaped lectures, computer programs, counseling, math anxiety workshops, algebra and trig review, calculus review) as well as tutoring options, all designed to help students through calculus. On request, they will come to your classroom and explain their services.

You can help your students by informing them of these services.

**Timing and Optional Sections**

A typical fall semester has 42 hours of lecture, 42 MWF and 28 TTh days, while the spring has 45 hours, 45 MWF and 30 TTh days (here, by one hour we mean 50 minutes -- thus in both cases there are three "hours" of lecture time per week). The following syllabus contains suggestions as to timing and includes approximately 35 hours. Even after including time for exams, etc., there will be some time for the optional topics, reviews, and/or additional depth in some areas.

**Syllabus**

- 1 Functions and Models (3 hours)
- 1.4 Exponential Functions
- 1.5 Inverse Functions and Logarithms

- 2 Limits and Derivatives (9 hours)
- 2.1 The Tangent and Velocity Problems
- 2.2 The Limit of a Function
- 2.3 Calculating Limits Using the Limit Laws
- 2.4 The Precise Definition of a Limit (optional)
- 2.5 Continuity
- 2.6 Limits at Infinity; Horizontal Asymptotes
- 2.7 Derivatives and Rates of Change
- 2.8 The Derivative of a Function

- 3 Differentiation Rules (10 hours)
- 3.1 Derivatives of Polynomials and Exponential Functions
- 3.2 The Product and Quotient Rules
- 3.3 Derivatives of Trigonometric Functions
- 3.4 The Chain Rule
- 3.5 Implicit Differentiation
- 3.6 Derivatives of Logarithmic Functions
- 3.7 Rates of Change in the Natural and Social Sciences (
*optional*) - 3.8 Exponential Growth and Decay (
*optional*) - 3.9 Related Rates
- 3.10 Linear Approximations and Differentials (
*optional*) - 3.11 Hyperbolic Functions (
*optional*)

- 4 Applications of Differentiation (9 hours)
- 4.1 Maximum and Minimum Values
- 4.2 The Mean Value Theorem
- 4.3 How Derivatives Affect the Shape of a Graph
- 4.4 Indeterminate Forms and L'Hospital's Rule
- 4.5 Summary of Curve Sketching (optional)
- 4.7 Optimization Problems
- 4.9 Antiderivatives

- 5 Integrals (4 hours)
- 5.1 Areas and Distances
- 5.2 The Definite Integral
- 5.3 The Fundamental Theorem of Calculus

## M408R Differential and Integral Calculus for Sciences

**Prerequisite and degree relevance: **An appropriate score on the mathematics placement exam or Mathematics 305G with a grade of at least B-. May not be counted by students with credit for Mathematics 408C, 408K, or 408N.

Web page: http://www.ma.utexas.edu/users/sadun/F16/M408R

Calc Lab: The Math Department Calculus Lab (see www.ma.utexas.edu/academics/undergraduate/calculus-lab/) is open starting on the second week of class. This is a joint TA session for all calculus classes taught at UT and will be staffed at all times by multiple TAs and undergraduate Learning Assistants. No matter what your question, you can always get help at Calc Lab.

Textbook (required): Calculus in Context, by Callahan et al, Available free online at www.math.smith.edu/Local/cicintro/. (If you want a hard copy, it costs about $80 new on Amazon, and a lot less used.)

Software (required): You will need to buy a copy of MATLAB. UT does have a site license and MATLAB is loaded on all of the math department computers. However, you will need to bring a laptop, with MATLAB loaded, to the discussion section, and occasionally to lecture. The student edition of MATLAB costs $50, or you can get it bundled with Simulink (a really good statistics package) for $99.

Scope of course: M408R is a 1-semester survey of calculus. As such, it covers more ground than the first semester of a 2-semester sequence, but with a very different emphasis. We will cover Chapters 1-6 of Callahan and part of Chapter 11.

Goals for the class:

Learning the key ideas of calculus, which I call the six pillars.

2. Track the changes (derivatives)

3. What goes up has to stop before it can come down (max/min)

4. The whole is the sum of the parts (integrals)

5. The whole change is the sum of the partial changes (fundamental theorem)

6. One variable at a time.

Learning how to analyze a scientific situation and model it mathematically.

Learning to analyze a mathematical model using calculus.

Learning how to apply the results of the model back into the real world.

Learning enough formulas and calculational methods to make other goals possible. There

are three questions associated with every mathematical idea in existence:

2. How do you compute it?

3. What is it good for?

Compared to most math classes, we're going to spend a lot more time on the first and third questions, but we still need to address the second. You can't spend all your time looking at the big picture! You need some practice sweating the details, too.

Exam schedule: There will be in-class midterm exams on Wednesday, September 21, Monday, October 17, and Monday, November 21. The final exam will be on Tuesday morning, December 13, 9-12. Exams are closed book, but you will be allowed a single sheet of handwritten (by you!) notes for the midterms and two sheets for the final exam.

Calculators: You are welcome to use whatever you want in class and for homework, but only very basic calculators will be allowed on exams. If it has a screen that shows anything besides numbers, or if it allows programs of any kind, or if it has built-in statistical functions like linear regression, it is not allowed. So don't waste your money on a fancy graphing calculator!! Save the expense and buy MATLAB for your laptop instead.

Classroom procedure: We will provide you with a variety of online learning resources on Quest, keyed to sections of the book. These learning modules were prepared by Bill Wolosensky and me, and are intended to explain and supplement the sections of the book that we are covering. You are expected to study the material and answer some fairly easy online questions before the lecture. Then, in a typical lecture session, we will discuss what you've studied (bring questions!) and you will work in teams on harder and more thought-provoking problems, while I circulate and talk with you about them.

Discussion sections and worksheets (10% of course grade): Much of your learning will come from working on tutorial worksheets on the different topics. These are intended to be done in groups of 3 or 4. You will form your groups on the first discussion day, Thursday, August 25. If you join the class later on, you will be assigned to an existing group. We may start on some tutorials in lecture, but the Tuesday-Thursday discussion sections are the main place where you will work on these. Attendance is required, and the worksheets are to be turned in at the end of the hour. If you are absent you will receive a 0 for the worksheet, and if you did not actively participate in your group's discussion, you may receive a reduced grade. I will drop your lowest 3 worksheet scores at the end of the semester. Mini-projects (6% of course grade) These are larger assignments to be done over a longer time scale. They should be done in the same groups of 3 or 4 that you use for the worksheets. I will drop your lowest score at the end of the semester. Individual homework (9% of course grade): You will also be given homework, mostly from the book, due roughly once every one or two weeks. Unlike the worksheets and mini-projects, these are to be done largely on your own. You are welcome to talk about these problems with me, or with your friends, but each student should turn in his or her own work, and what you turn in should reflect your understanding. Copying somebody else's solution is cheating. I will drop your lowest score at the end of the semester.

Preclass/Learning Modules (5%): Our online content delivery system is called Quest, which can be accessed by going to the page at quest.cns.utexas.edu, logging in, and selecting this class. You will be charged a one-time $30 fee to use this service, which is mandatory for this class. This is where you will find your learning modules, aka preclass assignments. There are learning modules, also known as pre-class assignments for each section of the book that we are covering. These include videos, text, and questions for you to answer. The questions are intended to be easy and are mostly intended to get you ready to learn the material in more depth in lecture. The questions are due at midnight the night before the class in which that section is scheduled to be taught. Your three lowest Quest scores will be dropped at the end of the semester. No late work will be accepted for any reason other than religious holidays. As noted elsewhere, I will drop some of the assignment scores to allow for the fact that stuff happens, much of it beyond our control. Please do not ask if I will accept a late assignment. I won't.

Academic honesty: The University is a place of honor and mutual respect, and students deserve to be treated with courtesy and trust. However, betraying that trust is dishonorable and unforgivable. On an evolutionary scale, cheaters belong somewhere between tapeworms and cockroaches. If you are caught cheating, you will be penalized as harshly as possible under the rules of UT. Most students are honest, honest students do not like cheaters, and they do report what they see.

Grading: Each midterm counts 15%. The final exam counts 25%. The pre-class homework counts 5%, the worksheets 10%, the individual homeworks 9% and the mini-projects count 6%. If you do badly on a midterm or miss a midterm for any reason, then I will substitute the final exam in its place. (E.g., if you bomb one midterm, then your grade will be based on 15% each for the other midterms and 40% for the final.) If you do badly on (or miss) two or more midterms, you're out of luck. The final exam will not substitute for the preclass, worksheet, homework or mini-projects. The final grade distribution is neither a straight scale nor a fixed curve. The cutoffs will be set at the end of the semester, based on overall class performance, with the following qualitative standard for the major grades (with obvious adjustments for plusses and minuses):

even in unusual settings.

A "B" means that you can do the standard problems we have done during the semester, but

struggle with novel applications.

A "C" means that you understand the techniques of the class well enough to handle a class that

has M408R as a prerequisite.

A "D" means that you have learned a substantial amount, but that you are not prepared to take

that successor course.

An "F" means that you have failed to grasp the essential concepts of the course.

Grading isn't an exact science, and I'm only going to adjust cutoffs. Nobody will leapfrog anybody else; if you have more points than your buddy, then your grade will be at least as good as your buddy's. Furthermore, a 90% average will guarantee you at least an A-, an 80% average a B-, and a

70% average a C-. My cutoffs are usually more generous than that, but each semester is unique.

Disabilities: The University of Texas at Austin provides upon request appropriate academic accommodations for qualified students with disabilities. For more information, contact the Office of the Dean of Students at 471-6259, 471-4641 TTY

Drop dates: The deadline for dropping the class without the course appearing on your transcript is September 9. After that date, a "Q" will appear on your record. The deadline for dropping, period, is November 1.

Religious Holidays: I have tried to schedule major class events to avoid religious holidays, and I apologize if I overlooked something. If you expect to miss class or miss an assignment because of a religious holiday, please let me know 14 days in advance, and you will be given the opportunity to make up the missed work within a reasonable time.

UT Core Requirements: This course may be used to fulfill the mathematics component of the university core curriculum and addresses the following three core objectives established by the Texas Higher Education Coordinating Board: communication skills, critical thinking skills, and empirical

and quantitative skills. Stress: If at any time you feel overwhelmed by your coursework, or by campus life, please contact

the Counselling and Mental Health Center

Student Services Bldg (SSB), 5th Floor

Hours: M--F 8 am-- 5 pm

512 471 3515

www.cmhc.utexas.edu

For what it's worth, I had serious anxiety issues myself about a dozen years ago. I know how hard it can be. A combination of counseling and medication helped me turn things around. There is no shame in seeking help, and the upside can be enormous.

Guns: Open carry of firearms, and concealed carry by people who do not hold a Licence to Carry (LTC) or a Concealed Handgun Licence (CHL), are forbidden throughout campus. In addition, LTC and CHL permit holders may not bring concealed firearms to my office (RLM 9.114). According to state law and UT policy, I do not have the authority to ban guns in class, in the TA office, or in CalcLab. However, I respectfully request that license holders do not bring their weapons to any of these places. The mere thought of an armed classroom scares the daylights out of your instructor, and out of many of your fellow students.

## M408S Integral Calculus for Science

**Text: Stewart, Calculus, Early Transcendentals, 8th Edition**

**Responsible Parties**: Jane Arledge, Kathy Davis, Ray Heitmann, December 2011.

**Prerequisite and degree relevance: **Mathematics 408C, 408K, or 408N with a grade of at least C-, or Mathematics 408R with a grade of at least B. Only one of the following may be counted: Mathematics 403L, 408L (or 308L), 408S.

Calculus is offered in three equivalent sequences at UT: an accelerated two-semester sequence, M 408C/D, and two three-semester sequences, M 408K/L/M and M 408N/S/M. The latter is restricted to students in the College of Natural Sciences.

Completion (with grades of C- or better) of one of these calculus sequences is required for a mathematics major. For some degrees, the two-semesters M 408N/S satisfies the calculus requirement. These two courses are also a valid prerequisite for some upper-division mathematics courses, including M 325K, 427K, 340L, and 362K.

**Course description: ** M 408S is the second-semester calculus course of the three-course calculus sequence. It is restricted to College of Natural Science Students. It is an introduction to the theory and applications of integral calculus of functions of one variable.

The syllabus for M 408S includes most of the basic topics of integration on functions of a single real variable: the fundamental theorem of calculus, applications of integrations, techniques of integration, sequences, and infinite series. The emphasis in this course is on problem-solving, not on the presentation of theoretical considerations. While the course includes some discussion of theoretical notions, these are supporting rather than primary.

**Overview and Course Goals**

The following pages comprise the syllabus for M 408S, and advice on teaching it. Calculus is a service course, and the material in it was chosen after interdepartmental discussions. Please do not make drastic changes (for example, skipping techniques of integration). You will do your students a disservice and leave them ill-equipped for subsequent courses.

This is not a course in the theory of calculus; the majority of the proofs in the text should not be covered in class. At the other extreme, some of our brightest math majors found their first passion in calculus; one ought not to bore them. Remember that 408N/S/M is the sequence designed for students who may not have taken calculus previously. Students who have seen calculus and have done well might be better placed in the faster M 408C/408D sequence.

**Resources for Students**

You can help your students by informing them of these services.

**Timing and Optional Sections**

A typical fall semester has 42 hours of lecture, 42 MWF and 28 TTh days, while a typical spring has 45 hours, 45 MWF and 30 TTh days (here, by one hour we mean 50 minutes -- thus in both cases there are three "hours" of lecture time per week). The following syllabus contains suggestions as to timing and includes approximately 36 hours of required material. Even after including time for exams, etc., there will be some time for the optional topics, reviews, and/or additional depth in some areas.

### Syllabus

- Ch. 5 Integrals (4 hours)
- 5.3 The Fundamental Theorem of Calculus (review)
- 5.4 Indefinite Integrals and the Net Change Theorem
- 5.5 The Substitution Rule

- Ch. 6 Applications of Integration (2 hours)
- 6.1 Areas between Curves
- 6.2 Volumes
- 6.3 Volumes by Cylindrical Shells (optional)

- Ch. 7 Techniques of Integration (9 hours)
- 7.1 Integration by Parts
- 7.2 Trigonometric Integrals (light)
- 7.3 Trigonometric Substitution
- 7.4 Integration of Rational Functions by Partial Fractions
- 7.5 Strategy for Integration
- 7.7 Approximate Integration (optional)
- 7.8 Improper Integrals

- Ch. 9 Differential Equations (optional -- not in special UT version of the book)
- 9.3 Separable Equations
- 9.4 Models for Population Growth

- Ch. 14 Partial Derivatives (1 hour)
- 14.3 Partial Derivatives

- Ch. 15 Multiple Integrals (4 hours)
- 15.1 Double Integrals over Rectangles
- 15.2 Double Integrals over General Regions

- Ch. 11 Infinite Sequences and Series (16 hours)
- 11.1 Sequences
- 11.2 Series
- 11.3 The Integral Test and Estimates of Sums
- 11.4 The Comparison Tests
- 11.5 Alternating Series
- 11.6 Absolute Convergence and the Ratio and Root Tests
- 11.7 Strategy for Testing Series
- 11.8 Power Series
- 11.9 Representations of Functions as Power Series
- 11.10 Taylor and Maclaurin Series
- 11.11 Applications of Taylor Polynomials (optional)

## M316 Elementary Statistical Methods Syllabus

**Prerequisite and degree relevance: **An appropriate score on the mathematics placement exam. M 316 is an elementary introduction to statistical methods for data analysis; knowledge of calculus is not assumed. Students with a background in calculus are advised to take M 362K plus either M 358K or M 378K instead. This course may not be counted toward the major requirement for the Bachelor of Arts with a major in mathematics or toward the Bachelor of Science in Mathematics. Students taking the course should have good basic algebra skills.

This course carries the Quantitative Reasoning flag. QR courses are designed to equip you with skills that are necessary for understanding the types of quantitative arguments you will regularly encounter in your adult and professional life. You should, therefore, expect a substantial portion of your grade to come from your use of quantitative skills to analyze real-world problems.

**Text:** **StatsPortal; The Basic Practice of Statistics, 6th edition(2010) by David S. Moore**

StatsPortal contains an interactive e-Book and numerous resources for students and instructors. For students: Learning Curve, statistical videos, Stats Tutor, applets, software manuals, online quizzes, etc. Resources for instructors include the e-Book, PowerPoint lecture slides, instructor's solution manual, printed test bank, i>clicker questions, grade book (which can be downloaded to Blackboard), extra exercises and solutions, etc.

To purchase StatsPortal and register your access code, go to http://courses.bfwpub.com/bps6e.php

Students can use the loose leaf version of the textbook packaged with StatsPortal for a nominal extra charge; the ISBN is 978-1-4641-2954-4. You can ask the Coop to order copies for you.

You may go to www.whfreeman.com/bps6e to browse some of the resources mentioned above.

**Responsible Party**: Evelyn Schultz, June 2012

**Topics:**

** Part I: Exploring Data**

- Chapter 1 Picturing Distributions with Graphs
- Chapter 2 Describing Distributions with Numbers
- Chapter 3 The Normal Distributions
- Chapter 4 Scatterplots and Correlation
- Chapter 5 Regression
- Chapter 6 Two-Way Tables (optional)
- Chapter 7 Exploring Data: Part I Review (May be assigned as reading.)

**Part II: From Exploration to Inference **

- Chapter 8 Producing Data: Sampling
- Chapter 9 Producing Data: Experiments

(Optional but strongly recommended: Commentary, Data Ethics. May be assigned as reading.) - Chapter 10 Introducing Probability (Section on Personal Probability is optional.)
- Chapter 11 Sampling Distributions
- (Optional: Chapter 12 General Rules of Probability)
- (Optional: Chapter 13 Binomial Distributions)
- Chapter 14 Confidence Intervals: The Basics
- Chapter 15 Tests of Significance: The Basics
- Chapter 16 Inference in Practice (more focus on Power and less on Type II error)
- Chapter 17 From Exploration to Inference: Part II Review (May be assigned as reading.)

**Part III: Inference about Variables**

- Chapter 18 Inference about a Population Mean
- Chapter 19 Two-Sample Problems (The section on details of the t approximation is optional, and so are the sections on avoiding the pooled two-sample t procedures and avoiding inference about standard deviations.)
- Chapter 20 Inference about a Population Proportion
- Chapter 21 Comparing Two Proportions
- Chapter 22 Inference about Variables: Part III Review (May be assigned as reading)

**Part IV: Inference about Variables **

- Chapter 23 Two Categorical Variables: The Chi-Square Test (Section on the goodness of fit optional.)
- (Optional: Chapter 24 Inference for Regression)
- (Optional: Chapter 25 One-Way analysis of variance: comparing several means)
- (Optional: Chapter 26 Non-parametric Tests)
- (Optional: Chapter 27 Statistical Process Control)
- (Optional: Chapter 28 Multiple regression)

**Comments for Instructors:**

If you choose to cover any of the optional chapters, save them (with the possible exception of the Commentary on Data Ethics) until the end of the semester. Don't try to do more than two of them. The Commentary on Data Ethics is recommended, with chapter 24 second priority. Note that chapters 12 and 13 are not needed for the rest of the course, with the exception of conditional probability.

The book is readable enough that, especially in chapters 1 – 9, you may want to cover some topics as reading assignments, to be followed by class discussion, rather than lecturing.

The material on inference (beginning with chapter 14) is more challenging for most students than in the earlier chapters. To allow adequate time for the material on inference, chapter 14 should be started just before or at the midpoint of the semester.

Some instructors require students to do a (usually group) project involving designing an experiment or observational study, carrying it out, and analyzing the results.

Chapters 20 and 21: The sections on more accurate confidence intervals should be covered, reflecting currently recommended changes in statistical practice.

Statistical applets. These can be used for in-class demonstrations of concepts if your classroom is equipped for computer projection. They are also available as a resource on StatsPortal.

Access to the website StatsPortal is bundled with new copies of the textbook.

## M316K Foundations of Arithmetic Syllabus

**Prerequisite and degree relevance: **Prerequisite is one of the following courses with a grade of C- or better:

• M 302: Introduction to Mathematics ("Math for Liberal Arts")

• M 303D: Applicable Mathematics

• M 305G or 505G: Elementary Functions and Coordinate Geometry ("Precalculus")

• M 316: Elementary Statistical Methods

This course is required for students preparing to teach elementary school.

**Text:** Beckmann

**Course Description:** An analysis, from an advanced perspective, of the concepts and algorithms of arithmetic, including sets; numbers; numeration systems; definitions, properties, and algorithms of arithmetic operations; and percents, ratios, and proportions. Problem-solving is stressed.

**Topics and Format:** The focus is on students working on Explorations supporting learning in the following sections of the textbook.

**Responsible party: ** Please contact Mark Daniels (mdaniels@math.utexas.edu ) about a detailed syllabi.

## M316L Foundations of Geometry, Statistics, and Probability Syllabus

**Prerequisite and degree relevance:** M316K with a grade of at least C. Restricted to students in a teacher preparation program. May not be counted toward the major requirement for the Bachelor of Arts, Plan I, degree with a major in mathematics or toward the Bachelor of Science in Mathematics degree. Credit for Mathematics 316L may not be earned after a student has received credit for any calculus course with a grade of C- or better unless the student is registered in the College of Education. This course is required for students preparing to teach elementary school, and for students in UTeach Liberal Arts planning to teach in the middle grades. It is also taken by some students preparing to teach middle grades mathematics.

**Text: Beckmann**

**Topics and Format**: The focus is on students working on Explorations supporting learning in the following sections of the textbook.

**More Detailed Syllabus for Instructors**: Instructors should contact Mark Daniels (mdaniels@math.utexas.edu) for details.

Responsible parties: Mark Daniels

## Upper Division Courses

## ACF329 Theory of Interest Syllabus

**Text: Vaaler & Daniel, Mathematical Interest Theory, Mathematical Association, Second Edition**

** Responsible Parties:** Shinko Harper, Milica Cudina, Alisa Havens, Jennifer Mann, January 2013

** Prerequisite and degree relevance:** M408D, M308L, M408L, or M408S with a grade of at least C-, or consent of instructor. This course covers the interest theory portion of the SOA/CAS Financial Mathematics exam (FM/2); this should be about 75-80% of the material on this professional exam, with the balance of the exam testing knowledge of elementary financial derivatives. Topics include nominal and effective interest and discount rates, general accumulation functions and force of interest, yield rates, annuities including those with non-level payment patterns, amortization of loans, sinking funds, bonds, duration, and immunization.

** Chapter 0 An Introduction to the Texas Instruments BA II Plus **(optional)

**Chapter 1 The Growth of Money (7 days)**

- 1.1 Introduction (optional)
- 1.2 What is interest?
- 1.3 Accumulation and Amount Functions
- 1.4 Simple Interest/Linear Accumulation Functions
- 1.5 Compound Interest (The usual case!)
- 1.6 Interest in Advance/The Effective Discount Rate
- 1.7 Discount Functions/The Time Value of Money
- 1.8 Simple Discount
- 1.9 Compound Discount
- 1.10 Nominal Rates of Interest and Discount
- 1.11 A Friendly Competition (Constant Force of Interest)
- 1.12 Force of Interest
- 1.14 Inflation

**Chapter 2 Equations of Value and Yield Rates (4-5 days)**

- 2.1 Introduction (optional)
- 2.2 Equations of Value for Investments Involving a Single Deposit made under Compound Interest
- 2.3 Equations of Value for Investments with Multiple Contributions
- 2.4 Investment Return
- 2.5 Reinvestment Considerations
- 2.6 Approximate Dollar Weighted Yield Rates (optional)
- 2.7 Fund Performance

**Chapter 3 Annuities (Annuities Certain) (11-12 days)**

- 3.1 Introduction (optional)
- 3.2 Annuities Immediate
- 3.3 Annuities Due
- 3.4 Perpetuities & 7.1 Common and Preferred Stock
- 3.5 Deferred Annuities and Values on any Date
- 3.6 Outstanding Loan Balances
- 3.7 Nonlevel Annuities
- 3.8 Annuities with Payments in Geometric Progression
- 3.9 Annuities with Payments in Arithmetic Progression
- 3.10 Yield Rate Examples Involving Annuities (optional)
- 3.11 Annuity Symbols for Nonintegral Terms (optional)
- 3.12 Annuities Governed by General Accumulation Functions (optional)
- 3.13 The Investment Year Method

**Chapter 4 Annuities with Different Payment and Conversion Periods (1-2 days)**

- 4.1 Introduction (optional)
- 4.2 Level Annuities with Payments Less Frequent Than Each Interest Period (optional)
- 4.3 Level Annuities with Payments More Frequent Than Each Interest Period (optional)
- 4.4 Annuities with Payments Less Frequent Than Each Interest Period and Payments in Arithmetic Progression (optional)
- 4.5 Annuities with Payments More Frequent Than Each Interest Period and Payments in Arithmetic Progression (optional)
- 4.6 Continuously Paying Annuities

**Chapter 5 Loan Repayment (2-3 days)**

- 5.1 Introduction (optional)
- 5.2 Amortized Loans and Amortization Schedules
- 5.3 The Sinking Fund Method
- 5.4 Loans with Other Repayment Patterns (optional)
- 5.5 Yield Rate Examples and Replacement of Capital (optional)

**Chapter 6 Bonds (5-6 days)**

- 6.1 Introduction (optional)
- 6.2 Bond Alphabet Soup and the Basic Price Formula
- 6.3 The Premium-Discount Formula
- 6.4 Other Pricing Formulas for Bonds
- 6.5 Bond Amortization Schedules
- 6.6 Valuing a Bond After Its Date of Issue (optional)
- 6.9 Callable Bonds

**Chapter 7 Stocks and Financial Markets (1 day)**

- 7.1 Common and Preferred Stock (cover dividend discount model with §3.4)
- 7.4 Selling Short; Selling Borrowed Stocks (optional)

**Chapter 8 Arbitrage, Term Structure of Interest Rates, and Derivatives (1-2 days)**

- 8.1 Introduction (optional)
- 8.3 The Term Structure of Interest Rates

**Chapter 9 Interest Rate Sensitivity (4-5 days)**

- 9.1 Overview
- 9.2 Duration
- 9.3 Convexity
- 9.4 Immunization (optional)
- 9.5 Other Types of Duration (optional)

## M325K Discrete Mathematics Syllabus

**Prerequisite and degree relevance:** Mathematics 408D, 408L, or 408S with a grade of at least C-. This is the first course that emphasizes understanding and creating proofs. Therefore, it provides a transition from the problem-solving approach of calculus to the entirely rigorous approach of advanced courses such as M365C or M373K. The number of topics required for coverage has been kept modest so as to allow adequate time for students to develop theorem-proving skills. Topics may include: fundamentals of logic and set theory; functions and relations; basic properties of integers, and elementary number theory; recursion and induction; counting techniques and combinatorics; introductory graph theory.

**Text:** Faculty have a choice among three recommended texts. The preferred texts are *Epp,* **Discrete Mathematics with Applications, 4th edition;** *Scheinerman,* **Mathematics: A Discrete Introduction, first edition.** A text in use before these was *Grimaldi,* **Discrete and Combinatorial Mathematics, current edition**. Of the three, Grimaldi is the most directed towards applications in Computer Science and Electrical Engineering. He also tends to integrate his applications directly into the flow of the text rather than discussing them separately.

Students in M325K are a mixture: some 25% Mathematics majors, 50% Electrical and Computer Engineering, and a mixture of majors such as Computer Science, Economics, etc. Some of the math majors may have changed their majors from Business or Engineering or Computer Science opting to try Mathematics since they have accumulated more hours in math than in any other area.

The course should serve our majors as a transition from early computational courses, to the more problem-solving, abstract and rigorous courses they will encounter in the BA and BS degree programs. It serves the EE's as a required course in discrete techniques; they are expected to deal with proof techniques in a discrete context. The course is not intended to be all rigor, but proofs and the cognitive skills required to read, comprehend and do proofs are a major theme. Students are expected to become familiar with the language and techniques of proof (converse, if and only if, proof by contradiction, etc); they should also see detailed, rigorous proofs presented in class. More importantly, they need to develop the ability to read and understand proofs on their own, and they must begin doing proofs; this cannot be slighted.

Discrete mathematics offers a variety of contexts in which the student can begin to understand mathematical techniques and appreciate the mathematical culture. Abstraction per se is not the goal; discrete mathematics offers very concrete computational contexts, and this can be exploited to develop a feeling for what it is that proofs, and proof techniques, say and do.

In the Epp text, one should include the topics below; this leaves time for the Instructor to cover additional topics of their choice. The instructor should focus on depth of understanding rather than breadth of coverage. However, subsequent courses will assume that students have seen induction and set theory in this course, so they must be covered.

**Chapter 1 The Logic of Compound Statements (4 - 5 days)**

- 1.1 Logical Form and Logical equivalence
- 1.2 Conditional Statements
- 1.3 Valid and Invalid Arguments

**Chapter 2 The logic of Quantified Statements (4 days)**

- 2.1 Introduction to Predicates and Quantified Statements I
- 2.2 Introduction to Predicates and Quantified Statements II
- 2.3 Statements Containing Multiple Quantifiers
- 2.4 Arguments with Quantified Statements

**Chapter 3 Elementary Number Theory and Statements of Proof (7 days, including 3.7)**

- 3.1 Direct Proof and Counterexample I: Introduction
- 3.2 Direct Proof and Counterexample II: Rational Numbers
- 3.3 Direct Proof and Counterexample III: Divisibility
- 3.4 Direct Proof and Counterexample IV: Division Into Cases
- 3.6 Indirect Argument: Contradiction and Contraposition

**Chapter 4 Sequences and Mathematical Induction (6 days)**

- 4.1 Sequences
- 4.2 Mathematical Induction I
- 4.3 Mathematical Induction II
- 4.4 Strong Mathematical Induction and Well Ordering

**Chapter 5 Set Theory (3-4 days; more if 5.4 is included)**

- 5.1 Basic Definitions of Set Theory
- 5.2 Properties of Sets
- 5.3 Disproofs, Algebraic Proofs, and Boolean Algebras
- 5.4 Russel's Paradox and the Halting Problem

**Responsible party:** Kathy Davis and Shinko Harper, August 2008

## M326K Foundations of Number Systems Syllabus

**Prerequisite and degree relevance:** Mathematics 408D, 408L, or 408S with a grade of at least C-. Restricted to students in a teacher preparation program or who have the consent of the instructor.

**Text: **optional: The Principles and Standards for School Mathematics, National Council of Teachers of Mathematics, 2000. (Also known as NCTM Standards.)

Number and operations, with emphasis on depth of understanding, mathematical communication, mathematical reasoning, mathematical representations, and pedagogical content knowledge in the context of number and operations. In this course, you may often need to re-examine things that have become “obvious” or automatic to you. For example,

• You know how to multiply. But can you help a student learn when to use multiplication in setting up an equation?

• You know how to divide fractions by inverting the divisor then multiplying. But can you explain to a questioning beginning algebra student why this procedure is legitimate?

• There are many math problems you can solve. But most mathematics problems have several correct methods of solution and many more incorrect methods of solution. Will you as a math teacher be able to decide which is which?

**Responsible party: **Kathy Davis and Shinko Harper, Spring 2010

## M328K Introduction to Number Theory Syllabus

INTRODUCTION TO NUMBER THEORY

**Prerequisite and degree relevance:** Mathematics 325K or 341 with a grade of at least C-.

This is a first course that emphasizes understanding and creating proofs; therefore, it must provide a transition from the problem-solving approach of calculus to the entirely rigorous approach of advanced courses such as M365C or M373K. The number of topics required for coverage has been kept modest so as to allow instructors adequate time to concentrate on developing the students theorem-proving skills.

A list of texts from which the instructor may choose is maintained in the text office.

The choice of text will determine the exact topics to be covered. The following subjects should definitely be included:

Divisibility: divisibility of integers, prime numbers and the fundamental theorem of arithmetic.

Congruences: including linear congruences, the Chinese remainder theorem, Euler's -function, and polynomial congruences, primitive roots.

The following topics may also be covered, the exact choice will depend on the text and the taste of the instructor.

Diophantine equations: (equations to be solved in integers), sums of squares, Pythagorean triples.

Number theoretic functions: the Mobius Inversion formula, estimating and partial sums n(x) of other number theoretic functions.

Approximation of real numbers by rationals: Dirichlet's theorem, continued fractions, Pell's equation, Liousville's theorem, algebraic and transcendental numbers, the transcendence of e and/or n.

## M329F Theory of Interest Syllabus

**Text: Vaaler & Daniel, Mathematical Interest Theory, Mathematical Association, Second Edition**

** Responsible Parties:** Shinko Harper, Milica Cudina, Alisa Havens, Jennifer Mann, January 2013

** Prerequisite and degree relevance:** Mathematics 408D, 308L, 408L, or 408S with a grade of at least C-. This course covers the content for the SOA Exam FM. Topics include nominal and effective interest and discount rates, general accumulation functions and force of interest, yield rates, annuities including those with non-level payment patterns, amortization of loans, sinking funds, bonds, spot and forward rates, interest rate swaps, duration, and immunization.

** Chapter 0 An Introduction to the Texas Instruments BA II Plus **(optional)

**Chapter 1 The Growth of Money (7 days)**

- 1.1 Introduction (optional)
- 1.2 What is interest?
- 1.3 Accumulation and Amount Functions
- 1.4 Simple Interest/Linear Accumulation Functions
- 1.5 Compound Interest (The usual case!)
- 1.6 Interest in Advance/The Effective Discount Rate
- 1.7 Discount Functions/The Time Value of Money
- 1.8 Simple Discount
- 1.9 Compound Discount
- 1.10 Nominal Rates of Interest and Discount
- 1.11 A Friendly Competition (Constant Force of Interest)
- 1.12 Force of Interest
- 1.14 Inflation

**Chapter 2 Equations of Value and Yield Rates (4-5 days)**

- 2.1 Introduction (optional)
- 2.2 Equations of Value for Investments Involving a Single Deposit made under Compound Interest
- 2.3 Equations of Value for Investments with Multiple Contributions
- 2.4 Investment Return
- 2.5 Reinvestment Considerations
- 2.6 Approximate Dollar Weighted Yield Rates (optional)
- 2.7 Fund Performance

**Chapter 3 Annuities (Annuities Certain) (11-12 days)**

- 3.1 Introduction (optional)
- 3.2 Annuities Immediate
- 3.3 Annuities Due
- 3.4 Perpetuities & 7.1 Common and Preferred Stock
- 3.5 Deferred Annuities and Values on any Date
- 3.6 Outstanding Loan Balances
- 3.7 Nonlevel Annuities
- 3.8 Annuities with Payments in Geometric Progression
- 3.9 Annuities with Payments in Arithmetic Progression
- 3.10 Yield Rate Examples Involving Annuities (optional)
- 3.11 Annuity Symbols for Nonintegral Terms (optional)
- 3.12 Annuities Governed by General Accumulation Functions (optional)
- 3.13 The Investment Year Method

**Chapter 4 Annuities with Different Payment and Conversion Periods (1-2 days)**

- 4.1 Introduction (optional)
- 4.2 Level Annuities with Payments Less Frequent Than Each Interest Period (optional)
- 4.3 Level Annuities with Payments More Frequent Than Each Interest Period (optional)
- 4.4 Annuities with Payments Less Frequent Than Each Interest Period and Payments in Arithmetic Progression (optional)
- 4.5 Annuities with Payments More Frequent Than Each Interest Period and Payments in Arithmetic Progression (optional)
- 4.6 Continuously Paying Annuities

**Chapter 5 Loan Repayment (2-3 days)**

- 5.1 Introduction (optional)
- 5.2 Amortized Loans and Amortization Schedules
- 5.3 The Sinking Fund Method
- 5.4 Loans with Other Repayment Patterns (optional)
- 5.5 Yield Rate Examples and Replacement of Capital (optional)

**Chapter 6 Bonds (5-6 days)**

- 6.1 Introduction (optional)
- 6.2 Bond Alphabet Soup and the Basic Price Formula
- 6.3 The Premium-Discount Formula
- 6.4 Other Pricing Formulas for Bonds
- 6.5 Bond Amortization Schedules
- 6.6 Valuing a Bond After Its Date of Issue (optional)
- 6.9 Callable Bonds

**Chapter 7 Stocks and Financial Markets (1 day)**

- 7.1 Common and Preferred Stock (cover dividend discount model with §3.4)
- 7.4 Selling Short; Selling Borrowed Stocks (optional)

**Chapter 8 Arbitrage, Term Structure of Interest Rates, and Derivatives (1-2 days)**

- 8.1 Introduction (optional)
- 8.3 The Term Structure of Interest Rates

**Chapter 9 Interest Rate Sensitivity (4-5 days)**

- 9.1 Overview
- 9.2 Duration
- 9.3 Convexity
- 9.4 Immunization (optional)
- 9.5 Other Types of Duration (optional)

## M427J Differential Equations with Linear Algebra Syllabus

** Prerequisite and degree relevance:** Mathematics 408D, 408L, or 408S with a grade of at least C-. Mathematics 427J and 427K may not both be counted.

** Course description: **The syllabus below contains 36-38 hours of required material and two 3 hour optional blocks. As the Fall semester contains 42 hours and the Spring 44 hours, one should adjust accordingly. Instructors should note that it is critical that the full length of allotted time be spent on Chapter 3 and little if any time can be devoted to optional topics.

**Text: ***Differential Equations and Their Applications,* by Martin Braun

**Chapter 1. First-order differential equations [4 hours]**

1.1 Introduction

1.2 First-order linear differential equations

1.4 Separable equations

1.9 Exact equations, and why we cannot solve very many differential equations

1.10 The existence-uniqueness theorem; Picard iteration

**Chapter 2. Second-order linear differential equations [11- 12 hours]**

2.1 Algebraic properties of solutions

2.2 Linear equations with constant coefficients

2.2.1 Complex roots

2.2.2 Equal roots; reduction of order

2.3 The nonhomogeneous equation

2.4 The method of variation of parameters

2.5 The method of judicious guessing

2.6 Mechanical vibrations (optional)

2.8 Series solutions

2.8.1 Singular points, Euler equations

2.8.2 Regular singular points, the method of Frobenius (optional)

* *

**Chapter 3. Systems of differential equations [15 hours]**

3.1 Algebraic properties of solutions of linear systems

3.2 Vector spaces

3.3 Dimension of a vector space

3.4 Applications of linear algebra to differential equations

3.5 The theory of determinants

3.6 Solutions of simultaneous linear equations

3.7 Linear transformations

3.8 The eigenvalue-eigenvector method of finding solutions

3.9 Complex roots

3.10 Equal roots

3.11 Fundamental matrix solutions; e^{At}

*Supplement*

3.A Matrix multiplication as linear combination of columns

3.B Vectors as arrows in **R*** ^{n}* and geometric meaning of operations (optional)

3.C Null and Column spaces

3.D Complete solution set of systems (RREF)

**Optional Chapter 4. Qualitative theory of differential equations [optional 3 hours]**

4.1 Introduction

4.2 Stability of linear systems

4.4 The phase-plane

4.7 Phase portraits of linear systems

**Chapter 5. Separation of variables and Fourier series [6-7 hours]**

5.1 Two point boundary-value problems

5.2 Introduction to partial differential equations

5.3 The heat equation; separation of variables

5.4 Fourier series

5.5 Even and odd functions

5.6 Return to the heat equation

**Optional Chapter 6. Sturm-Liouville boundary value problems [optional 3 hours]**

6.1 Introduction

6.2 Inner product spaces

6.3 Orthogonal bases, Hermitian operators

6.4 Sturm-Liouville theory

## M427K Advanced Calculus for Applications I Syllabus

**Prerequisite and degree relevance:** Mathematics 408D, 408L, or 408S with a grade of at least C-. Mathematics 427J and 427K may not both be counted.

**Course description:** M427K is a basic course in ordinary and partial differential equations, with Fourier series. It should be taken before most other upper division, applied mathematics courses. The course meets three times a week for lecture and twice more for problem sessions. Geared to the audience primarily consisting of engineering and science students, the course aims to teach the basic techniques for solving differential equations which arise in applications. The approach is problem-oriented and not particularly theoretical. Most of the time is devoted to first and second order ordinary differential equations with an introduction to Fourier series and partial differential equations at the end. Depending on the instructor, some time may be spent on applications, Laplace transformations, or numerical methods. Five sessions a week for one semester.

Note that some Engineering courses assume students have seen Laplace Transforms in M427K.

**Text: Boyce and DiPrima: Elementary Differential Equations and Boundary Value Problems. 10th Edition** The text is required for most sections; honors classes, computer supplement sections, or innovative sections may use other texts.

**Required Topics**

It will be impossible to cover everything here adequately. The core material which must be covered in selected sections from Chapters 1, 2, 3, 5, 10. Chapter 7 is so important that it ought to be covered, but be aware that most students have not already had matrix methods, and you will likely find yourself covering the 2 by 2 case. You might then do stability, etc. Numerical methods are becoming increasingly important, and covering this topic here is a good lead into the department's new computational science degree. Again, some engineering courses need their students to have seen some Laplace transforms. This will leave time for other topics, and you may choose to equations, applications. Whichever approach you take, you will have to carefully plan your sections and time to be spent on them.

**Resources**

If you are new to this course, you might talk to the senior faculty who teach this course regularly: Profs. Arbogast, Beckner, Bichteler, Gamba, Koch, Tsai, Uhlenbeck and others.

- Chapter I Introduction (2 - 3 weeks for Chapters 1 and 2)
- 1.1Some Basic Mathematical Models; Direction Fields
- 1.2 Solutions of Some Differential Equations 1.3 Classification of Differential Equations

- 1.4 Historical Remarks

- Chapter 2 First-Order Differential Equations(2 - 3 weeks for Chapters 1 and 2)
- 2.1 Linear Equations with Variable Coefficients
- 2.2 Separable Equations
- 2.3 Modeling with First Order Equations (optional)
- 2.4 Differences Between Linear and Nonlinear Equations
- 2.5 Autonomous Equations and Population Dynamics (optional)
- 2.6 Exact Equations and Integrating Factors
- 2.7 Numerical Approximations: Euler's Method (optional unless you do Ch 8)
- 2.8 The Existence and Uniqueness Theorem
- 2.9 First Order Difference Equations 115 (optional unless you do stability)

- Chapter 3 Second Order Linear Equations (2 - 3 weeks)
- 3.1 Homogeneous Equations with Constant Coefficients
- 3.2 Fundamental Solutions of Linear Homogeneous Equations
- 3.3 Complex Roots of the Characteristic Equation
- 3.4 Repeated Roots; Reduction of Order
- 3.5 Nonhomogeneous Equations; Method of Undetermined Coefficients
- 3.6 Variation of Parameters (optional)
- 3.7 Mechanical and Electrical Vibrations (optional)
- 3.8 Forced Vibrations (optional)

- Chapter 4 Higher Order Linear Equations (cover quickly)
- 4.1 General Theory of nth Order Linear Equations
- 4.2 Homogeneous Equations with Constant Coefficients
- 4.3 The Method of Undetermined Coefficients (optional)
- 4.4 The Method of Variation of Parameters

- Chapter 5 Series Solutions of Second Order Linear Equations (2 weeks)
- 5.1 Review of Power Series (optional)
- 5.2 Series Solutions Near an Ordinary Point, Part I
- 5.3 Series Solutions Near an Ordinary Point, Part II
- 5.4 Euler Equations, Regular Singular Points
- 5.5 Series Solutions Near a Regular Singular Point, Part I
- 5.6 Series Solutions Near a Regular Singular Point, Part II

- Chapter 6 The Laplace Transform (1 week: Important for some Engineers))
- 6.1 Definition of the Laplace Transform
- 6.2 Solution of Initial Value Problems
- 6.3 Step Functions
- 6.4 Differential Equations with Discontinuous Forcing Functions
- 6.5 Impulse Functions
- 6.6 The Convolution Integral

- Chapter 7 Systems of First Order Linear Equations (1 – 2 weeks)

- Chapter 8 Numerical Methods (1 week if covered) (optional)

- Chapter 9 Nonlinear Differential Equations and Stability

- Chapter 10 Partial Differential Equations and Fourier Series (3 weeks)
- 10.1 Two-Point Boundary Value Problems
- 10.2 Fourier Series
- 10.3 The Fourier Convergence Theorem
- 10.4 Even and Odd Functions
- 10.5 Separation of Variables; Heat Conduction in a Rod
- 10.6 Other Heat Conduction Problems (optional)
- 10.7 The Wave Equation; Vibrations of an Elastic String
- 10.8 Laplace's Equation (optional)
- Appendix A Derivation of the Heat Equation (optional)
- Appendix B Derivation of the Wave Equation (optional)

## M427L Advanced Calculus for Applications II Syllabus

**Prerequisite and degree relevance:** Mathematics 408D, 408L, or 408S with a grade of at least C-.

**Course description:** Topics include matrices, elements of vector analysis and calculus functions of several variables, including gradient, divergence, and curl of a vector field, multiple integrals, and chain rules, length and area, line and surface integrals, Greens theorem in the plane and space. If time permits, topics in complex analysis may be included. This course has three lectures and two problem sessions each week. It is anticipated that most students will be engineering majors. Five sessions a week for one semester.

**Text: Marsden & Tromba, Vector Calculus 5th edition**

- I THE GEOMETRY OF EUCLIDEAN SPACE (6 days)
- 1. I Vectors in two- and three-dimensional space
- 1.2 The inner product, length, and distance
- 1.3 Matrices, determinants, and the cross product
- 1.4 Cylindrical and spherical coordinates
- 1.5 n-dimensional Euclidean space

- 2 DIFFERENTIATION (5-6 days)
- (add discussion of linear maps, matrices) 2.1 The geometry of real-valued functions
- 2.2 Limits and continuity (assign to read)
- 2.3 Differentiation
- 2.4 Introduction to paths
- 2.5 Properties of the derivative
- 2.6 Gradients and directional derivatives

- 3 HIGHER-ORDER DERIVATIVES (3 days)
- 3.1 Iterated partial derivatives (briefly)
- 3.2 Taylor's theorem
- 3.3 Extrema of real-valued functions
- 3.4 Constrained extrema and Lagrange multipliers
- 3.5 The implicit function theorem(if time permits) 3,5

- 4 VECTOR-VALUED FUNCTIONS (5 days)
- 4.1 Acceleration and Newton's Second Law
- 4.2 Arc length
- 4.3 Vector fields
- 4.4 Divergence and curl

- 5 DOUBLE AND TRIPLE INTEGRALS (3 days)(cover first three sections in one lecture)
- 5.1 Introduction
- 5.2 The double integral over a rectangle
- 5.3 The double integral over more general regions
- 5.4 Changing the order of integration
- 5.6 The triple integral

- 6 THE CHANGE OF VARIABLES FORMULA (3 days)
- 6.1 The geometry of maps (not crucial)
- 6.2 The change of variables theorem (lightly)
- 6.3 Applications of double, triple integrals(if time permits)

- 7 INTEGRALS OVER PATHS AND SURFACES (7 days) (next chapter
- depends heavily on this) 7.1 The path integral
- 7.2 Line integrals
- 7.3 Parametrized Surfaces
- 7.4 Area of a surface
- 7.5 Integrals of scalar functions over surfaces
- 7.6 Surface integrals of vector functions

- 8 THEOREMS OF VECTOR ANALYSIS (5-6 days) (may reorder as (8.1, 8.4, 8.2, 8.3)
- 8.1 Green's theorem
- 8.2 Stokes' theorem
- 8.3 Conservative fields
- 8.4 Gauss' theorem

**Responsible party:** Kathy Davis 1998

## M339D Introduction to Financial Mathematics for Actuaries Syllabus

**Prerequisite and degree relevance:** Actuarial Foundations 329 or Mathematics 329F; and Mathematics 362K with a grade of at least C-. Moreover, the instructor advises that students will need a thorough understanding and operational knowledge of (at least) calculus, finite-stage-space probability, and the term structure of interest rates.

**Text: Robert L. McDonald, Derivative Markets, 2nd Edition (2015) Prentice Hall, ISBN 9780321280305**

**Responsible party**: Milica Cudina March 2015

**Description of the Course**: This course is intended to provide the mathematical foundations necessary to prepare for a portion of

(1) the joint SOA/CAS exam FM/2, as well as

(2) the SoA exam MFE and the financial economics portion of the CAS Exam 3.

Additionally, the course is aimed at building up the vocabulary and the techniques indispensable in the workplace at current financial and insurance institutions. This is not an exam-prep seminar. There is intellectual merit to the course beyond the ability to prepare for a professional exam.

The material exhibited includes elementary risk management, forward contracts, options, futures, swaps, the simple random walk, the binomial asset pricing model and its application to option pricing. The remainder of the Exam MFE/3F curriculum is exhibited in course M339W (also offered by the Department of Mathematics).

**Topics Covered:**

Orientation. Role of financial markets. Bid-ask spread. Commissions.

Standing assumptions. Conventions.

Outright purchase of an asset. Discrete dividends. Simple return.

Continuous-dividend-paying assets. Market Indices.

Short sales.

Static financial portfolios. Initial cost. Payoff.

Profit. Definition of long/short positions. Basic risk management. Forward contracts.

European call options (rationale, definition, implementation).

European call options (payoff/profit).

Hedging using European call options.

Caps, i.e., short intrinsic position hedged with a call.

Covered/naked option writing. Covered calls. European put options (definition).

Hedging using put options. Floors. Covered puts.

Parallels between classical property-insurance policies and put options.

Derivative securities.

Examples of “simplest” derivative securities: All-or-nothing options.

Review of finite probability spaces. Dynamic portfolios. Profit.

Arbitrage portfolio. Arbitrage.

Law of the unique price.

Prepaid forward contracts. Forward and prepaid forward pricing (stocks).

Annualized forward premium. Arbitrage and forwards’ pricing.

Commodity Swaps.

Futures.

Put-call parity.

Replicating portfolios. “Synthetic forward contracts”. Chooser options. Straddles.

Gap calls and puts. Gap-option parity.

American options. Options on futures contracts.

Options on currencies.

Exchange options.

Maximum option. Generalized put-call parity.

Option price bounds and monotonicity. Bull spreads.

Option price “slope” bounds. Bear Spreads.

Option price convexity. Butterfly Spreads. Speculating on volatility.

Strangles. Collars. Ratio Spreads. Equity-linked CDs.

Binomial asset-pricing model.

Derivative-pricing by replication. Risk-neutral probability.

The forward tree. Cox-Ross-Rubinstein binomial tree. Jarrow-Rudd binomial tree.

Two-period binomial pricing. Multiple binomial periods.

Early exercise. Bermudan options.

Pricing American options.

Properties of American-option prices.

Asian options and their binomial pricing.

Barrier options and their binomial pricing.

Compound options and their binomial pricing.

Binomial pricing of options on currencies.

Binomial pricing of options on futures contracts.

Interest-rate swaps.

## M339J Probability Models with Actuarial Applications Syllabus

**Prerequisite and degree relevance: **Mathematics 358K or 378K with a grade of at least C-. Please note that a thorough knowledge of calculus, probability, and statistics will be assumed.

**Course description:** Introductory actuarial models for life insurance, property insurance, and annuities. With Mathematics 349P, covers the syllabus for the professional actuarial exam on model construction.

**Textbook: Klugman,** S.A., Panjer, H.H. and Willmot, G.E.,** Loss Models: From Data to Decisions, (Fourth Edition)**, 2012

**Responsible party**: Alisa Havens Walch and Mark Maxwell, August 2014

**Topics Covered:**

[Chapters 3 – 9 have a combined weight of 15% - 20% of the SOA Exam C]

PART I INTRODUCTION

2 Random variables

2.1 Introduction

2.2 Key functions (cumulative distribution function, survival function, probability density function, probability mass function, and hazard rate (= force of mortality or failure rate)

3 Basic distributional quantities

3.1 Moments

3.2 Percentiles

3.3 Generating functions and sums of random variables

3.4 Tails of distributions

3.5 Measures of Risk

PART II ACTUARIAL MODELS

4 Characteristics of Actuarial Models

4.1 Introduction

4.2 The role of parameters

5 Continuous models

5.1 Introduction

5.2 Creating new distributions

5.3 Selected distributions and their relationships

5.4 The linear exponential family

6 Discrete Distributions

6.1 Introduction

6.2 The Poisson distribution

6.3 The negative binomial distribution

6.4 The binomial distribution

6.5 The (a, b, 0) class

6.6 Truncation and modification at zero

7 Advanced discrete distributions [Not Covered]

8 Frequency and severity with coverage modifications

8.1 Introduction

8.2 Deductibles

8.3 The loss elimination ratio and the effect of inflation for ordinary deductibles

8.4 Policy limits

8.5 Coinsurance, deductibles, and limits

8.6 The impact of deductibles on claim frequency

9 Aggregate loss models

9.1 Introduction

9.2 Model choices

9.3 The compound model for aggregate claims

9.4 Analytic results

9.5 Computing the aggregate claims distribution

9.6 The recursive method [Exclude 9.6.1]

9.7 The impact of individual policy modifications on aggregate payments

9.8 The individual risk model

[Chapters 10 – 12 have a combined weight of 20% - 25% of the SOA Exam C]

PART III CONSTRUCTION OF EMPIRICAL MODELS

10 Review of mathematical statistics

10.1 Introduction

10.2 Point estimation

10.3 Interval estimation

10.4 Tests of hypotheses

11 Estimation for complete data

11.1 Introduction

11.2 The empirical distribution for complete, individual data

11.3 Empirical distributions for grouped data

12 Estimation for modified data

12.1 Point estimation

12.2 Means, variances, and interval estimation

12.3 Kernel density models

12.4 Approximations for large data sets

**Calculators**

Any approved calculator can be used for this class (approved list: http://www.soa.org/Education/Exam-Req/exam-day-info/edu-calculators.aspx). You may use more than one calculator on this list.

**Actuarial Examinations**

In conjunction with M349P, M339J covers the content of SOA Exam C. Students are expected to be familiar with survival, severity, frequency and aggregate models, and use statistical methods to estimate parameters of such models given sample data. Students are further expected to identify steps in the modeling process, understand the underlying assumptions implicit in each family of models, recognize which assumptions are applicable in a given business application, and appropriately adjust the models for the impact of insurance coverage modifications.

## M339U Actuarial Contingent Payments I Syllabus

** Prerequisite and degree relevance: **Mathematics 362K with a grade of at least C-; credit with a grade of at least C- or registration for Actuarial Foundations 329 or Mathematics 329F; and credit with a grade of at least C- or registration for Mathematics 340L or 341. Please note that a thorough knowledge of calculus, probability, and interest theory will be assumed.

**Course description:** Intermediate actuarial models for life insurance, property insurance, and annuities.

**Textbook: Actuarial Mathematics for Life Contingent Risks**, 2nd Edition (2013) David C. Dickson, Mary R. Hardy, and Howard R. Waters, Cambridge University Press, ISBN 9781107044074

**Topics:**

Chapter 1 Introduction to life insurance (1 hour)

1.1 Summary

1.2 Background

1.3 Life insurance and annuity contracts

1.4 Other insurance contracts

1.5 Pension benefits

1.6 Mutual and proprietary insurers

Chapter 2 Survival models (4 hours)

2.1 Summary

2.2 The future lifetime random variable

2.3 The force of mortality

2.4 Actuarial notation

2.5 Mean and standard deviation of Tx

2.6 Curtate future lifetime

Chapter 3 Life tables and selection (5 hours)

3.1 Summary

3.2 Life tables

3.3 Fractional age assumptions

3.4 National life tables

3.5 Survival models for life insurance policyholders

3.6 Life insurance underwriting

3.7 Select and ultimate survival models

3.8 Notation and formulae for select survival models

3.9 Select life tables

3.10 Some comments on heterogeneity in mortality

3.11 Mortality trends

Chapter 4 Insurance benefits (8 hours)

4.1 Summary

4.2 Introduction

4.3 Assumptions

4.4 Variation of insurance benefits

4.5 Relatingand

4.6 Variable insurance benefits

4.7 Functions for select lives

Chapter 5 Annuities (9 hours)

5.1 Summary

5.2 Introduction

5.3 Review of annuities certain

5.4 Annual life annuities

5.5 Annuities payable continuously

5.6 Annuities payable 1/mthly

5.7 Comparison of annuities by payment frequency

5.8 Deferred annuities

5.9 Guaranteed annuities

5.10 Increasing annuities

5.11 Numerical illustrations

5.12 Functions for select lives

Chapter 6 Premium calculation (7 hours)

6.1 Summary

6.2 Preliminaries

6.3 Assumptions

6.4 The present value of future loss random variable

6.5 The equivalence principle

6.6 Gross premiums

6.7 Profit

6.8 The portfolio percentile premium principle

6.9 Extra risks

Chapter 7 Policy values (3 hours)

7.1 Summary

7.2 Assumptions

7.3 Policies with annual cash flows

For the suggested time devoted to each chapter, 1 hour corresponds to 50 minutes of actual class time. The total number of hours listed do not constitute an entire semester. They allow for review and examinations.

**Calculators**

Any approved calculator can be used for this class (approved list: http://www.soa.org/Education/Exam-Req/exam-day-info/edu-calculators.aspx). You may use more than one calculator on this list.

**Actuarial Examinations**

In conjunction with M339V, M339U covers the content of SOA Exam MLC and CAS Exam LC. Topics covered: life insurance, survival models, life tables, insurance benefits, annuities, and premium calculation. See https://www.soa.org/education/exam-req/edu-asa-req.aspx and http://www.casact.org/admissions/exams/ for further details regarding these exams.

## M339V Actuarial Contingent Payments II Syllabus

** Prerequisite and degree relevance: **Actuarial Foundations 329 or Mathematics 329F, and Mathematics 339U with a grade of at least C- in each. Please note that thorough knowledge of calculus, probability, interest theory, and Actuarial Contingent Payments I will be assumed.

**Text: **David C. Dickson, Mary R. Hardy, and Howard R. Waters, **Actuarial Mathematics for Life Contingent Risks**, 2nd Edition (2013) Cambridge University Press, ISBN 9781107044074

**Responsible party**: Mark Maxwell August 2014

**Description of the Course**: M 339V = M 389V Actuarial Contingent Payments II. Topics covered include Policy Values, Multiple State Models, Pensions, Interest Rate Risk, and Emerging Costs for Traditional Life Insurance.

This is an actuarial capstone course and students are expected to do some independent learning and improve verbal and written acumen. Three graded components of the course are 1) communication, 2) content, and 3) contribution to the class. This class carries the Independent Inquiry Flag. This class carries the Quantitative Reasoning flag.

Meets with M389V, the corresponding graduate-course number. Offered every spring semester only. This is a 3-credit course.

**Topics:**

Chapter 7 Policy values

7.4 Policy values for policies with cash flows at 1/mthly intervals

7.4.1 Recursions

7.4.2 Valuation between premium dates

7.5 Policy values with continuous cash flows

7.5.1 Thiele’s differential equation

7.5.2 Numerical solution of Thiele’s differential equation

7.6 Policy alterations

7.7 Retrospective policy values

7.7.1 Prospective and retrospective valuation

7.7.2 Defining the retrospective net premium policy value

7.8 Negative policy values

7.9 Deferred acquisition expenses and modified premium reserves

7.10 Notes and further reading

7.11 Exercises

Chapter 8 Multiple state models

8.1 Summary

8.2 Examples of multiple state models

8.2.1 The alive–dead model

8.2.2 Term insurance with increased benefit on accidental death

8.2.3 The permanent disability model

8.2.4 The disability income insurance model

8.3 Assumptions and notation

8.4 Formulae for probabilities

8.4.1 Kolmogorov’s forward equations

8.5 Numerical evaluation of probabilities

8.6 Premiums

8.7 Policy values and Thiele’s differential equation

8.7.1 The disability income insurance model

8.7.2 Thiele’s differential equation – the general case

8.8 Multiple decrement models

8.9 Multiple decrement tables

8.9.1 Fractional age assumptions for decrements

8.10 Constructing a multiple decrement table

8.10.1 Deriving independent rates from dependent rates

8.10.2 Deriving dependent rates from independent rates

8.11 Comments on multiple decrement notation

8.12 Transitions at exact ages

8.13 Markov multiple state models in discrete time

8.13.1 The Chapman–Kolmogorov equations

8.13.2 Transition matrices

8.14 Notes and further reading

8.15 Exercises

Chapter 9 Joint life and last survivor benefits

9.1 Summary

9.2 Joint life and last survivor benefits

9.3 Joint life notation

9.4 Independent future lifetimes

9.5 A multiple state model for independent future lifetimes

9.6 A model with dependent future lifetimes

9.7 The common shock model

9.8 Notes and further reading

9.9 Exercises

Chapter 10 Pension mathematics

10.1 Summary

10.2 Introduction

10.3 The salary scale function

10.4 Setting the DC contribution

10.5 The service table

10.6 Valuation of benefits

10.6.1 Final salary plans

10.6.2 Career average earnings plans

10.7 Funding the benefits [Not covered on SOA exam MLC]

10.8 Notes and further reading

10.9 Exercises

Chapter 11 Yield curves and non-diversifiable risk

11.1 Summary

11.2 The yield curve

11.3 Valuation of insurances and life annuities

11.3.1 Replicating the cash flows of a traditional non-participating product

11.4 Diversifiable and non-diversifiable risk

11.4.1 Diversifiable mortality risk

11.4.2 Non-diversifiable risk

11.5 Monte Carlo simulation [Not covered on SOA exam MLC]

11.6 Notes and further reading

11.7 Exercises

Chapter 12 Emerging costs for traditional life insurance

12.1 Summary

12.2 Introduction

12.3 Profit testing a term insurance policy

12.3.1 Time step

12.3.2 Profit test basis

12.3.3 Incorporating reserves

12.3.4 Profit signature

12.4 Profit testing principles

12.4.1 Assumptions

12.4.2 The profit vector

12.4.3 The profit signature

12.4.4 The net present value

12.4.5 Notes on the profit testing method

12.5 Profit measures

12.6 Using the profit test to calculate the premium

12.7 Using the profit test to calculate reserves

12.8 Profit testing for multiple state models

12.9 Notes

12.10 Exercises

Chapter 13 Participating and Universal Life insurance

13.1 Summary

13.2 Introduction

13.3 Participating insurance

13.3.1 Introduction

13.3.2 Examples

13.3.3 Notes on profit distribution methods

13.4 Universal Life insurance

13.4.1 Introduction

13.4.2 Key design features

13.4.3 Projecting account values

13.4.4 Profit testing Universal Life policies

13.4.5 Universal Life Type B

13.4.6 Universal Life Type A

13.4.7 No-lapse guarantees

13.4.8 Comments on UL profit testing

13.5 Comparison of UL and whole life insurance policies

13.6 Notes and further reading

13.7 Exercises

Calculators

Any approved calculator can be used for this class (approved list: http://www.soa.org/Education/Exam-Req/exam-day-info/edu-calculators.aspx). You may use more than one calculator on this list.

Actuarial Examinations

In conjunction with M339V, M339U covers the content of SOA Exam MLC and CAS Exam LC. Topics covered: life insurance, survival models, life tables, insurance benefits, annuities, and premium calculation. See https://www.soa.org/education/exam-req/edu-asa-req.aspx and http://www.casact.org/admissions/exams/ for further details regarding these exams.

## M339W Financial Mathematics for Actuarial Applications Syllabus

** Prerequisite and degree relevance: **Mathematics 339D with a grade of at least C-. Moreover, the instructor also advises that students will need a thorough understanding and operational knowledge of (at least) classical calculus, calculus-based probability (with emphasis on the normal distribution), the term structure of interest rates, and the principles of risk-neutral pricing in the binomial asset-pricing model.

**Text: Robert L. McDonald, Derivative Markets, 2nd Edition (2015) Prentice Hall, ISBN 9780321280305**

**Responsible party**: Milica Cudina March 2015

**Description of the Course**: This course is intended to provide the **mathematical foundations** necessary to prepare for a portion of the SOA Exam MFE and the "financial economics" portion of the CAS Exam 3.

Additionally, the course is aimed at building up the vocabulary and the techniques indispensable in the workplace at current financial and insurance institutions. **This is not an exam-prep seminar**.

The material exhibited includes: an in-depth study of the normal and log-normal distributions, the simple random walk, basics of stochastic calculus, the Samuelson (geometric Brownian motion) stock-price model and the Black-Scholes formula, analysis of option Greeks, market making, non-deterministic interest rate models (both discrete, and continuous-time), bond pricing, Monte-Carlo simulations. The remainder of the Exam MFE/3F curriculum is exhibited in course M339d (also offered by the Department of Mathematics).

**Topics:**

Orientation. Standing assumptions. Conventions.

Binomial interest rate models.

Black-Derman-Toy.

Review of uniform distribution. Random number generation.

Probability on the coin toss space. Simulation of the random walk.

Law of Large Numbers. Risk-neutral pricing by simulation (the binomial case).

Scaled random walk. 11.3: Proceeding to continuous time.

Normal and log-normal distributions.

Log-normal stock-price model.

Brownian motion.

Introduction to formal stochastic calculus for financial mathematics.

Stochastic integral ("definition", obstacles). Itô-Doeblin Lemma. Itô processes.

Samuelson's model for stock prices. Portfolio representation.

Sharpe ratio. The risk-neutral probability measure.

Black-Scholes PDE. Risk-neutral pricing.

Black-Scholes pricing formula. Price curve prior to expiration.

Black-Scholes pricing for options of futures, currencies, discrete-dividend-paying stocks.

Correlated assets. Exchange options. Black-Scholes pricing and exotic options.

Forward prices for powers of the underlying.

Implied volatility.

Greeks in the Black-Scholes pricing. A detailed look on the ∆. Option elasticity.

“Greeks” in the binomial tree. Market making and ∆−hedging.

Self-ﬁnancing portfolios. Overnight proﬁt/loss. Γ−hedging.

Market-making and bond-pricing. Duration-hedging.

The Ornstein-Uhlenbeck process. Continuous-time interest rate models.

Black formula.

Monte Carlo valuation.

Variance reduction methods. Control variate method.

## M340L Matrices and Matrix Calculations Syllabus

**Text: **David C. Lay,** Linear Algebra and its Applications, 4th ed.**

**Prerequisite and degree relevance:** Mathematics 408C, 408K, or 408N with a grade of at least C-. Only one of the following may count: Mathematics 340L, 341, Statistics and Data Sciences 329C, or Statistics and Scientific Computation 329C.

**Background:** M341 (Linear Algebra and Matrix Theory) and M340L (Matrices and Matrix Calculations) cover similar material. However, the emphasis in M340L is much more on calculational techniques and applications, rather than abstraction and proof. (M341 is the preferred linear algebra course for math majors and contains a substantial introduction to proof component.)

**Course Content:** Read the "Note to the Instructor" at the beginning of the book. The core of M340L is indeed the "core topics" listed on pages ix-x, plus sections 3.1 and 3.2. Various faculty members disagree strongly about which of the remaining "supplementary topics" and "applications" are most important; use your own judgment. You will probably have time for about half a dozen of these supplementary topics and applications.

The syllabus covers the essentials of all seven chapters of Lay, namely

- Linear Equations in Linear Algebra,
- Matrix Algebra,
- Determinants,
- Vector Spaces,
- Eigenvalues and Eigenvectors,
- Orthogonality and Least Squares, and
- Symmetric Matrices and Quadratic Forms.

Each section is designed to be covered in a single 50-minute lecture. However, in practice chapters 1 - 3 should be covered more quickly (a bit slower on the last 3 sections of chapter1), allowing more time for chapters 4-7. Most incoming M340L students are already quite adept at solving systems of equations, and it is important to move quickly at the beginning of the term to material that does challenge them, reserving time to tackle the more difficult vector space concepts of chapter 4. Many of the essential concepts, such as linear independence, are covered twice: once in chapter 1 for Rn, then again in chapter 4 for a general vector space.

**Computers:** Linear algebra lends itself extremely well to computerization, and there are many packages that students can use. Once students have learned the theory of row-reduction and matrix multiplication (which they pick up very quickly), they should be encouraged to use Maple, Matlab, Mathematica, or a similar package. There are also many "projects" on the departmental computers that students can learn from. Many concepts in the book, especially in the later chapters (e.g., understanding the long-time behavior of a dynamical system from its eigenvalues), can be absorbed quite easily through numerical experimentation.

*Revised by Gary Hamrick, June 2003*

## M341 Linear Algebra and Matrix Theory Syllabus

**Prerequisite and degree relevance:** Mathematics 408D, 408L, or 408S with a grade of at least C-. Restricted to mathematics majors. Only one of the following may count: Mathematics 340L, 341, Statistics and Data Sciences 329C, or Statistics and Scientific Computation 329C. Majors with a 'math' advising code must register for M341 rather than for M340L; majors without a 'math' advising code must register for M340L. Math majors must make a grade of at least C- in M341.

**Primary Text - Andrilli & Hecker, Elementary Linear Algebra fourth edition**

**Responsible Party**: Ray Heitmann, January 2008, July 2014

This course has three purposes and the instructor should give proper weight to all three. The students should learn some linear algebra - for most of them, this will be the only college linear algebra course they take. This is one of the first proof courses these students will take and they need to develop some proof skills. Finally, this is, for almost all students, the introductory course in mathematical abstraction and provides a necessary prerequisite for a number of our upper division courses. To teach this course successfully, the instructor should establish modest goals on all three fronts. On one hand, a student should not be able to pass this course simply by doing calculational problems well, but on the other hand, overly ambitious proof and abstraction goals simply discourage teacher and student alike.

To teach proofs, the instructor should cover Section 1.3 thoroughly to introduce various proof techniques. Afterward, a liberal (but not overwhelming) number of proofs should be sprinkled in the lectures, homework, and tests.

In teaching abstraction, it is critical to remember that almost no students are capable of becoming truly comfortable with it in a single semester; it is self-defeating to establish this as a goal. The study of abstract vector spaces is a unified treatment of various familiar vector spaces and students in this course should never be taken very far from the concrete. Linear algebra is the perfect subject for teaching students that abstraction can be a friend. For example, it underlines nicely how the solutions to a homogeneous system are better behaved than the solutions to a non-homogeneous system. However, amusing examples of unnatural algebraic systems that may or may not be vector spaces should be avoided.

A warning should be given concerning the calculational homework problems. The authors, intending the students to take full advantage of technology, have made no effort to make problems come out neatly.

**Suggested Coverage:**

**Chapter 1** Nine or ten lectures.

The first two sections provide necessary definitions for Section 1.3. The entire chapter should be covered. Generally, move quickly but cover 1.3 meticulously. Three or four lectures should be devoted to this section.**Chapter 2** Six or seven lectures.

Cover all sections but again move reasonably to have enough time for Chapters 4 and 5. **Chapter 3** Three lectures.

Row operations are easy for them and you can go quite quickly here. Cover Sections 3.1 and 3.2. Section 3.3 is optional - you might also choose to cover parts of this section. Section 3.4 is a fairly reasonable attempt to introduce eigenvalues before introducing linear transformations. It is an interesting and important part of this chapter, at least in my opinion. The instructor should cover at least part of this section, all if desired.**Chapter 4** Fourteen or fifteen lectures.

This chapter is the meat of the course and the instructor should plan to take a good deal of time here. Sections 4.1-4.6 should be covered thoroughly. Section 4.7 is optional and should probably be skipped to provide more time for Chapter 5.**Chapter 5** About five lectures.

In a perfect world, the entire chapter should be taught, but 5.5 is probably too much to hope for. Realistically, at least Sections 5.1 and 5.2 should be covered.

## M343K Introduction to Algebraic Structures Syllabus

**Prerequisite and degree relevance:** Either consent of Mathematics Advisor or two of the following courses with a grade of at least C- in each: Mathematics 325K or Philosophy 313K, Mathematics 328K, Mathematics 341. This course is designed to provide additional exposure to abstract rigorous mathematics on an introductory level. Students who demonstrate superior performance in M311 or M341 should take M373K instead of 343K. Those students whose performance in M311 or M341 is average should take M343K before taking M373K. Credit for M343K can NOT be earned after a student has received credit for M373K with a grade of at least C-.

**Course description:** Elementary properties of the integers, groups, rings, and fields are studied.

The number of topics should be kept modest to allow adequate time to concentrate on developing the students' theorem-proving skills. Some instructors will prefer to introduce groups before rings and some will reverse the order. In any case, below are some reasonable choices of topics. One should not try to cover all of these topics. It is very important to avoid superficial coverage of too many topics. All potential graduate students will take M373K, where it is possible to expect more and to do more.

**Topics:** Groups: Axioms, basic properties, examples, symmetry, cosets, Lagrange's Theorem, isomorphism. Homomorphisms, quotient groups, and the Fundamental Homomorphism Theorem.

**Optional:** Rings: Axioms, basic properties, examples, integral domains, and fields. rings and properties of fields.

**Optional:** More about polynomial

**Other options:** Groups acting on sets, characterization of the familiar number systems in terms of ring and field properties, and other applications of groups.

*Durbin July 2000*

## M343L Applied Number Theory Syllabus

**Prerequisite and degree relevance**: Mathematics 328K or 343K with a grade of at least C-.

**Text: **An Introduction to Mathematical Cryptography by Jill Pipher, Jeffrey Hoffstein, Joseph H. Silverman.

**Topics**: Basic properties of integers. Prime numbers and unique factorization. Congruences, Theorems of Fermat and Euler, primitive roots. Primality testing and factorization methods. Cryptography, basic notions. Public key cryptosystems. RSA. Implementation and attacks. Discrete log cryptosystems. Diffie-Hellman and the Digital Signature Standard. Elliptic curve cryptosystems. Symmetric cryptosystems, such as DES and AES.

**Responsible party: **Kathy Davis and Felipe Voloch, Spring 2010

## M346 Applied Linear Algebra Syllabus

**Prerequisite and degree relevance: **The prerequisite is M341 (or M311) or M340L with a grade of C- or better.

**Text:** Lorenzo Sadun, Applied Linear Algebra; the Decoupling Principle, second edition.

**Responsible Parties:** Charles Radin and Lorenzo Sadun, May, 2008.

We expect students to have a good feel for manipulating matrices, especially row reduction, but also taking determinants. We also expect students to have seen abstract vector spaces and linear transformations, but some rustiness is expected, and those topics should be reviewed. It is not assumed that students have seen eigenvalues and eigenvectors; those should be done from scratch.

This is a course in serious mathematics, not a cookbook. As such, results in lecture, and in the book, should generally be proved rigorously. However, it's not an intro-to-proof class and is aimed at an audience of engineers, economists, and physicists (as well as mathematicians), so *writing* proofs should only play a minor role in the problem sets and exams.**Detailed Syllabus: **This number of days in this syllabus is based on a TTh class.

** Chapter 1. The Decoupling Principle** (one day)

- Exploration: Beats

**Chapter 2. Vector Spaces and Bases** (two days)

- Vector Spaces
- Linear Independence, Basis, and Dimension
- Properties and Uses of a Basis
- Exploration: Polynomials
- Change of Basis
- Building New Vector Spaces from Old Ones
- Exploration: Projections

**Chapter 3. Linear Transformations and Operators** (three days)

- Definitions and Examples
- Exploration: Computer Graphics
- The Matrix of a Linear Transformation
- The Effect of a Change of Basis
- Infinite Dimensional Vector Spaces
- Kernels, Ranges, and Quotient Maps

**Chapter 4. An Introduction to Eigenvalues** (four days)

- Definitions and Examples
- Bases of Eigenvectors
- Eigenvalues and the Characteristic Polynomial
- The Need for Complex Eigenvalues
- Exploration: Circles and Ellipses
- When is an Operator Diagonalizable?
- Traces, Determinants, and Tricks of the Trade
- Simultaneous Diagonalization of Two Operators
- Exponentials of Complex Numbers and Matrices
- Power Vectors and Jordan Canonical Form

**Chapter 5. Some Crucial Applications** (five days)

- Discrete-Time Evolution: x(n) = Ax(n 1)
- Exploration: Fibonacci Numbers and Tilings
- First Order Continuous Time Evolution
- Second order Continuous Time Evolution
- Reducing Second Order Problems to First Order
- Exploration: Difference Equations
- Long Time Behavior and Stability
- Markov Chains and Probability Matrices
- Exploration: Random Walks
- Linear Analysis near Fixed Points of Nonlinear Problems
- Exploration: Nonlinear ODEs

**Chapter 6. Inner Products** (four days)

- Real Inner Products: Definitions and Examples
- Complex Inner Products
- Bras, Kets, and Duality
- Expansion in Orthonormal Bases: Finding Coefficients
- Projections and the Gram Schmidt Process
- Orthogonal Complements and Projections onto Subspaces
- Least Squares Solutions
- Exploration: Fourier Series
- Exploration: Curve Fitting
- The Spaces
*l*^{2}and L^{2} - Fourier Series on an Interval

**Chapter 7. Adjoints, Hermitian Operators, and Unitary Operators** (three days)

- Adjoints and Transposes
- Hermitian Operators
- Quadratic Forms and Real Symmetric Matrices
- Rotations, Orthogonal Operators, and Unitary Operators
- Exploration: Normal Matrices
- How the Four Classes are Related Exploration: Representations Of SU2

**Chapter 8. The Wave Equation** (four days)

- Waves on the Line
- Waves on the Half-Line: Dirichlet and Neumann Boundary Conditions
- The Vibrating String

## M348 Scientific Computation in Numerical Analysis Syllabus

**Prerequisite and degree relevance:** Computer Science 303E or 307, and Mathematics 341 or 340L with a grade of at least C-.

**Course description: **Introduction to mathematical properties of numerical methods and their applications in computational science and engineering. Introduction to object-oriented programming in an advanced language. Study and use of numerical methods for solutions of linear systems of equations, non-linear least-squares data fitting, numerical integration of multi-dimensional, non-linear equations and systems of initial value ordinary differential equations.

## M349P Applied Statistical Estimates Syllabus

**Prerequisite and degree relevance:** Mathematics 339J, and 341 or 340L, with a grade of at least C- in each. Please note that thorough knowledge of calculus, probability, and statistics will be assumed.

**Textbook:** Klugman, S.A., Panjer, H.H. and Willmot, G.E., **Loss Models: From Data to Decisions, Fourth Edition**, 2012

**Responsible party**: Alisa Havens Walch and Mark Maxwell, August 2014

**Description of the Course**: M 349P Probability Models with Actuarial Applications covers statistical estimation procedures for random variables and related quantities in actuarial models.

This is an actuarial capstone course and students are expected to do some independent learning and improve verbal and written acumen. Three graded components of the course are 1) communication, 2) content mastery, and 3) contribution to the class. This class carries the Independent Inquiry Flag. This class carries the Quantitative Reasoning flag.

Meets with M389P, the corresponding graduate-course number. Offered every spring semester only. This is a 3-credit course.

[Chapters 13 – 16 have a combined weight of 20% - 25% of the SOA Exam C]

PART IV PARAMETRIC STATISTICAL METHODS

13 Frequentist estimation

13.1 Method of moments and percentile matching

13.2 Maximum likelihood estimation

13.3 Variance and interval estimation

13.4 Non-normal confidence intervals

13.5 Maximum likelihood estimation of decrement probabilities

14 Frequentist Estimation for discrete distributions

14.1 Poisson

14.2 Negative binomial

14.3 Binomial

14.4 The (a, b, 1) class

14.5 Compound models [Not Covered]

14.6 Effect of exposure on maximum likelihood estimation

15 Bayesian estimation

15.1 Definitions and Bayes’ theorem

15.2 Inference and prediction

15.3 Conjugate prior distributions and the linear exponential family

15.4 Computational issues

16 Model selection

16.1 Introduction

16.2 Representations of the data and model

16.3 Graphical comparison of the density and distribution functions

16.4 Hypothesis tests

16.5 Selecting a model

[Chapters 17 – 19 have a combined weight of 20% - 25% of the SOA Exam C]

PART V CREDIBILITY

17 Introduction and Limited Fluctuation Credibility

17.1 Introduction

17.2 Limited fluctuation credibility theory

17.3 Full credibility

17.4 Partial credibility

17.5 Problems with the approach

17.6 Notes and References

18 Greatest accuracy credibility

18.1 Introduction

18.2 Conditional distributions and expectation

18.3 The Bayesian methodology

18.4 The credibility premium

18.5 The Buhlmann model

18.6 The Buhlmann-Straub model

18.7 Exact credibility

19 Empirical Bayes parameter estimation

19.1 Introduction

19.2 Nonparametric estimation

19.3 Semi-parametric estimation

[Chapter 20 has a weight of 5% - 10% of the SOA Exam C]

PART VI SIMULATION

20 Simulation

20.1 Basics of simulation

20.2 Simulation for specific distributions

20.3 Determining the sample size

20.4 Examples of simulation in actuarial modeling

**Calculators**

**Actuarial Examinations**

In conjunction with M339J, M349P covers the content of SOA Exam C. Students are expected to be familiar with survival, severity, frequency and aggregate models, and use statistical methods to estimate parameters of such models given sample data. Students are further expected to identify steps in the modeling process, understand the underlying assumptions implicit in each family of models, recognize which assumptions are applicable in a given business application, and appropriately adjust the models for the impact of insurance coverage modifications.

## M349R Applied Regression and Time Series Syllabus

**Prerequisite and degree relevance: **Mathematics 339J or 339U, and 358K or 378K, with a grade of at least C- in each.

** Textbook: **Bowerman and Koehler,

**Forecasting, Time Series, and Regression,**

**Fourth Edition**** Responsible parties**: Mark Maxwell and Gustavo Cepparo

**Description of the Course**** : **The purpose of this course is to provide students in actuarial science, statistics and applied disciplines with an introduction to simple and multiple regression methods for analyzing relationships among several variables, and to elementary time series analysis. The emphasis will be on fitting suitable models to data, evaluating models using numerical and graphical techniques and interpreting the results in the context of the original problem, as opposed to the derivation of mathematical properties of the models. At the end of this course, students will be able to analyze many kinds of data in which one variable of interest is thought to depend on, or at least be related to, several other measured quantities, and some kinds of data collected over time or in some other serial manner.

**Course Goals and Overview:**

Incoming Students should be very familiar with descriptive statistics, simple regression, the logic of statistical inference, hypothesis tests and confidence intervals for means and proportions. M349R is a computer intensive course starting with an introduction to R and gradually moving towards SAS. The focus of the course is on hands-on data analysis. Students will work on projects with real data, identifying and stating the problems, planning/solving and concluding/reflecting. The textbook will be supplemented with R/SAS code and additional topics.

**Timing**

A typical semester has 42-44 MWF days. The syllabus contains topics for 35 class days and an additional 6 class days with Optional Topics. There are 3 class days for midterms or review.

Calendar (Lecture by lecture) M349R (approximate calendar with 38 days three times a week and 6 days for Optional Topics)

1 |
The Univariate Model (as a base model) and Randomization (Two sample and Matched Pairs Test). |

2 |
One sample t and Checking conditions with Bootstrap distributions. |

3 |
The Bivariate Model vs Univariate Model. Simple Regression. The Least Squares estimator. |

4 |
Root Mean Square Error and Adequate Predictor. |

5 |
Inference on Regression and Residual Plots. |

6 |
Continue with Inference on Regression and Coefficient of Determination. |

7 |
Calculating Standard Errors for Confidence Intervals and Prediction Intervals |

8 |
Total Regression and Partial Regression (Correlation and Partial Correlation). Simpson’s Paradox. |

9 |
Multiple Regression and Interpreting Coefficients. |

10 |
Residual Plots (again) in the context of Multiple Regression |

11 |
Overall F-test and Individual t-tests. Dummy Variables |

12 |
Continue with Dummy Variable notation. One-way Anova from Regression and Traditional Approach. |

13 |
Interaction, Partial F-test. |

14 |
More Practice with Dummy Variables and Variance Covariance Matrix and Ancova. |

15 |
Continue with Ancova. |

16 |
Collinearity. |

17 |
Continue with Collinearity. |

18 |
Residual Analysis (Hat-values, DfFits, DfBetas, Studentized Residuals). |

19 |
Continue with Residual Analysis. |

20 |
Continue with Residual Analysis. |

21 |
Heteroskedasticity. |

22 |
Continue with Heteroskedasticity. |

23 |
Continue with Heteroskedasticity. |

24 |
Autocorrelation in Regression and in Time Series Regression. Dummy variables for Seasonal Models in Time Series Regression with AR(1) errors structure. |

25 |
An example of a Random Walk. The intercept model in TS Regression. |

26 |
Moving Average and Random Walk (Calculate: Expectation, Variance, Covariance and Correlation for MA(1), MA(2) and AR(1)) |

27 |
MA(1) and AR(1) (SAS). |

28 |
Correlograms (ACF and PACF). |

29 |
Estimation MLE and Method of Moments (MoM). |

30 |
Four steps of Arima Modeling (Backshift Notation) |

31 |
Four steps of Arima Modeling (Model Comparison) |

32 |
Intro to Seasonal Models (Box Jenkins Models) |

33 |
Continue with Seasonal (Multiplicative Backshift Notation) |

34 |
Continue with Seasonal |

35 |
Review Seasonal and Nonseasonal |

optional |
Two out of three Optional Topics (below): Intervention Models and Building a Transfer Function Model (if time permits). |

optional |
Intervention Models and Building a Transfer Function Model |

optional |
Intervention Models and Building a Transfer Function Model |

optional |
Linear Probability Model and Logistic Regression Model |

optional |
Linear Probability Model and Logistic Regression Model (if time permits) |

optional |
Delta Method for one and two parameters (Confidence Intervals and Hypothesis Testing) (if time permits) |

## M358K Applied Statistics Syllabus

**Prerequisite and degree relevance**: M362K with a grade of C- or better. This course is intended for students in the Probability and Statistics math major specialization, students planning to teach secondary mathematics, students working for a BA in mathematics, and (as space permits) students in the natural sciences. Students preparing for graduate work in mathematical statistics should take M378K instead of or after taking this course.

**Text**: The standard textbook is Moore and McCabe, **Introduction to the Practice of Statistics, Fifth Edition**. This will be supplemented with additional material.**Resources: **Instructors should contact Martha Smith for more details on the project, pacing, and supplemental material.

**Project:** Students will be expected to do a term project to apply the material studied in the course.

**Computer use: **Students are expected to use software (typically, Minitab) to create graphs and do statistical calculations. They should also be able to interpret software output.

**Syllabus**: keyed to Moore and McCabe:

Chapter 1: Looking at Data - Distributions

Sections 1 - 3, supplemented with additional activities and material.

Chapter 2: Looking at Data - Relationships

Sections 1 – 5, supplemented with additional material.

Chapter 3: Producing Data

Sections 1 – 4, supplemented with additional material, including the project proposal.

Chapter 4: Probability: The Study of Randomness.

Sections 1 – 5 (Mostly review from M362K.)

Chapter 5: Sampling Distributions

Sections 1 and 2, supplemented with class activities and material.

Chapter 6: Introduction to Inference

Sections 1 – 4, supplemented with class activities.

Chapter 7: Inference from Distributions

Sections 1 – 3, supplemented with additional material.

Chapter 8: Inference for Proportions

Sections 1 – 2

Chapter 9: Analysis of Two-Way Tables

Sections 9.1 – 9.3 (optional topics in 9.2 may be omitted), possibly supplemented with additional material. (Instructor may substitute Chapter 12: One-Way Analysis of Variance.)

Chapter 10: Inference for Regression

Sections 1 – 2, supplemented with derivations of formulas.

Syllabus written by Martha Smith, August 2008

## M360M Mathematics as Problem Solving Syllabus

**Prerequisite and degree relevance: **Mathematics 408D, 408L, or 408S with a grade of at least C- and written consent of instructor.

This is a course in problem-solving in mathematics, geared primarily toward prospective math teachers. The goal of the course is to improve problem-solving skills. Students will be solving problems in class and at home, in groups and individually. The focus of the course is on the problem-solving process. Students will gain familiarity with commonly used heuristics, learn to maintain good control of the problem-solving process, and will gain proficiency in presenting solutions in both oral and written form.

**Responsible Parties:** Kathy Davis and Altha Rodin, Spring 2010

## M361 Theory of Functions of a Complex Variable Syllabus

**Prerequisite and degree relevance:** Mathematics 427J, 427K, or 427L with a grade of at least C-.

**Course description:** M361 consists of a study of the properties of complex analytic functions. Students are mainly from physics and engineering, with some mathematics majors and joint majors. Representative topics are Cauchy's integral theorem and formula, Laurent expansions, residue theory and the calculation of definite integrals, analytic continuation, and asymptotic expansions. Rigorous proofs are given for most results, with the intent to provide the student with a reliable grasp of the results and techniques.

**Text:** a reasonable text is Brown and Churchill, Complex Variables and Applications, sixth edition.

**Topics:**

- Complex Numbers
- Analytic Functions
- Elementary Functions
- Integrals
- Series
- Residues and poles
- Applications of Residues

**Responsible party:** John Dollard 2001

## M361K Introduction to Real Analysis Syllabus

**Prerequisite and degree relevance:** Either consent of the Undergraduate Mathematics Faculty Advisor or two of the following courses with a grade of at least C- in each: Mathematics 325K or Philosophy 313K, Mathematics 328K, Mathematics 341. Students who have received a grade of C- or better in Mathematics 365C may not take Mathematics 361K.

**Course description:** This is a rigorous treatment of the real number system, of real sequences, and of limits, continuity, derivatives, and integrals of real-valued functions of one real variable.

**Text:** A reasonable text is **Introduction to Real Analysis** by Bartle and Sherbert. The course might cover the bulk of chapters one through six in that book.

**Topics:**

- The real number system: the axiomatic description of the real number system as the unique complete ordered field, with special emphasis on the completeness axiom; the elementary topology of the real line.
- Real sequences: the definition and elementary properties of sequential limits; subsequences and accumulation points; monotone sequences; inferior and superior limits; the Bolzano-Weierstrass theorem.
- Limits and continuity of functions: the definition and elementary properties of limits of functions, including the usual variations on the basic theme (e.g.,one-sided limits, infinite limits, limits at infinity); continuity; the fundamental facts concerning continuous functions on intervals (e.g., Intexmexliatc Value Theorem, Maximum-Minimum Theorem, continuity of inverse functions, uniform continuity on closed intervals).
- Differentiation: the definition and geometric significance of the derivative; differentiation rules; the Mean Value Theorem and its consequences; Taylor's Theorem; L'Hospital's rules; convexity.
- Riemann Integration: the definition and elementary properties of the Riemann integral; the integrability of continuous functions and monotone functions; the Fundamental Theorems of Calculus.

March 1989

## M362K Probability I Syllabus

**Prerequisite and degree relevance:** Mathematics 408D, 408L, or 408S with a grade of at least C-. Mathematics 362K and Statistics and Scientific Computation 321 may not both be counted.

**Course description:** This is an introductory course in the mathematical theory of probability, thus it is fundamental to further work in probability and statistics. Principles of set theory and a set of axioms for probability are used to derive some probability density and/or distribution functions. Special counting techniques are developed to handle some problems. Properties associated with a random variable are developed for the usual elementary distributions. Problem-solving is required, and some theorem proving can be done, but the course emphasizes computation and intuition.

**Suggested Textbook:** **A First Course in Probability, Eighth Edition**, by Sheldon Ross.

The following course outline refers to section numbers in Ross' book and assumes a MWF lecture format (it must be modified for TTh classes)

**Some Alternate Textbooks:**

- Charles M. Grinstead and J. Laurie Snell,
**Introduction to Probability, 2nd revised ed.**, AMS 1977. This book has an interesting style that is different from the more standard format of Ross. It introduces some important ideas in examples and exercises, so the instructor needs to know what not to omit. There is too much emphasis on computation for this course, but otherwise, it is very well written, with many good examples and exercises. - Saeed Ghahramani,
**Fundamentals of Probability**. Prentice Hall, 1996. Similar to Ross' text.

**Background:** M362K is required of all undergraduate mathematics majors, and it is a prerequisite for courses in statistics. However, many of the students are majoring in other subjects(e.g., computer science or economics), and have little preparation in abstract mathematics. Calculus skills (integration and infinite series) tend to be weak, even at this level. Similarly, you cannot expect students to have any background in proofs, and should not expect competence in this. The course tends to be relatively easier for the first three to four weeks, so some students get the wrong impression as to its difficulty. Clarifying this early for the students can avoid unpleasant surprises later.

**Course Content:** Emphasize problem solving and intuition. Some advanced concepts should be presented without proof, so as to devote more attention to the examples. Basic combinatorics: Counting principle, permutations, combinations. Basic concepts: Sample spaces, events, basic axioms and theorems of probability, finite sample spaces with equally likely probabilities. Conditional probability: Reduced sample space, independence, Bayes' Theorem. Random variables: Discrete and continuous random variables, discrete probability functions and continuous probability density functions, distribution functions, expectation, variance, functions of random variables. Special distributions: Bernoulli, Binomial, Poisson, and Geometric discrete random variables. Uniform, Normal, and Exponential continuous random variables. Approximation of Binomial by Poisson or Normal. Jointly distributed random variables: Joint distribution functions, independence, conditional distributions, expectation, covariance Sums of independent random variables: expectation, variance. Inequalities and Limit theorems: Markov's and Chebyshev's inequalities, Weak and Strong Law of Large Numbers, Central Limit Theorem.

- 1.1-1.4: 3 lectures, Limit this material to one week.
- 2.1-2.5; 2.7: 4 lectures, Do not get bogged down in 2.5; limit it to about one lecture.
- 3.1-3.4: 4 lectures, Students like tree diagrams for Bayes' Theorem, and need more help and examples to learn how to extract information from word problems.
- 4.1-4.5: 4 lectures.
- 4.6-4.7; 4.8.1: 3 lectures, Omit 4.6.2 and 4.7.1. One could delay 4.7 to 5.4.1. Sections 4.8.2 and 4.8.3 are optional.
- 5.1-5.5; 5.7: 7 lectures. Omit 5.5.1; Section 5.6.1 is optional.
- 6.1-6.5: 4 lectures.
- 7.1-7.2; 7.4: 2 lectures, Omit 7.2.1, 7.2.2. Sections 7.5, 7.7, 7.8 are optional, as is correlation.
- 8.1-8.4: 3 lectures, Do not let any optional material crowd out the limit theorems. Emphasize intuitive understanding of the Central Limit Theorem by examples, and omit the proof, especially if optional 7.7 is not covered. One or two topics are optional.

There are a wealth of examples in the text, so the instructor has time to present only some of them. The outline above allows room for 34 lectures, 3 in class exam days, and 3 review days, for a total of 40 days. A typical semester has 42 MWF class days in the fall and 44 in the spring, so a few days for make-up or optional material are provided. It is likely that an instructor will find no time for any of the optional material.

T. Arbogast, J. Luecke, and M. Smith, August 2008

## M362M Introduction to Stochastic Processes Syllabus

**Prerequisite and degree relevance: **Mathematics 362K with a grade of at least C-.

**Course description:** Introduction to Markov chains, birth and death processes, and other topics.

## M365C Real Analysis I Syllabus

**Prerequisite and degree relevance:** Either consent of the Undergraduate Mathematics Faculty Advisor or two of the following courses with a grade of at least C- in each: Mathematics 325K or Philosophy 313K, Mathematics 328K, Mathematics 341. Students who receive a grade of C- in one of the prerequisite courses are advised to take Mathematics 361K before attempting 365C. Students planning to take Mathematics 365C and 373K concurrently should consult a mathematics adviser.

**Course description:** This course is an introduction to Analysis. Analysis together with Algebra and Topology form the central core of modern mathematics. Beginning with the notion of limit from calculus and continuing with ideas about convergence and the concept of function that arose with the description of heat flow using Fourier series, analysis is primarily concerned with infinite processes, the study of spaces and their geometry where these processes act and the application of differential and integral to problems that arise in geometry, PDE, physics and probability. This should be a course in analysis rather than point-set topology; the latter is covered in M376K.

**Text:** An appropriate text is Rudin **"Principles of Mathematical Analysis"** and the course should roughly cover its first seven chapters. The main difference between M361K and M365C lies in the more abstract metric space point of view in the latter. A strong student should be able to handle M365C without first taking M361K.

**The real number system and Euclidean spaces:**the axiomatic description of the real number system as the unique complete ordered field; the complex numbers; Euclidean space IR.**Metric spaces:**elementary metric space topology, with special emphasis on Euclidcan spaces; sequences in metric spaces - limits, accumulation points, subsequences, etc.; Cauchy sequences and completeness; compacmess in metric spaces; compact sets in R; connectedness in metric spaces; countable and uncountable sets.**Continuity:**limits and continuity of mappings between metric spaces, with particular attention to real-valued functions def'med on subsets of IR; preservation of compactness and connectedness under continuous mapping; uniform continuity.**Differentiation on the line:**the definition and geometric significance of the derivative of a real-valued function of a real variable; the Mean Value Theorem and its consequences; Taylor's Theorem; L'Hospital's rules.**Riemann integration on the line:**the definition and elementary properties of the Riemann integral; existence theorems for Riemann integrals; the Fundamental Theorems of Calculus.**Sequences and series of functions:**uniform convergence, uniform convergence, and continuity, uniform convergence, and integration, uniform convergence and differentiation.

September 2008

## M365D Real Analysis II Syllabus

**Prerequisite and degree relevance:** Mathematics 365C, with a grade of at least C-.

**Course description:** A rigorous treatment of selected topics in real analysis, such as Lebesgue integration, or multivariable integration and differential forms.

**Possible Texts:** Spivak, **Calculus**. Ross, **Elementary Analysis; The Theory of Calculus**. Fulks,** Advanced Calculus**.

This is a continuation of 365C with emphasis on functions of several variables. The treatment should be reasonably simple (for example, it is inappropriate to use Banach space language). The teacher can select his own textbook and should weigh his/her choice carefully in light of the above remarks and not get too ambitious.

March 1989

## M367K Topology I Syllabus

**Prerequisite and degree relevance:** Mathematics 361K or 365C or consent of instructor.

**Course description:** This will be a first course that emphasizes understanding and creating proofs; therefore, it provides a transition from the problem-solving approach of calculus to the entirely rigorous approach of advanced courses such as M365C or M373K. The number of topics required for coverage has been kept modest so as to allow instructors adequate time to Concentrates on developing the students' theorem-proving skills. The syllabus below is a typical syllabus. Other collections of topics in topology are equally appropriate. For example, some instructors prefer to restrict themselves to the topology of the real line or metric space topology.

- Cardinality: 1-1 correspondence, countability, and uncountability.
- Definitions of topological space: basis, sub-basis, metric space.
- Countability properties: dense sets, countable basis, local basis.
- Separation properties:Hausdorff, regular, normal.
- Covering properties: compact, countably compact, Lindelof.
- Continuity and homeomorphisms: properties preserved by continuous functions, Urysohns Lemma, Tietze Extension Theorem.
- Connectedness: definition, examples, invariance under continuous functions.

Notes containing definitions, theorem statements, and examples have been developed for this course and are available. The notes include some topics beyond those listed above.

March 1989

## M367L Topology II Syllabus

**Prerequisite and degree relevance:** Mathematics 367K with a grade of at least C- or consent of instructor.

**Course description:** Various topics in topology, primarily of a geometric nature.

## M368K Numerical Methods for Applications Syllabus

**Prerequisite and degree relevance:** Mathematics 348 with a grade of at least C-. Only one of the following may be counted: Computer Science 367, Mathematics 368K, Physics 329.

**Course description: **Continuation of Mathematics 348. Topics include splines, orthogonal polynomials, and smoothing of data, iterative solution of systems of linear equations, approximation of eigenvalues, two-point-boundary value problems, numerical approximation of partial differential equations, signal processing, optimization, and Monte Carlo methods.

## M371E Learning Assistant Experience in Mathematics Syllabus

**Prerequisite and degree relevance: **Mathematics 408C, 408K, 408N, 408R, or equivalent, and consent of instructor.

**Course description: **Students assist instructors and TAs in mathematics courses. This is a hands-on experience in what it is like to teach and support students in the learning of mathematics in undergraduate courses. Students must attend classroom training and discussions and work in Calculus discussion sections or undergraduate classrooms where mathematics is being taught. One class hour and three hours of fieldwork in an undergraduate mathematics course a week for one semester.

## M372K PDE and Applications Syllabus

**Prerequisite and degree relevance:** Mathematics 427J or 427K with a grade of at least C-. One of M361K or M365C is also recommended.

**Course description:** Partial differential equations arise as basic models of flows, diffusion, dispersion, and vibrations. Topics include first and second order partial differential equations and classification, particularly the wave, diffusion, and potential equations, their origins in applications and properties of solutions, characteristics, maximum principles, Greens functions, eigenvalue problems, and Fourier expansion methods.

## M373K Algebraic Structures I Syllabus

**Prerequisite and degree relevance:** Either consent of the Undergraduate Mathematics Faculty Advisor or two of the following courses with a grade of at least C- in each: Mathematics 325K or Philosophy 313K, Mathematics 328K, Mathematics 341. Students who receive a grade of C- in one of the prerequisite courses are advised to take Mathematics 343K before attempting 373K. Students planning to take Mathematics 365C and 373K concurrently should consult a mathematics adviser.

**Course description:** M373K is a rigorous course in pure mathematics. The syllabus for the course includes topics in the theory of groups and rings. The study of group theory includes normal subgroups, quotient groups, homomorphisms, permutation groups, the Sylow theorems, and the structure theorem for finite abelian groups. The topics in ring theory include ideals, quotient rings, the quotient field of an integral domain, Euclidean rings, and polynomial rings.

This course is generally viewed (along with 365C) as the most difficult of the required courses for a mathematics degree. Students are expected to produce logically sound proofs and solutions to challenging problems.

**Text: **Herstein,** Topics in Algebra**

Material to be covered: Chapters 1, 2, 3 and if time permits some topics in Chapters 4 and 5. This includes properties of the .integers, including divisibility and prime factorization; properties of groups, including subgroups, homomorphisms, permutation groups, the Sylow theorems; properties of rings, including subrings and ideals, homomorphisms, domains, especially Euclidian, principal ideal and unique factorization domains, polynomial rings. If time permits: Fields, elementary properties of vector spaces including the concept of dimension, field extensions.

We will be glad to discuss any questions or listen to any comments which you may have now or during the term on the course, the text, or the syllabus.

## M373L Algebraic Structures II Syllabus

**Prerequisite and degree relevance:** Mathematics 373K with a grade of at least C-. M373L is strongly recommended for undergraduates contemplating graduate study in mathematics.

**Course description:** M373L is a continuation of M373K, covering a selection of topics in algebra chosen from field theory and linear algebra. Emphasis is on understanding theorems and proofs.

**Text:** Herstein, **Topics in Algebra**

Material to be covered: Chapters 4, 5, (sections 1-3), 6.

This includes elementary properties of vector spaces and fields, including bases an dimension, elementary properties of linear transformations, relations to matrices, change of bases, dual spaces, characteristic roots, canonical forms, inner product spaces, normal transformations, quadratic and bilinear forms.

If time permits: topics left up to the instructor.

## M374M Mathematical Modeling in Science and Engineering Syllabus

**Prerequisite and degree relevance: **Mathematics 427J or 427K, and 340L or 341, with a grade of at least C- in each; and some basic programming skills. Moreover, students will be expected to have some familiarity with the software package Matlab.

**Text: J. David Logan, Applied Mathematics, 4th Edition**

**Responsible party**: Oscar Gonzalez November 2015

**Description of the Course**: This course is for students interested in mathematical modeling and analysis. The goals are to develop tools for studying differential equation models that arise in applications and to illustrate how the derivation and analysis of models can be used to gain insight and make predictions about physical systems. Emphasis should be placed on examples and case studies, and a broad range of applications from the engineering and physical sciences should be considered.

**Topics:**

The following outline is a list of relevant concepts for each core topic. Instructors should carefully choose and balance the concepts depending on the case studies they have in mind. Note that a well-designed case study will likely occupy 1-2 class days, and can be continued in an associated homework assignment. The number of class days listed below is for a standard MWF schedule. A typical semester has 43 MWF days and the schedule below contains material for 41 days, allowing time for two midterm exams. The suggested text provides some coverage of all the core topics; instructors may find it necessary to employ supplementary material to increase the depth of coverage in areas of interest, and to support their case studies.

1) Dimensional analysis and scaling (6 days)

-Fundamental physical dimensions, units

-Dimensional vs dimensionless quantities

-Unit-free equations and their properties

-Buckingham Pi Theorem

-Characteristic scales for a function

-Transforming equations to dimensionless form

-Scaling to expose dominant/small effects

2) Dynamical systems in one dimension (4 days)

-Properties of solutions

-Phase line diagrams

-Equilibrium solutions

-Stability of equilibria

-Classification via linearization

-Classification via Lyapunov functions

-Bifurcation of equilibria

-Basic types of bifurcations, hysteresis

3) Dynamical systems in two dimensions (9 days)

-Properties of solutions

-Phase plane diagrams, nullclines, direction fields

-Equilibrium solutions, stability

-Stability in linear systems, eigenvalues

-Phase diagrams for linear systems

-Stability in nonlinear systems, linearization theorem

-Stability in nonlinear systems, Lyapunov theorem

-Bifurcations in linear and nonlinear systems

-Closed orbits and limit cycles, Hopf bifurcation

-Poincare-Bendixson theorem

4) Regular perturbation methods (6 days)

-Perturbed equations, regular vs singular

-Characteristics of regular problems

-Approximation via asymptotic series

-Regular method for algebraic equations

-Typical error bounds

-Regular method for differential equations

-Typical error bounds, issue of uniformity

-Poincare-Lindstedt method for oscillatory problems

5) Singular perturbation methods (5 days)

-Characteristics of singular algebraic problems

-Rescaling method for algebraic problems

-Characteristics of singular differential problems

-Boundary layers, failure of regular method

-Two-scale method for problems with boundary layers

-Idea of inner, outer and matched expansions

-WKB method for oscillatory, exponential problems

6) Calculus of variations (11 days)

-Function spaces and functionals

-Absolute extrema of a functional

-Local extrema of a functional, issue of norms

-Concept of admissible variations

-Necessary conditions for local extrema

-Fundamental lemma, Euler-Lagrange equations

-Fixed-endpoint and free-endpoint problems

-Multiple-function and higher-order problems

-Isoperimetric constraints, multiplier rule

-Convexity and sufficient conditions

## M378K Introduction to Mathematical Statistics Syllabus

**Prerequisite and degree relevance:** Mathematics 362K with a grade of at least C-. Same as Statistics and Data Sciences 378. Only one the following may be counted: Mathematics 378K, Statistics and Data Sciences 378, Statistics and Scientific Computation 378.

Students taking this course are usually majoring in mathematics, actuarial science, or one of the natural sciences. M362K, 358K, and 378K form the core sequence for students in statistics.

**Goals and level of course:** Goals are to give students some insight into the theory behind the standard statistical procedures and to prepare continuing students for the graduate courses. Within the limits of the prerequisites, students are expected to derive and apply the theoretical results as well as carry out some standard statistical procedures.

**Topics covered:**

- Moment-Generating Functions
- Gamma, Chi-squared, t- and F-distributions
- Sampling Distributions and the Central Limit Theorem
- Point Estimation (bias, mean square error, relative efficiency, consistency, sufficiency, Method of Moments, Method of Maximum Likelihood, Rao-Blackwell Theorem and Minimum Variance Unbiased Estimation). Examples should include cases where more than one estimator is possible. Examples involving max and min are probably the easiest to do.
- Confidence intervals (concepts; small and large sample CIs for means and differences of means; large sample CIs for proportions and differences of proportions; selecting sample sizes)
- Hypothesis testing (concepts; small and large sample for means and differences of means; large sample for proportions and differences of proportions; small sample for proportions)
- Errors and power (type I and II errors, power and Neyman Pearson Lemma, calculating sample sizes for desired error level or power)
- Likelihood ratio tests
- If time permits: Some selection from Order statistics, Chi-squared tests, non-parametric tests, least squares regression.

Detailed syllabus based on Wackerly et al (fifth edition): (Chapters 7 - 10 constitute the heart of the course)

- Review of Probability and Introduction to Statistics:
- Chapter 1, Sections 2.1, 2.2, 2.12, 3.1, 3.11, 3.12, 4.1, 4.2, 4.3, 4.10, 4.12, 5.1, 5.12, 6.1, 6.7

- Additional Probability Topics: Gamma and Chi-Squared Distributions: Section 4.6
- Moment Generating Functions: Sections 3.9, 4.9
- Probability Distributions of Functions of Random Variables: Sections 6.4, 6.5
- Probability Distributions of Max and Min: First part of Section 6.6

- Sampling Distributions and the Central Limit Theorem: All of Chapter 7
- Estimation All of Chapter 8
- Properties of point Estimators and Methods of Estimation Chapter 9, omitting Section 8
- Hypothesis Testing Chapter 10 (Section 9 optional)
- (Additional topics as time permits)

## M378N Generalized Linear Models Syllabus

**Prerequisite and degree relevance: **Mathematics 378K with a grade of at least C- or consent of instructor. Mathematics 375T (Topic: Generalized Linear Models) and 378N may not both be counted.

**Course description:** Extensions to ordinary least-squares regression, including Poisson regression, the lasso, mixed models, and ridge regression.

## Graduate Courses

## Algebra

It is assumed that students know the basic material from an undergraduate course in linear algebra and an undergraduate abstract algebra course. The first part of the Prelim examination will cover sections 1 and 2 below. The second part of the Prelim examination will deal with section 3 below.

**1. Groups: **Finite groups, including Sylow theorems, *p*-groups, direct products and sums, semi-direct products, permutation groups, simple groups, finite Abelian groups; infinite groups, including normal and composition series, solvable and nilpotent groups, Jordan-Holder theorem, free groups.

**References:** Goldhaber Ehrlich, Ch. I except 14; Hungerford, Ch. I, II; Rotman, Ch. I-VI, VII (first three sections).

**2. Rings and modules:** Unique factorization domains, principal ideal domains, modules over principal ideal domains (including finitely generated Abelian groups), canonical forms of matrices (including Jordan form and rational canonical form), free and projective modules, tensor products, exact sequences, Wedderburn-Artin theorem, Noetherian rings, Hilbert basis theorem.

**References: **Goldhaber Ehrlich, Ch. II, III 1,2,4, IV, VII, VIII; Hungerford, Ch. III except 4,6, IV 1,2,3,5,6, VIII 1,4,6.

**3. Fields:** Algebraic and transcendental extensions, separable extensions, Galois theory of finite extensions, finite fields, cyclotomic fields, solvability by radicals.

**References:** Goldhaber Ehrlich, Ch. V except 6; Hungerford, Ch. V, VI; Kaplansky, Part I.

**References:**

Goldhaber Ehrlich, *Algebra*, reprint with corrections, Krieger, 1980.

Hungerford, *Algebra*, reprint with corrections, Springer, 1989.

Isaacs, *Algebra, a Graduate Course*, Wadsworth, 1994.

Kaplansky, *Fields and Rings*, 2nd Edition, University of Chicago Press, 1972.

Rotman, *An Introduction to the Theory of Groups*, 4th Edition, W.C. Brown, 1995.

## Analysis

The objective of this syllabus is to aid students in attaining a broad understanding of analysis techniques that are the basic stepping stones to contemporary research. The prelim exam normally consists of eight to ten problems, and the topics listed below should provide useful guidelines and strategy for their solution. It is assumed that students are familiar with the subject matter of the undergraduate analysis courses M365C and M361. The first part of the Prelim examination will cover Real Analysis. The second part of the prelim examination will cover Complex Analysis.

**1. Measure Theory and the Lebesgue Integral**

Basic properties of Lebesgue measure and the Lebesgue integral on **R**^{n} (see [5], Ch. 1-4) and general measure and integration theory in an abstract measure space (see [5], Ch. 11-12; and especially [6], Ch. 1-2). L^{p} spaces (see [6], Ch. 3); convergence almost everywhere, in norm and measure; approximation in L^{p}-norm and L^{p}-L^{q} duality; integration in product spaces (see [6], Ch. 8) and convolution on **R**^{n}; and the concept of a Banach space, Hilbert space, dual space and the Riesz representation theorem.

**2. Holomorphic Functions and Contour Integration**

Basic properties of analytic functions of one complex variable (see [1], Ch. 4-5; [2], Ch. 4-7; [4], Ch. 4-8; or [6], Ch. 10-12 and 15). Integration over paths, the local and global forms of Cauchy's Theorem, winding number and residue theorem, harmonic functions, Schwarz's Lemma and the Maximum Modulus theorem, isolated singularities, entire and meromorphic functions, Laurent series, infinite products, Weierstrass factorization, conformal mapping, Riemann mapping theorem, analytic continuation, "little" Picard theorem.

**3. Differentiation**

The relationship between differentiation and the Lebesgue integral on a real interval (see [5], Ch. 5), derivatives of measures (see [6], Ch. 5), absolutely continuous functions and absolute continuity between measures, functions of bounded variation.

**4. Specific Important Theorems**

Students should be familiar with Monotone and Dominated Convergence theorems, Fatou's lemma, Egorov's theorem, Lusin's theorem, Radon-Nikodym theorem, Fubini-Tonelli theorems about product measures and integration on product spaces, Cauchy's theorem and integral formulas, Maximum Modulus theorem, Rouche's theorem, Residue theorem, and Fundamental Theorem of Calculus for Lebesgue Integrals. Students should be familiar with Minkowski's Inequality, Holder's Inequality, Jensen's Inequality, and Bessel's Inequality.

**References:**

1. L. Ahlfors, *Complex Analysis,* McGraw-Hill, New York, 1979.

2. J.B. Conway, *Functions of One complex Variable,* second edition, Springer-Verlag, New York, 1978.

3. G.B. Folland, *Real Analysis,* second edition, John Wiley, New York, 1999.

4. B. Palka, *An Introduction to Complex Function Theory,* second printing, Springer-Verlag, New York, 1995.

5. H.L. Royden, *Real Analysis,* Macmillan, New York, 1988.

6. W. Rudin, *Real and Complex Analysis,* third edition, McGraw-Hill, New York, 1987.

7. R. Wheeden and A. Zygmund, *Measure and Integral,* Marcel Dekker, New York, 1977.

### Syllabus for M365C -- Introduction To Analysis

**The real number system and Euclidean spaces: **The axiomatic description of the real number system as the unique complete ordered field; the complex numbers; Euclidean space ** R**.

**Metric spaces: **Elementary metric space topology, with special emphasis on Euclidean spaces; sequences in metric spaces --- limits, accumulation points, subsequences, etc.; Cauchy sequences and completeness; compactness in metric spaces; compact sets in ** R**; connectedness in metric spaces; countable and

uncountable sets.

**Continuity: **Limits and continuity of mappings between metric spaces, with particular attention to real-valued functions defined on subsets of ** R**; preservation

of compactness and connectedness under continuous mapping; uniform continuity.

**Differentiation on the line: **The definition and geometric significance of the derivative of a real-valued function of a real variable; the Mean Value Theorem and its consequences; Taylor's theorem; L'Hospital's rules.

**Riemann integration on the line: **The definition and elementary properties of the Riemann integral; existence theorems for Riemann integrals; the Fundamental Theorems of Calculus.

**Sequences and series of functions: **Uniform convergence, uniform convergence, and continuity, uniform convergence and integration, uniform convergence and differentiation.

(An appropriate text might be Rudin's *Principles of Mathematical Analysis*, and the course should cover roughly its first seven chapters.)

## Applied Math

It is assumed that students are familiar with the subject matter of the undergraduate analysis course M365C (see the Analysis section for a syllabus of that course) and an undergraduate course in linear algebra.

The Applied Math Prelim divides into these six areas. The first three are discussed in M383C and will be covered in the first part of the Prelim examination:

**1. Banach spaces:**

Normed linear spaces and convexity; convergence, completeness, and Banach spaces; continuity, open sets, and closed sets; continuous linear transformations; Hahn-Banach Extension Theorem; linear functionals, dual and reflexive spaces, and weak convergence; the Baire Theorem and uniform boundedness; Open Mapping and Closed Graph Theorems; Closed Range Theorem; compact sets and Ascoli-Arzelà Theorem; compact operators and the Fredholm alternative.

**2. Hilbert spaces:** Basic geometry, orthogonality, bases, projections, and examples; Bessel’s inequality and the Parseval Theorem; the Riesz Representation Theorem; compact and Hilbert-Schmidt operators; spectral theory for compact, self-adjoint and normal operators; Sturm-Liouville Theory.

**3. Distributions:** Seminorms and locally convex spaces; test functions and distributions; calculus with distributions.

These three areas are discussed in M383D and will be covered in the second part of the Prelim examination:

**4. The Fourier Transform and Sobolev Spaces:** The Schwartz space and tempered distributions; the Fourier transform; the Plancherel Theorem; convolutions; fundamental solutions of PDE’s; Sobolev spaces; Imbedding Theorems; the Trace Theorems for H^{s}.

**5. Variational Boundary Value Problems (BVP):** Weak solutions to elliptic BVP’s; variational forms; Lax-Milgram Theorem; Green’s functions.

**6. Differential Calculus in Banach Spaces and Calculus of Variations:** The Fréchet derivative; the Chain Rule and Mean Value Theorems; Banach’s Contraction Mapping Theorem and Newton’s Method; Inverse and Implicit Function Theorems, and applications to nonlinear functional equations; extremum problems, Lagrange multipliers, and problems with constraints; the Euler-Lagrange equation.

**References:**

The first four references cover most of the syllabus for the exam. The other references also cover some topics in the syllabus.

1. C. Carath'eodory, *Calculus of Variations and Partial Differential Equations of the First Order,* 2nd English Edition, Chelsea, 1982.

2. F.W.J. Olver, *Asymptotics and Special Functions,* Academic Press, 1974.

3. M. Reed and B. Simon, *Methods of Modern Physics,* Vol. 1, Functional analysis.

4. R.E. Showalter, *Hilbert Space Methods for Partial Differential Equations,* available at World Wide Web address http://ejde.math.swt.edu//mono-toc.html .

5. A. Avez, *Introduction to Functional Analysis,* *Banach Spaces, and Differential Calculus,* Wiley, 1986.

6. L. Debnath and P. Mikusi'nski, *Introduction to Hilbert Spaces with Applications,* Academic Press, 1990.

7. I.M. Gelfand and S.V. Fomin, *Calculus of Variations,* Prentice-Hall, 1963.

8. E. Kreyszig, *Introductory Functional Analysis with Applications,* 1978.

9. J.T. Oden and L.F. Demkowicz, *Applied Functional Analysis,* CRC Press, 1996.

10. W. Rudin, *Functional Analysis,* McGraw-Hill, 1991.

11. W. Rudin, *Real and Complex Analysis,* 3rd Edition, McGraw-Hill, 1987.

12. K. Yosida, *Functional Analysis,* Springer-Verlag, 1980.

## Numerical Analysis

The Prelim sequence is M387C and M387D. The first part of the Prelim examination will cover algebra and approximation and the second part of the Prelim examination will cover differential equations.

Principles of discretization of differential equations:

- ODEs: Stability and convergence theory, Stiff problems, Symplectic integrators
- FEM (finite element method) and FDM (finite difference method) for boundary value problems
- FEM for PDEs (main focus on elliptic problems): Basic theory, weak formulations, Lax-Milgram theorem, finite element spaces, approximation theory, a priori and a posteriori error estimates, practical algorithms, extensions, mixed methods, etc.
- FDM for PDEs (main focus on hyperbolic and parabolic problems): Lax equivalence theorem, Von Neumann and other stability analysis, nonlinear conservation laws, shocks, entropy, practical algorithms

Brief survey of other methods for PDEs:

- FVM, DG, Spectral and particle methods
- Applications: Elasticity (FEM), Fluids (FVM), and Waves (FDM)
- Solution of linear and nonlinear equations
- Solution of integral equations
- Eigenvalues
- Optimization
- Monte Carlo methods
- Fast Fourier, wavelet transforms, approximation theory
- Basic undergraduate numerical methods
- Interpolation, fixed point iterations, Newton's method for root finding
- Direct and iterative methods for solving linear equations
- Quadratures

Recommended texts:

- Dahlquist and Bjorck, Numerical methods. Dover
- Lambert, Numerical methods for ordinary differential systems. Wiley
- Gustafsson, Kreiss, and Oliger, Time-dependent problems and difference methods
- Iserles, A first course in the numerical analysis of differential equations, Cambridge
- Claes Johnson, Numerical solution of partial differential equations by the finite element method. Cambridge University Press

## Probability

**(The first part of the Prelim exam will deal with the material covered in M385C and the second part of the Prelim exam will deal with the material covered in M385D)**

#### 1. Theory of Probability I - M385C

- Prerequisites:
- Real Analysis (M365C or equivalent),
- Linear Algebra (M341 or equivalent),
- Probability (M362K or equivalent).

- R. Durrett, Probability: theory and examples, third ed., Duxbury Press, Belmont, CA, 1996. (required)
- D. Williams, Probability with martingales, Cambridge University Press, Cambridge, 1991. (recommended)

Literature:
- Syllabus:

(Note: all references are to Durrett's book)**Foundations of Probability:**- Random variables (Sections 1.1, 1.2): probability spaces, σ-algebras, measurability, continuity of probabilities, product spaces, random variables, distribution functions, Lebesgue-Stieltjes measures (without proof), random vectors, generation, a.s.-convergence
- Expected value (Section 1.3): abstract Lebesgue integration (without proofs), inequalities (Jensen, Cauchy-Schwarz, Chebyshev, Markov, Hölder, Minkowski), limit theorems (Fatou's lemma, monotone convergence and dominated convergence theorems), change-of-variables formula,
- Dependence (Section 1.4): independence, pairwise independence, Dynkin's - theorem, convolution of measures, Fubini's theorem, Kolmogorov's extension theorem (without proof)

**Classical Theorems:**

- Weak laws of large numbers (Sections 1.5, 1.6): the L
^{2}-weak law of large numbers, triangular arrays, Borel-Cantelli lemmas, modes of convergence, inequalities (Markov, Chebyshev, Jensen, Hölder), the weak law of large numbers - Central limit theorems (Sections 2.2, 2.3a, 2.3b, 2.3c, 2.4a, 2.9part ): weak convergence of distributions, the continuous mapping theorem, Helly's selection theorem, tightness, characteristic functions, the inversion theorem, continuity theorem, the central limit theorem, multivariate normal distributions

**Discrete-Time Martingale Theory:**- Conditional expectation (Sections 4.1a, 4.1b): Radon-Nikodym theorem (without proof), conditional expectation, filtrations, predictability and adaptivity
- Martingales (Sections 4.2, 4.4, 4.5, 4.6part, 4.7): martingale transforms, the optional sampling theorem, the upcrossing inequality, Doob's decomposition, Doob's inequality, Lp -convergence, maximum inequalities, L
^{2}-theory, uniform integrability, backwards martingales and the strong law of large numbers.

#### 2. Theory of Probability II - M385D

- Prerequisites:
- Graduate-level probability (M385C or equivalent).

- Literature:
- I. Karatzas and S. Shreve, Brownian motion and stochastic processes, second ed., Springer, 1991 (required)
- D. Revuz and M. Yor, Continuous martingales and stochastic processes, third ed., Springer, 1999 (recommended)

- Syllabus:

(Note: all references are to the book of Karatzas and Shreve)**Continuous-Time Martingale Theory:**- General theory of processes (Sections 1.1, 1.2): Continuous-time processes and filtrations, types of measurability (optional, predictable, progressive), continuous stopping/optional times
- Path regularity of martingales (Section 1.3 A): existence of RCLL modifications, usual conditions for filtrations
- Convergence and optional sampling (Section 1.3 A-C): martingale inequalities, convergence theorems, optional sampling, uniform integrability and martingale with a last element
- Quadratic variation (Section 1.5 or Section IV.1 in Revuz-Yor): quadratic variation for continuous martingales, local martingales, and localization, spaces of martingales
- Doob-Meyer decomposition (Section 1.4): no proof

**Brownian Motion:**- Definition, construction and basic properties (Sections 2.1, 2.2): construction via Kolomogorov extension theorem, Hölder regularity of paths (Kolmogorov-Centsov), Gaussian processes
- The canonical space (Section 2.4): weak convergence on C[0, infinity), invariance principle, Wiener measure
- Markov and strong Markov property of Brownian motion (Sections 2.5-2.8, selected topics): reflexion principle, density of hitting times, Brownian filtrations, Blumenthal zero-one law

**Stochastic Integration:**- Construction of the Stochastic Integral (Sections 3.1, 3.2): stochastic integration with respect to continuous local martingales, quadratic variation, and Itô isometry
- Itô formula (Section 3.3): Itô formula, exponential martingales, linear stochastic differential equations

**Applications (and extensions) of Itô's formula:**- Paul Léavy's characterization of Brownian motion (Section 3.3 B):
- Changes of measure (Section 3.5): Girsanov theorem, Brownian motion with drift Representations of martingales (Section 3.4): predictable representation property and Kunita-Watanabe decomposition, time-changed Brownian motions (Dambis-Dubins-Schwarz), Knight's theorem on orthogonal martingales
- Local time (Sections 3.6, 3.7): local time for Brownian motion and continuous semimartingales, Tanaka's formula, generalized Itô's formula for convex functions.

## Topology

It is assumed that students have a working knowledge of the equivalent of a one-semester course in general topology (for example, see the appended syllabus for the undergraduate course M367K). For the semester in differential topology, it will also be assumed that students know the basic material from an undergraduate linear algebra course. The first part of the Prelim examination will deal with Algebraic Topology and the second part will deal with Differential Topology.

**Algebraic Topology**

**1. Manifolds: **Identification (quotient) spaces and identification (quotient) maps; topological *n*-manifolds, including surfaces, ** S^{n}**,

**,**

*RP*^{n}**, and lens spaces.**

*CP*^{n}**2. Triangulated manifolds: **Representation of triangulated, closed 2-manifolds as connected sums of tori or projective planes.

**3. Fundamental group and covering spaces: **Fundamental group, functoriality, retract, deformation retract; Van Kampen's Theorem, classification of surfaces by abelianizing the fundamental group, covering spaces, path lifting, homotopy lifting, uniqueness of lifts, general lifting theorem for maps, covering transformations, regular covers, correspondence between subgroups of the fundamental group and covering spaces, computing the fundamental group of the circle, ** RP^{n}**, lens spaces via covering spaces.

**4. Simplicial homology: **Homology groups, functoriality, topological invariance, Mayer-Vietoris sequence; applications, including Euler characteristic, classification of closed triangulated surfaces via homology and via Euler characteristic and orientability; degree of a map between oriented manifolds, Lefschetz number, Brouwer Fixed Point Theorem.

** References:**

Armstrong, *Basic Topology,* Springer, 1983 (principal text).

Greenberg, *Lectures on Algebraic Topology,* W.A. Benjamin, 1967.

Massey, *Algebraic Topology, an Introduction,* 4th corrected printing, Springer, 1977.

Munkres, *Elements of Algebraic Topology*, Addison-Wesley, 1984.

**Differential Topology**

**1. Smooth mappings: **Inverse Function Theorem, Local Submersion Theorem (Implicit Function Theorem).

**2. Differentiable manifolds:** Differentiable manifolds and submanifolds; examples, including surfaces,** S^{n}**,

**,**

*RP*^{n}**and lens spaces; tangent bundles; Sard's Theorem and its applications; differentiable transversality; orientation.**

*CP*^{n}**3. Vector fields and differential forms:** Integrating vector fields; degree of a map, Brouwer Fixed Point Theorem, No Retraction Theorem, Poincare-Hopf Theorem; differential forms, Stokes Theorem.

**References:**

Guillemin Pollack, *Differential Topology*, Prentice-Hall, 1974 (basic reference).

Hirsch, *Differential Topology,* Springer, 1976.

Milnor, *Topology from the Differentiable Viewpoint,* University of Virginia Press, 1965.

Spivak, *Calculus on Manifolds*, Benjamin, 1965 (differentiation, Inverse Function Theorem, Stokes Theorem).

For the examples indicated we refer to the books of Greenberg, Hirsch, and Munkres.

### Syllabus for M367K -- Topology I

**Cardinality:** 1-1 correspondence, countability, and uncountability.

**Definitions of topological space:** Basis, sub-basis, metric space.

**Countability properties:** Dense sets, countable basis, local basis.

**Separation properties:** Hausdorff, regular, normal.

**Covering properties:** Compact, countably compact, Lindelof.

**Continuity and homeomorphisms:** Properties preserved by continuous functions, Urysohn's Lemma, Tietze Extension Theorem.

**Connectedness: **Definition, examples, invariance under continuous functions.

**Reference: **Munkres, *Topology: a First Course*, Prentice-Hall, 1975.

## M 380D (Ciperiani) Algebra

It is assumed that students know the basic material from an undergraduate course in linear algebra and an undergraduate abstract algebra course. The first part of the Prelim examination will cover sections 1 and 2 below. The second part of the Prelim examination will deal with section 3 below.

**1. Groups: **Finite groups, including Sylow theorems, *p*-groups, direct products and sums, semi-direct products, permutation groups, simple groups, finite Abelian groups; infinite groups, including normal and composition series, solvable and nilpotent groups, Jordan-Holder theorem, free groups.

**References:** Goldhaber Ehrlich, Ch. I except 14; Hungerford, Ch. I, II; Rotman, Ch. I-VI, VII (first three sections).

**2. Rings and modules:** Unique factorization domains, principal ideal domains, modules over principal ideal domains (including finitely generated Abelian groups), canonical forms of matrices (including Jordan form and rational canonical form), free and projective modules, tensor products, exact sequences, Wedderburn-Artin theorem, Noetherian rings, Hilbert basis theorem.

**References: **Goldhaber Ehrlich, Ch. II, III 1,2,4, IV, VII, VIII; Hungerford, Ch. III except 4,6, IV 1,2,3,5,6, VIII 1,4,6.

**3. Fields:** Algebraic and transcendental extensions, separable extensions, Galois theory of finite extensions, finite fields, cyclotomic fields, solvability by radicals.

**References:** Goldhaber Ehrlich, Ch. V except 6; Hungerford, Ch. V, VI; Kaplansky, Part I.

**References:**

Goldhaber Ehrlich, *Algebra*, reprint with corrections, Krieger, 1980.

Hungerford, *Algebra*, reprint with corrections, Springer, 1989.

Isaacs, *Algebra, a Graduate Course*, Wadsworth, 1994.

Kaplansky, *Fields and Rings*, 2nd Edition, University of Chicago Press, 1972.

Rotman, *An Introduction to the Theory of Groups*, 4th Edition, W.C. Brown, 1995.

## M 381D (Koch) Complex Analysis

The objective of this syllabus is to aid students in attaining a broad understanding of analysis techniques that are the basic stepping stones to contemporary research. The prelim exam normally consists of eight to ten problems, and the topics listed below should provide useful guidelines and strategy for their solution. It is assumed that students are familiar with the subject matter of the undergraduate analysis courses M365C and M361. The first part of the Prelim examination will cover Real Analysis. The second part of the prelim examination will cover Complex Analysis.

**1. Measure Theory and the Lebesgue Integral**

Basic properties of Lebesgue measure and the Lebesgue integral on **R**^{n} (see [5], Ch. 1-4) and general measure and integration theory in an abstract measure space (see [5], Ch. 11-12; and especially [6], Ch. 1-2). L^{p} spaces (see [6], Ch. 3); convergence almost everywhere, in norm and measure; approximation in L^{p}-norm and L^{p}-L^{q} duality; integration in product spaces (see [6], Ch. 8) and convolution on **R**^{n}; and the concept of a Banach space, Hilbert space, dual space and the Riesz representation theorem.

**2. Holomorphic Functions and Contour Integration**

Basic properties of analytic functions of one complex variable (see [1], Ch. 4-5; [2], Ch. 4-7; [4], Ch. 4-8; or [6], Ch. 10-12 and 15). Integration over paths, the local and global forms of Cauchy's Theorem, winding number and residue theorem, harmonic functions, Schwarz's Lemma and the Maximum Modulus theorem, isolated singularities, entire and meromorphic functions, Laurent series, infinite products, Weierstrass factorization, conformal mapping, Riemann mapping theorem, analytic continuation, "little" Picard theorem.

**3. Differentiation**

The relationship between differentiation and the Lebesgue integral on a real interval (see [5], Ch. 5), derivatives of measures (see [6], Ch. 5), absolutely continuous functions and absolute continuity between measures, functions of bounded variation.

**4. Specific Important Theorems**

Students should be familiar with Monotone and Dominated Convergence theorems, Fatou's lemma, Egorov's theorem, Lusin's theorem, Radon-Nikodym theorem, Fubini-Tonelli theorems about product measures and integration on product spaces, Cauchy's theorem and integral formulas, Maximum Modulus theorem, Rouche's theorem, Residue theorem, and Fundamental Theorem of Calculus for Lebesgue Integrals. Students should be familiar with Minkowski's Inequality, Holder's Inequality, Jensen's Inequality, and Bessel's Inequality.

**References:**

1. L. Ahlfors, *Complex Analysis,* McGraw-Hill, New York, 1979.

2. J.B. Conway, *Functions of One complex Variable,* second edition, Springer-Verlag, New York, 1978.

3. G.B. Folland, *Real Analysis,* second edition, John Wiley, New York, 1999.

4. B. Palka, *An Introduction to Complex Function Theory,* second printing, Springer-Verlag, New York, 1995.

5. H.L. Royden, *Real Analysis,* Macmillan, New York, 1988.

6. W. Rudin, *Real and Complex Analysis,* third edition, McGraw-Hill, New York, 1987.

7. R. Wheeden and A. Zygmund, *Measure and Integral,* Marcel Dekker, New York, 1977.

### Syllabus for M365C -- Introduction To Analysis

**The real number system and Euclidean spaces: **The axiomatic description of the real number system as the unique complete ordered field; the complex numbers; Euclidean space ** R**.

**Metric spaces: **Elementary metric space topology, with special emphasis on Euclidean spaces; sequences in metric spaces --- limits, accumulation points, subsequences, etc.; Cauchy sequences and completeness; compactness in metric spaces; compact sets in ** R**; connectedness in metric spaces; countable and

uncountable sets.

**Continuity: **Limits and continuity of mappings between metric spaces, with particular attention to real-valued functions defined on subsets of ** R**; preservation

of compactness and connectedness under continuous mapping; uniform continuity.

**Differentiation on the line: **The definition and geometric significance of the derivative of a real-valued function of a real variable; the Mean Value Theorem and its consequences; Taylor's theorem; L'Hospital's rules.

**Riemann integration on the line: **The definition and elementary properties of the Riemann integral; existence theorems for Riemann integrals; the Fundamental Theorems of Calculus.

**Sequences and series of functions: **Uniform convergence, uniform convergence, and continuity, uniform convergence, and integration, uniform convergence and differentiation.

(An appropriate text might be Rudin's *Principles of Mathematical Analysis*, and the course should cover roughly its first seven chapters.)

## M 382D (Sadun) Differential Topology

It is assumed that students have a working knowledge of the equivalent of a one-semester course in general topology (for example, see the appended syllabus for the undergraduate course M367K). For the semester in differential topology, it will also be assumed that students know the basic material from an undergraduate linear algebra course. The first part of the Prelim examination will deal with Algebraic Topology and the second part will deal with Differential Topology.

**Algebraic Topology**

**1. Manifolds: **Identification (quotient) spaces and identification (quotient) maps; topological *n*-manifolds, including surfaces, ** S^{n}**,

**,**

*RP*^{n}**, and lens spaces.**

*CP*^{n}**2. Triangulated manifolds: **Representation of triangulated, closed 2-manifolds as connected sums of tori or projective planes.

**3. Fundamental group and covering spaces: **Fundamental group, functoriality, retract, deformation retract; Van Kampen's Theorem, classification of surfaces by abelianizing the fundamental group, covering spaces, path lifting, homotopy lifting, uniqueness of lifts, general lifting theorem for maps, covering transformations, regular covers, correspondence between subgroups of the fundamental group and covering spaces, computing the fundamental group of the circle, ** RP^{n}**, lens spaces via covering spaces.

**4. Simplicial homology: **Homology groups, functoriality, topological invariance, Mayer-Vietoris sequence; applications, including Euler characteristic, classification of closed triangulated surfaces via homology and via Euler characteristic and orientability; degree of a map between oriented manifolds, Lefschetz number, Brouwer Fixed Point Theorem.

** References:**

Armstrong, *Basic Topology,* Springer, 1983 (principal text).

Greenberg, *Lectures on Algebraic Topology,* W.A. Benjamin, 1967.

Massey, *Algebraic Topology, an Introduction,* 4th corrected printing, Springer, 1977.

Munkres, *Elements of Algebraic Topology*, Addison-Wesley, 1984.

**Differential Topology**

**1. Smooth mappings: **Inverse Function Theorem, Local Submersion Theorem (Implicit Function Theorem).

**2. Differentiable manifolds:** Differentiable manifolds and submanifolds; examples, including surfaces,** S^{n}**,

**,**

*RP*^{n}**and lens spaces; tangent bundles; Sard's Theorem and its applications; differentiable transversality; orientation.**

*CP*^{n}**3. Vector fields and differential forms:** Integrating vector fields; degree of a map, Brouwer Fixed Point Theorem, No Retraction Theorem, Poincare-Hopf Theorem; differential forms, Stokes Theorem.

**References:**

Guillemin Pollack, *Differential Topology*, Prentice-Hall, 1974 (basic reference).

Hirsch, *Differential Topology,* Springer, 1976.

Milnor, *Topology from the Differentiable Viewpoint,* University of Virginia Press, 1965.

Spivak, *Calculus on Manifolds*, Benjamin, 1965 (differentiation, Inverse Function Theorem, Stokes Theorem).

For the examples indicated we refer to the books of Greenberg, Hirsch, and Munkres.

### Syllabus for M367K -- Topology I

**Cardinality:** 1-1 correspondence, countability, and uncountability.

**Definitions of topological space:** Basis, sub-basis, metric space.

**Countability properties:** Dense sets, countable basis, local basis.

**Separation properties:** Hausdorff, regular, normal.

**Covering properties:** Compact, countably compact, Lindelof.

**Continuity and homeomorphisms:** Properties preserved by continuous functions, Urysohn's Lemma, Tietze Extension Theorem.

**Connectedness: **Definition, examples, invariance under continuous functions.

**Reference: **Munkres, *Topology: a First Course*, Prentice-Hall, 1975.

## M 383D (Gamba) Methods of Applied Mathematics

It is assumed that students are familiar with the subject matter of the undergraduate analysis course M365C (see the Analysis section for a syllabus of that course) and an undergraduate course in linear algebra.

The Applied Math Prelim divides into these six areas. The first three are discussed in M383C and will be covered in the first part of the Prelim examination:

**1. Banach spaces:**

Normed linear spaces and convexity; convergence, completeness, and Banach spaces; continuity, open sets, and closed sets; continuous linear transformations; Hahn-Banach Extension Theorem; linear functionals, dual and reflexive spaces, and weak convergence; the Baire Theorem and uniform boundedness; Open Mapping and Closed Graph Theorems; Closed Range Theorem; compact sets and Ascoli-Arzelà Theorem; compact operators and the Fredholm alternative.

**2. Hilbert spaces:** Basic geometry, orthogonality, bases, projections, and examples; Bessel’s inequality and the Parseval Theorem; the Riesz Representation Theorem; compact and Hilbert-Schmidt operators; spectral theory for compact, self-adjoint and normal operators; Sturm-Liouville Theory.

**3. Distributions:** Seminorms and locally convex spaces; test functions and distributions; calculus with distributions.

These three areas are discussed in M383D and will be covered in the second part of the Prelim examination:

**4. The Fourier Transform and Sobolev Spaces:** The Schwartz space and tempered distributions; the Fourier transform; the Plancherel Theorem; convolutions; fundamental solutions of PDE’s; Sobolev spaces; Imbedding Theorems; the Trace Theorems for H^{s}.

**5. Variational Boundary Value Problems (BVP):** Weak solutions to elliptic BVP’s; variational forms; Lax-Milgram Theorem; Green’s functions.

**6. Differential Calculus in Banach Spaces and Calculus of Variations:** The Fréchet derivative; the Chain Rule and Mean Value Theorems; Banach’s Contraction Mapping Theorem and Newton’s Method; Inverse and Implicit Function Theorems, and applications to nonlinear functional equations; extremum problems, Lagrange multipliers, and problems with constraints; the Euler-Lagrange equation.

**References:**

The first four references cover most of the syllabus for the exam. The other references also cover some topics in the syllabus.

1. C. Carath'eodory, *Calculus of Variations and Partial Differential Equations of the First Order,* 2nd English Edition, Chelsea, 1982.

2. F.W.J. Olver, *Asymptotics and Special Functions,* Academic Press, 1974.

3. M. Reed and B. Simon, *Methods of Modern Physics,* Vol. 1, Functional analysis.

4. R.E. Showalter, *Hilbert Space Methods for Partial Differential Equations,* available at World Wide Web address http://ejde.math.swt.edu//mono-toc.html .

5. A. Avez, *Introduction to Functional Analysis,* *Banach Spaces, and Differential Calculus,* Wiley, 1986.

6. L. Debnath and P. Mikusi'nski, *Introduction to Hilbert Spaces with Applications,* Academic Press, 1990.

7. I.M. Gelfand and S.V. Fomin, *Calculus of Variations,* Prentice-Hall, 1963.

8. E. Kreyszig, *Introductory Functional Analysis with Applications,* 1978.

9. J.T. Oden and L.F. Demkowicz, *Applied Functional Analysis,* CRC Press, 1996.

10. W. Rudin, *Functional Analysis,* McGraw-Hill, 1991.

11. W. Rudin, *Real and Complex Analysis,* 3rd Edition, McGraw-Hill, 1987.

12. K. Yosida, *Functional Analysis,* Springer-Verlag, 1980.

## M 385D (Sirbu) Theory of Probability

**(The first part of the Prelim exam will deal with the material covered in M385C and the second part of the Prelim exam will deal with the material covered in M385D)**

#### 1. Theory of Probability I - M385C

- Prerequisites:
- Real Analysis (M365C or equivalent),
- Linear Algebra (M341 or equivalent),
- Probability (M362K or equivalent).

- R. Durrett, Probability: theory and examples, third ed., Duxbury Press, Belmont, CA, 1996. (required)
- D. Williams, Probability with martingales, Cambridge University Press, Cambridge, 1991. (recommended)

Literature:
- Syllabus:

(Note: all references are to Durrett's book)**Foundations of Probability:**- Random variables (Sections 1.1, 1.2): probability spaces, σ-algebras, measurability, continuity of probabilities, product spaces, random variables, distribution functions, Lebesgue-Stieltjes measures (without proof), random vectors, generation, a.s.-convergence
- Expected value (Section 1.3): abstract Lebesgue integration (without proofs), inequalities (Jensen, Cauchy-Schwarz, Chebyshev, Markov, Hölder, Minkowski), limit theorems (Fatou's lemma, monotone convergence and dominated convergence theorems), change-of-variables formula,
- Dependence (Section 1.4): independence, pairwise independence, Dynkin's - theorem, convolution of measures, Fubini's theorem, Kolmogorov's extension theorem (without proof)

**Classical Theorems:**

- Weak laws of large numbers (Sections 1.5, 1.6): the L
^{2}-weak law of large numbers, triangular arrays, Borel-Cantelli lemmas, modes of convergence, inequalities (Markov, Chebyshev, Jensen, Hölder), the weak law of large numbers - Central limit theorems (Sections 2.2, 2.3a, 2.3b, 2.3c, 2.4a, 2.9part ): weak convergence of distributions, the continuous mapping theorem, Helly's selection theorem, tightness, characteristic functions, the inversion theorem, continuity theorem, the central limit theorem, multivariate normal distributions

**Discrete-Time Martingale Theory:**- Conditional expectation (Sections 4.1a, 4.1b): Radon-Nikodym theorem (without proof), conditional expectation, filtrations, predictability and adaptivity
- Martingales (Sections 4.2, 4.4, 4.5, 4.6part , 4.7): martingale transforms, the optional sampling theorem, the upcrossing inequality, Doob's decomposition, Doob's inequality, Lp -convergence, maximum inequalities, L
^{2}-theory, uniform integrability, backwards martingales and the strong law of large numbers.

#### 2. Theory of Probability II - M385D

- Prerequisites:
- Graduate-level probability (M385C or equivalent).

- Literature:
- I. Karatzas and S. Shreve, Brownian motion and stochastic processes, second ed., Springer, 1991 (required)
- D. Revuz and M. Yor, Continuous martingales and stochastic processes, third ed., Springer, 1999 (recommended)

- Syllabus:

(Note: all references are to the book of Karatzas and Shreve)**Continuous-Time Martingale Theory:**- General theory of processes (Sections 1.1, 1.2): Continuous-time processes and filtrations, types of measurability (optional, predictable, progressive), continuous stopping/optional times
- Path regularity of martingales (Section 1.3 A): existence of RCLL modifications, usual conditions for filtrations
- Convergence and optional sampling (Section 1.3 A-C): martingale inequalities, convergence theorems, optional sampling, uniform integrability and martingale with a last element
- Quadratic variation (Section 1.5 or Section IV.1 in Revuz-Yor): quadratic variation for continuous martingales, local martingales, and localization, spaces of martingales
- Doob-Meyer decomposition (Section 1.4): no proof

**Brownian Motion:**- Definition, construction and basic properties (Sections 2.1, 2.2): construction via Kolomogorov extension theorem, Hölder regularity of paths (Kolmogorov-Centsov), Gaussian processes
- The canonical space (Section 2.4): weak convergence on C[0, infinity), invariance principle, Wiener measure
- Markov and strong Markov property of Brownian motion (Sections 2.5-2.8, selected topics): reflexion principle, density of hitting times, Brownian filtrations, Blumenthal zero-one law

**Stochastic Integration:**- Construction of the Stochastic Integral (Sections 3.1, 3.2): stochastic integration with respect to continuous local martingales, quadratic variation, and Itô isometry
- Itô formula (Section 3.3): Itô formula, exponential martingales, linear stochastic differential equations

**Applications (and extensions) of Itô's formula:**- Paul Léavy's characterization of Brownian motion (Section 3.3 B):
- Changes of measure (Section 3.5): Girsanov theorem, Brownian motion with drift Representations of martingales (Section 3.4): predictable representation property and Kunita-Watanabe decomposition, time-changed Brownian motions (Dambis-Dubins-Schwarz), Knight's theorem on orthogonal martingales
- Local time (Sections 3.6, 3.7): local time for Brownian motion and continuous semimartingales, Tanaka's formula, generalized Itô's formula for convex functions.

## M 387D (Engquist) Numerical Analysis: Algebra & Approximations

The Prelim sequence is M387C and M387D. The first part of the Prelim examination will cover algebra and approximation and the second part of the Prelim examination will cover differential equations.

Principles of discretization of differential equations:

- ODEs: Stability and convergence theory, Stiff problems, Symplectic integrators
- FEM (finite element method) and FDM (finite difference method) for boundary value problems
- FEM for PDEs (main focus on elliptic problems): Basic theory, weak formulations, Lax-Milgram theorem, finite element spaces, approximation theory, a priori and a posteriori error estimates, practical algorithms, extensions, mixed methods, etc.
- FDM for PDEs (main focus on hyperbolic and parabolic problems): Lax equivalence theorem, Von Neumann and other stability analysis, nonlinear conservation laws, shocks, entropy, practical algorithms

Brief survey of other methods for PDEs:

- FVM, DG, Spectral and particle methods
- Applications: Elasticity (FEM), Fluids (FVM), and Waves (FDM)
- Solution of linear and nonlinear equations
- Solution of integral equations
- Eigenvalues
- Optimization
- Monte Carlo methods
- Fast Fourier, wavelet transforms, approximation theory
- Basic undergraduate numerical methods
- Interpolation, fixed point iterations, Newton's method for root finding
- Direct and iterative methods for solving linear equations
- Quadratures

Recommended texts:

- Dahlquist and Bjorck, Numerical methods. Dover
- Lambert, Numerical methods for ordinary differential systems. Wiley
- Gustafsson, Kreiss, and Oliger, Time-dependent problems and difference methods
- Iserles, A first course in the numerical analysis of differential equations, Cambridge
- Claes Johnson, Numerical solution of partial differential equations by the finite element method. Cambridge University Press

## M 391C (Caffarelli) An Introduction to Diffusion Processes

This is a mathematics topics course taught by Dr. Luis Caffarelli. His research interests include non-linear analysis, partial differential equations, and their applications, calculus of variations, and optimization. Students outside of the Mathematics PhD program must get instructor approval to take this course.

## M 392C (Perutz) Gauge Theory

This will be a course about the Seiberg-Witten equations over 4-dimensional manifolds. We will use these equations to give examples of 4-dimensional homotopy types that admit no smooth manifold structure, and others that admit infinitely many: examples that show that 4-dimensional manifolds do not play by the rules that govern smooth manifolds of any other dimensions. Besides measuring the difference between homotopy theory and smooth topology, we shall also use the Seiberg-Witten equations to detect differences between smooth and symplectic topology in dimension 4.

What rules do in fact govern smooth 4-manifolds? The answer is, to date, the greatest mystery in geometric topology.

Methods used in the course will be from algebraic topology and differential geometry - for which the two topology prelims provide appropriate background - and geometric analysis - for which I will cover background material on elliptic operators. The course will complement the 4-manifolds course taught regularly by Prof. Gompf (which has a quite different flavor), and will provide training for students interested in Heegaard Floer and related theories for 3-manifolds.

Reference: S. K. Donaldson, The Seiberg-Witten equations and 4-manifold topology, Bulletin of the AMS, 1996; http://www.ams.org/journals/bull/1996-33-01/S0273-0979-96-00625-8/

## M 392C (Gordon) Knots / 3-Manifolds

The course will cover a topic in low-dimensional topology and knot theory. A possible topic is knot concordance, in both the smooth and topological categories.

## M 392C (Allcock) Lie Groups

These are groups which are also manifolds, named after Sophus Lie. Their structure theory and applications are never-endingly rich. We will go from the beginning of the theory to the classification of simple Lie groups and cover some basic representation theory and applications. This course is aimed at graduate students who have already taken both semesters of the graduate topology course (algebraic topology and differential topology), and the first semester of the graduate algebra prelim course (the group theory part). You will be expected to have this background on day one, so we can hit the ground running. There will be homework assigned every week or two.

The text will be Rossman's "Lie Groups: An Introduction through Linear Groups", supplemented by additional material.

## M 392C (Freed) Geometry/Topology/Physics

The Dirac operator is a first-order linear elliptic differential operator, originally introduced in Lorentz signature, but with many incarnations and applications in Riemannian geometry, differential topology, and beyond. On a compact manifold, or family of compact manifolds, a Dirac operator has many topological and geometric invariants. The most basic is the Fredholm index, which only depends on the kernel of the operator. The topological formula for the index is the Atiyah-Singer index theorem. Geometric invariants, such as the Atiyah-Patodi-Singer eta-invariant and determinant line bundle, are constructed from the entire spectrum of the Dirac operator, and there are geometric versions of the index theorem which pertain. These invariants have many contemporary applications in geometry and physics.

Students in the course will give many of the lectures; the instructor will provide materials and coaching.

## M 393C (Maggi) Partial Differential Equations II

This is the second semester of a year-long course which serves as an introduction to the modern mathematical treatment of linear and nonlinear partial differential equations. The beginning of the course will be devoted to studying existence and some properties of solutions (e.g. regularity) for linear equations of parabolic and hyperbolic type. Then we will discuss introductory topics in the theory of some important nonlinear equations (including but not limited to: nonlinear wave equations and nonlinear dispersive equations).

## M 393C (Patrizi) Viscosity Solutions and Applications

We first cover the basic theory of viscosity solutions: existence, uniqueness, and stability properties. Then, we discuss some applications: homogenization, mean curvature flow, regularity results for second order uniformly elliptic equations. If time permits, we will also study viscosity solutions for nonlocal operators and the Peierls-Nabarro model.