# Course Review: MATH 418

#### Probability

“I heard you had a lot of difficulty with the last homework. So you will be relieved to note that next homework you will have another opportunity to practice similar problems.”

Text:  A First Look at Rigorous Probability Theory, 2nd ed by J.S. Rosenthal

Prof Slade is very clear and keeps things simple. He also has no problem slowing down to answer questions from students if they are not following along. He has a dry sense of humor, that keeps the class interesting, and he can even by funny when he is talking in earnest.

Difficulty

After taking measure theory, several sections of the course can feel like review. This was a good thing for me, as I found MATH 420 a tad fast. There are a handful of new techniques that you learn in the homework and class, but there are not that many new concepts if you have some background in measure theory and probability. MATH 421 material also comes up in terms of weak convergence.

Key Concepts

Probability Triples

Random Variables

Distributions

Expectation

Borel-Cantelli

Modes of Convergence

Law of Large Numbers

Central Limit Theorem

Characteristic Functions

Hard Concepts

Tail Events: Kind of funny to think about. Also, include definitions of limit supremum and limit infimum for sequences of event which can be difficult to convert to statements about limits of random variables.

Weak Convergence: There are a lot of equivalent statements, and if you pick the wrong one it can be a mission to prove that convergence occurs.

Conclusion

Good review of measure theory, and gives you a mathematical foundation to elementary probability.

# Course Review: MATH 421

Real Analysis II

“If everyone is getting closer together, and you are going to Cleveland, then everyone’s going to Cleveland!”

Text: Real Analysis: Modern Techniques and Their Applications by Gerald B. Folland

Prof: Dr Brian Marcus

Prof Marcus is one of the kindest professors I know. He also explains each concept from multiple perspectives and breaks the concepts down to simple intuition. His notes are super useful and he has a great sense of humor.

Difficulty

The classes moves somewhat slower then 420, so we went a lot more deeper into the details of certain proofs and where they come from. This meant the classes were easier to follow (and in my view more enjoyable!). Further the homework was largely straightforward. The one challenge in this course was the material towards the latter half of the term was slightly more complex and were not assessed on homework. We also may not have covered the material generally covered in a similar course.

Key Concepts

Banach spaces

Hilbert spaces

Linear functionals

Hahn Banach Theorem

Open mapping theorem

Uniform boundedness

Riesz representation theorems

Hard Concepts

Weak Convergence: Can be hard to think about neighborhoods in this topology.

Hilbert Spaces: There are a lot of nifty tricks in Hilbert spaces that are not immediately obvious, and do not work in other spaces.

Conclusion

Really fun course and learnt a lot. Nice conclusion to my studies of analysis in some sense.

# Course Review: MATH 302

Introduction to Probability

“Donald claims that he won the popular vote if you subtract the 3 million illegal voters. Assuming that 3 million people did vote illegally, compute the probability that Donald is correct.”

Text: Introduction to Probability by David F. Anderson, Timo Seppalainen, and Benedek Valko

Prof:  Dr. Martin Lohmann

Dr. Lohmann’s lectures largely consisted of (usually) interesting examples. Some students found his accent and his handwriting a bit challenging to follow. This, combined with the fact that he defers posting of lecture notes, made the course harder than necessary for such students. I did not have any difficulty understanding what was being said, and the few times I found his handwriting became hard to read, he clarified immediately. Also, if you ask a stupid question, expect some deft sarcasm in response!

Difficulty

While I find probability counter-intuitive, the assignments were all quite doable. Occasionally, harder questions marked with a star were provided. The first midterm was quite tricky, however, we were given a practice midterm beforehand that was conceptually quite similar to the actual exam. The second midterm and final were both significantly easier and we also were given similar practice material.

Key Concepts

Counting

Sets

Discrete vs Continuous Probability Distributions

Mean, Variance and Covariance

Joint Distribution

Convergence in Probability/Distribution

Conditional Probability

Moment-generating functions

Hard Concepts

Counting: I always make incorrect assumptions w/regard to counting. I think the key to easier problems is to identify whether you are using replacement/no replacement and order/no order. For harder problems, it is often necessary to construct a bijection of sorts or use a symmetry argument.

Conclusion

Good to get some practice with counting and probability. About as much theory as you would expect from such a class.

# Course Review: MATH 215

Elementary Differential Equations I

“UBC is a very progressive place…Because you get to learn Linear Algebra before Differential Equations!”

Text: Notes on Diffy Qs: Differential Equations for Engineers, by Jiri Lebl

Prof: Dr Dan Coombs

Dr Dan Coombs has great British accent, and a wry sense of humour which helps to keep interest in the class. He tries to balance between tolerating conceptual questions and making progress in the more recipe-oriented curriculum. He spent a lot of effort restructuring the curriculum to be based on Linear Algebra, so as to make the class more conceptual and slightly less “formula-up-my-sleeve” math, though it still is.

Difficulty

The homework is really exhausting. The hand calculations have awful numbers in them, making them really tedious. The Matlab is … Matlab. As a CS student I thought Matlab would be a breeze, but that was not the case, as the language has a lot of quirks. The number of questions in a homework set is a lot considering the time each one takes. With the exception of the first homework, where we were given real world problems and had to come with models for them, I didn’t feel I got a lot out of the homework, except learning a few random facts about Matlab after trial and error.

Key Concepts

Modelling nature as a differential equation

First order linear equations

Linear systems of differential equations

Laplace transform

Non-linear systems

Hard Concepts

Partial fractions: Thought they were pretty easy, but had a really gross one on the final

Non-linear classification of fixed points: Can get a bit confused between different fixed points

Classification of 2nd order linear systems: If you don’t want to re-derive them, need to be able to recall them quickly.

Conclusion

Homework was a schlep. Interesting topic, but recipe-driven curriculum almost kills it. IMHO, focus should be modelling natural phenomenon. The problem with the recipe driven approach, even for non-math students, is that (1) Engineers will probably just use Wolfram/computer system to solve it anyway. (2) While it might be helpful for them to classify what can/cannot be solved etc, odds are if it is non-linear you will try your luck, or use a linear approximation anyway.

# Course Review: MATH 120

Honours Differential Calculus

“For any topic in mathematics, you just need to do a few ‘cute’ problems. Otherwise, your mind starts ‘turning'”

Text: Robert A. Adams and Christopher Essex: Calculus: Single Variable, 8th Edition, Pearson, Toronto, 2013.

Prof: Dr. Yue-Xian Li

The thing that I’ll remember the most from Dr. Li’s class were his bizarre one-line quips including:

“You should have asked me that question weeks ago! You are going to fail the midterm tomorrow!”

“How can you confuse a “gamma” with an “r”? The “gamma” looks like the thing you use to hang people!”

“I have some  videos I could show you… people driving on a bridge while it is collapsing… some really great videos”

“There is some theory behind this, but I don’t really understand the theory… let me show you how to get the solution though”

“My third-year students keeping making this mistake, but you guys are honours, so you shouldn’t have a problem”

Dr. Li is actually a mathematical biologist. He was alots of fun and very passionate. He was also really nice during office hours. The best part of his classes were when he deviated from the textbook a bit. That said, he took most of his examples  directly from the assigned readings in the textbook.

Difficulty

I am sure most people who takes this class are a bit apprehensive about the word honours. Is that apprehension justified? I am not so sure. I think that the key to this course is identifying the handful of additional/challenging concepts early on, and being prepared to put some extra effort into them. Both the midterms and the final were really fair, since the prof gave us very similar practice exams. I found the last few questions on the weekly homework pretty hard, though. I often had to think over the questions for a few days to understand them. I also tripped on an optimization question in the final, but I think that was just me…

Key Concepts

Limits

Derivatives

Mean-Value Theorem

Continuity

Hard Concepts

Epsilon-delta Definition of Limit: Arguably  the hardest concept to understand quickly in the course. Make sure you understand the “logic” of it before diving into the math.

Proofs: Proofs often involving mean value theorem on tests, but anything course-related for homework.

Differential  Equations: Lots of Physics often involved, could be hard if that’s not your strongest subject.

Derivatives of inverses: Some of these questions can be computationally difficult and you can get really confused if you don’t notice that a function evaluated at its inverse is x.

Continuous and Differentiable Functions: Often involving piecewise functions with parameters. Its best to go back to definitions when dealing with either or both of these properties.

Chain rule: If you don’t like computation, some chain rule questions could bring you down. Its just a matter of practice and accuracy though. Not really smarts.

Optimization and Related Rates; If geometry’s not your strongest point, these could be a challenge. Once again, practice is key. ( I didn’t practice optimization enough for my final)

Conclusion

Great course, but be prepared to put in extra effort.