# 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 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.

# Short Story: The Ideal Gift

Mr Pedro Sears was not happy. His manservant had just informed him that his chauffeur would be unable to take him to his friend Robert’s birthday party. Unfortunately, the car was still at the garage for repairs. This meant he would have to take what was euphemistically known as public transport. Pedro’s frail frame shivered involuntarily at the prospect.

As he proceeded down the wide-ish staircase to his front door, Pedro glowered at a rather dull painting of a bowl of supposedly “fresh” produce. In that instant, that painting had come to represent all his annoyances with his present predicament. The painting, quite out of character, glowered back at him with a ferocity that forced Pedro to look away. Pedro reached the front door. His manservant enveloped him in a thick black trench coat as he burst out into the almost equally thick, piercing rain.

Despite the downpour, Pedro was not the sole pedestrian that day en-route to the bus stop. A burly man in an attire that would have been more appropriate in the Bahamas, seem to be of the same purpose. Though he was a few blocks behind Pedro to start, he soon caught up with him and started walking in step. Pedro eyed the water dripping off the man’s khakhi trunk shorts suspiciously, before crossing over to the other side of the road.

Since he was on the wrong side of the road, Pedro only just caught the bus. After a mad sprint across the middle of the road he arrived just in time for the bus. He also arrived just in time for a muddy green tsunami that coincided with the bus’s traversal through the large puddle near the bus stop. “Eh…eh…eh!” Spluttered Pedro, as he felt the liquid seep through his formerly impenetrable coat and trickle down his spine. He staggered onto the bus, paid for his ticket, and headed for the sole vacant seat, that was covered by a greenish shawl. Pedro gave a large sigh of relief as he lowered his rear into it. The bus started move. At last he could relax.

And that was when he sat on the baby.

The amorphous green blob that Pedro had mistaken for a shawl proved to be particularly spongy. Yelping as if he had been stung, Pedro leapt to his feet. After regaining his senses and unfixing his clasp from his lower back, he looked around. He could not identify a likely mother, in what appeared to be bus load of rowdy university students. Feeling a sense of guilt at having concussed the unfortunate toddler, he decided to pick up the baby and take him to the bus driver, who could contact the relevant authorities.

Just then, the bus driver slammed on the brakes and Pedro lurched into the group students, who were now exiting the bus en masse. Still clutching the child, Pedro waved his unoccupied arm frantically, but it was no good. He was swept through the door and landed face first in the unforgivingly hard pavement. Pedro, or rather, his face, remained attached to the moist concrete for several minutes, as his body performed emergency repairs and the shawl carrying the baby started to roll down the nearby slope.

With excruciating pain, Pedro managed to unglue his face from the pavement orient it more favourably.  After affirming both his own and his unconscious companion’s continued existence in this realm, Pedro looked around in a daze, trying to simultaneously ascertain his location and recall the series of events that could have brought him here. Out of the corner of his eye, he spotted a neon sign in the familiar murky shade of green that seemed to be following him that day. Half in a trance, he proceeded towards the discount grocery store beneath the sign, half-dragging his new, blobby-green friend.

Pedro, at this point, wasn’t very sure why he was entering the store, but he vaguely recalled that he had intended to purchase a gift for some party he was attending. Upon entering, a giant snake of harried looking shoppers blocked his way. They all wore a hollow, scaly expression that Pedro was almost led to believe that they were in fact enacting a mime of a play featuring an anaconda. After slithering through the queue, he began to realize that this might not be the place for his purchase. However, upon noticing a huge display declaring “Our Gift to You!”, he decided to proceed further into the store.

Under the display, there was a barrel full of the greenest cabbages Pedro had ever seen.

Pedro cracked up. He lowered his free hand into the barrel and began to stroke the smooth flesh of the nearest cabbage. The cabbage seemed to absorb all Pedro’s woes in its succulent leaves. Pedro made a decision. He grabbed the cabbage and, still fondling it, joined the human snake, grinning from ear-to-ear.

After exiting the store, Pedro sped of down the street at top speed. He had asked the cashier for his location and was surprised to find that he was within two blocks of Robert’s residence. He didn’t even notice the mother who, after seeing Pedro in his swamped attire, had crossed to the opposite side of the road.

At long last, Pedro reached his destination. He heard the familiar voices of his friends as clambered up the cobbled stairway to the front-door. He rang the bell and after a few moments, Robert’s manservant opened the door.

“Hi…” Pedro said, his eyes shining.

The manservant eyes darted quickly from cabbage that Pedro was stroking nervously, to the baby swinging from his other hand. He took in the several layers of filth that coated Pedro’s skin and his soaking clothing. Finally, he stared right back into Pedro’s manic gleaming eyes and closed the door with a flourish.

Pedro stood at the door slightly dazed. He overheard the manservant explaining to Robert. “Just a tramp, sir. Trying his luck I suppose…”

He sat down on the stairway, placing the baby and the cabbage beside him. The baby had woken up and had started to gurgle. Pedro’s face broke into a relaxed smile. Resting either hands on the cabbage and the baby he slumped his back on a nearby banister. He had got the ideal gifts.