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Notes: Modern Algebra I

I took Modern Algebra I with Professor Matthias Beck in Fall 2015. These comprehensive notes were compiled using lecture notes and the textbooks,

Disclaimer: my notes are meant to be a toolbox while doing proofs and studying/practicing the course in general. There may contain typos or mistakes. Please feel free to let me know if you find any errors!

Topics Covered:

• Integers & the Euclidean algorithm
• Complex numbers, roots of unity & Cardano’s formula
• Modular arithmetic & commutative rings
• Polynomials, power series & integral domains
• Permutations & groups

Featured Image: Dodecahedron-Icosahedron Duality
Credit: Images from Algebra: Abstract and Concrete by Frederick M. Goodman

Notes: Mathematics of Optimization

I took this course with Professor Serkan Hosten in Fall 2016. These comprehensive notes were compiled using lecture notes and the textbook, Optimization Models by Giuseppe Calaore and Laurent El Ghaoui, Cambridge University Press.

Featured Image: Analysis explanation of why when optimization a linear objective function over a convex shape will lead to a optimal solution on the boundary of the feasible region.
Image Credit: Figure 08-19 in Optimization Models by Giuseppe Calaore and Laurent El Ghaoui, Cambridge University Press.

Disclaimer: my notes are meant to be a toolbox while doing proofs and studying/practicing the course in general. There may contain typos or mistakes. Please feel free to let me know if you find any errors!

Syllabus:

1. Optimization Models: modeling optimization problems as linear, quadratic, and semidenite programs,
2. Symmetric Matrices: using spectral decomposition of symmetric matrices for positive denite matrices and their role in optimization
3. Singular Value Decomposition: computing SVDs in the context of linear equations and optimization,
4. Least Squares: solving systems of linear equations and least squares problems,
5. Convexity: identifying key properties of convex sets and convex functions for optimization,
6. Optimality Conditions and Duality: developing criteria to identify optimal solutions and using dual problem, in particular, in the context of linear models, Linear and Quadratic Models: employing the geometry of linear and quadratic models for solution algorithms,
7. Semidefinite programs: modeling certain convex optimization problems as semidefinite programs

Fun fact: The subtitle for the site, “#teamnosleep” originated from my study group in this class. Homework assignments were intensely hard.

More on Website Infrastructure

After reading a bit on the WordPress.com support forums, I’ve learned that you can actually embed an entire PDF into a blog using google docs!

I’ll definitely be making posts soon that would embed some of my math notes, and experiment with how this site works. This seems common enough that I think I’ll continue sticking with google docs, which resolves an choice I mentioned in a previous post.

#caffeinated

Reflection: Following My Passions

When I was in my late teens to early twenties, I knew I liked math but I wanted to try everything that would require application of math: I jumped from applied math to physics to engineering. I loved the process of learning, and I took a lot of different classes, and by the time I was halfway through the second semester of Mechanical Engineering courses, I finally realized that I was only really interested in the math, and talking about the math.

Then today, I went digging through my time-capsules on the internet. I have blogs scattered across a lot of different platforms, and I found this post over on Hubpages that I wrote in 2009. I’m pretty sure this reaffirms that I’ve always wanted to teach math.

Looking back, I’m glad that I took a long, winding path. I needed to grow a lot spiritually and emotionally before I was ready to take on teaching. Hopefully I’ll maintain my capacity for growth in the upcoming years.

Teacher Evaluations & Reflection, Fall 2016

Teacher Evaluations are out at SFSU. 😀

From my students’ responses, I learned that I can improve in the following ways:

• Plan what I will write on the board in more detail instead of such a rough sketch,
• “Don’t let nerves cause mistakes” – definitely happened 2-3 times where I did a problem incorrectly because I tried to wing it on the board…
• More intensive examples that can tie different concepts together before the midterm (where they do see synthesized word problems),
• On Universal Design:
• Group work that involve manipulatives, geared for kinesthetic/tactile learners,
• Audio / Visual learners balance – I tend to write a conclusion and verbally say a paragraph of explanations.
• On Long Term Planning:
• More group work for inverse trig functions and beyond,
• Maybe building a story that can be used for the concept questions during class?
• Manage expectations earlier – students will need to work and figure out a lot of stuff on their own,
• Create systems that can help students organize all the information – give suggestions on how to take notes, maybe?
• Give more time to do Chapters 4-6, Trigonometry chapters of the book.
• On Class Policy:
• Attendance and participation should be recorded more in detail,

Looking forward to teaching next semester! I will teach one class of precalculus and TA one section of Calc II. I wonder how different TA’ing for Calculus II will feel. 🙂

So I wanted to look up some terms that I will commonly use on this blog over at Google Trends. I feel like I rediscovered the moon cycle…

The data shownabove compares the number of searches  of the terms algebra and geometry over time within the U.S., for the last 90 days, or approximately 3 months. Reading closely, we see that people search the terms “algebra” and “geometry” in a periodic way. Specifically, on the weekends, the amount of searches went down.

Great, now I have figured out that people don’t like to do homework on Saturday/Sunday… Well, we haven’t rigorously tested this theory with sufficient amount real data, since there’s no sample size or even an attempt at any calculation besides looking at a graph.

But, based on context clues I will bet that the null hypothesis would be rejected.

Website Infrastructure

The notes page is set up, next I aim to finish up organizing the precalculus resources and general non-math resources.

Now, the site contains links all my notes starting from Fall 2015, which send visitors directly to my google drive, but I wonder if that is secure/wise.

Maybe there is a better way to both host the files and allow for instant updates as I compile my LaTeX files. Right now, this blog links to individual files within my google drive which is synced to a directory on my laptop. When I compile on my computer, I work directly in that directory so that the file linked in google drive is automatically synced. I want the same functionality and convenience but I’m not sure if linking to the drive is secure, since it’s a personal account.

Time to do some non-math research~

Update: I hear that wordpress has great support, so I made a post at the forums. I wonder if there’ll be an easy solution?

Let $\mathscr{H}$ be the set of all humans.

Definition: If an element $x$ “cares for” a set $S$, it “cares for” every element in the set.

Remark: If you are a human, and you “care for” humanity, then you must “care for” yourself too.

Proof of Remark: Let $x$ denote you. $x \in \mathscr{H}; x$ “care for” $\mathscr{H} \implies x$ “care for” $x$.

Why a Blog?

So, why start a blog when there is approximately… $1.5 \times 10^{6}$ posts created per day? (Checked @ 8 AM 1/8/17)

For me, this website is a bit more than a collection of math notes and lists of resources. I am aiming to chronicle the challenges of being a student and a teacher at the same time. On the other hand, I want to also document the most fun parts of being in an intensive graduate program!

So this blog will be part survival guide, part chicken-soup-for-the-soul, and part scrapbook.

Oh. By the way, I survived the first semester of graduate school!