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Spectacular Books

Mathematicians suck at writing. This is part of the reasons why they went into math in the first place, because they suck at writing. Sucking at writing doesn’t mean that mathematicians don’t write books though; in fact there are tons of math books written by mathematicians.

The problem is most of them suck.

Half of them are for geniuses written by geniuses.

The other half are for geniuses written by borderline geniuses.

By far my two favorite books are William’s Probability with Martingales and Trefethen’s NLA book. Both British authors. Both written in a highly colloquial style.

It’s really too bad academics have this ego-stroking urge to write to the highest denominator rather than to, say grad students or undergrad.

God damn.

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The Tower of Babel

I initially intended to blog to make tons of money. That didn’t pan out well, so I’ve changed my goals to things more manageable. For one, it’s a tool for me to survey my mental fortitude at the time of writing. Posts tend to get depressing as I go through rough phases of life. Otherwise, it serves as a tool to memorialize my life.

Recently, I’ve realized my writing skills are decaying. Forcing myself to write helps to sustain what I’ve built up in my undergrad years. But this is all in English, and does little to help with communicating in Chinese. Truth is, the vast majority of my family only really speaks Chinese. Isn’t it ironic that the children of immigrants gradually lose the language that connects them with their roots?

I’m glad there are others who voice this opinion. I hope your rest is peaceful.

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Media and Entertainment

Two notes

  1. Just finished “The Things We Wish Were True,” which was one of the Kindle first selection. In a lot of aspects, it was very similar to A Spool of Blue Thread: white, American families with a lot of complexity lying behind their histories. The two books used those secrets to advance the plots, which mostly consisted of… fairly typical stuff in the case of TTWWWT, with a small twist at the end.

    Overall, the book just seems to move along so slowly with a writing style that’s not as beautiful as I wanted it to. Oh well.

  2. Batman Lego is pretty hilarious…
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Schur Complement and Minimal Energy Extension

(Note: this post is mainly for me to consolidate my thoughts)

In the framework of domain decomposition, consider creating the Schur complement which orthogonalizes interior nodes and the edges/vertex nodes. It turns out the norm of these functions which are orthogonal to the interior functions are minimal energy (i.e. L2 norm) extensions.

This can be seen in both a Hilbert space way or an optimization way. For the optimization way, write out the product for the mass matrix, and note that we can take a derivative to minimize one of the factors… now the Schur complement pops up naturally!

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Winter

Out of the bosom of the Air,
      Out of the cloud-folds of her garments shaken,
Over the woodlands brown and bare,
      Over the harvest-fields forsaken,
            Silent, and soft, and slow
            Descends the snow.

Even as our cloudy fancies take
      Suddenly shape in some divine expression,
Even as the troubled heart doth make
      In the white countenance confession,
            The troubled sky reveals
            The grief it feels.

This is the poem of the air,
      Slowly in silent syllables recorded;
This is the secret of despair,
      Long in its cloudy bosom hoarded,
            Now whispered and revealed
            To wood and field.

– Henry Wadsworth Longfellow
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Referencing Copies

A lot of times in numerical methods, I need to have a temporary variable as a time stepping tool or as an “incrementing” device without altering the original variable. A code snippet like

copy_u = u
for i in range(len(copy_u)):
    copy_u[i] = f(u)

But I gotta be more careful. There’s a deep difference between making a copy of a variable, and just creating a reference to a variable. Any change to copy_u might change u itself! Use np.copy or the copy module in Python!

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Simple Verification of Mass Matrix

Verifying a stiffness matrix isn’t too hard. See the following PDF file. Otherwise, once a stiffness matrix have been verified, the mass matrix can be too with a few lines more code.

Instead of a Poisson problem, discretize the heat equation such that the steady state is our Poisson problem with an easy time stepper (say Euler). Note that we will have to invert our mass matrix at each step and have a mass matrix multiply. Then we should see that the solution dotted with the force vector approaches the same value as above.

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Musings

As it turns out, the DG homework wasn’t that ridiculous. If you stripped everything down, and use a little bit of built-in numerical quadrature, then it was actually pretty fun. It turns out that programming all these solvers naturally lends itself to a abstract class with abstract methods paradigm with a few main functions (i.e. solve) that seems to be shared between all the problems.

On another note, I really fucked up meatloafs today. Didn’t season well (bland as ever) with too little bread and zero eggs (whoops, lesson learned). In the end, it was a bland piece of beef which fell apart very easily.

Finally, my god was the Aussie final spectacular. Federer’s backhand was immaculate, arguably better than 2006-2010 levels. Nadal’s passing shots were insane, forcing Federer to hit from the baseline.

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Kubo

I’ve wanted to watch “Kubo” for a few weeks now, but with the recent gif popping up… it just makes it so much cooler. How many people in the world have the technically and modelling know-how to do this… it’s mind blowing.

View post on imgur.com

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Bernstein Polynomials

This post won’t have too much math, but mainly musings.

After a four month hiatus from research, it seems my dive back into the world of unknown math is being squandered by my ineptitude. My documentation of the progress made, and the comments in my code seems to be greatly lacking. I’ve spent a good chunk of time trying to recall progress.

I think one of the cooler things Bernstein polynomials can prove is that polynomials are dense (uniformly too IRCC) in continuous functions on a bounded domain. Durrett has a proof in his book using a probabilistic view, which seems much cleaner than an algebraic/analysis proof such as here.