Euler's Method...

From Notes on Diffy Qs: Differential Equations for Engineers, by Jirí Lebl

"What if we want to find the value of the solution at some particular x? Or perhaps we want to produce a graph of the solution to inspect the behavior. In this section we will learn about the basics of numerical approximation of solutions.

The simplest method for approximating a solution is Euler’s method. It works as follows: We take x0 and compute the slope k = f (x₀; y₀). The slope is the change in y per unit change in x. We follow the line for an interval of length h on the x axis. Hence if y = y₀ at x₀, then we will say that y₁ (the approximate value of y at x₁ = x₀ + h) will be y₁ = y₀ + hk. Rinse, repeat! That is, compute x₂ and y₂ using x₁ and y₁." See Notes on Diffy Qs above (under graphic)

Topics: Differential Equations, Diversity in Science, Mathematics, Women in Science

Okay, this is the LAST time I'll talk about Hidden Figures (although I did order the book).

Not to spoil it for you, but Dr. Katherine Johnson (played by Taraji P. Henson) mentioned an "old method" of mathematics. What both the actress and NASA scientist referred to is something you're taught usually sophomore year in a STEM major. Euler's Method is named after Leonhard Euler, and it's used to numerically approximate differential equations, something in the movie and the embed below alludes to is now done by what we now know as computers (the laptop kind, not female mathematicians).

It is important to understand the steps, derivation and mathematics behind computer calculation. How do you KNOW it's right? I'm often challenged as to "when I ever use Calculus" at work. Most often they're right, I don't. There's a software package designed with the equations embed within them to literally SPIT out an answer. The program doesn't have imagination nor does it visualize an expected end result. "The answer" is the end of a calculation without any notion of its consequences if incorrect.

Part of its practicality is essentially how the study of mathematics and physics organizes one's thinking. I use systematic approaches to solving just about any problem in life. However in Hidden Figures, it was initially the NASA scientists and eventually Dr. Johnson knowing the mathematics and relying on human insight and intuition that averted catastrophe, not that it doesn't happen when launching humans on the top of essentially systematic staged bombs to achieve Earth orbit.

The old riddle "which came first: the chicken or the egg?" can easily be answered with regards to computers and humans. The Singularity will have a ways yet.

François Arago said of him (Euler) "He calculated just as men breathe, as eagles sustain themselves in the air" (Beckmann 1971, p. 143; Boyer 1968, p. 482). [1]

In a testament to Euler's proficiency in all branches of mathematics, the great French mathematician and celestial mechanic Laplace told his students, "Liesez Euler, Liesez Euler, c'est notre maître à tous" ("Read Euler, read Euler, he is our master in everything" (Beckmann 1971, p. 153). [2]

1, 2: Scienceworld.Wolfram.com: Euler
LA Times:
Meet the ‘Hidden Figures’ mathematician who helped send Americans into space, Amina Khan

More Terrifying Than Foreigners...

Image Source: CarloAlberto.org

Topics: Calculus, Differential Equations, Diversity, Politics

I think someone created a meme on Facebook from this incident. A little poking around the Internet revealed this WaPo article:

On Thursday evening, a 40-year-old man — with dark, curly hair, olive skin and an exotic foreign accent — boarded a plane. It was a regional jet making a short, uneventful hop from Philadelphia to nearby Syracuse.

Or so dozens of unsuspecting passengers thought.

The curly-haired man tried to keep to himself, intently if inscrutably scribbling on a notepad he’d brought aboard. His seatmate, a blond-haired, 30-something woman sporting flip-flops and a red tote bag, looked him over. He was wearing navy Diesel jeans and a red Lacoste sweater – a look he would later describe as “simple elegance” – but something about him didn’t seem right to her.

She decided to try out some small talk.

Is Syracuse home? She asked.

No, he replied curtly.

He similarly deflected further questions. He appeared laser-focused — perhaps too laser-focused — on the task at hand, those strange scribblings.

Skipping further into the article:

And then the big reveal: The woman wasn’t really sick at all! Instead this quick-thinking traveler had Seen Something, and so she had Said Something.

That Something she’d seen had been her seatmate’s cryptic notes, scrawled in a script she didn’t recognize. Maybe it was code, or some foreign lettering, possibly the details of a plot to destroy the dozens of innocent lives aboard American Airlines Flight 3950. She may have felt it her duty to alert the authorities just to be safe. The curly-haired man was, the agent informed him politely, suspected of terrorism.

The curly-haired man laughed.

He laughed because those scribbles weren’t Arabic, or another foreign language, or even some special secret terrorist code. They were math.

Yes, math. A differential equation, to be exact.

Had the crew or security members perhaps quickly googled this good-natured, bespectacled passenger before waylaying everyone for several hours, they might have learned that he — Guido Menzio — is a young but decorated Ivy League economist. And that he’s best known for his relatively technical work on search theory, which helped earn him a tenured associate professorship at the University of Pennsylvania as well as stints at Princeton and Stanford’s Hoover Institution.

Guido Menzio is a recipient of the prestigious Carlo Alberto medal, given to the best Italian economist under 40.

Did I mention he's Italian (not Arab/Near Eastern)? There are myriad other ethnic groups with "swarthy" complexions on the planet. Besides, here in the US, the threat of terrorism statistically may be of the homegrown and non-swarthy variety.

Majoring in a STEM field yields more than its own stereotypes: dorky, nerdy, socially awkward (I'm talking to you, "Big Bang Theory"), unattractive, but TERRORIST? I admit, the course we all affectionately called "DIFFY Qs" was daunting, but it didn't rewire any of us for violence. Have we gotten this addled in the brain?

So, in the era of not wanting actual experts in government or well, ANYTHING; in the era of racism, xenophobia, bombast, blatant lying, homophobia and misogyny as crass, carnival-barking political tactics (applicable when you've subordinated the processes a republic uses selecting its leaders into "reality TV," from a public caricature having more in common with "A Pimp Named Slickback") appealing to our lesser angels, we are SHOCKED that mathematics has literally become "Thoughtcrime"? Instead of a "shining city on a hill," we've become a dung heap infested with maggots in a junkyard - a joke! "Idiocracy" as a comedy was placed 500 years hence. That Apocalypse is now. All the hand-wringing about this being a strange election cycle is now solved. After telling Gorbachev to "tear down that wall," we're apparently intent on building another more lasting monument in the thick mortar paste of breathtaking stupidity. Sadly, the minions of this effort are PROUD of who and what they DON'T know. For them, ignorance is not just bliss: it's a belief system. And, you cannot explain science, mathematics, history, the economy or climate change... to a cult.

Rising xenophobia stoked by the presidential campaign, he (Guido) suggested, may soon make things worse for people who happen to look a little other-ish.

Washington Post:
Ivy League economist ethnically profiled, interrogated for doing math on American Airlines flight
Catherine Rampell

Blog break for ten days. Back on 23 May.