Wednesday, May 24, 2017

Social Media Math Memes

There's a simple math meme going around Facebook that asks people to find the next element in a sequence—if we're being honest, there are lots of these going around. This one, like most of the others, is fairly straight forward and easily solvable by a brute force attack. Of course, as a mathematician, I took the problem a bit further by finding a function that would work for any value of $n$, not just the one the meme was asking for.

I'll give you both the brute force solution and mine as well later in the post. What I want to talk about first is why memes like this are both important and valuable tools.

Meme Magic

Consider what the response would be if I posted a problem on social media that went something like this... "Given the sequence $5,12,21$, find the next two elements."

If you guessed that most people would pass over my post without a second glance, you would be right. However, if I were to restate the problem and put it in a meme then people will jump all over it. Why?

Before we answer that, lets look at how the problem was stated in the meme versus how I stated the problem above. 




If the goal is to get people thinking about math—and even working math problems for fun—then a meme like this is far more effective than stating the problem like it came out of a textbook. There are two things that make this meme effective. My original statement of the problem made it seem difficult, while the restated version seems simple.

The meme also offers a challenge, “97% Will fail this test.” While the statistic is likely inflated, the sentiment holds—most people will fail to solve the problem. Judging by the comments on the original post, it seems closer to 75% failing, but that’s still high for such a simple problem.
 
The good thing is that the meme not only got people thinking about math and working on a math problem, but also that it sparked discussions about math. But, I’m also interested in meme’s like this because they shed light on some actual problems in the United States—and probably around the world—when it comes to math education and attitudes about math.

The Brute Force Solution

Among those who got it wrong, most did so because they missed a subtle point in the problem—there’s a missing term. There should be a $4+7=32$ line between the $3+6=21$ and the $5+8=?$ line. By missing this term, those who were on the right track failed to solve the problem because they calculated the following… $$5+8=5+8+21=34$$
The brute force solution is fairly straight forward. For each step we increase the first term by one, then increase the second term by one, and finally add the previous result. Thus we get…
 
$$1+4=5\\2+5=2+5+5=12\\3+6=3+6+12=21\\4+7=4+7+21=32\\5+8=5+8+32=45$$ 
Many of those who failed saw the basic pattern, but failed to account for the missing term.

What the Wrong Answers Tell Us

What really prompted me to write this blog post was not those individuals who failed to solve the problem because they missed the $4+7$ term. The motivation for this post came from those who disagreed with the problem in the first place. These are the people who answered $5+8=13$, and usually with quite a bit of attitude.

Mathematics teaches us to think outside the box. It teaches us to look at the world a bit differently. So much so that mathematicians have been accused of practicing witchcraft in the past.

The good Christian should beware of mathematicians and all those who make empty prophecies. The danger already exists that mathematicians have made a covenant with the devil to darken the spirit and confine man in the bonds of Hell. – Saint Augustine 

It seems that attitude may well be alive and kicking in the US still today. There is a general suspicion of modern mathematics that may not be as widespread as suspicions surrounding evolution, but one that exists none the less. Don’t believe me? Then consider the following quote

Unlike the "modern math" theorists, who believe that mathematics is a creation of man and thus arbitrary and relative, A Beka Book teaches that the laws of mathematics are a creation of God and thus absolute....A Beka Book provides attractive, legible, and workable traditional mathematics texts that are not burdened with modern theories such as set theory. — Abeka.com 

There are so many things wrong with this quote and this attitude that it would take an entire post to discuss them all. For the moment it’s enough to point out that whoever wrote this knows precious little about mathematics, yet is somehow instructing generations of children in mathematics. The net result of this line of thinking is truly sad—people who will never be able to see the beauty in mathematics, because they were taught anything beyond the basics “is a creation of man and thus arbitrary and relative.”

That’s not something that I was ever taught, neither as an undergraduate nor as a graduate student. In fact, we are taught exactly the opposite. I can’t count the number of times I heard that “mathematics is the only body of knowledge that we can be 100% certain is completely true,” or something equivalent. Math is true whether or not you believe it. The integers are a ring but not a field—it doesn’t matter who you are, or what you think that fact remains a fact. There’s nothing arbitrary or relative about it.

Of course, the meme version of the problem is trying to make the problem accessible to everyone by assigning a new meaning to $+$ in this particular context. Which leads us into the general solution to the problem.

The General Solution to the Meme Problem

Anyone with a bit of mathematical sophistication can see that $+$ in the meme problem can be restated as a function $S:\mathbb{N}\to\mathbb{N}$. So now the problem looks like this…

$$S(1)=5\\S(2)=12\\S(3)=21\\S(4)=32\\S(5)=45$$
We’re still at the brute force level, but the problem looks more mathematical. 
The next step is to discover the function that making this thing tick. Notice that I said “discover” not create.

I know a couple things about this function. First of all, it’s recursive, i.e. to find $S(N)$ we have to first know $S(N-1)$, and to find $S(N-1)$ we must first know $S(N-2)$ and so on. We also know that for any $N\in\mathbb{N}$ we define $S(N)$ by adding three terms together.

From our brute force solution, we can quickly identify two of the three required terms.

$$S(N) = N+x+S(N-1)$$
A bit of basic observation shows us that we can define $x$ in terms of $N$ like this $x=N+3$ so that our function now looks like this…

$$S(N) = N+(N+3)+S(N-1)=2N+3+S(N-1)$$
But what about $S(1)$? It’s impossible to calculate $S(0)$ so we set it to what it must be, $S(0)=0$.
 
Finding the general solution is a fun exercise, and allows us write a simple computer program to calculate $S(N)$ for any $N\in\mathbb{N}$, if we were so inclined.

Summing it Up

Problems like this are a great way to get people who would never spend their time working on math problems, to work on math problems. But they also uncover a general suspicion from the public towards mathematics as well as the usual issues with mathematics—hatred of math, math is too hard, math is useless, etc.
 
Better education is certainly a first step, but it’s a dead horse we’ve been beating since I was in school. We need some fresh ideas. What can we do to overcome these issues?

Sunday, November 20, 2016

Men Don’t Read (Fiction)


This is version 2.0 of this post. After writing the first version where I pointed to the predictability, low ROI, and treating men like idiot I realized that I make a mistake. Most modern fiction is predictable and offers low ROI for most men. Likewise, much modern fiction puts men in supporting roles, even when they are the main characters, and talks down to us like we’re idiots. All my original points remain valid, but my mistake was taking them as the cause not the symptom. The cause is bigger, and the challenge to get men back into reading fiction is even bigger.
“Men don’t read (fiction) because the publishing industry marginalizes us.”
Women dominate the publishing industry. In a 2011 article in the Huffington Post, Jason Pinter points out “the fact that most editorial meetings tend to be dominated by women. Saying the ratio is 75/25 is not overstating things.” With a female dominated publishing industry it makes perfect sense why men don’t read as much as they did in the past.
My grandpa read fiction—I would say only but I seem to remember him reading a few biographies over the years. Most of these books were mystery, crime, and thriller novels; few of which would find their way into bookstores today. They almost always had a strong male lead character who thought and acted like a man. One of the most frustrating elements for me when reading Harlan Coben’s Tell No One was that the lead character (Dr. David Beck) didn’t act or think like a man. Throughout the book he was reliant on a cast of strong female character, and the one time he decided something on his own it was a disaster.
Men don’t want to read about how stupid and dependent on women we are. It’s no surprise that a female dominated industry would miss the mark by so much when attempting to market to men.
“Men read. Tons of them do. But they are not marketed to, not targeted, and often totally dismissed.” – Jason Pinter
As a man, I like strong female characters in books, but why do they have to come at the expense of the male characters. One of my favorite genre’s of music is Symphonic Metal, and if you know anything about Symphonic Metal you know that a strong female vocalist fronts the band most of the time. Despite this, Symphonic Metal—like all sub-genre's of metal—appeal to men at a higher rate than to women. Men aren’t opposed to strong women; we’re opposed to weak men.
The publishing industry seems oblivious to this concept, which is hard to understand since the Huffington Post article appeared in 2011, yet nothing has changed. Authors write books, particularly fiction, for women. Publisher market books to women. Where do men figure in the mix? The publishing industry marginalizes men with the self-perpetuating lie that men don’t read.
“Men aren’t opposed to strong women; we’re opposed to weak men.”
There’s a golden opportunity here for men to independently publish books interesting to men. If we present men with both fiction and nonfiction that speaks to men and our interests, I think we’ll be surprised at how many men start reading again. We have to solve this problem ourselves, we can’t wait for the (declining) traditional publishing world to solve it for us.

Wednesday, October 12, 2016

Time-Independent Schrödinger Equation in One Dimension

Time-independent Schrodinger Equation in one dimension


Schrödinger's Equation is one of the most important equations in modern physics. It's also fairly easy to derive (in one dimension) with some basic Calculus and a few fundamental concepts from Newtonian Mechanics and modern particle physics. So that's what we'll do here; derive the time-independent version of Schrödinger Equation in one dimension.

The approach I take in this post mimics the one taken here with additional details filled in since some observations made are not immediately apparent to non-physicists.

Preliminaries

There are several axioms and basic results we need to know in order for the rest of the derivation to make sense. We'll start with three basic relationships;

$E=KE+PE\\
E=hf\\
p=\frac{h}{\lambda}$

$h=6.62607004 × 10^{-34} m^2 kg / s$ is Planck's constant.

$f$ is the frequency of the wave, $\lambda$ is its wavelength.

$\omega=2\pi f$ is the wave's angular frequency.

We define the angular wavenumber as $k=\frac{2\pi}{\lambda}$.

We will also need the reduced Planck's constant $\hbar=\frac{h}{2\pi}$.

From Newtonian Mechanics, we remember that $KE=\frac{1}{2}mv^2$ where $m$ is the mass of the object and $v$ is its velocity. We also recall that we can write potential energy as $U$, thus transforming the energy equation above into;

$$E=\frac{1}{2}mv^{2}+U$$

We can write momentum a couple different ways. What we see above, $p=\frac{h}{\lambda}$ is de Broglie’s hypothesis. From Newtonian Mechanics, we recall that $p=mv$. It's worth noting that $E=hf$ is Planck’s hypothesis for the energy of a photon.

It will be useful later to have the energy equation written as $E=\frac{p^2}{2m}+U$. We can do this by starting from the classical equation for momentum as such;

$$p=mv\\
p^2=m^2v^2\\
\frac{p^2}{2m}=\frac{m^2v^2}{2m}\\
\frac{p^2}{2m}=\frac{mv^2}{2}$$

Choosing a Wave Function


The final preliminary piece we need is a wave function written in this form;

$$\Psi=e^{i(kx-\omega t)}$$

To derive this function we're going to remember Euler's formula

$$e^{ix}=\cos{x} + i\sin{x}$$.

If we just look at $kx$, then we see that $\cos{kx}=\cos\big(\frac{2\pi}{\lambda}x\big)$. If we take $\lambda=1$  we see this function looks like figure 1.

Figure 1 - $g(x)=\cos\big(\frac{2\pi}{\lambda}x\big)$

This is a standing wave and we need a wave function that takes time into account as well. But why do we choose to use $kx-\omega t$ in Euler's formula?

Let's start with the velocity of the wave. Like a lot of things in physics, being able to write velocity in a couple different ways turns out to be immensely useful. In this case, we know we can write the velocity as the distance $x$ over the time $t$ to get $v=\frac{x}{t}$. In terms of waves, we can also write the velocity as the frequency $f$ times the wavelength $\lambda$ to get $v=f\lambda$.

We need to be clever, so we're going to multiply this second version of the velocity equation by $\frac{2\pi}{2\pi}$ to get

$$v=\frac{2\pi f\lambda}{2\pi}$$

Then if we bring $\lambda$ to the denominator, we can rewrite $v$ as

$$v=\frac{2\pi f}{\frac{2\pi}{\lambda}}=\frac{\omega}{k}$$

Now, we get

$$v=\frac{\omega}{k}=\frac{x}{t}$$

So that

$$\omega t=kx\implies kx-\omega t=0$$

Thus $\cos(kx-\omega t)$ gives us a moving form of the wave we found in Figure 1, which is, obviously, $1$ whenever $\omega t=kx$.

Heading back to Euler's formula, and using $kx-\omega t$ we get

$$\cos(kx-\omega t)+i\sin(kx-\omega t)=e^{i(kx-\omega t)}$$

We call this $\Psi$ and thus arrive at

$$\Psi=e^{i(kx-\omega t)}$$

The Time-Independent Schrödinger's Equation

Starting with $\Psi=e^{i(kx-\omega t)}$, and since we want the time-independent version, we take the partial with respect to $x$ of $\Psi$ to get

$$\frac{\partial\Psi}{\partial x}=\frac{\partial}{\partial x}e^{i(kx-\omega t)}=ike^{i(kx-\omega t)}=ik\Psi$$

Taking the second partial with respect to $x$ we get

$$\frac{\partial^2\Psi}{\partial x^2}=(ik)^2\Psi$$

Now, remembering de Broglie’s hypothesis we have $p=\frac{h}{\lambda}$, but $\lambda=\frac{2\pi}{k}$. Thus we can rewrite

$$p=\frac{h}{\lambda}=\frac{h}{\frac{2\pi}{k}}=\frac{hk}{2\pi}$$

We remember that $\frac{h}{2\pi}=\hbar$ (reduced Planck's constant) and we have $p=\hbar k$. Thus $k=\frac{p}{\hbar}$.

Now, going back to our second partial of $\Psi$ we have

$$\frac{\partial^2\Psi}{\partial x^2}=(ik)^2\Psi=-\big(\frac{p^2}{\hbar^2}\big)\Psi$$

Now we can multiply both sides by $-\hbar^2$ to get

$$-\hbar\frac{\partial^2\Psi}{\partial x^2}=p^2\Psi$$

But, $E=\frac{p^2}{2m}+U$. If we multiply through by $\Psi$ we get

$$E\Psi=\frac{p^2\Psi}{2m}+U\Psi$$

But, $p^2\Psi=-\hbar\frac{\partial^2\Psi}{\partial x^2}$ so

$$E\Psi=\frac{p^2\Psi}{2m}+U\Psi=\frac{-\hbar}{2m}\frac{\partial^2\Psi}{\partial x^2}+U\Psi$$

And thus we arrive at the time-independent Schrödinger Equation in one dimension.

$$E\Psi=-\frac{\hbar}{2m}\frac{\partial^2\Psi}{\partial x^2}+U\Psi$$

Next time we'll look at the time-dependent Schrödinger Equation, which will be easier since we've already covered most of the physics requirements already.


Is Disdain for Science Fiction and Fantasy Rooted in Jealousy?



I was recently reading a pretty good post on Bryan Alexander's blog; Why do people still disdain science fiction and fantasy? In it, he points to five possible reasons for disdain (or outright hatred) of science fiction, fantasy, and horror; Elitism of taste, Criticism of quality, The charge of escapism, Perceived gender exclusivity, and Market segmentation. While he's right on all the points he makes, I think there's another issue at play here that was missed.

Science and The Humanities; Different Cultures

In a talk from April 10, 2010, in Los Angeles in which Ian McEwan was speaking with David Kipen, McEwan makes a poignant observation. He indicates that scientists (and by extension mathematicians) know the humanities very well, yet those in the humanities are largely ignorant of math and science. In the talk, he mentions that many math and science majors tend to take literature courses in their free time, whereas humanities majors never venture into the science classroom unless they do so kicking and screaming. There may be a bit of paraphrasing on my part there, but I think the sentiment remains.
There's seems to be a vast expanse separating science and the humanities; it's a one-way expanse, though. We know their stuff, but they don't know ours.

Is Disdain For Science-Fiction Rooted In Jealousy?

Science-Fiction and Fantasy are associated, rightly or wrongly, with nerd culture. We all know that math and science are also associated with nerd culture. So I offer a hypothesis; the literary types who display such a gathered of a Science-Fiction and Fantasy do so because of their underlying disdain for, and jealousy of, math and science.
It's easier to write off speculative fiction as "worthless" and the product of a bunch of unenlightened nerds than it is to learn some of our stuff. In a way, this is Alexander's Elitism of taste with the cause identified.
"They know our stuff, but we don't know their stuff."

It's A One Way Street

All the mathematicians and scientists I know have a solid understanding of literature, art, music, and history. Mathematicians and scientists are often seen defending the humanities from would-be cuts implemented by politicians who have about as much grasp of the importance of the humanities as they do for the importance of theoretical physics.
We can see this defense of the humanities in publications like Scientific American. Under Policy & Ethics this month (i.e. October 2016), the editors published an article STEM Education Is Vital--but Not at the Expense of the Humanities
What do you think? Is the disdain many enlightened and literary individuals hold towards Science-Fiction and Fantasy rooted in jealousy because some people "aren't good at math?"