Donald Dingerson Mathematician — Science Writer — Author
Math & Science Educator

The Quotient Ring

Introduction

Since moving my website and establishing a blog, I've posted things on science fiction, physics, and debunking conspiracy theories. What I haven't posted about yet is my own field of interest, namely commutative algebra. And what better place to start than with The Quotient Ring.

The Quotient Ring

While more generic rings (i.e. noncommutative rings) exist, I'm not really interested in them. So, when I talk about rings I always mean (unless otherwise stated) a commutative ring with unity.

Commutative rings are nice for a lot of reasons. An immediate benefit for us is that we can speak of ideals rather than left and right ideals. And, of course, having $1$ makes life a lot nicer.

Remember that for a commutative ring with unity $R$, a subset $I\subset R$ is an ideal of $R$ if, for all $x,y\in R$ the following properties hold.

• $x+y\in I$ ($I$ is closed under addition)
• $x\cdot y\in I$ ($I$ is closed under multiplication)
• $-x\in I$ (every element in $I$ has an additive inverse)

It should be obvious that $(I,+)$ is a subgroup of $R$. If it isn't obvious, it would be a good exercise to prove that $(I,+)$ is indeed a subgroup of $R$.

We can now define the quotient group $R/I=\{I+x\mid x\in R\}$ which is simply the group of cosets of $I$ under addition, but we don't just want a group--we want a ring. So, we can define multiplication for any $x,y\in R/I$ as follows. $$(I+x)\cdot(I+y)=I+(xy)$$ Now we have the Quotient Ring $R/I$.

Units and Nilpotent Elements

There are a lot of results we could prove about the quotient ring, but the one I'm interested right now concerns the preservation of units under the map $\phi:R\to R/I$ when $I\subset nil(R)$. Remember that $nil(R)=\{x\in R\mid x^n=0, n\in \mathbb{Z}, n>0\}$ is the nilradical of $R$, and the elements $x\in nil(R)$ are called nilpotent elements. In particular, I want to look at what a unit in $R/I$ can tell us about elements in $R$. This is formalized in the proposition below.

Proposition. For a commutative ring with unity $R$ and $I\subset nil(R)$ consider the natural map $\phi:R\to R/I$. If $\phi(a)$ is a unit in $R/I$ then $a$ is a unit in $R$.

Proof. Since $a$ is a unit in $R/I$ that means we have some $b\in R$ such that $ab=1-x\in R/I$ with $x\in I$. Technically, we should write $\phi(a)\phi(b)$, but the notation becomes cumbersome and I think the meaning is clear enough we can simplify the notation.

Now, I know that for some $n$ we have $x^n=0$, and since I want to show that $a$ times something is $1$, I'm going to be careful about how I choose that something. So, I take $\gamma=b(1+x+\cdots+x^{n-1})$. Then

$$a\gamma=ab(1+x+\cdots+x^{n-1})$$

But, $ab=(1-x)$ so we have

$$(1-x)(1+x+\cdots+x^{n-1})\\ 1+x+\cdots+x^{n-1}-x-x^2-\cdots-x^{n-1}-x^n$$

We immediately notice that everything cancels out except $1-x^n$. Since $x^n=0$ we are left with

$$a\gamma=1\in R$$

Thus $a$ is a unit in $R$.    q.e.d.

There isn't anything Earth shattering here, just a very basic introduction to the quotient ring and (what I think is) a nice little result.

I hope you found this interesting and informative. I'm going to be doing a lot more math and physics on my blog, so if you have any requests or suggests let me know in the comments below.

Debunking Conspiracy Theories

This weekend I found myself wondering if it was a waste of time trying to debunk conspiracy theories–particularly those related to science. However, when you consider that 26 Percent of Americans Say the Sun Revolves Around the Earth, it seems like we have a lot of work to do.

The most recent, and arguably the most concerning conspiracy theory is the reemergence of the Flat Earth movement. As you might expect, this movement is wrapped up with other equally false conspiracy theories; chemtrails, the moon landing hoax, geocentrism, etc. On both Twitter and Facebook, I’ve engaged Flat Earth believers, and on both platforms I wondered if I was simply wasting my time.

History Says No

If we consider that in 1999 only 18% of Americans believed the Sun revolved around the Earth, it becomes obvious that we’re losing ground. Part of this is due to the ability of conspiracy theorists to spread their disinformation easily online via blogs, self-publishing, and YouTube. Not only does this serve to increase the number of people who believe in these conspiracy theories, it discredits these legitimate outlets for anyone who is NOT spreading lies.

If history is our indicator, simply ignoring the conspiracy crowd won’t make them go away.

Just because it’s a waste of time trying to convince a believer that the Earth is round and orbits the sun, we cannot conclude that it’s a waste of time debunking their claims. Some people sitting on the fence could, and it seems have, concluded that a lack of debunking from the scientific community implies the conspiracy theorists are right.

Like so many other things in our modern world it comes down to education. There’s a reason that educated people tend not to believe in most conspiracies–education, particularly in STEM subjects, inoculates people against crazy theories.

The obvious long term solution is better STEM education, even for people who are not STEM majors. But this doesn’t mean we can stop debunking nonsense wherever we see it. If we do, we run the risk of 30%+ of Americans believing the Sun orbits the Earth within a few short years.

Current OS and Software--Hint It's Linux

We all use technology every day–this isn’t news to anyone, at least it shouldn’t be. With the majority of the world running either Windows or Mac OSX on their laptops, it’s sometimes surprising to find out there are people running things like Linux. So, I thought it would be an interesting post to discuss both why I use Linux and take a peek into the tools that I use to get work done.

Antergos Linux

Linux as a daily OS may seem strange to people who are indebted to either the Windows or Mac microcosm, but there is logic behind the madness.

I got my first taste of Unix in graduate school where the math department had a Sun lab complete with Macaulay, Matlab, and LaTeX. Anything I needed to do on the computer I did in the Sun lab. So, using Unix or Unix-like operating systems is second nature. It’s also much easier, for me, to have access to the Linux command line. There’s also the issue of free versus non-free.

Right now, I’m running Antergos Linux. It’s basically an installer for Arch Linux along with a few other niceties. Sure, you could get the same set up from Arch itself, but the time commitment is several orders of magnitude greater. The trade off for the easier (and faster) install is less customization. I can live with that.

A feature of Arch (and its derivatives like Antergos) is the rolling release cycle. This can be both a good thing and a bad thing. Arch updates can and do break things. For those unfamiliar with the idea of a rolling release verses a versioned release it simply means Arch (and its derivatives) don’t have version numbers. Whereas you have Ubuntu 16.10 and 17.04 and so forth, with Arch you simply have Arch–the OS updates regularly.

I’ve used Arch in the past, going through the manual install several times on various hardware configurations, but recently played around with Ubuntu, Mint, Fedora, and CentOS. While it’s true that Linux is Linux, there are differences in the different distributions. The reason I returned to an Arch based distro is that I simply like Arch better. That’s not to say I don’t like any of the others. I really liked Fedora, particularly the Scientific Spin, but there’s some weird issue with some of my hardware that would cause Fedora to refuse to boot. So after trying out some other distributions I landed back in Arch land via Antergos.

There are several things I like about Antergos Linux, not least of which is that I get an Arch install without the manual install process–which can be time consuming. I also like the custom Numix desktop and icon theme that comes standard with Antergos. Combined with Gnome and Dash-to-Dock installed by default, Antergos looks and works like a desktop, not a server.

Gnome doesn’t always get a lot of love, and in many ways it’s justified. In the case of Antergos though, Gnome is actually pretty good. Unlike Gnome on Ubuntu, it’s not sluggish–thought it does retain a bit of that OS X/mobile look and feel. But the more I use it, the more I like it.

This leads us into the next unique feature that I really like about Antergos. There’s only one ISO image and it includes several desktop environments to choose from. Other distributions require a different ISO for different DE’s. This makes trying out other desktop environments easier with Antergos as you only have one installer. Overall, I’m quite happy with my decision to install Antergos.

Software

I’ll close off with a quick look at some of the software I use on a, more or less, daily basis.

TeX Live, TeXstudio, and LaTeXML

For most writing, I use LaTeX and TeXstudio. Part of this is simply due to the amount of math I have to typeset. If you’ve ever tried to use a word processor to typeset math you’ll understand. However, I also use LaTeX for normal writing as well.

I do this partly out of habit, and partly because LaTeX produces documents that look much better than anything done on a word processor or WYSIWYG desktop publishing system. I use LaTeXML when I need to kick out XHTML with embedded math–such as for a blog post or epub book.

Ghostwriter

For general blogging, and short form writing, I use Ghostwriter. It’s free. It’s open source. It looks and works great. In fact, I’m using it to write this post right now. I particularly like the distraction free nature of the program.

With Ghostwriter I write my post in a combination of markdown and HTML, then export to full HTML before uploading to my blog.

Math and Science

There are a ton of them, so I’ll start with the basics and this list will grow over time.
MATH
Physics
Astronomy
Chemistry

What technology do you use? Let me know in the comment section.

The Earth is NOT Flat

 The Earth Seen From Apollo 17

The photo of Earth taken from Apollo 17 should suffice to prove that the Earth is indeed round. Alas, NASA is a fraud and so we simply can't trust anything they say or any photos they create.

Do we need NASA to tell us that the world is round? After all, Pythagoras looked up at the moon and realized that if it was round, the Earth was probably round as well. Aristotle, gives us a more scientific reason for why the Earth is round.
The evidence of the senses further corroborates this. How else would eclipses of the moon show segments shaped as we see them? As it is, the shapes which the moon itself each month shows are of every kind -- straight, gibbous, and concave -- but in eclipses the outline is always curved: and, since it is the interposition of the earth that makes the eclipse, the form of this line will be caused by the form of the earth's surface, which is therefore spherical.

Again, our observations of the stars make it evident, not only that the earth is circular, but that it is a circle of no great size. For quite a small change of position to south or north causes a manifest alteration of the horizon. There is much change, I mean, in the stars which are overhead, and the stars seen are different, as one moves northward or southward. Indeed there are some stars seen in Egypt and in the neighborhood of Cyprus which are not seen in the northerly regions; and stars, which in the north are never beyond range of observation, in those regions rise and set.

All of which goes to show not only that the earth is circular in shape, but also that it is a sphere of no great size: for otherwise the effect of so slight a change of place would not be so quickly apparent. Hence one should not be too sure of the incredibility of the view of those who conceive that there is continuity between the parts about the pillars of Hercules and the parts about India, and that in this way the ocean is one.
Maybe NASA just paid them off using their fully-functioning time machine.

All teasing aside, I find this resurgence of the Flat Earth movement particularly concerning for a number of reasons. The first and most obvious, is the simple fact that it isn't true.

An Extension of the Geocentric Movement

The Flat Earth movement is, more or less, centered around Evangelical circles--though not exclusively. But, prior to the resurgence of the Flat Earthers came the idea that the Bible is correct, and that the Earth is really the center of the universe.

The geocentric movement doesn't go as far as the Flat Earth movement though. But, some of the results the Flat Earthers use come from, or are drawn from, the "scientific work" done in the geocentric movement.

As a "scientific theory," modern geocentrism is an extension of creationism and Intelligent Design. I think we're starting to see a pattern here. However, if you expect to see me lay the full brunt of the blame at the feet of Abrahamic Fundamentalists, I'm about to disappoint you.

A Reaction Against Arrogance and Lies

This might be as big a problem as Biblical literalism.

It's safe to say that at least part of the people who buy into the Flat Earth movement do so by way of the conspiracy movement. I don't want to get off track here, but most people don't become conspiracy theorists because they think reptilians secretly rule the world. They do become conspiracy theorists because the start to recognize lies passed off as truth--WMDs as a pretext for invading Iraq.

While lies take people into the movement, arrogance fuels the fire. If there's middle ground to be found, this is where it exists. The public's skepticism against the scientific community is growing by the day, and it's stuff like this, combined with a generally arrogant attitude from the pop-science community that allows lies like the Flat Earth movement to perpetuate.

Solutions?

No the Earth isn't flat, immovable, and in the center of the universe--this is actually pretty easy to prove (see above). But right now science is losing the culture war, and in the case of "My Sex Junk" seems to be flailing to keep its head above water.

No real solution will come from any blog post. The best we can do on a blog is point out the problems, and get the discussion moving forward.

I'm a strong supporter of mathematics and science education, and see both as crucial elements for our future success. Which is why I'm writing A Brighter Future: Fixing Math & Science Education in America.

Time-Independent Schrödinger 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?"