a finite-state auto-Matan

A personal blog that is mostly about math, Jewish things, and linguistics.

[matan-matika]

Do u make these math animations? If so how? If not where are they from?

I make them all myself. I use Mathematica for the ones that make sense to automate (all the parts move in predictable ways) and draw shapes in Powerpoint for the ones that are too unpredictable to automate.

[matan-matika]

Why do Jewish studies departments give out Majors and Minors when they should be giving out Phrygian Dominants?

[matan-matika]

Australia's decision to halt the building of a synagogue over Islamist attack fears has provoked outrage

The decision has “effectively placed in jeopardy the future of Jewish life in Australia,” said one leader.

An Australian suburb recently banned the Jewish community from building a new synagogue.

[matan-matika]

A More Intuitive Topology

Topology is defined a bit oddly. Mostly,we know what sort of properties we want a "space" to have, but formalizing them to be defined using set theory is a little trickier.

The mixed metaphor I will use will be maps and lenses. It won't be perfect, since it’s hard have a good intuition for “infinitely many.” Also, many intuitions generalize from very nice topologies called "manifolds," and so won't generalize perfectly.

What we really want to do with space is figure out what is close together, what an individual area looks like, but we don't want to see individual points.

So, a topology should contain sets of points, and think of those as the areas we want.

Now, it can't just be any collection of sets, so let's think of some constraints we can impose on them.

1) If you can see two different areas, then you should be able to see what happens when you put both together.

In set theoretic terms, this means we want it to be closed under union.

There is no reason for it to just be a finite number. If we get infinite areas, when you put them all together, there is no reason for them to not also be like a space.

In fact, infinities let us make different shapes that we also want to be able to use.

So, it should be closed under arbitrary (both finite and infinite) union.

2) If you see two different areas, then you should also be able to only see areas where both are. You could think of putting the two filtering lenses on top of each other, or drawing only where the maps agree.

In set theory terms, we want it to be closed under intersection.

Unlike union, we don't want it to allow all infinities. Think of if you have circles getting smaller and smaller.

Then, if you could see the point they all share as an "area," then we can see any individual point too. This is ok for some topologies, but we don't want it to be a condition for all topologies.

That is, it should be closed under finite intersections (but not necessarily infinite ones).

Note that this makes an asymmetry between the operations, which is fun and interesting.

3) It must contain both the empty set and the entire space.

If we want to see areas that aren't near each other, for completeness, their intersection is the empty set.

If we take all of what we define as areas, they should cover the whole space, because we don't want to focus on anything we can't see.

[matan-matika]

Defining topology

This will be a formal definition. It comes from years of sloughing away explanations of what a topology is to get the most general form.

Definition: Assume you have a set X. A topology on X is a subset of the power set of X that contains the empty set and X, and is closed under union and finite intersection.

There are a lot of very dense words, so let's break it down.

Take a set X; a topology is a collection of subsets of X. For each subset of X, it is either in or out of the topology.

Nothing and everything must be sets in the topology.

If we choose a finite number of sets we know are in the topology, their intersection is too. If we choose any collection of sets in the topology, including any infinite collection of sets, then their union is also in the topology.

Having a slight asymmetry between intersection and union, since there is no guarantee of an infinite intersection being in the topology, leads to many important properties.

To make it make a bit more sense, here are all unique (up to relabeling of points) topologies on a set with 3 points.

(This one’s special. It only has the empty set and the entire set in the topology. That’s the trivial topology)

(This one’s special. The topology is the entire power set. This is called the discrete topology)

Here are sets that aren't topologies.

This one is not closed under finite intersection, because the orange and purple sets are in the collection, but the set containing the point in the middle isn't.

This one isn't closed under arbitrary union, because red and blue are in the collection, but their union isn't.

Some notation that will make talking about topology easier:

Definition: A topological space is a set X equipped with a topology on X, let's call it T. If we want to make sure that we are talking about the topological space X equipped with topology T, we notate it (X,T).

Definition: Let's say A is a subset of X. We say A is open in the topological space (X,T) if A is in T.

Note that A is not inherently "open" or "not open," it is only open relative to a specific topology. And any set with more than one element has many different topologies on it.

[matan-matika]

I feel that teaching topology is best summed up by a joke, which I've heard countless variations on.

Two men were sitting in Tsarist Odessa, and one was talking about his son studying in Vilna. They got talking about how his son sends a telegram home every week.

In this discussion, one of them starts wondering, how does a telegraph work?

To this, the other replies "imagine a large dog standing with its feet in Vilna and its front paws in Odessa. When your son pulls its tail in Vilna, it barks in Odessa."

"Ok," replies the first, "that makes sense. Sometimes, he sends his messages over wireless. How does that work?"

"well," the second responds, "imagine the same sort of setup, but without the dog"

Now, imagine the same sort of joke, except the Tsarist Empire is replaced by any topological space X, and the telegraph is an arbitrary function f:X->X.

[matan-matika]

There are usually two ways of teaching topology.

One is to try to get students comfortable working in particular nice examples of topological spaces, and then trying to abstract from there. The confusion from this methodis confusing the nice properties of those spaces with things true about topological spaces in general.

The other is to start with these weird abstract seeminly arbitrary definitions, and then showing how certain spaces fit into them. This is very non-intuitive, and frequently gets people confused.

I'll try to, with this next series, do a combination of the two.

[matan-matika]

On a scale from “1″ to “5 6 7 8!,” how much dance do you do?

[matan-matika]

Hey man, can I use your linear algebra gif for my homework in university?

As long as it’s properly cited!

[matan-matika]

I was talking to someone who is a college where math majors can graduate with a Bachelors of Arts (BA) degree or a Bachelors of Science (BS) degree. He had the following conversation with a friend who is majoring in Math.

A: If you take another Computer Science class, you can graduate with a BS.

Math major: I don’t want my degree to have anything as epistemologically unfounded as science! 

[matan-matika]

Mood: listening to “Zog nit keyn mol” on loop while reading through the US Constitution.

[matan-matika]

?אשת חיל, מי יבחר

[matan-matika]

כי התשובה יכול להיות “לא ארה”ב,” אני צריך להודיע למורות שלי שלימדו לי עברית

[matan-matika]

?אשת חיל, מי יבחר

[matan-matika]

I will be able to get back to math posting after the 8-day long Jewish holiday of Sukkot. I think my next series will be on topology.

Potential sequence of things: -Intro to topology (at least 4 posts) -Metric Spaces (multiple posts) -An interlude on some cool things about n-cubes (at least 3 posts) -Continuity -Homieomorphisms -Quotient spaces -Manifold basics -A baby look at Algebraïc topology (at least the fundamental group, maybe a few other coolnesses)

[matan-matika]

I’m a bit late, sorry.

[matan-matika]

I will be able to get back to math posting after the 8-day long Jewish holiday of Sukkot. I think my next series will be on topology.

Potential sequence of things: -Intro to topology (at least 4 posts) -Metric Spaces (multiple posts) -An interlude on some cool things about n-cubes (at least 3 posts) -Continuity -Homieomorphisms -Quotient spaces -Manifold basics -A baby look at Algebraïc topology (at least the fundamental group, maybe a few other coolnesses)

[matan-matika]

To anyone in High School in the United States of America:

If you are at all interested by the concept of Computational Linguistics, no matter what your skill level is, you should check out NACLO, the North American Computational Linguistics Olympiad.

Here’s their website:http://www.nacloweb.org/

Here is a past problem on their test, which I find to be a great introduction: http://www.nacloweb.org/resources/problems/2010/A.pdf

(Anyone else can look at their past problems, which are great, and very fun, but only US American high schoolers can take their test)

[matan-matika]

I’m rereading parts of “Being Colloquial in Esperanto,” and its example sentences are just as shitposty as I thought.

(here’s a link to it online, http://pages.ucsd.edu/~dkjordan/eo/colloq/colloq.html, but I first found it in print at a library)