Tag Archives: cantor set

Turducken Day 6, also, answer to Cantor set problem

12 Jun

Honestly we didn’t have that many leftovers: I packed up a bunch with my friends who came over to eat the turducken.

The best thing I made with the leftovers was turducken tacos, mostly because the fatty meat was a wonderful complement to the chipotle yogurt sauce inspired by Mark Bittman’s fish taco recipe.

Turducken tacos:

Leftover turducken, in heated flour tortillas, with lettuce, chopped radishes, cheddar cheese, and chipotle yogurt sauce.  Serve with a wedge of lime.

THIS SAUCE IS SO SO GOOD:

Mix:

2 c full-fat plain yogurt

2 minced garlic cloves

3 chopped up chipotle peppers in adobo sauce

Leave in a fridge for at least ten minutes: it just gets better the longer it sits.

Also, I made matzoh ball soup with the turducken broth leftover.  It was also quite tasty, though salty (must dilute the broth)

Don't DILLy-dally, eat it while it's hot!

Don’t DILLy-dally, eat it while it’s hot!

Finally, the answer to the Cantor set question from that other post:

This answer is straight from an exercise in Bruckner, Bruckner, and Thomson‘s book Real Analysis (exercise 1.1.3).

First, think about the number 0.5637.  In grade school we say this is five-tenths plus six-one-hundredths plus three one-thousandths plus seven ten-thousandths.  We can write this as a sum as $\latex \frac{5}{10}+\frac{6}{10^{-2}}+\frac{3}{10^{-4}}+\frac{7}{10^{-5}}$.  Switching to fancy math notation, we can write any number as a sum: \sum_{n=1}^{\infty} \frac{a_n}{10^{-n}}.  Here, \sum_{n=1}^k means that you take whatever’s after the sigma symbol, and start at n=1 and add.  So in our example, we have 0.5637 = \sum_{k=1}^4 a_k 10^{-k}, where $a_1=5, a_2=6, a_3=3, a_4=7$.

This decimal expansion is relevant.  I said earlier that the Cantor set has something to do with ternary expansions.  So in our decimal expansions, we let $a_k = 0,1, \ldots, 9$.  For a ternary expansion, we can only let $a_k = 0,1,2$.  And for our Cantor set, it’s all the numbers in [0,1] that don’t include the digit 1 in their ternary expansion.

Let’s use another set to get to our answer.  Let D = \{ x\in [0,1]: x=\sum_1^{\infty} \frac{j_n}{3^n}, j_n=0,1\}, that is, D is the points in the interval [0,1] with no 2 in their ternary expansion.

Pick any point y\in [0,1] and its ternary expansion \sum_1^{\infty} \frac{y_n}{3^n}.  We’re going to find numbers a,b\in D so that a+b=y, which will prove that D+D=[0,1].  If y_n = 2, let a_n=b_n=1.  If y_n=1, let a_n=1, b_n=0.  If y_n=0, let a_n=b_n=0.  This way we’ve defined sequences a_n,b_n, and if we let a= \sum_1^{\infty}\frac{a_n}{3^n}, b=\sum_1^{\infty}\frac{b_n}{3^n}, we’ve defined a,b\in D so that a+b=y, as desired.

This doesn’t quite finish our problem!  We wanted to show that C+C = [0,2].  But notice that $D=\frac{C}{2}$, that is, if some number x is in C, x/2 is in D, and if y is in D, then 2y is in C.  So if D+D=[0,1], then C+C = [0,2], and we’re done!

Brainstorming, also, Cantor sets

10 Jun

So I studied for those prelims for a few months (maybe two) on my own, reading old notes, highlighting things, rewriting relevant proofs, running through my homeworks, attempting extra exercises from the textbooks, the usual extreme studying.  I don’t know about you, but I am SO MUCH BETTER with other people when doing just about anything.  Knowing someone else will eat my baked goods makes my baked goods better.  Being accountable to a colleague makes me justify my ideas more and I write down fewer false things.

Math is a creative endeavor.  People seem to think mathematicians are somewhere between human calculator, engineer, and crazy person.  But we don’t get grouped with, say, artist, writer, poet, all that often.  I’m going to quote some wikipedia here:

Plato did not believe in art as a form of creation. Asked in The Republic,[18] “Will we say, of a painter, that he makes something?”, he answers, “Certainly not, he merely imitates.”[16]

Now, as with many creative things, there are certain tools you can use to help with math.  Going on walks is a good thing.  Eating breakfast.  Naps, even.  But the thing I love, and the thing that helped me pass my prelims (whoooo), is brainstorming.  A few weeks ago I read an article that a facebook friend posted (isn’t it funny how you know what I’m talking about when I say “facebook friend” rather than “friend”?  In this case, a guy I went to high school with) about how brainstorming is basically stupid and broken.  And, given the parameters that the article offers, it’s pretty right: get in a group, generate lots of ideas, don’t be critical.  I don’t have a lot to say about brainstorming (though Jonah Lehrer for the new yorker and a guy from Stanford’s d.school do and these are both fascinating), but I do have things to say about problem solving with a group.

Brainstorming, as this three-step outline is, isn’t the way solving problems with a group should be done.  He’s on point with steps one and two, but step three should be different: be critical.  Fight for your ideas.  Fight other people’s wrong ideas.  I work with other grad students a lot, and I often don’t realize that I’m completely wrong until I’ve been talking about an idea for a few minutes.  I need them to fight me in order to learn and understand and be better.

The idea behind brainstorming is that if you’re critical of others’ ideas, they won’t want to share them and will clam up.  I absolutely felt that way for the first year or so of graduate school- I would feel that my peers shot me (personally) down, and that I didn’t have any good ideas.  What finally changed my mind and made me more combative was realizing that I do have good ideas, sometimes.  Not all the time, probably not even 50% of the time, but sometimes I’m right, I’m absolutely right and I can prove it if you-just don’t shoot me down or interrupt me or dismiss me.  These last three things still happen, and I’ve learned to fight back in a nonconfrontational manner- it’s easier in my case because everyone in the room just wants to reach a solution.  Some people have egos (I do too), but we learn to listen to each other to the extent we’re capable, and to make each other listen when we can’t extend ourselves to do so.

Basically, I agree with this guy: “Innovation Is About Arguing” is the first bit of his title.

On a totally different topic, let’s talk about the Cantor set.  It’s a pretty cool subset of numbers with lots of unintuitive properties.  Building it is fairly straightforward.  Look at a number line.  Focus only on the segment between 0 and 1.  Cut the segment into thirds, so you’ll have notches at 1/3 and 2/3.  Delete the middle third (1/3,2/3) so you end up with two segments.  Do this over and over again, deleting the middle third from each segment you have at any one point.  Look at the picture

 

I DID NOT MAKE THIS PICTURE it is from wikipedia.

And you just do that forever.  A few cool things about this set:

  • It contains no open intervals, since we’d cut out the middle third from any interval that showed up.
  • Despite that, no points in the Cantor set are isolated: if you take a point in it and look in a teensy neighborhood (like, \pm .00000001 teensy), you’ll still find other points from the Cantor set.
  • It’s uncountable, which means literally that you can’t count it , even if you had infinite time.  Counting numbers are countable (1,2,3…), and so are numbers that can be written as fractions (1, 2, 1/2, 3/4…), but all the numbers between 0 and 1 aren’t (this is Cantor’s diagonalization argument, which I’ll blog about some time).  You can show that the Cantor set is uncountable by making a function from it to the interval [0,1], which hits all the numbers between 0 and 1.  If it hit all the numbers between 0 and 1, we say the map is surjective or onto [0,1].  If it’s surjective, then for every number in [0,1], there’s a number in the Cantor set that maps to it, so the Cantor set is bigger than or equal to [0,1] in size.  Since [0,1] is uncountable, this means Cantor set is too.
  • If you consider the set C+C = \{x+y : x,y\in C \}, you get an interval, which is way incredible and unexpected because it contains no intervals- it’s like putting together two sets of discrete bread crumbs and magically having a loaf.

You think about that last bullet point, and I’ll do a post later proving it.  Hint: know how we can think of any number as a decimal expansion?  Well, you can also use base 3 and do a ternary expansion, so 1/3 = 0.1, 2/3 = 0.2, 4/9 = 1/3+1/9 = 0.11, etc.  The Cantor set, by the way we defined it, is all the numbers between 0 and 1 that contain no “2”s in their ternary expansion.

Hasta later!

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