Convergence of an Interesting Sequence

Let

be a sequence of finite binary fraction in which each successive fraction is obtained by adjoining a 0 or a 1 to its predecessor. It may sound strange, but the sequence converges. Always.

Let . Then:

Therefore, is Cauchy and hence converges.

Would you have guessed?

Excuse me…

Driving home the other day, I heard an advertisement on the radio, that said “Tired of taking a square root of an isosceles triangle?”

Are we diseased? Is it this hard to look up and say at least something that has meaning?..

Well, that said, the new movie, titled 21, has also quite irritated my already twisted convolutions of the brain. I first saw the trailer sitting in a movie theatre, which started by showing a young man, who is extremely good with simple arithmetics. And I thought “Oh, nice, good fella”. But then the voice over frame said that this young man “was the brightest and most successful mathematics student at [some] university”…

And not that that can’t be true (it of course can… it’s a movie anyway…), but the idea that “the greatest mathematician” is always associated with “he who counts fastest” just bugs bujizas out of me.

I know many very clever people who are not generally fluent with numbers. In fact, doctor Bar-Natan (~) is surely one of the most clever and bright people I know; he has “no experience in computation whatsoever”…

So… ?

My Favourite Math Video!

Simple Proof of Mixed Derivative Theorem

Everybody knows the good old

Theorem (Mixed Derivative Theorem): If the function is in class (id est, twice continuously differentiable) in an open set containing , then:

In other words, the order of differentiation does not matter. The proof of this fact is not completely trivial, though.

I would like to give a much simpler proof of this fact, which I found to be quite interesting and somewhat elegant. It relies on the theorem of Fubini. Before I write out the proof, I state the Fubini’s theorem for a two-dimensional case.

Theorem (Theorem of Fubini): Let be two closed intervals. Let the function be continuous and integrable on . Then:

Here denotes really just a product of differentials $dx$ and $dy$, but the order is not specified and not assumed. I will not state the proof (which is fairly tricky and long). Also I would like to note that this is actually one of the corollaries of Fubini’s theorem. The function may not be continuous, but the theorem of Fubini would still hold, only in a slightly, but a little strange form. And even this generalisation of the state theorem is not exactly the theorem of Fubini, but is more properly known as “a theorem of Fubini type” for a few reasons.

We are now ready to prove the mixed derivative theorem.

Proof of Mixed Derivative Theorem: We need to show that

We construct the proof by finding a suitable contradiction. Suppose otherwise:

Since , there is a rectangle , such that

Since this difference is thus a nonnegative function on ,

Since both of the subintegral functions are continuous, we can apply the theorem of Fubini to obtain:

which is, of course, completely nonsensical. We are left to conclude that

Quod erat demonstrandum.

What does it take to compute 1/3 + 1/2 – 2/3?

I don’t know about you, but for me it first took about 3 to 4 minutes of thinking, then another minute of thinking about how stupid I am, then it took calling out the Run window to type “calc” in order to call out the calculator, then ridiculously typing in all those fractions and realising that the answer I get is a weird long number and that I cannot figure out what it is as a simple fraction, and then finally pulling out my normal calculator from the bag and punching in those fractions to get a good answer as a fraction. This was followed by another half a minute to a minute of thinking that there might be something wrong with me… =\ hmmm…

I am so greatful that neither mathematics nor physics are really about numbers. They just pretend that they are. :)

Update: The answer is 1/6, by the way, for those who encounter similar difficulties. :)

Canada’s Population

Have you ever wondered what the population in a county is at a specific time? Or have you ever wanted to calculate what the population of the country is going to be in, say, 3 minutes and 41 second?

Well, I don’t know about you, but I never have, up until a moment ago, when I encountered this “population clock”,

http://www.statcan.ca/english/edu/clock/population.htm

which applies to Canada only (being produced by the Statistics Canada), but I am sure there are similar things for most countries. In fact, from the moment I encountered it to the moment this post is being written, Canada’s population grew by 5 people! That’s five more mouths to feed! But, perhaps, that’s five more physicists, or engineers, or doctors, or may be teachers, mathematicians, firefighters, soldiers, who ever not? Or may be a bunch of high school dropouts, which still wouldn’t be worst case.

Of course, this clock applies to local time periods only, but it’s being updated sufficiently frequently, for they last updated it half a year ago. I mean there can’t be a dramatic change in population growth over such period, unless there’s sort of an unnatural reason for it, which would simply be a some kind of fluctuation.

Interesting :)

A Short Update and Motivation for Further Takeoff

I sincerely appologise for updating the blog thus rarely. The new semester has started from a tough note, and so it ought to take time for me to obtain a laminar flow, for I would really hate to see any turbulent behaviour.

As some of you may know (and those who have checked out my web page, http://individual.utoronto.ca/nikolaev, for more information), I organise the 2008 Canadian Undergraduate Mathematics Conference (CUMC) that will take place at the University of Toronto some time in July. I have recently found out that the 2008 Canadian Undergraduate Physics Conference will also take place at the University of Toronto some time in October. Knowing that I have gained much experience in organising the CUMC, I thought no longer and found people responsible for the organising the physics conference. I now organise both events. With any further questions, please contact me. If you would like to appear as one of the speakers, please let me know that, too!

Again, I regret that I haven’t been updating the blog too often. I will try to manage my time better so that I will leave a few minutes to scribble a note or two. I have had a few things I was ought to share with the blog, but found no time to actually write them; as a result, I now do not remember what they were…

Also, I have quite a few nice photographs of the University of Toronto campus to show. These will also hopefully appear reasonably shortly.

On another note, I mentioned that apparently University of Toronto Physics Department had a store (here and here). Well, I went there and I looked at it. Umm… I couldn’t find it. Perhaps I was either looking at the wrong place or it was closed. I should ask someone, I think. I’ll let you know.

Sets of measure zero

Despite common smouldering discontent about the concept of a measure zero, it is a very interesting and useful concept. I have been recently talking to a student and was surprised that the person hated the concept of measure zero, calling it completely useless. Hence I decided to clarify some things.

Consider a very easy case in . Recall that Riemann function

on, say, the interval 0 to 1.

Recall how hard it was to prove that this function is Riemann integrable? We had to take some and show that the limit of the function is everywhere 0 and… all different sorts of things.

Now consider a situation where we know the notion of a set with measure zero. That is, we can state the following

Definition: A set has or is of measure zero, if a cover of the set by an a (finite or not) countable system of intervals, the sum of lengths of which .

Then we can easily prove the fact that a set that consists of only the rational number between and is, in fact, a set of measure zero. Then use recall the Lebesgue criterion for Riemann integrability, which states

Theorem (Lebesgue Criterion for Riemann Integrability): A function defined on a closed interval is Riemann integrable, if and only if it is bounded on this interval and the set of points, where the function is discontinuous, is of measure zero.

We are now ready to solve the problem. We see that the Riemann function is discontinuous at every rational point . Then the set of points, where the function is discontinuous, contains only the rational points and hence it is of measure zero. By the Lebesgue criterion for Riemann integrability, the Riemann function is integrable on 0 to 1.

Easy? I thought so.

4 + 1 = infinity !?

While naively expecting to obtain 5 as a result of my calculation, I have, on the contrary, obtained an infinity. At the first proof-read, I wasn’t able to find what the mistake was, and so it had gotten me thinking: can 5 really equal infinity?

Well, the answer, perhaps surprisingly, is a definite yes! Indeed, just think about it: what if you only know how to count to 4?

That didn’t help me with the solution, by the way, so I agreed to follow the grey, uninteresting and boring convension: 5 \not = \infty… But just imagine how much brighter our lives would be, if that was true! Would it not?