That Math Blog

Weird Examples in Analysis #4:

Since the rationals are countable, let them be listed as such:

Now, for x in [0,1], let f(x) be:

Is this monotone? Is this continuous? If so, where? Is this Riemann integrable on alpha = x? If so, prove it. Bonus points for a 2 line dis/proof of the integrability. 

Ps — You can cite both Krantz and (Baby) Rudin. I have copies of both one hand.

hehe

Weiner space

TMB’s Weird Examples #3: Your turn to find the limit!

Consider this cringeworthy guy:

Tell me the limit:

Hint: 

The top one is bounded, the bottom one unbounded and monotonic increasing.

Good luck!

TMB’s Weird (Counter-?)Examples in Math #2: What’s the limit?

Wow. Looks daunting enough, don’t it? Holy hell, that’s evil. What could be worse?

Eep.

So, how we gonna do this? Well, we know that:

because

This holds for x=1, as well, since then it’s just values of multiples of pi in the argument afterwards. 

In fact, if you play around with it, noting that m! has the product of every value of m up to it in it, then any x which has some integer <= m as the denominator would make the cosine argument 0. So, we can say that as m goes to infinity, any rational number would make it 1. So the limit is 1 over the rationals. 

So, what if x is irrational? Then we see that the limit will be less than 1; then, we get a decreasing monotone function which goes to 0. 

Hence, the limit function is identically 0 over the irrationals and identically 1 on the rationals.

It’s also not continuous, not differentiable (well, duh, it’s not continuous), and it’s not integrable at all.

TMB’s Weird (Counter-?)Examples in Math #1: If x = 0, f(x) = 0, but what if x != 0?

Any bets on what this is going to be at any x != 0?

Well, first, we see that:

by pulling out the term. Then, this gives us a geometric series:

So, what’s this?

1. It’s 1.

This has been a sample weird example number 1. They will get weirder down the line. Might as well just make this a semi-regular thing. I’ve got more interesting ones than this.

How interested would you guys be in seeing weird examples of things in like….analysis or topology that do stuff?

Like, weirdass functions that go to 0 identically, or how we analyze shockwaves (hint — it can be modeled via the tanh function) or stuff like that?

I’ve got a metric shitton of stuff that’s like….continuous functions which you wouldn’t expect to be and whatnot. Been sitting on them and I don’t think I’ll be teaching an analysis class anytime soon (and calc students don’t care about them…)

If entropy always increases, how did the chicken ever create the ordered egg to begin with? A common explanation, advanced by the Austrian physicist Erwin Schrodinger in 1944 in a brief and charming book, “What is Life?”, is that living systems somehow borrow order from their environment and pay it back by somehow making the environment even more disordered that it would have otherwise been. That extra order corresponds to “negative entropy”, which the chicken can use to make the egg without violating the second law….. Chickens don’t access some storehouse of order to make the thermodynamic books balance: they use processes for which a thermodynamic model is inappropriate, and throw the books away because they don’t apply. 

The scenario in which an egg is created by borrowing entropy would be appropriate if the processes the chicken used were the time-reversal of an egg breaking up into its constituent molecules. At first sight, this is vaguely plausible, because molecules that eventually form the egg are scattered throughout the environment; they come together to form the chicken, where biochemical processes put them together in an ordered manner to form the egg. But that’s not how the chicken operates. Some molecules happen to end up in the egg and are conceptually labelled as part of it AFTER the process is complete. Other molecules could have done the same job — one molecule of calcium carbonate is just as good for making a shell as any other. So the chicken is not creating order from disorder. the order is assigned to the end result of the egg-making process — like shuffling a deck of cards into a random order then numbering them 1, 2, 3 and so on with a felt-tipped pen. Amazing — they’re in numerical order!

— “In Pursuit of the Unknown: 17 Equations That Changed the World”, Ian Stewart