Connectedness of R\QXQ

This one was just amazing.

Consider R\QXQ. Is it connected?

It turns out that it is polygonally path connected with at most two segments!!

And the proof is just two lines:


Consider two points p,q in R\QXQ.

Then, through p (and similarly for q) there is an infinity of lines that pass through only irrational points. Pick a line each for p and q. Call their intersection r. Then p->r->q  is the desired path.

Kolmogorrov's 0-1 Law

We learned the Kolmogorrov's 0-1 Law today in class.

And I had this (very very informal) intuition:

In the 0-1 Law we talk about the tail events, i.e., the events belonging to the tail sigma algebra. The tail sigma algebra is defined as

F_tail = \cap_{n >= 1}  \sigma(X_n, X_n+1, X_n+2, ... )

That is, it consists of events that are independent of finitely many initial X_n's. 

Obviously, F_tail \subset F := \sigma(X_1, X_2, ....), the "full" sigma algebra. 

In a way, to construct the F_tail, we 'drop' the initial X_n's one by one. For each X_n dropped, the sigma algebra becomes smaller but is still a sigma algebra. Note that this sigma algebra encodes lesser bits of information.

Thus, 'in the end', when you have 'dropped all the X's', you have a sigma algebra but you do not have any information bits left - and that sigma algebra is effectively the trivial sigma algebra. Thus any event in it must have probability 0 or 1!

Of course, this is very very vague and there is a LOT of handwaving in the arguments - but I thought the intuition was cool.

As a corollary, it would also be interesting to consider the effective information content of a probability measure on a sigma algebra. That is, in the tail sigma algebra above, the effective information content is 1 bit. 

Que mas and Como estas?

Well, cool things are not just math :-)

Taking lead from some of my Spanish-speaking friends, once I said "Que mas" to a Mexican.

He got confused.

Quite rightfully so - I asked him "What else?" before I spoke with him about anything else!

In Colombia, the typical informal greeting is "Que mas?", literally, "what else?", whereas in Mexico, you say "Como estas?" meaning "How are you"?

So though they speak the same language, some of the idioms are quite different.

Asymmetry of Unions and Intersections

[From my discussion with Prof. Green]

We know the usual result that most (if not all) theoretic identity using unions, intersections and complements has a corresponding dual with unions and intersections exchanged and complements unchanged. This indicates some kind of symmetry between the union and intersection operations. For example, the pairs of de'morgan's laws, distribution of union over intersection and vice versa.

But union and intersection are not completely symmetric in the following sense.

A.B <= A and A+B >= A

So union typically creates a bigger set and intersection typically creates a smaller set.

Now there is something called the smallest set - the empty set. You cannot get any smaller than that.

But Cantor's famous result implies that there is no biggest set; given any set, one can construct another set (its he power set) which is strictly bigger.

Thus we have some sort of asymmetry for unions and intersections.

So the question is, does this translate into an identity that is not valid when the unions and intersections are exchanged?

Welcome!

Don't know why I didn't think of this earlier...May be ideas like this come only when the mind relaxed by a four-day weekend :-) Anyways,

This blog will have some cool things that I came across during Ph.D.

Enjoy!