ChapterZero

I noticed that is approximately the same as . Is there a known proof (or am I just seeing things again)?

We know that and are and , respectively (assuming the function is nice enough to have a unique median). In general, what is ?

In the process of investigating the approximation error, I came across an interesting question. From numerical examples, it seems that , where but this baffles my intuition (which I’ll admit isn’t very sharp), especially for . Here is the Frobenius norm, and is the j-th column of .

Hopefully I’ll have time to look at this later. Besides being counterinuitive, it seems potentially useful: consider a vector of composite length , you could partition this vector into blocks of length then apply this inequality to get a family of bounds on the euclidean length of the vector. Exploit the fact that and you could derive some very weird bounds indeed.

My officemate and I have decided to take 30 minutes or so a week to inform each other about various mathematical topics– on the theory that teaching helps you to cement your own knowledge. I’ll be teaching him something relating to my research, and vice versa: he’ll probably be teaching me about Sobolev spaces, which I’ve wanted to learn about for a while now, so it should be an interesting experience. I’m debating between teaching him what I know of random matrix theory (which I’ve already gotten fuzzy on, despite having learned it last term), working my way through the Cauchy-Schwarz masterclass book (because I’m sure I’ll find those inequalities useful), or basic operator theory (which I know I’ll need down the road as my research becomes more theoretical).

Speaking of my research, I fell in love with the subject all over again. Remember my problem is to find bounds on the error in approximating a given matrix with a sparse matrix ; ideally, these bounds should be a function of only: the whole point of sparsification is to avoid making convoluted computations involving the original matrix ; if we had to make such computations to bound the error, we might as well just calculate it exactly. Anyhow, I saw anew how awesome it is that you *can* get useful bounds that are just a function of . I think it helped a lot that I got nice results

For what is it possible to have a -regular graph on vertices?

I thought about this a little while last night: it’s equivalent to asking how many hollow (the diagonal is zero) symmetric binary matrices satisfy for all . I haven’t seen a way to exploit this equivalence, so maybe there is another approach that would be more fruitful.

It turns out that if is in , , where depend on the dimension. I have no idea how to prove this in general, and don’t really care.

BUT, it makes for a nice problem in the one-dimensional case. Show that if (i.e. its second derivative exists and both it and its first and second derivatives are square integrable), then , and find .

I haven’t a clue how to proceed, but it looks like mighty fun.

Update: the proof is, in retrospect (always in retrospect, damn it), obvious.

Show that there is no metric on such that pointwise convergence and convergence in the metric coincide.

I finally read up to the Gelfand representation theorem in Murphy’s book. This is such a beautiful mathematical result that I couldn’t resist the urge to share it. I tried to write this synopsis so anyone who understands the Banach-Alaoglu theorem should have no problem following the gist of it.

Recall that a (complex) Banach algebra is a (complex) Banach space with a multiplication operation which is continuous w.r.t. the norm. The Gelfand representation theorem says that we can represent an abelian Banach algebra as the algebra of continuous functions on a locally compact Hausdorff space . The important case to consider is that of a unital Banach algebra, one with an identity element; in this case, we can take the space to be compact. For the remainder, will be a unital abelian complex Banach algebra.

The obvious question is, what is this space ? Can it be intrinsically connected to , or is the proof nonconstructive? It turns out that is a very natural space; in order to construct it, we need the concept of the spectrum of an algebra.

Let . We call the set the spectrum of . One can think of the spectrum as a generalization of the eigenvalues of a square matrix or the range of a bounded function. Recall that a (continuous) homomorphism between two algebras is a linear map which preserves the multiplication operation. A character on is a non-zero homomorphism between and . Let denote the set of characters on ; it turns out, magically, that for all .

One can show that is a weak* closed subset of the unit ball of A*, and in fact if we give the relative weak* topology, it is compact. From here it’s easy to guess that , the character space or spectrum of , is the appropriate . Given , define its Gelfand transform by . By the choice of topology on the character space, each such functional is continuous (and in fact, vanishes at infinity).

The Gelfand representation theorem specialized to unital abelian complex Banach algebras then says

The map is a norm-decreasing homomorphism, and for all . Furthermore, .

Given Rademacher variables , what can we say about
versus ?

Anything?