ChapterZero

Given Rademacher variables , what can we say about
versus ?

Anything?

I came across the following statement while reading Murphy’s book (Theorem 1.3.1): “If is a proper ideal in a Banach algebra, then is also proper.”

Quick recap: a Banach algebra is a Banach space with a multiplication operation, such that the norm is submultiplicative (). A prototypical (non-abelian) finite dimensional Banach space is that of all matrices with the spectral norm, and a prototypical (abelian) infinite dimensional Banach space is that of all continuous functions on with the sup norm. An ideal is a vector subspace that ‘absorbs’ under multiplication: if is in the space and is in the ideal, both and are in the ideal. A proper ideal is one strictly contained in the ambient space.

This statement caught my eye because it gives a way in which ideals topologically differ from vector subspaces. It is easy to come up with examples of proper subspaces whose closures are the entire space; e.g. take the set of differentiable functions in the above infinite dimensional Banach space: every continuous function on is the uniform limit of differentiable functions, so the closure of this subspace is the space itself. It’s harder for me to come up with ideals nontrivially exemplifying the above statement.

The question is: exactly how does one use the multiplicative closure property of ideals to prove the above?

Tonight’s been an insomniac night, just checking out a bunch of blogs. Feeding the addiction

Anyhoo, in the background, I’ve been youtubing various people. For the first time I listened to some of Pharell’s stuff: I’ve been a fan of his forever, because he demonstrably doesn’t think that to be successful in the music business you have to conform to the Jay-Z, Lil’ Wayne, Soulja Boy, etc. stereotypes. He’s being him, doing him, and it’s working for him, same as his bff Kanye.

But I’d never heard his music before; here’s a little sample:

If someone had played these for me, I would have thought the first two were parodies; the third is OK, but can Gwen Stefani have some more lines? On to someone else…

Update: I think in the first two songs he was trying to get the same offbeat feel as when he’s part of NERD; I like what I’ve seen of their stuff. Here’s a good one:

Occasionally I come across so-called feminist blog entries that strike me as particularly vile. Today, one of the articles in the latest Carnival of the Feminists linked directly into the blindered underworld of ‘feminists’ who ascribe everything to twisted male sexuality and the desire for dominance.

In this case, the nominal topic is transgenderism, but it quickly devolves into an attack solely on transwomen:

When transgendered folks get through the final stage of transitioning and reach “the end”, all that gender fluidity goes right out the window and solidifies into the crusty crud on the bottom of my boots.
…
Since you do not need a penis to pick up a hammer, since you do not need a vagina to vaccuum, or validate virility, vanquish vasselage, or *oh my!* vounch for volition, there is also no corresponding need to switch out body parts in order to express what is basically described as internal character, unless the purpose is merely to commodify one’s personality as a dainty girl might wear pink or a goth craves black. The genitalia have been reduced to the status of wardrobe accessory.

And since these folks insist they’re not swapping out genitalia for masturbatory fetish purposes, there too is no need to accessorize oneself with the preferred genitalia of their romantic partner unless they’re also prepared to reverse the surgery again, when they meet someone new who wants yet different genitals to play with.

So many things are wrong with this post, starting with the fact that she confuses transgenderism with transsexualism. One would think you’d take the time to familiarize oneself with the difference before you start spewing vitriol. There’s just too much to shake my head at to even begin deconstructing her ‘arguments’.

Shhh! Don’t tell anyone … I’m a Rihanna fan. I used to think she can’t sing, but I’m steadily losing that conviction– her voice still isn’t the greatest, but maybe she’s been taking lessons, or I’m being brainwashed by the media saturation.

At any rate, my stanness is not predicated on her singing talents: I still can’t listen to most of her songs unless they’re accompanied by the video (or should I say, unless they accompany a video? ). Mostly, it’s that I’m so damn proud of her; what other international stars do we Barbadians have claim to? The other part is that she is the sexiest high profile female musician I can think of, and she manages to stay on the right side of the thin line between sexy and trashy. A lot of her attractiveness is her incredible body, but her accent adds a final je ne sais quoi.

Actually, I guess I do know why her accent is so attractive: she comes across as laidback and approachable, the Barbadian girl-next-door.

One crucial feature that CiteULike lacks is the ability to remove entries en masse; the only way to remove several entries is to go to each entry’s page, click on delete, and confirm the deletion– that’s about 3 clicks per removal. Imagine if you want to remove *all* of your entries!

That was the problem I encountered as I attempted to clear my account of all the papers not related to my research, so I decided it’d be worth the time to write a little Greasemonkey script to remove all the entries. Unfortunately, in addition to being very out of practice with Javascript and the DOM, I’ve never used Greasemonkey before, so I ended up writing a two-script hack. The first script loads the first entry on my CiteULike library page, and the second script deletes any entry once its page is loaded; together they eliminate entries one by one until none are left. Inelegant, but much more inefficient than doing it by hand.

Here’s the code for the two scripts– just remember to retask them to point to your citeulike account, and to DISABLE under Greasemonkey or delete them after you’re done.

// ==UserScript==
// @name           First citeulike article
// @namespace      http://tangentspace.net
// @description    Loads first citeulike article
// @include        http://www.citeulike.org/user/swiftset
// ==/UserScript==
 
articlelist = document.evaluate("//div[@class='list']", 
document, null, XPathResult.UNORDERED_NODE_SNAPSHOT_TYPE, 
null).snapshotItem(0);
 
allarticles = document.evaluate("//a[@class='title']", 
articlelist, null, XPathResult.ORDERED_NODE_SNAPSHOT_TYPE, 
null);
 
document.location.href = allarticles.snapshotItem(0).href;

// ==UserScript==
// @name           Delete Citeulike Article on load
// @namespace      http://tangentspace.net
// @description    deletes any citeulike article on load of the article info page
// @include        http://www.citeulike.org/user/swiftset/article/*
// ==/UserScript==
 
allhidden = document.evaluate("//input[@name='user_article_id']", 
document, null, XPathResult.UNORDERED_NODE_SNAPSHOT_TYPE, 
null);
anelem = allhidden.snapshotItem(0);
 
document.location.href="http://www.citeulike.org/delete?user_article_id="
+anelem.value+"&"+"from=%2fuser%2fswiftset";

Recall the Schatten p-norm of a matrix : where are the singular values of . They’re interesting, among other reasons, because they’re unitarily invariant. Schatten and Von Neumann came up with these while investigating matrix analogues of the finite spaces— they’re the norms of the vectors of singular values.

Show that , where is the modulus of the matrix.

Note that the Schatten 2-norm (aka Frobenius norm aka Hilbert-Schmidt norm) can be easily calculated , and is actually induced by the inner product . When is positive, show that is also easily calculable: .

Can you show that Schatten -norms have the dual norms you’d expect from analogy with the usual norms? It’s fun

Can you think of any other interesting properties of the Schatten norms?

It seems that it’s a good idea to have some knowledge of operator space/ -algebra theory in my research area, so I’ve started reading some books on them. Honestly, I find this stuff boring– it’s only the applications that are interesting… that statement probably makes no sense, but lots of things I say don’t– but alternating my time between reading and doing actual research should make it bearable.

I spent today looking through several papers. Most of them were about finding graph sparsifiers by sampling– apparently this relates to a generalization of the sparsification problem I’ve been thinking about, where instead of looking for a sparsifier in the spectral norm, you look for one in the norm.

I snuck in a paper by Rudelson and Vershynin, to gain some familiarity with the more typical tools in this area, and came across the neat idea of the numerical rank of a matrix: . It’s clear that this satisfies , but it’s also stable in the sense that if is close to a low rank matrix, then is also low. I’m looking forward to seeing how they use this quantity.