Some of the books sitting on my desk that I’ve been planning to read for a long while– the ones most directly relevant to my nebulously envisioned future research:

• Asymptotic Theory of Finite Dimensional Normed Spaces. Milman and Schechtman
• Geometry and Probability in Banach Spaces. Schwartz
• Fundamentals of the Theory of Operator Algebras, Volume I. Kadison and Ringrose
• Operator Spaces. Effros and Ruan
• C*-algebras and Operator Theory. Murphy
• Classical Banach Spaces. Lindenstrauss and Tzafriri
• Introduction to Banach Spaces and Their Geometry. Beauzamy
• Topics in Banach Space Theory. Albiac and Kalton
• Banach Algebra Techniques in Operator Theory. Douglas
• A Short Course on Spectral Theory. Arveson

Anyone interested in studying collaboratively?– my aim is to learn operator theory and the geometry of banach spaces; I’ll be jumping around from book to book, instead of tackling them in their entirety– I’d like to have someone to keep me honest, engaged, and on track (nope, I’m not too good at doing that for myself). By the end of the year, I’d like to have a feel for the big concepts, like type and cotype, the Banach-Mazur distance, operator factorization, and a good handle on some of the big noncommutative inequalities: Khintchine, Gronthendieck, etc.

I’m currently reading (and rereading ) the first chapter Albiac and Kalton’s book on bases and basic sequences.

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• Taxi-cab geometry? (9/15/2004)
• Another plot twist (6/17/2008)
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Usually we prove that if a topological space is arcwise connected, it is connected by contradiction. We assume form a disjoint non-empty open cover of , pick elements from each, and use the path between these elements to construct our contradiction. The same idea holds if instead of having a path between each pair of elements, we have a path between each pair of nonempty open sets– let’s call this setwise connectedness (I made this up; if there’s a standard name, please let me know). Can you find an example of a space that is setwise connected, but not arcwise connected? When does setwise connectedness imply pathwise connectedness?

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• Lattice Problem (2/16/2005)
• Exactness of differential forms (7/28/2007)
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This week, I’ve been having smoothies for breakfast. Not only are they delicious, I’m probably finally getting something near my RDA of fruit– before this, I used to eat maybe one fruit a week on average. Not that I don’t like fruit, it’s that shopping for produce is a bitch. For smoothies, however, you only need frozen fruit, which you can get from Trader Joe’s (I particularly like theirs because they are unsweetened, preservative-free, and the strawberries don’t look force ripened). I wonder if there’s a nutritional difference between frozen and unfrozen fruits?

Anyhow, the agent of this change in my habits is the Magic Bullet blender that arrived sometime last week. I use it almost exclusively for making breakfast smoothies and yogurt-cookies-whiskey concoctions for after dinner snacks, but I imagine I could find many other uses for it. Besides that, it looks very cute sitting on the counter. Here are some pics I took the first time I used it:

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• Magic Bullet (9/27/2008)
• Successful culinary experiments (9/30/2008)
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Probably everyone but me caught that Brandi Carlile is a lesbian; it didn’t even cross my radar until today when I was walking home listening to The Story. The song Josephine puzzles me when I listen to that album– why is she singing about a woman– but I usually am not paying full attention. Since nothing else but the sidewalk was vying for my attention today, I picked up on how she doesn’t refer to gender in her love songs. A quick Google search confirmed my intimation (and now I know she makes many lesbians percolate). Now that I know this, I’m noticing a certain vocal resemblance to Melissa Etheridge. Whether this is chimerical or actual, I’m not yet certain.

Here’s Brandi covering Radiohead’s Creep– she’s fucking special :)– the version on her MySpace page is even better

I wanted to embed Melissa Etheridge’s cover of Refugee (IMO, in her hands, more poignant than it could be in Tom Petty’s) for comparison, but youtube disabled it.

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• Amusing covers (2/2/2008)
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A review of one of Demski’s attempts to slap a scientific facade on the theory of intelligent design, in addition to pointing out the failure of the attempt, raises a valid criticism of those who try to defend true science:

Nonetheless, there are several points intimately related to Dembski’s work that bear emphasizing. First, biologists in particular and scientists in general are horribly confused defenders of their field. When responding to attacks from non-scientists, rather than attempt the rigor that the geometry of induction and similar bodies of statistics provide, they fall back on Popperian incantations, trying to browbeat their opponents into acceding to the homily that if one follows certain magic rituals—the vaunted “scientific method”—then one is rewarded with The Truth. No mathematically precise derivation of these rituals from first principles is provided. The “scientific method” is treated as a first-category topic, opening it up to all kinds of attack. In particular, in defending neo-Darwinism, no admission is allowed that different scientific disciplines simply cannot reach the same level of certainty in their conclusions due to intrinsic differences in the accessibility of the domains they study.

This intrinsic lower certainty of neo-Darwinism than (for example) that of quantum electrodynamics means that there is legitimate room for disputation concerning the history of biology on Earth. So if Dembski had managed to use the geometry of induction properly to quantify that some search algorithm occurring in the biological world had, somehow, worked better than all but the fraction $10^{-50}$ (say) of alternative algorithms, then there would be a major mystery concerning the modern biological mantra. This would be true regardless of whether neo-Darwinists had performed the proper rituals in settling on that mantra.

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• The Structure of Evolutionary Theory (12/2/2004)
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It’s almost Nanowrimo month! That crept up on me …

Anyhow, here’s an interesting integration question which I have no idea how to be about. I believe is integrable over , where is an matrix, (when you integrate against some probability measure, of course, otherwise this is definitely not integrable). This quantity popped up in a tail bound I was attempting to make. More precisely, let be given by where denotes Schur (Hadamard) multiplication, then the Lipschitz constant of is . If we consider as a functional on Gaussian space, we have the deviation bound

Now if we consider to be a function of a random also, then the above bound is conditional on , so to get the desired inequality, we have to integrate out the Hence the question about integrability.

It seems unreasonable to expect there to be some first principles approach to calculating this, say for a Gaussian measure, but maybe there’s some clever argument that’ll get you there.

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• One direction in Khintchine’s inequality for Rademacher sums (9/26/2008)
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If is a nonnegative convex continuous function on such that then the Orlicz space consists of all measurable functions such that




I don’t know much about these spaces, except that certain classes of random variables are characterized by being in an Orlicz space (defined slightly differently– for one, the integration is over , and w.r.t. the appropriate probability measure, and I believe there’s another condition on that constrains its behavior at – but the idea is the same) for some Subexponential random variables (ones whose tails fall off faster than those of a mean-zero gaussian with a fixed arbitrary variance), for instance, are in the Orlicz space corresponding to I suppose that might be one cause of interest in these things, another is that they generalize the spaces.

An interesting question is, why are these Banach spaces?

Possibly relevant posts:

• A characterization of Schauder bases (10/2/2008)
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Today’ll be a good day. I’m going to use the Gaussian-Rademacher trick in the same way I mentioned Latala did, then try to use results on concentration of measure for functions of gaussian matrices to extend our bounds on the expectation of the spectral error in approximating a fixed matrix with some random matrix of independent entries to tail bounds on the probability distribution of that error. Also, I’ll try something along the same lines with our and error bounds.

But first, this equivalence got stuck in my head this morning. Let be a sequence of non-zero vectors in a Banach space; we say it is a Schauder basis for that space if every vector has a unique expansion of the form It’s true that is a Schauder basis for the closure of its span iff there is a constant such that for every and all scalars

Give it a shot.

Possibly relevant posts:

• What good are eigensystems? (9/16/2005)
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Ha ha! On Sunday, I made an excellent macaroni and cheese with porcini mushrooms, bacon, and panko breadcrumbs. I have to commit the blasphemy: this is much better than what my mom makes (or used to make before she effectively gave up cheese). I adapted this lighter recipe, replacing the milk with a mix of mushroom liquid, milk, and cream, adding mushrooms and bacon, and using a mix of gruyere and cheddar cheeses.

For the mornay sauce, I used a medium onion, 1/2 c. flour, a pint of heavy cream (which I originally acquired for the vaporware ginger cheesecake), a cup of the liquid from reconstituting the mushrooms, a cup of skim milk, 2 c. gruyere cheese, and 3 c. sharp cheddar. Reconstitute the mushrooms (I eyeballed it, no exact measurement: about as much mushrooms as bacon) and save some of the liquid for the bechamel. Cook the bacon and use the grease to saute the mushrooms. Chop the onion finely, saute in butter it until almost translucent: you don’t want it too cooked because … next you add the flour piecemeal and cook at the same heat for 4 minutes or so– I added chunks of butter as appropriate while adding in the flour, because this allowed me to be parsimonious with the initial amount of butter and helped me to avoid using more than necessary–, then add in the liquids and bring to a boil. Lower the heat and simmer until the sauce thickens. Now turn off the heat and fold in the cheese mixture.

Cook 1/2 lb. of noodles while doing all the above, until they’re tender (not al dente). Then fold in the cheese sauce, bacon, and mushrooms; add salt, pepper, and nutmeg to taste. Decant the mixture into a 9×9 pan– I lined mine with parchment paper so I didn’t have to deal with burnt-on sides and bottom, only the grease which leaked through. Melt a tablespoon of butter and mix in a cup of the breadcrumbs (and some parmesan if you have some), then sprinkle this mixture on top of the noodles. Bake in a 400 F oven until it’s done; I probably went about 40-50 minutes, but you should be able to say when from brownness or smell

Next time, I’ll try adding in some basil (this time I only had arugula on hand, and that seems like a weird thing to put in mac and cheese), more mushrooms, experiment with cheeses (use Emmental if I can find it), and try cavatapi pasta. But that’ll be a while; this is way too fatty to eat often.

This is so rich that a small portion will fill you right up, so it could serve as a meal by itself. I was too tired after all that work to actually do it, but I imagined some simple roasted carrots would help diminish the uneasiness you might feel from eating a meal consisting entirely of lipids. At any rate, you’d want to pair this with something simple.

The second culinary experiment. Today I made my first smoothie on the level of Jamba Juice. The secret which made this smoothie so much better than my other attempts was simple: replace the ice with frozen pineapple chunks. So, the recipe is simple: frozen strawberries and frozen raspberries in equal amounts, slightly more frozen pineapple, and OJ. Blend it all together well, and enjoy: fruity and sweet. *Really* sweet, the way I like it, like Jamba Juice’s Caribbean Passion (or whatever it’s called); you can control this by varying the amount of pineapple, or adding ice. Any ideas for other sweetening agents? I’ll try mango, but I’m a little worried that it isn’t sweet enough.

I feel a bit guilty for making that tonight, because my dinner was roasted stone fruit– peaches, nectarines, plums, and randomly apples– with vanilla bean ice cream. I figured having ice cream for dinner isn’t bad if you add fruit and avoid eating anything else after that. I had to throw out one of the peaches– which sucked, because peaches turned out to be the only fruit that agreed with the ice cream– because it turned out to be rotten, and the roasting recipe turned out to be wack. (Incidentally, why is it that every time a recipe states a certain amount of time, I end up having to double or triple it to get the desired results?) So instead of the roasting process as I received and used it, I’ll give what I have extrapolated should give better results: slice your fruits into thin chunks, not so thin that they’ll get waterlogged when you roast them, or when you put them on the ice cream, but not so fat that the insides will be raw when the outsides are well roasted. Sprinkle 2-3 tablespoons of white sugar on them (I used 2 plums, 2 nectarines, 1 apple, and 1 peach; extrapolate) and just enough lime juice to get them wet (you really don’t want to use too much lime juice, or have any free at the bottom of the pan, because then everything will taste like lime– yuck). The only point of the lime juice is to prevent the fruits from oxidizing; toss the fruit to coat it, then bake in a preheated 400 F oven until caramelized; toss them when halfway done and swish the juices to keep from burning. Ideally you’ll get a nice sauce from the caramelized sugar and the fruit juices, and nice roasted flesh. Bowl your ice cream and put the hot fruit on top. I wonder what else you can eat these with.

Disappointing as the fruit turned out to be, the ice cream was still ice cream, so following it with smoothies was a bit much.

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• Indian cooking (9/14/2006)
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