An application of the relative bound we derived allows us to show that if is a square matrix with all positive entries within some interval bounded away from zero, then the relative error in approximating it with a sparse matrix with entries where is the probability that can be made small with a That is, we can keep just nonzero entries in the matrix and still have a small relative error.
This result is interesting, but trivial in the sense that the norms are easy to calculate when the entries of a matrix are all nonnegative. Then again, it isn’t too trivial, because calculating the norm of the difference of two such matrices is not easy.
What’d I’d like to prove is similar behavior for the relative error in approximating a matrix that just has unit norm columns, or a circulant matrix, but finding a nontrivial, tightish lower bound for the norm of such matrices is proving elusive.
Where else can I hear someone say things like “you thought you were in love, so you were. that’s what love is,” and shortly thereafter, someone else say “she thinks she’s in love; even if it’s software instead of an emotion, it’s real to her.”
The humanoid Cylons are an embodied cognitive dissonance. If we judge by appearances, they can fall in love, they can be altruistic, they can be petty, they can be manipulative … they have quite human characteristics. Yet we’d like to think there is something more we can judge humanity by, some quality that makes us unique; but what is that exactly?
These are the kind of questions that speculative fiction is meant to address. So whoo-hoo for BSG.
(Based on the XKCD comic Substitute) You’re traveling in a straight line along the increasing x-axis at speed , a raptor starts anywhere in the plane and travels at unit speed, how long does it take to reach you?
The raptor’s motion is modeled with
that is, it moves toward you as you’re moving.
You can make this system slightly more manageable by changing to moving coordinates: transforms it to
where as usual.
Differential equations are not my bailiwick, so I don’t know if indeed as some people are saying, this doesn’t have a closed form solution. I’ve tried the tricks I remember seeing before: polar coordinates, finding DEs for independent functional combinations of , prayer … just kidding No success so far.
My grandma died yesterday evening, probably from complications due to Parkinsons; my dad left me a text this morning, and I didn’t see it until this evening. Before we left Barbados, she was a huge figure in my life, and I’ve always loved her for the things she did with and for me then. My sister and I used to spend more time with our grandparents than our parents; all of my memories from back then involve them, like the time my granddad let my sister keep a billy goat as a pet, and it butt me in the cleft of my lip so hard that my mouth was numb for the rest of the day. Or the way, when it got so hot that we could feet the asphalt through our slippers, grandma used to take us down to the beach and we’d run all the way back on the road, trying to get home fast enough that the fading cool from the ocean would keep our feet from burning. Even the forced trips to the Kingdom Hall, and the few islandwide Jehovah’s Witness assemblies were fun, because I got to be around my grandparents. My grandma was my favorite person back then: she had beautiful long hair that she used to wear down to the small of her back, a quiet air, and a comforting smell My fondest memories are of when we used to sit in the veranda enjoying the sea breeze, watching the tourists going in and out of the pub across the street, and chatting with my cousins. I’ll always remember her that way, happy and absorbed in her family.
The professor who taught me graph theory as an undergraduate was obsessed with fullerenes (math, chemistry). Every class he managed to mention that scientists just down the street at Rice had discovered them, and consequently won Nobel Prizes. It got stale.
He never mentioned that they were discovered accidentally in the course of an experiment originally designed to simulate the internal conditions in stars that form carbon. That’s a much more impressive story, in my opinion, and could be repeated more times without going stale.
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.
Filed under: Mathematics — Alex @ 9:56 am 10/12/2008
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?
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:
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.
bound we derived allows us to show that if 
is a square matrix with all positive entries within some interval bounded away from zero, then the relative error in approximating it with a sparse matrix with entries 
where 
is the probability that 
can be made 
small with a 
That is, we can keep just 
nonzero entries in the matrix and still have a small relative error.
norms are easy to calculate when the entries of a matrix are all nonnegative. Then again, it isn’t too trivial, because calculating the norm of the difference of two such matrices is not easy.
, a raptor starts anywhere in the plane and travels at unit speed, how long does it take to reach you? 
transforms it to
as usual. 
, prayer … just kidding 
No success so far.
is arcwise connected, it is connected by contradiction. We assume 
form a disjoint non-empty open cover of 
