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the fastest man alive
Interesting. As the 1990 Flash tv show was what got me into the Flash and comic books in general in the first place, I’m kind of stoked. As for what it might mean about the fate of my beloved Wally West, I dunno. We shall see.
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mission accomplished
About and links are done, as difficult as they were. I’ve also validated the entire site as xhtml 1.0 strict, although I’m sure if one looked hard enough, one could find a couple of entries in the archives that break it, thanks to various <p> tags without slashes. CSS validation isn’t quite there yet.
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my own mini-google
Search is up and running. I also fixed a little bug with the archive feature that was treating a month’s page like the regular front page and only showing ten entries rather than the entire month’s entries. So, the new site is essentially coded now, as the only features I have yet to add are about and links, which both require zero PHP/MySQL interaction. Makes me sad. Although I do have a couple ideas for possible future features. Future features, future features. Say that ten times fast.
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friedcheese.org through the ages
Archive is up and running. The major improvement over the previous version is that links to months (or in the case of 2005, years) which contain zero entries no longer exist. Also, the archive is merely an if statement within index.php, not its own separate page. Perhaps sometime I’ll mess around with some nifty CSS or Javascript expandable/collapsible menus or somesuch.
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tabbies!
Check out these uber CSS navigation tabs in the upper right hand corner. Took some rather sloppy PHP coding to finally get them to work right, but it looks pimp, right? For now, they don’t actually go anywhere special. Props to this site from where I ripped off the CSS code that made this possible.
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are ess ess
Feed me, biatch!
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pwned by timestamps
I’ve made a small change concerning how timestamps for articles are recorded. Before, it just kept track of the date. Now, it keeps track of the time as well. This is because entries were showing up out of order. Of course, all previous entries have defaulted to midnight, except for a couple (namely the last four), which I hand-tweaked.
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three blogs combine to become one! muahahaha
Well, I suppose my last entry was a bit off, as I’d forgotten about my short-lived math blog which existed in the middle there somewhere. My old blogs were divided between three different MySQL tables, so I’ve combined them into one. More functionality to come later. And then… content!
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test
So remember that major overhaul I referred to in the last entry, nearly two years ago? I think I might do it. Here’s the first attempt at an entry. Aside from the new look, I’ve also added titles, with the hopes of eventually implementing an RSS feed.
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Dot Product, Cross Product, Complex Product?
Alrighty, time for some actual math. First, a little word about who I am. I graduated a little over a year ago from Northwestern University with a B.A. in math. I kind of half-assedly looked for jobs as an actuary, but ended up moving back home to Ohio and began working as a retail clerk, eventually getting promoted to management. Gradually, I realized that maybe the corporate world of insurance wasn’t for me, and I decided to go back and get my PhD. So, if all goes according to plan, I’ll be doing that in the fall of 2005. Until then, it’s just me, my old math books, and this blog.
So I whipped out my algebra book from winter and spring quarters my senior year and started just doing problems the other day. The first section deals primarily with complex numbers, and it got me to thinking. When we graph complex numbers, we visualize them in 2-dimensional space. A complex number ceases to really be the sum a+bi, but rather the vector (a,b). However, any homework problems in either high school or college dealing with the multiplication of complex numbers didn’t deal with graphing the products on the Cartesian plane. So I did a little messing around. The product of the two complex numbers a+bi and c+di is ac-bd+i(ad+bc). Converting this to our little system of using vectors in 2-space, we have (ac-bd,ad+bc). And what does this look like geometrically? It’s actually quite lovely. First, we’ll take a look at the length of the vector. sqrt((ac-bd)^2^2) ends up simplifying to sqrt((a^2+b^2)(c^2+d^2)), which happens to be exactly the product of the lengths of our two original vectors (the ones we multiplied together). Imagine that. Okay, how about the angle? This is a little bit trickier, as you need to use an obscure arctangent identity I can’t remember ever learning, but fortunately found at this site after a quick Google search. arctan x arctan y = arctan ((x+y)/(1-xy)). It turns out that the angle between our product and the x-axis is simply the sum of the angles between our original two vectors and the x-axis. Fantastic!
So what does all this mean? Throughout my college coursework, the only products of two vectors that ever had any application were the dot and cross products. I guess this “complex product” serves mostly to just say, “Hey look! Multiplication of complex numbers makes a lot of sense geometrically!”
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