Living to the leeward side of a mountain range treats you to special clouds. One of my favorites is an altocumulus that creates a smooth fade from white to blue as seen here on the cloud's left edge—although the effect is (as usual) less striking in a pic than when you're just looking at it. Boulder, Colorado was the first area I lived in that had such clouds, and they were part of what endeared the place to me. Lone Pine has them too.
The climbing route my buddy is at the base of is new enough that it's not in the guidebook for the area. Unlike most of the routes in my neighborhood, we have no idea whose work this one is or what it's named. But to whoever developed it: thanks—fun route.
The tent belonged to other climbers visiting the area who we met later on and who had cool-looking dogs.
The upper-rightmost rock looks to me like a masked face in profile. Interesting, how looking at a photo can make me notice stuff I missed while I was taking it. Perception is so selective. This has been CPAC week, with various conservative specimens on full display. I have an other-than-political angle in posting today, but first a few CPAC moments:
I don't know how you're going to keep those numbers a secret.WOFF is a popular format for delivering fonts for use in web pages. It may be that the main motivation for WOFF wasn't technical but rather to make web fonts just a little harder to use for other purposes. (This isn't the first time that proprietary interest has caused proliferation of font formats. TrueType was created because Adobe was keeping a tight hold on PostScript Type 1.)
Knuth had an interesting comment on the economics of fonts:
Scientists are supported by the National Science Foundation to discover science, which benefits the human race. Artists, or font designers, are not supported by the National Font Foundation to develop fonts that are going to be beneficial to the human race. Fonts are beneficial to the human race, they just don't traditionally get supported that way. I don't know why. They're both important aspects of our life. It's just that one part has traditionally gotten funded by a royalty type mechanism and the other by public welfare grants for the whole country.Arizona has been in the news this week, culminating today with Governor Jan Brewer's veto of SB 1062. Der Spiegel put it bluntly with the headline Arizonas Gouverneurin stoppt Steinzeit-Gesetz gegen Schwule (Arizona's governor stops Stone Age law against gays).
Although the bill got attention for its relevance to discrimination against gay people, it's nominally about religious freedom. It was, of course, the work of Republicans—who like to bring up religion, especially when it lets them avoid mentioning homosexuality.
Assertion of religious belief can be the magic words that compel deference in the USA. If you were drafted during the Vietnam war and applied for conscientious objector status, you would be interviewed by a draft board. If you said you'd thought long and hard and had your own principled reasons why you objected to being a soldier, you could expect to be questioned further. But if you said "I was raised a Quaker" you'd get conscientious objector status with no questions asked.
SB 1062 would've said: to discriminate, just wave the flag of religion. (i.e., 1 + 2 + 3 + 4 + 5 + ...)
can be manipulated to have a finite sum, to wit: -1/12, a result that seems more than a little preposterous.
I've gotten used to counterintuitive results when infinities are involved but this is truly off the wall. I don't know quite what to make of it.
I can follow various methods that yield this -1/12 sum but I'm not in a position to comment on how meaningful or rigorous they are. I am disinclined to pronounce them worthless for several reasons, not least of which is that they seem to have practical applications in physics. I refer interested readers to the equation's Wikipedia page.
There is precedent for practical mathematics that lacked rigor when first devised. Leibniz did calculus by means of infinitesimals—numbers smaller than any positive real but larger than zero—a concept with no solid definition that struck quite a few people as preposterous. It wasn't until the 1960s (!) that infinitesimals were put on a sound footing.
Infinitesimals aren't the only way to formalize calculus. Epsilon-delta limits were made rigorous first, but that took a while too (roughly 150 years after Newton and Leibniz).