Ask me anything
Recent Responses
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Well, Rubik's cube is really a puzzle about group theory, as I explained a few years ago (http://unapologetic.wordpress.com/category/special-topics/rubiks-cube/ posts in reverse chronological order).
The only thing I can think of using vectors is to replace the sub-cubes with 27 vectors with {-1,0,+1} components. Rubik's group then acts to permute these vectors. But then it really doesn't matter that they're vectors, and it's not really using "vector algebra". -
Yeah, a direction field is just a particular kind of vector field. As for using tools from vector calculus, I don't think there are any that aren't already worked into the standard approaches to differential equations.
That said, a number of results in vector calculus -- particular the more advanced parts -- can be viewed as talking about how to solve particular partial differential equations. For instance, given a vector field we can ask if there's a function whose gradient is that vector field. This is a collection of three differential equations in three different variables that need to be solved simultaneously. -
In principle it does, but the practice is a bit more complicated. I'm not sure that there's a lot of value to be added in a video presentation for one thing.
For another, creating such videos takes a fair bit of effort and expense that I'm not willing to part with, given my need to hold down a real, private-sector job now, along with my other hobbies. Salman Khan had the benefit of a hedge fund analyst's income to buoy him when he started; we aren't all blessed with such resources. -
Largely, I've tried to go along in an order that only refers back to concepts I've previously discussed. Technically, nothing should be inaccessible to someone following from the beginning.
Unfortunately it's been years since I've even looked at any of the older stuff. If I were in a position that I could justify the effort and expense involved in curating the material into a book, I might do that, but I'm just not able to do that as things stand. -
Sure, so long as you're not interchanging the differentiation and integration over the same variable.
But "Feynman integration" is usually synonymous with path-integrals. Those don't usually have a sensible mathematical formalism underlying them, though in certain specialized cases a real mathematical sense can be made, and they do give physically interesting answers. -
Because those are more specialized concepts; a category is meant to be the most general concept capable of capturing the idea of function composition.
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The best I can suggest is the previous/next links. I admit it's not the greatest solution, but I don't really know of a better platform out there other than writing a whole book myself.
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It's probably a bug that arose when I split the weblog apart into the straight-math and more personal sides.
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Unfortunately, that would take quite a lot of effort to curate such a PDF. If I had a book offer or something it might be worth doing, or hiring an amanuensis, but as it is I'm inclined to just push forwards.
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I see the same words, and they're still meaningless. It's the sort of thing I hear when people are trying to co-opt mathematics and physics terms in order to make their woo-woo nonsense sound sciencey.
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Could they have determined what? There's no statement there, just gibberish.
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I'd have to say the latter, but it's hard to pin down exactly what they are. It's more that I recognized that my pay was falling at each position, while I was required to teach more and was given less time for research, and there was no guarantee that there would still be a position in a year.
I suppose the first criterion would be "willing to hire for more than a single year", followed by "provides time and resources (and culture) to encourage research", then "supports and effectively trains instructors, particularly when they have to teach below the calculus level". -
usually about an hour. As for learning, I read what I can but there's less and less time with the constraints of a real job.
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Actually, everything I've covered so far has been something that I believe should be within the knowledge of at least a masters-level student, and yes, at that point a broad view should be taken.
That said, I may actually be a little more specialized than I seem in my coverage. I take a particularly categorical -- or at least algebraic -- slant to every topic I cover. Even in analysis and measure theory I try to emphasize the algebraic nature over all those messy limits and error bounds.
Algebra is, to my mind, simply more fundamental than analysis, which we just don't yet have the right tools to deal with in a purely algebraic manner. And both of them are ultimately tools to solve geometric problems. -
I work for a small programming company. I'm out of academia because the job market there has collapsed and it's going to get much worse before it ever gets any better.
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I'm not going to do your homework
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Pure tachyons are like photons corresponding to a single frequency of light, and in terms of solutions of field equations they give "plane waves". But these solutions have infinite support: there is a nonzero field value over basically all of space and time.
But real signals obviously don't have this property, or we couldn't send them at one time, since they would have had to exist before that point. They also can't be spatially infinite because we only exist in a finite portion of space. So we get around this by adding up a bunch of plane waves with different frequencies and traveling in different directions. When we add them up we get a new solution of the wave equations called a "wave packet". And while the individual plane waves have one propagation velocity, the packet may propagate at a different velocity.
So it turns out that if you make wave packets with a tachyonic field, the packet velocity is either greater than or less than c, even though the wave velocity is by definition greater than c. Any information is actually sent at the packet velocity. And it turns out that if the packet velocity is greater than c, then it cannot be compactly supported. -
How much is your publishing company going to pay me for it?
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Given that I've been squeezed out of academia, nobody's doing any research with me.
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I don't know who first used that term.
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