April 05, 2016 at 01:54AM

Today I Learned: 1) ...a trick for bounding the possible number of real roots of a polynomial with relatively few terms (but possibly very high-order). In short (I'll give a specific example in a second) you want to look for the number inflection points in the function, which you do by taking the second derivative... and finding roots. If taking the second derivative gives you something that's easy to determine the root number for, then you're done. If not, then you recursively apply the same trick to figure out the number of roots of *that* polynomial.... Example: How many real roots does the equation ax^100 + bx + c = 0have? In general, figuring out how many real roots a high-order polynomial has is difficult. Here, though, you can put a bound on it pretty quickly. The key is that for the equation to have lots of roots, it has to squiggle up and down a lot so it crosses the y-axis a bunch. Each squiggle potentially gives you a new root, but it also requires an additional inflection point in the function, which is a location where the second derivative is zero. So, how many inflection points does ax^100 + bx + c have? To figure that out, take the second derivative and find its roots -- that gives you something like ax^98 = 0 (the constant becomes a new constant, but who cares what that constant is?). Now *that* equation is easy to solve -- x = 0. That means that the function ax^100 + bx + c has exactly one inflection point, which means it has at most three roots (try drawing out a polynomial with only one inflection point and it will pretty quickly become apparent why). What about ax^100 + bx^37 + cx^2 + dx + e = 0? When you take the second derivative, you get ax^98 + bx^35 + c = 0 -- not so easy to figure out how many inflection points this function has. But! You can use the same procedure on the new function, taking the second derivative and ending up with ax^96 + bx^33 = 0, or x^33(ax^63 + b) = 0. 0 is a root, as are the two 63rd roots of ax^63 + b = 0, so the original function can't have more than three inflection points, therefore the polynomial ax^100 + bx^37 + cx^2 + dx + e = 0 can't have more than 5 roots. This only puts an upper limit on the number of roots -- depending on the constants involved, you might or might not actually get y-axis crossings for those extra inflection points. But it's a nice way of bounding high-order polynomials with very few terms. Even better, you can use this technique to bound polynomials with known form but unknown order. As an example exercise for the reader, can you put an upper bound on the number of real roots in the polynomial ax^N + bx^(N-1) + cx + d = 0, for unknown whole number N>3? Also, for those real math majors out there, is this something I should have learned back in college or high school as a matter of course? It feels like the kind of thing that I might have just missed somewhere.... 2) PCR plates for qPCR don't have to be clear. Probably. Which makes sense, really -- both sample excitation and scanning is done from the top, so who cares what the sides are made out of? 3) It looks like Tesla Motors is now profitable! Sort of. Over the last couple of years*, Tesla has operated on revenues in the single-digit billions of dollars and still managed to lose between a few and a bunch of hundreds of millions each year. But! As of a few days ago, Tesla as about 276,000 reservations (!!!) for their Tesla 3, most of which were placed within the last week. That's almost $10 billion in sales on its own, which represents a couple years' worth of operation *on its own* at their current rate of spending. Of course, they don't actually *have that money* yet -- it's just promises to buy -- but it's still a lot of value that Tesla can pretty well count on seeing. Thanks to Andy Halleran for alerting me to this. Also, apologies for misinterpreting some of their financial numbers in an earlier conversation today -- it's not quite so ridiculously rosy as I thought for Tesla. * Tesla's a young enough and fast-growing enough company that looking back farther than "a few years" doesn't make sense to me.