Fun question:
"what are...non-trivial examples of mathematics where the parity of an integral parameter makes a crucial difference?"
https://mathoverflow.net/questions/447780/oddities-of-evenness
I expect many of these are related. I find the difference between odd-dimensional and even-dimensional rotation groups interesting and that has wide-ranging ramifications.
@dpiponi - in everyday life where parity makes the biggest difference is the number of socks in my drawer. But that's not what they wanted to know.
@johncarlosbaez It's not so bad if you have an odd number of socks that are identical though you want to rotate which ones you pick to keep the wear and tear evenly distributed.
@dpiponi @johncarlosbaez : I'm going to use this example to illustrate the fallacy of index funds. The S&P index lists the best performing socks. They regularly unlist the worn out socks, and replace them with brand new socks. It costs them nothing. An index fund, on the other hand, has to buy the socks that have been newly listed, and sell the worn out ones that are exiting the index. So it's buying high and selling low.
@BartoszMilewski @johncarlosbaez You're making a testable claim here: that when a stock is listed that it's high, which I have to interpret as meaning you expect it to perform worse, in the future, than other stocks. Do you have some data to justify this claim?
@dpiponi @johncarlosbaez I haven't done the calculations, just some intuitions from physics. I see the stock market as an ideal gas, and index funds an attempt to implement a Maxwell's demon.
@BartoszMilewski Isn't that analogy implicitly assuming that the system is in equilibrium? Economies tend to grow (in real terms) over the long haul (presumably due to population and productivity growth), so I don't think they are analogous to a system in equilibrium. @dpiponi @johncarlosbaez
@internic @dpiponi @johncarlosbaez In thermodynamics we assume that the system is always reasonably close to equilibrium, but there may be a heater attached to it, or more gas is pumped into it. After all, thermodynamics works for the internal combustion engine.
@BartoszMilewski I'm not saying thermodynamics can't be used as analogy at all (though I'm not convinced it's an apt one); however you were talking about Maxwell's demon, which is usually discussed in reference to extracting work from or creating disequilibrium in closed systems at equilibrium. I guess I'm saying that if anything the stock market is more analogous to a heat engine, where one can perfectly well extract work or bring previously equilibrated subsystems out of equilibrium (through compression or heating), because it is an open system, and part of a larger system that is not even approximately in equilibrium. @dpiponi @johncarlosbaez
@BartoszMilewski If the typical time scale from listing to de-listing of a given stock were small compared to the time scale over which the value of the index changes appreciably, then you might be able to think of it as quasi-static, but I think in actuality you're closer to the opposite regime. @dpiponi @johncarlosbaez