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Greg Kuperberg

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 Name Greg Kuperberg Member for 3 years Seen Jun 12 at 23:01 Website Location Davis, CA Age 45
I am a professor at UC Davis.

 May25 awarded ● Necromancer May14 awarded ● Enlightened May14 awarded ● Nice Answer Apr28 awarded ● Good Answer Apr18 awarded ● Nice Answer Mar28 accepted Element in the absolute Galois group of the rationals Mar21 awarded ● Guru Mar21 answered Can the expansion of a large integer in all bases consist of almost all zeroes? Mar9 awarded ● Good Answer Feb21 awarded ● Nice Answer Feb14 comment The value $\pm 1$ for the square root of Wilson’s theorem, ((p-1)/2)! mod pThese are interesting techniques for computing factorials in general, but it looks like they do not produce a better individual-case algorithm than what you can get from Mordell's identity and algorithms for the class number. On the other hand, the technique of computing a whole bunch of values at once is great. Feb14 revised The value $\pm 1$ for the square root of Wilson’s theorem, ((p-1)/2)! mod padded 177 characters in body Feb13 awarded ● Nice Question Feb13 comment The value $\pm 1$ for the square root of Wilson’s theorem, ((p-1)/2)! mod pOf course, in question 3 I want to exclude congruences that contradict that $p$ is 3 mod 4. Feb13 asked The value $\pm 1$ for the square root of Wilson’s theorem, ((p-1)/2)! mod p Feb8 awarded ● Nice Answer Feb6 comment Concise model of modern fiat money and its non-conservation@Steven First, it isn't necessarily puzzling to me any more. But in your analogy: Cars aren't sold with any specific demand to return them, but let's say that they were, i.e., that car registrations expired and all cars then had to be returned and junked. If every such registration were interpreted as a lease, then total car ownership would formally stay zero forever, even if actual cars increased. So, is it true that (almost) all dollars are borrowed and their total formal ownership is zero? Feb4 awarded ● Enlightened Feb4 awarded ● Nice Answer Feb3 comment Concise model of modern fiat money and its non-conservation@Steven - Yes, but if those two dollars come ultimately from another commercial bank that borrowed its money from the Fed as well, then all of the commercial banks collectively are in trouble unless either (a) monetary debt accumulates somewhere, or (b) the Fed from time to time borrows at a higher rate than it lends. Feb2 comment Concise model of modern fiat money and its non-conservation@Steven - But, consider the model in which the Fed only buys and sells Treasury bonds, and only buys and sells them with American dollars. Then it is not enough for the Fed to only loan money to commercial banks, because where can they get the principal and interest to pay it back? Either debt accumulates somewhere in the economy, or the Fed violates conservation of money by borrowing money at one rate and lending at a lower rate. Again, maybe seigniorage is not the right word for the latter; maybe there is a better word. (And maybe I can ask some economist on my campus.) Feb2 comment Concise model of modern fiat money and its non-conservation@Steven - I guess I might be confused on the definition. My understanding is that there is more than one type of seigniorage. I read this statement in Wikipedia: "Currently, under the rules governing monetary operations of major central banks (including the central bank of the USA), seigniorage on bank notes is simply defined as the interest payments received by central banks on the total amount of currency issued." So, maybe I am not sure that seigniorage is the best word to describe the real answer to my question. But..(see part 2) Jan29 comment How many values a polynomial map misses? Note that when I asked a related question, Bjorn gave a non-rigorous argument that there should be counterexamples. I think that "no actual answer" is too pessimistic; one can conjecture with evidence that the answer is no. mathoverflow.net/questions/9863/… Jan29 comment Concise model of modern fiat money and its non-conservation@arsmath If you look at Steven's first answer in the comments, he describes money creation as Federal lending. That was the answer that I already sort-of knew. But it requires seigniorage, not just loans, to have de jure non-conservation of money. The Fed has to borrow money at one interest rate and lend it at a lower interest rate. (Not necessarily at the same time, or to the same party.) Jan28 comment Concise model of modern fiat money and its non-conservation@Igor I agree that the rules of fiat currency policy with are in some sense trivial; certainly they're not all that complicated. But most good definitions in mathematics and in science are equally "trivial". In any case in economics and politics, people are often so busy discussing consequences that I don't see clear definitions. I'm happy that here I learned something that I was missing. Jan27 comment Concise model of modern fiat money and its non-conservation@Felipe - While I understand that it's a very basic question which does not strictly require formulas, it is a precursor to (at least me) understanding any mathematical model of a currency. Non-conservation of money is available to the central bank through seigniorage, so one can set up a differential equation in the sense of control theory to model the money supply. Such as the one that Steven outlined. I also agree that financial mathematics is not pure mathematics, but I hope that MO isn't only the latter. Jan27 comment Concise model of modern fiat money and its non-conservation@Steve - Sorry, I should say it this way: In the more restricted formula $M = P \cdot L(Y,i)$, I knew where $Y$ comes from, but I didn't know where $P$ comes from, and I also didn't know the real-life mechanism of $M_0$. But now I see how your equation could work as an answer to the second part --- $M_0$ is set by policy (and has no conservation property due to seigniorage) and your equation becomes a differential equation for $P$. Anyway I wish I could accept more than one answer, yours and Michael's, but MO only lets me choose one. Jan27 comment Concise model of modern fiat money and its non-conservation@arsmath Yes, but a Monopoly board is not an accurate model. One of the Fed's activities is to lend money to commercial banks, which raises the question of how they might ever pay it back. If all money could be traced back to the Fed lending money to commercial banks, they wouldn't be able to. Jan27 comment Concise model of modern fiat money and its non-conservation@armsath - If the Federal Reserve could hoard widgets, that's beside the point, because it doesn't. Every other party in the economy operates by certain accounting rules and practices. The rules and practices for a central bank with fiat money are different, and I asked because I didn't have a complete view of them. If you think that the complementary question would have been a better question, you could be right, but my thinking got stuck where it did and I wanted help from MO. Jan27 comment Concise model of modern fiat money and its non-conservation@Felipe - I could have voiced my own opinion of Murray Rothbard, but I chose not to. Not fair to vote to close my entire question because of this side discussion. Jan27 comment Concise model of modern fiat money and its non-conservationBesides, your analogy with manufacturing refrigerators is an explanation of money, but not fiat money. A central bank does not accumulate a hoard of refrigerators in exchange for issuing cash. Instead, it does something more circular which is still (usually) stable. Jan27 comment Concise model of modern fiat money and its non-conservationBecause I didn't know which financial rules make de jure non-conservation of an official currency possible. That was my real question. (De facto non-conservation of money is less surprising, but the two are related.) Jan27 comment Concise model of modern fiat money and its non-conservationThanks for this basic review. Your summary of the money demand equation was helpful and got me thinking along the right lines, but in the end Michael's answer looks a little closer to what I was missing in my thinking. In looking at your equation, I was stuck on where Y really comes from, i.e., how income is possible if all money is lent from the central bank. However, not all money is lent from the central bank due to seigniorage. (And some of it is lent by the central government and then spent, but I didn't think that that was the only non-conservation term.) Jan27 comment Concise model of modern fiat money and its non-conservation@Michael - Thanks for this answer. It does not give a formula, but it helped clear up my confusion, I think. I wrote another answer as "homework" and I would be happy to read your comments on it. Jan27 answered Concise model of modern fiat money and its non-conservation Jan27 awarded ● Nice Question Jan27 comment Concise model of modern fiat money and its non-conservation@quid I'm sorry for the hints of controversy, but the fact is that I'm learning from the earnest answers by Greinecker and Landsburg. I don't think that it's fair to close a question just because there are some ineffectual answers that I didn't want either. Jan27 comment Concise model of modern fiat money and its non-conservation@Felipe - It's not really my question how the Fed decides to expand the money supply, but rather, how it actually does so. I have no clear picture of an overall balance sheet of an economy's money. That is, I have seen such tabulations, involving things like M1 and so on, but I'm not sure what they mean. I can believe that other actors effectively change the money supply -- after all, you can save cash and bury it. I would hope that such activities are easy to list in a total balance sheet. Jan27 comment Concise model of modern fiat money and its non-conservation@S. Carnahan - Thanks! Your remark resembles part of an answer. But only part of one, by itself not clearly expressible as an equation. I also don't know how to define "seigniorage", and also the disclaimer "I am speaking from a standpoint of ignorance" doesn't help. But this is at least something. Jan27 comment Concise model of modern fiat money and its non-conservationI hope that it can be taken as a valid question in applied mathematics, although I realize that it does not work well as a pure mathematics question. For example, the rules of Monopoly are in fact mathematically rigorous (if not very interesting as mathematics) and are even worth discussing as an inaccurate toy model. Jan27 comment Concise model of modern fiat money and its non-conservationI'm sure that Hayashi and Matsui wrote an interesting paper,but they seem more interested in establishing a result about fiat money than describing what it is --- after all, this paper was written in 1994 and it does not look like it is meant just as an exposition. It looks more technical than what I had in mind. As for Murray Rothbard, I would much prefer a short explanation with equations than entire books with no equations. Jan27 asked Concise model of modern fiat money and its non-conservation Jan21 comment Volume-like property to upper bound lattice points in a convex bodyHere is a primitive bound for lattice polytopes (not necessarily centrally symmetric) that is already interesting. It looks like a lattice polytope can be triangulated by lattice simplices with no internal vertices. This yields $N \le (n+1)!(\text{Vol} K)$. This bound is apparently refined by a bound on Ehrhart coefficients due to Betke and McMullen. This estimate and the Betke-McMullen bound might already be what I'm looking for. I will leave the question open for now in case there are further ideas. Jan21 asked Volume-like property to upper bound lattice points in a convex body Jan18 awarded ● Good Answer Jan1 comment Algorithm for determining whether two polynomials have the same splitting fieldSo, does this rigorously establish a randomized polynomial time algorithm for the original question, assuming GRH? Dec26 awarded ● Nice Answer Dec23 awarded ● gr.group-theory