Henry Cohn
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Registered User
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19h |
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help with math homework, just need a push in right direction This is also the same problem as mathoverflow.net/questions/130897, which was asked and closed three hours earlier. |
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1d |
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Is there any proof that you feel you do not “understand”? The diagonal argument that fails is the explanation from pages 36-37 of Soare's book Recursively enumerable sets and degrees, right? I like that explanation, but I find the way Sipser approaches it in Introduction to the theory of computation even more compelling. |
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2d |
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Could the Jacobian conjecture be undecidable? @Steven Landsburg: For each fixed $n$ it is, but you can't quantify over $n$ in the first-order language of fields. |
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May 14 |
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Research on the structure of a non-Goldbach number? I think the issue is what "structure" means. Loosely interpreted, all research on the Goldbach conjecture deals with the structure of a hypothetical non-Goldbach number, where existence/non-existence is considered the most basic structural property of all. Of course that doesn't sound like what you're looking for, but saying more requires pinning down what really counts as structure. I agree with Johan that the answer to your question is probably "no", and structure of specific numbers probably just isn't relevant. However, it's not clear whether we're talking about the same thing. |
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May 14 |
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Reference for original paper (but translated to English) of Matiyasevich’s proof of Fibonacci relation being Diophantine? The English translation was published in Soviet Math. Dokl. 11 (1970), 354–358. I don't know of an online version, but if it's not in your library you can ask a librarian how to get it by interlibrary loan. |
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May 5 |
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Modern Mathematical Achievements Accessible to Undergraduates Regarding the 290-theorem, the proof is not accessible to undergraduates. (As written, it uses the Ramanujan conjecture for weight 2 cusp forms. There might be an undergraduate-accessible proof, but the one Bhargava and Hanke wrote up isn't it.) For the 15-theorem, I don't recall anything nearly as high-powered, so I think you could teach it to undergraduates, but you'd have to spend some time teaching them about genera of quadratic forms first. |
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May 5 |
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Collision resistance of hash functions after permuting one hash digest I can't think of anything so far that isn't shallow. Certainly given $H$ you can construct a $\pi$ for which you can find a permuted collision, and given any $\pi$ other than the identity you can modify $H$ (by composing it with another permutation) to produce a collision-resistant hash function that has a $\pi$-collision. So if $H$ or $\pi$ is chosen adversarially, you're in trouble. Presumably for most hash functions this works in practice if $H$ and $\pi$ are chosen independently in some sense, but I don't see how to formalize this offhand. For example, what if $\pi$ is chosen at random? |
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May 4 |
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the position of strings Mathoverflow is intended for research questions, so this question would be more appropriate for another site (see the FAQ list for suggestions). |
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May 3 |
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Irrationality of pi*e, pi^pi and e^(pi^2) Irrationality proofs generally aren't useful in any practical sense, but they can certainly be enlightening. |
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May 3 |
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Diagonalize the simultaneous matrices and its background This question doesn't make sense as stated, since "nonnegative definite" and "Hermitian" aren't defined for a general field. (Even if your field is the complex numbers, do you want to assume $A$ and $B$ commute, so you can take $P$ to be unitary?) See mathoverflow.net/questions/118680 for information about which fields have the property that every symmetric matrix is diagonalizable. |
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May 3 |
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Quantum algorithms for dummies P.S. Jordan also links to several survey papers in the navigation bar on the right. |
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May 3 |
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expectation of ln(1+e^x) Of course it's easy to write down the answer as the integral of $\log(1+e^x)$ times the density of the distribution. This lets you compute to much greater precision than you can get by simulation, but offhand I don't see any reason to think the integral can be evaluated in closed form. I'd recommend asking about this at math.stackexchange.com instead. |
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May 3 |
answered | Quantum algorithms for dummies |
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May 2 |
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What fields can be used for an inner product space? It's also worth keeping in mind that people often look at analogues of symmetric and even Hermitian inner products in cases without positive definiteness. (For example, to define unitary groups over finite fields.) This is certainly somewhat different from the real/complex/quaternionic case, but it's not like there's a sharp dividing line conceptually. Instead, you just give up more properties. |
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May 2 |
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What fields can be used for an inner product space? In the Hermitian case "field" is overly restrictive (the skew field of quaternions is important too). Even in the symmetric case, it can be convenient to look at rings: for example, integral lattices are often thought of as free abelian groups with integer-valued, positive-definite, symmetric bilinear forms. (Of course integers are real numbers, so this can be thought of as a special case of real inner products, but the restriction to a subring changes how things feel.) |
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May 2 |
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Why is it hard to prove that the Euler Mascheroni constant is irrational? I feel like this question points in the wrong direction. As far as I understand (I'm not an expert), there's nothing special about $\gamma$ that makes it particularly hard. Instead, just about all irrationality questions are hard by default, and it's $e$ and $\pi$ that are special in being unusually tractable. |
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May 1 |
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What are zeros of certain entire functions? What do you mean by "describe"? There are certainly some basic things you can say, for example because you've got an entire function of exponential type. How useful this might be depends on what you need to know. |
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Apr 22 |
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Is rigour just a ritual that most mathematicians wish to get rid of if they could? What's room101.mathoverflow.net? |
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Apr 21 |
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Is rigour just a ritual that most mathematicians wish to get rid of if they could? References to "thought police" aren't constructive. If you disagree with the closing, you can discuss it at the link Qiaochu gave above (meta.mathoverflow.net/discussion/1579). |
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Apr 20 |
awarded | ● Enlightened |
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Apr 20 |
awarded | ● Nice Answer |
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Apr 20 |
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Optimal 8-vertex isoperimetric polyhedron? @joriki: That's great! |
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Apr 20 |
accepted | Which hard mathematical problems do you have to solve to earn bitcoins ? |
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Apr 20 |
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What does a mathematician expect from mathematics education? What would you suggest as a good starting place for learning about Bass's ideas on granularity? |
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Apr 20 |
answered | Which hard mathematical problems do you have to solve to earn bitcoins ? |
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Apr 8 |
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How long can this string of digits be extended? @Will: I wrote a pari program to compute this. I'm on my cell phone right now but would be happy to send it to you (or anyone else) a bit later this evening. Send me an email if you're interested. |
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Apr 8 |
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How long can this string of digits be extended? Incidentally, 25 is pretty close to $10e$, in accordance with the heuristic Will Sawin gives. |
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Apr 8 |
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How long can this string of digits be extended? It's surprisingly quick and easy to find the longest such n by brute force (the number of possibilities of each length doesn't grow much, and for base 10 it never exceeds 2492). For example, for base 10 the longest is 3608528850368400786036725, which has length 25. However, this doesn't give a lot of insight into how $N(b)$ depends on $b$. |
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Apr 7 |
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Can we find a Groebner Basis? The leading term ideal $LT(I)$ does not generally determine the original ideal $I$ (think about principal ideals in one variable), while a Groebner basis does, so you can't find a Groebner basis for $I$ given just $LT(I)$. But maybe I am misunderstanding your question - what are you trying to do and why might it be possible? |
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Apr 6 |
awarded | ● Good Answer |
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Apr 5 |
accepted | Is there a “mathematical” definition of “simplify”? |
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Apr 5 |
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How to refer to a theorem that you have shown to be wrong This might be confusing to someone stumbling across it in a paper (it might be read as "this theorem was proved between 1983 and 1987 but published later"), but I love it. |
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Apr 5 |
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A way to get translations of foreign papers? See also Theorem 4.4.1 and equation (4.19) in "Finite Packing and Covering" by K. Böröczky - are they what you are looking for? There's a preview on books.google.com. |
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Apr 5 |
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A way to get translations of foreign papers? Both books say quite a bit about packing and covering on the 2-sphere. If I remember right, there is some stuff in "Lagerungen..." that didn't make it into "Regular Figures" (for example, the optimal 7-point code in S^2), but the material dealing with regular polyhedra is in both. So if you mean a bound in terms of trig functions that is sharp for a few regular polyhedra, I think you'll find it there. |
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Apr 5 |
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List of Charlatans in Mathematics Artie Prendergast-Smith makes an excellent point about the word "charlatan". Just to be clear, that word implies intentional dishonesty. |
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Apr 5 |
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A way to get translations of foreign papers? Regarding the specific case you mention, I don't know exactly what's in this paper, but Fejes Tóth discusses covering problems on spheres in his book "Regular Figures" (which was translated into English); see section II.2.5. If you read German you might also find information in "Lagerungen in der Ebene, auf der Kugel, und im Raum", but it sounds from your last sentence like you don't. I doubt you'll find a translation of this paper, so your best bet is if he included the contents in one of these books. |
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Apr 5 |
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List of Charlatans in Mathematics It's a tricky subject that needs to be handled very tactfully, but your article could make a valuable contribution and I would be interested in reading it. On the other hand, compiling a list like this is not a good fit for MO (it is likely to lead to extended discussion and controversy). |
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Apr 5 |
awarded | ● Nice Answer |
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Apr 5 |
revised |
Is there a “mathematical” definition of “simplify”? fixing wrong description of second function $f$ |
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Apr 5 |
revised |
Is there a “mathematical” definition of “simplify”? Oops, adding a "computable" |
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Apr 4 |
revised |
Is there a “mathematical” definition of “simplify”? added 247 characters in body |
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Apr 4 |
answered | Is there a “mathematical” definition of “simplify”? |
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Mar 20 |
awarded | ● Nice Answer |
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Mar 18 |
awarded | ● Yearling |
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Mar 2 |
accepted | Kissing Number of Spheres in Non-Euclidean Geometry |
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Mar 1 |
answered | Kissing Number of Spheres in Non-Euclidean Geometry |
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Feb 27 |
revised |
How many idempotent relations are there on an $n$-element set? fixed typo in title |
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Feb 3 |
awarded | ● Good Answer |
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Jan 29 |
awarded | ● Enlightened |
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Jan 29 |
awarded | ● Nice Answer |
