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Mathematics professor at Cambridge
Jul 1 |
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How to prove a known inequality from a book
My guess would be that to minimize the inner product, you make both vectors as "unflat" as possible (that is, you make sure that each one takes just two values and that the ratio of those two values is maximized) and as different as possible (that is, you make the big values of one vector appear where the small values of the other vector appear). Then all that's left to determine is the size of one set (where the big values of one vector occur), which ought to be doable. To prove all this, I would try a variational argument. |
Jun 30 |
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Name of a metric space concept
Oh yes -- it's like the uniqueness part of strict convexity but not the existence part. It's probably too much to hope that a metric space with Neil's property can be isometrically embedded into a strictly convex metric space in Bula's sense. |
Jun 30 |
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Name of a metric space concept
No time to check, but it looks related to Definition 2.6 in this paper: home.lu.lv/~ibula/lv/petnieciba/raksti/moravica.pdf |
Jun 30 |
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Zeros of functions involving polynomials and square roots
I can't think of any examples to show that the multivariable version of the question is more interesting than the single-variable version. If you fix all the variables but one, then the complex plane will, I think, split up into domains on which the resulting function is holomorphic, and therefore either constant or having isolated zeros. As Jacques Carette observes, both can occur. If you somehow add a condition to rule out constant parts, then you'll get measure zero. Or am I missing something? |
Jun 29 |
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Shortest closed curve to inspect a sphere
I presume you mean "closed curve in $\mathbb{R}^3$". |
Jun 29 |
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Generalizing a problem to make it easier
a multiple of U(x). But then I would have gained nothing by the generalization. |
Jun 29 |
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Generalizing a problem to make it easier
As it happens, the first example already appears on a subpage of the page linked to in the question: tricki.org/article/… (not that it would be reasonable to have expected you to check all subpages before giving your answer). As for your second example, it's not completely obvious what method you intend for evaluating S(x). Is it antidifferentiation? I myself would use the fact that S(x)(1-x) is of the form aT(x)+bU(x) where $T(x)=\sum kx^k$ and $U(x)=\sum x^k$ and that (1-x)T(x) is ... |
Jun 24 |
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Asymptotics for Ramsey Theory
I'm afraid I don't understand the approach you are suggesting, but the general class of difficult problem this one falls into is where any small example can be turned into a sequence of big examples, but the best small examples known are rather complicated, so one has no confidence in their being best possible or any hint of what the best possible would look like (and it's conceivable that the examples get more and more complicated as they get better and better). |
Jun 23 |
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Asymptotics for Ramsey Theory
Still very much open is what the minimum density actually is ... |
Jun 23 |
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Asymptotics for Ramsey Theory
One remark that I ought to have made is that the minimum number of monochromatic cliques is not what you might expect. You might think that the best example (when n is large) was a random colouring, but a famous result of Andrew Thomason is that if you 2-colour a graph and want to minimize the proportion of cliques of size 4 that are monochromatic, then there is a sequence of colourings that give a lower limiting density than 1/32 (which is what you get for a random colouring). This disproved a conjecture of Erdős. |
Jun 23 |
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Asymptotics for Ramsey Theory
Oops, when I said "original group" above I meant the set (not group) {1,2,...,n}. |
Jun 23 |
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Asymptotics for Ramsey Theory
PS Would it be useful if I expanded a bit on the regularity-lemma argument? |
Jun 23 |
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Asymptotics for Ramsey Theory
The trick with progressions is to work in the cyclic group of order n instead of the interval {1,2,...,n}. Then, at least if n is prime, the number of progressions of a given step size does not depend on the step size. Of course, not all progressions in the cyclic group correspond to progressions in the original group, but there are various tricks for dealing with this. (One, for instance, is to prove a lemma that every progression of length k in the cyclic group contains a "genuine" subprogression of length about $\sqrt{k}$, which comes from a pigeonhole argument.) |
Jun 22 |
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Digital Pen for Math: Your Experiences?
+1 for use of the word "quid" even if the rest of the answer had been uninteresting (which it wasn't). |
Jun 14 |
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How many definitions are there of the Jones polynomial?
I don't know enough to know whether you would count this as a genuinely different definition, but anyway I think Tutte deserves more publicity than he tends to get for his polynomial, which predates the Jones polynomial by many years. Given a knot, one can convert it into a graph and the Tutte polynomial of that graph restricts to the Jones polynomial: en.wikipedia.org/wiki/Tutte_polynomial, omup.jp/modules/papers/knot/chap01.pdf |
Jun 13 |
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Arithmetic closed subsets
Essentially the same question would be to describe sets of finitely supported non-negative-integer-valued sequences that are closed under the operation of adding disjointly supported sequences together (pointwise). Such sets don't seem to have a particularly interesting description. |
Jun 11 |
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Is this sequence of polynomials well-known?
Actually, I've just looked it up (though your second link takes me to the same page as the first) and I see that the answer is that even "unnatural" sequences such as those obtained by reading a triangle along its rows are included in OEIS, and that subsequences are recognised too. That makes it a more useful resource than I had realized. |
Jun 11 |
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Is this sequence of polynomials well-known?
Out of interest, what made you think that that sequence would yield something? And why not e.g. 1,1,1,1,3,2,1,7,12,6,1,15? |
Jun 2 |
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Examples of theorems misapplied to non-mathematical contexts
Wow, do you think he's making a bid for the Templeton Prize? |
Jun 1 |
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Why is differentiating mechanics and integration art?
The reason, I think, is that the analogy between the two situations isn't all that good. When you solve numerically you are trying to find a number. That coincides with what you are looking for when you solve a quadratic equation, whereas when you are antidifferentiating you are looking for a function, and numerical methods just give you function values. |