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Mathematics professor at Cambridge

Jun
30
comment 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
comment 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
comment Shortest closed curve to inspect a sphere
I presume you mean "closed curve in $\mathbb{R}^3$".
Jun
29
comment Generalizing a problem to make it easier
a multiple of U(x). But then I would have gained nothing by the generalization.
Jun
29
comment 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
comment 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
comment Asymptotics for Ramsey Theory
Still very much open is what the minimum density actually is ...
Jun
23
comment 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
comment Asymptotics for Ramsey Theory
Oops, when I said "original group" above I meant the set (not group) {1,2,...,n}.
Jun
23
comment Asymptotics for Ramsey Theory
PS Would it be useful if I expanded a bit on the regularity-lemma argument?
Jun
23
comment 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
comment 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
comment 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
comment 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
comment 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
comment 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
comment Examples of theorems misapplied to non-mathematical contexts
Wow, do you think he's making a bid for the Templeton Prize?
Jun
1
comment 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.
May
31
comment Why is differentiating mechanics and integration art?
I'm not sure I buy your argument that the linearization "is somehow a simplification". True, linear functions are simple, but the linearization we perform when differentiating varies from point to point in a nonlinear way, and it's the entire derivative we're interested in here rather than just the derivative at a point. (This is particularly clear if we look at functions of more than one variable, when the derivative is a decidedly more complicated object than the original function.)
May
25
comment nontrivial theorems with trivial proofs
Some might call that a trivial theorem with a non-trivial proof ...