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Mathematics professor at Cambridge

Mar 13 |
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Is the ratio Perimeter/Area for a finite union of unit squares at most 4?
Suppose you have an arbitrary union U of unit squares and another unit square S. Is it possible that the ratio of the length of the part of the boundary of S that is not in U to the area of S\U is greater than 4? I presume it is, or else there would be an easy inductive proof. Or is it that this is essentially the question one wants to answer? |

Mar 9 |
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Prime numbers with given difference
If you could solve the case N_1=1, N_2=3 I'd be very interested ... |

Mar 9 |
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Best way to teach concept of real numbers using a hands-on activity?
I too was going to suggest calculating square roots by means of successive approximation. I wouldn't go as far as to discuss the subtleties of the real number system, but this would allow them to feel it at an intuitive level. |

Mar 8 |
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Examples of inequality implied by equality.
It's Cauchy-Schwarz applied to the scalar product of X and the constant function 1. |

Mar 8 |
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Are there any important mathematical concepts without discrete analog?
I'd say the discrete analogue of a continuous function is one that is continuous in some quantitative way (such as being Lipschitz) on a finite metric space. If the finite metric space is one of a sequence of spaces with unbounded size, this can be very useful. |

Mar 8 |
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Are there any important mathematical concepts without discrete analog?
It amuses me that this answer has been accepted when discrete mathematicians would use analogues of every single one of these. I recommend a look at this post of Terry Tao: terrytao.wordpress.com/2007/05/23/… |

Mar 8 |
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Examples of inequality implied by equality.
It's hard to formalize Pete's question, since if A is always at most B then B-A=C for some C that is always non-negative. One needs a notion of "obviously non-negative", which is often provided by a number's being a sum of squares. |

Mar 7 |
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Criteria for accepting an invitation to become an editor of a scientific journal
How busy are you, how many journals are you already on, and how serious is this journal? |

Mar 7 |
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How many ways can we characterize gamma function?
I disagree. A characterization would be a set of properties that turned out to be uniquely satisfied by the gamma function. It's like the difference between defining the reals as Dedekind cuts and characterizing them as the unique complete ordered field. There is a characterization of the gamma function as the unique function that satisfies f(n)=(n-1)f(n-1) and has some other smoothness property. Unfortunately, I can't remember what that other property is. |

Mar 5 |
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Is there a good reason why a^{2b} + b^{2a} <= 1 when a+b=1?
An argument that backs this up to some extent is the fact that the maximum occurs at (0,1), (1/2,1/2) and (1,0). What's more, the place where the minimum occurs is, if my calculation is correct, the place where aloga = 1-a, which doesn't fill one with confidence that a slick solution exists. Even so, I don't completely rule it out. |

Mar 5 |
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Linear Algebra Proofs in Combinatorics?
Actually yes ... It's possible to see who Tricki articles are by if you look at their revision history. |

Mar 4 |
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Counting distinct undirected, partially labelled graphs
Sorry, I meant triples closed under the given rotation (some, but not all, of which give you triangles). If the vertices are 0,1,2,3,4,5, then some examples of such triples are 01,23,45 and 03,14,25, and also the equilateral triangles 02,24,40 and 13,35,51. In fact, there's one more triple, namely 14,25,30, and any graph with rotational symmetry of order 3 is a union of these triples, so there are 32 such graphs. |

Feb 27 |
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Can a connected planar compactum minus a point be totally disconnected?
Is there an example of a compact connected set such that no two points can be joined by a path? |

Feb 26 |
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Heuristic argument for the prime number theorem?
That may indeed be exactly what I am looking for. I'll have to digest it carefully to see. |

Feb 24 |
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Value of “of course” in the mathematical literature
Was it by any chance me? Probably not, but it is something I have often said. But I didn't think it up for myself -- I got it from David Preiss. It's useful not just for evaluating the work of others, but also one's own work. That is, if you've just written a proof of something but don't feel quite secure about it, look for the bits where you didn't give full detail. |

Feb 7 |
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alternative construction of the quotient group
I don't claim it's a good way of doing things, but one could (in desperation) argue that here one is defining quotients in just one situation (what you need to define a group in terms of generators and relations) and getting all quotients out of it. But I wouldn't want to go to the wall on this one ... |

Feb 3 |
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Why does the Riemann zeta function have non-trivial zeros?
That is a very useful comment -- thanks! |

Feb 2 |
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Why does the Riemann zeta function have non-trivial zeros?
That was another of the thoughts that lay behind my question. Somehow the fact that the distribution of primes can't be "better than random" feels like a fact that ought to have an elementary proof using some Parseval-like identity. I suspect the zeta function is sort of doing that (with a Mellin transform rather than a Fourier transform), but it doesn't appear to be saying something simple like, "That function has the same L_2 norm and trivially has L_2 norm at least the square root of n." |

Feb 1 |
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Why does the Riemann zeta function have non-trivial zeros?
That is a very nice argument, but it also has a magic flavour to it, since you somehow manage to bootstrap a very small error (arising from the fact that $\psi_0(x)$ is discontinuous) into a much bigger one (that the error term in PNT must be more like a square root). But perhaps the bootstrapping is done by the functional equation rather than your argument. |

Feb 1 |
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Why does the Riemann zeta function have non-trivial zeros?
It's precisely this issue -- why the error term in PNT isn't absolutely tiny -- that I want to understand. E.g. to prove that π(x) does not approximate $Li(x)$ to within $latex x^{1/3}$, the obvious method is to point to the zeros on the critical line. So I'm going round in circles. With the help of the functional equation one can say that if there are no zeros on or to the right of the critical line then there are none at all, but I don't count that as an intuitive argument. |