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After doing nearly all the coursework for a Ph.D. in math, I then did all the coursework for a Ph.D. in statistics and completed that degree.
Aug
23 |
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Important formulas in Combinatorics
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Aug
17 |
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Writing papers in pre-LaTeX era?
Where you say "already printed" do you mean "already typed"? ${}\qquad{}$ |
Aug
14 |
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How many rearrangements must fail to alter the value of a sum before you conclude that none do?
So $\|f-g\|<\infty$ where $\|\cdot\|$ is the sup-norm. ${}\qquad{}$ |
Aug
14 |
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How many rearrangements must fail to alter the value of a sum before you conclude that none do?
So for example $e_{3,1} - e_{3,3} + e_{3,5} - \cdots$ $= x_1 + x_2 + x_3 - x_1 x_2 x_3$ and $e_{4,1} - e_{4,3} + e_{4,5} -\cdots$ $= x_1+x_2+x_3+x_4 - x_1x_2x_3 - x_1x_2x_4 - x_1x_3x_4 - x_2x_3x_4$. Thus we have added four new terms in a certain order by incrementing $n$ from $3$ to $4$. ${}\qquad{}$ |
Aug
14 |
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How many rearrangements must fail to alter the value of a sum before you conclude that none do?
$\ldots\,{}$if I can show that (for the particular sequences $x_1,x_2,x_3,\ldots$ that are of interest) no such induced rearrangement changes the limit, is that enough to imply that no other rearrangement does it? Including, for example a rearragement that takes one of the many terms within $e_{n,5}$ and puts it after all the terms in $e_{n,9}$, etc.) I think I can show absolute convergence by a comparison test, but I'm not sure I want to do it that way in this case. ${}\qquad{}$ |
Aug
14 |
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How many rearrangements must fail to alter the value of a sum before you conclude that none do?
OK, here's a very narrow special case: $e_{n,k}$ is the sum of all products of $k$ of the variables $x_1,\ldots,x_k$. For example $e_{4,2} = x_1x_2 + x_1x_3 + x_1x_4 + x_2x_3 + x_2x_4 + x_3x_4$ (that implies $e_{n,k}=0$ when $k>n$ and $e_{0,0}=1$). So consider $\lim\limits_{n\to\infty}(e_{n,1} - e_{n,3} + e_{n,5} - \cdots)$ (for each $n$, only finitely many nonzero terms are here). Each rearrangement of $x_1,x_2,x_3,\ldots$ induces a rearrangement of the series whose limit is taken above. But not all rearrangements of the series whose limit is taken are induced in that way. So${}\,\ldots\ {}$ |
Aug
14 |
awarded | Nice Question |
Aug
14 |
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How many rearrangements must fail to alter the value of a sum before you conclude that none do?
@JoelDavidHamkins : Apparently one is not allowed to use more than five tags. ${}\qquad{}$ |
Aug
14 |
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Rearrangements that never change the value of a sum
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Aug
14 |
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How many rearrangements must fail to alter the value of a sum before you conclude that none do?
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Aug
14 |
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How many rearrangements must fail to alter the value of a sum before you conclude that none do?
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Aug
13 |
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How many rearrangements must fail to alter the value of a sum before you conclude that none do?
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Aug
13 |
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How many rearrangements must fail to alter the value of a sum before you conclude that none do?
This was inspired by a simpler problem: I have a particular somewhat well behaved sum and a particular somewhat well behaved class of bijections none of which alter its value. It would be convenient to deduce from that that no bijection alters its value. That's probably far easier than answering the question as I've phrased it here. ${}\qquad{}$ |
Aug
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asked | How many rearrangements must fail to alter the value of a sum before you conclude that none do? |
Aug
5 |
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Rearrangements that never change the value of a sum
Levi's duality: researchgate.net/publication/… ${}\qquad{}$ |
Aug
5 |
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Rearrangements that never change the value of a sum
Agnew R.P., Permutations preserving convergence of series, Proc. Amer. Math. Soc., 1955, 6(4), 563–564, Levi F.W., Rearrangement of convergent series, Duke Math. J., 1946, 13, 579–585, Pleasants P.A.B., Rearrangements that preserve convergence, J. London Math. Soc., 1977, 15(1), 134–142, Schaefer P., Sum-preserving rearrangements of infinite series, Amer. Math. Monthly, 1981, 88(1), 33–40, |
Aug
5 |
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Rearrangements that never change the value of a sum
$\ldots\,{}$ and another: projecteuclid.org/euclid.pjm/1102688295 ${}\qquad{}$ |
Aug
5 |
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Rearrangements that never change the value of a sum
I'd better record the following here before I lose track of it, even if I don't look at it before tomorrow: link.springer.com/article/10.2478%2Fs11533-012-0156-x#page-1 ${}\qquad{}$ |
Aug
5 |
awarded | Nice Question |
Aug
5 |
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Rearrangements that never change the value of a sum
I also find this: "Rearrangements that Preserve Convergence", Journal of the London Mathematical Society, volume s2-15, issue 1, pages 134-142. jlms.oxfordjournals.org/content/s2-15/1/134.full.pdf+html ${}\qquad{}$ |