User richard bonne - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-20T04:28:28Z http://mathoverflow.net/feeds/user/21700 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/130721/diophantine-equation-with-primitive-nth-root-of-unity Diophantine equation with primitive nth root of unity Richard Bonne 2013-05-15T14:01:59Z 2013-05-16T00:48:08Z <p>Fix an $n$th primitive root of unity $\xi$. I need to understand if we can characterize in an easy way all the solutions $k \in \mathbb{Z}$ of the equation $\left|1-\left(-\frac{\xi^k - 1}{\xi-1}\right)^n\right| = 1$ (note the complex modulus). Actually, I think that the only solutions are the trivial $k=an$, with $a \in \mathbb{Z}$. I know that similar problems can be really difficults, however I am not sure if this is the case.</p> <p>Thank you very much for any suggestion. :-)</p> http://mathoverflow.net/questions/115063/solved-cubic-thue-equation Solved cubic Thue equation Richard Bonne 2012-12-01T10:15:51Z 2012-12-22T15:15:50Z <p>Hi everybody. I need to know if the cubic Thue equation $x^3 + x^2y + 3xy^2 - y^3 = \pm 1$ is completely solved. I know that there are effective algorithms to solve any cubic Thue equation and that some of them are implemented in computer programs. However, I think that since the coefficients of that equation are small, it may have already been discussed in the literature. Thank you.</p> http://mathoverflow.net/questions/114707/the-diophantine-equation-x2-y2-z2-1 The diophantine equation X^2 - Y^2 - Z^2 = +- 1 Richard Bonne 2012-11-27T21:50:18Z 2012-11-28T14:04:19Z <p>Hi everybody. I'd like to know if the diophantine equation</p> <p>(1) $$X^2 - Y^2 - Z^2 = \pm 1$$</p> <p>has been studied and if the set of its solutions $(X,Y,Z)$ is known. I appreciate any reference. Thank you very much.</p> <p>P.S. If instead we look at the diophantine equation</p> <p>$$X^2 - Y^2 - Z^4 = \pm 1$$</p> <p>surely we can solve it imposing conditions on the solution of (1) so that Z be a square. However, is there a quicker method?</p> http://mathoverflow.net/questions/108864/irrationality-measure-of-formal-power-series/108901#108901 Answer by Richard Bonne for Irrationality measure of formal power series Richard Bonne 2012-10-05T09:19:42Z 2012-10-05T09:19:42Z <p>Note that thanks to Pade approximation we have $m_g(f) \geq 2g + 1$ for all $g \geq 1$ and $f \in \mathbb{Z}[[x]]$.</p> http://mathoverflow.net/questions/108864/irrationality-measure-of-formal-power-series Irrationality measure of formal power series Richard Bonne 2012-10-04T22:27:13Z 2012-10-05T09:19:42Z <p>Hi everybody. I'm looking for an analogue of irrationality measure for formal power series with integer coefficient, the elements of $\mathbb{Z}[[x]]$. For any $f \in \mathbb{Z}[[x]]$ and positive integer $g$, I thought to define something like $$m_g(f) := \sup_{p,q} \mbox{ord}(f - p / q)$$ where $p,q \in \mathbb{Z}[x]$ satisfy $\deg p, \deg q \leq g$ and $q \neq 0$; $\mbox{ord}(h) := n_0$ for any formal Laurent series $h = \sum_{n=n_0}^\infty a_n x^n$ with $a_{n_0} \neq 0$ and $\mbox{ord}(0) := +\infty$. Note that $m_g(f)$ is finite for all $g$ if and only if $f$ is irrational, otherwise $m_g(f) = +\infty$ for $g$ sufficently large. Do you have any references on this? Thank you.</p> http://mathoverflow.net/questions/107195/references-for-the-result-that-sqrtn-is-equidistributed-mod-1 References for the result that $\sqrt{n}$ is equidistributed mod 1 Richard Bonne 2012-09-14T16:40:14Z 2012-09-15T14:30:24Z <p>It is not difficult to show (even without Weyl criterion) that the sequence $\sqrt{n}$, $n=1,2,\ldots$ is equidistributed mod 1. However, I need a reference to this result. Can you help me? Thanks.</p> http://mathoverflow.net/questions/96992/references-for-the-poincare-cartan-forms References for the Poincaré-Cartan forms Richard Bonne 2012-05-15T12:05:12Z 2012-05-15T19:23:18Z <p>Hello, everybody. I'm looking for some reference about the Poincaré-Cartan form, I do not know how it is defined, I just know that it is used in Lagrangian mechanics but I have not found any references. Can you advise me a book that deals with the topic?</p> <p>In particular I would be interested to know who invented the forms of Poincaré-Cartan.</p> <p>Thank you very much.</p> http://mathoverflow.net/questions/93679/condition-to-ensure-that-the-product-of-closed-maps-be-closed Condition to ensure that the product of closed maps be closed Richard Bonne 2012-04-10T16:32:40Z 2012-04-11T00:53:51Z <p>If $f_i : X_i \to Y_i$ with $i=1,2,\ldots,n$ are closed maps between topological space it is known that their product map </p> <p>$$f : X_1 \times \cdots \times X_n \to Y_1 \times \cdots \times Y_n : (x_1, \ldots, x_n) \mapsto (f_1(x_1), \ldots, f_n(x_n))$$ doesn't need to be closed. However the question is: are there some nice conditions on $X_i$, $Y_i$ (like compactness, connection, Hausdorff...) such that $f$ will be closed? I have looked in "Bourbaki - General Topology Part 1,2" and I have found nothing about it. By the way I'm more interested to the case $X_1 = \cdots = X_n$, $Y_1 = \cdots = Y_n$. </p> <p>Thank you for your help!</p> http://mathoverflow.net/questions/89478/magnitude-of-the-sum-of-complex-i-u-d-random-variables-in-the-unit-circle Magnitude of the sum of complex i.u.d. random variables in the unit circle Richard Bonne 2012-02-25T11:23:36Z 2012-02-25T13:43:22Z <p>Hello everybody. I'm working about asymptotic estimates of</p> <p>$M_n = \left|\sum_{k=1}^n Z_k\right|$</p> <p>where $Z_1, Z_2, \ldots$ are independent uniformly distributed random variables on the complex unit circle. I found that the expectation $\textbf{E}[M_n^2] = n$ and the variance $\textbf{Var}[M_n^2] = n^2 - n$, so with Chebyshev's inequality I concluded that $M_n = o(n)$ almost surely as $n \to \infty$.</p> <p>It is possible to improve this estimate? I need something like $M_n = O(\sqrt{n})$ or $M_n = O(n^{1/2 + \varepsilon})$. Thanks.</p> http://mathoverflow.net/questions/130721/diophantine-equation-with-primitive-nth-root-of-unity Comment by Richard Bonne Richard Bonne 2013-05-15T20:43:41Z 2013-05-15T20:43:41Z I mean $\chi = \xi$. http://mathoverflow.net/questions/130721/diophantine-equation-with-primitive-nth-root-of-unity Comment by Richard Bonne Richard Bonne 2013-05-15T20:42:08Z 2013-05-15T20:42:08Z @Abhinav Kumar: Thanks a lot! You are right, so the problem now is if it is possible that $(-(\chi^k-1)/(\chi-1))^n = \pm 2$ (I think not) and, as you tell, this implies $\sqrt[n]{\pm 2} \in \mathbb{Q}(\chi)$. http://mathoverflow.net/questions/115063/solved-cubic-thue-equation/115084#115084 Comment by Richard Bonne Richard Bonne 2012-12-01T22:38:50Z 2012-12-01T22:38:50Z @Beenakker I know that a computer program like Mathematica can solve my equation, however I prefer to find some reference in the literature because I need to solve this equation in an article of mine - and I think that many referees do not like the use of Mathematica in this way. http://mathoverflow.net/questions/114707/the-diophantine-equation-x2-y2-z2-1/114720#114720 Comment by Richard Bonne Richard Bonne 2012-11-28T09:02:30Z 2012-11-28T09:02:30Z @GH Thank you! Your answer is very helpful. I have added a P.S. to my answer. http://mathoverflow.net/questions/108864/irrationality-measure-of-formal-power-series/108870#108870 Comment by Richard Bonne Richard Bonne 2012-10-05T07:41:42Z 2012-10-05T07:41:42Z @Gjergji Zaimi Thanks. I read that paper, however seems to me that they invented this notion of irrationality measure and no reference is given, about a general theory of it. http://mathoverflow.net/questions/107195/references-for-the-result-that-sqrtn-is-equidistributed-mod-1 Comment by Richard Bonne Richard Bonne 2012-09-14T17:27:34Z 2012-09-14T17:27:34Z @Rivin: See here <a href="http://www.isibang.ac.in/~sury/weyl.pdf" rel="nofollow">isibang.ac.in/~sury/weyl.pdf</a> http://mathoverflow.net/questions/96992/references-for-the-poincare-cartan-forms/97020#97020 Comment by Richard Bonne Richard Bonne 2012-05-16T15:31:15Z 2012-05-16T15:31:15Z All right. The strange thing is that my colleague had told me that the Poincar&#233;-Cartan form was invented after the mid-20th century, so I can definitely say that he is wrong. http://mathoverflow.net/questions/96992/references-for-the-poincare-cartan-forms/97020#97020 Comment by Richard Bonne Richard Bonne 2012-05-15T17:50:21Z 2012-05-15T17:50:21Z @Bryant So you can confirm that Poincar&#233;-Cartan forms was known (in it's modern form) before the twentieth century? http://mathoverflow.net/questions/96992/references-for-the-poincare-cartan-forms/97020#97020 Comment by Richard Bonne Richard Bonne 2012-05-15T16:50:34Z 2012-05-15T16:50:34Z @Bryant and Tortorella Thank you for your suggestions. However I forgot to specify that I would be interested to know who invented the forms of Poincar&#233;-Cartan (?). http://mathoverflow.net/questions/93679/condition-to-ensure-that-the-product-of-closed-maps-be-closed/93692#93692 Comment by Richard Bonne Richard Bonne 2012-04-10T18:41:55Z 2012-04-10T18:41:55Z So, if $X_1, \ldots, X_n$ are compacts, $Y_1, \ldots, Y_n$ are Hausdorff, $f_1, \ldots, f_n$ are closed and continuous then (for any $y \in Y_i$, $\{y\}$ is closed, $f_i^{-1}(y)$ is closed, $f_i^{-1}(y)$ is compact) $f$ is closed. Right? This will be a nice condiction for my purpouse. http://mathoverflow.net/questions/89478/magnitude-of-the-sum-of-complex-i-u-d-random-variables-in-the-unit-circle/89486#89486 Comment by Richard Bonne Richard Bonne 2012-02-25T14:07:14Z 2012-02-25T14:07:14Z OK, then I give up to prove $M_n = O(\sqrt{n})$, however I would be happy if there was I simpler proof that $M_n = o(n)$, which avoids the central limit theorem and the law of the iterated logarithm. http://mathoverflow.net/questions/89478/magnitude-of-the-sum-of-complex-i-u-d-random-variables-in-the-unit-circle/89479#89479 Comment by Richard Bonne Richard Bonne 2012-02-25T13:39:06Z 2012-02-25T13:39:06Z But how I can apply Hoeffding's inequality? It is true only for real random variables, not complex.