User lwins.gafield - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T07:57:53Z http://mathoverflow.net/feeds/user/22954 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/117146/for-sum-of-unit-roots For sum of unit-roots Lwins.Gafield 2012-12-24T14:52:51Z 2012-12-24T14:52:51Z <p>Define $\delta (0)=1$ and $\delta (x) = 0$ when $x \neq 0$.</p> <p>Can we find a "good" method or algorithm to work out $$S(n) = \sum_{k_1=0}^{n-1} \sum_{k_2=0}^{n-1} \cdots \sum_{k_n=0}^{n-1} \delta (\mathrm{e}^{2 \pi k_1 \mathrm{i} /n} + \mathrm{e}^{2 \pi k_2 \mathrm{i} /n} + \cdots + \mathrm{e}^{2 \pi k_n \mathrm{i} /n})$$</p> <p>with any positive integer $n$?</p> <p>(It's easy to count this when $n=p^l$ or $n=pq$. In particular, $S(p)=p!$ is obvious.)</p> http://mathoverflow.net/questions/115160/fourier-transform-for-entire-function Fourier Transform, for entire function Lwins.Gafield 2012-12-02T09:05:42Z 2012-12-03T19:29:41Z <p>On <a href="http://mathoverflow.net/questions/114875/" rel="nofollow">THIS</a> site, Alexandre used Fourier transform to solve the problem.</p> <p>If we use Fourier transform, how to define it to ensure any entire function has a FT?</p> <p>Classical FT is defined by $$\mathcal{F}[f] = F(\xi) = \frac{1}{\sqrt{2 \pi}}\int_{-\infty}^{+\infty}f(z)\mathrm{e}^{-\mathrm{i} \xi z} \mathrm{d} z.$$ This only work for $f \in L^1(\mathbb{R})$. (If improved, it can work for $f \in L^2(\mathbb{R})$.)</p> <p>I know $\mathcal{F}[\mathrm{e}^{sz}] = \sqrt{2 \pi} \delta(\xi - \mathrm{i}s)$, but I'm not sure about a general definition.</p> http://mathoverflow.net/questions/114875/on-equation-fz1-fzfz On equation f(z+1)-f(z)=f'(z) Lwins.Gafield 2012-11-29T12:03:38Z 2012-12-03T19:07:10Z <h2>Original Problem</h2> <p>If $f$ is an entire function such that $$f(z+1)-f(z)=f'(z)$$ for all $z$. Is there a non-trivial solution? ($f(z)=az+b$ is trivial)</p> <h2>And here is something uncertainty</h2> <p>If we use Fourier transform, how to define it to ensure any entire function has a FT? </p> <p>Classical FT is defined by $$\mathcal{F}[f] = F(\xi) = \frac{1}{\sqrt{2 \pi}}\int_{-\infty}^{+\infty}f(z)\mathrm{e}^{-\mathrm{i} \xi z} \mathrm{d} z.$$ This only work for $f \in L^1(\mathbb{R})$. (If improved, it can work for $f \in L^2(\mathbb{R})$.)</p> <p>I know $\mathcal{F}[\mathrm{e}^{sz}] = \sqrt{2 \pi} \delta(\xi - \mathrm{i}s)$, but I'm not sure about a general definition.</p> http://mathoverflow.net/questions/111617/a-special-case-of-catalans-conjecture/111726#111726 Answer by Lwins.Gafield for A special case of Catalan's conjecture Lwins.Gafield 2012-11-07T13:40:27Z 2012-11-07T13:40:27Z <p>I think this gives a nice elementary solution, too. See this: <a href="http://www.artofproblemsolving.com/Forum/viewtopic.php?f=56&amp;t=505773" rel="nofollow">http://www.artofproblemsolving.com/Forum/viewtopic.php?f=56&amp;t=505773</a></p> http://mathoverflow.net/questions/111617/a-special-case-of-catalans-conjecture A special case of Catalan's conjecture Lwins.Gafield 2012-11-06T04:55:16Z 2012-11-07T13:40:27Z <p>Solve equation $$y^p - (2^p-1)^x = 1$$ where $x,y>0 \in \mathbb{Z}$, $p \in \mathbb{P}$. Is there a elementary method to do it? Thanks. =)</p> http://mathoverflow.net/questions/111422/is-2-always-a-p-rm-th-power-nonresidue-modulo-2p-1 Is $2$ always a $p^{\rm th}$-power nonresidue modulo $2^p-1$? Lwins.Gafield 2012-11-04T03:07:21Z 2012-11-04T03:07:21Z <p>That is, is following inequality holds while $p \in \mathbb{P}$? $$x^p \neq 2 \ (\mathrm{mod} \ 2^p-1)$$ Certainly $x \in \mathbb{Z}$.</p> http://mathoverflow.net/questions/111250/equation-system-on-zx Equation system on Z[x] Lwins.Gafield 2012-11-02T07:07:05Z 2012-11-02T07:07:05Z <p>Can we solve this equation system when $n=2,3,4,\cdots \$ is given? $$F(x)+F((2^n-1)x) = (G(x))^nx^g$$ $$2F(2^{n-1}x) = (H(x))^nx^h$$ While $F,G,H \in \mathbb{Z}[x]. g,h \in \mathbb{N}.$ </p> http://mathoverflow.net/questions/110915/another-number-theory-question-about-polynomial Another Number Theory Question about Polynomial Lwins.Gafield 2012-10-28T16:38:06Z 2012-10-28T18:27:48Z <p>We know following theorem by Schur: Suppose that $f(x) \in \mathbb{Z}[x]$ is a polynomial such that exists an integer $m$ such that $f(n) = m^2$ for every integer $n$. Then $f(x) = g(x)^2$ for some $g(x) \in \mathbb{Z}[x]$.</p> <p>Is this true if I using $3,4,5 \cdots$ instead of $2$?</p> http://mathoverflow.net/questions/110904/a-number-theory-question-about-polynomial A Number Theory Question about Polynomial Lwins.Gafield 2012-10-28T13:55:41Z 2012-10-28T14:35:05Z <p>Now we have a Polynomial $P(n)$ on $\mathbb{Z}[x]$. It can't be wriiten as $P(n)=F(n)G(n)$ while $F(n),G(n) \neq 1$. Is it right that for any $P(n)$, there is a $n$ such that $P(n)$ is a prime? Is it right that for any $P(n)$, there is a $n$ such that $P(n)$ is not a square-number?</p> http://mathoverflow.net/questions/97280/a-problem-in-number-theorem-with-a-number-of-the-base-p A problem in number theorem with a number of the base p Lwins.Gafield 2012-05-18T06:53:45Z 2012-05-23T12:26:39Z <p>First we define a function $f(x,p)$, with $x$ a natural number and with $p$ a prime number.</p> <p>$f(x,p)$ stands for the location where the digit "$p-1$" first appears in the base-$p$ expansion of $x$. If the digit "$p-1$" doesn't appear, $f(x,p)$ will be equal to $-\infty$. That is, if $x=\sum_{i\geq 0} a_i p^i$ with $0\leq a_i &lt; p$, then $f(x,p)=\min\{i : a_i=p-1\}$. For example:</p> <ul> <li>$f((122)_3,3)=0$;</li> <li>$f((561603)_7,7)=2$;</li> <li>$f((123)_5,5)=-\infty$;</li> <li>$f((651634316331)_7,7)=3$.</li> </ul> <p>Now consider a positive integer $A=(a_m a_{m-1} \cdots a_1a_0)_p$ with $f(A,p)=m$, i.e., $a_m=p-1$, $a_0,a_1,\cdots,a_{m-1}\neq p-1$.</p> <p>My problem is how to prove that for any $h \in [0,2^{m-1}-1]\cap \mathbb{Z}$, which has at most $p-1$ digit "$1$" in binary representation. there is a $k \in \mathbb{Z}^+$ with $A-(p-1)k \geq 0$, satisfies that,</p> <p>$$2^{f(A-k,p)} \ \mathbf{xor}\ 2^{f(A-2k,p)} \ \mathbf{xor} \ \cdots \ \mathbf{xor} \ 2^{f(A-(p-1)k,p)} = h$$</p> <hr> <p>Edit: I (the one making the edit, not the OP) am not sure how to parse the problem. Is the following equivalent? </p> <blockquote> <p>Let $h_0,h_1,\dots, h_{m-2}\in \{0,1\}$ with $\sum h_i &lt; p$. Is there necessarily a positive integer $k$ with $A-(p-1)k\geq0$ and for $0\leq i \leq m-2$ $$\left| \{ j : 1\leq j &lt; p, f(A-jk,p)=i \} \right| \equiv h_i \pmod{2}?$$</p> </blockquote> <hr> <p>To Kevin O'Bryant,</p> <p>Yes, it's equivalent to my version.</p> http://mathoverflow.net/questions/96207/an-asymptotic-series-for-the-digamma-function An asymptotic series for the digamma function Lwins.Gafield 2012-05-07T12:04:18Z 2012-05-07T14:19:45Z <p>As we know, there is an asymptotic series for the digamma function when $z>0$ is a real number. $$\psi(z)=\ln z+\sum_{n=1}^{\infty}{\frac{B_n}{nz^n}}$$ $B_n$ is the first Bernoulli numbers.</p> <p>How to make a proof?</p> http://mathoverflow.net/questions/96207/an-asymptotic-series-for-the-digamma-function/96218#96218 Answer by Lwins.Gafield for An asymptotic series for the digamma function Lwins.Gafield 2012-05-07T14:19:45Z 2012-05-07T14:19:45Z <p>We can prove this, using Euler-Maclaurin Formula. Here is a introduction from Wikipedia. <a href="http://en.wikipedia.org/wiki/Euler%E2%80%93Maclaurin_formula" rel="nofollow">http://en.wikipedia.org/wiki/Euler%E2%80%93Maclaurin_formula</a></p> <p>This is a quite easy problem. To Admin, You may be able to consider deleting this question, thanks. ^_^</p> http://mathoverflow.net/questions/94226/a-random-walk-with-uniformly-distributed-steps A random walk with uniformly distributed steps Lwins.Gafield 2012-04-16T16:14:44Z 2012-04-28T21:38:47Z <p>The following problem has bothered me for a long time.</p> <p>Let us imagine a point on the real axis. At the beginning, it is located at point $O$. Then it will "walk" on the real axis randomly in the following way. For every step of the "walk", it will choose a real number $\Delta x$ uniformly from the interval $[-1,1]$, turn right, and move $\Delta x$ unit. Once it reaches the left side of the point $O$, it will "die" immediately.</p> <p>Our task is find out the probability of the point is alive after $n$ steps of "walk" $P_n$. I guess that $P_n=\frac{(2n)!}{4^n (n!)^2}$, but I can't prove this or explain why it is true.</p> http://mathoverflow.net/questions/95329/generating-permutations-with-given-set Generating Permutations with Given Set Lwins.Gafield 2012-04-27T08:07:19Z 2012-04-27T08:07:19Z <p>Let $S_n$ be the set of all permutations that act on $1...n$. You are given a subset $P\subseteq S_n$, and you are to compute the size of the set $G(P) \subseteq S_n$, where $G(P)$ meets the following requirements:</p> <ol> <li>$G(P)$ contains the identity permutation,</li> <li>$P \subseteq G(P)$,</li> <li>$G(P)$ is closed under taking inverses and multiplications (if $a, b \in G(P)$, then $ab \in G(P)$, if $a \in G(P)$, then $a^{-1} \in G(P)$),</li> <li>$G(P)$ is a minimal possible set that meets requirements 1,2 and 3.</li> </ol> <p>Is there a method or algorithm that can solve this?</p> http://mathoverflow.net/questions/94608/a-random-walk-with-uniformly-distributed-steps-ii A random walk with uniformly distributed steps II Lwins.Gafield 2012-04-20T07:12:39Z 2012-04-20T17:38:23Z <p>The problem is a improved version of this problem, <a href="http://mathoverflow.net/questions/94226/" rel="nofollow">http://mathoverflow.net/questions/94226/</a></p> <p>Let us imagine a point on the real axis. At the beginning, it is located at point $O$. Then it will "walk" on the real axis randomly. For every step of the "walk", it will choose a real number $\Delta x$ in interval $[l,r]$ equiprobably, and turn right and move $\Delta x$ unit. Once it move to the left side of the point $O$, it will "die" immediately.</p> <p>Our task is find out the probability of the point "live" after n steps of "walk" $P_n$. I have tried to solve it and found out a method to count $P_n$ with $\Theta (n^5)$ of time complexity, using fourier transform and something in complex analysis. But is there a more simple method? Or is there one which needs lower time complexity?</p> http://mathoverflow.net/questions/115160/fourier-transform-for-entire-function/115186#115186 Comment by Lwins.Gafield Lwins.Gafield 2012-12-11T07:03:16Z 2012-12-11T07:03:16Z Yet I'm doing this in $\mathbb{C}$, by that, FT and LT are equivalent. Thus, if we can define LT as your state, then, we can do FT as well. http://mathoverflow.net/questions/115160/fourier-transform-for-entire-function/115173#115173 Comment by Lwins.Gafield Lwins.Gafield 2012-12-02T11:33:08Z 2012-12-02T11:33:08Z Mmm.. Then, is a entire function $f$ have a Fourier transform $F$ (perhaps $F$ is a distribution) for sure? If not, then the answer for [THIS][1] is incomplete. [1]: <a href="http://mathoverflow.net/questions/114875/" rel="nofollow">mathoverflow.net/questions/114875</a> http://mathoverflow.net/questions/114875/on-equation-fz1-fzfz/114878#114878 Comment by Lwins.Gafield Lwins.Gafield 2012-12-02T07:11:47Z 2012-12-02T07:11:47Z Excellent, but how to define &quot;Fourier Transform&quot; to ensure that any possible entire function have its FT? Classical FT is only defined in $L^2(\mathbb{R})$. http://mathoverflow.net/questions/111617/a-special-case-of-catalans-conjecture/111624#111624 Comment by Lwins.Gafield Lwins.Gafield 2012-11-06T07:38:51Z 2012-11-06T07:38:51Z To Johan, obviously $(y-1,\frac{y^p-1}{y-1})=(y-1,p) \mid p$. So show that $p \nmid y-1$ is enough. It's trivial because $y \equiv 1 \pmod p \implies (2^p-1) \equiv 0 \pmod p$. http://mathoverflow.net/questions/111617/a-special-case-of-catalans-conjecture/111624#111624 Comment by Lwins.Gafield Lwins.Gafield 2012-11-06T07:31:33Z 2012-11-06T07:31:33Z I'm grateful for your patient. Very nice answer, I think. Thanks. =) http://mathoverflow.net/questions/111617/a-special-case-of-catalans-conjecture Comment by Lwins.Gafield Lwins.Gafield 2012-11-06T06:38:11Z 2012-11-06T06:38:11Z Dear Johan, sorry, I've modified it. Now $(1,2)$ is a solution. http://mathoverflow.net/questions/111617/a-special-case-of-catalans-conjecture Comment by Lwins.Gafield Lwins.Gafield 2012-11-06T05:54:39Z 2012-11-06T05:54:39Z Dear Will, but it's toooo hard to show that Catalan's conjecture is right. Can we find a easier proof? Dear Johan, there are only a solution $(x,y)=(1,2)$. http://mathoverflow.net/questions/111250/equation-system-on-zx Comment by Lwins.Gafield Lwins.Gafield 2012-11-02T10:06:01Z 2012-11-02T10:06:01Z And, $a,b&gt;0$. $n$ is given. &quot;we have&quot; means &quot;we can find&quot;. http://mathoverflow.net/questions/111250/equation-system-on-zx Comment by Lwins.Gafield Lwins.Gafield 2012-11-02T10:03:01Z 2012-11-02T10:03:01Z Mmm.. It's about this problem: for any $a+b=k^n$, we have a $l$ such that $F(a)+F(b)=l^n$, solve for $F$. Certainly $F \in \mathbb{Z}[x]$. http://mathoverflow.net/questions/111250/equation-system-on-zx Comment by Lwins.Gafield Lwins.Gafield 2012-11-02T09:13:00Z 2012-11-02T09:13:00Z I wanted to solve for all $(F,G,H,g,h)$. http://mathoverflow.net/questions/110904/a-number-theory-question-about-polynomial Comment by Lwins.Gafield Lwins.Gafield 2012-10-28T16:25:51Z 2012-10-28T16:25:51Z Dear Fran&#231;ois Brunault, thank you, that helps me a lot. \\ Dear Goldstern, +-1. \\ Dear Noam D. Elkies, I've been showed something interesting by you, thanks. http://mathoverflow.net/questions/110904/a-number-theory-question-about-polynomial Comment by Lwins.Gafield Lwins.Gafield 2012-10-28T14:22:28Z 2012-10-28T14:22:28Z Thanks. I've modified it. http://mathoverflow.net/questions/97280/a-problem-in-number-theorem-with-a-number-of-the-base-p Comment by Lwins.Gafield Lwins.Gafield 2012-05-21T04:40:03Z 2012-05-21T04:40:03Z To Karl Schwede, $\mathbf{xor}$ is always acting of base 2. http://mathoverflow.net/questions/97280/a-problem-in-number-theorem-with-a-number-of-the-base-p Comment by Lwins.Gafield Lwins.Gafield 2012-05-20T22:53:23Z 2012-05-20T22:53:23Z Thanks. I've fixed some error in the problem. http://mathoverflow.net/questions/97280/a-problem-in-number-theorem-with-a-number-of-the-base-p Comment by Lwins.Gafield Lwins.Gafield 2012-05-18T12:47:35Z 2012-05-18T12:47:35Z To David Roberts, This problem is closely related to how to find the identity of this function, $$F(x)=\mathbf{mex}(\{F(x-k) \ \mathbf{xor}\ F(x-2k)\ \mathbf{xor} \cdots \mathbf{xor} \ F(x-(p-1)k) \ | \ k \in \mathbb{Z}^+ , x-(p-1)k\geq 0\})$$