User gu yejun - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T23:13:26Z http://mathoverflow.net/feeds/user/4419 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/53194/riemann-hypothesis Riemann Hypothesis Gu Yejun 2011-01-25T05:11:38Z 2011-01-25T05:55:04Z <p>Professor Dyson once said quasiperiodic crystals are connected with Riemann Hypothesis. Does anyone have something deeper to help proving Riemann Hypothesis? </p> http://mathoverflow.net/questions/17395/what-does-gibbs-phenomenon-shows-the-nature-of-fourier-series What does Gibbs phenomenon shows the nature of Fourier Series Gu Yejun 2010-03-07T17:12:08Z 2010-08-27T14:54:16Z <p>As the title shows,we know that there is some points the series not approaching to the function.</p> <p>Now,take the convergence theorem into consideration.As there is some the first break-points，the series is still convergent.And,the Gibbs phenomenon always takes place on the first break-points.</p> <p>Why does Gibbs phenomenon take place?What does it show the nature of Fourier Series?</p> http://mathoverflow.net/questions/35576/the-implicit-function-theorem-in-subsets-of-r2 the implicit function theorem in subsets of R$^2$ Gu Yejun 2010-08-14T13:24:33Z 2010-08-14T14:20:11Z <p>As the implicit function theorem shows, if</p> <p>(i)Function F is continuous in the region D$\subseteq R^2$;</p> <p>(ii)F($x_0,y_0)=0,P_0(x_0,y_0)\in$D;</p> <p>(iii)There is a continuous partial derivative $F_y$(x,y)=0 in the region D;</p> <p>(iv)$F_y(x_0,y_0)\neq$0;</p> <p>Then there uniquely exists the function y=f(x) defined in the interval ($x_0-\alpha$,$x_0+\alpha$),that</p> <p>1 f(x$_0$)=$y_0$,(x,f(x))$\in$U(P$_0$)when x$\in$(x$_0$-$\alpha$,x$_0$+$\alpha$)and F(x,f(x))$\equiv$0;</p> <p>2 f(x) is continuous in ($X_0-\alpha$,$X_0+\alpha$).</p> <p>And what if the conditions are weaker than those above? That is (i)Function F is continuous in the region D$\subseteq R^2$;</p> <p>(ii)F($x_0,y_0)=0,P_0(x_0,y_0)\in$D;</p> <p>(iii)There is a continuous partial derivative $F_y$(x,y)=0 in the region D;</p> <p>In other words,what is the conclusion for the existence of implicit function in the branch or the subset of $R^2$?</p> <p>There have been a series conclusions in complex space，who can list some of them and what's the relationship with the situation of complex space and real space?</p> http://mathoverflow.net/questions/11761/why-sin-and-cos-in-the-fourier-series/17397#17397 Answer by Gu Yejun for Why sin and cos in the Fourier Series? Gu Yejun 2010-03-07T17:20:09Z 2010-03-08T02:42:45Z <p>because {cos x，cos 2x,...,cos nx,...sin x,sin 2x,...,sin nx,...}form a group of bases of an orthogonal.These bases have a lot of characterization.You can find them easily in any reference book. May it help!</p> http://mathoverflow.net/questions/15367/how-many-ways-can-we-characterize-gamma-function/17390#17390 Answer by Gu Yejun for How many ways can we characterize gamma function? Gu Yejun 2010-03-07T16:19:49Z 2010-03-07T16:19:49Z <p>maybe I can give you some help. Gamma function is also called the second Euler integral.</p> <p>Here comes some characterizations.</p> <p>a f(s)= $$t(x)=\int_{0}^{+\infty}{t^(s-1)}{exp(-t)}dt$$ s>0</p> <p>b f(s)=$$\lim n!n^s/[s(s+1)...(s+n)]$$ $$n\rightarrow +\infty$$</p> <p>c $$B（p，q）=\Gamma(p)\Gamma(q)/\Gamma(pq）$$ p>0 q>0</p> <p>d $$\Gamma(2s)=2^(2s-1）\Gamma(s)\Gamma(s+1/2)/\sqrt(2\pi)$$ s>0</p> <p>e $$\Gamma(s)\Gamma(1-s)=\pi/sin(s\pi)$$ 0 <p>May it help!</p> http://mathoverflow.net/questions/17268/how-to-weigh-the-conditions-given-in-a-proposition how to weigh the conditions given in a proposition Gu Yejun 2010-03-06T08:36:58Z 2010-03-07T03:22:58Z <p>As we can see,there are some conditions given in a proposition. If there are 2 propositions having approximate conclusions.Usually，we can name one propositions gives stronger conditions than the other. My question is that whether there is a good system to weigh the conditions given in a proposition.e.g. a condition "uniform convergence"is stronger than a condition"convergence". Can we give each condition a value to show their sharpness?</p> http://mathoverflow.net/questions/11761/why-sin-and-cos-in-the-fourier-series/17397#17397 Comment by Gu Yejun Gu Yejun 2010-08-17T15:36:08Z 2010-08-17T15:36:08Z which stands for a unit disk http://mathoverflow.net/questions/11761/why-sin-and-cos-in-the-fourier-series/17397#17397 Comment by Gu Yejun Gu Yejun 2010-08-17T15:35:12Z 2010-08-17T15:35:12Z I think the most basic reason is that cos$^2$s+sin$^2$=1 http://mathoverflow.net/questions/35576/the-implicit-function-theorem-in-subsets-of-r2 Comment by Gu Yejun Gu Yejun 2010-08-14T23:33:36Z 2010-08-14T23:33:36Z As for x$^2+y^2$=1,we can divide the space into two pieces,y&gt;o and y&lt;0,then there uniquely exists the function y=f(x) defined in the interval （1-$\alpha$,1],this is the key point. http://mathoverflow.net/questions/35576/the-implicit-function-theorem-in-subsets-of-r2 Comment by Gu Yejun Gu Yejun 2010-08-14T14:53:41Z 2010-08-14T14:53:41Z U stands for neighborhood，and $P_0=(x_0,y_0)$ http://mathoverflow.net/questions/17395/what-does-gibbs-phenomenon-shows-the-nature-of-fourier-series Comment by Gu Yejun Gu Yejun 2010-05-19T08:34:15Z 2010-05-19T08:34:15Z this means the Gibbs phenomenon always takes place on the first break-points. http://mathoverflow.net/questions/15367/how-many-ways-can-we-characterize-gamma-function/17390#17390 Comment by Gu Yejun Gu Yejun 2010-03-08T13:20:24Z 2010-03-08T13:20:24Z Maybe it is true.But the definition is also unique. http://mathoverflow.net/questions/15367/how-many-ways-can-we-characterize-gamma-function/17390#17390 Comment by Gu Yejun Gu Yejun 2010-03-07T16:38:49Z 2010-03-07T16:38:49Z actually ,I don't think there is a particularly difference between definitions and characterizations. That is my opinion！ http://mathoverflow.net/questions/17268/how-to-weigh-the-conditions-given-in-a-proposition/17311#17311 Comment by Gu Yejun Gu Yejun 2010-03-07T14:58:27Z 2010-03-07T14:58:27Z you have given me a new way to think about this question.Thanks for your help! http://mathoverflow.net/questions/17268/how-to-weigh-the-conditions-given-in-a-proposition Comment by Gu Yejun Gu Yejun 2010-03-07T14:50:07Z 2010-03-07T14:50:07Z to Andrej Bauer,I think ,in some way, it is meaningful. First, we can tell whether two propositions are equivalent ，iff they have the same value. Second, if one condition is necessary and sufficient ，then it must be stronger than a necessary condition.We have a direction to looking for the necessary and sufficient condition. And so on. http://mathoverflow.net/questions/17268/how-to-weigh-the-conditions-given-in-a-proposition Comment by Gu Yejun Gu Yejun 2010-03-07T14:44:09Z 2010-03-07T14:44:09Z thanks for Yemon Choi &amp; Mariano Su&#225;rez-Alvarez