User mathman - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-20T11:59:52Z http://mathoverflow.net/feeds/user/21933 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/97233/why-riemann-hypothesis-over-curves-is-easy-but-normal-hypothesis-for-the-rieman why Riemann Hypothesis over curves is easy but 'normal' hypothesis for the Riemann Zeta function $\zeta (s)$ is hard , Mathman 2012-05-17T16:39:44Z 2012-05-17T18:01:42Z <p>despite both zetas $\zeta (s,X)$ and $\zeta (s)$ have the same functional equation, the same Euler prodcut and the same Riemann-Weil formula </p> <p>why one of them is 'easy' and can be solved but the other is so hard?? the zeta one $\zeta (s)$</p> http://mathoverflow.net/questions/95309/how-did-connes-get-it-trace-formula How ..did Connes get it (trace formula) Mathman 2012-04-26T21:54:34Z 2012-04-26T22:17:30Z <p>i have been reading or at least trying to understand how Connes get the density (approximate) of states</p> <p>$N(E)= \frac{E}{2\pi}log \frac{E}{2\pi}- \frac{E}{2\pi}+ \frac{7}{8}+ \frac{1}{\pi}arg \zeta(1/2+iE)$</p> <p>from the Hamiltonian operator $H=xp$</p> <p>the 'smooth' part i know how it is evaluated , simply by computing $\frac{1}{2\pi}\int dx \int dp H(E-xp)$ and using a certina Maslov index</p> <p>However , how did he manage to get the oscillating part of the zeros ??? i mean $\frac{1}{\pi}Arg \zeta (1/2+iE)$</p> <p>i have been reading the approach <a href="http://www.alainconnes.org/docs/bookwebfinal.pdf" rel="nofollow">http://www.alainconnes.org/docs/bookwebfinal.pdf</a></p> http://mathoverflow.net/questions/93226/first-order-linear-differential-equation-with-boundary-conditions first-order linear differential equation with boundary conditions Mathman 2012-04-05T15:41:55Z 2012-04-24T00:18:44Z <p>let be the differential equation</p> <p>$-ixDf(x)-if(x)/2= E_{n}f(x)$</p> <p>with the boundary conditions $f(x)=f(p^{k}x)$ for 'p' prime and $k=...,-2,-1,0,1,2,...$</p> <p>is this possible to solve this eigenvalue problem ?? thanks</p> http://mathoverflow.net/questions/94644/trace-over-the-zeros-with-real-part-1-2-only Trace over the zeros with real part 1/2 Only Mathman 2012-04-20T14:31:13Z 2012-04-20T20:16:45Z <p>If RH is not true, we have that Weil's explicit formula still holds:</p> <p>$$\sum_{\gamma} h(\gamma) = h(i/2)+h(-i/2)-2 \sum_{n=1}^{\infty} \frac{ \Lambda(n)}{ \sqrt n}g(logn)+\frac{1}{2\pi} \int_{-\infty}^{\infty}h(r)\Psi(1/4+ir/2)dr.$$</p> <p>My question is if there are formulae that take into account ONLY the Riemann zeros with real part 1/2 and not the others.. thanks.</p> http://mathoverflow.net/questions/94512/understanding-zeta-function-regularization/94678#94678 Answer by Mathman for Understanding zeta function regularization Mathman 2012-04-20T19:35:13Z 2012-04-20T19:35:13Z <p>in general ONLY the quotient of 2 functional determinant is defined inside the Zeta regularization</p> <p>SEEK Elizalde zeta regularization theory :) this is the BEST book about the subject i learned ZR from it</p> <p>in general Zeta regualrization is valid to define QUOTIENT of functional determinants so the infinite constant goes out $\frac{ det(I-Az)}{det(I-Bz)}$</p> http://mathoverflow.net/questions/75300/is-there-some-explication-for-the-transformation-of-the-eigenvalues-in-selberg-tr/94645#94645 Answer by Mathman for Is there some explication for the transformation of the eigenvalues in Selberg trace formula Mathman 2012-04-20T14:35:34Z 2012-04-20T14:35:34Z <p>the 'Selberg zeros' appear as momenta of the Laplacian $p= \sqrt (E-1/4)$ NOT over the Eigenvalues.. a smilar thing happens with Poisson sum formula</p> <p>$\sum_{n\in Z}\delta (x-n)=\sum_{n\in Z}exp(2i\pi nx)$</p> <p>but the Energies are $E= n^{2}\pi^{2}$ this is the analogy between Selberg and Poisson</p> http://mathoverflow.net/questions/94236/selberg-trace-from-classical-physics selberg trace from classical physics Mathman 2012-04-16T18:00:48Z 2012-04-17T14:06:05Z <p>considering the Hamitlonian for the Selberg Operator $y^{2} ( \partial _{x}^{2}+ \partial _{y}^{2})$ given in the Hamiltonian form</p> <p>$H=g_{ab}p^{a}p^{b}$ with $ds^{2} = \frac{dx^{2}+dy^{2}}{y^{2}}$ being the metric</p> <p>can we obtain from this H the functional determinant expression for the selberg zeta ??</p> <p>can we obtain from the integral $d(E)= \iint \delta(E-H(x,y,p)$ the 'smooth' part of the eigenvalue countign fucniton for the Laplacian i mean the factor</p> <p>$\int_{0}^{E}dptanh(\pi p) p$ i mean if semiclassical physics can be applied to the problem of the Selberg zeta function.</p> http://mathoverflow.net/questions/93407/dilation-operator-green-function dilation operator green function Mathman 2012-04-07T11:26:34Z 2012-04-07T14:03:40Z <p>how can i solve $-ixDG(x,s)-iG(x,s)/2= \delta ( \frac{x}{s}-1)$</p> <p>i do not know , since it is a first odrder differntial operator, the formal solution i've found would be</p> <p>$G(x,s)= \sum_{n} \frac{u_{n}(x)u_{m}(s)}{\lambda _{n}}$</p> <p>where $-ixDu_{n}(x)-iu_{n}(x)/2= u_{n} \lambda _{n}$</p> <p>here G(x,s) is the analogue of the Green function for our operator :)</p> http://mathoverflow.net/questions/87041/looking-for-the-solution-of-first-order-non-linear-differential-equation-y-y/93418#93418 Answer by Mathman for Looking for the solution of first order non-linear differential equation ($y ′+y^{2}=f(x)$) without knowing a particular solution. Mathman 2012-04-07T13:20:01Z 2012-04-07T13:20:01Z <p>a Riccatti equation can be turned into $u''+f(x)u=0$ by the transformation $y= \frac{u'}{u}$ using the WKB ansatz we have the asymptotic solution to 'u' as</p> <p>$u(x)\sim C(f(x)^{-1/4}exp( -\int dx \sqrt -f(x))$</p> http://mathoverflow.net/questions/93158/what-is-the-connes-trace What is the Connes Trace ?? Mathman 2012-04-04T19:21:18Z 2012-04-04T19:21:18Z <p>in <a href="http://www.math.osu.edu/lectures/connes/zeta.pdf" rel="nofollow">http://www.math.osu.edu/lectures/connes/zeta.pdf</a> Connes gives the Trace</p> <p>$Tr(R_{\Lambda}U(h))=2h(1)log\Lambda + \sum_{Q_{p}}\int \frac{h(u^{-1})}{|1-u|}d^{*}u+o(1)$</p> <p>the second integral is over the p-adic and the Real place</p> <p>for the first integral connes gives $log\Lambda = \int d^{*}u$</p> <p>what does it mean ? what is teh connes operator exatly $R_{\Lambda}U(h)$</p> http://mathoverflow.net/questions/92740/invariance-under-dilations invariance under dilations Mathman 2012-03-31T09:46:07Z 2012-03-31T18:41:15Z <p>we have that the function (for suitable f) </p> <p>$F(x)= \sum_{-\infty}^{\infty}f(x+n)$ is INVARIANT under any integer traslation</p> <p>$y=x+n$ for integer 'n'</p> <p>however my question is can we find a lattice which is invariant under DILATIONS i mean under the transformation $y=qx$ for integer (positive) or rational 'q' ??</p> <p>so i am looking a formula like $F(x)= \sum f(qx)$ so F(x) is invariant under transformation of the form $y=qx$ thanks.</p> http://mathoverflow.net/questions/54501/riemann-zeta-function-connection-to-quantum-mechanics/90390#90390 Answer by Mathman for Riemann Zeta Function connection to Quantum Mechanics. Mathman 2012-03-06T18:48:44Z 2012-03-08T22:04:56Z <p>the Xi function appears as the functional determinant $\frac{ \xi(s)}{\xi(0)}= \frac{det(H+s(s-1)+1/4)}{det(H+1/4)}$ of a certain Hamiltonian with the potential $V^{-1}(x)= A \sqrt D n(x)$</p> <p>with $n(x) \pi = Arg\xi (1/2+ i \sqrt x)$</p> <p>here 'n' plays the role of Eigenvalue staircase $n(x)= \sum_{n=0}^{\infty}H(x-E_{n})$ and 'H(x)' is the Heaviside function.</p> http://mathoverflow.net/questions/96642/current-status-of-the-riemann-hypothesis/96658#96658 Comment by Mathman Mathman 2012-05-11T18:27:30Z 2012-05-11T18:27:30Z however is correct, see wu-sprung potential :) for more info http://mathoverflow.net/questions/94512/understanding-zeta-function-regularization/94515#94515 Comment by Mathman Mathman 2012-04-21T13:56:19Z 2012-04-21T13:56:19Z $det(A.B)=detA.detB$ in case its commutator is zero (i think) $[A,B]=0$ http://mathoverflow.net/questions/94644/trace-over-the-zeros-with-real-part-1-2-only/94648#94648 Comment by Mathman Mathman 2012-04-20T17:51:13Z 2012-04-20T17:51:13Z OK OK thanks.. i have seen it :) , however i am referring to the fact that there are only traces over ALL the zeros, i was wondering if there are traces which involve only the ZEROS on the line 1/2 to compare this trace with weil's one. http://mathoverflow.net/questions/93226/first-order-linear-differential-equation-with-boundary-conditions Comment by Mathman Mathman 2012-04-06T10:12:55Z 2012-04-06T10:12:55Z umm then perhaps my condition is too restrictive , how about only invariance under $f(x)= fp^{k}x)$ here 'p' means all the primes $p =2 ,3,5,7,.....$ and k=1,2,3,4,5,.. $is still$ f=0$?? thanks :) http://mathoverflow.net/questions/93226/first-order-linear-differential-equation-with-boundary-conditions Comment by Mathman Mathman 2012-04-05T21:39:16Z 2012-04-05T21:39:16Z aja.. however if we assume that$ f$is a DISTRIBUTION instead of a funciton is there a possibility to get a different result to$ f=0 $?? thanks for your answers http://mathoverflow.net/questions/93226/first-order-linear-differential-equation-with-boundary-conditions Comment by Mathman Mathman 2012-04-05T21:00:35Z 2012-04-05T21:00:35Z wouldn't the function$ F(x)= \sum_{q}f(qx) $with a sum taken over all the positive rational would satisfy the boundary conditions ?? with$ F(0)=0 $and$ \int_{0}^{\infty} F(x)dx =0 $http://mathoverflow.net/questions/93226/first-order-linear-differential-equation-with-boundary-conditions Comment by Mathman Mathman 2012-04-05T16:15:27Z 2012-04-05T16:15:27Z here D is the derivative operator with respect to 'x' and the boundary conditions apply to ALL the primes$ p=2,3,5,7,.... $http://mathoverflow.net/questions/54501/riemann-zeta-function-connection-to-quantum-mechanics/90390#90390 Comment by Mathman Mathman 2012-03-07T14:37:39Z 2012-03-07T14:37:39Z sorry.. A is a constant and$ D= \frac{d}{dx} $derivative operator , here$ \sqrt D \$ stands for the fractional derivative of N(x)