User rh - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T19:45:20Z http://mathoverflow.net/feeds/user/25947 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/128910/can-we-find-a-set-of-elliptic-curves-over-rationals-associated-with-f Can we find a set of elliptic curves over rationals associated with $f$?. RH 2013-04-27T09:23:14Z 2013-04-27T14:54:29Z <p>We know that for each elliptic curve over rationals, we can define the Dirichlet series of the Hasse–Weil $L$-function, i.e., the function associated with an elliptic curve over rationals.</p> <p>Then my <strong>question</strong> is: Consider a motivic $L$-functions $f$, can we find a set of elliptic curves over rationals associated with $f$?. Simply I ask about the inverse of the first statement in the motivation of this question.</p> http://mathoverflow.net/questions/127977/a-generalisation-of-the-birch-and-swinnerton-dyer-conjecture A generalisation of the Birch and Swinnerton-Dyer conjecture RH 2013-04-18T15:46:17Z 2013-04-18T16:14:31Z <p>We know that the grand Riemann hypothesis is a generalisation of the Riemann hypothesis and Generalized Riemann hypothesis. My question is about the existence of a similar generalisation of the Birch and Swinnerton-Dyer conjecture.</p> http://mathoverflow.net/questions/127462/how-i-can-choose-t-1-t-2-t-r-in-0-1r-such-that-fk-left1-2 How I can choose $(t_1,t_2,...,t_{r}) \in (0,1)^{r}$ such that $f^{(k)}\left(1-2\prod_{j=1}^{k}t_{j}\right)=0$? RH 2013-04-13T14:03:21Z 2013-04-15T15:27:13Z <p>Let $f:\mathbb{R} \to \mathbb{R}$ be a real analytic function. Assume that $f$ has simple trivial zeros at each nonpositive integer. Then, all the $k$-th derivatives $f^{(k)}$ of $f$ have necessarily infinitely many real zeros. Let us consider the functions: $f^{(k)}(1-2\prod_{j=1}^{k}t_{j})$ for $k=1,..,r$ and $(t_1,t_2,...,t_{r})\in (0,1)^{r}$. We know that these functions have also infinitely many real zeros.<br> My <strong>question</strong> is: How I can choose $(t_1,t_2,...,t_{k})∈(0,1)^{k}$ such that $f^{(k)}(1-2\prod_{j=1}^{k}t_{j})=0$ and $f^{(k+1)}(1-2 \prod_{j=1}^{k} s_{j})\neq 0$ that is, the real $(1 - 2\prod_{j=1}^{k}s_{j})$ is a simple root of the function $f^{(k)}$?</p> http://mathoverflow.net/questions/127115/does-h-have-infinitely-many-isolated-zeros Does $h$ have infinitely many isolated zeros? RH 2013-04-10T16:57:35Z 2013-04-13T16:49:54Z <p>Let $f:ℝ→ℝ$ be a real analytic function with infinitely many isolated zeros. Let us define the function: $$h(s₁,s₂,...,s_{r+1})=\prod_{k=1}^{r+1}f^{(k+1)}(\left(1-2\prod_{j=1}^{k}s_{j}\right)$$ Also, all the $k$-th derivatives of $f$ have infinitely many isolated zeros. Does $h$ have infinitely many <strong>isolated</strong> zeros? I know that $f′(1-2s₁)=0$ has infinitely many isolated zeros. </p> http://mathoverflow.net/questions/126720/the-kth-derivative-of-a-l-function-has-necessarily-infinitely-many-zeros The $k^{th}$ derivative of a L-function has necessarily infinitely many zeros RH 2013-04-06T17:34:42Z 2013-04-06T19:48:11Z <p>My current <strong>question</strong> is concerned with a reference (paper or book) containing a proof of this result: The $k^{th}$ derivative of a L-function has necessarily infinitely many zeros. </p> http://mathoverflow.net/questions/126243/the-hasse-weil-l-function-and-some-equations The Hasse-Weil L-function and some equations RH 2013-04-02T08:30:19Z 2013-04-02T09:17:13Z <p>Let $f$ be an analytic function verfifying </p> <p>$f(s)=\epsilon f(2-s)$</p> <p>where $\epsilon=\pm 1$. The expression of Hasse-Weil L-function $f$ is </p> <p>$$f(s)=N^{s/2}(2\pi)^{-s}\Gamma(s)\sum_{n=1}^{\infty}\frac{a_{n}}{n^{s}}$$</p> <p>where $N$ is an integer and $\Gamma(s)$ is the gamma function.</p> <p>Let $r$ be an integer. I have a set of equations of the form</p> <p>$$f^{(k)}\left(1-2\prod_{j=1}^{k}s_{j}\right)=f^{(k)}\left(1-2\prod_{j=1}^{k}t_{j}\right)$$ </p> <p>for all $k=1,...,r$. Here $f^{(k)}$ is the $k$-th derivative of $f$. </p> <p>Can I deduce that </p> <p>$$(t_1,t_2,\ldots,t_{r})=(s_1,s_2,\ldots,s_{r})$$</p> <p>under some conditions on the derivatives of $f$?</p> <p>The injectivity is not possible for this case. </p> http://mathoverflow.net/questions/116101/the-origin-of-the-root-number-wc-1-the-sign-of-the-functional-equation The origin of the root number $w(C/ℚ)=±1$ (the sign of the functional equation) RH 2012-12-11T17:08:02Z 2012-12-12T07:39:23Z <p>The motivation for this question is the same as in my previous question in MO: <a href="http://mathoverflow.net/questions/115179/real-root-1-of-the-hasse-weil-l-function-of-c-over" rel="nofollow">http://mathoverflow.net/questions/115179/real-root-1-of-the-hasse-weil-l-function-of-c-over</a></p> <p>I am just curious to know the origin of the root number $w(C/ℚ)=±1$ (the sign of the functional equation $f(s)=w(C/ℚ)f(2-s)=εf(2-s)$) of the curve $C$. I read several papers on this topic, but I cannot find where this root number come from. I wonder if this number has some relation with $a$ and $b$ in the equation of the curve $C$. </p> http://mathoverflow.net/questions/115891/hadamards-product-formula-for-the-derivative Hadamard's product formula for the derivative RH 2012-12-09T16:02:59Z 2012-12-09T21:52:17Z <p>Let $f$ be an entire function of order $ρ&lt;\infty$. Assume that $f$ does not vanish identically on $\mathbb{C}$. Then, we know that $f$ has a Hadamard's product formula</p> <p>$$f(s) =e^{g(s)}s^{r}\prod _ {k=1}^{\infty}\frac{s _ {k}-s}{s _ {k}} e^{s/s _ k}$$</p> <p>the integer $r$ is the order of vanishing of $f$ at $s=0$, the $s_{k}$ are the other zeros of $f$ listed with multiplicity, $g$ is a polynomial of degree at most $ρ$, and the product converges uniformly in bounded subsets of $ℂ$. My question is how I can deduce directely a Hadamard's product formula for the derivative $f^′$ from the one of the function $f$. </p> http://mathoverflow.net/questions/113962/a-rapidly-converging-series-of-the-hasseweil-l-function-associated-with-an-ellip A rapidly-converging series of the Hasse–Weil L-function associated with an elliptic curve over rationals RH 2012-11-20T16:30:10Z 2012-11-20T17:39:22Z <p>I know that for some L-series there is still a rapidly-converging series. My question is about the existence of a such a series for the Dirichlet series of the Hasse–Weil L-function associated with an elliptic curve over rationals. A google search do not gives important answers. </p> http://mathoverflow.net/questions/113841/solution-of-certain-forms-of-equations Solution of certain forms of equations RH 2012-11-19T16:15:03Z 2012-11-19T19:07:20Z <p>I ask about a possible method to find the solution of algebraic equations of the form</p> <p>$axⁿ+byⁿ+c=0$</p> <p>where $a,b,c,x,y$ are real constants and $n$ is an integer. Maybe there is a simple method, but I cannot find it.</p> http://mathoverflow.net/questions/112682/riemann-siegel-function-and-gamma-function Riemann Siegel function and gamma function RH 2012-11-17T14:41:41Z 2012-11-19T18:50:46Z <p>I ask about an idea to prove this formula:</p> <p>$Γ(1/2-iβ)=((√π)/(√(coshπβ)))exp(-i(2ϑ(β)+βln2π+arctan(tanh(1/2)πβ)))$</p> <p>where $ϑ(β)$ is the Riemann Siegel function.</p> http://mathoverflow.net/questions/74289/whats-the-application-of-hyperchaotic-system/112768#112768 Answer by RH for Whats the application of Hyperchaotic system? RH 2012-11-18T15:27:40Z 2012-11-18T15:27:40Z <p>It is well known that if two or more Lyapunov exponents of a dynamical system are positive throughout a range of parameter space, then the resulting attractors are hyperchaotic. The importance of these attractors is that they are less regular and are seemingly "almost full" in space, which explains their importance in fluid mixing [Scheizer &amp; Hasler, 1996, Abel et al., 1997, Ottino,1989; Ottino et al., 1992]. See:</p> <p>Abel. A, Bauer. A, Kerber. K, Schwarz. W, [1997] " Chaotic codes for CDMA application," Proc. ECCTD '97, 1, 306.</p> <p>Kapitaniak.T, Chua. L. O, Zhong. Guo-Qun, [1994] " Experimental hyperchaos in coupled Chua's circuits," Circuits,.Syst. I: Fund. Th. Appl. 41 (7), 499 -- 503.</p> <p>Ottino. J. M, [1989] " The kinematics of mixing: stretching, chaos, and transport," Cambridge: Cambridge University Press.</p> <p>Ottino. J. M, Muzzion. F. J, Tjahjadi. M, Franjione.J. G, Jana. S. C, Kusch. H. A, [1992] " Chaos, symmetry, and self-similarity: exploring order and disorder in mixing processes," Science. 257, 754--760.</p> <p>Scheizer. J, Hasler. M, [1996] " Multiple access communication using chaotic signals," Proc. IEEE ISCAS '96. Atlanta, USA, 3, 108.</p> <p>Thamilmaran. K, Lakshmanan. M, Venkatesan. A, [2004] " Hyperchaos in a Modified Canonical Chua's Circuit," Int. J. Bifurcation and Chaos. 14 (1),221--244.</p> http://mathoverflow.net/questions/111437/functional-equation-and-constant-functions Functional equation and constant functions RH 2012-11-04T09:02:40Z 2012-11-06T17:03:33Z <p>I ask about this claim: let $f$ be an entire function satisfying $f(s)=u(s)f(a-s)$. Assume that $s$ and $a-s$ are not zeroes of $f$ and $f (bar)(a-s)=f(s)$ in a region $D$ ($f(bar)$ is the conjugate of $f$). Then the module of $f(s)/f(bar)(a-s)$ is equal to $1$, implying that the module of $u(s)$ is also $1$. The question is: Does this result implies that in fact the function $u(s)$ is constant. Thank you in advance.</p> http://mathoverflow.net/questions/111623/weierstrass-factorization-theorem-in-several-variables Weierstrass factorization theorem in several variables RH 2012-11-06T06:11:16Z 2012-11-06T13:03:52Z <p>Can one indicate to me the Weierstrass factorization theorem in several variables (real or complex). In one complex variable the result is well known. Thank you in advance.</p> http://mathoverflow.net/questions/111391/the-dirichlet-series-of-the-hasseweil-l-function The Dirichlet series of the Hasse–Weil L-function RH 2012-11-03T16:30:12Z 2012-11-03T19:36:14Z <p>I have the following question: Is there is a paper claiming that the Dirichlet series of the Hasse–Weil $L$-function (associated with an elliptic curve over rationals) is of finite order. Thank you in advance. I cannot find any result in the net.</p> http://mathoverflow.net/questions/110330/functional-equation functional equation RH 2012-10-22T13:58:59Z 2012-10-22T21:32:30Z <p>Hi to all I have the following question: Let f be an analytic function satisfying the functional equation: f(z)=u(z)*f(a-z) where "a" is a real constant. Let g be another function satisfying the same functional equation. In this case I asking if f=g. Thank you in advance.</p> http://mathoverflow.net/questions/105550/functional-equation-of-the-alternating-zeta-function Functional equation of the alternating zeta function RH 2012-08-26T16:30:01Z 2012-08-26T18:09:07Z <p>Can one let me know about the functional equation of the alternating zeta function similar to the well known for the rieman function.</p> http://mathoverflow.net/questions/129113/is-the-weils-converse-theorem-cover-all-elliptic-curves Comment by RH RH 2013-04-29T16:40:36Z 2013-04-29T16:40:36Z I reformulate he question. http://mathoverflow.net/questions/127977/a-generalisation-of-the-birch-and-swinnerton-dyer-conjecture/127981#127981 Comment by RH RH 2013-04-18T16:46:41Z 2013-04-18T16:46:41Z @ Arijit: Thank you very much for clarification. http://mathoverflow.net/questions/127977/a-generalisation-of-the-birch-and-swinnerton-dyer-conjecture/127981#127981 Comment by RH RH 2013-04-18T16:18:26Z 2013-04-18T16:18:26Z @ Marc Palm: Does there is a set of special L-functions where this conjecture apply just like the case of the Generalized Riemann hypothesis. http://mathoverflow.net/questions/127462/how-i-can-choose-t-1-t-2-t-r-in-0-1r-such-that-fk-left1-2 Comment by RH RH 2013-04-15T15:19:53Z 2013-04-15T15:19:53Z Yes, thank you. Fixed. http://mathoverflow.net/questions/127462/how-i-can-choose-t-1-t-2-t-r-in-0-1r-such-that-fk-left1-2 Comment by RH RH 2013-04-15T13:41:32Z 2013-04-15T13:41:32Z @Lo&#239;c Teyssier: Ok I edit the question. But I notice that this is the first version. Something changes my text! http://mathoverflow.net/questions/127115/does-h-have-infinitely-many-isolated-zeros Comment by RH RH 2013-04-10T18:56:32Z 2013-04-10T18:56:32Z @Pietro Majer: I changed the function to be $f:ℝ→ℝ$. http://mathoverflow.net/questions/126548/about-some-bijections Comment by RH RH 2013-04-04T19:29:04Z 2013-04-04T19:29:04Z So, such bijections does not exists. http://mathoverflow.net/questions/126296/how-i-can-prove-that-the-hasse-weil-l-function-vanishes-at-non-positive-integer Comment by RH RH 2013-04-02T17:31:12Z 2013-04-02T17:31:12Z Thank you very much for clarification. http://mathoverflow.net/questions/126243/the-hasse-weil-l-function-and-some-equations Comment by RH RH 2013-04-02T09:17:44Z 2013-04-02T09:17:44Z Yes. You make things clearer. http://mathoverflow.net/questions/126243/the-hasse-weil-l-function-and-some-equations/126245#126245 Comment by RH RH 2013-04-02T09:03:17Z 2013-04-02T09:03:17Z and what about the case if one side is zero. http://mathoverflow.net/questions/126243/the-hasse-weil-l-function-and-some-equations/126245#126245 Comment by RH RH 2013-04-02T09:01:19Z 2013-04-02T09:01:19Z @ Marc Palm: So the functional equation has no role in this analysis. http://mathoverflow.net/questions/125704/about-the-boundedness-of-the-set-of-mordell-weil-ranks Comment by RH RH 2013-03-28T14:58:07Z 2013-03-28T14:58:07Z My question is about current works on the topic of finiteness of the Mordell-Weil rank r of E/K. http://mathoverflow.net/questions/117619/solving-a-functional-equation Comment by RH RH 2012-12-30T21:01:30Z 2012-12-30T21:01:30Z @ Yemon Choi: Thank you very much for valauable comments and constructive criticism!!!!! http://mathoverflow.net/questions/117619/solving-a-functional-equation Comment by RH RH 2012-12-30T13:36:20Z 2012-12-30T13:36:20Z But it is very hard to obtain a such a solution. The form is closely to a modular form of order $k$ but it is not. http://mathoverflow.net/questions/117619/solving-a-functional-equation Comment by RH RH 2012-12-30T13:10:33Z 2012-12-30T13:10:33Z Thank you for your comment. But Still have trouble with solving it with respect to $f$ not $s$.