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2022-06-11 05:47:36 +0200 edited question Automate the code for Topological recursion

Automate the code for Topological recursion Let me start by giving a brief idea of the recursion. This recursion generat

2022-06-10 08:44:23 +0200 received badge  Enthusiast
2022-06-08 02:26:18 +0200 edited question Automate the code for Topological recursion

Automate the code for Topological recursion Let me start by giving a brief idea of the recursion. This recursion generat

2022-06-08 02:25:36 +0200 edited question Automate the code for Topological recursion

Automate the code for Topological recursion Let me start by giving a brief idea of the recursion. This recursion generat

2022-06-08 02:23:54 +0200 edited question Automate the code for Topological recursion

Automate the code for Topological recursion Let me start by giving a brief idea of the recursion. This recursion generat

2022-06-08 02:22:29 +0200 edited question Automate the code for Topological recursion

Automate the code for Topological recursion Let me start by giving a brief idea of the recursion. This recursion generat

2022-06-08 02:21:31 +0200 edited question Automate the code for Topological recursion

Automate the code for Topological recursion Let me start by giving a brief idea of the recursion. This recursion generat

2022-06-08 02:20:13 +0200 edited question Automate the code for Topological recursion

Automate the code for Topological recursion Let me start by giving a brief idea of the recursion. This recursion generat

2022-06-08 02:19:37 +0200 asked a question Automate the code for Topological recursion

Automate the code for Topological recursion Let me start by giving a brief idea of the recursion. This recursion generat

2022-06-04 04:32:46 +0200 commented answer Computing with the residue in Sagemath

Thanks a lot, will follow your suggestion. I am wondering why my code didn't give the same answer. I will try to do it f

2022-06-04 04:24:05 +0200 marked best answer Computing with the residue in Sagemath

I have done the following computation to calculate $W12(z1,z2)$ I am expecting the result

$$ \begin{aligned} W_{1,2}(z_1 , z_2) &= Res_{z =0}\, K(z)\frac{1}{z-z_1} \left( W_{0,3}(z,-z,z_2) + W_{0,2}(z,z_2)W_{1,1}(-z)+ W_{0,2}(-z,z_2)W_{1,1}(z)\right) \\ &= 1/8\, {\frac { \left( 2\,a{{\it z1}}^{2}+5\,{a}^{2} \right) {{\it z2}}^ {4}+ \left( 2\,a{{\it z1}}^{4}+3\,{a}^{2}{{\it z1}}^{2} \right) {{\it z2}}^{2}+5\,{a}^{2}{{\it z1}}^{4}}{{{\it z1}}^{6}{{\it z2}}^{6}}} \end{aligned} $$ I got the correct result for W11, W03 with the code. For computing W12 I have to use the subsitution function, but at the end I not getting the correct result. I got the result I have computed in Maple. So please let me know what mistake I am making in the following code.

var('z,b,z1,z2,z3');

 assume(b > 0);

 y = lambda z: 2*arcsinh( z / (sqrt(2)*b) ) / sqrt(z^2 + 2*b^2)

K = lambda z: 1 / (z*(y(z) - y(-z)))

W02 = lambda z1, z2: 1/(z1 - z2)^2

W11 = lambda z1: ( K(z) / (z - z1) * (W02(z1,z2).substitute(z1==z, z2==-z))*derivative(-z,z) ).residue(z == 0).canonicalize_radical()

W03 = lambda z1, z2, z3 : ( K(z) / (z - z1) *( W02(z, z2)*W02(-z,z3)+W02(z, z3)*W02(-z,z2))*derivative(-z,z) 
                          ).residue(z == 0).canonicalize_radical()

W12 = lambda z1, z2: ( K(z) / (z - z1) *(W03(z1,z2,z2).substitute(z1==z , z2==-z, z3 ==z2)
                                         +W02(z1, z2).substitute(z1==z)*W11(z1).substitute(z1==-z)
                                         +W02(z1, z2).substitute(z1==-z)*W11(z1).substitute(z1==z))
                      * derivative(-z,z)).residue(z == 0).canonicalize_radical()
2022-06-01 14:09:37 +0200 commented question Computing with the residue in Sagemath

In my thesis https://bridges.monash.edu/articles/thesis/One-point_recursions_and_topological_recursion_for_enumerative_

2022-06-01 06:00:14 +0200 commented question Computing with the residue in Sagemath

It's a recursive definition $W_{g,n}(z_1 , \ldots ,z_n) = Res_{z=0} \left ( W_{g-1 , n+1}(z,-z,z2,\ldots z_n) +\sum_{g_

2022-05-31 06:27:18 +0200 commented answer Residue calculation in Sagemath

I was computing as you shown for W12, though having correct answer for W11 and W03 I am not getting correct answer for W

2022-05-31 06:25:21 +0200 asked a question Computing with the residue in Sagemath

Computing with the residue in Sagemath I have done the following computation to calculate $W12(z1,z2)$ I am expecting th

2022-05-31 06:11:53 +0200 commented answer Residue calculation in Sagemath

The following code like the way you told give me the right W11, W03. When I compute W12 it is not giving me the right an

2022-05-31 06:10:59 +0200 commented answer Residue calculation in Sagemath

The following code like the way you told give me the right W11, W03. When I compute W12 it is not giving me the right an

2022-05-24 05:17:19 +0200 commented answer Differential equation for function involving hyperbolic Sine

Also one thing to notice is that it never involves any $a$ in the coefficient of the ode.

2022-05-24 05:15:13 +0200 commented answer Differential equation for function involving hyperbolic Sine

I see thank. you.

2022-05-24 04:52:21 +0200 commented answer Differential equation for function involving hyperbolic Sine

I am bit confused about your claim that derivative of $K(z)$ is rational? I am having the log at the denominator.

2022-05-21 04:12:07 +0200 commented answer Differential equation for function involving hyperbolic Sine

Thanks a lot, I guess it should ODE not linear ODE.

2022-05-21 04:10:47 +0200 marked best answer Differential equation for function involving hyperbolic Sine

Let $y(z) = 2 \frac{sinh^{-1}(z/(2a)^{1/2})}{(z^2 +2a)^{1/2}}$

where $sinh(z)$ is the sine hyperbolic function.

$$ K(z):=\frac{1}{z(y(z) - y(-z))} \rightarrow 1/2\,{\frac {a}{{z}^{2}}}+1/6-{\frac {{z}^{2}}{90\,a}}+{\frac {{z}^{4} }{378\,{a}^{2}}}-{\frac {23\,{z}^{6}}{28350\,{a}^{3}}}+{\frac {263\,{z }^{8}}{935550\,{a}^{4}}}-{\frac {133787\,{z}^{10}}{1277025750\,{a}^{5} }}+{\frac {157009\,{z}^{12}}{3831077250\,{a}^{6}}}-{\frac {16215071\,{ z}^{14}}{976924698750\,{a}^{7}}}+{\frac {2689453969\,{z}^{16}}{ 389792954801250\,{a}^{8}}}+O \left( {z}^{18} \right) $$

How can I find the linear differential equation in $\frac{\partial}{\partial z }$ with coefficent in the polynomial ring $\mathbb{C}[z]$ that annihilates $K(z)$. I am unable to do it by hand I think some software in sagemath might help.

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2022-05-18 14:12:15 +0200 commented answer Residue calculation in Sagemath

Thanks will do that

2022-05-18 14:12:04 +0200 marked best answer Residue calculation in Sagemath

I have done the following calculation in maple, I want to know if we can do it in Sagemath and write a code using the recursive definition. We construct a family of meromorphic function $W_{g,n}(z_1, z_2, \ldots, z_n)$. We consturct it recursively. The intial data given is the following.

$$y(z) = 2 \frac{sinh^{-1}(z/(2a)^{1/2})}{(z^2 +2a)^{1/2}}$$ where $sinh(z)$ is sine hyperbolic funciton can be defined directly in maple as it is.

$$ K(z):=\frac{1}{z(y(z) - y(-z))} \rightarrow 1/2\,{\frac {a}{{z}^{2}}}+1/6-{\frac {{z}^{2}}{90\,a}}+{\frac {{z}^{4} }{378\,{a}^{2}}}-{\frac {23\,{z}^{6}}{28350\,{a}^{3}}}+{\frac {263\,{z }^{8}}{935550\,{a}^{4}}}-{\frac {133787\,{z}^{10}}{1277025750\,{a}^{5} }}+{\frac {157009\,{z}^{12}}{3831077250\,{a}^{6}}}-{\frac {16215071\,{ z}^{14}}{976924698750\,{a}^{7}}}+{\frac {2689453969\,{z}^{16}}{ 389792954801250\,{a}^{8}}}+O \left( {z}^{18} \right) $$ $K(z)$ we define using $sinh(z)$ and take the Taylor series expansion to study the few terms.

Let $W_{0,2}(z_1, z_2): = 1/(z_1 - z_2)^2$

Having this initial data we can construct a tower of $W_{g,n}(z_1, z_2, \ldots, z_n )$ as follows. Let me give few examples that I have computed by hand in maple.

Let's give some examples of how to generate families of functions $$W_{1,1} = Res_{z =0}\, K(z)\frac{1}{z-z_1} W_{0,2}(z,-z) = 1/24\,{\frac {{{\it z1}}^{2}+3\,a}{{{\it z1}}^{4}}} $$ Taking residue at $z =0$ means collecting the $1/z$ ceofficients. $$W_{0,3}(z_1, z_2 , z_3) = Res_{z =0}\, K(z)\frac{1}{z-z_1} \left(W_{0,2}(z_1,-z)W_{0,2}(z_2,z) + W_{0,2}(z_2,-z)W_{0,2}(z_1,z)\right)= {\frac {a}{{{\it z2}}^{2}{{\it z3}}^{2}{{\it z1}}^{2}}} $$

$$W_{1,2}(z_1 , z_2) =Res_{z =0}\, K(z)\frac{1}{z-z_1} \left( W_{0,3}(z,-z,z_2) + W_{0,2}(z,z_2)W_{1,1}(-z)+ W_{0,2}(-z,z_2)W_{1,1}(z)\right)= \ 1/8\,{\frac { \left( 2\,a{{\it z1}}^{2}+5\,{a}^{2} \right) {{\it z2}}^ {4}+ \left( 2\,a{{\it z1}}^{4}+3\,{a}^{2}{{\it z1}}^{2} \right) {{\it z2}}^{2}+5\,{a}^{2}{{\it z1}}^{4}}{{{\it z1}}^{6}{{\it z2}}^{6}}} $$ So let's define $W_{g,n}(z_1 , z_2, \ldots, z_n)$ in general by taking the residue of $$W_{g_1,n_1}(z,\ldots)W_{g_2 , n_2}(-z,\ldots) + W_{g-1 , n+1}()\tag{*}$$ along with the product $ K(z)\frac{1}{z-z_1}$. $W_{g_1,n_1}(z_1 , z_2, \ldots, z_{n_1})$ represent meromorphic functions in $n_1$ variables. And the sum is taken over all such possible combinations. And in case of $W_{g-1 , n+1}(z,-z,z_2 \ldots z_n)$.

In general, if we can write a code where we can give the initial data and then will compute $W_{g,n}(z_1, z_2, \ldots , z_n)$. It should be a recursive definition. I maple I cannot compute more than $(3,1)$ tuples. But with effective Dynamical programming, we definitely can do more. If someone shows me how to do the initial computation in sagemath, then I will try to write the general programme.

2022-05-17 01:24:24 +0200 received badge  Commentator
2022-05-17 01:24:24 +0200 commented answer Residue calculation in Sagemath

I meant $arcsinh( z/ \sqrt(2a))$ there is no square root over $z$ inside the arcsinh.

2022-05-16 09:13:28 +0200 edited question Residue calculation in Sagemath

Residue calculation in Sagemath I have done the following calculation in maple, I want to know if we can do it in Sagema

2022-05-16 09:13:21 +0200 commented answer Residue calculation in Sagemath

$$y(z) = 2 \frac{sinh^{-1}(z/(2a)^{1/2})}{(z^2 +2a)^{1/2}}$$ this is the function , so the sqroot on the numerator is on

2022-05-16 09:11:03 +0200 commented question Residue calculation in Sagemath

I have edited it now.

2022-05-16 09:10:36 +0200 edited question Residue calculation in Sagemath

Residue calculation in Sagemath I have done the following calculation in maple, I want to know if we can do it in Sagema

2022-05-16 08:54:00 +0200 commented question Differential equation for function involving hyperbolic Sine

Yes @Max Alekseyev you are right

2022-05-16 08:49:54 +0200 commented question Differential equation for function involving hyperbolic Sine

The last post https://ask.sagemath.org/question/62355/residue-calculation-in-sagemath/ I realise that if there is a recu

2022-05-16 08:48:31 +0200 commented question Differential equation for function involving hyperbolic Sine

Yes I have meant what you wrote. No it's homework question, it's part of a question I ask in last post, which I cannot s

2022-05-15 08:26:19 +0200 asked a question Differential equation for function involving hyperbolic Sine

Differential equation for function involving hyperbolic Sine Let $y(z) = 2 \frac{sinh^{-1}(z/(2a)^{1/2})}{(z^2 +2a)^{1/2

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2022-05-08 07:19:30 +0200 asked a question Residue calculation in Sagemath

Residue calculation in Sagemath I have done the following calculation in maple, I want to know if we can do it in Sagema

2022-05-08 06:34:11 +0200 asked a question Skew Schur function

Skew Schur function I have worked with the Schur function using the following command for example sage: Sym = Symmetri

2022-04-18 07:39:13 +0200 marked best answer Skew Schur function

I have worked with the Schur function using the following command for example

sage: Sym = SymmetricFunctions(QQ)
sage: P = Sym.p()  # to expand symmetric functions in power symmetric basis
sage: s = Sym.schur()
sage: P(s([2,2]))
1/12*p[1, 1, 1, 1] + 1/4*p[2, 2] + (-1/3)*p[3, 1]

Recently I need to work with the skew Schur function. I want to expand the shifted schur function $$ s_{\lambda}(x_1 +y , x_2 +y ,\ldots , x_n +y) $$ I am guessing that it can be the following identity $$ s_{\lambda}(x_1 +y , x_2 +y ,\ldots , x_n +y) = \sum_{\mu} s_{\lambda \setminus mu}(y) s_{\mu}(x_1 , x_2, \ldots, x_n) $$ For $y=1$ my idenity can be proven as I am sure there is an expression. The expression $s_{\lambda \setminus mu}(y)$ is skew Schur function. I am wondering if there are packages in sagemath where I can work with skew schur function and verify my identity is correct or not.