ASKSAGE: Sage Q&A Forum - RSS feedhttps://ask.sagemath.org/questions/Q&A Forum for SageenCopyright Sage, 2010. Some rights reserved under creative commons license.Tue, 22 Dec 2020 11:28:43 +0100Extending piecwise to 2 variables (or a variable and a parameter)https://ask.sagemath.org/question/54836/extending-piecwise-to-2-variables-or-a-variable-and-a-parameter/This piecewise function work correctly
score = piecewise([([i,i], ncand - i) for i in range(ncand)])
because `ncand` is defined earlier in my notebook. when I typeset say `score(1)` It returns what I expect. But I would like that ncand be a variable.CyrilleTue, 22 Dec 2020 11:28:43 +0100https://ask.sagemath.org/question/54836/Is it possible to substitute $U^\prime$ to $D_0(U)$ ?https://ask.sagemath.org/question/54555/is-it-possible-to-substitute-uprime-to-d_0u/The result of this code
var('x y p w0 g c')
assume(p>=0)
assume(p<=1)
U=function('U')(x)
EU(x,y,p)=p*U(x)+(1-p)*U(y)
show(EU(w0,g*w0-c,p))
show(diff(EU(w0,g*w0-c,p),w0))
uses $D_0(U)$ for the first order derivative of $U$. Is it possible to substitute $U^\prime$ to $D_0(U)$, to keep the path with textbook notations ?CyrilleFri, 04 Dec 2020 15:18:11 +0100https://ask.sagemath.org/question/54555/Substitute a function in a formal derivativehttps://ask.sagemath.org/question/54543/substitute-a-function-in-a-formal-derivative/This gives a nice result.
var('p w0 c g dc dg dp dw0')
EUa=function('EUa')(p,w0,c,g)
EUa_c=diff(EUa,c)
EUa_g=diff(EUa,g)
#
EUna=function('EUna')(p,w0,c,g)
EUna_g=diff(EUna,g)
EUna_c=diff(EUna,c)
##
dEUa=EUa_c*dc + EUa_g*dg
dEUna=EUna_c*dc + EUna_g*dg
###
show(EUa)
show(EUna)
show(dEUa)
show(dEUna)
sol=solve(dEUa==dEUna, dg)
sol=(sol[0]/dc).full_simplify()
show(sol)
But now, I would like to substitute to `EUa(p,w0,c,g) = p*U(w0)+(1-p)*U(0)` and `EUna(p,w0,c,g) = p*U(w0-c)+(1-p)*U(g*w0-c)` then, later change the unknown function `U(w)` to say `ln(w)` or `w^(1/2)`. I suppose I need to define first a function `U=function('U')(w)` after to define `w` as a variable. But all my tentatives fail.CyrilleThu, 03 Dec 2020 18:32:39 +0100https://ask.sagemath.org/question/54543/Formal implicit differentiationhttps://ask.sagemath.org/question/50991/formal-implicit-differentiation/As my question was poorly asked, I rewrite it.
I was searching for the code of first order and second order differentiation of an implicit function. Finally I found and post it.CyrilleFri, 24 Apr 2020 18:41:04 +0200https://ask.sagemath.org/question/50991/Composition of piecewise functionhttps://ask.sagemath.org/question/54466/composition-of-piecewise-function/[Remark: This is not a question but I do not know how to signal a mistake in the doc.
[http://fe.math.kobe-u.ac.jp/icms2010-dvd/SAGE/www.sagemath.org/doc/reference/sage/functions/piecewise.html](http://fe.math.kobe-u.ac.jp/icms2010-dvd/SAGE/www.sagemath.org/doc/reference/sage/functions/piecewise.html)
uses `Piecewise` which leads to an error, not `piecewise` which is good.]
Here is my question:
This code does exactly what I expect
A = 1000
g(x) = x^2
f = piecewise([[[1, 1], (A/12)],
[[2, 2], (A/(12 - 1) - 1*10)],
[[3, 3], (A/(12 - 2) - 2*10)]])
show(f(1), ", ", f(2), ", ", f(3))
fg = piecewise([[[1, 1], g(A/12)],
[[2, 2], g(A/(12 - 1) - 1*10)],
[[3, 3], g(A/(12 - 2) - 2*10)]])
show(fg(1), ", ", fg(2), ", ", fg(3))
but I have a lot of `g(x)` to compose with `f` so I would like
to know if there is a way to define a composition of functions
for piecewise functions (here defined on subset of `ZZ`).
There is also the problem where I have a function `h`
which is itself a piecewise function on the same set and I want
fgh = piecewise([[[1, 1], h(1)*g(A/12)],
[[2, 2], h(2)*g(A/(12 - 1) - 1*10)],
[[3, 3], h(3)*g(A/(12 - 2) - 2*10)]])
show(fg(1), ", ", fg(2), ", ", fg(3))CyrilleMon, 30 Nov 2020 16:53:05 +0100https://ask.sagemath.org/question/54466/sagemath fails to solve an equation and inequationhttps://ask.sagemath.org/question/54131/sagemath-fails-to-solve-an-equation-and-inequation/I do not understand why
var('R','p','x')
sol1=solve(R^(1/3)*p == x^(1/3),x)
show(sol1)
this work but this fails to isolate $x$
var('R','p','x')
sol1=solve(R^(1/3)*p <= x^(1/3),x)
show(sol1)
And that
var('R','p','x')
sol2=solve(0 == (R-x)^(1/3)*p + (1-p)* (-x)^(1/3),x)
show(sol2)
doesn't find $x$. And, as a consequence, of cours this :
var('R','p','x')
sol2=solve(0 <= (R-x)^(1/3)*p + (1-p)* (-x)^(1/3),x)
show(sol2)
fails. I have tried all simplification ways.CyrilleFri, 06 Nov 2020 12:58:08 +0100https://ask.sagemath.org/question/54131/From indexed variables to stringshttps://ask.sagemath.org/question/54026/from-indexed-variables-to-strings/ I can easily generate a list of variables with
xx = var("x", n=10, latex_name="x")
but I have many questions about this
1) `nc` is the number of variables needed. Say I need variables indexed from 10 to 20, not from 0 to 20. Am I obliged to
generate all 20 variables and let appart the 10 firsts (I have found nothing on options like n=, could it be a range ?).
2) if I have generated the following list `x=[x_1, x_2, x_3, x_4, x_5]` how may i transform it in a list of strings `x=['x_1', 'x_2', 'x_3', 'x_4', 'x_5']`
3) and what if I need to change the name of certain variable as ` 'x_3'-->'\\epsilon_2'` CyrilleSun, 25 Oct 2020 18:40:37 +0100https://ask.sagemath.org/question/54026/Creating sublist according countinuous intervalshttps://ask.sagemath.org/question/52066/creating-sublist-according-countinuous-intervals/I have a list of number say
`l = [1.25, 2.46, 3.99, 4, 5, 6, 7.12, 8.76 ,9.34, 10.42, 11.5, 12.]`
I have a list of real intervals
`Cl=[[0, 5),[5, 10),[10, 15)]`
How to create sublists of l according to the intervals in Cl ? If I was work in N there will be no difficulties.CyrilleThu, 18 Jun 2020 10:49:52 +0200https://ask.sagemath.org/question/52066/Replace all numbers < e-n by 0https://ask.sagemath.org/question/51784/replace-all-numbers-e-n-by-0/ As always two question in one.
F = random_matrix(RR,7,50)
import numpy as np
covMatrix = np.cov(F,bias=True)
Ω=matrix(covMatrix)
Λ=Ω.inverse()
Ω*Λ
1) Is there a way to ask Sama (nicname for SAgeMAth more elegant than SM -- a wink) to overcome the number less than e-n and replace it by 0 --- I have found a way : to use `change_ring(QQ)` but I want to know if therre is another way.
2) is there a way to ask Sama to display only n decimal digit.CyrilleSat, 06 Jun 2020 11:38:15 +0200https://ask.sagemath.org/question/51784/general implicit differentiationhttps://ask.sagemath.org/question/51519/general-implicit-differentiation/Recently i have been helped to write an implicit function differentiator (nice neologism isn't it). Here is the code
def implicit_derivative(V):
var("dw1, dw2")
V_w1 = diff(V, w1)
V_w2 = diff(V, w2)
# Differential
dV = V_w1 * dw1 + V_w2 * dw2
# Dérivée du premier ordre
sol=solve(dV==0, dw2)
impder=(sol[0]/dw1)
return impder
This work without difficulty for $V$ function of $w_1$ and $w_2$. But if my variables are $x$ and $y$ or say $\chi$ and $\zeta$. It will not work. I have not found the mechanism to define a general function not dependant of the name of its arguments. And here there is a second problem to find thway to associate the increase d... to its correlative argument that is if I use $\chi$ as the first variable $d\chi$ must substitute to $dw_1$.CyrilleSat, 23 May 2020 17:11:32 +0200https://ask.sagemath.org/question/51519/Strange behaviour in simplifying a functionhttps://ask.sagemath.org/question/51482/strange-behaviour-in-simplifying-a-function/ I am developping a long notebook to master Sage. But each time I fall on a problem. This time
p, D, w_0, x, I = var('p, D, w_0, x, I')
β = var('beta')
π = var('pi')
w00=8
D1=6
p1=0.5
A = matrix(SR, 2, 2, [[w_0, 1],[w_0-π,1]])
y = vector([w_0-D, w_0-D-π+I])
sol=A.solve_right(y)
sola=sol[0].full_simplify().function(D, π, w_0)
solb=sol[1].full_simplify().function(D, π, I, w_0)
f=(sola*x + solb).function(x, D, π, I, w_0)
g=f(x, D, β*I*(1-p), I, w_0).factor().collect(x).simplify().function(x, D, β, p, w_0)
h(x, D, p, w0)=g(x, D, 1, p, I, w_0).collect(D).collect(x).simplify()
show(f)
show(g)
show(h)
1) this is the main question : You can observe that `g` is independanr of `I`. But `h` is `g` with $\beta = 1$. So why `I` does reappear in h ?
2) How could we explain **SM** to write the slope of the 3 curves as I would write them in a sheet of paper ? CyrilleThu, 21 May 2020 19:57:45 +0200https://ask.sagemath.org/question/51482/Substitution in implicit functionhttps://ask.sagemath.org/question/51390/substitution-in-implicit-function/ This is the code to the implicit differentiaion of a function from the theoretical point of view
var("w1, w2, a, p")
EU(w1, w2, a, p)= p*w1^a+ (1-p)*w2^a
var("dw1, dw2")
V_w1 = diff(V, w1)
V_w2 = diff(V, w2)
# Differential
dV = V_w1 * dw1 + V_w2 * dw2
show("dV ="+latex(dV))
# Dérivée du premier ordre
sol=solve(dV==0, dw2)
show(sol[0]/dw1)
Now without to be obliged to rewrite all the commands, I would like to apply this procedure to the EU function or any other function. I have tried to use `dV.substitute_function(V, EU)` and `(sol[0]/dw1).right_hand_side()` but without success. I have an amplified problem for the second order derivative since I am nosure of my formulation due to the fact that one must substitute the first order derivative inside derivation.
CyrilleTue, 12 May 2020 18:09:05 +0200https://ask.sagemath.org/question/51390/Desattribution of a value to a variablehttps://ask.sagemath.org/question/51389/desattribution-of-a-value-to-a-variable/ Suppose I have attributed to `a` the value 3. But for the following operation I want to use anew `a` as a variable how should I operate, redefine `a` ?CyrilleTue, 12 May 2020 16:53:36 +0200https://ask.sagemath.org/question/51389/Full simplification of power doesn't workhttps://ask.sagemath.org/question/51359/full-simplification-of-power-doesnt-work/ x, a =SR.var('x a')
Uf1= x^a
ar1_x=diff(Uf1, x).full_simplify()
ar1_xx=diff(Uf1, x,2)
raar1 = -diff(Uf1, x,2)/diff(Uf1, x).full_simplify()
show(LatexExpr("U^{'}(x) ="+latex(ar1_x)))
show(LatexExpr("U^{''}(x) ="+latex(ar1_xx)))
show(LatexExpr("R_a(x) ="+latex(raar1)))
Why raar1 is not fully simplified ?CyrilleSun, 10 May 2020 00:22:33 +0200https://ask.sagemath.org/question/51359/Simplification failshttps://ask.sagemath.org/question/51052/simplification-fails/1) I have tried nearly all simplification methods, expansion, factorization, collect... on the following code for `Ra1`
var("x,a")
Ra1 = -diff(x^a, x,2)/diff(x^a, x)
show(Ra1)
but the self-evident simplification doesn't work. I wonder why. I also have tried to add assumption even I do not see why since it's only algebra
2) Is there a way to ask SM to factor on a variable ?CyrilleMon, 27 Apr 2020 11:18:46 +0200https://ask.sagemath.org/question/51052/Choosing the way a function is displayedhttps://ask.sagemath.org/question/50953/choosing-the-way-a-function-is-displayed/1) I have the following function `z=-(D*p-p*x - D + w0)/ (p-1)`. Is it possible to display it as `-(p/(1-p))*x + ((w0-D*(1-p)))/(1-p)`?
2) Can I select a part a this function --- the factor of `x`, `x` and the last part ?CyrilleThu, 23 Apr 2020 12:12:23 +0200https://ask.sagemath.org/question/50953/Strange behavior of bool()https://ask.sagemath.org/question/50942/strange-behavior-of-bool/Here is my code
var("w0, pi, D, I")
A = matrix(SR, 2, 2, [[w0, 1],[w0-pi,1]])
y = vector([w0-D, w0-D-pi+I])
sol=A.solve_right(y)
sola=sol[0].full_simplify().function(D, pi, I, w0)
solb=sol[1].full_simplify().function(D, pi, I, w0)
f(x, D, pi, I, w0)=sola*x + solb
show(f(x, D, pi, I, w0))
f_x = f.diff(x).full_simplify()
show(f_x)
f_I = f.diff(I).full_simplify()
show(f_I)
Now, I make some assumptions and I verify the sign of $f_x$ and $f_I$.
assume(I, "real")
assume(pi, "real")
assume(I > pi)
bool(f_x < 0)
assume(w0, "real")
assume(x, "real")
assume(w0 > x)
bool(f_I > 0)
What is weird is that the result doesn't change should I ask for `bool(f_x < 0)` or ` bool(f_x > 0)` and the same for $f_I$. I know that my question looks like an already asked question "symbolic functions and bool". But this time the expressions are elementary.CyrilleThu, 23 Apr 2020 06:41:02 +0200https://ask.sagemath.org/question/50942/Evaluation of a function and it's inverse with parametershttps://ask.sagemath.org/question/50878/evaluation-of-a-function-and-its-inverse-with-parameters/The following code works perfectly :
var('W1, W2, Ub, p')
a, x, y = SR.var('a,x,y')
W2=W
U(x)=x^a
assume(a, 'real')
assume(a, 'noninteger') # or alternatively 'integer'
assume(x, 'real')
assume(y, 'real')
assume(x > 0)
assume(y > 0)
U(x).substitute(x==y)
show(U(x))
V=solve(x == U(y), [y])[0].rhs().function(x)
show(V)
VV=((Ub-p*U(W2))/(1-p))^(1/a)
show(VV)
But now I want to be also able to attribute some values to the parameters and the variable to achieve an evaluation and plot according to the parameter values. I need the function and it's inverse. I have seen the methode `lambda x` in the documentation but it's end with an error.CyrilleMon, 20 Apr 2020 19:13:36 +0200https://ask.sagemath.org/question/50878/Computing the inverse of a function with parameter(s)https://ask.sagemath.org/question/50809/computing-the-inverse-of-a-function-with-parameters/ I have adapted the code shown in the answer of "Can Sagemath compute the inverse of a function?". In the original exemple, the function was $log(x)$ so there is no parameters. But I would like to do the same with $x^a$. But even if i insert an `assume` command, I finally fall on an error. Here is my code
var('x,y,a')
assume(0<=a<=1)
U(x) = x^a
V(x) = solve(x == U(y), y)[0].rhs()
show(V)
Is there a way to obtain the result $x^{1/a}$ ?CyrilleFri, 17 Apr 2020 12:29:27 +0200https://ask.sagemath.org/question/50809/Numerical solvehttps://ask.sagemath.org/question/50808/numerical-solve/When I use a dictionary with solve, the result is product with radicals. I have found no way in the documentation to obtain a decimal expension. But If I call the solution as `sol = solve()` it's not difficult to obtain directly the decimal expansion with `sol[0].rhs().n()`. So is there an unseen option in solve to obtain this and if not could it be implemented ?CyrilleFri, 17 Apr 2020 12:27:27 +0200https://ask.sagemath.org/question/50808/How to tell to sagemath to solve for the good roothttps://ask.sagemath.org/question/50160/how-to-tell-to-sagemath-to-solve-for-the-good-root/ Hello, this is a simple problem of optimization
> %display latex
>
> var('w, gamma, alpha, beta, h, theta,
> p, A, B'); U= w^(gamma); Pi =
> p*h^(theta)- w*h; N= (U)^alpha *
> (Pi-B)^beta show(U) show(Pi) show(N)
> show(N_w)
> N_w= factor(N.diff(w))
> assume(0 < alpha <1)
> assume(0 < beta <1)
>z=solve(N_w==0, w, solution_dict=True)
>show(z)
If I say no bulshit, with the assumptions only one solution is valid. But Sagemath display all solutions ? How could it be done automaticaly ?
The factoriZation shows that
> assume(0 < alpha <1) assume(0 < beta
> <1) z=solve(N_w==0, w,
> solution_dict=True)CyrilleWed, 04 Mar 2020 13:21:18 +0100https://ask.sagemath.org/question/50160/A last substitutionhttps://ask.sagemath.org/question/48033/a-last-substitution/The result of the following code is perfectly normal.
%display latex
var('A, x, y, l, alpha, beta, R, p_x, p_y');
assume(A>0,alpha>0,beta>0,x>0,y>0);
U=A*x^(alpha)*y^(beta);
U_x= U.diff(x);
U_y= U.diff(y);
tms=(U_x/U_y).canonicalize_radical();
var('x,y')
y= function('y');
tms1=(U_x/U_y).substitute(y=y(x)).canonicalize_radical();
tms1.diff(x)
courbtms=solve(tms1.diff(x)==0, y(x).diff(x),solution_dict=True)
courbtms
but it looks like this
$\left[\frac{\partial}{\partial x}y(x) = \frac{y(x)}{x}\right]$
It's the first time I see this type of brackets. I can extract the right member of this equation
courbtms[0][y(x).diff(x)]
but what I am not able to obtain is to substitue backward $y$ to $y(x)$? I have tried
courbtms[0][y(x).diff(x)].substitute[y(x)=z]
but it obviously doesn't work.CyrilleSun, 22 Sep 2019 22:29:22 +0200https://ask.sagemath.org/question/48033/Change of programmation of implicithttps://ask.sagemath.org/question/48022/change-of-programmation-of-implicit/ In the former question "Calculus with formal functions: substitution?", one can obtain implicit differentiation by the following code
var('f,x,y')
y = function('y',x)
f = function('f',x,y)
f.diff(x)
which gives
D[1](f)(x, y(x))*D[0](y)(x) +D[0](f)(x, y(x))
But I use the last version ofg SageMath and I have the following error
TypeError: function() takes exactly 1 positional argument (2 given)
Is there some new way to handle this code ?CyrilleSun, 22 Sep 2019 10:31:09 +0200https://ask.sagemath.org/question/48022/How to substitute a function in a derivativehttps://ask.sagemath.org/question/48007/how-to-substitute-a-function-in-a-derivative/ I know that if $U(x, y) = A x^\alpha y^\beta$, I have $U_x = \alpha A x^{\alpha-1} y ^\beta$. But by substitution, I can obtain $U_x = \alpha \left(\frac{U}{x}\right\)$. Is there a way to obtain this in Sagemath. I have tried
U_x= U.diff(x).subs(A*x^(alpha)*y^(beta)== U)
but this has no effect.
CyrilleSat, 21 Sep 2019 14:29:05 +0200https://ask.sagemath.org/question/48007/Simplification and implicit functionhttps://ask.sagemath.org/question/48010/simplification-and-implicit-function/I have two question in one :
1) I have this $z= \frac{\alpha x^{\alpha-1}y^\beta y^{-\beta-1}}{\beta x^\alpha}$. There are obvious simplification since this is equal to $\frac{\alpha}{\beta}
\frac{y}{x}$.
z.full_simplify() has no impact
z.expand() simplify on $y$ but not on $x$
So what can I do to obtain the good result.
2) how to take the result and consider it as an implicit function to obtain the derivative de $y$. as a function of xCyrilleSat, 21 Sep 2019 16:21:08 +0200https://ask.sagemath.org/question/48010/how to run maxima code in Sage?https://ask.sagemath.org/question/40184/how-to-run-maxima-code-in-sage/I have the following code in maxima to calculate the laplacian of a function in parabolic coordinates.
assume(r≥0)$
assume(theta≥0,theta≤2*π)$
load(vect)$
derivabbrev:true$
scalefactors(parabolic)$
declare(f,scalar)$
depends(f,rest(parabolic))$
ev(express(laplacian(f)),diff,expand,factor);
What is the proper way to run those Maxima commands in SageMath?
Thanks,
DanieldanielvolinskiSat, 16 Dec 2017 16:47:28 +0100https://ask.sagemath.org/question/40184/Gradient, Divergence, Curl and vector productshttps://ask.sagemath.org/question/10104/gradient-divergence-curl-and-vector-products/Are there implementations of vector product and the nabla operator yet? I can't find anything.KiMon, 21 Oct 2013 12:40:22 +0200https://ask.sagemath.org/question/10104/Nodal analysis in networkhttps://ask.sagemath.org/question/23624/nodal-analysis-in-network/Hi experts!
I have:
* Q nodes (Q stick-stick intersections)
* a list 'NODES'=[(x,y,i,j)_1,........, (x,y,i,j)_Q], where each element (x,y,i,j) represent the intersection point (x,y) of the sticks i and j.
* a matrix 'H' with QxQ elements {H_k,l}.
H_k,l=0 if nodes 'k' and 'l' aren't joined by a edge, and H_k,l = R_k,l = the electrical resistance associated with the union of the nodes 'k' and 'l' (directly proportional to the length of the edge that connects these nodes).
* a list 'nodes_resistances'=[R_1, ....., R_Q].
All nodes with 'j' (or 'i') = N+1 have a electric potential 'V' respect all nodes with 'j' or 'i' = N.
Now i must apply NODAL ANALYSIS for determinate the electrical current through each of the edges, and the net current. I have no ideas about how to do that. Can you help me?
Thanks a lot!
Best regards
mresimulatorTue, 29 Jul 2014 14:24:51 +0200https://ask.sagemath.org/question/23624/Social Network Analysishttps://ask.sagemath.org/question/10652/social-network-analysis/I would like to know if there is any features in Sage to analise social networks like:
> ## Social Network Analysis ##
Drawing on Mathematica's strong graph and network capabilities, Mathematica 9 introduces a complete and rich set of state-of-the art social network analysis functions. Access to social networks from a variety of sources, including directly from social media sites, and high level functions for community detection, cohesive groups, centrality, and similarity measures make performing network analysis tasks easier and more flexible than ever before.
----------
I mean exactly anlysis of the real networks not graph theory.bakantThu, 24 Oct 2013 06:57:31 +0200https://ask.sagemath.org/question/10652/Time series analysishttps://ask.sagemath.org/question/8086/time-series-analysis/Being new to TSA, I have some basic questions.
I have 5 minute interval sales data that I want to plot. The plot should have lower, higher and current estimate along with new 1 day prediction.
I understand I can use Holt Winters formula or ARIMA. Can someone help me achieve this (steps and methods) using SAGE?NIshantWed, 20 Apr 2011 09:31:25 +0200https://ask.sagemath.org/question/8086/