Hello

I wonder if there is a way to parallelize the function listed below.   It deals with a set of sets (dynamical systems).  As far as I can see the problem is with subs and simplify.  In some cases a division by zero shows up (the reason I added try..catch and had to use for). 

The procedure should remove the systems with imaginary coefficients and return the valid indices.  

Many thanks

Ed

Obs.:  The number of dynamical systems can easily be greater than 10,000.  Any speed improvement is welcome.

Example of one such a dynamical system is

[(-x*rho*xi[6]*(beta - 1)*sqrt(((rho + beta - 1)*sigma - beta + 1)^2*xi[8]^2/(beta - 1)^2) - xi[8]*(-2*y*sigma*xi[7]*xi[6]^2*(beta - 1)^2 + (sigma*rho + (sigma + 1)*(beta - 1))*x*rho*xi[6] + 2*sigma*z*rho^2*xi[8]))/(2*xi[6]*xi[8]*rho*(beta - 1)), (-(beta - 1)*(-y*sigma*xi[7]*xi[6]^2*(beta - 1)^2 + rho^2*(sigma*z*xi[8] + x^2))*xi[6]*sqrt(((rho + beta - 1)*sigma - beta + 1)^2*xi[8]^2/(beta - 1)^2) - (2*sigma*(beta - 1)^2*(y*xi[7]*(rho + beta - 1)*sigma/2 + x^2*rho + y*(beta - 1)*xi[7]/2)*xi[6]^3 + 2*x*z*rho*sigma*xi[8]*(beta - 1)^2*xi[6]^2 + (-z*xi[8]*(rho + beta - 1)*sigma^2 + ((2*beta^2*z - 3*beta*z + z)*xi[8] + x^2*rho - (beta - 1)*(-2*beta*y*xi[7] + x^2))*sigma + x^2*(beta - 1))*rho^2*xi[6] + 2*x*z*rho^3*sigma*xi[8])*xi[8])/(2*xi[8]*sigma*(beta - 1)*(rho^2 + (beta - 1)^2*xi[6]^2)*xi[6]*xi[7]), (-(-x*sigma*(beta - 1)^2*xi[6]^3 + (beta - 1)^2*(sigma*y*xi[7] + x^2)*xi[6]^2 - rho^2*sigma*x*xi[6] - sigma*z*rho^2*xi[8])*(beta - 1)*rho*sqrt(((rho + beta - 1)*sigma - beta + 1)^2*xi[8]^2/(beta - 1)^2) + xi[8]*(-2*y*sigma^2*xi[7]*(beta - 1)^4*xi[6]^4 + 2*(rho*(rho + beta - 1)*sigma/2 + (beta - 1)*((-beta + 1/2)*rho + y*(beta - 1)*xi[7]))*sigma*(beta - 1)^2*x*xi[6]^3 + (beta - 1)^2*(((-y*xi[7] + 2*z*xi[8])*rho + y*(beta - 1)*xi[7])*sigma^2 + (x^2*rho + (beta - 1)*(-2*beta*y*xi[7] - 2*beta*z*xi[8] + x^2 + y*xi[7]))*sigma - x^2*(beta - 1))*rho*xi[6]^2 + 2*(rho*(rho + beta - 1)*sigma/2 + (beta - 1)*((-beta + 1/2)*rho + y*(beta - 1)*xi[7]))*sigma*x*rho^2*xi[6] + xi[8]*sigma*z*((rho - beta + 1)*sigma - beta + 1)*rho^3))/(2*xi[8]^2*sigma*(beta - 1)*rho*(rho^2 + (beta - 1)^2*xi[6]^2))]

coefmodel

{beta = 8/3, rho = 28, sigma = 10}

and phi and coe equal to 

[[], [], []]

cleanSystems := proc(sys::list,phi::list,coe::list,coefmodel::{list,set})
description "This function applies coefficient values to a system and remove it if the coef is somehow imaginary":
local i,aaa:=table(),bbb:=table(),ccc:=table(),ind:=table():
for i from 1 to nops(sys) do
  try
   aaa[i]:=simplify(subs(coefmodel,sys[i])):
   if has(aaa[i],I) then
      aaa[i]:=NULL:
      bbb[i]:=NULL:
      ccc[i]:=NULL:
      ind[i]:=NULL:
   else
      bbb[i]:=simplify(subs(coefmodel,phi[i])):
      ccc[i]:=simplify(subs(coefmodel,coe[i])):
      ind[i]:=i:
   end if:
  catch:
   aaa[i]:=NULL:
   bbb[i]:=NULL:
   ccc[i]:=NULL:
   ind[i]:=NULL:
  end try:
end do:
aaa:=convert(aaa,list):print(nops(aaa)):
bbb:=convert(bbb,list):print(nops(bbb)):
ccc:=convert(ccc,list):print(nops(ccc)):
ind:=convert(ind,list):print(nops(ind)):
return(aaa,bbb,ccc,ind):
end proc:

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pde_ode.mw

The transformed boundary conditions are

Please and please, I have been having this challenge for some time now and I would be so so happy if I can get a solution to my challenge. I have been worried on:

how do I use these definitions (Volterra Integral Equation and Caputo definition) to solve

Helper, please. You can use any definition you are familiar with also. Only I need a fractional definition.

Thanks

My question is about the physics package: when transforming the metric tensor to new coordinates, sometimes the new coordinate can be expressed easily as a variable inside the new matrix entries,  but the new matrix is still displayed completely in terms of the old variables. How can I force it to display everything in terms of the new ones?

I have in internal procedure ppp that has an input prjpsn:=3. This set a Vector positions of x,y,z to either if prjpsn=1 

zpsn=1,   xpsn=2,  ypsn=3

or if prjpsn =3   

xpsn=1,   ypsn=2, zpsn=3

This then are used in another procedure. I can't get the values to transfere.  prjpsn would default to 3 in the module but can be set externally to 1 if needed.
The relevant parts are highlighted in green text. Cant get the worksheet to display.

Maple Worksheet - Errorprjpsn:=3


Failed to load the worksheet /maplenet/convert/Q_14-12-22_Module_internal_settings.mw .

Download Q_14-12-22_Module_internal_settings.mw

How do I calculate the conformal killing vectors of a metric tensor in GRTensorIII?

Why I get "Error, invalid subscript selector" error in my code?

y1 := Grid:-Seq(UP1(s, U, V, W, Phi, Xi, N, a, b, II, JJ, A, B, Dd, M, Ns), s = 1 .. 7);
UKt := add(y1[i], i = 1 .. 7);

Error, invalid subscript selector

As you can see y1 is defined without any problem and have 7 seqments, but the next line warns invalid subscript selector.

I have, for example,

> x(t) := sin(a)*cos(t) + cos(a)*sin(t)
            x := t -> sin(a) cos(t) + cos(a) sin(t)
> y(t) := combine(x(t))
            y := t -> combine(x(t))

>y(t);
            sin(a + t)

> x(t);
            sin(a) cos(t) + cos(a) sin(t)

# wanted but cannot do this: x(t) := combine(x(t))

Any workaround?

Thanks.

why does the command Matrix(2, 2, undefined) create the following

Matrix(2, 2, [[undefined(1, 1), undefined(1, 2)], [undefined(2, 1), undefined(2, 2)]])

instead of simple undefined for each item?

Example: After manipulating the following indefinite integral

Int(r^2, m)

I want to add ranges before evaluating the integral (for display purposes I keep the innert form). The op command or the IntegrationTools can be used for that (disassembling the expression and assembling it to new int expression) but require quite long code. Isn’t there a shorter  way to simply add ranges to the integration variable.

Update: Download Add_range.mw

Hi. I'm having a problem with the command phase portrait on maple when i'm trying to make a phase portrait of my autonomous differential equations. The problem is i can't get arrows on my phase portrait to see how the system behave with different intial value conditions, so my questions is how can i add arrows to the phase portrait? See below for a image of my code

Let a, b, c, d be real numbers. Define two geometric regions implicitly: a＋b＋c＋d＞0∧ab＋ac＋ad＋bc＋bd＋cd＞a²＋b²＋c²＋d², and abcd＞0∧abc＋abd＋acd＋bcd＞0. Suppose that somebody wants to prove that the former region is a subset of the latter one. Can we implement such an implication simply using certain ready-made commands in basic Maple? In other words, it is hoped that 

restart;
RealDomain:-simplify(forall([a, b, c, d], a + b​​​​​​​ + c + d > 0 and a*b + a*c + a*d + b*c + b*d + c*d > a^2 + b^2 + c^2 + d^2 implies a*b*c*d > 0 and a*b*c + a*b*d + a*c*d + b*c*d > 0));

Unfortunately, simplify just returns the statement unevaluated. So, how to address it in internal Maple? 

Note. The desired result (or return value) is true (see below).

1. restart; # Do not with(Logic): here
   RegularChains:-SemiAlgebraicSetTools:-QuantifierElimination(&A [d, c, b, a], ((d + c + b + a > 0) &and (b*a + c*a + d*a + c*b + d*b + d*c > a^2 + b^2 + c^2 + d^2)) &implies ((d*c*b*a > 0) &and (d*c*b + d*c*a + d*b*a + c*b*a > 0)));

2. restart;
   not SMTLIB:-Satisfiable(`not`(a + b + c + d > 0 and a*b + a*c + a*d + b*c + b*d + c*d > a^2 + b^2 + c^2 + d^2 implies d*c*b*a > 0 and a*b*c + a*b*d + a*c*d + b*c*d > 0)); # assuming real 

​​​​​​​​​​​​​​​​​​​​​Either ⒈ or ⒉ needs external calls. It seems that the Maple standard library functions in List of Commands still fail in simplifying expressions in the aforementioned form. Am I right?