mmcdara

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These are questions asked by mmcdara

Is it possible to transform relation (2) into relation (7) without using the hand-made relation (3) and the  Sum -> Int -> Sum trick?

restart

f := Product(x[i]^a*(1-x[i])^b, i)

Product(x[i]^a*(1-x[i])^b, i)

(1)

Lf := ln(f);

ln(Product(x[i]^a*(1-x[i])^b, i))

(2)

Sum(ln(x[i]^a*(1-x[i])^b), i)

Sum(ln(x[i]^a*(1-x[i])^b), i)

(3)

expand(%) assuming x[i] > 0, x[i] < 1, a > 0, b > 0

Sum(a*ln(x[i])+b*ln(1-x[i]), i)

(4)

eval(%, Sum=Int)

Int(a*ln(x[i])+b*ln(1-x[i]), i)

(5)

IntegrationTools:-Expand(%);

a*(Int(ln(x[i]), i))+b*(Int(ln(1-x[i]), i))

(6)

Lf = eval(%, Int=Sum)

ln(Product(x[i]^a*(1-x[i])^b, i)) = a*(Sum(ln(x[i]), i))+b*(Sum(ln(1-x[i]), i))

(7)

 

Download From2to7.mw

TIA

I recently answered a question concerning the Lane-Emden equation (see here LaneEmden) where the main topic was about finding its numerical solution.

The generic form of the Lane-Emden equation with parameter n is

LaneEmden := n -> (Diff(xi^2*(Diff(theta(xi), xi)), xi)) = -theta(xi)^n * xi^2

      d   /  2 / d            \\             n   2
n -> ---- |xi  |---- theta(xi)|| = -theta(xi)  xi 
      dxi \    \ dxi          //                  


I have just realized that I missed a "small" point in the original question: the OP ( @shashi598 ) wrote
"[...] Maple never comes out of evaluating [the] analytical solution when n=5 [...] ".
The important point here is that this solution (at least for some initial conditions) is known and simple (in the sense it doen't involve any special function).

So I tried for a few hours to verify this claim, and ended wondering myself if it might not be right?

Could you please tell me (I guess @shashi598 would be interested too in your return) if the differential equation LaneEmden(5) can be solved formally?
TIA.

Emden_equation.mw


EDITED:
After a little research it seems that very specigic method are used to build the analytic solution of the LaneEmden(n) (n not equal to 0, 1 and 5): serie expansions, homotopy, Adomian decomposition for instance.
I wasn't capable to find how the solution for LaneEmden(5) have been got for the first time (iseems to be atthe end of the 19th century).

(I would prefer a solution for Maple 2015, but answers relative to newer versions are welcome)

Is there a simple way to force the result -y(1) + y(2) without using one of these two tricks?

# how can I get the expression of
int(diff(y(x), x), x=1..2);
                      / d                  \
                   int|--- y(x), x = 1 .. 2|
                      \ dx                 /

# Trick 1
int(diff(y(x), x), x);
eval(%, x=2)-eval(%, x=1)
                              y(x)
                          -y(1) + y(2)

# Trick 2
J := Int(diff(y(x), x), x = 1..2): 
value(IntegrationTools:-Parts(J, 1));
                          -y(1) + y(2)

TIA

Note: I would -prefer an answer for Maple 2015, but I can accommodate an answer for a more recent version.

I have a function Gpdf from IR2 to IR+ of class C1 (this comes from the way this function is built).
Although its level curves are continuous, their display show discontinuities for some level values. 

The reason is that  Gpdf contains a term whose denominator vanishes and so, even if the left and right limits of Gpdf are the same at the vanishing point, the resulting plot is dicontinuous.

More details are given in the attached file Discontinuous_contours.mw.

I have tried to adjust the plotting grid, or even to superimpose contours drawn in domains containing no singularities, but I wasn't capable to get continuous drawings (see the attached file).

Do you have any idea to achieve this?

TIA

Could someone provide me with the references used when implementing the Statistics:-PredictiveLeastSquares function?

TIA

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