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?...

vv 13805
May 22 2020
0 0

@mmcdara 

I don't see any contradiction. A good PRNG (pseudorandom number generator) exists of course (in my opinion) in Maple. The question was about the existence of a statistical test in Maple able to discriminate the two lists.  

@Carl Love  Thanks. I was hoping t...

vv 13805
May 21 2020
0 0

@Carl Love 

Thanks. I was hoping that one of the general Statistics' tests can be used (more or less directly) to answer the question.
After all, the problem does not seem to be very special. Probably using run lengths the two sequences can be discriminated indeed.

@janhardo  If you need help, you s...

vv 13805
May 20 2020
0 0

@janhardo 

If you need help, you should post worksheets like this:


 

> 

 

> 

f := x -> x^sin(x);

proc (x) options operator, arrow; x^sin(x) end proc

 (1)
> 

g := x -> x^cos(x);

proc (x) options operator, arrow; x^cos(x) end proc

 (2)
> 

plot( [f,g], 1..15, color=[red,blue]);

 
> 

# area=?, please help.


Download 02xxx.mw

syntax...

vv 13805
May 17 2020
0 0

@ecterrab 

I don't quite understand about one and only one syntax:

In 1D input:  3!=6;    ==>  6 = 6
In 2D input:  3!=6;    ==>  3 <> 6

It's true that in 2D,  3!  =6;    ==>  6 = 6
but isn't this about syntax?

Classic...

vv 13805
May 15 2020
0 0

@janhardo 

If you apply what tomleslie said, you have nothing to learn, because Worksheet Mode + 1D-input is almost the same as Classic Maple (actually Classic Maple also had & has a 2D input). The disadvantage of the modern Worksheet Mode is that it starts slower and uses much more memory -- due to Java; but it's more flexible.

the "obvious" fact...

vv 13805
May 15 2020
0 0

@Preben Alsholm 

> 

restart;
local gamma;
beta:=1/6; gamma:=1; delta:=13/6;  # special case
 

gamma

  

1/6

  

1

  

13/6

 (1)
> 

##The system in the pdf-file. Right hand sides:
RHS:=[x*(1-x)-x*y,y*(delta-beta*y/x)-alpha*y/(gamma+y)];
ROsol:=solve(RHS=~0,{x,y});
RO:=op(indets(ROsol[2],specfunc(RootOf)));
E:=subs(ROsol[2],[x,y]);
 

[x*(1-x)-x*y, y*(13/6-(1/6)*y/x)-alpha*y/(1+y)]

  

{x = 1, y = 0}, {x = 1-RootOf(14*_Z^2+(-6*alpha+1)*_Z+6*alpha-13), y = RootOf(14*_Z^2+(-6*alpha+1)*_Z+6*alpha-13)}

  

RootOf(14*_Z^2+(-6*alpha+1)*_Z+6*alpha-13)

  

[1-RootOf(14*_Z^2+(-6*alpha+1)*_Z+6*alpha-13), RootOf(14*_Z^2+(-6*alpha+1)*_Z+6*alpha-13)]

 (2)
> 

x2y2,x3y3:=allvalues(E);

[29/28-(3/14)*alpha-(1/28)*(36*alpha^2-348*alpha+729)^(1/2), (3/14)*alpha-1/28+(1/28)*(36*alpha^2-348*alpha+729)^(1/2)], [29/28-(3/14)*alpha+(1/28)*(36*alpha^2-348*alpha+729)^(1/2), (3/14)*alpha-1/28-(1/28)*(36*alpha^2-348*alpha+729)^(1/2)]

 (3)
> 

da:=diff(E,alpha);
dxa,dya:=op(subs(RO=y,y=1-x,da));
J:=VectorCalculus:-Jacobian(RHS,[x,y]);
TR:=LinearAlgebra:-Trace(J);
 

da := [-(6*RootOf(14*_Z^2+(-6*alpha+1)*_Z+6*alpha-13)-6)/(28*RootOf(14*_Z^2+(-6*alpha+1)*_Z+6*alpha-13)-6*alpha+1), (6*RootOf(14*_Z^2+(-6*alpha+1)*_Z+6*alpha-13)-6)/(28*RootOf(14*_Z^2+(-6*alpha+1)*_Z+6*alpha-13)-6*alpha+1)]

  

dxa, dya := 6*x/(29-28*x-6*alpha), -6*x/(29-28*x-6*alpha)

  

Matrix(%id = 18446744074364419718)

  

19/6-2*x-y-(1/3)*y/x-alpha/(1+y)+alpha*y/(1+y)^2

 (4)
> 

TR0:=simplify(eval(TR, [x,y]=~x2y2));

(1/7)*((432*alpha^3+288*alpha^2-5508*alpha+1692)*(36*alpha^2-348*alpha+729)^(1/2)+2592*alpha^4-10800*alpha^3-45432*alpha^2+147708*alpha+45684)/((-29+6*alpha+(36*alpha^2-348*alpha+729)^(1/2))*(27+6*alpha+(36*alpha^2-348*alpha+729)^(1/2))^2)

 (5)
> 

alpha0:=solve(TR0); evalf(%);

-1/2+2*3^(1/2)

  

2.964101616

 (6)
> 

simplify(eval(diff(TR0,alpha), alpha=alpha0));

3/4

 (7)
> 

subs(x=x2y2[1], alpha=alpha0,  gamma*( (beta+delta)*x - beta )^2 / (alpha*x)^2):  
simplify(%); # the "obvious" fact: 3/4 <> 1/3

1/3

 (8)
> 

 

solve...

vv 13805
May 11 2020
0 0

@Carl Love 

Thanks. I know and I have also used SolveTools:-SemiAlgebraic. Both could be better and faster.

Risch...

vv 13805
May 07 2020

Thank you for this info!
I see that the Risch algorithm works indeed better when the radicals are expressed as RootOfs, and even in older versions (e.g. Maple 2015). So, I wonder why the conversion is not automatic (at least as an option in int).

parametric...

vv 13805
May 05 2020
0 0

@nm 

It is a standard parametric plot. plot( [sin(t),sin(t),  t=0..420]); So, x = sin(t), y = sin(t), t in the interval [0,420].
 

geometrically...

vv 13805
May 02 2020

Nice, vote-up.
Note that a geometric solution is much simpler:

1. An ellipse is the orthogonal projection of a circle.
2. A triangle inscribed in a circle has maximal area iff it is equilateral.
So, an animation is given by:

a:=4: b:=sqrt(3): B:=plot([a*cos(t),b*sin(t), t=0..2*Pi], scaling=constrained):
plots:-animate(plot,[[seq([a*cos(t+2*k*Pi/3),b*sin(t+2*k*Pi/3)], k=0..3)]],t=0..2*Pi, background=B)

 

 

ln(1+x)...

vv 13805
May 01 2020
0 0

@Carl Love 

I know, but for small x (otherwise ln1 is useless) even series is much better than calling ln.

@Carl Love My point is that evalf/l...

vv 13805
May 01 2020
0 0

@Carl Love 

My point is that evalf/ln1 should use a special algorithm for ln(1+x) (e.g. via series) being so much faster. BTW, Digits=64 in ln1 is enough for OP's example.

Digits=91?...

vv 13805
May 01 2020
0 0

@Carl Love 

Note that this way in OP's example the procedure evalf/ln1 uses Digits=91 (!). So, what it does is:

evalf[32](evalf[91](ln(1+exp(-64))));

More or less standard notations and defi...

vv 13805
April 29 2020
0 0

All these symbols are defined here: https://en.wikipedia.org/wiki/List_of_mathematical_symbols

It's a matter of preferences and definitions/notations, rather than ambiguity. Even N could represent {0,1,...} or {1,2,...}, but the exact definition is clear from the context, or reminded in the case of a serious textbook or article.

documented...

vv 13805
April 29 2020
0 0

@Carl Love 

It is unfortunate that they are not documented.

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