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help pages...

vv 13827
June 29 2017
1 1

OK, I have installed the update and everything seems to be OK, no problems detected (Win7, 64 bit).
The mentioned cmaple help problem is still present but it's not a very serious bug.

About your 2D example in the help page issue: the argument of value should be a label; you have probably unchecked "Show equation labels" In Tools > Options > Display. But even so, value(...) executes correctly after pressing Enter.
Switching 1D <--> 2D also works.

It is true that 1D <--> 2D  does nothing in the "How do I" page, but note that this is not actually a help page, but an embedded document. It seems that for all real help pages it works.

 

SmithForm...

vv 13827
June 26 2017
0 5

Maple has very good implementations for both, see  ?SmithForm,  ?HermiteForm

They even work in general Euclidean domains, see  ?LinearAlgebra[Generic]:-SmithForm

 

Eigenvectors...

vv 13827
June 25 2017
2 3

You just need:

Eigenvectors(K, evalf(M));

(omit evalf for a symbolic solution).

 

 

Example...

vv 13827
June 24 2017
2 2 
plot(cos(1/theta)/sqrt(1-theta^2), theta=0.2 .. 0.99,
     labels = ['theta', typeset(cos," (","1/",theta,") / ",sqrt(sort(1-theta^2,theta,ascending)))]);

Probably o solution using Typesetting exists but with more "acrobatics".
In my opinion Maple should provide a "LaTeX" option  for such situations:
label = ['theta', LaTeX(...)],    title = LaTeX(...)   etc.

 

assume...

vv 13827
June 16 2017
1 0

int(1/(x^2*sqrt(1-x^4/b^4)), x = a .. b) assuming a>0,b>a ;

 

The definition of the differential order...

vv 13827
June 14 2017
1 0

The definition of the differential order (DO) is recursive.
Suppose that we have a single indeterminate x.
E.g. if E is a (nonconstant) expression then DO(diff(E,x)) = DO(E) + 1  and  DO(h(E)) = DO(E), h being a function.
This approach seems to be natural.

Apollonius...

vv 13827
June 08 2017
5 4 
Apollo:=proc()
local A:=table([[0,-1],[-1/2,2],[1/2,2],[2*I/3,3]]), N:=4, fifth, gen;
fifth :=proc(i1,i2,i3,i4)
  local k5,z5;
  if N>500 then return fi;  
  k5 := 2*(A[i1][2]+A[i2][2]+A[i3][2])-A[i4][2];  
  z5 := (2*(A[i1][2]*A[i1][1]+A[i2][2]*A[i2][1]+A[i3][2]*A[i3][1])-A[i4][2]*A[i4][1])/k5;
  if k5<200 and not member([z5,k5],A) then N:=N+1; A[N]:=[z5,k5];
    gen(i1,i2,i3,N);
  fi;
end:
gen:= proc(i1,i2,i3,i4) fifth(i1,i2,i3,i4);fifth(i1,i2,i4,i3);
                        fifth(i1,i3,i4,i2);fifth(i2,i3,i4,i1) end;
gen(1,2,3,4);   #print(N);
plots:-display(seq(plottools:-circle([Re(A[i][1]),Im(A[i][1])],1/A[i][2]), i=1..N),
    axes=none, caption=cat("Apollonius gasket: ",N, " circles plotted"), size=[800,800])
end:

Apollo();

split...

vv 13827
June 07 2017
2 3


 

Maple does not know to manipulate indexed variables from a symbolic sum such as

> 

s:=sum(f[i](x[i]),i=1..n);

sum(f[i](x[i]), i = 1 .. n)

 (1)

For example, diff will act syntactically, giving wrong mathematical results

> 

diff(s,x[i]);  # wrong

sum(diff(f[i](x[i]), x[i]), i = 1 .. n)

 (2)
> 

diff(s,x[j]); # wrong

0

 (3)

To differentiate versus x[j]  we must split each such sum in sums independent of x[j]

 

In the previous example:

 

 

> 

sj:=sum(op(1,s),i=1..j-1) + sum(op(1,s),i=j+1..n) + subs(i=j, op(1,s));

sum(f[i](x[i]), i = 1 .. j-1)+sum(f[i](x[i]), i = j+1 .. n)+f[j](x[j])

 (4)
> 

diff(sj, x[j]);  # OK

diff(f[j](x[j]), x[j])

 (5)
> 

diff( x[j]*sj^2, x[j]);  # OK

(sum(f[i](x[i]), i = 1 .. j-1)+sum(f[i](x[i]), i = j+1 .. n)+f[j](x[j]))^2+2*x[j]*(sum(f[i](x[i]), i = 1 .. j-1)+sum(f[i](x[i]), i = j+1 .. n)+f[j](x[j]))*(diff(f[j](x[j]), x[j]))

 (6)

Note that sometimes such a split could be tricky, for example:

 

> 

sum(x[floor(i/3)]*x[i], i=3..n);

sum(x[floor((1/3)*i)]*x[i], i = 3 .. n)

 (7)
> 

 


 

Download split.mw

direct solution...

vv 13827
June 06 2017
0 0


 

> 

restart;
> 

f:=w__1 * x__1 + w__2 * x__2; constr:=x__1^a * x__2^b - y=0;

w__1*x__1+w__2*x__2

  

x__1^a*x__2^b-y = 0

 (1)
> 

extrema(f, constr, {x__1, x__2}, 's'):
> 

expand~(s[])  assuming positive;

{x__1 = a^(b/(a+b))*w__2^(b/(a+b))*y^(1/(a+b))/(w__1^(b/(a+b))*b^(b/(a+b))), x__2 = w__1^(a/(a+b))*b^(a/(a+b))*y^(1/(a+b))/(a^(a/(a+b))*w__2^(a/(a+b)))}

 (2)
> 

combine(%);

{x__1 = w__1^(-b/(a+b))*a^(b/(a+b))*b^(-b/(a+b))*w__2^(b/(a+b))*y^(1/(a+b)), x__2 = w__1^(a/(a+b))*a^(-a/(a+b))*b^(a/(a+b))*w__2^(-a/(a+b))*y^(1/(a+b))}

 (3)

It is possible to combine the middle terms but I think that this should not matter (i.e. Maple should not be used as a typesetting machine).

spacecurve...

vv 13827
June 05 2017
4 3

Catalan1 := (alpha,beta)->[alpha-sin(alpha)*cosh(beta), 1-cos(alpha)*cosh(beta), 4*sin((1/2)*alpha)*sinh((1/2)*beta)]:
p1:=plot3d(Catalan1(alpha,beta), alpha = 0 .. 2*Pi, beta = -3 .. 3, scaling = constrained, title = "Catalan's surface", titlefont = [Courier, bold, 14]):
with(plots):
p2:=spacecurve( [Catalan1(alpha,-3), Catalan1(alpha,3)],alpha=0..2*Pi,color=red,thickness=5):
display(p1,p2);

 

scalars...

vv 13827
June 05 2017
2 3


 

It seems that now the Vector and Matrix constructors accept only scalars.

But we may use delayed evaluation.

 

> 

Vector(2,(a) -> 'Matrix'(2,2,(b,c) -> m||a||b||c));

Vector(2, {(1) = Matrix(2, 2, {(1, 1) = m111, (1, 2) = m112, (2, 1) = m121, (2, 2) = m122}), (2) = Matrix(2, 2, {(1, 1) = m211, (1, 2) = m212, (2, 1) = m221, (2, 2) = m222})})

 (1)
> 

A:=<1,2;3,4>;B:=<2,3,4>;

A := Matrix(2, 2, {(1, 1) = 1, (1, 2) = 2, (2, 1) = 3, (2, 2) = 4})

  

Vector[column](%id = 18446744074182844158)

 (2)
> 

Vector([B,B]); #assemble

Vector(6, {(1) = 2, (2) = 3, (3) = 4, (4) = 2, (5) = 3, (6) = 4})

 (3)
> 

Vector([A,B]);

Error, (in Vector) initializer list contains elements of width > 1 and depth > 1

  
> 

Vector(['A','B']);

Vector(2, {(1) = Matrix(2, 2, {(1, 1) = 1, (1, 2) = 2, (2, 1) = 3, (2, 2) = 4}), (2) = Vector(3, {(1) = 2, (2) = 3, (3) = 4})})

 (4)
> 

Matrix(1,2,[[A,B]]);

Error, (in Matrix) this entry is too tall or too short: Vector(3, {(1) = 2, (2) = 3, (3) = 4})

  
> 

Matrix(1,2,[['A','B']])

Matrix(1, 2, {(1, 1) = Matrix(2, 2, {(1, 1) = 1, (1, 2) = 2, (2, 1) = 3, (2, 2) = 4}), (1, 2) = Vector(3, {(1) = 2, (2) = 3, (3) = 4})})

 (5)

Change...

vv 13827
June 03 2017
3 1

J:=int(ln(x)^n,x):
ch := t=ln(x):   IntegrationTools:-Change(J, ch):
simplify(eval(%, ch));

It will work...

vv 13827
June 01 2017
1 0

1. Don't use procedures. Use expressions:

SED2 := 32/3 * ...  

(if you really need procedures, for diff use unapply).

2. To have polynomials extract only the numerators:

SED2 := numer(SED2).

(Maybe you will have to take the derivatives before that, it depends on your intentions).

3. Use ":" instead of ";"  only after everything works OK,

 

Floats?...

vv 13827
June 01 2017
1 0

rand(-2 .. 2)  generates coefficients from the set {-2,-1,0,1,2}.

If you want floats (real numbers), use rand(-2.0 .. 2.0)

Numerically...

vv 13827
May 29 2017
0 1
> 

restart;
> 

Digits:=15;

15

 (1)

 

Solving numerically the recurrence

 

F(0,y) = 0,

F(x,y) = y/(y+1)*F(x-1,y)^((y+1)/y) + 1/(y+1)

 

and approximating  limit( F(n,n),  n = infinity)

 

No chance for a closed form (probably).

 

> 

F:=proc(x,y)
   if x=0 then return 0 fi;
   y/(y+1)*F(x-1,y)^((y+1)/y)+1/(y+1)
end:
> 

F(2,2);

(2/27)*3^(1/2)+1/3

 (2)
> 

F(1000,1000.);

.420589864597856

 (3)
> 

F(10^4,10.^4);

.420522651099579

 (4)
> 

F(10^5,10.^5);

Error, (in F) too many levels of recursion

  
> 

 

 

We must use a non-recursive version:

 

> 

f:=proc(x,y)
local r:=0,a, z:=(y+1)/y, w:=y/(y+1), y1:= 1/(y+1);
to x do r:=  w*r^z+y1 od;
r
end:
> 

f(2,2);

(2/27)*3^(1/2)+1/3

 (5)
> 

seq( [n, f(10^n,10.^n)],n=1..6);

[1, .428422578046699], [2, .421269413312481], [3, .420589864597856], [4, .420522651099579], [5, .420515940874970], [6, .420515270000679]

 (6)
> 

 

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