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Answer...

vv 13867
September 04 2025
3 5

The extra question is not clear enough (for me).

> 

restart;
> 

y:=x -> cos(a*x)+cos(x):
> 

#  Suppose y is periodic with period T. ==>
> 

y(T)=y(0);

cos(a*T)+cos(T) = 2

 (1)
> 

solve(op([1,1],%) =1, T, allsolutions) = solve(op([1,2],%) =1, T, allsolutions)

2*Pi*_Z1/a = 2*Pi*_Z2

 (2)
> 

a=solve(%, a);

a = _Z1/_Z2

 (3)
> 

# Impossible, `a` being irrational!
 

 

Download periodic-vv.mw

complexity...

vv 13867
August 21 2025
3 0

Maple does not use LeafCount for the complexity.

length~([e1,e2]);  # same complexity

                     [21, 21]

ifactors...

vv 13867
August 05 2025
2 1

Better use ifactors:

SumPrimeFactors := n -> add(map2(op,1,ifactors(n)[2])):

 

Normal behavior...

vv 13867
August 02 2025
3 0

Your matrix m  is positive semidefinite. But if we compute M := evalf(m) it will become indefinite (due to roundoff errors.)

On the other hand for such M  and a given permutation matrix P, the LDLt factorization of P^+ .M.P may not exist, so the behavior you describe is normal.

No solution...

vv 13867
July 22 2025
1 6

The error given by Maximize is clear:
Error, (in Optimization:-NLPSolve) no feasible point found for the linear constraints

Indeed, C_1, and C_2 imply:

C12:=op(C_1 union C_2):
solve(C12[1]);solve(C12[2]);

                  [36571.46530, infinity)

               (-infinity, 27143.30624]

so, disjoint intervals, no common point (i.e. no feasible point).

 

Re...

vv 13867
July 01 2025
1 0 
Re(term2) assuming real;
combine(%);

        

approx and exact...

vv 13867
June 26 2025
3 1

 

It is not possible to use an index this way:
a[i] := expr;
==> a table named a is constructed and the entry i is defined.
If i is symbolic, all the other entries such as a[1], a[2], ... remain undefined.

It is possible to use a procedure

a := i -> expr;

and use paranteses a(1), a(2), ...

 

Here is a method to define a function using your infinite product.

 

> 

restart
> 

# An approximation
> 

N:=100:
f:=t -> local n;  evalf(mul(2*cos(t/2^n)-1, n=1..N));

proc (t) local n; options operator, arrow; evalf(mul(2*cos(t/2^n)-1, n = 1 .. N)) end proc

 (1)
> 

plot(f, -5..5);

 
> 

# Exact computation; Maple is not able to obtain it. We have to use the brain or (maybe) IA.
> 

F:= t -> sin(3*t/2)/3/sin(t/2);

proc (t) options operator, arrow; (1/3)*sin((3/2)*t)/sin((1/2)*t) end proc

 (2)
> 

plot(F-f, -5..5)

 


Download inf-prod-vv.mw

OK, evaluation...

vv 13867
June 25 2025
2 1

A table has special evaluation (to a name). So, use:

eval(T1["1"]);

Also, print(T1["1"]); confirms.

The limit...

vv 13867
June 19 2025
2 0
> 

restart;
> 

limit( cos(Pi*sqrt(x^2+x+1)), x=infinity);

-1 .. 1

 (1)

 

The limit does not exist for a function, but it exists for a sequence.

 

> 

L := Limit(cos(Pi*sqrt(n^2+n+1)), n=infinity); # assuming n::integer

Limit(cos(Pi*(n^2+n+1)^(1/2)), n = infinity)

 (2)
> 

f := asympt(sqrt(n^2+n+1),n,2, oterm=false);

n+1/2+(3/8)/n

 (3)
> 

simplify(cos(f*Pi)) assuming n::posint;
> 

L=limit(%, n=infinity);

-sin((3/8)*Pi/n)*(-1)^n

  

Limit(cos(Pi*(n^2+n+1)^(1/2)), n = infinity) = 0

 (4)
> 

 

 

Kahan sums etc...

vv 13867
June 19 2025
1 2

See the 2018 post  add, floats, and Kahan sum - MaplePrimes.

Note that since 2021, the add command has the pairwise option, see Pairwise summation - Wikipedia.

 

Math subtlety...

vv 13867
June 16 2025
1 5
 

Let us check the provided manual solution.

 

> 

de := Diff(a(x)*Diff(u(x),x),x) = -1;

Diff(a(x)*(Diff(u(x), x)), x) = -1

 (1)
> 

bc := u(-1)=0, u(1)=0;

u(-1) = 0, u(1) = 0

 (2)
> 

a := x -> 1 + 2*Heaviside(x);

proc (x) options operator, arrow; 1+2*Heaviside(x) end proc

 (3)
> 

c := int(x/a(x),x=-1..1) / int(1/a(x),x=-1..1) ;

-1/4

 (4)
> 

U:=int((c-xi)/a(xi), xi=-1..x); # the "solution" u(x) = U

-(1/4)*x+(1/6)*x*Heaviside(x)-(1/2)*x^2+(1/3)*x^2*Heaviside(x)+1/4

 (5)
> 

eval(U, x=1), eval(U, x=-1);  # bc, OK

0, 0

 (6)

U is continuous everywhere, differentiable except at x=0.

> 

value(eval(lhs(de), u(x)=U)) assuming x<0;
value(eval(lhs(de), u(x)=U)) assuming x>0

-1

  

-1

 (7)

So, the provided solution U, is continuous in [-1,1],  not differentiable at 0 but C^2 in [-1,1] \ {0} and satisfies the de  here.
It seems to be a "generalized" solution, but the exact sense is not mentioned. Distributional sense maybe?

 

Unfortunately the documentation for dsolve is silent about this type of ode.
Is this solution acceptable for you?

Anyway, it is not a big surprise that dsolve has problems.

 

[edited]

 

Remove duplicates...

vv 13867
June 12 2025
1 0 
RD := (S::set) -> {ListTools:-MakeUnique([S[]], 1, LinearAlgebra:-Equal)[]}:

RD( {Matrix([1])} union {Matrix([1])} union {Matrix([1]), Matrix([2])} );

        

procedure...

vv 13867
June 08 2025
2 1

fn must be declared as procedure, not function

f1:=proc(t::numeric, fn::procedure)::numeric;

sin()  and sin(x)  do not have the type procedure; their type is function in Maple.

So, sin  or sin()  are functions mathematically, but for Maple, sin is a procedure and sin() gives an error because the procedure sin cannot be called with 0 arguments.
You may check whattype('sin()');

whattype(sin);
whattype(sin(x));

                           procedure
                           function

bottom left...

vv 13867
June 07 2025
3 2

The "interrupt current operation" button is in the bottom left corner of the screen, but it usually does not work.
The new interface has many bugs ...
Better use the "old" Screen Readers one. 

Optimization[Maximize]...

vv 13867
June 07 2025
2 1
> 

restart;
> 

f:=(x1,x2)->(x1-1/2)^4+x1*x2+2*x2^2-1;  
g:=(x1,x2)->(x1-sin(x1))^2+(x2-sin(x2))^2-1;

proc (x1, x2) options operator, arrow; (x1-1/2)^4+x1*x2+2*x2^2-1 end proc

  

proc (x1, x2) options operator, arrow; (x1-sin(x1))^2+(x2-sin(x2))^2-1 end proc

 (1)

Find the circle of maximum radius that is tangent to both curves f=0, g=0.

 

Here is an approach using Optimization:-Maximize

It gives local maxima.

For a global maxima we must play with the intervals for variables and/or their initialpoint (using rand(...) conditions).

 

> 

with(plottools):
> 

pfg:=plots:-implicitplot([f(x,y),g(x,y)], x=-2..2,y=-2..2);

 
> 

EQ:=
f(a,b)=0, g(c,d)=0, r2=(u-a)^2+(v-b)^2, r2=(u-c)^2+(v-d)^2,
D[1](f)(a,b) * (v-b) - D[2](f)(a,b) * (u-a) = 0,
D[1](g)(c,d) * (v-d) - D[2](g)(c,d) * (u-c) = 0:
rnd:=rand(-2. .. 2.):
> 

unassign('a','b','c','d','u','v','r2');
sol:=Optimization:-Maximize( r2, {EQ}, u=-2..-1, v=-1.5..2, a=-5..0, b=-1..0, c=-2..-1, d=-2..-1, r2=0..3,
     initialpoint={u=0.7, v=0.5} )[2];
assign(sol);r:=sqrt(r2):
plots:-display(pfg,circle([u,v],r), plot([[a,b],[u,v]],color=red),plot([[c,d],[u,v]],color=blue),
               scaling=constrained,title=("Local max radius"=r));

[a = HFloat(-0.3036274461741274), b = HFloat(-0.46927384815579104), c = HFloat(-1.910428572753095), d = HFloat(-1.1755927491183271), r2 = HFloat(0.8409103269146442), u = HFloat(-1.0), v = HFloat(-1.0659107504612695)]

  

 
> 

 

> 

unassign('a','b','c','d','u','v','r2');
solglobmax:=[r2=0]:
to 200 do
ok:=true;
try
  sol:=Optimization:-Maximize( r2, {EQ}, initialpoint={u=rnd(), v=rnd()} )[2];
  catch:
  ok:=false;
end try;
if not ok then next fi;  
if eval(r2,sol)>eval(r2,solglobmax) then solglobmax:=sol fi;
od:
assign(solglobmax);r:=sqrt(r2):

plots:-display(pfg,circle([u,v],r), plot([[a,b],[u,v]],color=red),plot([[c,d],[u,v]],color=blue),
               scaling=constrained, title=("Global max radius"=r));

Warning, limiting number of major iterations has been reached

  

Warning, limiting number of major iterations has been reached

  

Warning, limiting number of major iterations has been reached

  

Warning, limiting number of major iterations has been reached

  

Warning, limiting number of major iterations has been reached

  

Warning, limiting number of major iterations has been reached

  

Warning, limiting number of major iterations has been reached

  

Warning, limiting number of major iterations has been reached

  

Warning, limiting number of major iterations has been reached

  

Warning, limiting number of major iterations has been reached

  

 
> 

 

 

Download 2curves-vv.mw
(edited for Global max).

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