Advanced Search

vv

13790 Reputation

20 Badges

9 years, 304 days
Contact vv

MaplePrimes Activity


These are Posts that have been published by vv

Riemann hypothesis is false! (simple pro...

vv 13790 Maple
October 01 2017
2 2

# Riemann hypothesis is false! (simple proof)
 

restart;
assume( s>0, s<1/2, t>0 );
coulditbe(abs(Zeta(s+I*t))=0);
                              true

# Q.E.D.

Unfortunately coulditbe(Zeta(s+I*t)=0) returns FAIL, but our assertion is already demonstrated!

The moral: the assume facility deserves a much more careful implementation.

A geometric construction for the Summer ...

vv 13790 Maple
geometry fsolve
August 02 2017
1 2


 

A geometric construction for the Summer Holiday

 
Does every plane simple closed curve contain all four vertices of some square?
 This is an old classical conjecture. See:
https://en.wikipedia.org/wiki/Inscribed_square_problem

Maybe someone finds a counterexample (for non-analytic curves) using the next procedure and becomes famous!

 

> 

SQ:=proc(X::procedure, Y::procedure, rng::range(realcons), r:=0.49)
local t1:=lhs(rng), t2:=rhs(rng), a,b,c,d,s;
s:=fsolve({ X(a)+X(c) = X(b)+X(d),
            Y(a)+Y(c) = Y(b)+Y(d),
            (X(a)-X(c))^2+(Y(a)-Y(c))^2 = (X(b)-X(d))^2+(Y(b)-Y(d))^2,
            (X(a)-X(c))*(X(b)-X(d)) + (Y(a)-Y(c))*(Y(b)-Y(d)) = 0},
          {a=t1..t1+r*(t2-t1),b=rng,c=rng,d=t2-r*(t2-t1)..t2});  #lprint(s);
if type(s,set) then s:=rhs~(s)[];[s,s[1]] else WARNING("No solution found"); {} fi;
end:

 

Example

 

> 

X := t->(10-sin(7*t)*exp(-t))*cos(t);
Y := t->(10+sin(6*t))*sin(t);
rng := 0..2*Pi;

proc (t) options operator, arrow; (10-sin(7*t)*exp(-t))*cos(t) end proc

  

proc (t) options operator, arrow; (10+sin(6*t))*sin(t) end proc

  

0 .. 2*Pi

 (1)
> 

s:=SQ(X, Y, rng):
plots:-display(
   plot([X,Y,rng], scaling=constrained),
   plot([seq( eval([X(t),Y(t)],t=u),u=s)], color=blue, thickness=2));

 

A bug/typo with correct result!...

vv 13790
July 05 2017
1 2

The help page for  ?procedure has the following example.

> 

addList := proc(a::list,b::integer)::integer;
    local x,i,s;
    description "add a list of numbers and multiply by a constant";
    x:=b;
    s:=0;
    for i in a do
        s:=s+a[i];
    end do;
    s:=s*x:
end proc:
> 

sumList:=addList([1,2,3,4,5],2);

30

 (1)

Of course it's a bug (actually a typo) here:  s:=s+a[i];   should be   s:=s+i;

A strange/funny fact is that the result is correct but, for almost any other list the call will produce an error or a wrong answer.

SEEMOUS 2017...

vv 13790 Maple
sum
March 14 2017
0 3

Here is a problem from SEEMOUS 2017 (South Eastern European Mathematical Olympiad for University Students)
which Maple can solve (with a little help).

For k a fixed nonnegative integer, compute:

Sum( binomial(i,k) * ( exp(1) - Sum(1/j!, j=0..i) ), i=k..infinity );

(It is the last one, theoretically the most difficult.)

A compact Sudoku solver via Groebner...

vv 13790 Maple
February 20 2017
13 2

Sudoku is a well known Latin square type game, see https://en.wikipedia.org/wiki/Sudoku

Here is a Sudoku game and its (unique) solution:

A,Sol:=  # A = Sudoku matrix, 0 for each empty cell
Matrix(9, [
[0,0,3,0,9,0,1,0,0],
[0,5,0,3,0,0,7,0,0],
[1,0,2,0,0,5,0,6,4],
[0,1,0,0,2,0,9,0,0],
[2,0,0,6,0,3,0,0,1],
[0,0,7,0,8,0,0,3,0],
[7,6,0,9,0,0,8,0,5],
[0,0,8,0,0,7,0,9,0],
[0,0,4,0,6,0,2,0,0]]),
Matrix(9, [
[4,7,3,2,9,6,1,5,8],
[8,5,6,3,4,1,7,2,9],
[1,9,2,8,7,5,3,6,4],
[3,1,5,7,2,4,9,8,6],
[2,8,9,6,5,3,4,7,1],
[6,4,7,1,8,9,5,3,2],
[7,6,1,9,3,2,8,4,5],
[5,2,8,4,1,7,6,9,3],
[9,3,4,5,6,8,2,1,7]]);


The procedure which follows is a very compact Sudoku solver. It uses Groebner bases. I hope that you will like it.
The input is the Sudoku matrix and the solution matrix is returned.
Note that the Sudoku matrix must be valid and must have a unique solution.
(Otherwise, theoretically, the error "Invalid Sudoku matrix" should appear.)
Note also that the procedure may be very slow for some games or Maple may crash. This happened to me once with a very "hard" matrix.

I was impressed that Maple's implementation for Groebner bases works now so well for this problem!

A few years ago on this site: http://www.mapleprimes.com/questions/131939-Calculating-Groebner-Basis-For-Sudoku
it was an attempt to solve the problem with this method but it failed (due to wrong number of polynomials).

sudoku:=proc(A::'Matrix'(9,integer))
local x_A,x,Q,R,r, i,j,u,v,G;
Q:=proc(X,Y) normal((mul(X-i,i=1..9)-mul(Y-i,i=1..9))/(X-Y)) end;
x_A:=seq(seq( `if`(A[i,j]>0,x[i,j]-A[i,j],NULL),i=1..9),j=1..9);
R:={seq({seq(x[i,j],j=1..9)},i=1..9), seq({seq(x[i,j],i=1..9)},j=1..9),
    seq(seq({seq(seq(x[3*u+i,3*v+j],i=1..3),j=1..3)},u=0..2),v=0..2)};
G:=Groebner:-Basis({seq(seq(seq(Q(u,v),u=r minus {v}),v=r),r=R),x_A},'_vv');
if nops(G)<>81 then error "Invalid Sudoku matrix" fi;
eval(Matrix(9,symbol=x), `union`(map(u->solve({u}), G)[]));
end:

sudoku(A) < A; # Solving the previous game

# Let's solve another one:
A:=Matrix(9,9,[[0,0,0,4,0,0,0,8,0],[0,5,2,7,0,0,4,0,0],[3,0,0,0,0,0,0,0,0],[5,1,0,8,0,0,0,0,0],[0,0,0,5,0,0,6,7,0],[0,9,0,0,7,0,0,0,3],[2,4,0,0,0,5,0,0,0],[9,0,0,0,0,0,0,3,8],[0,0,0,0,0,0,9,4,0]]):
sudoku(A) < A;

Matrix   # A Sudoku matrix which crashes Maple!
(9,[[8,0,0,0,0,0,0,0,0],[0,0,3,6,0,0,0,0,0],[0,7,0,0,9,0,2,0,0],[0,5,0,0,0,7,0,0,0],[0,0,0,0,4,5,7,0,0],[0,0,0,1,0,0,0,3,0],[0,0,1,0,0,0,0,6,8],[0,0,8,5,0,0,0,1,0],[0,9,0,0,0,0,4,0,0]]):

 

 

3 4 5 6 7 Page 5 of 7