Robert Israel

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15 years, 333 days
University of British Columbia
Associate Professor Emeritus
North York, Ontario, Canada

MaplePrimes Activity


These are replies submitted by Robert Israel

It's possible to get all sorts of exact fitting curves.  But I don't understand your criteria for "even better". 

He's referring, I think, to the help page ?pdsolve,numeric,pdemethods, section Box.

I don't understand your statement that "the interpolation by a single polynomial is not possible in many cases when n>=2".  Surely it is possible.  For example, in the case n=2
with at most k distinct y values for each x value, you can think of interpolating a function of x whose values are polynomials of degree <= k in y.

I don't understand your statement that "the interpolation by a single polynomial is not possible in many cases when n>=2".  Surely it is possible.  For example, in the case n=2
with at most k distinct y values for each x value, you can think of interpolating a function of x whose values are polynomials of degree <= k in y.

Of course I should have loaded the numtheory package or replaced divisors by numtheory[divisors].
The number of odd divisors of m is odd if and only if the sum of all divisors of m is odd.
See A071860 in the Encyclopedia of Integer Sequences <http://oeis.org/A071860>.  In particular, from the formula given there, the number of sign changes up to N*Pi/2 is floor(sqrt(N))+floor(sqrt(N/2)).

> floor(sqrt(2011)) + floor(sqrt(2011/2));

75

Of course I should have loaded the numtheory package or replaced divisors by numtheory[divisors].
The number of odd divisors of m is odd if and only if the sum of all divisors of m is odd.
See A071860 in the Encyclopedia of Integer Sequences <http://oeis.org/A071860>.  In particular, from the formula given there, the number of sign changes up to N*Pi/2 is floor(sqrt(N))+floor(sqrt(N/2)).

> floor(sqrt(2011)) + floor(sqrt(2011/2));

75

I think there's a really interesting optical illusion going on here.  In Figure 2, although the length of the green segment is constant, it doesn't look constant.  To see that it is constant, what I would do is plot a circle as well as the segment.


> with(plots):
    P1:= plot([x^2,x^2+1],x=-2..2,colour=[black,red]):
    PL:= t -> display(plottools[circle]([t,t^2],1,colour=green),
plot([[t,t^2],[t,t^2+1]],colour=green));
   animate(PL, [t], t=-2..2,background=P1, title="");

@gtavictor : As a general rule, you should look at Maple's output, and certainly do so when something goes wrong.  If something appears there that should have been evaluated (e.g. the name of a command), that's a very significant clue about what has gone wrong: it may be that you have misspelled it, or neglected to load a package if it's in one, or failed to assign it a value.

@gtavictor : As a general rule, you should look at Maple's output, and certainly do so when something goes wrong.  If something appears there that should have been evaluated (e.g. the name of a command), that's a very significant clue about what has gone wrong: it may be that you have misspelled it, or neglected to load a package if it's in one, or failed to assign it a value.

> with(plots): with(plottools):
  p1:= rectangle([0,0],[3/2,1],colour=red):
  p2:= polygon([seq([197/240 + (1/12+1/24*(-1)^j)*sin(2*Pi*j/10-Pi/10),
      0.5+(1/12+1  /24*(-1)^j)*cos(2*Pi*j/10-Pi/10)],j=0..9)],colour=white):
  alpha:= arccos(32*(1/4^2 + 1/16^2 - 1/5^2)): dh1:= (Pi-alpha)/20:
  beta:= Pi - arccos(40*(1/16^2 + 1/5^2 - 1/4^2)): dh2:= (Pi-beta)/20:
  p3:= polygon([seq([1/2 + 1/4*cos(Pi+j*dh1),1/2+1/4*sin(Pi+j*dh1)],j=-20 .. 20),
              seq([9/16 + 1/5*cos(Pi-j*dh2),1/2+1/5*sin(Pi-j*dh2)],j=-20 .. 20)],colour=white):
  display(p3,p2,p1,scaling=constrained,axes=none);

interface(rtablesize) affects all interfaces (Standard, Classic and Command-line, AFAIK).  The only difference is that for Command-line the default is 25, for Standard and Classic it is 10.

Note also that you can use

> interface(rtablesize=infinity);

after which all Vectors and Matrices will be printed out in full.

The difference between Vectors and Arrays is not just how they are printed.  For an Array, the indices for a sub-Array are the same as the indices in the original.  For a Vector, they are re-numbered starting with 1.   For example:

> A:= Array([a,b,c,d,e]);
   B:= A[2..4];
   B[2], B[3];

b, c

For a Vector:

> V:= Vector([a,b,c,d,e]);
   W:= V[2..4];
   W[1], W[2];

b, c

interface(rtablesize) affects all interfaces (Standard, Classic and Command-line, AFAIK).  The only difference is that for Command-line the default is 25, for Standard and Classic it is 10.

Note also that you can use

> interface(rtablesize=infinity);

after which all Vectors and Matrices will be printed out in full.

The difference between Vectors and Arrays is not just how they are printed.  For an Array, the indices for a sub-Array are the same as the indices in the original.  For a Vector, they are re-numbered starting with 1.   For example:

> A:= Array([a,b,c,d,e]);
   B:= A[2..4];
   B[2], B[3];

b, c

For a Vector:

> V:= Vector([a,b,c,d,e]);
   W:= V[2..4];
   W[1], W[2];

b, c

You really need to ask a question if you expect an answer.  I assume you received the error message in the title and you want to know what caused it.  What were you trying to do?   How were you trying to do it?  It would help if you included actual code (in text please, not an image) that produces this error, or uploaded a worksheet where it occurs.

According to Jain and Snyder, "A Note on Factoring 0-1 Matrices", Linear and Multilinear Algebra 41 (1991) 193-198 <http://www.math.ohiou.edu/~jain/076.pdf> a positive semidefinite 0-1 matrix is equivalent to a direct sum of "all-ones" matrices and possibly a 0 matrix, where equivalence involves permuting the rows and columns with the same permutation.  These are in one-to-one correspondence with the number of ways of partitioning a set of n+1 labeled elements (where matrix element A[i,j] = 1 if and only if
i and j are in the same set of the partition and that set does not contain n+1).  The
number of such matrices is thus the Bell number combinat[bell](n+1).  See e.g.
<http://oeis.org/A000110>

According to Jain and Snyder, "A Note on Factoring 0-1 Matrices", Linear and Multilinear Algebra 41 (1991) 193-198 <http://www.math.ohiou.edu/~jain/076.pdf> a positive semidefinite 0-1 matrix is equivalent to a direct sum of "all-ones" matrices and possibly a 0 matrix, where equivalence involves permuting the rows and columns with the same permutation.  These are in one-to-one correspondence with the number of ways of partitioning a set of n+1 labeled elements (where matrix element A[i,j] = 1 if and only if
i and j are in the same set of the partition and that set does not contain n+1).  The
number of such matrices is thus the Bell number combinat[bell](n+1).  See e.g.
<http://oeis.org/A000110>

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