@Preben Alsholm Could you explain this more? What is the significance of t = 76.239? That appears to be the location of the rightmost local min of abs(y2-y3). Why doesn't b(t) record the local min at about t = 73 or the local maxes at about t = 74 and t = 78?

Why is the events criterion based the right sides of equations 2 and 3. Why not use the simpler left sides?

In other words, given a function x(t) on t = 0..1 and integers r >= 0 and N > 0, you want to find the (2*r+1)*N coefficient functions X[m,n](t), m= 0..2*r, n= 1..N, which in some sense minimize the error of the approximation

x(t) ~ sum(sum(X[n,m](t)*b[n,m](t), m= 0..2*r), n= 1..N)

where the b[n,m](t) are fixed basis functions given by

piecewise(n-1 <= N*t and N*t < n, piecewise(m <= r, cos(P*m*t), sin(P*(m-r)*t)))

with P = 2*Pi*N.

Note that I removed the treatment of m=0 as a special case because that's redundant.

@phil76600 You have a homogenous, linear, second-order differential equation with two initial conditions set to 0. The solution is thus the identically zero function. If you look closely at the x-axis of the plot, you'll see that the curve lies along it. Or you can set the plot option axes= boxed:

odeplot(sol, axes= boxed);

and then the curve y = 0 will be obvious.

If you want a different solution, then you need to use different initial conditions.

@janhardo I don't know what went wrong for you. You do have the correct path, because you would've received an error message if you did not. Just try the commands

showstat(AddFractions);

etc.

@Carl Love The .m file contains procedures named AddFractions, DrawFraction, FractionOptions, MultFractions, CompareFractions, and `convert/decimalfraction`. All of these are listable with showstat. I do not think that it has a procedure named mod, although I cannot tell you why mod shows up in the anames list.

@janhardo Send it to me by email. I'll send you an email that you can reply to.

Does OS X Mavericks use backslash as a directory name separator? I thought that only Windows used that. (Even Windows allows the forward slash instead.) Try replacing the double backslashes \\ with single forward slashes /.

@rit The identify command is extremely powerful and sophisticated. See ?idenitfy . But alas, it is not enough for the original problem as you posed it.

See my Answer to your previous question (just posted).

@Kitonum 

or 1/(1-exp(a*x)*erfc(a*x)).

@janhardo I am not familiar with that error message, and I have no idea what it means. Please attach the .m file so that we can have a better look at it.

@rit I don't know of any algorithm or Maple command for it. This is very close to what is done by the identify command. See test #9 at ?identify. However, that is for products, not sums. isolve returns an error. Here's a crude procedure for a brute force search over nonnegative integer exponents only. Obviously, this can be optimized:

PowerSearch:= proc(S, a, b)
     local i,j;
     if S <= 1 or a <= 1 or b <= 1 then
          error "Invalid input: All arguments must be greater than 1."
     end if;
     for i from 0 while a^i <= S do
          for j from 0 while a^i + b^j <= S do
               if S = a^i + b^j then return i,j end if
          end do
     end do;
     [][]  #No solution
end proc:

PowerSearch(191.715, 12.2, 3.5);

@Kitonum The unimodality is now proven (see above). There are at least two ways to make the fsolve work for large k. A simplistic way is to increase the value of Digits. Digits = 30 works for the k=50 case that you presented. A better way is to divide out a large positive factor, which, of course, doesn't change the zeros. It turns out that C(n,k) can be factored out of its own derivative with respect to k.

C:= (n,k,a,b)-> binomial(n,k)*a^(n-k)*b^k:
fsolve(simplify(D[2](C)(n,50,1,2)/C(n,50,1,2)), n= 50..infinity);

74.7487626691704

which agrees with your result.

@Axel Vogt Thanks. I have been thinking about this unimodality proof for a few days. To generalize, the generalized binomial coefficient function

C:= (n,k,a,b)-> binomial(n,k)*a^(n-k)*b^k

is unimodal as a function of k for 0 <= k <= n when a > 0 and b > 0. The derivative with respect to k is remarkably simple:

D[2](C)(n,k,a,b) = (Psi(n-k+1) - Psi(k+1) + ln(b/a))*C(n,k,a,b)

We can ignore the factor C(n,k,a,b) because it is strictly positive in the region of interest. We can ignore the shift by ln(a/b) because it does not depend on k. It remains prove that Psi(n-k+1) - Psi(k+1) is monotonic for 0 <= k <= n, which Axel did.

@oldstudent You can't use the plot command to plot equations. There are two options. The first is to subtract one side of the equation from the other and plot that, so that the x intercepts of the resulting plot are the roots of the equation:

plot(2*sin(2*x*Pi/180) - 1, x= 0..360);

The second is to plot separately the left side and right side of the equation, so that the x coordinates of the intersection points are the roots of the equation:

plot([2*sin(2*x*Pi/180), 1], x= 0..360);

You should get used to working in radians instead of degrees. All higher mathematics is done in radians, and nearly all mathematical software uses radians by default.