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comment General function for the expansion of a polynomial of operators
Yes—but in a comment the code does not format ...
comment Implementation of a matrix formula
Nice solution. However, the OP did not make it clear why he wanted to re-implement a built-in function. Yes, I can read what he asked for—but he did not explain why. Since your code is only as fast as the built-in code (not surprisingly—it would be really impressive if it was faster), the only reason he could have would be to implement something closely related that is not built-in.
comment Best way to emulate MATLAB's bsxfun using Map and similar functions
Can use Identity' instead of 1&.
comment Find six vectors with rational entries under constraints?
You can split the 6x6 problem into two sets of 3, which is easier to satisfy (or a 4+2). The most general is 5+1 (where the 5 rows are independent). Instead of solving over rationals, you can work with integers (just multiply the 6x6 matrix of rationals by the LCD to obtain a matrix of integers).
comment Find six vectors with rational entries under constraints?
Perhaps you should state the full problem — and also where and how it arises. Note that, the conditions you stated require that the vectors must be linearly dependent (as the sum of each columns adds to zero). It is trivial to obtain the full solution for the 2x2 problem, and the 3x3 problem can be solved in terms of pythagorean triples.
comment MMA breaks down when `DSolve` -ing a third-order linear ODE
I cannot see how the Weber function is LaplaceTransform-invariant, and I don't know what you mean about "in the physical sense". In the "mathematical sense" I played around in some special cases, and I looked in Tables of Laplace Transforms by Oberhettinger and Badii, which includes transforms of Weber functions, but your assertion is still not clear to me.
comment MMA breaks down when `DSolve` -ing a third-order linear ODE
I don't see how solutions to these equations, which appear to be unbounded for large Abs[x], could be LaplaceTransform-invariant functions. Do you have a reference for this?
comment Plot is discontinuous (it shouldn't be)
The leading term in the asymptotic expansion for x<0 is 2 Cos[2 Pi x]. For x>0, the expression is much more complicated but the leading term is A Cos[2 Sqrt[2] Pi x+d] where A~1.7 and d~0.5. The transition between these asymptotic behaviours is in over -3 < x < 3.
comment How to make the docked cell and the navigation toolbar in the Slide Show?
I note that this bug in Inherited does not appear to be fixed? Also, rather than using nested If's, how could you use Switch instead? All my attempts fail...
comment Finding the intersection of a curve with an interpolation function
Did you try increasing the number of PlotPoints or MaxRecursion options when dealing with the seismic signal? RootsInRange is not fully adaptive (you could try Ted Ersek's package instead). However, since you have the data, why not just subtract of the values of the intersecting function at the data ordinates and look for sign changes in successive data? This will be much more efficient that Interpolating a highly oscillatory function ...
comment Fitting ellipse to 5 given points on the plane
See also the demonstration Five Points Determine a Conic Section. An interesting related problem is that of Small Lattice Ellipses.
comment How does Mathematica understand branchcuts of the complex logarithm?
The paper Can your computer do complex analysis? is also relevant to this discussion.
comment Finding the intersection of a curve with an interpolation function
The RootSearch package <>; by Ted Ersek returns all the roots over a specified range as a list of replacement rules This package .A RootsInRange function was presented in "Finding Roots in an Interval" in The Mathematica Journal 7(2), 1998: should be sufficient: