Silvia
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 Oct 17 comment Expected number of people to not get shot? I'm not a mathematician, so I'm not that confident. And I think the expected number is much harder to get :) already +1 of course. Oct 17 comment Expected number of people to not get shot? Can we generalize the problem to $k$-dimension? Aug 23 comment Simplify $\tan^{-1}[(\cos x - \sin x)/(\cos x + \sin x)]$ define simplest please. Jul 21 comment Method for variable substitution in multiple summation Yes they are:( I used one of the explicit sum formulas of Chebyshev polynomial (mathworld.wolfram.com/ChebyshevPolynomialoftheFirstKind.html eq.15), which introduces those floor functions. Jul 21 comment Method for variable substitution in multiple summation @ Christian Blatter: Thanks very much for your answer. I understand the difficulties of counting lattice points in disk, but if considering the summation region bounded only by integer coefficients planes (like $\sum_k A_k x_k=B$ where $A_k, B\in \mathbb{Z}$), the problem might be much simpler and there might even exist a systematical technique for it. Besides, I'm thinking of using Iverson's bracket to avoid dealing with those tricky boundary conditions. Jul 21 comment Method for variable substitution in multiple summation @Patrick Da Silva: Yes I draw 2-dim or 3-dim axes too when I do the transformation by hand :D But if I have many levels of sum, this will be inefficient cause it's hard to draw high dimension, and those floor and ceiling functions and steps$\neq 1$ are bothering. I think I need to treat the whole summation region as a high dimension polyhedron and find a systematical routine to deal with it.