on page 13 of the paper here there is a proof in theorem 4 that all eigenvalues of this tridiagonal matrix, which has strictly positive entries down the subdiagonals, are simple. Unfortunately, I don't get the argument. Apparently, it is almost immediate to the editor that $ker(J-\lambda I)$ must...

Experimentally I discovered the limit below that says that $$\lim_{n\to\infty} \int_0^{\pi/2} \frac{1}{\displaystyle \cos\left(\frac{x}{2}\right)\left(\cos\left(\frac{x}{2}\right)-\cos\left(\frac{x}{2^2}\right)\right)\cdots \left(\cos\left(\frac{x}{2}\right)-\cos\left(\frac{x}{2^{2n+1}}\right)\r...

I wanted a function which is strongly positive (sharp gradient) in the ++ and positive in the +, and the same in minus

what is the solution of this integral:$$\int^1_0 \frac{-2(t+a)+(1-a)}{((t+a)^2+(1-a)^2)^2} dt$$ canyou help me? that is a part solultion of a question which I should to solve it!

There is a very interesting puzzle for fibonacci sequence You are climbing a stair case. It takes n steps to reach to the top. Each time you can either climb 1 or 2 steps. In how many distinct ways can you climb to the top? The answer is fibonacci sequence: F(n) = F(n-1) + F(n-2). The explanat...