# All Questions

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### Solution of $w_{xx}+w_{xy}+w_{yy}=0$

Consider the following pde, $w_{xx}+w_{xy}+w_{yy}=0$. How to prove that its solution has the form $w(x,y)=f(x)g(y)$? I realized that this is separation of variables, but I am having trouble in ...
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### Nonhomogeneous pde - problem with Fourier series for cosine

I've been trying to solve the following equation $$u_t - a^2 u_{xx} = tx, \ \ \ x \in (0, \pi), \ \ t>0$$ $$u_x(0,t)=u_x(\pi, t) = 0, \ \ \ t \ge 0$$ $$u(x, 0) = 1 \ \ \ x \in (0, \pi)$$ So I ...
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### the probability that there will be exactly two men between A and B?

If six men, among whom are A and B, stand in a circle, what is the probability that there will be exactly two men between A and B?
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### Proving if function is one to one?

f : (N × N) → N defined as: f(x, y) = [((x + y)(x + y + 1))/2] + y I know that f(x,y) = f(x',y') in order to be a one to one. So it'll be : ...
### equivalent of a $\mathrm{diag}(S)$ in matrix notation
Suppose I have a symmetric matrix $S\in\mathbb{R}^{n\times n}$ (positive semidefinite if this helps). Is there a way to write the transformation $\mathrm{diag}(S)$ using standard linear algebra ...