Let $\frac{x^2}{a^2}+\frac{y^2}{b^2} =1$ be an ellipse. Let $A = (x_0,y_0)$ be a point outside of an ellipse. Draw two lines that touch an ellipse, say $D_1 = (x_1,y_1),D_2 = (x_2,y_2)$. My question is that if I draw a line passing through $A$ and the middle point of $D_1$ and $D_2$ i.e., $(\fra...

I do not know the eg I am using can be on the site before so before . Actually, I was learning differential operators using a book given by our prof , and there was an example in it says like this> $\frac {d^2y}{dx^2}-3\frac{dy}{dx}+2y=e^{5x}$ now I have solved these kinds of question using the c...

Here's a question I thought of, that I don't know the answer to. Let $F_2$ be the free group on $\{a,b\}$, and $F_2^+$ be the subset where all the exponents are positive. For a set $S$, let $S^2=\{g^2:g\in S\}$. Is $\langle (F_2^+)^2\rangle$ (the group generated by $(F_2^+)^2$ the same as $\langl...

There are three main ways of interpreting the Fourier transform. Decomposition relative to eigenfunctions of the Laplacian On $\mathbb{R}^n$, the plane waves $E_\xi(x) = \exp( i \xi\cdot x)$ can be interpreted as generalized eigenfunctions of the Laplacian. That is, let $$ \triangle = \sum_{i=1}^...