# All Questions

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### Expectation of couples surviving after some time.

There are $2m$ persons forming $m$ couples who live together at a given time. Suppose that at some later time, the probability of each person being alive is $p$, independently of other persons. At ...
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### Surfaces, vector fields, and the Lie bracket.

$\textbf{Theorem:}$ Suppose that $X_1, \dots, X_k$ are vector field on a manifold $M$ and at a point $p \in M$, we have that $X_1(p), \dots, X_k(p) \in T_p(M)$ are linearly independent, then the Lie ...
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### Trace in finite Fields

(1) In $\mathbb{F}_{2^n}$ with odd n it should be shown that half of the Trace values are 0 and the other part is 1 with help of the additivity of the Trace. (2) Now n shall be even. Now the ...