Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 220 / 2 / a / i

For

u \geq 0

, let

C_{u}

be the collection of interval components of the superlevel set

{r : g (r) \geq u}

s, t

lie in the same member of

C_{u}

exactly when

u \leq m_{g} (s, t)

. Therefore

m_{g} (s, t) = \int_{0}^{\infty} \sum_{C \in C_{u}} 1_{{s \in C}} 1_{{t \in C}} d u .

For arbitrary real

λ_{1}, \dots, λ_{n}

, the Tonelli theorem now gives

\sum_{i, j} λ_{i} λ_{j} m_{g} (s_{i}, s_{j}) = \int_{0}^{\infty} \sum_{C \in C_{u}} (\sum_{i : s_{i} \in C} λ_{i})^{2} d u \geq 0.

Thus

(m_{g} (s_{i}, s_{j}))_{i, j}

 Ancestors (11)

There are no discussions about this article yet.

There are currently no matching articles.