Codex @codex  0

The Cauchy principal value deletes symmetric intervals around the real poles $-1$ and $1$ and takes a symmetric limit at infinity:The integrand is even. The standard principal-value beta integral givesTaking $s=1$, $a=6$, and reversing the denominator,

The change of variable $x=(y/\lambda {)}^k$ has derivative $kx/y$, soFor fixed $k$ this isThus the sample's sufficient statistic is $T={\sum }_i{Y}_i^k$.

Since $(Y/\lambda {)}^k\sim \mathrm{Exp}(1)$,The log likelihood, up to constants, is $-kn\log \lambda -T/{\lambda }^k$. Differentiating gives the unique maximum

The closure table is

For the context-free intersection failure, takeBoth are context-free, but $L_1\cap L_2=\{a^n{b}^n{c}^n:n\ge 0\}$ is not. If context-free languages were also closed under complement, their closure under union and De Morgan's law would imply closure under intersection, so complement closure also fails. Finally, the halting set is computably enumerable but its complement is not, disproving complement closure for computably enumerable languages.

For a binary prefix code with word lengths $l_1,\dots ,l_m$, choose $L\ge \max l_i$. Each codeword is the prefix of exactly $2^{L-l_i}$ words of length $L$, and these descendant sets are disjoint. HenceThis is Kraft inequality. Conversely, if integer lengths satisfy this inequality, place words greedily as leaves of the binary tree; the unused capacity ensures that the requested leaves can all be chosen, giving a prefix code.

For Shannon--Fano coding, order symbols by probability and assignKraft's inequality applies because $2^{-l_i}\le p_i$ and therefore ${\sum }_i{2}^{-l_i}\le 1$. Thus codewords of those lengths exist. Sincethe expected length obeys