So I'm looking at the proof of the ZBC lemma in Odifreddi's Classical Recursion Theory volume 2 page 808 and I don't see why $ 0' \oplus C$ produced computes $B'$ as claimed. The positive requirements try and code $$ P^C_e: \; x \in C^{[e]} \iff (\exists z > x) B^{[e]}(x) \not= B^{[e]}(z) $$ ...

The secant method for solving an equation $F(x)=0$ in one variable is much older than Newton's one. Recall that given two iterates $x_{k-1}$ and $x_k$, it provides an update $x_k$ by taking the intersection of the chord joining the graph points $(x_{k-1},F(x_{k-1}))$ and $(x_k,F(x_k))$, with the ...

I have two related questions about numerical methods for root solving: 1) $f: R \to R$ is continuous and piece-wise smooth, with $f(a)f(b) < 0$. $f$ has very high number of knot-points and computing analytic expression for $f'$ is not possible. To find a root of $f$ in $[a,b]$, I can try the ...

What is a fast algorithm to find the root of a strictly decreasing function? If the root is not exact I want to find a root such that the function value is positive to an error.

Given a positive integer $n$, the Hamming distance $d^H_n(x,y)$ of $x,y\in \{0,1\}^n$ is defined by $$d^H_n(x,y) = |\{k\in\{0,\ldots,n-1\}: x(k)\neq y(k)\}|.$$ Given an integer $n>0$ and a set $S\subseteq \{0,1\}^n$ with $|S| = n$, is it possible to find a map $f:S\to \{0,1\}^{n+1}$ such that $...