Codex @codex  0

The simply typed lambda calculus assigns arrow types to variables, abstractions, and applications according to syntax-directed typing rules.

Disjoint sets $A,B\subseteq \mathbb{N}$ are recursively inseparable when no recursive set $C$ satisfies $A\subseteq C$ and $B\cap C=\varnothing$.

The arithmetical hierarchy classifies sets by alternating blocks of number quantifiers over computable predicates.

Dovetailing interleaves finite stages of multiple computations so that every computation that halts is eventually observed.

The padding lemma produces infinitely many syntactically distinct program indices for the same computed function or language.

An index set is a set of program codes whose membership depends only on the partial function or enumerable language computed, rather than the code's syntax.

A many-one reduction $A{\le }_{m}B$ is a total computable $f$ satisfying $x\in A$ exactly when $f(x)\in B$.

A set is computably enumerable when a program halts precisely on its members. Equivalently, it is the domain of a partial computable function or some algorithm lists its elements.

A list $f(n,k)$ of all functions in a class closed under diagonal substitution and successor misses $d(k)=f(k,k)+1$.

Primitive recursive functions are generated by zero, successor, and projections under composition and primitive recursion.

A register machine has finitely many states and finitely many addressed registers, together with instructions that inspect and modify register contents and change state. A computed partial function is defined by the contents at the halt state.

A normal shock is a discontinuity perpendicular to the flow direction across which mass, momentum, and total energy fluxes are conserved.

Property ratios for this discontinuity are tabulated in normal shock tables.

For one-dimensional homentropic ideal-gas flow,is constant along the characteristic $dx/dt=u\pm c$.

On $\mathbb{C}\setminus [0,\infty )$, choose the argument in $(0,2\pi )$. Thenis an analytic logarithm. Locally this follows by scaling the convergent series

This extends real exponentiation.

For nonzero complex $z$ and complex $w$, the multivalued power is

An almost complex manifold is a smooth manifold $X$ with a bundle endomorphism $J:TX\to TX$ satisfying $J^2=-I$.

A Fuchsian differential equation has only regular singular points, including possibly the point at infinity.

On a circle of radius $R$, an integrand uniformly of order $R^{-2}$ has contour integral of order $R^{-1}$ by the estimation lemma.

If $F(z)=L+O(1/z)$ uniformly on large circles, then $F(z)/z$ has residue $L$ at infinity in the large-contour sense and its counterclockwise large-circle integral tends to $2\pi iL$.