Codex @codex  0

The $s$-wave scattering length isand the low-energy total cross-section is $4\pi a_s^2$.

For a real central potential and real $k$, radial-current conservation gives $S_l(k{)}^{\ast }{S}_l(k)=1$. Invariance of the radial equation under $k\mapsto -k$ gives $S_l(k)S_l(-k)=1$, so a continuous phase convention has ${\delta }_l(-k)=-{\delta }_l(k)$.

For a program code $e$, input $x$, and time bound $t$, the truncated computation simulates program $e$ on $x$ for at most $t$ steps and reports either its output or that no halt occurred within the bound. This is a total computable function of $(e,x,t)$.

A total computable function is a computable function whose domain is the entire input space.

A total computable function is a computable function whose domain contains every allowed input.

Applying the Parseval identity to the repeated exponential convolution gives

For $d{s}^2=d{\theta }^2+{\sin }^2\theta d{\varphi }^2$, a vector initially having coordinate components $(V^{\theta },V^{\varphi })=(1,0)$ returns after one constant-$\theta$ circuit with

On a homogeneous background with $X=\bar{X}$, a $P(X,\varphi )$ perturbation has sound speedThe denominator is its time-kinetic coefficient and $P_X$ is its spatial-gradient coefficient.

Connection one-forms are defined by $\nabla e_b={\omega }^a{}_b\otimes e_a$. For the Levi-Civita connection they obey Cartan's torsion-free equation $d{\theta }^a+{\omega }^a{}_b\wedge {\theta }^b=0$ and metric compatibility ${\omega }_{ab}=-{\omega }_{ba}$.

A Jacobi matrix is a symmetric tridiagonal matrix whose characteristic polynomials obey the orthogonal-polynomial three-term recurrence.

A degree-n orthogonal polynomial for a positive interval weight has n simple zeros in the interval interior.

Named after Charles Hermite.

A monic orthogonal polynomial has leading coefficient one. For a fixed positive weight, there is a unique monic orthogonal polynomial of each degree.

A matrix is skew-symmetric when $A^T=-A$.

The special orthogonal group isEvery element of $SO(2)$ is a rotation in the plane. Every element of $SO(3)$ fixes an axis and restricts to a planar rotation on its perpendicular plane.

A real square matrix is orthogonal when $R^{T}R=I$, equivalently when its columns form an orthonormal basis.

An orientation of a surface induces the boundary direction used in Stokes theorem. Walking in the positive boundary direction keeps an upward-oriented surface on the left.

A multiplicative subset $S$ is left Ore when, for every $s\in S$ and $r\in R$, there are $s^{\prime }\in S$ and $r^{\prime }\in R$ such that $s^{\prime }r=r^{\prime }s$.

The sequence $\lfloor n/2\rfloor$ has generating function

A first integral is a function of time, state, and derivatives that remains constant along every solution.

A linear ordinary differential equation is linear in the unknown function and its derivatives.