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If $c_1,\dots ,c_n$ are the columns of a matrix, the matrix has the columns property when $[n]$ has an ordered partition $B_1\bigsqcup \cdots \bigsqcup B_s$ such thatand, for $j>1$, ${\sum }_{i\in B_j}{c}_i$ belongs to the linear span of the columns indexed by $B_1\cup \cdots \cup B_{j-1}$.

A scattering resonance is a metastable state represented by a pole of the analytically continued scattering matrix. Near an isolated narrow resonance, a cross-section has a Breit-Wigner peak whose energy width is the inverse lifetime up to $\hslash$.

A pole at $k=i\kappa$ with $\kappa >0$ represents a normalizable bound state of energy $-{\hslash }^2{\kappa }^2/(2m)$. Resonance poles instead lie away from the imaginary axis on the analytically continued unphysical sheet.

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.

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