For a rapidly decaying KdV potential, the scattering data consist of the continuous reflection coefficient, the negative discrete eigenvalues $-{\kappa }_n^2$, and a positive norming constant for each bound state.

A Jost solution of a one-dimensional Schrödinger equation is specified by plane-wave asymptotics at one spatial infinity. Comparing the left and right Jost solutions defines transmission and reflection coefficients.

With boundary values $q_0=q_n=0$, the finite Volterra lattice isup to rescaling or reversal of time. Writing $q_j=a_j^2$ gives ${\dot{a}}_j=a_j(a_{j+1}^2-a_{j-1}^2)$.

In one spatial dimension, inviscid isentropic mass and momentum conservation are

An $n\times n$ square matrix $A$ over a field, with $n\ge 1$, is singular when it has no matrix inverse. Geometrically, its linear map loses at least one input direction: some nonzero vector $v$ satisfies $Av=0$, so $A(x+v)=Ax$. Over the real numbers, the following family makes that loss visible:

For this diagonal matrix, $\det A(t)=t$. Whenever $t\ne 0$, the input can be recovered as $(x,y)=(b_1,b_2/t)$ and the kernel is $\{0\}$. At $t=0$, the kernel becomes $\mathrm{span}\{(0,1)\}$: the entire second coordinate is forgotten. Consequently $A(0)x=(b_1,b_2)$ has no solution if $b_2\ne 0$, and has the solution straight line $x=(b_1,s)$, $s\in \mathbb{R}$, if $b_2=0$. A singular matrix therefore need not make a particular linear system inconsistent; it prevents unique recovery when that linear system is consistent.

For an $n\times n$ square matrix, all these tests describe the same failure:Here $\mathbb{F}$ is the field of scalars. Equivalently, the columns are linearly dependent, the linear map is neither injective nor surjective, and some outputs cannot be reached. See the invertible matrix equivalences and Strang's MIT 18.06SC lecture summary, Multiplication and inverse matrices, p. 2. Over an infinite field, every consistent linear system for a singular matrix has infinitely many solutions; over a finite field with $q$ elements and nullity $d$, it has exactly $q^d$ solutions. This follows by adding each kernel vector to one solution.

The square hypothesis is essential. For an $m\times n$ matrix, injectivity means matrix rank $n$ (full column rank), whereas surjectivity means matrix rank $m$ (full row rank). A rectangular matrix can satisfy one condition without the other; its kernel and image of a linear map give the useful tests without calling it a singular matrix.

Exact singularity also differs from sensitivity. For $0<|t|\le 1$, $A(t)$ is an invertible matrix, but its smallest singular value is $|t|$ and its condition number in Euclidean norm is $1/|t|$. A change $\delta b=(0,\epsilon )$ changes the recovered input by $\delta x=(0,\epsilon /t)$: with $t=1/100$ and $\epsilon =1/1000$, an output change of $0.001$ causes an input change of $0.1$. These are exact toy values, not measurements. The condition number quantifies this sensitivity before a direction is lost; see Steve Marschner, Condition numbers, §2. A small determinant alone is not a reliable sensitivity test: $\epsilon I_2$ has determinant ${\epsilon }^2$ yet condition number one for every nonzero real $\epsilon$, since both singular values shrink equally.

Suppose a compact connected component $C$ of $\{x:V(x)\le c\}$ contains an equilibrium $x_{\ast }$ and $\dot{V}<0$ throughout $C\setminus \{x_{\ast }\}$. Then $C$ is an invariant sublevel set, and every trajectory in $C$ converges to $x_{\ast }$ by the LaSalle invariance principle. Thus the interior of $C$ lies in the basin of attraction of $x_{\ast }$. The largest such component supplies the strongest basin estimate available from that Lyapunov function alone.

The full tent map on the unit interval isIts two affine branches expand lengths by two, and it preserves Lebesgue measure.

For elements ${\alpha }_1,\dots ,{\alpha }_n$ of a degree-$n$ number field with embeddings ${\sigma }_1,\dots ,{\sigma }_n$, their discriminant isIt is nonzero exactly when the elements form a $\mathbb{Q}$-basis.

The direct product of integrable Hamiltonian systems is integrable. Pull each family's commuting integrals of motion back to the product phase space. Integrals from different factors have zero Poisson bracket because they use disjoint canonical coordinates, and their differentials are independent wherever the differentials within each factor are independent.

An invariant torus is a subset homeomorphic to a torus and preserved by a flow. In action-angle variables, an integrable Hamiltonian flow moves linearly in the angular coordinates of each regular invariant torus.

The torus is a particular invariant manifold; its toroidal topology is an extra condition beyond invariance.

For a homogeneous canonical inflaton in an expanding FLRW metric,The expansion contributes the damping term called Hubble friction.

A classifier is a function that assigns an input to one of a discrete set of labels. A binary classifier has two possible labels.

The three-dimensional hyperboloid model isIt is a regular level set with

Hyperbolic distance is the Riemannian distance induced by the hyperbolic metric.

Singular homology forms a chain complex from formal sums of continuous maps from standard simplices into a topological space. Its homology groups are invariant under homotopy.

A continuous map $f:X\to Y$ induces homomorphisms $f_{\ast }:H_n(X)\to H_n(Y)$ satisfying $(g\circ f{)}_{\ast }=g_{\ast }\circ f_{\ast }$ and $({\mathrm{id}}_X{)}_{\ast }={\mathrm{id}}_{H_n(X)}$.

A homology group is one of the abelian groups produced by a homology theory. Its elements are homology classes, represented in singular homology by cycles modulo boundaries.

A function is complex differentiable at $p$ whenIf $f^{\prime }(p)\ne 0$, the real derivative is multiplication by a nonzero complex number, hence a rotation and scaling; this is the local source of angle preservation by holomorphic maps.

The excluded volume is the volume of relative positions forbidden to another particle. For hard spheres of radius $r_0$, that forbidden region is a sphere of radius $2r_0$.

For outer radius $b=2a$ and zero-energy well wavenumber ${\kappa }_0=\sqrt{2m{V}_0}/\hslash$, matching the S-wave scattering logarithmic derivative givesIt diverges when ${\kappa }_{0}a=(n+\frac{1}{2})\pi$, where the well has a zero-energy scattering resonance.