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Forsuppose the optimizer and multiplier vary smoothly and use $L=f-\lambda (g-b)$. Differentiating the optimum value and using stationarity gives

For a jump process with rate $W(n,r)$ from state $n$ to $n+r$, probability balance givesThe first term is inflow from $n-r$ and the second is outflow from $n$.

For jumps $n\to n+1$ at rate $\lambda$ and $n\to n-2$ at rate $\beta n^2$, the diffusion approximation hasThe stable deterministic population is $n_{\ast }=\sqrt{\lambda /(2\beta )}$. Its linear-noise stationary approximation is normal with mean $n_{\ast }$ and variance $3n_{\ast }/4$.

A soliton is a localized travelling wave whose shape is preserved by the evolution and whose interactions with other solitons are elastic up to phase shifts.

The inverse-scattering spectral problem associated with this KdV convention isFordifferentiation gives the Wronskian identity

A colored link assigns each component a color. In the trivial coloring all components have the same color, so a doubly slice realization uses one unknotted surface component.

A link is a smooth embedding of a finite disjoint union of circles in $S^3$, considered up to ambient isotopy. A one-component link is a knot.

The Seifert longitude is the zero-linking parallel of a knot on the boundary of its tubular neighborhood. It is the framing induced by a Seifert surface.

The Stevedore knot $6_1$ is a ribbon knot with Alexander polynomial $2t^2-5t+2$ up to a unit.

A surface knot is a smooth embedding of a closed connected surface in a four-manifold. In the narrow standard usage, a 2-knot is an embedded two-sphere in $S^4$.

It extends the embedding viewpoint of knot theory to two-dimensional domains.

An $m$-twist spin removes a trivial arc from a knot and spins the resulting knotted arc around the boundary of a three-ball in $S^4$, inserting $m$ full twists during one revolution.

The result is a surface knot with domain a two-sphere.

The Arf invariant is a $\mathbb{Z}/2$-valued concordance invariant obtained from the quadratic refinement of the mod-two intersection form on a Seifert surface.

An amphichiral knot is ambient-isotopic to its mirror. Positive and negative amphichirality distinguish whether the symmetry preserves or reverses the knot orientation.

The two-fold branched cover of a knot is the double cover of $S^3$ branched along the knot. Its first homology has a nonsingular linking form and, when finite, has order $\det K$.

Away from the knot this is a two-sheeted covering space; at the knot it has the standard local model of a branched covering.

Two knots are concordant when they cobound a smoothly embedded annulus in $S^3\times [0,1]$. Connected sum makes concordance classes into an abelian group.

A knot is the image $K=f(S^1)$ of a smooth embedding $f:S^1\hookrightarrow {\mathbb{R}}^3$, considered up to ambient isotopy. The circle $S^1$ supplies a closed loop with no free ends; the smooth embedding condition rules out self-intersections and requires a nonzero tangent vector in a regular parametrization. The equivalent convention $S^1\hookrightarrow S^3$ uses the three-sphere: removing a point outside the knot identifies the remaining space with three-dimensional Euclidean space by stereographic projection.

More precisely, knots $K_0,K_1$ are equivalent if there is a smooth family of diffeomorphisms $F_s$ of the ambient space, for $0\le s\le 1$, such that $F_0$ is the identity and $F_1(K_0)=K_1$. Every intermediate $F_s(K_0)$ remains a smooth embedding. This ambient isotopy allows bending, stretching and moving a loop, but neither cutting it nor letting strands cross through each other. We compare the embedded images rather than their parametrizations; an oriented knot additionally remembers a direction of travel.

For a worked example, start with $U(t)=(\cos t,\sin t,0)$, where $t$ is taken modulo $2\pi$, and useBoth scale factors stay positive, so each $F_s$ is a diffeomorphism. At $s=1$ the circle becomes the ellipse $(2\cos t,\frac{1}{2}\sin t,0)$, yet its knot type is still the unknot. Exact lengths and angles are therefore irrelevant to knot type; self-intersection is the obstruction that the deformation must avoid.

For a small boundary displacement and weak disturbance, evaluate boundary data on the undisturbed surface and discard products of small quantities.

For a free surface $z=\eta (x,y,t)$ and velocity $(u,v,w)$, material conservation of $z-\eta$ giveson the surface.

Each Killing vector field supplies a conserved quantity for freely moving test particles: it turns a geometric symmetry into a first integral of the geodesic equation. Let $\gamma (\lambda )$ be a geodesic with affine parameter $\lambda$, tangent vector $U^{\mu }=d{x}^{\mu }/d\lambda$, and ${\nabla }_{U}U=0$. Wherever a Killing vector field $K$ is defined along the geodesic,The statement applies to Riemannian manifolds and to timelike geodesics, null geodesics and spacelike geodesics in a Lorentzian manifold. It requires an affine parameter; for a massive test particle, proper time is one choice. See Tong, section 4.3.2, equation (4.32).

For a worked example in the Euclidean plane, take $\gamma (t)=(1+t,2-t)$ with affine parameter $t$ and velocity $U=(1,-1)$. The Euclidean Killing vector fields for translation and rotation giveThe rotational field along the geodesic is $K_{\mathrm{rot}}(\gamma (t))=(t-2,1+t)$: it changes from $(-2,1)$ at $t=0$ to $(-1,2)$ at $t=1$, yet its pairing with $U$ stays $-3$. With toy mass $m=1$ and $t$ interpreted as time, the first constant is the $x$ component of linear momentum, and the second is the signed $z$ component of angular momentum about the origin. Their numerical values use toy units. The geodesic is straight; the rotation flow orbits are circles.

This conservation law follows from metric compatibility and the Killing equation:The geodesic equation kills the acceleration term, and the symmetric product $U^{\mu }{U}^{\nu }$ selects the vanishing symmetric derivative. This tensor contraction is the whole proof outline; see Carroll, equation (5.43).

In a stationary spacetime, a chosen time-translation Killing vector field $K$ gives Killing energy. With metric signature $(-+++)$, $c=1$, and a massive test particle's four-momentum $p^{\mu }=m{U}^{\mu }$, the physical normalization isHere $\Phi$ generates the axial rotational symmetry when one exists. In coordinates with $K={\partial }_t$ and $\Phi ={\partial }_{\varphi }$, these are $E=-p_t$ and $L_z=p_{\varphi }$. Both are conserved quantities along the geodesic, even if the metric has a time-angle cross term. A future timelike $K$ makes $E$ positive for a future massive particle; the scale of $E$ depends on the normalization of $K$. Not every spacetime has a global timelike Killing vector field, and a stationary field can become spacelike in an ergoregion of a stationary spacetime. Conservation on the field's domain does not imply a universal global positive energy; see Tong, sections 4.3.1-4.3.2 on the timelike-domain qualification and conserved charges.

In a convex resource-allocation problem, tightening an upper bound can change its multiplier from zero to positive. The optimum then moves from a point where only the total-resource constraint is active to the intersection where both constraints are active.