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The $r$th falling factorial moment is $\mathbb{E}[(N{)}_r]$, where $(N{)}_r=N(N-1)\cdots (N-r+1)$.

Since births and deaths change the state by $+1$ and $-1$,

If $u\in L^{\infty }(0,T;H)$ and $u_t\in L^q(0,T;V^{\prime })$ for a dense continuous embedding $V\hookrightarrow H$ and some $q>1$, then $u$ has a representative in $C([0,T];H_{\mathrm{weak}})$. Indeed, its pairing with each element of a dense subset of $V$ is absolutely continuous, and the uniform $H$ bound extends continuity to every element of $H$.

The algebraic concordance group over a field $F$ is the Witt group of nonsingular Seifert forms over $F$, modulo metabolic forms.

A nonsingular Seifert form on a $2g$-dimensional space is metabolic when it vanishes on a $g$-dimensional subspace called a metabolizer.

The metabolizer is a half-dimensional subspace on which the Seifert form restricts to zero.

If an action has parameter dependence $I\propto \sqrt{m/|E|}$, adiabatic conservation implies $|E|\propto m$.

For a one-dimensional particle of energy $E$ between walls separated by $L$, the unnormalized action isAdiabatic conservation gives $E(t)L(t{)}^2=\text{constant}$.

The Chern-Simons current satisfies ${\partial }_{\mu }{K}^{\mu }=\mathrm{Tr}(F_{\mu \nu }{}^{\star }{F}^{\mu \nu })$. It is not gauge invariant even though its divergence is.

In four dimensions this current is dual to the degree-three Chern-Simons form.

The number of trees in a wired uniform spanning forest is almost surely constant. This follows from its tail triviality together with the fact that every component is infinite.

Wilson's algorithm rooted at infinity builds the wired uniform spanning forest on a transient graph by successively adding loop-erased random walks, run forever when they never hit the forest already constructed.

Because state vectors differing by a quantum phase represent the same ray, a unitary realization of a group action need only satisfy

A projective unitary representation is a projective representation whose chosen representatives are unitary operators. The phase factors disappear on rays.

The killing measure assigns the weight ${\kappa }_x$ of the transition from $x$ to a cemetery state.

For a subset of a Banach space, relative weak compactness is equivalent to every sequence having a weakly convergent subsequence. In particular, weak compactness and weak sequential compactness coincide.

On a smooth projective curve of genus $g$, every divisor of degree at least $2g+1$ is very ample. Riemann--Roch shows that its sections separate distinct points and tangent directions by comparing $L(D)$ with $L(D-P-Q)$ and $L(D-2P)$.

Applying the divergence theorem componentwise to $T_{ij}{v}_i$ yields a boundary traction term and a volume contraction.

A line integral integrates a scalar or tangential vector-field component along a curve.

Green’s theorem converts circulation around a positively oriented planar boundary into the area integral of scalar curl.

A surface integral uses the area element induced by a parametrisation to integrate over a surface.

Let $S$ be an oriented smooth surface with piecewise smooth boundary, and let $F$ be a continuously differentiable vector field on a neighborhood of $S$. With the compatible boundary direction determined by the right-hand rule,For $G=(-y,x,0)$ on the horizontal unit disc with upward normal, the left side is $2$ times its area, giving $2\pi$. The counterclockwise boundary has tangential field of magnitude one and circumference $2\pi$, giving the same line integral. Reversing the surface normal reverses the boundary direction and both signs. The theorem replaces a circulation calculation with a surface integral of the curl, whichever is easier.

For a continuously differentiable function $\Phi$ and a piecewise smooth parametric curve $\gamma$ from $a$ to $b$,The chain rule converts the integrand into $d(\Phi (\gamma (t)))/dt$, then the fundamental theorem of calculus supplies the endpoint difference. This is why a conservative vector field has a path independent line integral. For $\Phi (x,y)=x^2+y^2$, every path from $(0,0)$ to $(1,2)$ gives $5$, provided it lies in the domain where the potential is defined.