For an order-$m$ operator with smooth coefficients on a bounded piecewise smooth domain, integration by parts has no boundary remainder when at each boundary point either the entire boundary jet of $u$ through order $m-1$ vanishes or the entire such jet of $v$ vanishes. Every boundary summand pairs derivatives of the two functions of order at most $m-1$. It is insufficient to require only that, for each same multi-index $\beta$, one of ${\partial }^{\beta }u$ and ${\partial }^{\beta }v$ vanish: for $P={\partial }_x^2$, $u=x(a-x)\chi (y)$ and $v=1$ on a rectangle, all those same-index products vanish on the boundary but $\int (Pu)v=-2a\int \chi \ne 0$.

A flasque resolution is an exact sequence of sheaves of abelian groups whose terms ${\mathcal{I}}^j$ are flasque sheaves. Every such sheaf embeds by its section germs into the sheaf $U\mapsto {\prod }_{P\in U}{\mathcal{F}}_P$. Its restriction maps are projections, so it is flasque. Apply this construction successively to the cokernels to obtain a resolution. The sheaf cohomology of $\mathcal{F}$ is the cohomology of the cochain complex $\Gamma (X,{\mathcal{I}}^{\bullet })$. A quotient of two flasque sheaves in a short exact sequence of sheaves is flasque: the flasque kernel allows local lifts to be glued into a section of the middle sheaf over the smaller open set; extend that lift using flasqueness of the middle sheaf, and project. Consequently the resolution's successive cokernels are flasque when $\mathcal{F}$ is flasque, and exactness of sections shows that all its positive-degree cohomology vanishes.

For a short exact sequence of sheaves with flasque sheaf kernel, every section of the quotient on any open set lifts to the middle sheaf on that set. Use the Zorn lemma to choose a maximal partial lift, with chains glued by the sheaf gluing axiom. A new local lift differs from the old one on their overlap by a kernel section. Extend that difference by flasqueness, adjust the new lift, and glue, contradicting maximality unless its domain is the entire open set. Consequently a quotient of two flasque sheaves is flasque. With an injective resolution of sheaves, the long exact sequence in sheaf cohomology and repeated dimension shifts prove that flasque sheaves have zero positive-degree cohomology.

For a loss network with feasible occupancies $An\le C$, its stationary distribution is proportional to ${\prod }_r{\alpha }_r^{n_r}/n_r!$. It is the law of independent Poisson random variables conditioned on the resource constraints. With exponential holding times, detailed balance for a continuous-time Markov chain proves the formula directly.

For standard Brownian motion and $a>0$, $T_a=\inf \{t\ge 0:B_t\ge a\}$ is finite almost surely andThe Brownian reflection principle gives this distribution function. The Brownian scaling identity $T_{ca}\stackrel{d}{=}{c}^2{T}_a$ and the Strong Markov property show its probability distribution is a strictly stable distribution with index $1/2$.

With $D={\partial }_{\mu }{A}^{\mu }$, expand the Maxwell Lagrangian and its Feynman gauge term to findCommuting partial derivatives verifies the divergence identity. Both densities give $\square A^{\mu }=0$ under the usual variational boundary conditions. Their canonical momentum components differ: the original density gives $-F^{0\nu }-{\eta }^{0\nu }D$, whereas the subtracted density gives $-{\dot{A}}^{\nu }$. Thus a free-photon momentum expansion using the latter requires a stated boundary-term convention.

For a symplectic vector space $(V,\omega )$, contraction is the map $c:{\Lambda }^{2}V\to \mathbb{C}$ defined by $c(v\wedge w)=\omega (v,w)$. It is equivariant for the symplectic Lie algebra. In dimension four its five-dimensional kernel is irreducible of highest weight ${\omega }_2$, while the invariant inverse-form bivector spans a complementary trivial Lie algebra representation.

A small-scale spectral cutoff bounds the smoothing variance by $S_{\max }$. A sharp-k excursion remains Brownian up to this terminal variance, with total crossed fraction $\mathrm{erfc}({\delta }_c/\sqrt{2S_{\max }})<1$. The ordinary first-crossing density remains valid before the cutoff. Below the sharp-filter cutoff radius the variance is constant, so there are no new crossings and the idealized differential halo abundance vanishes. The missing fraction need not be artificially renormalized to one.

The eutectic temperature is the temperature of simultaneous liquid–two-solid phase coexistence in an eutectic system. In directional freezing of a solution away from the eutectic composition, a mushy layer can end at this temperature, where its remaining liquid solidifies.

For a Cartier divisor $D$ on an $n$-dimensional projective scheme, $\chi (X,{\mathcal{O}}_X(mD))$ is a polynomial in $m$ whose leading term is $D^n{m}^n/n!$. The intersection product uses the fundamental cycle, with the generic multiplicity of each component. Thus the statement also applies to nonreduced schemes.

The complementary error function describes the Gaussian integral area remaining beyond the endpoint. For a real $x$,It is an entire function for complex arguments when defined by $1-\mathrm{erf}(z)$. On the real line it decreases from two to zero, with $\mathrm{erfc}(0)=1$ and $\mathrm{erfc}(-x)=2-\mathrm{erfc}(x)$.^[ref]

For example, $\mathrm{erfc}(1)\approx 0.157299$ is a sizeable tail, whereas $\mathrm{erfc}(8)\approx 1.12243\times 10^{-29}$ is extremely small. In the tested floating-point arithmetic, a direct evaluation of $1-\mathrm{erf}(8)$ gives zero because the computed error function has already rounded to one. Evaluating the complementary error function directly preserves this tail. It is the useful quantity in an upper-tail normal distribution probability and in the decaying side of an error-function diffusion front.

For large positive real $x$, the leading asymptotic expansion isThe first corrections are $1-1/(2x^2)+3/(4x^4)-\cdots$ multiplying the leading term. This is an asymptotic expansion, not an assertion that summing arbitrarily many terms converges at a fixed $x$.^[ref] The complementary error function tail bound provides finite-$x$ inequalities instead.

The Erlang fixed point approximation treats resource blocking as independent and applies the Erlang loss formula to traffic screened by other resources. Its self-consistency equations have a unique fixed point under fixed routing, but can have several under alternative routing.

For a measure-preserving system on a standard probability space, an invariant probability measure is an integral of ergodic invariant probability measures. Its conditional measures given the invariant sigma-algebra are the ergodic components.

A probability measure-preserving transformation is strongly mixing when $\mu (T^{-n}A\cap B)\to \mu (A)\mu (B)$ for every pair of measurable sets $A,B$. Unlike weak mixing, this is an ordinary limit along all sufficiently large times.

A convex entropy $\eta$ is paired with any flux antiderivative satisfying $q^{\prime }(z)={\eta }^{\prime }(z)f^{\prime }(z)$ almost everywhere. Its admissibility inequality is ${\partial }_t\eta (u)+{\partial }_{x}q(u)\le 0$ distributionally, with the matching initial entropy trace. Affine entropies $\eta (z)=\pm z$ recover the weak formulation, while bounded classical solutions satisfy equality for smooth entropies and then by approximation for piecewise smooth convex entropies.

For a bounded entropy solution with its initial entropy inequalities, the displayed limit holds on every compact spatial set. Test a constant-level entropy with $(1-t/\delta {)}_{+}\rho (x)$. The flux remainder is $O(\delta )$, giving a bound by $\int \rho |u_0-k|$. A finite smooth partition and constant levels approximate the initial function in local $L^1$; the triangle inequality makes the limsup arbitrarily small. This averaged trace suffices for one-sided time mollifiers supported in $[\delta ,2\delta ]$.

For a stationary process associated with a finite-entropy measurable partition, $H_{\mu }({\xi }_0^{n-1})/n$ is non-increasing. The increments $c_k=H_{\mu }(\xi \mid {\bigvee }_{j=1}^k{T}^{-j}\xi )$ are non-increasing by conditioning reduces entropy, and the chain rule gives $H_{\mu }({\xi }_0^{n-1})={\sum }_{k=0}^{n-1}{c}_k$. Averages of a decreasing sequence are decreasing.

For a probability measure-preserving system and a countable measurable partition $\xi$ of finite entropy, $-n^{-1}\log \mu ({\xi }_0^{n-1}(x))$ converges almost everywhere and in $L^1$ to an invariant function whose integral is $h_{\mu }(T,\xi )$. For an ergodic transformation, the limit is the constant $h_{\mu }(T,\xi )$. In general it is $\mathbb{E}[I_{\mu }(\xi \mid {\bigvee }_{j=1}^{\infty }{T}^{-j}\xi )\mid \mathcal{I}]$. The martingale convergence theorem, maximal inequality for conditional information functions, and triangular ergodic averaging lemma prove this directly, without invertibility.

For a finite measurable partition $\xi$, put ${\mathcal{F}}_1=\sigma ({\bigvee }_{j\ge 1}{T}^{-j}\xi )$. Then $h_{\mu }(T,\xi )=H_{\mu }(\xi \mid {\mathcal{F}}_1)={\lim }_{n\to \infty }{H}_{\mu }(\xi \mid {\xi }_1^n)$. The backward entropy chain rule writes $H_{\mu }({\xi }_0^{N-1})$ as the sum of the first $N$ decreasing finite-future conditional entropies. Their averages have the same limit. Increasing conditioning fields converge by the martingale convergence theorem.

For $N$ equal weak shocks with individual pressure ratio $1+\delta$ and total ratio $P_s=(1+\delta {)}^N$, entropy production in a perfect-gas shock givesAt fixed $P_s>1$, this tends to zero as the compression is divided into increasingly many smaller weak shocks. A single finite normal shock wave instead gives the positive value $\log P_s-\gamma \log D(P_s)$. Gradual compression can consequently approach reversible isentropic flow while an abrupt compression generates finite entropy.