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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.

For the endomorphism bundle connection, $({\tilde{\nabla }}_X\varphi )(s)={\nabla }_X(\varphi s)-\varphi ({\nabla }_{X}s)$. Expanding two derivatives and subtracting ${\tilde{\nabla }}_{[X,Y]}$ givesThus its curvature form of a connection is the commutator action of the original curvature.

The j-invariant is unchanged by admissible changes of Weierstrass equation of an elliptic curve. With $b_2=a_1^2+4a_2$ and $b_4=a_1{a}_3+2a_4$, set $c_4=b_2^2-24b_4$ and $j=c_4^3/\Delta$, where $\Delta$ is the elliptic-curve discriminant. For good reduction over a local field, an integral equation with unit discriminant shows that $j$ lies in the valuation ring. Thus a negative valuation of $j$ proves bad reduction, even if the displayed equation has not been proved minimal.

For the Higgs doublet vacuum $(0,v{)}^T/\sqrt{2}$, the neutral gauge-boson mass matrix in $(W^3,B)$ is $(v^2/4)\left(\begin{array}{cc}{g}^2& -g{g}^{\prime }\\ -g{g}^{\prime }& g^{\prime 2}\end{array}\right)$. Its zero eigenvector is proportional to $(g^{\prime },g)$, the photon; the perpendicular vector is the Z boson. The charged fields $W^{\pm }=(W^1\mp i{W}^2)/\sqrt{2}$ have the stated W boson mass. Thus $m_W=m_Z\cos {\theta }_W$, where ${\theta }_W$ is the Weinberg angle. The unbroken charge generator annihilates the vacuum and explains the zero eigenvalue.