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For one-dimensional representations $\theta ,\phi$ of the upper-triangular subgroup $B\le S{L}_2({\mathbb{F}}_p)$,where ${\theta }^w(t)=\theta (w^{-1}tw)$.

BRST cohomology identifies physical states and observables with BRST-closed objects modulo BRST-exact ones. A change of gauge-fixing fermion changes the action by a BRST-exact term and therefore leaves BRST-cohomology classes unchanged when the measure has no BRST anomaly.

A gauge-fixing fermion is a Grassmann-odd functional whose BRST transformation $s\Psi$ supplies the gauge-fixing and ghost terms. Nilpotence gives $s(s\Psi )=0$ immediately.

For a twice differentiable integrable test function $f$ and Brownian motion $W$,is a martingale. This is the one-dimensional generator form of Dynkin's formula.

For $a,b>0$ and the convention $\sup \varnothing =0$,

For $a,y>0$,

For $a,b>0$,This follows from the Brownian reflection principle weighted by the Cameron-Martin theorem for a linear drift.

For a suitable twice differentiable function $u$ and an integrable stopping time $T$,

It is the Dynkin's formula with the Brownian motion generator $\frac{1}{2}\Delta$.

A body-centered cubic lattice consists of the points of a cubic lattice together with their body centers. Its reciprocal lattice is face-centered cubic.

The reciprocal lattice isIf $\Omega =a_1\cdot (a_2\times a_3)$, its primitive vectors areand satisfy $a_i\cdot b_j=2\pi {\delta }_{ij}$.

A lattice point is a vector $l=n_1{a}_1+n_2{a}_2+n_3{a}_3$ whose coefficients in a chosen primitive basis are integers.

For $u=\log |\varphi {|}^2$ and zeros of multiplicities $N_r$ at $z_r$, the Taubes equation is

This is the Bruhat decomposition associated with a BN-pair. The Bruhat decomposition is the disjoint unionassociated with a BN-pair.

The queue length in an $M/M/1$ queue is a birth-death process with constant arrival rate $\lambda$ and service rate $\mu$ while nonempty.

For constant birth rate $\lambda$ and death rate $\delta$ away from zero, the chain is transient when $\lambda >\delta$, null recurrent when $\lambda =\delta$, and positive recurrent when $\lambda <\delta$.

For unit jumps with birth and death rates $b(x),d(x)$, the Fokker--Planck drift and diffusion in the conventionare $u=b-d$ and $D=(b+d)/2$.

Approximating the population law by a Poisson law replaces its first two factorial moments by $\lambda$ and ${\lambda }^2$, closing a quadratic first-moment equation as a Riccati equation for $\lambda$.

For birth rates $b_n$, death rates $d_n$, and state probabilities $P_n$,

For birth rates $b_i$ and death rates $d_i$, a reversible measure satisfiesThus ${\pi }_{i+1}/{\pi }_i=b_i/d_{i+1}$, and it is an invariant distribution when the resulting measure is summable and the chain is nonexplosive.

For individual birth and death rates $\lambda ,\mu$ and one initial individual, the finite-time extinction probability satisfiesWhen $\lambda \ne \mu$,