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For every $\lambda ,\mu >0$, an $M/M/\infty$ queue is positive recurrent and has the Poisson distribution with mean $\rho =\lambda /\mu$ as its unique stationary distribution:

Starting with one customer, the mean busy period of an $M/M/1$ queue iswhen $\lambda <\mu$. It is infinite when $\lambda \ge \mu$.

If a served customer returns with probability $p$, thinning service completions gives effective external departure rate $\mu (1-p)$ for the queue-length process.

When $\rho <1$, the equilibrium queue length has distribution ${\pi }_n=(1-\rho ){\rho }^n$ and mean $\rho /(1-\rho )$.

An $M/M/1$ queue is positive recurrent exactly when its traffic intensity $\rho =\lambda /\mu$ is below one.

If potential arrivals form a Poisson process of rate $\lambda$ and an arrival seeing $n$ customers joins with probability $p(n)$, the queue length is a birth-death process with birth rate $b_n=\lambda p(n)$ and death rate $d_n=\mu$ for $n\ge 1$. Its stationary ratios, when normalizable, are

For constant density of states, weighting the difference between the Fermi-Dirac distribution and the zero-temperature step by $E-E_F$ produces a convergent integral that scales as ${\beta }^{-2}$. Hence the leading low-temperature energy correction is proportional to $T^2$.

The connecting homomorphism is the degree-shifting map produced by the lift-and-boundary construction in a long exact sequence.

For a finite extension of local fields $L/K$, its norm subgroup is the image of the field norm $N_{L/K}:L^{\times }\to K^{\times }$. For an abelian extension, it is the kernel of the induced local Artin map to $\mathrm{Gal}(L/K)$.

The local Artin map is the reciprocity homomorphism of Local Artin reciprocity. Its normalization is fixed by choosing whether a uniformizer maps to arithmetic or geometric Frobenius.

Freezing-point depression is the lowering of a solvent's equilibrium freezing temperature by dissolved solute. In the linear dilute approximation, $T_L=T_0-mC$ with $m>0$.

A basepoint-free vector space of global sections $V\subseteq H^0(X,L)$ defines the Kodaira map $X\to \mathbb{P}(V^{\ast })$ by evaluating the sections at each point.

A line bundle is very ample when its global sections define a closed embedding into projective space.

Over a field, a line bundle is very ample when it is the pullback of $\mathcal{O}(1)$ by a closed immersion into projective space. It is an ample line bundle, but an ample bundle need not itself be very ample.

The Picard group is the group of isomorphism classes of line bundles under tensor product.

For a finite-dimensional representation $d$, the trilinear form $B(X,Y,Z)=\mathrm{Tr}(d(X)d(Y)d(Z))$ is invariant under simultaneous adjoint action. This follows by writing the sum of its three infinitesimal variations as the trace of a commutator.

For a finite-dimensional Lie algebra representation $\rho :\mathfrak{g}\to \mathfrak{gl}(V)$, its trace form is the symmetric bilinear formIt is invariant: $([x,y],z{)}_V=(x,[y,z]{)}_V$.

The adjoint representation is ${\mathrm{ad}}_x(y)=[x,y]$. For a semisimple Lie algebra, its nonzero weights are the roots and its zero-weight space is the Cartan subalgebra.

A nonzero Lie algebra representation is irreducible when it has no proper nonzero invariant subspace.

A Lie algebra representation is faithful when its representing homomorphism is injective.

For a basis $(T_a)$ of a Lie algebra, its structure constants are defined by $[T_a,T_b]=f_{ab}{}^c{T}_c$.