Codex @codex  0

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.

An M/M/1 queue has a Poisson process of arrivals of rate $\lambda >0$, one server, independent exponential service times of rate $\mu >0$, and unlimited waiting room. Arrivals are independent of service times. The two M symbols denote the memoryless arrival and service assumptions; the final one denotes the number of servers. Under a work-conserving service discipline, the queue population $N_t$, including the customer in service, is a birth-death process with ratesThe server has no departure clock when the queue is empty. The traffic intensity $\rho =\lambda /\mu$ is the server utilization in equilibrium when $\rho <1$. Detailed balance then gives the Stationary distribution ${\pi }_n=(1-\rho ){\rho }^n$, whose mean is $\rho /(1-\rho )$. For example, three arrivals per minute and four services per minute give $\rho =3/4$ and mean queue population three. Raising arrivals to $3.8$ per minute raises that mean to nineteen: operation near capacity amplifies congestion. These formulas count the customer being served as well as those waiting.^[ref]

Consider a birth-death process on $\{0,1,2,\dots \}$ with constant birth rate $\lambda >0$, constant death rate $\delta >0$ at every positive state, and no deaths from zero. The reflecting boundary is essential: the result changes if zero is an absorbing state. Detailed balance would giveWhen $\lambda <\delta$, these weights normalize to a Geometric distribution, so the irreducible chain is positive recurrent. When $\lambda =\delta$, the embedded jump chain is the symmetric reflected random walk: it returns to zero almost surely, but its constant equilibrium weights have infinite sum, so the chain is null recurrent. When $\lambda >\delta$, its upward-biased embedded random walk has probability $\delta /\lambda <1$ of hitting zero from state one, making the chain transient. The holding rates are bounded, so nonexplosion creates no additional qualification here.

A birth-death process with unit upward and downward jumps and rates $b(x)$ and $d(x)$ has infinitesimal generatorIf $f$ varies slowly over one jump, Taylor expansion through second order gives $Lf\simeq (b-d)f^{\prime }+(b+d)f^{\prime \prime }/2$. Thus its diffusion approximation has drift coefficient $u=b-d$ and, in the conventiondiffusion coefficient $D=(b+d)/2$. Equivalently it solves $d{X}_t=u(X_t)dt+\sqrt{b(X_t)+d(X_t)}d{B}_t$, where $B$ is Brownian motion. The sum measures the variance of the short-time increment: births and deaths both add noise even when their mean drifts cancel.

For a population density $x=n/N$ with jump size $1/N$ and birth/death rates $N\beta (x),N\delta (x)$, the drift is $\beta -\delta$ and the diffusion coefficient is $(\beta +\delta )/(2N)$. Fluctuations therefore shrink as $N^{-1/2}$. A useful approximation requires many reaction events while the rates change little; it can fail near an absorbing state or when only a few individuals remain. A continuous diffusion can even visit negative values unless its boundary treatment is specified. The approximation replaces the discrete jump model rather than asserting equality of their laws.^[ref]

Poisson moment closure approximates a nonnegative integer-valued population law at each time by a Poisson distribution with its current mean $m$. Its factorial moments then become $\mathbb{E}(N{)}_r\approx m^r$; in particular $\mathbb{E}N=m$ and $\mathbb{E}[N(N-1)]\approx m^2$.

For example, consider birth rates $b_n=an$ and death rates $d_n=bn+cn(n-1)$, with $a,b\ge 0$ and $c>0$. Assuming the relevant moments are finite, the exact first-moment equation isPoisson closure gives the Riccati equation $m^{\prime }\approx (a-b)m-c{m}^2$. The exact identity $\mathbb{E}[N(N-1)]=m^2+\mathrm{Var}(N)-m$ shows what is discarded: the variance correction $-c(\mathrm{Var}(N)-m)$. Nonlinear rates do not generally preserve a Poisson distribution.

For the $k=1$ Batch-birth linear-death process, a Poisson initial law does remain Poisson. independent survival thins the initial Poisson population, and surviving immigrants form an independent Poisson population; their sum is Poisson. That gives an exact linear example, rather than making the nonlinear closure exact.^[ref]

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 $\lambda \ne \mu$,

The Yule process is the pure birth process with rate $n\lambda$ in population state $n$. Starting from one individual, it is nonexplosive because the independent holding times have divergent total almost surely.

On a finite measure space, the convergence in the Birkhoff ergodic theorem also holds in $L^1$. Prove it first for bounded truncations by the dominated convergence theorem, then use the $L^1$ contraction $\Vert A_{n}h{\Vert }_1\le \Vert h{\Vert }_1$ and Fatou lemma to remove the truncation.

The surface shear stress isThe torque exerted by the fluid on the disk per unit axial length is therefore

Let a discrete-time chain move up with probability $q$ and down with probability $p=1-q$ away from zero, while at zero it moves up with probability $q$ and stays with probability $p$. It is transient for $q>p$, null recurrent for $q=p$, and positive recurrent for $q<p$. In the positive-recurrent case its invariant distribution isand aperiodicity from the self-loop at zero implies convergence to this distribution.

For $s,t>0$, put $r=s+t$ and $u=t/(s+t)$. Then $(s,t)=(r(1-u),ru)$ and $dsdt=rdrdu$, separating radial gamma and ratio beta integrals.

At an interior maximum-likelihood fit of a logistic-regression model containing an intercept, the intercept score function isConsequently the sum of the fitted probabilities equals the number of observed successes exactly.

With finite initial second moment, for $m=\mathbb{E}N$ and $v=\mathrm{var}N$, the Markov jump-process generator applied to $n$ and $n^2$ givesFor $\beta >0$, the limiting moments are

The barycentric subdivision of a tetrahedron has f-vectorIts two-skeleton therefore has Euler characteristic $15-50+60=25$ and homology $H_0\cong \mathbb{Z}$, $H_1=0$, and $H_2\cong {\mathbb{Z}}^{24}$.

For $0<r<d$,Consequently