The dipolar flux-tube area has $A(r)=A_{\star }(r/R_{\star }{)}^3$. Applying transonic accretion in a power-law tube with $n=3$ and a polytropic equation of state givesHere $c_0$ and ${\rho }_0$ are the reservoir sound speed and mass density, and $c_s$ is the sound speed at the sonic point. The mass accretion rate isFor two equal polar caps of surface angular radius ${\theta }_{\star }\ll 1$, $A_{\star }\simeq 2\pi R_{\star }^2{\theta }_{\star }^2$. One loaded cap gives half the total rate. The finite-radius transonic branch exists for $1<\gamma <7/5$, must cross outside the star ($r_s>R_{\star }$), and requires ${\theta }_{\star }^2{r}_s/R_{\star }\ll 1$ for the sonic region to remain a narrow flux tube.

The inner radius at which a stellar magnetic dipole field disrupts a surrounding accretion disk. In a free fall pressure-balance estimate with fixed aspect ratio $ϵ=H/r$, $r_m\sim ({ϵ}^2{\mu }^4/(GM{\dot{M}}^2){)}^{1/7}$. Geometry and inflow structure change its numerical coefficient.

A rapidly spinning magnetosphere adds angular momentum to inflowing matter outside the corotation radius, inhibiting accretion and potentially ejecting gas. The centrifugal onset condition does not by itself ensure that every parcel escapes to infinity.

The spin at which the net material and magnetic torque on an accreting star vanishes. The corotation radius is typically comparable to the magnetospheric truncation radius, but the precise torque balance depends on the coupling and any outflow.

In this model, strong magnetic braking of a binary star inflates the mass-losing donor above the cataclysmic-variable period gap. A reduction in the torque near the fully convective star transition allows thermal contraction and detachment. Gravitational-wave emission from a binary system continues to shrink the orbit until contact returns below the gap. The model requires a braking decrease and a previously inflated donor, not disappearance of stellar magnetism. Knigge's review describes this evolutionary picture and observational tests.

Suppose the columns $a_i$ of a matrix over the rational numbers satisfy the columns condition with blocks $B_1,\dots ,B_s$. Choose rational coefficients ${\lambda }_{ij}$ such thatChoose a positive integer $c$ clearing all denominators and $p\ge \max |c{\lambda }_{ij}|$. For generators of any full positive M-p-c set, the coordinateslie in that set and satisfy $Ax=0$. Thus the monochromatic m-p-c set theorem supplies the sufficiency direction of Rado's theorem.

Fix a uniformizer $\pi$ of a local field $K$ and a Lubin–Tate formal group $\mathcal{F}$. Its nth torsion field is generated by all its ${\pi }^n$-torsion points. A primitive point ${\lambda }_n$ generates the Lubin–Tate torsion as a free rank-one ${\mathcal{O}}_K/{\pi }^n$-module; therefore $K_{\pi ,n}=K({\lambda }_n)$. If the residue field has size $q$, the Eisenstein layers of Lubin–Tate torsion give $[K_{\pi ,n}:K]=(q-1)q^{n-1}$ and total ramification.

The Herbrand function is a continuous strictly increasing piecewise linear function on the nonnegative reals. Its inverse is denoted ${\psi }_{L/K}$. It reparametrizes the lower ramification numbering to the upper ramification numbering.

Under fixed routing, each call type always requests the same collection of resources. The link-route incidence matrix records those requirements. This differs from alternative routing, in which a rejected direct request can try another route.

For calls arriving at rate ${\nu }_r$ with mean holding time $1/{\mu }_r$, offered traffic is the dimensionless load ${\alpha }_r={\nu }_r/{\mu }_r$, often measured in Erlangs. The Erlang loss formula distinguishes this offered traffic from the mean traffic actually carried.

The Erlang loss formula gives the blocking probability for a single resource with $C$ identical circuits, no waiting room, independent Poisson arrivals of rate $\lambda$ and independent exponential holding times of mean $1/\mu$. Each admitted call occupies one circuit. Write offered traffic $a=\lambda /\mu$.

The occupied-circuit count is a birth-death process on $0,\dots ,C$, with upward rate $\lambda$ below capacity and downward rate $n\mu$. Detailed balance gives $\lambda \pi (n)=(n+1)\mu \pi (n+1)$, hence $\pi (n)\propto a^n/n!$. By Poisson arrivals see time averages, the arrival blocking probability isThe resource has a finite state space even when $a>C$; excess calls are rejected rather than accumulating.^[ref]

For $C=2$, $a=1$, the weights are $1,1,1/2$, giving blocking $1/5$ and carried load of an Erlang loss resource $a[1-E(a,C)]=4/5$. A useful recursion avoids large factorials:The insensitivity of loss networks extends the occupancy formula to independent general holding times with the same mean under the standard loss-model assumptions.

Alternative routing lets a call try another resource path when its preferred route is blocked. Overflow onto longer paths can create feedback and multiple solutions of the Erlang fixed point approximation, even though an exact finite irreducible occupancy Markov chain has a unique stationary distribution.

When the full Lorentz-invariant phase-space measure treats $n$ identical final particles as labeled, each physical final configuration is counted $n!$ times. The unlabelled relativistic scattering cross-section therefore includes $1/n!$. For two identical scalars, this divides the full-angle result by two; equivalently, integrate over a region containing only one representative of each exchanged pair.

This total decay-width formula applies in the parent's rest frame when the squared scattering amplitude is independent of direction and the daughters are distinct. For identical daughters an additional symmetry factor is needed.

For $E>0$ and the positive-energy branch, the root Jacobian of the Dirac delta function gives this identity. In a massless collinear parton approximation, it produces $\delta (x-\xi )/(P\cdot q)$.

A logistic-normal count model takes $Y_i\mid {\theta }_i\sim \mathrm{Bin}(n_i,{\mathrm{logit}}^{-1}{\theta }_i)$ and ${\theta }_i=x_i^T\beta +\lambda Z_i+{\eta }_i$, where $Z$ is a stationary unit-variance autoregressive process of order one with coefficient $a$ and the independent ${\eta }_i$ have normal distribution with variance $v$. The marginal latent variance is ${\lambda }^2+v$, and its off-diagonal covariance is ${\lambda }^2{a}^{|i-j|}$. Random success probabilities produce overdispersion relative to the binomial distribution.

A regression for a categorical response with more than two possible outcomes, parametrizing log probability ratios relative to a baseline category by linear predictors. It preserves categories that a binary logistic regression would merge.

For scales $U,L,D$, thermal balance estimates $w\sim f_0{U}^2/(N^{2}D)$. Density-weighted continuity gives horizontal ageostrophic flow of order $Lw/\min (D,H)$. It is small relative to $U$ only ifThis condition supplements small Rossby number and small relative variation $\beta L/|f_0|$ of the Coriolis parameter.

When the indicated limit exists continuously and the hulls are generated by its past, it defines the trace of the Loewner chain. The hull $K_t$ includes the curve up to time $t$ and any regions it disconnects from infinity. Therefore swallowing a point by $K_t$ is different from visiting that point by the Loewner trace. For SLE, the continuous trace is part of its basic existence theory.

A nonnegative regression kernel supplies weights $K((x_i-x)/h)$ in a local weighted least squares fit. Unlike a probability density function, its integral need not equal one, because rescaling every weight does not change the fitted coefficients.