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A loss network admits a call only when all its required resources have enough free capacity. Rejected calls do not queue. Under fixed routing and independent Poisson processes, its exact occupancy law has a product-form stationary distribution of a loss network.

The link-route incidence matrix records how much resource $j$ is used by one call or flow on route $r$. For unit requirements $A_{jr}\in \{0,1\}$; the product $Ax$ gives resource throughputs from route flows.

A Wardrop equilibrium assigns positive traffic only to routes with minimum delay for their source-sink pair. With fixed demands and continuous increasing link delays, it minimizes the Beckmann potential. Strictly increasing delays make link throughputs unique, while route flows can remain nonunique.

A random access network lets stations make transmission attempts independently using a shared medium. A collision can prevent all attempted packets from being delivered; slotted ALOHA adjusts attempt probabilities using slot feedback.

A fluid model replaces discrete counts by continuous quantities whose evolution follows averaged large-scale drifts. A heuristic drift calculation motivates such a model; proving a fluid limit requires a separate convergence argument.

A flow-level model treats each document transfer as a flow whose rate depends on the current occupancy and resource capacities. With independent Poisson processes for arrivals and independent unit-mean exponential distributions for sizes, a route's total completion rate equals its allocated aggregate service.

For $-u^{\prime \prime }+u=f$ with homogeneous Dirichlet boundary conditions, the conforming piecewise-linear hat functions on a uniform grid give a diffusion stiffness matrix $S$ and a mass matrix $M$. Their nonzero entries are $S_{ii}=2/h$, $S_{i,i+1}=-1/h$, $M_{ii}=2h/3$, and $M_{i,i+1}=h/6$, with symmetric counterparts. The sum is a positive-definite matrix because its coefficient quadratic form is $\int ((u_h^{\prime }{)}^2+u_h^2)$. For forcing $f(x)=x$, the load at $x_i$ is $h{x}_i$.

A stellar polytrope with polytropic index one has $\nabla \Phi =-2K\nabla \rho$ in hydrostatic equilibrium. Combining this with the Poisson equation for Newtonian gravity gives the Helmholtz equationThe spherical solution regular at the origin is $\rho (r)={\rho }_c\sin (kr)/(kr)$. Its first zero gives $R=\pi /k=\sqrt{\pi K/(2G)}$, independent of central mass density, and $\bar{\rho }/{\rho }_c=3/{\pi }^2$.

For a spherical stellar polytrope of finite radius and zero surface pressure, the specific enthalpy is $(n+1)P/\rho$. Hydrostatic equilibrium gives $h+\Phi =-GM/R$, hence $(n+1)\int PdV=-G{M}^2/R-2\Omega$. Eliminating the pressure integral with the stellar virial theorem proves the displayed formula. For a gas adiabatic stellar polytrope, $n=1/(\gamma -1)$ and $U=G{M}^2/[(5\gamma -6)R]$; a general polytropic index need not have this relation to the specific-heat ratio.

For chemically uniform, gas-pressure-supported stars with homologous dimensionless profiles, mass conservation and hydrostatic pressure support equation give ${\rho }_c\propto M/R^3$ and $T_c\propto M/R$. Suppose opacity scales as $\kappa \propto {\rho }^a{T}^b$ and specific nuclear energy generation as $ϵ\propto {\rho }^c{T}^d$. ThenThe second scaling follows from radiative diffusion in a star, $L\propto R{T}_c^4/({\kappa }_c{\rho }_c)$. Equating the two giveswhen the denominator is nonzero. Fixed composition and self-similar profiles are essential; these are relations between idealized models rather than universal stellar laws.

A fully convective star has convection throughout nearly its entire interior. Very low-mass red dwarfs are fully convective; the transition to this structure is used in the disrupted magnetic braking model.

A radiative envelope transports most of its energy by radiative diffusion. Its stratified entropy profile gives a different rapid mass-loss response from a convective envelope.

A convective envelope is an outer stellar region in which convection carries a substantial fraction of the energy and mixes material. Its nearly adiabatic response to rapid mass loss is important for the dynamical stability of binary mass transfer.

For a grey atmosphere with $T^4=\frac{3}{4}{T}_{\mathrm{eff}}^4(\tau +2/3)$ and $P^2=P_0^2\log (1+3\tau /2)$, logarithmic differentiation givesA monatomic perfect gas of uniform composition has adiabatic temperature gradient $2/5$. The Schwarzschild criterion therefore predicts the first marginally unstable layer at ${\tau }_c=\frac{2}{3}(e^{4/5}-1)$, and convection for larger optical depth. Beyond that onset, the assumed purely radiative profile is no longer a self-consistent transport solution.

Stellar convective instability occurs when an adiabatic process displaces a fluid element and buoyancy amplifies the displacement. For uniform composition the Schwarzschild criterion gives instability when the actual temperature gradient exceeds the adiabatic temperature gradient; composition gradients instead require the Ledoux criterion.

A statistical test is a measurable rule for deciding whether to reject a null hypothesis. Allowing randomisation, it is represented by a function $\phi (x)\in [0,1]$ giving the conditional rejection probability after observing $x$. A deterministic test takes only values zero and one. Under a simple null, $E_0\phi$ is its size of a statistical test; under an alternative, $E_1\phi$ is its statistical power. Randomisation on a likelihood ratio equality set allows a Neyman-Pearson lemma threshold test to attain a prescribed size even for discrete data.

For a splitting $A=D+L+U$, successive over-relaxation updatesIt reduces to the Gauss-Seidel method at $\omega =1$. If $A$ is symmetric positive definite, the Householder-John theorem applies to $M=D/\omega +L$, $N=(1/\omega -1)D-U$, since $M^T+N=(2/\omega -1)D$ is positive definite for $0<\omega <2$.

With $A=D+L+U$, the Gauss-Seidel method solves $(D+L)x^{(k+1)}=b-U{x}^{(k)}$. It uses each newly computed coordinate immediately in the remaining triangular solve. It is successive over-relaxation with parameter one, and converges from every start when $A$ is a symmetric positive-definite matrix.

To solve a linear system $Ax=b$, split its matrix into the diagonal matrix $D$ and the strictly lower and upper parts $L,U$, retaining their actual signs. Every diagonal entry must be nonzero. A sweep updates coordinates in order:The Jacobi method uses the old vector throughout; the Gauss-Seidel method reuses each fresh coordinate. Thus the update order is part of the method.^[ref]

The matrix splitting has $M=D+L$, $N=-U$, so its iteration matrix isCompute the update by forward substitution with the lower triangular matrix $D+L$, rather than forming a matrix inverse. For a sparse matrix, a sweep only visits stored entries and can overwrite the working vector. Dependencies on fresh coordinates constrain parallel execution. This makes the Gauss-Seidel method an inexpensive solver component, including in a multigrid method.^[ref]

For Gaussian initial conditions and the Einstein-de Sitter second-order standard perturbation theory density kernel, the unsmoothed density contrast obeysThere are three choices of the second-order field and two connected Wick contractions; the angular average of $F_2$ is $5/7+(2/7)/3=17/21$. Thus the reduced skewness is $6(17/21)=34/7$. Smoothing changes this result, and the unsmoothed calculation requires a finite variance.

For plane-parallel pressureless matter in an Einstein-de Sitter universe, before shell crossing, the symmetrized growing-mode standard perturbation theory density kernel isIn particular $F_{2,s}=(q_1+q_2{)}^2/(2q_1{q}_2)$. An unsymmetrized representation of the one-dimensional beta mode-coupling kernel is $(q_1+q_2)/q_1$; its symmetric part is the expression above.