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A Gaussian process is a stochastic process whose every finite vector of values has a multivariate normal distribution. Its law is determined by its mean and covariance functions.

A Lévy process starts at zero, has independent stationary increments, is stochastically continuous, and is conventionally taken to have càdlàg paths.

A stochastic process $X$ is stochastically continuous when $X_s\to X_t$ in probability as $s\to t$.

Stochastic calculus extends integration and differential calculus to stochastic processes such as Brownian motion and semimartingales.

Processes $X_n$ converge uniformly on compacts in probability to $X$ when, for every $T<\infty$ and $\epsilon >0$,

The natural filtration of a stochastic process $X$ records its history:possibly completed and made right-continuous when the usual conditions are required.

The graphical Stirling number ${\left\{\begin{array}{c}n\\ k\end{array}\right\}}_G$ counts partitions of the vertices of a graph $G$ into $k$ nonempty independent sets. Equivalently, it counts proper colourings with $k$ unlabeled nonempty colour classes.

The mass-luminosity relation connects a star's mass to its luminosity. Its slope depends on opacity, nuclear burning, pressure support, composition, and evolutionary state.

In a radiative stellar region, photons carry luminosity down the temperature gradient. Dimensional scaling of the radiative-diffusion equation gives $L\sim R{T}_c^4/({\kappa }_c{\rho }_c)$.

Homologous stellar models have the same dimensionless radial profiles after mass, radius, density, pressure, and temperature are scaled by characteristic values.

A stellar polytrope obeys $P=K{\rho }^{1+1/n}$ for constant $K$ and polytropic index $n$. Its dimensionless density profile satisfies the Lane-Emden equation.

The CNO cycle catalyses hydrogen fusion through carbon, nitrogen, and oxygen nuclei. Near $2\times 10^7\mathrm{K}$, its specific energy generation is approximately proportional to $\rho T^{17}$ and is therefore strongly concentrated near a star's centre.

The proton–proton chain converts hydrogen to helium and dominates hydrogen burning in stars near and below the Sun's mass. Near $2\times 10^7\mathrm{K}$, its specific energy generation is roughly proportional to $\rho T^4$.

If Steiner symmetrization preserves area and cannot lower the perimeter of a minimizer, equality forces an axis of symmetry in the chosen direction. Applying every direction gives symmetry axes in all directions.

For a convex planar domain between graphs $u_{-}$ and $u_{+}$, the Euclidean triangle inequality giveswhich is the pointwise perimeter comparison with its symmetrization.

Every perpendicular chord keeps its length under Steiner symmetrization, so Fubini's theorem preserves the total area.

Constitutional supercooling occurs when liquid ahead of a solidification phase boundary lies below its local liquidus. In a saline liquid with $T_L=-mC$ and $m>0$, salt rejection enriches the interface liquid and depresses its liquidus; as the concentration decreases away from the interface, the liquidus rises. If it rises faster than the actual temperature, a supercooled interval appears. This can make a planar solidification phase boundary unstable.

The Neumann solution places a planar phase boundary at $h(t)=2\lambda \sqrt{\kappa t}$. Similarity reduction of the heat equation gives temperature profiles in terms of the error function, while the Stefan condition determines $\lambda$ implicitly.

A phase boundary is an interface separating distinct phases of matter. Its temperature, velocity, and flux jumps are constrained by phase equilibrium and conservation laws.

The Stefan condition equates latent heat generated by motion of a phase boundary to the jump in conductive heat flux across it. With interface speed $V$ and normal directed from solid to liquid, one common sign convention is