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Fick's laws describe diffusion through a flux law and conservation of mass. The first law says that diffusive flux points down the concentration gradient:Combining it with local conservation gives
$C_t=\nabla \cdot (D(C)\nabla C)$.

The porous medium equation $u_t=\Delta (u^m)$ with $m>1$ is a nonlinear partial differential equation whose diffusivity vanishes at $u=0$. Unlike the heat equation, it has compactly supported solutions with finite propagation speed.

The injective diagonal map $(x_n)\mapsto (x_n/n)$ on ${\ell }^2$ is bounded, but its inverse on the range is unbounded; the range is correspondingly not closed.

A block matrix is block diagonal if all off-diagonal blocks vanish. It represents independent actions on a direct sum of vector spaces. Its determinant is the product of the diagonal-block determinants.

Between weak freeze-out and deuterium formation, beta decay lowers the surviving free-neutron fraction.

Use natural units and a positive nuclear binding energy ${\epsilon }_D=m_n+m_p-m_D$. Chemical equilibrium of $n+p\leftrightarrow D+\gamma$ requires ${\mu }_D={\mu }_n+{\mu }_p$. The nonrelativistic Maxwell--Boltzmann number density givesHere $g_D=3$ and $g_n=g_p=2$ count the spin states. Taking $m_n\simeq m_p\simeq m_N$ and $m_D\simeq 2m_N$, then writing $X_i=n_i/n_B$, $n_B=\eta n_{\gamma }$ and using the photon number density yields the displayed abundance. The coefficient is approximately $8.14$. Omitting the spin-state ratio $3/4$ gives the rougher prefactor $16\zeta (3)/\sqrt{\pi }$. A convention $B=m_D-m_n-m_p<0$ instead writes the exponential as $e^{-B/T}$. The small baryon-to-photon ratio suppresses stable deuterium until the temperature is well below the binding scale, producing the deuterium bottleneck.

A deterministic automaton is irreducible when every state is accessible and every pair of distinct states is distinguishable.

Two states are indistinguishable when every continuation is accepted from both or rejected from both. Distinguishable states admit at least one suffix with different acceptance outcomes.

A state is accessible when some input word takes the initial state to it. Removing inaccessible states preserves the accepted language.

For a transition function $\delta :Q\times \Sigma \to Q$, defineThe accepted language is $\{w:\hat{\delta }(q_0,w)\in F\}$.

The variety of $m\times n$ matrices of rank at most one has dimension $m+n-1$. For $m,n\ge 2$, its only singular point is the zero matrix.

The projective locus wherehas rank one is cut out by its three $2\times 2$ minors. It is a smooth surface: on each of the charts $y_0\ne 0$, $y_1\ne 0$, $y_3\ne 0$, and $y_4\ne 0$, the equations eliminate three coordinates and leave two free affine coordinates; these charts cover the locus.

For a pressureless fluid withthe first-order density contrast $\delta =\delta \rho /\bar{\rho }$ obeys

The linear growth factor multiplies the growing mode of a linear cosmological density perturbation, so $\delta (\mathbf{x},t)=D(t)\delta (\mathbf{x},t_0)/D(t_0)$ when scale-independent growth applies.

A graph is locally finite when every vertex has finite degree.

The maximum degree $\Delta (G)$ is the greatest degree of a vertex of $G$.

It is the maximum of the degree of a vertex over all vertices.

The minimum degree $\delta (G)$ is the least degree of a vertex of $G$.

The minimum degree is $\delta (G)={\min }_{v\in V(G)}\mathrm{deg}(v)$, the least degree of a vertex in the graph.

A universal vertex is adjacent to every other vertex of its graph.

In a finite graph with at least one edge, the degree-biased vertex distribution assigns probability proportional to the degree of a vertex. It is the distribution of an endpoint on a specified side of a uniformly random edge.

Projective degree belongs to an embedding rather than to the abstract variety. A line and a smooth conic in ${\mathbb{P}}^2$ are both abstractly ${\mathbb{P}}^1$ but have degrees one and two.