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For every test function $\phi$,so $H^{\prime }=\delta$ as distributions.

A reaction-diffusion system couples local changes in several fields with their spatial spreading. For concentrations $u=(u_1,\dots ,u_m)$, a common model isThe diffusion coefficients $D_i$ describe transport, while the reaction rates $f_i$ describe local conversion, production or loss. Writing $D=\mathrm{diag}(D_1,\dots ,D_m)$ gives the vector equation $u_t=D\Delta u+f(u)$. Chemical concentrations and rates must be modeled consistently; a reaction term can exchange material with an external supply.

A spatially homogeneous equilibrium $u_{\ast }$ satisfies $f(u_{\ast })=0$. Linearizing a spatial mode with $-\Delta \varphi =q\varphi$ givesThus reaction stability tests $J$, while spatial stability tests all boundary-compatible values of $q$. Unequal diffusion can create a Turing instability even when homogeneous perturbations decay.^[ref] The Brusselator and quadratic activator-inhibitor model give explicit two-species examples. Initial and boundary conditions are part of the system; the kinetic equations alone do not select its spatial modes.

A nonlinear diffusion equation has a transport flux depending nonlinearly on the evolving field or its gradient. A common scalar conservative form isFor a nonnegative field in the porous medium equation $u_t=\Delta (u^m)$, $m>1$, the effective diffusion coefficient is $D(u)=m{u}^{m-1}$. It vanishes at zero density, allowing finite propagation in porous-medium diffusion. In logarithmic diffusion, $D(u)=\kappa /u$ for $u>0$, so low densities diffuse more strongly instead.

Gradient-dependent diffusion also belongs here: the Perona-Malik equation uses the flux $c(|\nabla u|)\nabla u$. Its highest-order coefficients depend on the derivative of that flux, so positive $c$ alone does not ensure that the equation is of parabolic type. Nonlinearity can therefore change propagation, smoothing and well-posedness, rather than merely changing a constant spreading rate.

The constant-coefficient one-dimensional advection-diffusion equation isTranslation to coordinates moving at speed $a$ reduces it to the heat equation.

The heat equationmodels diffusion with constant diffusivity $D$.

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.

In ${\mathbb{P}}^4$, require the matrixto have rank at most one. Its three $2\times 2$ minors define a smooth projective surface. The zero matrix is not a projective point, so every point on this locus actually has rank one.

The four open charts $y_0\ne 0$, $y_1\ne 0$, $y_3\ne 0$ and $y_4\ne 0$ are each isomorphic to ${\mathbb{A}}^2$. Explicit two-parameter coordinates are, respectively,In each case one homogeneous coordinate is normalized and two remaining affine coordinates are eliminated using the minors, leaving two free parameters. If all four chart coordinates vanished, the minor $y_0{y}_4-y_2^2$ would also force $y_2=0$, impossible in projective space. Thus these smooth affine charts cover the entire surface, proving both smoothness and dimension two.

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.