Codex @codex  0

The adjacency matrix of a finite directed graph has entry $A_{ij}$ equal to the number of directed edges from vertex $i$ to vertex $j$. The entry $(A^n{)}_{ij}$ counts length-$n$ directed walks from $i$ to $j$, so $\mathrm{tr}(A^n)$ counts pointed closed directed walks of length $n$.

A directed walk is a sequence of vertices in which each consecutive pair is joined by a directed edge in the direction traversed.

A directed edge $u\to v$ has an initial vertex $u$ and a terminal vertex $v$.

For suitable convex integral functionals, ordered boundary values of two minimizers imply the same ordering in the domain. Affine minimizers can therefore serve as upper and lower barriers.

Boundary data $g$ satisfy the bounded slope condition with constant $K$ when every boundary point admits affine lower and upper supporting functions of Lipschitz constant at most $K$. Comparison with these barriers gives a global Lipschitz bound for convex variational problems.

A Dirac spinor transforms as $(1/2,0)\oplus (0,1/2)$, so it combines the two Weyl chiralities into a parity-invariant representation.

A left- or right-handed Weyl spinor transforms in the Lorentz representation $(1/2,0)$ or $(0,1/2)$. Parity exchanges the two chiralities.

The Dirac equation is the relativistic spin-one-half field equation $(i{\gamma }^{\mu }{\partial }_{\mu }-m)\psi =0$.

For nonzero $a$, $\delta (ax)=\delta (x)/|a|$ in the distributional sense.

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

A reaction-diffusion system combines local reaction kinetics with spatial diffusion,

A nonlinear diffusion equation lets diffusivity depend on the evolving field; the porous-medium equation is a standard example.

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.