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Expanding an angular distribution in multipoles converts the Boltzmann equation into a hierarchy in which free streaming couples neighboring multipoles and collisions damp or source selected moments.

The first Brillouin zone is a fundamental cell in reciprocal space. For a one-dimensional lattice of spacing $a$, one conventional choice is

The delta-comb Kronig-Penney potential is

For a potential of period $a$, the Floquet matrix $M(E)$ maps Cauchy data through one cell:For a real Schrodinger equation, $\det M=1$.

The nearly-free electron model treats a weak periodic potential by degenerate perturbation theory. At a Bragg crossing, a Fourier coefficient $V_n$ couples the two degenerate plane waves and opens an energy gap of width $2|V_n|$.

An energy band is a continuous branch $E_j(k)$ of the dispersion relation of a periodic quantum system.

A periodic potential satisfies $V(x+a)=V(x)$ for some lattice period $a$. Its Fourier coefficients couple plane waves whose wavevectors differ by reciprocal-lattice vectors.

The Kerr black hole is the asymptotically flat stationary axisymmetric vacuum black hole with mass $M$ and angular momentum $J=Ma$.

A Kerr black hole is the Kerr metric with mass $M>0$ and spin parameter satisfying $|a|\le M$ in geometrized units; its outer horizon is $r_{+}=M+\sqrt{M^2-a^2}$.

Black-hole thermodynamics relates horizon geometry to thermodynamic laws, assigning temperature and entropy to a stationary black hole.

A Penrose diagram conformally compactifies a spacetime while preserving causal directions, placing null infinity, timelike infinity, spacelike infinity, horizons, and singularities at finite coordinate locations.

An event horizon is the null boundary of the causal past of future null infinity. Its global definition makes it respond teleologically to matter that falls in later.

If coaxial loops of radii $R$ and $2R$, separated by $D$, carry currents $I$ and $2I$ in opposite senses, their on-axis fields cancel between the loops at distance $D/3$ from the smaller loop.

A circular loop of radius $R$ carrying electric current $I$ has, on its axis,where the sign of $\hat{z}$ is fixed by the right-hand rule. At its centre, $B(0)={\mu }_{0}I/(2R)$.

A pinned filament of length $L$ and bending modulus $A$ first buckles under compressive force $F_c={\pi }^{2}A/L^2$.

The worm-like chain models an inextensible filament with bending energy $A\int {\kappa }^{2}ds/2$ and thermal tangent fluctuations.

Resistive-force theory approximates the local viscous force on a slender filament by separate drag coefficients multiplying its velocity components parallel and perpendicular to the tangent.

Cytoplasmic streaming is directed cytoplasmic flow within a cell, often driven by molecular motors acting along the boundary and used to enhance intracellular transport.

Microcirculation is blood flow through arterioles, capillaries, and venules, where vessel dimensions are comparable with cellular length scales.

For a nonnegative integer $n$ and commuting quantities $x,y$,In expanding the $n$ factors, obtain $x^k{y}^{n-k}$ by selecting the $k$ positions that contribute $x$. There are $\left(\genfrac{}{}{0ex}{}{n}{k}\right)$ such choices. For example,The coefficients are row five of Pascal's triangle, in reverse order under the displayed $x^k{y}^{n-k}$ convention; the row is symmetric. Setting $x=y=1$ gives $1+5+10+10+5+1=32$, the number of subsets of a five-element set. Each member has two inclusion choices, agreeing with the rule of product and the cardinality of the power set.

For nonnegative integers with $n_1+\cdots +n_r=n$, the multinomial coefficientcounts arrangements of a multiset containing $n_j$ copies of symbol $j$.