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A path-connected region is multiply connected when some closed path cannot contract to a point within the region. An annulus and the complement of a solid torus are examples. The accessible region in the ideal magnetic Aharonov-Bohm effect has this obstruction. Such a region is not a simply connected space.

For orbital angular momentum $\ell =1$ and spin $s=1/2$, considerThe quantum number $m_j=m_{\ell }+m_s$ is conserved. The stretched states with $m_j=\pm 3/2$ have energies $K\pm 2a$. The remaining eigenvalues follow from two $2\times 2$ blocks with off-diagonal entries $\sqrt{2}K$:The subscripts denote $m_j=\pm 1/2$. For $K>0$, the zero-field multiplets are $j=3/2$ at $K$ and $j=1/2$ at $-2K$. Their weak-field Zeeman splitting is $g_{j}a{m}_j$, with Landé g-factors $4/3$ and $2/3$, respectively. When $a\gg |K|$, the Paschen-Back effect gives approximate product states $(m_{\ell },m_s)$ and energies $a(m_{\ell }+2m_s)+2K{m}_{\ell }{m}_s$. Mixing within a fixed $m_j$ causes an avoided crossing; different $m_j$ sectors can cross because the Hamiltonian does not couple them.

The particle flux of neutrinos through a specified surface; an incident-energy distribution may also be specified.

In diffusion-limited cluster aggregation, mobile clusters undergoing Brownian motion attach with probability close to one when they encounter another cluster. Irreversible first-contact sticking tends to trap particles in exposed outer branches, producing open aggregates. In the common three-dimensional isotropic cluster model, the mass fractal dimension is roughly $1.8$. This is different from depositing individual random walkers onto one fixed cluster. Rotational motion, anisotropic attraction and bond restructuring can change the morphology.

In reaction-limited cluster aggregation, encounters occur frequently compared with successful attachment: an activation or interaction barrier gives a small sticking probability. Repeated unsuccessful approaches allow clusters to sample more relative configurations before bonding, commonly producing denser aggregates than diffusion-limited cluster aggregation. Typical three-dimensional isotropic irreversible cluster models give mass fractal dimension around $2.1$. This structural contrast does not require previously formed bonds to break and restructure; it can arise from unsuccessful encounters alone. ^[ref]

Poisson's equation is $\Delta u=f$, where $\Delta$ is the Laplace operator and $f$ a prescribed source. It generalizes the Laplace equation, recovered when $f=0$. Boundary conditions are needed to determine a unique solution; Neumann boundary conditions can leave an additive constant.

For electrostatics in vacuum, the convention is $\Delta V=-\rho /{ϵ}_0$. For Newtonian gravity, $\Delta \Phi =4\pi G\rho$. The opposite signs reflect the specified force and potential conventions. A simple mathematical example is $u(x)=x^2/2$ on $[0,1]$, which solves $u^{\prime \prime }=1$ with endpoint values zero and $1/2$. On a periodic domain the source must have zero mean, by integrating the equation.

For a state of integer total angular momentum $J$, natural parity means $\pi =(-1{)}^J$: $0^{+},1^{-},2^{+},3^{-},\dots$. The other sign is called unnatural parity. The terminology is useful when a spin-zero positive-parity parent emits a spin-zero positive-parity particle: angular momentum conservation fixes relative orbital angular momentum $L=J$, and parity conservation then forces natural parity for the daughter.

For a prime number $p$ and integers $0\le k\le n$, the p-adic valuation of the binomial coefficient $\left(\genfrac{}{}{0ex}{}{n}{k}\right)$ equals the number of arithmetic carries in the base-$p$ addition of $k$ and $n-k$.^[ref] Indeed, subtract the two denominator factorial valuations in the Legendre formula:Each arithmetic carry accounts for one unit of this quotient by subadditivity of the base-p digit sum. In particular, the binomial coefficient is divisible by $p$ exactly when at least one carry occurs.

For $n=50,k=17,p=3$, the factorial counts giveThe matching base-p expansions are $17=(0122{)}_3$ and $33=(1020{)}_3$. Their units add to two; the next column gives $2+2=4=1+3$, sending one carry into the nines column. That column then gives $1+0+1=2$, and no further carry occurs. The result is $(1212{)}_3=50$. The base-p digit sums check the same result: $(5+3-6)/(3-1)=1$. Consequently $\left(\genfrac{}{}{0ex}{}{50}{17}\right)$ is divisible by three but not by nine.

Magnetite is the iron oxide $\mathrm{F}{\mathrm{e}}_3{\mathrm{O}}_4$, commonly described as containing both iron(II) and iron(III) in an inverse-spinel structure. Its magnetic order is ferrimagnetic.

The unequal magnetic contributions of its sublattices leave a net magnetization. This makes magnetite a useful connection between oxidation-state chemistry, crystal structure, and bulk magnetism.

Respiratory uncoupling allows proton return to bypass ATP synthase, dissipating the electrochemical gradient. Electron transport and oxygen consumption can continue while less ATP is recovered.^[ref]

In the thermodynamic example under chemiosmosis, reducing both the activity ratio to one and the electric potential difference to zero removes the driving contribution. The hypothetical ATP-forming conversion then costs $+45\mathrm{kJ}\mathrm{mo}{\mathrm{l}}^{-1}$. Continued donor oxidation would not restore coupling unless it rebuilt a usable electrochemical gradient. This distinction makes respiratory uncoupling useful conceptually: abundant oxygen consumption can coexist with poor recovery of ATP.

Chemiosmosis couples transmembrane ion movement down an electrochemical gradient to a chemical transformation. For mitochondrial ATP synthesis, protons return from the intermembrane space to the mitochondrial matrix through ATP synthase.^[ref]

An original thermodynamic example separates the electrical and chemical contributions. At fixed temperature $T$, use ideal dilute thermodynamic activities $a$ and the electric potential $\psi$ on each side. The molar electrochemical potential difference for inward transfer of a singly charged proton isHere $R$ is the molar gas constant, $F$ the Faraday constant, and $\ln$ the natural logarithm. The sign convention is destination minus origin: a negative difference makes inward transfer favourable. Both lower proton activity and a more negative electric potential inside favour inward movement.

Stipulate $T=300\mathrm{K}$, $a_{\mathrm{matrix}}/a_{\mathrm{outside}}=0.1$, ${\psi }_{\mathrm{matrix}}-{\psi }_{\mathrm{outside}}=-0.12\mathrm{V}$, $R=8.314\mathrm{J}\mathrm{mo}{\mathrm{l}}^{-1}{\mathrm{K}}^{-1}$ and $F=96485\mathrm{C}\mathrm{mo}{\mathrm{l}}^{-1}$. These illustrative inputs give approximately $-5.743$ and $-11.578\mathrm{kJ}\mathrm{mo}{\mathrm{l}}^{-1}$ for the chemical and electrical terms, hence $-17.321\mathrm{kJ}\mathrm{mo}{\mathrm{l}}^{-1}$ per transferred proton. For a hypothetical device coupling three inward protons per ATP to an ATP-formation cost of $45\mathrm{kJ}\mathrm{mo}{\mathrm{l}}^{-1}$,The sign follows by adding changes in the state function Gibbs free energy. It establishes thermodynamic feasibility for this chosen model, not its reaction rate. Three is a stipulated coupling count, not a universal ATP synthase stoichiometry; transport costs and leaks can also consume the available free energy.