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For real $0<k<1$, the period lattice of $\mathrm{sn}(u,k)$ is generated byThe periods arise by composing the monodromy reflections of its inverse elliptic integral of the first kind.

If ${\phi }_n(x,t)\sim c_n(t)e^{-{\kappa }_{n}x}$ as $x\to +\infty$, the time equation $Q=0$ givesand hence $c_n(t)=c_n(0)e^{4{\kappa }_n^{3}t}$.

The translation length of an isometry $g$ of a metric space is $\ell (g)={\inf }_{x}d(x,gx)$. For a hyperbolic tree isometry, it is attained exactly on the axis of a tree isometry.

A hyperbolic tree isometry has positive translation length and preserves a unique bi-infinite geodesic.

An elliptic tree isometry fixes a vertex. Its fixed-point set is a nonempty convex subtree.

Every simple module over a commutative ring $R$ is isomorphic to $R/\mathfrak{m}$ for a maximal ideal $\mathfrak{m}$; its annihilator is therefore maximal.

A minimal deterministic finite automaton is the result of DFA minimization. An irreducible deterministic automaton has the fewest states among deterministic automata for its language and is unique up to isomorphism. Its states are the Myhill--Nerode classes.

For $\omega =e^{2\pi i/n}$ and $1\le j<n/2$, defineThese representations are irreducible and pairwise nonisomorphic. They give all two-dimensional irreducibles: $(n-1)/2$ of them for odd $n$, and $n/2-1$ for even $n$.

A nonlinear irreducible character has degree greater than one.

A linear character is the character of a one-dimensional representation and therefore has degree one.

A radiative-convective boundary separates a region transporting energy mainly by radiation from a region where convection carries an appreciable fraction. In a homogeneous, optically thick fluid, compare the radiative temperature gradient required to carry the flux with the adiabatic temperature gradient. When the required gradient is steeper, a displaced parcel can remain warmer and more buoyant than its surroundings, making the radiative state unstable under the Schwarzschild criterion.

A cooling giant planet commonly has a convective deep interior beneath a radiative layer. Stellar heating can push their boundary deeper and slow the interior's heat loss. Multiple convective regions are possible, and composition gradients modify the stability test; the boundary is therefore not a universal fixed pressure or an assurance that everything below it convects.

A semi-grey irradiated atmosphere uses separate averaged opacities for incoming stellar light and outgoing planetary thermal radiation. Their ratio $\gamma ={\kappa }_{\mathrm{vis}}/{\kappa }_{\mathrm{IR}}$ compares where the two energy streams interact with matter. Large visible opacity relative to infrared opacity deposits stellar heat at a relatively small thermal optical depth; sufficiently strong high-altitude heating can produce an atmospheric thermal inversion.^[ref]

The model connects irradiation temperature, internal effective temperature of a planet and heating depth to an atmospheric pressure-temperature profile. It is useful for interpreting broad thermal structure, but averaging each band suppresses spectral windows and molecular lines that matter in detailed atmospheric retrieval.

The inverse limit is the module of compatible families

Transposing a nonzero invariant bilinear form on a finite-dimensional irreducible complex representation gives a scalar multiple $B^T=\lambda B$. Transposing twice yields ${\lambda }^2=1$, so the form is symmetric or alternating.

A Kripke model is a poset of worlds with a persistent valuation of atoms. Conjunction and disjunction are forced pointwise, while $w⊩\alpha \to \beta$ exactly when every $v\ge w$ that forces $\alpha$ also forces $\beta$.

The forcing relation is the intuitionistic instance of Kripke semantics; persistence and the order on worlds determine its implication clause.

The Curry-Howard correspondence identifies propositions with types and proofs with typed programs; implication corresponds to a function type, conjunction to a product type, and falsity to the empty type.

A Heyting algebra is a bounded distributive lattice with an implication $a\Rightarrow b$ characterized by $x\le (a\Rightarrow b)$ exactly when $x\wedge a\le b$.

A lattice is a partially ordered set in which every pair has a greatest lower bound, called its meet, and a least upper bound, called its join.

Given finitely many intervals $J_1,\dots ,J_r$, their directed covering graph has an directed edge $J_i\to J_j$ whenever $F(J_i)\supseteq J_j$.

A cosmic ray is a high-energy charged particle propagating through space. Galactic cosmic-ray electrons emit synchrotron and inverse-Compton radiation, while cosmic-ray nuclei produce gamma rays through collisions with gas.

In modern terminology, a primary cosmic ray arriving at Earth's atmosphere is usually a proton or another atomic nucleus, with an electron component as well. Atmospheric collisions generate secondary cosmic rays and air showers, including muons, electrons and photons. A gamma ray can also initiate an air shower, although it is electrically neutral. Consequently, detecting atmospheric ionization does not establish that an incoming primary cosmic ray passed through the detector. Magnetic fields bend charged particles, while gamma rays provide a more direct directional connection to their sources.^[ref]

The Hess cosmic-ray balloon experiment asks how atmospheric shielding and separation from the ground affect ionization. Its power comes from changing the environment around an enclosed detector instead of identifying each incoming particle.