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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.

The radiative-convective boundary is where radiative transport first requires a temperature gradient steeper than the adiabatic gradient, causing convection below.

A semi-grey atmosphere uses separate mean visible and infrared opacities. Their ratio $\gamma ={\kappa }_{\mathrm{vis}}/{\kappa }_{\mathrm{IR}}$ controls where stellar energy is deposited.

The inverse limit is the module of compatible families

Transposing a nonzero invariant bilinear form on an 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.

The neutral-hydrogen 21-centimeter line is the ground-state hyperfine transition of atomic hydrogen. Its brightness and Doppler shift map atomic-gas column density and line-of-sight velocity.

Interstellar dust absorbs and scatters short-wavelength starlight and reradiates the absorbed energy thermally in the infrared.

A molecular cloud is a cold dense region of the interstellar medium in which hydrogen is predominantly molecular and stars can form.

For $0<p<1/2$ and $\epsilon >0$, every intersecting family is, up to ${\mu }_p$-measure $\epsilon$, contained in the upward closure of a set family generated by an intersecting family on a bounded set of coordinates.

The upward closure of $\mathcal{B}$ is $\overline{\mathcal{B}}=\{A:\text{some }B\in \mathcal{B}\text{ satisfies }B\subseteq A\}$.

Two set families $\mathcal{A}$ and $\mathcal{B}$ are cross-intersecting when $A\cap B\ne \varnothing$ for every $A\in \mathcal{A}$ and $B\in \mathcal{B}$.