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The algebraic multiplicity of an eigenvalue is its multiplicity as a root of the characteristic polynomial. In Jordan form it is the sum of the sizes of all blocks carrying that eigenvalue.

If $F$ and $G$ have no explicit time dependence and satisfy $\{F,H\}=\{G,H\}=0$, the Jacobi identity gives $\{\{F,G\},H\}=0$. Their Poisson bracket is therefore also conserved.

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.

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.