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A deterministic automaton is irreducible when every state is accessible and every pair of distinct states is distinguishable.

Two states are indistinguishable when every continuation is accepted from both or rejected from both. Distinguishable states admit at least one suffix with different acceptance outcomes.

A state is accessible when some input word takes the initial state to it. Removing inaccessible states preserves the accepted language.

For a transition function $\delta :Q\times \Sigma \to Q$, defineThe accepted language is $\{w:\hat{\delta }(q_0,w)\in F\}$.

The variety of $m\times n$ matrices of rank at most one has dimension $m+n-1$. For $m,n\ge 2$, its only singular point is the zero matrix.

The projective locus wherehas rank one is cut out by its three $2\times 2$ minors. It is a smooth surface: on each of the charts $y_0\ne 0$, $y_1\ne 0$, $y_3\ne 0$, and $y_4\ne 0$, the equations eliminate three coordinates and leave two free affine coordinates; these charts cover the locus.

For a pressureless fluid withthe first-order density contrast $\delta =\delta \rho /\bar{\rho }$ obeys

The linear growth factor multiplies the growing mode of a linear cosmological density perturbation, so $\delta (\mathbf{x},t)=D(t)\delta (\mathbf{x},t_0)/D(t_0)$ when scale-independent growth applies.

A graph is locally finite when every vertex has finite degree.

The maximum degree $\Delta (G)$ is the greatest degree of a vertex of $G$.

It is the maximum of the degree of a vertex over all vertices.

The minimum degree $\delta (G)$ is the least degree of a vertex of $G$.

The minimum degree is $\delta (G)={\min }_{v\in V(G)}\mathrm{deg}(v)$, the least degree of a vertex in the graph.

A universal vertex is adjacent to every other vertex of its graph.

In a finite graph with at least one edge, the degree-biased vertex distribution assigns probability proportional to the degree of a vertex. It is the distribution of an endpoint on a specified side of a uniformly random edge.

Projective degree belongs to an embedding rather than to the abstract variety. A line and a smooth conic in ${\mathbb{P}}^2$ are both abstractly ${\mathbb{P}}^1$ but have degrees one and two.

Adjoining projective coordinates that vanish identically does not change the homogeneous coordinate ring or Hilbert polynomial. A linear embedding of projective space therefore preserves the degree of an embedded variety.

For a general linear form $\ell$ not vanishing identically on a projective curve, the exact sequenceshows that the hyperplane-section length is $P_C(m)-P_C(m-1)=\mathrm{deg}C$.

The Hilbert function of a finitely generated graded algebra agrees in all sufficiently large degrees with a unique polynomial. For a projective variety of dimension $r$, its leading term is $\mathrm{deg}(V)m^r/r!$.

A nonzero polynomial of degree $d$ over a field has at most $d$ distinct roots. Indeed, each root supplies a linear factor by polynomial division, and induction on $d$ gives the bound.

If $S$ is compact and $V$ is a smooth vector field, thenis smooth and well-defined by the Hopf-Rinow theorem. The maps ${\varphi }_t(p)={\exp }_p(tV(p))$ form a smooth homotopy from the identity to $\varphi$, so ${\mathrm{deg}}_2(\varphi )=1$.

The constructible hierarchy is defined byfor limit ordinals $\lambda$. Its union is the constructible universe $L$.