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In a finite chain, a Communicating class that is not closed is transient: after it is left, it need not be visited again and is eventually left almost surely.

A Communicating class is closed when no transition from it can reach a state outside it.

Two states of a Markov chain communicate when each can be reached from the other with positive Probability in finitely many steps.

The necessity direction in Rado's theorem colors an integer using initial data from a p-adic valuation. Applying partition regularity and grouping the coordinates of a monochromatic solution by valuation yields the blocks in the columns property; reduction modulo successively higher powers of $p$ supplies the required linear dependences.

Rado's theorem states that a rational matrix is partition regular if and only if it has the columns property.

A string operator creates excitations at its endpoints. Multiplying by a successive segment moves an endpoint because the shared intermediate excitation is toggled twice.

The catastrophic disruption threshold is the impact energy per unit target mass required both to shatter a body and to disperse enough fragments against self-gravity.

A differential power-law size distribution has $n(D)dD=k{D}^{-\alpha }dD$ objects with sizes between $D$ and $D+dD$.

A local state classifier of $\mathcal{C}$ is a colimit of the inclusion of the wide subcategory containing all objects and only monomorphisms into $\mathcal{C}$.

A category is filtered when every finite diagram in it admits a cocone. Equivalently, it is nonempty, every pair of objects maps to a common object, and every pair of parallel arrows becomes equal after postcomposition.

Limits of shape $\mathcal{I}$ commute with colimits of shape $\mathcal{J}$ when, for every $D:\mathcal{I}\times \mathcal{J}\to \mathcal{C}$, the canonical comparisonis an isomorphism whenever both sides exist.

For a closed subspace $F$ of a Hilbert space,The closest point $P_{F}x$ gives the first component, and differentiating $\Vert x-P_{F}x-tu{\Vert }^2$ at zero shows that the remainder is orthogonal to every $u\in F$.

Let $V$ be a real Banach space and $T:V\to V^{\ast }$ be linear with $T_v(v)\ge 0$ for every $v$. If $v_n\to 0$ and $T_{v_n}\to f$ in $V^{\ast }$, positivity of $T_{v_n+tw}(v_n+tw)$ and passage to the limit givefor every real $t$ and every $w$. Taking small $t$ of either sign forces $f(w)=0$. Thus the graph of $T$ is closed, and the closed graph theorem makes $T$ continuous.

On incomplete $c_{00}$ with the ${\ell }^2$ norm, let $S(x_n)=(x_n/n)$ and let $T$ be inclusion into ${\ell }^2$. Their images agree, but $S^{-1}T(e_n)=n{e}_n$ is unbounded.

Let $E,X,F$ be Banach spaces, let $S:E\to F$ and $T:X\to F$ be bounded linear operators, and assume $S$ is injective and $\mathrm{im}T\subseteq \mathrm{im}S$. Then $R=S^{-1}T:X\to E$ is a well-defined linear operator with closed graph. If $x_n\to x$ and $R{x}_n\to z$, continuity of both $S$ and $T$ givesInjectivity gives $z=Rx$. The closed graph theorem therefore makes $R$ bounded.^[ref]

It is not necessary for $S^{-1}$ to be bounded on its whole range. On ${\ell }^2$, take $S(x_n)=(x_n/n)$ and $T(x_n)=(x_n/n^2)$. Then $S^{-1}$ is unbounded, but $R(x_n)=(x_n/n)$ is bounded. The boundedness assumption on $T$ is essential to the closed-graph argument; it cannot be omitted from the hypotheses.

For a closed graph, the projection $\Gamma (T)\to X$ is a bounded bijection between Banach spaces. Its bounded inverse followed by projection to $Y$ is $T$.

A differential form is exact when it is the exterior derivative of a form of one lower degree. Since $d^2=0$, every exact form is closed.

On the multiplicity-one summand $V_{2n-2j}\subset V_n\otimes V_n$, interchange of tensor factors acts by $(-1{)}^j$. A highest-weight vector exhibiting the sign is

An isolated singularity is essential when its Laurent series has infinitely many nonzero negative-power terms.

A function has a pole of order $k$ at $a$ when $(z-a{)}^{k}f(z)$ extends holomorphically and is nonzero at $a$.

A pole is the negative-order case in the zeros and poles classification of a meromorphic function.