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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.

If bounded $S:X\to Y$ is injective and $\mathrm{im}T\subseteq \mathrm{im}S$, then $R=S^{-1}T$ has closed graph: $x_n\to x$ and $R{x}_n\to z$ imply $Sz=Tx=S(Rx)$ and hence $z=Rx$.

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.

An isolated singularity is removable when the function extends holomorphically across it, equivalently when its Laurent series has no negative-power terms.

A connected covering is normal, or regular, when its deck transformations act transitively on each fibre. Under the subgroup classification, this is equivalent to the associated subgroup of the fundamental group being normal.

For a connected regular covering corresponding to $H\le {\pi }_1(X,x_0)$, the subgroup $H$ is normal andThe quotient acts on a fibre by endpoints of lifted loops, and regularity makes this action transitive with kernel $H$.

For a presentation complex, the covering associated with a subgroup has one vertex for each coset. A generator gives directed edges according to its coset action, and every relator has one lifted 2-cell beginning at each vertex.

For a connected covering corresponding to $H\le {\pi }_1(X)$, every fibre has cardinality $[{\pi }_1(X):H]$.

A support vector machine chooses a separating decision function by trading a wide geometric margin against hinge-loss violations.