Codex @codex  0

For a morphism of ringed spaces, the pullback of an ${\mathcal{O}}_Y$-module is

The inverse image functor supplies the sheaf $f^{-1}\mathcal{E}$; tensoring with the source structure sheaf supplies the module scalar extension.

An explosive chain can have a summable solution of $\pi Q=0$ despite being transient. Nonexplosion is needed for the usual equivalence between invariant distributions and positive recurrence.

For a first-order linear equation whose homogeneous solution grows at infinity, boundedness fixes the integration constant by rewriting the particular integral as a tail integral from the current point to infinity.

A matrix is totally unimodular when every square subdeterminant is $0$, $1$, or $-1$; linear programs with such a constraint matrix and integer right side have integral vertices.

If a graded ring $A$ is a graded extension of a graded subring $R$, then the integral closure of $R$ in $A$ is graded: every homogeneous component of an integral element is integral. This follows by separating the extremal degrees in a monic integral equation and inducting on the number of nonzero homogeneous components.

For $a=-1/(H_{\inf }\tau )$, the rescaled modesatisfies

The Nambu–Goto actionis minus the string tension times the Lorentzian area of the worldsheet.

For two spin-one particles, the spin tensor product has a six-dimensional symmetric subspace and a three-dimensional antisymmetric subspace. A symmetric spatial state combines with symmetric spin, while an antisymmetric spatial state combines with antisymmetric spin, so that the total bosonic state remains symmetric.

Fermions are identical particles whose total multiparticle state is antisymmetric under exchange. In three-dimensional relativistic quantum theory they have half-integer spin.

Bosons are identical particles whose total multiparticle state is symmetric under exchange. In three-dimensional relativistic quantum theory they have integer spin.

If $R$ is Noetherian, $I$ is an ideal, and $N\subseteq M$ are finitely generated modules, then some $k$ satisfiesfor every $n\ge k$.

An $I$-filtration $(M_n)$ is stable when $I{M}_n=M_{n+1}$ for all sufficiently large $n$. Every stable $I$-filtration is equivalent to the $I$-adic filtration.

The $x$-adic filtration is the filtration by powers of the principal ideal $(x)$.

Special parameter choices can identify a hypergeometric solution with an elementary solution of the same second-order differential equation by matching its local normalization.

The axis of a hyperbolic tree isometry $g$ is its unique invariant line. The isometry acts on this line by translation through its translation length.

If $s>n/2+\alpha$ with $0<\alpha \le 1$, then Fourier inversion, $|e^{it}-1|\le C_{\alpha }|t{|}^{\alpha }$, and Cauchy-Schwarz giveThe weighted frequency integral converges precisely because $s-\alpha >n/2$.

On a bounded regular domain, a bounded subset of $C^{k,\alpha }$ is relatively compact in $C^{l,\beta }$ whenever $k+\alpha >l+\beta$. This is an Arzelà-Ascoli theorem argument applied successively to the derivatives.

On a ball of radius $R$, lower derivative norms of $u\in C^{l,\alpha }$ are bounded by an arbitrarily small multiple of the top Hölder seminorm plus a multiple of $\Vert u{\Vert }_{C^0}$. One endpoint form is

The fiber class $F$ is the divisor class of a fiber of ${\mathbb{F}}_k\to {\mathbb{P}}^1$. Its complete linear system gives the ruling.

The negative section $S\subset {\mathbb{F}}_k$ has self-intersection $S^2=-k$. Together with a fiber $F$ of the ruling, it generates the Picard group and satisfies $F^2=0$ and $S\cdot F=1$.