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If local compactness makes every subsequential local limit agree with the global weak limit and the $L^2$ mass is uniformly tight, weak convergence upgrades to strong convergence.

For fixed $0<\theta <\pi$,

For a contour enclosing $t$,

For the Lefschetz operator of a Kähler manifold and its adjoint, the Kähler identities include

For the squared distance $r^2(x)=|x{|}^2$, a minimal $n$-submanifold satisfies ${\Delta }_M{r}^2=2n$. On a compact boundaryless manifold, $r^2$ attains a maximum where the Laplacian is nonpositive, a contradiction for $n>0$.

For an elliptic curve over ${\mathbb{F}}_q$, the Frobenius isogeny sends $(x,y)$ to $(x^q,y^q)$. If $a=q+1-\#E({\mathbb{F}}_q)$, then

For $\varphi \in \mathrm{End}(E)$, its trace is the integer characterized byEquivalently, $\mathrm{tr}(\varphi )=1+\mathrm{deg}\varphi -\mathrm{deg}(1-\varphi )$.

The dual of an isogeny $\varphi :E\to E^{\prime }$ of degree $n$ is the unique isogeny $\hat{\varphi }:E^{\prime }\to E$ satisfying

For an isogeny $\varphi :E\to E^{\prime }$, its kernel is denoted $E[\varphi ]$. In characteristic zero its cardinality equals $\mathrm{deg}\varphi$.

Every orbit of an irrational rotation of the circle visits a half-open interval $I$ with limiting frequency equal to its length. One proof sandwiches the orbit of an arbitrary point between inner and outer intervals along a nearby orbit for which the Birkhoff ergodic theorem holds.

The total influence isThis Fourier-Walsh transform identity uses $\{-1,1\}$-valued $f$ and the flip-probability convention for influence of a variable. For $\{0,1\}$-valued $f$, the right-hand side is multiplied by four. Indeed Parseval's identity applied to the discrete derivative of a Boolean function gives $\Vert D_{i}f{\Vert }_2^2={\sum }_{S\ni i}\hat{f}(S{)}^2$; sum over coordinates and apply the chosen normalization.

Implicit differentiation differentiates an identity $F(x,y(x))=0$ along a differentiable branch and uses the chain rule to solve for its derivative:The partial derivatives $F_x,F_y$ treat $x,y$ as independent coordinates before evaluating on the branch. The implicit function theorem justifies the branch's local existence and regularity under its hypotheses; differentiating alone does not establish either. For the unit circle at $(3/5,4/5)$, this calculation yields the tangent line $y-4/5=-(3/4)(x-3/5)$.

For an abelian variety,is the subgroup of translation-invariant line bundles.

The dual of the tangent-normal sequence is the short exact sequence of holomorphic vector bundles

A Dolbeault partial connection obeys ${\bar{\partial }}_E(fs)=\bar{\partial }f\otimes s+f{\bar{\partial }}_{E}s$. Its square is a tensorial $(0,2)$-form with values in $\mathrm{End}E$; it vanishes for the canonical partial connection of a holomorphic bundle.

A Rodrigues formula constructs an orthogonal polynomial by differentiating a weight multiplied by a power. The Rodrigues formula for Legendre polynomials uses a derivative of $(x^2-1{)}^l$. For Generalized Laguerre polynomials,

The local parameter $z=-x/y$ at the identity of a Weierstrass curve expresses addition as a formal group law. Over a local field it describes points reducing to the identity.

Over a characteristic-zero field, a one-dimensional commutative formal group law has a unique strict isomorphism to the additive formal group, called its formal logarithm. It satisfies ${\log }_F(F(X,Y))={\log }_F(X)+{\log }_F(Y)$.

If $N$ is Poisson with mean $\lambda$, then $\mathbb{E}[(N{)}_r]={\lambda }^r$; in particular, $\mathbb{E}N=\lambda$ and $\mathbb{E}[N(N-1)]={\lambda }^2$.

Let $\nu :\tilde{C}\to C$ be the normalization of a nodal curve. Its rational inverse $C⇢\tilde{C}$ cannot extend over the node: the two branches give two distinct points of $\tilde{C}$, whereas a morphism can assign only one image to the node.