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Contracting two Levi-Civita symbols gives

For three-dimensional vectors,The corresponding identity in ${\mathbb{R}}^n$ is

Contracting one index gives

The critical threshold is the boundary between levels at which an unbounded superlevel component can occur and levels at which every superlevel component is almost surely finite.

The polynomial solution of polynomial degree $l$ normalized by $P_l(1)=1$ is the Legendre polynomial $P_l$.

The generalized Laguerre polynomial $L_n^{(\alpha )}$ is orthogonal on $[0,\infty )$ for the weight $x^{\alpha }{e}^{-x}$ when $\alpha >-1$.

For a symmetric bandwidth-$h$ averaging kernel and a density with bounded first derivative, the interior pointwise bias is at most a constant times $h\Vert f^{\prime }{\Vert }_{\infty }$. Near a support boundary, an uncorrected symmetric kernel can instead have nonvanishing bias.

For one observation from the logistic distribution, the size-$\alpha$ test of $H_0:\theta \le 0$ against $H_1:\theta >0$ rejects when

For a compact Kähler manifold,and every class has a unique harmonic representative of each type.

This is the compact Kähler specialization of Hodge theory.

The Fubini-Study form is the standard Kähler form on complex projective space. With the integral normalization, $[{\omega }_{FS}/(2\pi )]=c_1(\mathcal{O}(1))$.

The two-form is associated with the Fubini-Study metric.

The Lefschetz operator of a Kähler manifold is exterior multiplication by its Kähler form,Its formal adjoint is denoted by $\Lambda =L^{\ast }$ and lowers the bidegree of a differential form by $(1,1)$.

This wedge operator participates in the Kähler identities; the theorem on integral $(1,1)$ classes is a different result.

For a representative ${\dot{x}}_i\in N$, the Hecke parameter isThe Iwahori-Hecke algebra of a BN-pair is obtained from the generic algebra by $a_i\mapsto q_i\cdot 1_k$.

An equation can define a function without giving an explicit formula for it. The implicit function theorem tells us when the constraint $F(x,y)=0$ determines $y$ uniquely from nearby $x$, and makes the resulting function continuously differentiable. It is useful when solving an equation explicitly is cumbersome or impossible: the partial derivatives already reveal whether local recovery is possible.

Let $F$ be a continuously differentiable map from an open subset of ${\mathbb{R}}^m\times {\mathbb{R}}^n$ to ${\mathbb{R}}^n$. Suppose $F(x_0,y_0)=0$ and the partial derivative $D_{y}F(x_0,y_0)$ is an invertible linear map. There are open neighbourhoods $U$ of $x_0$ and $V$ of $y_0$ and a unique continuously differentiable function $g:U\to V$ such that, for $(x,y)\in U\times V$,Thus the nearby level set is a graph of a function. The derivative of this graph isThe matrix inverse acts on the left; the order matters in several dimensions.

For example, take the unit circle, written as the level set $F(x,y)=x^2+y^2-1=0$. At $(x_0,y_0)=(3/5,4/5)$, $F_y=2y_0=8/5\ne 0$, so the local upper branch is $y=g(x)=\sqrt{1-x^2}$. Implicit differentiation givesThis tangent line slope is available directly from the constraint. The negative square root branch is elsewhere on the same circle; it does not destroy uniqueness near the selected upper point.

At $(1,0)$, however, $F_y=0$ and the circle cannot be represented locally as one function $y=g(x)$: just to the left of $x=1$ there are two nearby values of $y$. The circle is still smooth there. Since $F_x=2\ne 0$, use the other coordinate instead: $x=h(y)=\sqrt{1-y^2}$, with $h^{\prime }(0)=0$. A failed coordinate choice is different from a geometric singularity.

The derivative condition is sufficient, not necessary for every possible implicit graph. For $F(x,y)=y^3$, the entire zero set is the smooth graph $y=0$, even though $F_y$ vanishes there. In this case the defining equation itself has a degenerate derivative; the theorem's test gives no conclusion.

The connection to the inverse function theorem is particularly elegant. Augment the constraint to the map $H(x,y)=(x,F(x,y))$, whose Jacobian matrix isIts determinant is $\det D_{y}F$. When this is nonzero, locally invert $H$ and set its second output to zero. The recovered second input is $g(x)$. In this way an implicit constraint becomes an ordinary inverse function problem. ^[ref]

A local diffeomorphism is a differentiable map that restricts near every point to a diffeomorphism onto an open subset of its target. The local inverse functions need not combine into a single global inverse function: different parts of the domain of a function can map onto the same output region.

For continuously differentiable maps between open subsets of ${\mathbb{R}}^n$, being a local diffeomorphism is equivalent to having an invertible derivative at every point. The inverse function theorem proves one direction. For the other, the chain rule for each local inverse function gives two inverse linear maps. The periodic angle example under the inverse function theorem has this property everywhere, even though it has infinitely many inputs for every point of the punctured plane. The noninjective map with constant Jacobian determinant gives another useful distinction: even a constant nonzero Jacobian determinant need not give global injectivity.

Every local diffeomorphism is an open map. To see this, take an open subset of the source and intersect it with the neighbourhoods where the map is a diffeomorphism. Each local image is open, and their union is the full image. Thus nearby outputs remain available locally, despite the possible global ambiguity.

A recurrent jump chain with infinite invariant measure $\mu$ can yield a positive-recurrent continuous-time chain when ${\sum }_i{\mu }_i/q_i<\infty$.

Even when the jump chain has invariant probability $\mu$, the continuous-time invariant measure ${\mu }_i/q_i$ may have infinite mass if holding rates become too small, producing infinite mean return time.

Every continuous injective function from a real interval to the real line is strictly monotone. A continuous bijection $\mathbb{R}\to \mathbb{R}$ therefore has a continuous inverse.

If a continuous $g:[0,na]\to \mathbb{R}$ satisfies $g(na)-g(0)=na$, then the incrementssatisfy ${\sum }_{k=0}^{n-1}h(ka)=0$. Continuity forces $h$ to vanish somewhere on $[0,(n-1)a]$.

If a continuous real-valued function on a real or complex vector space has opposite signs at two points, restricting it to the line segment between them and applying the intermediate value theorem gives a zero.

The wild inertia group is the first ramification group $G_1$. It is the unique Sylow subgroup of the inertia group for the residue characteristic.