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For a connected reductive algebraic group and a Weyl group element $w$, this locally closed subset of the flag variety consists of flags in relative position $w$ to their Frobenius image. For rational $B$ it is ${\mathcal{L}}_F^{-1}(B\dot{w}B)/B$, using the twisted right action identity of the Lang map. Its dimension is the length of $w$.

For a rational maximal torus $T$, choose a Borel subgroup containing it with unipotent radical $U$. The alternating compactly supported étale cohomology of ${\mathcal{L}}_F^{-1}(U)$, in the appropriate $T^F$-isotypic component, defines the virtual character of $G^F$ $R_T^G(\theta )$. It is independent of the chosen Borel. When the Borel subgroup is rational, this is ordinary induction from its rational points after inflation of a group representation from $T^F$.

For prime $p$, the two cusps of ${\Gamma }_0(p)$ have widths one and $p$. At infinity the modular discriminant product gives the first pair of orders. At zero use $S\tau =-1/\tau$ and the parameter $q_0=e^{2\pi i\tau /p}$. The weight-twelve transforms are $\Delta {|}_{12}S=\Delta (\tau )$ and $\Delta (p\tau ){|}_{12}S=p^{-12}\Delta (\tau /p)$, giving orders $p$ and one in $q_0$.

Two factorial contrasts are aliased if their columns are proportional on the selected fractional factorial design. A word $G$ that has fixed sign $s_G$ gives ${\chi }_{AG}=s_G{\chi }_A$. Conversely, averaging a character over the fraction vanishes unless its word belongs to the defining contrast subgroup, so every alias class is exactly a subgroup coset. In a half fraction each class has two distinct words; their separate coefficients cannot be identified without assumptions or additional runs.

Restriction of differential forms to the finitely many open and closed components gives an isomorphism of cochain complexes ${\Omega }^{\ast }(⨆M_j)\cong {⨁}_j{\Omega }^{\ast }(M_j)$. Piecewise forms glue because there are no overlaps, and the exterior derivative acts componentwise. Taking the quotient of closed differential forms by exact differential forms proves the displayed de Rham cohomology isomorphism. For an infinite disjoint union, forms and cohomology give products instead.

There are no exact forms in degree zero. A smooth function has zero exterior derivative precisely when it is locally constant, so $H_{\mathrm{dR}}^0(M)$ is the space of locally constant functions. On a connected nonempty smooth manifold, it is $\mathbb{R}$. An overlapping cover by two connected open subsets therefore does not give the direct-sum formula for de Rham cohomology of a finite disjoint union: restriction sends a constant to equal constants on both pieces.

For a holomorphic submersion, relative differential forms are forms modulo those containing a base differential. The relative exterior derivative gives a complex whose fibrewise hypercohomology is complex cohomology for a smooth proper family. Its filtration by degree defines the Hodge filtration; retaining exactly one base differential in the absolute complex constructs the Gauss–Manin connection.

If a continuous vector local martingale has bracket matrix $A_{t}I$ with $A_{\infty }=\infty$, inverse-clock time change gives bracket matrix $rI$. The Lévy characterization of multidimensional Brownian motion then gives Brownian motion in the common time-changed filtration. A continuous local martingale is constant on flat intervals of its bracket, so inverses using $A_t\ge r$ and $A_t>r$ give the same martingale values. The filtration at the earlier inverse is smaller; adaptation and independence from the larger past transfer the Brownian property to it. Nonzero initial values are retained by adding the initial vector back.

For a continuous local martingale $M$ with divergent bracket $K$ and a continuous drift $D$ satisfying $|D_t-D_s|\le C(K_t-K_s)$, the process $y+M+D$, $|y|<1$, exits $(-1,1)$ almost surely. In unit bracket time a Brownian increment of absolute size greater than $C+2$ forces exit within one unit, giving a geometric survival bound. The inverse bracket clock transfers this finite exit to ordinary time.

// Target: probability-and-statistics.bigb

Let $M$ be a zero-starting continuous local martingale and let $T<\infty$ be a stopping time with $[M{]}_{t\wedge T}\le a$ and $[M{]}_T=a$ for a deterministic $a$. The complex exponential $\exp (iu{M}_{t\wedge T}+u^2[M{]}_{t\wedge T}/2)$ is a bounded local martingale, hence a true martingale. Its terminal expectation is one, giving the characteristic function $\mathbb{E}{e}^{iu{M}_T}=e^{-u^{2}a/2}$. Conditional optional sampling between two bracket levels also gives independent Gaussian increments in inverse-bracket time; Gaussian one-time marginals alone do not establish that independence.

For independent standard Brownian motions $B,W$, Itô formula gives $dY=e^W\cos BdW-e^W\sin BdB$: the two second-order drifts cancel. Independence gives zero cross quadratic covariation, so the displayed bracket follows. The process itself is not Brownian because its bracket is random. Its inverse-bracket time change is Brownian motion starting at one, in the time-changed filtration; subtracting one gives standard Brownian motion. The Divergence of a driftless Brownian exponential clock guarantees that the inverse is finite at every new time.

If $a=N_{L/K}c$, let $L$ act on itself by multiplication and let the extra generator act by $c\sigma$. Its $n$th power is multiplication by $a$, and these operators give the cyclic algebra as ${\mathrm{End}}_K(L)$. Conversely, if the algebra is a matrix algebra on an $n$-dimensional vector space, its embedded degree-$n$ field $L$ makes that space one-dimensional over $L$. The extra generator is then $c\sigma$ in an $L$-basis, so its $n$th power forces $a=Nc$.

The cycle class of an algebraic cycle is its image under the cycle class map. For a codimension-$r$ cycle on a smooth complex projective variety, it belongs to $H^{2r}(X,\mathbb{Z})$ and is the Poincare dual of its fundamental homology class. The map factors through the Chow group, but a Chow class retains finer information than its cohomological cycle class.

For a unit-speed geodesic $\gamma (t)={\exp }_p(tv)$, a finite cut time $c(v)$ is the last time to which its segment from $p$ minimizes Riemannian distance. Its endpoint is the Riemannian cut point along that geodesic. It lies in the cut locus; it need not be a cut point in the topological sense. The segment up to the cut time still minimizes by continuity of distance.

For the symplectic Lie algebra ${\mathfrak{sp}}_{2n}$, the defining crystal basis is the chain $1,2,\dots ,n,\bar{n},\dots ,\bar{1}$, with weights ${\epsilon }_1,\dots ,{\epsilon }_n,-{\epsilon }_n,\dots ,-{\epsilon }_1$. Its successive lowering-edge colors are $1,2,\dots ,n-1,n,n-1,\dots ,1$. Every step subtracts its color's simple root. Applying the crystal tensor-product rule to two copies gives a complete grid description of the tensor-square crystal. For $n\ge 2$, its only highest vertices are $1\otimes 1$, $1\otimes 2$, and $1\otimes \bar{1}$, of highest weights $2{\omega }_1$, ${\omega }_2$, and zero. For $n=1$, the middle vertex is absent and only the other two highest vertices remain.

For the B2 root system with ${\alpha }_1={\epsilon }_1-{\epsilon }_2$ long and ${\alpha }_2={\epsilon }_2$ short, the spin fundamental weight is ${\Lambda }_2=({\epsilon }_1+{\epsilon }_2)/2$. Its four-dimensional highest-weight representation has the minuscule weight orbit $\frac{1}{2}(\pm {\epsilon }_1\pm {\epsilon }_2)$, each of multiplicity one. The crystal basis is the chain $++\stackrel{2}{\to }+-\stackrel{1}{\to }-+\stackrel{2}{\to }--$, where signs denote the coordinates. Each Kashiwara operator subtracts its indicated simple root; the sl2 Lie algebra strings have length one. This fixes every edge and gives a connected highest-weight graph with the correct character.

With long-first simple roots ${\alpha }_1={\epsilon }_1-{\epsilon }_2$, ${\alpha }_2={\epsilon }_2$, the Adjoint representation has highest weight ${\epsilon }_1+{\epsilon }_2=2{\Lambda }_2$. Its crystal basis has one vertex $x_{\beta }$ for every root and two distinct zero-weight vertices $z_1,z_2$. The color-one strings are $x_{{\epsilon }_1}\to x_{{\epsilon }_2}$, $x_{{\alpha }_1}\to z_1\to x_{-{\alpha }_1}$, and $x_{-{\epsilon }_2}\to x_{-{\epsilon }_1}$. The color-two strings are $x_{{\epsilon }_1+{\epsilon }_2}\to x_{{\epsilon }_1}\to x_{{\epsilon }_1-{\epsilon }_2}$, $x_{{\epsilon }_2}\to z_2\to x_{-{\epsilon }_2}$, and $x_{-{\epsilon }_1+{\epsilon }_2}\to x_{-{\epsilon }_1}\to x_{-{\epsilon }_1-{\epsilon }_2}$. All other edges vanish. One construction takes the component generated by $1\otimes 2$ in the tensor square of the crystal of the defining odd-orthogonal representation; the crystal tensor-product rule produces exactly these ten vertices. The equivariant isomorphism ${\Lambda }^2{\mathbb{C}}^5\cong {\mathfrak{so}}_5$ identifies its highest weight and representation.

At a point of local degree $e$, the critical multiplicity is $e-1$. At an ordinary finite point this is the order of vanishing of the derivative. The invariant local-coordinate definition also works at poles and infinity. For $R(z)=z^d$, zero and infinity both have multiplicity $d-1$, so a critical-point count must count multiplicities: there are two distinct critical points but total multiplicity $2d-2$.

Local degrees multiply under composition: $e_p(S\circ R)=e_p(R)e_{R(p)}(S)$, directly by substituting the two local power-series expansions. ConsequentlyIn particular every critical point of $R$ remains critical for each positive iterate, and every critical point of an iterate reaches a critical point of $R$ in fewer than $n$ steps. This covers infinity as well as finite points. For polynomials at finite points the same conclusion follows from $(P^n{)}^{\prime }(z)={\prod }_{j=0}^{n-1}{P}^{\prime }(P^j(z))$.

A critical orbit is the forward sequence $p,R(p),R^2(p),\dots$ starting at a critical point of a rational map. The postcritical set records the forward images, usually from the first iterate onward, of all critical points; a single critical orbit and the union of all such images are distinct notions. For quadratic polynomials there is one finite critical point, zero, and its orbit controls connectedness of the filled Julia set and membership in the Mandelbrot set. A bounded orbit can be preperiodic to a repelling cycle, as $0\mapsto i\mapsto -1+i\mapsto -i\mapsto -1+i$ for $z^2+i$; boundedness then does not imply Fatou membership.