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The special orthogonal Lie algebra consists of the infinitesimal transformations preserving a nondegenerate symmetric bilinear form.

For a nondegenerate alternating matrix $J$, the symplectic Lie algebra is ${\mathfrak{sp}}_{2n}=\{A:AJ+J{A}^T=0\}$.

The special linear Lie algebra ${\mathfrak{sl}}_n$ consists of the trace-zero $n\times n$ matrices with the commutator bracket.

The even signed symmetric group is the index-two subgroup of signed permutations with an even number of sign changes. It is the Coxeter group of type $D_n$.

For a layer $0<z<h$ with rigid lower boundary and traction-free upper boundary, an SH mode
$u_y=Y(z)e^{i(\kappa x-\omega t)}$ hasand dispersionIts cutoff is ${\omega }_n=c_s{q}_n$, its phase speed is $\omega /\kappa$, and its group speed is $c_s^2\kappa /\omega$.

A continuous semimartingale has a decomposition in which both the local-martingale and finite-variation parts are continuous.

For $\omega \in S^1\setminus \{1\}$, the Levine-Tristram signature is the signature of the Hermitian matrixIt is locally constant away from unit roots of the Alexander polynomial of a knot.

At $\omega =-1$, this becomes the signature of a knot.

With outward normal $N(x)=x$, the unit sphere has $A(X,Y)=-⟨X,Y⟩N$. The Gauss equation therefore gives sectional curvature one on every tangent two-plane.

Ricci curvature is the trace of the Riemann curvature tensor in its first and third arguments. For a unit tangent vector $X$ and an orthonormal basis $X,e_2,\dots ,e_n$,

For a Brownian motion with drift $X$, the process $X_T-X_{T-t}$ on $[0,T]$ has the same law as $X_t$. Thereforehave the same distribution, which equates the corresponding geometric-Brownian lookback payoffs.

For the payoff $S_T^p$, risk-neutral valuation givesIts delta hedge holds $p{V}_t/S_t$ units of the risky asset.

If a theory has arbitrarily large finite nonrigid models, expand its language by a unary function $f$, axiomatize that $f$ is an automorphism, require $f$ to move some element, and add sentences demanding at least $n$ elements for every $n$. Every finite fragment has a sufficiently large finite nonrigid model, so compactness supplies an infinite nonrigid model.

For a unit-speed geodesic and a fixed-endpoint variation, the second variation of length is the index form of the normal component of its variation vector field.

A ribbon knot is the boundary of a ribbon disk.

For lower speed $U$, stationary upper tangential speed, upper normal speed $-V$, and equal endpoint pressures, define $⟨f⟩=L^{-1}{\int }_0^{L}f(x)dx$. Thenwhere

The Yoneda lemma gives natural bijectionsand their contravariant analogues.

Strong convergence on every bounded region combines with a tail estimate uniform in the sequence to give global strong convergence.

On bounded $\Omega$ with $V\in L^{\infty }(\Omega )$, weak convergence in $H_0^1(\Omega )$ makes the Dirichlet term lower semicontinuous and, by Rellich compactness, makes the potential term $\int V{u}^2$ continuous. Thusis weakly lower semicontinuous.

Weak convergence on a bounded domain gives pointwise convergence of Fourier transforms and dominated convergence on bounded frequency balls. A uniform derivative bound controls the complementary high-frequency tails.

For $u\in H^1({\mathbb{R}}^d)$, Plancherel gives