Total variation flow is the $L^2$ gradient flow of the total variation seminorm on a domain. Formally it is $u_t=\mathrm{div}(\nabla u/|\nabla u|)$, but the subgradient formulation handles zero gradients and jumps. Adding quadratic fidelity gives a strictly convex stationary total variation denoising problem.

When gravity and expansion can be neglected and $R$ is constant, the photon monopole satisfiesThe first damping contribution comes from photon-baryon velocity slip; the second comes from the photon quadrupole. This is a local form of Silk damping.

For the expansion $\Theta (\mu )={\sum }_{\ell }(-i{)}^{\ell }{\Theta }_{\ell }{P}_{\ell }(\mu )$ without a $(2\ell +1)$ factor, the leading polarization relation is $E_2=-\sqrt{6}{\Theta }_2/4$. The photon Boltzmann hierarchy then gives ${\Theta }_2=-8k{\Theta }_1/(9\dot{\tau })$. This small photon quadrupole supplies the shear-viscosity part of Silk damping.

The velocity slip is the difference between the bulk velocities of photons and baryons. Thomson scattering suppresses it at a rate $\Gamma (1+R^{-1})$, where $R=3{\rho }_b/(4{\rho }_{\gamma })$ and $\Gamma =-\dot{\tau }>0$. Its first correction in the tight-coupling approximation produces heat-conduction damping of acoustic waves.

The thermal timescale compares the internal energy stored in a disc annulus with its local heating or cooling. In a gas-pressure-supported alpha disk, $t_{\mathrm{th}}\sim (\alpha {\Omega }_K{)}^{-1}$, whereas the viscous timescale is longer by approximately $(R/H{)}^2$. A local thermal perturbation is therefore studied at fixed surface density of a disk, rather than fixed steady-state mass accretion rate.

A tensor field is a smooth section of a vector bundle formed from tensor products of the tangent bundle and cotangent bundle. With $k$ covector factors and $l$ vector factors it belongs toIts coefficients in any local frame vary smoothly. Different conventions list the two counts in different orders, so the factor order should be specified. Tensor contractions use the canonical evaluation of a covector on a vector and are independent of the frame.

For residue field ${\mathbb{F}}_q$, these two properties characterize the Teichmuller lift on every residue class, including zero. The Hensel lemma applies to $T^q-T$, whose derivative reduces to $-1$. Uniqueness implies $[ab]=[a][b]$, $[0]=0$, $[1]=1$, and $[a^p]=[a{]}^p$. In mixed characteristic this multiplicative section need not be additive.

For $K/{\mathbb{Q}}_p$ finite, let $x_n^{p^n}=x$ in its finite residue field, and choose arbitrary lifts $y_n\in {\mathcal{O}}_K$. With normalized discrete valuation,Applying this binomial estimate repeatedly to $y_n$ and $[x_n]$ proves the displayed bound, and hence convergence to $[x]$.

On closed Riemann surfaces of genus at least two, if a Teichmüller map $f_0:X\to Y$ has unit-area source differential $q$ and dilatation $K_0$, any quasiconformal map $f:X\to Y$ in the same homotopy class satisfies $K_0\le {\int }_X|q||1+{\mu }_{f}q/|q|{|}^2/(1-|{\mu }_f{|}^2)$. The plus sign corresponds to a Teichmüller map with ${\mu }_{f_0}=k_0|q|/q$ and $K_0=(1+k_0)/(1-k_0)$. This fundamental inequality yields Teichmüller's uniqueness theorem by the pointwise triangle inequality and its equality case. See Gardiner and Hu, §5, equation (11).

On a closed genus-at-least-two Riemann surface, a Teichmüller map uniquely minimizes maximal dilatation in its homotopy class among maps to the same target. The Reich–Strebel inequality forces equality of Beltrami coefficients for an extremal competitor; their conformal difference is homotopic to the identity and hence is the identity. In genus one, translations must be factored out.

For a transverse Taylor swimming sheet below a flat no-slip boundary condition at height $d$, the first-order amplitude $f(y)$ satisfies $({\partial }_y^2-1{)}^{2}f=0$, $f(0)=1$, $f^{\prime }(0)=f(d)=f^{\prime }(d)=0$. Writing $\Delta ={\sinh }^{2}d-d^2$ givesThe mean boundary velocity determines Taylor-sheet swimming speed, yieldingThis is greater than the unbounded value $1/2$ for every $d>0$. In a narrow gap it scales as $3/d^2$, with the small-amplitude calculation requiring $ϵ\ll d$ as well as $ϵ\ll 1$. Taylor's swimming sheet near a soft boundary recovers this rigid-wall limit while studying how compliance changes propulsion.

A Taylor expansion of the no-slip boundary condition about the flat sheet makes the second-order mean tangential velocity $⟨u_2(0)⟩=-⟨y_1{\partial }_y{u}_1(0)⟩$. The mean mode of Stokes flow is linear in height when no pressure gradient is imposed. In an unbounded fluid, bounded velocity excludes mean shear; in a confined fluid, the force-free condition excludes it. The remaining mean velocity is uniform and equals the swimming speed in the sheet frame. Thus the leading speed can be found from the first-order field without solving the oscillatory second-order field.

At a material interface without surface tension, continuity of displacement and of the pressure evaluated on the displaced interface givesThe first follows from the kinematic boundary condition $ik(U-c)\hat{\eta }=\hat{w}$. The second uses $\hat{p}={\rho }_0[(U-c){\hat{w}}^{\prime }-U^{\prime }\hat{w}]/(ik)$ and hydrostatic displacement $[\hat{p}-g\rho \hat{\eta }]=0$. These conditions apply to density, velocity and vorticity jumps within the Boussinesq approximation.

For a mode with nonreal phase velocity $c$, let $V=U-c$, choose a continuous branch of $V^a$, and put $\hat{w}=V^{a}q$. Under impermeable boundary conditions, multiplying the transformed Taylor–Goldstein equation by $V^{2a}\overline{q}$ and applying integration by parts givesChoosing $a=1/2$ gives the Miles–Howard theorem; choosing $a=1$ makes the real phase velocity of an unstable mode a weighted mean of $U$.

Over Real projective space ${\mathbb{RP}}^n$, the real tautological line bundle isOn the standard chart $U_i=\{[x]:x_i\ne 0\}$, let $s_i([x])=x/x_i$. A local trivialization sends $([x],t)$ to $([x],t{s}_i([x]))$, with inverse $([x],v)\mapsto ([x],v_i)$. These maps prove that this is a real line bundle. Its unit sphere bundle is the antipodal map quotient $S^n\to {\mathbb{RP}}^n$. Its mod-two Euler class of a real line bundle is the degree-one generator of the mod-two cohomology ring of real projective space.

A subspace $W$ of a symplectic vector space $(V,\Omega )$ is isotropic when $\Omega {|}_{W\times W}=0$. It has at most half the dimension of $V$; equality makes it a Lagrangian subspace. This use of isotropic concerns an alternating bilinear form, rather than an isotropic quadratic form.

The symplectic orthogonal complement of a subspace $W$ is $W^{\omega }=\{v:\omega (v,w)=0\text{ for every }w\in W\}$. The nondegenerate bilinear form gives $\dim W^{\omega }=\dim V-\dim W$. An isotropic subspace of a symplectic vector space satisfies $W\subset W^{\omega }$; a Lagrangian subspace satisfies equality.

If $T$ preserves a nondegenerate bilinear form $\omega$ that is alternating and satisfies $T^2=I$ on a finite-dimensional vector space over a field of characteristic different from two, thenThe two eigenspaces are orthogonal for $\omega$, since $\omega (v_{+},v_{-})=\omega (T{v}_{+},T{v}_{-})=-\omega (v_{+},v_{-})$. Each restricted form is nondegenerate: a vector annihilating its own eigenspace also annihilates the other and hence all of $V$. Both dimensions are even, say $2a$ and $2b$. Consequently $\mathrm{tr}T=2a-2b=\dim V-4b$ as an integer obtained from the eigenspace dimensions. Over the real numbers this gives the congruence $\mathrm{tr}T\equiv \dim V(\mathrm{mod}4)$.

The symplectic area of a compact symplectic surface $(\Sigma ,\omega )$ is ${\int }_{\Sigma }\omega$, using the orientation determined by the symplectic form.

The surface divergence of an ambient vector field is the tangential trace ${\nabla }_s\cdot \mathbf{v}=({\delta }_{ij}-n_i{n}_j){\partial }_j{v}_i$. For a tangent vector field it is the intrinsic divergence. For $\mathbf{v}=\mathbf{n}$ it gives the sum of the two principal curvatures; a sphere with outward normal has ${\nabla }_s\cdot \mathbf{n}=2/a$.