For propagation along $x$, the Alfvén right eigenvector has $\delta \rho =\delta p=\delta u_x=\delta B_x=0$, ${\mathbf{B}}_{\perp }\cdot \delta {\mathbf{u}}_{\perp }=0$ and $\delta {\mathbf{B}}_{\perp }=-s\sqrt{{\mu }_0\rho }\delta {\mathbf{u}}_{\perp }$. Its speed is $u_x+s{B}_x/\sqrt{{\mu }_0\rho }$, $s=\pm 1$. When the transverse background field vanishes, both transverse polarizations are allowed.

A finite-amplitude ideal-MHD Alfvén wave can rotate the transverse magnetic field at constant magnitude, with constant mass density, pressure and longitudinal components. Its transverse velocity obeys ${\mathbf{u}}_{\perp }=\mathbf{C}-s{\mathbf{B}}_{\perp }/\sqrt{{\mu }_0\rho }$ and its speed is constant along the wave. The constant magnetic pressure prevents compressive acceleration.

A torsional Alfvén wave couples azimuthal velocity to a toroidal magnetic field around a fixed poloidal magnetic field. Axisymmetric induction gives ${\partial }_t{B}_{\varphi }=R{\mathbf{B}}_p\cdot \nabla \Omega$ and magnetic tension gives $\rho R^2{\mu }_0{\partial }_t\Omega ={\mathbf{B}}_p\cdot \nabla (R{B}_{\varphi })$. Their combination is the displayed wave equation. Its local dispersion relation is ${\omega }^2=(\mathbf{k}\cdot {\mathbf{B}}_p{)}^2/({\mu }_0\rho )$.

Let $\mu$ be a Borel probability measure on the compact convex set $K$, and fix $f\in C(K,\mathbb{R})$. the sublinear functional $p(g)=\mu (\overline{g})$ has a linear supporting functional taking the value $p(f)$ at a specified continuous $f$, by the Hahn-Banach theorem. It is positive and normalized because $p$ has these values on constants. The Riesz-Markov-Kakutani representation theorem gives the probability $\nu$. Testing affine functions and their negatives shows that $\mu$ and $\nu$ have the same barycenter.

For smooth spatial fields $E^i,C^i{}_j$, define a torsion-free connection on $\mathbb{R}\times {\mathbb{R}}^n$ by ${\Gamma }^i{}_{00}=-E^i$, ${\Gamma }^i{}_{0j}={\Gamma }^i{}_{j0}=-C^i{}_j/2$ and all other coefficients zero. The time geodesic equation is $\ddot{t}=0$. For $\dot{t}\ne 0$, use $t$ as affine parameter; the spatial geodesic equation becomes $d^2{x}^i/d{t}^2=E^i+C^i{}_{j}d{x}^j/dt$. In dimension three, $C^i{}_j=2{ϵ}^i{}_{kj}{B}^k$ realizes $\ddot{\mathbf{x}}=\mathbf{E}+2\mathbf{B}\times \dot{\mathbf{x}}$. Such a connection need not preserve a metric; a spacetime description of these trajectories only needs affine geometry.

The difference of two affine connections is a smooth tensor field of type $(1,2)$. Linearity over smooth functions holds in the first slot by the connection axioms. In the second slot, the two extra terms $X(f)Y$ cancel, so $\Delta (X,fY)=f\Delta (X,Y)$. This tensoriality makes the value depend only on $X_p,Y_p$. In coordinates its components are the differences of the Christoffel symbols, although either collection of connection coefficients individually is not a tensor.

For a Riemannian metric on an ambient manifold and an embedded submanifold, let $\pi :TN{|}_M\to TM$ be orthogonal projection. Project the restriction of a connection to an embedded submanifold to obtain $D_{X}Y=\pi {\nabla }_{X}Y$. Since $\pi Y=Y$, the Leibniz rule and linearity over smooth functions hold, making $D$ an affine connection. If the ambient connection is the Levi-Civita connection, pairing with tangent vectors shows that $D$ is a metric connection; projecting the zero-torsion identity shows it is a torsion-free connection. By uniqueness, $D$ is the induced Levi-Civita connection.

A one-form $k$ with ${\nabla }_b{k}_a=0$. If $\ell =d\varphi$ satisfies ${\nabla }_b{\ell }_a+{\ell }_a{\ell }_b=0$, then $k=e^{\varphi }\ell =d(e^{\varphi })$ is parallel.

The torsion of an affine connection is the displayed antisymmetric tensor. It measures the failure of the antisymmetrized covariant derivative to agree with the Lie bracket of vector fields. With derivative-last coefficients ${\nabla }_{\mu }{Y}^{\rho }={\partial }_{\mu }{Y}^{\rho }+{\Gamma }^{\rho }{}_{\nu \mu }{Y}^{\nu }$, its components are ${\mathcal{T}}^{\rho }{}_{\mu \nu }={\Gamma }^{\rho }{}_{\nu \mu }-{\Gamma }^{\rho }{}_{\mu \nu }$. Some conventions use the negative component tensor. Geodesic equations depend on the symmetric connection coefficients, so they alone cannot detect torsion.

Given a metric tensor and an affine connection, $N_{abc}={\nabla }_c{g}_{ab}$ measures failure of metric compatibility. It is symmetric in $a,b$. Some authors instead define nonmetricity as $-\nabla g$, so formulas must specify the sign. Together with the torsion tensor, it determines the difference from the Levi-Civita connection.

Relative to a nondegenerate metric tensor, any affine connection is uniquely the Levi-Civita connection plus a contorsion tensor contribution and a disformation tensor contribution. In derivative-last notation, with $T_{ab}{}^c={\Gamma }^c{}_{ab}-{\Gamma }^c{}_{ba}$ and $N_{abc}={\nabla }_c{g}_{ab}$, the lowered difference is $A_{abc}=\frac{1}{2}(T_{abc}+T_{bca}-T_{cab}+N_{abc}-N_{bca}-N_{cab})$. This follows by solving $T_{abc}=A_{abc}-A_{bac}$ and $N_{abc}=-A_{acb}-A_{bca}$. The difference of affine connections is a tensor.

For an affine connection, let ${\gamma }_X$ be the affinely parametrized geodesic starting at $p$ with velocity $X$. Define the exponential map by the displayed formula for initial velocities whose geodesics exist until parameter one. It is defined near zero, has differential equal to the identity there, and obeys ${\exp }_p(tX)={\gamma }_X(t)$ wherever defined. This construction requires no metric and extends the metric exponential map.

For a complex semisimple Lie algebra $\mathfrak{g}\subseteq \mathrm{End}(V)$ and $X\in \mathfrak{g}$, both parts of its Additive Jordan decomposition lie in $\mathfrak{g}$. The polynomial semisimple part of $\mathrm{ad}X$ shows $X_s$ normalizes $\mathfrak{g}$. Split $\mathrm{End}(V)=\mathfrak{g}\oplus M$ by the Weyl complete reducibility theorem; the $M$-component of $X_s$ centralizes $\mathfrak{g}$. On each irreducible summand of $V$ it is scalar by the Schur lemma. Its trace is zero because $\mathfrak{g}$ is a perfect Lie algebra and the nilpotent part has zero trace, so those scalars vanish. Nonsemisimple subalgebras may fail this property: $\mathbb{C}(I+N)$ with $N\ne 0$ nilpotent contains neither Jordan part of $I+N$.

For a homogeneous dry ideal gas in a hydrostatic adiabat, $dT/dz=-g/C_p$, where $z$ increases upwards and $C_p$ is the specific heat capacity at constant pressure per unit mass. It follows from $dT/dP=1/(\rho C_p)$ and hydrostatic equilibrium. The altitude version of the Schwarzschild criterion is $dT/dz<-g/C_p$ for instability; equality is marginal. Moisture, composition gradients and nonideal equations of state modify the criterion.

An additive idempotent ultrafilter satisfies $\mathcal{U}+\mathcal{U}=\mathcal{U}$. On positive integers it is necessarily nonprincipal, since a principal ultrafilter at $n$ adds to itself to give the principal ultrafilter at $2n$. The idempotent-ultrafilter star-set lemma converts this algebraic property into a recursive construction of finite sums. If zero is included, the trivial principal idempotent at zero must be excluded for that increasing-sequence application.

In an abelian category, the pullback in a category of an epimorphism $p:B\to C$ along $k:C^{\prime }\to C$ is the kernel of the epimorphism $[p,-k]:B\oplus C^{\prime }\to C$. If $v:C^{\prime }\to D$ annihilates its projection, $[0,v]$ annihilates that kernel and therefore factors through $[p,-k]$. Restriction to $B$ forces the factor to vanish, giving $v=0$ and proving the projection epic. This uses the definition's normality of epimorphisms.

In an abelian category, a morphism is an epimorphism if and only if its categorical cokernel object is zero. For the reverse implication, any difference of maps annihilating the morphism factors through its cokernel and is therefore zero. The cokernel invariance under pushout then proves both preservation and reflection of epimorphisms by pushout.

The logarithm of the Euler product and $3+4\cos \theta +\cos 2\theta \ge 0$ give $\zeta (\sigma {)}^3|\zeta (\sigma +it){|}^4|\zeta (\sigma +2it)|\ge 1$. At $\sigma =1+a/{\log }^9|t|$, upper bounds $\zeta (\sigma )\ll {\log }^9|t|/a$ and $|\zeta (\sigma +2it)|\ll \log |t|$ imply the lower bound $|\zeta |\gg a^{3/4}{\log }^{-7}|t|$. The Cauchy estimate for derivatives gives $|{\zeta }^{\prime }|\ll {\log }^2|t|$ nearby, allowing a sufficiently small leftward displacement of order ${\log }^{-9}|t|$. This elementary region is weaker than the classical logarithmic region but gives an explicit reciprocal bound by simple estimates.

For $t\ge 2$ and $0<\eta \le 1/4$, suppose $|\zeta |\le M$, $M\ge 2$, on radius-$\eta$ discs centred at $1+\eta /4+it$ and $1+\eta /4+2it$. Every zero $\beta +it$ has the displayed gap. The reciprocal Euler product bounds both centre values below by a constant times $\eta$. The local logarithmic-derivative lemma gives errors $B/\eta$, $B=1+\log (M/\eta )$. The three-four-one zero-free-region argument then implies $4/(\sigma -\beta )-3/(\sigma -1)\ll B/\eta$. For a zero close to one, choose $\sigma =1+6(1-\beta )$, making the left side $1/(14(1-\beta ))$. Zeros farther away already satisfy the bound. Thus a logarithmic upper bound for zeta translates into a quantitative zero-free width.

For sufficiently large $|t|$, the Riemann zeta function has no zeros in the displayed region, for a fixed positive $c$. To derive it from the Richert bound for the Riemann zeta function, take $\eta \asymp (\log \log t/\log t{)}^{2/3}$. The logarithm of the maximum on the two discs in the Landau zero-free-region theorem is $O(\log \log t)$, as is $\log (1/\eta )$. Dividing $\eta$ by this logarithmic factor gives the displayed width.