For real boundary amplitude $C$ and $q=(1+i)/\delta$, the leading electric current density is $j_z=qC{e}^{-qy}/{\mu }_0$. Averaging products of real harmonic fields gives $\overline{\mathbf{F}}=\mathrm{Re}[(\nabla \times \mathbf{B})\times {\mathbf{B}}^{\ast }]/(2{\mu }_0)$. Its leading normal component is the displayed positive force. The first candidate tangential contribution is proportional to $\mathrm{Re}(q/q^{\ast })C{C}^{\prime }=0$, so retaining the small normal field does not introduce a tangential force of that order.

In a thin magnetic skin layer next to a rigid no-slip boundary condition, vertical pressure balance gives $p=P-C^2{e}^{-2y/\delta }/(4{\mu }_0)$. With negligible inertia in the layer, the tangential Stokes flow balance is $\rho \nu u_{yy}=p_x$. Matching zero leading shear outside the layer gives $u=U(1-e^{-2y/\delta })$ and the displayed effective tangential slip. The physical wall still has zero velocity; the slip is a boundary condition for the outer flow extrapolated through the unresolved layer. It points toward larger squared boundary-field amplitude.

For constant mass density $\rho$, kinematic viscosity $\nu >0$, ideal induction, zero boundary velocity and tangent magnetic field, put $E={\int }_D[\rho |\mathbf{v}{|}^2/2+|\mathbf{B}{|}^2/(2{\mu }_0)]dV$. Multiplying the momentum equation by $\mathbf{v}$ and integrating cancels pressure work and advective flux. The magnetic-energy derivative is minus the integral of Lorentz force density dotted with velocity, canceling the same work in the kinetic-energy equation. Integration by parts then gives the displayed identity. For incompressible flow with zero boundary velocity, $\int |\nabla \mathbf{v}{|}^2=\int |\nabla \times \mathbf{v}{|}^2$.

For a planar magnetic field $\mathbf{B}=({\psi }_y,-{\psi }_x,0)$, ideal induction advects the Cartesian magnetic flux function: $D_t\psi =0$. Incompressible flow also preserves the area inside each material contour. During relaxation, the arms of a magnetic separatrix can approach each other while their field-line connectivity remains fixed. A transition of width $ϵ$ with a finite change of tangential field has electric current density of order ${ϵ}^{-1}$ and can approach a current sheet. This is a possible limiting mechanism, not a claim that every planar equilibrium is singular.

For azimuthal flow $u_{\varphi }=s({\Omega }_0+\omega s^2)$, a planar magnetic flux mode $s^{m}g(t)e^{im\varphi -im{\Omega }_{0}t-i{s}^{2}f(t)}$ obeys the resistive induction equation if $\dot{f}=m\omega -4i\eta f^2$ and $\dot{g}/g=-4i\eta (m+1)f$. With $f(0)=0$ and $g(0)=1$, their solution is $f=(m\omega /\mu )\tanh \mu t$ and $g=(\cosh \mu t{)}^{-(m+1)}$. Choose $\mathrm{Re}\mu >0$. For positive $\eta ,\omega$, the fixed-position field decays asymptotically at rate $(m+1)\sqrt{2\eta m\omega }$. Differential rotation creates small spatial scales, accelerating magnetic diffusion.

Turbulent magnetic diffusivity is an effective transport coefficient produced by correlated small-scale velocity and magnetic field. It supplements microscopic magnetic diffusivity in a mean-field dynamo description. Its identification requires a specified mean-field geometry and averaging procedure; in a planar flux problem, the correlation $\overline{w{a}^{\prime }}=-{\eta }_t{B}_{0}f(z)$ describes transport relative to a prescribed uniform field.

The magnetic Prandtl number compares kinematic viscosity with magnetic diffusivity. It is the ratio of the viscous and magnetic diffusion coefficients; equivalently it is the magnetic Reynolds number divided by the Reynolds number when the same speed and length scale are used.

The Mach cone forms the geometric envelope of acoustic wave wavefronts from a source moving at a supersonic Mach number $M>1$. In the source reference frame the ray-velocity sphere is centered at $-M{c}_0{\mathbf{e}}_x$ and has radius $c_0$. Its tangent rays make angle ${\sin }^{-1}(1/M)$ to the backward flight direction; all other emitted acoustic rays lie inside that Mach cone.

The Mach cone is the geometric envelope of the spheres in moving-point acoustic wavefronts. To obtain its axial boundary without a diagram, put $s=-\tau >0$ and impose the geometric envelope conditions $F=0$ and ${\partial }_{s}F=0$ onThe partial derivative condition gives $x=-(U^2-c^2)s/U$. Substituting into $F=0$ gives $y^2=c^2(U^2-c^2)s^2/U^2$, henceFor $M=2$, the Mach cone section has slope magnitude $1/\sqrt{3}$, consistent with a Mach angle of $30^{\circ }$. If emission begins at a finite time, the reached region is limited by the oldest wavefront; the infinite cone describes the steady geometric boundary obtained with an indefinitely long source history. In a uniformly moving gas, advection shifts each emission's sphere by the gas velocity times its age; working with the source's relative velocity through the gas restores the same construction.

For $F={\mathrm{vol}}_{012}\wedge d{H}^{-1}$ and $F=dA$, choose $A=(1-H^{-1}){\mathrm{vol}}_{012}$. A parallel probe then has $q=-1$ in the supermembrane action, so its induced-volume term $-T_2{H}^{-1}$ and potential term $-T_2(1-H^{-1})$ add to a position-independent constant. This proves the cancellation without a weak-field approximation. An oppositely charged probe has no such cancellation.

For $H\sim R^6/r^6$, changing radial coordinate to $z=R^3/(2r^2)$ gives an AdS metric of radius $R/2$ times a round sphere of radius $R$. The Freund-Rubin compactification flux has magnitude $6/R$ in the AdS volume form. This is the regular positive-charge membrane throat, rather than the limit of an arbitrary singular harmonic multipole.

// Target: supersymmetry.bigb

In a classical risk model, an adjustment coefficient $R$ makes $\lambda (M_X(R)-1)-cR=0$. The random-sum transform identity and independent increments show that $\mathbb{E}[Z_t\mid {\mathcal{F}}_s]=Z_s$. This nonnegative martingale starts at one. At the ruin stopping time $\tau$, $Z_{\tau }>e^{Ru}$. Applying the optional stopping theorem only to $\tau \wedge t$ yields $\mathbb{P}(\tau \le t)\le e^{-Ru}$; passage to increasing horizons proves the Lundberg inequality without an unjustified expectation identity at an unbounded stopping time.

Aristarchus of Samos used the half-lit Moon to estimate the relative distances of the Sun and Moon. In the spherical illumination model a half-lit disk means phase angle $\alpha =\pi /2$, so the Sun–Moon–Earth triangle is right-angled at the Moon. Keeping the Sun at a finite distance, let $d_{EM}$ and $d_{ES}$ be the distances from Earth to the Moon and Sun. Then elongation $ϵ$ at Earth givesThe $87^{\circ }$ elongation attributed to Aristarchus of Samos gives a ratio about $19.1$, consistent with his conclusion that the Sun was between $18$ and $20$ times farther away. The example shows how lunar phase geometry becomes a distance measurement, but the ratio is extremely sensitive to an angle close to $90^{\circ }$. Unlike the distant-Sun approximation, this finite triangle has elongation slightly less than $90^{\circ }$ at half illumination. ^[ref].

A synodic month is the synodic period of the Moon relative to the Sun, about $29.53059$ days on average. A lunation is commonly measured from one new Moon to the next. Actual lunations vary, so the constant-speed relation derived in lunar phase is useful for understanding the mean cycle rather than predicting exact dates. ^[ref].

A sidereal month is the Moon's orbital period relative to distant stars, about $27.32166$ days. It is shorter than the synodic month because the sunward direction changes as Earth travels around the Sun. The lunar spin period associated with synchronous rotation matches the sidereal month in the mean. ^[ref].

The lunar illuminated fraction is the directly sunlit projected area of the apparent lunar disk divided by its total projected area. For a sphere with parallel sunlight and a distant observer, outside a lunar eclipse, it is $k=(1+\cos \alpha )/2$, with phase angle $\alpha$ at the Moon. It measures geometry rather than the fraction of full-Moon radiant flux. ^[ref].

Elongation is the apparent angular separation of a celestial body from the Sun, measured at the observer. For a lunar phase, the elongation is the Sun–Earth–Moon angle; in the distant-Sun approximation it is supplementary to the phase angle. A given unsigned elongation can occur on either side of the Sun in the sky.

The astronomical phase angle is the angle at an illuminated body between the directions to its light source and to its observer. For a lunar phase it is the Sun–Moon–observer angle at the Moon, rather than the elongation measured at Earth. A small phase angle gives a nearly full apparent disk in the spherical model.

A retroreflector returns incident light approximately toward its source. Corner-cube arrays make the lunar laser ranging round trip possible.

For a triangular-lattice walk in a sixty-degree wedge with rate $1/6$ on interior edges and doubled outward boundary rates, the horizontal position is conditionally uniform on $\{0,\dots ,k\}$ given its current vertical level and past levels. Within level $k$, the reflecting horizontal generator is symmetric, while vertical killing has constant rate $2/3$, so the uniform law survives any holding period. Each column of the upward or downward transition-rate block has sum $1/3$ for $k>0$, yielding a uniform conditional law at the next level. The induced birth-death process has rates ${\lambda }_k=(k+2)/(3(k+1))$ and ${\mu }_k=k/(3(k+1))$. Its probability of ever hitting zero from level $k$ is $1/(k+1)$, so it is transient.

A sphere of radius $R$ settling in a stationary cylindrical tube of radius $R+d$, with $d\ll R$ and zero net through-flow, has leading speedThe neck gap is $H(z)=d+z^2/(2R)$. In the sphere frame, leading bypass flux per circumferential length is $q\simeq UR/2$; wall-motion transport is smaller by $d/R$. The pressure-driven flux through a parabolic lubrication gap gives $\Delta p=9\pi \mu UR\sqrt{2R}/(4d^{5/2})$. Pressure dominates the sphere drag, $F\simeq \pi R^2\Delta p$, and balancing $F$ against $4\pi R^3\Delta \rho g/3$ proves the speed. Tube end conditions allowing appreciable net flow change the bypass balance and can give other drag regimes.