The potentials of a point charge $q$ arewith every source quantity evaluated at the retarded time.

For an irreducible complex character $\chi$ of a finite group $G$ and subgroup $H$, write its restriction as ${\sum }_i{d}_i{\psi }_i$. Orthogonality gives ${\sum }_i{d}_i^2=|H{|}^{-1}{\sum }_{h\in H}|\chi (h){|}^2\le |G|/|H|$, because $|G{|}^{-1}{\sum }_G|\chi {|}^2=1$. Equality holds exactly when $\chi$ vanishes outside $H$, since the omitted sum consists of nonnegative terms.

For an exterior $(p+1):p$ mean-motion resonance, use mean longitude in $\varphi =(p+1)\lambda -p{\lambda }_p-\varpi$, $z=e\exp (i\varphi )$, $x=(a-a_r)/a_r$ and $B=3p{n}_p/2$. Given $\dot{e}=-C\sin \varphi -Ae$ and $\dot{\varpi }=(C/e)\cos \varphi$, the leading Keplerian detuning gives $\dot{\varphi }=-Bx-(C/e)\cos \varphi$. Differentiating $z$ and using $e^{i\varphi }(\sin \varphi +i\cos \varphi )=i$ proves the displayed equation. It retains resonant forcing and eccentricity damping without the coordinate singularity at $e=0$. True longitude adds a fast orbital ripple and does not yield this autonomous averaged equation directly.

For a fixed dimensionless Hermitian operator $A$ driven by $H(t)=\Omega (t)A$, all instantaneous generators commute. The exact unitary operator is $U(T)=\exp (-i\theta A)$, so the area, rather than the detailed envelope, fixes the rotation. For $A^2=I$, it is $\cos \theta I-i\sin \theta A$. This convention uses the Quantum Hamiltonian coefficient directly; a convention with $\hslash {\Omega }_{R}A/2$ instead has $\theta =\frac{1}{2}\int {\Omega }_{R}dt$.

Flux expulsion is redistribution of magnetic flux out of the interiors of persistent circulating flows into their edges and surrounding regions. In two-dimensional incompressible flow, a magnetic flux function obeys ${\partial }_{t}A+\mathbf{u}\cdot \nabla A=\eta {\nabla }^{2}A$. At large but finite magnetic Reynolds number, circulation stretches gradients of $A$, enabling resistive smoothing and weak interior magnetic field with strong field near the circulation boundary. Finite magnetic diffusivity is essential to the eventual rearrangement; exact ideal magnetic flux freezing alone does not justify a flux-free steady cell. Magnetic back reaction can limit the process.

A magnetic skin layer is the region penetrated by an oscillating magnetic field in a conducting material. With constant magnetic permeability $\mu$ and electrical conductivity $\sigma$, define the magnetic diffusivityHere $\eta$ has units ${\mathrm{m}}^2{\mathrm{s}}^{-1}$; it is not the electrical resistivity $1/\sigma$. Negligible material motion reduces the resistive induction equation to magnetic diffusion.

Take the conductor to occupy $y>0$ and use time dependence $e^{i\omega t}$ with $\omega >0$. For a tangential magnetic field, the one-dimensional equation isThe solution decaying into the conductor hasThus the skin depth $\delta$ is both the amplitude decay length and the distance for one radian of phase lag. Raising frequency by a factor of four halves $\delta$. ^[ref]

If the boundary amplitude varies slowly as $C(x)$, the solenoidal magnetic-field constraint additionally requires, to leading order,Indeed ${\partial }_x{\hat{B}}_x+{\partial }_y{\hat{B}}_y=0$. This normal component is smaller by the ratio of $\delta$ to the along-boundary variation length; a predominantly tangential field need not be exactly tangential.

A demonstration described by Hermann A. Haus and James R. Melcher drives a winding on a magnetic core passing through a slit in an aluminium block. A magnetometer probe samples the field through a measuring slit. Comparing 60 and 240 Hz tests the predicted factor-of-two change in penetration depth. ^[ref]

Magnetic reconnection changes connectivity between magnetic field lines. It requires departure from smooth magnetic flux freezing, for example through finite resistivity or other nonideal terms concentrated in a current sheet. Ideal viscous magnetic relaxation dissipates mechanical energy while retaining frozen connectivity, so viscosity alone does not provide reconnection.

The absolute residue degree is the degree of the residue field over its prime field. For a tower above $p$, degrees multiply: $[k_{\mathfrak{q}}:{\mathbb{F}}_p]=[k_{\mathfrak{q}}:k_{\mathfrak{p}}][k_{\mathfrak{p}}:{\mathbb{F}}_p]$. In a Galois extension over $\mathbb{Q}$, the decomposition group has a cyclic quotient group of this order by its inertia group. This can exclude residue degrees even when the whole extension has that degree: $S_3$ has no cyclic quotient group of order six.

For $V(r)=c/r^2$ with $c>0$, nonzero physical angular momentum $h$ and positive energy $E$, put $\beta =\sqrt{1+2mc/h^2}>1$ and $r_{\min }=\sqrt{(c+h^2/(2m))/E}$. Direct integration of the effective potential radial equation gives the displayed orbit. It has one radial minimum and two straight-line asymptotes separated by polar angle $\pi /\beta$. The repulsive scattering deflection is $\pi -\pi /\beta$; no bounded or circular orbit exists at finite positive energy.

A representation of a complex Banach algebra is a complex-linear multiplicative action on a vector space. For a unital action require $\pi (1)=I$; a nonunital algebra can be handled by extending the action to its unitization. A nonzero action is algebraically irreducible when it has no nonzero proper invariant vector subspace, equivalently when it defines a simple module.

For $N=2^t$, the binary cyclic codes are exactly the nested ideals generated by $(X+1{)}^j$ for $0\le j\le N$. They have dimensions $N-j$ and canonical check polynomials $(X+1{)}^{N-j}$. The endpoint ideals are the entire word space and the zero code.

// Target: algebra.bigb

If the generator polynomial of a cyclic code has distinct roots ${\alpha }_i$ with multiplicities $m_i$, a word belongs to the code exactly when $c^{[\ell ]}({\alpha }_i)=0$ for $0\le \ell <m_i$, where $c^{[\ell ]}$ is its Hasse derivative. The corresponding rows have entries $\left(\genfrac{}{}{0ex}{}{j}{\ell }\right){\alpha }_i^{j-\ell }$ for $j\ge \ell$ and zero otherwise. Expansion in a basis of the splitting field gives equations over the original finite field; an independent selection has rank ${\sum }_i{m}_i$.

The exact separated-kick response is $f_p=f_T\Delta x/(C+\Delta x)$ for circumference $C=2\pi R$. Its relative correction to the independent-kick estimate is controlled by $\Delta x/C$. For a localized well with force-induced speed bounded by $ϵv_T$, $\Delta x/C\le (w/C){ϵ}^2/(1-ϵ)$. Thus high speed is sufficient when the support width is below one circumference. The local straight-force approximation also requires a narrow trap on the curved track.

A blocking kernel assigns a conditional probability to coarse configurations given microscopic configurations. Multiplying each microscopic Boltzmann weight by the kernel and summing over microscopic configurations defines the coarse Hamiltonian. The normalization preserves the partition function exactly when the coarse operator space is not truncated. In $D$ dimensions a blocking factor $b$ changes the lattice spacing to $ba$ and the number of sites to $N/b^D$. Generated interactions and additive constants must be retained for the transformation to remain exact.

For dimensionless free energy per site $F(u,C)=f(u)+\beta C$, an exact blocking kernel preserves the partition function and gives $F(u,C)=b^{-D}F(u^{\prime },C^{\prime })$. Thus $f(u)=b^{-D}f(u^{\prime })+g(u)$, where $g(u)=\beta [b^{-D}{C}^{\prime }-C]$ is the additive contribution from the eliminated degrees of freedom. Repeating this equality gives the displayed sum. After subtracting analytic backgrounds, the singular free-energy density obeys the homogeneous scaling hypothesis for critical phenomena, subject to marginal or dangerous-variable qualifications.

A renormalized coupling is an interaction parameter defined by a specified finite renormalization condition or subtraction scheme at a scale $\mu$. Observables remain independent of that arbitrary scale when its running and the loop corrections are included consistently. The coupling itself, and higher-order coefficients in its renormalization-group beta function, can depend on the scheme; they should not be identified without qualification with a regulator-independent bare coefficient.

For a finite-alphabet IID source of entropy $H$, fixed-length block codes of rate $R<H$ have reconstruction success probability tending to zero. At most $2^{nR}$ strings can be correctly reconstructed. Their typical probability is at most $2^{nR}{2}^{-n(H-\delta )}$, and the atypical probability tends to zero; choose $0<\delta <H-R$.

// Target: algebra.bigb

For $\kappa \ll 1$ and $\lambda \kappa \gg 1$, the displayed formula retains both inner transverse shear stress and outer normal-velocity resistance. When ${\kappa }^{-1}\ll \lambda \ll {\kappa }^{-3}$, almost immobile tangential interfaces create a pressure-driven parabolic flow and $s\sim \Delta \rho g{h}_0{\kappa }^2/(3\mu )$. When $\lambda {\kappa }^3\gg 1$, the exterior's normal viscous dissipation dominates and $s\sim \Delta \rho g/(2\lambda \mu |k|)$. The exact symmetric three-layer Stokes flow dispersion relation isThe term proportional to ${\lambda }^2{\kappa }^3$ in its denominator prevents extrapolating a fixed-$\lambda$ long-wave expansion uniformly to every large-$\lambda$ scaling.

For metric signature $(+---)$, a stationary perfect fluid follows a timelike Killing vector field $K$ with four-velocity $U=K/N$. Its Killing-flow four-acceleration is $U^b{\nabla }_b{U}_a=-{\nabla }_a\log N$. The relativistic Euler equation and stationarity of the pressure give the displayed hydrostatic equilibrium equation. This uses stationarity along $K$ and does not require hypersurface orthogonality.

For $d{s}^2=(1+2\varphi )d{t}^2-d{\mathbf{x}}^2$ with static weak $\varphi$, normalization gives $u^0=1-\varphi +O({\varphi }^2,v^2)$ and ${\Gamma }^i{}_{00}={\partial }_i\varphi$. Around a homogeneous state, the first-order equations are ${\partial }_t\delta n+n_0\nabla \cdot \mathbf{v}=0$, ${\partial }_t\delta \rho +({\rho }_0+p_0)\nabla \cdot \mathbf{v}=0$, and $({\rho }_0+p_0){\partial }_t\mathbf{v}=-\nabla \delta p-({\rho }_0+p_0)\nabla \varphi$. The inertial density is enthalpy density even though the flow is slow. Only a further nonrelativistic equation of state, $p\ll \rho \simeq mn$, replaces it by mass density.