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An element that can always be omitted from a generating family. These are exactly the elements of the Frattini subgroup: a proper generated subgroup is contained in a maximal subgroup, while an element outside a maximal subgroup generates the group together with it.

Let $\Phi =\Phi (G)$ in a finite group. If $H⊴G$ contains $\Phi$ and $H/\Phi$ is a nilpotent group, every Sylow subgroup $P$ of $H$ has $P\Phi ⊴G$. The Frattini argument gives $G=(P\Phi )N_G(P)=\Phi N_G(P)$, and the defining non-generator property of $\Phi$ forces $N_G(P)=G$. Hence $H$ is a nilpotent group.

At linear order in $ϵ=GM/R$ and the leading rotation term, this coordinate change takes the interior shell metric to Minkowski spacetime. Consequently the retained interior Riemann curvature tensor vanishes although inertial axes rotate relative to distant reference axes. A freely moving particle has $\mathbf{X}={\mathbf{X}}_0+\mathbf{V}T$, while its original coordinates obey $d^2\mathbf{x}/d{t}^2=2\mathbf{\omega }\times d\mathbf{x}/dt$ to first rotation order. Terms of order ${\omega }^2$ would include centrifugal acceleration and are outside this approximation. A gyroscope at rest to the retained order has its parallel-transported spin precessing at $\mathbf{\omega }$ in the original coordinates.

Fourier weight is the squared Fourier-Walsh transform coefficient mass on specified levels or a specified family of subsets. Parseval's identity identifies the total weight with $\mathbb{E}|f{|}^2$, and the nonconstant weight with variance. A decision tree of depth at most $s$ has no weight above level $s$.

With the $e^{-ikx}$ convention, exponential damping on the positive half-line gives $1/(a+ik)$. Its even part tends to $\pi \delta (k)$ and its odd part to $-i\mathrm{PV}(1/k)$. Thus the Heaviside step function transform requires a Cauchy principal value, not an ordinary reciprocal function at zero.

The distributional derivative of $\log |x|$ is the principal-value reciprocal distribution. Its transform is $-i\pi \mathrm{sgn}k$. Dividing the derivative-transform relation by $ik$ gives the displayed finite-part inverse-absolute-value distribution, with an undetermined delta term fixed by the finite-part convention.

A constant-coefficient translation-invariant grid update is diagonalized spatially by discrete Fourier modes. Its scalar or matrix multiplier is the displayed symbol. In a semidiscrete evolution, the analogous symbol is a continuous-time generator and its exponential propagates each mode. Discrete Parseval identity converts uniform modal bounds to discrete L2 norm bounds.

On the unit circle, the arclength-measure transform satisfies $|\hat{\sigma }(x)|\lesssim (1+|x|{)}^{-1/2}$. This follows by removing angular neighborhoods of width $|x{|}^{-1/2}$ around the two stationary points and integrating by parts on the remainder. Gaussian positivity for even extension moments bounds the weighted fourth moment for any bounded $f$ by that for $f=1$. Radial integration of $(1+r{)}^{-2}{e}^{-\pi r^2/R^2}$ with area element $rdr$ costs only $\log R$, proving the displayed local estimate. Modulated disjoint caps and random signs convert it into the planar Kakeya neighborhood lower bound.

A well-mixed layer of depth $h$ losing heat through an unstable cold upper boundary has bulk excess temperature $\theta =T-T_0$. With four-thirds convective heat-transfer law flux $q=Ck[g\beta /(\nu \kappa ){]}^{1/3}{\theta }^{4/3}$, its heat capacity per area is $kh/\kappa$. Thus $\dot{\theta }=-K{\theta }^{4/3}$ with $K=C\kappa [g\beta /(\nu \kappa ){]}^{1/3}/h$, giving the displayed cooling law. It excludes the initial conductive transient and the late low-Rayleigh conduction regime.

If $(A,\mathfrak{m})$ is a local ring, then $B=A[[x]]$ is local with maximal ideal $I=\mathfrak{m}B+xB$ and residue field $A/\mathfrak{m}$. A series $f=\sum a_n{x}^n$ is a unit if and only if $a_0$ is a unit. Necessity follows by taking constant coefficients of an inverse; sufficiency follows from the recursion $b_0=a_0^{-1}$ and $b_n=-a_0^{-1}{\sum }_{i=1}^n{a}_i{b}_{n-i}$ for its inverse. Thus the nonunits are precisely series with $a_0\in \mathfrak{m}$, which is exactly $I$, since $f=a_0+xg$. The constant-coefficient map followed by reduction modulo $\mathfrak{m}$ has kernel $I$ and image $A/\mathfrak{m}$.

A formal power series with zero constant coefficient has a compositional inverse over $R$ exactly when its linear coefficient is a unit. Starting with $b_1=a_1^{-1}$, the degree-$k$ coefficient of $f(b_{1}T+\cdots +b_k{T}^k)$ is $a_1{b}_k$ plus an expression already determined by lower coefficients, so recursion gives a unique inverse. For a homomorphism of formal group laws, cancellation in the homomorphism identity proves that this inverse is also a formal-group homomorphism.

The formal power series ring $R={\mathbb{Z}}_p[[T]]$ is complete for two useful topologies. In the $p$-adic topology, convergence means that all coefficient errors are uniformly divisible by arbitrarily large powers of $p$. In the $(p,T)$-adic topology, only finitely many coefficients are controlled at each precision. The latter topology is compact, since $R/(p,T{)}^n$ is finite and $R$ is its inverse limit. The former is finer and is not compact: the series $T^n$ converge to zero in the latter but remain pairwise separated modulo $p$ in the former. Coleman norm contraction uses the former; compact interpolation arguments use the latter.

For a commutative Noetherian ring, the ideals of lowest possible coefficients of an ideal in $R[[X]]$ form an ascending chain. Choose lifts of finite generating sets through the stabilization degree. Successive coefficient cancellation expresses any series in the ideal using these finitely many lifts. The multiplier series converge coefficientwise, so this generates the actual ideal without first assuming it is closed. Iterating gives the result for finitely many variables.

The general ideal magnetic-energy balance has outward-surface contribution ${\mu }_0^{-1}\oint [(u\cdot B)B-B^{2}u]\cdot dS$ and volume contribution $-\int u\cdot (J\times B)dV$. The latter vanishes for a force-free magnetic field. For an axisymmetric boundary rotating tangentially, $u\cdot B=\Omega f(\psi )$ and the directed flux of a thin tube is $2\pi d\psi$. Pair its exit and entry endpoints to obtain the displayed energy injection, counting each boundary endpoint once. Equal endpoint angular velocities inject no magnetic energy.

For a stationary quenched intrinsic curvature with power spectrum $S_{\kappa }(q)$ and finite second spectral moment, the bulk deficit satisfiesThe moment is $⟨({\kappa }_0^{\prime }{)}^2⟩$ for mean-square differentiable curvature. Infinite-moment disorder and finite endpoint boundary layers can change the exponent, so this is a smooth bulk law rather than a universal law for all random polymers.

For a finitely supported nonnegative function $\mu$ of sum one, let $F_t=\{x:\mu (x)>t\}$. Then ${\int }_0^{\infty }|F_t|dt=1$ and ${\int }_0^{\infty }|F_t△g{F}_t|dt=\Vert \mu -\lambda (g)\mu {\Vert }_1$. If the sum of these latter integrals over a finite set of translations is less than $\epsilon$, some nonempty level set has the sum of its relative boundaries less than $\epsilon$. This converts the Reiter condition into finite Følner sets without assuming that the group is finitely generated.

A spectral-parameter-dependent divergence identity equivalent to a partial differential equation. Integrating it over a space-time region gives a global relation between transformed initial and boundary data. For the half-line free Schrödinger equation, take $P=e^{-ikx+i{k}^{2}t}q$ and $Q=e^{-ikx+i{k}^{2}t}(-kq+i{q}_x)$.

Let $I_C[F]=(2\pi {)}^{-1}{\int }_C{e}^{ikx-(k^2-i\alpha k)t}F(k)dk$. The drift-diffusion half-line global relation and the half-line drift null contour identity give the formula in the secondary title. Its real-space kernel is the difference of two drifted Gaussian heat kernels, and its boundary-forcing kernel is $x(2\sqrt{\pi }{t}^{3/2}{)}^{-1}\exp [-(x+\alpha t{)}^2/(4t)]$. This kernel is an approximate identity in time as $x\downarrow 0$, which verifies the Dirichlet boundary condition.

The reflected global relation expresses the unknown value trace in terms of the prescribed derivative trace. Multiplying the half-line drift null contour identity by $(ik+\alpha )/(ik)$ removes the reflected time-dependent transform. Its pole at $k=0$ lies below the contour and is not crossed. The formula concerns the derivative $q_x(0,t)$, rather than the outward derivative $-q_x(0,t)$.

For $\lambda >0$, put $a=k-\lambda /k$, $b=k+\lambda /k$. The spectral integral uses the positive real and positive imaginary rays, both oriented outward. On the first ray its density is ${\rho }_x=i{\int }_0^{\infty }{e}^{-iax}[bq(x,0)-q_y(x,0)]dx$; on the second it is ${\rho }_y={\int }_0^{\infty }{e}^{by}[aq(0,y)-i{q}_x(0,y)]dy$. The global relation ${\rho }_x+{\rho }_y=0$ holds in the third spectral quadrant. Primitives from the corner and from each infinite boundary end give an additive Riemann-Hilbert problem. Their limits are $-q$ at infinity and $q$ at zero, which explains the factor $1/(4\pi i)$.