Suppose $f(a+h)=f(a)+Lh+r(h)$ and $|r(h)|/\Vert h{\Vert }_1\to 0$. If $c\Vert h{\Vert }_1\le \Vert h{\Vert }_2\le C\Vert h{\Vert }_1$, then $|r(h)|/\Vert h{\Vert }_2\le c^{-1}|r(h)|/\Vert h{\Vert }_1\to 0$. The boundedness of $L$ is also preserved, and the same argument in reverse proves equivalence of differentiability. The derivative is the same linear map, not a norm-dependent replacement.

A subset generates a finite p-group precisely when its images span the Frattini quotient. Indeed $⟨S⟩\Phi (G)=G$ implies $⟨S⟩=G$, since otherwise a maximal subgroup contains both factors. Minimal generating sets correspond to vector-space bases in the quotient.

For a vector bundle carrying a smooth nondegenerate symmetric bilinear form of constant metric signature, a pseudo-orthonormal frame is a smooth local frame of a vector bundle whose Gram matrix is the constant diagonal signature matrix $\eta$. It identifies the metric bundle locally with a fixed model ${\mathbb{R}}^{p,q}$. Two such frames differ by a smooth map into the corresponding orthogonal group. Positive-definite signature is included as the ordinary orthonormal case. A global pseudo-orthonormal frame is a section of a fiber bundle of the orthonormal frame bundle and trivializes that principal bundle.

factor subsets multiply by symmetric difference; their sign columns multiply pointwise. The subgroup generated by independent defining words has $2^k$ elements. On a selected fractional factorial design, every word in this subgroup is constant, with sign determined by the chosen generator signs. The cosets classify identical or opposite contrast columns.

The resolution of a regular fraction is the shortest nonidentity word in its defining contrast subgroup. Two effects alias only when their word product belongs to that subgroup. Thus resolution VI separates all main effects and two-factor interactions from one another and from three-factor interactions, though some three-factor interactions alias in complementary pairs.

On the unit square with exact Dirichlet boundary data, let $D_{9}U=B_{h}f$ and suppose the normalized residual on the exact smooth solution has supremum norm at most $\epsilon$. The error satisfies $D_{9}e=-\tau$ and vanishes on the boundary. The barrier $q=[x(1-x)+y(1-y)]/4$ has $D_{9}q=-1$, $q\ge 0$ on the boundary, and maximum $1/8$. Apply the discrete maximum principle to $e-\epsilon q$ and $-e-\epsilon q$, obtaining $|e|\le \epsilon q$. Thus $\Vert e{\Vert }_{\infty }\le \epsilon /8$, and fourth-order residual gives fourth-order supremum norm nodal convergence. The proof needs a uniform smoothness bound and a compatible boundary discretization.

The nine-point finite-difference stencil for the Laplacian has expansion $D_{9}u=\Delta u+h^2{\Delta }^{2}u/12+h^4(u_{xxxxxx}+u_{yyyyyy}+5u_{xxxxyy}+5u_{xxyyyy})/360+O(h^6)$. If $\Delta u=f$, differentiating this equation gives $u_{xxxxyy}+u_{xxyyyy}=f_{xxyy}$ and $u_{xxxxxx}+u_{yyyyyy}={\Delta }^{2}f-3f_{xxyy}$. Hence the corrected compact equationhas sixth-order normalized consistency of a numerical method. Source derivatives can be supplied analytically; numerical approximation of $\Delta f$ must be fourth order and of the fourth derivatives second order. With exact square-grid Dirichlet boundary data and uniformly smooth solution, the discrete maximum principle inverse bound converts this residual into sixth-order nodal convergence. The unknown stencil stays compact even if more distant known source values are used.

For a random restriction, first fix the live set $J$ and average over the unbiased assignment $a$ outside it. The coefficient at $U\subset J$ is ${\sum }_{B\subset J^c}\hat{f}(U\cup B){\chi }_B(a)$. Orthogonality eliminates cross terms in its squared expectation. Averaging over $J$ gives the displayed identity. The original Fourier weight is weighted by a binomial distribution survival probability, not simply preserved.

This uniform bound is the actual fixed-time stability requirement for a vector-valued periodic grid evolution. Eigenvalue moduli alone are insufficient for nonnormal matrices. A unit Jordan block with an order-one off-diagonal entry grows like $n$, while an entry proportional to $\Delta t$ can give bounded growth on $n\Delta t\le T$. Uniform eigenvector conditioning or a suitable discrete energy estimate can establish the bound. Mesh-dependent diagonalizations cannot be treated as automatically harmless.

For a scalar constant-coefficient grid update, Fourier transformation turns $S$ into multiplication by its symbol $G$. Parseval identity proves the displayed norm equality; necessity is obtained by concentrating Fourier data near a maximizing frequency. Contractivity for every step is equivalent to $|G|\le 1$. Mesh-uniform fixed-time stability, on $nk\le T$, permits $\sup |G|\le 1+Ck$: powers are bounded by $e^{CT}$. Conversely a uniform bound at $n=\lfloor T/k\rfloor$ gives this $1+O(k)$ restriction as $k\to 0$. For matrix symbols, use the existing uniform power bound for matrix Fourier symbols rather than only eigenvalue moduli.

A bound tending to zero with one factor's Gowers U3 norm, uniformly over all finite abelian groups, is false. On $G={\mathbb{F}}_2^m$, choose a random independent sign at each point. A cube with eight distinct vertices has sign product of expectation zero. The probability of a collision among its vertices is at most $\left(\genfrac{}{}{0ex}{}{8}{2}\right)/|G|$, since each equality of distinct vertices imposes a nonzero linear equation in the increments. Thus some sign function $f$ has $\Vert f{\Vert }_{U^3}^8\le 28/|G|$. But the four-term arithmetic progression is $(x,x+d,x,x+d)$, so choosing factors $(f,1,f,1)$ gives correlation identically one. As $m\to \infty$, the norm tends to zero. Bounded torsion, as in cyclic groups, avoids this obstruction.

For a complete ring with an adic topology defined by $I$, the formal spectrum has the prime ideals containing $I$ as its points and structure sheaf ${\underleftarrow{\lim }}_n{\mathcal{O}}_{\mathrm{Spec}(A/I^{n+1})}$. On a principal open subset $D(f)$ its sections form the completed localization of an adic ring, not generally the ordinary localization $A_f$.

// Target: ringed-space.bigb

For a closed subscheme with ideal sheaf $\mathcal{I}$ in a locally Noetherian scheme, the completion is the formal scheme on $|Y|$ with topological structure sheaf ${\underleftarrow{\lim }}_n({\mathcal{O}}_X/{\mathcal{I}}^{n+1}){|}_{|Y|}$. On a Noetherian affine scheme $\mathrm{Spec}A$, it is $\mathrm{Spf}({\underleftarrow{\lim }}_{n}A/I^{n+1})$. Its adic topology remembers every infinitesimal neighbourhood.

// Target: commutative-algebra.bigb

For a local ring $(A,\mathfrak{m})$ with residue field $k$, put $B=A[[x]]$ and $I=\mathfrak{m}B+xB$. There is a natural $k$-vector space isomorphism $I/I^2\cong (\mathfrak{m}/{\mathfrak{m}}^2)\oplus k$. The inverse sends $(\bar{a},\bar{b})$ to the class of $a+bx$. It is well-defined because changing $a$ by ${\mathfrak{m}}^2$ or $b$ by $\mathfrak{m}$ changes the result by $I^2$. It is surjective since every $f\in I$ is congruent modulo $x^{2}B\subseteq I^2$ to its constant and linear terms. As $I^2={\mathfrak{m}}^{2}B+x\mathfrak{m}B+x^{2}B$, those coefficients of an element of $I^2$ lie in ${\mathfrak{m}}^2$ and $\mathfrak{m}$, respectively. This proves injectivity. Thus adjoining one formal variable increases finite embedding dimension by one.

For smooth positive $f$, let $L=f^{-1}\nabla \cdot (f\nabla )+k^2$. Its unweighted bilinear transpose is the displayed operator. If $v$ satisfies $\alpha v+\beta {\partial }_{n}v=0$, its transpose boundary condition is $\alpha u+\beta ({\partial }_{n}u-u{\partial }_n\log f)=0$. At parameters for which the corresponding boundary-value inverses exist, their Green kernels satisfy $G_L({\mathbf{r}}_2,{\mathbf{r}}_1)=G_{L^{\mathrm{T}}}({\mathbf{r}}_1,{\mathbf{r}}_2)$. Equivalently $G_{L^{\mathrm{T}}}(\mathbf{r},{\mathbf{r}}_2)=f(\mathbf{r})G_L(\mathbf{r},{\mathbf{r}}_2)/f({\mathbf{r}}_2)$, exhibiting weighted acoustic Green-function reciprocity.

The formal adjoint of an infinite matrix does not specify the domain of the true adjoint. For the weighted difference generator $(Qf{)}_n=4^n(f_{n-1}/2-f_n)$, its formal positive-eigenvalue solution is $h_n=2^{-n}{h}_0{\prod }_{k<n}(1+\alpha 4^{-k})$. This is square summable but does not belong to the true adjoint domain of the maximal operator: testing against $f_n=2^{-n}$ contradicts the adjoint identity. The nonzero limiting boundary term is the obstruction.

// Target: analysis.bigb

With positive stable buoyancy destruction $B$ and positive dissipation $ϵ$, this normalized sink fraction need not equal the Flux Richardson number $B/P$ away from equilibrium. If $R=(B+ϵ)/P$, then $J={\mathrm{Ri}}_f/R$. In the fixed-correlation equilibrium model for stratified turbulence, a minimum of $R=a/q+bq$ has $B=ϵ$, hence $J=1/2$. The actual flux Richardson number there is $R_{\min }/2$ and equals $1/2$ only when that minimum also satisfies the equilibrium condition $R=1$.

At zero frequency the graded-layer transfer equation integrates to $\exp [L\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right)]$, where $L=\frac{1}{2}\log (Z_b/Z_a)$. Eliminating the outgoing wave amplitude at the left endpoint gives the displayed incoming-to-outgoing scattering matrix. It is real orthogonal, so it preserves total incoming and outgoing flux. It coincides with continuity of velocity and traction at a sharp seismic impedance interface; the profile between the endpoints enters at nonzero frequency.

If the rate functions are bounded and Lipschitz on an invariant compact state space, the Poisson time-change representation of a Markov chain has drift $F={\sum }_j{v}_j{\beta }_j$. Each centered Poisson term is uniformly small after division by $n$. The Gronwall inequality then bounds the distance to $\dot{z}=F(z)$ by the initial error and the centered counting error, multiplied by $e^{LT}$. A Poisson maximal concentration bound gives exponentially decreasing error probabilities for a fixed horizon and fixed error tolerance.

For a density-dependent Markov jump process with finitely many jump vectors, bounded nonnegative rates and Lipschitz drift $F(x)={\sum }_{\ell }\ell {\beta }_{\ell }(x)$, an initial state converging in probability to $x_0$ gives uniform convergence in probability on every finite horizon to the ODE $\dot{x}=F(x)$, $x(0)=x_0$. The compensated martingale has expected maximal squared size $O(N^{-1})$ by the Doob L2 maximal inequality; Gronwall's inequality transfers that estimate to the solution error.

// Target: probability-and-statistics.bigb