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Finite local subdivision matrices for the scaled difference scheme track derivative variations in nested parameter intervals. A product-norm bound below one proves uniform contraction for every interval and therefore gives a lower bound on derivative continuity. A product eigenvalue of modulus at least one is an obstruction when its mode is reachable from the data and visible in the limit: generic data excite it, so the corresponding derivative differences fail to vanish. Harmless polynomial or unobservable modes must be removed before applying such an upper bound. Borderline smoothness needs direct analysis, not rounding a numerical exponent. Stability and convergence of the underlying scheme must be checked rather than inferred from mask width.

A mixed tensor has $k$ defining indices and $l$ dual indices. The SU(3) action applies $U$ to every upper index and $U^{-1}$ to every lower index. Young symmetrizers select permutation types, while invariant Kronecker deltas take traces and invariant alternating tensors convert an antisymmetric pair of defining indices into a dual index, or conversely. Repeated fundamental and dual tensor rules decompose it with multiplicities: $(p,q)\otimes (1,0)=(p+1,q)\oplus (p-1,q+1)\oplus (p,q-1)$ and the conjugate rule, omitting negative labels.

The central element $\omega I$ of SU(3), with $\omega =e^{2\pi i/3}$, acts on an irreducible representation of Dynkin labels $(a,b)$ as ${\omega }^{a+2b}$. This residue is its triality. In a tensor realization the representation has $a+2b$ fundamental boxes, and a complete column of three is a singlet; removing such columns preserves the residue. Tensor products add trialities and complex conjugation reverses them.

For a nonvanishing adjoint Higgs field, put $n=\Phi /|\Phi |$. If an oriented orthonormal internal basis satisfies $[e_a,e_b]=\kappa {\epsilon }_{abc}{e}_c$, the displayed tensor is gauge invariant. In a gauge with $n=e_3$, write $A_j=a_j^a{e}_a$; the transverse quadratic terms cancel, leaving ${\mathcal{F}}_{jk}={\partial }_j{a}_k^3-{\partial }_k{a}_j^3$. Hence it is a closed differential form wherever $\Phi \ne 0$. Under trace normalization of su(2) with $-\mathrm{tr}(XY)$, ${\kappa }^2=2$ and the commutator term has coefficient $1/2$. With $-2\mathrm{tr}(XY)$, its coefficient is one. Omitting this normalization factor generally destroys closure.

A global idempotent on a scheme is zero or one in every local ring. Its one-locus is both open and closed. Conversely a clopen subset determines a global idempotent by gluing one on it and zero on its complement. Under this correspondence, decompositions into nonzero orthogonal idempotents correspond to disjoint clopen decompositions. When all connected components are open, the primitive idempotents correspond exactly to those components. Without orthogonality the assertion fails in characteristic two: $(1,0)=(1,1)+(0,1)$ in ${\mathbb{F}}_2\times {\mathbb{F}}_2$.

For the circle with center distance $a=\gamma U/(\gamma +1)$ and radius $R=U/(\gamma +1)$, the most deflected ray from the origin is tangent. The right triangle formed by the origin, center and tangency point gives $\sin {\delta }_{\max }=R/a=1/\gamma$. This is a velocity-vector deflection angle, not the angle of the shock normal.

Take $U$ to be the kernel of a $t$-row Vandermonde matrix on distinct coordinates. For each $(r+t)$-set $T$, $U$ intersects its coordinate subspace in dimension $r$. Wedge a basis of that intersection: its coefficient on every $r$-subset of $T$ is nonzero, because projection onto that subset is an isomorphism by independence of the complementary $t$ columns. Modulo the exterior power of $U$, this gives a dependence involving every clique edge with nonzero coefficient. Strong saturation expresses every missing-edge vector in the span of present ones. Since all coordinate-edge vectors span the quotient, edge count is at least its dimension.

In a uniform sharp-front approximation, an inner ${\mathrm{He}}^{++}$ zone of radius $R_2$ lies inside a ${\mathrm{He}}^{+}$ shell ending at $R_1$, within the hydrogen ionization front $R_H$. For nuclear densities $n_H,n_{He}$, the Electron densities are $n_H+2n_{He}$, $n_H+n_{He}$ and $n_H$ in the three zones. Thus the hydrogen recombination integral is $(4\pi /3){\beta }_H{n}_H[n_H{R}_H^3+n_{He}(R_1^3+R_2^3)]$. The helium integrals are $(4\pi /3){\beta }_{HeI}{n}_{He}(n_H+n_{He})(R_1^3-R_2^3)$ and $(4\pi /3){\beta }_{HeII}{n}_{He}(n_H+2n_{He})R_2^3$. These are balanced against species-specific effective photon budgets, not three independently reused cumulative stellar photon counts.

For an isolated spherical distribution, shell assembly gives gravitational potential energy $W=-{\int }_0^{\infty }GM(r)r^{-1}dM(r)$. Use $S^{\prime }=-2\pi r\rho$ and $M^{\prime }=-2r{S}^{\prime }$ to write $W=2G\int M{S}^{\prime }dr=-2G\int S{M}^{\prime }dr=4G\int rS{S}^{\prime }dr$. A second integration by parts gives the displayed negative quadratic functional. The boundary terms $MS$ and $r{S}^2$ must vanish; finite-mass regular profiles with finite binding energy satisfy the usual sufficient decay conditions.

For planar Brownian motion started at zero, $\log |B_t|$, considered from time one onward, is a local martingale because $\log |x|$ is harmonic off the polar point zero. Every fixed-time absolute moment is finite, but $\mathbb{E}{X}_t=\frac{1}{2}\log t+\mathbb{E}\log |B_1|$ increases. It is therefore not a true martingale. Finite moments at individual times do not establish class DL.

// Target: probability-and-statistics.bigb

A uniform positive quasi-geostrophic potential vorticity anomaly in a sphere of stretched radius $R$ has the displayed regular, decaying streamfunction. Both it and its radial derivative match at the sphere. In its equatorial plane the horizontal relative vorticity is $2Q_0/3$ inside and $-Q_0{R}^3/(3r^3)$ outside. The exterior potential-vorticity anomaly remains zero because its vertical stretching contribution compensates that relative vorticity. A stretched sphere corresponds to a flattened ellipsoid in physical coordinates when $N\gg |f|$.

For an oriented smooth surface and compatible boundary orientation, apply Stokes theorem to $\varphi \mathbf{k}$ for an arbitrary constant vector $\mathbf{k}$. Since $\nabla \times (\varphi \mathbf{k})=\nabla \varphi \times \mathbf{k}$, its surface integral is $\mathbf{k}\cdot {\int }_{S}d\mathbf{S}\times \nabla \varphi$. The boundary integral is $\mathbf{k}\cdot \oint \varphi d\mathbf{x}$. Equality for every $\mathbf{k}$ proves the displayed vector identity. Reversing both orientations preserves the identity.

Deleting $q$ fitted mean parameters improves the Akaike information criterion exactly when the displayed condition holds, since the criterion difference is $n\log ({\mathrm{RSS}}_{\mathrm{reduced}}/{\mathrm{RSS}}_{\mathrm{full}})-2q$. For one parameter, the equivalent partial F-test statistic is below $(n-p)(e^{2/n}-1)$, where $p$ is the full mean-model dimension. This is not the usual fixed-level significance-test cutoff. Fits must use the same observations and response transformation.

Weak equal-mass small-angle gravitational encounters give a single-star transverse kick $2Gm/(bu)$ and a relative-velocity kick twice as large. With encounter rate $2\pi bdbnu$, the single-star squared-velocity diffusion rate is the displayed $D_v$, while the relative rate is $4D_v$. A chosen reference squared velocity divided by either rate defines a relaxation estimate; specifying which rate and reference are used is essential to a numerical prefactor. If $u=\sqrt{2}\sigma$, the convention $3{\sigma }^2/(4D_v)$ gives $3{\sigma }^3/(16\pi \sqrt{2}n{G}^2{m}^2\ln \Lambda )$. The single-star convention $3{\sigma }^2/D_v$ is four times larger. Velocity-distribution averaging further changes numerical coefficients, while the scaling and Coulomb logarithm in stellar dynamics persist. The factor $n$ must be number density, not a dimensionless population count.

Half the peak-to-peak line-of-sight orbital velocity range of a star in a single-companion Kepler orbit. It equals $K=2\pi a_{\star }\sin i/[P\sqrt{1-e^2}]$, where $a_{\star }$ is the stellar barycentric semi-major axis, $P$ the orbital period, $i$ the sky-plane orbital inclination, and $e$ the orbital eccentricity. The Doppler spectroscopy formula follows from Kepler's third law for the relative orbit, $a^3=G(M_{\star }+M_p)P^2/(4{\pi }^2)$.

In a well-mixed closed-box model of galactic chemical evolution, gas consumed into locked stellar mass supplies a natural cumulative mass coordinate $\delta$. The mean stellar metallicity is ${\overline{Z}}_{\ast }(\delta )={\delta }^{-1}{\int }_0^{\delta }{Z}_{\mathrm{birth}}(s)ds$, giving the displayed derivative relation. If the birth abundance increases linearly, $Z_{\mathrm{birth}}=b\delta$, its cumulative mean is half the current maximum. More generally the maximum of any nonnegative abundance distribution cannot lie below its mean. A fixed initial mass function and the same survival selection are required to identify this mass-weighted distribution with a star-count distribution.

For indicator test functions $g$, the equation has a solution with $\Vert \Delta f{\Vert }_{\infty }\le 1$. One proof uses an immigration--death process with immigration rate $\mu$ and per-particle death rate one. Its potential $u(k)=-{\int }_0^{\infty }(\mathbb{E}g(Z_t^k)-\mathbb{E}g(Z))dt$ satisfies its generator equation, and $f(k)=u(k)-u(k-1)$. Coupling two additional initial particles bounds the second difference of $u$ by ${\int }_0^{\infty }2e^{-2t}dt=1$.

For indicators $I_a$ with means $p_a$, let $B_a$ contain $a$ and suppose $I_a$ is independent of the entire family outside $B_a$. Put $\mu ={\sum }_a{p}_a$, $b_1={\sum }_a{\sum }_{b\in B_a}{p}_a{p}_b$ and $b_2={\sum }_a{\sum }_{b\in B_a\setminus \{a\}}\mathbb{E}(I_a{I}_b)$. The Poisson Stein equation and its bound $\Vert \Delta f{\Vert }_{\infty }\le 1$ give the displayed total variation distance estimate. Stronger Stein factors can improve the bound when $\mu$ is large.

Let $W={\sum }_i{I}_i$ and $\lambda ={\sum }_i{p}_i>0$, where each indicator random variable $I_i$ is independent of all indicators outside $\{i\}\cup J_i$ jointly. Write $V_i={\sum }_{j\in J_i}{I}_j$ and $W_i=W-I_i-V_i$. For a test-event solution of the Poisson Stein equation, independence givesTelescoping each difference and using $\Vert \Delta g{\Vert }_{\infty }\le \min (1,{\lambda }^{-1})$ bounds the total variation distance by that factor times ${\sum }_i{p}_i^2+{\sum }_i{\sum }_{j\in J_i}(p_i{p}_j+\mathbb{E}{I}_i{I}_j)$. Directed dependence neighborhoods are allowed. If $\lambda =0$, the count is identically zero.

With a fixed axial seed field and prescribed localized differential rotation, the winding source is time independent. Early induction gives $B_{\varphi }\sim B_0\Omega t$, while the steady balance has $B_{\varphi }\sim B_0\Omega {\ell }^2/\eta$. Their comparison gives the magnetic diffusion time, not merely the rotation period. It diverges as conductivity becomes infinite. In an unbounded domain, this is a local equilibration estimate on scale $\ell$, not finite-time exact convergence everywhere. Lorentz force feedback may invalidate the prescribed-flow approximation sooner.