For the negative binomial distribution with mean $\mu >0$ and size $\theta >0$, the one-observation score function in the mean is $u_{\mu }=\theta (y-\mu )/[\mu (\mu +\theta )]$. Its size derivative is $(y-\mu )/(\mu +\theta {)}^2$, whose expectation is zero. Thus the expected Fisher information matrix is diagonal in $(\mu ,\theta )$. Regular interior maximum-likelihood estimators have zero asymptotic cross covariance, without necessarily being independent in finite samples.

For scalar interest parameter $\psi$, choose new nuisance coordinates $\eta$ and keep $\psi$ fixed. The transformed interest score function is $U_{\psi }+U_{\lambda }^{\top }{\partial }_{\psi }\lambda$, so the displayed differential equation makes its Fisher information cross-block with the nuisance score zero. Under smoothness and nonsingularity it has a local solution from initial coordinates on a fixed interest slice. Orthogonality is local expected-information decoupling; it implies first-order asymptotic independence of fitted blocks, not exact independence or nuisance-free conditional inference.

For a regular interest/nuisance parameterization with parameter orthogonality, this nuisance-information adjustment approximates a conditional likelihood construction and reduces first-order nuisance effects. It is related to modified profile likelihood but is not a universal substitute for its sample-coordinate Jacobian. When the constrained and full nuisance fits coincide in the relevant ancillary coordinates, that Jacobian is one and the two expressions agree. For normal variance with unknown mean, it replaces the profile's $n$ by the residual $n-1$ in the logarithmic variance term.

In a normal linear model whose predictor columns are orthogonal to the intercept, two coefficient blocks are orthogonal statistical parameters precisely when the cross-block Fisher information is zero, equivalently the displayed design cross-product vanishes. Then $X^{\mathsf{T}}X$ is block diagonal, so the corresponding least-squares estimators have zero covariance and, being jointly Gaussian, are independent. Their fitted subspaces are orthogonal and their extra sums of squares do not depend on which block is added first.

In a regular common-support statistical model with probability density function $a(\lambda ,y)e^{\lambda t(y;\psi )}$, the score function in $\psi$ is $\lambda t_{\psi }$. Its zero expectation gives $\mathbb{E}{t}_{\psi }=0$ for $\lambda \ne 0$. The mixed log-likelihood derivative is $t_{\psi }$, so the cross Fisher information matrix block vanishes. At zero the same conclusion follows by continuity when the regular derivatives extend there. This proves parameter orthogonality; it does not establish exact independence of fitted parameters.

A factor loading is the coefficient of latent factor $F_j$ in observed variable $X_i$. In the orthogonal factor model, $\mathrm{Cov}(X_i,F_j)=L_{ij}$. If $X_i$ has variance one, this is also its correlation with the variance-one factor. Its sign and size depend on the factor-coordinate convention; factor rotation preserves the implied covariance while changing individual loadings.

The communality is the variance of the shared-factor contribution to variable $i$. In an orthogonal factor model, $\mathrm{Var}(X_i)=h_i^2+{\psi }_i$, where ${\psi }_i$ is its specific variance. For standardized variables, $h_i^2\le 1$. It measures variance shared through the modeled factors, not variance in a chosen principal component.

An orthogonal rotation of factors leaves $L{L}^T$ unchanged and hence preserves the model covariance. It proves rotational nonidentifiability of the unrestrained factor loadings. Choose an orientation for interpretability, for example varimax rotation, or impose identifying restrictions in a confirmatory model. An oblique transformation permits correlated factors: if $L$ becomes $LH$, their covariance becomes $H^{-1}{H}^{-T}$, preserving the same shared covariance.

This collects the gradient, divergence, curl and Laplacian formulas in cylindrical and spherical coordinates. Coordinate unit vectors vary with position, so taking their derivatives is essential when translating vector calculus operators from Cartesian coordinates. Curl in spherical coordinates is one of these formulas.

The weighted strain density in Ornstein-Uhlenbeck multiplicative amplification evolves under ${\mathcal{L}}_{n}p=(\kappa /2)p^{\prime \prime }+{\tau }^{-1}(\sigma p{)}^{\prime }+n\sigma p$. Setting $p=e^{-{\sigma }^2/(2\kappa \tau )}\psi$ eliminates its first derivative. Shifting and scaling to $x=(\sigma -\kappa n{\tau }^2)/\sqrt{\kappa \tau }$ gives ${\psi }_{xx}+[1+\kappa n^2{\tau }^3-2\gamma \tau -x^2]\psi =0$. The quantum harmonic oscillator levels yield the displayed eigenvalues. Its ground state is the dominant positive mode; higher sign-changing modes may occur in a transient expansion.

In global coordinates the de Sitter spacetime metric is $-d{\tau }^2+{\cosh }^2\tau g_{S^3}$. A global frame of a vector bundle orthonormal for the round metric $Y_1,Y_2,Y_3$ on $S^3$ gives the displayed global pseudo-orthonormal frame. For example, identify $S^3$ with the unit quaternions and use $qi,qj,qk$. Choosing the spatial orientation and the future direction consistently gives a section of a fiber bundle for the oriented, time-oriented frame bundle. The principal bundle trivialization by a global section then proves that bundle is trivial.

On the orbit of a face $\tau ⪯\sigma$, the monomial ${\chi }^{\mu }$, with $\mu \in {\sigma }^{\vee }\cap M$, vanishes exactly when $\mu \notin {\tau }^{\perp }$. Its reduced zero locus is consequently the union of those orbit closures. The principal subscheme need not be reduced: $V(x^2)$ and $V(x)$ have the same support but different ideals.

// Target: geometry-and-topology.bigb

Let $Z_j$ be the sum of $j$ independent identically distributed real increments. For a finite convex function $f$, integrability of $f(Z_j)$ and $f(Z_{j+1})$ implies integrability of $f(Z_j+x)$ for every fixed $x$. Its negative part is controlled by negative part of a finite convex function is Lipschitz. For $x>0$, if the increment law is unbounded above, take an independent increment $\eta$ with $p=\mathbb{P}(\eta \ge x)>0$. On this event convexity gives $f(Z_j+x{)}^{+}\le f(Z_j{)}^{+}+f(Z_j+\eta {)}^{+}$, so the expectation is bounded by $\mathbb{E}f(Z_j{)}^{+}+p^{-1}\mathbb{E}f(Z_{j+1}{)}^{+}$. If increments are bounded above by $M$, use $Z_j\le jM$ and bound the positive part by $f(Z_j{)}^{+}+f(jM+x{)}^{+}$. For $x<0$, use the symmetric lower-tail argument. This supplies translated integrability without assuming integrable increments or a growth bound on $f$.

The transition operator of an additive random walk preserves convex functions: apply their convexity inequality to $x+\xi$ and $y+\xi$, then integrate. The pointwise maximum of convex functions preserves convexity too, so induction proves convexity of the Snell envelope Bellman functions. Expectations are allowed to be extended-valued until finiteness is justified. Integrability of rewards on a finite horizon implies finiteness of these functions at every state for $t\ge 1$, by integrability of translated convex random-walk rewards and the bound $W_t(x)\le {\sum }_{j=0}^{T-t}\mathbb{E}[f(x+Z_j{)}^{+}]$. At time zero only $X_0=0$ is observed; if $W_0$ is infinite elsewhere, its finite value at zero has a constant convex extension representing the same Snell envelope. Full statewise Bellman recursion at time zero requires an additional translated-integrability hypothesis.

In this temporal sense, the optical coherence length is $L_c=c{\tau }_c$, where $c$ is the vacuum speed of light and ${\tau }_c$ is an explicitly chosen optical coherence time. It measures a delay as a vacuum-equivalent path difference; it does not measure transverse spatial optical coherence. ^[ref].

For the stipulated $1\mathrm{GHz}$ profiles, the $1/e$ convention gives $L_c\approx 9.54269\mathrm{cm}$ for the Lorentzian and $15.8896\mathrm{cm}$ for the Gaussian. In a vacuum Michelson interferometer, changing one mirror position by $x$ changes the round-trip delay by $2x/c$, so the mirror travel to either of these delays is $L_c/2$. In a medium, convert physical travel using the actual group delay, which can depend on frequency. For a narrow line centered at vacuum wavelength ${\lambda }_0$, differentiating $\nu =c/\lambda$ gives $\Delta \nu \approx c\Delta \lambda /{\lambda }_0^2$; this small-width approximation must use consistently defined spectral widths.

Allow terminal positions of an infinite game of perfect information to carry a losing-player label. A clopen set of infinite winning plays still gives a determined infinite game. At a position where all infinite continuations have one winner, the other player can win only by forcing a terminal loss for that winner. The usual winning-position attractor in an infinite game decides this reachability game. The tree of remaining undecided positions is well-founded: every infinite branch would eventually have its clopen payoff decided by a prefix. Assign the already decided winners upwards by well-founded recursion. Terminal labels are essential; a terminal node created by a covering need not be lost by the player whose turn would have come next.

For $0<S<1$ and $V=1-S$, the translational branch of one-sided kinetic solidification dispersion has $\sigma =(S/V-\Gamma ){\alpha }^2+O({\alpha }^4)$. Expand $p=V+(\sigma +{\alpha }^2)/V+O({\alpha }^4)$ and substitute to prove this limit. At zero growth, $p_0=[V+\sqrt{V^2+4{\alpha }^2}]/2$, and the destabilizing right side has sign ${\alpha }^2(SV/p_0^2-\Gamma )$. For nonnegative real $\sigma$, the residual has derivative at least one, so there is one growing root precisely when $\Gamma <SV/p_0^2$. Thus instability exists exactly when $\Gamma <S/V$, with unstable band ${\alpha }^2<p_c(p_c-V)$, $p_c=\sqrt{SV/\Gamma }$ for $\Gamma >0$. At threshold the next term is $\sigma =-2S{\alpha }^4/V^3+O({\alpha }^6)$. Surface energy removes short-wave growth even when long waves remain unstable.

The cubic follows by eliminating $\sigma =p^2-Vp-{\alpha }^2$ from one-sided kinetic solidification dispersion, with $V=1-S$. For $\Gamma =0$, the interface branch tends to $\sigma =SV-SV/\alpha +O({\alpha }^{-2})$. For fixed $0<\Gamma <1$, substituting $p=\sqrt{1-\Gamma }\alpha +d$ gives $2d-V=S\Gamma /(1-\Gamma )$, hence $\sigma =-\Gamma {\alpha }^2+[S\Gamma /\sqrt{1-\Gamma }]\alpha +O(1)$. At $\Gamma =1$ the balance is $p^3\sim S{\alpha }^2$, giving $\sigma =-{\alpha }^2+S^{2/3}{\alpha }^{4/3}+o({\alpha }^{4/3})$. For $1<\Gamma <1+2S/V$, $p\to S/(\Gamma -1)>V/2$ and $\sigma =-{\alpha }^2+p^2-Vp+o(1)$. Stronger capillarity can remove the real isolated interface mode from the chosen thermal branch; it does not create growth. These distinctions explain why replacing $p$ by $\alpha$ is not a controlled fixed-capillarity limit.

The initial cost of the one-period quadratic hedge is $\mathbb{E}[{\rho }^{\ast }\xi ]$. This pricing weight satisfies $\mathbb{E}[{\rho }^{\ast }R]=1$ and $\mathbb{E}[{\rho }^{\ast }{S}_1]=S_0$. Among square-integrable weights satisfying these equations it has the smallest squared norm:For another admissible weight $\rho$, its difference from ${\rho }^{\ast }$ is orthogonal to $1$ and all coordinates of $S_1$, hence to ${\rho }^{\ast }$. The Pythagorean theorem in an inner-product space gives $\mathbb{E}[{\rho }^2]=\mathbb{E}[({\rho }^{\ast }{)}^2]+\mathbb{E}[(\rho -{\rho }^{\ast }{)}^2]$. This weight can be signed; positivity, and hence interpretation as a state-price density, requires additional assumptions.

If $S_1$ has a multivariate normal distribution with mean $\mu$ and invertible covariance $V$, and $g$ has bounded gradient, Gaussian integration by parts gives $\mathrm{Cov}(S_1,g(S_1))=V\mathbb{E}[\nabla g(S_1)]$. Consequently the risky holdings of its one-period quadratic hedge equal the expected payoff gradient. The initial cost is $R^{-1}\mathbb{E}[g(S_1)]+(S_0-R^{-1}\mu )\cdot \mathbb{E}[\nabla g(S_1)]$. Bounded gradient ensures linear growth and square integrability, even if the payoff is not bounded.