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Uniform geometric ergodicity strengthens geometric ergodicity by bounding the prefactor uniformly over the starting state. If $P(x,\cdot )\ge \epsilon \mu (\cdot )$ globally and $\mu$ is invariant, thenWrite $P=\epsilon \Pi +(1-\epsilon )R$, where $\Pi$ draws independently from $\mu$, to obtain the bound.

The phase integral converts a monochromatic geometric albedo into the corresponding reflected power fraction. The bolometric Bond albedo is the incident-spectrum-weighted average of $q_{\lambda }{A}_{g,\lambda }$.

Given a Jacobi field with initial values $J(0)=a$ and $D_{t}J(0)=b$, vary the initial point along a curve with tangent $a$ and the initial velocity with covariant derivative $b$. The resulting geodesic variation has the same Jacobi initial values and therefore the same field by uniqueness of the linear equation. Smooth dependence on initial conditions gives a common parameter interval around any fixed compact geodesic segment, even on an incomplete manifold.

For a forcing name $\tau$, let $U$ consist of pairs $(\rho ,p)$ where $(\rho ,r)\in \tau$ for some $r\ge p$. Every ground-model subset $\nu$ of $U$ is a name whose value is contained in ${\tau }^G$. Every subset ${\sigma }^G\subseteq {\tau }^G$ has an equivalent such name: retain those $(\rho ,p)\in U$ with $p⊩\rho \in \sigma$. The atomic membership truth lemma for forcing proves equality of values. Collect all these names from the ground-model power set ${\mathcal{P}}^M(U)$ into a single outer name, pairing each with every condition. Its value is the full internal power set, without presupposing that power set in the extension.

Let $M_{n+1}=M_n[G_n]$ for a fixed atomless forcing order $\mathbb{P}$, and $N={\bigcup }_{n<\omega }{M}_n$. If a set $A\in N$ were its internal power set of $\mathbb{P}$, choose $n$ with $A\in M_n$. The next generic filter $G_n$ belongs to $N$, so $G_n\in A$; transitivity of $M_n$ implies $G_n\in M_n$, contradicting generic filter for an atomless order is new. The chain is increasing but need not be an elementary chain.

A Jordan chain of a linear operator $T$ for an eigenvalue $\lambda$ is a list of nonzero vectors satisfying $(T-\lambda I)v_1=0$ and $(T-\lambda I)v_j=v_{j-1}$ for $2\le j\le r$. These vectors are linearly independent: in a vanishing linear combination, applying the largest relevant power of $T-\lambda I$ isolates the last coefficient times $v_1$, then descending induction removes all coefficients. On their span the operator has one Jordan block. This is different from a generalized eigenfunction that solves $(T-\lambda I)u=0$ outside the original function space.

A generalized additive mixed model supplements smooth population effects with latent random effects. Given these effects, responses follow a specified response family; integrating them out induces dependence between related observations.

Let $T$ be a standard Young tableau, $s_i=(ii+1)$, and $d=c_T(i+1)-c_T(i)$. A suitable Gelfand–Tsetlin basis has, for an admissible pair $R=s_{i}T$ with tableau length increasing,For a nonadmissible swap the scalar is $+1$ in a row and $-1$ in a column. The diagonal coefficient follows from the Young–Jucys–Murphy element relation $s_i{X}_i{s}_i+s_i=X_{i+1}$, and $s_i^2=1$ forces the product of off-diagonal coefficients. One global normalization is $v_T=P_T{\pi }_T{v}_{T_0}$ from the row-reading tableau: a reduced admissible path makes this vector nonzero and gives coefficient one on every length-increasing edge.

Generalized Gaussian white noise over a real Hilbert space $H=L^2$ is an isonormal Gaussian process $W(h)$ with covariance $⟨h,g{⟩}_H$. Its law $P_0$ is carried on a larger observation space; the identity is not the covariance of an infinite-dimensional $H$-valued Gaussian measure because it is not a trace-class operator. A deterministic signal $h\in H$ defines a translated observation law $P_h$ with Radon-Nikodym derivativeIn an orthonormal basis, $W_m(h)={\sum }_j{m}_j{h}_j$ is a stochastic series with total variance $\Vert h{\Vert }_H^2$, not an inner product of two $H$-valued observations. The formula follows from the Cameron-Martin theorem for a Gaussian measure in its generalized white-noise version, or from finite-dimensional Gaussian likelihood ratios. On ${\mathbb{R}}^2$, generalized noise can be realized on tempered distributions; an unweighted global negative Sobolev space is not automatically a suitable almost-sure support on an unbounded domain. The stochastic series avoids imposing that unsupported regularity.

If a linear functional $Z=\int a(\mathbf{r})V(\mathbf{r})d\mathbf{r}$ of a real Gaussian random field exists in mean square, it is a Gaussian random variable. Completing the square in its density givesFor a complex linear functional $\varphi$, apply this to $Z=2\mathrm{Re}\varphi$; then $\frac{1}{2}\mathrm{Var}Z=\mathbb{E}|\varphi -\mathbb{E}\varphi {|}^2+\mathrm{Re}\mathbb{E}(\varphi -\mathbb{E}\varphi {)}^2$.

For a centered Gaussian measure on a real Hilbert space with injective covariance operator of a Gaussian measure $\Sigma$, its Cameron-Martin space is $E_{\mu }=\mathrm{Ran}{\Sigma }^{1/2}$, with inner product $⟨h,g{⟩}_{E_{\mu }}=⟨{\Sigma }^{-1/2}h,{\Sigma }^{-1/2}g⟩$. In an orthonormal basis satisfying $\Sigma e_j={\lambda }_j{e}_j$, it consists of exactly the vectors $h$ for which ${\sum }_j{h}_j^2/{\lambda }_j<\infty$. It is complete in this stronger norm even if its range is not closed in the ambient norm. Its importance is that it describes precisely the translations that preserve the null sets of the Gaussian measure, as in the Cameron-Martin theorem for a Gaussian measure. For generalized Gaussian white noise over $L^2$, the corresponding Cameron-Martin space is L2 space itself.

For $\mu =\mathcal{N}(0,\Sigma )$ with injective covariance operator of a Gaussian measure, its translated law ${\mu }_h=\mathcal{N}(h,\Sigma )$ is an equivalent probability measure to $\mu$ if and only if $h\in E_{\mu }$, the Cameron-Martin space of a Gaussian measure. For such $h$,The series has mean-square convergence and converges almost surely under $\mu$, with normal distribution $\mathcal{N}(0,\Vert h{\Vert }_{E_{\mu }}^2)$. Its exponential density has expected value one by the moment-generating function of a normal distribution. Finite-dimensional projections give the formula by ratios of multivariate normal densities; convergence of these likelihood ratios gives the infinite-dimensional result. The expression $⟨h,x{⟩}_{E_{\mu }}$ is only formal when the sample is outside $E_{\mu }$. For $h\notin E_{\mu }$, translation gives mutually singular measures.

The Dawson function obeys $F_D^{\prime }+2z{F}_D=1$ by differentiation. Its large-real-argument asymptotic expansion begins $F_D(z)=1/(2z)+1/(4z^3)+O(z^{-5})$. This inverse-power behavior contrasts with the separate exponentially growing factors in its definition.

For $m=Au+\eta$, with independent $u\sim \mathcal{N}(0,I_d)$ and $\eta \sim \mathcal{N}(0,{\delta }^2{I}_k)$, the multivariate normal distribution of the Bayesian posterior has posterior mean $b_{\delta }=(A^{T}A+{\delta }^2{I}_d{)}^{-1}{A}^{T}m$ and covariance matrix $C_{\delta }={\delta }^2(A^{T}A+{\delta }^2{I}_d{)}^{-1}$. The singular value decomposition shows that the variance on a right singular vector of nonzero singular value $s$ is ${\delta }^2/(s^2+{\delta }^2)$, while the variance along the null space is one. Thus, for fixed data, the mean tends to the minimum-norm least-squares solution and the covariance tends to the orthogonal projection onto the null space. Unobserved components retain their prior distribution even as the noise vanishes.

Let $m\mid u,\gamma \sim \mathcal{N}(Au,{\gamma }^{-1}{I}_k)$, let the prior distribution of the precision be the shape-rate Gamma distribution $\mathrm{Gamma}(\alpha ,\beta )$, and use a constant improper prior for $u\in {\mathbb{R}}^d$. Assume $A$ has zero null space. Write $B=A^{T}A$, $u_{\ast }=B^{-1}{A}^{T}m$, $a=\alpha +k/2$, $b_0=\beta +\Vert m-A{u}_{\ast }{\Vert }^2/2$, and $s=\alpha +(k-d)/2>0$. The proper joint posterior density is proportional toIts conditional distributions are $u\mid \gamma ,m\sim \mathcal{N}(u_{\ast },{\gamma }^{-1}{B}^{-1})$ and $\gamma \mid u,m\sim \mathrm{Gamma}(a,\beta +\Vert m-Au{\Vert }^2/2)$. Its marginal precision law is $\mathrm{Gamma}(s,b_0)$. Integration of the displayed kernel gives $(2\pi {)}^{d/2}\Gamma (s)/(\sqrt{\det B}{b}_0^s)$, proving propriety despite the improper prior.

A joint maximum a posteriori estimate relative to $dud\gamma$ exists exactly when $a>1$, and is $(u_{\ast },(a-1)/b_0)$. If $a=1$, the supremum is approached at $\gamma =0$ but not attained; if $a<1$, the density is unbounded there. Separate marginal maximization gives $u_{\ast }$ for the unknown and $(s-1)/b_0$ for the precision when $s>1$. This illustrates that joint and marginal modes need not agree, and that a proper posterior distribution need not have a mode in its open parameter space.

For $\Re c>\Re b>0$ and a fixed branch away from the cut,Expand $(1-zs{)}^{-a}$ for $|z|<1$ and integrate termwise using the beta function to recover the Gauss hypergeometric function series. Analytic continuation extends the identity in $z$.

For an involutive distribution of constant rank $r$, induct on $r$. Straighten a nonzero section by the flow-box theorem, and choose a frame ${\partial }_1,Y_2,\dots ,Y_r$ with no ${\partial }_1$ component in the remaining fields. Involutivity implies ${\partial }_{1}Y=AY$ for their column $Y$. The invertible solution of ${\partial }_{1}B=-BA$, initialized by $B=I$ on $x_1=0$, makes $Z=BY$ independent of $x_1$. Its restriction to the transverse slice is involutive of rank $r-1$, so induction provides coordinates spanning it. Extending these coordinates independently of $x_1$ gives $D=\mathrm{span}({\partial }_1,\dots ,{\partial }_r)$. Conversely, fields tangent to an integral manifold have tangent brackets because they preserve the ideal of smooth functions vanishing on it.

The Fox derivative of a word in a free group obeys $\partial (uv)/\partial x=\partial u/\partial x+u\partial v/\partial x$ and $\partial x^{-1}/\partial x=-x^{-1}$.

For $u(x)=\frac{1}{2}\log (1+x^2)$, the angular-frequency transform is the tempered distributionThe numerator cancels the singularity at zero and makes this integral absolutely convergent. Differentiating $u$ and using $\widehat{(1+x^2{)}^{-1}}=\pi e^{-|\xi |}$ determines this expression up to a Dirac delta distribution. That delta coefficient is zero: testing with the expanding Gaussian ${\phi }_n(x)=(2\sqrt{\pi }{)}^{-1}{e}^{-x^2/(4n)}$ makes both the proposed expression and $⟨u,{\hat{\phi }}_n⟩$ tend to zero, while ${\phi }_n(0)$ stays fixed. Changing the subtraction convention would change the delta coefficient.

For $D_j=-i{\partial }_j$, the Fourier transform of a tempered distribution satisfiesThe base cases follow from $⟨D_{j}u,\phi ⟩=i⟨u,{\partial }_j\phi ⟩$ and integration by parts in the Fourier integral. Iterating gives the multi-index formulas. The $D$ convention matters: for the unscaled derivative, $\widehat{{\partial }_{j}u}=i{\xi }_j\hat{u}$.