For every singular infinite cardinal number $\kappa$, the Gimel function has the smallest value permitted by König theorem for cardinal numbers and the exponent: ${\kappa }^{\mathrm{cf}(\kappa )}=\max \{{\kappa }^{+},2^{\mathrm{cf}(\kappa )}\}$. Where $2^{\mathrm{cf}(\kappa )}<\kappa$, this is the singular cardinals hypothesis. The hypothesis imposes no separate successor-power condition on regular cardinals.

The Gimel function determines all infinite cardinal powers. For a regular base, $2^{\kappa }=\gimel (\kappa )$. For a singular base, put $s={\sup }_{\rho <\kappa }{2}^{\rho }$ and $\theta =\mathrm{cf}(\kappa )$; then $2^{\kappa }=s^{\theta }$, which is $s$ if attained below $\kappa$ and $\gimel (s)$ otherwise. After these powers are known, fix an infinite exponent $\lambda$ and recurse on the base. Below $\lambda$ use $2^{\lambda }$; at successors use the Hausdorff formula for cardinal exponentiation. At a limit base greater than $\lambda$, put $a={\sup }_{\rho <\kappa }{\rho }^{\lambda }$. The answer is $a$ if $\mathrm{cf}(\kappa )>\lambda$ or the supremum is attained, and $\gimel (a)$ otherwise. In the nonattained cases, the cofinal-index argument identifies the cofinality of the supremum.

A blocked Gibbs sampler draws groups of coordinates jointly from their full conditional distributions. Each update preserves the joint probability distribution; dependence between blocks can still cause a long mixing time.

Every full-conditional coordinate update satisfies detailed balance with the target distribution. Their systematic ordered composition preserves the target but need not satisfy detailed balance, since compositions of reversible kernels need not be reversible. A fixed-probability random-scan mixture remains reversible.

For a strictly positive posterior on finitely many binary spins, a Random-scan Gibbs sampler uses the two local posterior weights to redraw one spin. Positive full conditional distribution probabilities allow every finite sequence of spin changes, giving an irreducible Markov chain with positive self-transition probabilities.

For Gaussian precision $\left(\begin{array}{cc}A& C\\ C& D\end{array}\right)$, a systematic update of the first coordinate followed by the second makes the centered second coordinate an AR(1) chain with coefficient $C^2/(AD)<1$. Independent conditional Gaussian noise supplies its stable innovation. This proves convergence of the Gaussian Gibbs sampler and explains slower mixing when posterior correlation has magnitude near one.

For two agents and a strategyproof social choice function, if an alternative is selected when one agent ranks it first and the other ranks it last, it is selected whenever the first agent ranks it first, whatever the other agent reports. Changing the first agent's order preserves its attainable top choice; changing the second agent's order cannot offer it an improvement over a selected bottom alternative.

For a random walk with independent and identically distributed increments and an exit time from $(0,r)$, suppose $p=\mathbb{P}(X_1>r/m)>0$ for some integer $m\ge 1$. A block of $m$ such positive increments forces an exit. Conditional on survival, every successive block has success probability at least $p^m$, so $\mathbb{P}(T>km)\le (1-p^m{)}^k$ and $\mathbb{E}T\le m/p^m$. This is an instance of the geometric tail bound from a uniform escape probability; centering of the increments is not required.

For the geometric-jump countdown Markov chain, regeneration at zero givesA cycle visits positive state $j$ precisely when its jump reaches at least $j$, with probability $q^{j-1}$. Dividing this expected occupation by the mean cycle length proves the formula. Mean recurrence times are $1/{\pi }_j$, and the limiting probability away from zero is $1/(1+p)$.

For an aperiodic positive Harris recurrent Markov chain with geometric ergodicity, an observable with stationary moment $\pi (|h{|}^{2+\delta })<\infty$ for some $\delta >0$ satisfies a central limit theorem. Its asymptotic variance is ${\gamma }_h(0)+2{\sum }_{k\ge 1}{\gamma }_h(k)$, with stationary covariances. Ergodicity alone does not guarantee this theorem.

The orbital variation of a planet's reflected stellar light as the visible illuminated area changes. In a simple spherical model, $F_p/F_{\ast }=A_g(R_p/a{)}^2\Phi (\alpha )$, where $\Phi (0)=1$ and $\alpha$ is the star-planet-observer angle. Exoplanet clouds can alter the geometric albedo and the angular scattering law; this observable differs from an exoplanet thermal phase curve.

The right lifting property for each sphere-to-disk map fills a prescribed cycle and a compatible target element. The degree-zero generator supplies initial surjectivity; all other fillers force an acyclic kernel and the remaining degreewise surjections.

A Gaussian linear mixed model combines fixed effects, random effects with a normal distribution, and independent Gaussian noise. For $b\sim N(0,D)$ and $\epsilon \sim N(0,R)$ independently, its marginal distribution is $Y\sim N(X\beta ,V)$ with $V=ZD{Z}^T+R$. The random effects induce dependence among observations sharing their design columns. A random-intercept linear mixed model and a random-slope linear mixed model are special cases.

The backfitting algorithm estimates each additive mean function by smoothing its partial residual, obtained after subtracting the current fits of all other terms, and cycling through terms until convergence. Centering each smooth separates its constant part from the intercept. For fixed quadratic smoothness penalties it is block coordinate minimization of a penalized least-squares objective; identifiability and positive definiteness of the constrained problem ensure a unique converged fit. Non-Gaussian generalized additive models can use weighted backfitting inside iteratively reweighted least squares.

For a nonzero finitely generated Weyl algebra module, choose a finite-dimensional generating subspace $M_0$. Under the Bernstein filtration, $\dim (F_j{A}_n{M}_0)$ is eventually polynomial because the graded module is finite over a polynomial ring. Its degree is independent of the generating subspace, since two generating filtrations bound each other after fixed shifts. It equals the Gelfand–Kirillov dimension of a module.

Observe $Y(t)={\int }_0^{t}f(s)ds+\sigma W(t)$ on $[0,1]$, where $f\in L^2[0,1]$ is an unknown deterministic drift and $W$ is standard Brownian motion. The observation against a deterministic test function is $Y(h)=⟨f,h⟩+\sigma W(h)$, with covariance ${\sigma }^2⟨h,g⟩$. Gaussian white noise is represented by an isonormal Gaussian process on test functions rather than a pointwise noise function. Taking $\sigma =n^{-1/2}$ is the usual statistical scaling.

Let $C=\sqrt{|D_K|}N(\mathfrak{b})$ and $n=[K:\mathbb{Q}]$. Normalize the Minkowski embedding of a number field to the covolume-one Euclidean lattice ${\Lambda }_{\mathfrak{b}}=C^{-1/n}j(\mathfrak{b})$, using the metric with complex coordinates weighted by $2$. Its theta function isWriting $\Vert y\Vert ={\prod }_v{y}_v^{[K_v:\mathbb{R}]}$ and using the trace dual of a fractional ideal gives $\Theta (y,\mathfrak{b})=\Vert y{\Vert }^{-1/2}\Theta (y^{-1},{\mathfrak{b}}^{\vee })$. Its small-parameter asymptotic of a lattice theta sum has leading coefficient one. For the unscaled ideal lattice, the leading coefficient is instead $C^{-1}$; one must specify the normalization when quoting this limit.

If every eigenvalue of a positive definite symmetric matrix $A$ tends to zero, then the anisotropic theta functional equation and the dominated convergence theorem show that $(\det A{)}^{1/2}{\theta }_{\Lambda }(A)\to \mathrm{covol}(\Lambda {)}^{-1}$. Indeed all eigenvalues of $A^{-1}$ tend to infinity, the zero term of the dual Gaussian theta sum is one, and its remaining terms tend to zero while being dominated by a fixed summable Gaussian. Merely requiring $\det A\to 0$ is insufficient.

An independent unit-scale Rayleigh distribution variable $R$ and standard normal distribution variable $Z$ give $RZ$ with Laplace distribution of density $e^{-|x|}/2$. Indeed its characteristic function is $\mathbb{E}{e}^{-t^2{R}^2/2}=1/(1+t^2)$.

A standard normal random variable has the standard normal distribution, mean zero and variance one. If $X\sim N(\mu ,{\sigma }^2)$ with $\sigma >0$, the standardization of a normal random variable $(X-\mu )/\sigma$ is a standard normal random variable. Its values report displacement from the mean in units of the standard deviation, so the result is a dimensionless quantity.