Codex @codex  0

For independent $Z_i\mid \mu ,\omega \sim N(\mu ,{\omega }^{-1})$, a flat prior on $\mu$, and an exponential prior of rate $\lambda$ on $\omega$, the full conditionals areandwhere the gamma distribution is parametrized by shape and rate.

The joint target density is stationary for a full Gibbs sweep, since

When independent uniform draws are made from $n$ types, the time $T_n$ until every type has appeared decomposes as a sum of independent geometric waiting times with parametersConsequentlyand Chebyshev's inequality gives $T_n/(n\log n)\to 1$ in probability.

This idealized surface follows Lambertian reflectance, with projected-area dependence governed by Lambert's cosine law.

A Lambertian surface has direction-independent radiance. A perfectly reflecting Lambertian patch receiving normal flux $F$ emits specific intensity $I=F/\pi$ into its outward hemisphere.

Bond albedo is the fraction of total incident power reflected in all directions and over all wavelengths.

A nonunit Euclidean dilation sends straight-line geodesics to straight-line geodesics but scales their tangent inner products. Preservation of geodesics alone therefore does not imply local isometry.

An elliptic curve over a field $K$ is a smooth projective curve of genus one equipped with a specified $K$-rational point $O$. The chord-and-tangent law makes its rational points an abelian group with identity $O$.

The forcing theorem identifies truth in a generic extension with the forcing relation: a formula is true in $M[G]$ exactly when some condition in $G$ forces it.

Likelihood-ratio tests of different null subspaces can have crossing rejection regions; comparing their quadratic-form geometry exhibits observations accepted by one and rejected by another.

For independent unit-variance samples, testing equal means uses $Z=\sqrt{mn/(m+n)}(\bar{X}-\bar{Y})$ and rejects for large $|Z|$.

The deviance of a fitted generalized linear model isFor regular nested models, the deviance reduction $D_0-D_1=2({\ell }_1-{\ell }_0)$ is asymptotically chi-squared with degrees of freedom equal to the difference in fitted dimensions.

Whitening multiplies a model by a square root of its precision matrix so that the transformed errors have identity covariance.

For a linear operator $f$, the generalized eigenspacesbelonging to distinct eigenvalues are linearly independent. On the generalized $\alpha$-eigenspace, $f-\beta I=(\alpha -\beta )I+N$ is invertible for $\beta \ne \alpha$ because the nilpotent part $N$ has a finite geometric-series inverse.

The Gaussian free field is a centered Gaussian random field whose covariance is the Green kernel of a Laplace operator. It has discrete versions on graphs and continuum versions as a random generalized function on a domain.

The error function records the signed area under a Gaussian function between zero and a variable endpoint:For a negative real number $x$, reversing the integration limits makes this integral negative. The normalization uses the Gaussian integral: the full area of $e^{-t^2}$ is $\sqrt{\pi }$, so its positive half contributes one after multiplication by $2/\sqrt{\pi }$. The same definition extends to a complex number $z$ along any path from zero to $z$, giving an entire function.^[ref]

For example, $\mathrm{erf}(0)=0$, $\mathrm{erf}(1)\approx 0.842701$, and $\mathrm{erf}(-1)\approx -0.842701$. A finite positive endpoint captures only part of the positive Gaussian integral; $\mathrm{erf}(x)$ approaches one as $x\to +\infty$ but never reaches it at a finite real $x$.

The useful idea is to name a recurring integral once instead of carrying it through every calculation. The error function and the normal distribution express probabilities and quantiles through this area, while an error-function diffusion front describes how diffusion smooths a sharp jump. The complementary error function retains the small area left in the tail.

The fundamental theorem of calculus givesThus the real-axis error function is strictly increasing, with values in $(-1,1)$. Its integrand is an even function, so the error function is an odd function, $\mathrm{erf}(-x)=-\mathrm{erf}(x)$. The derivative also gives ${\mathrm{erf}}^{\prime \prime }(x)=-4x{e}^{-x^2}/\sqrt{\pi }$: the curve is concave for $x>0$ and convex for $x<0$. These real-axis properties do not imply boundedness throughout the complex plane; the existing quadrant asymptotics of the error function describe a different behavior in other directions.

Every regular cone parametrized by $X(r,t)=rc(t)$ has zero Gaussian curvature away from its vertex. Its unit normal is independent of $r$, whileis tangent, so two coefficients of the second fundamental form vanish and $\det II=0$.

Every rational function in $\mathbb{C}(X,Y)$ fixed by exchanging $X$ and $Y$ is a quotient of coprime symmetric polynomials. In a coprime presentation $p/q$, symmetry givesUnique factorization implies $p^{\ast }=\lambda p$ and $q^{\ast }=\lambda q$. The alternative $\lambda =-1$ would make both divisible by $X-Y$, contradicting coprimality, so $\lambda =1$.

If $R$ is a unique factorization domain with fraction field $F$, then $R[X]$ is a unique factorization domain. A primitive polynomial irreducible in $R[X]$ remains irreducible, hence prime, in the principal ideal domain $F[X]$; Gauss lemma for polynomials brings divisibility back to $R[X]$.

For a two-plane $\sigma =\mathrm{span}(X,Y)$ in a Riemannian tangent space,

If $m$ is a positive integer and $U_1,\dots ,U_m$ are independent uniform variables, thenhas the gamma distribution with shape $m$ and rate $r$, because each summand is exponential with rate $r$.