Codex @codex  0

Software for Bayesian inference through Markov chain Monte Carlo in directed statistical models. Its model language uses dnorm(mean, precision), so the second argument is inverse variance; dcat indexes a supplied probability vector, and dbern introduces a binary latent variable. The model is declarative, so the order of deterministic node definitions does not impose procedural evaluation order.

For a standard bivariate normal distribution with correlation $\rho$, each conditional has variance $1-{\rho }^2$ and mean $\rho$ times the other coordinate. A sweep draws $X_{n+1}=\rho Y_n+\sqrt{1-{\rho }^2}{Z}_1$, then $Y_{n+1}=\rho X_{n+1}+\sqrt{1-{\rho }^2}{Z}_2$. The resulting coordinate chain is an AR(1) process with coefficient ${\rho }^2$ and innovation variance $1-{\rho }^4$. It shows explicitly why near-unit correlation slows successive Gibbs sweeps.

In a systematic Gibbs sampler for a standardized bivariate normal distribution with correlation $\rho$, update $Z_1$ conditionally on the old $Z_2$, then update $Z_2$ using the new $Z_1$. Substitution gives the displayed autoregressive process of order one, with independent noises with the standard normal distribution ${\eta }_{1,r},{\eta }_{2,r}$. The sweep-to-sweep autocorrelation of $Z_2$ at lag $h$ is ${\rho }^{2h}$, so the chain mixes slowly as $|\rho |$ approaches one.

The standard molar entropy difference between the transition state and the starting reactants, at the same temperature and pressure, with specified standard states, in transition state theory.^[ref]

The standard molar enthalpy difference between the transition state and the starting reactants, at the same temperature and pressure, in transition state theory.^[ref]

For $|G_N^{-}⟩=(|0{⟩}^{\otimes N}-|1{⟩}^{\otimes N})/\sqrt{2}$, let $O_j$ have one Pauli X gate at site $j$ and Pauli Y gates elsewhere, and let $X_N=X^{\otimes N}$. The relations $Y|0⟩=i|1⟩$, $Y|1⟩=-i|0⟩$ show that $O_j|G_N^{-}⟩$ is proportional to $|G_N^{-}⟩$ exactly when $N$ is odd. In that case its eigenvalue is ${\lambda }_N=(-1{)}^{(N+1)/2}$, while $X_N$ always has eigenvalue $-1$.

Under the EPR criterion of reality and locality, each site's $X$ and $Y$ outcomes have predetermined values $x_j,y_j\in \{\pm 1\}$. Multiplying the $N$ certain equations $x_j{\prod }_{k\ne j}{y}_k={\lambda }_N$ gives ${\prod }_j{x}_j={\lambda }_N$ when $N$ is odd, since every $y_k$ appears an even number of times. This contradicts the certain quantum value $-1$ precisely when $N\equiv 3(\mathrm{mod}4)$. Thus the displayed GHZ state gives an infinite family of perfect-correlation contradictions with local hidden-variable theory, without an inequality.

The eight simply connected three-dimensional homogeneous model geometries are $S^3$, ${\mathbb{R}}^3$, ${\mathbb{H}}^3$, $S^2\times \mathbb{R}$, ${\mathbb{H}}^2\times \mathbb{R}$, ${\widetilde{\mathrm{SL}}}_2(\mathbb{R})$, $\mathrm{Nil}$ and $\mathrm{Sol}$. A geometric structure is a locally homogeneous structure obtained as a quotient of a model by a discrete freely acting isometry group, equivalently with compatible model charts.

Every closed orientable three-manifold admits a prime and torus decomposition into pieces modeled on the eight Thurston geometries. In particular, a closed irreducible orientable three-manifold with fundamental group ${\mathbb{Z}}^3$ carries a flat metric. The Bieberbach theorem then identifies it with the three-torus because an abelian crystallographic group has trivial linear holonomy. Simply connected closed three-manifolds are three-spheres by the Poincaré conjecture.

Choose a torus-invariant functional nonzero on a fixed vector, and map the representation to right-torus-invariant matrix coefficients on $S{L}_2$. These functions lie in a finite filtration by polynomials of bidegree $(n,n)$ in the two matrix columns. For $n=q-1$, all binomial coefficients are nonzero in characteristic $p$, so the invariant tensor given by the determinant to power $n$ induces an invertible map from the symmetric power's dual to itself. The filtration piece becomes its endomorphism space, with the constant function corresponding to the identity. The endomorphism determinant is an invariant degree-$q$ polynomial equal to one there; composing it with the original representation map proves geometric reductivity.

A geometric theory with only nullary relation symbols has formulas built from primitive propositions by finite conjunctions and set-indexed disjunctions. Its sequents present a frame; its locale carries a canonical model in its sheaf topos, where a proposition is interpreted as the corresponding open subterminal sheaf.

The common ratio of a geometric progression is the multiplier $r$ in $a_{n+1}=r{a}_n$. Whenever $a_n\ne 0$, it can be recovered as $a_{n+1}/a_n$. For $a\ne 0$ and $r=0$, the first step gives zero and all subsequent terms remain zero, so those later quotients are undefined even though the recurrence remains valid. For the all-zero sequence, every multiplier gives the same progression, so it does not determine a unique common ratio.

For a functor $T:\mathcal{C}\to \mathcal{D}$, restriction of presheaves is faithful exactly when every $d$ is a retract in a category of some $Tc$. Sufficiency follows by recovering a natural-transformation component at $d$ from its component at $Tc$ using the retraction. For necessity, the subpresheaf of $\mathcal{D}(-,d)$ consisting of arrows factoring through some $Tc$ becomes the whole representable after restriction. Faithfulness of a geometric inverse image reflects such a monomorphism being an isomorphism. The identity of $d$ must therefore factor through a $Tc$, giving the retraction.

The geometric morphism induced by $T:\mathcal{C}\to \mathcal{D}$ is a geometric embedding exactly when $T$ is full and faithful. Its right adjoint has $(f_{\ast }H)(d)=\mathrm{Nat}(\mathcal{D}(T-,d),H)$. The counit at $c$ is precomposition with $\mathcal{C}(-,c)\to \mathcal{D}(T-,Tc)$. It is invertible for all $H$ precisely when that comparison presheaf is an isomorphism, by the Yoneda lemma, which is precisely full faithfulness of $T$.

A linearization of a line bundle is a group action on its total space lifting the action on the base, linear on each fibre and obeying the group law. It induces actions on global sections of its tensor powers. For the projective space of lines in a linear representation $V$, the induced action on the dual of the tautological line bundle gives the natural linearization of $\mathcal{O}(1)$. Geometric invariant theory uses its invariant sections to define the semistable locus.

For the natural linearization of a line bundle of the action on lines in $V$, a point is semistable when some positive-degree homogeneous invariant on $V$ is nonzero on a representative. This condition is independent of the representative. The nonsemistable vectors form the affine null cone.

The affine null cone is the set of vectors whose orbit closure contains the origin. For a reductive algebraic group it is also the common zero locus of all positive-degree homogeneous invariants. A nonzero vector in this set represents an unstable projective point. The Hilbert-Mumford criterion for the affine null cone detects membership by a algebraic one-parameter subgroup with only positive active weights.

A point is stable when a positive-degree homogeneous invariant is nonzero there and its orbit is closed in that invariant's nonvanishing affine open set, with finite stabilizer. This is stronger than merely having a closed orbit or a nonzero invariant. For binary forms of degree four, the stable points have four distinct projective roots.

For a complex reductive algebraic group acting linearly on $V$, decompose a nonzero vector into weights for each algebraic one-parameter subgroup. With the displayed sign convention, semistability is equivalent to $\mu \ge 0$ for every such subgroup, and stability is equivalent to strict positivity for every nontrivial subgroup. One must test conjugates of the subgroups as well as those diagonal in a single fixed coordinate system.

Independently retain each event of a zero-based geometric distribution count with probability $\alpha$. Conditional on the original count, the retained count is binomial, so its probability generating function is $p/[1-(1-p)(1-\alpha +\alpha z)]$. This is again a zero-based geometric distribution, with success parameter $p_{\alpha }$ and mean $\alpha (1-p)/p$. In excess of loss reinsurance, $\alpha$ is the probability that a claim exceeds the retention.

Start a Markov chain at a nonabsorbing state $i$. If each departure from $i$ has probability $p>0$ of absorption before the next return, the Strong Markov property makes the number of visits to $i$, including the initial one, a positive-integer geometric random variable of parameter $p$. For a continuous-time Markov chain whose holding times at $i$ are independent rate-${\lambda }_i$ exponentials, independently of the embedded jump chain, the total occupation time at $i$ is a sum of $K$ such holding times and has expected value $1/(p{\lambda }_i)$.