Let $F(k)={\int }_0^{\infty }{e}^{-ikx}{q}_0(x)dx$ and $G_0(k,t)={\int }_0^t{e}^{w(k)s}{g}_0(s)ds$. Using the roots of cubic dispersion symmetry with negative transport, put $\mathcal{T}F=[(k-{\nu }_2)F({\nu }_1)+({\nu }_1-k)F({\nu }_2)]/({\nu }_1-{\nu }_2)$. The Fokas method formula is an initial real-line integral minus an integral over $\partial D_{+}$ of $e^{ikx-wt}[\mathcal{T}F+(3k^2+1)G_0]$, both divided by $2\pi$. The boundary is oriented with $D_{+}$ on the left. Its unknown final-state transform has zero integral by analyticity and upper-half-plane decay.

A physical unitary gate used without coherent control gives the same quantum channel for $V$ and $e^{i\gamma }V$:Interleaving any preparations, known operations and measurements preserves this equality. Equivalently, each fixed measurement branch with $k$ uses gains only the overall factor $e^{ik\gamma }$, which cancels in its probability. Thus unrestricted repeated uncontrolled uses do not determine the common phase of the eigenvalues. A supplied controlled unitary gate is a different resource: $|0⟩⟨0|\otimes I+|1⟩⟨1|\otimes V$ changes by a relative control phase when $V$ is replaced by $e^{i\gamma }V$. Such controlled access permits quantum phase estimation to measure that phase relative to the identity branch.

For $|z|>1$, both logarithms in the display can use their principal logarithms. The real part is $\log (|z+1{|}^2/|z-1{|}^2)$ and the derivative is $2/(z+1)-2/(z-1)$. The two residues cancel on a loop surrounding both points. Equivalently $f=4{\sum }_{m\ge 0}{z}^{-(2m+1)}/(2m+1)$ is globally single-valued outside the unit circle, although individual logarithms of $z+1$ and $z-1$ are not.

// Destination: complex-dynamics.bigb

The critical region is the range of control parameters near a thermodynamic critical point where correlated order parameter fluctuations invalidate a simple mean-field approximation. For short-range scalar quartic theory below four dimensions, the dimensionless Ginzburg criterion ratio grows as $g{\xi }^{4-D}$ in fixed microscopic units. The range where this ratio is of order one or larger is fluctuation dominated. Its width depends on nonuniversal microscopic coefficients even though the limiting critical exponents can be universal.

For the dimensionless scalar Landau-Ginzburg theory with stiffness $c$, quadratic coefficient $r<0$ and quartic term $u{\varphi }^4/4$, the mean-field approximation gives ${\varphi }_0^2=|r|/u$ and correlation length $\xi \asymp (c/|r|{)}^{1/2}$. Restricting the Gaussian Fourier transform covariance $(c{q}^2+2|r|{)}^{-1}$ to $|q|\lesssim {\xi }^{-1}$ estimates the variance of the field averaged over a correlation volume as $c^{-d/2}|r{|}^{(d-2)/2}$, up to a dimension-dependent constant. Dividing by ${\varphi }_0^2$ proves the displayed Ginzburg criterion ratio. It diverges below four dimensions, vanishes above four and is marginal at four. Short-distance fluctuations renormalize microscopic coefficients and should not be confused with this long-distance critical test.

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 Thurston geometries are the simply connected, three-dimensional homogeneous model spacesThe first three have constant positive, zero and negative sectional curvature. The product models and the three remaining Lie-group models allow additional locally homogeneous metrics. A complete geometric structure is a quotient of the entire model by a discrete freely acting group of isometries.

Compatible model charts define a local geometric structure; to recover a quotient of the entire model one also needs completeness. An open unit ball with its usual Euclidean metric has compatible Euclidean charts but is incomplete, since a radial geodesic reaches its missing boundary in finite length. This distinguishes local coordinate geometry from a complete geometric piece in the geometrization theorem.

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.