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Several chains from dispersed initial states, trace plots, autocorrelation, effective sample sizes, and the rank-normalized split potential scale reduction factor $\hat{R}$ assess mixing and disagreement between chains. An $\hat{R}$ close to one is useful evidence but cannot prove convergence or prove posterior propriety. Comparing chains that explore different modes is especially important; monitor each scientifically important observable and its Monte Carlo uncertainty. A long apparent plateau can also occur when an improper-target sampler is drifting toward a boundary.

The hit-and-run sampler moves within a full-dimensional compact convex set by choosing a uniform direction, then a uniform point on the chord through the current point in that direction. Its invariant law is the uniform distribution on the set. Starting in the interior keeps the chain in the interior almost surely. To define absolutely continuous transitions also at boundary points, redraw directions whose chords have zero length. Without this convention, a corner can have a positive holding atom.

For nonempty $X$, the topological mapping cone has this long exact sequence in homology, with ordinary homology for $X,Y$ and reduced homology for $C_f$. In degree zero, the augmentation of $X$ is accounted for by the cone vertex. The sequence follows from the Mayer–Vietoris theorem applied to neighborhoods of the cone and $Y$.

For a cellular map between nonempty CW complexes, the reduced cellular chain complex of its topological mapping cone has differentialHere $C_{-1}(X)=0$ and the degree-zero group identifies a vertex $y$ with the reduced chain $y-v$, where $v$ is the cone vertex. Thus this is the mapping cone of $f_{\#}$, in the displayed ordering of the summands.

For a continuous function $f:{\mathbb{Z}}_p\to {\mathbb{Q}}_p$, its nth Mahler coefficient isThis is the explicit iterate of the forward difference operator. These coefficients are exactly those of its Mahler expansion.

The Mahler coefficients tend to zero, so the displayed series defines a continuous function and has value zero at zero. The Pascal's identity and uniform convergence give $\Delta Jf=f$. Thus the forward difference operator is surjective on continuous functions on the p-adic integers. Its kernel consists of constants: period one implies agreement on the dense nonnegative integers, and continuity then implies constancy. This selects the unique discrete antiderivative vanishing at zero.

There is also a direct construction on locally constant functions. If $h$ has period $M=p^r$, put $a_j=h(j)$, $S={\sum }_{i=0}^{M-1}{a}_i$, and $P_j={\sum }_{i<j}{a}_i$. For $x=j+My$ with $0\le j<M$ and $y\in {\mathbb{Z}}_p$, setThis is a continuous function on each residue class. Increasing $j$ gives $Jh(x+1)-Jh(x)=a_j$; at $j=M-1$, the next point has residue zero and quotient $y+1$, giving the same identity. The ultrametric inequality gives $\Vert Jh{\Vert }_{\infty }\le \Vert h{\Vert }_{\infty }$, and $Jh(0)=0$. The construction is independent of the chosen period, since two normalized discrete antiderivatives agree on the nonnegative integers and then on the p-adic integers by continuity. It is linear on the locally constant functions, which form a dense subset in the supremum norm. Completeness therefore extends it to every continuous function, retaining $\Delta J=I$ and the norm bound. This proves surjectivity without first using the Mahler theorem.

If a positive vapour-tip pressure magnitude $p_T$ sets $h_f\sim p_T{l}_p^4/B$, matching $h_f/l_p^2$ to the interior curvature ${\kappa }_i$ gives $l_p\sim (B{\kappa }_i/p_T{)}^{1/2}$. The viscous peeling of an elastic plate law then yields $\dot{R}\sim B^{3/2}{\kappa }_i^{7/2}/(\mu p_T^{1/2})$. For constant injected volume flux $Q$, $V=Qt$ and ${\kappa }_i\sim V/R^4$, henceOrder-one coefficients require solving the local peeling profile. The scaling assumes a front region much shorter than the intrusion radius.

With the axisymmetric clamped-plate deflection and fixed front curvature ${\kappa }_f>0$, a magma intrusion hasThese relations describe the advancing fracture branch, with roof bending rather than hydrostatic pressure controlling the interior load.

For a uniformizer $\pi$ of a non-Archimedean local field $K$ with residue field of cardinality $q$, a Lubin–Tate series is a series $f(T)\in {\mathcal{O}}_K[[T]]$ satisfyingIt determines the associated two-variable Lubin–Tate formal group. A useful example is $f(T)=\pi T+T^q$. For $K={\mathbb{Q}}_p$ and $\pi =p$, the choice $(1+T{)}^p-1$ gives the formal multiplicative group and the Lubin–Tate cyclotomic example.

For a finite simple graph with at least one vertex, the complement theta number has the equivalent semidefinite program formulationsandThe Schur complement and $U=tZ-J$ prove the equivalence; any feasible $t$ is at least one, so division by $t$ is valid. For a graph with an edge, $U/(t-1)$ is the Gram matrix of a strict vector coloring. Consequently $\bar{\vartheta }(G)$ is the strict vector chromatic number, and an ordinary $k$-graph colouring gives $\bar{\vartheta }(G)\le k$ by assigning the colors the vertices of a regular simplex.

For a family of first-order structures, take free filters on the index set as worlds, ordered by inclusion, and product sequences as parameters. Force an atom when its coordinate truth set belongs to the filter. Conjunction is local, implication and negation quantify over filter extensions, and a disjunction is forced when every extension has a further extension forcing one disjunct. Existence is witnessed by a product sequence and universality holds for all sequences at all extensions. With the axiom of choice, induction on formulas identifies forcing with membership of the classical coordinate truth set. Equivalently, every ultrafilter extension satisfies the formula in its ultraproduct. Thus the semantics validates classical logic. A local disjunction rule would fail this identification: a set and its complement can both be absent from a free filter, while their union is the whole index set.

A local solution is a locally defined stochastic process taking values in a specified open domain $U$ and satisfying the stochastic differential equation on every stopped interval before its lifetime. For $dX=b(X)dt+\sigma (X)dB$, the integral equation holds after each announcing stop, with the requisite local drift and noise integrability. A maximal local solution of a stochastic differential equation cannot be extended while remaining in $U$; a finite lifetime can be a boundary hit rather than divergence to infinity.

For a multiplicative subset $S$ of a commutative ring $A$, localization preserves short exact sequences of $A$-modules. Surjectivity follows by lifting a numerator. If a fraction maps to zero, some element of $S$ annihilates its image, so multiplying its numerator by that element puts it into the original kernel. For an injective original map, the same criterion shows its localized kernel is zero. Thus $S^{-1}A$ is a flat module over $A$.

For a long potential approximately constant over the short-scale region, local-type primordial non-Gaussianity gives ${\phi }_s=(1+2f_{\mathrm{NL}}{\phi }_{G,L}){\phi }_{G,s}$, to first order in the long mode and ignoring short-mode squares. A linear cosmological Poisson equation therefore givesto the same order. The variance is conditional on the long mode.

If the linear density contrast obeys $\delta (\mathbf{k})=\alpha (k)\phi (\mathbf{k})$, then local-type primordial non-Gaussianity contributesIn a squeezed bispectrum configuration and for a nearly scale-invariant potential, the leading term is $4f_{\mathrm{NL}}{P}_m(k_L)P_m(k_S)/\alpha (k_L)$. This excludes the standard perturbation theory in cosmology contribution from nonlinear gravity.

A local isometry from a nonempty complete Riemannian manifold to a connected Riemannian manifold is a surjective covering map. Lift radial geodesics in a normal ball using completeness; the exponential map at each inverse-image centre maps a tangent ball diffeomorphically onto a sheet. Uniqueness of the geodesic initial-value problem makes these sheets disjoint and exhaustive. Surjectivity follows by lifting piecewise geodesic paths. Conversely a Riemannian covering of a complete target is complete, by lifting complete target geodesics and using the Hopf-Rinow theorem. If the target is disconnected, the first assertion gives a covering only of the union of components met by the map.

Near a point where a smooth vector field is nonzero, there are coordinates in which it is $\partial /\partial x_1$. Flow a transverse coordinate slice by the local flow; the inverse function theorem makes the resulting map a coordinate chart. This is the first step of the coordinate proof of the Frobenius theorem.

For an equilibrium process, a path and its physically time-reversed path have equal probability after including their equilibrium starting weights. Conditional on their endpoints, this meansThis is the path version of detailed balance. A time-even order parameter is read backward without changing sign, while a transport flux changes sign. The Onsager--Machlup path probability implements this reversal in a coarse-grained stochastic description.

An isotropic phase has no preferred spatial direction. In a liquid crystal, its orientational nematic order parameter vanishes, although the molecules themselves may be anisotropic.

Apply an orientation-reversing homeomorphism of $S^3$ to a link and transport its component orientations. Reflection through the projection plane produces a link diagram by switching every crossing.