Both the ensemble position density transported by the guidance equation and the squared wave amplitude satisfy the same probability continuity equation. Subject to existence and uniqueness of that transport, an initial Born rule distribution stays a Born rule distribution. This is a preservation statement, not a derivation that every initial ensemble is already in equilibrium.

If the order of a finite abelian group $\Delta$ is invertible in the coefficient ring and all its character values belong to that ring, its character idempotents are orthogonal and sum to one. They decompose any module into character components. In a cyclotomic tower with odd $p$, the Teichmüller character provides these components integrally over ${\mathbb{Z}}_p$.

In a cycle square, each triple of consecutive vertices is a clique. A proper three-colouring therefore repeats with period three and closes exactly when the length is divisible by three. For all other cycle lengths at least four colours are needed; the five-cycle square is a complete five-vertex graph.

If every graph component has at most $(k+1)n/[k(2k+1)]$ vertices, assigning whole graph components to two sides gives an empty graph cut with the displayed bound. To prove it, maximize the smaller side's weight $S\le n/2$. If $S<kn/(2k+1)$, every component on the larger side has weight at least $n-2S$, because moving a smaller one would improve the smaller side. There are at least $k+1$ such components, since their total exceeds $k$ times the permitted maximum weight. Thus $n-S\ge (k+1)(n-2S)$, a contradiction. The proof also works for arbitrary positive weights.

A family of positively many $s$-cliques forces a logarithmic balanced graph blow-up, which can also contain a matching of that many members of the family. To induct on $s$, prune faces with few extensions, apply induction to the remaining $(s-1)$-faces, and use a matching of those faces as one side of a bipartite incidence graph. The common neighbourhood from bipartite density estimate selects logarithmically many disjoint faces with polynomially many common extension vertices.

Normalize the conjugated cube average by the number of integer cubes whose vertices all lie in the interval. Equivalently, extend $f$ by zero to a cyclic group of order greater than $8N$ and divide its Gowers U3 norm by that of the interval's indicator function. A quadratic phase then has norm one. Omitting the denominator gives a different but uniformly equivalent interval normalization.

For functions bounded by one on a cyclic group of prime order greater than three, three applications of Cauchy-Schwarz bound the four-term arithmetic progression average by the smallest Gowers U3 norm of its factors. Zero extension yields the corresponding interval bound with an absolute factor. This makes Gowers U3 norm a measure of uniformity relevant to four-term arithmetic progression counts.

For a nonnegative all-integer row $x_B+{\sum }_j{a}_j{x}_j=b$ with $a_j>0$, the Gomory fractional cut requires ${\sum }_j\{a_j\}{x}_j\ge \{b\}$. The original row gives ${\sum }_j{a}_j{x}_j\le b$. If a constant $\kappa$ satisfies $\{a_j\}\le \kappa a_j$ for every $j$ and $\kappa b<\{b\}$, these inequalities contradict the cut. A single row therefore certifies integer infeasibility. Slacks may be used as integer variables when the original constraint coefficients and right sides are integral and the decision variables are integral.

Curvature depresses the equilibrium melting temperature of solid convex into liquid. Here $\mathcal{K}$ is the sum of principal curvatures, equal to twice a common convention for mean curvature, and $T_m$ is absolute temperature. The coefficient uses solid-liquid surface energy, mass density and latent heat. This stabilizes fine corrugations of a solidification front.

Fix $k\ge 1$, ${\gamma }_0>1/(k+1)$, and ${\gamma }_0\le \gamma \le 1-\omega (n)/\log n$ with $\omega (n)\to \infty$. A binomial random graph has one giant component of the displayed order and all other graph components are tree components of orders at most $k$, with high probability. The expected value and variance of the isolated vertex count concentrate it around $n^{1-\gamma }\to \infty$. A Cayley formula upper bound for connected sets excludes component sizes $k+1$ through a small fixed fraction of $n$; the empty-graph cut bound excludes the remaining sizes up to $n/2$. An extra-edge count excludes cyclic small graph components. The expected value of the number of vertices in small nonisolated tree components is $O(nd{e}^{-2d})=o(n{e}^{-d})$, where $d=np$. The Markov inequality completes the count of vertices outside the unique large graph component.

Suppose $\epsilon =o(1)$ and $\epsilon \ge n^{-1/6}$. The breadth-first exploration of a binomial random graph has deterministic drift $F(t)=\epsilon t-t^2/(2n)+O({\epsilon }^{3}n+\epsilon )$ up to $3\epsilon n$ steps. Its weighted martingale has variance $O(\epsilon n)$. For $h=\sqrt{\epsilon }+(n{\epsilon }^3{)}^{-1/8}$, the Doob L2 maximal inequality makes its maximum smaller than $h{\epsilon }^{2}n/6$ with high probability. Then $A_t>0$ throughout $[h\epsilon n,(2-h)\epsilon n]$, giving a graph component of order $(2-o(1))\epsilon n$.

For the upper expectation bound, let $K=\lceil {\epsilon }^{-3}\rceil$. The dominating binomial branching process has branching survival probability $(2+o(1))\epsilon$ by the binomial branching survival correction. Its branching process conditioned on extinction has mean $1-\epsilon +O({\epsilon }^2+\epsilon /n)$ and total-progeny expected value $O(1/\epsilon )$. Therefore $\mathbb{E}{N}_{\ge K}\le n\rho +O(n/(\epsilon K))$. Since $L_1\le K+N_{\ge K}$ and $K=o(\epsilon n)$, this yields the matching upper bound. Controlling rare large components is necessary to conclude an expected value asymptotic from a typical-size statement.

A geometric morphism is surjective when its inverse-image functor is faithful, equivalently conservative for geometric inverse images. Reflection of isomorphisms of subobjects already proves faithfulness by applying the inverse image to equalizers of parallel arrows. The surjection-embedding factorization of a geometric morphism separates this property from a fully faithful direct image.

A geometric embedding is a geometric morphism whose direct-image functor is full and faithful. Its inverse image presents a left-exact reflective subcategory of the target topos. It is the second part of the surjection-embedding factorization of a geometric morphism.

For $f:\mathcal{E}\to \mathcal{F}$, the comonad $G=f^{\ast }{f}_{\ast }$ is Cartesian, and its coalgebra topos $\mathcal{D}$ gives a factorization $f=i\circ p$. Here $p^{\ast }$ is the faithful forgetful functor and $i^{\ast }$ is the comparison functor. Its right adjoint sends $(A,a)$ to $\mathrm{Eq}(f_{\ast }a,{\eta }_{f_{\ast }A})$; applying $f^{\ast }$ identifies the comparison counit with the equalizer $a:A\to GA$ of $Ga,{\delta }_A$, proving that $i_{\ast }$ is full and faithful.

A geometric morphism is essential when its inverse image has an additional left adjoint. For a functor of small categories, precomposition on presheaves has left Kan extension and Right Kan extension as its two adjoints. Precomposition preserves finite limits pointwise, so this is an essential geometric morphism.

The inverse image is a finite-limit-preserving left adjoint with right adjoint $f_{\ast }$. It preserves arbitrary colimits, and hence images as well as finite limits. These properties preserve the interpretation of geometric formulas. It need not preserve Heyting implication or every universal quantifier.

Every Grothendieck topos has a unique geometric morphism to sets, up to isomorphism. Its inverse image is $\Delta S={\coprod }_{s\in S}1$ and its direct image is $\Gamma A=\mathrm{Hom}(1,A)$. A geometric inverse image out of sets must have this form because it preserves coproducts and the terminal object. A local topos is one for which $\Gamma$ itself is an inverse image.

Geometric morphisms into a sheaf topos correspond to continuous flat functors from its site into the domain topos, with continuity meaning that covers become jointly epimorphic families. Pulling back sheafified representables gives the functor; the tensor construction gives the inverse image in the other direction. This representation theorem should not be confused with the separately named Diaconescu theorem about choice and excluded middle.

A geometric formula is built from atoms by finite conjunction, arbitrary small disjunction and existential quantification. Empty conjunction and disjunction give truth and falsity. In a Grothendieck topos, these constructions use finite limits, unions and images. Formula contexts have finitely many variables.

A set of sequents between geometric formulas over a many-sorted signature. Its geometric syntactic category and geometric syntactic topology yield its classifying topos. A model satisfies a sequent when the interpreted antecedent subobject is contained in the consequent in the indicated context.