For the pumped Gross–Pitaevskii equation with gain term $i(\gamma -\Gamma n)\psi$, put $\psi =\sqrt{n}\exp (im\varphi /\hslash )$. The physical superfluid velocity is $\nabla \varphi$. The imaginary part gives the displayed number balance, while the real part gives $m{\varphi }_t+m|\nabla \varphi {|}^2/2+gn-{\hslash }^2{\nabla }^2\sqrt{n}/(2m\sqrt{n})=0$. These Madelung equations hold away from density zeros. The source term permits stationary density with nonzero current divergence; local gain need not balance local loss pointwise.

The tangent bundle of a complex torus is holomorphically trivial with a flat Hermitian metric. Apply the curvature formula for a holomorphic subbundle to $TV\subset TM{|}_V$: its determinant dual $K_V$ has semipositive curvature. For a closed smooth hypersurface, adjunction for a smooth submanifold identifies $K_V$ with the restriction of its divisor line bundle. Thus a positive ambient line bundle remains positive on the hypersurface after tensoring by any nonnegative power of the restricted divisor bundle.

Let $Y$ be a embedded submanifold of a complex manifold. If its tangent spaces are invariant under the almost complex structure induced by a complex atlas, choose ambient holomorphic coordinates making its tangent space at one point ${\mathbb{C}}^k\times 0$. The first $k$ coordinates restrict to a local real diffeomorphism by the inverse function theorem, so $Y$ is locally the graph of a smooth function $F$. Invariance of the tangent graph gives $dF(iv)=idF(v)$, the Cauchy-Riemann equations. Thus $F$ is holomorphic and these graph coordinates give the induced complex submanifold structure. This proves the result directly, without needing the general integrability theorem as an unproved step.

A closed analytic subset of a complex manifold is nonsingular when every point has a neighbourhood on which the subset is the zero set of a holomorphic map of full rank along that zero set. The holomorphic implicit function theorem then makes it a local coordinate subspace. This includes a closed complex submanifold, whereas an arbitrary open piece of a complex submanifold need not be globally closed in its ambient manifold.

For every real $x>0$,For a short proof, put $I(x)={\int }_x^{\infty }{e}^{-t^2}dt$. Integration by parts givesThe last integral is positive and strictly smaller than $I(x)/x^2$. Using these two inequalities gives the upper and lower bounds after multiplication by $2/\sqrt{\pi }$. Dividing both bounds by $e^{-x^2}/(\sqrt{\pi }x)$ also proves the leading asymptotic equivalence by the squeeze theorem. For small positive $x$, the bounds remain true but can be loose; they are especially useful when the Gaussian integral tail is already small.

Capacity of a disjoint union can strictly exceed the sum of the capacities. On the five-element subsets of a twenty-point set, join distinct sets when their intersection is odd. Linear polynomials over ${\mathbb{F}}_2$ bound its capacity by twenty; squarefree quadratic polynomials over ${\mathbb{F}}_3$ bound its complement's capacity by $\left(\genfrac{}{}{0ex}{}{20}{2}\right)=190$. The complement-pair lower bound is $2\sqrt{\left(\genfrac{}{}{0ex}{}{20}{5}\right)}>210$, establishing strict nonadditivity.

The displayed distance uses the oscillation length of a compactly supported Hamiltonian isotopy. On a closed manifold one may normalize each Hamiltonian to have zero mean. Additive time-dependent constants do not change the flow or its oscillation. Pullback by a symplectomorphism preserves extrema, so this distance is bi-invariant. Nondegeneracy requires an additional geometric argument; it does not follow merely from the definition as an infimum of path lengths.

For finite-dimensional complex linear operators with $C=[A,B]$ commuting with both $A$ and $B$, differentiating $e^{-vB}A{e}^{vB}$ gives the constant derivative $C$. Hence $e^{-B}A{e}^B=A+C$, and conjugating the matrix exponential gives $e^{-B}{e}^{-A}{e}^B=e^{-A-C}$. Since $[A,C]=0$, multiplication by $e^A$ proves the identity. The commutator order determines the minus sign.

The free commutative monoid on $d$ generators consists of the nonnegative integral sums of those generators, with unique coefficients. Its semigroup algebra over a commutative ring is the polynomial ring in $d$ variables. In a smooth toric chart, the dual-cone lattice points form such a monoid, yielding affine-space coordinates.

For a height-one CM elliptic curve formal group and compatible ${\mathfrak{p}}^n$-division parameters $t_n$, an integral Coleman series interpolates a norm-compatible sequence of local units $u_n$. Its norm relation is multiplication of its translates by the $\mathfrak{p}$-torsion, with coefficient Frobenius where required by the unramified base. Coates–Wiles homomorphisms and the Frobenius-corrected logarithm turn this local series into critical values and measures.

On $R={\mathbb{Z}}_p[[T]]$, this injective substitution reduces modulo $p$ to $f(T)\mapsto f(T^p)$. The ring is a free module of rank $p$ over $\phi (R)$, with basis $1,(1+T),\dots ,(1+T{)}^{p-1}$. Prove this first modulo $p$, where the change of basis from $1,T,\dots ,T^{p-1}$ is triangular and invertible, and then lift coefficients successively modulo powers of $p$.

The determinant norm for the finite free extension $R/\phi (R)$ defines $\mathcal{N}$ on units. After adjoining the p-th roots of unity, the norm is the displayed product of conjugates. A Coleman power series of a norm-compatible unit sequence satisfies $\mathcal{N}f=f$. For symmetric cyclotomic units, the product relation follows directly from ${\prod }_{{\xi }^p=1}(1-{\xi }^a{X}^a)=1-X^{ap}$ when $p\nmid a$.

For a norm-compatible sequence of local units, its Coleman power series $f_u\in {\mathbb{Z}}_p[[T]{]}^{\times }$ defines these integral additive homomorphisms. In the formal coordinate $z=\log (1+T)$ they are the derivatives of $\log f_u(e^z-1)$; the constant term of the logarithm is irrelevant. A Galois element with cyclotomic value $a$ multiplies the kth derivative by $a^k$. This is the multiplicative-formal-group version of the Coates–Wiles homomorphism.

Such a sequence $u=(u_n)$ satisfies $N_{n/n-1}{u}_n=u_{n-1}$ at every successive layer. The inverse limit uses the field norm, rather than inclusion of units from the lower field. In a p-power cyclotomic tower a Coleman power series interpolates the sequence, and its Coleman norm operator fixes the interpolating series.

Let $V_{\infty }$ be the norm limit of local principal units in the $p$-power cyclotomic field tower, for odd $p$. The Coleman power series $f_u$ gives the Frobenius-corrected logarithm $p^{-1}\log (f_u^p/\varphi f_u)$, whose inverse Amice transform is supported on ${\mathbb{Z}}_p^{\times }$. This is the middle map in the displayed exact sequence; the last map is the first moment $\mu \mapsto \int xd\mu$. Its kernel consists of norm-compatible $p$-power roots of unity. The logarithmic difference equation has one resonant degree, namely degree one in $\log (1+T)$, accounting for the last term. Complex conjugation acts by $-1$ on both copies of ${\mathbb{Z}}_p(1)$. Taking its even part therefore identifies the local principal-unit limit in the real tower with ${\mathbb{Z}}_p[[{\mathbb{Z}}_p^{\times }/\{\pm 1\}]]$.

For odd $p$ and a norm-compatible sequence of local units in ${\mathbb{Q}}_p({\mu }_{p^{n+1}})$, there is a unique $f_u\in {\mathbb{Z}}_p[[T]{]}^{\times }$ interpolating all its entries, and the Coleman norm operator fixes it. To construct it, choose integral unit polynomials $F_j$ interpolating entry $u_j$, and take a convergent subsequence of $N^j{F}_{2j}$ in the compact $(p,T)$-adic unit space. The congruence $NF\equiv F(\mathrm{mod}p)$ and Coleman norm contraction show that its evaluation at any fixed level tends to the desired entry. Uniqueness follows from the Weierstrass preparation theorem: a nonzero integral power series has only finitely many zeros in the open unit disk. The norm relation on evaluations then proves norm fixation. Principal units correspond to $f_u(0)\equiv 1(\mathrm{mod}p)$.

With $⟨{\mathrm{ad}}_{\xi }^{\ast }\ell ,\eta ⟩=-⟨\ell ,[\xi ,\eta ]⟩$, the displayed form on a coadjoint orbit is well-defined: changing a generator by an element of the stabilizer changes no pairing. It is nondegenerate on orbit tangents, and its closedness follows from the Jacobi identity. This sign agrees with the positive Lie-Poisson bracket when Hamiltonian vector fields act by $X_f(g)=\{g,f\}$. A negative Lie-Poisson convention reverses the orbit form as well.

A surjective ring homomorphism $A\to B=A/I$ identifies the spectrum of a ring $B$ with the closed subset $V(I)$ and induces a surjective map of structure sheaves. Conversely, the direct image sheaf of ${\mathcal{O}}_{\mathrm{Spec}B}$ under an affine morphism is the sheaf associated with the $A$-module $B$: its sections on $D(a)$ are $B_{\varphi (a)}$. The cokernel of the structure-sheaf map is therefore associated with the module cokernel of $A\to B$. If the sheaf map is surjective, all localizations of that cokernel vanish and the module cokernel is zero. This proves the converse without assuming global sections preserve arbitrary sheaf surjections.

A nonempty flat closed immersion of finite presentation into a connected scheme is an isomorphism. On a chart it is $A\to A/I$ with $I$ finitely generated. At a prime containing $I$, the quotient is a finite flat module over the local ring, hence free of rank one, so $I$ vanishes there. Finite generation extends this vanishing to a neighbourhood. Thus the image is open as well as closed; connectedness makes it all of the target, and the ideal is everywhere zero. Without finite presentation the conclusion can fail.

For finitely many orthonormal local trivializations $t_i$ and a subordinate partition of unity ${\rho }_i$, the displayed map embeds a rank-k complex vector bundle isometrically in $X\times {\mathbb{C}}^{kr}$. Outside the support of each ${\rho }_i$ its component is zero. Its continuous image planes give a finite-dimensional Grassmannian map and identify the original bundle with a tautological pullback.