An embedded positive domain which is a bigon or rectangle, has convex alternating alpha and beta corners, and contains no other selected coordinate in its interior, has Maslov index one. A bigon has Euler measure $1/2$ and two corner averages $1/4$; a rectangle has Euler measure zero and four corner averages $1/4$. For the usual regular complex structure, its holomorphic disk count modulo translation is $\pm 1$. For a bigon the Riemann map to the strip is unique up to translation. For a rectangle, the tautological correspondence represents the disk in the symmetric product by a branched double cover of the strip onto the rectangular domain: its conformal modulus fixes the branch parameter, and the only remaining freedom is translation. The resulting isolated regular disk gives one point in the oriented moduli space. The differential coefficient is $\pm U^{n_z}$, so a basepoint-free domain contributes a unit.

Over $\mathbb{Z}$, holomorphic disk moduli spaces in a Heegaard Floer chain complex must be oriented compatibly with gluing: the orientation of a concatenated disk class is the product determinant-line orientation of its two factors. Boundary orientations of compactified one-dimensional moduli spaces then make broken disks cancel, yielding ${\partial }^2=0$. Isolated regular disks contribute $+1$ or $-1$. Changing the chosen orientation at an intersection generator changes the corresponding row and column signs of the differential and yields an isomorphic integral chain complex. Cobordism triangle counts require compatible orientation choices as well; an overall orientation ambiguity accounts for the usual sign ambiguity of the closed invariant.

In normal coordinates $q={\exp }_{p}x$, write the Riemannian volume form as $J(p,x)dx$ and let $r=|x|$. With the heat operator ${\partial }_t-{\Delta }_q$, the Gaussian times a power-series amplitude has its singular terms cancelled by the displayed radial equations for $i\ge 1$. The initial coefficient is $w_0=J^{-1/2}$, andMultiplying the equation by $r^{i-1}{J}^{1/2}$ gives the derivative of $r^i{J}^{1/2}{w}_i$. Smoothness forces the integration constant to vanish. The integral on a fixed compact interval proves smoothness across the diagonal by induction; it yields $w_i(p,p)={\Delta }_q{w}_{i-1}(p,p)/i$. A truncated heat parametrix therefore has residual $-g{t}^k{\Delta }_q{w}_k$.

The normalized pullback is a unitary map from $L^2$ of the quotient onto the deck-invariant subspace of $L^2$ of the cover. Local isometry and the sheet-counting integration formula preserve the Dirichlet quadratic form. Therefore the Friedrichs extension and its heat semigroup intertwine with this map. A finite sum of lifted Riemannian heat kernels gives the quotient kernel; independence of lifts follows by reindexing the deck group. This formulation avoids a uniqueness claim for unrestricted heat-equation solutions on a noncompact or incomplete manifold.

On a closed two-dimensional Riemannian manifold, the first three integrated scalar heat invariants are $A$, $\frac{1}{3}\int KdA$, and $\frac{1}{15}\int K^{2}dA$, where $K$ is Gaussian curvature. If its scalar spectrum agrees with that of a hyperbolic surface of area $A$, these integrals are $A,-A/3,A/15$. Expanding the displayed square then forces $K=-1$ everywhere. Thus constant negative curvature is spectrally determined in this setting, although the global hyperbolic metric need not be determined up to isometry.

For the positive scalar Laplace-Beltrami operator on a closed manifold with a Riemannian metric, the coefficient of $t^2$ in the normalized heat trace is the displayed integral. The local coefficient additionally has a scalar-curvature divergence term whose integral vanishes. In dimension two, $R=2K$, $|\mathrm{Ric}{|}^2=2K^2$, $|\mathrm{Rm}{|}^2=4K^2$, giving $a_2=\frac{1}{15}{\int }_M{K}^{2}dA$. Together with volume and total Gaussian curvature, it determines the variance of curvature and detects constant curvature.

For positive real numbers $x_i$ and nonnegative weights with ${\sum }_i{w}_i=1$, the weighted harmonic mean isEqual weights give the harmonic mean, while a zero weight excludes that input. For travel at positive constant speeds $v_i$ over positive leg lengths $d_i$, the weights are fractions of total distance:For $30\mathrm{km}$ at $30km/h$ and $90\mathrm{km}$ at $60km/h$, the weights are $1/4,3/4$, elapsed time is $5/2$ hours, and the average speed is $48km/h$. Weighting the speeds by their time fractions instead gives the same result as a weighted arithmetic mean: $(2/5)30+(3/5)60=48$. The two weight systems describe the same journey in different coordinates and must not be interchanged. ^[ref].

Restriction first views a $G^F$-module as a $P^F$-module, then takes the subspace fixed by the unipotent radical. The resulting $L^F$-module is adjoint to Harish-Chandra induction by ordinary Frobenius reciprocity. Over characteristic zero, averaging makes invariants exact and identifies them with coinvariants.

For rational parabolic subgroups $P=LU$ and $Q=MV$, define $\mathcal{S}(M,L)=\{x:M\cap {}^{x}L\text{ contains a maximal algebraic torus of }G\}$. Then ${}^{\ast }{R}_M^G{R}_L^G$ is the sum over $M^F\backslash \mathcal{S}(M,L{)}^F/L^F$ of $R_{M\cap {}^{x}L}^M{}^{\ast }{R}_{M\cap {}^{x}L}^{{}^{x}L}\mathrm{ad}x$, with the intersection parabolics understood. It follows from ordinary finite-group Mackey restriction formula, taking $V^F$-invariants, and the compatible-Levi intersection lemma for parabolics.

The Brownian representation of half-plane capacity is $\mathrm{hcap}(K)={\lim }_{y\to \infty }y{\mathbb{E}}_{iy}\mathrm{Im}{B}_T$. The harmonic capacity from infinity in the upper half-plane uses $\mathrm{cap}(K)={\lim }_{y\to \infty }\pi y{\mathbb{P}}_{iy}(B_T\in K)$. Since $0\le \mathrm{Im}{B}_T\le \mathrm{rad}(K)1_{\{B_T\in K\}}$, the displayed bound follows. The normalization factor $\pi$ cannot be omitted. Independently, enclosure in a half-disc and monotonicity of half-plane capacity give the sharp bound $\mathrm{hcap}(K)\le \mathrm{rad}(K{)}^2$.

A nowhere-zero tangent vector field on $S^n$ can be normalized to a unit field $w(x)$ perpendicular to $x$. Then $\cos (\pi t)x+\sin (\pi t)w(x)$ is a homotopy from the identity to the antipodal map. Their mapping degrees are $1$ and $(-1{)}^{n+1}$, respectively, contradicting homotopy invariance of mapping degree when $n$ is positive and even. For $S^0$ the tangent spaces are zero, so every tangent field is zero directly.

Let a divisible ordered Abelian group be the union of a strictly increasing sequence of convex subgroups $C_1=0,C_2,\dots$, with each ordered quotient $C_{n+1}/C_n$ isomorphic to the additive real line. Convexity makes every $C_n$ divisible, so these are rational vector subspaces. Choose a rational-linear section for each quotient. Every element then has a unique finite sum of section values. Its sign is the sign of the coefficient of the largest active index, because convexity makes the corresponding ordered quotient decide its sign. The coefficients give an embedding, indeed an ordered-group isomorphism, into the Hahn group with positive integer indices in the maximum-support convention. No compatibility of a separate real scalar multiplication is needed for this group embedding.

Choose points $a_v\in F^r$ and polynomials $f_v$ in a finite-dimensional polynomial space $M$, with $f_v(a_v)\ne 0$ and $f_v(a_w)=0$ for distinct nonadjacent vertices. Evaluation produces a fitting matrix of rank at most ${\dim }_{F}M$. Equivalently, tensoring the polynomial spaces gives linearly independent polynomials for every independent code in a strong power, proving $c(G)\le {\dim }_{F}M$.

For a Hadamard differentiable functional with continuous linear derivative $L$, define $R_t(h)=(\Phi (s+th)-\Phi (s))/t-L(h)$. For every compact $K$ and positive $ϵ$, some positive $\delta ,a$ make $|R_t(h)|<ϵ$ whenever $\mathrm{dist}(h,K)<\delta$ and $0<|t|<a$. If this failed, choose $t_j\to 0$, directions within $1/j$ of $K$ and remainders at least $ϵ$. Compactness gives a convergent subsequence of nearby points of $K$, hence convergent directions. The derivative definition and continuity of $L$ give remainder tending to zero, a contradiction. With a tight weak limit, the Portmanteau theorem makes random directions lie near a suitable compact set with arbitrarily large probability. Thus this lemma proves the functional delta method without assuming separability of the entire normed space or uniform differentiability on its whole unit ball.

A coverage specifies covering families on a small category, stable under refinement along arrows: for a covering family into $C$ and $f:B\to C$, there is a covering family into $B$ whose composites with $f$ factor through the original family. Passing to generated sieves and imposing maximality and local transitivity gives its associated Grothendieck topology, with the same sheaves on a site. One can equivalently specify the saturated covering sieves directly. Rigidity refers to these covering sieves, so it is unchanged by replacing a family presentation by its generated topology.

On the opposite of the category of finitely presented unital commutative rings, a basic covering family is $A\to A[f_i^{-1}]$, where finitely many $f_i$ generate the unit ideal. Include the empty cover of the zero ring. Localization commutes with base change, so these families define a coverage on a category. Their sheaf topos classifies the coherent theory of local rings. This coverage is subcanonical: representable functors obey the usual localization gluing equations.

The dense sieves form the double-negation Grothendieck topology. Its sheaf topos is a Boolean topos: the union of a subobject with its pseudocomplement is dense and thus becomes the whole object after sheafification. This need not be the atomic topology, which requires every nonempty sieve to be dense.

For a finite-energy Gross–Pitaevskii solitary wave, write $E=E_z+E_{\perp }+E_{\mathrm{pot}}$ with $E_z=\frac{1}{2}\int |{\psi }_z{|}^2$. The longitudinal dilation ${\psi }_{\lambda }(x_{\perp },z)=\psi (x_{\perp },\lambda z)$ leaves its renormalized momentum of a condensate invariant and changes energy to $\lambda E_z+{\lambda }^{-1}(E_{\perp }+E_{\mathrm{pot}})$. Stationarity of $E-Up$ at $\lambda =1$ implies $E_z=E_{\perp }+E_{\mathrm{pot}}$ and hence the displayed identity. This is a directional Pohozaev identity, valid when the integrals and variation boundary terms converge.

In a product symplectic manifold $S^2(a)\times T^{2n-2}$, every compatible almost complex structure admits a J-holomorphic sphere in class $F$ through each point. Here $S^2(a)$ has area $a$, and the torus carries a constant symplectic form. All spherical classes are integer multiples of $F$, so a curve in class $F$ cannot split into positive-area nonconstant spheres. For the product complex structure the fibre through a point is unique and regular. Its one-point count modulo two is one; generic transversality and the compact one-dimensional parameterized moduli spaces preserve this count. Approximation and Gromov compactness for spheres remove the genericity requirement. This gives the area-$a$ curve used in the proof of the Gromov non-squeezing theorem.

For a grey atmosphere with $T^4=T_e^4(2+3\tau )/4$, an ideal gas and $\kappa ={\kappa }_0\rho T^{17}$, hydrostatic equilibrium in optical depth integrates to $P^2=512\mathcal{R}g[1/8-(2+3\tau {)}^{-3}]/(9\mu {\kappa }_0{T}_e^{16})$, taking $P(0)=0$. The logarithmic radiative gradient is $[(2+3\tau {)}^3-8]/48$. Equating it to the monatomic adiabatic temperature gradient $2/5$ gives $(2+3{\tau }_b{)}^3=136/5$ and the displayed onset temperature. Beyond this point the radiative profile must be replaced by convective transport.