Chordal SLE at parameter four has driver $2{\beta }_t$ in half-plane-capacity time. The Itô formula makes the imaginary part of $\log (g_t(z)-W_t)$ drift-free, a feature used in the SLE4 coupling with a Gaussian free field. The trace is simple and avoids fixed nonzero boundary points.

Expanding the covariant scalar kinetic term gives $ie{A}_{\mu }(\varphi {\partial }^{\mu }{\varphi }^{\ast }-{\varphi }^{\ast }{\partial }^{\mu }\varphi )$. With $D=\partial +ieA$, the vertex for an incoming charged particle of momentum $p$ and an outgoing one of momentum $p^{\prime }$ is $-ie(p+p^{\prime }{)}^{\mu }$. The antiparticle has the opposite charge-flow sign. The quadratic term $e^2{A}^2{\varphi }^{\ast }\varphi$ gives the separate seagull vertex $2i{e}^2{\eta }^{\mu \nu }$. On equal-mass external shells, contracting the three-point momentum with the photon momentum gives zero, the tree scalar quantum electrodynamics Ward identity.

At order $e^2$, charged-scalar particle-antiparticle scattering has $s$ annihilation and $t$ photon exchange. For the convention $D=\partial +ieA$ and incoming momenta $p_1,p_2$, outgoing $p_3,p_4$, its amplitude isEqual external masses turn the numerators into $u-t$ and $s-u$. The scalar electrodynamics three-point vertex gives conserved currents, allowing the Coulomb-gauge propagator between conserved currents identity to replace each propagator by the covariant one. The seagull vertex has only two scalar legs and does not give a four-scalar tree at this order.

For stationary liquid below a salt-free solid and an upward melting interface $a(t)=2A\sqrt{Dt}$, the complementary error function salt profile and flux balance $D{C}_z(a)=-\dot{a}{C}_i$ give the displayed concentration ratio. For $A>0$ it lies below one. At large $A$ it behaves as $e^{-A^2}/(2\sqrt{\pi }A)$.

In a planar no-flow model with pure ice, a stable cold brine below freshwater can support an ice layer whose upper phase boundary freezes freshwater while its lower boundary melts into diluted brine. Heat conduction and salt diffusion obey different similarity scales. Local liquidus equilibrium, salt conservation and two Stefan conditions determine both boundary positions. The heat lost into the brine supports net ice growth even while some basal ice melts.

The effective dimensionless mass is $e^{-t}m(t)/\mu$. It vanishes precisely when this integrated exponent tends to minus infinity, for positive initial mass. An eventual bound $\delta \le 1-\eta$ is sufficient. Replacing the massive correlator by its massless limit additionally requires regularity uniform along the coupling trajectory.

If a finite extension of the P-adic numbers contains a primitive $p$th root of unity $\zeta$, the identity $p={\prod }_{j=1}^{p-1}(1-{\zeta }^j)$ gives $e=(p-1)v_L(1-\zeta )\ge p-1$. The ratios of these factors reduce to the nonzero integers $j$ in the residue field. Thus an extension with $e<p-1$ has only roots of unity of order prime to $p$.

The logarithm of a risk ratio is zero at equal risks, negative when the first risk is smaller, and positive when it is larger. For independent binomial event counts $d_1,d_0$ from samples $n_1,n_0$, its usual delta-method variance estimate is $1/d_1-1/n_1+1/d_0-1/n_0$. This approximation requires nonzero counts; zero-event arms need another procedure. Log estimates are convenient for inverse-variance meta-analysis, and exponentiation returns to the risk-ratio scale.

A fraction of invariant elements is fixed by every automorphism. If that fraction is integral over $R^G$, its monic equation also makes it integral over $R$. A normal domain therefore places it in $R$, and fixedness places it in $R^G$. This proves integral closedness of the invariant subring in its own fraction field. The proof works for any group and in every characteristic; no averaging or finiteness of the group is needed for normality.

The local rigid worldsheet supersymmetry identity $\delta \mathcal{L}={\partial }_{\mu }{J}^{\mu }$ does not by itself prove preservation of every spin structure. If $X$ is periodic on a spatial circle and $\psi$ is antiperiodic, a constant $ϵ$ gives antiperiodic $\delta X=i\overline{ϵ}\psi$. It does not preserve the bosonic boundary condition. Compatible periodic fields admit the constant transformation; other sectors must be handled with the local superconformal formulation and their global spin data.

For a nonnegative smooth bump function supported in an open set, multiplying a Riemannian metric by $1+t\eta$ increases volume there for $t>0$ while leaving the metric elsewhere fixed. For distinct regular compact domains a perturbation supported in the interior of one outside the other breaks equal volume, and therefore breaks any isometry between the domains. The perturbations tend to the original metric in the smooth topology.

The Levi-Civita connection of an oriented Riemannian manifold preserves its Riemannian volume form. In a positively oriented orthonormal frame its connection matrix is skew-symmetric. Differentiating the wedge of the dual coframe gives minus the trace of that matrix times the volume form, which is zero. Equivalently parallel transport preserves the metric and orientation, so it preserves the normalized volume form.

A basic function for a Riemannian submersion is a function pulled back from the base. It is constant along each fiber. Its Riemannian gradient is the horizontal lift of a vector field through a submersion of the base gradient. Constancy on connected fiber components alone need not imply global descent when fibers are disconnected.

For a Riemannian submersion with totally geodesic submanifolds as fibers, the positive Laplace-Beltrami operator commutes with pullback of functions. Horizontal terms in the Riemannian Hessian are pulled back from the base, and vertical terms vanish. Minimal fibers suffice because only the trace of the vertical second fundamental form enters.

For closed manifolds with the product Riemannian metric, the positive Laplace-Beltrami operator is the sum of the two factor operators. Products of their orthonormal eigenbases form a complete basis, and eigenvalues add. Hence the product eigenspace at $\lambda$ is the displayed finite orthogonal sum of tensor products. The discrete assertion requires a discrete spectral setting.

A Killing tensor is a symmetric covariant tensor satisfying the displayed equation for the Levi-Civita connection. Its contraction with $r$ copies of a geodesic velocity is constant along every affinely parametrized geodesic. The converse follows because the resulting symmetric derivative is determined by its homogeneous polynomial on tangent vectors. Rank one recovers the Killing vector field after metric duality.

A Killing-Yano tensor is a differential form satisfying the displayed equation for the Levi-Civita connection. Its covariant derivative is totally antisymmetric. Contraction with the velocity of an affinely parametrized geodesic gives a form that is carried by parallel transport along that geodesic. This differs from a Killing tensor, which is symmetric.

A conical singularity is a failure of smoothness at the apex of a metric cone. If $\vartheta$ has period $P$, small circles have circumference-to-radius ratio $aP$. The two-dimensional apex is smooth precisely when this equals $2\pi$. Euclidean black-hole regularity condition applies this test to the imaginary-time and radial plane.

A Riemannian manifold is homogeneous when its isometry group acts transitively on points. For example, a Lie group with a left-invariant metric is homogeneous under left translations. Homogeneity gives the same local metric-ball geometry at every point, which yields the completeness of homogeneous Riemannian manifolds. It does not imply two-point homogeneity: that stronger property also controls directions and pairs at equal distance.

Let $C$ be a smooth projective curve over a field $k$ with $H^0(C,{\mathcal{O}}_C)=k$. For a closed point $P$ and a line bundle $L$, the exact sequencehas a quotient of $k$-dimension $[k(P):k]$. Additivity of Euler characteristic therefore gives $\chi (L)-\chi (L(-P))=[k(P):k]$, not necessarily one. Iterating for positive and negative divisor coefficients proveswhere $g={\dim }_k{H}^1(C,{\mathcal{O}}_C)$. Over an algebraically closed field, every closed point has residue degree one and this is the familiar coefficient-sum degree.^[ref]

Serre duality identifies $h^1({\mathcal{O}}_C(D))$ with $h^0({\mathcal{O}}_C(K-D))$, givingThe elementary point modifications supply the Euler-characteristic formula; duality is the additional input turning it into the usual section-counting Riemann-Roch theorem.