The scalar Ward identity relates the charged-scalar photon vertex to differences of inverse scalar propagators. At tree level, contraction with photon momentum $q=p^{\prime }-p$ usesIncluding the appropriate charge and vertex conventions makes the right side the difference of inverse free propagators. The full identity uses the dressed vertex and full inverse propagators instead.

Scalar quantum electrodynamics also contains a two-photon contact interaction from the square of the gauge covariant derivative. Its coefficient is tied to the one-photon and kinetic terms by gauge invariance. Consequently, checks of scalar scattering must include both sequential one-photon attachments and the contact term. Renormalization conventions preserving the symmetry retain these relations between the coefficients.

For a continuous self-map $f:X\to X$, the mapping torus has the Wang sequence even when $f$ is not a homeomorphism. Cover the time circle by a seam neighborhood and an interval away from the seam. Their inverse images retract to $X$: the seam neighborhood is a mapping cylinder, and the other is a product. Their intersection retracts to two copies of $X$. The Mayer–Vietoris sequence map on homology is, after consistent choices of signs,Invertible changes of coordinates reduce it to the direct sum of $\mathrm{id}-f_{\ast }$ and an isomorphism. Canceling the isomorphism summands gives the Wang sequence. This proof does not assume a fiber bundle structure for a noninvertible $f$.

Suppose a nonempty compact $(n+1)$-manifold with a volume form has $n$ pointwise independent volume-preserving vector fields that commute and are tangent to its boundary. If restriction $H_{\mathrm{dR}}^1(M)\to H_{\mathrm{dR}}^1(\partial M)$ is injective, the boundary must have at least two connected components.

Indeed $\eta ={\iota }_{X_1}\cdots {\iota }_{X_n}\omega$ is nowhere zero. Cartan's magic formula and commutation show $d\eta =0$, while boundary tangency gives $j^{\ast }\eta =0$. Injectivity in de Rham cohomology makes $\eta =df$. On a connected boundary $f$ is constant; a maximum or minimum differing from that boundary value would be an interior critical point, contradicting $df\ne 0$. Empty boundary is also impossible.

For inward accretion rate $\dot{M}>0$, the steady conservation of angular momentum equation is $\mathcal{G}=\dot{M}(h-h_{\mathrm{in}})$ when the inner torque vanishes. The viscous torque in an accretion disk is $\mathcal{G}=2\pi qh\bar{\nu }\Sigma$, where $h=r^2\Omega$ and $q$ is the orbital shear parameter. ThusThe familiar Keplerian accretion disk follows when $q=3/2$ and $h\propto r^{1/2}$. The general form also applies to the non-Keplerian Paczyński-Wiita circular orbit rotation law.

For $a=-1-{\alpha }^{\prime }s/4$, $b=-1-{\alpha }^{\prime }t/4$ and $a+b+c=1$, the amplitude has generic simple poles at $a=-n$. The gamma function residue and recurrence giveThe residue polynomial has degree $2n$, consistent with closed-string exchange up to spin $2n$. When $a=-n+{\delta }_a$ and $b=-m+{\delta }_b$, the reciprocal factor $1/\Gamma (a+b)$ removes a putative double pole: the leading ratio is ${\left(\genfrac{}{}{0ex}{}{n+m}{n}\right)}^2(1/{\delta }_a+1/{\delta }_b)$. Thus channel singularities must be counted using the kinematic constraint, not by multiplying independent numerator poles.

The radiation-to-gas pressure ratio compares radiation pressure with gas pressure. It is distinct from plasma beta, which compares gas pressure with magnetic pressure. Both are sometimes denoted by $\beta$, so the convention must be stated.

In harmonic stellar gravity, an ideal gas with a polytropic equation of state of index $n>0$ has a quadratic temperature profile and finite semi-thickness $H^2=2(n+1)\mathcal{R}{T}_0/{\Omega }^2$. This differs from the polytropic vertical structure of a self-gravitating disk.

The time for a vertical pressure or gravity response. A sound-crossing time and a small vertical oscillation period have this scale in a thin disk. It need not equal a damping time.

For unit vectors $u,v$ and a random normal $a$ with independent standard normal distribution coordinates,The Gaussian distribution is invariant under orthogonal transformations. Projecting onto the plane spanned by $u,v$ therefore gives a uniformly distributed direction. If the angle between $u,v$ is $\theta$, the sign-disagreement directions form two sectors of total angle $2\theta$ out of $2\pi$. The endpoint cases $u=v$ and $u=-v$ give probabilities zero and one directly.

A strict vector $k$-coloring requires equality $⟨v_i,v_j⟩=-1/(k-1)$ on every edge. The least admissible $k$ is the complement theta number for a graph with an edge. Allowing merely an inequality defines the potentially smaller vector chromatic number.

The projective bundle of a real or complex vector bundle $E\to X$ has fiber the space of one-dimensional subspaces of $E_x$. It carries a tautological bundle $\lambda \subset p^{\ast }E$; in the real case this is the projectivization of a real vector bundle.

The same construction applies to an algebraic vector bundle over an algebraic variety: local trivializations glue products with projective space. In the lines convention its universal tautological bundle is a line bundle; the alternative projectivization by quotients convention parametrizes one-dimensional quotient spaces. Both conventions give dimension $\dim X+\mathrm{rank}(E)-1$ and a proper morphism to the base.

Complexification replaces each fiber of a real vector bundle by its tensor product with $\mathbb{C}$ over $\mathbb{R}$. The complexification of a real vector bundle of rank one is a complex line bundle. For the tautological bundle over Real projective space, this complex line bundle squares to the trivial vector bundle.

A local trivialization of a rank-$r$ vector bundle $\pi :E\to M$ is a bundle isomorphism ${\pi }^{-1}(U)\cong U\times {\mathbb{R}}^r$ over an open set $U$, linear on each fiber. Overlapping trivializations differ by $(p,v)\mapsto (p,g(p)v)$ for a smooth $G{L}_r(\mathbb{R})$-valued function $g$. A tangent bundle chart uses the derivative of its base manifold chart for this fiber identification.

A morphism of vector bundles over one base is a smooth map between their total spaces that commutes with projection to the base and is linear on each fiber. In local vector bundle trivializations it is multiplication by a smoothly varying matrix.

The rank of a vector bundle at a point is the dimension of its fiber as a vector space. A vector bundle trivialization identifies nearby fibers with the same vector space, so this rank is locally constant. A rank-$r$ bundle has fibers of dimension $r$ everywhere.

Given $\Vert f_{\delta }-f\Vert \le \delta$, the residual method minimizes a penalty $J(u)$ subject to $\Vert Au-f_{\delta }\Vert \le \delta$. Every exact solution is feasible, so a reconstruction $u_{\delta }$ has penalty no larger than that of any J-minimizing solution.

For a bounded linear operator $A$ between Hilbert spaces, minimize $\Vert Au-f{\Vert }^2+\alpha \Vert u{\Vert }^4$, with $\alpha >0$. The objective is coercive and weakly lower semicontinuous, and its strictly convex norm penalty gives a unique minimizer. Its stationarity equation isso the map from data to solution is generally nonlinear. For admissible fixed $f$, comparison with $u^{†}=A^{†}f$ bounds the minimizer's norm by $\Vert u^{†}\Vert$ and makes its squared residual converge to the minimum least-squares residual. Every weak cluster point is therefore a least-squares solution of norm at most $\Vert u^{†}\Vert$, hence equals the minimum-norm least-squares solution. Weak lower semicontinuity then forces convergence of norms, and the Radon-Riesz theorem gives strong convergence. This supplies a nonlinear regularization of an inverse problem.

For completeness, the fixed-$\alpha$ data-to-solution map is continuous. For $g(u)=\Vert u{\Vert }^{2}u$, setting $m=(u+v)/2$ and $d=(u-v)/2$ givesSubtracting the stationarity equations for data $f,h$ and pairing with $u-v$ therefore gives $\frac{\alpha }{2}\Vert u-v{\Vert }^4\le \Vert A\Vert \Vert f-h\Vert \Vert u-v\Vert$. Thuswhich proves continuity directly.

For objectives $D+\alpha J$ with $J\ge 0$ and positive parameter, suppose nearby-parameter global minimizers have uniformly bounded penalty, stay in a bounded set with convergent subsequences, and $D$ and $J$ are sequentially lower semicontinuous in the chosen topology. Their optimality inequalities and lower semicontinuity show that every subsequential limit minimizes the limiting objective. Uniqueness makes the regularized solution continuous in the parameter. Existence alone does not make an arbitrary selection among multiple global minimizers continuous.

For a nonnegative regulariser $J$, the optimal objective value is non-decreasing and concave in the positive regularization parameter. If $J(0)=0$ and $D(0)<\infty$, comparison with zero makes it finite and a locally Lipschitz function away from parameter zero.

Exact fitting in the small-parameter limit requires data that can be approximated by forward images of elements in the regulariser's effective domain. Membership merely in the operator range is insufficient when an extended-real regulariser imposes a hard constraint.