On the Riemann sphere, three holomorphic reparameterization ghost zero modes correspond to the three infinitesimal Möbius transformations. Three $c$ insertions saturate their Grassmann integral. In a basis $1,z,z^2$, their integral is the Vandermonde determinant, giving $z_{12}{z}_{13}{z}_{23}$ up to an ordering-dependent sign. Including the antiholomorphic factor gives $|z_{12}{z}_{13}{z}_{23}{|}^2$, the Jacobian in sphere gauge fixing for four string vertices.

With gauge covariant derivative $D=d+A$, a Wilson loop is the matrix trace in a specified group representation of parallel transport around a closed curve. The holonomy of a connection changes by conjugation under gauge equivalence of principal connections, so its trace is gauge-invariant. It detects global connection data even when the local gauge curvature is zero.

For boost matrices $A=cI+s\mathbf{n}\cdot \mathbf{\sigma }$ and $A^{\prime }=c^{\prime }I+s^{\prime }{\mathbf{n}}^{\prime }\cdot \mathbf{\sigma }$, the Pauli matrix multiplication law gives the displayed anti-Hermitian part. If both rapidities are nonzero and the axes are noncollinear, it cannot vanish, while either lift of a pure boost is Hermitian. Thus an additional Wigner rotation is necessary. A zero boost is a degenerate exception.

Truncate the unscaled entries of a Wigner matrix at a fixed threshold and subtract their expected values. The difference between original and truncated entries is $N^{-1/2}(Z_C-\mathbb{E}{Z}_C)$ with $Z_C=Y{1}_{|Y|\ge C}$, so its variance is bounded by $N^{-1}\mathbb{E}[Y^2{1}_{|Y|\ge C}]$. The centering term cannot be discarded for an asymmetric entry law.

Every open set of finite Lebesgue measure in Euclidean space is covered, up to a null set, by countably many pairwise disjoint open balls contained in it. At each stage cover a compact subset of the open remainder, choose a packing with the Wiener covering lemma, and remove its closed balls. A fixed positive fraction of the remaining measure is removed each time, while sphere boundaries are null.

For odd Grassmann variables, $(\chi \psi )={\chi }^{\alpha }{\psi }_{\alpha }$ is symmetric under exchanging the two spinors: the Grassmann sign cancels the sign of the antisymmetric epsilon contraction. With ${ϵ}^{12}=1$, lower components $\chi =(a,b)$ and $\psi =(c,d)$ give $(\chi \psi )=bc-ad$. Commuting numerical spinors instead give an antisymmetric contraction.

With $K_i=M_{0i}$ and $M_{\mu \nu }=i{\gamma }_{\mu \nu }/2$ in the supplied mostly-minus index convention, the two Weyl spinor blocks have equal rotation generators $J_i={\sigma }_i/2$ and opposite boost generators. A complex-linear intertwiner must commute with every SU(2) rotation and hence be scalar by the Schur lemma, but no nonzero scalar intertwines the opposite boosts. This proves inequivalence of the two complex representations; parity exchanges them.

A distinguished polynomial over ${\mathbb{Z}}_p$ is monic and has every non-leading coefficient divisible by $p$. The Weierstrass preparation theorem expresses any nonzero element of ${\mathbb{Z}}_p[[T]]$ as a power of $p$ times a unit times a distinguished polynomial.

The spectral measure of a weakly stationary process is the finite nonnegative measure whose Fourier coefficients are its autocovariances. Its total mass is the process variance. A time-series spectral density exists exactly when this measure is absolutely continuous with respect to Lebesgue measure.

An absolutely summable sequence of coefficients defines $Y_t={\sum }_s{a}_s{X}_{t-s}$ in mean square. It remains weakly stationary and has covariance ${\gamma }_k^Y={\sum }_{s,u}{a}_s{a}_u{\gamma }_{k-s+u}$. Its frequency response is $\alpha (\omega )={\sum }_s{a}_s{e}^{is\omega }$.

Starting from the full finite bimatrix game, sequentially remove actions weakly dominated by current mixed strategies, with a strict improvement somewhere. At least one Nash equilibrium survives on the final action sets. Take an Nash equilibrium of the reduced game and restore actions in reverse. A restored action is no better against the fixed surviving opponent strategy than its dominating mixture of already available actions, so it introduces no profitable deviation. Any self-weight in a dominating mixture can be removed and renormalized because strict improvement forces that weight below one. Using dominance tested only against an arbitrary initially restricted opponent set does not guarantee an Nash equilibrium of the original full game.

A bounded sequence in the space of continuous functions on a compact space that has a discontinuous pointwise limit cannot lie in a weakly compact set. Compactness would give a weakly convergent subnet, and continuous point-evaluation functionals would identify its continuous limit with the prescribed pointwise limit. No assumption of sequential compactness is needed.

The continuous dual of a separable Banach space is separable in its weak-star topology, even when it is not norm separable. Banach-Alaoglu theorem and weak-star metrizability of the dual ball make the dual unit ball a separable compact metric space. The countable union of positive integer multiples of a countable dense subset of that ball is dense in the whole dual. This does not imply global weak-star metrizability.

If a neutral-current vertex has prefactor $g/\cos {\theta }_W$ and chiral coefficients $g_L,g_R$, then $g_L{P}_L+g_R{P}_R=c_V-c_A{\gamma }^5$, with $c_V=(g_L+g_R)/2$ and $c_A=(g_L-g_R)/2$. For one Standard Model charged lepton, $(c_V,c_A)=(-1/4+{\sin }^2{\theta }_W,-1/4)$; for an active neutrino, $(c_V,c_A)=(1/4,1/4)$. A prefactor $g/(2\cos {\theta }_W)$ instead requires doubled coefficients. Mixing these conventions creates a factor-of-four error in squared amplitudes.

Zero values at both endpoints of a length-$L$ string allow the D'Alembert formula to be written with opposite travelling waves and a $2L$-periodic profile. This represents reflection with sign reversal at each fixed endpoint. For positive-time boundary conditions, periodicity concerns the arguments visited by the string solution; unused profile values can be defined by periodic extension.

Interfaces between ice floes and open water reflect and redirect surface gravity waves. The effective attenuation depends on floe thickness, diameter, spatial arrangement and wave period. For very small diameter relative to wavelength, a floe responds nearly with the water and scatters weakly. Individual-floe interference and the number of interfaces per unit distance prevent a universal monotonic diameter law.

Localizing an infinitesimal symmetry inside a normalized path integral turns current conservation into a distributional identity. The variation of each inserted field contributes a delta function at that insertion. The displayed sign uses $\delta S=-\int ({\partial }_{\mu }{ϵ}^a)j_a^{\mu }$; the opposite current convention reverses the left side. Measure anomalies or noninvariant boundary states can add terms.

The two photon orders in a tree-level charged-fermion amplitude must be summed. Contracting one photon vertex with its momentum writes the contracted slash as a difference of inverse Dirac propagators. The external Dirac equations remove the on shell inverse factors; the two remaining terms have equal size and opposite sign. Thus the sum vanishes under the momentum replacement although individual diagrams generally do not. This implements the gauge equivalence of photon polarization vectors.

In an axisymmetric thin disc with no vertical mass or torque flux, integrate mass and conservation of angular momentum over height. With fixed height-independent $l=r^2\Omega$, the equations are ${\partial }_t\Sigma +r^{-1}{\partial }_r(r\Sigma v_r)=0$ and ${\partial }_t(\Sigma l)+r^{-1}{\partial }_r(r\Sigma v_{r}l-r^3\bar{\nu }\Sigma {\Omega }^{\prime })=0$. Their difference gives $r\Sigma v_r{l}^{\prime }=(r^3\bar{\nu }\Sigma {\Omega }^{\prime }{)}^{\prime }$. The density-weighted viscosity of a disk supplies the integrated stress.

Within an instantaneous vertically averaged transport closure, the local Keplerian viscous diffusion equation has effective diffusion coefficient $3{\partial }_{\Sigma }(\bar{\nu }\Sigma )$. A positive coefficient smooths short radial Fourier modes in the allowed thin-disc wavelength range; a negative coefficient gives the negative-diffusion criterion for viscous disk instability. A zero coefficient is linearly degenerate and does not establish strict decay. This local response criterion alone does not settle thermal, self-gravitating or boundary-driven stability.