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At least one finite prime ramifies in a Zp-extension; otherwise the entire tower would lie in a finite Hilbert class field. Ramification can occur only over $p$: local class field theory makes inertia away from $p$ an image of a unit group with finite maximal pro-$p$ quotient, whereas ${\mathbb{Z}}_p$ has no nontrivial finite subgroup. Nonzero inertia is open in ${\mathbb{Z}}_p$, so it becomes total after passing to a sufficiently high finite layer.

A structural-zero component in a count mixture produces a count identically equal to zero. If its mixture weight is $\pi$ and the ordinary component also assigns mass $p_0$ to zero, the observed zero probability is $\pi +(1-\pi )p_0$. Thus an observed zero does not reveal the latent component membership. A shared component for a repeated count profile makes the whole profile zero and induces dependence between its components.

Buy one maturity-$(T-1)$ bond and short $1+f$ maturity-$T$ bonds. Reinvest the first bond's unit maturity payoff in $1/P_{T-1}(T)$ maturity-$T$ bonds. This self-financing portfolio pays $r_T-f$ at $T$. The replication uses only the adjacent maturities and does not require completeness of the whole bond market.

For a real scalar coupled through $-g\bar{\psi }\varphi \psi$, the scalar-fermion Feynman vertex is $-ig$. Two-fermion scattering has scalar exchange in the $t$ and $u$ channels with a relative minus from exchanging identical external fermions. Fermion-antifermion scattering has $t$ exchange and $s$ annihilation, again with a relative fermionic minus for a fixed state ordering. Fermion-scalar scattering has the two orderings along one open fermion line, using internal momenta $p_1+p_2$ and $p_1-p_4$; these have the same sign. The Mandelstam variables specify the propagator denominators.

For one four-component Dirac field with scalar interaction $-y\bar{\psi }\psi \varphi$ and the mostly-plus propagator convention, the fermion loop numerator is $4[M^2-k\cdot (k-p)]$. Reduce it to a bubble and tadpole: $-4y^2[(2M^2+p^2/2)J-I]$. Dimensional regularization gives pole parts $J=2/(\epsilon 16{\pi }^2)$ and $I=-2M^2/(\epsilon 16{\pi }^2)$. The local pole then requires both wavefunction renormalization and a scalar mass counterterm.

In signature $(-,+)$, choose real gamma matrices ${\gamma }^0=\left(\begin{array}{cc}0& -1\\ 1& 0\end{array}\right)$, ${\gamma }^1=\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right)$ and ${\gamma }_{\ast }=-{\gamma }^0{\gamma }^1$. The Dirac equation then gives $({\partial }_{\tau }+{\partial }_{\sigma }){\psi }_{+}=0$ and $({\partial }_{\tau }-{\partial }_{\sigma }){\psi }_{-}=0$. Positive chirality is right-moving and negative chirality is left-moving in this orientation. Reversing the definition of ${\gamma }_{\ast }$ exchanges the labels without changing the physics.

Contracting an antighost with the cubic FP ghost term of the BRST operator produces the bilinear generators. They give $[L_m,b_n]=(m-n)b_{m+n}$ and $[L_m,c_n]=-(2m+n)c_{m+n}$. Their zero-mode additive constant depends on the vacuum and normal ordering convention; it must agree with the corresponding $c_0$ term in the charge.

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.