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For a nonzero Fourier mode in synchronous gauge, write $h_{ij}=h{\delta }_{ij}/3+h_s({\hat{k}}_i{\hat{k}}_j-{\delta }_{ij}/3)$. Under the passive displacement ${\xi }^{\mu }=(T,i{\hat{k}}_{i}L)e^{i\mathbf{k}\cdot \mathbf{x}}$, cancellation of spatial scalar shear requires $h_s+2kL=0$, and cancellation of the shift requires $kT-L^{\prime }=0$. The resulting cosmological Newtonian potential is $\Phi =(h_s^{\prime \prime }+\mathcal{H}{h}_s^{\prime })/(2k^2)$ and the spatial potential is $\Psi =-(h-h_s)/6-\mathcal{H}{h}_s^{\prime }/(2k^2)$. Both remain unchanged under residual synchronous-gauge freedom, for which $T^{\prime }+\mathcal{H}T=0$ and $L^{\prime }=kT$.

A smooth finite-energy monopole in an $SU(2)$ Yang-Mills theory with an adjoint Higgs field uses the normalized Higgs direction at infinity to define a map $S_{\infty }^2\to S^2$. Its nonzero degree supplies the magnetic topological sector. Unlike a singular Abelian Dirac monopole, the non-Abelian gauge and Higgs fields can have a regular core. In the zero-potential limit the Bogomolny-Prasad-Sommerfield monopole equations are $B_i=\pm D_i\Phi$.

Localizing a parameter of a global symmetry produces its Noether current times a derivative of that parameter. Introduce a gauge field with an inhomogeneous transformation and couple it to the Noether current to cancel this variation. Recompute the remaining variation and add the required interaction and transformation terms. Closure of the local symmetry algebra constrains the completion. For supersymmetry, its supercurrent couples to the gravitino, and the translation in the supersymmetry algebra requires a vierbein.

A holomorphic gauge coupling is the coefficient of the gauge kinetic F-term in a local Wilsonian effective action written in holomorphic variables. In a common normalization its real part is $1/g_h^2$. The continuous perturbative shift of its topological angle, together with holomorphy, restricts its renormalization-group beta function to one-loop order. Wave-function renormalization and anomalous canonical field rescalings distinguish it from the physical gauge coupling.

// Target: supersymmetry.bigb

With the reversed convention in curvature-sign convention in Killing derivative identities, choose the representative $h_{00}=2\Phi$, $h_{0i}=V_i$, $h_{ij}=2\Psi {\delta }_{ij}+h_{ij}^{\mathrm{TT}}$. The linearized Einstein components are $G_{00}^{(1)}=-2\Delta \Psi$, $G_{0i}^{(1)}=-2{\partial }_i\dot{\Psi }-\Delta V_i/2$ and $G_{ij}^{(1)}=({\partial }_i{\partial }_j-{\delta }_{ij}\Delta )(\Phi -\Psi )-2{\delta }_{ij}\ddot{\Psi }-({\partial }_i{\dot{V}}_j+{\partial }_j{\dot{V}}_i)/2+\square h_{ij}^{\mathrm{TT}}/2$. On all space with regular decaying fields, or for nonzero spatial Fourier modes, the vacuum constraints remove the scalar and vector parts, leaving two radiative gravitational wave polarizations. Singular Coulomb fields, spatial boundaries and homogeneous modes are not covered by that removal argument.

For a matrix-valued gauge field $A$, let $F=dA+A\wedge A$ be its gauge field strength; products below combine matrix multiplication with the wedge product of differential forms. The five-form can be written $Q_5=\mathrm{Tr}(A(dA{)}^2+\frac{3}{2}{A}^{3}dA+\frac{3}{5}{A}^5)$. Applying the exterior derivative, using its graded product rule and the graded cyclic property of the matrix trace, gives $d{Q}_5=\mathrm{Tr}(F^3)$. The six-form is an invariant polynomial in the gauge curvature, while its local potential depends on the connection.

The Euler--Mascheroni constant is

On $(0,\infty )$, the Gamma function is the unique positive function $f$ satisfying $f(1)=1$, $f(x+1)=xf(x)$, and the condition that $\log f$ is a convex function. This last condition concerns the natural logarithm of $f$, rather than $f$ itself, and removes the interpolation freedom left by the Gamma function recurrence.^[ref]

Rotating the integral defining the Gamma function toward the positive imaginary axis introduces the factor $i{e}^{-\pi n/2}$. After scaling by $n$, the remaining phase is $\log s-s$, whose unique stationary point is $s=1$ with second derivative $-1$. The stationary phase method gives $e^{-in-i\pi /4}\sqrt{2\pi /n}$ for this scaled integral, producing the displayed asymptotic. The boundary contour integral must be Abel-regularized; its off-saddle tails are smaller by integration by parts.

The beta-gamma identity gives $\Gamma (z/2{)}^2=B(z/2,z/2)\Gamma (z)$ for $\mathrm{Re}z>0$. A zero of $\Gamma (z)$ would therefore force zeros at every $z/2^m$. But $w\Gamma (w)=\Gamma (1+w)\to 1$ as $w\to 0$ in the right half-plane, a contradiction. The Gamma function recurrence then extends nonvanishing to every point where the meromorphic gamma function is finite. Nonpositive integers are poles, not zeros.

For positive real $a$, differentiating the gamma function integral under the integral sign gives the displayed formula; successive derivatives insert higher powers of $\log t$. Dominated differentiation on compact positive-$a$ intervals is justified by exponential decay at infinity and integrability of $t^{a-1}|\log t{|}^n$ at zero. At $a=1$, these derivatives give moments used in logarithmic endpoint asymptotic expansions.

For a Galton-Watson process, replacing an individual's reproduction by a one-descendant uneventful step with probability $q<1$ changes its offspring probability generating function to the displayed mixture. Its fixed-point equation reduces to $G(z)=z$, so the smallest fixed point and hence the extinction probability of a branching process are unchanged. The mean is $q+(1-q)G^{\prime }(1)$, which also preserves the subcritical/critical/supercritical classification. The excluded case $q=1$ never reproduces or dies and need not share the extinction probability.

Let $R$ be an integrally closed domain and $f\in R[X]$ monic, with discriminant outside a prime ideal $P$. The roots stay distinct after reduction. In the splitting algebra choose a prime above $P$. Its decomposition group maps onto the Galois group of the residue splitting field; its inertia group fixes every reduced root and is trivial because reduction bijects the distinct root sets. Thus the residue Galois group embeds in the generic group as a permutation group. For finite residue fields, irreducible factor degrees give the Frobenius cycle type.

For an abelian variety $A$ over a field $K$, its Weil–Châtelet group is the Galois cohomology group $H^1(K,A(K^{\mathrm{sep}}))$, classifying torsors under $A$. Such an algebraic-group torsor becomes $A$ after passing to a separable closure but need not have a $K$-point. A chosen geometric point identifies the torsor with $A$ and its Galois descent with a cocycle; changing that point adds a coboundary. The zero class is precisely the torsor with a rational point. For an elliptic curve, these torsors are genus-one curves with a specified action and Jacobian. Over a number field the kernel of restriction to every completion is the Tate–Shafarevich group.

Every red-blue finite coloring of the pairs of positive integers has either a blue complete arithmetic progression of any prescribed finite length $m$, or two disjoint $m$-term arithmetic progressions with all cross-pairs red. To see the geometry, assign $(a,d)$ a chosen red pair of indices $(r,s)$ in its progression, assuming the first alternative fails. Apply the Gallai theorem for an integer lattice to the fixed finite patternIn a monochromatic copy $(a_0,d_0)+qF$ with common label $(r,s)$, the selected red pairs are preciselyThe copy lies in the positive quadrant, so $d_0>q(m-1)$. Consequently the two displayed arithmetic progressions are disjoint. Translating the finite pattern before applying the lattice theorem permits its initially negative coordinates.

For a finite pattern $\{v_1,\dots ,v_k\}$ in a real vector space, pull a finite colouring back to words through the displayed sum map. A combinatorial line with active coordinates J maps to $a+|J|\{v_1,\dots ,v_k\}$. The Hales-Jewett theorem therefore forces a monochromatic homothetic copy. Integer patterns remain in the integer lattice; choosing b sufficiently large gives the positive-lattice version.

Every colouring of the positive integers, even with infinitely many colours, has an arithmetic progression of any prescribed finite length on which the colouring is constant or injective. Encode a progression by the equivalence relation of equality of its colours. This gives a finite colouring of its starting point and step. A Gallai theorem for an integer lattice copy of a suitable two-dimensional pattern makes either all indices inequivalent, or one equal-colour pair range independently over two whole progressions. In the second case, one of those progressions is monochromatic.

A survey window function describes the finite spatial coverage and weights of a galaxy survey. Multiplication by this window in position space mixes Fourier modes by convolution in Fourier space. A dimension of extent $L$ cannot robustly measure wavelengths much larger than $L$; the fundamental wavenumber of a periodic box of that size is $2\pi /L$.

Sampling with a homogeneous Poisson point process of mean comoving number density $\bar{n}$ adds a constant $1/\bar{n}$ to the dimensional galaxy power spectrum. Subtracting its expectation leaves sampling uncertainty. Correlated tracer populations and survey weights modify this simple formula; survey depth alone does not specify $\bar{n}$.

Redshift-space distortions arise because observed galaxy redshifts include peculiar velocity as well as cosmological redshift. Distances inferred solely from expansion therefore distort clustering along the line of sight. Coherent large-scale motion and small-scale velocity dispersion produce different effects on the inferred matter power spectrum.