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A pivot element is the current diagonal entry used to eliminate entries below it in Gaussian elimination, after any row interchange. At step $k$ it is $a_{kk}^{(k-1)}$, where $A^{(k-1)}$ is the current reduced matrix. Computing elimination multipliers requires this entry to be nonzero. A zero pivot element can reflect an unsuitable row order even for an invertible matrix, as Pivoted LU decomposition illustrates.

For $Y_s=\lambda s+W_s$ and fixed $t>0$, subtracting $(s/t)Y_t$ leaves the Brownian bridge $W_s-(s/t)W_t$. It is independent of $(\lambda ,Y_t)$, since it is independent of $\lambda$ and has zero covariance with $W_t$ in their jointly Gaussian law. The observed path is determined by this bridge and its endpoint, so the bridge gives no further information about $\lambda$. Hence conditioning on the full observation filtration at $t$ gives the same posterior as conditioning on $Y_t$ alone.

For the Gaussian Brownian drift filter with positive definite $V_0$, define $Z_t=\det (I+t{V}_0{)}^{1/2}\exp [-m_t^T{V}_t^{-1}{m}_t/2+m_0^T{V}_0^{-1}{m}_0/2]$. Because $V_t^{\prime }=-V_t^2$ and $d{m}_t=V_{t}d{\hat{W}}_t$, the Itô formula gives $d\log Z_t=-m_t^{T}d{\hat{W}}_t-|m_t{|}^{2}dt/2$. Thus $Z$ is a positive stochastic exponential, hence a nonnegative local martingale and a supermartingale. For a singular prior, the inverse-matrix expression is undefined; the same process can instead be defined by $Z_t=({\mathbb{E}}_{\lambda \sim N(m_0,V_0)}\exp [{\lambda }^T{Y}_t-t|\lambda {|}^2/2]{)}^{-1}$, with $Y_t$ treated as fixed inside this Gaussian integral.

For a nontrivial Dirichlet character modulo a prime $p$ and $a\not\equiv 0(\mathrm{mod}p)$, set $\tau ={\sum }_{y\ne 0}\chi (y)e(ay/p)$. Substituting $y=tz$ in $|\tau {|}^2$ gives ${\sum }_{t\ne 0}\chi (t){\sum }_{z\ne 0}e(a(t-1)z/p)$. The inner sum is $p-1$ at $t=1$ and $-1$ otherwise. Since character orthogonality gives ${\sum }_{t\ne 0}\chi (t)=0$, the result is $p$.

The Gauss hypergeometric function is the normalized analytic solution at zero,initially for $|z|<1$ and $c\notin \{0,-1,-2,\dots \}$, followed by analytic continuation where possible.

The ordinary hypergeometric function is ${}_2{F}_1(a,b;c;z)={\sum }_{n\ge 0}(a{)}_n(b{)}_n{z}^n/((c{)}_{n}n!)$ near zero for admissible $c$. It solves the Gauss hypergeometric equation and extends by analytic continuation.

Multiply a solution with parameters $(c-a,c-b;c)$ by $(1-z{)}^{c-a-b}$. Its local exponents become exactly those of the Gauss hypergeometric equation with parameters $(a,b;c)$: those at one shift by $c-a-b$, and those at infinity shift by its negative. Direct differentiation gives the same differential equation. Near zero, choose the factor with value one; uniqueness of the analytic normalized solution, for $c$ not a nonpositive integer, gives the identity. Compatible analytic continuation extends it to other domains and parameters where the normalized functions are defined.

The inverse branch of the $n$th iterate of the Gauss continued-fraction map on a fixed digit cylinder has the displayed form, with continued fraction convergents $p_j/q_j$. Its derivative has absolute value $(q_n+q_{n-1}y{)}^{-2}$. Since $q_{n-1}\le q_n$, the maximum-to-minimum derivative ratio is at most four. The density of the Gauss measure changes this ratio by at most a further factor two. The cylinders shrink to irrational points, so this uniform distortion bound and the Lebesgue density theorem prove ergodicity of the map.

On each branch with digit $a=\lfloor 1/x\rfloor$, the displayed extension of the Gauss continued-fraction map is invertible almost everywhere: $x=(a+x^{\prime }{)}^{-1}$ and $y=(y^{\prime }{)}^{-1}-a$, with $a=\lfloor 1/y^{\prime }\rfloor$. Its invariant joint probability density function is $[(\log 2)(1+xy{)}^2{]}^{-1}$. The branch Jacobian determinant is $[x^2(a+y{)}^2{]}^{-1}$, which cancels the transformation of $1+xy$. Integrating out $y$ gives the Gauss measure density.

For a hypersurface with unit normal vector $n$, put $\epsilon =g(n,n)=\pm 1$, $P=1-\epsilon n\otimes n^{♭}$, and $K(X,Y)=g({\nabla }_{X}n,Y)$ for tangent vectors. The induced metric is nondegenerate precisely when the normal is non-null. With $R(X,Y)Z=[{\nabla }_X,{\nabla }_Y]Z-{\nabla }_{[X,Y]}Z$ and $R_{abcd}=g(e_a,R(e_c,e_d)e_b)$, the Gauss formula is ${\nabla }_{X}Y=D_{X}Y-\epsilon K(X,Y)n$. Substituting twice into the ambient Riemann curvature tensor and projecting tangentially gives the displayed Gauss equation. Its normal projection gives $g(n,R(X,Y)Z)=-(D_{X}K)(Y,Z)+(D_{Y}K)(X,Z)$, the curved-ambient Codazzi equation. A null hypersurface needs additional geometric data because its induced metric is degenerate.

On a contractible Cartesian patch of a flat background, zero linearized Riemann curvature operator says that the connection 1-forms ${\Gamma }^{(1)a}{}_{bc}d{x}^c$ are closed. The Poincaré lemma writes them as derivatives of a matrix-valued potential. Symmetry of the two lower affine connection indices then supplies a second potential, giving ${\Gamma }^{(1)a}{}_{bc}={\partial }_b{\partial }_c{\xi }^a$. Linearized metric compatibility implies that $h_{ab}-2{\partial }_{(a}{\xi }_{b)}$ is constant. Its symmetric constant part is absorbed by an affine vector field. Thus the metric perturbation is locally a coordinate change. Global topology and imposed boundary conditions can obstruct the corresponding global gauge choice.

A spatial transformation is a symmetry of a gauge-field configuration when its pullback lies in the gauge orbit of the original configuration. It need not fix a particular representative. For a planar vortex, rotation changes its winding phase and a constant $U(1)$ transformation compensates. For a hedgehog ansatz for a monopole, a spatial rotation is compensated by the corresponding internal adjoint $SU(2)$ rotation. Gauge-invariant densities remain invariant without choosing a special gauge.

A gauge generator is the representation matrix of a basis element of the Lie algebra of a continuous gauge group on a specified set of fields. For Hermitian generators $T^a$, an infinitesimal transformation is $\delta \varphi =i{\alpha }^a{T}^a\varphi$, and $[T^a,T^b]=i{f}^{ab}{}_c{T}^c$. Gauge invariance of a superpotential consequently requires $W_i(T^a{)}^i{}_j{\varphi }^j=0$ for each generator.

A gauge condition $F(x)=0$ is regular at a representative $x_0$ if its derivative along the gauge orbit is an invertible matrix. The inverse function theorem then makes the gauge-condition values valid local coordinates along that orbit. A unique zero without this regularity is insufficient: the translation action on the line with $F(x)=x^3$ has a unique zero but a zero derivative there. Multiple regular intersections give a Gribov ambiguity rather than a globally unique slice.

For the Abelian Higgs mechanism with scalar phase $\chi$ and vector mass $m$, take ${\mathcal{L}}_{\mathrm{gf}}=-(\partial \cdot A+\xi m\chi {)}^2/(2\xi )$ when the scalar kinetic mixing is $-m{A}_{\mu }{\partial }^{\mu }\chi$. The cross term cancels this mixing after integration by parts. The resulting vector propagator isAt fixed finite $\xi$ it decays as $1/k^2$. Would-be Goldstone bosons and the required Faddeev-Popov ghosts remain in intermediate calculations; BRST symmetry identifies the physical states and ensures cancellation of unphysical modes. The limit $\xi \to \infty$ gives the unitary gauge propagator and is not uniform in ultraviolet momentum.

For the ordinary three-dimensional cross product, $D_{\mu }\Phi ={\partial }_{\mu }\Phi +e{A}_{\mu }\times \Phi$ requires $F_{\mu \nu }={\partial }_{\mu }{A}_{\nu }-{\partial }_{\nu }{A}_{\mu }+e{A}_{\mu }\times A_{\nu }$. This follows from $[A\times ,B\times ]=(A\times B)\times$. The same curvature then transforms covariantly under local rotations. Using a minus sign in its quadratic term while retaining the plus sign in $D$ breaks local gauge invariance: a pure gauge for $D$ has $F_{+}=0$ but generally $F_{-}=-2e{A}_{\mu }\times A_{\nu }\ne 0$. Reversing both signs is an alternative consistent convention.

In complexified four-dimensional space-time, antisymmetry giveswhere both gauge curvature two-component spinors are symmetric. The two summands are the two duality components. In the convention where the primed symmetric part is self-dual, the Anti-self-dual Yang-Mills equations are ${\phi }_{A^{\prime }{B}^{\prime }}=0$. Equivalently ${\pi }^{A^{\prime }}{\pi }^{B^{\prime }}{F}_{A{A}^{\prime }B{B}^{\prime }}=0$ for every primed two-component spinor $\pi$, since a quadratic polynomial vanishing for every two-component spinor has all symmetric coefficients zero.

For the Killing form or another invariant bilinear form on a Lie algebra,The connection terms cancel by invariance. Thus, for compactly supported variations, a gauge covariant derivative can be moved to the other factor in an integral with a minus sign, just as an ordinary derivative.

// Target: quantum-field-theory.bigb

Embed $C_4\times C_4$ and $Q_8\times C_2$ in $S_{16}$ by their regular representations. Both have one element of order one, three of order two, and twelve of order four. An order-$r$ element in a regular action has cycle type $r^{16/r}$. Their intersections with every conjugacy class of $S_{16}$ consequently have the same size, so they are Gassmann equivalent. The first group is abelian and the second is not, so their underlying groups are not isomorphic. Finite-cover constructions from the Sunada theorem therefore give isospectral manifolds with different fundamental groups.

In a finite general linear group, the permutation representations on nonzero vectors and on nonzero dual vectors have equal characters: an element fixes $q^{\dim \ker (t-I)}-1$ vectors in each action because a matrix and its transpose have the same rank. In characteristic two the stabilizers of a nonzero vector and a hyperplane are therefore almost conjugate subgroups. In dimension three over ${\mathbb{F}}_2$, both have order $24$ and index $7$ in a group of order $168$, but they are not conjugate: a point stabilizer preserves no plane. A triangle cover construction for Sunada surfaces with generator orders $(7,7,7)$ gives quotient surfaces of genus three.

Feynman slash notation contracts a four-vector with the gamma matrices. With mostly-minus signature, the Clifford algebra gives $/p^2=p^{2}I$, so $(/p-m{)}^{-1}=(/p+m)/(p^2-m^2)$ away from the mass shell.