The integrated mean squared error is $\mathbb{E}{\int }_D(\hat{m}(x)-m(x){)}^{2}dx$ over a specified domain $D$. The Tonelli theorem permits integration of the pointwise bias-variance decomposition of mean squared error.

The source-free Maxwell Lagrangian gives $S=-\frac{1}{4}\int F_{\mu \nu }{F}^{\mu \nu }{d}^{4}x$ in mostly-minus Lorentzian signature. Its Euclidean action is the displayed positive quadratic functional of the electromagnetic field tensor. Variation, with boundary terms vanishing, yields ${\partial }_{\mu }{F}_{\mu \nu }=0$.

For a harmonic function $w$ smooth inside a bounded domain and $C^1$ on its closure, $\Delta |Dw{|}^2=2|D^{2}w{|}^2\ge 0$. The weak maximum principle for elliptic operators and continuity of the gradient imply ${\sup }_{\Omega }|Dw|={\sup }_{\partial \Omega }|Dw|$.

For density $r(x-\theta {)}^{r-1}$ on $[\theta ,\theta +1]$, $0<r<1$, the interior likelihood function increases without bound as $\theta$ approaches the sample minimum. The extended-likelihood estimator is that minimum. An arbitrary finite reassignment of the density at the endpoint can eliminate an attained finite maximum, so the endpoint convention must be stated.

A real matrix is nonnegative when every entry is nonnegative. This entrywise condition differs from being a positive semidefinite matrix: the symmetric matrix $\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right)$ is nonnegative but has eigenvalues $1$ and $-1$.

A rational matrix is a matrix whose entries are rational numbers. Multiplying by a common denominator produces an integer matrix and preserves its homogeneous solution set.

A square real matrix is doubly stochastic if its entries are nonnegative and every row and column sums to $1$. Its positive-entry support satisfies the condition of the Hall marriage theorem: for a row subset $S$, $|S|\le |N(S)|$ by summing column capacities. Consequently its support contains a permutation matrix.

The Hermite normal form is a canonical triangular form for an integer matrix under integer elementary row operations or, in the column convention, column operations. In the row convention for a nonsingular rank-two matrix, positive diagonal entries and reduction of the upper-right entry modulo the lower diagonal entry give the row Hermite normal form in rank two.

A simple zero of an advection coefficient can support an interior layer. For the specific homogeneous equationset $z=(x-x_{\ast })/\epsilon$. The two derivative terms then have the same order, giving $Y^{\prime \prime }+2z{Y}^{\prime }=0$. Hence $Y^{\prime }=C{e}^{-z^2}$ andconnects $Y(-\infty )=A$ and $Y(+\infty )=B$. Its physical width is $O(\epsilon )$. More generally, $a(x_{\ast })=0$, $a^{\prime }(x_{\ast })\ne 0$ in ${\epsilon }^2{y}^{\prime \prime }+a(x)y^{\prime }=0$ gives the same width, while the sign of $a^{\prime }(x_{\ast })$ determines whether the nonconstant Gaussian derivative decays.

An order-one reaction term cannot be hidden in this balance: adding $c(x)y=f(x)$ gives the leading inner equation $Y^{\prime \prime }+a^{\prime }(x_{\ast })z{Y}^{\prime }+c(x_{\ast })Y=f(x_{\ast })$. An arbitrary pair of constants need not match that equation. The error function connection above requires the extra terms to vanish or be negligible at leading order; reversing diffusion also changes the bounded matching structure.

A uniformly integrable martingale $M_n$ has almost sure convergence and convergence in L1 to an integrable random variable $M_{\infty }$, and $M_n=\mathbb{E}[M_{\infty }\mid {\mathcal{F}}_n]$.

A martingale satisfying ${\sup }_n\mathbb{E}|M_n{|}^p<\infty$, for $1<p<\infty$, has almost sure convergence and convergence in Lp to a limit $M_{\infty }\in L^p$. Also $M_n=\mathbb{E}[M_{\infty }\mid {\mathcal{F}}_n]$. The Doob Lp maximal inequality makes ${\sup }_n|M_n|$ integrable to the power $p$; the martingale convergence theorem gives the almost-sure limit, and the dominated convergence theorem gives convergence in the Lp norm. Boundedness in $L^1$ alone does not imply convergence in L1.

A marginally irrelevant operator has a coupling that is unchanged to linear order by the renormalization group, but flows slowly toward zero once nonlinear terms are included. For an increasing coarse-graining length with logarithmic scale $\ell$, a simple example isTo leading order, $g(\ell )=g_0/(1+a{g}_0\ell )$, so the perturbation decays as the inverse logarithm of length rather than as a power of length.

The coupling leaves the limiting universality class unchanged, but can produce logarithmic corrections and slow convergence in finite systems. Ignoring it can make an apparent critical exponent drift with the observation scale. The direction of the flow and the sign of the initial coupling matter: the same local formula sends negative $g$ away from zero rather than making every perturbation irrelevant.

A marginal density is a probability density function for one component of a joint probability distribution. Integrating the joint probability density over the other coordinates gives it; for two components, $f_X(x)=\int f_{X,Y}(x,y)dy$.

For Grassmann-odd spinors in two dimensions with $\bar{\psi }={\psi }^{T}C$, $C^T=-C$ and $({\gamma }^{\mu }{)}^{T}C=-C{\gamma }^{\mu }$, moving components past each other gives $\bar{\chi }ϵ=\overline{ϵ}\chi$ and $\bar{\chi }{\gamma }^{\mu }{\gamma }^{\nu }ϵ=\overline{ϵ}{\gamma }^{\nu }{\gamma }^{\mu }\chi$. The transpose of a two-gamma product reverses its order; the Grassmann sign and the antisymmetry of $C$ cancel. For $\delta \psi =B{\gamma }^{\mu }{\partial }_{\mu }Xϵ$, this also gives $\delta \bar{\psi }=-B\overline{ϵ}{\gamma }^{\mu }{\partial }_{\mu }X$. These identities fix the boundary variation in worldsheet supersymmetry; commuting spinors would not obey the same interchange rule.

Approximate a continuous function on ${\mathbb{Z}}_p$ uniformly by a function constant on residue classes modulo $p^r$. On this finite space, translation $S$ satisfies $S^{p^r}=I$, so ${\Delta }^{p^r}=(S-I{)}^{p^r}$ has every matrix coefficient divisible by $p$. Hence $\Vert {\Delta }^{j{p}^r}\Vert \le p^{-j}$, and the Mahler coefficients of a locally constant function tend to zero. Since $|a_n(f)-a_n(g){|}_p\le \Vert f-g{\Vert }_{\infty }$, uniform approximation proves the assertion for every continuous function. This completes the coefficient-decay part of the Mahler theorem.

For a vertical magnetic field in a homogeneous shearing sheet, a horizontal normal mode with growth rate $s$ and vertical Alfvén frequency ${\omega }_a=k{v}_{Az}$ satisfiesThe two roots for $s^2$ are real. For $q>0$, a growing magnetic mode exists precisely when $0<{\omega }_a^2<2q{\Omega }^2$. Magnetic tension is strong enough to couple radially displaced elements, yet weak enough to allow their separation to increase. Ogilvie's lecture on magnetorotational instability derives this relation and its finite-thickness boundary conditions.

For ideal magnetohydrodynamics at constant mass density $\rho$, a shearing sheet with velocity $-q\Omega x{\mathbf{e}}_y+{\mathbf{v}}_h(z,t)$ and magnetic field ${\mathbf{B}}_h(z,t)+B_z{\mathbf{e}}_z$ obeysHorizontal invariance and vanishing vertical velocity eliminate nonlinear advection. These equations retain Coriolis acceleration, orbital shear and magnetic tension. The vertical pressure adjusts to balance gravity and magnetic pressure.

For the magnetically charged dilaton black hole, the metric-dual one-form of $k={\partial }_t$ is $k^{♭}=-Vdt$. With orientation $dt\wedge dr\wedge d\theta \wedge d\varphi$, its Hodge star operator gives $\star d{k}^{♭}=-V^{\prime }{R}^2\sin \theta d\theta \wedge d\varphi$. Therefore the Komar mass is $R^2{V}^{\prime }/2=r_{+}/2-P^2/r$. Its radial dependence is allowed because the Ricci tensor is nonzero; the divergence of $d{k}^{♭}$ is sourced by $R_{ab}{k}^b$. At the horizon $M_K(r_{+})=(r_{+}-r_{-})/2=\kappa A_H/(4\pi )$.

Rotate any geodesic into an equatorial plane of the magnetically charged dilaton black hole. Conserved energy and angular momentum are $E=V\dot{t}$ and $L=R^2\dot{\varphi }$. Normalizing its tangent to $g(\dot{x},\dot{x})=-ϵ$, with $ϵ=1,0,-1$ for timelike, null and spacelike geodesics, gives ${\dot{r}}^2=E^2-V(ϵ+L^2/R^2)$. Thus its radial equation is ${\dot{r}}^2/2+\tilde{V}=0$. Radial null rays have $\dot{r}=\pm E$ and use the same Schwarzschild tortoise coordinate with $2M$ replaced by $r_{+}$.

For $r_{-}<r_{+}$, an ingoing chart with $v=t+r_{\ast }$ has metric $-Vd{v}^2+2dvdr+R^{2}d{\Omega }_2^2$. In its future interior, $\nabla r={\partial }_v+V{\partial }_r$ is future timelike, so every future causal curve decreases $r$ and cannot reach future null infinity. The singularity $r=r_{-}$ is spacelike. A Kruskal–Szekeres coordinates extension has the same qualitative Penrose diagram as Schwarzschild spacetime, with two exterior regions, a future black-hole interior and a past white hole interior. The past interior has the same radial interval but is not a black hole region; specifying the radial interval alone does not identify the future component. For $r_{-}>r_{+}$, the physical exterior $r>r_{-}$ has $V>0$ everywhere and outgoing null rays escape from arbitrarily near a timelike naked singularity.