A second-order stationary complex random field on the line has uncentered correlation $C(\zeta )=\mathbb{E}[E(z+\zeta )\overline{E(z)}]$. When this correlation is continuous, its Fourier transform is a finite positive measure $F$ characterized by $C(\zeta )=\int e^{i\nu \zeta }F(d\nu )$. Its total mass is $\mathbb{E}|E(z){|}^2$. If it has a density, $S(\nu )=(2\pi {)}^{-1}\int C(\zeta )e^{-i\nu \zeta }d\zeta$. A constant mean $m$ contributes $|m{|}^2{\delta }_0$; the remaining measure is the covariance spectrum. This uncentered convention is useful for coherent and diffuse wave fields.

In the zero-sum game with row payoff matrix $\left(\begin{array}{cc}0& 1\\ 0& 0\end{array}\right)$, the second row is weakly dominated by the first. Nevertheless second row and first column form a Nash equilibrium: both rows tie against that column, and both columns tie against that row. Weak dominance therefore cannot be used to discard all possible equilibrium strategies without further qualifications.

For an infinite-dimensional normed space, the weak-star open unit ball of its dual is not a Baire space. The smaller closed norm balls are weak-star closed and have empty relative interior: a nonzero functional annihilating any finitely many tested vectors can move the norm beyond the smaller radius while remaining below one. Their countable union is the open norm ball. Thus this topology is not topologically complete, regardless of separability.

For the linear program maximizing $c^{T}x$ with $Ax\le b$, $x\ge 0$, a vector $y\ge 0$ with $A^{T}y\ge c$ proves $c^{T}x\le y^{T}b$ for every feasible $x$. If a feasible $x$ attains that bound, it is optimal. This is weak duality written as a directly checkable certificate; adding nonnegative multiples of constraints suffices to verify it. The equivalent reversed-inequality certificate applies to a minimization program.

The weak gradient is the vector of first-order weak derivatives of a function. Its $i$th component is specified by $\int u{\partial }_i\phi =-\int ({\partial }_{i}u)\phi$ for compactly supported test functions. For a differentiable function with appropriate local integrability it agrees with the classical gradient.

A sequence can converge as distributions off one point while retaining a nonconvergent multiple of a Dirac delta function at that point. If $u_j=v+c_j{\delta }_p+o_{{\mathcal{D}}^{\prime }}(1)$ with $c_j$ nonconvergent, restriction to the punctured domain tends to $v$, but a test nonzero at $p$ detects the obstruction to global convergence.

Every bounded sequence in a Hilbert space has a weakly convergent subsequence. The closed span of a sequence is separable; diagonal convergence of coordinates along an orthonormal basis gives a bounded limiting linear functional, represented by a vector of the Hilbert space. Applied to a bounded family of mollifications, this proves that their strong Lebesgue limit inherits the weak derivative bound.

A spin-zero parent decaying through a chiral weak charged current to a charged lepton and a massless neutrino cannot supply the helicities selected by a massless interaction. A charged-lepton mass insertion makes the amplitude proportional to $m_{\ell }$ and the decay width proportional to $m_{\ell }^2$. A spin-one W boson does not suffer this suppression.

The weak charged current pairs each neutrino with its corresponding charged lepton. Thus elastic muon-neutrino/electron scattering has only tree-level neutral-current exchange, while electron-neutrino and electron-antineutrino scattering also have crossed and annihilation charged-current diagrams. A contact term pairing a muon neutrino directly with an electron would violate this Standard Model flavour assignment.

For two outgoing massless leptons, ${\ell }_{\mu }=\bar{u}(q_1){\gamma }_{\mu }(1-{\gamma }^5)v(q_2)$ is transverse to $q=q_1+q_2$. This follows from the massless Dirac equation and anticommutation of the chirality matrix with the gamma matrices. Thus the $q^{\mu }{f}_{-}(q^2)$ term in a pseudoscalar-to-pseudoscalar form factor does not contribute to this scattering amplitude. A nonzero charged-lepton mass spoils exact transversality and restores a mass-suppressed scalar-form-factor contribution.

For a real wavefront error $h={\sum }_k{a}_k{s}_k$, its squared spatial root mean square is $⟨h^2⟩={\sum }_{k,l}{a}_k{a}_l⟨s_k{s}_l⟩$. Orthogonal unit-peak sine modes have mean square $1/2$, so $h_{\mathrm{rms}}^2=\frac{1}{2}{\sum }_k{a}_k^2$, whereas the coefficient norm is $\sqrt{{\sum }_k{a}_k^2}$. On a finite optical pupil orthogonality needs checking; independent uniform random wave phases give the same $1/2$ diagonal after wave phase averaging. This normalization matters when a speckle contrast of a sinusoidal wavefront error is specified by peak wave amplitude.

For the unit-speed wave equation, the integral of $(u_t^2+|\nabla u{|}^2)/2$ over $B_r(x_0)$ at time $T\ge 0$ is at most its initial integral over $B_{r+T}(x_0)$. On a shrinking ball the outward energy flux minus the loss from its moving boundary is $-((u_t-{\partial }_{n}u{)}^2+|{\nabla }_{\tan }u{|}^2)/2$, hence nonpositive. This estimate establishes the domain of dependence and finite propagation speed without any assumptions at spatial infinity.

For an individual monochromatic component in $c=c(Y)$, conserved $\omega ,k$ give $\ell =\sqrt{{\omega }^2/c^2-k^2}$. With amplitude independent of $X,T$, wave-action transport for a variable-speed wave conserves $c^2\ell A^2$ on a positive-$\ell$ branch, hence $A=\mu /(c\sqrt{\ell })$. The divergence at $\ell =0$ signals failure of the outer WKB approximation.

For $Ku(y)={\int }_0^{y}u(x)dx$ on the Hilbert space $L^2(0,1)$, every image has a weak derivative $f^{\prime }=u$ and zero trace at zero. Conversely every such Sobolev space element is this integral of its weak derivative. The range is dense because it contains smooth functions supported in $(0,1)$, but it is not closed: a step function can be approximated in $L^2$ by continuous ramps and is not itself in $H^1$. The Moore–Penrose inverse of an operator has this range as its domain and differentiates there.

On $(0,1/2)$ use the forward difference and on $(1/2,1)$ the backward difference, with $0<h<1/2$. This defines a bounded linear operator on $L^2(0,1)$ and $\Vert D_h\Vert \le \sqrt{6}/h$. For $f$ in the range of the Volterra integration operator, these differences are local averages of $f^{\prime }$ and converge to $f^{\prime }$ in $L^2$ by continuity of translations. Hence $D_h$ gives a linear regularization of differentiation. No pointwise evaluation of an arbitrary $L^2$ equivalence class is needed: translated functions are defined almost everywhere.

A lower-viscosity basal sediment layer changes the plate coupling in viscous mountain spreading over an underthrust plate. With mountain thickness $H$ and sediment depth $d$, equal densities give one hydrostatic pressure gradient $G=\rho g{\partial }_x(H+d)$. Fluxes are $q_s=Ud-G(d^{2}H/2+d^3/3)/{\mu }_s$ and $q_m=UH-G(d{H}^2+d^{2}H/2)/{\mu }_s-G{H}^3/(3\mu )$. Separate conservation of mass laws determine each layer. Steady zero mountain flux and sediment influx $U{d}_{\infty }$ give contact height $3d_{\infty }$ and surface slope $2{\mu }_{s}U/(9\rho g{d}_{\infty }^2)$.

For a Keplerian disk with evolution ${\partial }_t\Sigma =(2\pi r{)}^{-1}{\partial }_r\mathcal{F}+S$ and $\mathcal{F}=6\pi r^{1/2}{\partial }_r(r^{1/2}\bar{\nu }\Sigma )$, take zero viscous torque at $r_1>0$ and zero flux outside the source region at $r_2>r_1$. The steady state gives ${\mathcal{F}}^{\prime }=-2\pi rS$, and its outer condition gives the displayed flux. Integrating once more from the zero-torque inner boundary condition yieldsThe equation determines the viscosity-density product, so a thermodynamic viscosity closure can be imposed afterward. For nonnegative integrable $S$, it is nonnegative and the inner accretion rate is exactly the total supplied mass per unit time.

For a Keplerian accretion disk with $\bar{\nu }\propto r^{5/2}$, dimensionless $g=\bar{\nu }\Sigma /({\bar{\nu }}_0{\Sigma }_0)$ satisfies $g_{\tau }=4x^{5/2}{g}_{xx}+6x^{3/2}{g}_x$. In $y=x^{-1/2}$ this is $g_{\tau }=y{g}_{yy}$. The ansatz $g=G(y/\tau )$ gives $G=A+B{e}^{-y/\tau }$; finite disk mass selects $A=0$. Its nonzero profile has logarithmically infinite disk angular momentum on an infinite radial domain.

A Keplerian disk with surface density $\Sigma \sim r^{-p}$ at infinity has finite outer mass for $p>2$ and finite outer angular momentum for $p>5/2$, since the relevant integrals are $\int \Sigma rdr$ and $\int \Sigma r^{3/2}dr$. The borderline $p=5/2$ has logarithmically divergent angular momentum even though its mass converges.

The conduit equation evolves the positive cross-sectional area $A$ of a viscous buoyant conduit. Its scaling uses axial length $a_0/\sqrt{8\lambda }$ and velocity $\Delta \rho g{a}_0^2/(8\lambda {\mu }_o)$, where $\lambda ={\mu }_i/{\mu }_o$. Linearization at $A=1$ gives $\omega =2k/(1+k^2)$: phase velocity is upward but group velocity changes sign at $|k|=1$. Nonlinear elevation waves obey the solitary-wave amplitude-speed relation for the conduit equation.