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Independent claim Poisson processes of rates ${\lambda }_i$ merge into a Poisson process of rate $\lambda ={\sum }_i{\lambda }_i$, by the Superposition theorem for Poisson point processes. The merged claim law is a mixture distribution of the individual claim laws with weights ${\lambda }_i/\lambda$. Equivalently, multiplication of the individual compound-Poisson transforms produces $\exp (\lambda [{\sum }_i({\lambda }_i/\lambda )M_i(r)-1])$.

For nonnegative matter and coasting-fluid densities, $0\le {\Omega }_m\le 1$. Since $a\le {\Omega }_m+(1-{\Omega }_m)a\le 1$ on the normalized integration interval, the age integrand lies between $\sqrt{a}$ and one. The age of a flat matter-coasting-fluid universe therefore decreases monotonically with the matter fraction and stays between the pure-fluid endpoint ages.

A closed half-space consists of a hyperplane and one of its two sides in a real affine space. Replacing the weak inequality by a strict inequality gives an open half-space. Half-spaces are convex sets, and finite intersections of closed half-spaces are linear polyhedra.

An adiabatic perturbation obeys ${\Delta }_r=(4/3){\Delta }_c$. In a radiation background this makes the cold-matter density equation ${\Delta }_c^{\prime \prime }+{\Delta }_c^{\prime }/\tau =4{\Delta }_c/{\tau }^2$, whose growing solution is ${\tau }^2$. In a matter background it instead becomes ${\Delta }_c^{\prime \prime }+2{\Delta }_c^{\prime }/\tau =6{\Delta }_c/{\tau }^2$, with the same growing conformal-time exponent and a different scale-factor dependence.

For a nontrivial additive character $\psi$ on a local field, the character annihilator of an additive subgroup $H$ is $H^{\perp }=\{y:\psi (xy)=1\text{ for all }x\in H\}$. If ${\mathcal{O}}_F^{\perp }={\pi }^c{\mathcal{O}}_F$, then $({\pi }^n{\mathcal{O}}_F{)}^{\perp }={\pi }^{c-n}{\mathcal{O}}_F$. This is a character pairing construction, distinct from the annihilator of a module.

A completely additive arithmetic function satisfies $f(mn)=f(m)+f(n)$ for all positive $m,n$, equivalently $f(p^k)=kf(p)$. The total number of prime factors is an example.

A strongly additive arithmetic function satisfies $f(p^k)=f(p)$ for every prime $p$ and $k\ge 1$, so $f(n)={\sum }_{p\mid n}f(p)$. The prime omega function is an example.

For an additive arithmetic function, the model mean is $A_f(x)={\sum }_{p^k\le x}f(p^k)(1-1/p)/p^k$. The exact uniform expectation differs by $O(x^{-1}{\sum }_{p^k\le x}|f(p^k)|)$.

An acoustic monopole represents unsteady mass or volume injection. A moving-interface mass-flux defect contributes ${\partial }_t(Q{\delta }_s)$ to the density wave equation.

An acoustic dipole represents an applied force density. A surface force defect contributes $-{\partial }_i(L_i{\delta }_s)$ to the density wave equation.

An acoustic quadrupole has forcing ${\partial }_i{\partial }_j{T}_{ij}$. The acoustic compact-source approximation reduces its leading far-field density to $n_i{n}_j{\ddot{S}}_{ij}/(4\pi c_0^{4}r)$ in three dimensions, with $S_{ij}=\int T_{ij}{d}^{3}y$ evaluated at retarded time.

An accretion-disk boundary layer connects nearly Keplerian rotation to a stellar surface rotating at a different angular speed. For a slowly rotating star, dissipation of the incoming orbital kinetic energy gives $L_{\mathrm{BL}}\simeq GM\dot{M}/(2r_{\ast })$, comparable to the standard thin-disk luminosity. If the emitting area scales as $r_{\ast }H$, its blackbody effective temperature scales as $T_{\mathrm{BL}}\sim (r_{\ast }/H{)}^{1/4}{T}_{\mathrm{in}}$. The coefficient depends on geometry, and stellar rotation reduces the available energy.

Let $K:U\to V$ be a bounded linear operator from a real Banach space to a real Hilbert space, and let $J$ be a proper lower-semicontinuous convex absolutely one-homogeneous functional on $U$. The normalization and source relation imply $\lambda =J(u_{\lambda })$ by the Euler identity for a convex one-homogeneous functional. For the norm penalty $J(u)=\Vert u\Vert$ on a Hilbert space, its nonzero-point subgradient is $u/\Vert u\Vert$. Hence $K^{\ast }K{u}_{\lambda }=u_{\lambda }/\Vert u_{\lambda }{\Vert }^2$: $v=u_{\lambda }/\Vert u_{\lambda }\Vert$ is a right singular vector with singular value $\sigma =1/\Vert u_{\lambda }\Vert$, and $K{u}_{\lambda }$ is the corresponding left singular vector. Thus the definition recovers the ordinary operator singular-vector relation using the one-homogeneous norm penalty. Uniqueness of the vector from its data requires extra hypotheses on $K$ and $J$.

The planetary contribution to absolute vorticity is supplied by the rotating frame. Its local vertical component is the Coriolis parameter $f=2\Omega \sin \varphi$. On a beta plane, motion with northward velocity $v$ changes this contribution at the rate $\beta v$.

The trivial absolute value on a field is zero at zero and one elsewhere. Its metric topology is discrete. Its valuation ring and residue field are both the field itself. This is why rational absolute-value classification and nontrivial discrete-value compactness criteria need a nontriviality hypothesis.

The metric completion of a field for an absolute value on a field is formed from Cauchy sequences modulo sequences tending to zero. Addition and multiplication extend continuously, and every nonzero completed element is invertible, so the result remains a field. The absolute value on a field extends uniquely. A place of a number field determines the corresponding completion $K_v$.

The absolute frequency is the generally complex angular frequency at the physically selected absolute wavenumber. Its imaginary part is the fixed-position exponential growth rate under the convention $e^{-i\omega t}$. Marginal zero exponential growth may still carry an algebraic prefactor.

If $\sum a_n$ converges and $b_n$ is bounded with $\sum |b_{n+1}-b_n|<\infty$, then $\sum a_n{b}_n$ converges. For $T_j={\sum }_{k=m}^j{a}_k$, summation by parts givesThe first factor tends uniformly to zero by the Cauchy sequence criterion, and the second has a uniform bound. A bounded monotone sequence has finite total variation, so monotone multipliers are included. Eventual monotonicity also suffices, since finitely many initial terms do not affect convergence. Absolute convergence of $\sum a_n$ is unnecessary.

For a positive lapse function $\alpha$, shift vector ${\beta }^i$ and spatial induced metric ${\gamma }_{ij}$, the block metric has $g_{00}=-{\alpha }^2+{\gamma }_{ij}{\beta }^i{\beta }^j$ and $g_{0i}={\gamma }_{ij}{\beta }^j$. Its block determinant is $g=-{\alpha }^2\det \gamma$. Therefore the metric volume tensor contracted with the future unit normal restricts to the spatial volume tensor: $n^{\mu }{ϵ}_{\mu ijk}=\sqrt{\det \gamma }[ijk]$.

In fixed spatial coordinates set $a_{\mathrm{loc}}=(\det h{)}^{1/6}$ and ${\gamma }_{ij}=a_{\mathrm{loc}}^{-2}{h}_{ij}$, so $\det \gamma =1$. This volume-shape decomposition is useful for separating isotropic expansion and anisotropic shape. The determinant-defined scale depends on the coordinate volume convention.