The mean lifetime number of offspring is the birth rate integrated against the probability of surviving to each age. For a continuous age model it is ${\int }_0^{\infty }b(a)\exp [-{\int }_0^a\mu (s)ds]da$. The Euler-Lotka equation compares this with one to locate positive population growth.

Integrating $dt=da/(aH)$ gives $H_0{t}_0={\int }_0^1\sqrt{a}da/\sqrt{{\Omega }_m+(1-{\Omega }_m)a}$. For $b=1-{\Omega }_m$ in the interior of the physical density range, this is $b^{-1}-{\Omega }_m{b}^{-3/2}\mathrm{arsinh}\sqrt{b/{\Omega }_m}$. The pure matter and pure coasting endpoints have ages $2/(3H_0)$ and $1/H_0$.

Non-abelian additive combinatorics studies product sets and configuration counts in noncommutative groups. Fourier analysis on a finite group and quasirandom groups connect small-dimensional group representations with multiplicative mixing.

For a real-valued function $H$ on a nonempty finite Cartesian product $X\times Y$, its normalized cut norm is the maximum of $|{\mathbb{E}}_{x,y}H(x,y)1_A(x)1_B(y)|$ over $A\subseteq X,B\subseteq Y$. It measures all rectangular subset discrepancies. The convention using arbitrary test functions bounded by one is equivalent within a factor of four, by decomposing their positive and negative parts into indicator functions.

A corner consists of these three points in the integer grid, all contained in the set being studied. The displacement may be positive or negative, but must be nonzero, so the points are distinct. The corners theorem guarantees a corner in every subset of positive fixed density of a finite subset of a sufficiently large square grid.

The product set of two sets of real numbers is $A\cdot B=\{ab:a\in A,b\in B\}$. It is a multiplicative counterpart of a sumset, not the Cartesian product $A\times B$.

An aggregate claims model separates the number $N$ of claims from their sizes $X_j$, with total $S={\sum }_{j=1}^N{X}_j$ and the empty sum equal to zero. Under independent and identically distributed random variables for claim sizes, independent of $N$, conditional expectation gives tractable moments and transforms.

Reinsurance transfers part of an insurer’s claim liability to another insurer. If aggregate claims are $S$, a retained payout $g(S)$ with $0\le g(S)\le S$ leaves the reinsurer with $S-g(S)$. Quota share reinsurance retains a fixed fraction, whereas aggregate stop loss reinsurance retains losses only up to a fixed aggregate threshold.

The classical risk model has surplus $U_t=u+ct-{\sum }_{j=1}^{N_t}{X}_j$, where $N_t$ is a Poisson process of rate $\lambda$, independent claim sizes are positive, and premiums flow in at constant rate $c$. The relative safety loading is $\rho =c/(\lambda \mathbb{E}{X}_1)-1$. Ruin is the first time surplus becomes negative.

Given a latent risk parameter $\Theta$, observations are independent and identically distributed random variables with conditional expectation $m(\Theta )$ and conditional variance $v(\Theta )$. The Bühlmann credibility premium estimates $m(\Theta )$ by an affine map of past observations under mean squared error. Its population parameters are $m=\mathbb{E}m(\Theta )$, expected process variance $v=\mathbb{E}v(\Theta )$, and variance of hypothetical means $a=\mathrm{Var}(m(\Theta ))$.

A stratified atmosphere can reflect low-frequency acoustic waves while permitting sufficiently high-frequency waves to escape. In a plane-parallel isothermal atmosphere, the angular acoustic cutoff is approximately $c_s/(2H_{\rho })$, where $H_{\rho }$ is the density scale height. Its detailed form depends on the atmospheric profile and on the wave-variable approximation.

With the time convention $e^{-i\omega t}$, pressure amplitude $p$ and background density ${\rho }_0$, the acoustic energy flux is $I_x=(2{\rho }_0\omega {)}^{-1}\mathrm{Im}(\overline{p}{\partial }_{x}p)$. A transverse plane wave mode with positive axial wavenumber $\kappa$ carries flux $\kappa |p{|}^2/(2{\rho }_0\omega )$. For the reduced pressure $p=e^{ikx}E$, the parabolic wave equation gives $\kappa =k-{\nu }^2/(2k)$, approximated by $k$ at leading paraxial order. Stationary illumination uses the infinite-aperture spectral normalization: replace $|p{|}^2$ by the corresponding spectral power per unit transverse length. Other acoustic amplitude conventions change the dimensional prefactor.

If a unit incident plane wave has scattered field ${\psi }_s=e^{ikR}{f}_{\infty }(\hat{\mathbf{r}},{\hat{\mathbf{r}}}_0)/R+O(R^{-2})$, the differential wave scattering cross-section isIt is the far-field limit of $R^2$ times the scattered-to-incident energy flux ratio. The common physical amplitude normalization cancels when the incident and scattered waves are in the same background medium. The formula assumes that the $1/(4\pi )$ Green-function factor, if present, is already included in the far-field pattern.

The incident wave is the prescribed background solution that illuminates a scatterer. It solves the unperturbed Helmholtz equation in scalar time-harmonic models. The scattered contribution is defined by subtracting this field from the total field; reflected waves from a planar boundary are included unless a separate reference reflection is explicitly removed.

The scattered wave is the part of the total solution produced by the scatterer relative to the prescribed incident wave. It obeys an outgoing radiation condition. For a rough reflecting surface, this convention includes both specular flat-surface reflection and the roughness-dependent correction. Its coherent field is its ensemble mean, which need not equal the full mean intensity.

For an operator $c^{-2}{\partial }_t^2-a^{-1}\nabla \cdot (a\nabla )$ with positive stationary coefficients and reciprocal boundary conditions, its delta-normalized Green function satisfiesMultiplying its frequency-domain equation by $a$ and using the bilinear Green second identity proves the relation. Both Green functions must use the same outgoing or causal prescription. Exchanging source and receiver preserves time lag, and involves no complex conjugation.

The energy ${\int }_{\Omega \setminus K}(u-g{)}^2+\alpha {\int }_{\Omega \setminus K}|\nabla u{|}^2+\beta {\mathcal{H}}^1(K)$ balances data fidelity, within-region smoothness and image edge length. In its relaxed SBV space formulation the image edge set is $J_u$. Clipping to the bounded data range, the SBV compactness theorem and lower semicontinuity prove existence; the full segmentation problem is not strictly convex.

A modular machine of modulus $m>1$ acts on pairs in ${\mathbb{N}}^2$. Each instruction $(a,b,c,R)$ or $(a,b,c,L)$ has $0\le a,b<m$ and $0\le c<m^2$, and at most one instruction is assigned to each residue pair $(a,b)$. The two transition types are $(mu+a,mv+b)\mapsto (m^{2}u+c,v)$ and $(mu+a,mv+b)\mapsto (u,m^{2}v+c)$. These finite arithmetic operations can encode a Turing machine; a designated terminal configuration can have a nonrecursive halting set.

Encode the cells immediately left of the head in a reversed positional stack $L$, and the current and right cells in $R$. Quotient and remainder by the alphabet base read and remove cells; multiplication by that base pushes a cell. The resulting local tape transitions are primitive recursive, which makes them directly implementable on Church numerals.

A zero-dimensional model is a finite-dimensional integral with bosonic and Grassmann variables, rather than a field depending on spacetime. Odd graded derivations preserve its action and integration measure. Such models expose supersymmetric Ward identities and supersymmetric localization without spacetime dynamics.