For the Airy equation with opposite dispersive sign, $q_t-q_{xxx}=0$ on $x>0$, let $Q$ be the initial Half-range Fourier transform and $F_j(k,t)={\int }_0^t{e}^{i{k}^{3}s}{\partial }_x^{j}q(0,s)ds$. Three integrations by parts give the displayed global relation. The spatial transform is holomorphic for $\mathrm{Im}k<0$, whereas the finite-time spectral boundary transforms are entire functions. The symmetry $k\mapsto \alpha k$, ${\alpha }^3=1$, preserves their time exponent.

// Target: analysis.bigb

For $Q_T+Q_{XXX}+a{Q}_{XX}+b{Q}_X+cQ=0$, an exponential gauge removes the second derivative. With ${\kappa }^2=a^2/3-b>0$ and $\sigma =-c+ab/3-2a^3/27$, the remaining equation is $q_t+q_{xxx}-q_x=0$. Negative $a$ improves spatial decay of the transformed initial data; positive $a$ can destroy the convergence of its Half-range Fourier transform.

For dispersion $w(k)=-i(k^3+k)$, the other roots of $w(\nu )=w(k)$ satisfy ${\nu }^2+k\nu +k^2+1=0$. In $D_{+}=\{\mathrm{Im}k>0,\mathrm{Re}w(k)<0\}$ both roots lie in the lower half-plane. Interpolating the unknown linear polynomial $G_2+i\nu G_1$ at them eliminates two boundary traces. The resulting symmetric expression has removable singularities where the roots coalesce.

Each policy has an independent Bernoulli random variable $B_i$ of probability $q_i$, and its independent severity $Y_i$ has expected value ${\mu }_i$ and variance ${\sigma }_i^2$. independent policies give $\mathbb{E}T={\sum }_i{q}_i{\mu }_i$ and $\mathrm{Var}(T)={\sum }_i[q_i{\sigma }_i^2+q_i(1-q_i){\mu }_i^2]$. Conditioning on the claim indicator proves the formulas. Keeping the policy label preserves the dependence between the occurrence probability and the severity law; replacing all labels by a single mixture is an approximation unless the relevant laws agree.

The monetary amount of an individual insurance claim. Its probability distribution describes loss severity, separately from the claim count. In an independent random sum of independent claims, severity mean $\mu$ and expected count combine to give aggregate mean $\mathbb{E}N\mu$.

The number of insurance claims in a specified exposure period. Its probability generating function combines with the moment-generating function of an independent claim size through the random-sum transform identity. A Poisson distribution, geometric distribution, or negative binomial distribution provides a possible count model, with different dispersion and zero-claim probabilities.

Insurance exposure measures the amount of opportunity for claims, such as the number of policy-years observed. With known positive insurance exposures $w_j$ and a common annual claim frequency $\Theta$, a simple model iswhere the claim counts $K_j$ are conditionally independent. The insurance exposures are fixed inputs here; the conditional independence assumption concerns the claim counts.

Put $W={\sum }_j{w}_j$ and $K={\sum }_j{K}_j$. Multiplying the conditional probability generating functions gives $K\mid \Theta \sim \mathrm{Poisson}(W\Theta )$. The conditional likelihood has the formAll dependence on $\theta$ is through the total claim count $K$ and total insurance exposure $W$; with the insurance exposures fixed, $K$ is a sufficient statistic. This is the exposure aggregation used in Poisson-gamma conjugacy with unequal exposures.

The normalized claim count $K_j/w_j$ has conditional expectation $\Theta$ and conditional variance $\Theta /w_j$. Thus its conditional expectation does not change with insurance exposure, while its conditional variance decreases. For example, two groups with $2$ and $3$ policy-years at annual frequency $0.4$ have expected claim counts $0.8$ and $1.2$; together they have a Poisson distribution claim count of mean $2$. The count model and aggregation follow from the Poisson distribution formula; the likelihood and example are derived here.^[ref]

The age of a flat matter-Lambda universe is $H_0{t}_0=2\mathrm{arsinh}\sqrt{(1-{\Omega }_m)/{\Omega }_m}/(3\sqrt{1-{\Omega }_m})$. The large-argument expansion $\mathrm{arsinh}z=\log (2z)+O(z^{-2})$ gives the first limit. Its small-argument expansion $\mathrm{arsinh}z=z+O(z^3)$ gives the second. The divergence at the pure-vacuum endpoint reflects the absence of a finite Big Bang time in the expanding flat de Sitter spacetime patch.

For an expanding pressureless matter universe with positive curvature, zero cosmological constant and present scale factor one, set $u_0=\arccos (2/{\Omega }_0-1)\in (0,\pi )$. The closed matter-dominated Friedmann solution then gives the displayed age of an FLRW universe. Its flat limit is $2/(3H_0)$, despite the individually vanishing numerator and denominator. To see this, write $\delta ={\Omega }_0-1$; then $u_0=2\arctan \sqrt{\delta }$ and $u_0-\sin u_0=(4/3){\delta }^{3/2}+O({\delta }^{5/2})$.

For an expanding pressureless matter universe with zero cosmological constant, fixed current Hubble parameter $H_0$ and current matter cosmological density parameter $\Omega >0$, the age of a dust universe without a cosmological constant is $H_0{t}_0={\int }_0^1\sqrt{x}/\sqrt{\Omega +(1-\Omega )x}dx$. For every $0<x<1$ the integrand decreases strictly as $\Omega$ increases. Thus open models with $0<\Omega <1$ are older than the flat model, and closed expanding models with $\Omega >1$ are younger. The flat value is $2/3$ and the empty Milne universe limit is $1$. Specifying what is held fixed matters: at fixed present scale factor and energy density, the integral $t_0={\int }_0^{a_0}da/\sqrt{C/a-k}$ instead increases with $k$, provided all three models reach $a_0$ on their expanding branches.

An affine combination is a weighted point $P={\sum }_{i=0}^k{\lambda }_i{P}_i$ with ${\sum }_{i=0}^k{\lambda }_i=1$. The coefficients can be negative. The sum is meaningful in an affine space without selecting a preferred origin: relative to any point $O$, it means $P=O+{\sum }_i{\lambda }_i(P_i-O)$. Affine maps preserve these weighted points. ^[ref]

For an original example in the Euclidean plane, take $A=(0,0)$, $B=(2,0)$ and $C=(0,2)$. Then $P=\frac{1}{4}A+\frac{1}{2}B+\frac{1}{4}C=(1,\frac{1}{2})$ lies inside their triangle. Its coefficients are nonnegative, making it a convex combination. In contrast, $R=-\frac{1}{2}A+\frac{3}{2}B=(3,0)$ is beyond $B$: it is an affine combination but not a convex combination of these three points. Negative weights allow extrapolation, rather than only interpolation inside the convex hull.

For the affine map $T(x,y)=(2x+y+1,y-2)$, the same weights produce $T(P)=(\frac{7}{2},-\frac{3}{2})$ from $T(A),T(B),T(C)$, and produce $T(R)=(7,-2)$ from $T(A),T(B)$. Thus one can compute a weighted point before transforming the vertices or afterwards, obtaining exactly the same point.

This is useful whenever an object is built from weighted control points: its construction commutes with an affine map, so transforming the control points transforms the constructed object as well. The sum of the weights is the essential condition; no assumption about lengths or angles is needed.

A translation and positive uniform scaling of a finite configuration $F$ in a real vector space. It preserves the ratios and directions of differences between its points. In an integer lattice, the Gallai theorem for an integer lattice guarantees a monochromatic homothetic copy of every finite pattern in any finite colouring, with positive integer scaling factor.

The points with integer coordinates in a real d-dimensional affine space with a chosen origin and coordinate basis. Translations by integer vectors and dilations by positive integers preserve this set. The Gallai theorem for an integer lattice concerns finite colourings of these points and homothetic copies of finite configurations of them.

Points $p_0,\dots ,p_k$ in an affine space are affinely independent if the only affine dependence displayed above has all coefficients zero, using any choice of origin to read the vector equation. Since the coefficients sum to zero, replacing the origin leaves that equation unchanged. Substituting ${\lambda }_0=-{\sum }_{i=1}^k{\lambda }_i$ shows that this condition is equivalent to linear independence of $p_1-p_0,\dots ,p_k-p_0$. It also says that an affine combination of the points has unique coefficients. In ${\mathbb{R}}^n$, at most $n+1$ points can be affinely independent; $n+1$ such points span a full-dimensional simplex.

For unaffected genetic carriers of a disease with complete penetrance and recessive inheritance, suppose parental marker-disease haplotype phases are equally likely and the marker is fully informative. Write $u=\theta (1-\theta )$ and $v=1/2-u$, where $\theta$ is the recombination fraction. Conditional on sharing a parental marker copy, the probability that both siblings inherit that parent's disease allele is $[(1-\theta {)}^2+{\theta }^2]/2=v$; conditional on inheriting different marker copies it is $\theta (1-\theta )=u$. Independent paternal and maternal transmissions therefore give $L_0=u^2,L_1=uv,L_2=v^2$. Conditional instead on both siblings being affected, the IBD probabilities are $s^2,2s(1-s),(1-s{)}^2$, where $s=2u\in [0,1/2]$.

// Target: biology.bigb

An elastic collision with a wall moving toward the particle at speed $V$ changes the particle speed by approximately $2V$ when recoil is negligible. The collision rate is approximately $v/\ell$ for $v\gg V$, and $\dot{\ell }=-2V$, yielding $\dot{v}/v=-\dot{\ell }/\ell$. For a freely moving finite wall of mass $M$, the neglected recoil changes the increment by order $(m/M)v$, so relative accuracy also requires $(m/M)(v/V)\ll 1$.

// Target: classical-mechanics.bigb

In a slowly changing spherical gravitational potential, angular momentum is exactly conserved and the radial action is an adiabatic invariant. An initially circular orbit remains circular in the adiabatic limit, so $J_r=0$ and $L^2=GrM(r)$ give the displayed shell relation. If dark-matter shells do not cross, their enclosed dark mass is also conserved. A density profile alone does not specify this orbital model: eccentric-orbit populations require their full action distribution.

Over a field $K$ containing ${\mathbb{F}}_q$, a q-linearized polynomial isThe Frobenius endomorphism gives $L(u+v)=L(u)+L(v)$ and $L(cu)=cL(u)$ for $c\in {\mathbb{F}}_q$, so its polynomial function is ${\mathbb{F}}_q$-linear. If $a_0\ne 0$, its formal derivative is the nonzero constant $a_0$. Thus it has exactly $q^d$ distinct roots of a polynomial in an algebraic closure, forming a vector space of dimension $d$ over ${\mathbb{F}}_q$.

The chosen coefficient field need not contain all those roots. For instance, $X^{q^2}-X$ has $q^2$ roots in ${\mathbb{F}}_{q^2}$ but only $q$ in ${\mathbb{F}}_q$. The Galois group of the splitting field over $K$ acts faithfully and linearly on the full root vector space, embedding in $GL(d,q)$. Faithfulness holds because the root vectors generate the splitting field.

If a subset $A$ of a cyclic group has normalized additive energy at most ${\alpha }^4+ϵ$, every nonconstant Fourier coefficient on a finite abelian group of its indicator has modulus at most ${ϵ}^{1/4}$. Expanding the count of $a+c=2b$ gives the frequency product $\widehat{1_A}(r)\widehat{1_C}(r)\widehat{1_B}(-2r)$. The Cauchy-Schwarz inequality and Parseval identity on a finite group bound its nonconstant part by the displayed quantity. Character doubling has multiplicity $\gcd (2,N)$, so no assumption that the modulus is odd is needed. The error is a normalized density error multiplied by $N^2$.

Two equal spins $j$ have the unique total-spin-zero stateIt is invariant under joint rotations.

A singlet state is the one-dimensional rotation representation; this construction supplies the angular-momentum-zero case for two equal spins.