For two segregating biallelic loci, encode the allele on a random biological population haplotype by indicator variables $X,Y$. Their covariance is the linkage disequilibrium coefficient $D$. The squared correlation is $r^2=D^2/[p_A(1-p_A)p_B(1-p_B)]$, between zero and one. It measures how well one marker predicts the other and is undefined if either marker is monomorphic.

// Target: biology.bigb

Genetic recombination erodes linkage disequilibrium by reshuffling alleles between haplotypes, while preserving marginal allele frequencies in the idealized model. Let $f_{AB,t}$ denote the haplotype frequency in the gamete pool of generation $t$, let $p,q$ denote its marginal allele frequencies, and let $r\in [0,1/2]$ be the constant recombination fraction between two autosomal genetic loci. Assume random mating within one population, nonoverlapping generations, fair Mendelian segregation, equal reciprocal gamete production, the same $r$ in both sexes, and a deterministic population large enough to neglect genetic drift. Exclude natural selection, mutation and gene flow. ThenHere $p$ and $q$ remain constant: recombining changes which alleles occur together, not their total numbers. The other haplotype frequencies change by $+r{D}_t$ for $Ab,aB$ and $-r{D}_t$ for $ab$. Consequently the joint probability distribution stays within its allowed bounds and approaches the product of its marginal distributions when $r>0$.

For the original positive example $p=0.60,q=0.40,D_0=0.12$, a recombination fraction $r=0.10$ gives $D_1=0.108$ and $D_{10}\approx 0.04184$. The negative example $D_0=-0.12$ has exactly the opposite sequence: genetic recombination reduces the magnitude without reversing the sign. With $r=0$, the initial linkage disequilibrium persists in this model; even independent assortment at $r=1/2$ only halves $D$ each generation. A population association does not vanish after one generation merely because the genetic loci are unlinked.

The geometric sequence gives a generation scale: for $r>0$, the continuous interpolation reaches $|D_0|/2$ at $t_{1/2}=\log (1/2)/\log (1-r)$. Actual generations are integers; the first at or below half is $\lceil t_{1/2}\rceil$. For small $r$, $t_{1/2}\approx (\log 2)/r$. Because the denominator of allelic correlation stays fixed, $r_{\mathrm{LD},t}^2=(1-r{)}^{2t}{r}_{\mathrm{LD},0}^2$; its decay is faster than the magnitude of $D_t$.

A short derivation uses random mating to assign frequency $2f_{AB}{f}_{ab}$ to the double heterozygote $AB/ab$ and $2f_{Ab}{f}_{aB}$ to $Ab/aB$. Relative to $r=0$, each loses or gains an $AB$ gamete fraction $r/2$, respectively. Their net contribution is $r(f_{Ab}{f}_{aB}-f_{AB}{f}_{ab})=-rD$. The other genotypes produce no net change in $f_{AB}$, proving the recurrence and, by iteration, its solution.

This temporal model does not identify elapsed generations from physical separation alone. The recombination fraction depends on genetic recombination, including recombination hotspots, while population stratification, natural selection, gene flow or genetic drift can create or maintain linkage disequilibrium. Repeated sampling across time would also require accounting for sample size and haplotype estimation. A cross-sectional plot of association against distance between genetic loci is therefore a different observation from the idealized trajectory plotted here.

For positive linkage disequilibrium $D$, take $D_{\max }=\min (p_A(1-p_B),(1-p_A)p_B)$; for negative $D$, take $D_{\max }=\min (p_A{p}_B,(1-p_A)(1-p_B))$ and retain its negative sign in $D^{\prime }$. Thus $|D^{\prime }|\le 1$. Its magnitude measures the fraction of the largest disequilibrium possible at the specified allele frequencies. High $|D^{\prime }|$ need not imply high allelic correlation, so it does not alone guarantee good marker prediction.

For two genetic loci with two alleles each, linkage equilibrium is independence of their states on sampled haplotypes. It concerns association between genetic loci; Hardy-Weinberg equilibrium concerns pairing of alleles into diploid genotypes.

The Newtonian limit of general relativity describes slowly moving matter in a weak, approximately static gravitational field. With metric signature $(-,+,+,+)$ and $x^0=ct$, identify the Newtonian gravitational potential $\Phi$ byNeglecting time derivatives and terms of higher order in particle velocity divided by the speed of light, the spatial geodesic equation becomesThis recovers Newtonian gravitational acceleration rather than merely matching a metric coefficient.

For nonrelativistic matter, $T_{00}\simeq \rho c^2$ dominates the stress-energy tensor. With negligible anisotropic stress, the weak-field Einstein tensor gives $G_{00}\simeq 2{\nabla }^2\Phi /c^2$. Matching $G_{\mu \nu }=\kappa T_{\mu \nu }$ to the Poisson equationtherefore fixes $\kappa =8\pi G/c^4$. Setting $c=1$ gives the commonly written $h_{00}=-2\Phi$ and $\kappa =8\pi G$. Relativistic pressure, rapid time variation or strong gravitational fields requires terms omitted in this limit.

For a static spherical point source, the weak-field solution in isotropic radius $\rho$ has $g_{tt}=1-2GM/\rho$ and spatial metric $-(1+2GM/\rho )(d{\rho }^2+{\rho }^{2}d{\Omega }^2)$. The isotropic-to-areal weak-field radius change gives the displayed metric with $r_S=2GM$. Its validity requires $r_S/r\ll 1$. Treating it as an exact metric near $r\sim r_S$ can turn omitted terms into large corrections; it does not establish horizons or strong-field photon dynamics.

For static scalar perturbations with no mixed time-space or spatial shear terms, the displayed metric tensor uses signature $(+---)$. With the reversed convention in curvature-sign convention in Killing derivative identities, its first-order Einstein tensor is $\delta G_{00}=2\Delta C$, $\delta G_{0i}=0$, and $\delta G_{ij}={\partial }_i{\partial }_j(A+C)-{\delta }_{ij}\Delta (A+C)$. These follow by substituting the metric into the first-order Christoffel symbols and Ricci tensor, discarding products of perturbations. A static source with the dust stress-energy tensor obeys $\Delta C=-4\pi G\rho$ and the decaying spatial equation makes $A=-C$. For a point mass the Newtonian potential of a point mass gives $A=-GM/r$, $C=GM/r$. The spatial potential is necessary to determine light propagation; a time potential alone is insufficient.

For signature $(+---)$, decompose a metric perturbation into $h_{00}=2A$, $h_{0i}={\partial }_{i}B+B_i$ and $h_{ij}=2C{\delta }_{ij}+2({\partial }_i{\partial }_j-{\delta }_{ij}\Delta /3)E+{\partial }_i{E}_j+{\partial }_j{E}_i+h_{ij}^{\mathrm{TT}}$, where the spatial vectors are divergence-free and $h^{\mathrm{TT}}$ is a transverse-traceless tensor. Under a passive coordinate shift ${\xi }^0=\tau$, ${\xi }^i={\partial }_{i}L+L_i$, with ${\partial }_i{L}_i=0$, the changes are $\delta A=-\dot{\tau }$, $\delta B=\dot{L}-\tau$, $\delta C=\Delta L/3$, $\delta E=L$, $\delta B_i={\dot{L}}_i$, and $\delta E_i=L_i$. The displayed combinations and $h^{\mathrm{TT}}$ are therefore gauge invariant. The decomposition is unique after appropriate spatial decay or mode restrictions; homogeneous and boundary modes require separate conventions.

An inhomogeneous linear recurrence has a prescribed forcing sequence $g_n$ rather than zero on the right. The difference of two solutions satisfies the associated homogeneous linear recurrence relation, so every solution is a particular solution plus a homogeneous solution.

For two solutions of the linear recurrence relation $x_{n+2}+b_n{x}_{n+1}+c_n{x}_n=0$, their discrete Wronskian obeys $W_{n+1}=c_n{W}_n$. Substitute the recurrence into $p_{n+1}{q}_{n+2}-p_{n+2}{q}_{n+1}$: the $b_n$ terms cancel and the remaining expression is $c_n(p_n{q}_{n+1}-p_{n+1}{q}_n)$. Iteration gives $W_{n+1}=W_1{\prod }_{m=1}^n{c}_m$, including when some $c_m$ vanish. A nonzero discrete Wronskian proves linear independence of the two solutions.

For $dx=ydt$, $dy=udt+dW$ and terminal cost $x(T{)}^2$ plus running cost $u^2$, the displayed change of state gives $dz=(T-t)udt+(T-t)dW$. A quadratic value function has coefficient $P(t)=[1+(T-t{)}^3/3{]}^{-1}$ and additive noise cost $\log [1+(T-t{)}^3/3]$. Minimizing its Hamilton-Jacobi-Bellman equation gives the displayed control. The gain is unchanged if the additive noise is removed, illustrating certainty equivalence.

Given finitely encoded linear inequalities, a maximum feasible subsystem is a largest collection that can be simultaneously satisfied. Its decision version asks whether some $k$ inequalities admit a common solution. This is NP-complete, despite polynomial time feasibility testing for each fixed collection. For a graph, use nonnegative variables $x_v$, constraints $x_v\ge 1$ for vertices, and $M>|V|$ copies of $x_u+x_v\le 1$ for each edge. A subsystem of size $M|E|+k$ must retain a copy of every edge constraint and at least $k$ vertex constraints; those vertices form an independent set. Conversely an independent set of size $k$ supplies such a feasible subsystem by its indicator vector.

The ellipsoid method repeatedly asks a separation oracle about the center of a containing ellipsoid and replaces that ellipsoid by a smaller one containing its intersection with a separating half-space. The volume decreases by a dimension-dependent factor at each iteration. For rational polyhedra, the exact feasibility theorem has polynomial bit complexity in the dimension and coefficient encoding bounds when separation is polynomial time. It includes lower-dimensional feasible sets through rational precision bounds or suitable relaxations; one must not declare infeasibility just because the original feasible set has zero volume.

A nonempty linear polyhedron with integer inequality coefficients and right sides bounded by $U\ge 1$ contains a point $x_i=p_i/q_i$ with the displayed bounds. Intersect with an orthant containing a feasible point. Moving along directions annihilating the active constraints, until another constraint becomes active, produces a vertex: the orthant prevents a whole nontrivial line from remaining feasible. At that vertex, choose $n$ independent active inequalities, including coordinate constraints as necessary. Cramer's rule and the Hadamard determinant inequality bound both determinants by $n^{n/2}{U}^n\le (nU{)}^n$. The resulting rational witness has polynomial binary length and is verifiable in polynomial time, proving that integer linear feasibility is in NP.

The set of minimizers of a linear objective on a linear polyhedron. It is a face: if a strict convex combination of two feasible points minimizes the objective, both points must also minimize it. Hence an extreme point of an optimal face is an extreme point of the original polyhedron. Successively minimizing further linear objectives restricts to nested faces, whose vertices remain original vertices. This preserves uniform rational denominator bounds in exact feasibility-oracle optimization.

A surplus variable converts a lower-bound inequality $a^{T}x\ge b$ into $a^{T}x-z=b$, with $z\ge 0$. This is the lower-bound counterpart of a slack variable. When the original coefficients, right-hand side and decision variables are integer, the surplus variable is integer as well; that fact permits it to appear in an all-integer Gomory fractional cut row.

This form of linear programming additionally has the uniform vector as a strictly positive feasible point and known optimum zero. A homogeneous self-dual embedding of a linear program, followed by elimination of unrestricted Lagrange multipliers and normalization by the sum of nonnegative coordinates, yields this form without presupposing feasibility of the original program. The known start and optimum support projective interior-point methods. Polynomial bit-complexity statements additionally require finite rational input encoding; bounded arbitrary real coefficients alone do not provide it.

For a stimulus-conditioned Inhomogeneous Poisson process, the event log-likelihood is the displayed expression. A causal filter expansion $\lambda (t)=\exp [b+{\sum }_a{w}_a(s\ast b_a)(t)]$ can be fitted by maximum likelihood estimation, with a smoothness or size penalty and held-out validation. The integral term penalizes predicted spikes that did not occur; fitting only stimulus values at observed spikes omits this essential part.

For $h_{ij}=\mathrm{diag}(L,0,0)$, the determinant is zero for every $L$, even when one perturbation component is arbitrarily large. Linearized metric theory instead requires the eigenvalues of $h$ relative to the background spatial metric to be small, equivalently all its components in a background orthonormal frame to be small. The determinant of $1+h$ then expands as $1+\mathrm{tr}h+O(h^2)$, and the volume element as $1+\mathrm{tr}h/2+O(h^2)$.

Mass conservation in a Lagrangian dust displacement gives $1+\delta =\det (I+{\nabla }_q\psi {)}^{-1}$, hence $\delta =-{\nabla }_q\cdot \psi$ to first order. The perturbing gravitational potential determines the longitudinal part of the force, so the displayed equation holds for the curl-free displacement under the usual homogeneous boundary convention. A divergence-free transverse displacement instead satisfies ${\ddot{\psi }}_T+2H{\dot{\psi }}_T=0$. Taking the divergence gives $\ddot{\delta }+2H\dot{\delta }-4\pi G\bar{\rho }\delta =0$ without assuming the full displacement is longitudinal. The background Poisson equation with positive source requires $\nabla {\Phi }_0=+4\pi G\bar{\rho }r/3$.