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Haplotype estimation reconstructs which alleles share a chromosome from observed genotypes and other information, such as pedigrees. A doubly heterozygous genotype can be $AB/ab$ or $Ab/aB$; a phase-known genotype distinguishes them. Uncertainty in that reconstruction must propagate to estimated haplotype frequencies and linkage disequilibrium.

A haplotype frequency is the proportion of chromosome copies carrying specified alleles together across chosen genetic loci. Its marginal distributions give allele frequencies; its departures from products of those frequencies are linkage disequilibrium.

Liouville's theorem states that a smooth Hamiltonian flow preserves canonical phase-space volume. With coordinates $(q_i,p_i)$, the divergence of the phase-space velocity isprovided the mixed derivatives agree. This also holds for a smooth explicitly time-dependent Hamiltonian.

An evolving patch may stretch, fold or shear while retaining its volume, much as an incompressible fluid element can change shape. For a harmonic oscillator, an ellipse of initial positions and momenta rotates around the equilibrium without shrinking. Dissipative attraction toward a point therefore cannot be represented by a smooth finite-dimensional Hamiltonian flow of those same closed-system variables alone. In statistical mechanics, volume preservation supports the transport description of a phase-space distribution; a coarse observational description can still lose visible structure.

For a critical closed bosonic string in 25 noncompact spatial dimensions, the smoothed string oscillator rest-mass density behaves as $\rho (M)\sim C{M}^{-26}{e}^{{\beta }_{H}M}$, with ${\beta }_H=4\pi \sqrt{{\alpha }^{\prime }}$. At large $M$, the centre-of-mass momentum integral of $e^{-\beta \sqrt{M^2+p^2}}$ is asymptotic to $(2\pi M/\beta {)}^{25/2}{e}^{-\beta M}$, up to the convention for phase-space measure. Thus the high-mass tail of the one-string canonical partition function is proportional to ${\int }_{M_0}^{\infty }{M}^{-27/2}{e}^{-(\beta -{\beta }_H)M}dM$. This converges at $\beta ={\beta }_H$ but fails for $\beta <{\beta }_H$. Expanding about the endpoint gives twelve finite derivatives and a nonanalytic term proportional to $(\beta -{\beta }_H{)}^{25/2}$; the thirteenth derivative diverges. Consequently exponential state growth does not imply divergence of the partition function at the threshold itself. The discussion assumes the separate infrared instability of the bosonic tachyon has been regulated and does not claim a stable bosonic thermal vacuum.

Euclidean finite temperature compactifies time to a circle of radius $R_{\beta }=\beta /(2\pi )$. The string oscillator ground states with zero momentum number and winding $w=\pm 1$ have the displayed mass from the bosonic string mass spectrum. They become massless at ${\beta }_H=4\pi \sqrt{{\alpha }^{\prime }}$ and tachyonic at higher temperature. This thermal winding mode diagnoses breakdown of the perturbative low-temperature string gas at the Hagedorn temperature. The endpoint and order of a transition require interactions and a suitable stable vacuum; the ordinary zero-winding bosonic tachyon is already unstable independently of this thermal threshold.

Under the promise of either no marked entry or one marked entry in an $N$-element Boolean quantum oracle search space, run the Grover search algorithm for the displayed number of iterations, measure a candidate, and make one additional oracle query to check it. Since $|(2k+1)\theta -\pi /2|\le \theta$, the marked-entry success probability is at least ${\cos }^2\theta =1-1/N$. For $N\ge 4$ this is at least $3/4$. If no entry is marked the verification never accepts, so there is no false positive. The query count is $k+1=O(\sqrt{N})$ because $\theta \ge 1/\sqrt{N}$. Measuring a candidate without verification would not determine whether a marked entry exists.

Let $|s⟩=H^{\otimes n}|0^n⟩$, let $U_f$ be the marked-state phase oracle, and replace reflection about $|0^n⟩$ by reflection about a known computational basis vector $|y⟩$. Define $T_y={⨂}_{j=0}^{n-1}{Z}_j^{y_j}$, so $T_y|x⟩=(-1{)}^{x\cdot y}|x⟩$ and $H^{\otimes n}|y⟩=T_y|s⟩$. The resulting Grover iterate obeysbecause the two diagonal operators $T_y,U_f$ commute. Thus the good and bad uniform vectors are replaced by their signed vectors $T_y|g⟩,T_y|b⟩$, with exactly the same Grover rotation angle. Initializing in $H^{\otimes n}|y⟩$ gives $G_y^r|s_y⟩=T_y{G}^r|s⟩$. Computational-basis measurement probabilities are unchanged because $T_y$ only multiplies amplitudes by signs.

Take normalized states with real positive overlap $a\in (0,1)$. The eigenvectors proportional to $|s⟩\pm |w⟩$ have energies $\Delta (1\pm a)$. With $\hslash =1$, their relative phase changes by $\pi$ after the displayed time, giving $e^{-iH{t}_{\ast }}|s⟩=-i{e}^{-i\Delta t_{\ast }}|w⟩$. For a uniform search state and one marked basis vector in dimension $D$, $a=D^{-1/2}$, so the time is proportional to $\sqrt{D}$. Implementing the marked projector assumes access to a search oracle; the formula does not supply oracle information for free.

For both oracle phases equal to $\pi$, the good/bad matrix is $\left(\begin{array}{cc}\cos 2\chi & \sin 2\chi \\ -\sin 2\chi & \cos 2\chi \end{array}\right)$. Acting $r$ times on the initial vector $(\sin \chi ,\cos \chi )$ yields success ${\sin }^2((2r+1)\chi )$. For $0<\chi \le \pi /4$, the first crossing of one half is the displayed integer, since the rounded angle stays between $\pi /4$ and $3\pi /4$. With ${\sin }^2\chi =m/N\ll 1$ this is asymptotic to $(\pi /8)\sqrt{N/m}$. Iterating indefinitely is incorrect: the success oscillates after its first maximum.

In group theory, order can mean the cardinality of a group or the least positive $n$ for which a group element $g$ satisfies $g^n=e$. If no such $n$ exists, the element has infinite order. For a finite group, Lagrange's theorem implies that each element order divides the group order.

A group has the maximal condition on subgroups when every ascending sequence of subgroups eventually stabilizes. Equivalently, every subgroup is a finitely generated group. If a subgroup were not finitely generated, repeatedly adjoining an element outside the previously generated subgroup would give a strictly ascending sequence. Conversely, generators of the union of an ascending sequence all lie in one term, making that term the whole union. This condition is preserved by group extensions because each subgroup is an extension of its intersection with the kernel by its image in the quotient.

The Prüfer rank bounds the number of generators of a group needed for each finitely generated subgroup, without requiring the ambient group to be finitely generated. The additive group $\mathbb{Q}$ has Prüfer rank one: any finitely generated subgroup has a common denominator and is cyclic, but no finite collection generates all rational numbers. Prüfer rank cannot increase under taking subgroups or quotients; for quotients, lift a finite generating set and apply the rank bound to the subgroup generated by its lifts.

For a subgroup $H$ of finite index, choose representatives $t$ of right cosets $Ht$. Write $tg=h(t,g)t_g$ with $h(t,g)\in H$. The transfer is $V_{G,H}(g)={\prod }_{t}h(t,g)(\mathrm{mod}{H}^{\prime })$. Replacing $t$ by $a_{t}t$ multiplies each factor by $a_t$ on the left and $a_{t_g}^{-1}$ on the right; these cancel in the abelianization because $t\mapsto t_g$ is a permutation. The identity $h(t,g_1{g}_2)=h(t,g_1)h(t_{g_1},g_2)$ proves the group homomorphism property after multiplying all factors.

The impulse response records the response to a unit impulse, with specified initial or boundary conditions. In continuous time the impulse is a Dirac delta function; in discrete time it is the Kronecker delta $\delta [n]$. For a linear time-invariant system with zero initial state, the response to a general input is its convolution with the impulse response. The impulse response of a causal filter is zero before the impulse.

For the five-sample moving average, a discrete impulse produces $h[n]=1/5$ at $n=0,1,2,3,4$ and zero otherwise. The response lasts five samples, so this is a finite impulse response system. Its frequency response is the finite sum of the five delayed phase contributions: one short input experiment determines how every sustained sinusoid is filtered.

With the positive logarithmic convention, a Green function with pole $p$ on a noncompact Riemann surface is positive and harmonic away from $p$, has local form $-\log |\zeta |$ plus a harmonic function in a coordinate $\zeta (p)=0$, and is minimal among positive functions with these properties. It is obtained, when finite, as the increasing limit of zero-boundary Green functions on an exhaustion by compact bordered subdomains. On the unit disk it isIts sign convention is $-\Delta G=2\pi {\delta }_p$ in a conformal coordinate, with the opposite overall sign from conventions using $\log |\zeta |$ as the singular part.

For a monic second-order operator, choose a homogeneous solution $u$ satisfying the left Robin boundary condition and a solution $v$ satisfying the right decay condition. With $W=u{v}^{\prime }-u^{\prime }v\ne 0$, the displayed kernel is continuous and its first derivative has jump one. It therefore satisfies $LG=\delta$ and the two boundary conditions. The operator's first derivative term generally makes $W$ nonconstant and the resulting Green function nonsymmetric in the unweighted variables.

For the three-dimensional reference Green function $g(r)=1/(4\pi br)$, the distributional mixed Hessian $K={\nabla }_x{\nabla }_{x^{\prime }}g$ has the displayed contact and principal-value parts. Since the spherical mean of $n\otimes n$ is $I/3$, its ordinary dipole part has zero angular integral. Radial weighting therefore isolates the contact term, provided boundary truncation is absent or negligible in the bulk limit.

For $L=d^2/d{x}^2+(2/x)d/dx+k^2$ on $0<x<1$, impose finiteness at zero and zero value at one. Set $H=xG$; then $H^{\prime \prime }+k^{2}H=\xi \delta (x-\xi )$ with ordinary zero-endpoint conditions. The displayed Green function follows from the left/right sine solutions, where $x_{<}=\min (x,\xi )$ and $x_{>}=\max (x,\xi )$. Its derivative jumps by one. It exists for $k\notin \pi \mathbb{Z}\setminus \{0\}$; at zero take the removable limit $G=-\xi x_{<}(1-x_{>})/x$. At nonzero integer multiples of pi the radial problem has a nonzero homogeneous solution and a Fredholm solvability condition. Integrating against the constant forcing gives $y=k^{-2}[1-\sin (kx)/(x\sin k)]$, with $y=(x^2-1)/6$ at $k=0$.

For $L=d^2/d{x}^2-2/x^2$ on $x>0$, the homogeneous solutions $x^2,x^{-1}$ satisfy respectively the zero and infinity boundary conditions. Their Wronskian is $-3$. The displayed Green's function is continuous at $x=\xi$ and its first derivative has jump one, so $L_{x}G=\delta (x-\xi )$ in the distribution sense. Integration against a compactly supported forcing gives the unique solution decaying at both boundaries.

An internal wave is an oscillation within a stably stratified fluid, where a displaced parcel experiences restoring buoyancy. Such waves can propagate through the interior of an ocean or atmosphere, unlike waves confined to a free surface. The local buoyancy frequency $N$ measures the stratification's restoring strength.

For an inviscid nonrotating fluid in the Boussinesq approximation with constant $N>0$, write horizontal and vertical velocities as $u,w$, pressure divided by reference density as $p$, and the buoyancy perturbation as $b$. The linear two-dimensional equations areA plane wave proportional to $e^{i(kx+mz-\omega t)}$ therefore obeysThis is the nonrotating internal-gravity-wave dispersion relation. It gives $|\omega |\le N$ and depends on the direction of the wavevector, rather than its magnitude. Consequently group velocity and phase velocity need not point together; in this uniform idealization the group velocity is perpendicular to the wavevector. Rotation, compressibility and varying stratification modify the relation. Internal waves can transport energy away from their source without transporting each oscillating parcel along the entire propagation path.