Simulated annealing accepts worse feasible solutions to escape a local minimum. Minimize an objective function $E(x)$, analogous to energy. Effective temperature $T>0$ has the same units as $E$, absorbing the Boltzmann constant; it controls exploration, rather than a material's thermodynamic temperature.^[ref]

Given current feasible point $x$, draw $y$ from a proposal distribution $q(x,y)$. For a symmetric proposal distribution, $q(x,y)=q(y,x)$, use the Metropolis–Hastings acceptance probabilityAccept with this conditional probability; on rejection retain $x$. Every downhill or equal-cost proposal is accepted. Draw a fresh uniform random variable $U$ on $(0,1)$ and accept if $U<a_T(x,y)$. The off-diagonal transition probability $P_T(x,y)=q(x,y)a_T(x,y)$ includes proposing $y$.^[ref]

For a maximization problem with objective function $f$, set $E=-f$: every improving proposed neighbor is accepted, and a worsening move of loss $\Delta =f(x)-f(y)>0$ has conditional probability $\exp (-\Delta /T)$ of acceptance, under the same symmetry assumption. Likelihood-power annealing uses precisely this sign convention for a log-likelihood. Retain the best feasible solution seen, since the current state can worsen even after a global minimum has been visited.

If the proposal distribution is asymmetric, use the full Metropolis–Hastings acceptance probability, for $q(x,y)>0$,A missing reverse proposal gives zero acceptance. Even a downhill proposal need not be accepted with probability one in this case. Choosing uniformly among a variable number of neighbors is generally asymmetric: $q(x,y)=1/d(x)$ and $q(y,x)=1/d(y)$ give the correction $d(x)/d(y)$, where $d$ is vertex degree.^[ref]

The useful idea is to separate exploring from insisting on improvement: an uphill move can open a route to a better feasible solution. Its usefulness depends on the proposal distribution, the scale of the objective function, and the cooling schedule for simulated annealing. Finite practical schedules carry no general guarantee of finding a global minimum.

Tabu search keeps a short-term memory of recently reversed moves or visited attributes and prohibits them temporarily, reducing cycling in local search. It can select the best admissible neighbor even when that neighbor worsens the objective; an aspiration rule permits a tabu move when it improves the best known solution. For allocations, transfers and exchanges maintain feasibility and recent reverse transfers can be tabu.

For $-W^{\prime \prime }+2x{W}^{\prime }=EW$, the power series coefficients obey $(n+2)(n+1)a_{n+2}=(2n-E)a_n$. The condition $W(0)=0$ makes every even coefficient zero. A nonzero odd series terminates precisely when $E=2d$ with $d$ an odd positive integer. A polynomial of degree $d$ must have this value of $E$ by comparison of leading coefficients, so these are all possibilities. With $W^{\prime }(0)=1$, the first three polynomial solutions are $x$, $x-2x^3/3$, and $x-4x^3/3+4x^5/15$, with $E=2,6,10$. They are scalar multiples of the odd Hermite polynomials.

A solution of $b^{\prime \prime }+(\Lambda -{\xi }^2)b=0$ decaying at both real infinities has $b=e^{-{\xi }^2/2}{H}_n(\xi )$ with a Hermite polynomial. Substitution gives $H^{\prime \prime }-2\xi H^{\prime }+(\Lambda -1)H=0$; its power-series recurrence terminates at degree $n$ when $\Lambda =2n+1$. The square-integrable Hermite functions give the complete oscillator spectrum, including the ground state $n=0$. This quantization also determines global modes of a quadratically confined Ginzburg-Landau equation after removing drift.

A complex-valued function on a group is positive-definite when the displayed inequality holds for every finite choice of points and coefficients. This includes positive semidefinite, rather than necessarily strictly positive definite, Gram matrices. In particular $p(e)\ge 0$ and $|p(x)|\le p(e)$. On an Abelian group, every convolution square $u\ast u^{\ast }$ with $u\in C_c(G)$ is continuous and positive-definite: its quadratic form is the integral of $|{\sum }_i{z}_{i}u(t+x_i){|}^2$.

For $f\in L^1({\mathbb{R}}^d)$ and a positive threshold $\lambda$, take maximal dyadic cubes with mean of $|f|$ exceeding $\lambda$. They are disjoint, have total volume at most $\Vert f{\Vert }_1/\lambda$, and their means are at most $2^d\lambda$. Replace $f$ inside each selected cube by its mean to obtain $g$; write $b_Q=(f-f_Q)1_Q$. Then $\Vert g{\Vert }_{\infty }\le 2^d\lambda$, $\Vert g{\Vert }_1\le \Vert f{\Vert }_1$, $\int b_Q=0$, and ${\sum }_Q\Vert b_Q{\Vert }_1\le 2\Vert f{\Vert }_1$. The outside bound follows from the Lebesgue differentiation theorem. This separates a bounded part from localized cancellation errors.

For $0<p<\infty$, an analytic Hardy space controls the integral means of a holomorphic function as it approaches a domain boundary. On the disk, $H^p$ requires ${\sup }_{0<r<1}\int |F(r{e}^{i\theta }){|}^{p}d\theta <\infty$; on the upper half-plane it requires ${\sup }_{y>0}{\int }_{\mathbb{R}}|f(x+iy){|}^{p}dx<\infty$. For $p=\infty$ use a supremum norm. These analytic spaces differ from the harmonic Hardy space of the upper half-space; the Hardy space of the circle is the boundary Hilbert-space realization of the disk $H^2$ space.

This supremum function takes a supremum over a cone approaching the boundary point $x$; the positive number $a$ is its aperture. For an analytic upper-half-plane $H^1$ function, it belongs to $L^1$. To prove this without first assuming boundary absolute continuity, shift the function upwards, apply subharmonic Poisson comparison to its square root, and use the strong $L^2$ bound for the Hardy-Littlewood maximal function. Fatou lemma removes the shift.

A positive Borel measure near a domain boundary is a Carleson measure when its mass in boundary neighborhoods is bounded by a constant times their boundary measure. On the unit disc, this becomes the displayed bound for every Carleson box. The equivalent hyperbolic geodesic cap version follows from the geodesic-cap and Carleson-box comparison. The Carleson embedding theorem on the disk explains why this geometric growth condition controls integration of analytic Hardy spaces.

If $k(x)=\varphi (|x|)\ge 0$ is decreasing in radius and has integral one, then $|k_t\ast f(x)|\le Mf(x)$, where $k_t(x)=t^{-d}k(x/t)$ and $M$ is the centered Hardy-Littlewood maximal function of $|f|dx$. The layer cake representation expresses $k_t$ as a positive mixture of ball indicators; each corresponding integral is bounded by its ball volume times $Mf(x)$. Integrating these volumes gives exactly one. This applies to the Poisson kernel for the upper half-space and supplies a weak type (1,1) domination for its boundary convergence.

A haplotype block is a chromosomal region with limited historical genetic recombination, strong local linkage disequilibrium and comparatively few common haplotypes. Its definition is operational and population-dependent; it does not mean that recombination never occurs within it.

// Target: biology.bigb

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.