Gene prediction infers candidate genes and their boundaries from genomic sequence and supporting data. Models can recognize coding patterns and splice signals, while transcript and comparative evidence help constrain possible structures. Predictions can miss short genes, confuse duplicated regions or propose unsupported splice forms. Experimental transcript evidence can improve a gene model, but observation of a transcript does not alone establish that it produces a functional protein. ^[ref]

Gauss-Legendre quadrature approximates an integral on a finite interval using nodes obtained from roots of a Legendre polynomial. With $n$ nodes it integrates every polynomial of degree at most $2n-1$ exactly.

The quadratic fluctuation eigenvalues are $(n\pi /\mathcal{T}{)}^2+{\omega }^2$. Dividing by the free-particle determinant gives ${\prod }_{n\ge 1}[1+(\omega \mathcal{T}/n\pi {)}^2{]}^{-1/2}=\sqrt{\omega \mathcal{T}/\sinh (\omega \mathcal{T})}$. The short-time free kernel fixes the absolute normalization.

For nonnegative $k_1,\dots ,k_r$ with sum $n$, the Gaussian multinomial coefficient isIt is a polynomial in $q$ and at $q=1$ gives the ordinary multinomial coefficient. For a prime power $q$, it counts flags in ${\mathbb{F}}_q^n$ whose successive dimension increments are $k_1,\dots ,k_r$: choose each new subspace in the quotient by the previous one, using Gaussian binomial coefficients. For $(k_1,k_2,k_3)=(1,1,1)$, the value is $(1+q+q^2)(1+q)$, giving 21 complete flags over ${\mathbb{F}}_2$.

Let the spherically symmetric volume charge density be $\rho (r)=A{r}^p$, with $p>-3$ so the origin contains finite charge. Integration gives $Q(r)=4\pi A{r}^{p+3}/(p+3)$. Gauss's law yields the radial electric field $\mathbf{E}=A{r}^{p+1}\hat{\mathbf{r}}/[{ϵ}_0(p+3)]$ for $r>0$. The field tends to zero at the centre if $p>-1$; finite enclosed charge alone does not guarantee a regular central field.

A gauge theory whose left- and right-chiral fermions transform differently under its gauge group. The weak charged current gives a physical example. A chiral representation assignment must satisfy the relevant gauge anomaly cancellation conditions to define a consistent quantum theory.

A theory embedding the Standard Model gauge factors in a larger gauge group. Particular embeddings can relate couplings and matter multiplets or predict proton decay. The unification hypothesis alone does not specify a unique mass spectrum or an experimentally established extension.

A pair field transforms as $\Psi \mapsto e^{iq\chi /\hslash }\Psi$ when $\mathbf{A}\mapsto \mathbf{A}+\nabla \chi$. Its phase gradient alone is gauge dependent; ${\mathbf{p}}_s=\hslash \nabla \theta -q\mathbf{A}$ is invariant. With $\varphi \mapsto \varphi -{\partial }_t\chi$, the temporal combination $\hslash \dot{\theta }+q\varphi$ is also invariant. The supercurrent is $\mathbf{j}=q{\rho }_s{\mathbf{p}}_s/m_{\ast }$. A negative electronic pair charge is consistent with either sign of current once this convention is retained throughout.

A linear gap penalty assigns a cost proportional to gap length: $g(\ell )=e\ell$ for $e>0$. One gap of length three and three gaps of length one therefore incur the same total cost, before considering alignment substitutions. An affine gap penalty distinguishes opening from extending a gap and can favour fewer longer gaps. The chosen cost is part of the alignment model rather than an experimentally measured universal price of an insertion. ^[ref]

An affine gap penalty charges $o$ for the first position of a gap and $e$ for each additional position. With $o>e$, one long gap costs less than several separate gaps of the same total length. Some software instead uses $o+e\ell$; numerical settings must be interpreted with their convention.

An empirical scaling between a disk galaxy's total baryonic mass and its characteristic flat rotation speed. The ideal isolated deep-MOND limit predicts $v_f^4=G{M}_b{a}_0$. Observational tests must specify the speed estimator, distances, gas masses and stellar mass-to-light ratios; the empirical relation alone does not determine a unique theory of gravity.

A gas-budget model in which inflow competes with star formation, gas return and outflows. A zero instantaneous gas derivative defines a balance nullcline; it need not define a fixed point of the evolving stellar and black-hole masses.

A galaxy with negligible present star formation can still gain stars through mergers. Stellar mass assembly and in-situ star formation therefore describe different histories; quenching the latter does not freeze the former.

A merging galaxy cluster exhibiting the bullet Cluster dark-matter separation: peaks of weak gravitational lensing mass lie nearer the galaxies than the displaced X-ray gas. Combining the components supports a dominant less-collisional gravitating component under the usual gravitational model.^[ref]

An inward flow driven by loss of thermal pressure as hot gas undergoes radiative cooling. A classical unheated cluster model predicts substantial central cooling and deposition of cold material. active-galactic-nucleus feedback, transport and other heating can reduce the actual inflow; an X-ray cooling luminosity is not automatically a measured mass-deposition rate.

A collective perturbation of surface density and gravitational potential in a galactic disk. Material can cross its spiral-shaped crests; the pattern angular speed need not equal the local material angular speed. A tightly wound fluid approximation has $(\omega -m\Omega {)}^2={\kappa }^2+c_s^2{k}^2-2\pi G\Sigma |k|$, while a collisionless stellar disk requires its own response function.

The beta-2 adrenergic receptor is a G protein-coupled receptor whose ligand-dependent conformational populations have been studied by NMR spectroscopy and electron paramagnetic resonance.^[ref]

A ligand that inhibits receptor activation by occupying the binding site used by an activating ligand.