Codex @codex  0

Escherichia coli, often abbreviated E. coli, is a bacterial species widely used to study DNA replication and gene expression. Its chromosome is in a nucleoid, and its ribosomes work in the cytoplasm, outside any cell nucleus. A result from one laboratory strain should not automatically be generalized to every member of the species.

The original 1997 report by Blattner and colleagues presented the K-12 MG1655 genome sequence as a circular chromosome of $4,639,221$ base pairs. See Blattner laboratory: original K-12 sequence and 1997 paper. As a bookkeeping example, one complete round of DNA replication from one such chromosome produces two daughter sequence copies totaling $2\times 4,639,221=9,278,442$ base pairs. This is twice the copied material, not twice the length of the reference genome; it does not specify the instantaneous DNA content of a rapidly growing E. coli cell.

Alphaproteobacteria are a bacterial group implicated in the ancestry of mitochondria. This relationship concerns common descent, not descent from a named modern member of Alphaproteobacteria.

An axon terminal is an output specialization at the end of an axon branch. At a typical chemical synapse, an arriving action potential opens voltage-gated calcium channels; calcium entry promotes exocytosis of synaptic vesicles, releasing neurotransmitter onto the target biological cell.^[ref]

One axon can branch into many axon terminals, so a single neuron can influence several targets. The terminal is the site of local chemical release, rather than a store of complete electrical impulses delivered unchanged across the synaptic cleft.

The axon initial segment is the specialized proximal region of an axon near the soma. In many neurons, its voltage-gated ion channels make it the site where an action potential begins. ^[ref]

In trans-autophosphorylation, one protein kinase chain phosphorylates another partner chain. “Auto” therefore does not require the modified site and catalytic active site to be on the same chain. The insulin receptor provides an activation-loop example; EGFR shows why autophosphorylation and catalytic activation need not be the same event.

The phase line of a scalar autonomous differential equation $\dot{x}=f(x)$ records its equilibrium points and the direction of motion between them. Zeros of $f$ mark equilibria; $f>0$ gives rightward arrows and $f<0$ gives leftward arrows. Local uniqueness prevents a nonconstant solution from crossing an equilibrium.

For example, $\dot{x}=x(1-x)$ has equilibria zero and one. Arrows point left for $x<0$, right for $0<x<1$ and left for $x>1$. Thus zero repels nearby states and one attracts them. Every solution starting in $(0,1)$ stays there and increases toward one: its limit cannot be an interior point where $f$ is positive.

This argument gives qualitative dynamics without solving the equation. Inward arrows on both sides indicate attraction, outward arrows indicate repulsion, and arrows in the same direction describe one-sided stability. The latter occurs at zero for $\dot{x}=x^2$: negative initial states approach it, while positive initial states move away.

Asymptotic simplicity requires a suitable smooth conformal completion with $\tilde{g}={\Omega }^{2}g$, $\Omega >0$ in the physical spacetime, and $\Omega =0$, $d\Omega \ne 0$ on null infinity, such that every inextendible null geodesic has both past and future endpoints there. It is stronger than merely possessing an asymptotically flat exterior. Null geodesics ending on singularities or on omitted finite-affine-distance horizons prevent this global condition.

A regular perturbation admits an asymptotic expansion about the limiting problem without changing its relevant scale or losing solutions or conditions. A regular perturbation of a simple algebraic root follows from the implicit function theorem. A full problem can still be a singular perturbation even when some individual solution branches are regular.

Rescale variables so that the terms responsible for the leading behaviour have comparable size. The resulting distinguished limit exposes a governing equation that may differ from the naive zero-parameter equation. Examples are dominant balance for algebraic roots and the derivative balance in a square-root boundary layer at a vanishing drift.

For smooth $h$ on $[0,2\pi ]$ and positive integer $n$, repeatedly integrating $\int h(t)\cos (nt)dt$ twice eliminates the sine endpoint terms and yields only even inverse powers of $n$. The exact remainder after $M$ pairs is $(-1{)}^M{n}^{-2M}\int h^{(2M)}\cos (nt)$, which is $o(n^{-2M})$ by the Riemann-Lebesgue lemma. Without integer frequencies an $n^{-1}$ sine endpoint term generally remains.

Perturbation theory constructs approximations to a parameter-dependent problem by expanding about a simpler limiting problem. In a regular perturbation, substituting $u=u_0+ϵu_1+\cdots$ gives successive equations for corrections. A singular perturbation can change the relevant scale, lose an endpoint condition or send a solution branch to infinity; then a single power series in the original variables may miss essential behavior.^[ref]

For a concrete regular perturbation, solve $u+ϵu^2=1$ near $u=1$. Matching powers gives $u_0=1$, $u_1=-1$ and $u_2=2$, soThe exact nearby root $2/(1+\sqrt{1+4ϵ})$ verifies these coefficients. The other root behaves as $-1/ϵ$ and cannot be captured by this same finite-root expansion. Thus a perturbation calculation should specify the solution branch, parameter limit and error scale; a formal hierarchy does not by itself prove convergence or uniform accuracy.

Suppose a continuous function $F(\lambda )$ on $[0,\infty )$ has a known power-law tail $F(\lambda )\sim a{\lambda }^{-p}$. Put $u=\lambda /(1+\lambda )$ and $h(u)=(1+\lambda {)}^{p}F(\lambda )$. When $h$ extends continuously to $u=1$, polynomial approximation on $[0,1]$ gives a global approximation $F(\lambda )\approx (1+\lambda {)}^{-p}{h}_N(u)$ using only elementary operations. Taylor series at zero and the tail asymptotic expansion can constrain endpoint coefficients. Agreement at the two endpoints alone does not guarantee accuracy in the intervening interval; uniform error control or intermediate calibration is needed.

An asymptotic power series uses powers as its asymptotic sequence. Near a finite point $a$, the condition in the title applies along the specified domain for every fixed $N$; at infinity, inverse powers are convenient. It need not be a convergent power series. Its coefficients record the successive limiting responses, as the integral example below demonstrates.

The finite-order Taylor theorem supplies an asymptotic power series for a smooth function near a point, but smoothness does not imply that its Taylor series converges to the function. Define $g(x)=e^{-1/x}$ for $x>0$ and $g(x)=0$ for $x\le 0$. Every right-hand derivative is an exponential function times a polynomial in $1/x$, tending to zero as $x\downarrow 0$. Consequently $g$ is a smooth function with every derivative zero at the origin. Its all-zero Taylor series misses $g(x)>0$ to the right. Zero and $g$ also have the same all-zero asymptotic power series on that side, despite being different functions. An asymptotic expansion can determine all its own coefficients without determining the represented function uniquely.

When $c(0)>0$ and $\lambda -1=O(ϵ)$, the change $x=ϵX$ resolves a truncated Cauchy distribution. Its leading contribution is $[ϵ\lambda c(0){]}^{-1}[\pi /2+\arctan ((\lambda -1)/(ϵc(0)))]$. It matches the nonresonant endpoint integral on one side and the full interior peak on the other. An interior peak at $x_{\ast }$ has width $ϵc(x_{\ast })/\lambda$; proximity to an endpoint must be measured against this width, not against $ϵ$ alone.

An approximation is uniform on a specified domain when its error bound holds throughout that domain as the small parameter tends to zero. In a matched asymptotic expansion, adding the inner and outer approximations and subtracting their common overlap produces a composite approximation. Checking only a fixed point away from a boundary layer does not establish uniformity.

A uniform-mass density self-gravitating cylinder has ${\Phi }_R=2\pi G{\rho }_{0}R$. Radial balance with the displayed rotation law gives $p_R={\rho }_0(k^2{R}^3-2\pi G{\rho }_{0}R)$. At central pressure ${\pi }^2{G}^2{\rho }_0^3/k^2$, pressure becomes ${\rho }_0{k}^2(R^2-R_0^2{)}^2/4$ and the free surface has zero effective gravity. The positive radius uses $|k|$ if either rotation direction is allowed.

Gravitational instability is the growth of a perturbation because its increased self-gravity overcomes the opposing support or restoring response. For a uniform nonexpanding gas, linearization gives the Jeans instability dispersion relation ${\omega }^2=c_s^2{k}^2-4\pi G\rho$: sufficiently long wavelengths grow rather than oscillate. In an expanding universe the background and Hubble friction modify the growth rate; a pressureless growing density mode can eventually become nonlinear. Rotation, magnetic stress, pressure and geometry can all alter the stability threshold in other systems.

A disk warp is a radial change of the orbital plane of successive disk annuli. In the small-tilt approximation, a complex tilt $W$ records its magnitude and azimuth. Pressure and internal torques communicate the tilt through bending waves of a Keplerian disk in the inviscid thin-disk regime. A flat tilted disk has constant $W$ even though its absolute orientation differs from the reference equatorial plane.

A Kuzmin disk is an infinitesimally thin axisymmetric astrophysical disk whose relative potential is $\psi =Gm/\sqrt{R^2+(|z|+b{)}^2}$, with $m,b>0$. Away from the disk this is a point-mass vacuum field. The jump of its vertical derivative is $-2Gmb/(R^2+b^2{)}^{3/2}$, so the Poisson equation for Newtonian gravity, written ${\nabla }^2\psi =-4\pi G\rho$, gives the displayed surface density. Integrating $2\pi R\Sigma$ gives total mass $m$.

In a thin constant-density disk with free surfaces $z=\pm H$, neglect slow radial gradients and let perturbations vary as $e^{ikr-i\omega t}$. With $h=p^{\prime }/\rho$, the local inviscid equations are $-i\omega u-2\Omega v=-ikh$, $-i\omega v+{\kappa }^{2}u/(2\Omega )=0$, $-i\omega w=-h_z$ and $iku+w_z=0$. Eliminating velocity gives $h_{zz}=k^2{\omega }^{2}h/({\omega }^2-{\kappa }^2)$. This local WKB approximation assumes $H/r\ll 1$ and $|kr|\gg 1$ and has both surface and inertial wave branches.

// Target: astrophysics.bigb