A fork that has lost the integrity or organization needed for ordinary progression, potentially including a DNA break. Restoration requires more than correction of a base mismatch.

In a positional notation of integer radix $b\ge 2$, a repunit is a repdigit whose numerical digits are all $1$. A length-$k$ repunit, with $k\ge 1$, is the geometric seriesFor example, $R_{10}(4)=1111$, while $R_2(4)=(1111{)}_2=15$. The same string has different values in different radices.

Appending one numerical digit gives $R_b(k+1)=b{R}_b(k)+1$. Thus remainders modulo a positive integer $n$ obey $r_{k+1}=(b{r}_k+1)\mathrm{mod}n$, starting with $r_0=0$. This recurrence relation avoids constructing huge integers. For decimal repunits modulo $7$, the successive remainders are $1,4,6,5,2,0$; hence the six-digit repunit is divisible by $7$.

A repunit divisible by $n$ exists exactly when $b,n$ are coprime integers, and one then exists with at most $n$ digits. Necessity follows because a common prime factor of $b,n$ divides every nonconstant term of $R_b(k)$, leaving remainder one. For sufficiency, examine $R_b(1),\dots ,R_b(n)$ modulo $n$. If none is zero, the pigeonhole principle gives equal remainders at two indices $i>j$. Their difference is $b^j{R}_b(i-j)$. Cancelling the invertible factor $b^j$ modulo $n$ produces a divisible repunit of length $i-j\le n$.

For $C\to B+t$ with massless $t$, use $p_C=p_B+p_t$ in the B rest frame. The mass-shell identity gives $E_t=(m_C^2-m_B^2)/(2m_B)$ and ${\mathbf{p}}_C={\mathbf{p}}_t$. The denominator differs from the familiar daughter-momentum formula in the parent's rest frame. Units here have $c=1$.

A moving feature can have inferred transverse speed ${\beta }_{\mathrm{app}}c$ greater than $c$ because its second signal starts closer to the observer. The light-travel delay gives ${\beta }_{\mathrm{app}}=\beta \sin \theta /(1-\beta \cos \theta )$, whose maximum is $\Gamma \beta$ at $\cos \theta =\beta$. No local material or signal speed exceeds $c$.

Coherent intrinsic luminosity variation on observed time $\Delta t_{\mathrm{obs}}$ bounds the characteristic emitting radius by a light-crossing argument. relativistic beaming shortens the observed interval by Doppler factor $D$, while cosmological redshift lengthens it by $1+z$. This bound assumes that the variation samples the emitting region causally; geometry, independent subregions and propagation-driven variation can change its interpretation.

Rapidly varying high-energy sources can appear too dense in target photons to transmit their observed gamma rays. For an approximately isotropic stationary region with target-band luminosity $L_2$, radius $R$ and target photon energy ${ϵ}_2$, the target number density is $n\sim L_2/(4\pi R^{2}c{ϵ}_2)$. photon-photon pair production then gives optical depth $\tau \sim n\sigma R$. A luminosity variability size bound can make this very large. relativistic beaming both increases the inferred rest-frame size and reduces the inferred rest-frame radiation density, allowing escape. This radiation-opacity argument is distinct from gravitational stellar compactness and topological compactness.

A coarse-grained description in which quadratic inflaton oscillations have zero mean pressure and lose energy at constant effective decay rate $\Gamma$ to a relativistic component. Adding the equations preserves total energy accounting with the appropriate expansion work. Efficient transfer occurs around $H\sim \Gamma$, but producing particles and achieving thermal equilibrium are different steps. The potential alone does not specify the decay rate or thermalization interactions.

Nonperturbative particle production after cosmic inflation, for example by parametric resonance caused by an oscillating inflaton. Large occupation numbers can grow before a thermal bath exists. Backreaction, rescattering and subsequent thermalization affect the outcome, so an effective perturbative decay rate need not describe every stage.^[ref]

An Escherichia coli transcription unit encoding lactose-use proteins. Expression integrates LacI-dependent repression, allolactose induction, catabolite activator protein activation and inducer exclusion.

Inhibition of uptake of an inducing substrate by a preferred-carbon-source regulatory system. Glucose-dependent inhibition of lactose permease reduces formation of allolactose and reinforces lac operon repression.

Post-transcriptional regulation controls gene expression after RNA synthesis through processes such as RNA splicing, export, localization, biological translation and degradation. Two cells can produce the same initial transcript yet accumulate different mature products if processing or stability differs. Nonsense-mediated decay illustrates selective removal based partly on the context of biological translation termination. Transcript abundance alone therefore cannot distinguish altered transcription from altered lifetime. ^[ref]

The ratio $n=c/v_{\mathrm{ph}}$ compares vacuum light speed with the phase velocity at a specified frequency. In a dispersive material it differs from the group index; the Cherenkov radiation cone uses the phase index.

The mean kinetic energy removed per recombination is $⟨E{⟩}_{\mathrm{rec}}=\int E{f}_T(E){\sigma }_{\mathrm{rec}}(E)v(E)dE/\int f_T(E){\sigma }_{\mathrm{rec}}(E)v(E)dE$. The capture-rate weighting generally differs from averaging an unselected Maxwell distribution, so $⟨E{⟩}_{\mathrm{rec}}=3k_{B}T/2$ is only an additional approximation.

Absorption of momentum $\mathbf{p}$ followed by emission of ${\mathbf{p}}^{\prime }$ changes a free atom's momentum by $\mathbf{p}-{\mathbf{p}}^{\prime }$. For isotropic emission of fixed magnitude, its mean vector is zero. The energy change then separates into a directional work term and two positive recoil terms. If absorption directions are also isotropic and uncorrelated with the initial momentum, the work term averages away. Equal incoming and outgoing photon magnitudes give two recoil energies per cycle.

A receptor for fibroblast growth factors. Its outputs include, but are not restricted to, MEK and ERK activity.

Platelet-derived growth factor receptor, a receptor tyrosine kinase activated by PDGF. Its characteristic ligand-induced phosphotyrosines require an appropriate detection reagent.

A member of the ErbB receptor family. Different ligands and receptor combinations have distinct signalling properties.