For source electrons $S$, sky electrons $B$, dark-current rate $d$, total exposure $t$, $M$ reads and rms read noise $r$, independent variances add to $S+B+dt+M{r}^2$. calibration can remove reproducible means but cannot remove the science exposure's random photon or dark-current fluctuations. A finite background estimate and imperfect calibration add further uncertainty.

A photon of energy $E$ fully absorbed in silicon produces on average $E/w$ electron-hole pairs, with variance $FE/w$ rather than necessarily a Poisson variance. Thus ${\sigma }_E=\sqrt{FwE}$ and Gaussian FWHM is $2\sqrt{2\ln 2}{\sigma }_E$. Representative silicon values $w\simeq 3.65$ eV and $F\simeq 0.12$ give about $156$ eV FWHM at $10$ keV. read noise in split-event pixels, trapping and gain errors broaden that limit; optical pixel size alone does not specify the energy resolution.^[ref]

A symmetry transformation acts on localized atomic orbitals both by moving their centres and by transforming their angular shapes. Only centres left in place can contribute to the trace of the corresponding group representation. At each fixed centre, sum the diagonal coefficients of the orbital transformation: a reversed p orbital contributes $-1$, whereas a fixed s orbital contributes $+1$. For Cartesian displacement vectors at a fixed atom, use the trace of the full polar-vector matrix, not merely the number of fixed components. This rule generates characters of a representation without writing its entire matrix.

Adding an inverse-cube force can preserve radial motion while rescaling angular motion. For potential $-A/r+B/r^2$ and angular momentum $L$, the radial effective potential equals that of $-A/r$ with effective angular momentum $\sqrt{L^2+2mB}$ when this is real.

Large-scale changes in chromatin-associated regulatory states and DNA methylation during a developmental transition. Loss of methylation is not by itself a measurement of reduced transcription.

A description of the normal developmental descendants of identified embryonic regions or cells. It reports realised fate, not the entire potential revealed by experimental transplantation.

A provisional developmental commitment demonstrated operationally by adopting a fate in an appropriate neutral environment. It is weaker than cell determination.

A commitment demonstrated by retaining a fate in a conflicting environment that is viable and supplies the alternative developmental cues.

Context-dependent capacity to change differentiated identity or developmental potential. Demonstrating a transition requires distinguishing conversion of a defined starting cell from selection of a rare pre-existing population.

Persistence of a developmental choice. cell specification tests behaviour in a neutral environment; cell determination tests persistence in a viable conflicting environment.

The centered Cauchy densitiesform a convolution semigroup: $K_a\ast K_b=K_{a+b}$. With Fourier convention ${\hat{K}}_a(t)=\int e^{itx}{K}_a(x)dx$, the characteristic function is $e^{-a|t|}$. Multiplying characteristic functions proves the semigroup identity. Equivalently, the sum of independent centered Cauchy random variables has scale equal to the sum of their scales; for example, scales two and three combine to scale five.

This family permits an exact deconvolution calculation for a particular right-hand side. If $a>b>0$, the integral equationhas probability-density solution $f=K_{a-b}$. If $a=b$, the probability-measure solution is the Dirac measure at zero rather than an ordinary density. If $a<b$, a probability-measure solution would have characteristic function $e^{(b-a)|t|}$, which exceeds one in modulus for nonzero $t$ and is impossible.

Subtracting scales works because both sides belong to this convolution family. For a general right-hand side, Fourier deconvolution instead multiplies its transform by $e^{b|t|}$, potentially amplifying high-frequency errors. The semigroup identity does not make arbitrary Cauchy-kernel integral equations stable or solvable by a scale subtraction.

The number $N(n,k)=\frac{1}{n}\left(\genfrac{}{}{0ex}{}{n}{k}\right)\left(\genfrac{}{}{0ex}{}{n}{k-1}\right)$ counts Dyck paths of semilength $n$ with exactly $k$ peaks, for $1\le k\le n$. Summing over $k$ gives the $n$th Catalan number.

The number of walks with $a$ up-steps and $b$ down-steps, $a\ge b$, that never go below zero is $\left(\genfrac{}{}{0ex}{}{a+b}{b}\right)-\left(\genfrac{}{}{0ex}{}{a+b}{b-1}\right)$. Reflecting the initial segment through the first step below zero pairs forbidden walks with walks having $b-1$ down-steps. Setting $a=b=n$ gives the Catalan number $C_n$.

A catalytically inactive form of Cas9 retaining guide-directed DNA binding. Recruitment of regulatory domains makes it a platform for CRISPR interference and CRISPR activation.

The monarch butterfly is Danaus plexippus, a butterfly whose larvae feed on milkweeds. Some populations undertake long-distance seasonal migration. Larval host dependence and adult movement link different habitats within one life cycle.^[ref]

For uniform composition, upward coordinate $z$ and downward acceleration $g>0$, a pressure-equilibrated adiabatic parcel in a hydrostatic ideal gas has $\ddot{\zeta }=-N^2\zeta$ with $N^2=g[(d\ln p/dz)/\gamma -d\ln \rho /dz]$. A positive value gives restoring buoyancy; for a calorically perfect gas it is also $(g/c_p)ds/dz$. Heat or composition exchange requires a modified stability test.

The pressure response to compression at fixed entropy, $K_S=\rho (\partial P/\partial \rho {)}_S$. Seismic compressions are commonly approximately adiabatic; a background density gradient follows the corresponding derivative only when composition and entropy assumptions also apply.

For an isotropic linearly elastic solid, equal compressive principal stresses $-p$ produce three equal normal strains $-(1-2\nu )p/E$. Adding them gives the displayed volume strain, or bulk modulus $K=E/[3(1-2\nu )]$. It is a small-strain relation using the pressure change, not an arbitrary absolute pressure reference.

Formation and growth of a daughter-cell bud. A visible bud is evidence of initial polarized cell growth but does not establish normal cytokinesis or long-term proliferative capacity.