A symmetry consisting of rotation about an axis and translation parallel to that axis. A $2_1$ operation combines a half-turn with half a lattice period; applying it twice gives a complete lattice translation. It is a spatial symmetry, distinct from a screw dislocation.

A symmetry combining reflection in a plane with translation parallel to that plane. An $a$ glide translates by half an $a$ period, and a $c$ glide by half a $c$ period. Specifying the reflection trace alone loses the translation and can falsely suggest an ordinary mirror.

The finite orthogonal group obtained by quotienting a crystal space group by its translation subgroup. In three dimensions there are 32 crystallographic point-group types, constrained by preservation of a lattice. It describes macroscopic rotational and reflection symmetry, not the fractional translations of the full space group.

A small bivalved crustacean. Sexually dimorphic fossil carapaces can provide indirect evidence about reproductive investment.

For equilibrium association of $q$ isolated amphiphiles into one micelle, equality of chemical potentials gives the displayed relation between dimensionless thermodynamic activities. Large aggregation number makes aggregate abundance rise rapidly with monomer activity. A finite-aggregate model gives a concentration crossover, rather than an exactly discontinuous thermodynamic transition.

With $\mu ={\mu }_1^0+k_{B}T\ln x_1$, large aggregates become abundant when monomer chemical potential approaches the favorable aggregate standard value ${\mu }_{\ast }^0$. This gives $x_{1,\ast }\simeq \exp [({\mu }_{\ast }^0-{\mu }_1^0)/(k_{B}T)]$ in the large-aggregation approximation. For one finite preferred size $n$, equal molecular populations $x_n=x_1$ instead give $\ln x_{1,\ast }=[n({\mu }_n^0-{\mu }_1^0)/(k_{B}T)-\ln n]/(n-1)$.

A lineage-recording approach coupling inherited CRISPR barcode edits with single-cell transcriptomes. It can relate cell identity to developmental ancestry while retaining the limitations of incomplete barcode recovery and edit homoplasy.^[ref]

An RNA molecule whose complementary sequence directs a targeting complex to a nucleic-acid sequence. In CRISPR genome editing, guide complementarity and a compatible protospacer adjacent motif jointly constrain targeting.

A short target-adjacent DNA motif recognized by a CRISPR-associated nuclease. The motif requirement depends on the nuclease; the commonly used Cas9 from Streptococcus pyogenes recognizes NGG.

A guide-directed DNA nuclease. Target recognition depends on a guide RNA and an appropriate protospacer adjacent motif; cleavage is followed by cellular repair, so cleavage alone does not specify the final allele.

Guide-directed decrease of gene expression using nuclease-inactive DNA targeting. In mammalian cells, dCas9 fused to a repressor can reduce transcription without requiring a DNA break.

Guide-directed recruitment of transcriptional activation machinery to increase endogenous gene expression. Unlike a promoter-replacing knock-in, it need not change the target sequence.

A guide-directed bacterial DNA-recognition and cleavage system. Catalytically inactive dCas9 retains targeting functions used in imaging and transcriptional regulation.

Sequence-targeted fluorescent localization of chromosomal sites using a nuclease-inactive targeting system. Colocalization with a protein is evidence of shared location, not necessarily direct molecular contact.

Two equal small-angle pendulums coupled by a spring have displacement patterns $(1,1)$ and $(1,-1)$, with squared angular frequencies $g/l$ and $g/l+2k/m$. An initial displacement of only the first pendulum excites both patterns equally.

For $x_2=(a/2)[\cos {\omega }_{1}t-\cos {\omega }_{2}t]$, the slowly varying signed factor is $a\sin [({\omega }_2-{\omega }_1)t/2]$. The absolute amplitude envelope repeats in $2\pi /|{\omega }_2-{\omega }_1|$, half the period of that signed factor.

For a normalized bound eigenstate of a factorized Coulomb radial Hamiltonian, the displayed squared norm must be nonnegative. At fixed negative $E$, increasing $l$ indefinitely would make it negative. The intertwining chain must therefore terminate with zero norm, giving $E=-{\hslash }^2/[2m{a}^2(L+1{)}^2]$. The positivity contradiction does not apply to the positive-energy continuum; domain preservation and normalizability are necessary assumptions.

In the thin-gap, laminar approximation, a rotating disc of radius $a$ separated by $d$ from a stationary disc shears a Newtonian fluid. The resisting torque is $\pi \eta \omega a^4/(2d)$, neglecting edge effects and secondary flows. This gives a way of inferring dynamic viscosity from a measured torque.

For a comoving emitter and receiver, photon energy falls as $1/(1+z)$ and arrival intervals grow by $1+z$. Their combination reduces bolometric flux by $(1+z{)}^2$ in addition to the area dilution.

In a Friedmann equation written $H^2=8\pi G\rho /3+\Lambda /3-k{c}^2/a^2$, the symbol $\Lambda$ has units of inverse time squared. If ${\Lambda }_{\mathrm{geom}}$ is the geometrical constant measured in inverse length squared, $\Lambda =c^2{\Lambda }_{\mathrm{geom}}$. Natural units hide this distinction.