For velocity $\beta c$ at angle $\theta$ toward the observer, ${\delta }_D=[\Gamma (1-\beta \cos \theta ){]}^{-1}$. Frequencies transform as $\nu ={\delta }_D{\nu }^{\prime }$ before cosmological redshift. Specific-intensity transformation follows invariance of $I_{\nu }/{\nu }^3$.

An approaching jet can be enhanced and its receding counterpart suppressed by different Doppler factors. A discrete moving emitter with rest-frame power-law spectrum $F_{\nu }\propto {\nu }^{-s}$ has the usual boost ${\delta }_D^{3+s}$, whereas a steady continuous-jet convention changes the exponent. Absorption and environmental asymmetry can also affect the contrast.

The photon phase-space occupation is a Lorentz scalar. For radiation propagating through vacuum, this makes specific intensity divided by cubed frequency invariant under a change of inertial observer. With Doppler factor $D$, $I_{\nu }(\nu )=D^3{I}_{{\nu }^{\prime }}^{\prime }(\nu /D)$. This describes corresponding ray directions; it does not assert that an angularly integrated radiation field is isotropic in both frames.

For an unresolved, isotropically emitting rest-frame blob with power-law spectrum $F_{\nu }^{\prime }\propto {\nu }^{-\alpha }$, the observed spectral flux density at a fixed frequency is enhanced by $D^{3+\alpha }$. The factors follow from Lorentz-invariant specific intensity and the frequency shift. The bolometric or corresponding-logarithmic-bin luminosity transforms with $D^4$. A spatially steady continuous jet has a different equal-arrival-time sampling and can have exponent $2+\alpha$ instead.

An addition whose regiochemical outcome is opposite to the conventional Markovnikov orientation. hydroboration followed by oxidation places OH at the less substituted alkene carbon.

The conventional addition orientation in which the incoming hydrogen goes to the alkene carbon already bearing more hydrogens, in the ordinary applicable unsymmetrical-alkene setting.

Bias toward sequence represented by a chosen reference genome, for example when divergent reads map less successfully or non-reference sequence has no alignment target.

Addition of hydrogen to an unsaturated substrate. Catalyst and conditions determine which unsaturated groups are reduced; the mere presence of H2 does not guarantee chemoselectivity.

Reduction at a carbonyl group, for example conversion of a ketone to an alcohol. In an enone, 1,2- and 1,4-reduction produce different functional-group patterns.

Removal of a sulfone substituent by a reducing process, replacing its carbon–sulfur connection with a carbon–hydrogen connection under an appropriate work-up. Compatibility with other reducible groups must be considered.

A shift of spectral features to shorter wavelengths and higher frequencies. Relative approach produces a kinematic blueshift through the Doppler effect; gravitational and cosmological contributions can also affect an observed shift.

For a real number $x$, its cube root is the unique real number $y$ satisfying $y^3=x$. The function $y\mapsto y^3$ is strictly increasing, so negative inputs also have a unique real cube root.

For coprime integers $b,c>1$, a Stoneham number isFor example, ${\alpha }_{2,3}=1/(3\cdot 2^3)+1/(9\cdot 2^9)+1/(27\cdot 2^{27})+\cdots$. The rapidly growing exponents separate long blocks in its base-$b$ expansion.

These constants are useful as explicit constructions in the theory of normal numbers: the Stoneham number ${\alpha }_{b,c}$ is normal in base $b$ under the stated coprimality hypothesis. Normality means every finite block of base-$b$ digits has the expected limiting frequency; it is much stronger than merely being irrational.

A positive real number is a real number $x$ with $x>0$. Zero is excluded; allowing zero gives a nonnegative real number. Both $1/2$ and $\sqrt{2}$ are positive.

The sum, product, and reciprocal of positive real numbers are positive. Consequently the positive real numbers form a group under multiplication, and the natural logarithm converts that multiplication into addition: $\log (xy)=\log x+\log y$. Positivity is also the domain restriction that makes the real logarithm well-defined.

A relation between an individual’s behaviour and social conditions. Baseline differences and differences in responsiveness are separate components.

A curved arrow whose tail marks an electron pair and whose head marks its destination in a proposed reaction step. The pair may start as a lone pair or a bond pair; a full arrowhead represents two electrons.

The dimensionless density contrast $({\rho }_h-{\rho }_l)/({\rho }_h+{\rho }_l)$, conventionally using heavy and light densities. In the inviscid, zero-surface-tension heavy-over-light Rayleigh-Taylor instability, the growth rate is $s=\sqrt{\mathcal{A}gk}$. State the signed ordering if labels are not heavy/light.

A binary tree enumerating every positive reduced fraction exactly once. Begin with boundary fractions $0/1$ and $1/0$; between neighboring fractions $a/b<c/d$, insert their mediant $(a+c)/(b+d)$ and recurse into the two new intervals. The determinant relation $bc-ad=1$ is preserved, and the new denominator is $b+d$.

Coherent absorption from one applied beam and stimulated emission into another, transferring an atom between two long-lived states. The energy transfer uses the frequency difference and the momentum transfer the wavevector difference. It is distinct from an absorption-absorption process.