For positive $a,b$, fixed real $p,q$, and $f(a,b)=a^p{b}^q$, change the inputs to $a(1+h),b(1+j)$ with $h,j$ small. The ratio of the new value to the old one is exactly $(1+h{)}^p(1+j{)}^q$. Applying the binomial series to each factor and multiplying proves the displayed Taylor expansion. Positive inputs and $1+h,1+j>0$ make arbitrary real exponents well defined. A percentage change of $k$ in $a$ means $h=k/100$, not $h=k$. The mixed term $pqhj$ is required even though there is no such term in either one-variable expansion separately.

The cold predominantly atomic neutral phase of interstellar gas, with characteristic temperatures of order $10^2\mathrm{K}$. It is denser than the warm neutral phase at comparable thermal pressure. Its existence depends on radiation, grain heating and atomic-line cooling; one universal density or temperature does not define every environment.

Diffuse mostly neutral atomic gas with temperatures below about $10^4\mathrm{K}$. It can share a pressure range with the cold neutral medium where the thermal equilibrium curve has two stable branches.

Diffuse ionized gas near $10^4\mathrm{K}$, maintained by ionization and heating balanced by radiative losses. It is physically distinct from warm gas that remains mostly neutral despite a similar kinetic temperature.

Dilute shock-heated interstellar plasma with characteristic temperatures around $10^6\mathrm{K}$. supernova energy injection and subsequent cooling are important to its evolution; temperature alone does not determine its volume filling factor.

A useful optically thin model subtracts volumetric heating from radiative cooling. At equilibrium its net cooling vanishes. Pressure-confined temperature perturbations are governed by the derivative of net cooling per unit mass at fixed pressure, so a negative derivative is unstable. The heating law, species abundances and radiation field must be specified before predicting an equilibrium curve. thermal conduction and finite sound-crossing times limit the isobaric approximation.

For $\ddot{\theta }+2{\omega }_0\dot{\theta }+{\omega }_0^2\theta =F$ switched on from rest, $\theta (t)=(F/{\omega }_0^2)[1-(1+{\omega }_{0}t)e^{-{\omega }_{0}t}]$. critical damping is the fastest nonoscillatory step response at fixed natural frequency within the ordinary second-order model.

A motional carrier transition changes an atom's internal state without changing its oscillator quantum number. Its matrix element from the motional ground state contains the factor $e^{-{\eta }^2/2}$. A first blue sideband is smaller in amplitude by $\eta$ and changes the motional energy by one trap quantum. Conditional energy changes must be distinguished from averages over all motional outcomes.

For laboratory velocity $v\hat{r}+w\hat{\theta }$ and field $B\hat{z}$, $F_{B,r}=qBw$ and $F_{B,\theta }=-qBv$. Their powers are opposite. The field-dependent guide-reaction power is $-qBvw$; the total ideal radial reaction for bead mass $m_b$ and constant expansion speed additionally contributes the ordinary centripetal term $-m_{b}v{w}^2/R$.

A vertical field linking a conducting loop on an incline contributes flux $Bl\cos \theta s$. For zero resistance and zero initial current, flux conservation gives $LI=Bl\cos \theta s$. The uphill magnetic force component then makes the downhill coordinate oscillate about $s_e=mgL\sin \theta /(B^2{l}^2{\cos }^2\theta )$. Release from $s=0$ first stops at $2s_e$, not at the force-free point $s_e$. The energy is stored in the inductor, not dissipated.

Danaus plexippus is the scientific species name of the monarch butterfly. Using the binomial connects observations made under different common names to the same biological species.^[ref]

Subtract the phases $2\pi {\mathbf{g}}_1\cdot \mathbf{r}$ and $2\pi {\mathbf{g}}_2\cdot \mathbf{r}$ of two projected periodic structures. Their relative phase repeats along the difference vector after distance $D=1/|{\mathbf{g}}_1-{\mathbf{g}}_2|$. For magnitudes $1/d_1$ and $1/d_2$ separated by angle $\theta$, this gives $D=d_1{d}_2/\sqrt{d_1^2+d_2^2-2d_1{d}_2\cos \theta }$. At zero rotation it reduces to $d_1{d}_2/|d_1-d_2|$; equal spacings give $d/[2|\sin (\theta /2)|]$. Finite specimen extent limits how many periods can be observed.

Let identical spatially incoherent stellar brightness profiles of equal integrated flux be displaced by projected angles $\pm \beta /2$ along an interferometer baseline. The Fourier shift property makes their normalized signed coherence $g(Y)=g_{\star }(Y)\cos (\pi \beta Y/\lambda )$. The observed visibility is its absolute value. The slow single-star envelope depends on each star's brightness profile, while the oscillatory factor records projected separation. Unequal flux prevents complete cancellation at the binary minima.

The ratio of RNA concentration inside an assembly to that outside, with accessible volumes and molecule detection specified. A high concentration ratio does not mean a large fraction of all cellular molecules is recruited.

The indefinite special unitary group preserves a Hermitian form of signature $(p,q)$ and has determinant one: $M^{†}JM=J$, where $J=\mathrm{diag}(I_p,-I_q)$. With both $p,q$ positive it is noncompact, unlike the group preserving a positive-definite form. Its Lie algebra satisfies $X^{†}J+JX=0$ and $\mathrm{Tr}X=0$.

A matrix whose entries are complex numbers. For an endomorphism of ${\mathbb{C}}^n$, Hermitian matrices and unitary matrices are distinguished by their behavior under the conjugate transpose.

Matrices of the same shape are added entry by entry. For fixed dimensions over a field, this operation and scalar multiplication make the matrices a vector space. A matrix product has a different dimension-matching rule: its inner dimensions must agree, whereas addition requires the entire shapes to agree.

For an impulse vector $\mathbf{J}=M\Delta \dot{\mathbf{q}}$ delivered at an unstrained equilibrium, project the velocity jump onto mass-orthogonal normal modes. Multiplying $M\Delta \dot{\mathbf{q}}$ by ${\mathbf{u}}_j^T$ gives modal velocity coefficient ${\mathbf{u}}_j^T\mathbf{J}/({\mathbf{u}}_j^{T}M{\mathbf{u}}_j)$. Its kinetic energy is the displayed expression, including a translational zero mode if one is present. A single endpoint kick in an equal-mass free three-mass spring chain puts fractions $1/3,1/2,1/6$ into translation and the two internal modes.