For an isotropically emitting rest-frame blob moving with Doppler factor $D$, take the observed target-band spectral flux density to satisfy $F_{\nu }\propto {\nu }^{-\alpha }$ over the relevant frequencies. The sign convention for $\alpha$ matters. At fixed cosmological redshift, absorb fixed distance and redshift factors into a reference optical depth ${\tau }_0$.

A light-crossing estimate gives $R^{\prime }\propto D$ for fixed observed variability time, while a corresponding logarithmic frequency bin has observed-to-rest-frame luminosity boost $D^4$. At a fixed observed high-energy photon energy, the internal pair-production target energy in a relativistic blob shifts the observed target energy by $D^2$. Combining these factors giveswhere $F_{\log }=\nu F_{\nu }$ and ${ϵ}_{20}$ is the reference target energy at $D=1$. For example, $\alpha =1$ gives $D^{-6}$, whereas a locally flat $F_{\nu }$ gives $D^{-4}$. The power-law assumption must cover the target energies swept by changing $D$. Spectral breaks, different emission geometries and external stationary illumination require a different calculation; $D^{-6}$ is not a universal pair-opacity law.

For a prompt nuclear level fed in a steady decay cascade, incoming and outgoing transition populations balance. Quoted photon intensities must represent efficiency-corrected emission counts; internal conversion, alternative particle decays and unobserved feeds must either be included or explicitly excluded. Equal photon counts suggest a serial cascade, while a feed intensity equal to the sum of two branches suggests splitting. intensities alone do not determine the chronological order of equal-count lines.

Named for Victor Weisskopf, a Weisskopf estimate is a benchmark electromagnetic nuclear transition strength based on a simple single-particle size model. It supplies a scale against which enhancement or hindrance is measured, rather than a universal rate. The partial rate contains both the photon-energy power $E_{\gamma }^{2L+1}$ and the actual reduced nuclear strength. Electric and magnetic multipoles at different energies cannot be universally ranked by order alone.^[ref]

Extended stellar structures drawn out by time-dependent gravitational interactions between galaxies. Close, prograde encounters can coherently perturb rotating discs. Restricted particle calculations demonstrate that the structures need not be evidence of direct star-star impacts.^[ref]

Self-gravitating energy is nonlocal. Shell assembly assigns $-GM(<r)dM/r$ and counts pairs once; the density $\rho \Phi /2$ is another total-energy allocation. Removing a test parcel from a fixed potential instead requires $-\Phi dM$. These prescriptions need not give the same radial feedback threshold, even though their units agree.

Gas lifted out of a galactic disk can remain bound and return rather than escape into the intergalactic medium. This recycling carries metals and angular momentum between the disk and its surroundings and delays the availability of gas for subsequent star formation.

For an Eddington-luminous hole with $M_{\mathrm{BH}}=\beta M_s$ and wind mass flux $\lambda L$, define $A=\lambda \beta (4\pi G{m}_{p}c/{\sigma }_T)$. The gas budget ${\dot{M}}_g=\Phi -ϵM_g-A{M}_s$ has positive instantaneous balance only for $M_s<\Phi /A$. A positive constant-gas trajectory is incompatible with ongoing ${\dot{M}}_s=ϵM_g$ unless the secular black-hole coupling is neglected.

In a system where gas and stellar masses both evolve, the locus ${\dot{M}}_g=0$ gives the instantaneous gas-balance nullcline. Motion along the other state variables generally changes the equilibrium reservoir, so the nullcline is not itself an exact constant-gas trajectory.

With constant inflow $\Phi$, no mass return or outflow, and $\Psi =ϵM_g$, mass conservation gives $M_g=\Phi /ϵ+(M_0-\Phi /ϵ)e^{-ϵt}$. Positive $ϵ$ makes the gas equilibrium stable, with perturbations decaying on timescale ${ϵ}^{-1}$. The star formation rate tends to the inflow, not necessarily to zero.

A common name for several burrowing frogs. Their tadpoles include systems used to investigate environmentally induced resource-use phenotypes.

Used here for the living-frog crown group, including the common ancestor of living frogs and every descendant.

Let $S$ be a real symmetric matrix and $A$ a real antisymmetric matrix of the same size. Invariance of the trace under transpose and the cyclic trace identity giveHence $\mathrm{tr}(SA)=0$. Equivalently these two matrices are orthogonal for the Frobenius inner product, since $⟨S,A{⟩}_F=\mathrm{tr}(S^{T}A)=\mathrm{tr}(SA)$. Together with the unique symmetric and antisymmetric parts of a matrix, this is an orthogonal decomposition of the real square matrices.

An acylium ion is a positively charged acyl species, commonly written $\mathrm{RC}{\mathrm{O}}^{+}$, with charge and bonding described by resonance forms $\mathrm{R}-\mathrm{C}\equiv {\mathrm{O}}^{+}$ and $\mathrm{R}-{\mathrm{C}}^{+}=\mathrm{O}$. Its carbon is strongly electrophilic.

Acylium-like species explain carbonyl-group transfer in Friedel-Crafts acylation. Unlike an ordinary alkyl carbocation, the acylium ion is stabilized by oxygen donation and usually does not undergo the same skeletal rearrangements. The precise reactive species can include a counterion or Lewis-acid complex, so the isolated ion drawing is a mechanistic model.

The self-gated second-harmonic spectrogram has the displayed ideal form. Its delay marginal is proportional to the intensity autocorrelation by the Parseval identity. Absolute time and wave phase are undetermined; $A(t)$ and $A^{\ast }(-t)$ also give the same record. Finite crystal acceptance and spatial walk-off modify the ideal gate.

An optical Kerr gate rotates a probe's polarization in proportion to a delayed replica's intensity in the weak-gate regime. Resolving the gated spectrum gives a pulse-characterization record with a third-order gate law, distinct from a second-harmonic gate.

For $N$ bonds of length $a$, fixed forward angle $\theta$ and uniform independent torsions, conditional averaging gives $⟨{\mathbf{a}}_i\cdot {\mathbf{a}}_j⟩=a^2{\alpha }^{|j-i|}$. Consequently $⟨R^2⟩=N{a}^2+2a^2{\sum }_{k=1}^{N-1}(N-k){\alpha }^k$. Dividing by the independent-step result $N{a}^2$ and summing the geometric series gives the displayed $C_N$, for $\alpha \ne 1$. For fixed $|\alpha |<1$, $C_N\to C_{\infty }=(1+\alpha )/(1-\alpha )$.

The angle is measured between forward vectors; a chemical internal bond angle is supplementary. At $\alpha =1$, the finite chain is a rigid rod with $C_N=N$, so taking the infinite-chain limit before the rigid limit would be invalid. The Kuhn length of a freely rotating chain is asymptotically $a{C}_{\infty }$.

With independently uniform torsion, consecutive forward tangent correlations multiply by $\cos \vartheta$. After $n$ bonds the correlation is $(\cos \vartheta {)}^n$. Matching it to $e^{-nb/{\ell }_p}$ gives the displayed decay length for positive correlation. This discrete orientational definition need not equal half the equivalent Kuhn length exactly.

A change from predominantly ductile deformation and fracture to predominantly brittle fracture as temperature decreases under specified loading and specimen conditions. A quoted transition temperature depends on its criterion: absorbed impact energy, fracture appearance and fracture toughness need not give the same number. In ferritic steel, irradiation hardening can raise the transition temperature.

Brittle crack propagation associated with separation along particular crystallographic planes. Cleavage facets and river patterns can help identify the mechanism, while establishing its cause requires material, loading and exposure evidence.

Crack propagation predominantly along grain boundaries. Boundary cavities, chemical segregation and environmental attack are possible contributors; fracture appearance alone does not establish which caused failure.