For two inviscid incompressible half-spaces, lower density/velocity ${\rho }_1,U_1$, upper density/velocity ${\rho }_2,U_2$, downward gravity $g$ and surface tension $\sigma$, a mode $e^{i(kx-\omega t)}$ with $k>0$ obeysA negative right-hand side produces exponential growth. The gravitational term recovers Rayleigh-Taylor instability for heavy-over-light ordering; the shear term can destabilize otherwise stable stratification. Finite layer thickness, viscosity, compressibility and magnetic tension lie outside this interface model.

For unchanged fluid, gravity and capillary geometry, equilibrium rise is proportional to the cosine of the contact angle measured through the liquid. Changing from angle $\theta$ to complete wetting therefore multiplies the height by $1/\cos \theta$. If contamination also changes liquid surface tension or density, the contact-angle ratio alone does not fix the new height.

For $S_n^{+}={\prod }_{m<n}(1-2n_m)c_n$ and periodic spins, the end exchange term carries the global fermion parity $P=(-1{)}^{N_f}$. Replacing the end bond by a uniform hopping expression requires $c_{N+1}=-P{c}_1$: even parity gives antiperiodic fermions, odd parity periodic fermions. Each resulting quadratic spectrum must be restricted to states of the parity used in its boundary condition.

For two isotropic parabolic subbands, the joint density of states per area and per independent channel is $\mu /(2\pi {\hslash }^2)\Theta (\hslash \omega -E_{\mathrm{edge}})$. The constant is a two-dimensional phase-space effect. Degeneracy and optical matrix elements determine the actual step weights.

For a smooth two-dimensional transition energy $D(\mathbf{k})$ and a regular level $D=E$, the coarea identity gives $\int d^{2}k\delta (D(\mathbf{k})-E)={\int }_{D=E}dl/|\nabla D|$. Critical points require a limiting or distributional treatment; the apparent small contour at a parabolic edge is compensated by its small energy gradient.

Parabolic electron-hole continua with $d$ unconfined directions have cumulative state count proportional to $(E-E_{\mathrm{th}}{)}^{d/2}$. Differentiating gives the threshold joint density proportional to $(E-E_{\mathrm{th}}{)}^{d/2-1}$: a square root for bulk, a step for a well, and an inverse square root for a wire. A fully confined dot instead has discrete lines. Optical matrix elements, occupations, excitons and broadening determine the actual absorption beyond this counting argument.

For a cloud supported by an effective speed of sound $v$ and confined by a surface pressure $P$, dimensional analysis fixes the gravitational instability mass scale to $v^4/(G^{3/2}{P}^{1/2})$. The dimensionless coefficient depends on the equilibrium model and the convention used for the instability threshold.

Equate the free-fall time of a uniform sphere $\sqrt{3\pi /(32G\rho )}$ to a radius-based sound-crossing time $R/c_s$, then substitute $R$ in $M=4\pi \rho R^3/3$. This gives the displayed mass. It has the scaling of a Jeans mass, but its coefficient differs from the convention using a sphere of diameter equal to the Jeans length. Equality of these two times is an approximate support comparison, not the exact marginal mode of an infinite-medium Jeans instability.

Replacing the sound speed in a Jeans mass estimate by the Alfvén speed $B/\sqrt{{\mu }_0\rho }$ gives the displayed magnetic-support mass scale. A sphere whose diameter is the corresponding wavelength gives $C={\pi }^{5/2}/6$; the crossing-time cloud support mass instead gives $C=(4\pi /3)(3\pi /32{)}^{3/2}$. The coefficient is model dependent, and a real magnetic field supplies anisotropic stresses rather than isotropic gas pressure.

During isotropic magnetic flux freezing, $B\propto {\rho }^{2/3}$, so $M_B$ and the number of such masses in a fixed material cloud remain constant. This is another expression of a conserved mass-to-flux ratio. It contrasts with the decreasing thermal Jeans mass of isothermal fragmentation, and does not rule out motion or fragmentation along field lines.

For real $z$, the probabilities $J_m(z{)}^2$ over integer angular-momentum modes satisfy ${\sum }_m{m}^2{J}_m(z{)}^2=z^2/2$. Differentiate the unit-modulus function $e^{iz\cos \theta }$ and use Parseval to obtain the identity; its zero first moment follows from the symmetry of opposite orders.

Spin reversal in exchange-split bands creates an electron-hole excitation with energy ${ϵ}_{k+q}-{ϵ}_k+2M$ for a majority-to-minority flip in the simple uniform model with $M>0$. Its allowed continuum depends on occupied initial and empty final states. At $q=0$ all such allowed transitions have exchange energy $2M$; finite momentum changes the thresholds. A collective ferromagnetic magnon can remain gapless below this continuum when spin rotation is exact. The continuum is not the same excitation as a longitudinal amplitude mode.

Use the isothermal Lane-Emden equation variables $r=a\xi$, $a^2=c_T^2/(4\pi G{\rho }_c)$ and $\rho ={\rho }_c{e}^{-\Psi }$. Truncation at fixed dimensionless radius ${\xi }_b$ gives $R=a{\xi }_b$ and $M=4\pi {\rho }_c{a}^3{\xi }_b^2{\Psi }^{\prime }({\xi }_b)$. Division proves the displayed relation. A common temperature, composition and dimensionless truncation therefore imply $M\propto R$, while the external pressure generally varies along that homologous sequence. Arbitrary changes of truncation at fixed temperature do not retain a common coefficient. A finite regular cloud needs an outer boundary such as that of a pressure-confined isothermal sphere.

For the cubic-density model, ${\kappa }_T=(n\partial p/\partial n{)}^{-1}$ gives ${\kappa }_T(n_c,T)=1/[n_c{k}_B(T-T_c)]$. Divergence from above is a mean-field critical prediction. The negative value below $T_c$ signals an unstable homogeneous branch rather than an equilibrium negative compressibility.

For sequential on-shell two-body decays $C\to B+t$ and $B\to A+s$, with massless $s,t$, evaluate their energies in the B rest frame. Their pair mass is $2E_s{E}_t(1-\cos \theta )$, so its squared invariant mass ranges from zero to the displayed endpoint. The mass ordering is $m_C>m_B>m_A>0$. spin correlations can alter the distribution inside these limits, but cannot move the kinematic endpoints while the same on-shell decay chain applies.

Adding two null photon four-momenta gives the displayed invariant mass, with opening angle $\theta$ measured in the same frame as the energies. A nonzero mass of the pair does not give either photon a rest mass. At fixed energies it vanishes for parallel photons and is greatest for opposite directions. A reconstructed mass peak must be assessed with background, resolution and identification controls before assigning a parent particle.

An interval contained in another interval.

Interstellar gas occupies distinct thermal and ionization regimes because heating, cooling and compression depend on density, radiation and composition. Cold and warm neutral gas can coexist over a finite pressure range, while ionized gas, molecular clouds and hot shock-heated regions add further components. turbulence, magnetic stress and finite cooling times mean that these are dynamically exchanging phases, not necessarily a common global equilibrium.

Measurement using interfering light fields. A phase difference changes the detected intensity, allowing measurements of displacement, optical path length or a spectrum. The round-trip phase in a Michelson interferometer gives one explicit path-to-phase calibration; Fourier transform spectroscopy recovers spectral information from an interferogram.