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Airy wave theory linearizes the motion of small-amplitude surface gravity waves in an inviscid, incompressible, irrotational fluid. At depth $h$, its dispersion relation is ${\omega }^2=gk\tanh (kh)$; deep water is the limit $kh\to \infty$.

The excess depth-integrated, time-averaged momentum flux of a surface gravity wave is its radiation stress. In deep water $c_g/c_p=1/2$, so $S_{xx}=\mathcal{E}/2={\rho }_{w}g{a}^2/4$. The force per transverse width on a reflecting object is the incident plus reflected minus transmitted radiation stress. Reflection increases force because the reflected wave carries momentum in the opposite direction.

For depth $h=\alpha x^p$ with $p>0$, stationary surface gravity waves conserve frequency $\omega$ and alongshore wavenumber $K$. Near shore their dispersion becomes ${\omega }^2\sim g{\kappa }^{2}h$, so $\kappa \sim \omega x^{-p/2}/\sqrt{g\alpha }$. The shoreward ray has $y^{\prime }\sim -K\sqrt{g\alpha }{x}^{p/2}/\omega$, giving the displayed integrated scaling. Rays approach normally, while equal-phase crests approach parallel to the shore and their shore-normal spacing shrinks as $x^{p/2}$. These are asymptotics within the ray approximation; for $p<2$, its slow-variation assumption eventually fails arbitrarily close to the endpoint.

For a tank with side lengths $a,b$, depth $h$, impermeable walls and negligible surface tension, the small-amplitude surface gravity wave normal modes have horizontal factors $\cos (m\pi x/a)\cos (n\pi y/b)$ and $k_{mn}^2=(m\pi /a{)}^2+(n\pi /b{)}^2$. The velocity potential has vertical factor $\cosh (k_{mn}(z+h))$, satisfying the bottom Neumann boundary condition. Combining the linearized free-surface boundary conditions gives ${\omega }_{mn}^2=g{k}_{mn}\tanh (k_{mn}h)$. The uniform $m=n=0$ surface elevation is excluded when the total water volume is fixed. Taking one horizontal index zero gives the two-dimensional tank modes.

For $k_{mn}h\ll 1$, the shallow-water approximation gives ${\omega }_{mn}\sim \sqrt{gh}{k}_{mn}$; for $k_{mn}h\gg 1$, the deep-water gravity wave relation ${\omega }_{mn}^2\sim g{k}_{mn}$ follows. An oscillating-wall forcing of a rectangular tank couples to these same normal modes and becomes resonant at their natural frequencies.

A surface gravity wave breaks when its smooth free-surface motion becomes unstable and overturns or forms a strongly dissipative front. In the nondispersive shallow-water approximation, intersecting compressive characteristic curves signal failure of a smooth simple wave; a hydraulic bore replaces it at the integral-model level. Dispersion can instead produce an oscillatory bore, while physical breaking involves turbulence and energy loss beyond the inviscid equations.

An optically thin spherical source with volume emissivity $j(r)=j_0{r}^{-p}$ has surface brightnessSubstitute $z=Ru$ in the line-of-sight integral and evaluate ${\int }_{-\infty }^{\infty }(1+u^2{)}^{-p/2}du$ using the beta function. The condition $p>1$ is convergence at large distance. A core or finite outer boundary changes the projection near that boundary; the stated scaling applies between those scales.

A Sérsic profile models projected galactic light, with index $n$ controlling curvature and $b_n$ chosen so that $R_e$ encloses half the luminosity. Its logarithmic slope is $d\ln I/d\ln R=-(b_n/n)(R/R_e{)}^{1/n}$. The $n=1$ case is exponential; $n=4$ gives the traditional de Vaucouleurs form. Unlike a fixed power law, its slope steepens outward.

In light-cone coordinates $x^{\pm }$ on two-dimensional Minkowski spacetime, add two real odd Grassmann variables ${\theta }^{\pm }$. With left Grassmann derivatives, the supercharges and supersymmetric covariant derivatives can be represented asThe identity $\{{\partial }_{\theta },\theta \}=1$ gives $Q_{\pm }^2=i{\partial }_{\pm }$, $D_{\pm }^2=-i{\partial }_{\pm }$, $\{Q_{+},Q_{-}\}=\{D_{+},D_{-}\}=0$, and $\{Q_s,D_t\}=0$ for every $s,t$. A Lorentz boost acts as $x^{\pm }\mapsto e^{\pm \omega }{x}^{\pm }$, ${\theta }^{\pm }\mapsto e^{\pm \omega /2}{\theta }^{\pm }$. Thus $D_{\pm }$ have boost weights $\mp 1/2$, and $D_{-}{D}_{+}$ is a Lorentz scalar. Since it commutes with both supercharges, an equation $i{D}_{-}{D}_{+}\Phi =W(\Phi )$ for an even scalar superfield is Lorentz invariant and supersymmetry covariant.

A local fermionic gauge symmetry of supersymmetric brane actions. Since the kappa symmetry projector has half rank, the symmetry removes half the embedding fermions. The remaining physical fermions pair with the bosonic worldvolume degrees of freedom.

The supervielbein is the local frame of a superspace geometry. Pulling its bosonic-frame components back along a brane embedding gives ${\Pi }_i^a={\partial }_i{Z}^{\mathcal{M}}{E}_{\mathcal{M}}{}^a$, and hence the induced worldvolume metric $h_{ij}={\Pi }_i^a{\Pi }_j^b{\eta }_{ab}$. Fermionic coordinates remain present through their dependence on $Z$.

// Target: string-theory.bigb

The displayed one-forms give an invertible triangular change from the coordinate coframe on flat Majorana superspace. A rigid supertranslation has $dϵ=0$, and the variation of $d{X}^m$ cancels that of $i\bar{\theta }{\gamma }^{m}d\theta$. Their Maurer-Cartan equation is $d{\Pi }^m=id\bar{\theta }{\gamma }^{m}d\theta$. The bosonic frame transforms as a Lorentz vector and the odd frame as a Majorana spinor.

Collapse of a massive stellar core can launch an explosion that ejects material processed by hydrostatic and explosive burning. Whether an explosion succeeds, and how much material escapes instead of falling back, depend on the core and envelope structure. A failed supernova can retain much of the nucleosynthetic core; initial stellar mass alone is not a complete prediction.

A hypothetical explosion in which a degenerate carbon-oxygen core within a single evolved star grows toward the Chandrasekhar mass and ignites explosive carbon burning while retaining an envelope. This differs from accretion onto an already exposed white dwarf in standard Type Ia supernova scenarios. Evolutionary calculations by Doherty et al. explain the strong dependence of this proposed channel on envelope mass loss; their adopted winds remove the carbon-oxygen envelope before this outcome. It is a proposed possibility under different assumptions, rather than an established endpoint for normal $5M_{\odot }$ stars.

A four-dimensional N=1 chiral multiplet consists off shell of a complex scalar $\varphi$, a Weyl fermion ${\psi }_{\alpha }$, and a complex auxiliary scalar $F$. A chiral superfield packages their supersymmetry transformations. Eliminating the auxiliary scalar leaves two bosonic and two fermionic physical polarizations; an allowed supersymmetric mass gives equal scalar and fermion masses.

The gravitino mass parameter in four-dimensional supergravity is the displayed expression. In a zero-energy supersymmetry breaking vacuum with only F-term breaking, $K_{i\bar{j}}{F}^i\overline{F^j}=3m_{3/2}^2/{\kappa }^2$. Writing this norm as ${\Lambda }_{\mathrm{SUSY}}^4$ gives $m_{3/2}={\Lambda }_{\mathrm{SUSY}}^2/(\sqrt{3}{m}_p)$. It is small relative to ${\Lambda }_{\mathrm{SUSY}}$ when the breaking scale is below the Planck mass. The super-Higgs mechanism supplies the longitudinal gravitino states.

Let $k^i$ generate a holomorphic gauge transformation, with $k^i{\partial }_{i}W=-{\kappa }^{2}rW$ and moment map $\mathcal{P}=i(k^i{K}_i-r)$. Substitution of the Kähler covariant derivative of a superpotential gives the displayed identity. Consequently, a finite point with $W\ne 0$ and all $D_{i}W=0$ also has $\mathcal{P}=0$. Pure D-term breaking with vanishing chiral auxiliary fields can therefore evade this argument only where $W=0$ or outside its hypotheses. In conventional two-derivative matter-coupled supergravity, a constant Fayet–Iliopoulos term gauges an R-symmetry and constrains the transformation of $W$; a neutral superpotential and an independently assigned nonzero constant shift are not generally consistent. The gauge-covariance restriction is derived in Van Proeyen's discussion of Fayet–Iliopoulos terms and R-symmetry.

The normal component of a superfluid describes the momentum and entropy carried by excitations rather than the phase-gradient condensate current. Its velocity need not equal the superfluid velocity. In phenomenological condensate equations, prescribing its motion supplies a reference flow for dissipation; a full two-component fluid theory would also evolve its momentum and energy.

These are two responses of one liquid, not separable kinds of atoms. Their fractions need not equal the Bose-Einstein condensation fraction.^[ref]

With left Grassmann derivatives, define ${\mathcal{P}}_{\mu }=-i{\partial }_{\mu }$, ${\mathcal{Q}}_{\alpha }=-i{\partial }_{{\theta }^{\alpha }}-{\sigma }_{\alpha \dot{\gamma }}^{\mu }{\bar{\theta }}^{\dot{\gamma }}{\partial }_{\mu }$ and ${\bar{\mathcal{Q}}}_{\dot{\beta }}=i{\partial }_{{\bar{\theta }}^{\dot{\beta }}}+{\theta }^{\gamma }{\sigma }_{\gamma \dot{\beta }}^{\mu }{\partial }_{\mu }$. The anticommutators of the derivative parts and of the coordinate parts vanish. Each mixed derivative/coordinate pair contributes $-i{\sigma }^{\mu }{\partial }_{\mu }$ by the graded product rule, giving the displayed mixed anticommutator. Equal-chirality anticommutators vanish, and all these operators commute with translations. Under Lorentz transformations their spinor indices transform in conjugate Weyl spinor representations, completing the minimal Super-Poincaré algebra. A convention for left derivatives is essential to the signs.

A supertranslation is generated by translations and supercharges, excluding the Lorentz transformations. For flat Majorana superspace with ${\Pi }^m=d{X}^m+i\bar{\theta }{\gamma }^{m}d\theta$, the rigid odd transformation $\delta \theta =ϵ$, $\delta X^m=-i\overline{ϵ}{\gamma }^m\theta$ leaves ${\Pi }^m$ invariant. Its commutator gives an ordinary translation.

A supersymmetric domain wall contributes a boundary or tensorial charge to the extended anticommutator of supercharges. It is not the Lorentz-scalar central charge of an extended particle algebra. The wall's energy per unit area can cancel this charge in half the real charge combinations, giving the BPS tension bound and half-supersymmetry preservation. This does not contradict the rank classification for the ordinary algebra without wall charges.