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.

For a Planck photon distribution, the hot-electron Kompaneets equation gives $\Delta n=yx{e}^x(e^x-1{)}^{-2}[x\coth (x/2)-4]$, where $x=h\nu /(k_B{T}_{\gamma })$. The equivalent thermodynamic temperature perturbation has the displayed form, while the Rayleigh-Jeans law gives $\Delta T/T=-2y$. The distortion changes sign at $x\simeq 3.830$. This first-order expression assumes a small Compton y parameter and nonrelativistic thermal electrons.

Bulk motion of a scattering electron cloud relative to the cosmic microwave background gives a first-order Doppler temperature shift. With positive $v_{\parallel }$ directed away from the observer, the displayed sign applies, and $\tau$ is the Thomson scattering optical depth. In the nonrelativistic, single-scattering limit it has the spectrum of a small blackbody temperature change, unlike the thermal Sunyaev-Zeldovich effect.

For a uniform spherical cluster of depth $\ell \propto \theta d_A$, thermal bremsstrahlung gives received bolometric flux $S_X\propto n_e^2{T}_e^{1/2}{\ell }^3/d_L^2$, while the Rayleigh-Jeans law limit of the thermal Sunyaev-Zeldovich effect gives $|\Delta T|\propto n_e{T}_e\ell$. Eliminating $n_e$ and using $d_L=(1+z{)}^2{d}_A$ yields the displayed angular diameter distance. A model for the density profile fixes its numerical coefficient. Clumping, nonsphericity and temperature structure can bias this distance because the two observables weight the gas differently.

Represent each element of $S_1+\cdots +S_n$ by the lexicographically least tuple with that sum. Every coordinate projection of the resulting tuple set is itself lexicographically least for its partial sum: replacing it by a smaller representative would improve the original tuple without changing its sum. Consequently projected cardinalities are at most the corresponding partial sumset cardinalities. Projection inequalities, including the box theorem, therefore give sumset inequalities.

For $0<s_1<\cdots <s_n$, adding the largest integer creates at least $n$ new sums of subsets. If $P={\sum }_{i<n}{s}_i$ is the previous maximum, the values $P+s_n$ and $P+s_n-s_i$ for $i<n$ are distinct new sums above $P$. Induction gives the displayed bound. Equality holds for $s_i=i$, whose subset sums fill the interval of integers from zero to $n(n+1)/2$.

Given numbers $s_1,\dots ,s_n$, the sum associated with a subset $A$ of indices is ${\sum }_{i\in A}{s}_i$. The empty subset gives zero. The set of all these values is the sumset $\{0,s_1\}+\cdots +\{0,s_n\}$.

For a subset $A$ of density $\alpha$ in a finite cyclic group, put $F=1_A\ast 1_A\ast 1_A$ using normalized convolution on a finite group. Its average is ${\alpha }^3$, so choose $t_0$ with $F(t_0)\ge {\alpha }^3$. Retain the large spectrum where $|\widehat{1_A}|\ge {\alpha }^2/8$. The Parseval identity on a finite group gives at most $64{\alpha }^{-3}$ frequencies. Their phases change by at most $\alpha /4$ on the associated Bohr set, contributing at most ${\alpha }^3/4$ to the change of $F$. The discarded frequencies contribute at most another ${\alpha }^3/4$, since $\sum |\widehat{1_A}{|}^2=\alpha$. Therefore $F(t_0+b)\ge {\alpha }^3/2>0$ on the Bohr set, proving the containment.

For finite sets $A,B\subseteq G$ in an abelian group $G$, the sumset representation function isIt counts ordered pairs in the Cartesian product, so $(a,b)$ and $(b,a)$ are different when $a\ne b$, even though commutativity makes their sums equal. It is zero outside the sumset, andFor $A=\{0,1,2,3\}$, its values at $s=0,1,\dots ,6$ are $1,2,3,4,3,2,1$. For $B=\{0,1,4,10\}$, its values at the increasing elements of $B+B$ are $1,2,1,2,2,1,2,2,2,1$. Both lists total sixteen, but the first concentrates more ordered pairs on fewer sums. This distinction is what additive energy quantifies.

A telescoping sum cancels intermediate sequence terms, leaving boundary terms. For a finite sequence of real numbers $a_0,\dots ,a_n$, the displayed identity follows because every $a_1,\dots ,a_{n-1}$ occurs once with each sign. Applying it to an infinite series also requires checking the limit of the boundary terms; finite cancellation alone does not prove that the series is a convergent series.^[ref]

For $n\ge 1$, expanding the finite sum givesNo monotone sequence assumption is required. Consider the original sequence $4,9,2,7,5$. Its successive finite differences are $5,-7,5,-2$, soThe large intermediate rise and fall disappear from the net change. This is useful whenever one knows successive changes rather than every absolute value: adding the changes reconstructs the final value from the initial one. It also checks a cumulative-update calculation without requiring all increments to be positive.

A finite difference $\Delta a_{k-1}=a_k-a_{k-1}$ measures one discrete change. The identity ${\sum }_{k=1}^n\Delta a_{k-1}=a_n-a_0$ is the discrete counterpart of the fundamental theorem of calculus: accumulation of changes returns the endpoint difference. The related summation by parts transfers a finite difference from one factor of a product to another, with explicit boundary terms.