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A Pöschl-Teller potential is a solvable one-dimensional potential built from ${\mathrm{sech}}^{2}x$. The attractive memberis reflectionless when $n$ is a positive integer.

Partner Hamiltonians reverse the order of two first-order factors and share corresponding positive-energy states.

In Planck units, chiral multiplets in four-dimensional $\mathcal{N}=1$ supergravity have

Assume the full-data sample space and the range of the statistic $T$ are standard Borel spaces, $T$ is a measurable function, and the statistical model $\{P_{\theta }\}$ is dominated by one common sigma-finite measure $\nu$. Write $f_{\theta }=d{P}_{\theta }/d\nu$ for its densities. Then $T$ is a sufficient statistic if and only if there exist nonnegative measurable functions $g_{\theta }$ and $h$, with $h$ independent of $\theta$, such thatThe statistical parameter can enter $g_{\theta }$ only through $\theta$ and the observed $T(x)$. ^[ref]

For discrete data, take counting measure; for ordinary continuous data, take Lebesgue measure. Independence, identical sampling laws, smooth dependence on $\theta$, and a fixed positive support of a probability distribution are not requirements of this theorem. Parameter-dependent support must nevertheless be included in $f_{\theta }$, usually through an indicator function, and must fit the displayed factorization. The sufficient statistic for a symmetric uniform sample illustrates exactly this: its indicator function depends on the unknown scale through the observed absolute maximum. The null sets in the theorem concern versions of densities; the factors are not generally themselves normalized probability distributions.

For a discrete sample space, the cancellation behind the theorem is particularly transparent. On a group with positive probability, factorization givesConversely, multiply a common conditional probability mass function by $P_{\theta }(T=T(x))$ to obtain the required factorization. For continuous data, a sum over a group is not a density integral over the original Lebesgue measure: the group may have measure zero. Use the regular conditional distribution formulation and the Fisher-Neyman factorization theorem instead of that discrete calculation.

A sufficient statistic is minimal sufficient when it is a function of every other sufficient statistic, up to null sets. Its level sets form the coarsest sufficient partition of the sample space.

For independent observations $X_i\sim N(\mu ,{\sigma }^2)$ with known variance, ${\sum }_i{X}_i$ is minimal sufficient for $\mu$. Factorization proves sufficiency, while the likelihood ratio for two samples is independent of $\mu$ exactly when their sums agree.

A differential operator is self-adjoint when integration by parts makes its inner products symmetric on its domain.

A regular Sturm-Liouville eigenvalue problem has the formon a bounded interval with self-adjoint boundary conditions and positive weight $w$.

The dominant energy condition requires the energy flux measured by every future-directed timelike observer to be future-directed and causal. Equivalently, $T_{ab}{u}^a{v}^b\ge 0$ for every pair of future-directed causal vectors $u$ and $v$ under the $(-,+,+,+)$ convention.

The weak energy condition requiresfor every timelike vector $X$, so every observer measures nonnegative energy density.

For ${\nabla }_{\alpha }{\nabla }^{\alpha }\varphi =m^2\varphi$, considerCovariant conservation for all solutions holds exactly when

The null energy condition requires $T_{\mu \nu }{k}^{\mu }{k}^{\nu }\ge 0$ for every null vector $k$. Through the Einstein field equations this is equivalent to $R_{\mu \nu }{k}^{\mu }{k}^{\nu }\ge 0$ when the cosmological term is absent.

For $(u,v)=(y,ax)$, the stream function $\psi =(y^2-a{x}^2)/2$ gives hyperbolic streamlines and a saddle when $a>0$, while $a<0$ gives elliptical streamlines and a centre.

A binary star is a gravitationally bound pair orbiting their common centre of mass.

Stellar evolution is the change of a star's structure and composition as nuclear reactions, energy transport, mass loss, and interactions proceed.

A main-sequence star is in a long-lived phase of core hydrogen burning.

Stellar nuclear fusion converts light nuclei into more tightly bound nuclei and supplies most main-sequence stellar luminosity.

Stellar structure follows from mass conservation, hydrostatic equilibrium, energy generation, and energy transport together with an equation of state and opacity law.

Flooring an exponential variable of rate $\theta$ produces a geometric variable on the nonnegative integers with success probability $1-e^{-\theta }$.

For a regular scalar model with score function $S_{\theta }$ and Fisher information $I(\theta )={\mathbb{E}}_{\theta }{S}_{\theta }^2$, an estimator $T$ with mean $\psi (\theta )$ satisfiesIndeed, differentiating ${\mathbb{E}}_{\theta }T=\psi (\theta )$ gives ${\mathrm{Cov}}_{\theta }(T,S_{\theta })={\psi }^{\prime }(\theta )$, and the Cauchy-Schwarz inequality gives the bound.