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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.

The risk of an estimator is its expected loss as a function of the unknown parameter. Under quadratic loss,

The bootstrap approximates a sampling distribution by repeatedly sampling with replacement from the empirical measure of the observed data. In the ordinary nonparametric version, observations are assumed independent and identically distributed; a bootstrap sample has the same size as the original statistical sample. A statistic, such as a sample mean or sample median, is recomputed on each resample. Its variation across resamples estimates how the original statistic would vary across repeated sampling from the unknown probability distribution.^[ref]

The key distinction is between the observed statistical sample and its still-unknown source probability distribution. Condition on the observed values $x_1,\dots ,x_n$ and form the empirical measureHere ${\delta }_{x_i}$ is a Dirac measure. Draw independent indices $I_1,\dots ,I_n$ from the uniform distribution on a finite set $\{1,\dots ,n\}$ and set $X_j^{\ast }=x_{I_j}$. Each observed slot has probability $1/n$ on every draw. If equal values occupy several slots, their total probability is the sum of those slot probabilities; distinct observed values need not be equally likely. A star denotes this conditional resampling experiment. There are $n^n$ equally likely ordered index sequences, although bootstrap count vectors and statistics can combine many such sequences.

A complicated sampling distribution need not stop statistical inference when recomputing the statistic is easy. The bootstrap reuses that calculation on simulated datasets. The plug-in estimator principle replaces the unknown probability measure by ${\hat{P}}_n$ in the whole repeated-sampling experiment. Whether that substitution approximates the target sampling distribution needs an appropriate bootstrap consistency theorem for the sample mean or a result for the particular statistic. A larger number of resamples reduces Monte Carlo error; it cannot repair the bootstrap failure for a uniform endpoint, an unrepresentative statistical sample or ignored statistical dependence.^[ref]

Quadratic discriminant analysis uses class-conditional Gaussian distributions with class-dependent covariance matrices. Their log density ratio is a quadratic polynomial in the observation.

LDA assumes class-conditional Gaussian distributions with a shared covariance matrix, producing affine log odds.

For jointly Gaussian $(S,Y)$ with $\mathrm{cov}(S)=V$,

The logistic location family has density and distribution function