A stellar stream is a correlated collection of stars commonly originating in the disruption of a bound progenitor. Phase mixing can erase spatial contrast while leaving coherence in orbital integrals. Such fossil debris constrains a galaxy's past accretion, although not every stream is from a galaxy: star clusters also produce streams.

Intracluster light comes from stars bound to a cluster rather than an individual galaxy. Tidal stripping and disruption of member systems can create extended diffuse light and substructure. Distinguishing this component from the outer envelope of the central galaxy depends on the adopted dynamical or photometric definition.

Kelvin-Helmholtz contraction releases gravitational binding energy as a self-gravitating body cools and shrinks. It powers young stars and giant planets before nuclear burning or degeneracy dominates their thermal evolution.

A stellar remnant is the object left after a star finishes its principal nuclear-burning evolution, such as a white dwarf, neutron star or black hole. Stellar remnants retain mass while often contributing little optical light, affecting an old stellar population's mass-to-light ratio.

The time for stellar nuclear fusion to consume the accessible fuel at characteristic luminosity $L$. If a fraction $f$ of a star's mass $M$ supplies fuel abundance $X$ and released energy per unit fuel mass $E_H$, hydrogen burning gives $t_{\mathrm{nuc}}\sim fXM{E}_H/L$. Accessible core fuel, rather than the entire star, sets the lifetime unless mixing is complete. This is longer than a structural Kelvin-Helmholtz cooling time when the nuclear reservoir greatly exceeds the gravitational contraction reservoir.

After substantial envelope loss on the Asymptotic giant branch, a low- or intermediate-mass star's exposed core heats while its radius shrinks, often at approximately constant luminosity during continued shell burning. It may ionize the expelled envelope as a planetary nebula. After the remaining nuclear fuel is depleted, its luminosity decreases and it joins a white dwarf cooling track. Evolutionary tracks on the Hertzsprung-Russell diagram connect the central-star temperature and luminosity with core mass and elapsed time.

Measurable planar sets with equal one-dimensional slice lengths in one direction have equal areas.

A spherical solid of radius $a$ growing inside a fixed cavity of radius $R$ expels volume at rate $4\pi (1-{\rho }_s/{\rho }_l)a^2\dot{a}$. The Darcy flux outside the cavity gives $p-p_m=\mu (1-{\rho }_s/{\rho }_l)a^2\dot{a}/(\Pi R)$. The common-pressure melting-point slope and quasi-steady spherical thermal conduction give ${\rho }_{s}La\dot{a}=k(T_i-T_{\infty })$. Define $x=a/R$, $\tau =tk(T_m-T_{\infty })/(R^2{\rho }_{s}L)$ and $K=L^2\Pi /[k{T}_m\mu (1/{\rho }_s-1/{\rho }_l{)}^2]$. Elimination yields $(x+x^2/K)x_{\tau }=1$ and the displayed zero-seed integral. The pressure rise is $p^{\ast }x/(x+K)$, with $p^{\ast }=L(T_m-T_{\infty })/[T_m(1/{\rho }_s-1/{\rho }_l)]$. Easy drainage gives square-root growth; low permeability gives an initial square-root regime followed by cube-root growth and pressure saturation. The model stops at $x=1$ and uses a uniform exterior conductivity.

For a one-phase spherical Stefan problem with melting-temperature boundary value $T_s-\Gamma /a$ and supercooling $\Delta T=T_s-T_{\infty }>0$, use the latent-to-sensible heat ratio $S=\mathcal{L}/(c_p\Delta T)$. Large $S$ separates heat diffusion from slow interface motion. The exterior harmonic temperature is $T=T_{\infty }+(\Delta T-\Gamma /a)a/r$. The Stefan condition gives the displayed radius equation with $a_c=\Gamma /\Delta T$. This is the large inverse-Stefan number convention; the common sensible-to-latent Stefan number is $1/S$.

Consider a planar binary alloy with solid at $x<a(t)$ and liquid at $x>a(t)$, equal constant phase mass densities, no bulk flow, constant thermal properties and constant solute diffusion coefficients $D_s,D_l>0$. A similarity solution places the interface at $a(t)=2\lambda \sqrt{D_{l}t}$. In the liquid, far-field concentration $C_{\infty }$ and constant interfacial concentration $C_a$ giveThis solves the diffusion equation, has $C_l(a,t)=C_a$ and tends to $C_{\infty }$ far ahead. The solid has its own diffusion profile, with coordinate $x/(2\sqrt{D_{s}t})$. Similarly, thermal profiles use each phase's thermal diffusivity, not automatically $D_l$.

Local liquidus equilibrium gives $T_a=T_m-m{C}_a$, with slope magnitude $m>0$, and the solid-liquid segregation coefficient $k_D$ gives $C_s(a,t)=k_D{C}_a$. For the normal directed toward the liquid, the heat and solute jump conditions areThe first is the Stefan condition; the second conserves solute rejected by the moving boundary. Together with the far-field thermal data, these determine $\lambda$ and $C_a$. For $k_D=0$ and negligible solid solute flux, the second reduces to salt rejection. Neglecting solid diffusion is an additional model limit: finite diffusion on both sides can matter near a nearly stationary interface $\lambda =0$.

A collection of independent identically distributed normal random variables. Orthogonal coordinates in the standardized Gaussian vector separate the sample mean from the residual sum of squares, proving their independence and giving the normal and chi-squared distributions used for exact inference.

Control limits bound the sampling outcomes expected under a specified reference process. An observation outside a limit is a statistical hypothesis testing signal to investigate the reference model. These are limits for observations under that model, rather than a confidence interval constructed around each observed estimate. Their calibration may be pointwise, simultaneous or sequential.

A hospital mortality funnel plot places each hospital's observed mortality proportion against its case volume. Under an independent binomial distribution model with common target p, the standard error is $\sqrt{p(1-p)/n_i}$, so uncertainty bands contract as volume grows. This is a provider-performance plot, distinct in purpose and axes from a meta-analysis funnel plot. A signal requires investigation of data quality and patient case mix before drawing a causal conclusion.

A CUSUM accumulates evidence for a change in a monitored probability distribution. With $W_t$ a log-likelihood ratio increment favoring a specified changed distribution, reset the accumulated score at zero and signal when it exceeds a chosen boundary $h$. For $C_0=0$, $C_t$ is the largest nonnegative sum of successive increments ending at $t$. This permits repeated detection of a change with an unknown onset, rather than a single fixed-start test.

For two independent samples of size $n$ with event probabilities $p_0,p_1$, put $\delta =p_0-p_1$, $\bar{p}=(p_0+p_1)/2$, $s_0^2=2\bar{p}(1-\bar{p})/n$ and $s_1^2=[p_0(1-p_0)+p_1(1-p_1)]/n$. The normal approximation to a two-sided level-$\alpha$ test has powerFor rare events, independent Poisson distributions approximate the counts. Conditional on their total, the null count allocation is binomial distribution with probability one half; under the alternative the allocation probability is $p_0/(p_0+p_1)$. Averaging its exact rejection probability over the alternative total supplies a discrete power calculation.

// Target: probability-theory.bigb

A statistical hypothesis test is a decision rule calibrated to distinguish a null hypothesis $\theta \in {\Theta }_0$ from a specified alternative hypothesis. A randomized statistical test gives a rejection probability $\varphi (x)\in [0,1]$ for possible data $x$. Its power function of a statistical test is ${\mathbb{E}}_{\theta }\varphi (X)$, and a level-$\alpha$ test satisfiesThe supremum is the size of a statistical test. Controlling it limits type I error; a useful test also has high statistical power against scientifically relevant alternatives.

For ten independent Bernoulli trials with success probability $p$, test $H_0:p\le 1/2$ against $H_1:p>1/2$ by rejecting for at least nine successes. The upper-tail probability increases with $p$, so the largest null rejection probability occurs at $p=1/2$ and equalsIt is therefore a conservative level-0.05 test. Rejecting at eight successes would have size $56/1024>0.05$. This illustrates why discrete calibration need not attain the advertised level exactly. A p-value is a null-tail probability for the observation, not a probability that the null hypothesis is true. The decision also does not measure the size or practical importance of an effect.

The score is the gradient of the log-likelihood, $S_n(\theta )={\nabla }_{\theta }{\ell }_n(\theta )$.

A power transformation replaces positive data by a member of a parameterized family of powers, commonly to improve the fit of a normal linear model.

These describe constancy or variation of conditional error variance across observations. A homoscedastic regression model has $\mathrm{Var}({\epsilon }_i\mid X)={\sigma }^2$; a heteroscedastic model allows this variance to depend on the observation or its covariates.

A scientific control supplies a comparison or procedure that distinguishes an intended effect from other influences on an experiment or observational measurement. Negative controls should show no target effect; positive controls should show a known effect. A negative control outcome applies this logic to an outcome that the exposure cannot causally affect.