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A loss cone consists of low-angular momentum stellar trajectories whose next pericentre enters a central absorbing region. For capture radius $r_c$ in a black hole-dominated potential, the Newtonian near-parabolic estimate is $J_{\mathrm{lc}}^2\simeq 2G{M}_H{r}_c$. At radius $r$ and speed $v$, its angular half-width is $J_{\mathrm{lc}}/(rv)$. Two-body relaxation can refill this depleted part of phase space.

In a thin ideal-gas grey atmosphere with $T^4=T_e^4(1/2+3\tau /4)$, constant enclosed mass $M$ and luminosity $L$, and $\kappa ={\kappa }_0\rho T^{13}$, divide the stellar hydrostatic equation by radiative diffusion in a star. This gives $PdP/dT=16\pi acGM\mathcal{R}{T}^{-9}/(3{\kappa }_{0}L\mu )$. Integrating from $P=0$, $T^4=T_e^4/2$, yieldsLogarithmic derivative gives $\nabla =T^8/T_e^8-1/4$. For a monatomic gas gas, the Schwarzschild criterion reaches equality at $\nabla =2/5$, hence $T/T_e=(13/20{)}^{1/8}$. The purely radiative profile cannot be continued past this point as a convectively stable model.

The abundance of chemical elements heavier than helium in astronomical material. A metal mass fraction measures their combined mass; spectroscopic abundance ratios are other measures of metallicity and need not be numerically identical to that fraction.

For a stationary axisymmetric tracer with isotropic velocity dispersion, vanishing tilted stress, density $\nu \propto e^{-R/R_d}$, and ${\sigma }^2\propto e^{-R/R_{\sigma }}$, the radial Jeans equation gives the displayed expression. More generally its right side is $-{\sigma }^{2}d\ln (\nu {\sigma }^2)/d\ln R$. It requires a nonnegative streaming-speed square; the density and dispersion laws must describe the same tracer and the chosen height or consistent vertical average.

Reflection symmetry makes $\overline{v_R{v}_z}$ odd in height, so its value at the midplane vanishes while its derivative need not vanish. The radial Jeans equation includes this derivative in the stellar asymmetric drift. For a velocity ellipsoid locally aligned with spherical coordinates, $\overline{v_R{v}_z}\simeq ({\sigma }_R^2-{\sigma }_z^2)z/R$, and its midplane contribution is $-({\sigma }_R^2-{\sigma }_z^2)$. Dropping a mixed moment at the symmetry plane does not justify dropping its derivative.

For a small, uniform surface at temperature $T$, enclosed by much larger black walls all at temperature $T_s$, the outward net heat transfer by thermal radiation isAssume an opaque grey body that is a diffuse thermal emitter, with the same emissivity $ϵ$ and radiometric absorptance at the surface's current temperature. Its emitting area $A$ sees only the enclosure: the view factor to the walls is one, with no self-viewing surface, opening or intervening obstruction. The intervening medium must neither absorb nor emit appreciable thermal radiation; a vacuum supplies a useful idealization. The enclosure is maintained at uniform $T_s$. The small-body assumption makes that reservoir description physically reasonable; genuinely black isothermal walls supply the incoming irradiance even without an infinite area ratio. Positive ${\dot{Q}}_{\mathrm{rad}}$ means net loss by the surface; negative means net gain. This expression counts thermal radiation alone, not all heat transfer.^[ref]

The reasoning is a two-way energy balance. Black walls supply irradiance $E=\sigma T_s^4$, so the surface absorbs $P_{\mathrm{abs}}=ϵAE$ and emits $P_{\mathrm{emit}}=ϵA\sigma T^4$. Subtract these to obtain ${\dot{Q}}_{\mathrm{rad}}$. For an opaque surface the remaining fraction $1-ϵ$ of incoming radiant flux is reflected; counting emitted plus reflected outgoing radiant flux minus total incoming radiant flux gives the same result. Kirchhoff's law of thermal radiation justifies equal spectral absorption and emission factors; the grey body assumption lets one factor be taken outside both spectral integrals.

Choose $ϵ=0.8$, $A=0.01{\mathrm{m}}^2$, $T=500\mathrm{K}$ and $T_s=300\mathrm{K}$. These are stipulated model inputs, not measured material properties. Using $\sigma \approx 5.670374419\times 10^{-8}\mathrm{W}{\mathrm{m}}^{-2}{\mathrm{K}}^{-4}$ givesThe incoming irradiance is $459.300\mathrm{W}{\mathrm{m}}^{-2}$, corresponding to $4.5930\mathrm{W}$ incident on the surface; only $80\%$ is absorbed. At $T=T_s=300\mathrm{K}$, emitted and absorbed powers are both $3.6744\mathrm{W}$, so radiative equilibrium has zero net exchange despite continuing emission. Changing only emissivity to $0.2$ reduces both one-way powers and the net loss by a factor of four, giving $6.1694\mathrm{W}$ net. A low-emissivity coating therefore suppresses both radiative cooling to colder walls and radiative heating from hotter walls.

Near equal temperatures, write $T=T_s+\Delta T$. Taylor's theorem givesFor constant emissivity, the radiative heat transfer coefficient $h_{\mathrm{rad}}$ is approximately $4.8992\mathrm{W}{\mathrm{m}}^{-2}{\mathrm{K}}^{-1}$ at $T_s=300\mathrm{K}$ and $ϵ=0.8$. This local linear approximation is useful for small temperature differences; the $200\mathrm{K}$ difference in the worked example needs the full fourth-power expression.

A cooling experiment requires a wider first law of thermodynamics balance. In a uniform-temperature model without a heater,where $C(T)$ is the sample's heat capacity and the last two terms are signed outward convection and thermal conduction losses. A vacuum suppresses convection but does not eliminate thermal conduction through supports or sensor leads. A cooling slope therefore cannot establish emissivity unless these paths and heat capacity are independently constrained. Multiple wall temperatures, incomplete view factors, wavelength-selective materials or appreciable intervening absorption require a more complete radiative transfer calculation.

Write the Steane code basis as uniform quantum superpositions over a binary linear code $D$ and a disjoint coset $t+D$, where $D$ is the even-weight part of the length-seven Hamming code. For $Z(w)={⨂}_{j=1}^7{Z}_j^{w_j}$, diagonal Pauli Z gates cannot connect the two cosets. Within coset $t_a+D$, the overlap isIf $w\notin D^{\perp }$, translation by a word $x_0\in D$ with $w\cdot x_0=1$ negates this sum, so it vanishes. Since the dual code $D^{\perp }$ is the length-seven Hamming code of minimum Hamming distance three, every nonzero $w$ of Hamming weight one or two has vanishing overlap. Taking $u,v$ from $\{0,e_1,\dots ,e_7\}$ proves the displayed identity. The eight two-dimensional error images are orthogonal, so their projective measurement determines the error syndrome without revealing the logical state; applying the identified phase flip again recovers every code state. This proves nondegenerate quantum error-correcting code behaviour for the identity and all single-site phase flips, including their coherent linear span.

Cluster analysis groups observations using similarities or dissimilarities without supplied class labels. A distance choice specifies which features and scales matter. Agglomerative hierarchical clustering merges groups recursively, producing partitions at different resolutions rather than one intrinsically determined number of groups.

Learning structure from data without supplied target labels. Examples include extracting principal components, clustering and learning representations that efficiently reconstruct inputs.

Generalization is the ability of a fitted prediction rule to perform well on new observations under a specified target probability distribution. It is assessed through prediction error rather than training error. The target distribution and use of data are part of the question: dataset shift can change performance, and data leakage can make an apparent assessment misleading.

Supervised learning fits a prediction rule from training data containing both inputs and associated target responses. Regression analysis predicts numerical responses; classification in statistical learning predicts class labels. A loss function specifies which prediction errors matter, and generalization concerns performance beyond the observations used to fit the rule.

The Pearson chi-squared test compares observed categorical counts with null expected counts using the displayed statistic. Its applications include Pearson chi-squared goodness-of-fit test for specified probabilities and Pearson chi-squared test of independence for contingency tables; the fitted restrictions determine the limiting degrees of freedom.

For paired binary observations $(C_i,M_i)$, let $b$ count pairs $(1,0)$ and $c$ pairs $(0,1)$. Under the null hypothesis of equal marginal event probabilities, these discordant outcomes have equal probabilities. Conditional on $b+c$, $b$ has the binomial distribution $\mathrm{Bin}(b+c,1/2)$, giving an exact test. For enough discordant pairs, $(b-c)/\sqrt{b+c}$ has an approximate standard normal distribution, or its square has an approximate one-degree-of-freedom chi-squared distribution. Concordant pairs provide no information about the direction of the marginal difference. If there are no discordant pairs, there is no directional information and the divided statistic is not computed. The pairing and independence between pairs are essential; the usual independent-two-sample formula is not a paired test.

A rejection region is the set of sample outcomes on which a hypothesis test rejects its null hypothesis. Its probability under the null determines the type I error rate.

For independent multivariate normal observations with positive-definite covariance, this statistic is the maximum squared studentized mean contrast. With unbiased sample covariance matrix $S$ and $n>p$, its null distribution obeys $(n-p)T^2/(p(n-1))\sim F_{p,n-p}$. This follows from independence of the Gaussian sample mean and the residual Wishart distribution.

The alternative hypothesis specifies the parameter values or distributions against which a null hypothesis is tested. If the null is $\theta \in {\Theta }_0$, the alternative is $\theta \in {\Theta }_1$ with ${\Theta }_0\cap {\Theta }_1=\varnothing$. For example an upper one-sided comparison uses $\theta >{\theta }_0$, while a two-sided comparison uses $\theta \ne {\theta }_0$. The alternative determines the relevant rejection tail. A p-value is computed under the null; it is not the probability that the alternative is false.

A one-sided test targets deviations in one specified direction, such as $\theta >{\theta }_0$. If larger values of a statistic $T$ indicate this direction, its p-value is an upper-tail null probability $P_0(T\ge t_{\mathrm{obs}})$. The direction should be chosen from the scientific question before examining the data. An upper normal test uses $1-\Phi (z)$; a lower test uses $\Phi (z)$.

A two-sided test detects departures in either direction from the null hypothesis, for example $\theta \ne {\theta }_0$. With a continuous symmetric null statistic centred at zero, its p-value is $P_0(|T|\ge |t_{\mathrm{obs}}|)=2P_0(T\ge |t_{\mathrm{obs}}|)$. This is the usual coefficient test printed by Gaussian regression summaries, unlike a directional one-sided comparison.

A result is statistically significant at a specified significance level $\alpha$ when its valid p-value is at most $\alpha$ and the corresponding null hypothesis is rejected. Significance does not measure effect size, establish practical importance, or give the probability that the null is true. Software stars encode thresholds for $p$-values; they do not supply an additional statistic.

A sequential probability ratio test accumulates the log-likelihood ratio for two specified simple statistical hypotheses until it reaches an upper or lower boundary. An upper crossing favors the alternative, a lower crossing favors the null, and observations continue between them. Error-calibrated boundaries are approximately $\log [(1-\beta )/\alpha ]$ and $\log [\beta /(1-\alpha )]$; discrete overshoot means these approximations need calibration.