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The Hernquist model has relative potential $\psi =GM/(r+a)$ and density $\rho =Ma/[2\pi r(r+a{)}^3]$. It has enclosed mass $M{r}^2/(r+a{)}^2$, a central $r^{-1}$ cusp and an outer $r^{-4}$ tail. Since $r\rho =a{\psi }^3/(2\pi G^3{M}^2)$, the half-anisotropic distribution function is $f(E,L)=3a{E}^2/(4{\pi }^3{G}^3{M}^{2}L)$ for $0<E<GM/a$. Its moments are $⟨v_r^2⟩=\psi /4$ and each tangential moment $\psi /8$. Isotropic and other admissible anisotropic distributions can also support this same density; the gravitational potential does not select a unique velocity anisotropy.

The spherical Plummer model has $\psi =GM/\sqrt{a^2+r^2}$ and $\rho =3M{a}^2/[4\pi (a^2+r^2{)}^{5/2}]$. It is the $p=2$ hypervirial model. In unit scales its isotropic distribution is $24\sqrt{2}{E}^{7/2}/(7{\pi }^3)$ for positive relative energy.

For fixed-composition monatomic ideal gas plus equilibrium radiation pressure, specific internal energy is $u=3\mathcal{R}T/2+a_{\mathrm{r}}{T}^4/\rho$. An isentropic process satisfies $du=Pd\rho /{\rho }^2$, giving ${\Gamma }_3-1=(8-6\beta )/(24-21\beta )$ andThe stellar gas-pressure fraction must be allowed to change along the adiabat. The pure-radiation and pure-gas limits give ${\Gamma }_1={\Gamma }_2=4/3$ and $5/3$, respectively. Partial ionization or different gas heat capacities change these formulas.

The leading separation between successive high-order low-degree stellar acoustic mode frequencies is the inverse round-trip acoustic travel time, $\Delta \nu \simeq [2{\int }_0^{R}dr/c_s{]}^{-1}$. For homologous stars it scales approximately as $(M/R^3{)}^{1/2}$ and therefore measures mean density.

The Stefan-Boltzmann constant converts the fourth power of absolute temperature into blackbody radiant exitance:Here $k_B$ is the Boltzmann constant, $h$ the Planck constant and $c$ the speed of light. In the current International System of Units, these three constants have exact defining values, so $\sigma$ is exact as an expression in them and $\pi$. The displayed finite decimal is an approximation, not an exact decimal or a measured uncertainty. The 2022 recommended-constants table lists an ellipsis and an exact uncertainty for $\sigma$.^[ref]

A dimensional check uses $[k_B]=\mathrm{J}{\mathrm{K}}^{-1}$, $[h]=\mathrm{J}\mathrm{s}$ and $[c]=\mathrm{m}{\mathrm{s}}^{-1}$. Consequently $[k_B^4/(h^3{c}^2)]=\mathrm{J}{\mathrm{s}}^{-1}{\mathrm{m}}^{-2}{\mathrm{K}}^{-4}=\mathrm{W}{\mathrm{m}}^{-2}{\mathrm{K}}^{-4}$, exactly the units needed in the Stefan–Boltzmann law.

For an equilibrium photon gas, the energy density is $u=a_{\mathrm{rad}}{T}^4$ with radiation constant $a_{\mathrm{rad}}=4\sigma /c$. The radiation pressure is $p_{\mathrm{rad}}=u/3=4\sigma T^4/(3c)$. These describe the radiation filling an equilibrium cavity; $u$ has units $\mathrm{J}{\mathrm{m}}^{-3}$, whereas the outward radiant exitance through a small opening is $cu/4$. The factor $1/4$ comes from averaging propagation directions and their projected area over a radiation field with isotropy, not from reducing the opening's emissivity.

The Pearson chi-squared test of homogeneity tests whether several independent categorical samples have the same category probabilities. Under the null, expected cell counts equal the product of the corresponding row and column totals divided by the total sample size. The Pearson chi-squared statistic for contingency tables $\sum (O-E{)}^2/E$ has an asymptotic chi-squared distribution with $(r-1)(c-1)$ degrees of freedom, when expected counts are sufficiently large. The sampling design differs from a Pearson chi-squared test of independence, though the statistic has the same form.

A least-favourable configuration maximizes a test procedure’s type I error probability over allowed null hypothesis parameters. For one-sided normal comparisons with a fixed covariance matrix, coupling by common centered variables shows monotonicity in the means. For the familywise error rate, mixed true and false null configurations must also be checked.

A scan statistic searches a family of candidate locations or subsets and reports the largest local statistic. Under a null hypothesis, a union bound converts a tail estimate for each candidate into a bound for the maximum; no independence between local statistics is required. Structured alternatives often permit a chi-squared testing lower bound using the overlap geometry of two candidates.

If $q={\sigma }^{2}p$ and $q^{\prime }=2bp$, then $pLu=(q{u}_x{)}^{\prime }/2$. Insert compact cutoffs ${\chi }_R$, integrate the backward equation in time, and integrate by parts twice to move spatial derivatives onto ${\chi }_R$. The error is bounded by $t\Vert u{\Vert }_{\infty }(C{R}^{-2}\Vert q{\Vert }_1+C{R}^{-1}\Vert q^{\prime }{\Vert }_1)/2$. When these norms are finite it vanishes, proving $\int u(t)p=\int fp$ without assuming derivative behavior at infinity.

For the sum-zero standard representation of the symmetric group, the vectors $v_j=e_1+\cdots +e_j-j{e}_{j+1}$ form a Gelfand–Tsetlin basis. The Young–Jucys–Murphy element $X_i$ has eigenvalue $i-1$ for $i\le j$, $-1$ for $i=j+1$, and $i-2$ for $i>j+1$. Normalizing by $\sqrt{j(j+1)}$ gives an orthonormal basis corresponding to tableaux of shape $(n-1,1)$ whose second-row entry is $j+1$.

Put $C^0={⨁}_i\mathrm{Hom}(X_i,Y_i)$ and $C^1={⨁}_{\rho :i\to j}\mathrm{Hom}(X_i,Y_j)$. The differential has components $g_{\rho }{h}_i-h_j{f}_{\rho }$. Its kernel is ${\mathrm{Hom}}_Q(X,Y)$ and its cokernel is ${\mathrm{Ext}}_Q^1(X,Y)$. In a vertexwise splitting of an extension, the off-diagonal arrow blocks change by these coboundaries when the splitting changes.

In normalization $Q=T_3+Y$, the left quark doublet has $Y=1/6$, right up/down singlets have $2/3,-1/3$, the left lepton doublet has $-1/2$, the right charged lepton has $-1$, and the Higgs doublet has $1/2$. The alternative convention $Q=T_3+\hat{Y}/2$ uses $\hat{Y}=2Y$. Up-type Yukawa interactions use the conjugate doublet of hypercharge $-1/2$.

For $H=\frac{1}{2}\sum p_j^2/h_j^2-\psi$, $l=p_{\varphi }$ and a Stäckel potential, multiplying the separated Hamilton-Jacobi equation by $\lambda -\mu$ gives $2(\tau +\alpha )(\tau +\beta )p_{\tau }^2-(\alpha -\beta )l^2/[2(\tau +\alpha )]-G(\tau )-E\tau =-I$ for both $\tau =\lambda ,\mu$. Eliminating $E$ gives $I=\frac{1}{2}[\mu v_{\lambda }^2+\lambda v_{\mu }^2+(\lambda +\mu +\alpha )v_{\varphi }^2]-[\mu G(\lambda )-\lambda G(\mu )]/(\lambda -\mu )$. Physical components are $v_j=h_j{\dot{q}}_j$ and $p_j=h_j{v}_j$. Separation proves conservation locally; direct Poisson brackets extend it across regular coordinate patches.

For fixed $E,l,I$, separated motion obeys $p_{\tau }^2=[E\tau +G(\tau )-I+(\alpha -\beta )l^2/(2(\tau +\alpha ))]/[2(\tau +\alpha )(\tau +\beta )]$. Allowed intervals are where this is nonnegative, with turning points and coordinate-fold endpoints treated separately. A generic nonresonant bound orbit with $l\ne 0$ circulates azimuthally and densely samples a projected volume bounded by confocal spheroids and hyperboloids. It is not uniformly distributed in volume. Resonant or zero-angular-momentum orbits can project to lower-dimensional regions; integrability alone does not imply that every orbit fills a three-dimensional shell.

The steady upstream film of an infinite blade has $h_x=-3(h-1)/h^3$. Thus $X(h)=h^3/9+h^2/6+h/3+\log (h-1)/3$. The head is fixed by under-blade leakage balancing the incoming film. The deep pile has a cubic profile, while the actual upstream transition has an exponential tail towards $h=1$.

For a rounded blade with local parabolic lubrication gap $d=ϵ+x^2/(2\delta )$, $ϵ\ll \delta$, the Couette-Poiseuille flow in a thin gap yields a floor-driven flux $2ϵ/3$ and pressure-driven leakage proportional to ${ϵ}^{5/2}\Delta p/\sqrt{\delta }$. Positive $J$ is from the upstream pile towards the rear.

A long blade with negligible under-blade leakage diverts incoming fluid laterally. A deep-pile profile $h^3=9[x_N(y)-x]$ and integrated flux conservation give $d(\int h^{4}dx)/dy=-12y$, hence $x_N=[126({\alpha }^2-y^2){]}^{3/7}/9$. This leading approximation excludes the nose and blade-tip transition regions.

The risk of an estimator under squared Euclidean error is its expected squared distance from the true parameter. Pointwise risk domination can be strict even when the supremum risk agrees, because the supremum can be approached at parameters tending to infinity.

Given finitely many points in a right triangle with hypotenuse endpoints $a,b$, there is a path from $a$ to $b$ visiting them all with sum of squared edge lengths at most $\Vert a-b{\Vert }^2$. Repeated altitude subdivision reduces to cells containing one point. Joining the child paths preserves the squared-cost bound by the Pythagorean theorem; deleting an auxiliary right-angle vertex preserves it by the law of cosines.

The square class of $a\in K^{\ast }$ is its image in the square-class group of a field $K^{\ast }/K^{\ast 2}$. Two nonzero elements have the same square class exactly when their quotient is a square. Every element of this quotient has order dividing $2$. Over the rational numbers, a square class has a unique signed squarefree integer representative.