This logarithmic area integral measures the accumulated pullback hyperbolic metric area of a holomorphic disc map. Its nonnegative kernel makes it increasing in $R$. Schwarz-Pick theorem bounds $H_g(z)$ by $2/(1-|z{|}^2)$, which controls its characteristic through a critical-point Jensen identity.

For a barotropic fluid in hydrostatic equilibrium along a Killing vector, assume $\rho +p>0$. Integrating $dp/(\rho +p)=-d\log N$ gives the displayed first integral, constant on each connected fluid region. For the equation of state $p=w\rho$ with constant $w>0$, it gives $\rho N^{(1+w)/w}=C_1$.

The equation of state of isotropic blackbody radiation is $p=\rho /3$. The barotropic hydrostatic first integral gives $\rho N^4=C$. The Stefan–Boltzmann law $\rho \propto T^4$ then yields the Tolman–Ehrenfest relation $TN=\mathrm{constant}$. Here $N=\sqrt{|K^a{K}_a|}$ for the stationary timelike Killing vector field; its normalization only changes the constants.

For a hydrogen-only gas with $g_e=g_p=2$, $g_H=4$ and $m_H\simeq m_p$, the Saha equation is $(1-X_e)/X_e^2=n_B[2\pi /(m_{e}T){]}^{3/2}{e}^{I/T}$. Defining ${\eta }_{\gamma }=n_B/n_{\gamma }$ and using $n_{\gamma }=2\zeta (3)T^3/{\pi }^2$ gives coefficient $2\zeta (3){\eta }_{\gamma }/{\pi }^2$ multiplying $(2\pi T/m_e{)}^{3/2}{e}^{I/T}$. If instead ${\eta }_s=n_B/s$, with entropy density $s=(2{\pi }^2/45)g_{\ast s}{T}^3$, that coefficient is $(2{\pi }^2/45)g_{\ast s}{\eta }_s$. The two expressions agree because $s/n_{\gamma }={\pi }^4{g}_{\ast s}/[45\zeta (3)]$. The reference entropy sector must be specified, especially after particles have decoupled.

In a thin nondegenerate hydrogen-burning white-dwarf envelope, fix gravity $g$ and use $P=g\Sigma$, $\rho =B\Sigma /T$. With opacity $\kappa ={\kappa }_0\rho T^{-2}$ and specific heating $ϵ={ϵ}_0\rho T^{15}$, set $y=T^7$, $x={\Sigma }^2/2$. Then $y^{\prime }=AF$ and $F^{\prime }=-{ϵ}_{0}B{y}^2$, where $A=21{\kappa }_{0}B/(4ac)$ and ${\omega }^2=A{ϵ}_{0}B$. Radiative-zero surface and zero core-flux base conditions give $(y^{\prime }{)}^2=(2{\omega }^2/3)(y_b^3-y^3)$. Thus $T_b\propto F_s^{2/21}$ and $F_s=12I^3/(A{\omega }^2{\Sigma }_b^6)$, with $I={\int }_0^1(1-t^3{)}^{-1/2}dt$. At fixed column the specific heating varies as $T^{14}$ while radiative cooling varies as $T^7$, giving thin-shell instability in this approximation.

For a smooth rectangular channel, conservation of discharge and Bernoulli equation head gives $(1-F^2)h_x=F^{2}h{b}_x/b-H_x$. A regular hydraulic control has $F=1$ and the displayed geometric compatibility condition. At fixed discharge the critical-head envelope is $T(x)=H(x)+3h_c(x)/2$, with $h_c=(Q^2/(g{b}^2){)}^{1/3}$. A simple smooth control requires a local maximum of $T$: $T_x=0$ and $T_{xx}<0$. A local minimum cannot be crossed at the critical head because the real depth branches disappear nearby.

For a prescribed positive head-loss rate $\gamma (x)$, the steady energy balance is $(E+H{)}_x=-\gamma$. The accumulated loss is added to the bed elevation when applying hydraulic control compatibility: $H_x+\gamma =h_c{b}_x/b$. A local loss law alone does not determine the accumulated loss between a remote inlet and the control; that integral must also be specified.

For loss of fluid carrying its local axial velocity, the inviscid Bernoulli equation head stays constant while discharge decreases. On a flat bed a regular control requires a positive local maximum of $q=Q/b$. In the parabolic-width channel $b=b_0(1+x^2/L^2)$, this maximum occurs at $x_c=-S{L}^2/(Q_0+\sqrt{Q_0^2+S^2{L}^2})$, with $q_c=(Q_0+\sqrt{Q_0^2+S^2{L}^2})/(2b_0)$. Its critical depth is $(q_c^2/g{)}^{1/3}$. A control exists only when the prescribed total head equals the maximum critical-head envelope and the reach includes that point while retaining positive discharge.

A light shallow layer flows below a roof of height $H(x)$ and width $W(x)$ through a deeper approximately motionless ambient. Hydrostatic equilibrium and inviscid steady Bernoulli equation give the displayed specific energy for constant volume flux $Q$ and reduced gravity $g^{\prime }$. At fixed position, $E_h=g^{\prime }(1-{\mathrm{Fr}}^2)$ and $h_c=[Q^2/(g^{\prime }{W}^2){]}^{1/3}$. A roof depression raises the minimum admissible energy, just as a bed elevation does for a heavy lower layer.

A weir is an obstruction across a channel that can regulate volume flux by creating a hydraulic control. In a shallow water model a sufficiently smooth crest has a critical section, with upstream flow subcritical and downstream flow supercritical under free discharge. Losses, submergence and geometry determine whether this ideal control applies.

A sphere map $f:S^n\to X$ is sent to $f_{\ast }[S^n]\in H_n(X;\mathbb{Z})$. After tensoring with $\mathbb{Q}$ this gives the rational Hurewicz homomorphism. It is natural in $X$, sends Whitehead products to zero, and is an isomorphism in the first nonzero homotopy degree of a simply connected space by the Hurewicz theorem.

For standard normal distribution location estimation, an influence function $h$ obeys $\mathbb{E}[Zh(Z)]=1$. Under $|h|\le C$, minimize $\mathbb{E}{h}^2$ by projecting $ax$ onto $[-C,C]$, choosing $a$ to meet that identity. For $C>\sqrt{\pi /2}$ this gives the displayed Huber score normalized by $2\Phi (K)-1$, with $C=K/(2\Phi (K)-1)$. Pointwise minimality of $h^2-2axh$ proves global optimality after integration. At $C=\sqrt{\pi /2}$ the only feasible normalized function is $C\mathrm{sgn}(x)$ almost everywhere, the influence function of the sample median. The endpoint is a limit of rescaled clipped scores, not the identically zero score obtained by substituting $K=0$ literally.

Write $S(\theta )={\sum }_{x\le n}e(\theta x^3)$ and $I_j={\int }_0^1|S{|}^j$. Character orthogonality gives $I_2=n$. For $I_4$, fix the difference $D=x^3-y^3$. If $D=0$, there are $n^2$ choices for the two equal pairs. If $D\ne 0$, any solution $z^3-w^3=D$ has $z-w\mid D$; fixing this divisor gives a quadratic equation for $w$, hence at most two choices. The subpower bound for the divisor function yields $I_4{\ll }_{\eta }{n}^{2+\eta }$.

The twice-differenced inequality in Weyl differencing reads $|S{|}^4\le 2n{\sum }_v{c}_{v}e(\theta v)$, with $c_v$ counting triples $(h,l,x)$ for which $v=3hl(2x+h+l)$ and all four shifted arguments are in $[1,n]$. Thus $c_0\ll n^2$, and $c_v{\ll }_{\eta }{n}^{\eta }$ for $v\ne 0$: the product $hl$ divides $v$, and the remaining equation determines $x$. The Fourier coefficients $r(v)$ of $|S{|}^4$ are nonnegative counts, with $r(0)=I_4$ and ${\sum }_{v}r(v)=n^4$. Multiply the inequality by $|S{|}^4$ and integrate. It follows that $I_8\ll n(n^2{I}_4+n^{\eta }{n}^4){\ll }_{\eta }{n}^{5+\eta }$, after choosing the divisor-bound exponents small enough.

For a fixed rank-$m$ constraint matrix $B$, apply Hotelling's T-squared statistic to the transformed observations $B{X}_j$. Under $B\mu =0$, $(n-m)T_B^2/(m(n-1))\sim F_{m,n-m}$ for $n>m$. If the displayed contrasts are dependent, use a basis of their row space and replace $m$ by the rank.

Under a nonsingular affine transformation $Y=AX+b$, transform the null mean by the same map. Then the centered sample mean becomes $A(\bar{X}-{\mu }_0)$ and the sample covariance matrix becomes $AS{A}^T$. Since $(AS{A}^T{)}^{-1}=A^{-T}{S}^{-1}{A}^{-1}$, the quadratic form in Hotelling's T-squared statistic is unchanged. Its value and test decision are consequently invariant under changes of units, origin and nonsingular coordinates.

At a fixed horizontal wavenumber magnitude, the steady neutral curve in the horizontal-field magnetoconvection dispersion relation is minimized by $m=0$. The disturbance is then constant along the imposed magnetic field; it induces no field bending and incurs no magnetic tension penalty. Minimizing $(k^2+{\pi }^2{)}^3/k^2$ gives the displayed stationary threshold for every $Q\ge 0$. These convection rolls have axes parallel to the imposed field. This assertion concerns steady onset.

// Target: fluid-mechanics.bigb

Restrict the horizontal-field magnetoconvection dispersion relation to $k=0$, $x=m^2>0$. The steady neutral curve becomes $R={\pi }^{2}Q+(Q+3{\pi }^2)x+x^2+3{\pi }^4+{\pi }^6/x$. Its unique minimum satisfies $x^2(Q+3{\pi }^2+2x)={\pi }^6$, since $R^{\prime \prime }(x)=2+2{\pi }^6/x^3>0$. Expanding at large Chandrasekhar number gives $x\sim {\pi }^3{Q}^{-1/2}$ and the stated asymptotic expansion. Unlike field-aligned rolls, these disturbances bend the field, and their stationary threshold grows with field strength.

// Target: fluid-mechanics.bigb

Triangulate ${\Delta }^q\times I$ by the simplices with vertices $(v_0,0),\dots ,(v_i,0),(v_i,1),\dots ,(v_q,1)$ and coefficients $(-1{)}^i$. Composing with a homotopy between f and g gives a degree-one map P on singular chains. Shared faces cancel; the remaining faces give the displayed chain homotopy identity. Hence homotopic maps induce the same map on singular homology.

For homologous, chemically uniform, gas-pressure-supported radiative stars with specific energy generation $ϵ\propto X\rho T^{13}$ and composition-dependent constant opacity, stellar homology gives $L_{\mathrm{nuc}}\propto X{\mu }^{13}{M}^{15}{R}^{-16}$ and $L_{\mathrm{rad}}\propto {\mu }^4{M}^3/\kappa$. Equating them gives the displayed scalings. Hydrogen depletion changes both mean molecular weight and electron-scattering opacity; holding composition fixed gives $R\propto M^{3/4}$ and $L\propto M^3$. The power-law burning approximation applies only while hydrogen fuel remains.

For a fixed-mass fully mixed hydrogen burning star with $\mu =4/(3+5X)$ whose radiative stellar homology gives $L\propto {\mu }^7$, put $s=3+5X$. The luminosity is $L=L_0(s_0/s{)}^7$ and fuel conservation gives $M{E}_H\dot{X}=-L$. Hence $d(s^8)/dt=-40L_0{s}_0^7/(M{E}_H)$, producing the displayed feedback law. This applies both to CNO homology with density-dependent inverse-cubic opacity and to radiative homology with fifth-power hydrogen burning and inverse-cubic opacity, at fixed metal mass fraction in the latter. Its physical domain ends at $X=0$, before the formal denominator vanishes. The idealization holds the coefficient normalizations fixed while stellar composition changes through the mean molecular weight and fuel abundance.