Gravity darkening is the variation of surface brightness and effective temperature with effective gravity over a rotating or tidally distorted star. Lower equatorial gravity generally corresponds to cooler equatorial regions. The radiative gravity-darkening law applies to the idealized uniformly rotating, radiatively transporting, composition-uniform case; its exponent is not universal for every stellar envelope.

On a regular equipotential surface, $f,f^{\prime },\rho$ and ${\nabla }^2\varphi$ are constant, but $|\nabla \varphi {|}^2$ generally varies. Consequently a flux $\mathbf{F}=f(\varphi )\nabla \varphi$ cannot ordinarily satisfy local radiative equilibrium $\nabla \cdot \mathbf{F}=\rho ϵ$ at all latitudes. Special cases such as $f^{\prime }=0$ or spherical symmetry evade this obstruction. Stellar meridional circulation supplies heat advection in the leading hydrostatic treatment.

Slow circulation in planes containing the rotation axis can transport entropy through a rotating star while the leading mechanical balance remains nearly hydrostatic equilibrium. Its steady mass flux satisfies the continuity equation. It is distinct from assuming an exactly motionless radiative interior, and from turbulent convection.

The divergence theorem and Poisson equation give ${\int }_S\nabla \varphi \cdot d\mathbf{S}=4\pi Gm-2{\Omega }^{2}V$ on a bounding equipotential surface. Equipotential luminosity conservation with stellar circulation then fixes $f=L/(4\pi Gm-2{\Omega }^{2}V)$. Combining $dT/d\varphi =-f/\chi$, $dP/d\varphi =-\rho$ and radiative conductivity gives the displayed logarithmic temperature-pressure gradient. Both appearances of mass mean the enclosed mass $m$; only on the outer boundary can both be replaced by the total mass.

This approximation represents the gravitational potential by a central point mass while retaining the centrifugal potential. A surface through polar radius $R_p$ satisfies $R_p/r+q(r/R_p{)}^2{\sin }^2\theta /2=1$, where $q={\Omega }^2{R}_p^3/(GM)$. The inner closed surface is regular below $q=8/27$; at the limiting value its equatorial radius is $3R_p/2$ and equatorial effective gravity vanishes. The approximation is useful for illustrating oblateness and gravity darkening, rather than modeling the density distribution of every rotating star.

Assume an exponentially decreasing temperature profile while imposing wall heat flux, impermeability and the exact integrated heat budget. For $f=C(1-e^{-\eta /d})$, the wall condition gives $C=d^2$ and $\int (f^{\prime }{)}^{2}d\eta =1$ gives $d^3=2$. This yields the displayed integral approximation and discharge $\int wdy\simeq 2^{2/3}\kappa z/\delta$. It satisfies the boundary conditions and exact flux budget, but not the differential equation pointwise; its mass-flux coefficient is approximate.

Schauder estimates control Hölder norms of derivatives of a solution to an elliptic differential equation. For a second-order uniformly elliptic operator $L=a^{ij}{D}_{ij}+b^i{D}_i+c$ with $C^{0,\alpha }$ coefficients, where $0<\alpha <1$, interior estimates gain two derivatives over a $C^{0,\alpha }$ forcing, while estimates up to a boundary also require regular boundary data.

On a bounded $C^{2,\alpha }$ domain, a standard global Dirichlet problem estimate isHere $g$ denotes a $C^{2,\alpha }$ extension of the prescribed boundary data. For homogeneous data choose $g=0$. The constant depends on ellipticity, domain geometry, coefficient bounds and $\alpha$; it is independent of the particular solution. The $C^0$ term allows a nontrivial homogeneous kernel and can be removed only with an additional uniqueness estimate.^[ref]

The boundary term is essential. On the unit disk, $u_n(r,\theta )=r^n\cos (n\theta )$ is harmonic with supremum one but boundary second tangential derivative of size $n^2$. Its global $C^{2,\alpha }$ norm cannot be bounded using only zero forcing and that fixed supremum. An interior Schauder estimate remains valid on a smaller disk, whose distance from the boundary controls the constant. This separates interior regularity from control of rapidly oscillating boundary data.

The quadratic form of a matrix only sees its symmetric part. Thus $\lambda |\xi {|}^2\le {\xi }^{T}A\xi \le \Lambda |\xi {|}^2$ need not bound the skew part of a measurable coefficient matrix. For $A=\left(\begin{array}{cc}1& k\\ -k& 1\end{array}\right)$ the quadratic form is exactly $|\xi {|}^2$, regardless of $k$. On a square, take $k(x)=1/|x_1|$ off the line $x_1=0$, with a finite arbitrary value on that line. Two compactly supported smooth functions that equal $x_2$ and $x_1$ near the origin have $(A\nabla u)\cdot \nabla v=k$ there; the purported energy pairing is not integrable. A Lax-Milgram theorem proof therefore requires bounded full coefficients or another hypothesis that makes the bilinear form bounded. Symmetry supplies the missing matrix bound from the quadratic upper bound.

Exhaust a simply connected hyperbolic Riemann surface in potential theory by compact bordered Jordan domains $R_n$. Their Green functions on a Riemann surface $G_n(\cdot ,p)$ have logarithmic poles and zero boundary values. Exponentiating their harmonic conjugates produces proper degree-one maps $f_n:R_n\to \mathbb{D}$ with $f_n(p)=0$. If $G_n=-\log |\zeta |+c_n+o(1)$, normalize $f_n^{\prime }(p)=e^{-c_n}>0$. Domain monotonicity and a global positive Green function bound $c_n$ increasingly from above. Thus Montel theorem gives a nonconstant limit $F$, which is injective by the locally uniform limit of univalent functions theorem. The Schwarz lemma makes $F^{\prime }(p)$ extremal among normalized disk-valued maps. If $F$ omitted $a\ne 0$, taking a global holomorphic square root of a disk automorphism applied to $F$, then renormalizing at $p$, would multiply its derivative by $(1+|a|)/(2\sqrt{|a|})>1$. Thus $F$ is onto and gives the desired biholomorphism.

The level-two principal congruence subgroup $\Gamma (2)\subset {\mathrm{PSL}}_2(\mathbb{R})$ consists of integer determinant-one matrices congruent to the identity modulo two, modulo the scalar sign. It acts freely and properly discontinuously on the complex upper half-plane. An ideal quadrilateral with vertices $-1,0,1,\infty$ has its paired sides identified by $\tau \mapsto \tau +2$ and $\tau \mapsto \tau /(1-2\tau )$. The quotient has genus zero and three punctures, so it is conformally the sphere with three points removed. The invariant metric $|d\tau |/\mathrm{Im}\tau$ descends to its curvature-minus-one hyperbolic metric.

// Destination: analysis.bigb

A generating family consists of sets of size at most $r$ whose $r$-element supersets are exactly the prescribed uniform set family. Small-support generators express dependence on a bounded prefix rather than on the full ground set. Taking the entire family itself as generators always works, but extremal intersection problems admit much smaller supporting intervals.

Every nonzero Noetherian module contains a uniform module: otherwise repeatedly split a nonuniform nonzero submodule into two disjoint nonzero submodules, retaining one and collecting the other, to build an infinite direct sum. Repeatedly adding a uniform submodule disjoint from the current finite sum must likewise stop. The final finite sum is an essential submodule, since any disjoint nonzero submodule would allow another addition.

The uniform dimension of a module is the supremum of the numbers of nonzero submodules that can occur in an internal direct sum. Finite uniform dimension is equivalent to having no infinite internal direct sum of nonzero submodules. A finite direct sum of uniform modules contained as an essential submodule gives that number as the dimension. The finite uniform decomposition of a Noetherian module therefore gives finite uniform dimension.

On a finite time horizon, an adapted process is of class D if its values at all stopping times in that horizon form a uniformly integrable family. Deterministic boundedness implies this property. The stopping-time uniformity is stronger than having an integrable value at each fixed time.

An $r$-uniform hypergraph has the extension property when, for every finite vertex set $S$ and every subfamily $\mathcal{A}\subseteq \left(\genfrac{}{}{0ex}{}{S}{r-1}\right)$, there is a vertex $v\notin S$ such that $F\cup \{v\}$ is a hyperedge exactly for $F\in \mathcal{A}$, among all $F\in \left(\genfrac{}{}{0ex}{}{S}{r-1}\right)$. Every prescribed pattern has infinitely many witnesses: enlarge $S$ by any finite set of witnesses already found and prescribe the original pattern on its old $(r-1)$-subsets, with arbitrary choices on the new ones. For $r=2$, this says that any finite adjacency and nonadjacency prescription has a new realizing vertex.

Choose each $r$-element subset of the natural numbers as a hyperedge independently with probability $p$, where $0<p<1$. This defines a countable random uniform hypergraph. For any finite pattern in the hypergraph extension property, an outside vertex realizes it with probability $q=p^{|\mathcal{A}|}(1-p{)}^{|(\genfrac{}{}{0ex}{}{S}{r-1})\setminus \mathcal{A}|}>0$. Distinct outside vertices use disjoint edge decisions, so the probability that the first $N$ candidates all fail is $(1-q{)}^N\to 0$. Taking a countable intersection over all finite patterns proves the extension property almost surely. The bounds $0<p<1$ are necessary for all patterns to have positive probability.

The link at a fixed vertex $v$ of an $r$-uniform hypergraph has vertex set $V(H)\setminus \{v\}$ and the displayed $(r-1)$-element hyperedges. For a universal countable uniform hypergraph, its link has the hypergraph extension property: include $v$ in the finite set of vertices and prescribe only those $r$-edges containing $v$, completing the remaining prescription arbitrarily. Thus the link is a universal countable $(r-1)$-uniform hypergraph. In particular the link at a fixed vertex in the three-uniform case is the Rado graph. This operation retains a specified vertex; deleting the maximum from every edge is a different operation.

For a tripartite three-uniform hypergraph on nonempty finite parts $X,Y,Z$, let $h:X\times Y\times Z\to \{0,1\}$ be its hypergraph edge indicator, let $p=\mathbb{E}h$ and put $g=h-p$. In the eighth-power convention, $\alpha$-quasirandomness meansAll averages are uniform and independent, with repeated sampled vertices of a hypergraph allowed. This expression is nonnegative because it equals ${\mathbb{E}}_{x_0,x_1,y_0,y_1}({\mathbb{E}}_z{\prod }_{i,j}g(x_i,y_j,z){)}^2$. It measures correlations across the eight faces of a tripartite octahedral configuration. If instead the norm itself is used as the quasirandomness parameter, the parameter in this definition is its eighth power.

A property of uniform hypergraphs is hereditary if every induced subhypergraph of a hypergraph with that property also has the property. It is understood to be invariant under relabelling the vertex set. For example, excluding a specified induced hypergraph is hereditary. This permits counting restrictions to smaller vertex sets using the same counting function.

A uniform Newtonian gravitational field inside one spherical void can be cancelled at a point by the repulsive field representing another removed sphere. The latter field has a Jacobian matrix proportional to $I-3\hat{r}{\hat{r}}^T$: its axial eigenvalue is negative and both transverse eigenvalues positive. The equilibrium is restoring along the axis but unstable transversely. A zero force is thus not enough for stable rest.