Write $f={\sum }_k{\beta }_{k,n}{b}_{k,n}$ in the Bernstein basis. Representing the same polynomial at degrees $n$ and $n+1$ gives the displayed convex combinations for internal indices; the endpoint coefficients are copied. Therefore the minimum coefficient cannot decrease under degree elevation. In the unnormalized basis $x^k(1-x{)}^{n-k}$, the elevated coefficients are $c_k^{\prime }=c_{k-1}+c_k$ with the corresponding endpoint convention, so nonnegativity certificates are preserved.

If a polynomial $f$ is strictly positive on $[0,1]$, take $g_n=B_n^{-1}f$. By inverse Bernstein approximation on a fixed-degree polynomial space, eventually $g_n>0$ on the interval. Thereforehas strictly positive coefficients. The converse implication to nonnegativity follows immediately because each basis term is nonnegative. Strict positivity is essential for the general existence result: a nonzero polynomial vanishing at an interior point cannot have a nonnegative-coefficient representation in this basis, whose individual terms are positive throughout the open interval.

Let $R$ be the number of rejections by the Benjamini-Hochberg procedure at level $\alpha$, and let $R_i$ be the rejection count after replacing $p_i$ by zero. Then $R_i\ge 1$ andIf $i$ was rejected, lowering its p-value does not change any ordered p-value beyond rank $R$, so $R_i=R$. Conversely, if $p_i\le \alpha R_i/m$, the other $R_i-1$ rejected values together with $p_i$ satisfy the original threshold; monotonicity gives $R=R_i$. For a valid true-null p-value independent of the others, conditioning on $R_i$ bounds the expectation of the right side by $\alpha /m$. Summing over true nulls proves false discovery rate at most $m_0\alpha /m$.

Suppose an unknown has prior distribution $\Pi$ and its conditional observation law $P_u$ has jointly measurable Radon-Nikodym derivative $L(u,m)$ relative to a fixed observation law $P_0$. If $L>0$ for $\Pi (du)P_0(dm)$-almost every pair, thendefines the posterior distribution for almost every observation. Indeed, $\int L(u,m)P_0(dm)=1$ for almost every $u$, so Tonelli theorem gives $\int Z(m)P_0(dm)=1$. Hence $Z$ is finite almost everywhere, and positivity of $L$ and Fubini's theorem give $Z>0$ almost everywhere. The data law is $Z{P}_0$, and integration of the proposed posterior against this data law recovers the joint law. This proves the conditional distribution property and also shows that the data law is an equivalent probability measure to $P_0$.

The nonoscillatory thermal branch of the radially stratified shearing sheet spectrum. Its decoupled small-buoyancy limit is a diffusing scalar perturbation.

A stationary axisymmetric magnetic field in ideal purely rotational flow requires constant angular velocity along the magnetic field lines of its poloidal magnetic field. Otherwise axisymmetric magnetic winding generates a time-dependent toroidal component.

For Keplerian rotation, initial $B_R\propto R^{-1}$ and thermal pressure $p\propto R^{-2}$, axisymmetric magnetic winding gives $B_{\varphi }=-(3/2)\Omega t{B}_R$. If the initial plasma beta is ${\beta }_p>1$, the formal equality of magnetic pressure and gas pressure occurs atThe magnetic-to-thermal pressure ratio decreases with radius, so magnetic pressure dominates inside this radius. In a disk with a finite radial range, an equality radius exists in the disk only when this formal radius lies within that range.

An atmospheric thermal inversion has outward-increasing temperature. In a semi-grey irradiated atmosphere, define downward thermal optical depth $\tau$, constant opacity ratio $\gamma ={\kappa }_{\mathrm{vis}}/{\kappa }_{\mathrm{IR}}>0$, irradiation temperature $T_{\mathrm{irr}}$ and internal effective temperature of a planet $T_{\mathrm{int}}$. In the isotropic-irradiation closure, write the incident mean irradiance as $f_r{\sigma }_{\mathrm{SB}}{T}_{\mathrm{irr}}^4$, with $f_r>0$ explicitly specifying the averaging convention. Common choices are $1/4$ for a whole-sphere average and $1/2$ for a dayside average.

The analytic radiative equilibrium profile in this closure isDifferentiation givesSince $\tau$ increases inward, an atmospheric thermal inversion occurs when this derivative is negative. Thus $\gamma >1$ is necessary in this closure; a nonzero intrinsic radiative flux also requires sufficiently strong irradiation. For $T_{\mathrm{int}}>0$, any inversion is restricted toprovided the logarithm is positive. The $T_{\mathrm{int}}=0$ limit gives the threshold $\gamma >1$. These coefficients depend on the angular closure; a collimated beam has an incidence-angle dependence.^[ref]

A symmetric triangle with direct calls and two-link overflow routes has Erlang fixed point approximation equation $B=E(\lambda [1+2B(1-B)],C)$. For $C=1000$ and $\lambda =950$, this has at least three solutions, certified by the intermediate value theorem using signs at $B=0,0.01,0.1,0.4$.

A simplified conservative model with a fully convective donor gives long-lived stable Roche-lobe overflow only for donor-to-accretor mass ratio $q<2/3$. This follows from the Roche-lobe radius response exponent and the donor's adiabatic stellar radius response exponent $-1/3$. The boundary is model-dependent; radiative donors, evolved core structure and mass or angular-momentum loss change it. Van Rensbergen and collaborators' Algol population study includes reported Algol mass ratios above $0.7$ and discusses observational uncertainties.

During binary mass-ratio reversal, an initially more massive donor star loses enough material that its companion becomes the more massive member. The stars retain their different prior nuclear evolution, resolving the Algol paradox. Conservative binary mass transfer shrinks the orbit before equal masses and widens it after reversal, because $a\propto (M_d{M}_a{)}^{-2}$ at fixed total mass and orbital angular momentum.

The Algol paradox is the apparently inconsistent evolutionary ordering in an Algol binary: the less massive star is more evolved, whereas higher-mass isolated main sequence stars normally exhaust core fuel sooner. Binary mass-ratio reversal resolves it by identifying the present donor as the initially more massive star.

The Alfvén frequency is the frequency of an Alfvén wave with wavevector $\mathbf{k}$ in a uniform magnetic field $\mathbf{B}$ and mass density $\rho$. For a vertical field and vertical wavenumber $k$, its square is ${\omega }_a^2=k^2{B}_z^2/({\mu }_0\rho )$. Its competition with orbital shear determines the ideal magnetorotational dispersion relation.

The Alfvén radius characterizes where a magnetized stellar wind crosses its Alfvén surface. For a roughly radial wind, $v_{\mathrm{wind}}\simeq B/\sqrt{{\mu }_0\rho }$ and ${\dot{M}}_w\simeq 4\pi R_A^2\rho v_{\mathrm{wind}}$. The wind's angular-momentum lever arm is of order $R_A$: its torque is $\dot{J}\sim -{\dot{M}}_w\Omega R_A^2$, up to geometry factors. The precise wind invariant uses the cylindrical radius of the surface crossing, rather than necessarily its spherical radius.

Named after George Biddell Airy, the standard Airy functions solve the Airy ordinary differential equation. Their Wronskian is $\mathrm{Ai}(Z){\mathrm{Bi}}^{\prime }(Z)-{\mathrm{Ai}}^{\prime }(Z)\mathrm{Bi}(Z)=1/\pi$, so they form a fundamental solution pair, including for complex $Z$.

For the Airy ordinary differential equation $y^{\prime \prime }=xy$, a power series obeys $a_2=0$ and $a_{m+3}=a_m/[(m+3)(m+2)]$. Setting its first two coefficients to $(1,0)$ or $(0,1)$ givesEmpty products are one. The ratio test proves convergence on the whole complex plane. Termwise differentiation verifies the equation, and the initial normalization gives Wronskian one. By the Abel identity, the Wronskian remains one everywhere, so these solutions span the whole solution space.

Connections ${\nabla }^1,{\nabla }^2$ on $T{M}_1,T{M}_2$ give a connection onby taking the direct sum of their pullback connections. In product coordinates its Christoffel symbols have the two factor blocks and zero mixed blocks. On arbitrary fields, the ordinary derivatives of their coefficients are still taken in both factors. On fields lifted separately from the factors it satisfies ${\nabla }_{Y_1+Y_2}(X_1+X_2)={\nabla }_{Y_1}^1{X}_1+{\nabla }_{Y_2}^2{X}_2$.

A SU(2) representation descends to the SO(3) group exactly when $-I$ acts trivially. In ${\mathrm{Sym}}^n({\mathbb{C}}^2)$ it acts by $(-1{)}^n$, so descent holds precisely for even $n$, or integer spin $j=n/2$. This is the inflation of a group representation criterion for the kernel $\{\pm I\}$. Half-integer-spin representations instead remain representations of the covering group.

Substitution of the adiabatic superbubble similarity solution into spherical mass conservation gives

If the similarity profiles account for all kinetic and internal energy inside an adiabatic strong shock, letThe injected energy $\dot{E}t$ equals $4\pi {\rho }_1{\dot{R}}^2{R}^{3}I$. Consequently conservation of energy fixesAn interior region not included in the profiles must have its energy added separately; a radiatively cooled shell changes this energy budget.