For blackbody star and planet observed at one common wavelength sufficiently long for the Rayleigh-Jeans law to hold for both,This follows by substituting $B_{\lambda }=2c{k}_{B}T/{\lambda }^4$ in the thermal eclipse depth. A radiative flux ratio alone does not determine the radius without a planetary brightness temperature and a stellar temperature. Spectral peaks interpreted with Wien's displacement law must refer to $F_{\lambda }$, not $F_{\nu }$ or $\lambda F_{\lambda }$.

For blackbodies with $T_p<T_s$, the contrast $r_{\lambda }=(R_p/R_s{)}^2(e^{x_s}-1)/(e^{x_p}-1)$ increases with wavelength, where $x_i=hc/(\lambda k_B{T}_i)$. Indeed $d\log r/d\log \lambda =G(x_p)-G(x_s)>0$, with $G(x)=x/(1-e^{-x})$ increasing for $x>0$ because $1-(1+x)e^{-x}>0$. The contrast approaches $(R_p/R_s{)}^2{T}_p/T_s$ in the Rayleigh-Jeans law limit. This behavior differs from the peaked absolute blackbody flux.

The troposphere is the lowest main layer of the terrestrial atmosphere, with weather and generally decreasing temperature with altitude. Local inversions and moist convection complicate a single lapse-rate description.

The terrestrial stratosphere lies above the troposphere and has an upward-warming region associated with ultraviolet heating by the ozone layer.

The mesosphere lies above the stratosphere and below the thermosphere. Its typical terrestrial temperature decreases upward.

The thermosphere is a high atmospheric region where absorption of energetic stellar radiation can produce an upward-rising kinetic temperature. Its very low density distinguishes kinetic temperature from a large thermal energy content.

If $L,L^{\prime }$ are field extensions of $K$, choose a $K$-basis of $L$ containing $1$. After tensoring with $L^{\prime }$, that basis expresses $L{\otimes }_K{L}^{\prime }$ as a nonzero direct sum of copies of $L^{\prime }$, and $1\otimes 1$ is nonzero. A prime ideal of this nonzero ring supplies a point used in the point-lifting property of a scheme fibre product.

Interchanging homogeneous objects of degrees $p$ and $q$ introduces $(-1{)}^{pq}$. In particular, moving a degree-minus-one differential past a degree-$p$ factor introduces $(-1{)}^p$. This convention produces the differential on the tensor product of chain complexes.

Insert one factor difference at a time: the $j$th term is $U_m\cdots U_{j+1}(U_j-V_j)V_{j-1}\cdots V_1$. Multiplication by the surrounding unitary operators preserves the spectral norm, so the triangle inequality gives the displayed dimension-independent bound. No commutation assumption is required.

For normal displacement $ϵb\sin (k(x-t))$, the first-order decaying streamfunction is $b(1+ky)e^{-ky}\sin (k(x-t))$. It obeys zero tangential material velocity and normal velocity $-kb\cos (k(x-t))$ on the reference boundary. Its second-order mean swimming coefficient in the convention of far-field velocity $U{\mathbf{e}}_x$ is $k^2{b}^2/2$.

For tangential displacement $ϵa\sin (k(x-t))$, the first-order decaying streamfunction is $-kay{e}^{-ky}\cos (k(x-t))$. It obeys tangential material velocity $-ka\cos (k(x-t))$ and zero normal velocity on the reference boundary. Its second-order mean swimming coefficient is $-k^2{a}^2/2$.

A transversely waving Taylor swimming sheet can have a Navier slip boundary condition relating surface tangential velocity to shear and a slip length $\gamma$. The small-amplitude parameters are $ϵ=k{y}_0$ and $\delta =k\gamma$. The first-order velocity is unchanged by slip, but evaluation of shear at the displaced surface changes the second-order mean velocity.

A Peano zero through order $r$ means that the function admits the zero Taylor polynomial with remainder $o(|t{|}^r)$ at $x_0$. This does not assert the existence of ordinary iterated derivatives on a neighborhood. For example, multiplying $t^{r+1}$ by a bounded continuous function with dense points of nondifferentiability preserves this estimate while destroying neighborhood differentiability.

Its fibre over a point of quaternionic projective space is the quaternionic line represented by that point. As a real vector bundle it has rank four, and its unit sphere bundle is $S^{4n+3}\to {\mathbb{HP}}^n$. Right multiplication by a chosen imaginary unit equips it with complex rank two. Under the complex inclusion into quaternionic projective space, it restricts to $L\oplus \overline{L}$, where $L$ is the complex tautological bundle. Thus its Euler class restricts to $-c_1(L^{\ast }{)}^2$.

Over complex projective space, its fibre at a line $\ell \subseteq {\mathbb{C}}^{n+1}$ is $\ell$ itself. It is a holomorphic line bundle contained in the trivial bundle. Its dual is the hyperplane line bundle.

On a finite product of standard probability spaces, for any measurable event $A$ of positive probability,Consequently $\mathbb{P}(A)\mathbb{P}(d_T(X,A)\ge s)\le e^{-s^2/4}$. Here $d_T$ is the Talagrand convex distance. The distance-event conclusion can be read with outer probability when necessary.

A symplectic form on a $2n$-dimensional smooth manifold determines the orientation in which the nowhere-zero volume form ${\omega }^n/n!$ is positive. Consequently every symplectic manifold is orientable. In dimension two this forbids a symplectic form on the Möbius strip.

For a Hamiltonian group action, let $c$ be a regular value of the moment map fixed by the coadjoint action, and assume the action on ${\mu }^{-1}(c)$ is free and proper. The quotient is a symplectic manifold of dimension $\dim M-2\dim G$. Its unique reduced symplectic form is characterized by ${\pi }^{\ast }{\omega }_c={\iota }^{\ast }\omega$, where $\iota$ includes the level set and $\pi$ is the quotient projection. The kernel of the restricted two-form consists precisely of the orbit directions. More general noncentral levels use their stabilizer group; the stated central-level version suffices for circle group reduction.

The assertion that there is no Suslin line, equivalently that every complete dense linear order without endpoints satisfying the interval countable chain condition is separable. A Suslin tree gives a counterexample.

The orthogonal projector onto the tangent plane of a surface with unit normal vector $\mathbf{n}$. It removes the normal component of a vector and defines the surface gradient ${\nabla }_s=P\nabla$.