An astrophysical cooling function packages radiative losses of a plasma into a temperature-dependent coefficient. In the hydrogen-density-squared convention, the volume energy loss is $n_H^2\Lambda (T)$, so the coefficient has units $\mathrm{erg}\mathrm{c}{\mathrm{m}}^3{\mathrm{s}}^{-1}$. Its dependence on ionization state, chemical abundances and radiation field must be specified; a low-density collisional-equilibrium curve is not universal.

This model solves a vertical atmospheric pressure-temperature profile consistent with heating, radiative transfer and a convective adjustment where the radiative gradient exceeds the adiabatic temperature gradient. Irradiation, intrinsic flux, gravity and opacity determine the radiative-convective boundary. A one-dimensional model describes a column or average rather than resolving all atmospheric winds.

A planet can block stellar radiation during part of a circumplanetary orbit. Then the full-orbit constant-flux averages for radiation pressure and Poynting–Robertson drag require modification. The detailed shadow duty cycle depends on the orbital inclination, stellar angular extent and dust radius.

Choose a radial mollifier $\rho \in C_c^{\infty }$ with integral one and support in the unit ball, and a radial cutoff function $\chi \in C_c^{\infty }$ equal to one there. For a radial tempered distribution $T$, the functionsare radial Schwartz functions. Smoothness comes from convolution of a tempered distribution with a Schwartz function, and compact support comes from the cutoff. Pairing with $\phi$ gives $⟨f_j,\phi ⟩=⟨T,{\check{\rho }}_{1/j}\ast (\chi (\cdot /j)\phi )⟩$. With $q_m(\phi )={\max }_{|\beta |\le m}{\sup }_x(1+|x|{)}^m|{\partial }^{\beta }\phi (x)|$, the cutoff tail and the mean value theorem giveThe continuity estimate for $T$ therefore proves $f_j\to T$ even in the strong dual topology. Inserting a cutoff is essential because convolution alone need not give rapid decay.

For ideal axisymmetric radial displacements in a Keplerian shearing sheet, the hydrodynamic squared oscillation frequency is $N_r^2+{\kappa }^2$, with radial epicyclic frequency $\kappa =\Omega$. Exponential instability occurs when $N_r^2+{\kappa }^2<0$. This is the radial special case of rotating-fluid convective stability, rather than the complete set of three-dimensional Solberg–Høiland criteria.

On the Boolean hypercube $\{-1,1{\}}^d$ with uniform measure, each coordinate is a Rademacher random variable. Products of these sign coordinate functions are Walsh characters. This use of Bernoulli function is distinct from the fluid-mechanical Bernoulli function.

A Rademacher sum is a linear combination of independent Rademacher random variables, with scalar or normed-space-valued coefficients. Its moments are controlled by the Khintchine inequality in the scalar case and the Kahane-Khintchine inequality in a general normed vector space.

A module retraction onto a submodule $U\stackrel{i}{\to }X$ is a module homomorphism $r:X\to U$ with $ri=I_U$. It gives $X=i(U)\oplus \ker r$: write $x=i(r(x))+(x-i(r(x)))$. Thus a nonzero proper submodule admitting a retraction contradicts indecomposability. Extending a projection from a larger submodule can produce such a retraction through the long exact sequence of Ext groups.

Treatment coding sets one reference level in a regression factor to zero and represents each other level by an indicator column. With two interacting factors, a main-effect coefficient compares levels at the other factor's reference level; an interaction coefficient is a difference of these contrasts. A joint nested-model F-test can be significant even when some individual coefficient tests are not.

If an array specified in each orthonormal basis has invariant Frobenius inner product with every Cartesian second-rank tensor, it transforms as such a tensor. For a component change $R$, invariance says $(R^{T}A\prime R-A):B=0$ for every matrix $B$. Nondegeneracy of the Frobenius inner product forces the difference to vanish. Testing every tensor, rather than a single selected tensor, is essential.

If contraction of an array with every symmetric second-rank tensor is a vector in every orthonormal frame, its part symmetric in the contracted slots obeys the rank-three tensor transformation law. The difference between its proposed transformation and actual components is symmetric and contracts to zero against every symmetric matrix, so it vanishes. The antisymmetric part is invisible, by the blindness of symmetric contraction tests to antisymmetric arrays, and need not be tensorial.

The quotient norm measures the smallest norm among representatives of a coset. Its infimum need not be attained. Nevertheless, for every $\epsilon >0$, an element $z\in X/Y$ has a representative $v\in X$ with $\Vert v\Vert <\Vert z\Vert +\epsilon$. The quotient map is contractive.

If a Banach space $X$ has separable dual and $q:X\to X/Y$ is a quotient by a closed subspace, every weakly null sequence in the quotient's closed unit ball has a subsequence with approximate weakly null lifts of norm at most $3$. Choose representatives of norm below $3/2$, pass to a common bidual weak-star limit, and subtract convex blocks whose quotient images tend to zero in norm. Convex-block cancellation of a weak-star limit gives weak nullity and the triangle inequality gives the bound $3$.

The cell structure gives one integral cohomology generator in each degree divisible by four through $4n$. The Gysin sequence of a sphere bundle for the quaternionic tautological line bundle makes multiplication by its Euler class an isomorphism between successive such degrees. Choose $u$ to be the negative of that Euler class; its powers generate the displayed cohomology ring.

Coordinate inclusion of complex vectors sends a complex line to its quaternionic span. If two such vectors span the same quaternionic line, a nonzero complex coordinate forces their scalar ratio to be complex, so this map is injective. The pulled-back quaternionic tautological line bundle splits as $L\oplus \overline{L}$ with complex first Chern classes $-x,x$. Its top Chern class, and thus Euler class, is $-x^2$. With $u$ the negative Euler generator, $i^{\ast }u=x^2$ and $i^{\ast }(u^k)=x^{2k}$. The map on infinite projective spaces is injective on cohomology; a finite inclusion loses the degrees with $2k>n$.

For an equal-part biregular graph with part size $n$ and degree $\delta n$, let $M$ be its biadjacency matrix and $H=M-\delta J$, where $J$ is the all-ones matrix. The constant row and column sums make $H$ annihilate constants on both sides. Since $M{M}^{\mathsf{T}}={\delta }^{2}nJ+H{H}^{\mathsf{T}}$ and the two summands have zero products, their squared matrix trace splits as displayed. The bipartite four-cycle density counts the squared matrix trace of $M{M}^{\mathsf{T}}$ divided by $n^4$.

Let $H=M-\delta J$ be the centered biadjacency matrix of an equal-part biregular graph, and put $s=\Vert H{\Vert }_{\mathrm{op}}$. For subsets $A,B$ of the two parts, their edge discrepancy is $1_A^{\mathsf{T}}H{1}_B$. The Cauchy-Schwarz inequality proves the displayed bound. If the part size is $n$, centering the two indicator vectors improves it to $sn\sqrt{\alpha (1-\alpha )\beta (1-\beta )}$. The centered four-cycle identity for a biregular graph shows that $t_4(G)\le {\delta }^4(1+c^4)$ implies $s\le c\delta n$ for $c\ge 0$.

With basic zonal velocity $U(z)$, basic horizontal buoyancy gradient $b_y=-f_0{U}_z$, and $b^{\prime }=f_0{\psi }_z^{\prime }$, adiabatic buoyancy evolution at a flat impermeable wall givesA normal mode thus obeys $(U-c){\hat{\psi }}^{\prime }-U_z\hat{\psi }=0$ at the wall. This dynamical condition, rather than vanishing streamfunction, sets the frequency of an Eady edge wave.

On an integral scheme, a quasi-coherent sheaf is torsion-free when its module on every affine chart is a torsion-free module over that chart's integral domain. It embeds into its generic fibre. Therefore a torsion-free sheaf with zero generic fibre is zero; locally free sheaves and their subsheaves are torsion-free.

On an affine variety or affine scheme with coordinate ring $A$, the global section functor and $M\mapsto \tilde{M}$ are inverse equivalences between modules and quasi-coherent sheaves. On every principal open subset, sections are the appropriate localization of a module. Localization detects exactness at maximal ideals, so these functors are exact even for modules which are not finitely generated.