Jeans escape is collisionless atmospheric escape by particles in the high-speed tail of a thermal distribution at the exobase. The Jeans escape parameter $\lambda =GMm/(k_{B}T{r}_e)$ controls the exponential suppression, and the Jeans escape flux is proportional to $(1+\lambda )e^{-\lambda }$. There is no exact positive-temperature threshold below which the tail vanishes. For $\lambda$ of order a few or smaller, a purely hydrostatic tail model can fail and hydrodynamic atmospheric escape becomes important.

Nonthermal atmospheric escape supplies particle energy through processes other than the local thermal distribution. Examples include dissociation producing fast atoms, ion pickup by a stellar-wind electric field, and sputtering caused by impacting energetic particles. Radiation pressure can also accelerate material in an extended exosphere.

For two uniform angular motions with $P_b<P_c$, an inner exoplanet transit at zero and next outer exoplanet transit at $\tau$, the next conjunction solves $(n_b-n_c)t+n_c\tau =2\pi$. Hence its longitude modulo $2\pi$ isFor an eccentric outer body this is only an approximation. Let $f_L,M_L$ be its line-of-sight anomalies and ${\Delta }_e(t)=f(t)-M(t)-(f_L-M_L)$. Exact true alignment instead requires $(n_b-n_c)t+n_c\tau -{\Delta }_e(t)=2\pi$. The correction follows directly from uniform mean anomaly and nonuniform true anomaly.

A commutative Artinian ring has finite composition length as a module over itself. It has finitely many maximal ideals, and its Jacobson radical $J$ is nilpotent. The Chinese remainder theorem makes $R/J$ a finite product of fields; each Artinian module $J^i/J^{i+1}$ has finite-dimensional components over those fields. Adding their lengths along the finite radical filtration proves the assertion, and in particular the ascending chain condition.

A nonzero element of a number field is totally positive when its image is positive under every real field embedding. In a totally real number field, the principal fractional ideals having a totally positive generator form the subgroup used to define the narrow ideal class group. Signs of different generators are related by the unit signature map.

An associated prime of an $R$-module $M$ is a prime ideal equal to the annihilator ${\mathrm{Ann}}_R(m)$ of some nonzero $m\in M$. The set of these primes is denoted ${\mathrm{Ass}}_R(M)$. Over a Noetherian ring, any maximal member among the annihilators of nonzero elements is prime: if $abm=0$ and $bm\ne 0$, maximality forces ${\mathrm{Ann}}_R(bm)={\mathrm{Ann}}_R(m)$, so $am=0$. Hence every nonzero module has an associated prime.

The effective dimension for a specified anisotropic rescaling is the total scaling weight of its coordinate measure. With $D_{\perp }$ directions of weight 1 and $D_{\parallel }$ directions of weight $\theta$, $d_{\mathrm{eff}}=D_{\perp }+\theta D_{\parallel }$. The uniaxial Gaussian Lifshitz point has $\theta =1/2$, so $d_{\mathrm{eff}}=D-1/2$. This is not a topological or Hausdorff dimension.

A closed conic enlargement of an oscillatory integral amplitude support is obtained by taking the closure of its frequency-direction projection in $X\times S^{k-1}$ and lifting that set radially to nonzero frequencies. It records limiting spatial locations and frequency directions, including limits reached only at arbitrarily large frequency. Ordinary support in $X\times {\mathbb{R}}^k$ need not record those limits. This distinction is necessary in singular support bounds for an oscillatory integral.

The alternating conjugacy class splitting criterion gives seven conjugacy classes in $A_6$. Their cycle types and sizes are $1^6:1$, $2^2{1}^2:45$, $31^3:40$, $3^2:40$, $42:90$, and two $51$ classes of size 72. Representatives of the split classes are a 5-cycle and its square. Their sizes sum to 360. A normal subgroup must be a union of these classes containing the identity, so these explicit sizes can prove simplicity using Lagrange's theorem.

A zenith blind spot is a region in which an altazimuth mount cannot follow a target because the required axis motion exceeds a drive constraint. A finite slew across an exact zenith crossing gives a lost-track interval; a full forbidden footprint additionally depends on the target trajectory, acceleration and the control strategy. It need not be circular.

In the exact constant-opacity, pure-radiation pressure diffusion model, dynamic viscosity is height-independent. Imposing an additional exact pointwise relation $\mu \propto p(z)$ forces constant pressure, incompatible with nonzero vertical hydrostatic equilibrium away from the midplane. The characteristic-pressure alpha disk prescription can still be used for order-of-magnitude scalings; it is not an exact simultaneous vertical solution.

An almost complex structure $J$ gives the $+i$ and $-i$ eigenbundles of its complex-linear extension. Their projections are $(I-iJ)/2$ and $(I+iJ)/2$. For an integrable structure the first is the holomorphic tangent bundle.

A rational parametrization expresses the coordinates of an algebraic curve as rational functions of a parameter, with a rational inverse on a dense open subset. It identifies the function field with $K(t)$ and the smooth projective normalization of an algebraic curve with the projective line. A nonconstant parametrizing map without an inverse is a weaker notion; even this cannot exist for an elliptic curve in characteristic zero.

For an algebra for a monad $(X,a)$, its action $a:TX\to X$ satisfies $a{\eta }_X=1_X$.

Every algebra for a monad $(A,a)$ is the coequalizer, in the Eilenberg-Moore category, of $F(TA)\rightrightarrows F(A)$ with underlying arrows ${\mu }_A$ and $Ta$, followed by $a:F(A)\to (A,a)$. The pair has common section $F{\eta }_A$. If an algebra morphism $h:F(A)\to (B,b)$ equalizes the pair, the unique induced algebra morphism is $h{\eta }_A:(A,a)\to (B,b)$. This explicit proof does not require general colimits of algebras.

Suppose nonnegative numbers satisfy $a_0{b}_0\le c_0{d}_0$, $a_1{b}_1\le c_1{d}_1$, and $a_0{b}_1,a_1{b}_0\le c_1{d}_0$. Set $x=a_0{b}_1$, $y=a_1{b}_0$, $M=c_1{d}_0$, $N=c_0{d}_1$. Then $x,y\le M$ and $xy\le MN$. For $M>0$, $(M-x)(M-y)\ge 0$ gives $x+y\le M+xy/M\le M+N$; for $M=0$, $x=y=0$. Adding the two diagonal bounds proves $(a_0+a_1)(b_0+b_1)\le (c_0+c_1)(d_0+d_1)$. This lets the four functions theorem sum out one Boolean coordinate while preserving its pointwise hypothesis.

In the ABL rule, probabilities depend on the full intermediate quantum measurement instrument. A binary Lüders rule quantum measurement preserves coherence within its unresolved complement, whereas a fully resolved projective measurement removes it. Summing fine-grained probabilities after quantum measurement does not generally reproduce the coarse-grained experiment. Different postselection success rates are part of the distinction.

Among retained payouts $g(S)$ with $0\le g(x)\le x$ and the same expected value as $\min (S,M)$, aggregate stop loss reinsurance minimizes the variance. Pointwise $(g(x)-M{)}^2\ge (\min (x,M)-M{)}^2$, and subtracting the identical squared distance of their common expected value from $M$ proves the claim. Equality requires equal payouts almost surely.

If $S$ has exponential distribution with expected value $\mu$, the tail integral formula for moments gives $\mathbb{E}\min (S,M)=\mu (1-e^{-m})$ and $\mathrm{Var}(\min (S,M))={\mu }^2(1-2m{e}^{-m}-e^{-2m})$, where $m=M/\mu$. At matching retained expected value, the excess variance under quota share reinsurance is $2{\mu }^2{e}^{-m}(m-1+e^{-m})\ge 0$.

In a vector space over a field $k$, an affine line is a translate of a one-dimensional vector subspace. Its direction is that vector subspace, so multiplying $v$ by a nonzero scalar preserves the direction. Over a finite field of size $q$ it contains exactly $q$ distinct points. This usage concerns affine subsets of vector spaces, rather than the affine line as an affine scheme.