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For a binomial distribution with $p_n\to 0$ and $n{p}_n\to \infty$, the Bennett inequality at $C$ standard deviations tends to $2e^{-C^2/2}$, whereas the Hoeffding inequality tends to two. The raw Bennett inequality bound is eventually smaller for any fixed $C>0$; after clipping at one it is nontrivial in the limit only when $C>\sqrt{2\log 2}$.

If $a\ge 2$ is an integer with irrational ${\log }_{10}a$, then the leading digit $d$ of $a^n$ has limiting frequency ${\log }_{10}(1+1/d)$. Indeed the logarithmic fractional parts are the orbit of zero under an irrational rotation of the circle, so everywhere interval frequency under an irrational rotation applies. Thus for $a=2$, digit seven has frequency $(\log 8-\log 7)/\log 10$.

The equation $f_{\bar{z}}=\mu f_z$ prescribing the Beltrami coefficient of a quasiconformal map. Two homeomorphic solutions on the same source differ by a biholomorphism between their targets.

For complex $C(K)$ on a compact Hausdorff space, the extreme points of its dual unit ball are precisely unimodular multiples of Dirac measures. A measure whose variation measure has mass on two disjoint sets splits as a nontrivial convex combination of normalized restrictions and is not extreme. This description is the key to the Banach–Stone theorem.

A steady top-hat plume model supplied with kinematic buoyancy flux $f_0>0$ but zero source volume flux and momentum flux hasThe Batchelor entrainment hypothesis and momentum balance yield these constants. Its Froude number satisfies $w^2/(g^{\prime }b)=5/(8\alpha )$. The Boussinesq approximation requires $g^{\prime }/g\ll 1$ and therefore fails at the ideal point source itself.

For a stationary unit-variance normal distribution autoregression with coefficient $a\in (-1,1)$, its vector of length $T\ge 2$ has precision matrix $J$ that is tridiagonal: the two endpoint diagonal entries are $1/(1-a^2)$, interior entries are $(1+a^2)/(1-a^2)$, and adjacent off-diagonal entries are $-a/(1-a^2)$. An independent observation $r=\lambda Z+\eta$, with covariance $vI$, updates precision and information vector toThe conditional law is multivariate normal distribution $N(Q^{-1}h,Q^{-1})$.

For $X_t-\mu =\varphi (X_{t-1}-\mu )+{\epsilon }_t$, the autocovariance equation is $\gamma (1)=\varphi \gamma (0)$. Replacing both autocovariances by sample sums divided by the same $n$ yields ${\hat{\varphi }}_{YW}=\hat{\gamma }(1)/\hat{\gamma }(0)$, the lag-one sample autocorrelation function.

For $|\varphi |<1$, independent $N(0,{\sigma }^2)$ innovations and stationary initial variance ${\sigma }^2/(1-{\varphi }^2)$, the full likelihood function isConditioning on $X_1$ gives a different likelihood.

For a stationary autoregressive process of order one with mean $\mu$, autoregressive coefficient $\varphi$ and innovation variance ${\sigma }^2$, an interior observation conditional on its neighbors hasThe Markov property makes the same law valid when all other observations are also conditioned on.

For a known zero mean, ${\hat{\gamma }}_n(h)=n^{-1}{\sum }_{t=1}^{n-h}{Y}_t{Y}_{t+h}$ for $0\le h<n$. Weakly stationary process assumptions give expectation $(1-h/n)\gamma (h)$. Dividing by $n-h$ instead gives an unbiased estimate at each individual lag, although the resulting collection need not form a positive semidefinite covariance sequence.

If a generic filter $G$ for an atomless forcing order belonged to its ground model, then the complement $\mathbb{P}\setminus G$ would be a dense ground-model set. A condition outside $G$ is already there; a condition in $G$ has two incompatible strengthenings, at least one outside its directed filter. Genericity would then require meeting the complement, a contradiction.

A non-atomic measure contains no atom of a measure. For a finite Borel measure on a Polish space, this is equivalent to giving every singleton mass zero. For a probability measure on $\mathbb{R}$, it is also equivalent to continuity of its cumulative distribution function.

For an isothermal thin atmosphere with number density $n(z)=n_0{e}^{-z/H}$ and extinction cross-section ${\sigma }_{\lambda }$, a tangent ray near radius $R_p+z$ has height $z+{\ell }^2/(2R_p)$ at distance $\ell$ along the ray. The resulting Gaussian integral gives slant optical depthThe effective transit height is near ${\tau }_{\lambda }\sim 1$, so $z(\lambda )=H\log {\sigma }_{\lambda }+\mathrm{constant}$ if composition is constant. This explains the scattering slope of a transmission spectrum.

A sufficiently cold layer condenses a volatile or refractory carrier and limits its transport to higher layers. Water depletion above the terrestrial tropopause and titanium-bearing condensates in some hot Jupiter models illustrate this process.

For an increasing filtration of a module $M_n$ compatible with a filtered algebra $A$, set $\mathrm{gr}M={⨁}_n{M}_n/M_{n-1}$. Multiplication on the quotients makes it a graded module over $\mathrm{gr}A$. Its cumulative homogeneous dimensions equal $\dim M_n$ for a finite-dimensional exhaustive nonnegative filtration of a module.

An odd map between spheres is a continuous map $f:S^n\to S^m$ satisfying $f(-x)=-f(x)$. Thus it is equivariant for the two antipodal maps. It descends to a continuous map of Real projective spaces.

A nonempty fixed-point set of an anti-symplectic involution is a Lagrangian submanifold. Local linearization identifies its tangent space with the $+1$ eigenspace of the derivative.

An exoplanet cloud deck divides the projected atmospheric annulus into cloudy and clear radial zones. Their contributions to the exoplanet transmission spectrum add as projected areas multiplied by $1-e^{-{\tau }_{\lambda }}$ for each zone's slant optical depth.

A slit of width $x$ at the focus of a collimator subtends $x/f_{\mathrm{coll}}$. Its projected width is $x{f}_{\mathrm{cam}}\cos \alpha /(f_{\mathrm{coll}}\cos \beta )$. Dividing this width by the grating dispersion gives $\Delta \lambda =xd\cos \alpha /(m{f}_{\mathrm{coll}})$. Thus the spectral resolving power is $R=m\lambda f_{\mathrm{coll}}/(xd\cos \alpha )$. Omitting the anamorphic factor gives a denominator $\cos \beta$ instead; these agree in the Littrow configuration.