On the fixed finite-dimensional space of polynomials of degree at most $d$, uniform convergence of $B_n$ on each monomial and equivalence of norms in finite dimensions imply ${\epsilon }_n=\Vert B_n-I{\Vert }_{\mathrm{op}}\to 0$. A Neumann series then gives $\Vert B_n^{-1}-I{\Vert }_{\mathrm{op}}\le {\epsilon }_n/(1-{\epsilon }_n)$ for sufficiently large $n$. Consequently the displayed convergence holds. A strictly positive polynomial has positive minimum on the compact interval, so its inverse Bernstein approximants are also strictly positive for all sufficiently large $n$.

For $m\ge 1$ and $n\ge 2$, the Bernstein polynomial of the monomial $x^m$ satisfiesIt follows by applying $k\left(\genfrac{}{}{0ex}{}{n}{k}\right)=n\left(\genfrac{}{}{0ex}{}{n-1}{k-1}\right)$ and expanding $k=(k-1)+1$ with the binomial theorem. The coefficients sum to one; only the first tends to one as $n\to \infty$. Induction on $m$ proves uniform convergence of these monomial approximants, starting from $B_n(1)=1$.

The Bennett rate function has Taylor expansion $\varphi (u)=u^2/2-u^3/6+O(u^4)$ near zero. It is the optimized exponent in the Chernoff bound based on the moment-generating function estimate for the Bennett inequality.

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 pi-suppressed kinematic buoyancy flux $f_0=b^{2}w{g}^{\prime }>0$ but zero source volume flux and momentum flux hasThe full-cross-section buoyancy flux is $\pi f_0$. The 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.