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In a regular scalar model under the null, Taylor expansion about the MLE givesA uniform law of large numbers for the observed information makes
$-{\ell }_n^{\prime \prime }(\tilde{\theta })/n$ and the Fisher information used in the Wald statistic converge to the same positive limit. Their ratio therefore converges in probability to one.

For a regular $p$-parameter model, the Wald statistic for a candidate $\theta$ isUnder the true parameter it converges in distribution to ${\chi }_p^2$.

Forthe amplification polynomial isAll modes have unit-modulus roots only when $\rho =1$ and $0\le \mu \le 1$.

For the centered spatial second difference, the Crank-Nicolson diffusion scheme has amplification factorIt is stable for every $\mu \ge 0$.

For a schemethe Fourier ansatz gives the amplification polynomialIts roots are the amplification factors of that Fourier mode.

For a one-step translation-invariant recurrence, the amplification factor is the multiplier satisfyingThe Parseval identity converts the pointwise bound $|H|\le 1$ into non-growth of the discrete $2$-norm.

A decretion disk is fed near its inner edge and transports mass outward. Rapidly rotating Be stars can sustain gaseous decretion disks whose outer material is removed by a companion, radiation, or a wind.

For vertically integrated kinematic viscosity $\bar{\nu }$, the viscous torque across a cylinder of radius $r$ is

A $k$-connected graph has a cycle of length at least $\min \{|G|,2k\}$.

A closed vector subspace of a topological vector space is both a vector subspace and a closed set. The kernel of a continuous linear map is a closed vector subspace.

The cross product $a\times b$ of two three-dimensional vectors is perpendicular to both, has magnitude $|a||b|\sin \theta$, and is oriented by the right-hand rule.

Intervals together with their label complements shatter three ordered points, but cannot realize an alternating labeling on four, so their VC dimension is three.

Indicators of real intervals have VC dimension two: two ordered points can be labeled arbitrarily, while an interval containing the two outer points of a triple must contain the middle point.

The class of lower orthants ${\prod }_{j=1}^p(-\infty ,a_j]$ in ${\mathbb{R}}^p$ has VC dimension at most $p$. Among any $p+1$ points, choose a coordinate maximizer for each of the $p$ coordinates. A remaining point cannot be excluded while all those maximizers are included.

If $VC(H)=D<\infty$, then $s(H,n)\le {\sum }_{j=0}^D\left(\genfrac{}{}{0ex}{}{n}{j}\right)$.

The first variation of a functional $L$ at $y$ in the direction $\eta$ is

The covariance measures the average product of two centered deviations. For real square-integrable random variables $X$ and $Y$,Deviations with the same sign contribute positively; deviations with opposite signs contribute negatively. Their magnitudes and probabilities determine the net result. Taking $Y=X$ recovers the variance. The definitions and algebraic properties are presented in Orloff and Bloom, MIT 18.05, Covariance and Correlation, sections 2–2.1.

For a worked example, let $X,Z$ be independent random variables, each equally likely to be $-1$, $0$ or $1$, and put $Y=2X+Z$. All nine pairs $(X,Z)$ have probability $1/9$. The expected values are zero, while linearity of covariance and the vanishing covariance of independent random variables giveThe shared $X$ term makes large $X$ accompany large $Y$ on average. Replacing $2X$ by $-2X$ reverses the covariance to $-4/3$ without changing either variance.

The covariance is useful whenever deviations are added or subtracted. Expanding the square givesFor the worked model, these are $20/3$ and $4/3$. Positive covariance increases variation in the sum while reducing variation in the difference: the same shared fluctuation reinforces one combination and partly cancels in the other. The identity applies beyond independent random variables.

The size of a covariance depends on the measurement scale. For constants $a,b,c,d$, linearity of covariance gives $\mathrm{Cov}(aX+b,cY+d)=ac\mathrm{Cov}(X,Y)$. Adding offsets has no effect; rescaling either measurement rescales the covariance. The Pearson correlation coefficient divides by both standard deviations to describe the standardized linear relationship.

The standard deviation is the nonnegative square root ${\sigma }_X=\sqrt{\mathrm{Var}(X)}$ of the variance.

For independent square-integrable random variables,

For grammars $G=(\Sigma ,V,P,S)$ and $G^{\prime }=(\Sigma ,V^{\prime },P^{\prime },S^{\prime })$ with disjoint variable sets, introduce a new variable $S_0$ and setIf both input grammars are variable based, then $L(H)=L(G)L(G^{\prime })$.