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The Newlander-Nirenberg theorem says that a smooth almost complex structure is integrable if and only if its Nijenhuis tensor vanishes.

The alephs enumerate the infinite initial ordinals: ${\aleph }_0=\omega$, successor indices select the next initial ordinal, and limit indices select the least initial ordinal above all earlier values.

The $i$th $k$-ary projection returns its $i$th input:

The successor function is

The zero function has the constant valueDummy variables give a zero function of any finite arity.

Radial null curves comprise the ingoing family $v=\text{constant}$ and the outgoing familyOutside the horizon the outgoing family moves to larger $r$; inside it both future-directed families move to smaller $r$.

At a distribution with positive density $f$ at its median $m$, the median influence function is $\mathrm{sgn}(x-m)/(2f(m))$ away from $m$.

Gross-error sensitivity is the supremum of the absolute influence function over contamination points.

For nearly constant inflaton potential and constant $H_{\inf }$, the spatial Fourier transform obeys

For a nonempty set $X$ of words, the propertyis a nontrivial extensional property of unary partial computable functions: the nowhere-defined function fails it, while any total constant function with value in $X$ satisfies it. The Rice theorem therefore makes its index set undecidable.

For $G=C_{11}⋊C_5$ with faithful action, the abelianization $G/C_{11}\cong C_5$ supplies five linear characters. The two orbits of size five on the nontrivial characters of $C_{11}$ induce two irreducible characters of degree five. Their squared degrees satisfyso these seven characters are all the irreducible characters of $G$.

Indistinguishability is a right-invariant equivalence relation. The quotient has states $[q]$, transition $[q]\mapsto [\delta (q,a)]$, initial state $[q_0]$, and accepting classes represented by accepting states. It accepts the original language and has no two distinct indistinguishable states.

The indicator vector of a subset $E\subseteq \{1,\dots ,n\}$ has coordinate $1$ on $E$ and coordinate $0$ outside $E$.

An independent and identically distributed sequence combines two conditions: its members are independent random variables, and every member has the same probability distribution. For a finite sequence $X_1,\dots ,X_n$ with common law $\nu$, this means that for all measurable sets $B_1,\dots ,B_n$,For an infinite sequence, require this condition for every finite collection of distinct indices. Equal marginal distributions supply the common law; independence supplies the product. Neither condition supplies the other. Grinstead and Snell call the combination an independent trials process.^[ref]

Choose a Bernoulli distribution with success probability $1/2$. Four independent draws produce all sixteen binary patterns with probability $1/16$ each. If instead $X_1=X_2=X_3=X_4=Z$ for one fair Bernoulli random variable $Z$, every member still has that same Bernoulli distribution, but only patterns 0000 and 1111 occur. Conversely, independently drawn Bernoulli random variables with success probabilities $1/4,3/4,1/4,3/4$ have a product joint probability distribution but unequal marginal distributions.

The combination is useful because one probability distribution describes each observation, while a product measure describes the full IID sample. In particular, an IID sample of finite-variance, real-valued observations has a predictable sample mean: its expected value stays at the population expected value, while its variance of a sample mean decreases as $1/n$. This is a mechanism behind Monte Carlo integration and the finite-variance weak law of large numbers. Whether real observations satisfy the independence and common-law assumptions is a separate modelling question.

If $X_i$ are independent centered square-integrable random variables in a Hilbert space, then $\mathbb{E}⟨X_i,X_j⟩=0$ for $i\ne j$, and hence

An ordinal $\delta >2$ is multiplicatively closed when $\beta \gamma <\delta$ for all $\beta ,\gamma <\delta$. These are exactly

For every ordinal $\gamma$ and $\lambda ={\omega }^{\alpha }$, there are unique ordinals $\beta ,\delta$ such thatChoose the greatest initial multiple $\lambda \beta \le \gamma$; if its remainder were at least $\lambda$, the next multiple would still fit. Uniqueness follows because every remainder below $\lambda$ lies before $\lambda (\beta +1)$.

A nonzero ordinal $\delta$ is additively closed when $\beta +\gamma <\delta$ for all $\beta ,\gamma <\delta$. These are exactly the ordinals ${\omega }^{\alpha }$: sufficiency follows from Cantor normal form, while necessity follows by splitting any leading coefficient or nonzero tail below $\delta$ into two smaller summands.

Under uniform length on $[0,1]$, the position $K$ of the first occurrence of a fixed decimal digit hasThe set of expansions in which the digit never occurs has length zero, and

As a quadrature rule, the trapezoidal rule replaces an integrand by the linear interpolation joining its two endpoint values:For nonnegative endpoint values, this is the area of a trapezoid; in general it is a signed area. The quadrature nodes are $a,b$, each with quadrature weight $(b-a)/2$. Unlike the midpoint quadrature rule, it uses endpoint samples, which can be reused between adjacent panels.

Every affine function $f(x)=\alpha x+\beta$ agrees with its linear interpolation, so the trapezoidal rule integrates it exactly. On $[0,2]$, $f(x)=2x+1$ gives $T(f)=(2/2)(1+5)=6=I(f)$. On $[0,1]$, the polynomial $f(x)=x^2$ instead gives $T(f)=1/2$, exceeding $I(f)=1/3$ by $1/6$. Its degree of precision of a quadrature rule is exactly one.

If $f\in C^2[a,b]$, its signed approximation error satisfiesA convex function lies below its endpoint chord, so the trapezoidal rule overestimates its integral; a concave function gives the reverse inequality. The error formula assumes that the second derivative exists and is a continuous function throughout the closed interval.^[ref] A short derivation is the identityobtained by two integrations by parts. The nonnegative Peano kernel has integral $(b-a{)}^3/12$, and the weighted mean value theorem for integrals gives the formula.

The trapezoidal rule for an ordinary differential equation averages vector fields at the old and new states and is A-stable.

This time-stepping use applies the same trapezoidal rule to the integral of the derivative in an ordinary differential equation. That connection explains its place under implicit Runge-Kutta methods; its quadrature behavior and the nonlinear issue in trapezoidal rule fails B-stability concern different uses of the same endpoint averaging.