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

The successor function on the natural numbers is the function $S(n)=n+1$. With $0$ as the initial natural number, its repeated application produces $0,S(0),S(S(0)),\dots$.

In recursion theory, the successor function is an initial function of recursion theory. Together with the other initial functions and the permitted operations, it helps generate primitive recursive functions. For example, addition can be specified recursively by $a+0=a$ and $a+S(b)=S(a+b)$. This description builds addition from a base case and a successor step, matching the structure used in mathematical induction.

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 $r$.

Let $F$ have a unique median $m$ and a continuous probability density function $f$ with $f(m)>0$. The influence function of a quantile at $p=1/2$ givesFor contamination exactly at $m$, the generalized-inverse median remains $m$ and the influence function is zero. Thus its value is determined everywhere by taking $\mathrm{sgn}(0)=0$.

The response is bounded by $1/(2f(m))$, so the gross-error sensitivity is finite. It has nonzero constant tails, giving an infinite rejection point of an influence function, and its jump gives infinite local-shift sensitivity. For a standard normal distribution, the bound is $\sqrt{\pi /2}$. These three different sensitivity measures describe different aspects of the same estimator.

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

Consider a small scalar field fluctuation $\delta \varphi$ on a prescribed FLRW metric, neglecting its feedback on the metric tensor. Linearizing the canonical field equation about homogeneous ${\varphi }_0(t)$ gives, in units $c=\hslash =1$,Here $\mathbf{k}$ is the comoving wavevector of the spatial Fourier transform. For constant Hubble parameter $H_{\inf }$ and a negligibly small effective mass term $V^{\prime \prime }$, this reduces toA nearly constant value of the inflaton potential alone does not justify dropping its second derivative; the mass assumption is separate. At large physical wavenumber $k/a$, the mode oscillates, whereas at $k/a\ll H_{\inf }$ the massless equation admits a constant mode and a decaying mode. A full comoving curvature perturbation calculation also includes metric perturbations and specifies a coordinate condition such as Newtonian gauge.

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.

The Karoubi envelope freely splits idempotents. Its objects are pairs $(A,e)$ and a morphism $(A,e)\to (B,f)$ is a map $u:A\to B$ satisfying $u=fue$.

A steady axisymmetric magnetohydrodynamic wind follows nested magnetic-flux surfaces and has conserved mass loading, field-line angular velocity, angular momentum, entropy, and Bernoulli quantities along each surface.