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A predicate on possibly partial lists is admissible when it holds at the limit of every increasing chain of finite approximations on which it holds. Increasing here means replacing a bottom value by more defined information while retaining already defined constructors and values. This closure condition permits a property established for finite approximations to pass to an infinite list. The predicate “the list is finite” lacks this closure: every finite prefix ending in an undefined tail is finite in extent, but their limit may have infinitely many constructors.

For an admissible predicate on lazy lists $P$, prove $P(\perp )$ and $P([])$, and prove $P(xs)\Rightarrow P(x:xs)$ for every possibly partial head $x$. Constructor induction then establishes $P$ on every finite partial approximation, and admissibility extends it to limits, including infinite lists. Without the bottom value case and the limit condition, ordinary finite structural induction does not justify a claim about infinite data.^[ref]

With fixed endpoint values on a finite interval, the displayed derivative coercivity makes a finite-energy strong local minimizer of a variational integral also a local minimizer in $L^1$. Otherwise choose better competitors converging in $L^1$. The upper energy bound controls their derivatives in $L^p$, and a common endpoint controls their function norms. Compact Sobolev embedding gives a uniformly convergent subsequence; its limit is the $L^1$ limit. This contradicts strong local minimality. Convexity of the integrand is unnecessary for this neighborhood-transfer argument.

A comparison of strings that disregards a chosen distinction between uppercase and lowercase characters. A simple convention compares characterwise lowercase forms. Full Unicode case folding is a distinct normalization operation and can change character counts.

A geometrically irregular attractor need not have chaotic exponential separation of nearby trajectories. Quasiperiodically forced systems provide primary examples in which strangeness and a positive largest Lyapunov exponent do not coincide. Consequently a fractal-looking numerical plot alone cannot establish chaos. ^[ref]

Elastic strain is the reversible part of strain that determines stress through the elastic constitutive equation. In a small-strain model with plastic and thermal effects, subtract the plastic strain and thermal strain from total strain to obtain it. A stress-free heated body can therefore have nonzero total strain but zero elastic strain; after plastic loading, unloading can likewise remove elastic strain while leaving permanent deformation.

Doubling the volume of a unit cube requires constructing its new side length $\sqrt[3]{2}$. Its minimal polynomial $X^3-2$ is irreducible by Eisenstein's criterion at two, so this length has degree three. Every constructible number belongs to a finite tower of quadratic extensions, whose total degree is a power of two. The tower law for field extensions would force the degree three to divide that power of two, a contradiction. Thus no such straightedge and compass construction exists.

Consider the constructible angle whose cosine is $5/6$ and sine is $\sqrt{11}/6$. If one fifth of this angle were constructible, its cosine $c$ would satisfy the displayed polynomial, by the identity $\cos 5\theta =16{\cos }^5\theta -20{\cos }^3\theta +5\cos \theta$. The identity follows by taking the real part of $(\cos \theta +i\sin \theta {)}^5$ and substituting ${\sin }^2\theta =1-{\cos }^2\theta$.

The polynomial is irreducible by Eisenstein's criterion at five, so $c$ has degree five over the rational numbers. The initial angle field $\mathbb{Q}(\sqrt{11})$ already has degree two, and adjoining any finite collection of constructible numbers gives total degree over the rational numbers a power of two. Such a field cannot contain the degree-five element $c$, by the tower law for field extensions. Hence this constructible angle cannot be divided into five equal angles by straightedge and compass construction. Some particular angles can be quintisected; the obstruction disproves a construction valid for all angles.

Distinct coplanar straight lines with proportional direction vectors. They have a common perpendicular direction and a constant positive separation.

On a positive compact interval, both even-power and odd-power complex polynomial families are uniformly dense in the continuous-function algebra. The even family is a conjugation-closed unital algebra separating points, since squaring is injective on the interval; the Stone-Weierstrass theorem proves its density. Equivalently use the homeomorphism $t\mapsto t^2$ and polynomial approximation on $[a^2,b^2]$. For the odd family, given a target $f$, first approximate $f/t$ by an even polynomial $p(t^2)$. Then $tp(t^2)$ is odd and its error is at most $b$ times the first error. Division is continuous because the interval avoids zero. Thus the odd family can be dense even though it is not a unital subalgebra.

On $[0,1]$, polynomials in $t^2$ form a unital algebra closed under complex conjugation separating points, so the Stone-Weierstrass theorem makes them dense in the continuous complex functions. Their approximation can preserve both endpoint values exactly. Given $\Vert a-p(t^2){\Vert }_{\infty }<\eta$, add $[a(0)-p(0)](1-t^2)+[a(1)-p(1)]t^2$. This is still an even polynomial, fixes both endpoints, and changes the approximation by at most $\eta$, since its correction is a convex combination of the endpoint errors. Taking $\eta <\epsilon /2$ yields any prescribed uniform error $\epsilon$.

For an axisymmetric flow without azimuthal velocity, the spherical Stokes streamfunction gives $u_r={\Psi }_{\theta }/(r^2\sin \theta )$ and $u_{\theta }=-{\Psi }_r/(r\sin \theta )$. These formulas enforce incompressibility. The only possibly nonzero component of vorticity isThus the displayed equation is precisely the condition for irrotational flow, away from coordinate poles, with limits interpreted through regularity of the physical velocity field. It differs from the scalar spherical Laplace equation: $\Psi$ is a Stokes streamfunction, not a velocity potential.

The unbounded two-dimensional Stokes flow past a no-slip circular cylinder has no solution approaching a nonzero uniform stream. With radius and speed scaled to one, the forced angular mode has stream function $\psi =f(r)\sin \theta$. The biharmonic equation gives $f=A{r}^3+Br\log r+Cr+D/r$. Uniform flow at infinity forces $A=B=0$, $C=1$; impermeability forces $D=-1$, but then $f^{\prime }(1)=2$ violates no slip. The failure belongs to a nonuniform small-Reynolds number limit. In a distant outer region inertia must be retained through an Oseen approximation or the full Navier-Stokes equations, matched to an inner Stokes region. A finite outer wall changes the problem and removes this particular infinity obstruction.

Begin with positive counts of two colours. At each step select one ball uniformly and replace it together with one new ball of its colour. If the total initial count is $m$, the blue proportion satisfies $X_{n+1}-X_n=(I_{n+1}-X_n)/(m+n+1)$, where the next blue indicator has conditional mean $X_n$. Thus $X_n$ is a bounded martingale and converges almost surely and in $L^2$ to a random proportion. If uniforms generate the draws, the update from time $n$ to $n+1$ must use the fresh $U_{n+1}$ relative to ${\mathcal{F}}_n=\sigma (U_1,\dots ,U_n)$.

An estimation error is the difference between an estimator and the statistical parameter it estimates. It is a random variable before observing the sample. Its mean is the bias of the estimator, and its mean square is the sum of the variance and squared bias. Standardizing an estimation error by its standard error often yields an approximate standard normal distribution, allowing construction of a Wald confidence interval.