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 strange nonchaotic attractor combines geometrically irregular attraction with no positive Lyapunov exponent for the dynamics under consideration. Systems driven at two incommensurate frequencies supplied the original examples.^[ref]

One useful framework is a skew productThe base is an irrational rotation of the circle. An attracting invariant graph of a skew product obeys $h(\theta +\omega )=F_{\theta }(h(\theta ))$; the geometric question concerns how irregular that graph is. For a differentiable one-dimensional fiber map, the fiber Lyapunov exponent along it is, when the limit exists,A negative value describes average contraction of perturbations within the fiber. The base rotation has a neutral exponent, so a negative fiber exponent does not mean that every exponent of the full forced system is negative. Under appropriate boundedness and integrability assumptions for this triangular derivative system, the full largest exponent is zero rather than positive.

These equations specify a framework, not a claim that every choice of $F_{\theta }$ produces a strange attractor. For instance, $F_{\theta }(x)=ax$ with $|a|<1$ has the smooth attracting graph $h=0$. A negative fiber exponent alone therefore establishes neither geometric strangeness nor a discontinuous graph.

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.

A starch granule is a discrete particle in which a plant stores starch, with amylose and amylopectin organized into regions of different molecular order. Granule shape and internal organization vary with biological source.

A starch granule is consequently more than a pile of independent dissolved carbohydrate chains. Heating in water can disrupt its order during starch gelatinization, changing swelling and accessibility to enzymes.

Amylose is the largely linear component of starch, consisting principally of glucose residues joined by α(1→4) glycosidic bonds. This connectivity allows chain conformations different from the branched architecture of amylopectin.

Hydrolysis of an amylose chain can yield shorter carbohydrates such as maltose. The comparison illustrates how a polymer's properties depend on both the identity of its repeating unit and the pattern of its chemical bonds.^[ref]

Amylopectin is the highly branched glucose polymer in starch. Most residues are connected by α(1→4) glycosidic bonds, with α(1→6) glycosidic bonds at branch points.

The branching permits chain segments to form ordered regions within a starch granule. Amylopectin therefore contributes both to storage of glucose units and to the material properties altered during starch gelatinization.^[ref]

Starch gelatinization is the disruption of ordered structure in starch granules when starch is heated with sufficient water. Water uptake, swelling, and loss of molecular order change the material's flow and optical properties.

The transition depends on temperature, water availability, and starch composition; it is not simply the melting of a pure crystalline substance. It explains why heating a starch suspension can thicken it, linking the organization of amylose and amylopectin with a macroscopic texture.

A counting trait that increments the original argument to insertion assumes that subsequent traits store that argument unchanged. A later capitalization trait breaks the assumption: the counter can record lowercase a while the document stores uppercase A. To maintain counts of actual document contents across such a composition, inspect the stored character after delegation or constrain the legal mixin order. Correctness for one base implementation does not establish correctness under every trait combination.

A pointer from an activation's stack frame to the activation providing its immediate lexical environment. A static link follows lexical enclosure rather than necessarily pointing to the caller. Calls through a programming closure obtain the environment from the passed closure.

A pointer from an activation's stack frame to its caller's stack frame, used to restore the caller when the activation returns. The dynamic link is independent of a static link to the lexically enclosing environment.

A processor register or equivalent value designating a stable reference location within the current stack frame. Parameters, local variables and saved links can be accessed at fixed offsets from the frame pointer even if the evaluation stack changes height.