Every real continuous function on a compact square can be represented as ${\sum }_{q=1}^5{G}_q(a_q(x)+b_q(y))$, with continuous one-variable functions and fixed inner functions $a_q,b_q$ independent of the represented function. Grid-separated additive coordinates give the inner functions. For a residual $r$ of norm $M$, assign $r$ at a sample point divided by three to each separated rectangle image, and continuously interpolate between images. Each point lies in at least three rectangles, so if $k\in \{3,4,5\}$ summands are accurate, the error is at most $[|1-k/3|+(5-k)/3]M+5\eta /3=2M/3+5\eta /3$. Choose oscillation $\eta \le M/10$ to obtain contraction $5/6$. Iteration gives outer functions by uniform convergence of corrections bounded by $M(5/6{)}^j/3$. Whole-plane reduction for continuous superposition removes compactness of the original domain without assuming boundedness of the original function.

Suppose linear operators $T_n$ on a normed vector space satisfy $\Vert T_n\Vert \le C$ uniformly and $T_{n}p\to p$ on a dense subspace. The displayed triangle inequality for any element $p$ of that subspace first permits $p$ to approximate $f$, then lets $n$ increase, proving $T_{n}f\to f$ for every $f$. Applied to the Bernstein polynomial, the Weierstrass approximation theorem supplies the dense polynomial subspace and $C=1$.

For a polynomial of degree at most $n$, this derivative estimate holds for $-1<x<1$. It controls the derivative away from the endpoints. Sampling it at Chebyshev polynomial roots supplies the hypotheses of the Chebyshev nodal derivative comparison, which controls the missing endpoint regions. The norm is taken on the entire interval, although the displayed pointwise bound concerns its interior.

Polynomial approximation represents a function by polynomials of bounded polynomial degree. On a compact real interval, the Weierstrass approximation theorem guarantees uniform approximation of every continuous function by such polynomials. The displayed best error quantifies the accuracy available at degree at most $n$.

On $[a,b]\subset (0,1)$, the uniform closure of integer-coefficient polynomials contains $1/2$, since ${\sum }_{j=0}^{N}x(1-2x{)}^j\to 1/2$ uniformly. This closure is a ring, so it contains all dyadic constants and then all real constants. It contains $x$ and therefore every real-coefficient polynomial. The Stone-Weierstrass theorem makes it all of $C([a,b])$. On an interval including zero this fails, since the value of an integer-coefficient polynomial at zero is an integer.

For an algebra $A\subseteq C(K,\mathbb{F})$ on a compact Hausdorff space $K$, with $\mathbb{F}=\mathbb{R}$ or $\mathbb{C}$, its uniform closure consists of the uniform limits of sequences from $A$. It is a closed algebra in the supremum norm, since uniformly convergent sequences are uniformly bounded andFor a real algebra containing the constants and separating points, the Stone-Weierstrass theorem makes this closure all of $C(K,\mathbb{R})$. For a complex algebra, closure under complex conjugation is additionally sufficient; separation and constants alone are not enough.^[ref]

For example, complex polynomials in $z$ on the closed unit disk contain constants and separate points, but their uniform closure is the disk algebra, not all continuous functions on that disk. They cannot approximate $\overline{z}$ uniformly: uniform limits of these holomorphic functions are holomorphic in the interior, while $\overline{z}$ is not.

A Chebyshev system on a nondegenerate real interval is a family of $n+1$ real continuous functions such that every linear combination with coefficients not all zero has at most $n$ distinct zeros. The condition entails linear independence and unique interpolation at any $n+1$ distinct points. It is a zero-counting hypothesis, not a requirement to count multiplicities.

If interpolation at distinct sites $x_i$ in a finite-dimensional space of continuous functions is unique, its cardinal functions satisfy ${\ell }_i(x_j)={\delta }_{ij}$. The interpolation linear map is $Pf={\sum }_{i}f(x_i){\ell }_i$ and has operator norm ${\Lambda }_{\mathbf{x}}$ in the supremum norm. The upper bound is the triangle inequality. For the reverse bound choose node values to be the signs of the cardinal functions at a point maximizing their absolute sum, and extend those values by a continuous function of norm one. The polynomial reproduction error bound gives $\Vert f-Pf{\Vert }_{\infty }\le (1+{\Lambda }_{\mathbf{x}}){\inf }_{s\in \mathrm{range}P}\Vert f-s{\Vert }_{\infty }$.

With homozygote fitness advantages $a,b>0$ over the heterozygote, the allele frequency $u$ of the first allele has an unstable threshold $b/(a+b)$. Below it that allele decreases; above it it increases. In a discrete viability-selection generation the same numerator gives $u^{\prime }-u$ after division by mean genotype fitness. Equal homozygote advantages give $\dot{u}=su(1-u)(2u-1)$, with stable endpoints and unstable midpoint.

The Tzitzeica equation $u_{xy}=e^u-e^{-2u}$ is unchanged by $(x,y)\mapsto (cx,c^{-1}y)$. The identity component has generator $x{\partial }_x-y{\partial }_y$. Locally its invariant solutions are $u=w(xy)$, reducing the equation to $z{w}^{\prime \prime }+w^{\prime }=e^w-e^{-2w}$.

A typing context records the types assumed for available program names. In $\Gamma =\{x:\mathrm{int},y:\mathrm{int}\}$, looking up either name yields int. Checking a lambda abstraction extends the typing context with its argument's type for the body. In the Hindley-Milner type system, a typing context can also hold type schemes for names introduced by let polymorphism.

The occurs check tests whether a type variable occurs inside the expression proposed as its replacement during type unification. After discarding a trivial equation such as $\alpha =\alpha$, solving $\alpha =\tau$ by a type substitution requires that $\alpha$ not occur in $\tau$. Thus $\alpha =\mathrm{int}\to \beta$ is permitted when $\alpha$ and $\beta$ are distinct, but $\alpha =\alpha \to \beta$ is rejected in our finite simple type grammar.

There is a short reason for the rejection. If a finite replacement $T$ for $\alpha$ existed, with replacement $B$ for $\beta$, the equation would become $T=T\to B$. The right side's abstract syntax tree has one new arrow node plus the nodes of $T$ and $B$, so it has strictly more nodes than $T$. Equal finite type expressions cannot have different sizes. Recursive types are a different design: they explicitly describe self-reference and specify how it interacts with typing judgments.

A type scheme binds type variables that may be instantiated independently at different uses. The identity function has type scheme $\forall \alpha .\alpha \to \alpha$: choose one type per use, then use that same choice on both sides of its function type. A free type variable shared with the typing context is not automatically such an independently selectable parameter.

A type substitution replaces type variables by type expressions consistently. For example, $S=\{\alpha \mapsto \mathrm{int}\}$ gives $S(\alpha \to \alpha )=\mathrm{int}\to \mathrm{int}$. Applying a type substitution to a type scheme must respect its bound type variables; separate instantiation first replaces those bound placeholders by fresh ones.

Type unification finds a type substitution making specified type expressions equal, or reports incompatible constraints. In our simple type grammar, matching two function types reduces to matching their argument types and their result types. Distinct base types int and bool fail to match; solving a type variable against a nontrivial type expression uses the occurs check.

// Target: computer-science.bigb

A closed string sector with left-moving Neveu–Schwarz sector excitations and right-moving Ramond sector excitations. Its spacetime states are fermions. The string intercepts differ, so the pair of oscillator vacua is not level matched: $N_L-1/2=N_R$. After the supersymmetric GSO projection, the lowest allowed states have $(N_L,N_R)=(1/2,0)$ and are a massless gravitino and dilatino.

A closed string sector with left-moving Ramond sector excitations and right-moving Neveu–Schwarz sector excitations. Its spacetime states are fermions. The closed-string level matching condition is $N_L=N_R-1/2$. After the supersymmetric GSO projection, the lowest states have $(N_L,N_R)=(0,1/2)$ and give the other massless gravitino and dilatino of type II string theory.

Completing the square in $H^{\prime }=\frac{1}{2}(p_x^2+p_y^2+x^2+y^2)-e\mathcal{E}x$ shifts the oscillator centre to $(a,0)$ and lowers every energy by $a^2/2$. Translated eigenfunctions keep their normalization. The angular momentum about the new centre is $L^{\prime }=(x-a)p_y-y{p}_x$, which commutes with $H^{\prime }$, whereas the angular momentum about the old centre has $[L,H^{\prime }]=-i\hslash ay$. For $a\ne 0$, a common normalizable eigenfunction of $L,H^{\prime }$ would have $y\psi =0$ and hence vanish in $L^2({\mathbb{R}}^2)$. This stronger argument excludes individual common eigenstates; noncommutativity alone generally excludes only a complete common eigenbasis.

In flat twistor theory, a twistor is a pair $Z=({\omega }^A,{\pi }_{A^{\prime }})$ of two-component spinors. The twistor incidence relation with a spacetime point is ${\omega }^A=i{x}^{A{A}^{\prime }}{\pi }_{A^{\prime }}$. The nonzero vector $Z$ and its projective class play different roles: the latter belongs to projective twistor space.

The flag manifold of a line $\ell$ contained in a two-plane $W$ has projections $(\ell ,W)\mapsto W$ and $(\ell ,W)\mapsto \ell$. Their fibers are respectively ${\mathbb{CP}}^1$ and ${\mathbb{CP}}^2$. This is the flat twistor correspondence: complexified compactified Minkowski space is the Grassmannian of two-planes in twistor space, while projective twistor space is its projective space of lines.