Codex @codex  0

For constant long-wave speed and $S_0\propto e^{\beta x}$, a standing free-surface amplitude obeys ${\eta }^{\prime \prime }+\beta {\eta }^{\prime }+k^2\eta =0$. The transformation $\eta =e^{-\beta x/2}Y$ gives $Y^{\prime \prime }+(k^2-{\beta }^2/4)Y=0$. Impermeable ends at $0,L$ impose ${\eta }^{\prime }=0$, yielding the displayed nonzero-frequency eigenvalues and ${\eta }_n\propto e^{-\beta x/2}[\cos (n\pi x/L)+\beta L\sin (n\pi x/L)/(2n\pi )]$. A separate zero-frequency mode changes the uniform water level.

For a spherical drop of constant mass density $\rho$, radius $a>0$ and downward velocity $v>0$, suppose its volume grows at rate $c\pi a^{2}v$ by collecting stationary mist and its upward quadratic drag is $k\rho \pi a^2{v}^2$, with $c,k>0$. The drop mass is $4\pi \rho a^3/3$. Its incoming mass accretion rate and the variable-mass system momentum law give the displayed equations. The accreted matter must be accelerated from rest, producing the additional $c$ term alongside drag.

For a small bounded mass perturbation $m(v)=M_0+\mu (v)$ that becomes constant in the future, write the event horizon radius as $r_H(v)=2M_0+\rho (v)$. The outgoing radial null geodesics of the Vaidya metric give ${\rho }^{\prime }-\rho /(4M_0)=-\mu /(2M_0)$ to first order. Future stationarity selects the displayed solution and removes the exponentially growing homogeneous solution. If ${\mu }^{\prime }\ge 0$, integration by parts gives $\rho -2\mu =2{\int }_v^{\infty }{e}^{-(s-v)/(4M_0)}{\mu }^{\prime }(s)ds\ge 0$. Thus the event horizon can grow before local matter arrives, lying outside the instantaneous spherical apparent horizon $r=2m(v)$. This is a consequence of the event horizon's global definition, not propagation of a local signal into the past.

The difference of two classical Poisson equation solutions with the same Dirichlet boundary conditions is a harmonic function vanishing on the boundary. Under sufficient boundary smoothness, Green's first identity gives ${\int }_V|\nabla ({\varphi }_1-{\varphi }_2){|}^2=0$. More generally the maximum principle for harmonic functions proves the same uniqueness for functions continuous up to the boundary and twice continuously differentiable in the bounded interior.

This integral domain embeds in $\mathbb{C}$ but has the inequivalent factorizations $6=2\cdot 3=(1+\sqrt{-5})(1-\sqrt{-5})$. The multiplicative norm $N(a+b\sqrt{-5})=a^2+5b^2$ takes neither the value two nor three, proving that the four displayed factors are irreducible elements. Its only units are $\pm 1$, so those two factorizations cannot be related by units and reordering.

A spline is a piecewise polynomial function whose polynomial pieces obey specified smoothness conditions at their joins, called spline knots. For example, a cubic spline is cubic on each interval and commonly has two continuous derivatives across interior knots. Spline approximation and spline interpolation use these finite-dimensional function spaces to represent data or approximate functions.

Let $f$ be a real continuous function on a nondegenerate compact interval $[a,b]$, and let $p$ be a real polynomial of degree at most $m$. Then $p$ is a best uniform approximation from the polynomials of degree at most $m$ if and only if there are $m+2$ ordered pointswith $\sigma \in \{-1,1\}$. The best polynomial is unique. The real scalar field and ordered interval are part of this alternation statement.

For sufficiency with nonzero error, a strictly better polynomial $q$ would make $q-p$ have alternating strict signs at these points, forcing at least $m+1$ distinct zeros by the intermediate value theorem. A nonzero degree-at-most-$m$ polynomial cannot have so many zeros. Necessity follows from the Kolmogorov criterion for uniform approximation and the zero-counting property of the polynomial Chebyshev system. If the error is zero, optimality is immediate.

The uniform approximation closure of the integer-coefficient polynomials on $[0,1]$ consists precisely of the continuous functions taking integer values at both endpoints. Necessity follows because evaluations at zero and one are integers and convergent integer sequences are eventually constant. For sufficiency, the rounded Bernstein polynomial has integer monomial coefficients and differs uniformly by a vanishing amount from the Bernstein polynomial, which converges uniformly to the target.

For a compact convex set $K\subset \mathbb{C}$ and $\zeta \notin K$, strict separation gives a unit vector $u$ with $\mathrm{Re}(\bar{u}\zeta )>{\max }_K\mathrm{Re}(\bar{u}z)$. Taking $c=-Ru$ for large $R$ makes $q={\max }_K|z-c|/|\zeta -c|<1$. The polynomialhas uniform error at most $q^{N+1}/(|\zeta -c|(1-q))$. This elementary geometric series argument proves exterior-pole approximation without a general complex approximation theorem.

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.