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Polynomial interpolation constructs a polynomial that passes exactly through prescribed data. Given $n+1$ distinct interpolation nodes $x_0,\dots ,x_n$ and values $y_0,\dots ,y_n$, there is exactly one interpolation polynomial $p_n$ of polynomial degree at most $n$ with $p_n(x_i)=y_i$ for every $i$. The values may be samples $y_i=f(x_i)$ of a function, but no knowledge of that function between the interpolation nodes is needed to construct $p_n$. The Lagrange polynomial and Newton polynomial are two representations of this same answer.^[ref]

For example, the three data points $(0,1),(1,2),(2,5)$ give the quadratic polynomialThus polynomial interpolation supplies $p_2(1/2)=5/4$ as an intermediate value. Whether this approximates an unknown underlying function well is a separate question about polynomial interpolation error.

Both hypotheses in the uniqueness statement matter. The Lagrange cardinal polynomials prove existence constructively. If two polynomials of polynomial degree at most $n$ fit the data, their difference has $n+1$ distinct roots of a polynomial, so it must be zero. Repeated interpolation nodes with conflicting values have no solution, and repeated equal values provide no additional independent condition; prescribed derivatives instead lead to Hermite interpolation. Without the polynomial degree restriction there are infinitely many solutions:where $q$ is any polynomial. By the factor theorem, this describes every polynomial fitting the same data. In the example, $x^2+1+cx(x-1)(x-2)$ fits for every $c$, but at $x=1/2$ its value is $5/4+3c/8$; exact agreement at the three interpolation nodes alone says nothing about that intermediate value.

The usefulness of polynomial interpolation is that data become an explicit function whose values, derivatives, and integrals can be calculated. Differentiating or integrating the Lagrange cardinal polynomials produces differentiation weights from nodal evaluations or a quadrature rule. Exact passage through observations is the defining constraint. In polynomial regression, linear least squares instead chooses a polynomial degree and minimizes a sum of squared residuals, generally without passing through every observation. Interpolating noisy values reproduces their errors at the interpolation nodes and can amplify them between interpolation nodes; polynomial interpolation does not itself smooth measurement noise. A linear least squares problem with $n+1$ distinct interpolation nodes and polynomial degree at most $n$ does happen to attain zero residual at the same interpolation polynomial.

For $f\in C^2[a,b]$, the composite midpoint rule with mesh width $h$ has local error $O(h^3)$ and global error $O(h^2)$.

For bounded matrices,Commutativity removes this splitting error, though first-order substeps such as Backward Euler method retain their own quadratic local errors.

For a real monic quadratic $z^2+a_{1}z+a_2$, both roots lie in the closed unit disk whenwith the usual simplicity requirement for roots on the unit circle.

An arithmetic operation is one scalar addition, subtraction, multiplication, or division. Counting such operations gives a basic model of an algorithm's computational cost.

A trapped surface is a closed spacelike codimension-two surface whose two future-directed orthogonal null congruences both have negative expansion.

In four-dimensional spacetime, “closed” here means compact without boundary; “spacelike” means that the induced metric tensor is positive definite. Let $k_{+},k_{-}$ be the two future null vector normals and $q_{ab}$ the induced metric tensor. Their null expansions areHere $dA$ is an infinitesimal transverse area element and ${\lambda }_{\pm }$ an affine parameter along the respective normal null geodesics. A future trapped surface has ${\theta }_{+}<0$ and ${\theta }_{-}<0$ everywhere: even the outward-directed light front initially loses area. Multiplying either future normal by a positive function multiplies its initial null expansion by that function, preserving the sign. A small inward-converging patch is insufficient: the entire closed surface must satisfy both strict inequalities.

The Penrose singularity theorem states that a globally hyperbolic spacetime with a noncompact Cauchy hypersurface, a closed trapped surface, and the null convergence condition is future null-geodesically incomplete. Time reversal replaces the trapped surface by an anti-trapped surface and gives past incompleteness.

In his 1965 paper Gravitational Collapse and Space-Time Singularities, Roger Penrose used trapped surfaces and global causal structure to address whether nonspherical perturbations could avert a breakdown during gravitational collapse. Quasars supplied the opening astrophysical motivation. The paper used metric signature $(+,-,-,-)$ and a stronger timelike local-energy assumption; the statement above uses the modern null convergence condition.^[ref]

For this version, take a smooth connected four-dimensional spacetime with a time orientation. A Cauchy hypersurface meets every inextendible causal curve exactly once; its existence expresses global hyperbolicity. The hypersurface must be noncompact, meaning not compact; this is a global spatial condition, not an assumption about the size of the collapsing body. The null convergence condition is $R_{ab}{k}^a{k}^b\ge 0$ for every null vector $k$. Through the Einstein field equations it follows from the null energy condition; a cosmological constant drops out of the null contraction.^[ref]

The conclusion is that at least one future-inextendible null geodesic has a finite upper endpoint of its affine parameter. It neither asserts infinite mass density or divergent curvature at every endpoint, nor says that such an endpoint is visible. Locating an event horizon requires additional global information; the theorem is not a horizon-finding rule for arbitrary spacetimes. Weak cosmic censorship conjecture is an additional issue.^[ref]

The supermassive black hole account describes the motivating observational evidence from stellar orbits at the Galactic centre. Such measurements infer a compact gravitating mass. They do not observe a trapped surface, measure a geodesic's complete future, or experimentally prove the Penrose singularity theorem.

In an antisymmetric vibration of a symmetric linear triatomic molecule, the outer atoms move oppositely and the central atom does not couple to that mode.

For $g_{\mu \nu }={\phi }^2{\eta }_{\mu \nu }$,For a null curve, the last term drops from the geodesic equation; the remaining acceleration is parallel to the tangent and can be removed by reparametrization.

A metric is conformally flat when it is locally a positive scalar multiple of a flat metric. Multiplication by the conformal factor preserves null cones and unparametrized null geodesics.

A conformally flat metric is locally a positive scalar multiple of a flat metric. It makes its domain a conformally flat manifold.

Nonlinear Schrödinger blowup analysis studies whether a solution of the Focusing nonlinear Schrodinger equation remains bounded in its natural Sobolev space or develops an unbounded gradient norm in finite time.

The Emden-Fowler transformation is a change of variables for scaling-invariant radial equations, including the Lane-Emden equation. The Emden-Fowler transformation replaces a radial variable $r$ by logarithmic time $s=\log r$ and rescales the dependent variable by a power of $r$. It converts many scale-invariant radial equations into autonomous ordinary differential equations.

A proper two-sided ideal $P$ of a noncommutative ring is prime when $AB\subseteq P$ for two-sided ideals $A,B$ implies $A\subseteq P$ or $B\subseteq P$.

Ore localization extends localization to suitable multiplicative subsets of noncommutative rings.

An injective module has the extension property for homomorphisms defined on submodules.

The Jacobson radical is the intersection of all maximal left ideals, equivalently the intersection of the annihilators of all simple left modules.

Morita-equivalent rings have equivalent module categories. In particular, $R$ and ${\mathrm{Mat}}_n(R)$ are Morita equivalent.

A left primitive ring has a faithful simple left module.

The first Weyl algebra is $A_1(k)=k⟨x,y⟩/(yx-xy-1)$, the algebra of polynomial-coefficient differential operators in one variable.

A ring is right Noetherian when every ascending chain of right ideals stabilizes.