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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 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.

A ring is left Noetherian when every ascending chain of left ideals stabilizes, equivalently when every left ideal is finitely generated.

Suspension rheology relates particle concentration, particle pressure, shear stress, deformation rate, and fluid migration in a mixture of particles and a liquid.

Viscoelasticity combines viscous dissipation with elastic storage and relaxation of deformation.

A power-law fluid has shear stress $\tau =K|\dot{\gamma }{|}^{n-1}\dot{\gamma }$. It is shear thinning for $n<1$, Newtonian for $n=1$, and shear thickening for $n>1$.

A yield-stress fluid behaves as a rigid material below a critical shear stress and flows once that stress is exceeded.

A local field is a nondiscrete locally compact topological field. A non-Archimedean local field is a finite extension of ${\mathbb{Q}}_p$ or of ${\mathbb{F}}_p((t))$.