The rocket exhaust speed is the magnitude of the emitted material's velocity relative to the instantaneous rocket. In the ideal rearward-ejection convention of the rocket equation, a rocket with inertial velocity $v$ emits material with inertial velocity $v-u$ to first order. A constant $u$ does not make $v-u$ constant.

Valuative criteria test properties of scheme morphisms by extending a lift from the generic point of the spectrum of a valuation ring to the whole spectrum. Under the usual finiteness hypotheses, uniqueness characterizes separatedness, existence characterizes universal closedness, and existence with uniqueness characterizes properness. The valuative criterion for separatedness and valuative criterion for properness are the respective specific statements.

For a continuous map $f:X\to Y$, the inverse-image functor takes a sheaf on $Y$ to the sheafification of its local inverse-image presheaf on $X$. It is left adjoint to the direct image functor. For sheaves of modules on ringed spaces, extension of scalars gives the separate pullback of a sheaf of modules.

For a continuous map $f:X\to Y$, the direct-image functor sends a sheaf $\mathcal{F}$ on $X$ to the sheaf $V\mapsto \mathcal{F}(f^{-1}V)$ on $Y$, and sends sheaf morphisms to their induced maps on sections. Its output is a direct image sheaf.

An element $x$ of a ring is nilpotent when $x^n=0$ for some positive integer $n$. A field has no nonzero nilpotent elements.

The opposite ring has the same additive group and identity as a ring $R$, with multiplication $a{\cdot }_{\mathrm{op}}b=ba$. Left modules over it are right modules over $R$. Forming the opposite exchanges left and right chain conditions and reverses the order of composition when interpreting endomorphisms as right multiplications. For an associative algebra with its scalar action retained, this construction gives the opposite algebra.

A commutative ring of prime characteristic $p$ is perfect in the Frobenius sense when its Frobenius endomorphism is an automorphism. It is necessarily reduced: if $a$ is nilpotent, some $p$-power iterate is zero, and injectivity gives $a=0$. This use of perfect is distinct from the homological notion of a perfect ring. The Absolute Frobenius morphism of a scheme is an isomorphism exactly when all its local rings are perfect in this sense, because it is the identity on points and is an isomorphism on every stalk.

The ring with a single element, which is both its additive and multiplicative identity. If a unital ring has $0=1$, then every element $r$ equals $r1=r0=0$, so it is this ring. Under the usual definition it is excluded from integral domains and fields. The quotient ring $\mathbb{Z}/(1)$ is an example.

The matrix ring consists of all $n\times n$ matrices over a ring $R$, using entrywise addition and matrix multiplication. Its identity element is the identity matrix. Even when $R$ is a commutative ring, this ring need not have commutative multiplication.

Suppose $A\subseteq B$ are unital rings and $B$ is a finitely generated module over $A$ on the right. If $A$ is a right Noetherian ring, then $B_A$ is a Noetherian module. Every right ideal of $B$ is an $A$-submodule, hence has finitely many generators over $A$; these also generate it over $B$. Thus $B$ is a right Noetherian ring. The same argument works on the left under the corresponding hypotheses. Finite-index group ring extensions give an important example, using coset representatives as a finite module basis.

For an additive contravariant left-exact functor, apply it to a projective resolution of $M$. Reversal of the arrows gives a cochain complex. Comparison maps lift using projectivity, and any two lifts are chain homotopic, so its cohomology is independent of the resolution. The functors are contravariant, vanish on projective modules in positive degrees, and send $0\to M^{\prime }\to M\to M^{\prime \prime }\to 0$ to $0\to F(M^{\prime \prime })\to F(M)\to F(M^{\prime })\to R^{1}F(M^{\prime \prime })\to \cdots$. Degreewise split resolutions from the horseshoe lemma yield this sequence.

The finite intersections of maximal right ideals of a right Artinian ring have a minimal member by the descending chain condition. Intersecting it with any other maximal right ideal changes nothing, so it equals the Jacobson radical. If that radical is zero, the diagonal map from the regular module into a finite direct sum of its simple quotients is injective. A submodule of a finite direct sum of simple modules is semisimple and has finite length. This proves semisimplicity of the regular module without assuming Noetherianity first.

Let $K/k$ be an infinite-dimensional field extension. For the triangular ring in the display, the right regular module has length three: $e_{1}T$ has its off-diagonal simple submodule and simple quotient, and $e_{2}T$ is simple. Thus $T$ is right Artinian. Its off-diagonal ideal is, on the left, the infinite-dimensional $k$-vector space $K$. Every $k$-subspace determines a left ideal, so an ascending sequence of finite-dimensional subspaces violates left Noetherianity. For example take $K=k(z)$ and the spans of $1,z,\dots ,z^m$.

A positive linear functional on the space of continuous compactly supported functions of a locally compact Hausdorff space has a unique representing positive Radon measure. Positivity gives boundedness on fixed compact supports: choose a compactly supported continuous cutoff $p\ge 1$ on the support $K$, then $|I(f)|\le \Vert f{\Vert }_{\infty }I(p)$ for real $f$ supported in $K$. The resulting measure is finite on compact sets but need not have finite total mass.

A (pseudo-)Riemannian manifold is geodesically complete when every inextendible affinely parametrized geodesic is defined for all real parameter values. In the positive-definite case, the Hopf-Rinow theorem identifies this with metric completeness. Null-only completeness in spacetime is a separate weaker condition.

On a connected Riemannian manifold of dimension at least three, pointwise independence of sectional curvature from the tangent two-plane forces the curvature to be constant. The contracted Bianchi identity differentiates the relation $\mathrm{Ric}=(n-1)Kg$ and gives $(n-2)dK=0$.

A geodesic polygon is a closed polygonal curve whose sides are geodesic segments. A geodesic triangle has three sides. Surface angle excess is governed by the Gauss-Bonnet theorem.

A Riemannian metric satisfying the displayed equation for a vector field $X$ and constant $\lambda$. A gradient Ricci soliton has $X=\nabla f$; it is steady when $\lambda =0$.

A minimal submanifold has vanishing mean curvature vector, equivalently zero first variation of volume for compactly supported normal variations. A minimal surface is its two-dimensional case. A calibrated submanifold satisfies the stronger property of minimizing volume in its relative homology class.

// Target: general-relativity.bigb

For $1<\sigma \le 2$, the three-four-one inequality for Euler products gives $|1/\zeta (\sigma +it)|\le \zeta (\sigma {)}^{3/4}|\zeta (\sigma +2it){|}^{1/4}$. For $|t|\ge 1$, the fractional-part continuation formula for the Riemann zeta function, truncated at $H\asymp 2+|t|$, gives ${\zeta }^{(j)}(\sigma +it)=O_j({\log }^{j+1}(2+|t|))$. Thus the reciprocal bound even holds with exponent $1/4$ instead of $1/2$ in this range. On bounded-height compact sets, three-four-one product proof of zeta boundary nonvanishing and removal of the reciprocal's zero at the pole give uniform bounds. Differentiating $1/\zeta$ repeatedly yieldsThe range restriction matters: $1/\zeta (\sigma )\to 1$ as $\sigma \to \infty$, so the first displayed bound cannot hold uniformly for all $\sigma >1$. A global version replaces $(\sigma -1{)}^{-3/4}$ with $1+(\sigma -1{)}^{-3/4}$.