For an incident particle of rest mass $m_1$ and total energy $E_1$ hitting a stationary target of mass $m_2>0$, four-momentum conservation gives $W^2=m_1^2{c}^4+m_2^2{c}^4+2m_2{c}^2{E}_1$. Applying the relativistic collision mass threshold gives the displayed necessary condition. The incoming energy must also satisfy $E_1\ge m_1{c}^2$; other conserved quantum numbers or reaction dynamics can impose further restrictions.

With a common excess hazard, let ${\overline{h}}_B={\sum }_i{Y}_i{h}_B^{(i)}/{\sum }_i{Y}_i$ be the known background hazard averaged over the current risk set. Estimate the excess cumulative hazard by subtracting $\int {\overline{h}}_B$ from the Nelson–Aalen estimator. Applying the product-limit construction gives the displayed estimator: the continuous background correction multiplies the ordinary Kaplan–Meier estimator. Its event jump is $1-d_j/r_j$, while between events it grows at rate ${\overline{h}}_B{\hat{F}}_E$. Hence the unconstrained curve can increase, even if the true excess hazard is nonnegative. Replacing its jump factor by $\exp (-d_j/r_j)$ gives the alternative exponential cumulative-hazard estimator; these versions are close for small event fractions but are not identical.

A relative homology class is represented by a singular chain whose boundary lies in $A$. Two representatives give the same class when their difference is a boundary operator image plus a chain in $A$. Thus a path with endpoints in $A$ can represent a class even when its endpoints are different.

Let $A$ be a nonempty union of disjoint embedded circles in a closed oriented topological surface ${\Sigma }_g$, and let $c$ be the number of components after cutting along them. Excision and Poincare-Lefschetz duality identify the relative homology with the complementary-degree cohomology of the cut surface. Its components have boundary, so the relative groups are free, with the displayed ranks; $H_0$ and higher groups vanish. The rank of the circle classes in $H_1({\Sigma }_g)$ is $|{\pi }_0(A)|+1-c$. For a single circle these groups detect whether its class is zero, but for genus at least two cannot distinguish a disk boundary from an essential separating circle.

The signed relative error is $\epsilon =(x_{\mathrm{approx}}-x)/x$ for a nonzero reference $x$. Its absolute value is the relative error. For a positive reference, a negative signed relative error means underestimation, and a positive one means overestimation. For a negative reference that interpretation reverses, so retain the displayed definition when interpreting its sign.

For an inclusion $A\subseteq X$, the restriction cochain map fits into an exact triangle with the relative cochain complex. Its cohomology gives $\cdots \to H^i(X,A)\to H^i(X)\to H^i(A)\stackrel{\delta }{\to }{H}^{i+1}(X,A)\to \cdots$. The same construction with derived sections of a sheaf complex gives relative hypercohomology and is natural under restrictions of pairs.

A recommended atomic weight applicable to normal materials, rather than the result for every possible enriched sample. The tabulated neon value $20.1797(6)$ and the arithmetic result above need not agree in every last displayed digit. Some chemical elements have intervals of standard atomic weight because their isotopic abundances vary across normal materials.^[ref]

Set $y=r_{+}/|Q|\ge 1$. Then $M/|Q|=(y+y^{-1})/2$ and $|Q|T=(y^{-1}-y^{-3})/(4\pi )$. Differentiation gives a unique maximum at $y=\sqrt{3}$, proving the displayed mass and temperature. The curve is zero at extremality and asymptotic to $1/(8\pi M)$ at large mass.

In the ideal thermal-cutoff model where charged particles of mass $m$ are emitted only for $T>m$, a sufficiently large initial mass keeps charge fixed throughout evaporation exactly when $m$ is at least the fixed-charge Reissner-Nordstrom temperature maximum. Neutral radiation then approaches an extremal charged remnant of area $4\pi Q^2$. This conclusion does not include nonthermal electric-field discharge, nor establish finite-time attainment.

For a quartic inflaton potential, an endpoint field of order the unreduced Planck mass leaves density of order $\lambda M_P^4$. If prompt, efficient reheating converts it to equilibrium radiation, $\rho =({\pi }^2/30)g_{\ast }{T}_R^4$ gives the displayed scaling. With the potential-only endpoint estimate ${\varphi }_e=M_P/\sqrt{\pi }$, the coefficient is $[15/(2{\pi }^4){]}^{1/4}$. End-of-inflation kinetic energy changes that coefficient by order one, and delayed transfer or incomplete thermalization requires a reheating model rather than the potential alone.

Let $(A,\mathfrak{m})$ be a regular local ring of Krull dimension $d$. Its formal power series ring $B=A[[x]]$ is Noetherian and local, with maximal ideal $I=\mathfrak{m}B+(x)$ and embedding dimension $d+1$. Since $I$ is generated by $d+1$ elements, the Krull height theorem gives $\dim B\le d+1$. For the reverse inequality choose a strict prime chain ${\mathfrak{p}}_0⊊\cdots ⊊{\mathfrak{p}}_d=\mathfrak{m}$ in $A$. Finite generation of each ${\mathfrak{p}}_i$ gives $B/{\mathfrak{p}}_{i}B\cong (A/{\mathfrak{p}}_i)[[x]]$, an integral domain, so each extended ideal is prime. The inclusions remain strict by contraction to $A$, and $\mathfrak{m}B⊊I$ adds one more step since $x\notin \mathfrak{m}B$. This proves $\dim B=d+1$, equal to its embedding dimension, and hence regularity.

In a regular local ring of dimension $d$, a regular system of parameters is a list in the maximal ideal whose images form a residue-field basis of $\mathfrak{m}/{\mathfrak{m}}^2$. The Nakayama lemma shows that it generates the maximal ideal. At a smooth closed point over an algebraically closed field, these are the local parameters in all dimensions, extending the single local parameter on a smooth algebraic curve.

For a regular value $q$ of a smooth map $f:M\to N$, let $F=f^{-1}(q)$. The differential of a smooth map identifies the quotient vector bundle $TM{|}_F/TF$ with the fixed target tangent space $T_{q}N$ over every point of $F$. After choosing a Riemannian metric, that quotient identifies with the normal bundle. Hence the normal bundle is trivial even when the target manifold has no global orientation.

A nonempty regular level set $f^{-1}(0)$ for a real smooth function has both positive and negative values immediately beside any of its points, by the inverse function theorem. On its complement the function never vanishes, so the positive and negative regions are disjoint nonempty open sets partitioning that complement. Hence the complement is disconnected. This is a global necessary condition beyond local embeddedness or coorientation.

For fixed $r,\ell \ge 3$ and $rn$ even, a prescribed graph cycle in the configuration model has two ordered distinct half-edges at each of its vertices. Therefore its expected count is $(n{)}_{\ell }[r(r-1){]}^{\ell }/[2\ell {\prod }_{j=0}^{\ell -1}(rn-2j-1)]$. Conditioning on the chosen cycle leaves degrees $r-2$ at its vertices and $r$ elsewhere; forbid further edges along that cycle. The bounded-degree pairing avoidance estimate changes by $o(1)$, so the simplicity probability conditional on the cycle divided by the unconditional probability tends to one. The displayed expectation therefore holds for the uniform simple graph, not just the unconditioned pairing model.

Let $p$ be prime and set $s=2p-2$. If a bipartite graph has edge-disjoint perfect matchings $M_0,\dots ,M_{\sigma -1}$ on nested nonempty vertex sets, and $|V(M_{i+s})|>s|V(M_i)|/(s+1)$, then the union of $s+1$ consecutive matchings in a graph has maximum degree at most $2p-1$ and more than $(p-1)|V(M_i)|$ edges. The prime-regular subgraph from Boolean polynomial constraints gives a $p$-regular subgraph. Its perfect matching decomposition gives a $k$-regular subgraph for every $k\le p$. If none exists, every $s$ steps shrink the vertex set by factor at most $s/(s+1)$, bounding $\sigma$ by a constant times $\log n$.

The support of an object $A$ in a regular category is the subterminal object appearing as the image of a morphism in a regular category of $A\to 1$. A regular functor preserves this construction. In the Category of sets, the support is the empty set if $A$ is empty and a singleton otherwise.

A capital regular category, in the convention of Definitions 1.1–1.3, is a regular category whose well-supported objects are well-pointed objects in a category. It does not require arbitrary objects to be detected by global points. A partially ordered set with finite meets, regarded as a category, is capital: its only well-supported object is the top element. Such a category can have many proper subterminal objects, all with no points. This example distinguishes the definition from the stronger convention that the global-sections functor reflects every isomorphism.

For a small category that is a regular category, use a sort for each object, an operation for each morphism, and regular theory axioms describing finite limits and regular epimorphisms. Its models are regular functors into the Category of sets. For a proper subobject $m:S\hookrightarrow A$, the sequent asserting $\forall x:A\exists y:S,m(y)=x$ cannot be derivable, since its categorical interpretation would make $m$ a cover and hence invertible. The chase completeness for regular logic supplies a model where it fails. It is essential to allow empty sorts.

The boundary simplicial complex of the cross-polytope has vertices $\pm e_i$, and its faces choose at most one vertex from each opposite pair. Radial projection identifies its geometric realization with the round sphere $S^n$. It is a regular antipodal triangulation of a sphere and is useful as a target for signed vertex labels: a vertex map is a simplicial map precisely when no edge receives opposite labels.