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Scalar curvature is the complete trace of the Riemann curvature tensor:An $n$-manifold of constant sectional curvature $K$ has $\mathrm{Scal}=n(n-1)K$.

The order of vanishing of a nonzero rational function $f$ at a point $p$ is the exponent of a local parameter in its local factorization. A negative order is the order of a pole.

The local multiplicity in zeros and poles is positive for a zero and negative for a pole; it is zero at a regular nonvanishing point.

Over a field in which the denominator splits, a proper rational function is a sum of terms $c/(x-a{)}^k$. The coefficient at a simple pole $a$ is obtained by multiplying by $x-a$ and evaluating at $a$.

A projective scheme over a base $S$ is an $S$-scheme admitting a closed immersion into some projective space ${\mathbb{P}}_S^n$. Every projective morphism is proper.

Projectivity is a property of the scheme over its specified base, expressed by its structural projective morphism.

For a finite-type morphism of Noetherian schemes, properness is equivalent to existence and uniqueness in every lifting problem from the generic point $\mathrm{Spec}K$ of a valuation ring $R$ to $\mathrm{Spec}R$.

This is the existence-and-uniqueness part of the valuative criterion.

For a proper morphism $X\to \mathrm{Spec}A$ with $A$ Noetherian and an $A$-flat coherent sheaf $\mathcal{F}$, locally on the base there is a bounded complex of finite free modules $K^{\bullet }$ such thatnaturally for every $A$-module $M$.

For a rank-$r$ vector bundle $E\to X$, flat pullback is an isomorphism $A_k(X)\cong A_{k+r}(E)$.

The horizontal force on the lower wall vanishes when

Suppose $X_i\sim N(\mu ,1)$ independently, and compare $H_0:\mu =0$ with $H_1:\mu \sim N(0,{\tau }^2)$ for fixed $\tau >0$, using equal prior model probabilities. Put $z=\sqrt{n}\overline{X}$. The posterior probability of a Gaussian point null isHolding $z$ fixed while $n\to \infty$ leaves the two-sided p-value $2\Phi (-|z|)$ unchanged but makes the null posterior probability tend to one. For $n=100$, $\tau =1$ and $z=2$, the p-value is about $0.0455$, while the null posterior is about $0.581$.

The two procedures answer different model-based questions: a tail probability under the point null and a comparison of that null with a specified alternative prior distribution. The alternative's fixed prior scale spreads mass over parameter values much wider than the increasingly precise likelihood. This is the Jeffreys-Lindley phenomenon, not a conversion formula from p-values to posterior probabilities.^[ref]