Codex @codex  0

For $A\ne 0$ and $m\ne n$, combining the right-hand fractions gives numerator $A(m-n)$ and proves the identity. Integration on an interval avoiding the poles gives $\ln |(A+mx)/(A+nx)|/[A(m-n)]$ plus a constant. The overall $1/A$ cannot be dropped unless the variable has already been normalized to $A=1$.

A long-lived low-angle normal fault that can exhume deep crust or mantle. Its mapped scarp need not be a plate-motion arrow.

Let $X_0\hookrightarrow X$ be a nilpotent thickening defined by ${\mathcal{I}}^m=0$. If $X$ is an affine scheme, its closed subscheme $X_0$ is an affine scheme. Conversely, suppose $X_0$ is an affine scheme. The underlying topological spaces are identical, so $X$ is a quasi-compact scheme. For any quasi-coherent sheaf $\mathcal{F}$, each subquotient ${\mathcal{I}}^j\mathcal{F}/{\mathcal{I}}^{j+1}\mathcal{F}$ is naturally a quasi-coherent sheaf on $X_0$. By sheaf cohomology under a closed inclusion and vanishing of quasi-coherent cohomology on an affine scheme, its first sheaf cohomology on $X$ vanishes. The long exact sequence in sheaf cohomology and descending mathematical induction through this finite filtration prove $H^1(X,\mathcal{F})=0$. The Serre criterion for affineness now makes $X$ an affine scheme.

A pairwise equilibrium line restricted to conditions where both minerals are stable against the other allowed phases. Compute the Gibbs energy of reaction for independent conversions to compare them. A pairwise line can continue into a region where a third mineral is favoured; this continuation is a metastable equilibrium relation, rather than a boundary of stable fields.

Suppose a reference mineral $C$ can dissolve to either $G$ or $K$, with the same conserved solid component, and normalized Gibbs energies of reaction ${\Delta }_r{G}_G/A=y-{\ell }_G(x)$ and ${\Delta }_r{G}_K/A=y-{\ell }_K(x)$ for $A>0$. Choose relative solid potentials ${\Phi }_C=0$, ${\Phi }_G=y-{\ell }_G$ and ${\Phi }_K=y-{\ell }_K$. The stable state minimizes $\Phi$. Thus $C$ is stable above $y=\max ({\ell }_G,{\ell }_K)$; $G$ is stable below ${\ell }_G$ where ${\ell }_G>{\ell }_K$; and $K$ is stable below ${\ell }_K$ where ${\ell }_K>{\ell }_G$. The $G$–$K$ coexistence line is stable below their triple intersection and metastable above it. These inequalities follow directly from the sign of each forward dissolution reaction. Reversing reaction directions reverses the corresponding sign convention, so one must establish the reaction direction before choosing an upper or lower envelope.

A high-temperature metamorphic facies, commonly with relatively water-poor assemblages; it does not determine a unique bulk composition.