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Historical formulation name; composition is identified by pharmacopoeial edition.

If $w$ is a nonvanishing solution of $-(P{w}^{\prime }{)}^{\prime }+Qw=0$, expanding the right-hand side gives the displayed identity. Integrating expresses a scalar second variation as a square plus a boundary term. A simple zero of $w$ at a fixed endpoint causes no difficulty for Lipschitz endpoint-vanishing variations, since $\phi =O(x-a)$ makes the boundary term tend to zero. This identity is the scalar form of a ground-state transform for a weighted Dirichlet energy.

Let $T$ be a self-map of a nonempty closed subset of a Banach space, with the displayed bounds. Then $T$ has a unique fixed point $p$ and every orbit $x_n=T^n{x}_0$ converges to it. Indeed, $\Vert x_{n+1}-x_n\Vert \le {\alpha }_n\Vert x_1-x_0\Vert$ makes the orbit Cauchy by summing successive increments. The first bound makes $T$ continuous, so the limit is fixed. Since ${\alpha }_n\to 0$, the bounds also imply uniqueness. The tail sum gives $\Vert p-x_n\Vert \le {\sum }_{k\ge n}{\alpha }_k\Vert x_1-x_0\Vert$ for $n\ge 1$. A Volterra integral equation supplies factorially decreasing bounds even when the first iterate is not a contraction mapping.

For a simple module $M$ and a central element $z$ of an algebra $A$, the image $zM$ is a submodule. If it is nonzero, it equals $M$, and iterating gives $z^{r}M=M$ for every $r$. This is impossible when $z$ is nilpotent. Hence $zM=0$. It follows, for example, that the only simple module for $k[X]/(X^r)$ is the one-dimensional quotient on which $X$ acts by zero.

The multiplicative field norm $N(m+n\sqrt{-7})=m^2+7n^2$ is an integer. Units have field norm one and are exactly $\pm 1$. A nontrivial factorization of 2 would force both factors to have field norm two, which the displayed quadratic form cannot represent. Thus 2 is an irreducible element. It is not a prime element, since it divides $(1+\sqrt{-7})(1-\sqrt{-7})=8$ but neither factor. This is an example in a nonmaximal quadratic order, rather than the full ring of integers of $\mathbb{Q}(\sqrt{-7})$; the precise ring matters for factorization.

Historical formulation name; composition is identified by pharmacopoeial edition.

Historical formulation name; composition is identified by pharmacopoeial edition.

Historical formulation name; composition is identified by pharmacopoeial edition.

Historical formulation name; composition is identified by pharmacopoeial edition.

Historical formulation name; composition is identified by pharmacopoeial edition.

Historical formulation name; composition is identified by pharmacopoeial edition.

In the 1864 British standard, the fresh wet hydrated ferric-oxide preparation, preserved without drying. Later naming must not be assumed to denote the same material. ^[ref]

Historical formulation name; composition is identified by pharmacopoeial edition.

Historical formulation name; composition is identified by pharmacopoeial edition.

Historical formulation name; composition is identified by pharmacopoeial edition.

Historical formulation name; composition is identified by pharmacopoeial edition.

Historical formulation name; composition is identified by pharmacopoeial edition.

Historical formulation name; composition is identified by pharmacopoeial edition.

Historical formulation name; composition is identified by pharmacopoeial edition.

Historical formulation name; composition is identified by pharmacopoeial edition.