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For a bounded skew-adjoint operator on a complex Hilbert space, the operator exponential converges in norm because its term norms are bounded by $\Vert T{\Vert }^n/n!$. Continuity of the adjoint operator gives $(e^T{)}^{\ast }=e^{T^{\ast }}=e^{-T}$. Absolute convergence permits the Cauchy product of $e^T$ and $e^{-T}$: its degree-$n$ coefficient is $T^n{\sum }_{j=0}^n(-1{)}^j/[j!(n-j)!]$, zero for $n>0$ and the identity for $n=0$. Both multiplication orders therefore give the identity, proving unitarity. The boundedness hypothesis makes this elementary power-series proof valid without domain issues for unbounded generators.

Historical pharmaceutical preparation; its composition and clinical use must be identified by edition. ^[ref]

Historical pharmaceutical preparation; its composition and clinical use must be identified by edition. ^[ref]

For two rods of length $a$ and mass $m$, one hinged to a fixed point and the other to its free end, use the absolute rod angles from the downward vertical. The small-oscillation mass matrix is $m{a}^2\left(\begin{array}{cc}4/3& 1/2\\ 1/2& 1/3\end{array}\right)$ and the stiffness matrix is $(mga/2)\mathrm{diag}(3,1)$. Their generalized eigenvalue equation gives $7{\kappa }^2-42\kappa +27=0$ for $\kappa =a{\omega }^2/g$. The low-frequency mode has angles of the same sign and the high-frequency mode has opposite signs.

Ordinal addition, multiplication and exponentiation are continuous in their right argument by recursion. For fixed $\beta$, $\beta +\alpha$ is normal for every $\beta$, $\beta \alpha$ for $\beta >0$, and ${\beta }^{\alpha }$ for $\beta \ge 2$. Reversing the roles changes the conclusion: $\alpha +\beta$ is not normal for $\beta >0$, while $\alpha \beta$ and ${\alpha }^{\beta }$ are normal only for fixed $\beta =1$. Large principal-ordinal limits expose the failures of continuity; constant zero or one cases fail strict increase.

A Gaussian error is an error term with a normal distribution. Independent Gaussian errors of equal variance turn least squares into maximum likelihood and make standardized linear coefficient estimators exactly normal. Zero mean and common variance alone do not imply Gaussian errors.

A strictly distance-decreasing map decreases the distance between every pair of distinct points. Unlike a contraction mapping, it need not have a single Lipschitz constant below one. Two distinct fixed points would contradict strict decrease, so there is at most one. Existence can fail even on a complete metric space: on $[0,\infty )$, $T(x)=x+1/(1+x)$ has no fixed point, while the mean value theorem and $0\le T^{\prime }(x)<1$ show strict distance decrease.

A field equipped with a Non-Archimedean absolute value is a non-Archimedean valued field. Its metric is $d(x,y)=|x-y|$ and satisfies the ultrametric inequality. The P-adic numbers give the basic mixed-characteristic examples. Completeness means that every Cauchy sequence for this metric has a limit in the field.

The sets $R=\{x:|x|\le 1\}$ and $\mathfrak{m}=\{x:|x|<1\}$ are its valuation ring and maximal ideal. Multiplicativity and the strong triangle inequality prove that $R$ is a ring and $\mathfrak{m}$ is an ideal. A nonzero element of $R$ is invertible in $R$ exactly when its absolute value is one, since $|x^{-1}|=|x{|}^{-1}$. Thus all nonunits are precisely $\mathfrak{m}$, proving that it is the unique maximal ideal and that $R/\mathfrak{m}$ is a field. These definitions concern a valuation, independently of whether the field admits an ordering.

A nilpotent group is finite exactly when its abelianization is finite; finite generation is not a separate hypothesis. Induct on the nilpotency class $c$. The abelian case is immediate. Otherwise $G/{\gamma }_c(G)$ has smaller class and the same abelianization, so is finite. Let $H=G$, $K={\gamma }_{c-1}(G)$. Their commutator subgroup $[H,K]={\gamma }_c(G)$ is central. Hence its commutator map is a homomorphism in each variable. Both centralizer quotients are finite because their kernels contain ${\gamma }_c(G)$. Finitely many pairs of coset representatives therefore generate the abelian group $[H,K]$, and its exponent divides the exponent of $H/C_H(K)$. It is finite, completing the induction.

An element of an associative algebra is square-zero if its square vanishes. Every such element is a nilpotent element. These elements are preserved by an algebra isomorphism, so their structure distinguishes algebras. Their set need not be a vector space: the matrix units $E_{12}$ and $E_{21}$ are square-zero, while their sum has square equal to the identity in $M_2(K)$. In a square-zero ideal, by contrast, every pairwise product vanishes and all its elements are square-zero. In $K[t]/(t^3)$ the square-zero elements are exactly the multiples of $t^2$, over every field: $(yt+z{t}^2{)}^2=y^2{t}^2$ forces $y=0$.

An open neighbourhood of a point $x$ in a topological space $(X,\mathcal{T})$ is an open set $U$ containing $x$. A general neighbourhood need only contain an open neighbourhood; it need not itself be open. A finite intersection of open neighbourhoods of the same point is again an open neighbourhood, by the topology axioms.

From $P$, temporarily assume $\neg P$. Negation elimination rule gives logical falsity; discharge that temporary assumption by negation introduction to obtain $\neg \neg P$. This is valid in Intuitionistic propositional logic, unlike unrestricted double-negation elimination.

Introduce exclusive or from either $P,\neg Q$ or $\neg P,Q$. To eliminate $P\oplus Q$, derive the same conclusion $K$ in two subproofs, one under $P,\neg Q$ and the other under $\neg P,Q$, then discharge both pairs of case assumptions. These rules express exactly the two disjuncts of the displayed disjunctive normal form. Conjunction introduction and disjunction introduction derive that formula in each elimination branch; conversely disjunction elimination and conjunction elimination supply the two introduction cases. This proves equivalence without adding an unproved truth-table inference to a formal derivation.

Derive each side of a logical biconditional under a temporary assumption of the other, then discharge both assumptions. The surrounding assumptions $\Gamma$ remain. This is equivalent to deriving the two implications and combining them as the two directions of equivalence.

The reflexivity rule for first-order equality: any well-formed term $t$ satisfies $t=t$ without additional premises. Together with the equality elimination rule, it derives symmetry: from $s=t$ and $s=s$, substitute in the first designated argument position of $u=s$ to obtain $t=s$. Transitivity follows from $s=t$ and $t=u$ by substitution in the second designated argument position of $s=v$.

Adding unused assumptions to the context of a logical derivation. The original derivation still proves the same conclusion because it need not use every available assumption. When eigenvariables are used, rename them first if necessary to preserve the eigenvariable condition for any newly added assumptions. This explains how disjunction elimination with different branch contexts can be written using a common context containing all their assumptions.

Reading an inference rule backwards to replace a desired conclusion by sufficient subgoals. A conjunction goal can be replaced by its two conjuncts using the conjunction introduction rule; an available disjunction can instead lead to two case subgoals using the disjunction elimination rule. Different choices change the organization and sharing of subproofs. Scope and assumption discharge conditions remain obligations even when a rule is applied backwards during proof search.

Historical pharmaceutical preparation; its composition and clinical use must be identified by edition. ^[ref]

For a diffusion-controlled mushy layer of constant solid fraction $\varphi$, a sharp moving edge $h=2\lambda \sqrt{\kappa t}$ reduces the thermal heat equation to ${\theta }^{\prime \prime }+2\eta {\theta }^{\prime }=0$ and the exterior solute diffusion equation to ${\mathcal{C}}^{\prime \prime }+2Le\eta {\mathcal{C}}^{\prime }=0$, where $Le=\kappa /D$ is the Lewis number. Wall values select an error function in the mush; far-field decay selects complementary error functions in the liquid. Continuity fixes their amplitudes in terms of the interfacial temperature and $\lambda$. The Stefan condition, solute conservation and marginal equilibrium at a mush-liquid boundary give three algebraic/transcendental equations for these two constants and $\varphi$. Derivation follows by ${\eta }_t=-\eta /(2t)$ and ${\eta }_z=1/(2\sqrt{\kappa t})$, followed by one integration $Y^{\prime }=K{e}^{-a{\eta }^2}$. Dividing interface equations can discard degenerate branches, so the undivided equations must be retained when a denominator vanishes. Physical admissibility additionally requires $\lambda >0$ and $0\le \varphi \le 1$.