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Let $R$ be a Euclidean domain with Euclidean function $\nu$, and let $I$ be a nonzero ideal. Choose $b\in I\setminus \{0\}$ minimizing the nonnegative integer $\nu (b)$. For $a\in I$, divide $a=qb+r$ with $r=0$ or $\nu (r)<\nu (b)$. Since $r=a-qb\in I$, the latter case contradicts minimality. Hence every $a$ is a multiple of $b$, giving $I=(b)$. The zero ideal is $(0)$, so every ideal is a principal ideal and $R$ is a principal ideal domain.

For a field $F$, polynomial division makes the one-variable polynomial ring $F[X]$ a Euclidean domain with Euclidean function equal to the polynomial degree. For a divisor $g\ne 0$, subtract a suitable scalar multiple of $X^{\mathrm{deg}f-\mathrm{deg}g}g$ to cancel the leading coefficient of the current remainder. Its degree strictly decreases. Iteration terminates at $f=qg+r$, where $r=0$ or $\mathrm{deg}r<\mathrm{deg}g$. Division by the leading coefficient is valid because it is nonzero in a field.

The least equality on lambda terms closed under lambda application and lambda abstraction and containing beta equivalence and eta conversion, with bound names identified by alpha equivalence.

A congruence relation on a first-order structure is an equivalence relation compatible with each interpreted function and relation. Equivalent input tuples give equivalent function outputs, and an atomic relation has the same truth value on equivalent input tuples. This compatibility makes the functions and relations well-defined on the quotient set. In a Henkin term model with equality, membership of a formal equality in the maximal consistent theory gives precisely the equivalence relation used for this quotient.

An automaton transition that consumes no input symbol. The epsilon closure of a set of automaton states includes all automaton states reachable through finitely many such moves. Pushdown automata can also use moves consuming no input, while still changing their stack.

Take two identical normalized entangled qubit pairs ${\sum }_{r=0}^1{b}_r|{\xi }_r{\eta }_r⟩$. Project their two $\eta$ systems onto $|\Phi ⟩=2^{-1/2}{\sum }_k|{\eta }_k{\eta }_k⟩$. The overlap with $|{\eta }_r{\eta }_s⟩$ is ${\delta }_{rs}/\sqrt{2}$, so the unnormalized remote state is ${\sum }_r{b}_r^2|{\xi }_r{\xi }_r⟩/\sqrt{2}$. Its squared norm gives success probability $\frac{1}{2}{\sum }_r|b_r{|}^4$, and its normalized Schmidt coefficients are $b_r^2/({\sum }_s|b_s{|}^4{)}^{1/2}$. At least two survive whenever the original pair was entangled. This is a conditional entanglement swapping result; disregarding the local measurement outcome leaves the remote reduced state unchanged. The coefficients are squared amplitudes, not their absolute squares, because the two input kets multiply before the Bell-state contraction.

Use antiplane energy $\frac{\mu }{2}\int |\nabla u{|}^2$ and normalize the tip singularity as $u=k\sqrt{r}\sin (\theta /2)$ plus a less singular term. The energy release rate is the small-circle flux $\int [\mathcal{W}{n}_x-\mu u_r{u}_x],r,d\theta$. Here $\mathcal{W}=\mu k^2/(8r)$, $u_r=k\sin (\theta /2)/(2\sqrt{r})$ and $u_x=-k\sin (\theta /2)/(2\sqrt{r})$. Integrating over $(-\pi ,\pi )$ gives the displayed coefficient. The full three-dimensional antiplane stress has squared Frobenius norm $2{\mu }^2|\nabla u{|}^2$, so its leading norm ahead of the tip is $\mu |k|/\sqrt{2r}$. These constants depend on the normalization of the stress intensity factor.

A variable used as an arbitrary parameter in a natural deduction subproof, subject to an eigenvariable condition. For universal introduction, it must be free in no undischarged assumption. For existential elimination, a temporary witness must also be free in no conclusion exported from its subproof. These restrictions distinguish an arbitrary object from one constrained by a special hypothesis.

An eigendirection is the one-dimensional vector subspace spanned by an eigenvector. Multiplying the eigenvector by a nonzero scalar leaves this direction unchanged. In a planar linear dynamical system, a negative real eigenvalue gives a direction of exponential approach to the equilibrium point, while a positive one gives exponential departure. For a hyperbolic saddle equilibrium, the nonlinear stable manifold and unstable manifold are tangent to the corresponding eigendirections.

The classical cystic-hydatid organism; historical usage often covers a complex of organisms distinguished more finely in current classification.

A larval cyst; human hydatid cysts commonly involve the liver or lungs.

On $(0,1]$ with Lebesgue probability and the usual dyadic filtration, each displayed support is a dyadic atom. Its next split has one child of doubled height and one of zero height, preserving the conditional average. Hence the sequence is a martingale from time one. To extend it to the trivial time-zero sigma-algebra one must set $X_0=1$: applying the same displayed formula at zero instead gives $1_{(0,1/2]}$, which is not adapted and has the wrong expected value. The repaired martingale tends to zero almost surely while its expected values remain one, so it is not uniformly integrable.

For a uniformly integrable bounded sequence of derivatives on a compact interval and a fixed initial value, its primitives are uniformly bounded and equicontinuous. The increment over any short interval is bounded by the integral of the absolute derivative on that interval. The Arzelà-Ascoli theorem gives relative uniform compactness. If the derivatives converge weakly in $L^1$, tests by interval indicators identify every pointwise limit with $u(x)=\alpha +{\int }_a^{x}v$. Hence every convergent subsequence has this same limit and the primitives converge uniformly. The limit belongs to $W^{1,1}$ and retains any common terminal value. Uniform integrability alone gives no essential-supremum bound on its derivative, so this compactness statement does not assert a Lipschitz limit.

In a metric space $X$, let the nonempty compact set $K$ lie in the open set $\Omega$. If $F=X\setminus \Omega$ is nonempty, the distance to a set $d(x,F)$ is a continuous function of $x$: the triangle inequality gives $|d(x,F)-d(y,F)|\le d(x,y)$. Since $F$ is a closed set, this distance is strictly positive on $K$. The extreme value theorem attains its minimum on $K$, soIf $F$ is empty, the extended-real convention gives $d(K,F)=+\infty$. This result provides a single positive neighbourhood radius that stays inside $\Omega$ at every point of $K$. Compactness is essential: in the real line $K=\{1/n:n\ge 1\}\subset (0,2)$ has zero distance between two sets from $(-\infty ,0]\cup [2,\infty )$, although each point of $K$ has positive distance to a set from that complement.

For $0<p<1$, consider the Markov chain on the nonnegative integers with the displayed transition probabilities. From zero it starts a new upward run; from each positive state it either increments or resets. Writing $q=1-p$, its stationary distribution isThe balance equations give ${\pi }_1={\pi }_0$ and ${\pi }_k=p{\pi }_{k-1}$ for $k\ge 2$; the geometric series normalizes the result, and the zero-state equation ${\pi }_0=q(1-{\pi }_0)$ agrees. It is an irreducible Markov chain and is positive recurrent. Equivalently, its first return time to zero is one plus a geometric random variable with success probability $q$, so its mean recurrence time is $1+1/q$. Return paths of lengths two and three also make it aperiodic.

For the five-point discretization $L_h$ of $-\Delta$ on the unit square, the displayed quadratic polynomial has $L_{h}B=1$ exactly, nonnegative boundary values and $0\le B\le 1/8$. If $L_{h}e=\tau$ and $e$ has zero boundary values, the discrete maximum principle applied to $\pm e-\Vert \tau {\Vert }_{\infty }B$ gives $|e|\le \Vert \tau {\Vert }_{\infty }B\le \Vert \tau {\Vert }_{\infty }/8$. The same estimate holds for $L_h+I$, since $(L_h+I)B=1+B\ge 1$. This turns a uniform local truncation error bound into a pointwise global error bound without inverting the full matrix.

A finite elliptic stencil has boundary-reaching connectivity when every unknown node has a directed path of strictly positive neighbor coefficients to a prescribed boundary node. For $L{\psi }_p={\sum }_{r\ne p}{c}_{p,r}{\psi }_r-c_{p,p}{\psi }_p$, with $c_{p,r}\ge 0$ and $c_{p,p}\ge {\sum }_{r\ne p}{c}_{p,r}$, a nonnegative interior maximum of a subsolution $L\psi \ge 0$ propagates along every positive-weight edge. The path condition carries it to the boundary, proving the discrete maximum principle. Connectedness of the underlying geometric domain alone does not imply this graph property.

For a monotone discrete elliptic operator, a nonnegative mesh function $\varphi$ with $L_h\varphi \ge d>0$ can bound errors through the discrete maximum principle. If $L_{h}e=\tau$ and the boundary error is zero, the subsolutions $\pm e+(\Vert \tau {\Vert }_{\infty }/d)\varphi$ yield $\Vert e{\Vert }_{\infty }\le (\Vert \tau {\Vert }_{\infty }/d){\max }_{\partial \Omega }\varphi$. On different subregions one may use different lower bounds for $L_h\varphi$ and different truncation-error bounds, taking the maximum of their ratios. Nonnegativity, or an explicit subtraction of the minimum when $L_{h}1=0$, is essential for the unshifted estimate.

In a finite nonempty directed set, repeatedly take an internal common upper bound of the element reached so far and the next element. The final member is above every element of the directed set and is therefore its greatest element and least upper bound.