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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.

Echinococcus multilocularis is a tapeworm whose larval tissue stage grows through many small interconnected compartments, commonly involving the liver. This growth pattern differs from the more bounded cyst classically associated with Echinococcus granulosus.^[ref]

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.

A digital counter following a Gray code, so adjacent output states differ in exactly one bit. The two-bit cycle $00,01,11,10$ is implemented by D flip-flops with next-state functions $G^{+}=H$, $H^{+}=\neg G$.

Coupled digital counters sharing one clock, with a slower stage enabled by a terminal-state decode of a faster stage. Both stages sample the old state at the same clock edge. An enable changes whether a stage advances; it does not create a separate clock from a combinational decode.

A digital comparator processes one pair of bits per clock cycle, storing a comparison result in a finite-state machine. For least-significant-first unsigned input, a new unequal pair overrides the stored result, while equal bits preserve it: the newest unequal pair has the greatest place value seen so far.

If $W$ is an invariant subspace for a finite-dimensional diagonalizable linear map $T$, its restriction is diagonalizable over the same field. The minimal polynomial of $T{|}_W$ divides that of $T$: an annihilating polynomial for $T$ still annihilates the restriction, and polynomial division by its minimal polynomial proves divisibility. A divisor of a product of distinct linear polynomials remains a product of distinct linear polynomials, proving the assertion. This is the key step in simultaneous diagonalization of commuting linear maps.

For a matrix over a field $F$, diagonalizability requires an eigenbasis with entries in $F$ and eigenvalues in $F$. Passing to a larger field can supply missing roots of the characteristic polynomial. For example,is not a diagonalizable matrix over the rational numbers, since the quadratic factor has no rational root. It is a diagonalizable matrix over the real numbers and complex numbers, since its three eigenvalues are distinct. More generally, diagonalizability is equivalent to the minimal polynomial splitting into distinct linear factors over the specified field.

For an integer matrix, reduction modulo a prime number is a different operation from extension of its coefficient field, and can merge eigenvalues. In the same example modulo $5$, ${\chi }_A(t)=(t-1)(t-3{)}^2$, while $\ker (A-3I)$ is spanned by $(1,3,4{)}^T$. Thus the geometric multiplicity of $3$ is one and its algebraic multiplicity is two, preventing diagonalizability even though all eigenvalues belong to the finite field.

A finite-dimensional linear operator over a field is diagonalizable exactly when its minimal polynomial splits into distinct linear factors. For the forward implication, a polynomial annihilates a diagonal matrix precisely when it vanishes at every distinct diagonal entry. Conversely, if $m_T={\prod }_j(t-{\lambda }_j)$ with distinct roots, the Chinese remainder theorem yields a direct sum of the kernels of $T-{\lambda }_{j}I$. Each kernel is an eigenspace, so concatenating their bases diagonalizes $T$.

Let $S$ send successive standard basis vectors of $K^n$ to the next one, wrapping the last to the first. Its minimal polynomial is $X^n-1$: this polynomial annihilates it, and the cyclic vectors $e_1,S{e}_1,\dots ,S^{n-1}{e}_1$ form a basis, excluding any annihilator of smaller degree. The diagonalizability criterion using the minimal polynomial gives the displayed equivalence. A splitting polynomial is square-free exactly when the characteristic of a field does not divide $n$, as follows by taking its derivative $n{X}^{n-1}$ and checking common roots. Splitting over the specified field is a separate requirement. For $n=3$, the shift is diagonalizable over the complex numbers and ${\mathbb{F}}_7$, but not over ${\mathbb{F}}_2$ or ${\mathbb{F}}_3$.

A definable set has parameter-free definability when its defining first-order formula uses only the original first-order language, with no additional constants naming elements. Constants already present in the language are allowed. Every such set is invariant under all automorphisms of a first-order structure. For example, zero in the constant-free additive integers is defined by $x+x=x$, while the singleton containing one is not parameter-free definable because negation is an additive automorphism moving it. Invariance is a necessary condition in every structure, and becomes sufficient for finite sets in a countable omega-saturated model.