The living inner parasite layer of an echinococcal hydatid cyst, distinguished from its laminated layer and the surrounding host tissue.

A brood capsule is a small internal structure produced by the germinal tissue of a hydatid cyst, within which protoscolices develop. It distinguishes that internal reproductive organization from the simpler bladder architecture of a coenurus.^[ref]

For real $a_r$, put $h(z)={\sum }_{r=1}^n{a}_r{z}^r$. The quadratic form equals ${\int }_0^{1}h(x{)}^{2}dx$, proving its nonnegativity. Apply the centred logarithmic keyhole identity bound to the entire function $h^2$. Since $|h(e^{i\theta }{)}^2|=|h(e^{i\theta }){|}^2$, orthogonality of the finite Fourier sum gives ${\int }_0^{2\pi }|h(e^{i\theta }){|}^{2}d\theta =2\pi {\sum }_r{a}_r^2$. Multiplication by the contour estimate's factor $1/2$ proves the upper bound. The matrix $(1/(r+s+1))$ is the principal submatrix of a Hilbert matrix obtained by deleting its first row and column.

The reduction of a real square matrix to upper Hessenberg form uses orthogonal similarity transformations. A Householder reflector at column $k$ acts on rows $k+1,\dots ,n$ to annihilate entries below row $k+1$, and its matching right action preserves similarity. Previous zeros are preserved. Rank-one application of each reflector gives total $O(n^3)$ work. Subsequent shifted systems with $H-\lambda I$ can be factored in $O(n^2)$ work each because only one entry per column lies below the diagonal.

When Gaussian elimination without row interchanges is defined, an upper Hessenberg matrix needs at most one elimination per pivot column, because only its first subdiagonal can be nonzero. The LU decomposition has a unit lower bidiagonal matrix $L$ and an upper triangular matrix $U$. If the input matrix is also symmetric, it is a tridiagonal matrix and $U$ is upper bidiagonal.

The nonzero-pivot condition is essential: $\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right)$ is upper Hessenberg and invertible but has no such unpivoted LU decomposition. A pivoted factorization can still be computed, with its permutations recorded explicitly.

A set of universally quantified equality-free first-order clauses is unsatisfiable if and only if some finite set of its ground instances is propositionally unsatisfiable. Equivalently, such a set is satisfiable if and only if it has a Herbrand model. General first-order sentences can first undergo Skolemization. A fresh constant is added if needed to make the Herbrand universe nonempty; first-order equality requires its corresponding equality treatment.

For a satisfiable equality-free universal logical conjunction of Horn clauses, intersecting the sets of true ground atoms of its Herbrand models gives another Herbrand model, contained in every one of them. A ground atom is true in this least Herbrand model exactly when it holds in every Herbrand model. Positive existential sentences are true in the least Herbrand model exactly when they follow from the Horn clauses; allowing negated atoms makes this equivalence fail. For example, the least Herbrand model of $\forall x(P(x)\to Q(x))$ over a single constant $a$ makes both predicates empty and satisfies $\exists x\neg P(x)$, whereas the first-order structure making both predicates universal is a countermodel to that consequence.

A Henkin witness axiom names a witness to an existential sentence using a fresh constant symbol. Freshness is part of the consistency-preservation argument in the Henkin construction.

A full set of first-order sentences has existential witnesses among names in its own first-order language: if it proves an existential sentence, it proves some closed-term instance of the matrix. For a deductively closed set this may be stated by membership of the first-order sentences. In a function-free first-order language the witness is a constant symbol. A first-order consistent complete full first-order theory has a canonical term model.

For $0<a<1$ in the unit disk, the Möbius transformation $(z-a)/(1-az)$ takes $a$ to zero and the left semicircle to the arc centered at $-1$ with half-angle $\arctan ((1-a^2)/(2a))$. Brownian exit from the centered disk is uniform in angle, so conformal invariance of harmonic measure gives the displayed probability. Scaling and the reflection identity for Brownian exit from a half-disc give probability $4\arctan (1/r)/\pi$ of exiting the radius-$r$ circle from start one before reaching the imaginary axis.

For a Gödel sentence $G\leftrightarrow \neg \square G$ with the Hilbert-Bernays-Löb derivability conditions, fix a refutable contradiction sentence $\perp$. Since $\perp \to G$ is provable, $\square \perp \to \square G$, giving $G\to \neg \square \perp$. For the reverse direction, $\square G$ implies both $\square \square G$ and $\square \neg \square G$, hence $\square \perp$. Contraposition gives $\neg \square \perp \to \neg \square G\to G$. This is an internal equivalence of sentences, not a rule asserting the theory's own consistency.

Assuming PA is consistent, the theory $S=\mathrm{PA}+\neg {\mathrm{Con}}_{\mathrm{PA}}$ is consistent by second incompleteness. Its false axiom $G=\neg {\mathrm{Con}}_{\mathrm{PA}}$ is provable, so the standard semantic biconditional $G\leftrightarrow \neg {\mathrm{Pr}}_S(┌G┐)$ is true with both sides false. Thus a true biconditional alone does not make an arbitrary sentence a true canonical Gödel sentence. A canonical Pi-1 construction adds the necessary syntactic or sound-base information. Truth, internal equivalence and canonical construction must be distinguished.

Let $Z=\exp (X-⟨X⟩/2)$ and $dQ=Z_{T}dP$. The Girsanov theorem first makes $N=M-⟨M,X⟩$ a continuous local Q-martingale. Boundedness of $Y$ gives $⟨X{⟩}_T\le C^{2}T$ and ${\mathbb{E}}_P{Z}_T^2\le e^{C^{2}T}$: write $Z_T^2=\mathcal{E}(2X{)}_T{e}^{⟨X{⟩}_T}$ and apply Novikov's condition to $2X$.

The quadratic covariation bound gives ${\sup }_{s\le T}|⟨M,X{⟩}_s|\le C\sqrt{T}⟨M{⟩}_T^{1/2}$. For a square-integrable martingale $M$, both its maximal square moment and its expected terminal bracket are finite. Cauchy-Schwarz inequality with the density $Z_T$ therefore yields ${\mathbb{E}}_Q{\sup }_{s\le T}|N_s|<\infty$. The dominated limit of its stopped conditional identities proves the true martingale property. Mere P-integrability of $M_T$ does not suffice.

Let $W$ be a P-Brownian motion on $[0,1]$ and tilt by the bounded constant drift integrand one, so $dQ=e^{W_1-1/2}dP$. Then $W_s^Q=W_s-s$ is Q-Brownian motion, either by Girsanov theorem or direct Gaussian exponential tilting. Set $a=1/2$,The density of $W_a$ under P is ${\pi }^{-1/2}{e}^{-x^2}$, so ${\mathbb{E}}_{P}H={\pi }^{-1/2}\int (1+x^2{)}^{-1}dx<\infty$. Independence of future Brownian increments makes $M$ a continuous true P-martingale. Under Q, that density is multiplied by $e^{x-1/4}$, so ${\mathbb{E}}_{Q}H=\infty$.

For $s>a$, its compensated local martingale isThe future Q-Brownian increment is independent of the positive variable $H$ and has positive finite mean absolute value. Hence the displayed variable has infinite expected absolute value under Q and is not a true martingale. This shows why the general Girsanov theorem promises a local martingale, while a true-martingale conclusion needs an additional integrability condition such as Girsanov compensation with square-integrable input.

Let $\mathbb{P}(X=n)=p(1-p{)}^n$ for $n\ge 0$ and $Y\mid X=n\sim \mathrm{Binomial}(n,q)$, with $0<p,q<1$. Write $D=p+q-pq$. Summing the joint probability mass function with the differentiated geometric series identity ${\sum }_{n=k}^{\infty }\left(\genfrac{}{}{0ex}{}{n}{k}\right)t^{n-k}=(1-t{)}^{-k-1}$ gives $\mathbb{P}(Y=k)=(p/D)((1-p)q/D{)}^k$. By Bayes' theorem,Thus $X-k$, given $Y=k$, has the failure-count negative binomial distribution for $k+1$ successes and success parameter $D$. Its conditional expectation is $k+(k+1)(1-D)/D$ for $X$ itself. The shift by $k$, the number of retained successes, must be retained when identifying the posterior distribution.

One step of restarted GMRES searches along the current residual. In the displayed nonsingular system, the first residual minimization is ${\min }_{\alpha }[(1-\alpha {)}^2+(1+\alpha {)}^2]$, whose minimizer is zero. Restarting then repeats the same problem forever and never moves. Unrestarted GMRES instead solves exactly at step two, since $b,Ab$ span the entire space. Thus a smaller search-space cost can come at the price of stagnation; nonsingularity alone does not guarantee progress for one-step restarting.

Let $X_i=\theta +E_i$ for $n\ge 2$ independent and identically distributed observations, with $E_i$ having an exponential distribution of unknown scale $\beta >0$. Test the null hypothesis $\theta =0$ within $\theta \ge 0$. For nondegenerate data, the likelihood function isMaximizing over the scale gives $\hat{\beta }(\theta )=\overline{X}-\theta$, and the resulting profile likelihood increases with $\theta$. Thus the unrestricted maximum-likelihood estimates are $\hat{\theta }=X_{(1)}$ and $\hat{\beta }=\overline{X}-X_{(1)}$, while the null estimate is $\tilde{\beta }=\overline{X}$. The likelihood ratio is consequently

Under the null, put $A=n{X}_{(1)}$ and $D={\sum }_i(X_i-X_{(1)})$. The memoryless property gives independent gamma distributions of shapes $1$ and $n-1$ and common scale $\beta$. For a direct verification, fix which observation is the minimum and write its value as $z$ and the other observations as $z+e_j$. Their joint density is proportional to $e^{-nz/\beta }{\prod }_j{e}^{-e_j/\beta }$, which factors into an exponential density for the minimum and independent exponential densities for the excesses. The minimum's index does not affect the resulting laws of $A,D$.

By beta sampling by gamma ratios, $D/(A+D)\sim \mathrm{Beta}(n-1,1)$. ThereforeAn exact test size $\alpha$ is obtained by rejecting for $R\le {\alpha }^{n/(n-1)}$. Equivalently,Although the alternative adds one endpoint parameter, a regular one-parameter Wilks theorem calibration is incorrect: the support depends on the endpoint, and the null lies on its boundary. In particular, $-2\log R$ tends to a chi-squared distribution with two statistical degrees of freedom, not one.

Put $\kappa ={\aleph }_{\omega }$. ZFC gives ${\kappa }^{\kappa }=2^{\kappa }$. König's theorem and the shift of product coordinates give $\kappa <{\prod }_n{\aleph }_n\le {\kappa }^{{\aleph }_0}\le 2^{\kappa }$. The single instance $2^{\kappa }={\kappa }^{+}$ of the generalized continuum hypothesis therefore fixes all three displayed values. The strict lower bound explains why countable exponentiation cannot leave this singular cardinal unchanged.

Let $G=⟨x,y\mid y^2=1⟩$. For each $n\ge 2$, a free basis of a subgroup of rank $n$ isA covering-space proof uses $r=n-1$ spheres double-covering the real projective plane, with antipodal points $a_j,b_j$ over the wedge point. Add lifted $x$ edges in a cyclic order through the $a_j$, and an $x$ loop at each $b_j$. This is a connected $2r$-sheeted cover of a circle wedged with a projective plane. After collapsing one connecting arc in each simply connected sphere, its fundamental group is that of a graph with $r$ vertices and $2r$ edges, hence free of rank $r+1$.

A spanning tree along the first $r-1$ cyclic edges gives the displayed words: the closing cycle is $x^r$, and the loop reached through $a_j,b_j$ is $x^{j}yxy{x}^{-j}$. Injectivity of the induced fundamental-group map of a covering map proves they form a free basis in $G$.

For $I(u)={\int }_a^{b}f(x,u,u^{\prime })dx$ on $W^{1,\infty }$, continuity of $f,f_u,f_v$ on $[a,b]\times {\mathbb{R}}^2$ gives a bounded Fréchet derivative in the displayed form. Fixing $u$ and restricting $\Vert w{\Vert }_{W^{1,\infty }}$ to a bounded neighborhood confines all intermediate pairs $(u+tw,u^{\prime }+t{w}^{\prime })$ to a compact range up to null sets. Uniform continuity of the partial derivatives bounds the integrated first-order remainder by $(b-a)\omega (\Vert w\Vert )(\Vert w{\Vert }_{\infty }+\Vert w^{\prime }{\Vert }_{\infty })$, where $\omega (s)\to 0$. This is little-oh of the Sobolev norm. The result needs no continuous derivative representative for $u$; essential boundedness suffices.