Codex @codex  0

A bladder-like larval form with an invaginated scolex, occurring in the life cycles of some Taenia.

The broad fish tapeworm, historically described under different generic names.

A larval bladder with multiple protoscolices, distinct from a single-scolex cysticercus and the brood capsule architecture of a typical hydatid cyst.

The developing attachment region of certain larval tapeworms, capable of giving rise to the scolex of an adult in a suitable definitive host.

Let $X_k$ be independent and identically distributed random variables with expected value zero, variance one, and a moment-generating function finite in a neighborhood of zero. Each row of deterministic real weights $a_{n,k}$ is finite. If ${\max }_k|a_{n,k}|\to 0$ and ${\sum }_k{a}_{n,k}^2\to v<\infty$, then the weighted sums converge in distribution to a normal distribution with mean zero and variance $v$; $v=0$ denotes the constant zero.

By the quadratic expansion of a cumulant-generating function, write $K(u)=u^2/2+r(u)$ with $r(u)=o(u^2)$. For a fixed real $t$, put ${\eta }_n={\sup }_{0<|u|\le |t|{\max }_k|a_{n,k}|}|r(u)|/u^2$, with the supremum taken as zero if its range is empty. Then ${\eta }_n\to 0$ andThus the log moment-generating function of the sum tends to $v{t}^2/2$. The convergence theorem for moment-generating functions proves the conclusion. The supremum is the step that makes the remainder estimate uniform across the row; a separate little-o assertion for each term would not justify summing an increasing number of terms.

For the comparison map $\pi :\beta X\to K$, the fibers of compactification comparison maps identity ${\pi }^{-1}(X)=X$ pulls a G-delta set presentation of $X$ back from $K$. Conversely if $\beta X\setminus X={\bigcup }_n{F}_n$ with $F_n$ closed compact, then $K\setminus X={\bigcup }_n\pi (F_n)$. Each image is closed and misses $X$. Thus $X$ is a G-delta set in one Hausdorff compactification exactly when it is in every one.

Write the dense space $X$ as ${\bigcap }_n{G}_n$ in compact Hausdorff $K$. Every $G_n$ is dense and open. A dense open subset $U_n$ of $X$ has the form $X\cap O_n$ with $O_n$ dense and open in $K$. The compact Baire category theorem makes ${\bigcap }_n(G_n\cap O_n)={\bigcap }_n{U}_n$ dense in $K$, hence in $X$. Therefore every Čech-complete space is a Baire space. The rationals are not a Baire space, as deleting their points one at a time gives dense open sets with empty intersection.

If a theory without finite models is categorical in an infinite cardinal at least the language size, it is complete. Otherwise its two consistent extensions deciding a sentence oppositely have infinite models. The Löwenheim-Skolem theorems produce models of both of the specified cardinality, contradicting their forced isomorphism. Categoricity includes existence at that cardinality, so it is not vacuous.

For any infinite cardinal $\kappa$, insert a dense order without endpoints of size $\kappa$ into an irrational cut of the named rationals. Compare it with an order of the same size having a dense tail above every named rational. Both models satisfy the complete named rational diagram by quantifier elimination, but only the second realizes the type of being above every named rational. This is an isomorphism invariant over the constants, even though it is an infinite type rather than one defining sentence. Thus the diagram is categorical in no cardinal with a model.

With Fibonacci indexing $F_0=F_1=1$, powers of $A=\left(\begin{array}{cc}2& 1\\ 1& 1\end{array}\right)$ satisfy $A^n=\left(\begin{array}{cc}{F}_{2n}& F_{2n-1}\\ F_{2n-1}& F_{2n-2}\end{array}\right)$ for $n\ge 1$. The base case is immediate. Multiplication by $A$ advances each entry by the two-index Fibonacci recurrence, proving the identity by induction.

Two sets are equinumerous when a bijection exists between them. This is equality of their cardinalities.

For an infinite well-orderable cardinal number $\kappa$, each strict total order is a subset of ${\kappa }^2$, giving at most $2^{\kappa }$ possibilities by the square of an infinite cardinal. For a lower bound, order $\kappa$ two-element blocks by the ordinal index and independently reverse each block according to a subset of $\kappa$. Transport along $\kappa \times 2\cong \kappa$. Distinct subsets give distinct orders. The usual axiom of choice development supplies these product identities; for well-orderable cardinals they are already provable in ZF.

The cardinal sum is the cardinality of the disjoint union of sets of the specified sizes. Using initial ordinals, this is $|\{(i,\xi ):i\in I,\xi <{\kappa }_i\}|$. The tags distinguish overlapping summands. The empty sum is zero.

The cardinal product is the cardinality of the set of functions $f$ on the index set with $f(i)<{\kappa }_i$ at every coordinate. The empty product is one, because it contains the empty function. A zero factor makes a nonempty-index product zero. In ZFC, a family of nonzero cardinals has nonzero product by the axiom of choice.

Cardinal exponentiation counts functions from a set of size $\lambda$ to one of size $\kappa$. The currying law for cardinal exponentiation gives $({\kappa }^{\lambda }{)}^{\mu }={\kappa }^{\lambda \cdot \mu }$. These are cardinal operations, distinct from ordinal exponentiation. For example the Generalized continuum hypothesis specifies $2^{\kappa }={\kappa }^{+}$ at infinite cardinals.

Starting from $[0,1]$, remove the central fraction $1/(2^n+2)$ of every stage-$n$ interval. The total length obeys $L_{n+1}/L_n=(2^n+1)/(2^n+2)$, which telescopes to the displayed value. The limiting intersection has Lebesgue outer measure at most $1/3$ by covering it with the stage intervals and slightly enlarging them to open intervals. This argument uses no integration.

For $\xi =x+e^y$, $\eta =x-e^y$, the Jacobian determinant is $-2e^y$ and the transformed domain is $\xi >\eta$. The chain rule gives $u_{yy}=e^{2y}(U_{\xi \xi }-2U_{\xi \eta }+U_{\eta \eta })+e^y(U_{\xi }-U_{\eta })$, while $u_y=e^y(U_{\xi }-U_{\eta })$. The drift terms cancel, leaving $U_{\xi \eta }=0$. Thus the general solution is $F(x+e^y)+G(x-e^y)$; its characteristic curves are the level sets of the two new coordinates.

For a second-order hyperbolic partial differential equation, the usual reduction to a mixed derivative uses twice continuously differentiable characteristic coordinates. Continuity of the principal coefficients alone does not ensure this regularity, even locally. Consider $R=0$, $S=1/2$ and $T=|x|$ near the origin; the discriminant $S^2-RT$ is the positive constant $1/4$. A characteristic coordinate must satisfyThe two characteristic families have first integrals $x$ and $y-x|x|/2$. Locally, any coordinate transverse to the first family has the form $\eta =h(y-x|x|/2)$ with $h^{\prime }\ne 0$. If this were twice continuously differentiable, its restriction at $x=0$ would make $h$ twice continuously differentiable. But its second $x$ derivative approaches $-h^{\prime }(y)$ from the right and $h^{\prime }(y)$ from the left, a contradiction. Independent characteristic coordinates cannot both choose the other family. Thus no twice continuously differentiable characteristic coordinate system exists near the origin.

With sufficiently smooth coefficients, the two distinct characteristic direction fields do yield smooth local coordinates. If $J$ is their Jacobian determinant and $B$ is the transformed mixed coefficient, then $B^2=J^2(S^2-RT)>0$. Division by $2B$ gives the mixed-derivative canonical form, including all lower-order terms produced by the chain rule. This is a local result; global intersections of characteristic curves can create separate obstructions.