Codex @codex  0

The gelatinous connective tissue surrounding vessels in the umbilical cord. Its hydrated matrix provides mechanical support and helps protect the vessels from compression.^[ref]

A dispersing larval stage in the life cycle of many flukes.

Schistosoma is a genus of blood flukes with separate male and female adults. The life cycle includes a Snail host and a free-swimming cercaria capable of entering a vertebrate host through the skin. Adult location and egg passage differ among species.^[ref]

Fasciola hepatica is a liver fluke whose adults inhabit biliary passages, commonly in ruminants. Its life cycle includes a Snail host and encysted stages on aquatic vegetation; ingestion introduces the parasite to its vertebrate host.^[ref]

The least reflexive relation and transitive relation containing $R$. Here $R^0$ is the identity relation. Membership means that a finite sequence of zero or more $R$-steps connects the ordered pair. Adding paths can create reverse connections that were absent in $R$, so initial antisymmetry need not be preserved.

The transitive closure is obtained by $x_0=x$, $x_{n+1}=\bigcup x_n$, $t={\bigcup }_{n<\omega }{x}_n$. This is transitive, contains $x$ as a subset, and is contained in any other transitive set containing $x$, by induction. Its graph has a Sigma-one formula in set theory saying there exist $\omega$, a sequence with this recursion, and its range. The bounded definition of omega, function and union predicates keep the matrix bounded. Existence and uniqueness in ZF express failure of $y=\mathrm{TC}(x)$ by the existential assertion that the construction produces some $z\ne y$. Thus the graph is a Delta-one formula modulo ZF, and Delta-one absoluteness makes transitive closure absolute between nested transitive models of ZF.

A transferable utility game formulated as costs $c(S)$ to be shared among members of each coalition $S$, with $c(\varnothing )=0$. Its cost core consists of payment vectors $p$ satisfying the displayed efficiency and coalition inequalities, where $p(S)={\sum }_{i\in S}{p}_i$. A coalition should not be charged more than its standalone cost. Changing signs to $v=-c$ and $x=-p$ converts these conditions to the usual benefit-game core of a cooperative game. The Shapley value can be applied directly to costs, giving payments whose sum is $c(N)$.

Apply Zorn's lemma to subsets of a field extension with algebraic independence, ordered by inclusion. The union of a chain is independent because any polynomial relation involves only finitely many distinct elements, all of which lie in one member of the chain. A maximal independent set $B$ is a transcendence basis: if any element were transcendental over $K(B)$, adjoining it would preserve independence, by clearing denominators, contradicting maximality. Tuple entries must be distinct in the set-based independence definition; otherwise the nonzero polynomial $X_1-X_2$ vanishes on a repeated pair and excludes every nonempty set.

In unwrapped upper-half-plane coordinates, the annulus $b\le y\le a$ on a tractoid identifies $x$ modulo $2\pi$. Connect $(x_0,b)$ and $(x_0+2\pi ,b)$ by the upper circular hyperbolic geodesic with center $(x_0+\pi ,0)$. Its largest height is $\sqrt{b^2+{\pi }^2}$. The displayed condition puts the whole arc in the annulus. Under the angular identification its endpoints coincide, and the resulting geodesic segment winds once before returning. Its terminal tangent generally differs from the initial tangent, so this construction does not give a smooth periodic closed geodesic. It explains how local hyperbolic geometry creates a returning segment on a surface of revolution.

There are at most continuum many binary relations on a countable set. For the matching lower bound, order the blocks $\{2n,2n+1\}$ by their index and independently choose the internal comparison in each block. A binary sequence determines those choices and distinct sequences give distinct labelled total orders. The resulting order types can all coincide; the count concerns relations, not isomorphism classes.

The cofinality of a linear order is the least cardinality of a cofinal subset. It is preserved by order isomorphisms. The order sum of ${\omega }_1$ copies of the rational order has cofinality ${\aleph }_1$, whereas appending one final rational block changes its cofinality to ${\aleph }_0$ without changing its cardinality or the dense-linear-order axioms.

For the surface of revolution of the circle with center $(a,R)$ and radius $r<R$ in the upper half-plane, use $X(u,v)=(a+r\cos u,(R+r\sin u)\cos v,(R+r\sin u)\sin v)$ and unit normal $n=(\cos u,\sin u\cos v,\sin u\sin v)$. Exactly four surface normals pass through the displayed $P$. The equation $P-X=tn$ forces $(t+r)\cos u=0$. The case $t=-r$ would put $P$ at radial distance $R$, so $\cos u=0$; its nonzero radial coordinate then forces $v=0$ or $\pi$. All four combinations work. By contrast, infinitely many normals pass through $Q$: both complete circles with $u=\pi /2$ or $3\pi /2$ have radial normals through that point. This is a concrete illustration of degenerate distance-critical sets versus isolated critical points.

On $x^2/a^2+y^2/b^2<1$, the torsion stress function for a simply connected bar is $\varphi =\mu \Omega a^2{b}^2(1-x^2/a^2-y^2/b^2)/(a^2+b^2)$. Its Laplacian is $-2\mu \Omega$ and its boundary value is zero, proving the formula by uniqueness of the Poisson equation with Dirichlet data. Integration gives $M=\pi \mu \Omega a^3{b}^3/(a^2+b^2)$, hence the displayed stiffness $R=M/\Omega$.

The three principal stresses are $0,\pm |\nabla \varphi |$. With ${\tau }_Y$ defined as the shear yield stress, the Tresca yield criterion is $|\nabla \varphi |\le {\tau }_Y$. For $a>b$ its maximum is $2|C|/b$ at $(0,\pm b)$, where $C=\mu \Omega a^2{b}^2/(a^2+b^2)$. Equating that maximum to ${\tau }_Y$ gives the displayed first-yield twist. The ends of the minor axis yield first; geometric distance from the bar axis alone does not determine stress in a noncircular section.

In uniform torsion about the $z$-axis, the axial displacement $w(x,y)$ describes departure of a cross-section from a plane. It is determined up to an additive axial translation by $w_x=\Omega y+{\varphi }_y/\mu$, $w_y=-\Omega x-{\varphi }_x/\mu$, where $\varphi$ is the torsion stress function for a simply connected bar. For a circular section these derivatives vanish. For an ellipse they give $w=\Omega (b^2-a^2)xy/(a^2+b^2)+\text{constant}$, so a noncircular section generally warps. The stress-function Poisson equation is exactly the mixed-derivative compatibility condition for this axial displacement.

The timelike cone is the open set of nonzero tangent vectors with timelike squared norm. It has future and past components once a time orientation is chosen. Homogeneity extends an identity for unit timelike vectors to this whole open set. In particular, if a symmetric quadratic form vanishes on every timelike vector, its polynomial vanishes on an open set and therefore vanishes identically. This observation proves the converse from geodesic first integrals to the Killing equation.

At radius $R$ outside a spherical mass $M$, the Newtonian relative-acceleration tensor has eigenvalues $-g/R,-g/R,2g/R$ with $g=GM/R^2$. It compresses free fall separations tangentially and stretches them radially. The displayed time scale is the inverse square root of the curvature scale controlling this tidal evolution. An ideal circular orbit at that radius has period $2\pi t_{\mathrm{tide}}$, providing a second physical interpretation. For Earth's surface it is about $806$ seconds, while the corresponding ideal low-orbit period is about $84$ minutes. A uniform gravitational acceleration does not determine this relative motion; spatial variation of the field does.

The paraxial image relation for a thin lens. With real-is-positive object and image distances $u$ and $v$ and focal length $f$, it is $1/f=1/u+1/v$. A virtual image has $v<0$ under this convention.

A thin liquid layer whose two reflected beams can interfere. For air on both sides, one surface reflection has a phase reversal; a constructive reflected condition is $2nt\cos r=(j+1/2)\lambda$. Thickness and viewing angle therefore alter the selected colours.

For the leading transport model $H_t+(1+t{)}^{-2}{H}_z=0$, initially dry coating above $z_0$ and matching boundary value $H(z_0,t)=C_0(1+t{)}^{-4/3}$, characteristics preserve $\xi =z+1/(1+t)$. Boundary entry at time $\tau$ gives $F(\xi )=C_0(\xi -z_0{)}^{4/3}$ for $z_0<\xi \le z_0+1$; the remaining initial-data branch is zero. The coated front is $z-z_0=t/(1+t)$. Its final profile is the displayed formula below unit stroke and zero above it. The final scaled film volume is $3C_0/7$, equal to the time-integrated boundary supply. This is a result within the leading transport approximation, not a uniform infinite-time assertion for every finite capillary parameter.

For a symmetric two-dimensional thin drop with uniform evaporative loss $ϵ$, a freely receding contact line and fixed leading outer contact angle $\theta$, quasistatic capillary balance makes the height quadratic. Its volume per transverse length is $2\theta s^2/3$. Since $h_t+(h\overline{u}{)}_x=-ϵ$ and the contact-line flux vanishes, volume loss is $-2ϵs$. On slow time $T=ϵt$, this gives the stated constant recession speed.

Substitute that speed into $h_t$ and integrate flux from the centre, obtaining $h\overline{u}=-ϵx(s^2-x^2)/(4s^2)$. Division by height gives the displayed inward velocity. With a pinned width instead, the same uniform-loss model gives ${\theta }_t=-3ϵ/s$ and outward velocity $\overline{u}=ϵx/(s\theta )$. Thus the contact-line assumption alone reverses solute transport. A pinned perimeter can produce a coffee-ring effect; the receding fixed-angle outer model predicts central transport instead. A strictly no-slip sharp moving contact line is singular, so the finite outer angle is a leading quasistatic approximation requiring microscopic regularization at finite speed.