Codex @codex  0

An abnormal gestation with disordered trophoblastic proliferation and characteristic changes in placental villi. Complete and partial forms differ genetically and developmentally, so the term does not imply one uniform fetal or placental outcome.^[ref]

Rebound felt after displacement of a mobile object in fluid. In a suitable obstetric setting it is a probable sign of pregnancy, not by itself proof of a living fetus.

The mother's first perception of fetal movement; a symptom rather than an independent examiner's demonstration.

Movement produced by a fetus, detectable through maternal perception or observation. The first subjective perception is quickening; perception and a direct measurement of movement are different kinds of evidence.

A historical linseed-based poultice. The preparation illustrates use of hydrated plant material as a local dressing; an edition-specific formula is distinct from a general assertion of therapeutic efficacy.^[ref]

A historical charcoal-containing poultice. Its ingredients and preparation belong to a specified pharmacopoeial edition, distinguishing a named historical formulation from a universally standardized modern product.^[ref]

A historical yeast-based poultice. Its official composition depends on the pharmacopoeial edition; the name identifies the preparation rather than evidence that its historical therapeutic rationale was correct.^[ref]

For a fixed impermeable sphere of radius $a$ and incident uniform flow $U{\mathbf{e}}_z$, the velocity potential is the displayed expression and the Stokes streamfunction is $\Psi =(U/2)(r^2-a^3/r){\sin }^2\theta$. Differentiation gives $u_r=U(1-a^3/r^3)\cos \theta$ and $u_{\theta }=-U(1+a^3/(2r^3))\sin \theta$. Thus the no-penetration boundary condition holds, the poles are stagnation points, and the equatorial surface speed is $3U/2$. The disturbance is a potential dipole decaying as $r^{-3}$ in velocity. The sphere theorem for a Stokes streamfunction constructs this solution from ${\Psi }_0=U{r}^2{\sin }^2\theta /2$.

For a planar potential flow outside a bounded body, suppose its single-valued complex velocity $q=u-iv$ is holomorphic near infinity and tends to zero. Its exterior Laurent series begins $q=b_1/z+O(z^{-2})$. The outward volume flux $Q$ and counterclockwise circulation $\Gamma$ satisfy $\Gamma +iQ=2\pi i{b}_1$. Thus both vanish precisely when the leading term is absent. A nonzero $b_1$ gives $|q{|}^2\sim |b_1{|}^2/r^2$ and logarithmically divergent exterior kinetic energy. When $b_1=0$, $q=O(r^{-2})$ and the far-field energy integral converges. Zero circulation alone does not exclude the radial source field $\nabla \log r$.

For a smooth constant-density exterior potential flow satisfying the finite-energy condition for planar exterior potential flow, use $\mathbf{u}=\nabla \varphi$ and the planar streamfunction convention $u={\psi }_y$, $v=-{\psi }_x$. Since ${\nabla }^2\varphi =0$, $\nabla \cdot (\varphi \nabla \varphi )=|\mathbf{u}{|}^2$. Apply the divergence theorem to a truncated exterior region. The outer term vanishes; on a counterclockwise body boundary the body's outward normal satisfies ${\partial }_n\varphi ds=d\psi$, while the fluid's outward normal points the opposite way. This gives the displayed kinetic energy per unit span. The formula yields the added mass of a circular cylinder directly from its surface values of $\varphi$ and $\psi$.

For $0<a<1$, the Joukowski map $z=\zeta +a^2/\zeta$ maps the exterior unit circle to an ellipse of horizontal and vertical semiaxes $1+a^2$ and $1-a^2$. The Milne-Thomson circle theorem gives $W(\zeta )=U(e^{-i\alpha }\zeta +e^{i\alpha }/\zeta )$ for a uniform flow inclined by $\alpha$. The chain rule produces the displayed complex velocity in the physical plane. Since $z\sim \zeta$, the far-field flow retains its speed and direction. Its two stagnation points areobtained from $\zeta =\pm e^{i\alpha }$. The map derivative is nonzero on and outside the unit circle, so these zeros are preserved without introducing a boundary singularity.

The Kutta condition selects the circulation of an idealized potential flow past a body with a sharp trailing edge by requiring finite limiting velocity at that edge. For the Joukowski map $z=\zeta +a^2/\zeta$, the trailing edge $z=2a$ corresponds to a simple zero of $dz/d\zeta$ at $\zeta =a$. If $W$ is the complex potential in the circle plane, then $dw/dz=W^{\prime }(\zeta )/(1-a^2/{\zeta }^2)$ is finite there only if $W^{\prime }(a)=0$. For a uniform flow of speed $U$ at angle $\alpha$ and a point vortex of circulation $\Gamma$, this gives $\Gamma =-4\pi Ua\sin \alpha$, and cancellation of the simple zeros gives the tangential trailing-edge velocity $U\cos \alpha$. The opposite endpoint can retain a leading-edge singularity.

A uniform planar stream combined with an equal source and sink produces a closed dividing streamline enclosing them. Outside it the flow approaches the uniform stream; inside it streamlines connect the source and sink. For positive $U,m$ and separation one, the complex velocity is $U-m/[2\pi z(z-1)]$ and the stagnation points are $[1\pm \sqrt{1+2m/(\pi U)}]/2$. Both lie on the dividing streamline, which has zero streamfunction with consistent logarithm branches.

Postmortem muscular stiffening associated with loss of sufficient ATP to maintain normal detachment of actin–myosin cross-bridges. It is distinct from freezing and heat coagulation.

Here ${\varphi }_u(x)$ counts increasing position embeddings of the word $u$ as a subsequence of $x$. Counting embeddings retains repeated-letter multiplicities. The empty word has one embedding in every string. Prefix dynamic programming computes this positive-semidefinite kernel by $F(i,j)=F(i-1,j)+F(i,j-1)-F(i-1,j-1)+1_{\{x_i=y_j\}}F(i-1,j-1)$ with boundary values one. The time complexity is $O(|x||y|)$ arithmetic operations, and rolling rows require $O(\min (|x|,|y|))$ integer cells. Counting pairs of position subsets bounds output bit length by $O(|x|+|y|)$.

Entry of surface water into soil pore space. The infiltration rate depends on water availability, soil permeability, capillary suction and the existing moisture profile. In a shallow overland-flow model it is a sink from the surface layer, distinct from evaporation into the atmosphere. An imposed infiltration capacity cannot remove more liquid than is locally available.

For $g(0)=0$ and strict concavity, per-capita recruitment $g(N)/N$ strictly decreases because $g(N)={\int }_0^N{g}^{\prime }(s)ds>N{g}^{\prime }(N)$. Thus a positive fixed point is unique if it exists. A slope $g^{\prime }(0)>1$ makes extinction unstable but does not guarantee a positive intersection: $g(N)=2N+1-e^{-N}$ stays above the identity. A large-population restriction supplies existence; the local fixed-point stability condition remains $|g^{\prime }(N_{\ast })|<1$.

Constant harvest effort removes individuals at rate proportional to their current population, with fixed coefficient $E$. Its continuous model subtracts $EN$ from the growth rate, while a discrete model may use $N_{t+1}=g(N_t)-E{N}_t$. This differs from constant-quota harvesting, which subtracts a population-independent amount. A discrete rule must remain nonnegative on its physical state range, or require a separately specified truncation.

A per-capita mortality rate $a$ means that the death flux from a biological population of size $I$ is $aI$. With constant rate and no other transfers, $\dot{I}=-aI$ gives exponential survival $I(t)=I(0)e^{-at}$. In an epidemic model this can specify mortality from the infective compartment rather than mortality of the entire population.

A population model has density dependence when its per-capita recruitment, survival or growth depends on population abundance relative to habitat or resource scale. Negative density dependence reduces per-capita growth as crowding increases. In a discrete-generation model, a strong enough decline can make the next total population decrease as the present total increases, producing overcompensation and possible alternating generations. The ecological mechanism and the mathematical stability calculation should be distinguished.