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.

Products of partial breakdown or modification of starch, comprising a range of chain lengths rather than one molecule of fixed composition.

A storage fructan of various plants, chemically distinct from the glucose-based starch polymers.

A fructan is a carbohydrate built largely from linked fructose residues. Inulin is an example; its principal fructose-to-fructose linkages differ from those in other fructan families.

The type of glycosidic bond determines which enzymes can break down a fructan. Knowing only that a plant polymer contains fructose does not specify its digestibility or chain architecture.^[ref]

Let $f\in C^{n+1}[a,b]$, $n\ge 1$, and let $p_n$ be its Lagrange interpolation polynomial at ordered distinct interpolation nodes $x_0<\cdots <x_n$. Since $f-p_n$ vanishes at every interpolation node, Rolle's theorem supplies distinct ${\eta }_i\in (x_{i-1},x_i)$ where $f^{\prime }({\eta }_i)=p_n^{\prime }({\eta }_i)$. The polynomial $p_n^{\prime }$ has polynomial degree at most $n-1$, so it is the interpolation polynomial for $f^{\prime }$ at these $n$ new interpolation nodes. Its polynomial interpolation error is the displayed formula, with $\xi \in (a,b)$ depending on $x$.

Consequently $|f^{\prime }(x)-p_n^{\prime }(x)|\le M{\prod }_i|x-{\eta }_i|/n!\le M(b-a{)}^n/n!$, where $M=\max |f^{(n+1)}|$. The existence of the interpolation nodes is enough for this bound; their explicit locations generally depend on $f$. This proof avoids differentiating an unspecified intermediate point in the original polynomial interpolation error formula.

Every root of the derivative of a nonconstant complex polynomial lies in the convex hull of its roots. If a critical point $\beta$ is itself a root, this is immediate. Otherwise factorization and the product rule giveAfter taking complex conjugates, set $w_k=|\beta -{\alpha }_k{|}^{-2}>0$ to obtain $\beta ={\sum }_k{w}_k{\alpha }_k/{\sum }_k{w}_k$. This is a convex combination, proving the theorem, with roots counted according to their multiplicities. The same formula shows that roots strictly inside an open half-plane or a convex set that is open keep all critical points there. For a degree-one polynomial the derivative has no zeros and the assertion is vacuous.

The real polynomials of degree at most $k$, together with the zero polynomial, form a vector space of dimension $k+1$ with basis $1,x,\dots ,x^k$. Polynomial spaces of lower degree are nested vector subspaces.

Distal symmetric polyneuropathy is a pattern of polyneuropathy affecting distal regions on both sides in a broadly similar distribution. In a length-dependent form, symptoms typically begin in the feet before extending proximally.

The pattern can constrain the mechanism and suggest a systemic process, but it is not itself one cause. Sensory findings, weakness and autonomic involvement must be characterized separately.

For $x^{\prime \prime }+{\omega }^{2}x=\gamma \cos \theta$, $\dot{\theta }=\omega \ne 0$, use the section $\theta =0$ modulo $2\pi$ and $T=2\pi /|\omega |$. Exact resonance produces the secular term $\gamma t\sin (\omega t)/(2\omega )$, giving the displayed return translation. Nonzero forcing makes every section orbit unbounded. At zero forcing the map is the identity. The assumption $\omega \ne 0$ is essential: at zero frequency no transverse phase section exists and the motion is constant-acceleration or free-particle motion.

A patchy pattern of pneumonia centered on airways and adjacent lung tissue. Its distribution differs from confluent lobar consolidation without requiring one specific pathogen.

A pattern of pneumonia involving confluent consolidation of a substantial part of a lung lobe. The distribution is a pathological description rather than a unique identification of the causative organism.

The liver-like firmness of consolidated lung in classical descriptions of pneumonia. It is an appearance, not conversion into liver tissue.^[ref]

Accumulation of pus in a pre-existing anatomical cavity. In pleural disease, it occupies the space around the lung rather than a newly formed cavity within lung tissue, distinguishing pleural empyema from a pulmonary abscess.

The costodiaphragmatic recess is the inferior pleural recess where costal and diaphragmatic pleura meet. The lung expands toward it during deeper inspiration. Its extent therefore reflects the difference between the pleural envelope and the lung border at a particular breathing phase.

In small-strain elastoplasticity, total infinitesimal strain tensor splits into an elastic part and a generally irreversible plastic part. The elastic constitutive equation determines stress from $e^{\mathrm{e}}$, while a yield condition and a plastic flow law determine the evolution of $e^{\mathrm{p}}$. Removing external load need not remove plastic strain, so using Hooke's law on total strain after yielding is insufficient.

A yield criterion specifies the admissible elastic stress region, possibly depending on internal hardening variables $q$. Plastic loading occurs on its boundary. The Tresca yield criterion uses maximum principal-stress difference, while the Von Mises yield criterion uses a quadratic deviatoric-stress invariant. Thus a numerical stress difference must be compared to a yield stress using the convention of the selected criterion.