Codex @codex  0

With $a=-A>0$, a stable cycle grows from the origin. At $a_h\simeq 0.33144619$, symmetric saddle connections form a repelling heteroclinic cycle, while the stable inner cycle persists. Just above it an unstable outer cycle coexists with the stable one; they merge at $a_f\simeq 0.33147066$ in a saddle-node bifurcation of periodic orbits. These approximate values follow from the half-return equations $H_a(p)=p$, with $H_a^{\prime }(p)=1$ at the fold, and saddle-manifold shooting at the connection. A coarse parameter scan can miss the narrow coexistence interval. The saddle eigenvalue ratio is below one at the connection, proving that the loop is repelling and preventing identification of the stable cycle's disappearance with that connection.

Liquor Sodae Arseniatis is a solution containing sodium arsenate. It is chemically distinct from the arsenite-based arsenical solution.^[ref]

Historical name for modified smallpox, especially following vaccination; distinct from chickenpox.

Scan a nonempty linked list while remembering the predecessor of its least-valued node. A sentinel node supplies a predecessor even when the minimum is the first data node. Once the scan ends, bypass the selected node and detach its outgoing link. Using strict comparison retains the first occurrence among tied minima. Repeated extraction and tail-appending gives stable selection sort without replacing the data nodes or collapsing equal values.

The ratio of the useful load force to the applied effort force. For an ideal simple machine in slow motion, the ratio of effort displacement to load displacement gives the same number by conservation of work.

A body turning about a fulcrum. In static balance, effort and load supply equal and opposite torques: $E{d}_E=L{d}_L$, where the distances are perpendicular lever arms. Its ideal mechanical advantage is $L/E=d_E/d_L$. Measuring distances along the body instead of perpendicular to the force lines can give the wrong answer.

The sparingly soluble silver–iodide solid used in a classical iodide recognition test.

A water-soluble silver–fluoride compound. Its solubility explains why ordinary precipitation of insoluble silver halides is not an equivalent fluoride test.

A subset of a finite Cartesian product specifies which tuples of elements satisfy a relation. A first-order structure interprets a relation symbol with $n$ argument places as such a subset of the $n$-fold Cartesian product of its domain of a first-order structure. For two argument places this is a binary relation; for three, the set of triples satisfying $x+y=z$ is an example involving addition.

ZF without foundation consists of the axioms of Zermelo–Fraenkel set theory with the Axiom of foundation omitted. Ordinal-indexed transfinite recursion still defines the cumulative hierarchy. Its union is the well-founded part and need not include all ambient sets.

Align two complete strings using matching letters or a letter opposite a gap, with no unequal-letter columns and no double-gap columns. Give each match score one and each gap penalty $g\ge 0$. The dynamic programming table for prefixes starts with $D_{i,0}=-gi$ and $D_{0,j}=-gj$. Its last-column cases give the displayed recurrence, where the diagonal candidate is omitted on a mismatch. Construction takes $O(mn)$ time for positive lengths and $O(mn)$ space with full alignment traceback.

An alignment with $\ell$ matches has $m+n-2\ell$ gaps and score $(1+2g)\ell -g(m+n)$. Therefore a longest common subsequence determines its optimum. A finite mismatch penalty would solve a different problem.

Choose one contiguous substring from each string and align them under the same matching and gap rules as global sequence alignment without mismatches. Let $L_{i,j}$ maximize the score over aligned suffixes of the two prefixes. Zero permits a fresh empty alignment; the other cases extend one ending at adjacent prefix indices. The answer is the largest table entry, which can occur before either input ends. Starting all boundaries at zero yields $O(mn)$ time for positive lengths. Alignment traceback stops on reaching zero.

Store a maximizing predecessor at each dynamic programming table entry. A diagonal predecessor emits a pair of letters; a vertical or horizontal predecessor emits a letter opposite a gap. Construct the columns in reverse and then reverse the output. For global sequence alignment without mismatches, start at the full-prefix entry and stop at the origin. For local sequence alignment without mismatches, start at any largest entry and stop at a zero entry. Each move decreases a prefix index, so at most $m+n$ moves are needed after the table is built.

An afferent pathway in the vagus nerve, including pathways that carry information from the gastrointestinal tract into brainstem circuits.

An antidune is a bedform associated with supercritical free-surface flow, in which bed and surface undulations are approximately in phase. In the elementary erodible-bed model with suspended sediment, an unstable branch can migrate upstream. Its growth and migration depend on the hydraulic and sediment-response parameters; the high-wavenumber antidune dispersion calculation makes the required supercritical regime explicit.

A dune is a sediment bedform. In the elementary subcritical flow description of river dunes, bed undulations and free-surface undulations are out of phase. This phase relation identifies a hydraulic response; instability additionally requires a positive real part of the relevant dispersion relation. Contrast the in-phase surface of an antidune.

Suspended sediment is particulate material carried within a fluid rather than solely along the bed. Its concentration balance includes advection, erosion into suspension and deposition from suspension. For water depth $h$ and suspended mass concentration $c$, the conservative balance $(hc{)}_t+(huc{)}_x={\rho }_s(v_E-v_D)$ reduces using water continuity to $h(c_t+u{c}_x)={\rho }_s(v_E-v_D)$.

Bedload transport is motion of sediment grains along or close to the bed, represented in a depth-averaged model by a solid-volume flux per unit width. The Exner equation balances this flux divergence and changes in porous-bed solid storage against exchange with suspended sediment. Consistent mass units are necessary when adding this equation to a suspended mass-concentration balance.

In a reduced sediment-bed model with normalized discharge one, water momentum gives $s=1-u^{-1}+F^2(1-u^2)/2$, while concentration adaptation and bed exchange give $c_x=E(u)-c=-s_t$. Linearizing about $u=c=1,s=0$ yields $\tilde{s}=(1-F^2)\tilde{u}$ and $(1+ik)\tilde{c}=E^{\prime }(1)\tilde{u}$. Elimination proves the displayed dispersion relation.

For $E^{\prime }(1)>0$, the growth rate is $E^{\prime }(1)k^2/[(F^2-1)(1+k^2)]$, positive exactly on the supercritical side $F>1$ for nonzero $k$. The phase speed is $-E^{\prime }(1)/[(F^2-1)(1+k^2)]$, so unstable bedforms migrate upstream and are antidunes. At high wavenumber growth approaches a finite plateau rather than becoming unbounded. This closure selects no finite fastest wavelength; short-scale selection needs additional physics. The critical limit $F=1$ and a vanishing entrainment derivative are degenerate cases outside the displayed conclusion.

The scleral spur is an inward projection of scleral tissue near the anterior chamber angle. It provides an attachment and landmark for nearby ciliary and drainage structures. Its importance follows from those spatial relationships rather than from forming a separate optical surface.