Codex @codex  0

Exchange two equally long disjoint segments of an array, preserving the order within each segment. Applying one swap at each corresponding offset exchanges length-$n$ blocks in $n$ swaps and constant auxiliary storage. Disjointness ensures that an earlier swap does not overwrite an original value needed by a later pair.

A curved free interface of a liquid near a solid wall. Its shape reflects surface tension, contact angle and hydrostatic pressure.

A sulfuric-acid monoester involved in a traditional account of producing diethyl ether from ethanol.

A sugar capable of reducing a suitable oxidizing reagent under the stated test conditions; glucose is an example, but a positive test is not specific to it.

A type $S$ is a supertype of $T$ when $T$ is a subtype of $S$. A lower type bound requires an admissible type argument to lie above a specified type in this subtyping relation.

A lower bound requires an admissible type argument to be a supertype of a specified type. In Scala, [V >: U] means $U<:V$. For an immutable software class covariant in U, a method accepting a new output value cannot consume U directly in a contravariant method-parameter position. Instead, add[V >: U](x: V) accepts a possibly wider value and returns a fresh instance parameterized by V. The original narrower instance remains unchanged. The variance-position rule and subtyping justify this widening; a bound alone does not authorize mutating an existing instance through a widened reference.^[ref]

In positional subtraction, a higher place supplies one unit of its place value when the current place cannot subtract the requested digit. In binary representation, subtraction of one, a low bit toggles at every operation and the borrow continues through each zero bit, stopping at the first one bit. Thus decrementing a binary counter toggles position $i$ exactly when all less significant bits were zero. This is the downward-counting counterpart of a carry propagating through one bits during incrementing.

The singular equation $2x{w}^{\prime \prime }-w^{\prime }+\lambda x^{2}w=0$ becomes $W^{\prime \prime }+(2\lambda /9)W=0$ under $z=x^{3/2}$, $w(x)=W(z)$. Its zero-endpoint finite-energy quotient transforms as $2{\int }_0^1{x}^{-1/2}(w^{\prime }{)}^{2}dx/{\int }_0^1{x}^{1/2}{w}^{2}dx=(9/2){\int }_0^1(W^{\prime }{)}^{2}dz/{\int }_0^1{W}^{2}dz$. The Poincaré inequality therefore proves the sharp minimum $9{\pi }^2/2$, attained by $w=\sin (\pi x^{3/2})$. This coordinate change turns an apparently singular weighted variational problem into the usual interval energy problem. The admissible eigenfunction is $O(x^{3/2})$, making the integration-by-parts boundary term $w{x}^{-1/2}{w}^{\prime }$ vanish and both energy integrals converge.

For data Natural = Zero | Succ Natural, replace Zero by a base value and Succ by a step function: foldNat g a Zero = a and foldNat g a (Succ n) = g (foldNat g a n). On a finite total natural number $n$ this iterates $g$ exactly $n$ times.

A layered biomineral in shells and pearls. Repeating mineral–organic layers can produce interference colour, while oriented surface grooves can also diffract light.^[ref]

A free streamline is a streamline that forms an unknown boundary of a fluid region, typically against a region of prescribed pressure. In steady inviscid potential flow with gravity and surface tension neglected, the Bernoulli equation makes its speed constant when its pressure is constant. It is simultaneously a level set of the stream function. These two boundary conditions take particularly simple forms in the complex potential of a two-dimensional potential flow and the hodograph plane.