Archives of Medicine, volume 1, pp. 100–102. Contemporary account of extraction, solubility, melting behaviour and equivalent-weight analysis. The early analytical claims must be distinguished from later revisions of purity and identity.

For a sorted array $a[\mathrm{0..}N)$, find the unique boundary $p\in [0,N]$ with $a[\mathrm{0..}p)\le x<a[p..N)$. A binary search maintains $0\le l\le r\le N$, an excluded prefix $a[\mathrm{0..}l)\le x$, and an excluded suffix $x<a[r..N)$. At an interior midpoint $m$, replace $l$ by $m+1$ when $a(m)\le x$, and otherwise replace $r$ by $m$. Sortedness preserves the loop invariant; the width $r-l$ is a decreasing loop variant. Termination at $l=r$ gives the boundary, including $p=0$ and $p=N$. The time complexity is $O(\log (N+1))$ with constant-cost comparisons and indexed access. Unlike searching for $x+1$, this formulation does not risk integer overflow at the largest representable key.

Find both the lower-bound search boundary $i$ and the upper-bound search boundary $j$ of a key $x$ in a sorted array. Every entry in $[i,j)$ is at least and at most $x$, so all are equal to $x$. Values immediately outside this interval, when present, are strictly smaller or strictly larger; the interval is consequently maximal. An absent key gives $i=j$ at its insertion position. Two binary searches give total time complexity $O(\log (N+1))$ under constant-cost comparison and indexed access.

In a nonempty rotation of a strictly increasing array, keep an inclusive interval containing its minimum. Compare its midpoint with its right endpoint. If the midpoint is larger, the minimum lies strictly to the right of the midpoint; otherwise the midpoint-to-right segment is increasing and the minimum lies at or left of the midpoint. The interval at least halves until one index remains. The cyclic predecessor of this minimum holds the maximum, so both extrema are found with $O(\log n)$ comparisons. Distinctness makes the two-way comparison decisive.

In a nondecreasing array, the first position whose entry is at least $x$ is found by maintaining a half-open unresolved interval $[l,h)$. Entries before $l$ are smaller than $x$; entries from $h$ onward are at least $x$. Initially $l=c_0,h=c_1$. At midpoint $m$, replace $l$ by $m+1$ if $A[m]<x$, and otherwise replace $h$ by $m$. Monotonicity preserves the loop invariant, and $h-l$ decreases. On exit $l=h$ is the first qualifying position, or $c_1$ if no such entry exists. Repeated values and endpoint cuts require no additional case.

A base that dissolves in water, giving an alkaline solution.

A B+ tree node split repairs an overflowing B+ tree node by distributing its entries between two nodes and adding a separating downlink to its parent. A parent overflow can propagate the split upward; splitting the root creates a new root one level higher.^[ref]

In a simple leaf convention, the parent separator is a copy of the smallest key in the new right leaf. If a leaf with capacity three receives a fourth key, $[2,5,9,14]$ can split into $[2,5]$ and $[9,14]$, with separator nine. Nine remains in the right leaf as an indexed record. The leaf-neighbour links must also be updated.

An internal split instead promotes a separator between child groups; it need not retain that separator as an ordinary key in either new internal node. Distinguishing promotion from the leaf copy rule prevents missing records or inconsistent child ranges. Exact occupancy and separator conventions are implementation choices. Every split replaces one level by two nodes at that same level, so existing root-to-leaf depths remain equal; only a root split increases all depths together.

A plasma cell is an antibody-secreting differentiated B cell. Extensive rough endoplasmic reticulum and a developed Golgi apparatus support the production and secretion of large amounts of antibody. Recognition of a particular antigen is inherited from the activated B cell lineage; cell differentiation changes the secretory programme rather than inventing specificity from scratch. Long-lived plasma cells can maintain antibody production after the initiating response has subsided. ^[ref]

An authentication goal in which both participants establish the identity and relevant participation of the other. Freshness, agreement on parameters and completed participation are separate properties that a protocol should specify.

The wavelength-dependent distribution of sunlight, including absorption lines used to investigate the solar atmosphere.

An astrophysicist studies astrophysics through mathematical theory, observation or experiment. The field determines the relevant physical systems and measurements, while the individual biographies below connect particular contributions with their scientific ideas.

An associative array represented by a list of key-value pairs. Looking up a key scans entries until a matching key is found, giving linear worst-case search cost in the number of entries when key comparisons have unit cost.

A program that translates assembly language into machine code. A pseudo-instruction can expand into several instructions, for example to construct a constant that cannot fit one immediate operand.

A contiguous segment whose elements all have the same value. Its longest suffix after scanning a new element extends the previous constant suffix when the adjacent values agree, and otherwise starts again with length one. Keeping the largest suffix length seen gives a linear time longest-segment scan.

Here a bitonic contiguous segment may be weakly increasing then weakly decreasing, or the reverse. Equivalently, remove zero signs from its adjacent differences: the remaining signs change at most once. This convention includes monotone and constant segments. A plateau can lie at the shared boundary between two directions, so maximal bitonic segments can overlap across that entire plateau.