For a string $s$, the value ${\pi }_i$ is the length of the longest proper prefix of $s_0\cdots s_i$ that is also its suffix. Repeatedly following these lengths lists all proper borders of a string; this is the failure-link information used by the Knuth-Morris-Pratt algorithm.

A border of a string is a string that is both a prefix and a suffix. A proper border is shorter than the whole string; the empty string is often included. The string ababa has proper nonempty borders a and aba. A border of length $b$ in a length-$m$ string corresponds to an overlap displacement $m-b$. The Knuth-Morris-Pratt algorithm uses a chain of such borders to decide how much of a partial match survives a mismatch, avoiding comparison of characters already known to agree.

The Knuth-Morris-Pratt algorithm searches for a pattern using its proper prefix-suffix overlaps. After matching $j$ characters, a mismatch replaces $j$ by the length of the longest proper border of a string for that matched prefix, repeating until the next character can extend it or $j=0$. The text index never moves backward. Precomputing prefix failure links takes $O(m)$ time; during scanning each increase in $j$ can pay for a later decrease, giving $O(n+m)$ total character work and $O(m)$ auxiliary storage. A dense Knuth-Morris-Pratt automaton is a related implementation with alphabet-dependent preprocessing and storage.^[ref]

A transition to a shorter suffix match after an attempted extension fails. In a Knuth-Morris-Pratt automaton, the prefix function determines the next shorter matching prefix.

The Burrows-Wheeler transform appends a unique smallest terminator to a string, sorts its cyclic rotations in lexicographic order, and records the last column. It is reversible because the first and last columns contain the same characters and the last-to-first mapping preserves the occurrence order of each character. The resulting letter arrangement supports both compression and backward search.^[ref]

In a sorted cyclic rotation matrix, the last-to-first mapping sends the $k$th occurrence of a character in the last column to its $k$th occurrence in the first column. For one-based rows, $\mathrm{LF}(r)=C(L_r)+\mathrm{Occ}(L_r,r)$, where $C(c)$ counts smaller characters and $\mathrm{Occ}(c,r)$ counts $c$ in the first $r$ entries of the last column. It moves a rotation's start one character backwards cyclically. Equal characters retain the lexicographic order of their following sequences, which proves the occurrence-order rule.^[ref]

Backward search finds a query's matching interval in a suffix array by processing the query from right to left. With inclusive one-based interval $[l,r]$, prepending $c$ changes it to $[C(c)+\mathrm{Occ}(c,l-1)+1,C(c)+\mathrm{Occ}(c,r)]$. An empty interval means no match. This works through the occurrence-order property of the last-to-first mapping.^[ref]

An FM-index is a compressed text index supporting backward search through the Burrows-Wheeler transform and occurrence counts. Character rank queries locate a matching interval; sampled suffix array positions then allow the actual text positions to be recovered. Query length, number of matches and the chosen sampling density control different parts of the search cost.^[ref]

Energy loss of a moving particle to electronic excitation and ionization. It is distinct from direct momentum transfer to target nuclei; electronic energy deposition can also couple to atomic motion in a material-dependent way.

Energy loss of a moving particle through collisions transferring momentum to target nuclei. Recoiling target atoms can initiate displacement cascades.

A 1,3-Diaxial interaction is the close nonbonded contact between axial groups separated by one ring atom in a chair. Replacing a carbon-bearing axial hydrogen by an oxygen changes this contact, so cyclohexane substituent preferences cannot be transferred unchanged to dioxanes.

Energetic destabilization associated with unfavourably close nonbonded contacts or deformation needed to avoid those contacts.

Geometric isomers share connectivity but differ in spatial relationships such as cis/trans or facial/meridional coordination. A reflected chiral facial complex instead gives an enantiomer.

Different spatial arrangements of a fixed ligand connectivity, including cis and trans arrangements in square-planar or octahedral complexes.

A nitrogen bonded to four distinct substituents is stereogenic. It cannot undergo ordinary three-coordinate amine inversion without a bond-breaking process.

A phosphorus centre with an appropriate set of four distinguishable substituent directions can be stereogenic. A phosphate triester with three different alkoxy groups illustrates phosphorus chirality.

A stem cell derived from early embryonic tissue with capacity to generate many differentiated cell types under suitable conditions.

A stem cell contributing sustained differentiated descendants to the intestinal lining. Marker expression is suggestive; long-term lineage tracing tests renewal and lineage output.

Production of descendants that retain the originating biological population's developmental capacity. For a stem-cell biological population, self-renewal must accompany suitable differentiated output; persistence of a label alone does not establish both properties.