Molar concentration is amount of substance of a named chemical species per final chemical solution volume: $c=n/V$. State whether the count describes dissolved species or a conserved analytical concentration.

Reference wave phases label complex amplitudes without changing measured powers. If $\mathbf{b}=S\mathbf{a}$, choose diagonal matrices $D_{\mathrm{in}},D_{\mathrm{out}}$ whose entries have modulus one, and set ${\mathbf{a}}^{\prime }=D_{\mathrm{in}}\mathbf{a}$, ${\mathbf{b}}^{\prime }=D_{\mathrm{out}}\mathbf{b}$. Then $S^{\prime }=D_{\mathrm{out}}S{D}_{\mathrm{in}}^{-1}$ and each component has unchanged squared modulus. For a beam splitter scattering matrix, transforming $S$ without transforming input complex amplitudes changes the stated physical input.

Compilation performed during execution, often producing machine code for subsequent execution.

Consider the program expression x*(2+3)-4 in a deliberately small programming language. Its only program identifier is x, supplied as an integer input. Its literal values are nonnegative decimal numerals, interpreted as integers. Whitespace is ignored between lexical tokens. Its context-free grammar is:

expression ::= term (("+" | "-") term)*
term       ::= factor ("*" factor)*
factor     ::= numeral | "x" | "(" expression ")"
numeral    ::= digit digit*
digit      ::= "0" | "1" | "2" | "3" | "4"
             | "5" | "6" | "7" | "8" | "9"

Here * outside quotation marks means zero or more repetitions. Repeated operators are grouped to the left, specifying operator associativity; term gives multiplication higher operator precedence than addition and subtraction. Parentheses override that grouping. There is no unary minus, variable assignment, function call, or side effect in this language. Evaluation obtains each left operand before its right operand and uses exact integer arithmetic, with no fixed-width overflow.

Lexical analysis recognizes x, *, (, 2, +, 3, ), -, 4. Parsing constructs the concrete abstract syntax tree Sub(Mul(Name(x), Add(Int(2), Int(3))), Int(4)). Parentheses have done their work by making Add a child of Mul; they need not survive as nodes. Constant folding changes this to Sub(Mul(Name(x), Int(5)), Int(4)). It cannot replace Name(x) by a literal value because the input is supplied at execution. The resulting function of the input is $f(x)=5x-4$.

A compiler for our stack machine emits LOAD x; PUSH 5; MUL; PUSH 4; SUB. LOAD x pushes the current input; PUSH n pushes the integer literal value n. Each binary instruction removes the top item as its right operand, then the next item as its left operand, and pushes the result. The instruction sequence is an intermediate representation that an interpreter can execute; no machine code has been emitted. A compact binary encoding could make it bytecode, but these printed instruction names are only a readable notation.

Why does this compilation preserve the result? A structural induction supplies a short proof outline. Translating a literal value or x appends its value to any initial stack. For a binary abstract syntax tree node, first translating the left child and then the right child appends the two operand values; the matching instruction replaces them by exactly the original operation's result. Previously present stack items remain intact. Constant folding also preserves the result because $2+3=5$ in the specified integer arithmetic. Together these facts explain agreement for every integer input, rather than merely agreement for one trial.

An independence system consists of a finite set $E$ and a nonempty family $\mathcal{I}$ of subsets such that $I\in \mathcal{I}$ and $J\subseteq I$ imply $J\in \mathcal{I}$. This is the hereditary condition: deleting chosen elements keeps a selection admissible. In particular, the empty set belongs to $\mathcal{I}$. An independence system becomes a matroid when it also satisfies the matroid independence axioms for exchange. The hereditary nonmatroid greedy obstruction shows why deletion alone does not justify a greedy algorithm.

A kinetochore is a protein assembly at a centromere that attaches a chromosome to microtubules of the spindle apparatus. Each replicated chromosome has sister kinetochores; their attachment and subsequent separation help distribute the copied material. A kinetochore is therefore the attachment machinery, while the centromere is its chromosome site.^[ref]

The kinetochore ablation experiment tests this mechanical role directly. Counting kinetochores as independent chromosomes before sister separation would confuse duplicated attachment machinery with ploidy.

A double thymidine block is a synchronization strategy in which two thymidine-induced interruptions of DNA synthesis, separated by release, enrich a cultured-cell biological population near the transition into S phase. Excess thymidine perturbs nucleotide balance rather than acting as a physical barrier. Cells resume progression after release with a distribution of timings, so synchronization is enrichment rather than perfect identity of cell states. The perturbation can itself affect replication and stress responses; measurements after release must account for this changed starting condition. ^[ref]

Tartaric acid has two hydroxy groups and two carboxylic acid groups. Its meso compound form illustrates why two stereocenters need not make a chiral molecule.

Pyruvic acid is a three-carbon α-keto acid, $\mathrm{C}{\mathrm{H}}_3\mathrm{COCOOH}$. At ordinary cellular pH, its conjugate base, pyruvate, predominates.

The pyruvate carbon skeleton is a branch point connecting glycolysis with acetyl-CoA formation, lactic acid formation, and alanine metabolism. The different names distinguish protonation states without implying unrelated carbon pathways.

A carbon nanotube is a cylindrical carbon nanostructure with a graphitic bonding framework. Single-walled tubes can be described by the way a graphene-like sheet wraps around the cylinder; multiwalled tubes contain nested shells.

Diameter and wrapping geometry influence electronic and optical properties, including whether an idealized tube behaves as metallic or semiconducting. Thus carbon composition alone cannot predict the behavior of a particular nanotube.