An algorithm's running time is its computational work on an input, measured either as elapsed time on a specified system or as an operation count under a cost model. Time complexity expresses how that count grows with input size. For binary search in a random-access sorted array, halving the candidate interval needs $O(\log n)$ comparisons. The same index-based code on a linked representation need not have that cost. Worst-case, average-case and amortized bounds quantify different questions; a measured duration alone does not prove any asymptotic guarantee.

An algorithm's memory usage is the storage it occupies under a specified accounting convention. Total storage can include input and output, while auxiliary space complexity counts extra working memory; recursion stacks, temporary arrays and cached results must be included when relevant. An iterative binary search needs constant auxiliary storage, but a recursive implementation can keep logarithmically many stack frames. Reporting $O(n)$ memory without specifying whether $n$ counts bits, elements or machine words can hide a substantial difference. Peak live storage and cumulative allocated storage are also different measurements.

Bit parallelism represents many Boolean states in the bits of a machine word and updates them together with bitwise operations. For example, if a bit at position $j$ records whether sum $j$ is reachable, adding an item of weight $w$ updates the state mask to mask | (mask << w), retaining old sums and adding shifted ones. A fixed-width operation processes up to the word width in parallel. An arbitrary-precision integer spanning many words still costs time proportional to the affected storage; using one language-level integer does not make every operation constant-time.

The four-sum problem asks whether four selected numbers sum to a target $t$, with the selection rules part of the input specification. In the four-list version, choose one value from each of $A,B,C,D$. Store sums $a+b$ and look for $t-(c+d)$, a meet-in-the-middle algorithm. For lists of length $n$, hashing gives expected $O(n^2)$ time and $O(n^2)$ storage; sorting pair sums gives deterministic comparison-based alternatives. In a single-list version requiring four distinct positions, stored pairs must also satisfy the index-disjointness condition. Equal values and equal positions are different restrictions.

A sweep-line algorithm processes geometric events in the order a conceptual line crosses a plane. It keeps an event queue and a status describing objects intersecting the current line. For horizontal and vertical line segments, sweep left to right: insert a horizontal segment at its left endpoint, query active heights at a vertical segment, and delete horizontals at their right endpoints. A balanced search structure supports the updates and range reporting. Equal-coordinate event ordering determines whether endpoint contacts count. The method reduces a two-dimensional problem to a changing one-dimensional ordering rather than physically sampling every line position.

A heuristic algorithm uses a selective search or approximation to obtain a useful result efficiently, without a general guarantee of finding the exact optimum.

A quantum algorithm specifies quantum state preparation, a sequence of quantum gates and quantum measurements, and any classical computation controlling them. For a pure quantum state $|\psi ⟩={\sum }_x{\alpha }_x|x⟩$, computational-basis measurement returns $x$ with probability $|{\alpha }_x{|}^2$, as prescribed by the Born rule. David Deutsch formulated a universal quantum computer using unitary operators and measurement.^[ref]

For example, one Hadamard gate prepares $H|0⟩=(|0⟩+|1⟩)/\sqrt{2}$. Computational-basis measurement gives either bit with probability $1/2$. Applying another Hadamard gate before measuring instead gives $H^2|0⟩=|0⟩$: the amplitudes for outcome one cancel by quantum interference. This illustrates why manipulating probability amplitudes matters; merely preparing a superposition does not reveal every component.

An analysis of algorithms must state the counted resources: quantum gate count, number of qubits, quantum state preparation cost and repetitions needed for the required probability of a correct result. Quantum query complexity counts quantum oracle calls separately; one such call need not be one elementary quantum gate.

Transfer of a hydrogen nucleus from a donor to an acceptor. Electron-pair bookkeeping distinguishes transfer of ${\mathrm{H}}^{+}$ from transfer of a hydride equivalent.

Deprotonation removes a proton from a chemical species, changing its electric charge by $-1$. It can expose a donor lone pair or change donor charge without changing the metal oxidation state.

A Lewis acid-base adduct forms when an electron-pair donor binds an acceptor through a coordinate covalent bond. A ring nitrogen can bind $\mathrm{GaC}{\mathrm{l}}_3$ without displacement of a chlorine substituent from phosphorus.

A Lewis acid accepts an electron pair to form a coordinate covalent bond. A Lewis base supplies that pair. This definition covers reactions without proton transfer: $\mathrm{B}{\mathrm{F}}_3+\mathrm{N}{\mathrm{H}}_3⟶{\mathrm{F}}_3\mathrm{B}\leftarrow \mathrm{N}{\mathrm{H}}_3$ pairs the empty acceptor orbital of boron with the nitrogen lone pair.

Metal cations commonly act as Lewis acids by coordinating ligands. Coordination to a carbonyl oxygen can increase the electrophilicity of the carbonyl carbon, explaining many catalytic effects. proton transfer concerns proton donation; Lewis acidity instead describes electron-pair acceptance, so the two classifications answer different questions.

A qualitative matching framework for Lewis acids and bases. Hard species favour interactions with compact, less polarizable partners, whereas soft species favour more polarizable partners. It is not an ordering based solely on atomic electronegativity.

A Lewis base donates an electron pair to a Lewis acid, forming a coordinate covalent bond in an adduct.

A carbon acid is a chemical acid that loses a proton from a carbon-hydrogen bond, producing a carbanion or a resonance-stabilized related species. The name refers to the atom bearing the acidic hydrogen, not to every compound containing carbon.

For example, deprotonation of acetone next to its carbonyl group gives an enolate. Delocalization of the negative charge makes this hydrogen much more acidic than an ordinary alkane hydrogen. Carbon acidity therefore explains how carbon nucleophiles arise in reactions such as the aldol reaction, while the actual acid strength depends strongly on substituents and solvent.

Radiation transports energy away from a source through propagating fields or emitted particles. Electromagnetic radiation includes radio waves, visible light, X-rays and gamma rays; alpha particles, beta particles and neutrons provide particle-radiation examples. The word alone does not specify an energy, a hazard, or a detection method.

An experiment must distinguish the incident field from its interaction with the detector. A photodetector measures a light-induced response, while an ionization chamber collects charge created in a gas. Neither instrument responds equally to every kind of radiation. Ionizing radiation is classified by its ability to remove bound electrons under the relevant interaction conditions, rather than simply by whether the source is radioactive.

Irradiation is exposure of a material or system to radiation. The source, particle flux, energy distribution and duration specify what arrives; the material's interactions determine what is transmitted, scattered or absorbed. Exposure to light can excite a molecule, whereas sufficiently energetic ionizing radiation can remove an electron or produce more extensive changes.

Incident energy and deposited energy are different quantities. A sample can transmit most of an incident beam, so the absorbed dose cannot generally be inferred from incident power and duration without an absorption model. The same distinction matters when an irradiation experiment compares materials with different thicknesses or compositions.