The planar molecule ${\mathrm{H}}_2\mathrm{CO}$. With the C=O bond along $z$ and the molecule in the $xz$ plane, its $C_{2v}$ vibrational representation is $3A_1+2B_1+B_2$. Interchanging the $x$ and $y$ axes interchanges the $B_1$ and $B_2$ labels; physical activity is unchanged.

The conjugated aldehyde $\mathrm{C}{\mathrm{H}}_2=\mathrm{CHCHO}$. A carbon nucleophile can add at its carbonyl carbon or, in a conjugate addition, at the terminal beta carbon. Atom mapping distinguishes the two products.

Benzaldehyde is ${\mathrm{C}}_6{\mathrm{H}}_5\mathrm{CHO}$, an aldehyde whose carbonyl carbon is attached directly to a phenyl group. It combines aromatic-ring chemistry with reactions of an aldehyde carbonyl.

Addition of an appropriate nucleophile at the carbonyl carbon gives a new tetrahedral centre. Because benzaldehyde has no ordinary enolizable alpha carbon, its role in an aldol reaction commonly differs from that of an enolate-forming partner.

An Aho-Corasick automaton augments a trie with a failure link from each nonroot state to its longest proper suffix that is also a trie prefix. Terminal states identify completed pattern strings; an output link points to the next terminal state along the failure link chain. The root represents the empty string. These structures support the Aho-Corasick algorithm.^[ref]

For pattern strings ab, bab, bc, the state bab has failure link to terminal ab, so both occurrences are reported after reading bab. If the next character is c, no edge leaves bab on c; following failure links reaches b, whose edge leads to bc. Reporting only the deepest terminal state would miss the simultaneous ab match.

A completed transition table stores the final destination for every state and alphabet character, giving one table lookup per input character. A sparse representation stores only trie edges and follows failure links on demand. They implement the same state invariant; the relevant space and lookup costs depend on the alphabet representation.

PhysiCell is an open-source agent-based modeling framework for simulating interacting cells in a spatial microenvironment. Individual cell agents carry phenotypes and interact mechanically, while diffusing substrates are represented by continuum transport fields. Linking these descriptions permits cell behavior to depend on the local environment and to modify that environment in turn.

The original publication demonstrates multicellular tumor simulations and describes cell cycling, death, volume, mechanics and motility models. The scientific value is the coupling of scales; a simulated pattern remains conditional on the specified rules and parameters, rather than establishing biological causation by itself.^[ref]

An aerosol particle able to activate into a growing cloud droplet at the supersaturation attained by a cloud parcel. Activity depends on particle dry size, hygroscopicity and interfacial properties. The number active at a specified supersaturation differs from total aerosol number.

A change in cloud microphysics or cloud properties caused by aerosol, including droplet activation, precipitation changes and adjustments of water content or coverage. The Twomey effect isolates a fixed-water-content optical response; it does not by itself establish the sign or magnitude of every cloud adjustment.

A particle's tendency to take up water vapor, which depends on composition, solubility and ambient humidity. In a common parameterization, a dimensionless $\kappa$ summarizes soluble-water uptake for use with dry diameter in Köhler theory. A single composition parameter does not remove the importance of particle size or parcel supersaturation.

Atmospheric particles containing sulfate, commonly in sulfuric-acid/water or neutralized-salt mixtures. Particle size, water uptake, acidity and accompanying material determine scattering, absorption, cloud activation and heterogeneous chemistry; sulfate mass alone does not specify radiative forcing.

Growth by collision and attachment of existing particles. Coagulation reduces particle number while conserving particulate material in a closed system without other sources or sinks. Larger particles can have a different scattering-per-mass efficiency and faster sedimentation, so increasing precursor injection need not increase albedo proportionally.

Adsorption in which material can form more than one layer above a surface. The number of adsorbed particles per underlying site can exceed one and differs from the fraction of sites having any adsorbate.

Chemisorption is adsorption involving chemical bonding between an adsorbate and a surface. It changes the electronic interaction more specifically than a description based only on weak physical attraction.

For example, a reacting molecule can bind to a metal surface before undergoing a catalytic transformation. Bond formation can impose activation barriers and site selectivity, but chemisorption is not universally irreversible. Distinguishing binding strength, adsorption rate, and subsequent reaction is necessary when interpreting a surface experiment.

The probability that an impinging species is retained at a surface under specified conditions. Incident flux alone does not fix net surface accumulation when sticking is incomplete or desorption occurs.

The Brunauer-Emmett-Teller multilayer adsorption model, distinguishing first-layer binding from higher layers. Suitable portions of an adsorption isotherm can estimate accessible surface area.

Measurement of desorbing species during a controlled temperature ramp.