For a real-frequency lossless reciprocal periodic line, a cell transfer matrix has determinant one. If half its trace lies outside $[-1,1]$, the Bloch multipliers are real reciprocal numbers rather than unit-modulus propagation factors. The physical interior field decays and carries no net power to infinity. A real reference line is then totally reflected by a semi-infinite structure; a finite structure can still tunnel.

Binding at one site changes binding at another. Positive cooperativity can produce a steep occupancy response. Several independent sites alone do not establish cooperativity.

Coupling between equilibria such that favouring one state changes another measured equilibrium. For example, preferential ligand binding to an assembled state can shift an assembly equilibrium; the coupling follows from consistent free-energy differences around the thermodynamic cycle.

A molecule binding away from the principal ligand site and changing its affinity or efficacy. The direction and size of modulation can depend on the orthosteric ligand and measured response.

$\mathrm{C}{\mathrm{H}}_3-\mathrm{C}\equiv \mathrm{CH}$, a terminal alkyne.

The simplest aryne, ${\mathrm{C}}_6{\mathrm{H}}_4$. Its extra in-plane bonding interaction is distinct from the out-of-plane aromatic pi system.

The hydrocarbon $\mathrm{HC}\equiv \mathrm{C}-\mathrm{C}{\mathrm{H}}_2-\mathrm{C}{\mathrm{H}}_2-\mathrm{C}\equiv \mathrm{CH}$. Its triple-bond fragments are linear, but the entire carbon skeleton need not be. The extended anti conformation has an inversion centre at the central bond midpoint and belongs to the C2h molecular point group.

An anion formed by removing the terminal proton of an alkyne, $\mathrm{RC}\equiv {\mathrm{C}}^{-}$. It can act as a carbon nucleophile, for example in addition to a ketone.

An alkene bearing a nitro group can undergo conjugate addition. Addition at its beta carbon generates a nitronate that can subsequently react at carbon.

Styrene is vinylbenzene, ${\mathrm{C}}_6{\mathrm{H}}_5\mathrm{CH}=\mathrm{C}{\mathrm{H}}_2$. The vinyl double bond is conjugated with the aromatic ring.

Addition polymerization consumes that double bond to form polystyrene while leaving phenyl groups as side substituents. This makes styrene a useful example of how a small molecular change, converting a pi bond into backbone sigma bonds, creates a macromolecular material.

A terminal alkene has a carbon-carbon double bond at the end of its carbon framework, commonly written $\mathrm{RCH}=\mathrm{C}{\mathrm{H}}_2$. The terminal carbon is a methylene group.

It differs from an internal alkene whose double-bond carbons both continue into carbon substituents. This distinction affects substitution, possible geometrical isomerism, and the regioselectivity of reactions across the double bond.

A six-membered carbocycle with one carbon–carbon double bond.

An alkene bearing two thioalkoxy groups on one carbon, ${\mathrm{R}}_2\mathrm{C}=\mathrm{C}(SR\prime {)}_2$. pi conjugation to a carbonyl can allow addition–elimination at that beta carbon with thiolate departure.

A compound containing two carbon–carbon double bonds; conjugation is a separate structural qualification.

Bicyclo[2.2.1]hept-2-ene, a strained bicyclic alkene.

$\mathrm{C}{\mathrm{H}}_2=\mathrm{CHN}{\mathrm{O}}_2$, an electron-poor alkene that can serve as a dienophile. Conversion of a resulting secondary nitro group to a ketone can connect a Diels–Alder reaction to a Nef reaction.

Multiplying numerator and denominator of a fraction by the same nonzero expression can remove a square root from the denominator. For instance,when $a\ge 0$ and both denominators are nonzero. This uses the difference-of-squares identity $(\sqrt{a}+b)(\sqrt{a}-b)=a-b^2$. If $a=b^2$, the original expression may still be defined even though the rationalized quotient is not, so that case must be handled separately. The related rationalization of a square-root difference applies the same multiplication to a numerator.

Factorization expresses an object as a product of simpler factors. Integer factorization decomposes an integer; polynomial factorization decomposes a polynomial. The permitted factors depend on the ambient algebraic structure.