For an array $a$, its difference array stores $d_0=a_0$ and $d_i=a_i-a_{i-1}$ for $i>0$. The original entries are prefix sums of $d$. Adding $v$ on indices $[l,r]$ requires only $d_l+=v$ and, when present, $d_{r+1}-=v$.

An aromatic ring is a cyclic conjugated system with electronic stabilization associated with aromaticity. For a simple planar monocyclic pi system, the familiar criterion is $4n+2$ pi electrons, together with continuous orbital overlap.

Benzene is the standard six-pi-electron example. A polygonal drawing alone does not establish aromaticity: geometry, conjugation, and electronic state matter. Aromatic stabilization helps explain why many such rings undergo substitution while preserving their conjugated framework.

A monocyclic, approximately planar, continuously conjugated pi system with $4n+2$ pi electrons has a closed-shell aromatic configuration in the elementary Hückel model, where $n$ is a nonnegative integer. Fused systems and heterocycles require a specified orbital and resonance description rather than counting every drawn double bond in each polygon.

An aromatic hydrocarbon; the term is also commonly used for the carbocyclic aromatic donor portion of an organometallic arene complex.

A mathematical mean assigns a representative value to several quantities. A real mean of $x_1,\dots ,x_n$ normally lies between their minimum and maximum and returns $x$ when every input equals $x$. Different means preserve different operations: the arithmetic mean averages sums, while the geometric mean averages positive multiplicative factors.

For $2$ and $8$, the arithmetic mean is $5$ and the geometric mean is $4$. If equal distances are traveled at speeds $2$ and $8$, the average speed is instead their harmonic mean, $2/(1/2+1/8)=16/5$. The useful choice depends on which quantity is being added: distances, times, or logarithms of growth factors.

A proapoptotic BH3-only BCL-2-family protein induced in some p53 responses, contributing to mitochondrial apoptosis.

A proapoptotic BH3-only BCL-2-family protein that can be induced by p53. Its effects depend on the survival-protein context.

A BCL-2-family effector promoting mitochondrial outer-membrane permeabilization during apoptosis. Its activation differs from merely increasing expression of an apoptotic gene.

Hybridoma technology combines an antibody-producing primary cell with an immortal cell line to establish continued production of a chosen antibody. Fusion-product survival and antigen specificity require separate selection and screening steps.

Antibody phage display displays antibody fragments on phage particles carrying their encoding sequences. Binding selection enriches candidates while maintaining genotype–phenotype linkage, followed by testing in soluble and intended therapeutic formats.

An antibody linked to a cytotoxic payload for targeted delivery to antigen-bearing biological cells. Binding, uptake, linker stability and drug release all influence selectivity.

An antibody possessing catalytic activity. Selection against a stable transition-state analogue can produce rate enhancement, but affinity for an analogue alone does not guarantee rapid turnover or product release.

Survival of a subpopulation during bactericidal exposure while the broader population remains susceptible. Regrown survivors can recover the original susceptibility; persistence is not synonymous with every persistent infection.

The fraction of accepted input power radiated by an antenna. With consistent feed definitions, antenna gain is radiation efficiency times antenna directivity. Ohmic losses reduce the efficiency, while feed mismatch is usually tracked separately.

For a narrow angular power pattern $\exp [-(u^2+w^2)/(2{\alpha }^2)]$ about an equatorial beam centre, the solid angle measure is locally $dudw$. Extending the small lobe to the plane gives its beam solid angle ${\Omega }_A\simeq 2\pi {\alpha }^2$. Both sides of the azimuthal lobe must be included. A lossless antenna then has peak antenna gain $G_0\simeq 2/{\alpha }^2$ and peak antenna effective area $A_0\simeq {\lambda }^2/(2\pi {\alpha }^2)$.

For a uniformly illuminated circular aperture of diameter D, normalized received power is $[2J_1(x)/x{]}^2$ with $x=\pi D\sin \theta /\lambda$. The first null has $\sin \theta =1.21967\lambda /D$ and the half-power point has $\sin \theta \simeq 0.5145\lambda /D$. A pointing tolerance must name the tolerated power loss; the first-null angle is not a good low-loss tolerance. These estimates require an ideal aperture pattern and are only approximate for a dish a few wavelengths across.