A gene related to another through duplication. Comparing paralogues as if they were orthologues can mislead inference about lineage-specific sequence change.

The RRS1-RPS4 receptor pair is a coupled immune-receptor system studied in Arabidopsis thaliana. RRS1 includes a WRKY domain resembling targets of pathogen activity, providing an integrated decoy linked to recognition and signaling by the pair.

The mechanism illustrates how a plant can monitor an attack on a host-like structure rather than recognizing only one isolated foreign pattern. It also explains why transferring or losing one receptor need not preserve the behavior of the complete paired NLR system.^[ref]

A ceramic uranium oxide used as nuclear fuel. Its thermal conductivity, gas retention and microstructure evolve with temperature and irradiation.

Oxide nuclear fuel containing more than one metal oxide; in the usual power-reactor context it refers to uranium–plutonium oxide fuel. Its initial averaged composition does not preclude local variations or actinide redistribution in oxide fuel during service.

The number ratio of oxygen atoms to metal atoms in an oxide. Departures from a reference stoichiometry change defect populations, oxidation states and chemical potentials; these can influence transport and compatibility.

A plutonium oxide, $\mathrm{Pu}{\mathrm{O}}_2$, used as a component of mixed oxide nuclear fuel. Its use requires attention to decay heat, chemical compatibility and the distribution of fissile material; no fabrication recipe is specified.

The chromosomal origin of replication in E. coli, where DnaA-dependent initiation assembles bidirectional replication machinery.

A cultured three-dimensional intestinal epithelial structure reproducing selected stem-cell and cell differentiation behaviours. Exogenous growth factors and matrix replace important environmental inputs, so successful growth does not demonstrate signal-independent development.

A vinylic halide has a halogen atom bonded directly to a carbon of a carbon-carbon double bond. Vinyl chloride is the simplest common example.

Ordinary substitution at a saturated carbon cannot be transferred unchanged to this geometry and bonding. The distinction between a vinylic halide and an allylic halide is therefore mechanistically useful, even when both molecules contain an alkene and a halogen.

An allylic halide has a halogen atom bonded to a carbon adjacent to a carbon-carbon double bond. The halogen-bearing carbon is not itself one of the two alkene carbons.

The adjacent π system can stabilize developing charge or participate in delocalized intermediates. Allylic halides can therefore undergo substitution with patterns different from an otherwise comparable saturated alkyl halide.

A six-membered alternating boron–oxygen ring, commonly $(\mathrm{RBO}{)}_3$. Three-coordinate boron has a vacant p orbital that accepts oxygen lone-pair donation. This stabilizing delocalization does not imply the same aromaticity or equal bond polarity as benzene.

An aluminium–oxygen compound containing Al–O–Al connectivity, often represented compositionally as $(\mathrm{RAlO}{)}_{\mathrm{n}}$. Electron-deficient aluminium can achieve four-coordinate environments through cage assembly and multiply bridging oxygen.

A first-order ordinary differential equation contains the first derivative of its unknown function and no higher derivative. When solved for that derivative it has the displayed form. With an initial condition, local existence and uniqueness follow under appropriate regularity assumptions on $F$; the order alone does not guarantee either property.

A constant function $y(x)=c$ is an equilibrium solution of $y^{\prime }=F(x,y)$ on an interval when $F(x,c)=0$ throughout that interval. Dividing by a factor such as $y-c$ during separation of variables excludes this solution, so it must be checked separately. For an autonomous differential equation, the constant state is an equilibrium point.

The Hardy hierarchy assigns increasingly fast-growing functions to ordinals, once fundamental sequences for limit ordinals have been chosen. A common convention isfor nonzero limits $\lambda$, where $\lambda [n]$ increases toward $\lambda$. The ordinal notation system and its fundamental sequences are part of the definition.

For finite $k$, this gives $H_k(n)=n+k$. Taking $\omega [n]=n$ gives $H_{\omega }(n)=2n$. At larger ordinals the recursive nesting produces much faster growth. The hierarchy compares the strength of termination arguments and induction principles in proof theory; changing conventions can shift the exact formulas at particular indices.

An upper record value in a sequence is an observation strictly larger than all its predecessors; the first observation is a record by convention. For $(3,1,4,4,2,6)$, the strict upper record values are $3,4,6$ at times $1,3,6$. A tied maximum is not a new strict record. For independent and identically distributed continuous observations, the $n$th observation is a record with probability $1/n$, since each position is equally likely to hold the maximum of the first $n$ values. By linearity of expectation, the expected number of records by time $n$ is the harmonic number $H_n$.

For distinct observations $X_1,\dots ,X_n$, the relative rank of the latest observation among those seen so far is $R_n=1+\#\{i<n:X_i<X_n\}$, taking rank one as smallest. With values $(4,1,3)$ the successive ranks are $(1,1,2)$. Under independent and identically distributed continuous observations, symmetry gives $\mathbb{P}(R_n=j)=1/n$ for $j=1,\dots ,n$. Ranking largest as one is another common convention. Ties require a stated rule, such as random tie-breaking. Relative ranks make sequential selection possible without knowing the underlying distribution.