A mitochondrion-related organelle class associated with anaerobic metabolism and, in typical examples, hydrogen and ATP production. Substrate-level phosphorylation can supply energy without a conventional oxygen-terminated respiratory chain.^[ref]

A highly reduced mitochondrion-related organelle retaining selected metabolic functions rather than oxidative ATP production. Functions differ among lineages, including iron-sulfur cluster maturation in Giardia and sulfate activation in Entamoeba histolytica.^[ref]

A microtubule-organizing structure in relevant animal cells. Positioning centrosomes relative to a niche can prepare the orientation of a later mitotic spindle. Budding yeast instead has a spindle pole body.

Its microtubule-nucleating material helps organize arrays without requiring every filament to remain permanently attached. A centrosome and a centriole are different structural levels: centrioles can be components of a centrosome, while the surrounding material carries much of the organizing activity. ^[ref]

The maintained combination of membrane composition, resident proteins, lumenal conditions, geometry and trafficking functions that distinguishes a compartment. Its molecules can turn over while this functional organization persists.

A membrane-bounded organelle with pathways including oxidative metabolism. Many matrix proteins are imported after cytosolic synthesis, including folded substrates.

Oxidative reactions in the compartment can generate hydrogen peroxide, which associated enzymes help handle. Separation from the cytosol permits a specialized chemical environment. Protein import and organelle growth must still be coordinated with maintenance of the enclosing membrane. ^[ref]

A distributed Bragg reflector uses a periodic refractive index structure to make successive weak reflections add coherently. Near the design wavelength, constructive wave interference produces strong reflection even when a single interface reflects only a small fraction of incident radiant power.

A quarter-wave Bragg stack alternates layers with optical thickness ${\lambda }_0/4$ at normal incidence. Measuring transmission across wavelength reveals a rejected band rather than a perfectly reflecting surface at every frequency. Such mirrors provide wavelength selection in an optical resonator; their reflection band shifts with incidence angle and material dispersion.

A structure with spatially periodic dielectric response, giving optical bands and, under suitable conditions, forbidden frequency intervals.

A designed spatial arrangement of dielectric or other electromagnetic material controls optical propagation, confinement or emission. Examples include waveguides, distributed Bragg reflectors, photonic crystals and defect cavities. Spatial interference and mode boundary conditions act on electromagnetic states; they are distinct from the electronic band structure of the material itself.

A group of diseases in which altered cells acquire uncontrolled growth and invasive potential. Many form solid tumours, while some blood cancers do not; metastasis describes successful spread to another site.^[ref]

Intramolecular olefin metathesis connecting two alkene sites into a ring. Terminal-alkene substrates can release ethene; concentration and preorganization influence competition with chain growth.

Step-growth polymerization in which terminal diene groups undergo intermolecular olefin metathesis. Removing a volatile alkene coproduct can drive conversion.

A four-membered ring containing one metal and three carbon atoms. Its formation and alternate fragmentation exchange alkylidene groups during olefin metathesis.

An area-normalized interface resistance, ${\rho }_c$ in resistance times area. For nonlinear contacts specify the bias-dependent differential quantity.

A pad-spacing method separating contact and sheet resistance. For uniform ohmic contacts, $R(d)=2R_c+R_{\square }d/W$; the transfer length is $L_T=\sqrt{{\rho }_c/R_{\square }}$. A rectifying Schottky junction requires bias-dependent analysis.

A basic oxygen–ozone mechanism combining oxygen photodissociation, three-body ozone formation, ozone photodissociation and $\mathrm{O}+{\mathrm{O}}_3\to 2{\mathrm{O}}_2$. The last reaction removes two units of odd oxygen; a catalytic cycle can add another loss pathway.

A positive integer $N$ that is both the sum and the product of one multiset of $k$ positive integers. For factors at least $2$ with product $P$, sum $S$, and count $t$, appending $P-S$ ones gives $k=t+P-S$ without changing the product. For $k\ge 2$, the example $2,k$ with $k-2$ ones proves that the smallest product-sum value is at most $2k$.

A positive integer $n$ such that every positive integer at most $n$ is a sum of distinct divisors of $n$. For example, $6$ is practical because its divisors $1,2,3,6$ represent every value from $1$ through $6$.

For positive integers $a_1,\dots ,a_k>1$ having greatest common divisor one, the Frobenius number is the largest integer not expressible as ${\sum }_i{a}_i{x}_i$ with nonnegative integer coefficients $x_i$. A coin interpretation allows unlimited copies of denominations $a_i$, rather than a fixed number of coins.

For denominations $4$ and $7$, the value is $17$. The next four values are representable: $18=4+7+7$, $19=4+4+4+7$, $20=5\cdot 4$ and $21=3\cdot 7$. Repeatedly adding $4$ then covers every larger integer. The two-generator Frobenius theorem proves the formula $g(a,b)=ab-a-b$ for two coprime denominations.

The hypotheses exclude two different degeneracies. If all denominations have a common divisor greater than one, infinitely many positive amounts are impossible, so no largest impossible amount exists. If denomination one is present, every nonnegative integer is possible; the usual positive-gap problem disappears. Some conventions assign the latter case the value $-1$.

A Harshad number in base $b\ge 2$ is a positive integer divisible by its base-p digit sum in that base. In base ten, $18$ is a Harshad number because $1+8=9$ divides $18$. It is strong when the quotient by its base-p digit sum is prime, and right-truncatable when every nonempty prefix is a Harshad number.

The arithmetic derivative on positive integers is specified by $p^{\prime }=1$ for each prime number $p$ and the product rule $(ab{)}^{\prime }=a^{\prime }b+a{b}^{\prime }$. These conditions force $1^{\prime }=0$. For a prime factorization $n={\prod }_i{p}_i^{e_i}$ they giveRepeated use of the product rule proves necessity. Conversely, the fundamental theorem of arithmetic makes this formula well-defined, and the addition of prime exponents under multiplication verifies the product rule. Thus it proves existence as well as uniqueness.^[ref]

For example, $12^{\prime }=12(2/2+1/3)=16$. Despite its name, the arithmetic derivative does not preserve addition: $(2+3{)}^{\prime }=5^{\prime }=1$, whereas $2^{\prime }+3^{\prime }=2$. Its analogy with an ordinary derivative preserves the product rule while sacrificing the usual sum rule.