An example is modern Greek which may write the phoneme in six different ways: ⟨ ι ⟩, ⟨ η ⟩, ⟨ υ ⟩, ⟨ ει ⟩, ⟨ οι ⟩, and ⟨ υι ⟩ ( although the last is rare ).

If F and G are ( covariant ) functors between the categories C and D, then a natural transformation η from F to G associates to every object X in C a morphism in D such that for every morphism in C, we have ; this means that the following diagram is commutative:

The two functors F and G are called naturally isomorphic if there exists a natural transformation from F to G such that η < sub > X </ sub > is an isomorphism for every object X in C.

The above definition imposes no constraints on η ( 0 ), even though it is assumed that η ( k ) tends to zero as k tends to zero.

If we set, then η is continuous at 0.

The role of Q in the first proof is played by η in this proof.

The need to define Q at g ( a ) is analogous to the need to define η at zero.

Eta ( uppercase Η, lowercase η ) ) is the seventh letter of the Greek alphabet.

The lower-case letter η is used as a symbol in:

* Statistics, η < sup > 2 </ sup > is the " partial regression coefficient ".

* Economics, η is the elasticity.

* Oceanography, η is the measurement ( usually in metres ) of sea-level height above or below the mean sea-level at that same location.

* η is the dimensionless efficiency of the turbine

The general name metallocene is derived from ferrocene, ( C < sub > 5 </ sub > H < sub > 5 </ sub >)< sub > 2 </ sub > Fe or Cp < sub > 2 </ sub > Fe, systematically named bis ( η < sup > 5 </ sup >- cyclopentadienyl ) iron ( II ).

Using the nomenclature of " hapticity ", the equivalent bonding of all 5 carbon atoms of a cyclopentadienyl ring is denoted as η < sup > 5 </ sup >, pronounced " pentahapto.

η is the pump efficiency, and may be given by the manufacturer's information, such as in the form of a pump curve, and is typically derived from either fluid dynamics simulation ( i. e. solutions to the Navier-stokes for the particular pump geometry ), or by testing.

There is a natural homomorphism η: G → G < sup >^</ sup >, and the image of G under this homomorphism is dense in G < sup >^</ sup >.

The homomorphism η is injective if and only if the group G is residually finite ( i. e.,

The homomorphism η is characterized by the following universal property: given any profinite group H and any group homomorphism f: G → H, there exists a unique continuous group homomorphism g: G < sup >^</ sup > → H with f

The symbol η ( eta ) may be used instead of Z for wave impedance to avoid confusion with electrical impedance.

The symbol η ( Greek " eta ") is typically used to denote a linear predictor.

The flat space metric ( or Minkowski metric ) is often denoted by the symbol η and is the metric used in special relativity.

He also offers a conversion table ( see Cohen, 1988, p. 283 ) for eta squared ( η < sup > 2 </ sup >) where 0. 0099 constitutes a small effect, 0. 0588 a medium effect and 0. 1379 a large effect.

* Experimental particle physics, η stands for pseudorapidity.

* Neural network backpropagation, η stands for the learning rate.

* Electronics, η stands for the ideality factor of a bipolar transistor, and has a value close to 1. 000.

* Power electronics, η stands for the efficiency of a power supply, defined as the output power divided by the input power.

Many of these trivial errors occurred in the Byzantine period, following a change in script from uncial to minuscule, and many were ' homophonic ' errors, when scribes accidentally substituted homophones for words in the textequivalent in English to substituting ' right ' for ' write ', except that there were more opportunities for Byzantine scribes to make these errors because the Greek letters η, ι, οι and ει were pronounced similarly in the Byzantine period.

Some metallocene complexes of actinides have been reported where there are three cyclopendadienyl ligands for a monometallic complex, all three of them bound η < sup > 5 </ sup >.

A structural trend for the series MCp < sub > 2 </ sub > involves the variation of the M-C bonds, which elongate as the valence electron count deviates from 18 .< ref > Kevin R. Flower, Peter B. Hitchcock " Crystal and molecular structure of chromocene ( η < sup > 5 </ sup >- C < sub > 5 </ sub > H < sub > 5 </ sub >)< sub > 2 </ sub > Cr " Journal of Organometallic Chemistry, 1996, volume 507, p. 275-277..

Let F and G be a pair of adjoint functors with unit η and co-unit ε ( see the article on adjoint functors for the definitions ).

* For each object X in C, ( F ( X ), η < sub > X </ sub >) is an initial morphism from X to G. That is, for all f: X → G ( Y ) there exists a unique g: F ( X ) → Y for which the following diagrams commute.

If F and G are functors between the categories C and D, then a natural transformation η from F to G associates to every object X in C a morphism between objects of D, called the component of η at X, such that for every morphism we have:

If, for every object X in C, the morphism η < sub > X </ sub > is an isomorphism in D, then η is said to be a ( or sometimes natural equivalence or isomorphism of functors ).

More strongly, if one wishes to prove that X and G ( X ) are not naturally isomorphic, without reference to a particular isomorphism, this requires showing that for any isomorphism η, there is some A with which it does not commute ; in some cases a single automorphism A works for all candidate isomorphisms η, while in other cases one must show how to construct a different A < sub > η </ sub > for each isomorphism.

One then restricts the maps to only those maps that commute with these isomorphism ( restricts to the naturalizer of η ), in other words, restrict to the maps that do not change the bilinear form: The resulting category, with objects finite-dimensional vector spaces with a nondegenerate bilinear form, and maps linear transforms that respect the bilinear form, by construction has a natural isomorphism from the identity to the dual ( each space has an isomorphism to its dual, and the maps in the category are required to commute ).

# for all g ∈ P, η restricts a linear isomorphism of T < sub > g </ sub > P with ( η is an absolute parallelism on P ).

# for all p in P, the restriction of η defines a linear isomorphism from the tangent space T < sub > p </ sub > P to.

A linear predictor η

η is expressed as linear combinations ( thus, " linear ") of unknown parameters β.

The coefficients of the linear combination are represented as the matrix of independent variables X. η can thus be expressed as