Functors often describe " natural constructions " and natural transformations then describe " natural homomorphisms " between two such constructions.

The chain rule says that the composite of these two linear transformations is the linear transformation, and therefore it is the function that scales a vector by f ′( g ( a )) g ′( a ).

* The citrate then goes through a series of chemical transformations, losing two carboxyl groups as CO < sub > 2 </ sub >.

Under Galilean transformations, the time t < sub > 2 </ sub > − t < sub > 1 </ sub > between two events is the same for all inertial reference frames and the distance between two simultaneous events ( or, equivalently, the length of any object, | r < sub > 2 </ sub > − r < sub > 1 </ sub >|) is also the same.

Under Lorentz transformations, the time and distance between events may differ among inertial reference frames ; however, the Lorentz scalar distance s between two events is the same in all inertial reference frames

Shape transformations defined by keyframes cannot accurately show how the shape will be transformed in between the two keyframes.

For example, a symmetry group encodes symmetry features of a geometrical object: it consists of the set of transformations that leave the object unchanged, and the operation of combining two such transformations by performing one after the other.

All programming languages have some primitive building blocks for the description of data and the processes or transformations applied to them ( like the addition of two numbers or the selection of an item from a collection ).

* Chiral symmetries involve independent transformations of these two types of particle.

The time differential between two reunited clocks is deduced through purely uniform linear motion considerations, as seen in Einstein's original paper on the subject, as well as in all subsequent derivations of the Lorentz transformations.

He is best known today for the two Topper novels, comic fantasy fiction involving sex, much drinking and supernatural transformations.

Affine transformations in two real dimensions include:

In fact, since any matrix may be written as the product of unitary and positive Hermitian matrices, any sequence of linear propagation effects, no matter how complex, can be written as the product of these two basic types of transformations.

Additionally, every pair of adjoint functors comes equipped with two natural transformations ( generally not isomorphisms ) called the unit and counit.

This forms a category since for any functor F there is an identity natural transformation ( which assigns to every object X the identity morphism on F ( X )) and the composition of two natural transformations ( the " vertical composition " above ) is again a natural transformation.

* 2-cells between two 1-cells ( functors ) and are the natural transformations from to.

What cannot easily be expressed without the language of natural transformations is how homology groups are compatible with morphisms between objects, and how two equivalent homology theories not only have the same homology groups, but also the same morphisms between those groups.

* Since NP-complete problems transform to each other by polynomial-time many-one reduction ( also called polynomial transformation ), all NP-complete problems can be solved in polynomial time by a reduction to H, thus all problems in NP reduce to H ; note, however, that this involves combining two different transformations: from NP-complete decision problems to NP-complete problem L by polynomial transformation, and from L to H by polynomial Turing reduction ;

The electric potential and the magnetic vector potential together form a four vector, so that the two kinds of potential are mixed under Lorentz transformations.

The coordinates of points in the plane are two dimensional vectors in R < sup > 2 </ sup >, so rigid transformations are those that preserve the distance measured between any two points.

With every bijection from the space to itself two coordinate transformations can be associated:

In geometry, two subsets of a Euclidean space have the same shape if one can be transformed to the other by a combination of translations, rotations ( together also called rigid transformations ), and uniform scalings.

where N < sub > 2 </ sub >( 0 ) is the number of excited atoms at time t = 0, and τ < sub > 21 </ sub > is the lifetime of the transition between the two states.

The bottom two rows ' columns contain electron neutrino ( ν sub e ) and electron ( e ), muon neutrino ( ν sub μ ) and muon ( μ ), and tau neutrino ( ν sub τ ) and tau ( τ ), and Z sup 0 and W sup ± weak force.

An Allan deviation of 1. 3 × 10 < sup >− 9 </ sup > at observation time 1 s ( i. e. τ = 1 s ) should be interpreted as there being an instability in frequency between two observations a second apart with a relative root mean square ( RMS ) value of 1. 3 × 10 < sup >− 9 </ sup >.

Almost all Franciscan churches have painted a tau with two crossing arms, both with stigmata, the one of Jesus and the other of Francis ; usually members of the Secular Franciscan Order wear a wooden τ in a string with three knots around the neck.

In particle physics, a kaon (, also called a K meson and denoted < ref group = nb > The positively charged kaon used to be called τ < sup >+</ sup > and θ < sup >+</ sup >, as it was supposed to be two different particles until the 1960s.

Let ( either the open or closed sets does not matter ) τ < sub > 1 </ sub > and τ < sub > 2 </ sub > be two topologies on a set X such that τ < sub > 1 </ sub > is contained in τ < sub > 2 </ sub >:

Let τ < sub > 1 </ sub > and τ < sub > 2 </ sub > be two topologies on a set X.

Let τ < sub > 1 </ sub > and τ < sub > 2 </ sub > be two topologies on a set X and let B < sub > i </ sub >( x ) be a local base for the topology τ < sub > i </ sub > at x ∈ X for i

In aqueous solution histamine exists in two tautomeric forms, N < sup > π </ sup >- H-histamine and N < sup > τ </ sup >- H-histamine.

For example, it is equivalent to Kendall's τ correlation coefficient if one of the variables is binary ( that is, it can only take two values ).

The summations after the coefficients 60 and 140 are the first two Eisenstein series, which are modular forms when considered as functions G < sub > 4 </ sub >( τ ) and G < sub > 6 </ sub >( τ ), respectively, of τ = ω < sub > 2 </ sub >/ ω < sub > 1 </ sub > with Im ( τ ) > 0.

One Jacobi theta function ( named after Carl Gustav Jacob Jacobi ) is a function defined for two complex variables z and τ, where z can be any complex number and τ is confined to the upper half-plane, which means it has positive imaginary part.

Dirac also predicted a reaction + → +, where an electron and a proton annihilate to give two photons.

A homomorphism between two Boolean algebras A and B is a function f: A → B such that for all a, b in A:

The remaining two components of curl result from cyclic permutation of indices: 3, 1, 2 → 1, 2, 3 → 2, 3, 1.

As an example of a conditional proof in symbolic logic, suppose we want to prove A → C ( if A, then C ) from the first two premises below:

Despite its rare use, Italian orthography allows the circumflex accent ( î ) too, in two cases: it can be found in old literary context ( roughly up to 19th century ) to signal a syncope ( fêro → fecero, they did ), or in modern Italian to signal the contraction of ″- ii ″ due to the plural ending-i whereas the root ends with another-i ; e. g., s. demonio, p. demonii → demonî ; in this case the circumflex also signals that the word intended is not demoni, plural of " demone " by shifting the accent ( demònî, " devils "; dèmoni, " demons ").

Model example: if U and V are two connected open subsets of R < sup > n </ sup > such that V is simply connected, a differentiable map f: U → V is a diffeomorphism if it is proper and if

Given two groups G and H and a group homomorphism f: G → H, let K be a normal subgroup in G and φ the natural surjective homomorphism G → G / K ( where G / K is a quotient group ).

In mathematics, given two groups ( G, *) and ( H, ·), a group homomorphism from ( G, *) to ( H, ·) is a function h: G → H such that for all u and v in G it holds that

In complete analogy, one can define a right group action of G on X as a function X × G → X by the two axioms:

In this setting, a homomorphism ƒ: A → B is a function between two algebraic structures of the same type such that

A function f: X → Y between two topological spaces ( X, T < sub > X </ sub >) and ( Y, T < sub > Y </ sub >) is called a homeomorphism if it has the following properties:

Let V and W be vector spaces over the same field K. A function f: V → W is said to be a linear map if for any two vectors x and y in V and any scalar α in K, the following two conditions are satisfied:

Suppose two curves γ < sub > 1 </ sub >: (- 1, 1 ) → M and γ < sub > 2 </ sub >: (- 1, 1 ) → M with γ < sub > 1 </ sub >( 0 )

In this case, the limit of a diagram F: J → C can be constructed as the equalizer of the two morphisms

* The inclusion functor Ab → Grp creates limits but does not preserve coproducts ( the coproduct of two abelian groups being the direct sum ).

A more formal definition: two subsets A and B of Euclidean space R < sup > n </ sup > are called congruent if there exists an isometry f: R < sup > n </ sup > → R < sup > n </ sup > ( an element of the Euclidean group E ( n )) with f ( A ) = B. Congruence is an equivalence relation.

In terms of phrase structure rules, phrasal categories can occur to the left of the arrow while lexical categories cannot, e. g. NP → D N. Traditionally, a phrasal category should consist of two or more words, although conventions vary in this area.

A homomorphism between two algebras A and B is a function h: A → B from the set A to the set B such that, for every operation f < sub > A </ sub > of A and corresponding f < sub > B </ sub > of B ( of arity, say, n ), h ( f < sub > A </ sub >( x < sub > 1 </ sub >,..., x < sub > n </ sub >)) = f < sub > B </ sub >( h ( x < sub > 1 </ sub >),..., h ( x < sub > n </ sub >)).

If ƒ: A → B is a homomorphism between two algebraic structures ( such as homomorphism of groups, or a linear map between vector spaces ), then the relation ≡ defined by

A ring is a set R equipped with two binary operations +: R × R → R and ·: R × R → R ( where × denotes the Cartesian product ), called addition and multiplication.