OAlg.Structure.Exponential

Multiplicative structures with a power function (^).

Exponential

class (Multiplicative f, Number (Exponent f)) => Exponential f where

Multiplicative structures with a partially defined power function with numbers as exponents.

Properties

1. For all f and a holds:

   1. If start f == end f or a is an element of [-1,1] then f^a is valid.
   2. If start f /= end f and a is not an element of [-1,1] then f^a is not valid and its evaluation will end up in a NotExponential-exception.
2. For all f holds: f^1 == f.
3. For all f holds: f^(-1) == invert f.
4. For all f and a with start f == end f and a not in [-1,1] holds: start (f^a) == start f and end (f^a) == end f.
5. For all f, a and b with start f == end f holds: f^(a*b) == (f^a)^ b.
6. For all f with start f == end f holds: f^0 == one (end f).
7. For all f, a and b with start f == end f holds: f^(a + b) == f^a * f^b.
8. For all a and p holds: (one p)^a == one p.
9. For all f, g and a with start f == end f, start g == end g start f == start g and f * g == g * f holds: (f * g)^a == f^a * g^a.

Note

1. The phrase ..a is an element of [-1,1].. for the properties of ^ is meant to be: isOne a or isMinusOne a.
2. If -1 is an instance of Exponent f (see minusOne) then f has to be Cayleyan.

Associated Types

type Exponent f

the exponent.

Methods

(^) :: f -> Exponent f -> f infixl 9

the power of a factor to an exponent.

Instances

Galoisian x => Exponential (GL2 x)

Defined in OAlg.Entity.Matrix.GeneralLinearGroup

Associated Types

type Exponent (GL2 x)

Methods

(^) :: GL2 x -> Exponent (GL2 x) -> GL2 x

Oriented x => Exponential (GLT x)

Defined in OAlg.Entity.Matrix.GeneralLinearGroup

Associated Types

type Exponent (GLT x)

Methods

(^) :: GLT x -> Exponent (GLT x) -> GLT x

Oriented x => Exponential (ColTrafo x)

Defined in OAlg.Entity.Matrix.Transformation

Associated Types

type Exponent (ColTrafo x)

Methods

(^) :: ColTrafo x -> Exponent (ColTrafo x) -> ColTrafo x

Oriented a => Exponential (RowTrafo a)

Defined in OAlg.Entity.Matrix.Transformation

Associated Types

type Exponent (RowTrafo a)

Methods

(^) :: RowTrafo a -> Exponent (RowTrafo a) -> RowTrafo a

Entity x => Exponential (ProductSymbol x)

Defined in OAlg.Entity.Product.ProductSymbol

Associated Types

type Exponent (ProductSymbol x)

Methods

(^) :: ProductSymbol x -> Exponent (ProductSymbol x) -> ProductSymbol x

(Entity i, Ord i) => Exponential (Permutation i)

Defined in OAlg.Entity.Sequence.Permutation

Associated Types

type Exponent (Permutation i)

Methods

(^) :: Permutation i -> Exponent (Permutation i) -> Permutation i

Multiplicative c => Exponential (Inv c)

Defined in OAlg.Structure.Exponential

Associated Types

type Exponent (Inv c)

Methods

(^) :: Inv c -> Exponent (Inv c) -> Inv c

(Oriented x, Typeable p, p ~ Point x) => Exponential (Dim x p)

Defined in OAlg.Entity.Matrix.Dim

Associated Types

type Exponent (Dim x p)

Methods

(^) :: Dim x p -> Exponent (Dim x p) -> Dim x p

(Oriented a, Integral r) => Exponential (Product r a)

Defined in OAlg.Entity.Product.Definition

Associated Types

type Exponent (Product r a)

Methods

(^) :: Product r a -> Exponent (Product r a) -> Product r a

opower :: (Entity p, Number r) => Orientation p -> r -> Orientation p

the power of an orientation by an number.

Note opower fulfill the properties of Exponential for any number structure.

Real

class Multiplicative f => Real f where

reals.

Methods

power :: Number r => f -> r -> f

Instances

Entity p => Real (Orientation p)

Defined in OAlg.Structure.Exponential

Methods

power :: Number r => Orientation p -> r -> Orientation p