algebra-4.3: Constructive abstract algebra

Safe HaskellSafe
LanguageHaskell98

Numeric.Covector

Contents

Synopsis

Documentation

newtype Covector r a Source #

Linear functionals from elements of an (infinite) free module to a scalar

Constructors

Covector 

Fields

  • ($*) :: (a -> r) -> r
     

Instances

RightModule r s => RightModule r (Covector s m) Source # 

Methods

(*.) :: Covector s m -> r -> Covector s m Source #

LeftModule r s => LeftModule r (Covector s m) Source # 

Methods

(.*) :: r -> Covector s m -> Covector s m Source #

Monad (Covector r) Source # 

Methods

(>>=) :: Covector r a -> (a -> Covector r b) -> Covector r b #

(>>) :: Covector r a -> Covector r b -> Covector r b #

return :: a -> Covector r a #

fail :: String -> Covector r a #

Functor (Covector r) Source # 

Methods

fmap :: (a -> b) -> Covector r a -> Covector r b #

(<$) :: a -> Covector r b -> Covector r a #

Applicative (Covector r) Source # 

Methods

pure :: a -> Covector r a #

(<*>) :: Covector r (a -> b) -> Covector r a -> Covector r b #

(*>) :: Covector r a -> Covector r b -> Covector r b #

(<*) :: Covector r a -> Covector r b -> Covector r a #

Monoidal r => Alternative (Covector r) Source # 

Methods

empty :: Covector r a #

(<|>) :: Covector r a -> Covector r a -> Covector r a #

some :: Covector r a -> Covector r [a] #

many :: Covector r a -> Covector r [a] #

Monoidal r => MonadPlus (Covector r) Source # 

Methods

mzero :: Covector r a #

mplus :: Covector r a -> Covector r a -> Covector r a #

Monoidal r => Plus (Covector r) Source # 

Methods

zero :: Covector r a #

Additive r => Alt (Covector r) Source # 

Methods

(<!>) :: Covector r a -> Covector r a -> Covector r a #

some :: Applicative (Covector r) => Covector r a -> Covector r [a] #

many :: Applicative (Covector r) => Covector r a -> Covector r [a] #

Apply (Covector r) Source # 

Methods

(<.>) :: Covector r (a -> b) -> Covector r a -> Covector r b #

(.>) :: Covector r a -> Covector r b -> Covector r b #

(<.) :: Covector r a -> Covector r b -> Covector r a #

Bind (Covector r) Source # 

Methods

(>>-) :: Covector r a -> (a -> Covector r b) -> Covector r b #

join :: Covector r (Covector r a) -> Covector r a #

Idempotent r => Idempotent (Covector r a) Source # 
Abelian s => Abelian (Covector s a) Source # 
Additive r => Additive (Covector r a) Source # 

Methods

(+) :: Covector r a -> Covector r a -> Covector r a Source #

sinnum1p :: Natural -> Covector r a -> Covector r a Source #

sumWith1 :: Foldable1 f => (a -> Covector r a) -> f a -> Covector r a Source #

Monoidal s => Monoidal (Covector s a) Source # 

Methods

zero :: Covector s a Source #

sinnum :: Natural -> Covector s a -> Covector s a Source #

sumWith :: Foldable f => (a -> Covector s a) -> f a -> Covector s a Source #

Coalgebra r m => Semiring (Covector r m) Source # 
Coalgebra r m => Multiplicative (Covector r m) Source # 

Methods

(*) :: Covector r m -> Covector r m -> Covector r m Source #

pow1p :: Covector r m -> Natural -> Covector r m Source #

productWith1 :: Foldable1 f => (a -> Covector r m) -> f a -> Covector r m Source #

Group s => Group (Covector s a) Source # 

Methods

(-) :: Covector s a -> Covector s a -> Covector s a Source #

negate :: Covector s a -> Covector s a Source #

subtract :: Covector s a -> Covector s a -> Covector s a Source #

times :: Integral n => n -> Covector s a -> Covector s a Source #

CounitalCoalgebra r m => Unital (Covector r m) Source # 

Methods

one :: Covector r m Source #

pow :: Covector r m -> Natural -> Covector r m Source #

productWith :: Foldable f => (a -> Covector r m) -> f a -> Covector r m Source #

(Idempotent r, IdempotentCoalgebra r a) => Band (Covector r a) Source # 
(Rig r, CounitalCoalgebra r m) => Rig (Covector r m) Source # 
(Ring r, CounitalCoalgebra r m) => Ring (Covector r m) Source # 
(Commutative m, Coalgebra r m) => Commutative (Covector r m) Source # 
Distinguished a => Distinguished (Covector r a) Source # 

Methods

e :: Covector r a Source #

Complicated a => Complicated (Covector r a) Source # 

Methods

i :: Covector r a Source #

Infinitesimal a => Infinitesimal (Covector r a) Source # 

Methods

d :: Covector r a Source #

Hamiltonian a => Hamiltonian (Covector r a) Source # 

Methods

j :: Covector r a Source #

k :: Covector r a Source #

Hyperbolic a => Hyperbolic (Covector r a) Source # 

Methods

cosh :: Covector r a Source #

sinh :: Covector r a Source #

Trigonometric a => Trigonometric (Covector r a) Source # 

Methods

cos :: Covector r a Source #

sin :: Covector r a Source #

Coalgebra r m => RightModule (Covector r m) (Covector r m) Source # 

Methods

(*.) :: Covector r m -> Covector r m -> Covector r m Source #

Coalgebra r m => LeftModule (Covector r m) (Covector r m) Source # 

Methods

(.*) :: Covector r m -> Covector r m -> Covector r m Source #

Covectors as linear functionals

counitM :: UnitalAlgebra r a => a -> Covector r () Source #

comultM :: Algebra r a => a -> Covector r (a, a) Source #

multM :: Coalgebra r c => c -> c -> Covector r c Source #

antipodeM :: HopfAlgebra r h => h -> Covector r h Source #

convolveM antipodeM return = convolveM return antipodeM = comultM >=> uncurry joinM

convolveM :: (Algebra r c, Coalgebra r a) => (c -> Covector r a) -> (c -> Covector r a) -> c -> Covector r a Source #