Numeric.Covector

Documentation

newtype Covector r a

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)

Methods

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

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

Methods

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

Monad (Covector r)

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)

Methods

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

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

Applicative (Covector r)

Methods

pure :: a -> Covector r a

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

liftA2 :: (a -> b -> c) -> Covector r a -> Covector r b -> Covector r c

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

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

Monoidal r => Alternative (Covector r)

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)

Methods

mzero :: Covector r a

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

Monoidal r => Plus (Covector r)

Methods

zero :: Covector r a

Additive r => Alt (Covector r)

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)

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)

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)

Abelian s => Abelian (Covector s a)

Additive r => Additive (Covector r a)

Methods

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

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

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

Monoidal s => Monoidal (Covector s a)

Methods

zero :: Covector s a

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

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

Coalgebra r m => Semiring (Covector r m)

Coalgebra r m => Multiplicative (Covector r m)

Methods

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

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

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

Group s => Group (Covector s a)

Methods

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

negate :: Covector s a -> Covector s a

subtract :: Covector s a -> Covector s a -> Covector s a

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

CounitalCoalgebra r m => Unital (Covector r m)

Methods

one :: Covector r m

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

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

(Idempotent r, IdempotentCoalgebra r a) => Band (Covector r a)

(Commutative m, Coalgebra r m) => Commutative (Covector r m)

(Rig r, CounitalCoalgebra r m) => Rig (Covector r m)

Methods

fromNatural :: Natural -> Covector r m

(Ring r, CounitalCoalgebra r m) => Ring (Covector r m)

Methods

fromInteger :: Integer -> Covector r m

Trigonometric a => Trigonometric (Covector r a)

Methods

cos :: Covector r a

sin :: Covector r a

Hyperbolic a => Hyperbolic (Covector r a)

Methods

cosh :: Covector r a

sinh :: Covector r a

Distinguished a => Distinguished (Covector r a)

Methods

e :: Covector r a

Infinitesimal a => Infinitesimal (Covector r a)

Methods

d :: Covector r a

Complicated a => Complicated (Covector r a)

Methods

i :: Covector r a

Hamiltonian a => Hamiltonian (Covector r a)

Methods

j :: Covector r a

k :: Covector r a

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

Methods

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

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

Methods

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

Covectors as linear functionals

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

unitM :: CounitalCoalgebra r c => Covector r c

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

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

invM :: InvolutiveAlgebra r h => h -> Covector r h

coinvM :: InvolutiveCoalgebra r h => h -> Covector r h

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

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