Safe Haskell	Safe
Language	Haskell98

Numeric.Covector

Contents

Covectors as linear functionals

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 # 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) 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 #

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

(Rig r, CounitalCoalgebra r m) => Rig (Covector r m) Source #
Methods fromNatural :: Natural -> Covector r m Source #
(Ring r, CounitalCoalgebra r m) => Ring (Covector r m) Source #
Methods fromInteger :: Integer -> Covector r m Source #
Trigonometric a => Trigonometric (Covector r a) Source #
Methods cos :: Covector r a Source # sin :: Covector r a Source #
Hyperbolic a => Hyperbolic (Covector r a) Source #
Methods cosh :: Covector r a Source # sinh :: Covector r a Source #
Distinguished a => Distinguished (Covector r a) Source #
Methods e :: Covector r a Source #
Infinitesimal a => Infinitesimal (Covector r a) Source #
Methods d :: Covector r a Source #
Complicated a => Complicated (Covector r a) Source #
Methods i :: Covector r a Source #
Hamiltonian a => Hamiltonian (Covector r a) Source #
Methods j :: Covector r a Source # k :: 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 #

unitM :: CounitalCoalgebra r c => Covector r c Source #

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

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

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

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