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Language	Haskell98

Numeric.Covector

Contents

Covectors as linear functionals

Synopsis

newtype Covector r a = Covector {
- ($*) :: (a -> r) -> r
}
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 :: (Algebra r c, Coalgebra r a) => (c -> Covector r a) -> (c -> Covector r a) -> c -> Covector r a

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) #
Instance details Defined in Numeric.Covector Methods (*.) :: Covector s m -> r -> Covector s m #
LeftModule r s => LeftModule r (Covector s m) #
Instance details Defined in Numeric.Covector Methods (.*) :: r -> Covector s m -> Covector s m #
Monad (Covector r) #
Instance details Defined in Numeric.Covector 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) #
Instance details Defined in Numeric.Covector Methods fmap :: (a -> b) -> Covector r a -> Covector r b # (<$) :: a -> Covector r b -> Covector r a #
Applicative (Covector r) #
Instance details Defined in Numeric.Covector 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) #
Instance details Defined in Numeric.Covector 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) #
Instance details Defined in Numeric.Covector Methods mzero :: Covector r a # mplus :: Covector r a -> Covector r a -> Covector r a #
Monoidal r => Plus (Covector r) #
Instance details Defined in Numeric.Covector Methods zero :: Covector r a #
Additive r => Alt (Covector r) #
Instance details Defined in Numeric.Covector 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) #
Instance details Defined in Numeric.Covector 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 # liftF2 :: (a -> b -> c) -> Covector r a -> Covector r b -> Covector r c #
Bind (Covector r) #
Instance details Defined in Numeric.Covector 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) #
Instance details Defined in Numeric.Covector
Abelian s => Abelian (Covector s a) #
Instance details Defined in Numeric.Covector
Additive r => Additive (Covector r a) #
Instance details Defined in Numeric.Covector Methods (+) :: Covector r a -> Covector r a -> Covector r a # sinnum1p :: Natural -> Covector r a -> Covector r a # sumWith1 :: Foldable1 f => (a0 -> Covector r a) -> f a0 -> Covector r a #
Monoidal s => Monoidal (Covector s a) #
Instance details Defined in Numeric.Covector Methods zero :: Covector s a # sinnum :: Natural -> Covector s a -> Covector s a # sumWith :: Foldable f => (a0 -> Covector s a) -> f a0 -> Covector s a #
Coalgebra r m => Semiring (Covector r m) #
Instance details Defined in Numeric.Covector
Coalgebra r m => Multiplicative (Covector r m) #
Instance details Defined in Numeric.Covector 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) #
Instance details Defined in Numeric.Covector 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) #
Instance details Defined in Numeric.Covector 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) #
Instance details Defined in Numeric.Covector
(Commutative m, Coalgebra r m) => Commutative (Covector r m) #
Instance details Defined in Numeric.Covector
(Rig r, CounitalCoalgebra r m) => Rig (Covector r m) #
Instance details Defined in Numeric.Covector Methods fromNatural :: Natural -> Covector r m #
(Ring r, CounitalCoalgebra r m) => Ring (Covector r m) #
Instance details Defined in Numeric.Covector Methods fromInteger :: Integer -> Covector r m #
Trigonometric a => Trigonometric (Covector r a) #
Instance details Defined in Numeric.Coalgebra.Trigonometric.Class Methods cos :: Covector r a # sin :: Covector r a #
Hyperbolic a => Hyperbolic (Covector r a) #
Instance details Defined in Numeric.Coalgebra.Hyperbolic.Class Methods cosh :: Covector r a # sinh :: Covector r a #
Distinguished a => Distinguished (Covector r a) #
Instance details Defined in Numeric.Algebra.Distinguished.Class Methods e :: Covector r a #
Infinitesimal a => Infinitesimal (Covector r a) #
Instance details Defined in Numeric.Algebra.Dual.Class Methods d :: Covector r a #
Complicated a => Complicated (Covector r a) #
Instance details Defined in Numeric.Algebra.Complex.Class Methods i :: Covector r a #
Hamiltonian a => Hamiltonian (Covector r a) #
Instance details Defined in Numeric.Algebra.Quaternion.Class Methods j :: Covector r a # k :: Covector r a #
Coalgebra r m => RightModule (Covector r m) (Covector r m) #
Instance details Defined in Numeric.Covector Methods (*.) :: Covector r m -> Covector r m -> Covector r m #
Coalgebra r m => LeftModule (Covector r m) (Covector r m) #
Instance details Defined in Numeric.Covector 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 #