Numeric.Coalgebra.Hyperbolic

Documentation

class Hyperbolic r where

Minimal complete definition

cosh, sinh

Methods

cosh :: r

sinh :: r

Instances

Hyperbolic HyperBasis

Methods

cosh :: HyperBasis

sinh :: HyperBasis

Hyperbolic HyperBasis'

Methods

cosh :: HyperBasis'

sinh :: HyperBasis'

Rig r => Hyperbolic (Hyper r)

Methods

cosh :: Hyper r

sinh :: Hyper r

Rig r => Hyperbolic (Hyper' r)

Methods

cosh :: Hyper' r

sinh :: Hyper' r

Rig r => Hyperbolic (HyperBasis -> r)

Methods

cosh :: HyperBasis -> r

sinh :: HyperBasis -> r

Rig r => Hyperbolic (HyperBasis' -> r)

Methods

cosh :: HyperBasis' -> r

sinh :: HyperBasis' -> r

Hyperbolic a => Hyperbolic (Covector r a)

Methods

cosh :: Covector r a

sinh :: Covector r a

data HyperBasis

Constructors

Cosh

Sinh

Instances

Bounded HyperBasis

Methods

minBound :: HyperBasis

maxBound :: HyperBasis

Enum HyperBasis

Methods

succ :: HyperBasis -> HyperBasis

pred :: HyperBasis -> HyperBasis

toEnum :: Int -> HyperBasis

fromEnum :: HyperBasis -> Int

enumFrom :: HyperBasis -> [HyperBasis]

enumFromThen :: HyperBasis -> HyperBasis -> [HyperBasis]

enumFromTo :: HyperBasis -> HyperBasis -> [HyperBasis]

enumFromThenTo :: HyperBasis -> HyperBasis -> HyperBasis -> [HyperBasis]

Eq HyperBasis

Methods

(==) :: HyperBasis -> HyperBasis -> Bool

(/=) :: HyperBasis -> HyperBasis -> Bool

Data HyperBasis

Methods

gfoldl :: (forall d b. Data d => c (d -> b) -> d -> c b) -> (forall g. g -> c g) -> HyperBasis -> c HyperBasis

gunfold :: (forall b r. Data b => c (b -> r) -> c r) -> (forall r. r -> c r) -> Constr -> c HyperBasis

toConstr :: HyperBasis -> Constr

dataTypeOf :: HyperBasis -> DataType

dataCast1 :: Typeable (* -> *) t => (forall d. Data d => c (t d)) -> Maybe (c HyperBasis)

dataCast2 :: Typeable (* -> * -> *) t => (forall d e. (Data d, Data e) => c (t d e)) -> Maybe (c HyperBasis)

gmapT :: (forall b. Data b => b -> b) -> HyperBasis -> HyperBasis

gmapQl :: (r -> r' -> r) -> r -> (forall d. Data d => d -> r') -> HyperBasis -> r

gmapQr :: (r' -> r -> r) -> r -> (forall d. Data d => d -> r') -> HyperBasis -> r

gmapQ :: (forall d. Data d => d -> u) -> HyperBasis -> [u]

gmapQi :: Int -> (forall d. Data d => d -> u) -> HyperBasis -> u

gmapM :: Monad m => (forall d. Data d => d -> m d) -> HyperBasis -> m HyperBasis

gmapMp :: MonadPlus m => (forall d. Data d => d -> m d) -> HyperBasis -> m HyperBasis

gmapMo :: MonadPlus m => (forall d. Data d => d -> m d) -> HyperBasis -> m HyperBasis

Ord HyperBasis

Methods

compare :: HyperBasis -> HyperBasis -> Ordering

(<) :: HyperBasis -> HyperBasis -> Bool

(<=) :: HyperBasis -> HyperBasis -> Bool

(>) :: HyperBasis -> HyperBasis -> Bool

(>=) :: HyperBasis -> HyperBasis -> Bool

max :: HyperBasis -> HyperBasis -> HyperBasis

min :: HyperBasis -> HyperBasis -> HyperBasis

Read HyperBasis

Methods

readsPrec :: Int -> ReadS HyperBasis

readList :: ReadS [HyperBasis]

readPrec :: ReadPrec HyperBasis

readListPrec :: ReadPrec [HyperBasis]

Show HyperBasis

Methods

showsPrec :: Int -> HyperBasis -> ShowS

show :: HyperBasis -> String

showList :: [HyperBasis] -> ShowS

Ix HyperBasis

Methods

range :: (HyperBasis, HyperBasis) -> [HyperBasis]

index :: (HyperBasis, HyperBasis) -> HyperBasis -> Int

unsafeIndex :: (HyperBasis, HyperBasis) -> HyperBasis -> Int

inRange :: (HyperBasis, HyperBasis) -> HyperBasis -> Bool

rangeSize :: (HyperBasis, HyperBasis) -> Int

unsafeRangeSize :: (HyperBasis, HyperBasis) -> Int

Hyperbolic HyperBasis

Methods

cosh :: HyperBasis

sinh :: HyperBasis

MonadReader HyperBasis Hyper

Methods

ask :: Hyper HyperBasis

local :: (HyperBasis -> HyperBasis) -> Hyper a -> Hyper a

reader :: (HyperBasis -> a) -> Hyper a

(Commutative k, Semiring k) => Coalgebra k HyperBasis

the hyperbolic trigonometric coalgebra

Methods

comult :: (HyperBasis -> k) -> HyperBasis -> HyperBasis -> k

Semiring k => Algebra k HyperBasis

the trivial diagonal algebra

Methods

mult :: (HyperBasis -> HyperBasis -> k) -> HyperBasis -> k

(Commutative k, Semiring k) => Bialgebra k HyperBasis

(Commutative k, Semiring k) => CounitalCoalgebra k HyperBasis

Methods

counit :: (HyperBasis -> k) -> k

Semiring k => UnitalAlgebra k HyperBasis

Methods

unit :: k -> HyperBasis -> k

(Commutative k, Group k, InvolutiveSemiring k) => HopfAlgebra k HyperBasis

Methods

antipode :: (HyperBasis -> k) -> HyperBasis -> k

(Commutative k, Group k, InvolutiveSemiring k) => InvolutiveCoalgebra k HyperBasis

Methods

coinv :: (HyperBasis -> k) -> HyperBasis -> k

(Commutative k, Group k, InvolutiveSemiring k) => InvolutiveAlgebra k HyperBasis

Methods

inv :: (HyperBasis -> k) -> HyperBasis -> k

Rig r => Hyperbolic (HyperBasis -> r)

Methods

cosh :: HyperBasis -> r

sinh :: HyperBasis -> r

data Hyper a

Constructors

Hyper a a

Instances

Monad Hyper

Methods

(>>=) :: Hyper a -> (a -> Hyper b) -> Hyper b

(>>) :: Hyper a -> Hyper b -> Hyper b

return :: a -> Hyper a

fail :: String -> Hyper a

Functor Hyper

Methods

fmap :: (a -> b) -> Hyper a -> Hyper b

(<$) :: a -> Hyper b -> Hyper a

Applicative Hyper

Methods

pure :: a -> Hyper a

(<*>) :: Hyper (a -> b) -> Hyper a -> Hyper b

liftA2 :: (a -> b -> c) -> Hyper a -> Hyper b -> Hyper c

(*>) :: Hyper a -> Hyper b -> Hyper b

(<*) :: Hyper a -> Hyper b -> Hyper a

Foldable Hyper

Methods

fold :: Monoid m => Hyper m -> m

foldMap :: Monoid m => (a -> m) -> Hyper a -> m

foldr :: (a -> b -> b) -> b -> Hyper a -> b

foldr' :: (a -> b -> b) -> b -> Hyper a -> b

foldl :: (b -> a -> b) -> b -> Hyper a -> b

foldl' :: (b -> a -> b) -> b -> Hyper a -> b

foldr1 :: (a -> a -> a) -> Hyper a -> a

foldl1 :: (a -> a -> a) -> Hyper a -> a

toList :: Hyper a -> [a]

null :: Hyper a -> Bool

length :: Hyper a -> Int

elem :: Eq a => a -> Hyper a -> Bool

maximum :: Ord a => Hyper a -> a

minimum :: Ord a => Hyper a -> a

sum :: Num a => Hyper a -> a

product :: Num a => Hyper a -> a

Traversable Hyper

Methods

traverse :: Applicative f => (a -> f b) -> Hyper a -> f (Hyper b)

sequenceA :: Applicative f => Hyper (f a) -> f (Hyper a)

mapM :: Monad m => (a -> m b) -> Hyper a -> m (Hyper b)

sequence :: Monad m => Hyper (m a) -> m (Hyper a)

Distributive Hyper

Methods

distribute :: Functor f => f (Hyper a) -> Hyper (f a)

collect :: Functor f => (a -> Hyper b) -> f a -> Hyper (f b)

distributeM :: Monad m => m (Hyper a) -> Hyper (m a)

collectM :: Monad m => (a -> Hyper b) -> m a -> Hyper (m b)

Representable Hyper

Associated Types

type Rep (Hyper :: * -> *) :: *

Methods

tabulate :: (Rep Hyper -> a) -> Hyper a

index :: Hyper a -> Rep Hyper -> a

Traversable1 Hyper

Methods

traverse1 :: Apply f => (a -> f b) -> Hyper a -> f (Hyper b)

sequence1 :: Apply f => Hyper (f b) -> f (Hyper b)

Foldable1 Hyper

Methods

fold1 :: Semigroup m => Hyper m -> m

foldMap1 :: Semigroup m => (a -> m) -> Hyper a -> m

toNonEmpty :: Hyper a -> NonEmpty a

Apply Hyper

Methods

(<.>) :: Hyper (a -> b) -> Hyper a -> Hyper b

(.>) :: Hyper a -> Hyper b -> Hyper b

(<.) :: Hyper a -> Hyper b -> Hyper a

Bind Hyper

Methods

(>>-) :: Hyper a -> (a -> Hyper b) -> Hyper b

join :: Hyper (Hyper a) -> Hyper a

MonadReader HyperBasis Hyper

Methods

ask :: Hyper HyperBasis

local :: (HyperBasis -> HyperBasis) -> Hyper a -> Hyper a

reader :: (HyperBasis -> a) -> Hyper a

RightModule r s => RightModule r (Hyper s)

Methods

(*.) :: Hyper s -> r -> Hyper s

LeftModule r s => LeftModule r (Hyper s)

Methods

(.*) :: r -> Hyper s -> Hyper s

Eq a => Eq (Hyper a)

Methods

(==) :: Hyper a -> Hyper a -> Bool

(/=) :: Hyper a -> Hyper a -> Bool

Data a => Data (Hyper a)

Methods

gfoldl :: (forall d b. Data d => c (d -> b) -> d -> c b) -> (forall g. g -> c g) -> Hyper a -> c (Hyper a)

gunfold :: (forall b r. Data b => c (b -> r) -> c r) -> (forall r. r -> c r) -> Constr -> c (Hyper a)

toConstr :: Hyper a -> Constr

dataTypeOf :: Hyper a -> DataType

dataCast1 :: Typeable (* -> *) t => (forall d. Data d => c (t d)) -> Maybe (c (Hyper a))

dataCast2 :: Typeable (* -> * -> *) t => (forall d e. (Data d, Data e) => c (t d e)) -> Maybe (c (Hyper a))

gmapT :: (forall b. Data b => b -> b) -> Hyper a -> Hyper a

gmapQl :: (r -> r' -> r) -> r -> (forall d. Data d => d -> r') -> Hyper a -> r

gmapQr :: (r' -> r -> r) -> r -> (forall d. Data d => d -> r') -> Hyper a -> r

gmapQ :: (forall d. Data d => d -> u) -> Hyper a -> [u]

gmapQi :: Int -> (forall d. Data d => d -> u) -> Hyper a -> u

gmapM :: Monad m => (forall d. Data d => d -> m d) -> Hyper a -> m (Hyper a)

gmapMp :: MonadPlus m => (forall d. Data d => d -> m d) -> Hyper a -> m (Hyper a)

gmapMo :: MonadPlus m => (forall d. Data d => d -> m d) -> Hyper a -> m (Hyper a)

Read a => Read (Hyper a)

Methods

readsPrec :: Int -> ReadS (Hyper a)

readList :: ReadS [Hyper a]

readPrec :: ReadPrec (Hyper a)

readListPrec :: ReadPrec [Hyper a]

Show a => Show (Hyper a)

Methods

showsPrec :: Int -> Hyper a -> ShowS

show :: Hyper a -> String

showList :: [Hyper a] -> ShowS

Idempotent r => Idempotent (Hyper r)

Abelian r => Abelian (Hyper r)

Partitionable r => Partitionable (Hyper r)

Methods

partitionWith :: (Hyper r -> Hyper r -> r) -> Hyper r -> NonEmpty r

Additive r => Additive (Hyper r)

Methods

(+) :: Hyper r -> Hyper r -> Hyper r

sinnum1p :: Natural -> Hyper r -> Hyper r

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

Monoidal r => Monoidal (Hyper r)

Methods

zero :: Hyper r

sinnum :: Natural -> Hyper r -> Hyper r

sumWith :: Foldable f => (a -> Hyper r) -> f a -> Hyper r

(Commutative k, Semiring k) => Semiring (Hyper k)

(Commutative k, Semiring k) => Multiplicative (Hyper k)

Methods

(*) :: Hyper k -> Hyper k -> Hyper k

pow1p :: Hyper k -> Natural -> Hyper k

productWith1 :: Foldable1 f => (a -> Hyper k) -> f a -> Hyper k

Group r => Group (Hyper r)

Methods

(-) :: Hyper r -> Hyper r -> Hyper r

negate :: Hyper r -> Hyper r

subtract :: Hyper r -> Hyper r -> Hyper r

times :: Integral n => n -> Hyper r -> Hyper r

(Commutative k, Rig k) => Unital (Hyper k)

Methods

one :: Hyper k

pow :: Hyper k -> Natural -> Hyper k

productWith :: Foldable f => (a -> Hyper k) -> f a -> Hyper k

(Commutative k, Semiring k) => Commutative (Hyper k)

(Commutative r, Group r, InvolutiveSemiring r) => InvolutiveSemiring (Hyper r)

(Commutative r, Group r, InvolutiveSemiring r) => InvolutiveMultiplication (Hyper r)

Methods

adjoint :: Hyper r -> Hyper r

(Commutative r, Rig r) => Rig (Hyper r)

Methods

fromNatural :: Natural -> Hyper r

(Commutative r, Ring r) => Ring (Hyper r)

Methods

fromInteger :: Integer -> Hyper r

Rig r => Hyperbolic (Hyper r)

Methods

cosh :: Hyper r

sinh :: Hyper r

(Commutative r, Semiring r) => RightModule (Hyper r) (Hyper r)

Methods

(*.) :: Hyper r -> Hyper r -> Hyper r

(Commutative r, Semiring r) => LeftModule (Hyper r) (Hyper r)

Methods

(.*) :: Hyper r -> Hyper r -> Hyper r

type Rep Hyper

type Rep Hyper = HyperBasis