Math.VectorSpace.ZeroDimensional

Documentation

data ZeroDim s :: * -> *

Constructors

Origin

Instances

Eq (ZeroDim s)

Methods

(==) :: ZeroDim s -> ZeroDim s -> Bool

(/=) :: ZeroDim s -> ZeroDim s -> Bool

Show (ZeroDim s)

Methods

showsPrec :: Int -> ZeroDim s -> ShowS

show :: ZeroDim s -> String

showList :: [ZeroDim s] -> ShowS

Monoid (ZeroDim s)

Methods

mempty :: ZeroDim s

mappend :: ZeroDim s -> ZeroDim s -> ZeroDim s

mconcat :: [ZeroDim s] -> ZeroDim s

AdditiveGroup (ZeroDim s)

Methods

zeroV :: ZeroDim s

(^+^) :: ZeroDim s -> ZeroDim s -> ZeroDim s

negateV :: ZeroDim s -> ZeroDim s

(^-^) :: ZeroDim s -> ZeroDim s -> ZeroDim s

AdditiveGroup s => InnerSpace (ZeroDim s)

Methods

(<.>) :: ZeroDim s -> ZeroDim s -> Scalar (ZeroDim s)

VectorSpace (ZeroDim s)

Associated Types

type Scalar (ZeroDim s) :: *

Methods

(*^) :: Scalar (ZeroDim s) -> ZeroDim s -> ZeroDim s

HasBasis (ZeroDim s)

Associated Types

type Basis (ZeroDim s) :: *

Methods

basisValue :: Basis (ZeroDim s) -> ZeroDim s

decompose :: ZeroDim s -> [(Basis (ZeroDim s), Scalar (ZeroDim s))]

decompose' :: ZeroDim s -> Basis (ZeroDim s) -> Scalar (ZeroDim s)

AffineSpace (ZeroDim s)

Associated Types

type Diff (ZeroDim s) :: *

Methods

(.-.) :: ZeroDim s -> ZeroDim s -> Diff (ZeroDim s)

(.+^) :: ZeroDim s -> Diff (ZeroDim s) -> ZeroDim s

Semimanifold (ZeroDim k)

Associated Types

type Needle (ZeroDim k) :: *

type Interior (ZeroDim k) :: *

Methods

(.+~^) :: Interior (ZeroDim k) -> Needle (ZeroDim k) -> ZeroDim k

fromInterior :: Interior (ZeroDim k) -> ZeroDim k

toInterior :: ZeroDim k -> Maybe (Interior (ZeroDim k))

translateP :: Tagged * (ZeroDim k) (Interior (ZeroDim k) -> Needle (ZeroDim k) -> Interior (ZeroDim k))

(.-~^) :: Interior (ZeroDim k) -> Needle (ZeroDim k) -> ZeroDim k

semimanifoldWitness :: SemimanifoldWitness (ZeroDim k)

PseudoAffine (ZeroDim k)

Methods

(.-~.) :: ZeroDim k -> ZeroDim k -> Maybe (Needle (ZeroDim k))

(.-~!) :: ZeroDim k -> ZeroDim k -> Needle (ZeroDim k)

pseudoAffineWitness :: PseudoAffineWitness (ZeroDim k)

Num' s => LinearSpace (ZeroDim s)

Associated Types

type DualVector (ZeroDim s) :: *

Methods

dualSpaceWitness :: DualSpaceWitness (ZeroDim s)

linearId :: ZeroDim s +> ZeroDim s

idTensor :: ZeroDim s ⊗ DualVector (ZeroDim s)

sampleLinearFunction :: (TensorSpace w, (* ~ Scalar (ZeroDim s)) (Scalar w)) => (ZeroDim s -+> w) -+> (ZeroDim s +> w)

toLinearForm :: DualVector (ZeroDim s) -+> (ZeroDim s +> Scalar (ZeroDim s))

fromLinearForm :: (ZeroDim s +> Scalar (ZeroDim s)) -+> DualVector (ZeroDim s)

coerceDoubleDual :: Coercion * (ZeroDim s) (DualVector (DualVector (ZeroDim s)))

trace :: (ZeroDim s +> ZeroDim s) -+> Scalar (ZeroDim s)

contractTensorMap :: (TensorSpace w, (* ~ Scalar w) (Scalar (ZeroDim s))) => (ZeroDim s +> (ZeroDim s ⊗ w)) -+> w

contractMapTensor :: (TensorSpace w, (* ~ Scalar w) (Scalar (ZeroDim s))) => (ZeroDim s ⊗ (ZeroDim s +> w)) -+> w

contractTensorFn :: (TensorSpace w, (* ~ Scalar w) (Scalar (ZeroDim s))) => (ZeroDim s -+> (ZeroDim s ⊗ w)) -+> w

contractLinearMapAgainst :: (LinearSpace w, (* ~ Scalar w) (Scalar (ZeroDim s))) => Bilinear (ZeroDim s +> w) (w -+> ZeroDim s) (Scalar (ZeroDim s))

applyDualVector :: Bilinear (DualVector (ZeroDim s)) (ZeroDim s) (Scalar (ZeroDim s))

applyLinear :: (TensorSpace w, (* ~ Scalar w) (Scalar (ZeroDim s))) => Bilinear (ZeroDim s +> w) (ZeroDim s) w

composeLinear :: (LinearSpace w, TensorSpace x, (* ~ Scalar w) (Scalar (ZeroDim s)), (* ~ Scalar x) (Scalar (ZeroDim s))) => Bilinear (w +> x) (ZeroDim s +> w) (ZeroDim s +> x)

tensorId :: (LinearSpace w, (* ~ Scalar w) (Scalar (ZeroDim s))) => (ZeroDim s ⊗ w) +> (ZeroDim s ⊗ w)

applyTensorFunctional :: (LinearSpace u, (* ~ Scalar u) (Scalar (ZeroDim s))) => Bilinear (DualVector (ZeroDim s ⊗ u)) (ZeroDim s ⊗ u) (Scalar (ZeroDim s))

applyTensorLinMap :: (LinearSpace u, TensorSpace w, (* ~ Scalar u) (Scalar (ZeroDim s)), (* ~ Scalar w) (Scalar (ZeroDim s))) => Bilinear ((ZeroDim s ⊗ u) +> w) (ZeroDim s ⊗ u) w

Num' s => TensorSpace (ZeroDim s)

Associated Types

type TensorProduct (ZeroDim s) w :: *

Methods

scalarSpaceWitness :: ScalarSpaceWitness (ZeroDim s)

linearManifoldWitness :: LinearManifoldWitness (ZeroDim s)

zeroTensor :: (TensorSpace w, (* ~ Scalar w) (Scalar (ZeroDim s))) => ZeroDim s ⊗ w

toFlatTensor :: ZeroDim s -+> (ZeroDim s ⊗ Scalar (ZeroDim s))

fromFlatTensor :: (ZeroDim s ⊗ Scalar (ZeroDim s)) -+> ZeroDim s

addTensors :: (TensorSpace w, (* ~ Scalar w) (Scalar (ZeroDim s))) => (ZeroDim s ⊗ w) -> (ZeroDim s ⊗ w) -> ZeroDim s ⊗ w

subtractTensors :: (TensorSpace (ZeroDim s), TensorSpace w, (* ~ Scalar w) (Scalar (ZeroDim s))) => (ZeroDim s ⊗ w) -> (ZeroDim s ⊗ w) -> ZeroDim s ⊗ w

scaleTensor :: (TensorSpace w, (* ~ Scalar w) (Scalar (ZeroDim s))) => Bilinear (Scalar (ZeroDim s)) (ZeroDim s ⊗ w) (ZeroDim s ⊗ w)

negateTensor :: (TensorSpace w, (* ~ Scalar w) (Scalar (ZeroDim s))) => (ZeroDim s ⊗ w) -+> (ZeroDim s ⊗ w)

tensorProduct :: (TensorSpace w, (* ~ Scalar w) (Scalar (ZeroDim s))) => Bilinear (ZeroDim s) w (ZeroDim s ⊗ w)

tensorProducts :: (TensorSpace w, (* ~ Scalar w) (Scalar (ZeroDim s))) => [(ZeroDim s, w)] -> ZeroDim s ⊗ w

transposeTensor :: (TensorSpace w, (* ~ Scalar w) (Scalar (ZeroDim s))) => (ZeroDim s ⊗ w) -+> (w ⊗ ZeroDim s)

fmapTensor :: (TensorSpace w, TensorSpace x, (* ~ Scalar w) (Scalar (ZeroDim s)), (* ~ Scalar x) (Scalar (ZeroDim s))) => Bilinear (w -+> x) (ZeroDim s ⊗ w) (ZeroDim s ⊗ x)

fzipTensorWith :: (TensorSpace u, TensorSpace w, TensorSpace x, (* ~ Scalar u) (Scalar (ZeroDim s)), (* ~ Scalar w) (Scalar (ZeroDim s)), (* ~ Scalar x) (Scalar (ZeroDim s))) => Bilinear ((w, x) -+> u) (ZeroDim s ⊗ w, ZeroDim s ⊗ x) (ZeroDim s ⊗ u)

coerceFmapTensorProduct :: Functor p => p (ZeroDim s) -> Coercion * a b -> Coercion * (TensorProduct (ZeroDim s) a) (TensorProduct (ZeroDim s) b)

wellDefinedVector :: ZeroDim s -> Maybe (ZeroDim s)

wellDefinedTensor :: (TensorSpace w, (* ~ Scalar w) (Scalar (ZeroDim s))) => (ZeroDim s ⊗ w) -> Maybe (ZeroDim s ⊗ w)

Num' s => FiniteDimensional (ZeroDim s)

Associated Types

data SubBasis (ZeroDim s) :: *

Methods

entireBasis :: SubBasis (ZeroDim s)

enumerateSubBasis :: SubBasis (ZeroDim s) -> [ZeroDim s]

subbasisDimension :: SubBasis (ZeroDim s) -> Int

decomposeLinMap :: (LSpace w, (* ~ Scalar w) (Scalar (ZeroDim s))) => (ZeroDim s +> w) -> (SubBasis (ZeroDim s), DList w)

decomposeLinMapWithin :: (LSpace w, (* ~ Scalar w) (Scalar (ZeroDim s))) => SubBasis (ZeroDim s) -> (ZeroDim s +> w) -> Either (SubBasis (ZeroDim s), DList w) (DList w)

recomposeSB :: SubBasis (ZeroDim s) -> [Scalar (ZeroDim s)] -> (ZeroDim s, [Scalar (ZeroDim s)])

recomposeSBTensor :: (FiniteDimensional w, (* ~ Scalar w) (Scalar (ZeroDim s))) => SubBasis (ZeroDim s) -> SubBasis w -> [Scalar (ZeroDim s)] -> (ZeroDim s ⊗ w, [Scalar (ZeroDim s)])

recomposeLinMap :: (LSpace w, (* ~ Scalar w) (Scalar (ZeroDim s))) => SubBasis (ZeroDim s) -> [w] -> (ZeroDim s +> w, [w])

recomposeContraLinMap :: (LinearSpace w, (* ~ Scalar w) (Scalar (ZeroDim s)), Functor f) => (f (Scalar w) -> w) -> f (DualVector (ZeroDim s)) -> ZeroDim s +> w

recomposeContraLinMapTensor :: (FiniteDimensional u, LinearSpace w, (* ~ Scalar u) (Scalar (ZeroDim s)), (* ~ Scalar w) (Scalar (ZeroDim s)), Functor f) => (f (Scalar w) -> w) -> f (ZeroDim s +> DualVector u) -> (ZeroDim s ⊗ u) +> w

uncanonicallyFromDual :: DualVector (ZeroDim s) -+> ZeroDim s

uncanonicallyToDual :: ZeroDim s -+> DualVector (ZeroDim s)

(Fractional' s, SemiInner s) => SemiInner (ZeroDim s)

Methods

dualBasisCandidates :: [(Int, ZeroDim s)] -> Forest (Int, DualVector (ZeroDim s))

tensorDualBasisCandidates :: (SemiInner w, (* ~ Scalar w) (Scalar (ZeroDim s))) => [(Int, ZeroDim s ⊗ w)] -> Forest (Int, DualVector (ZeroDim s ⊗ w))

symTensorDualBasisCandidates :: [(Int, SymmetricTensor (Scalar (ZeroDim s)) (ZeroDim s))] -> Forest (Int, SymmetricTensor (Scalar (ZeroDim s)) (DualVector (ZeroDim s)))

symTensorTensorDualBasisCandidates :: (SemiInner w, (* ~ Scalar w) (Scalar (ZeroDim s))) => [(Int, SymmetricTensor (Scalar (ZeroDim s)) (ZeroDim s) ⊗ w)] -> Forest (Int, SymmetricTensor (Scalar (ZeroDim s)) (ZeroDim s) +> DualVector w)

type Scalar (ZeroDim s)

type Scalar (ZeroDim s) = s

type Basis (ZeroDim s)

type Basis (ZeroDim s) = Void

type Diff (ZeroDim s)

type Diff (ZeroDim s) = ZeroDim s

type Interior (ZeroDim k)

type Interior (ZeroDim k) = ZeroDim k

type Needle (ZeroDim k)

type Needle (ZeroDim k) = ZeroDim k

type DualVector (ZeroDim s)

type DualVector (ZeroDim s) = ZeroDim s

data SubBasis (ZeroDim s)

data SubBasis (ZeroDim s) = ZeroBasis

type TensorProduct (ZeroDim s) v

type TensorProduct (ZeroDim s) v = ZeroDim s