Copyright	(c) Justus Sagemüller 2016
License	GPL v3
Maintainer	(@) jsag $ hvl.no
Stability	experimental
Portability	portable
Safe Haskell	Safe
Language	Haskell2010

Math.VectorSpace.ZeroDimensional

Description

Documentation

data ZeroDim s :: * -> * #

Constructors

Origin

Instances

Eq (ZeroDim s)
Methods (==) :: ZeroDim s -> ZeroDim s -> Bool Source # (/=) :: ZeroDim s -> ZeroDim s -> Bool Source #
Show (ZeroDim s)
Methods showsPrec :: Int -> ZeroDim s -> ShowS Source # show :: ZeroDim s -> String Source # showList :: [ZeroDim s] -> ShowS Source #
Semigroup (ZeroDim s)
Methods (<>) :: ZeroDim s -> ZeroDim s -> ZeroDim s Source # sconcat :: NonEmpty (ZeroDim s) -> ZeroDim s Source # stimes :: Integral b => b -> ZeroDim s -> ZeroDim s Source #
Monoid (ZeroDim s)
Methods mempty :: ZeroDim s Source # mappend :: ZeroDim s -> ZeroDim s -> ZeroDim s Source # mconcat :: [ZeroDim s] -> ZeroDim s Source #
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) :: * # Methods (.+~^) :: ZeroDim k -> Needle (ZeroDim k) -> ZeroDim k # (.-~^) :: 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) Source #
Associated Types type DualVector (ZeroDim s) :: * Source # Methods dualSpaceWitness :: DualSpaceWitness (ZeroDim s) Source # linearId :: ZeroDim s +> ZeroDim s Source # idTensor :: ZeroDim s ⊗ DualVector (ZeroDim s) Source # sampleLinearFunction :: (TensorSpace w, (* ~ Scalar (ZeroDim s)) (Scalar w)) => (ZeroDim s -+> w) -+> (ZeroDim s +> w) Source # toLinearForm :: DualVector (ZeroDim s) -+> (ZeroDim s +> Scalar (ZeroDim s)) Source # fromLinearForm :: (ZeroDim s +> Scalar (ZeroDim s)) -+> DualVector (ZeroDim s) Source # coerceDoubleDual :: Coercion * (ZeroDim s) (DualVector (DualVector (ZeroDim s))) Source # trace :: (ZeroDim s +> ZeroDim s) -+> Scalar (ZeroDim s) Source # contractTensorMap :: (TensorSpace w, (* ~ Scalar w) (Scalar (ZeroDim s))) => (ZeroDim s +> (ZeroDim s ⊗ w)) -+> w Source # contractMapTensor :: (TensorSpace w, (* ~ Scalar w) (Scalar (ZeroDim s))) => (ZeroDim s ⊗ (ZeroDim s +> w)) -+> w Source # contractTensorFn :: (TensorSpace w, (* ~ Scalar w) (Scalar (ZeroDim s))) => (ZeroDim s -+> (ZeroDim s ⊗ w)) -+> w Source # contractLinearMapAgainst :: (LinearSpace w, (* ~ Scalar w) (Scalar (ZeroDim s))) => Bilinear (ZeroDim s +> w) (w -+> ZeroDim s) (Scalar (ZeroDim s)) Source # applyDualVector :: Bilinear (DualVector (ZeroDim s)) (ZeroDim s) (Scalar (ZeroDim s)) Source # applyLinear :: (TensorSpace w, (* ~ Scalar w) (Scalar (ZeroDim s))) => Bilinear (ZeroDim s +> w) (ZeroDim s) w Source # composeLinear :: (LinearSpace w, TensorSpace x, (* ~ Scalar w) (Scalar (ZeroDim s)), (* ~ Scalar x) (Scalar (ZeroDim s))) => Bilinear (w +> x) (ZeroDim s +> w) (ZeroDim s +> x) Source # tensorId :: (LinearSpace w, (* ~ Scalar w) (Scalar (ZeroDim s))) => (ZeroDim s ⊗ w) +> (ZeroDim s ⊗ w) Source # applyTensorFunctional :: (LinearSpace u, (* ~ Scalar u) (Scalar (ZeroDim s))) => Bilinear (DualVector (ZeroDim s ⊗ u)) (ZeroDim s ⊗ u) (Scalar (ZeroDim s)) Source # applyTensorLinMap :: (LinearSpace u, TensorSpace w, (* ~ Scalar u) (Scalar (ZeroDim s)), (* ~ Scalar w) (Scalar (ZeroDim s))) => Bilinear ((ZeroDim s ⊗ u) +> w) (ZeroDim s ⊗ u) w Source # useTupleLinearSpaceComponents :: (* ~ ZeroDim s) (x, y) => (LinearSpace x -> LinearSpace y -> (* ~ Scalar x) (Scalar y) -> φ) -> φ Source #
Num' s => TensorSpace (ZeroDim s) Source #
Associated Types type TensorProduct (ZeroDim s) w :: * Source # Methods scalarSpaceWitness :: ScalarSpaceWitness (ZeroDim s) Source # linearManifoldWitness :: LinearManifoldWitness (ZeroDim s) Source # zeroTensor :: (TensorSpace w, (* ~ Scalar w) (Scalar (ZeroDim s))) => ZeroDim s ⊗ w Source # toFlatTensor :: ZeroDim s -+> (ZeroDim s ⊗ Scalar (ZeroDim s)) Source # fromFlatTensor :: (ZeroDim s ⊗ Scalar (ZeroDim s)) -+> ZeroDim s Source # addTensors :: (TensorSpace w, (* ~ Scalar w) (Scalar (ZeroDim s))) => (ZeroDim s ⊗ w) -> (ZeroDim s ⊗ w) -> ZeroDim s ⊗ w Source # subtractTensors :: (TensorSpace (ZeroDim s), TensorSpace w, (* ~ Scalar w) (Scalar (ZeroDim s))) => (ZeroDim s ⊗ w) -> (ZeroDim s ⊗ w) -> ZeroDim s ⊗ w Source # scaleTensor :: (TensorSpace w, (* ~ Scalar w) (Scalar (ZeroDim s))) => Bilinear (Scalar (ZeroDim s)) (ZeroDim s ⊗ w) (ZeroDim s ⊗ w) Source # negateTensor :: (TensorSpace w, (* ~ Scalar w) (Scalar (ZeroDim s))) => (ZeroDim s ⊗ w) -+> (ZeroDim s ⊗ w) Source # tensorProduct :: (TensorSpace w, (* ~ Scalar w) (Scalar (ZeroDim s))) => Bilinear (ZeroDim s) w (ZeroDim s ⊗ w) Source # tensorProducts :: (TensorSpace w, (* ~ Scalar w) (Scalar (ZeroDim s))) => [(ZeroDim s, w)] -> ZeroDim s ⊗ w Source # transposeTensor :: (TensorSpace w, (* ~ Scalar w) (Scalar (ZeroDim s))) => (ZeroDim s ⊗ w) -+> (w ⊗ ZeroDim s) Source # fmapTensor :: (TensorSpace w, TensorSpace x, (* ~ Scalar w) (Scalar (ZeroDim s)), (* ~ Scalar x) (Scalar (ZeroDim s))) => Bilinear (w -+> x) (ZeroDim s ⊗ w) (ZeroDim s ⊗ x) Source # 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) Source # coerceFmapTensorProduct :: Functor p => p (ZeroDim s) -> Coercion * a b -> Coercion * (TensorProduct (ZeroDim s) a) (TensorProduct (ZeroDim s) b) Source # wellDefinedVector :: ZeroDim s -> Maybe (ZeroDim s) Source # wellDefinedTensor :: (TensorSpace w, (* ~ Scalar w) (Scalar (ZeroDim s))) => (ZeroDim s ⊗ w) -> Maybe (ZeroDim s ⊗ w) Source #
Num' s => FiniteDimensional (ZeroDim s) Source #
Associated Types data SubBasis (ZeroDim s) :: * Source # Methods entireBasis :: SubBasis (ZeroDim s) Source # enumerateSubBasis :: SubBasis (ZeroDim s) -> [ZeroDim s] Source # subbasisDimension :: SubBasis (ZeroDim s) -> Int Source # decomposeLinMap :: (LSpace w, (* ~ Scalar w) (Scalar (ZeroDim s))) => (ZeroDim s +> w) -> (SubBasis (ZeroDim s), DList w) Source # decomposeLinMapWithin :: (LSpace w, (* ~ Scalar w) (Scalar (ZeroDim s))) => SubBasis (ZeroDim s) -> (ZeroDim s +> w) -> Either (SubBasis (ZeroDim s), DList w) (DList w) Source # recomposeSB :: SubBasis (ZeroDim s) -> [Scalar (ZeroDim s)] -> (ZeroDim s, [Scalar (ZeroDim s)]) Source # recomposeSBTensor :: (FiniteDimensional w, (* ~ Scalar w) (Scalar (ZeroDim s))) => SubBasis (ZeroDim s) -> SubBasis w -> [Scalar (ZeroDim s)] -> (ZeroDim s ⊗ w, [Scalar (ZeroDim s)]) Source # recomposeLinMap :: (LSpace w, (* ~ Scalar w) (Scalar (ZeroDim s))) => SubBasis (ZeroDim s) -> [w] -> (ZeroDim s +> w, [w]) Source # recomposeContraLinMap :: (LinearSpace w, (* ~ Scalar w) (Scalar (ZeroDim s)), Functor f) => (f (Scalar w) -> w) -> f (DualVector (ZeroDim s)) -> ZeroDim s +> w Source # 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 Source # uncanonicallyFromDual :: DualVector (ZeroDim s) -+> ZeroDim s Source # uncanonicallyToDual :: ZeroDim s -+> DualVector (ZeroDim s) Source # tensorEquality :: (TensorSpace w, Eq w, (* ~ Scalar w) (Scalar (ZeroDim s))) => (ZeroDim s ⊗ w) -> (ZeroDim s ⊗ w) -> Bool Source # dualFinitenessWitness :: DualFinitenessWitness (ZeroDim s) Source #
(Fractional' s, SemiInner s) => SemiInner (ZeroDim s) Source #
Methods dualBasisCandidates :: [(Int, ZeroDim s)] -> Forest (Int, DualVector (ZeroDim s)) Source # tensorDualBasisCandidates :: (SemiInner w, (* ~ Scalar w) (Scalar (ZeroDim s))) => [(Int, ZeroDim s ⊗ w)] -> Forest (Int, DualVector (ZeroDim s ⊗ w)) Source # symTensorDualBasisCandidates :: [(Int, SymmetricTensor (Scalar (ZeroDim s)) (ZeroDim s))] -> Forest (Int, SymmetricTensor (Scalar (ZeroDim s)) (DualVector (ZeroDim s))) Source # 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) Source #
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 Needle (ZeroDim k)
type Needle (ZeroDim k) = ZeroDim k
type DualVector (ZeroDim s) Source #
type DualVector (ZeroDim s) = ZeroDim s
data SubBasis (ZeroDim s) Source #
data SubBasis (ZeroDim s) = ZeroBasis
type TensorProduct (ZeroDim s) v Source #
type TensorProduct (ZeroDim s) v = ZeroDim s