Math.Algebra.Polynomial.Monomial.Exterior.Indexed

Exterior monomials where the variable set looks like {x_1, x_2, ... , x_N}

The internal representation is a bit vector

Exterior monomials

newtype Ext (var :: Symbol) (n :: Nat)

Exterior monomials of the variables x1,x2,...,xn. The internal representation is a bit vector encoded as an Integer

Constructors

Ext Integer

Instances

Eq (Ext var n)

Defined in Math.Algebra.Polynomial.Monomial.Exterior.Indexed

Methods

(==) :: Ext var n -> Ext var n -> Bool

(/=) :: Ext var n -> Ext var n -> Bool

Ord (Ext var n)

Defined in Math.Algebra.Polynomial.Monomial.Exterior.Indexed

Methods

compare :: Ext var n -> Ext var n -> Ordering

(<) :: Ext var n -> Ext var n -> Bool

(<=) :: Ext var n -> Ext var n -> Bool

(>) :: Ext var n -> Ext var n -> Bool

(>=) :: Ext var n -> Ext var n -> Bool

max :: Ext var n -> Ext var n -> Ext var n

min :: Ext var n -> Ext var n -> Ext var n

Show (Ext var n)

Defined in Math.Algebra.Polynomial.Monomial.Exterior.Indexed

Methods

showsPrec :: Int -> Ext var n -> ShowS

show :: Ext var n -> String

showList :: [Ext var n] -> ShowS

KnownSymbol var => Pretty (Ext var n)

Defined in Math.Algebra.Polynomial.Monomial.Exterior.Indexed

Methods

pretty :: Ext var n -> String

prettyInParens :: Ext var n -> String

data SgnExt (var :: Symbol) (n :: Nat)

Signed exterior monomial

Constructors

SgnExt !Sign !(Ext var n)

Instances

Eq (SgnExt var n)

Defined in Math.Algebra.Polynomial.Monomial.Exterior.Indexed

Methods

(==) :: SgnExt var n -> SgnExt var n -> Bool

(/=) :: SgnExt var n -> SgnExt var n -> Bool

Ord (SgnExt var n)

Defined in Math.Algebra.Polynomial.Monomial.Exterior.Indexed

Methods

compare :: SgnExt var n -> SgnExt var n -> Ordering

(<) :: SgnExt var n -> SgnExt var n -> Bool

(<=) :: SgnExt var n -> SgnExt var n -> Bool

(>) :: SgnExt var n -> SgnExt var n -> Bool

(>=) :: SgnExt var n -> SgnExt var n -> Bool

max :: SgnExt var n -> SgnExt var n -> SgnExt var n

min :: SgnExt var n -> SgnExt var n -> SgnExt var n

Show (SgnExt var n)

Defined in Math.Algebra.Polynomial.Monomial.Exterior.Indexed

Methods

showsPrec :: Int -> SgnExt var n -> ShowS

show :: SgnExt var n -> String

showList :: [SgnExt var n] -> ShowS

KnownNat n => PartialMonoid (SgnExt var n)

Defined in Math.Algebra.Polynomial.Monomial.Exterior.Indexed

Methods

pmUnit :: SgnExt var n

pmAdd :: SgnExt var n -> SgnExt var n -> Maybe (SgnExt var n)

pmSum :: [SgnExt var n] -> Maybe (SgnExt var n)

KnownSymbol var => Pretty (SgnExt var n)

Defined in Math.Algebra.Polynomial.Monomial.Exterior.Indexed

Methods

pretty :: SgnExt var n -> String

prettyInParens :: SgnExt var n -> String

unExt :: Ext v n -> Integer

extVar :: KnownSymbol var => Ext var n -> String

Name of the variables

nOfExt :: KnownNat n => Ext var n -> Int

Number of variables

nOfMbExt :: KnownNat n => Maybe (Ext var n) -> Int

nOfSgnExt :: KnownNat n => SgnExt var n -> Int

nOfMbSgnExt :: KnownNat n => Maybe (SgnExt var n) -> Int

emptyness

emptyExt :: KnownNat n => Ext v n

emptySgnExt :: KnownNat n => SgnExt v n

isEmptyExt :: Ext v n -> Bool

normalization

isNormalExt :: KnownNat n => Ext v n -> Bool

conversion

extFromList :: KnownNat n => [Index] -> Maybe (SgnExt v n)

extToList :: Ext v n -> [Index]

extFromSet :: KnownNat n => Set Index -> Maybe (Ext v n)

extToSet :: Ext v n -> Set Index

creation

variableExt :: KnownNat n => Index -> Ext v n

multiplication

mulExt :: KnownNat n => Ext v n -> Ext v n -> Maybe (SgnExt v n)

mulExtCoeff :: (Num c, KnownNat n) => Ext v n -> Ext v n -> Maybe (Ext v n, c)

mulSgnExt :: KnownNat n => SgnExt v n -> SgnExt v n -> Maybe (SgnExt v n)

productExt :: (KnownNat n, Foldable f) => f (Ext v n) -> Maybe (SgnExt v n)

v1 :: Ext "a" 7

a :: SgnExt "a" 7

b :: SgnExt "a" 7

prop_graded_anticomm :: KnownNat n => Ext v n -> Ext v n -> Bool

prop_graded_anticomm_sgn :: KnownNat n => SgnExt v n -> SgnExt v n -> Bool

degree

degreeExt :: Ext v n -> Int

degreeSgnExt :: SgnExt v n -> Int

Permutations

newtype Permutation

Constructors

Permutation (UArray Int Int)

Instances

Eq Permutation

Defined in Math.Algebra.Polynomial.Monomial.Exterior.Indexed

Methods

(==) :: Permutation -> Permutation -> Bool

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

Ord Permutation

Defined in Math.Algebra.Polynomial.Monomial.Exterior.Indexed

Methods

compare :: Permutation -> Permutation -> Ordering

(<) :: Permutation -> Permutation -> Bool

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

(>) :: Permutation -> Permutation -> Bool

(>=) :: Permutation -> Permutation -> Bool

max :: Permutation -> Permutation -> Permutation

min :: Permutation -> Permutation -> Permutation

toPermutationUnsafe :: [Int] -> Permutation

sortingPermutationAsc :: Ord a => [a] -> Permutation

isEvenPermutation :: Permutation -> Bool

numberOfInversionsMerge :: Permutation -> Int

Returns the number of inversions, using the merge-sort algorithm. This should be O(n*log(n))