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Some computations. An important thing: everything is mathematically typed in Axiom.
(1) -> 1+1 (1) 2 Type: PositiveInteger (2) -> integrate(1/x^(1/3),x) 3+-+2 3\|x (2) ------ 2 Type: Union(Expression Integer,...)
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Axiom is capable of nesting a matrix within a matrix as can be seen from this screenshot:
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Some more complicated computations:
)cl all All user variables and function definitions have been cleared. Word := OrderedFreeMonoid(Symbol) (1) OrderedFreeMonoid Symbol Type: Domain poly:= XPR(Integer,Word) (2) XPolynomialRing(Integer,OrderedFreeMonoid Symbol) Type: Domain p:poly := 2 * x - 3 * y + 1 (3) 1 + 2x - 3y Type: XPolynomialRing(Integer,OrderedFreeMonoid Symbol) q:poly := 2 * x + 1 (4) 1 + 2x Type: XPolynomialRing(Integer,OrderedFreeMonoid Symbol) p + q (5) 2 + 4x - 3y Type: XPolynomialRing(Integer,OrderedFreeMonoid Symbol) p * q 2 (6) 1 + 4x - 3y + 4x - 6y x Type: XPolynomialRing(Integer,OrderedFreeMonoid Symbol) (p +q)^2 -p^2 -q^2 - 2*p*q (7) - 6x y + 6y x Type: XPolynomialRing(Integer,OrderedFreeMonoid Symbol) M := SquareMatrix(2,Fraction Integer) (8) SquareMatrix(2,Fraction Integer) Type: Domain poly1:= XPR(M,Word) (9) XPolynomialRing(SquareMatrix(2,Fraction Integer),OrderedFreeMonoid Symbol) Type: Domain m1:M := matrix [[i*j**2 for i in 1..2] for j in 1..2] +1 2+ (10) | | +4 8+ Type: SquareMatrix(2,Fraction Integer) m2:M := m1 - 5/4 + 1 + |- - 2 | | 4 | (11) | | | 27| | 4 --| + 4+ Type: SquareMatrix(2,Fraction Integer) m3: M := m2**2 +129 + |--- 13 | | 16 | (12) | | | 857| |26 ---| + 16+ Type: SquareMatrix(2,Fraction Integer) pm:poly1 := m1*x + m2*y + m3*z - 2/3 + 2 + + 1 + +129 + |- - 0 | |- - 2 | |--- 13 | | 3 | +1 2+ | 4 | | 16 | (13) | | + | |x + | |y + | |z | 2| +4 8+ | 27| | 857| | 0 - -| | 4 --| |26 ---| + 3+ + 4+ + 16+ Type: XPolynomialRing(SquareMatrix(2,Fraction Integer),OrderedFreeMonoid Symbol) qm:poly1 := pm - m1*x + 2 + + 1 + +129 + |- - 0 | |- - 2 | |--- 13 | | 3 | | 4 | | 16 | (14) | | + | |y + | |z | 2| | 27| | 857| | 0 - -| | 4 --| |26 ---| + 3+ + 4+ + 16+ Type: XPolynomialRing(SquareMatrix(2,Fraction Integer),OrderedFreeMonoid Symbol) qm**3 (15) + 8 + + 1 8+ +43 52 + + 129 + |- -- 0 | |- - -| |-- -- | |- --- - 26 | | 27 | | 3 3| | 4 3 | | 8 | 2 | | + | |y + | |z + | |y | 8| |16 | |104 857| | 857| | 0 - --| |-- 9| |--- ---| |- 52 - ---| + 27+ + 3 + + 3 12+ + 8 + + + 3199 831 + + 3199 831 + + 103169 6409 + |- ---- - --- | |- ---- - --- | |- ------ - ---- | | 32 4 | | 32 4 | | 128 4 | 2 | |y z + | |z y + | |z | 831 26467| | 831 26467| | 6409 820977| |- --- - -----| |- --- - -----| | - ---- - ------| + 2 32 + + 2 32 + + 2 128 + + +3199 831 + +103169 6409 + +103169 6409 + |---- --- | |------ ---- | |------ ---- | | 64 8 | 3 | 256 8 | 2 | 256 8 | | |y + | |y z + | |y z y |831 26467| | 6409 820977| | 6409 820977| |--- -----| | ---- ------| | ---- ------| + 4 64 + + 4 256 + + 4 256 + + +3178239 795341 + +103169 6409 + +3178239 795341 + |------- ------ | |------ ---- | |------- ------ | | 1024 128 | 2 | 256 8 | 2 | 1024 128 | | |y z + | |z y + | |z y z |795341 25447787| | 6409 820977| |795341 25447787| |------ --------| | ---- ------| |------ --------| + 64 1024 + + 4 256 + + 64 1024 + + +3178239 795341 + +98625409 12326223 + |------- ------ | |-------- -------- | | 1024 128 | 2 | 4096 256 | 3 | |z y + | |z |795341 25447787| |12326223 788893897| |------ --------| |-------- ---------| + 64 1024 + + 128 4096 + Type: XPolynomialRing(SquareMatrix(2,Fraction Integer),OrderedFreeMonoid Symbol)
Axiom can do rule substitution to help simplify complex results as this screenshot shows: