#Polinomok: megalkotjuk a polinomgyűrűt:
R.<x> = PolynomialRing(QQ)
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#utána dolgozunk benne (a fent megadott x jelöli a határozatlant)
f = x^2 + 1
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g = x^2 - 1
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f*g
x^4 - 1 x^4 - 1 |
factor(_)
(x - 1) * (x + 1) * (x^2 + 1) (x - 1) * (x + 1) * (x^2 + 1) |
#Akár ideálokat is létrehozhatunk
I = f * R
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I
Principal ideal (x^2 + 1) of Univariate Polynomial Ring in x over Rational Field Principal ideal (x^2 + 1) of Univariate Polynomial Ring in x over Rational Field |
x^3+x in I
True True |
K.<y> = R.quotient_by_principal_ideal(f)
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K
Univariate Quotient Polynomial Ring in y over Rational Field with modulus x^2 + 1 Univariate Quotient Polynomial Ring in y over Rational Field with modulus x^2 + 1 |
y^2
-1 -1 |
# Egy másik példa polinomgyűrűre
R.<x> = PolynomialRing(GF(3))
R
Univariate Polynomial Ring in x over Finite Field of size 3 Univariate Polynomial Ring in x over Finite Field of size 3 |
K.<t> = R.quotient_ring(x^2-2)
K
Univariate Quotient Polynomial Ring in t over Finite Field of size 3 with modulus x^2 + 1 Univariate Quotient Polynomial Ring in t over Finite Field of size 3 with modulus x^2 + 1 |
K.is_field()
True True |
K.is_finite()
True True |
K2.<t> = GF(9)
K2.multiplication_table()
* a b c d e f g h i +------------------ a| a a a a a a a a a b| a c d e f g h i b c| a d e f g h i b c d| a e f g h i b c d e| a f g h i b c d e f| a g h i b c d e f g| a h i b c d e f g h| a i b c d e f g h i| a b c d e f g h i * a b c d e f g h i +------------------ a| a a a a a a a a a b| a c d e f g h i b c| a d e f g h i b c d| a e f g h i b c d e| a f g h i b c d e f| a g h i b c d e f g| a h i b c d e f g h| a i b c d e f g h i| a b c d e f g h i |
K.<y> = NumberField(x^2+5)
K
Number Field in y with defining polynomial x^2 + 5 Number Field in y with defining polynomial x^2 + 5 |
R = ZZ[y]
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I = (6)* R
I
Fractional ideal (6) Fractional ideal (6) |
I.is_principal()
True True |
I.factor()
(Fractional ideal (2, y0 + 1))^2 * (Fractional ideal (3, y0 + 1)) * (Fractional ideal (3, y0 + 2)) (Fractional ideal (2, y0 + 1))^2 * (Fractional ideal (3, y0 + 1)) * (Fractional ideal (3, y0 + 2)) |
#Vektorok, mátrixok
M = Matrix([[1,2],[3,4],[5,6],[7,8]])
v = vector([-1, 7])
(M, v, M*v)
( [1 2] [3 4] [5 6] [7 8], (-1, 7), (13, 25, 37, 49) ) ( [1 2] [3 4] [5 6] [7 8], (-1, 7), (13, 25, 37, 49) ) |
Matrix(2,2,[1, 2, 5, 6]).inverse()
[-3/2 1/2] [ 5/4 -1/4] [-3/2 1/2] [ 5/4 -1/4] |
Matrix(2, range(10, 14))
[10 11] [12 13] [10 11] [12 13] |
M = Matrix(2, 2, [a, b, c, d])
Traceback (click to the left of this block for traceback) ... NameError: name 'a' is not defined Traceback (most recent call last):
File "<stdin>", line 1, in <module>
File "_sage_input_68.py", line 10, in <module>
exec compile(u'open("___code___.py","w").write("# -*- coding: utf-8 -*-\\n" + _support_.preparse_worksheet_cell(base64.b64decode("TSA9IE1hdHJpeCgyLCAyLCBbYSwgYiwgYywgZF0p"),globals())+"\\n"); execfile(os.path.abspath("___code___.py"))
File "", line 1, in <module>
File "/tmp/tmpeE84oS/___code___.py", line 3, in <module>
exec compile(u'M = Matrix(_sage_const_2 , _sage_const_2 , [a, b, c, d])
File "", line 1, in <module>
NameError: name 'a' is not defined
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M = Matrix(SR, 2, var('a, b, c, d'))
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M.inverse()
[-b*c/((b*c/a - d)*a^2) + 1/a b/((b*c/a - d)*a)] [ c/((b*c/a - d)*a) -1/(b*c/a - d)] [-b*c/((b*c/a - d)*a^2) + 1/a b/((b*c/a - d)*a)] [ c/((b*c/a - d)*a) -1/(b*c/a - d)] |
M.det()
a*d - b*c a*d - b*c |
M.charpoly()
x^2 + (-a - d)*x + a*d - b*c x^2 + (-a - d)*x + a*d - b*c |
M = Matrix(ZZ, 2, [0, 1, 2, 0])
M
[0 1] [2 0] [0 1] [2 0] |
M.charpoly()
x^2 - 2 x^2 - 2 |
_.factor()
x^2 - 2 x^2 - 2 |
hilbert = Matrix(5, 5, [[1/(i+j-1) for i in [1..5]] for j in [1..5]])
hilbert
[ 1 1/2 1/3 1/4 1/5] [1/2 1/3 1/4 1/5 1/6] [1/3 1/4 1/5 1/6 1/7] [1/4 1/5 1/6 1/7 1/8] [1/5 1/6 1/7 1/8 1/9] [ 1 1/2 1/3 1/4 1/5] [1/2 1/3 1/4 1/5 1/6] [1/3 1/4 1/5 1/6 1/7] [1/4 1/5 1/6 1/7 1/8] [1/5 1/6 1/7 1/8 1/9] |
#csoportok
G = SymmetricGroup(4)
G
Symmetric group of order 4! as a permutation group Symmetric group of order 4! as a permutation group |
show(G.cayley_graph())
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G.order()
24 24 |
G.is_abelian()
False False |
G.center()
Subgroup of (Symmetric group of order 4! as a permutation group) generated by [()] Subgroup of (Symmetric group of order 4! as a permutation group) generated by [()] |
G.subgroups()
[Permutation Group with generators [()], Permutation Group with generators [(1,2)(3,4)], Permutation Group with generators [(1,3)(2,4)], Permutation Group with generators [(1,4)(2,3)], Permutation Group with generators [(3,4)], Permutation Group with generators [(2,3)], Permutation Group with generators [(2,4)], Permutation Group with generators [(1,2)], Permutation Group with generators [(1,3)], Permutation Group with generators [(1,4)], Permutation Group with generators [(2,4,3)], Permutation Group with generators [(1,2,3)], Permutation Group with generators [(1,4,2)], Permutation Group with generators [(1,3,4)], Permutation Group with generators [(1,3)(2,4), (1,4)(2,3)], Permutation Group with generators [(3,4), (1,2)(3,4)], Permutation Group with generators [(2,3), (1,4)(2,3)], Permutation Group with generators [(2,4), (1,3)(2,4)], Permutation Group with generators [(1,2)(3,4), (1,3,2,4)], Permutation Group with generators [(1,3)(2,4), (1,4,3,2)], Permutation Group with generators [(1,2,4,3), (1,4)(2,3)], Permutation Group with generators [(3,4), (2,4,3)], Permutation Group with generators [(3,4), (1,3,4)], Permutation Group with generators [(1,2), (1,2,3)], Permutation Group with generators [(1,2), (1,4,2)], Permutation Group with generators [(1,2), (1,3)(2,4), (1,4)(2,3)], Permutation Group with generators [(1,2)(3,4), (1,3)(2,4), (1,4)], Permutation Group with generators [(1,2)(3,4), (1,3), (1,4)(2,3)], Permutation Group with generators [(2,4,3), (1,3)(2,4), (1,4)(2,3)], Permutation Group with generators [(2,4,3), (1,2), (1,3)(2,4), (1,4)(2,3)]] [Permutation Group with generators [()], Permutation Group with generators [(1,2)(3,4)], Permutation Group with generators [(1,3)(2,4)], Permutation Group with generators [(1,4)(2,3)], Permutation Group with generators [(3,4)], Permutation Group with generators [(2,3)], Permutation Group with generators [(2,4)], Permutation Group with generators [(1,2)], Permutation Group with generators [(1,3)], Permutation Group with generators [(1,4)], Permutation Group with generators [(2,4,3)], Permutation Group with generators [(1,2,3)], Permutation Group with generators [(1,4,2)], Permutation Group with generators [(1,3,4)], Permutation Group with generators [(1,3)(2,4), (1,4)(2,3)], Permutation Group with generators [(3,4), (1,2)(3,4)], Permutation Group with generators [(2,3), (1,4)(2,3)], Permutation Group with generators [(2,4), (1,3)(2,4)], Permutation Group with generators [(1,2)(3,4), (1,3,2,4)], Permutation Group with generators [(1,3)(2,4), (1,4,3,2)], Permutation Group with generators [(1,2,4,3), (1,4)(2,3)], Permutation Group with generators [(3,4), (2,4,3)], Permutation Group with generators [(3,4), (1,3,4)], Permutation Group with generators [(1,2), (1,2,3)], Permutation Group with generators [(1,2), (1,4,2)], Permutation Group with generators [(1,2), (1,3)(2,4), (1,4)(2,3)], Permutation Group with generators [(1,2)(3,4), (1,3)(2,4), (1,4)], Permutation Group with generators [(1,2)(3,4), (1,3), (1,4)(2,3)], Permutation Group with generators [(2,4,3), (1,3)(2,4), (1,4)(2,3)], Permutation Group with generators [(2,4,3), (1,2), (1,3)(2,4), (1,4)(2,3)]] |
(1, 2, 3) in G
True True |
H = G.subgroup([G((1,2,3)), G((1,2))])
H
Subgroup of (Symmetric group of order 4! as a permutation group) generated by [(1,2), (1,2,3)] Subgroup of (Symmetric group of order 4! as a permutation group) generated by [(1,2), (1,2,3)] |
H.order()
6 6 |
H.is_normal()
False False |
g = G((1,2))
g
(1,2) (1,2) |
G = DihedralGroup(4); G
Dihedral group of order 8 as a permutation group Dihedral group of order 8 as a permutation group |
for g in G:
print([g, order(g)])
[(), 1] [(2,4), 2] [(1,2)(3,4), 2] [(1,2,3,4), 4] [(1,3), 2] [(1,3)(2,4), 2] [(1,4,3,2), 4] [(1,4)(2,3), 2] [(), 1] [(2,4), 2] [(1,2)(3,4), 2] [(1,2,3,4), 4] [(1,3), 2] [(1,3)(2,4), 2] [(1,4,3,2), 4] [(1,4)(2,3), 2] |
#Feladat: keressük meg S4 normálosztóit, illetve D4 konjugáltosztályait.
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#Gráfok
#éllistával
d = {0: [1,4,5], 1: [2,6], 2: [3,7], 3: [4,8], 4: [9], \
5: [7, 8], 6: [8,9], 7: [9]}
G = Graph(d); G
Graph on 10 vertices Graph on 10 vertices |
G.show()
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#Rajzoljuk ki a következő gráfot: csúcsai az {1, 2, 3, 4, 5} kételemű részei,
#élek pedig a diszjunkt csúcsok közt mennek.
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#Plotok
plot(sin(x), (x, -5, 5))
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a = animate([circle((i,i), 1-1/(i+1), hue=i/10) for i in srange(0,2,0.2)], xmin=0,ymin=0,xmax=2,ymax=2,figsize=[2,2])
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a.show()
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var('x')
x0 = 0
f = sin(x)*e^(-x)
p = plot(f,-1,5, thickness=2)
dot = point((x0,f(x=x0)),pointsize=80,rgbcolor=(1,0,0))
@interact
def _(order=(1..12)):
ft = f.taylor(x,x0,order)
pt = plot(ft,-1, 5, color='green', thickness=2)
html('$f(x)\;=\;%s$'%latex(f))
html('$\hat{f}(x;%s)\;=\;%s+\mathcal{O}(x^{%s})$'%(x0,latex(ft),order+1))
show(dot + p + pt, ymin = -.5, ymax = 1)
Click to the left again to hide and once more to show the dynamic interactive window |
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