3745.sagews

#Polinomok: megalkotjuk a polinomgyűrűt:
R.<x> = PolynomialRing(QQ)

#utána dolgozunk benne (a fent megadott x jelöli a határozatlant)
f = x^2 + 1

g = x^2 - 1

f*g

x^4 - 1

factor(_)

(x - 1) * (x + 1) * (x^2 + 1)

#Akár ideálokat is létrehozhatunk
I = f * R

I

Principal ideal (x^2 + 1) of Univariate Polynomial Ring in x over Rational Field

x^3+x in I

True

K.<y> = R.quotient_by_principal_ideal(f)

K

Univariate Quotient Polynomial Ring in y over Rational Field with modulus x^2 + 1

y^2

-1

# Egy másik példa polinomgyűrűre
R.<x> = PolynomialRing(GF(3))
R

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

K.is_field()

True

K.is_finite()

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

K.<y> = NumberField(x^2+5)
K

Number Field in y with defining polynomial x^2 + 5

R = ZZ[y]

I = (6)* R
I

Fractional ideal (6)

I.is_principal()

True

I.factor()

(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) )

Matrix(2,2,[1, 2, 5, 6]).inverse()

[-3/2 1/2] [ 5/4 -1/4]

Matrix(2, range(10, 14))

[10 11] [12 13]

M = Matrix(2, 2, [a, b, c, d])

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"))' + '\n', '', 'single') 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])' + '\n', '', 'single') File "", line 1, in <module> NameError: name 'a' is not defined

M = Matrix(SR, 2, var('a, b, c, d'))

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)]

M.det()

a*d - b*c

M.charpoly()

x^2 + (-a - d)*x + a*d - b*c

M = Matrix(ZZ, 2, [0, 1, 2, 0])
M

[0 1] [2 0]

M.charpoly()

x^2 - 2

_.factor()

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]

#csoportok
G = SymmetricGroup(4)
G

Symmetric group of order 4! as a permutation group

show(G.cayley_graph())

G.order()

24

G.is_abelian()

False

G.center()

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)]]

(1, 2, 3) in G

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)]

H.order()

6

H.is_normal()

False

g = G((1,2))
g

(1,2)

G = DihedralGroup(4); G

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]

#Feladat: keressük meg S4 normálosztóit, illetve D4 konjugáltosztályait.

#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

G.show()

#Rajzoljuk ki a következő gráfot: csúcsai az {1, 2, 3, 4, 5} kételemű részei,
#élek pedig a diszjunkt csúcsok közt mennek.

#Plotok
plot(sin(x), (x, -5, 5))

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])

a.show()

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)

order  [removed] [removed] [removed] [removed]