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Project: Testing 18.04
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Kernel: SageMath 9.1
version()
'SageMath version 9.1, Release Date: 2020-05-20'
x = var('x')
eq = 6*x^6 - 7*x^5 - 7*x^4 + 7*x^2 + 7*x - 6
sol = solve(eq, x)
sol
[x == (2/3), x == -1, x == (3/2), x == -1/2*I*sqrt(3) - 1/2, x == 1/2*I*sqrt(3) - 1/2, x == 1]
show(sol)

plot(eq, (x, -1.1, 1.6))

eq = 6*x^6 - 7*x^5 - 7*x^4 + 7*x^2 + 7*x - 6
complex_plot(eq, (-1.5, 2.1), (-1.5, 1.5))
Image in a Jupyter notebook

Polyhedral geometry

There is now a catalog for common polyhedral cones, e.g.

cones.nonnegative_orthant(5)
5-d cone in 5-d lattice N

New features for polyhedra:

sage: P = polytopes.cube(intervals='zero_one') # obtain others than the standard cube
sage: P = matrix([[0,1,0],[0,1,1],[1,0,0]])*P  # linear transformations
sage: it = P.face_generator()                  # a (fast and efficient) face generator
sage: next(it)
A 3-dimensional face of a Polyhedron in ZZ^3 defined as the convex hull of 8 vertices
sage: next(it)


sage: f = next(it)
sage: f.normal_cone()                          # normal cone for faces

sage: P.an_affine_basis()


sage: P = polytopes.hypercube(4)
sage: P.flag_f_vector(0,3)                     # flag_f_vector is exposed
64
P.plot()

Integral curves over finite fields

A.<x,y> = AffineSpace(GF(16),2)
C = Curve(y^3 + x^3*y + x); C  # Klein quartic
Affine Plane Curve over Finite Field in z4 of size 2^4 defined by x^3*y + y^3 + x
C.function_field()
Function field in y defined by y^3 + x^3*y + x
C.genus()
3
C.closed_points()
[Point (x, y), Point (x + (z4), y + (z4^3 + z4^2)), Point (x + (z4^2), y + (z4^3 + z4^2 + z4 + 1)), Point (x + (z4^3), y + (z4^2 + z4)), Point (x + (z4 + 1), y + (z4^3 + z4)), Point (x + (z4^2 + z4), y + (z4^2 + z4 + 1)), Point (x + (z4^2 + z4), y + (z4^3 + 1)), Point (x + (z4^2 + z4), y + (z4^3 + z4^2 + z4)), Point (x + (z4^3 + z4^2), y + (z4^2 + z4 + 1)), Point (x + (z4^2 + 1), y + (z4^3)), Point (x + (z4^3 + z4), y + (z4^2 + z4 + 1)), Point (x + (z4^2 + z4 + 1), y + (z4^2 + z4)), Point (x + (z4^2 + z4 + 1), y + (z4^3 + z4 + 1)), Point (x + (z4^2 + z4 + 1), y + (z4^3 + z4^2 + 1)), Point (x + (z4^3 + z4^2 + z4 + 1), y + (z4^2 + z4))]
p1, p2 = _[:2]
P1 = p1.place()
P2 = p2.place()
D = 5 * P1 - P2
D.basis_function_space()
[(x + z4)/x, 1/x^2*y + (z4^2 + z4)/x]
f1, f2 = D.basis_function_space()
f1.zeros()
[Place (x + z4, y^2 + (z4^3 + z4^2)*y + z4^2 + z4 + 1), Place (x + z4, y + z4^3 + z4^2)]

Generator for cube-connected cycles

graphs.CubeConnectedCycle(3).plot()
Image in a Jupyter notebook

Manifolds

More functionalities in index notation for tensors

E.<x,y> = EuclideanSpace()
v = E.vector_field(-y, x)
t = E.tensor_field(0, 2, [[1, x], [-2*y, x^2]])
v['j']*(t['_ij'] + t['_ji']) == v.contract(2*t.symmetrize())
True
v
Vector field on the Euclidean plane E^2
t
Tensor field of type (0,2) on the Euclidean plane E^2
v['j'], t['_ij'],  t['_ji']
(X^j, X_ij, X_ji)
t.symmetrize()
Field of symmetric bilinear forms on the Euclidean plane E^2










@interact
def func(k = slider(0, 10, .1), j = (-10, 10), l = range_slider(-5, 5)):
    print("k: %s" % k)
    print("j: %s" % j)
    print("l: [%s, %s]" % l)



var('t y')
plot_slope_field(y - t, (t,0,10), (y,0,6), plot_points=25)
Image in a Jupyter notebook




G = cellular_automata.GraftalLace([2,0,3,3,6,0,2,7])
G
Graftal Lace Cellular Automata with rule 20336027
G.evolve(42)
G.plot()
Image in a Jupyter notebook


init = 200*[1]
init[100] = 0
init[101] = 0
ECA = cellular_automata.Elementary(151, width=200, initial_state=init)
ECA
Elementary cellular automata with rule 151 and initial state [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]
ECA.evolve(200)
ECA.plot()
Image in a Jupyter notebook