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Kernel: SageMath 9.2

Sage 9.2 on CoCalc

https://wiki.sagemath.org/ReleaseTours/sage-9.2

In [1]:

version()

'SageMath version 9.2, Release Date: 2020-10-24'

In [29]:

pretty_print_default(True)

In [41]:

x,n = var('x,n')

taylor((1+x)^n, n, 1)

\renewcommand{\Bold}[1]{\mathbf{#1}}{\left(n - 1\right)} {\left(x + 1\right)} \log\left(x + 1\right) + x + 1

In [53]:

ex1 = taylor((1+x)^n, n, 1) - (1+n*x)
ex1

\renewcommand{\Bold}[1]{\mathbf{#1}}2 \, x \log\left(x + 1\right) - 2 \, x + 2 \, \log\left(x + 1\right)

In [54]:

ex1.subs({log(x+1) : x})

\renewcommand{\Bold}[1]{\mathbf{#1}}2 \, x^{2}

In [52]:

simplify(taylor((1+x)^n, n, 2))

\renewcommand{\Bold}[1]{\mathbf{#1}}\frac{1}{2} \, x^{2} \log\left(x + 1\right)^{2} + x^{2} \log\left(x + 1\right) + x \log\left(x + 1\right)^{2} + x^{2} + 2 \, x \log\left(x + 1\right) + \frac{1}{2} \, \log\left(x + 1\right)^{2} + 2 \, x + \log\left(x + 1\right) + 1

In [21]:

upper = taylor((1+x)^n, n, 1)
simplify(upper - 1 + n*x)

(n - 1)*(x + 1)*log(x + 1) + n*x + x

In [25]:

n = 2
p0 = plot(lambda x : (1+x) ^ n, (x,-1,2) , color='green')
p1 = plot(lambda x : f(x=x, n=n), (x, -1, 2))
p2 = plot(lambda x : 1+n*x, (x, -1, 2), color='red')
show(p0 + p1 + p2)

In [3]:

#%%python
import sys
print(sys.executable)
print([1..10])

/ext/sage/sage-9.2/local/bin/python3
[1, 2, 3, 4, 5, 6, 7, 8, 9, 10]

In [3]:

var('x y')
ex1 = 2*x+1
print(ex1)
print(ex1.subs(x = 3))
print(ex1.subs(x = pi))
ex2 = ex1.subs(x = (x + y - 1) / 2)
print(ex2)

2*x + 1
7
2*pi + 1
x + y

In [4]:

@interact
def f(x=[1,3,4,5,4,2,3,1], y = (-10, 10, .1)):
    print(f"x={x:} y={y:}")

NEW: 3D Animations!

In [0]:

x, y = var('x, y')

def build_frame(t):
    """Build a single frame of animation at time t."""
    e = parametric_plot3d([sin(2*x - t), sin(x + t), x],
                          (x, 0, 2*pi), color='red')
    b = parametric_plot3d([cos(x + t), -sin(x - t), x],
                          (x, 0, 2*pi), color='green')
    return e + b

frames = [build_frame(t) for t in (0, pi/32, pi/16, .., 2*pi)]
animate(frames, delay=5).interactive(projection='orthographic')

eigenvalues with errors using Arb

In [5]:

from sage.matrix.benchmark import hilbert_matrix
mat = hilbert_matrix(5).change_ring(CBF)
mat.eigenvalues()

<ipython-input-5-175dab4ddc5a>:3: FutureWarning: This class/method/function is marked as experimental. It, its functionality or its interface might change without a formal deprecation.
See https://trac.sagemath.org/30393 for details.
  mat.eigenvalues()

[[1.56705069109823 +/- 8.92e-15] + [+/- 5.27e-15]*I, [0.2085342186110 +/- 2.01e-14] + [+/- 5.27e-15]*I, [3.28792877e-6 +/- 7.64e-15] + [+/- 5.27e-15]*I, [0.00030589804015 +/- 6.67e-15] + [+/- 5.27e-15]*I, [0.01140749162342 +/- 5.68e-15] + [+/- 5.27e-15]*I]

Polyomino tilings

In [6]:

from sage.combinat.tiling import Polyomino
H = Polyomino([ (-1, 1), (-1, 4), (-1, 7), (0, 0), (0, 1), (0, 2),
                (0, 3), (0, 4), (0, 5), (0, 6), (0, 7), (0, 8), (1, 1), (1, 2),
                (1, 3), (1, 4), (1, 5), (1, 6), (1, 7), (1, 8), (2, 0), (2, 2),
                (2, 3), (2, 5), (2, 6), (2, 8)])
H.show2d()

In [7]:

%time solution = H.self_surrounding(10, ncpus=2)

CPU times: user 1.35 s, sys: 248 ms, total: 1.6 s
Wall time: 5.09 s

In [8]:

G = sum([p.show2d() for p in solution], Graphics())
G

Manifolds: diff function for exterior derivatives

In [0]:

M = Manifold(2, 'M')
X.<x,y> = M.chart()
f = M.scalar_field(x^2*y, name='f')
diff(f)

In [0]:

diff(f).display()

In [12]:

a = M.one_form(-2*x*y, x, name='a'); a.display()
diff(a).display()

da = (2*x + 1) dx/\dy

Differential Weyl algebra

In [13]:

W.<x,y> = DifferentialWeylAlgebra(QQ)
dx, dy = W.differentials()
dx.diff(x^3)

3*x^2

In [14]:

(x*dx + dy + 1).diff(x^4*y^4 + 1)

5*x^4*y^4 + 4*x^4*y^3 + 1

Temperley-Lieb diagrams now have unicode

In [15]:

from sage.combinat.diagram_algebras import TL_diagram_ascii_art

TL = [(-15,-12), (-14,-13), (-11,15), (-10,14), (-9,-6),
      (-8,-7), (-5,-4), (-3,1), (-2,-1), (2,3), (4,5),
      (6,11), (7, 8), (9,10), (12,13)]
TL_diagram_ascii_art(TL, use_unicode=True)

 ⚬ ⚬ ⚬ ⚬ ⚬ ⚬ ⚬ ⚬ ⚬ ⚬ ⚬ ⚬ ⚬ ⚬ ⚬
 │ ╰─╯ ╰─╯ │ ╰─╯ ╰─╯ │ ╰─╯ │ │ 
 │         ╰─────────╯     │ │ 
 │                 ╭───────╯ │ 
 ╰───╮             │ ╭───────╯
     │     ╭─────╮ │ │ ╭─────╮
 ╭─╮ │ ╭─╮ │ ╭─╮ │ │ │ │ ╭─╮ │ 
 ⚬ ⚬ ⚬ ⚬ ⚬ ⚬ ⚬ ⚬ ⚬ ⚬ ⚬ ⚬ ⚬ ⚬ ⚬

some calculus

In [14]:

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]

In [15]:

show(sol)

\renewcommand{\Bold}[1]{\mathbf{#1}}\left[x = \left(\frac{2}{3}\right), x = \left(-1\right), x = \left(\frac{3}{2}\right), x = 1\right]

In [16]:

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

In [17]:

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))

In [18]:

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

Generator for cube-connected cycles

In [19]:

graphs.CubeConnectedCycle(3).plot()

Manifolds

More functionalities in index notation for tensors

In [0]:

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())

In [0]:

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v['j'], t['_ij'],  t['_ji']

In [0]:

t.symmetrize()

In [0]:

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@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)

In [0]:

In [0]:

var('t y')
plot_slope_field(y - t, (t,0,10), (y,0,6), plot_points=25)

In [0]:

In [0]:

Cellular Automata

https://doc.sagemath.org/html/en/reference/dynamics/cellular_automata.html

In [0]:

G = cellular_automata.GraftalLace([2,0,3,3,6,0,2,7])
G

In [0]:

G.evolve(42)
G.plot()

In [0]:

In [0]:

init = 200*[1]
init[100] = 0
init[101] = 0
ECA = cellular_automata.Elementary(151, width=200, initial_state=init)
ECA

In [0]:

ECA.evolve(200)
ECA.plot()

In [0]:

In [0]:

print("Scatter plot and line")   # http://www.packtpub.com/article/plotting-data-sage

# python function
def noisy_line(m, b, x):
    ret  = m * x + b + 0.5 * (random() - 0.5)
    return ret

slope     = 1.0
intercept = -0.5

x_coords = [random() for t in range(50)]
y_coords = [noisy_line(slope, intercept, x) for x in x_coords]

sp = scatter_plot(list(zip(x_coords, y_coords)))

sp += line([(0.0, intercept), (1.0, slope+intercept)], color='red')

sp.show( figsize=[6,2])

Torus

In [0]:

from sage.plot.plot3d.shapes import Torus
inner_radius = .3; outer_radius = 1
show(Torus(outer_radius, inner_radius, color='orange'), aspect_ratio=1, spin=3)

In [0]:

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Sage 9.2 on CoCalc

NEW: 3D Animations!

eigenvalues with errors using Arb

Polyomino tilings

Manifolds: diff function for exterior derivatives

Differential Weyl algebra

Temperley-Lieb diagrams now have unicode

some calculus

Generator for cube-connected cycles

Manifolds

Cellular Automata

Product

Resources

Company

Real-time collaboration for Jupyter Notebooks, Linux Terminals, LaTeX, VS Code, R IDE, and more, all in one place. Commercial Alternative to JupyterHub.

Sage 9.2 on CoCalc

NEW: 3D Animations!

eigenvalues with errors using Arb

Polyomino tilings

Manifolds: diff function for exterior derivatives

Differential Weyl algebra

Temperley-Lieb diagrams now have unicode

some calculus

Generator for cube-connected cycles

Manifolds

Cellular Automata

Real-time collaboration for Jupyter Notebooks, Linux Terminals, LaTeX, VS Code, R IDE, and more,
all in one place. Commercial Alternative to JupyterHub.