FrédéricC's profile - activity

Like this

sage: e = EtaGroup(8).basis()[0]                                                
sage: e                                                                         
Eta product of level 8 : (eta_2)^-4 (eta_4)^12 (eta_8)^-8
sage: e._rdict                                                                  
{2: -4, 4: 12, 8: -8}

Adding elements one by one, you need to define a successor function : see https://doc.sagemath.org/html/en/refe...

Maybe using RecursivelyEnumeratedSet ?

Like this maybe

sage: A = algebras.Free(QQ, 'abc')                                                                                                             
sage: elt = 4 * A.monomial(Word('abc')) + 6*A.monomial(Word('a'))                                                                                 
sage: data = [(w.to_word(), cf) for w, cf in elt]                                                                                             
sage: A.sum_of_terms((w[1:] + w[:1], cf) for w, cf in data)                                                                                   
6*a + 4*b*c*a

Use solve([tan(x) + tan(2*x) - tan(3*x) == 0], x, to_poly_solve='force', algorithm='sympy')

Hello everyone,

I have installed Sage on Ubuntu 16.04 from the pre-built tarball binary exactly as described in the installation guide but I am left with some newbie questions.

After browsing the web for several hours, I still cannot manage to get the module graphs up and running in the terminal. First I couldn't even import graphs, but after running sage --pip install graphs in terminal I can do

sage: import graphs

without ModuleNotFound error, but the command

sage: G = graphs.EmptyGraph()

leads to

AttributeError: module 'graphs' has no attribute 'EmptyGraph'

Moreover, I'd like to add a new package (sage-drg by Janos Vidali, which I cannot link) but it remains unclear on how to do this. Where do I unpack the zip? How do I let Sage know where to find this package? Is this even possible if I didn't built from source?

Just in case, you may like to use sage: from sage.graphs.all import *

There is no need to install any package, as graphs are included in sage itself. You have probably installed https://pypi.org/project/graphs/ which has nothing to do with sage.

You can use parent(t) to know in what t lives. In your case, each element of the list is a matrix.

Have a look at sage: U.set_latex_options?

Arrows names are there but not displayed by default. You need to ask for them to be displayed.sage: U.plot(edge_labels=True)

you can just copy-paste if you want

sage: data = ["v1","v2","v3","v4","v5","v6","v7","v8"], [["v1","v1","a"],["v1","
....: v2","b"],["v2","v3","c"],["v3","v4","d"],["v4","v5","e"],["v5","v5","f"],
....: ["v3","v6","g"],["v6","v7","h"],["v7","v8","i"],["v8","v3","j"]]          
sage: DiGraph(data, format="vertices_and_edges",loops=True)                     
Looped digraph on 8 vertices

EDIT: here is a more procedural way

def from_QPA(Q): 
    arrows = [(v.SourceOfPath(), v.TargetOfPath(), v) for v in Q.ArrowsOfQuiver() ] 
    elements = Q.VerticesOfQuiver() 
    return DiGraph([elements, arrows], loops=True, format='vertices_and_edges')

then

sage: libgap.LoadPackage("QPA")                                                 
true
sage: Q = libgap.Quiver( ["v1","v2","v3"], [["v1","v2","a1"],["v2","v3","a2"],["v3","v1","a3"]] )
sage: from_QPA(Q)                                                               
Looped digraph on 3 vertices

but

sage: (e^x)^y - e^(x*y)                                                                                                                     
    y       
⎛ x⎞     x⋅y
⎝ℯ ⎠  - ℯ   
sage: simplify(_)                                                                                                                           
0

Is this a bug ?

sage: var('a b c x y z')
(a, b, c, x, y, z)
sage: assert( (e^x)^y == e^(x*y) )
---------------------------------------------------------------------------
AssertionError                            Traceback (most recent call last)
/Users/marks/mysage/sage-9.0/local/lib/python3.7/site-packages/sage/all_cmdline.py in <module>()
----> 1 assert( (e**x)**y == e**(x*y) )

AssertionError: 
sage: assert( (e^2)^3 == e^(2*3) )
sage:

Side remark : sage: P.is_cyclotomic_product() is useful to recognize polynomials with all their roots on the unit circle.

Are you working on dessins d'enfants?

Here is a simple idea for an approximate plot.

sage: x = polygen(QQ, 'x')                                           
sage: f = 1/729*(2*x**2-3*x+9)**3*(x+1)                           
sage: N = 50               
sage: point2d(root for k in range(N+1) for root in (f - k / N).complex_roots())

The function pre01 below builds on this idea. It gives a good idea of the desired plot and runs fast.

def pre01(f, n=50, style='dots', **opt):
    r"""
    Return the preimage of the unit interval under this rational function.

    INPUT:

    - ``f`` -- a rational function

    - ``n`` -- optional (default: 50) -- roughly how many points
      in the unit interval [0, 1] to use for preparing the plot

    Roughly a third or the points are inverse powers of two,
    roughly a third are one minus inverse powers of two,
    and roughly a third are linearly spaced along the interval.

    EXAMPLES::

        sage: x = polygen(QQ, 'x')
        sage: f = 1/729 * (2 * x**2 - 3 * x + 9)**3 * (x + 1)
        sage: pre01(f, n=50, style='dots', color='firebrick').show(figsize=5)
        sage: pre01(f, n=50, style='line', color='firebrick').show(figsize=5)

        sage: z = polygen(QQ, 'z')
        sage: num = -(z^4-6*z^3+12*z^2-8*z)*(z-1)^3*(z-3) 
        sage: den = (2*z-3)*(z-2)^3*z
        sage: g = num/den
        sage: pre01(g, n=50, style='dots', color='firebrick').show(figsize=5)
        sage: pre01(g, n=50, style='line', color='firebrick').show(figsize=5)

        sage: Qz.<z> = QQ[]
        sage: cc = [81, -432, 972, -1200, 886, -400, 108, -16, 1]
        sage: num = -Qz(cc)*(z-2)^6*(z-4)*z
        sage: den = (6*z^4-48*z^3+140*z^2-176*z+81)*(z-1)^4*(z-3)^4
        sage: h = num/den
        sage: pre01(h, n=50, style='dots', color='firebrick').show(figsize=5)
        sage: pre01(h, n=50, style='line', color='firebrick').show(figsize=5)
    """
    nn = n // 3 + 2
    num, den = f.numerator(), f.denominator()
    tt = [0] + [ZZ(2)**-k for k in (0 .. nn)]
    tt.extend([1 - t for t in tt[3:]])
    tt.extend(QQ((k, nn)) for k in (1 .. nn - 1))
    tt = sorted(set(tt))
    cols = {'color': 'teal', 'axes': True, 'axes_color': 'grey',
            'axes_label_color': 'grey', 'tick_label_color': 'grey'}
    dop = {'aspect_ratio': 1, 'size': 4, 'zorder': 10}
    lop = {'aspect_ratio': 1, 'edge_thickness': 2}
    for k, v in opt.items():
        if 'color' in k or 'axis' in k or 'axes' in k or 'tick' in k:
            cols[k] = v
        elif k == 'size':
            dop[k] = v
        elif k == 'edge_thickness':
            lop[k] = v
        else:
            dop[k] = lop[k] = v
    if style == 'dots':
        zz = (root for t in set(tt) for root in (num - t*den).complex_roots())
        G = point2d(zz, color=cols['color'], **dop)
    elif style == 'line':
        tzz = {t: (num - t*den).complex_roots() for t in tt}
        D = Graph()
        for k, t in enumerate(tt[1:-1], start=1):
            s, u = tt[k-1], tt[k+1]
            szz, uzz = tzz[s], tzz[u]
            for z in tzz[t]:
                zs = min((abs(z - w), w) for w in szz)[1]
                D.add_edge(z, zs, k-1)
                zu = min((abs(z - w), w) for w in uzz)[1]
                D.add_edge(z, zu, k)
        D.set_pos({z: (z.real(), z.imag()) for z in D})
        ecol = {cols['color']: D.edges()}
        G = D.plot(vertex_labels=False, vertex_size=0, edge_colors=ecol, **lop)
    else:
        raise ValueError(f"style should be 'dots' or 'line', got :'{style}'")
    G.axes(cols['axes'])
    G.axes_color(cols['axes_color'])
    G.axes_label_color(cols['axes_label_color'])
    G.tick_label_color(cols['tick_label_color'])
    return G

Three examples of using this function follow.

Quick starting example of this answer:

x = polygen(QQ, 'x')
f = 1/729 * (2 * x**2 - 3 * x + 9)**3 * (x + 1)
a_dots = pre01(f, n=50, style='dots', color='teal')
a_line = pre01(f, n=50, style='line', color='teal')
a_dots_lines = graphics_array([a_dots, a_line], ncols=2)
a_dots_lines.show(figsize=5)

First example in the question:

z = polygen(QQ, 'z')
num = -(z^4-6*z^3+12*z^2-8*z)*(z-1)^3*(z-3) 
den = (2*z-3)*(z-2)^3*z
b_dots = pre01(num/den, n=50, style='dots', color='steelblue')
b_line = pre01(num/den, n=50, style='line', color='steelblue')
b_dots_lines = graphics_array([b_dots, b_line], ncols=1)
b_dots_lines.show(figsize=5)

Second example in the question:

z = polygen(QQ, 'z')
num = -(z^8-16*z^7+108*z^6-400*z^5+886*z^4-1200*z^3+972*z^2-432*z+81)*(z-2)^6*(z-4)*z
den = (6*z^4-48*z^3+140*z^2-176*z+81)*(z-1)^4*(z-3)^4
c_dots = pre01(num/den, n=50, style='dots', color='firebrick')
c_line = pre01(num/den, n=50, style='line', color='firebrick')
c_dots_lines = graphics_array([c_dots, c_line], ncols=1)
c_dots_lines.show(figsize=5)

Each of those three examples runs in under a second per plot style.

Here is a start

sage: S=SymmetricFunctions(QQ).s()                                              
sage: S([4]).tensor(S([3]))                                                     
s[4] # s[3]

I have created https://trac.sagemath.org/ticket/31065

This is a bug, indeed. Comes from using "f.variables()" in minimize_constrained.

The documentation says that you need to ask for the edge labels : G.show(edge_labels=True)

Read the documentation ?

I've downloaded the Sage 9.2 Ubuntu 20.04 tarball and got it up and running. I'm having problems accessing the help documentation when I have a jupyter notebook open. When I click on any help topic, there is an error message: "404 Not Found. The page you are requesting does not exist" How can I fix this? Thanks for any help.

Hi, I naively created a chess board with a matrix_plot.

m = matrix_plot([[1,0,1,0,1,0,1,0],
                 [0,1,0,1,0,1,0,1],
                 [1,0,1,0,1,0,1,0],
                 [0,1,0,1,0,1,0,1],
                 [1,0,1,0,1,0,1,0],
                 [0,1,0,1,0,1,0,1],
                 [1,0,1,0,1,0,1,0],
                 [0,1,0,1,0,1,0,1],
                 [1,0,1,0,1,0,1,0]]
                  )

Shown alone, it looks almost like a chess board, but it has axes and white padding on the periphery

 (m).show(aspect_ratio=1)

I then naively tried to add a line to that plot -

Q = line([(-0.5,0),(1,1)])
(m+Q).show()

Somehow I have lost the whole matrix plot . This is version 8.7 in a jupyter notebook

Q = line([(-0.5,0),(1,1)])
(m+Q).show()

i only get 2 rows - the rest is truncated.

Question - how to graph lines on top of this matrix plot without truncating the apprearance?

Merci! Pat

Suggestion : find in which directory your notebook files lives, and un-archive them (they are just compressed archives). You may see the pictures there.

You can accept my answer, maybe ? Also consider sponsoring sage : https://github.com/sponsors/sagemath

This documentation link is obsolete and not official. Use only doc.sagemath.org

For 1, use P+Q where P and Q are two plots. Or use a sequence of functions as argument of plot.

Some code for the r-number, not tested.

def r_number(P): 
    H = P.hasse_diagram() 
    M = Matroid(H.to_undirected()) 
    b = FreeModule(GF(2), tuple(H.edges(labels=False))).basis() 
    vecs = [sum(b[e] for e in ci).to_vector() for ci in M.circuits()] 
    return matrix(vecs).rank()

Remains only to do the same thing for the subspace that defines s.

EDIT: here is a tentative, not tested either

def s_number(P):
    H = P.hasse_diagram()
    b = FreeModule(GF(2), tuple(H.edges(labels=False))).basis()
    vecs = []
    for x, y in P.relations():
        chains = H.all_paths(x, y, report_edges=True)
        if len(chains) <= 1:
            continue
        v0 = sum(b[e] for e in chains[0]).to_vector()
        vecs += [v0 + sum(b[e] for e in ci).to_vector() for ci in chains[1:]]
    return matrix(vecs).rank()

G.edges is a method, use list(G.edges()) for the list of edges. Store this in a variable before the loop.

Can this be computed using some kind of homology group ?

Maybe something like this

sage: def banana(n): 
....:     for P in posets(n): 
....:         if not P.is_connected(): 
....:             continue 
....:         H = P.hasse_diagram() 
....:         if H.to_undirected().is_tree(): 
....:             continue 
....:         dg = H.transitive_closure() 
....:         if any(dg.degree(x) == n - 1 for x in dg): 
....:             continue 
....:         yield P

https://doc.sagemath.org/html/en/prep... and https://wiki.sagemath.org/interact/gr... maybe

see also https://ask.sagemath.org/question/415...

try load(r"C:\Users\vika\Desktop\example.sage") ?

In my laptop for 200 digit numbers

sage: time p=next_prime(10^200)                                                 
CPU times: user 1.04 s, sys: 591 µs, total: 1.04 s
Wall time: 1.04 s
sage: time p=next_probable_prime(10^200)                                        
CPU times: user 20.7 ms, sys: 0 ns, total: 20.7 ms
Wall time: 21.8 ms

What return exacty next_probable_prime? When is desired to use it?

Like this

sage: M = matrix(2, 2, [1, 2, 3, 4])                                                   
sage: sum(M.coefficients())                                                     
10

I propose a fix here : https://trac.sagemath.org/ticket/30965

Please look at the code and tell us where it is wrong. Open-source, we are.