Sage 4.0.1 released

Sage 4.0.1 was released on June 06, 2009. For the official, comprehensive release note, please refer to sage-4.0.1.txt. The Sage trac server contains all tickets merged in this release. The following points are some of the foci of version 4.0.1:

  • Nested lists as nicely formatted HTML tables.
  • Update FLINT and MPIR to latest upstream releases.
  • Speed optimization for algebra, basic arithmetics, and the GAP interface.
  • A tool for understanding pickling.

The following 3 people made their first contribution in this release:

  1. Dan Christensen
  2. David Perkinson
  3. Joanna Gaski

In version 4.0.1, doctest coverage increased by 0.3%, while an additional 177 functions have been added. The overall weighted coverage score is now at 77.2% and the total number of functions is at 21,988.

The upcoming version 4.0.2 is a bug fix release for ironing out any outstanding bugs in 4.0.1 and cleaning up mergeable patches. This version is scheduled to be released before or during the MEGA 2009 conference, Barcelona, June 15–19, 2009.

After releasing version 4.0.rc0, Michael Abshoff took a break from release management. This is a well-deserved break since he has been managing over 33 high-quality releases over a period of about two years. Well done, Michael! A standing ovation to you. Mike Hansen and William Stein managed the release of Sage versions 4.0 and 4.0.1. The release managers for the upcoming 4.0.2 version are Nick Alexander and Craig Citro. You can read Michael’s announcement at this sage-devel thread, and the announcement of the 4.0.2 release managers at this sage-release thread.

Many of you know by now that Sage is on Facebook. You can become a fan of Sage by joining the Sage Math Facebook group. So far, the group boasts over 500 fans. Spurred by the success of this group, I’ve created the corresponding Sage Math LinkedIn group. After a week since its creation, the group now has about 6 members. Clearly we need more people to join this new group (wink, wink, nudge, nudge ;-).

The Annual Spies Sage Development Prize first started in 2008. The 2008 recipient was Michael Abshoff. It’s that time of the year again when the Sage community recognizes outstanding contributions from an individual to the development of Sage. Although not officially announced yet, this year’s winner is Mike Hansen. Expect to get an official announcement soon on sage-devel. Congratulations, Mike!

The following are some of the major features in this release, categorized under various headings:

Algebra

  • Factoring rational functions (Soroosh Yazdani) — New method factor() in the class FractionFieldElement of sage/rings/fraction_field_element.pyx to return the factorization of self over the base ring. Here’s an example for working with this new method:
    sage: K.<x> = QQ["x"]
    sage: f = (x^3 + x) / (x-3)
    sage: f.factor()
    (x - 3)^-1 * x * (x^2 + 1)
    
  • Faster basis_matrix() for ambient modules (John Cremona) — The speed-up can be up to 376x faster than previously. The following timing statistics were obtained using the machine sage.math:
    # BEFORE
    
    sage: K = FreeModule(ZZ, 2000)
    sage: %time I = K.basis_matrix()
    CPU times: user 292.74 s, sys: 20.11 s, total: 312.85 s
    Wall time: 312.90 s
    
    
    # AFTER
    
    sage: K = FreeModule(ZZ, 2000)
    sage: %time I = K.basis_matrix()
    CPU times: user 0.41 s, sys: 0.43 s, total: 0.84 s
    Wall time: 0.83 s
    
  • Optimize the construction of Lagrange interpolation polynomials (Minh Van Nguyen) — Rewrite the method lagrange_polynomial() in the class PolynomialRing_field of sage/rings/polynomial/polynomial_ring.py for generating the n-th Lagrange interpolation polynomial. The method now provides two new options:
    • algorithm — (default: divided_difference) If algorithm=”divided_difference”, then use the method of divided difference. If algorithm=”neville”, then use a variant of Neville’s method to recursively generate the n-th Lagrange interpolation polynomial. This adaptation of Neville’s method is more memory efficient than the original Neville’s method, since the former doesn’t generate the full Neville table resulting from Neville’s recursive procedure. Instead the adaptation only keeps track of the current and previous rows of the said table.
    • previous_row — (default: None) This is only relevant if used together with algorithm=”neville”. Here “previous row” refers to the last row in the Neville table that was obtained from a previous computation of an n-th Lagrange interpolation polynomial using Neville’s method. If the last row is provided, then use a memory efficient variant of Neville’s method to recursively generate a better interpolation polynomial from the results of previous computation.

    There’s also the new method divided_difference() to compute the Newton divided-difference coefficients of the n-th Lagrange interpolation polynomial. The following are some timing statistics obtained using sage.math. When the results of previous computations are fed to lagrange_polynomial in order to produce better interpolation polynomials, we can gain an efficiency of up to 42%.

    # BEFORE
    
    # using the definition of Lagrange interpolation polynomial
    sage: R = PolynomialRing(QQ, 'x')
    sage: %timeit R.lagrange_polynomial([(0,1),(2,2),(3,-2),(-4,9)])
    1000 loops, best of 3: 1.71 ms per loop
    sage: R = PolynomialRing(GF(2**3,'a'), 'x')
    sage: a = R.base_ring().gen()
    sage: timeit("R.lagrange_polynomial([(a^2+a,a),(a,1),(a^2,a^2+a+1)])")
    625 loops, best of 3: 233 µs per loop
    
    # without using precomputed values to generate successively better interpolation polynomials
    
    sage: R = PolynomialRing(QQ, 'x')
    sage: timeit("R.lagrange_polynomial([(0,1),(2,2)])");
    625 loops, best of 3: 571 µs per loop
    sage: # add two more points
    sage: timeit("R.lagrange_polynomial([(0,1),(2,2),(3,-2),(-4,9)])");
    125 loops, best of 3: 2.29 ms per loop
    sage: 
    sage: R = PolynomialRing(GF(2**3,'a'), 'x')
    sage: a = R.base_ring().gen()
    sage: timeit("R.lagrange_polynomial([(a^2+a,a),(a,1)])")
    625 loops, best of 3: 76.1 µs per loop
    sage: # add another point
    sage: timeit("R.lagrange_polynomial([(a^2+a,a),(a,1),(a^2,a^2+a+1)])")
    625 loops, best of 3: 229 µs per loop
    sage:
    sage: R = PolynomialRing(QQ, 'x')
    sage: points = [(random(), random()) for i in xrange(100)]
    sage: time R.lagrange_polynomial(points);
    CPU times: user 1.21 s, sys: 0.00 s, total: 1.21 s
    Wall time: 1.21 s
    sage: # add three more points
    sage: for i in xrange(3): points.append((random(), random()))
    ....: 
    sage: time R.lagrange_polynomial(points);
    CPU times: user 1.28 s, sys: 0.01 s, total: 1.29 s
    Wall time: 1.29 s
    sage: # add another 100 points
    sage: for i in xrange(100): points.append((random(), random()))
    ....: 
    sage: time R.lagrange_polynomial(points);
    CPU times: user 5.87 s, sys: 0.02 s, total: 5.89 s
    Wall time: 5.89 s
    
    
    # AFTER
    
    # using the method of divided-difference
    sage: R = PolynomialRing(QQ, 'x')
    sage: %timeit R.lagrange_polynomial([(0,1),(2,2),(3,-2),(-4,9)])
    1000 loops, best of 3: 827 µs per loop
    sage: R = PolynomialRing(GF(2**3,'a'), 'x')
    sage: a = R.base_ring().gen()
    sage: timeit("R.lagrange_polynomial([(a^2+a,a),(a,1),(a^2,a^2+a+1)])")
    625 loops, best of 3: 111 µs per loop
    
    # using precomputed values to generate successively better interpolation polynomials
    
    sage: R = PolynomialRing(QQ, 'x')
    sage: timeit("R.lagrange_polynomial([(0,1),(2,2)], neville=True)");
    625 loops, best of 3: 332 µs per loop
    sage: p = R.lagrange_polynomial([(0,1),(2,2)], neville=True);
    sage: # add two more points
    sage: timeit("R.lagrange_polynomial([(0,1),(2,2),(3,-2),(-4,9)], neville=True, previous_row=p)");
    625 loops, best of 3: 1.41 ms per loop
    sage:
    sage: R = PolynomialRing(GF(2**3,'a'), 'x')
    sage: a = R.base_ring().gen()
    sage: timeit("R.lagrange_polynomial([(a^2+a,a),(a,1)], neville=True)");
    625 loops, best of 3: 36.4 µs per loop
    sage: p = R.lagrange_polynomial([(a^2+a,a),(a,1)], neville=True);
    sage: # add another point
    sage: timeit("R.lagrange_polynomial([(a^2+a,a),(a,1),(a^2,a^2+a+1)], neville=True, previous_row=p)");
    625 loops, best of 3: 131 µs per loop
    sage:
    sage: R = PolynomialRing(QQ, 'x')
    sage: points = [(random(), random()) for i in xrange(100)]
    sage: time R.lagrange_polynomial(points, neville=True);
    CPU times: user 1.26 s, sys: 0.00 s, total: 1.26 s
    Wall time: 1.26 s
    sage: p = R.lagrange_polynomial(points, neville=True);
    sage: # add three more points
    sage: for i in xrange(3): points.append((random(), random()))
    ....: 
    sage: time R.lagrange_polynomial(points, neville=True, previous_row=p);
    CPU times: user 0.09 s, sys: 0.00 s, total: 0.09 s
    Wall time: 0.08 s
    sage: p = R.lagrange_polynomial(points, neville=True, previous_row=p)
    sage: # add another 100 points
    sage: for i in xrange(100): points.append((random(), random()))
    ....: 
    sage: time R.lagrange_polynomial(points, neville=True, previous_row=p);
    CPU times: user 4.62 s, sys: 0.00 s, total: 4.62 s
    Wall time: 4.62 s
    

Basic Arithmetic

  • Speed overhaul for digits, exact_log and ndigits (Joel B. Mohler) — Speed-up for the cases where the method exact_log can conveniently be computed by log 2 estimation. In some cases, time efficiency can be up to 927x faster than previously. The following timing statistics were obtained using the machine sage.math:
    # BEFORE
    
    sage: n = 5^1000
    sage: m = 2975982357823879528793587928793592
    sage: %timeit n.exact_log(m)
    1000 loops, best of 3: 205 µs per loop
    sage: n = 5^50
    sage: m = 33
    sage: %timeit n.exact_log(m)
    10000 loops, best of 3: 29.6 µs per loop
    sage: def zlog(m, n, k):
    ....:     for i in xrange(0, m/1000):
    ....:         a = ZZ.random_element(n) + 2
    ....:         b = ZZ.random_element(k)
    ....:         c = a^b
    ....:         for i in xrange (0, 1000):
    ....:             c.exact_log(a)
    ....:             
    sage: time zlog(100000, 2^100, 100)
    CPU times: user 22.59 s, sys: 0.12 s, total: 22.71 s
    Wall time: 22.70 s
    sage: time zlog(100000, 100, 100)
    CPU times: user 3.45 s, sys: 0.02 s, total: 3.47 s
    Wall time: 3.47 s
    
    
    # AFTER
    
    sage: n = 5^1000
    sage: m = 2975982357823879528793587928793592
    sage: %timeit n.exact_log(m)
    1000000 loops, best of 3: 221 ns per loop
    sage: n = 5^50
    sage: m = 33
    sage: %timeit n.exact_log(m)
    1000000 loops, best of 3: 526 ns per loop
    sage: def zlog(m, n, k):
    ....:     for i in xrange(0, m/1000):
    ....:         a = ZZ.random_element(n) + 2
    ....:         b = ZZ.random_element(k)
    ....:         c = a^b
    ....:         for i in xrange (0, 1000):
    ....:             c.exact_log(a)
    ....:             
    sage: time zlog(100000, 2^100, 100)
    CPU times: user 1.96 s, sys: 0.02 s, total: 1.98 s
    Wall time: 1.99 s
    sage: time zlog(100000, 100, 100)
    CPU times: user 0.05 s, sys: 0.01 s, total: 0.06 s
    Wall time: 0.05 s
    

Calculus

  • Deprecate the function numerical_sqrt() (Robert Bradshaw, John H. Palmieri) — The function numerical_sqrt() in sage/misc/functional.py is now deprecated. Users are advised to instead use sqrt().

Combinatorics

  • Sets enumerated by exploring a search space with a (lazy) tree or graph structure (Nicolas Thiery) — Extend the sage.combinat.backtrack library with other generic tools for constructing large sets whose elements can be enumerated by exploring a search space with a (lazy) tree or graph structure. The following generic utilities have been added:
    • SearchForest: Depth first search through a tree described by a “child” function.
    • GenericBacktracker: Depth first search through a tree described by a “child” function, with branch pruning, etc.
    • TransitiveIdeal: Depth first search through a graph described by a “neighbours” relation.
    • TransitiveIdealGraded: Breath first search through a graph described by a “neighbours” relation.
  • The Sloane sequence A000008 (Joanna Gaski) — The Sloane sequence A000008 is concerned with the number of ways of making change for n cents where one is restricted to using only coins of denominations 1, 2, 5, and 10 cents. This is contained in the new class A000008 in sage/combinat/sloane_functions.py. Here are some examples on using this class:
    sage: a = sloane.A000008; a
    Number of ways of making change for n cents using coins of 1, 2, 5, 10 cents.
    sage: a(0)
    1
    sage: a(1)
    1
    sage: a(13)
    16
    sage: a.list(14)
    [1, 1, 2, 2, 3, 4, 5, 6, 7, 8, 11, 12, 15, 16]
    
  • Read ext_rep format of combinatorial designs (Carlo Hamalainen) — The new module sage/combinat/designs/ext_rep.py is an API to the abstract tree represented by an XML document containing the External Representation of a list of block designs. The relevant combinatorial designs are read from the online database at the design theory database. This module also provides the related I/O operations for reading and writing ext-rep files or data. The parsing is based on expat.
  • Dynkin diagram ASCII art for reducible Cartan types (Dan Bump) — Here are some examples on such ASCII art:
    sage: CartanType("F4xA2").dynkin_diagram()
    
    O---O=>=O---O
    1   2   3   4
    O---O
    5   6
    F4xA2
    sage: t = CartanType("A2xB2xF4")
    sage: dd = DynkinDiagram(t); dd
    
    O---O
    1   2
    O=>=O
    3   4
    O---O=>=O---O
    5   6   7   8
    A2xB2xF4
    
  • Speed-up computation in symmetric algebra group (Dan Christensen) — The previous code essentially reimplemented the multiplication in the group algebra. Now it accumulates the symmetrizers and antisymmetrizers separately, and then does one multiplication at the end. This probably results in the same number of operations, but it avoids creating many intermediate objects. The following timing statistics were obtained using the machine sage.math:
    # BEFORE
    
    sage: from sage.combinat.symmetric_group_algebra import e
    sage: time dummy = e([[1,2,3,4], [5,6,7]])
    CPU times: user 1.91 s, sys: 0.31 s, total: 2.22 s
    Wall time: 3.15 s
    sage: time e([[1,2,3,4,5],[6,7,8],[9,10],[11]]);
    # hangs for over 6 hours
    
    
    # AFTER
    
    sage: from sage.combinat.symmetric_group_algebra import e
    sage: time dummy = e([[1,2,3,4], [5,6,7]])
    CPU times: user 0.12 s, sys: 0.05 s, total: 0.17 s
    Wall time: 0.32 s
    sage: time e([[1,2,3,4,5],[6,7,8],[9,10],[11]]);
    CPU times: user 31.20 s, sys: 0.53 s, total: 31.73 s
    Wall time: 31.73 s
    
  • Improve speed of combinatorial algebra multiplication (Dan Christensen) — The speed-up concerns the method multiply() of the class CombinatorialAlgebra in sage/combinat/combinatorial_algebra.py. In some cases, the efficiency can be up to 3x. The following timing statistics were obtained using the machine sage.math:
    # BEFORE
    
    sage: from sage.combinat.symmetric_group_algebra import e
    sage: Y=[[1,2,3,4],[5,6]]
    sage: time eY = e(Y)
    CPU times: user 0.46 s, sys: 0.08 s, total: 0.54 s
    Wall time: 0.74 s
    sage: time eY2 = eY*eY
    CPU times: user 1.47 s, sys: 0.00 s, total: 1.47 s
    Wall time: 1.47 s
    
    
    # AFTER
    
    sage: from sage.combinat.symmetric_group_algebra import e
    sage: Y = [[1,2,3,4], [5,6]]
    sage: time eY = e(Y)
    CPU times: user 0.05 s, sys: 0.02 s, total: 0.07 s
    Wall time: 0.22 s
    sage: time eY2 = eY*eY
    CPU times: user 1.24 s, sys: 0.00 s, total: 1.24 s
    Wall time: 1.24 s
    

Graphics

  • Mesh lines for 3-D plots (Bill Cauchois) — One can produce 3-D plots with mesh lines as follows:
    sage: plot3d(lambda x,y: exp(x+y*I).real(), (-2, 2.4), (-3, 3), mesh=True, zoom=1.3)
    


  • Centering of contour and density plots (Jason Grout) — The following example shows a “spotlight” on the origin:
    sage: x, y = var('x,y')
    sage: density_plot(1/(x^10+y^10), (x, -10, 10), (y, -10, 10))
    


    This plots concentric circles centered at the origin:

    sage: x, y = var('x,y')
    sage: contour_plot(x^2 + y^2 - 2, (x,-1,1), (y,-1,1)).show(aspect_ratio=1)
    


    The following example plots a disk centered at the origin:

    sage: x, y = var('x,y')
    sage: region_plot(x^2 + y^2 < 1, (x,-1,1), (y,-1,1)).show(aspect_ratio=1)
    


Interfaces

  • Improving the GAP interface by pre-compiling certain regular expressions (Simon King) — The speed-up in the GAP interface is now up to 37% faster than previously. The following timing statistics were obtained using the machine sage.math:
    # BEFORE
    
    sage: G = SymmetricGroup(7)
    sage: g = G._gap_()
    sage: l = g.Elements()
    sage: time L = [gap.eval(l.name() + '[%d]^2' % (i)) for i in xrange(1, 7.factorial() + 1)]
    CPU times: user 1.90 s, sys: 0.16 s, total: 2.06 s
    Wall time: 2.08 s
    
    
    # AFTER
    
    sage: G = SymmetricGroup(7)
    sage: g = G._gap_()
    sage: l = g.Elements()
    sage: time L = [gap.eval(l.name() + '[%d]^2' % (i)) for i in xrange(1, 7.factorial() + 1)]
    CPU times: user 1.07 s, sys: 0.22 s, total: 1.29 s
    Wall time: 1.31 s
    

Miscellaneous

  • Wrapping Sage or Python objects as Sage elements (Nicolas Thiery) — New class ElementWrapper in sage/structure/element_wrapper.py for wrapping Sage or Python objects as Sage elements, with reasonable default implementations of repr, cmp, hash, etc. The typical use case is for trivially constructing new element classes from pre-existing Sage or Python classes, with a containment relation. Here’s an example on using ElementWrapper:
    sage: o = ElementWrapper("bla", parent=ZZ); o
    'bla'
    sage: isinstance(o, sage.structure.element.Element)
    True
    sage: o.parent()
    Integer Ring
    sage: o.value
    'bla'
    
  • A tool for understanding Python pickles (Carl Witty) — The new module sage/misc/explain_pickle.py has a function called explain_pickle that takes a pickle and produces Sage code that will evaluate to the contents of the pickle. The combination of explain_sage to produce Sage code and sage_eval to evaluate the code should be a 100% compatible implementation of cPickle’s unpickler. That is, explain_sage explains a pickle by producing source code such that evaluating the code is equivalent to loading the pickle. Feeding the result of explain_pickle to sage_eval should be totally equivalent to loading the pickle with cPickle. Here are some examples on using explain_pickle:
    sage: explain_pickle(dumps({('a', 'b'): [1r, 2r]}))
    {('a', 'b'):[1r, 2r]}
    sage: explain_pickle(dumps(RR(pi)), in_current_sage=True)
    
    from sage.rings.real_mpfr import __create__RealNumber_version0
    from sage.rings.real_mpfr import __create__RealField_version0
    __create__RealNumber_version0(__create__RealField_version0(53r, False, 'RNDN'), '3.4gvml245kc0@0', 32r)
    sage: s = 'hi'
    sage: explain_pickle(dumps((s, s)))
    ('hi', 'hi')
    sage: explain_pickle(dumps((s, s)), pedantic=True)
    
    si = 'hi'
    (si, si)
    sage: explain_pickle(dumps(5r)
    ....: )
    5r
    sage: explain_pickle(dumps(22/7))
    
    pg_make_rational = unpickle_global('sage.rings.rational', 'make_rational')
    pg_make_rational('m/7')
    sage: explain_pickle(dumps(22/7), in_current_sage=True)
    
    from sage.rings.rational import make_rational
    make_rational('m/7')
    sage: explain_pickle(dumps(22/7), default_assumptions=True)
    
    from sage.rings.rational import make_rational
    make_rational('m/7')
    
  • S-box calling when m \neq n (Martin Albrecht) — An S-box takes m input bits and transforms them into n output bits. This is called an m \times n S-box. The case of invoking an S-box with m \neq n is now supported:
    sage: S = mq.SBox(3, 0, 1, 3, 1, 0, 2, 2)
    sage: S(0)
    3
    sage: S([0,0,0])
    [1, 1]
    

Modular Forms

  • Membership testing for modular forms subspaces (David Loeffler) — One can test such membership as follows:
    sage: M = ModularForms(17, 4)
    sage: S = M.cuspidal_submodule()
    sage: M.0 == S.0
    True
    sage: M.0 in S
    True
    

Notebook

  • Show nested lists as HTML tables (Wilfried Huss) — One can produce such HTML tables as follows:
    sage: functions = [sin(x), cos(x), tan(x), acos(x)]
    sage: t = [[f, taylor(f, x, 0, 10)] for f in functions]
    sage: html.table([["Function", "Series"]] + t, header = True)
    


    One can also place graphic objects into the table:

    sage: f = 1/x*sin(x)
    sage: t = [["Function", "Plot"],[f, plot(f, x, -4*pi, 4*pi)]]
    sage: html.table(t, header = True)
    


  • Limit the number of worksheet snapshots (Rob Beezer) — Reduce the unlimited growth of snapshots of worksheets when using the notebook.

Number Theory

  • Galois action (David Loeffler) — For example, one can now perform computations similar to the following:
    sage: F.<z> = CyclotomicField(7)
    sage: G = F.galois_group()
    sage: phi = G.random_element()
    sage: phi(z)
    z^4
    
  • Period lattices for elliptic curves over \mathbf{C} (John Cremona) — For elliptic curves over number fields, period lattice for complex embeddings is supported, using the complex AGM (Gauss’ Arithmetic-Geometric Mean) method to compute the basis. Here’s an example:
    sage: K.<a> = NumberField(x^3 - 2)
    sage: E = EllipticCurve([0, 1, 0, a, a])
    sage: emb = K.embeddings(ComplexField())[0]
    sage: E.period_lattice(emb)
    
    Period lattice associated to Elliptic Curve defined by y^2 = x^3 + x^2 + a*x + a over Number Field in a with defining polynomial x^3 - 2 with respect to the embedding Ring morphism:
      From: Number Field in a with defining polynomial x^3 - 2
      To:   Algebraic Field
      Defn: a |--> -0.6299605249474365? - 1.091123635971722?*I
    
  • Move the algebraic_closure method from RLF to LazyField (Nick Alexander).

Numerical

  • Solving the subset sum problem for super-increasing sequences (Minh Van Nguyen) — New module sage/numerical/knapsack.py for solving knapsack problems. The class Superincreasing in that module can be used to solve the subset sum problem for super-increasing sequences. Here are some examples:
    sage: from sage.numerical.knapsack import Superincreasing
    sage: L = [1, 2, 5, 21, 69, 189, 376, 919]
    sage: seq = Superincreasing(L)
    sage: seq
    Super-increasing sequence of length 8
    sage: seq.is_superincreasing()
    True
    sage: Superincreasing().is_superincreasing([1,3,5,7])
    False
    

Packages

  • Update the ATLAS spkg to version atlas-3.8.3.p3.spkg (William Stein).
  • Update FLINT to version 1.2.5 latest upstream release (Michael Abshoff, Mike Hansen).
  • Update the GAP spkg to version gap-4.4.10.p12.spkg (William Stein).
  • Update MPIR to version 1.2 latest upstream release (William Stein).
  • Update the optional SageTeX spkg to version 2.1.1 (Dan Drake).

Symbolics

  • Simplify when multiplying by exponential expressions (Burcin Erocal, Mike Hansen) — Here are some examples:
    sage: x, y = var("x,y")
    sage: exp(x) * exp(y)
    e^(x + y)
    sage: x^y * exp(x+y) * exp(-y)
    x^y*e^x
    sage: x^y * exp(x+y) * (x+y) * (2*x + 2*y) * exp(-y)
    2*(x + y)^2*x^y*e^x
    sage: A = exp(I*pi/5)
    sage: t = A*A*A*A; t
    e^(4/5*I*pi)
    sage: t*A
    -1
    

Topology

  • Change facets from an attribute to a method in simplicial complexes (John Palmieri).

A big thank you to all the Sage bug report/patch authors who made my life as a release tour author easier through your comprehensive and concise documentation. There are too many to list here; you know who you are. A release tour can also be found on the Sage wiki.

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  1. omologos
    9 May 2010 at 3:13 am

    Great work everyone!

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