Sage 4.1.1 Release Tour

Algebra

• FIXME: summarize #6510

Basic Arithmetic

• FIXME: summarize #6159

• FIXME: summarize #2737

Calculus

• FIXME: summarize #5996

Combinatorics

• FIXME: summarize #6519

• FIXME: summarize #6621

• FIXME: summarize #5790

Cryptography

• FIXME: summarize #6454

Documentation

• FIXME: summarize #4460

Elliptic Curves

• #6381 (bug in integral_points when rank is large):

The function integral_x_coords_in_interval() for finding all integral points on an elliptic curve defined over the rationals whose x-coordinate lies in an interval is now more efficient when the interval is large.

• FIXME: summarize #6407

Graphics

• FIXME: summarize #6098

• FIXME: summarize #5649

• FIXME: summarize #5651

Graph Theory

• FIXME: summarize #6355

• FIXME: summarize #6540

• FIXME: summarize #6552

• FIXME: summarize #6578

• FIXME: summarize #5793

Interfaces

• FIXME: summarize #6395

• FIXME: summarize #6591

Linear Algebra

• FIXME: summarize #5081

• FIXME: summarize #6553

• FIXME: summarize #6554

• Elementwise (Hadamard) product of matrices (Rob Beezer) (Trac #6301)

  • Given matrices A and B of the same size, C = A.elementwise_product(B) creates the new matrix C, of the same size, with entries given by C[i,j]=A[i,j]*B[i,j]. The multiplication occurs in a ring defined by Sage's coercion model, as appropriate for the base rings of A and B (or an error is raised if there is no sensible common ring). In particular, if A and B are defined over a noncommutative ring, then the operation is also noncommutative. The implementation is different for dense matrices versus sparse matrices, but there are probably further optimizations available for specific rings. This operation is often call the Hadamard product.

    sage: G = matrix(GF(3),2,[0,1,2,2]) 
    sage: H = matrix(ZZ,2,[1,2,3,4]) 
    sage: J = G.elementwise_product(H) 
    sage: J 
    [0 2] 
    [0 2] 
    sage: J.parent() 
    Full MatrixSpace of 2 by 2 dense matrices over Finite Field of size 

Modular Forms

• FIXME: summarize #6606

• FIXME: summarize #6071

• FIXME: summarize #6534

Notebook

• FIXME: summarize #5653

Number Theory

• #6457 (Intersection of ideals in a number field)

Intersection of ideals in number fields is now implemented.

• FIXME: summarize #6045

• FIXME: summarize #6396

• FIXME: summarize #6458

Numerical

• FIXME: summarize #6506

• FIXME: summarize #4571

Packages

• FIXME: summarize #6558

• FIXME: summarize #6380

• FIXME: summarize #6443

• FIXME: summarize #6445

• FIXME: summarize #6451

• FIXME: summarize #6453

• FIXME: summarize #6528

• FIXME: summarize #6143

• FIXME: summarize #6438

• FIXME: summarize #6493

• FIXME: summarize #6563

• FIXME: summarize #6602

• FIXME: summarize #6302

• new optional package p_group_cohomology (Simon A. King, David J. Green)

  • Compute the cohomology ring with coefficients in GF(p) for any finite p-group, in terms of a minimal generating set and a minimal set of algebraic relations. We use Benson's criterion to prove the completeness of the ring structure.
  • Compute depth, dimension, Poincare series and a-invariants of the cohomology rings.
  • Compute the nil radical
  • Construct induced homomorphisms.
  • The package includes a list of cohomology rings for all groups of order 64.
  • With the package, the cohomology for all groups of order 128 and for the Sylow 2-subgroup of the third Conway group (order 1024) was computed for the first time. The result of these and many other computations (e.g., all but 6 groups of order 243) is accessible in a repository on sage.math.

  Examples:

  • Data that are included with the package:

    sage: from pGroupCohomology import CohomologyRing
    sage: H = CohomologyRing(64,132) # this is included in the package, hence, the ring structure is already there
    sage: print H
    
    Cohomology ring of Small Group number 132 of order 64 with coefficients in GF(2)
    
    Computation complete
    Minimal list of generators:
    [a_2_4, a 2-Cochain in H^*(SmallGroup(64,132); GF(2)),
     c_2_5, a 2-Cochain in H^*(SmallGroup(64,132); GF(2)),
     c_4_12, a 4-Cochain in H^*(SmallGroup(64,132); GF(2)),
     a_1_0, a 1-Cochain in H^*(SmallGroup(64,132); GF(2)),
     a_1_1, a 1-Cochain in H^*(SmallGroup(64,132); GF(2)),
     b_1_2, a 1-Cochain in H^*(SmallGroup(64,132); GF(2))]
    Minimal list of algebraic relations:
    [a_1_0*a_1_1,
     a_1_0*b_1_2,
     a_1_1^3+a_1_0^3,
     a_2_4*a_1_0,
     a_1_1^2*b_1_2^2+a_2_4*a_1_1*b_1_2+a_2_4^2+c_2_5*a_1_1^2]
    sage: H.depth()
    2
    sage: H.a_invariants()
    [-Infinity, -Infinity, -3, -3]
    sage: H.poincare_series()
    (-t^2 - t - 1)/(t^6 - 2*t^5 + t^4 - t^2 + 2*t - 1)
    sage: H.nil_radical()
    
    a_1_0,
    a_1_1,
    a_2_4

  • Data from the repository on sage.math:

    sage: H = CohomologyRing(128,562) # if there is internet connection, the ring data are downloaded behind the scenes
    sage: len(H.gens())
    35
    sage: len(H.rels())
    486
    sage: H.depth()
    1
    sage: H.a_invariants()
    [-Infinity, -4, -3, -3]
    sage: H.poincare_series()
    (t^14 - 2*t^13 + 2*t^12 - t^11 - t^10 + t^9 - 2*t^8 + 2*t^7 - 2*t^6 + 2*t^5 - 2*t^4 + t^3 - t^2 - 1)/(t^17 - 3*t^16 + 4*t^15 - 4*t^14 + 4*t^13 - 4*t^12 + 4*t^11 - 4*t^10 + 4*t^9 - 4*t^8 + 4*t^7 - 4*t^6 + 4*t^5 - 4*t^4 + 4*t^3 - 4*t^2 + 3*t - 1)

  • Some computation from scratch, involving different ring presentations and induced maps:

    sage: tmp_root = tmp_filename()
    sage: CohomologyRing.set_user_db(tmp_root)
    sage: H0 = CohomologyRing.user_db(8,3,websource=False)
    sage: print H0
    
    Cohomology ring of Dihedral group of order 8 with coefficients in GF(2)
    
    Computed up to degree 0
    Minimal list of generators:
    []
    Minimal list of algebraic relations:
    []
    
    sage: H0.make()
    sage: print H0
    
    Cohomology ring of Dihedral group of order 8 with coefficients in GF(2)
    
    Computation complete
    Minimal list of generators:
    [c_2_2, a 2-Cochain in H^*(D8; GF(2)),
     b_1_0, a 1-Cochain in H^*(D8; GF(2)),
     b_1_1, a 1-Cochain in H^*(D8; GF(2))]
    Minimal list of algebraic relations:
    [b_1_0*b_1_1]
    
    sage: G = gap('DihedralGroup(8)')
    sage: H1 = CohomologyRing.user_db(G,GroupName='GapD8',websource=False)
    sage: H1.make()
    sage: print H1  # the ring presentation is different ...
    
    Cohomology ring of GapD8 with coefficients in GF(2)
    
    Computation complete
    Minimal list of generators:
    [c_2_2, a 2-Cochain in H^*(GapD8; GF(2)),
     b_1_0, a 1-Cochain in H^*(GapD8; GF(2)),
     b_1_1, a 1-Cochain in H^*(GapD8; GF(2))]
    Minimal list of algebraic relations:
    [b_1_1^2+b_1_0*b_1_1]
    
    sage: phi = G.IsomorphismGroups(H0.group())
    sage: phi_star = H0.hom(phi,H1)
    sage: phi_star_inv = phi_star^(-1) # ... but the rings are isomorphic
    sage: [X==phi_star_inv(phi_star(X)) for X in H0.gens()]
    [True, True, True, True]
    sage: [X==phi_star(phi_star_inv(X)) for X in H1.gens()]
    [True, True, True, True]

  • An example with an odd prime:

    sage: H = CohomologyRing(81,8) # this needs to be computed from scratch
    sage: H.make()
    sage: H.gens()
    
    [1,
     a_2_1, a 2-Cochain in H^*(SmallGroup(81,8); GF(3)),
     a_2_2, a 2-Cochain in H^*(SmallGroup(81,8); GF(3)),
     b_2_0, a 2-Cochain in H^*(SmallGroup(81,8); GF(3)),
     a_4_1, a 4-Cochain in H^*(SmallGroup(81,8); GF(3)),
     b_4_2, a 4-Cochain in H^*(SmallGroup(81,8); GF(3)),
     b_6_3, a 6-Cochain in H^*(SmallGroup(81,8); GF(3)),
     c_6_4, a 6-Cochain in H^*(SmallGroup(81,8); GF(3)),
     a_1_0, a 1-Cochain in H^*(SmallGroup(81,8); GF(3)),
     a_1_1, a 1-Cochain in H^*(SmallGroup(81,8); GF(3)),
     a_3_2, a 3-Cochain in H^*(SmallGroup(81,8); GF(3)),
     a_5_2, a 5-Cochain in H^*(SmallGroup(81,8); GF(3)),
     a_5_3, a 5-Cochain in H^*(SmallGroup(81,8); GF(3)),
     a_7_5, a 7-Cochain in H^*(SmallGroup(81,8); GF(3))]
    sage: len(H.rels())
    59
    sage: H.depth()
    1
    sage: H.a_invariants()
    [-Infinity, -3, -2]
    sage: H.poincare_series()
    (t^4 - t^3 + t^2 + 1)/(t^6 - 2*t^5 + 2*t^4 - 2*t^3 + 2*t^2 - 2*t + 1)
    sage: H.nil_radical()
    
    a_1_0,
    a_1_1,
    a_2_1,
    a_2_2,
    a_3_2,
    a_4_1,
    a_5_2,
    a_5_3,
    b_2_0*b_4_2,
    a_7_5,
    b_2_0*b_6_3,
    b_6_3^2+b_4_2^3

Symbolics

• FIXME: summarize #6195

• FIXME: summarize #6243