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| = Sage 9.2 Release Tour = in progress (2020) <<TableOfContents>> == Python 3 transition completed == [[ReleaseTours/sage-9.0|SageMath 9.0]] was the first version of Sage running on Python 3 by default. [[ReleaseTours/sage-9.1|SageMath 9.1]] continued to support Python 2. === Support for Python 2 removed === Sage 9.2 has removed support for Python 2. The Sage library now makes use of Python language and library features that are only available in Python 3.6 or newer; and large amounts of compatibility code have been removed. However, note that this is unrelated to the minimal requirements for a source installation of the Sage distribution: Sage 9.2 is still able to build on a system that only provides Python 2.x or Python 3.5 or older. In this case, the SageMath distribution builds its own copy of Python 3. === Support for Python 3.6 and 3.8 added === Sage 9.2 has added support for Python 3.8 in [[https://trac.sagemath.org/ticket/27754|#27754]]. Sage 9.2 has also added support for Python 3.6. This allows Sage to use the system Python on some older Linux distributions that are still in widespread use in scientific computing, including `centos-8` and `fedora-{26,27,28}` (although Python 3.7.x packages are also available for these). See [[https://trac.sagemath.org/ticket/29033|#29033]] for more details. Hence, Sage 9.2 conforms to (and exceeds) [[https://numpy.org/neps/nep-0029-deprecation_policy.html|NumPy Enhancement Proposal 29]] regarding Python version support policies. If no suitable system Python, version 3.6, 3.7, or 3.8 is found, Sage installs its own copy of Python 3 from source. The version of Python shipped with the Sage distribution has been upgraded from 3.7.3 to 3.8.5. === For developers: Using Python 3.6+ features in sagelib === [[https://trac.sagemath.org/ticket/29756|Meta-ticket #29756]] provides a starting point for a discussion of new features of the Python language and standard library to bring them to systematic use in sagelib. All features provided by Python 3.6 can be used immediately; features introduced in Python 3.7 or later will require backporting or a decision to drop support Python 3.6. === More details === * [[https://trac.sagemath.org/query?keywords=~python3&milestone=sage-9.2&or&component=python3&milestone=sage-9.2&or&keywords=~py3&milestone=sage-9.2&groupdesc=1&group=status&max=1500&col=id&col=summary&col=keywords&col=component&col=time&col=changetime&col=author&col=reviewer&order=component|Trac tickets with keyword/component python3 in milestone 9.2]] * See [[Python3-Switch]] for more details. == Package upgrades == The removal of support for Python 2 has enabled major package upgrades. Major user-visible package upgrades below... === matplotlib === Dropping Python 2 support allowed us to make a major jump from matplotlib 2.2.5 to 3.3.1. See matplotlib's [[https://matplotlib.org/3.3.0/users/prev_whats_new/whats_new_3.0.html|release notes for 3.0]], [[https://matplotlib.org/3.3.0/users/prev_whats_new/whats_new_3.1.0.html|3.1]], [[https://matplotlib.org/3.3.0/users/prev_whats_new/whats_new_3.2.0.html|3.2]],[[https://matplotlib.org/3.3.0/users/prev_whats_new/whats_new_3.3.0.html|3.3]]. In addition to improved output, this update will likely enable Sage developers to implement new features for plotting and graphics. === rpy2 and R === The [[https://pypi.org/project/rpy2/|rpy2 Python package]] is the foundation for [[https://doc.sagemath.org/html/en/reference/interfaces/sage/interfaces/r.html|SageMath's interface]] to [[https://www.r-project.org/|R]]. Dropping Python 2 support allowed us to make the major upgrade from 2.8.2 to 3.3.5 in [[https://trac.sagemath.org/ticket/29441|#29441]]; see the [[https://rpy2.github.io/doc/latest/html/changes.html#release-3-3-1|release notes]] for details. We only did a minor upgrade of R itself in the Sage distribution, to 3.6.3, the latest in the 3.6.x series. Of course, if R 4.0.x is installed in the system, Sage will use it instead of building its own copy. The SageMath developers are eager to learn from users how they use the SageMath-R interface, and what needs to be added to it to become more powerful. Let us know at [[https://groups.google.com/d/msg/sage-devel|sage-devel]]. === sphinx === Sage uses [[https://www.sphinx-doc.org/en/master/|Sphinx]] to build its [[https://doc.sagemath.org/html/en/index.html|documentation]]. Sage 9.2 has updated Sphinx from 1.8.5 to 3.1.2; see [[https://www.sphinx-doc.org/en/master/changes.html#release-3-1-2-released-jul-05-2020|Sphinx release notes]] for more information. === SymPy === [[https://www.sympy.org/en/index.html|SymPy]] has been updated from 1.5 to 1.6 in [[https://trac.sagemath.org/ticket/29730|#29730]]. See the [[https://github.com/sympy/sympy/wiki/Release-Notes-for-1.6|Release notes for version 1.6]]. === IPython, Jupyter notebook, JupyterLab === Dropping support for Python 2 allowed us to upgrade IPython from 5.8.0 to 7.13.0 in [[https://trac.sagemath.org/ticket/28197|#28197]]. See the [[https://ipython.readthedocs.io/en/stable/whatsnew/version6.html|release notes for the 6.x]] and [[https://ipython.readthedocs.io/en/stable/whatsnew/version7.html|7.x series]]. We have also upgraded the Jupyter notebook from 5.7.6 to 6.1.1 in [[https://trac.sagemath.org/ticket/26919|#26919]]; see the [[https://jupyter-notebook.readthedocs.io/en/stable/changelog.html|notebook changelog]] for more information. Besides, the pdf export of Jupyter notebooks has been fixed, so that LaTeX-typeset outputs are now rendered in the pdf file ([[https://trac.sagemath.org/ticket/23330|#23330]]). [[https://jupyterlab.readthedocs.io/en/stable/|JupyterLab]] is now fully supported as an optional, alternative interface [[https://trac.sagemath.org/ticket/30246|#30246]], including [[https://doc.sagemath.org/html/en/prep/Quickstarts/Interact.html|interacts]]. To use it, install it first, using the command `sage -i jupyterlab_widgets`. Then you can start it using `./sage -n jupyterlab`. === Normaliz === The optional package [[https://www.normaliz.uni-osnabrueck.de/|Normaliz]], a tool for computations in affine monoids, vector configurations, lattice polytopes, rational cones, and algebraic polyhedra has been upgraded from 3.7.2 to 3.8.8, and !PyNormaliz to version 2.12. The upgrade [[https://github.com/Normaliz/Normaliz/releases|adds]] support for incremental ("dynamic") computations, the computation of automorphism groups and refined triangulations of cones and polyhedra, and limited support for semiopen cones and polyhedra. To install Normaliz and !PyNormaliz, use `sage -i pynormaliz`. === Other package updates === * [[https://trac.sagemath.org/ticket/29141|Meta-ticket #29141: Upgrades and other changes that require dropping py2 support]] * [[https://trac.sagemath.org/query?summary=~update&milestone=sage-9.2&or&milestone=sage-9.2&summary=~upgrade&groupdesc=1&group=status&max=1500&col=id&col=summary&col=component&col=time&col=changetime&col=author&col=reviewer&col=keywords&order=component|Upgrade tickets, milestone 9.2]] === For developers: Upgrading packages === Upgrading Python packages in the Sage distribution from PyPI has again become easier, thanks to [[https://trac.sagemath.org/ticket/20104|#20104]]. You can now do: {{{ $ sage --package update-latest matplotlib Updating matplotlib: 3.3.0 -> 3.3.1 Downloading tarball to ...matplotlib-3.3.1.tar.bz2 [...............................................................] }}} When you do this, please remember to check that the `checksums.ini` file has an `upstream_url` in the format `upstream_url=https://pypi.io/packages/source/m/matplotlib/matplotlib-VERSION.tar.gz`. (This is not needed for `updated-latest` to work, but helps with automated tests of the upgrade ticket -- see [[https://wiki.sagemath.org/ReleaseTours/sage-9.1#For_developers-1|Sage 9.1 release tour]] on this topic.) === For packagers: Changes to packages === The packages `giacpy_sage` and `sage_brial` have been merged into `sagelib` as `sage.libs.giac` and `sage.rings.polynomial.pbori`. The Sage library is now built out of the directory `build/pkgs/sagelib/src/`. A pip-installable source distribution (sdist) can be built using the script `build/pkgs/sagelib/spkg-src` ([[https://trac.sagemath.org/ticket/29411|#29411]], [[https://trac.sagemath.org/ticket/29950|#29950]]). The scripts in `src/bin/` are now installed by sagelib's `setup.py` ([[https://trac.sagemath.org/ticket/21559|#21559]]). Also several scripts have been moved to `build/bin/`, and some obsolete scripts have been removed ([[https://trac.sagemath.org/ticket/29825|#29825]], [[https://trac.sagemath.org/ticket/27171|#27171]]). Many build-related functions of the main Sage script, `src/bin/sage` (installed as `sage`), have been moved to a script `build/bin/sage-site` (not installed) in [[https://trac.sagemath.org/ticket/29111|#29111]]. It is hoped that downstream distribution packaging is able to use this cleaned up script instead of replacing it with an ad-hoc distribution-specific script -- so that users can rely on a consistent interface. Contributions of further clean ups and refactoring of the script are welcome. == Graphics == === New features === * Specify the rectangle in which to draw a matrix using the new `xrange` and `yrange` options of `matrix_plot`. For example, to draw a matrix in [0,1]×[0,1] instead of the default [-0.5,4.5]×[-0.5,4.5]: `matrix_plot(identity_matrix(5), xrange=(0, 1), yrange=(0, 1))`. [[https://trac.sagemath.org/ticket/27895|27895]] (Markus Wageringel) * Set the initial camera orientation in Three.js plots using the new `viewpoint` option. Pass it a list/tuple of the form `[[x,y,z],angle]`, such as that provided by the existing `Get Viewpoint` option accessible from the menu button in the lower-right corner of a Three.js plot. [[https://trac.sagemath.org/ticket/29192|29192]] (Paul Masson) * Save a 3D graphics object directly to an HTML file that uses the Three.js viewer, similar to how you would save a PNG image: `G.save('plot.html')`. [[https://trac.sagemath.org/ticket/29194|29194]] (Joshua Campbell) * Produce an interactive 3D animation that you can pan, rotate, and zoom while the animation is playing using the Three.js viewer. A slider and buttons for controlling playback are included on the page by default. To use this new feature construct an animation as you normally would, passing a list of still frames to the `animate` function, then call the `interactive` method. For example: {{{ #!python def build_frame(t): """Build a single frame of animation at time t.""" e = parametric_plot3d([sin(x-t), 0, x], (x, 0, 2*pi), color='red') b = parametric_plot3d([0, -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') }}} [[https://trac.sagemath.org/ticket/29194|29194]] (Joshua Campbell) === Implementation improvements === * Points are now sampled exponentially when `scale='semilogx'` or `scale='loglog'` is specified. This decreases the number of points necessary for an accurate plot (and also increases the chance that the default number of points will produce an acceptable plot). [[https://trac.sagemath.org/ticket/29523|29523]] (Blair Mason) * Points and lines are now ignored in STL 3D export. Moreover disjoint union of surfaces can be saved. [[https://trac.sagemath.org/ticket/29732|29732]] (Frédéric Chapoton) * Three.js has been upgraded to version r117. [[https://trac.sagemath.org/ticket/29809|29809]] (Paul Masson) * Long text is no longer clipped in Three.js plots. Multi-line text is not yet supported but is in the works. [[https://trac.sagemath.org/ticket/29758|29758]] (Joshua Campbell) * JSmol's telemetry functionality has been disabled. It will no longer phone home when, for example, using `viewer='jmol'` in a Jupyter notebook. [[https://trac.sagemath.org/ticket/30030|30030]] (Joshua Campbell) * SVG export has been added to the javascript graph display tool: {{{G.show(method='js')}}} [[https://trac.sagemath.org/ticket/29807|29807]] === For developers === * Clarified that example Three.js plots in the documentation should use the `online=True` viewing option. [[https://trac.sagemath.org/ticket/30136|30136]] (Paul Masson) == Linear and multilinear algebra == === One free module constructor to rule them all === Sage has several specialized implementation classes for free modules and vector spaces. The factory functions `FreeModule` and `VectorSpace` select the appropriate class depending on the base ring and other parameters: {{{ #!python sage: FreeModule(ZZ, 10) Ambient free module of rank 10 over the principal ideal domain Integer Ring sage: FreeModule(FiniteField(5), 10) Vector space of dimension 10 over Finite Field of size 5 sage: QQ^10 is VectorSpace(QQ, 10) True }}} The free modules (vector spaces) created here have a distinguished standard basis indexed by `range(rank)`. In Sage 9.2, these factory functions have been extended in [[https://trac.sagemath.org/ticket/30194|#30194]] so that they cover two more cases: 1. If a sequence or family of indices is passed instead of the rank (dimension), then a [[https://doc.sagemath.org/html/en/reference/combinat/sage/combinat/free_module.html#sage.combinat.free_module.CombinatorialFreeModule|CombinatorialFreeModule]] is created instead. These modules underly SageMath's facilities for [[https://doc.sagemath.org/html/en/reference/combinat/sage/combinat/__init__.html|algebraic combinatorics]]. {{{ #!python sage: U = FreeModule(AA, ['x', 'y', 'z']); U Free module generated by {'x', 'y', 'z'} over Algebraic Real Field sage: V = VectorSpace(QQ, ZZ); V sage: V.basis() Lazy family (Term map from Integer Ring to Free module generated by Integer Ring over Rational Field(i)) _{i in Integer Ring} sage: QQ^SymmetricGroup(4) Free module generated by Symmetric group of order 4! as a permutation group over Rational Field }}} 2. If the factory function is invoked with the parameter `with_basis=None`, then a free module of the given rank ''without'' distinguished basis is created. {{{ #!python sage: W = FreeModule(AA, 3, with_basis=None); W 3-dimensional vector space over the Algebraic Real Field sage: W.category() Category of finite dimensional vector spaces over Algebraic Real Field sage: W.tensor_module(2, 2) Free module of type-(2,2) tensors on the 3-dimensional vector space over the Algebraic Real Field }}} It is represented by an instance of the class [[https://doc.sagemath.org/html/en/reference/tensor_free_modules/|FiniteRankFreeModule]] from `sage.tensor.modules`. These modules are the foundation for the multilinear algebra developed by the !SageManifolds project. === Connecting FiniteRankFreeModule and free modules with distinguished basis === Given a basis of a `FiniteRankFreeModule`, the new method `isomorphism_with_fixed_basis` ([[https://trac.sagemath.org/ticket/30094|#30094]]) constructs an isomorphism from the `FiniteRankFreeModule` to a free module in the category `ModulesWithBasis`. By default, it uses a `CombinatorialFreeModule`: {{{ #!python sage: V = FiniteRankFreeModule(QQ, 3, start_index=1); V 3-dimensional vector space over the Rational Field sage: basis = e = V.basis("e"); basis Basis (e_1,e_2,e_3) on the 3-dimensional vector space over the Rational Field sage: phi_e = V.isomorphism_with_fixed_basis(basis); phi_e Generic morphism: From: 3-dimensional vector space over the Rational Field To: Free module generated by {1, 2, 3} over Rational Field sage: phi_e(e[1] + 2 * e[2]) e[1] + 2*e[2] }}} === Eigenvalues and eigenvectors === * Experimental functions for computing eigenvalues and eigenvectors in arbitrary precision (via [[http://arblib.org/acb_mat.html#eigenvalues-and-eigenvectors|Arb]]) including error bounds have been added. [[https://trac.sagemath.org/ticket/30393|#30393]] {{{ sage: from sage.matrix.benchmark import hilbert_matrix sage: mat = hilbert_matrix(3).change_ring(CBF) sage: mat.eigenvalues() [[1.40831892712365 +/- 7.16e-15] + [+/- 2.02e-15]*I, [0.12232706585391 +/- 6.49e-15] + [+/- 2.02e-15]*I, [0.00268734035577 +/- 5.60e-15] + [+/- 2.02e-15]*I] }}} * Solving [[https://en.wikipedia.org/wiki/Eigendecomposition_of_a_matrix#Generalized_eigenvalue_problem|generalized eigenvalue problems]] `Av = λBv` for two square matrices `A`, `B` over `RDF` or `CDF` is now supported (via [[https://docs.scipy.org/doc/scipy/reference/generated/scipy.linalg.eig.html|SciPy]]) [[https://trac.sagemath.org/ticket/29243|#29243]]. A generalized eigenvalue `λ` is defined to be a root of the polynomial `det(A - λ B)` if this polynomial is not constantly zero. {{{ sage: A = matrix.identity(RDF, 2) sage: B = matrix(RDF, [[3, 5], [6, 10]]) sage: D, V = A.eigenmatrix_right(B); D # tol 1e-14 [0.07692307692307694 0.0] [ 0.0 +infinity] sage: λ = D[0, 0] sage: v = V[:, 0] sage: (A * v - B * v * λ).norm() < 1e-14 True sage: A.eigenvalues(B, homogeneous=True) [(0.9999999999999999, 13.000000000000002), (0.9999999999999999, 0.0)] }}} === Other improvements === Sage 9.2 has also merged a number of improvements to `sage.tensor.modules`: [[https://trac.sagemath.org/ticket/30094|#30094]], [[https://trac.sagemath.org/ticket/30169|#30169]], [[https://trac.sagemath.org/ticket/30179|#30179]], [[https://trac.sagemath.org/ticket/30181|#30181]], [[https://trac.sagemath.org/ticket/30194|#30194]], [[https://trac.sagemath.org/ticket/30250|#30250]], [[https://trac.sagemath.org/ticket/30251|#30251]], [[https://trac.sagemath.org/ticket/30254|#30254]], [[https://trac.sagemath.org/ticket/30255|#30255]], [[https://trac.sagemath.org/ticket/30287|#30287]] == Polyhedral geometry == === New features === It is now possible to choose which backend to use to compute regions of hyperplane arrangements [[https://trac.sagemath.org/ticket/29506|29506]]: {{{ #!python sage: R.<sqrt5> = QuadraticField(5) sage: H = HyperplaneArrangements(R, names='xyz') sage: x,y,z = H.gens() sage: A = H(sqrt5*x+2*y+3*z, backend='normaliz') sage: A.backend() 'normaliz' sage: A.regions()[0].backend() # optional - pynormaliz 'normaliz' }}} It is now possible to compute the slack matrix of a polyhedron [[https://trac.sagemath.org/ticket/29838|29838]]: {{{ #!python sage: P = polytopes.cube(intervals='zero_one') sage: P.slack_matrix() ⎛0 1 1 1 0 0⎞ ⎜0 0 1 1 0 1⎟ ⎜0 0 0 1 1 1⎟ ⎜0 1 0 1 1 0⎟ ⎜1 1 0 0 1 0⎟ ⎜1 1 1 0 0 0⎟ ⎜1 0 1 0 0 1⎟ ⎝1 0 0 0 1 1⎠ }}} It is now possible to apply an affine transformation on a polyhedron [[https://trac.sagemath.org/ticket/30327|30327]]: {{{ #!python sage: M = random_matrix(QQ,3,3) sage: v = vector(QQ,(1,2,3)) sage: F = AffineGroup(3, QQ) sage: f = F(M, v); f [ 0 0 -2] [1] x |-> [ 0 1 0] x + [2] [ -1 -1 1/2] [3] sage: cube = polytopes.cube() sage: f * cube A 3-dimensional polyhedron in QQ^3 defined as the convex hull of 8 vertices sage: f(cube) # also works A 3-dimensional polyhedron in QQ^3 defined as the convex hull of 8 vertices }}} === Implementation improvements === * It is now possible to set up polyhedra with both Vrep and Hrep in the following constructions: * Linear transformation [[https://trac.sagemath.org/ticket/29843|29843]] * Polar [[https://trac.sagemath.org/ticket/29569|29569]] * Product [[https://trac.sagemath.org/ticket/29583|29583]] * The generation of regions of hyperplane arrangement has been improved [[https://trac.sagemath.org/ticket/29661|29661]] * Ehrhart related functions are now cached [[https://trac.sagemath.org/ticket/29196|29196]] * Obtaining incidence matrix and combinatorial polyhedron is much faster for integer and rational polyhedra [[https://trac.sagemath.org/ticket/29837|29837]], [[https://trac.sagemath.org/ticket/29841|29841]] * The test coverage for the [[http://match.stanford.edu/reference/discrete_geometry/index.html#backends-for-polyhedra|various backends for polyhedral computations]] has been improved by using `_test_...` methods to the abstract base class `Polyhedron_base`, in addition to doctests. See [[https://trac.sagemath.org/ticket/29842|Meta-ticket #29842 Run a more stable test suite on polyhedra]]. * The face lattice can be obtained in reasonable time and no longer leaks memory [[https://trac.sagemath.org/ticket/28982|28982]] There are also some bug fixes and other improvements. For more details see the [[https://trac.sagemath.org/wiki/SagePolyhedralGeometry#release_9.2|release notes for optimization and polyhedral geometry software interactions in Sage]]. == Combinatorics == === Reduction from Dancing links to SAT or MILP === It is now possible to solve an instance of an [[https://en.wikipedia.org/wiki/Exact_cover|exact cover problem]] using a reduction from a dancing links instance to SAT [[https://trac.sagemath.org/ticket/29338|29338]] or MILP [[https://trac.sagemath.org/ticket/29955|29955]]: {{{ #!python sage: from sage.combinat.matrices.dancing_links import dlx_solver sage: rows = [[0,1,2], [3,4,5], [0,1], [2,3,4,5], [0], [1,2,3,4,5]] sage: d = dlx_solver(rows) sage: d.one_solution() [1, 0] sage: d.one_solution_using_sat_solver('cryptominisat') [2, 3] sage: d.one_solution_using_sat_solver('glucose') [2, 3] sage: d.one_solution_using_sat_solver('glucose-syrup') [2, 3] sage: d.one_solution_using_sat_solver('picosat') [4, 5] sage: d.one_solution_using_milp_solver() [0, 1] sage: d.one_solution_using_milp_solver('Gurobi') [0, 1] }}} === Polyomino tilings === It is now possible to find a surrounding of a polyomino with copies of itself, see [[https://trac.sagemath.org/ticket/29160|29160]]. This is based on the dancing links solver in Sage. This is motivated by the [[https://en.wikipedia.org/wiki/Heesch%27s_problem|Heesch's problem]]. An example is below: {{{ sage: from sage.combinat.tiling import Polyomino sage: 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)]) sage: H.show2d() }}} {{attachment:H.png}} {{{ sage: %time solution = H.self_surrounding(10, ncpus=8) CPU times: user 1.69 s, sys: 1.08 s, total: 2.77 s Wall time: 3.85 s sage: G = sum([p.show2d() for p in solution], Graphics()) sage: G }}} {{attachment:G.png}} === Fully commutative elements of Coxeter groups === It is now possible by [[https://trac.sagemath.org/ticket/30243|30243]] to enumerate and work with the fully commutative elements of a Coxeter group. Methods to compute *star operations* and plot the *heaps* of such elements are also included. {{{ #!python sage: A3 = CoxeterGroup(['A', 3]) sage: FCA3 = A3.fully_commutative_elements() sage: FCA3.category() Category of finite enumerated sets sage: FCA3.list() [[], [1], [2], ... [1, 3, 2], [1, 2, 3], [2, 1, 3, 2]] sage: B8 = CoxeterGroup(['B', 8]) sage: FCB8 = B8.fully_commutative_elements() sage: len(FCB8) # long time (7 seconds) 14299 sage: B6 = CoxeterGroup(['B', 6]) sage: FCB6 = B6.fully_commutative_elements() sage: w = FCB6([1, 6, 2, 5, 4, 6, 5]) sage: w.coset_decomposition({5, 6}) ([6, 5, 6], [1, 2, 4, 5]) sage: w.star_operation({5,6}, 'lower') [1, 5, 2, 4, 6, 5] sage: FCB6([3, 2, 4, 3, 1]).plot_heap() }}} {{attachment:heap.png}} == Commutative algebra == === Laurent polynomials === Rings of Laurent polynomials now support ideal creation and manipulation [[https://trac.sagemath.org/ticket/29512|29512]]: {{{ sage: L.<x,y,z> = LaurentPolynomialRing(QQ, 3) sage: I = L.ideal([(x+y+z)^3+x*y, x^2+y^2+z^2]) sage: I.groebner_basis() (y^4 + 4*x*y*z^2 + y^2*z^2 + 2*x*z^3 + 2*y*z^3 - z^4 + 3/2*x*y*z + 1/4*x*z^2 + 1/4*y*z^2 - 1/4*z^3 + 1/8*x*y, x*y^2 - y^3 + 3*x*y*z + x*z^2 - z^3 + 1/2*x*y, x^2 + y^2 + z^2) sage: (x^3+y^3+z^3) in I False sage: x + x^-1*y^2 + x^-1*z^2 in I True }}} === Motivic multiple zetas === The ring of motivic multiple zeta values has been implemented, using algorithms of Francis Brown. It allows to compute at least up to weight 12 [[https://trac.sagemath.org/ticket/22713|22713]]. {{{ sage: Multizeta(1,2)**2 12*ζ(1,1,1,3) + 6*ζ(1,1,2,2) + 2*ζ(1,2,1,2) sage: Multizeta(1,2)==Multizeta(3) True sage: Multizeta(2,3,4).n(100) 0.0029375850405618494701189454256 }}} The numerical evaluation is based on PARI implementation. === Power series === There is new method to compute the coefficients in the Jacobi continued fraction expansion of a power series [[https://trac.sagemath.org/ticket/29789|29789]]. {{{ sage: t = QQ[['t']].0 sage: f = sum(factorial(n)*t**n for n in range(20)).O(20) sage: f.jacobi_continued_fraction() ((-1, -1), (-3, -4), (-5, -9), (-7, -16), (-9, -25), (-11, -36), (-13, -49), (-15, -64), (-17, -81)) }}} === Ring homomorphisms === For many polynomial ring homomorphisms, the methods `inverse`, `is_invertible`, `is_injective`, `is_surjective`, `kernel` and `inverse_image` have been implemented. This covers not only polynomial rings, but also quotient rings, number fields and Galois fields. [[https://trac.sagemath.org/ticket/9792|#9792]] [[https://trac.sagemath.org/ticket/29723|#29723]] {{{ sage: R.<x,y,z> = QQ[] sage: sigma = R.hom([x - 2*y*(z*x+y^2) - z*(z*x+y^2)^2, y + z*(z*x+y^2), z], R) sage: tau = sigma.inverse(); tau Ring endomorphism of Multivariate Polynomial Ring in x, y, z over Rational Field Defn: x |--> -y^4*z - 2*x*y^2*z^2 - x^2*z^3 + 2*y^3 + 2*x*y*z + x y |--> -y^2*z - x*z^2 + y z |--> z sage: (tau * sigma).is_identity() True }}} The method `inverse_image` can be used to test whether an element is contained in a subalgebra: {{{ sage: R.<s,t> = PolynomialRing(QQ) sage: S.<x,y,z,w> = PolynomialRing(QQ) sage: f = S.hom([s^4, s^3*t, s*t^3, t^4], R) sage: f.inverse_image(R.ideal(0)) Ideal (y*z - x*w, z^3 - y*w^2, x*z^2 - y^2*w, y^3 - x^2*z) of Multivariate Polynomial Ring in x, y, z, w over Rational Field sage: f.inverse_image(s^3*t^4*(s+t)) x*w + y*w sage: f.inverse_image(s^2*t^2) ... ValueError: element s^2*t^2 does not have preimage }}} == Manifolds == === diff function for exterior derivatives === It is now possible to invoke '''diff''' to compute the differential (exterior derivative) of a differentiable form ([[https://trac.sagemath.org/ticket/29953|#29953]]). For instance, for a scalar field: {{{ sage: M = Manifold(2, 'M') sage: X.<x,y> = M.chart() sage: f = M.scalar_field(x^2*y, name='f') sage: diff(f) 1-form df on the 2-dimensional differentiable manifold M sage: diff(f).display() df = 2*x*y dx + x^2 dy }}} and for a 1-form: {{{ sage: a = M.one_form(-y, x, name='a'); a.display() a = -y dx + x dy sage: diff(a) 2-form da on the 2-dimensional differentiable manifold M sage: diff(a).display() da = 2 dx/\dy }}} === Dot and cross products of vector fields along a curve === The methods '''dot_product()''', '''cross_product()''' and '''norm()''' can be now be used for vector fields defined along a differentiable map, the codomain of which is a Riemannian manifold ([[https://trac.sagemath.org/ticket/30318|#30318]]). Previously, these methods worked only for vector fields ''on'' a Riemannian manifold, i.e. along the identity map. An important subcase is of course that of a curve in a Riemannian manifold. For instance, let us consider a helix ''C'' in the Euclidean space E^3^ parametrized by its arc length ''s'': {{{ sage: E.<x,y,z> = EuclideanSpace() sage: R.<s> = RealLine() sage: C = E.curve((2*cos(s/3), 2*sin(s/3), sqrt(5)*s/3), (s, -oo, +oo), ....: name='C') sage: C.display() C: R --> E^3 s |--> (x, y, z) = (2*cos(1/3*s), 2*sin(1/3*s), 1/3*sqrt(5)*s) }}} The tangent vector field ''T=C' '' has a unit norm since the parameter ''s'' is the arc length: {{{ sage: T = C.tangent_vector_field() sage: T.display() C' = -2/3*sin(1/3*s) e_x + 2/3*cos(1/3*s) e_y + 1/3*sqrt(5) e_z sage: norm(T) Scalar field |C'| on the Real interval (0, 6*pi) sage: norm(T).expr() 1 }}} We introduce the unit normal vector ''N'' via the derivative of ''T'': {{{ sage: T_prime = R.vector_field([diff(T[i], s) for i in E.irange()], dest_map=C, ....: name="T'") sage: N = T_prime / norm(T_prime) sage: N.display() -cos(1/3*s) e_x - sin(1/3*s) e_y }}} and we get the binormal vector ''B'' as the cross product of ''T'' and ''N'': {{{ sage: B = T.cross_product(N) sage: B Vector field along the Real number line R with values on the Euclidean space E^3 sage: B.display() 1/3*sqrt(5)*sin(1/3*s) e_x - 1/3*sqrt(5)*cos(1/3*s) e_y + 2/3 e_z }}} We can then form the '''Frenet-Serret''' frame: {{{ sage: FS = R.vector_frame(('T', 'N', 'B'), (T, N, B), ....: symbol_dual=('t', 'n', 'b')) sage: FS Vector frame (R, (T,N,B)) with values on the Euclidean space E^3 }}} and check that it is orthonormal: {{{ sage: matrix([[u.dot(v).expr() for v in FS] for u in FS]) [1 0 0] [0 1 0] [0 0 1] }}} The Frenet-Serret formulas, expressing the '''curvature''' and '''torsion''' of ''C'', are obtained as: {{{ sage: N_prime = R.vector_field([diff(N[i], s) for i in E.irange()], ....: dest_map=C, name="N'") sage: B_prime = R.vector_field([diff(B[i], s) for i in E.irange()], ....: dest_map=C, name="B'") sage: for v in (T_prime, N_prime, B_prime): ....: v.display(FS) ....: T' = 2/9 N N' = -2/9 T + 1/9*sqrt(5) B B' = -1/9*sqrt(5) N }}} === Orientability of manifolds and vector bundles === It is now possible to define an orientation [[https://doc.sagemath.org/html/en/reference/manifolds/sage/manifolds/differentiable/manifold.html#sage.manifolds.differentiable.manifold.DifferentiableManifold.orientation|on a differentiable manifold]] and [[https://doc.sagemath.org/html/en/reference/manifolds/sage/manifolds/vector_bundle.html#sage.manifolds.vector_bundle.TopologicalVectorBundle.orientation|on a vector bundle]] ([[https://trac.sagemath.org/ticket/30178|#30178]]). [[https://doc.sagemath.org/html/en/reference/manifolds/sage/manifolds/manifold.html#sage.manifolds.manifold.TopologicalManifold.orientation|Orientations of topological manifolds]] have also been introduced, according to [[http://www.map.mpim-bonn.mpg.de/Orientation_of_manifolds|this definition]]. === Euclidean spaces as metric spaces === Euclidean spaces have been endowed with a distance function and have been set in the category of complete metric spaces ([[https://trac.sagemath.org/ticket/30062|#30062]]): {{{ sage: E.<x,y> = EuclideanSpace() sage: p = E((1,0)) # the point of coordinates (1,0) sage: q = E((0,2)) # the point of coordinates (0,2) sage: d = E.dist # the distance function sage: d(p,q) sqrt(5) sage: p.dist(q) sqrt(5) sage: E.category() Join of Category of smooth manifolds over Real Field with 53 bits of precision and Category of complete metric spaces }}} === Bundle connections === Bundle connections have been improved ([[https://trac.sagemath.org/ticket/30208|#30208]]) and their action on vector fields and sections has been implemented ([[https://trac.sagemath.org/ticket/30209|#30209]]). === Internal code improvements and bug fixes === Many improvements/refactoring of the code have been performed in this release: * [[https://doc.sagemath.org/html/en/reference/manifolds/manifold.html|topological part]]: [[https://trac.sagemath.org/ticket/30266|#30266]], [[https://trac.sagemath.org/ticket/30267|#30267]], [[https://trac.sagemath.org/ticket/30291|#30291]] * [[https://doc.sagemath.org/html/en/reference/manifolds/diff_manifold.html|differentiable part]]: [[https://trac.sagemath.org/ticket/30228|#30228]], [[https://trac.sagemath.org/ticket/30274|#30274]], [[https://trac.sagemath.org/ticket/30280|#30280]], [[https://trac.sagemath.org/ticket/30285|#30285]], [[https://trac.sagemath.org/ticket/30288|#30288]] In addition, various bugs have been fixed: [[https://trac.sagemath.org/ticket/30108|#30108]], [[https://trac.sagemath.org/ticket/30112|#30112]], [[https://trac.sagemath.org/ticket/30191|#30191]], [[https://trac.sagemath.org/ticket/30289|#30289]], [[https://trac.sagemath.org/ticket/30401|#30401]]. == Algebra == === Lie Conformal Algebras === Implemented Lie conformal algebras and superalgebras. Here are some examples of their usage: {{{ sage: V = lie_conformal_algebras.Virasoro(QQ); V The Virasoro Lie conformal algebra over Rational Field sage: V.inject_variables() Defining L, C sage: L.bracket(L) {0: TL, 1: 2*L, 3: 1/2*C} sage: L.T(2).bracket(L) {2: 2*TL, 3: 12*L, 5: 10*C} sage: V = lie_conformal_algebras.NeveuSchwarz(QQ) sage: V.some_elements() [L, G, C, TG, TG + 4*T^(2)G, 4*T^(2)G] sage: W = lie_conformal_algebras.FreeFermions(QQbar, 2); W The free Fermions super Lie conformal algebra with generators (psi_0, psi_1, K) over Algebraic Field sage: W.inject_variables() Defining psi_0, psi_1, K sage: psi_0.bracket(psi_1.T()) {} sage: psi_0.bracket(psi_0.T()) {1: K} sage: psi_0.is_even_odd() 1 }}} For documentation on implemented features see [[https://doc.sagemath.org/html/en/reference/algebras/sage/algebras/lie_conformal_algebras/lie_conformal_algebra.html|Lie Conformal Algebra]]. For a list of implemented examples see [[https://doc.sagemath.org/html/en/reference/algebras/sage/algebras/lie_conformal_algebras/examples.html|Lie Conformal Algebra Examples]]. === Differential Weyl algebra === The action of differential operators from the Weyl algebra on polynomials has been implemented [[https://trac.sagemath.org/ticket/29928|#29928]]: {{{ sage: W.<x,y> = DifferentialWeylAlgebra(QQ) sage: dx, dy = W.differentials() sage: dx.diff(x^3) 3*x^2 sage: (x*dx + dy + 1).diff(x^4*y^4 + 1) 5*x^4*y^4 + 4*x^4*y^3 + 1 }}} == Improved Unicode support == === Unicode identifiers === Python 3 made much improved support for Unicode available, and Sage 9.2 has merged several Unicode improvements. Note that Python does not allow ''arbitrary'' Unicode characters in identifiers but only [[https://docs.python.org/3/reference/lexical_analysis.html#identifiers|word constituents]]. So before you get excited about using emojis... note that they cannot be used: {{{ #!python sage: K.<🍎,🥝> = QQ[] SyntaxError: invalid character in identifier }}} However, we can use letters from various alphabets. The updated IPython allows us to type them using [[https://ipython.readthedocs.io/en/stable/api/generated/IPython.core.completer.html|latex and unicode tab completion]]: {{{ #!python sage: μ, ν, ξ = 1, 2, 3 # type \mu<TAB>, # \nu<TAB>, ... sage: SR('λ + 2λ') 3*λ sage: var('α', domain='real') α sage: Ш = EllipticCurve('389a').sha() # type \CYR<TAB> CAP<TAB> # LET<TAB> SHA<TAB><ENTER> sage: Ш Tate-Shafarevich group for the Elliptic Curve defined by y^2 + y = x^3 + x^2 - 2*x over Rational Field sage: GelʹfandT͡setlinPattern = GelfandTsetlinPattern # type \MODIFIER LETTER # PRIME<TAB><ENTER> # for the romanized soft mark sage: ГельфандЦетлинPattern = GelʹfandT͡setlinPattern sage: ГельфандЦетлинPattern([[3, 2, 1], [2, 1], [1]]).pp() 3 2 1 2 1 1 sage: 四次方(x) = x^4 sage: 四次方(3) 81 }}} We can use math accents... {{{ #!python sage: a = 1 sage: â = 2 # type a\hat<TAB><ENTER> sage: ā = 3 # type a\bar<TAB><ENTER> sage: a, â, ā (1, 2, 3) sage: s(t) = t^3; s t |--> t^3 sage: ṡ = diff(s, t); ṡ # type s\dot<TAB><ENTER> t |--> 3*t^2 sage: s̈ = diff(ṡ, t); s̈ # type s\ddot<TAB><ENTER> t |--> 6*t }}} ... and have fun with modifier letters: {{{ #!python sage: ℚ̄ = QQbar # type \bbQ<TAB>\bar<TAB> sage: %display unicode_art sage: A = matrix(ℚ̄, [[1, 2*I], [3*I, 4]]); A ⎛ 1 2*I⎞ ⎝3*I 4⎠ sage: Aᵀ = A.transpose() # type A\^T<TAB><ENTER> sage: Aᵀ ⎛ 1 3*I⎞ ⎝2*I 4⎠ sage: Aᴴ = A.conjugate_transpose() # type A\^H<TAB><ENTER> sage: Aᴴ ⎛ 1 -3*I⎞ ⎝-2*I 4⎠ sage: C = Cone([[1, 1], [0, 1]]) sage: Cᵒ = C.dual(); Cᵒ # type C\^o<TAB><ENTER> 2-d cone in 2-d lattice M }}} But note that Python [[https://stackoverflow.com/questions/34097193/identifier-normalization-why-is-the-micro-sign-converted-into-the-greek-letter|normalizes identifiers]], so the following variants are ''not'' distinguished: {{{ #!python sage: AT == Aᵀ, AH == Aᴴ, Co == Cᵒ (True, True, True) sage: ℚ = QQ # type \bbQ<TAB><ENTER> sage: ℚ Rational Field sage: Q = 42 sage: ℚ 42 sage: F = 1 sage: 𝐹, 𝐅, 𝓕, 𝕱, 𝗙, 𝘍, 𝙁, 𝙵 # type \itF<TAB>, \bfF<TAB>, # \scrF<TAB>, \frakF<TAB>, # \sansF<TAB>, ... (1, 1, 1, 1, 1, 1, 1, 1) }}} We have also added a few Unicode aliases for global constants and functions. {{{ #!python sage: π pi sage: _.n() 3.14159265358979 sage: Γ(5/2) 3/4*sqrt(pi) sage: ζ(-1) -1/12 }}} See [[https://trac.sagemath.org/ticket/30111|Meta-ticket #30111: Unicode support]] for more information. === Unicode characters allowed in tensor index notation === Greek letters (and more generally any Unicode non-digit word-constituent character) are now allowed in index notation for tensors ([[https://trac.sagemath.org/ticket/29248|#29248]]). For instance, taking the trace of a type-(1,1) tensor field: {{{ sage: E.<x,y> = EuclideanSpace() sage: t = E.tensor_field(1, 1, [[x, 1], [0, y]]) sage: t['^μ_μ'] Scalar field on the Euclidean plane E^2 sage: t['^μ_μ'] == t.trace() True }}} === Unicode art === * [[https://trac.sagemath.org/ticket/30119|#30119]] Implemented a general function for writing integers as unicode sub/superscripts. * In [[https://trac.sagemath.org/ticket/29205|#29205]], some Lie algebra elements now have better unicode support: {{{ sage: L = LieAlgebra(QQ, cartan_type="A2", representation='matrix') sage: unicode_art(L.an_element()) ⎛ 1 1 0⎞ ⎜ 1 0 1⎟ ⎝ 0 1 -1⎠ sage: L = lie_algebras.Heisenberg(QQ, 2) sage: unicode_art(sum(L.basis())) p₁ + p₂ + q₁ + q₂ + z sage: L = lie_algebras.VirasoroAlgebra(QQ) sage: unicode_art(L.an_element()) d₋₁ + d₀ - 1/2 + c sage: L = LieAlgebra(QQ, cartan_type=['A',1,1]) sage: unicode_art(L.an_element()) ( alpha[1] + alphacheck[1] + -alpha[1] )⊗t⁰ + ( -alpha[1] )⊗t¹ + ( alpha[1] )⊗t⁻¹ + c + d sage: L.<x,y> = LieAlgebra(QQ) sage: unicode_art(x.bracket(y)) [x, y] sage: L = LieAlgebra(QQ, cartan_type=['A',2], representation="compact real") sage: unicode_art(L.an_element()) ⎛ i i + 1 i + 1⎞ ⎜i - 1 i i + 1⎟ ⎝i - 1 i - 1 -2*i⎠ }}} * As part of [[https://trac.sagemath.org/ticket/29696|#29696]], Temperley-Lieb diagrams now have unicode (and ascii) art: {{{ sage: from sage.combinat.diagram_algebras import TL_diagram_ascii_art sage: 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)] sage: TL_diagram_ascii_art(TL, use_unicode=False) o o o o o o o o o o o o o o o | `-` `-` | `-` `-` | `-` | | | `---------` | | | .-------` | `---. | .-------` | .-----. | | .-----. .-. | .-. | .-. | | | | .-. | o o o o o o o o o o o o o o o sage: TL_diagram_ascii_art(TL, use_unicode=True) ⚬ ⚬ ⚬ ⚬ ⚬ ⚬ ⚬ ⚬ ⚬ ⚬ ⚬ ⚬ ⚬ ⚬ ⚬ │ ╰─╯ ╰─╯ │ ╰─╯ ╰─╯ │ ╰─╯ │ │ │ ╰─────────╯ │ │ │ ╭───────╯ │ ╰───╮ │ ╭───────╯ │ ╭─────╮ │ │ ╭─────╮ ╭─╮ │ ╭─╮ │ ╭─╮ │ │ │ │ ╭─╮ │ ⚬ ⚬ ⚬ ⚬ ⚬ ⚬ ⚬ ⚬ ⚬ ⚬ ⚬ ⚬ ⚬ ⚬ ⚬ }}} == Configuration and build changes == === Initial configuration with ./configure required === Sage 9.1 introduced [[https://wiki.sagemath.org/ReleaseTours/sage-9.1#Portability_improvements.2C_increased_use_of_system_packages|informational messages at the end of a ./configure run]] regarding system packages. To make sure that these messages are not overlooked, Sage 9.2 no longer invokes `./configure` when you type `make` in an unconfigured source tree. See [[https://groups.google.com/d/msg/sage-devel/9gOkmF6rSjY/wEV4WBQABwAJ|sage-devel: require "./configure" before "make"]], [[https://trac.sagemath.org/ticket/29316|#29316]]. === Support for gcc/gfortran 10.x added === All standard Sage packages have been upgraded in Sage 9.2 so that they build correctly using gcc/gfortran 10.x. The Sage `./configure` script therefore now accepts these compiler versions. === System package information for more distributions === System package information has been added for [[https://www.gentoo.org/|Gentoo Linux]], [[https://www.freebsd.org/|FreeBSD]], [[https://voidlinux.org/|Void Linux]], and [[https://nixos.org/|NixOS]]. === For developers: Changes to the build system of sagelib === Let's talk about `src/setup.py`. The build system of the Sage library is based on `distutils` (not `setuptools`); it is implemented in the package `sage_setup`. In particular, it implements its own version of source code discovery methods similar to [[https://setuptools.readthedocs.io/en/latest/setuptools.html#using-find-packages|setuptools.find_packages]]: `sage_setup.find.find_python_sources`. Because of source discovery, developers can add new Python modules and packages under `src/sage/` simply by creating files and directories; it is not necessary to edit `setup.py`. Prior to Sage 9.2, the situation was different for Cython extensions. They had to be listed in `src/module_list.py`, either one by one, or using glob patterns such as `*` and `**`. Sage 9.2 has eliminated the need for `src/module_list.py` by extending `sage_setup.find.find_python_sources`; it now also finds Cython modules in the source tree (Trac [[https://trac.sagemath.org/ticket/29701|#29701]]). Some Cython modules need specific compiler and linker flags. Sage 9.2 has moved all of these flags from `Extension` options in `src/module_list.py` to `distutils:` directives in the individual `.pyx` source files, see [[https://trac.sagemath.org/ticket/29706|#29706]] and [[https://cython.readthedocs.io/en/latest/src/userguide/source_files_and_compilation.html#compiler-directives|Cython documentation]]. Sage 9.2 has also changed the mechanism for conditionalizing a Cython extension module on the presence of a Sage package. Consider the module [[https://git.sagemath.org/sage.git/tree/src/sage/graphs/graph_decompositions/tdlib.pyx?id=55c3fbc565fd7884f3df9555de83dd326ace276e|sage.graphs.graph_decompositions.tdlib]] as an example. Prior to Sage 9.2, this module was declared as an `OptionalExtension`, conditional on the SPKG `tdlib`, in `src/module_list.py`. The new mechanism is as follows. [[https://git.sagemath.org/sage.git/tree/src/setup.py?id=55c3fbc565fd7884f3df9555de83dd326ace276e#n53|src/setup.py]] maps the SPKG name `tdlib` to the "distribution name" `sage-tdlib`. At the top of the Cython source file [[https://git.sagemath.org/sage.git/tree/src/sage/graphs/graph_decompositions/tdlib.pyx?id=55c3fbc565fd7884f3df9555de83dd326ace276e|src/sage/graphs/graph_decompositions/tdlib.pyx]], there is a new directive `sage_setup: distribution = sage-tdlib`. Now the source discovery in [[https://git.sagemath.org/sage.git/tree/src/sage_setup/find.py?id=55c3fbc565fd7884f3df9555de83dd326ace276e#n61|sage_setup.find.find_python_sources]] includes this Cython module only if the SPKG `tdlib` is installed and current. == Cleaning == * [[https://trac.sagemath.org/ticket/29636|#29636: Delete changelog sections from all SPKG information files]]; they were deprecated in favor of using Trac years ago. The contributions of Sage developers maintaining SPKGs are documented by our [[http://www.sagemath.org/changelogs/index.html|historical changelogs]]. * Removing support for Python 2 allowed us to remove several backport packages in [[https://trac.sagemath.org/ticket/29754|#29754]] * We also removed the deprecated SageNB (superseded a long time ago by the Jupyter notebook) in [[https://trac.sagemath.org/ticket/29754|#29754]] and several of its dependencies. For converting old Sage worksheet files (*.sws), the script `sage -sws2rst` is available. (In Sage 9.0 and 9.1, it was available only in Python 2 builds of Sage; in [[https://trac.sagemath.org/ticket/28838|#28838]], it was ported to Python 3.) * Support for installing "old-style Sage packages" (`.spkg` files), [[https://trac.sagemath.org/ticket/19158|deprecated in Sage 6.9]], has been removed in [[https://trac.sagemath.org/ticket/29289|#29289]], after making the last two missing packages, `cunningham_tables` and `polytopes_db_4d`, available as normal optional Sage packages. Users who wish to package their own Sage code for distribution may find a [[https://wiki.sagemath.org/SageMathExternalPackages|list of external packages]] helpful, many of which follow best practices in packaging. == Availability of Sage 9.2 and installation help == Sage 9.2 has not been released yet. See [[https://groups.google.com/forum/#!forum/sage-release|sage-release]] for announcements of beta versions and release candidates. * See [[https://groups.google.com/forum/#!forum/sage-devel|sage-devel]] for development discussions. == More details == * [[https://trac.sagemath.org/query?status=needs_info&status=needs_review&status=needs_work&status=new&summary=~Meta&col=id&col=summary&col=status&col=type&col=priority&col=milestone&col=component&order=priority|Open Meta-Tickets]] * [[https://trac.sagemath.org/query?milestone=sage-9.2&groupdesc=1&group=status&max=1500&col=id&col=summary&col=author&col=reviewer&col=time&col=changetime&col=component&col=keywords&order=component|Trac tickets with milestone 9.2]] |
