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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. '''Sage 9.2 has removed support for Python 2.''' * [[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 === Support for system Python 3.6 added === 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. === 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 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. === 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. == 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 3.2.1. 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]]. 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|Trac #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 === 1.8.5 -> 3.1.2 === IPython and Jupyter notebook === 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|Trac #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|Trac #26919]]; see the [[https://jupyter-notebook.readthedocs.io/en/stable/changelog.html|notebook changelog]] for more information. === 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|Trac #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.) == 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|Trac #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|Trac #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] }}} === 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]]. 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}} == 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)) }}} == 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 }}} === Unicode characters allowed in index notations === 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 }}} === 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]]. == Configuration and build changes == 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|Trac #29316]]. 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. === 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|Trac #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|Trac #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|Trac #29754]] * We also removed the deprecated SageNB (superseded a long time ago by the Jupyter notebook) in [[https://trac.sagemath.org/ticket/29754|Trac #29754]] and several of its dependencies. * 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|Trac #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]] |
