diff --git a/experimental/BasisLieHighestWeight/src/UserFunctions.jl b/experimental/BasisLieHighestWeight/src/UserFunctions.jl index 1e2db62f0eb3..044ee7cdc4ff 100644 --- a/experimental/BasisLieHighestWeight/src/UserFunctions.jl +++ b/experimental/BasisLieHighestWeight/src/UserFunctions.jl @@ -6,7 +6,7 @@ together with their index. Operators $f_\alpha$ of negative roots are shown as the coefficients of the corresponding positive root. w.r.t. the simple roots $\alpha_i$. -# Example +# Examples ```jldoctest julia> basis_lie_highest_weight_operators(:B, 2) 4-element Vector{Tuple{Int64, Vector{QQFieldElem}}}: @@ -407,7 +407,7 @@ A birational sequence of type `Vector{Vector{Int}}` is a sequence of weights in `monomial_ordering` describes the monomial ordering used for the basis. If this is a weighted ordering, the height of the corresponding root is used as weight. -# Example +# Examples ```jldoctest julia> bases = basis_coordinate_ring_kodaira(:G, 2, [1,0], 6; monomial_ordering = :invlex) 6-element Vector{Tuple{MonomialBasis, Vector{ZZMPolyRingElem}}}: @@ -510,7 +510,7 @@ The the birational sequence used consists of all operators in descening height o The monomial ordering is fixed to `degrevlex`. -# Example +# Examples ```jldoctest julia> bases = basis_coordinate_ring_kodaira_ffl(:G, 2, [1,0], 6) 6-element Vector{Tuple{MonomialBasis, Vector{ZZMPolyRingElem}}}: diff --git a/experimental/LieAlgebras/src/LieAlgebra.jl b/experimental/LieAlgebras/src/LieAlgebra.jl index f54cc983f6fa..50f4fdf09476 100644 --- a/experimental/LieAlgebras/src/LieAlgebra.jl +++ b/experimental/LieAlgebras/src/LieAlgebra.jl @@ -854,7 +854,7 @@ Return the abelian Lie algebra of dimension `n` over the field `R`. The first argument can be optionally provided to specify the type of the returned Lie algebra. -# Example +# Examples ```jldoctest julia> abelian_lie_algebra(LinearLieAlgebra, QQ, 3) Linear Lie algebra with 3x3 matrices diff --git a/experimental/LieAlgebras/src/LieAlgebraModule.jl b/experimental/LieAlgebras/src/LieAlgebraModule.jl index 76ece8a7b749..a91e6b99998a 100644 --- a/experimental/LieAlgebras/src/LieAlgebraModule.jl +++ b/experimental/LieAlgebras/src/LieAlgebraModule.jl @@ -1412,7 +1412,7 @@ Compute the dimension of the simple module of the Lie algebra `L` with highest w using Weyl's dimension formula. The return value is of type `T`. -# Example +# Examples ```jldoctest julia> L = lie_algebra(QQ, :A, 3); @@ -1445,7 +1445,7 @@ sorted ascendingly by the total height of roots needed to reach them from `hw`. See [MP82](@cite) for details and the implemented algorithm. -# Example +# Examples ```jldoctest julia> L = lie_algebra(QQ, :B, 3); @@ -1477,7 +1477,7 @@ together with their multiplicities. This function uses an optimized version of the Freudenthal formula, see [MP82](@cite) for details. -# Example +# Examples ```jldoctest julia> L = lie_algebra(QQ, :A, 3); @@ -1513,7 +1513,7 @@ Computes all weights occurring in the simple module of the Lie algebra `L` with together with their multiplicities. This is achieved by acting with the Weyl group on the [`dominant_character`](@ref dominant_character(::LieAlgebra, ::Vector{<:IntegerUnion})). -# Example +# Examples ```jldoctest julia> L = lie_algebra(QQ, :A, 3); @@ -1557,7 +1557,7 @@ This function uses Klimyk's formula (see [Hum72; Exercise 24.9](@cite)). The return type may change in the future. -# Example +# Examples ```jldoctest julia> L = lie_algebra(QQ, :A, 2); @@ -1599,7 +1599,7 @@ with extremal weight `x*w`, together with their multiplicities. Instead of a Weyl group element `x`, a reduced expression for `x` can be supplied. This function may return arbitrary results if the provided expression is not reduced. -# Example +# Examples ```jldoctest julia> L = lie_algebra(QQ, :A, 2); diff --git a/experimental/LieAlgebras/src/RootSystem.jl b/experimental/LieAlgebras/src/RootSystem.jl index d304d93bc3ba..a206eb82f0d9 100644 --- a/experimental/LieAlgebras/src/RootSystem.jl +++ b/experimental/LieAlgebras/src/RootSystem.jl @@ -1468,7 +1468,7 @@ Compute the dimension of the simple module of the Lie algebra defined by the roo with highest weight `hw` using Weyl's dimension formula. The return value is of type `T`. -# Example +# Examples ```jldoctest julia> R = root_system(:B, 2); @@ -1512,7 +1512,7 @@ sorted ascendingly by the total height of roots needed to reach them from `hw`. See [MP82](@cite) for details and the implemented algorithm. -# Example +# Examples ```jldoctest julia> R = root_system(:B, 3); @@ -1575,7 +1575,7 @@ with highest weight `hw`, together with their multiplicities. This function uses an optimized version of the Freudenthal formula, see [MP82](@cite) for details. -# Example +# Examples ```jldoctest julia> R = root_system(:B, 3); @@ -1667,7 +1667,7 @@ Computes all weights occurring in the simple module of the Lie algebra defined b with highest weight `hw`, together with their multiplicities. This is achieved by acting with the Weyl group on the [`dominant_character`](@ref dominant_character(::RootSystem, ::WeightLatticeElem)). -# Example +# Examples ```jldoctest julia> R = root_system(:B, 3); @@ -1720,7 +1720,7 @@ This function uses Klymik's formula. The return type may change in the future. -# Example +# Examples ```jldoctest julia> R = root_system(:B, 2); @@ -1845,7 +1845,7 @@ with extremal weight `x*w`, together with their multiplicities. Instead of a Weyl group element `x`, a reduced expression for `x` can be supplied. This function may return arbitrary results if the provided expression is not reduced. -# Example +# Examples ```jldoctest julia> R = root_system(:B, 3); diff --git a/experimental/MatrixGroups/src/MatrixGroups.jl b/experimental/MatrixGroups/src/MatrixGroups.jl index 4c34885c092f..a87a7f0b6abb 100644 --- a/experimental/MatrixGroups/src/MatrixGroups.jl +++ b/experimental/MatrixGroups/src/MatrixGroups.jl @@ -19,7 +19,7 @@ end Compute the JuliaMatrixRep of `m` in GAP. -# Example +# Examples ```jldoctest julia> m = matrix(ZZ, [0 1 ; -1 0]); @@ -43,7 +43,7 @@ A nice monomorphism from `G` to a GAP matrix group `G2` over a finite field is stored in `G`, such that calculations in `G` can be handled automatically by transferring them to `G2`. -# Example +# Examples ```jldoctest julia> m1 = matrix(QQ, [0 1 ; -1 0]); diff --git a/experimental/Schemes/src/Resolution_structure.jl b/experimental/Schemes/src/Resolution_structure.jl index f376766a36a4..add4991b964a 100644 --- a/experimental/Schemes/src/Resolution_structure.jl +++ b/experimental/Schemes/src/Resolution_structure.jl @@ -55,7 +55,7 @@ end Return a `CartierDivisor` on the `domain` of `f` which is the exceptional divisor of the sequence of blow-ups `f`. -# Example +# Examples ```jldoctest julia> R,(x,y) = polynomial_ring(QQ,2); @@ -90,7 +90,7 @@ end Return a `WeilDivisor` on the `domain` of `f` which is the exceptional divisor of the sequence of blow-ups `f`. -# Example +# Examples ```jldoctest julia> R,(x,y) = polynomial_ring(QQ,2); diff --git a/experimental/Schemes/src/Resolution_tools.jl b/experimental/Schemes/src/Resolution_tools.jl index 7b0e66e6eb6a..5c594d410de3 100644 --- a/experimental/Schemes/src/Resolution_tools.jl +++ b/experimental/Schemes/src/Resolution_tools.jl @@ -19,7 +19,7 @@ Return a tuple `M`, `v`, `M2`, `m` where !!! note The intersection matrix referred to in textbooks is `M2`, as these usually restrict to the case of algebraically closed fields, but computations are usually performed over suitable subfields, e.g. `QQ` instead of `CC`. -# Example +# Examples ```jldoctest julia> R,(x,y,z) = polynomial_ring(QQ,3); diff --git a/src/AlgebraicGeometry/Schemes/CoveredProjectiveScheme/CoveredProjectiveScheme.jl b/src/AlgebraicGeometry/Schemes/CoveredProjectiveScheme/CoveredProjectiveScheme.jl index 9cc120243fc0..24e4e3c7b5d6 100644 --- a/src/AlgebraicGeometry/Schemes/CoveredProjectiveScheme/CoveredProjectiveScheme.jl +++ b/src/AlgebraicGeometry/Schemes/CoveredProjectiveScheme/CoveredProjectiveScheme.jl @@ -132,7 +132,7 @@ end Given a ring `R`, return the empty relative projective scheme over the empty covered scheme over `R`. -# Example +# Examples ```jldoctest julia> R, (x,y,z) = QQ[:x, :y, :z]; diff --git a/src/AlgebraicGeometry/Schemes/Divisors/WeilDivisor.jl b/src/AlgebraicGeometry/Schemes/Divisors/WeilDivisor.jl index 1f0138973197..601983acca84 100644 --- a/src/AlgebraicGeometry/Schemes/Divisors/WeilDivisor.jl +++ b/src/AlgebraicGeometry/Schemes/Divisors/WeilDivisor.jl @@ -67,7 +67,7 @@ end Given an ideal sheaf `I` of pure codimension ``1``, return the weil divisor $D = 1 ⋅ I$ with coefficients in the integer ring. -# Example +# Examples ```jldoctest julia> P, (x, y, z) = graded_polynomial_ring(QQ, [:x, :y, :z]); diff --git a/src/AlgebraicGeometry/Schemes/ProjectiveSchemes/Objects/Attributes.jl b/src/AlgebraicGeometry/Schemes/ProjectiveSchemes/Objects/Attributes.jl index affb0dd306ba..230e1ef06252 100644 --- a/src/AlgebraicGeometry/Schemes/ProjectiveSchemes/Objects/Attributes.jl +++ b/src/AlgebraicGeometry/Schemes/ProjectiveSchemes/Objects/Attributes.jl @@ -24,7 +24,7 @@ base_scheme(P::AbsProjectiveScheme) =base_scheme(underlying_scheme(P)) On a projective scheme ``P = Proj(S)`` for a standard graded finitely generated algebra ``S`` this returns ``S``. -# Example +# Examples ```jldoctest julia> S, _ = grade(QQ[:x, :y, :z][1]); @@ -52,7 +52,7 @@ homogeneous_coordinate_ring(P::AbsProjectiveScheme) = homogeneous_coordinate_rin On ``X ⊂ ℙʳ_A`` this returns ``r``. -# Example +# Examples ```jldoctest julia> S, _ = grade(QQ[:x, :y, :z][1]); @@ -89,7 +89,7 @@ On a projective scheme ``P = Proj(S)`` with ``S = P/I`` for a standard graded polynomial ring ``P`` and a homogeneous ideal ``I`` this returns ``P``. -# Example +# Examples ```jldoctest julia> S, _ = grade(QQ[:x, :y, :z][1]) (Graded multivariate polynomial ring in 3 variables over QQ, MPolyDecRingElem{QQFieldElem, QQMPolyRingElem}[x, y, z]) @@ -133,7 +133,7 @@ end On ``X ⊂ ℙʳ_A`` this returns ``ℙʳ_A``. -# Example +# Examples ```jldoctest julia> S, _ = grade(QQ[:x, :y, :z][1]); @@ -229,7 +229,7 @@ end On ``X ⊂ ℙʳ_A`` this returns the homogeneous ideal ``I ⊂ A[s₀,…,sᵣ]`` defining ``X``. -# Example +# Examples ```jldoctest julia> R, (u, v) = QQ[:u, :v]; @@ -265,7 +265,7 @@ from the `homogeneous_coordinate_ring` to the `coordinate_ring` of the affine co Note that if the base scheme is not affine, then the affine cone is not affine. -# Example +# Examples ```jldoctest julia> R, (u, v) = QQ[:u, :v]; @@ -388,7 +388,7 @@ On ``X ⊂ ℙʳ_A`` this returns a vector with the homogeneous coordinates ``[s₀,…,sᵣ]`` as entries where each one of the ``sᵢ`` is a function on the `affine cone` of ``X``. -# Example +# Examples ```jldoctest julia> R, (u, v) = QQ[:u, :v]; diff --git a/src/AlgebraicGeometry/Surfaces/duValSing.jl b/src/AlgebraicGeometry/Surfaces/duValSing.jl index ac05374d06a9..2701295df1b8 100644 --- a/src/AlgebraicGeometry/Surfaces/duValSing.jl +++ b/src/AlgebraicGeometry/Surfaces/duValSing.jl @@ -3,7 +3,7 @@ Return whether the given ``X`` has at most du Val (surface) singularities. -# Example: +# Examples ```jldoctest julia> R,(x,y,z,w) = QQ[:x, :y, :z, :w] (Multivariate polynomial ring in 4 variables over QQ, QQMPolyRingElem[x, y, z, w]) @@ -67,7 +67,7 @@ Return whether the given ``X`` has at most du Val (surface) singularities at the **Note**: For the ideal ``I`` in a ring ``R``, `dim(R/I) = 0` is asserted -# Example: +# Examples ```jldoctest julia> R,(x,y,z,w) = QQ[:x, :y, :z, :w] (Multivariate polynomial ring in 4 variables over QQ, QQMPolyRingElem[x, y, z, w]) @@ -147,7 +147,7 @@ If ``X`` has a least one singularity which is not du Val, the returned vector co **Note**: For the ideal ``I`` in a ring ``R``, `dim(R/I) = 0` is asserted -# Example: +# Examples ```jldoctest julia> R,(x,y,z,w) = QQ[:x, :y, :z, :w] (Multivariate polynomial ring in 4 variables over QQ, QQMPolyRingElem[x, y, z, w]) diff --git a/src/Combinatorics/EnumerativeCombinatorics/partitions.jl b/src/Combinatorics/EnumerativeCombinatorics/partitions.jl index 167a46a32e99..5b2300393463 100644 --- a/src/Combinatorics/EnumerativeCombinatorics/partitions.jl +++ b/src/Combinatorics/EnumerativeCombinatorics/partitions.jl @@ -489,7 +489,7 @@ order. The implemented algorithm is "partb" in [RJ76](@cite). -# Example +# Examples The number of partitions of 100 where the parts are from {1, 2, 5, 10, 20, 50}: ```jldoctest julia> length(collect(partitions(100, [1, 2, 5, 10, 20, 50]))) diff --git a/src/Combinatorics/Matroids/quantum_automorphism_groups.jl b/src/Combinatorics/Matroids/quantum_automorphism_groups.jl index 08d635904bd1..08aadb3d513c 100644 --- a/src/Combinatorics/Matroids/quantum_automorphism_groups.jl +++ b/src/Combinatorics/Matroids/quantum_automorphism_groups.jl @@ -10,7 +10,7 @@ The relations are: - idempotent relations: `n^2` relations - relations of type `u[i,j]*u[i,k]` and `u[j,i]*u[k,i]` for `k != j`: `2*n*n*(n-1)` relations -# Example +# Examples ```jldoctest julia> S4 = quantum_symmetric_group(4); diff --git a/src/PolyhedralGeometry/Cone/properties.jl b/src/PolyhedralGeometry/Cone/properties.jl index 6e56e125921c..3b0ce6a5e991 100644 --- a/src/PolyhedralGeometry/Cone/properties.jl +++ b/src/PolyhedralGeometry/Cone/properties.jl @@ -379,7 +379,7 @@ lineality_dim(C::Cone) = pm_object(C).LINEALITY_DIM::Int Facet degrees of the cone. The degree of a facet is the number of adjacent facets. In particular a general $2$-dimensional cone has two facets (rays) that meet at the origin. -# Example +# Examples Produce the facet degrees of a cone over a square and a cone over a square pyramid. ```jldoctest julia> c = positive_hull([1 1 0; 1 -1 0; 1 0 1; 1 0 -1]) diff --git a/src/PolyhedralGeometry/Polyhedron/properties.jl b/src/PolyhedralGeometry/Polyhedron/properties.jl index c400ea1bce88..3b9ae0fd1def 100644 --- a/src/PolyhedralGeometry/Polyhedron/properties.jl +++ b/src/PolyhedralGeometry/Polyhedron/properties.jl @@ -900,7 +900,7 @@ codim(P::Polyhedron) = ambient_dim(P) - dim(P) Number of vertices in each facet. -# Example +# Examples ```jldoctest julia> p = johnson_solid(4) Polytope in ambient dimension 3 with EmbeddedAbsSimpleNumFieldElem type coefficients @@ -929,7 +929,7 @@ end Number of incident facets for each vertex. -# Example +# Examples ```jldoctest julia> vertex_sizes(bipyramid(simplex(2))) 5-element Vector{Int64}: diff --git a/src/PolyhedralGeometry/Polyhedron/standard_constructions.jl b/src/PolyhedralGeometry/Polyhedron/standard_constructions.jl index 29e51665b074..99a1c3e228b3 100644 --- a/src/PolyhedralGeometry/Polyhedron/standard_constructions.jl +++ b/src/PolyhedralGeometry/Polyhedron/standard_constructions.jl @@ -1353,7 +1353,7 @@ $(n+1)$-space. SIM-bodies are defined in [GK14](@cite), but the input needs to be descending instead of ascending, as used in [JKS22](@cite), i.e. `alpha` has parameters $(a_1,\dots,a_n)$ such that $a_1 \geq \dots \geq a_n \geq 0$. -# Example +# Examples To produce a $2$-dimensional SIM-body, use for example the following code. Note that the polytope lives in $3$-space, so we project it down to $2$-space by eliminating the last coordinate. @@ -1392,7 +1392,7 @@ We use the facet description given in section 9.2. of [Zie95](@cite). Note that in polymake, this function has an optional Boolean parameter `group`, to also construct the symmetry group of the polytope. For details, see [CSZ15](@cite). -# Example +# Examples Produce the $2$-dimensional associahedron is a polygon in $\mathbb{R}⁴$ having $5$ vertices and $5$ facets. ```jldoctest @@ -1455,7 +1455,7 @@ binary_markov_graph_polytope(observation::AbstractVector{<:Base.Integer}) = Produce the $d$-dimensional dwarfed cube as defined in [ABS97](@cite). -# Example +# Examples The $3$-dimensional dwarfed cube is illustrated in [Jos03](@cite). ```jldoctest @@ -1487,7 +1487,7 @@ end Produce a $d$-dimensional dwarfed product of polygons of size $s$ as defined in [ABS97](@cite). It must be $d\geq4$ and even as well as $s\geq 3$. -# Example +# Examples ```jldoctest julia> p = dwarfed_product_polygons(4,3) Polytope in ambient dimension 4 @@ -1522,7 +1522,7 @@ as defined in [SS12](@cite). Note that in polymake, this function has an optional Boolean parameter `group`, to also construct the symmetry group of the simplex. -# Example +# Examples The $3$-dimensional lecture hall simplex: ```jldoctest julia> S = lecture_hall_simplex(3) @@ -1576,7 +1576,7 @@ Produce a $d$-dimensional cyclic polytope with $n$ points. Clearly $n\geq d$ is It is a prototypical example of a neighborly polytope whose combinatorics completely known due to Gale's evenness criterion. The coordinates are chosen on the trigonometric moment curve. -# Example +# Examples ```jldoctest julia> C= cyclic_caratheodory_polytope(4,5) Polytope in ambient dimension 4 @@ -1628,7 +1628,7 @@ Produce the hypersimplex $\Delta(k,d)$, that is the the convex hull of all $0/1$ - `no_facets::Bool`: If set equal to `true`, facets of the underlying `polymake` object are not computed. - `no_vif::Bool`: If set equal to `true`, vertices in facets of the underlying `polymake` object are not computed. -# Example +# Examples ```jldoctest julia> H = hypersimplex(3,4) Polytope in ambient dimension 4 @@ -1714,7 +1714,7 @@ Here we use the description as a deformed product due to [AZ99](@cite). For $g=0$ we obtain a Klee-Minty cube, in particular for $e=g=0$ we obtain the standard cube. -# Example +# Examples The following produces a $3$-dimensional Klee-Minty cube for $e=\frac{1}{3}$. ```jldoctest julia> c = goldfarb_cube(3,1//3,0) @@ -1785,7 +1785,7 @@ Produce a $d$-dimensional hypertruncated cube with symmetric linear objective fu - `k`: cutoff parameter - `lambda`: scaling of extra vertex -# Example +# Examples ```jldoctest julia> H = hypertruncated_cube(3,2,3) Polytope in ambient dimension 3 @@ -1823,7 +1823,7 @@ Warning: Some of the $k-$cyclic polytopes are not simplicial. Since the components are rounded, this function might output a polytope which is not a $k-$cyclic polytope! More information see [Sch95](@cite). -# Example +# Examples To produce a (not exactly) regular pentagon, type this: ```jldoctest julia> p = k_cyclic_polytope(5,[1]) @@ -1873,7 +1873,7 @@ Produce a `d`-dimensional polytope of maximal Gomory-Chvatal rank $\Omega(d/\log integrally infeasible. With symmetric linear objective function $(1,1..,1)$. Construction due to Pokutta and Schulz, see [PS11](@cite). -# Example +# Examples ```jldoctest julia> c = max_GC_rank_polytope(3) Polytope in ambient dimension 3 @@ -1936,7 +1936,7 @@ end Create an $8$-dimensional polytope without rational realizations due to Perles. See [Gru03](@cite). -# Example +# Examples ```jldoctest julia> perles_nonrational_8_polytope() Polytope in ambient dimension 8 with EmbeddedAbsSimpleNumFieldElem type coefficients @@ -2013,7 +2013,7 @@ pitman_stanley_polytope(y::AbstractVector{<:IntegerUnion}) = pitman_stanley_poly Produce a `d`-dimensional del-Pezzo polytope, which is the convex hull of the cross polytope together with the all-ones vector. All coordinates are plus or minus one. -# Example +# Examples ```jldoctest julia> DP = pseudo_del_pezzo_polytope(4) Polytope in ambient dimension 4 @@ -2039,7 +2039,7 @@ Produce a `d`-dimensional $0/1$-polytope with `n` random vertices. Uniform distr # Optional Argument -`seed::Int`: Seed for random number generation -# Example +# Examples ```jldoctest julia> s = rand01_polytope(2, 4; seed=3) Polytope in ambient dimension 2 @@ -2074,7 +2074,7 @@ points in the cube $[0,\texttt{b}]^{\texttt{d}}$. # Optional Argument -`seed`: Seed for random number generation. -# Example +# Examples ```jldoctest julia> r = rand_box_polytope(3, 10, 3, seed=1) Polyhedron in ambient dimension 3 @@ -2192,7 +2192,7 @@ normally distributed in the unit ball. -`seed`: controls the outcome of the random number generator; fixing a seed number guarantees the same outcome -`precision`: number of bits for MPFR sphere approximation -# Example +# Examples ```jldoctest julia> rnp = rand_normal_polytope(2,4; seed=42, precision=4) Polytope in ambient dimension 2 @@ -2421,7 +2421,7 @@ Construct the vertex figure of the vertex `n` of a bounded polytope. The vertex Value $0$ would let the hyperplane go through the chosen vertex, thus degenerating the vertex figure to a single point. Value $1$ would let the hyperplane touch the nearest neighbor vertex of a polyhedron. Default value is $\frac{1}{2}$. -# Example +# Examples To produce a triangular vertex figure of a $3$-dimensional cube in the positive orthant, do: ```jldoctest julia> T = vertex_figure(cube(3), 8) diff --git a/src/Serialization/Upgrades/main.jl b/src/Serialization/Upgrades/main.jl index d1f87eafa96b..28a905ca1d5f 100644 --- a/src/Serialization/Upgrades/main.jl +++ b/src/Serialization/Upgrades/main.jl @@ -18,7 +18,7 @@ from any Oscar code and should rely solely on `julia` core functionality so to avoid conflicts with any future Oscar code deprecations. -# Example +# Examples ``` push!(upgrade_scripts_set, UpgradeScript( v"0.13.0", diff --git a/src/TropicalGeometry/groebner_basis.jl b/src/TropicalGeometry/groebner_basis.jl index 743ff3595f35..a465ad4c62c9 100644 --- a/src/TropicalGeometry/groebner_basis.jl +++ b/src/TropicalGeometry/groebner_basis.jl @@ -209,7 +209,7 @@ end Return an integer vector `wSim` so that the (tropical) Groebner basis of an ideal `I` with respect to `w` corresponds to the standard basis of its simulation with respect to `wSim`. If `perturbation!=nothing`, also returns a corresponding `perturbationSim`. -# Example +# Examples ```jldoctest julia> nuMin = tropical_semiring_map(QQ,2); @@ -305,7 +305,7 @@ end Given a weight vector `wSim` on the simulation ring, return weight vector `w` on the original polynomial ring so that a standard basis with respect to `wSim` corresponds to a (tropical) Groebner basis with respect to `w`. -# Example +# Examples ```jldoctest julia> nuMin = tropical_semiring_map(QQ,2); diff --git a/src/TropicalGeometry/semiring_map.jl b/src/TropicalGeometry/semiring_map.jl index da27e5449635..bfa732e1548c 100644 --- a/src/TropicalGeometry/semiring_map.jl +++ b/src/TropicalGeometry/semiring_map.jl @@ -59,7 +59,7 @@ polynomial_rings_for_initial(nu::TropicalSemiringMap) = nu.polynomial_rings_for_ Return a map `nu` from `K` to the min (default) or max tropical semiring `T` such that `nu(0)=zero(T)` and `nu(c)=one(T)` for `c` non-zero. In other words, `nu` extends the trivial valuation on `K`. -# Example +# Examples ```jldoctest julia> nu = tropical_semiring_map(QQ) # arbitrary rings possible Map into Min tropical semiring encoding the trivial valuation on Rational field @@ -122,7 +122,7 @@ end Return a map `nu` from `QQ` to the min (default) or max tropical semiring `T` such that `nu(0)=zero(T)` and `nu(c)=+/-val(c)` for `c` non-zero, where `val` denotes the `p`-adic valuation. Requires `p` to be a prime. -# Example +# Examples ```jldoctest julia> nu_2 = tropical_semiring_map(QQ,2) Map into Min tropical semiring encoding the 2-adic valuation on Rational field @@ -192,7 +192,7 @@ end Return a map `nu` from rational function field `Kt` to the min (default) or max tropical semiring `T` such that `nu(0)=zero(T)` and `nu(c)=+/-val(c)` for `c` non-zero, where `val` denotes the `t`-adic valuation with uniformizer `t`. Requires `t` to be non-constant and have denominator `1`. -# Example +# Examples ```jldoctest julia> Kt,t = rational_function_field(QQ,"t"); diff --git a/src/utils/docs.jl b/src/utils/docs.jl index 612f6724be39..1061acb32a25 100644 --- a/src/utils/docs.jl +++ b/src/utils/docs.jl @@ -60,7 +60,7 @@ end Fixes all doctests for the given function `f`. -# Example +# Examples The following call fixes all doctests for the function `symmetric_group`: ```julia julia> Oscar.doctest_fix(symmetric_group) @@ -89,7 +89,7 @@ end Fixes all doctests for all files in Oscar where `path` occurs in the full pathname. -# Example +# Examples The following call fixes all doctests in files that live in a directory called `Rings` (or a subdirectory thereof), so e.g. everything in `src/Rings/`: ```julia