Alphabeta MathArticle items · 04 — Stereographic projection and its inverse are explicit in coordinates
Presents 27 items from the article and its immediate prerequisites, covering published library prerequisites.
1 · Prerequisites
- Stereographic projection identifies the Riemann sphere with the unit two-sphere
- Artin monoid and Artin group presentations, and the canonical monoid-to-group map
- Coxeter matrices, the presented Coxeter group, reduced words, length, and standard parabolic subgroups
- Monoid homomorphism and group homomorphism
- Group and abelian group
- The subgroup ⟨ S ⟩ generated by a subset, the cyclic subgroup ⟨ g ⟩, and cyclic groups
- A homomorphism that kills a normal subgroup factors uniquely through the quotient group
- Von Dyck's theorem: maps of generators that satisfy the relators extend uniquely from a presented group
- Group presentation by generators and relations
- Relators and relations; finitely generated, finitely related, and finite presentations
- The normal closure of a subset of a group
- The quotient group G/N and coset product (gN)(hN)=ghN
- Third isomorphism theorem for groups: (G/K)/(N/K)≅ G/N
- If K is normal in G, N is normal in G and K⊆ N, then N/K is normal in G/K
- Correspondence theorem: subgroups of G/N correspond to subgroups of G containing N, with normality preserved
- First isomorphism theorem for groups: G/ker f congimf
- Group isomorphisms, automorphisms and the set Aut(G)
- Free group on a set of generators
- Reduced words form the free group on an alphabet
- Semigroup and monoid
- The braid group by Artin presentation
- The Artin presentation is complete for geometric braids
- The image of a group homomorphism is a subgroup and its kernel is a normal subgroup
- The Axiom of Choice
- Coxeter diagrams: edges, labels, components and finite type
- Support, intrinsic parabolic presentations, minimal coset representatives and length additivity, with the type-A identification
- The real Coxeter form, its radical, reflections, and form-preserving maps
- The canonical reflection homomorphism, roots, reflections, and the positive cone
- The external direct product G× H with componentwise multiplication
- G× H is a group with identity (e_G,e_H), coordinatewise inverses, and homomorphic coordinate projections
- Internal direct products of finitely many normal subgroups
- Linear combination of a finite list, and the span span(S) as the smallest linear subspace containing S
- Linear subspace of a vector space
- Internal direct sum V = ⨁_i<n Uᵢ: the sum is everything and each summand meets the sum of the others only in 0_V
- The sum U + W of two linear subspaces and the sum ∑_i<n Uᵢ of a finite family
- Linear map between vector spaces over the same field
- Kernel and image of a linear map
- Basis of a vector space: a linearly independent spanning subset; and ordered basis: an injective finite list whose image is a basis
- Finite-dimensional vector space, and its dimension dim_F V; infinite-dimensional means having no finite basis
- Bilinear forms, and symmetric, skew-symmetric, and alternating bilinear forms
- Bilinear forms on V correspond linearly and bijectively to linear maps V→ V^*
- Positive and negative definiteness, the inertia (p,q,r), rank p+q, and signature p-q of a real symmetric bilinear or quadratic form
- Real and complex inner-product spaces and their induced length
- Finite sums and finite products, by recursion
- Laws of finite sums and finite products
- The cardinality | A| of a finite set
- Quarter-turn values and shifts by pi/2 and pi
- Signs, monotonicity intervals, and ranges of sine and cosine
- Standard parabolic subgroups, descent-free one- and two-sided representatives, parabolic and reflection subgroups
- Length parity, exchange, two-letter deletion, and faithfulness of the signed reflection action
- The well-ordering principle
- The natural numbers ℕ (von Neumann)
- The root-length criterion and faithfulness of the canonical reflection representation
- The inversion formula |N(w)|=ℓ(w), the root-reflection dictionary and strong exchange
- The dual action, chambers, faces, and root hyperplanes
- The dual action, the faces, and the rank-two chamber tiling
- Linear functionals and the algebraic dual V^*=L(V,F)
- Left and right cosets gH and Hg of a subgroup
- The principle of mathematical induction
- Rank-nullity: dim_F V=nullityT+rankT
- Rank and nullity of a linear map with finite-dimensional domain
- A finite list v : n → V is an ordered basis if and only if every x ∈ V equals ∑_i<n λᵢ vᵢ for exactly one λ : n → F; those scalars are the coordinates of x in that ordered basis
- V = ⨁_i<n Uᵢ if and only if every v ∈ V is ∑_i<n uᵢ with uᵢ ∈ Uᵢ in exactly one way; equivalently, if and only if the sum is V and ∑_i<n uᵢ = 0_V with uᵢ ∈ Uᵢ forces every uᵢ = 0_V
- Coordinate columns [v]_B and matrices [T]_B^C of linear maps relative to ordered bases
- [S∘ T]_B^D=[S]_C^D[T]_B^C
- Rectangular matrix multiplication and the identity matrix Iₙ, including zero-sized shapes
- The addition formulas for sine and cosine
- Parity and the Pythagorean identity for sine and cosine
- The zero sets of sine and cosine and the least positive common period 2 pi
- The reals form a totally ordered field
- The standard list e : n → Fⁿ with eᵢ(i) = 1_F and eᵢ(j) = 0_F for j ≠ i is an ordered basis of Fⁿ; hence dim_F Fⁿ = n, and F⁰ is the zero space with basis ∅ and dimension 0
- The vector space F^X of all functions X → F with pointwise operations, and Fⁿ as the case X = n = {0, 1, …, n-1}
- Sine and cosine defined by their real power series
- The matrix, left and right radicals, rank, and nondegeneracy of a bilinear form on a finite-dimensional space
- Invertible linear maps, linear isomorphisms, and inverse linear maps
- The space L(V,W) of linear maps with pointwise addition and scalar multiplication
- Identity maps and composites of linear maps are linear
- The invertible elements of a monoid form a group under the restricted operation
- A linear map L:ℝᵐ→ℝⁿ in Euclidean coordinates
- The Euclidean inner product ⟨ x,y⟩ = ∑_k<n xₖ yₖ on ℝⁿ
- A norm on a real vector space, the induced metric, and the dictionary with the metric axioms
- Cauchy-Schwarz |⟨ x,y⟩| ≤ ‖ x‖₂‖ y‖₂ with its equality case, the triangle inequality for ‖·‖₂, the parallelogram law and polarisation
- Vector space over a field
- The geometric representation on the simple-root basis over a common splitting field, and the root set
- If dim_F V = n and U is a linear subspace of V, then U is finite-dimensional, dim_F U ≤ n, and dim_F U = n if and only if U = V
- Eigenvalues, eigenvectors, eigenspaces E_λ(T)=ker(T-λ I), and the spectrum σ_F(T) of an endomorphism
- The basis-independent characteristic polynomial χ_T of an endomorphism of a finite-dimensional space, including χ_T=1 in dimension zero
- Eigenvectors belonging to pairwise distinct eigenvalues are linearly independent
- A characteristic polynomial that splits into distinct linear factors forces diagonalisability
- The order |G| of a finite group and the order ord(g) of an element, with ord(g) = ∞ when no positive power of g is the identity
- Powers gⁿ: natural exponents in a monoid and integer exponents in a group, with g⁰ = e
- The group μₙ(K) of n-th roots of unity in a field, and primitive n-th roots of unity
- μₙ(K) is cyclic of order dividing n, and has a primitive n-th root of unity exactly when its order is n
- Divisibility in ℤ: d ∣ a when a = dq for some integer q
- For A∈ Mₙ(F), the characteristic polynomial is χ_A(x)=det(xIₙ-A) when n≥1, with χ_A(x)=1 for the unique 0×0 matrix
- χ_A(x) is monic of degree n; for n≥1 its xⁿ⁻¹ coefficient is -tr(A) and its constant coefficient is (-1)ⁿ det(A), while χ_0×0=1
- Subgroup
- One-step subgroup test: a nonempty H ⊆ G is a subgroup iff gh⁻¹ ∈ H for all g, h ∈ H; the identity and the inverses of H are then those of G
- Refinements, locally finite families, point-finite families, and star refinements
- A point lies in the closure of A iff every basic neighbourhood of it meets A; the closure is the smallest closed superset and equals A together with its derived set
- Topology on a set, open and closed sets, clopen sets, the closed-set axiomatisation, and the coarser/finer comparison
- The product set ∏_i ∈ I Xᵢ of functions choosing a point in each factor, the projections, the box topology, and the product topology as the initial topology of the projections; the empty product is a one-point space
- ℝⁿ as the set of functions n → ℝ, and d₁, d₂, d_∞ are metrics on it
- The metric topology: a set is open when every one of its points has a ball around it inside the set; closed means open complement
- Open ball, closed ball and sphere in a metric space
- Metrizable space: a topological space whose topology is induced by some metric; metrizability is topological, the metric is not
- The absolute value makes ℝ a metric space: d(x,y) = |x-y| is a metric, its open balls are the intervals (x-r, x+r), and it is unbounded
- Topologically, uniformly and Lipschitz equivalent metrics on a set
- Lipschitz equivalence implies uniform equivalence implies topological equivalence
- Square roots exist: a unique √a ≥ 0 with (√a)² = a; the positives are {x² : x ≠ 0}
- Intervals of ℝ: the nine order-convex forms, nondegeneracy, and length
- Maximum and minimum of a set
- Every nonempty finite set of reals has a maximum and a minimum
- Every natural-number-indexed list of nonempty sets has a choice function on its family of values
- Squaring is monotone on the nonnegatives
- Squares of nonzero elements are positive
- Basic properties of the absolute value
- Canonical naturals are positive and strictly increasing
- Sign rules for products and monotonicity of multiplication
- Nonnegativity of a metric is a consequence of the other axioms, not an axiom
- Basis and subbasis for a topology, and the topology generated by a family of sets
- The union ⋃ x of a set, and the binary union a ∪ b := ⋃ {a,b}
- The intersection ⋂ x of a nonempty set, the binary intersection a ∩ b := ⋂{a,b}, and disjointness
- The unordered pair {x,y} and the singleton {x} = {x,x}
- There is exactly one set with no elements, written ∅
- The Axiom of Extensionality: ∀ x ∀ y (∀ z (z ∈ x ↔ z ∈ y) → x = y)
- Word length of a group element with respect to a generating set
- In a group e⁻¹ = e, (g⁻¹)⁻¹ = g and (gh)⁻¹ = h⁻¹g⁻¹, the order of the last product being essential
- Class-sized category theory in ZFC: definable-class schemas, small and locally small categories, and why CAT is not formed
- Covariant functor, identity functor, composite functor, and contravariant functor
- f : A → B is a bijection if and only if there is a function g : B → A with g ∘ f = Δ_A and f ∘ g = Δ_B; such a g is unique, equals the inverse relation f⁻¹, and is itself a bijection
- Isomorphism, groupoid, and connected category
- Small, locally small, and large categories
- Under the Axiom of Choice, a functor with small target is an equivalence exactly when it is fully faithful and essentially surjective
- A monoid is a one-object category, and a group is a one-object category in which every morphism is invertible
- Higher homotopy group by based cubes
- Higher homotopy classes form groups and are abelian above degree one
- Continuity may be checked on any open cover, and on any finite closed cover; composites of continuous maps are continuous
- Precomposition and postcomposition by continuous maps preserve homotopies, including their relative form
- Smooth manifolds and their smooth charts
- Smooth atlases
- Manifold charts, coordinate domains, and coordinate functions
- Smoothly compatible charts and the smoothness of Euclidean transition maps
- An open subset of a smooth manifold has a canonical restricted smooth structure
- Hausdorff space: distinct points have disjoint open neighbourhoods; every metrizable space is Hausdorff and the indiscrete topology on two points is not
- Second countability: an at most countable basis for the topology
- ℚⁿ is a countable dense subset of ℝⁿ, and rational open boxes form a countable basis
- Arbitrary-rank Coxeter presentations, length, reflections, and Bruhat order
- The Bruhat graph by length-increasing reflection chains, the Bruhat order, inversion symmetry, and reflection parity
- The subword characterization of Bruhat order and its independence of the reduced expression
- Irreducible affine Coxeter type: the corank-one form, the radical quotient, and the affine slice
- Positive radical, corank one, positive definiteness of proper submatrices, and domination exclusions
- The standard affine diagrams A-tilde, B-tilde, C-tilde, D-tilde, E-tilde, F-tilde and G-tilde
- Euclidean simplices with the same facet-normal Gram matrix are similar facet to facet
- The affine slice: faithful isometric action, the alcove simplex, and its facet reflections
- Crystallographic alcove diagrams: the affine list realized by Weyl types A–G
- Enumeration of the connected positive semidefinite corank-one diagrams
- Finiteness criterion: W is finite exactly when the Coxeter form is positive definite
- Affine root hyperplanes, coroot translations, alcoves, and the affine reflection group
- Highest-root dominance and the fundamental alcove
- Alcove transitivity, the affine Coxeter presentation, and the length function
- The geometric simplex spanned by affinely independent vertices
- Convex closures and hulls of finitely many compact convex sets
- Affine reflections: translation form, involutivity, local finiteness, and Wₐ=Q^∨⋊ W
- The Weyl group is finite and faithful
- A normed space is locally compact if and only if it is finite-dimensional
- In a locally compact metric space every point has arbitrarily small compact closed balls, hence a neighbourhood base of compact sets
- A chosen algebraic basis identifies a finite-dimensional normed space with a coordinate space
- Root, coroot, weight, and coweight lattices
- Double-angle and quadratic power-reduction identities
- Cofunction, supplementary, quarter-turn, and reflection identities for the six trigonometric functions
- Pi as twice the smallest positive zero of cosine
- The Tits cone, its interior, and the negative-root set of a functional
- Root sign coherence and the action of simple reflections on positive roots
- Exclusions for positive definite diagrams: trees, valency, labels, chains and arms
- Sylvester's criterion: a real symmetric n× n matrix with n≥1 is positive definite if and only if all leading principal minors are positive
- Deleted-row-and-column minors, cofactors, the cofactor matrix and the adjugate over a commutative ring
- The determinant is the unique normalized alternating multilinear function on the columns
- Graph isomorphisms, automorphisms and graph complements
- Tₙ(cosθ)=cos(nθ) and Uₙ(cosθ)sinθ=sin((n+1)θ) for every n∈ℕ
- Chebyshev polynomials of the first and second kinds by their three-term recurrences
- Pi is the first positive zero of sine
- The finite Viete cosine product and its positive nested-radical factors
- Laplace expansion computes the determinant along every row and every column over a commutative ring
- The finite reflection arrangement, its chambers, the spherical chamber complex, and the coset face poset
- The geometric inversion set N(w) of an element of a Coxeter group
- The finite chamber tiling, the face-stabiliser identification, and the spherical Coxeter complex as a triangulation of the sphere
- span(S) is exactly the set of linear combinations of finite lists of elements of S, and span(∅) = {0_V}
2 · Logical flowchart
Arrows show item dependencies. A grouped prerequisite node points to items that use one or more of its members; their exact links are listed above.
3 · Items
ExampleLibrary prerequisite
Stereographic projection and its inverse are explicit in coordinates
LemmaLibrary prerequisite
Universal properties of the Artin monoid and group, the projection onto the Coxeter group, and the quotient by the squares
LemmaLibrary prerequisite
Disconnected diagrams, direct products, and comparison of invariant forms
LemmaLibrary prerequisite
Descent reduction, minimum-length elements, and the additive factorization in a double coset
LemmaLibrary prerequisite
The parabolic intersection W_I∩ dW_Jd⁻¹ for d∈^IW^J
LemmaLibrary prerequisite
The rank-two half-space alternative and the chamber-length induction (Pₙ), (Qₙ)
LemmaLibrary prerequisite
Reflections: involutivity, form invariance, fixed hyperplane, and exact rank-two order
LemmaLibrary prerequisite
Descent of the reflection representation, unit root norms, and conjugation of reflections
LemmaLibrary prerequisite
Every Euclidean linear map has a unique matrix and satisfies ‖Lh‖₂≤ K‖h‖₂ for some K≥0
LemmaLibrary prerequisite
A finite-dimensional vector space over an infinite field is not a finite union of proper subspaces
LemmaLibrary prerequisite
The rank-two block computation, exact dihedral orders, the signed reflection action, and ambient reducedness
LemmaLibrary prerequisite
The intersection of a nonempty family of subgroups of G is a subgroup of G
LemmaLibrary prerequisite
Locally finite families remain locally finite after taking closures, closure commutes with their union, and a locally finite union of closed sets is closed
LemmaLibrary prerequisite
For n ≥ 1 the product topology on n copies of the usual topology of ℝ is the metric topology of d_∞ on ℝⁿ, and hence also of d₁ and d₂, so ℝⁿ as a product and ℝⁿ as a metric space are one space
LemmaLibrary prerequisite
⋃ ∅ = ∅, ⋃ {a} = a, ⋃ {a,b} = a ∪ b, ⋂ {a} = a, and ⋂ {a,b} = a ∩ b
LemmaLibrary prerequisite
Word length is defined on every element and satisfies the subadditivity, inversion and vanishing laws
PropositionLibrary prerequisite
A functor is an isomorphism of categories exactly when its object and morphism maps are bijective
PropositionLibrary prerequisite
Under the Axiom of Choice, a connected small groupoid is equivalent to the automorphism group of any one of its objects
PropositionLibrary prerequisite
Higher homotopy basepoint transport and moving homotopies
PropositionLibrary prerequisite
Higher homotopy groups are functorial and based homotopy invariant
PropositionLibrary prerequisite
Open subsets of Euclidean space have the standard smooth structure
TheoremLibrary prerequisite
Finite-generator standard parabolics and the Bruhat subword property at arbitrary rank
TheoremLibrary prerequisite
Classification of affine Coxeter diagrams and their Euclidean simplex realization
TheoremLibrary prerequisite
Chamber collisions, point stabilizers, and the intersection rule
TheoremLibrary prerequisite
Classification of finite Coxeter systems, including the H and dihedral families
TheoremLibrary prerequisite
The longest element as the opposition of the chamber, and longest elements of finite parabolics
TheoremLibrary prerequisite