Alphabeta MathArticle items · 06 — Finite phase tori, local lifts and the flat metric
Presents 28 items from the article and its immediate prerequisites, covering dependency levels 0–1.
1 · Prerequisites
- The quotient topology of a surjection, quotient maps, saturated sets, and the quotient of a space by an equivalence relation with its canonical projection
- 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
- t↦(cos t,sin t) is a bijection from [0,2π) onto the real unit circle
- The zero sets of sine and cosine and the least positive common period 2 pi
- The derivatives of sine and cosine are cosine and minus sine
- Smooth atlases
- A product of finitely many compact spaces is compact in the product topology
- Heine-Borel by bisection: every closed bounded interval [a,b] is compact
- A continuous image of a compact space is compact; a continuous real-valued map on a nonempty compact space attains a maximum and a minimum; and a continuous bijection from a compact space to a Hausdorff space is a homeomorphism
- Integer part: for every real x there is exactly one integer m with m ≤ x < m + 1
- Cauchy-Schwarz |⟨ x,y⟩| ≤ ‖ x‖₂‖ y‖₂ with its equality case, the triangle inequality for ‖·‖₂, the parallelogram law and polarisation
- ℚⁿ is a countable dense subset of ℝⁿ, and rational open boxes form a countable basis
- The product, Euclidean-metric and norm topologies on ℝⁿ agree, and for n=1 they agree with the real-line topology
- Topological manifolds without boundary: Hausdorff, second-countable, and locally Euclidean spaces
- Smooth manifolds and their smooth charts
- Each smooth atlas is contained in a unique maximal smooth atlas
- Riemannian metric and riemannian manifold
- Piecewise c one curve on a manifold
- Riemannian speed and length
- Riemannian length is independent of piecewise c one subdivision
- Riemannian distance on a connected manifold
- Every open cover of a compact metric space has a Lebesgue number: a δ > 0 such that every nonempty subset of diameter less than δ lies inside a single member of the cover
- A continuous piecewise-C¹ path is rectifiable and its length is the sum of the speed integrals over its pieces
- Every endpoint chord is no longer than the arc: ‖γ(b)-γ(a)‖₂≤ L(γ)
- Every path-connected space is connected, and every path component lies inside a component
- The Axiom of Countable Choice (AC_ω)
- Lebesgue measurable sets, the family L(ℝⁿ), and the restricted set function λₙ
- Assuming countable choice, L(ℝⁿ) is a sigma-algebra containing every elementary set and λₙ is a complete measure extending elementary volume
- A box in ℝⁿ with parameters aᵢ≤ bᵢ is Lebesgue measurable of measure ∏_i<n(bᵢ-aᵢ), whichever of its faces are included
- Lebesgue outer measure, Lebesgue measurability and Lebesgue measure are unchanged by translation
- For a nonzero real c, dilation by c multiplies Lebesgue outer measure by |c|ⁿ, and reflection in the origin preserves it
- The one-dimensional torus and its normalized Haar integral
- Probability measures and probability spaces
- Random elements and real random variables
- Law or distribution of a random element
- The Borel sigma-algebra of a topological space
- Assuming countable choice, every Borel subset of ℝⁿ is Lebesgue measurable
- Lebesgue measure on ℝⁿ is invariant under every orthogonal linear map
- Sigma-algebras
- Measures on sigma-algebras
- Measurable spaces and measurable sets
- Complete measure spaces
- Finite and countable subadditivity of measures
- The complex exponential by its power series
- Complex sine, cosine, hyperbolic sine, and hyperbolic cosine from the complex exponential
- exp(x+iy)=eˣ(cos y+i sin y), |exp(x+iy)|=eˣ, and e^iπ+1=0
- exp(z+w)=exp z exp w, and the complex exponential extends the real exponential
- Complex sine, cosine, hyperbolic sine, and hyperbolic cosine are entire with their standard derivatives
- Sum, product, quotient, and chain rules from difference quotients
- Conjugation is an involutive real-field automorphism, zz̄=|z|², and modulus is definite, multiplicative, and subadditive
- Equivalence relation, equivalence class, and the quotient set A/∼
- The polynomial ring R[xᵢ:i∈ I] as finitely supported coefficient families on monomials
- Intermediate value theorem, by bisection with a canonical left-half rule: a continuous function on [a,b] takes every value between f(a) and f(b)
- On an interval I, for f continuous on I and differentiable at every interior point: f' ≥ 0 throughout gives f nondecreasing, f' > 0 gives f increasing, f' ≤ 0 and f' < 0 give the two decreasing forms; conversely a nondecreasing f has f' ≥ 0 and a nonincreasing f has f' ≤ 0 wherever it is differentiable, and no strict converse is claimed
- Finite real interval operations enclose their exact values
- Universal property of a polynomial ring on an arbitrary family of indeterminates
- Intervals of ℝ: the nine order-convex forms, nondegeneracy, and length
- Continuity of f : A → ℝ at a point of A and on A: the ε-δ condition, its agreement with lim_x → c f(x) = f(c) at a limit point, and continuity at an isolated point
- Sign rules for products and monotonicity of multiplication
- The derivative f'(c) = lim_x → c f(x) - f(c)/x - c of f : A → ℝ at a point c ∈ A that is a limit point of A, and differentiability on a set
- The Euclidean inner product ⟨ x,y⟩ = ∑_k<n xₖ yₖ on ℝⁿ
- The tangent space to a regular level set
- Real and complex inner product spaces, with the inner product linear in the first argument
- The vector space M_m × n(F) := F^ m × n of m by n matrices over a field, with entrywise operations
- The total (Fréchet) derivative Df(a) as the linear first-order approximation with o(‖h‖₂) remainder
- Cᵏ Euclidean maps and diffeomorphisms
- Metric space: d(x,y) = 0 iff x = y, symmetry, and the triangle inequality; pseudometric and ultrametric
- 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
- Local and strict local extrema for scalar fields on Euclidean open sets
- 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
- Continuity of a map of topological spaces at a point and globally
- Nonnegativity of a metric is a consequence of the other axioms, not an axiom
- Every nonempty set bounded below has an infimum
- Limits at +∞ and -∞, and infinite limits at a point
- Symmetric matrices are Euclidean under the Frobenius norm
- The operator norm is zero on the zero domain and otherwise is max_‖ v‖=1‖ Tv‖
- Convex and strictly convex functions on Euclidean convex sets
- Lipschitz map, α-Hölder map for rational 0 < α ≤ 1, and contraction
- Real spectral theorem: a self-adjoint endomorphism of a finite-dimensional real inner product space has an orthonormal eigenbasis
- In an orthonormal basis, self-adjoint means conjugate-transpose symmetry and normal means commuting with the conjugate transpose
- The Gram matrix G(v₀,…,vᵣ₋₁)=(⟨ vᵢ,vⱼ⟩)_i,j<r and Gram determinant, with empty value 1
- Row space, column space, nullspace, row rank, column rank and matrix rank
- The trace tr(A) as the sum of the diagonal entries
- Rectangular matrix multiplication and the identity matrix Iₙ, including zero-sized shapes
- Laws of finite sums and finite products
- ∑_i ∈ S∑_j ∈ T aᵢⱼ = ∑_(i,j) ∈ S × T aᵢⱼ = ∑_j ∈ T∑_i ∈ S aᵢⱼ for finite index sets S and T
- Row rank equals column rank, and both equal the number of pivots
- 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
- 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 transpose A^T of a matrix
- Subspace topology: the traces of the open sets, its closed sets and its bases, the continuity of the inclusion, and the characteristic property of a map into a subspace
- Embedded submanifolds and slice charts
- A regular level set is an embedded submanifold
- Slice-chart restrictions form a smooth atlas
- Cʳ and smooth maps between smooth manifolds
- Cᵏ maps and multi-index derivative notation in Euclidean space
- If all partial derivatives exist on a neighbourhood and are continuous at a point, then the map is totally differentiable there with Jacobian derivative
- Identity maps and composites of smooth maps are smooth
- Critical points and critical values of a smooth function
- The differential of a smooth real-valued function
- The tangent space of a regular level set is the kernel
- The tangent space of Euclidean space
- Coordinate formula for the differential
- The norm ‖ v‖=√⟨ v,v⟩ induced by a real or complex inner product
- Finite rectangular matrices over a commutative ring, their entries, rows and columns
- A finite sum in a commutative monoid indexed by an arbitrary finite set
- The induced length is a norm
- Every finite-dimensional normed space is Banach
- Hilbert space
- The adjoint T^*:W→ V is characterised by ⟨ Tv,w⟩_W=⟨ v,T^*w⟩_V
- Linear map between vector spaces over the same field
- Basis of a vector space: a linearly independent spanning subset; and ordered basis: an injective finite list whose image is a basis
- Eigenvalues, eigenvectors, eigenspaces E_λ(T)=ker(T-λ I), and the spectrum σ_F(T) of an endomorphism
- Vector space over a field
- Real and imaginary parts, complex conjugation, and modulus
- Huygens’s clocks · Bennett, Schatz, Rockwood and Wiesenfeld, Proc. R. Soc. Lond. A 458 (2002), Sections 1 and 2(a), printed pp. 563–566
- Order, continuity, range, and the product, quotient, and reciprocal laws for the natural logarithm
- The natural logarithm has derivative 1/x and equals the integral from 1 to x of 1/t
- Sums, scalar multiples, products and quotients: (f+g)'(c) = f'(c) + g'(c), (α f)'(c) = α f'(c), (fg)'(c) = f'(c)g(c) + f(c)g'(c), and (f/g)'(c) = (f'(c)g(c) - f(c)g'(c))/g(c)² when g(c) ≠ 0
- For a natural n ≥ 1 the function x ↦ xⁿ is differentiable everywhere with derivative ι(n) x^ n-1; for n = 0 it is the constant 1, with derivative 0; for a natural n ≥ 1 the function x ↦ x⁻ⁿ is differentiable at every x ≠ 0 with derivative -ι(n) x⁻ⁿ⁻¹; consequently every polynomial function is differentiable at every real, with the derivative computed term by term
- A function differentiable at c is continuous at c
- The logarithm grows more slowly than every positive real power
- The natural logarithm as the inverse of the exponential function
- The exponential addition formula exp(x+y)=exp(x)exp(y)
- Real powers for positive bases, with the zero-base positive-exponent convention
- Pairwise and mutual independence of finite-valued random variables
- Bernoulli random variables and binomial random variables as sums of independent Bernoulli trials
- The set [A]ᵏ of k-element subsets and the binomial coefficient C(n, k) := | [n]ᵏ|
- The factorial n! and the falling factorial n^k̲, defined by recursion in ℕ
- The Binomial count has its usual probability mass function
- C(n, k) k! (n-k)! = n! for k ≤ n; hence C(n, k) k! = n^k̲, the quotient n!/(k!(n-k)!) is a natural number, and C(n, k) = C(n, n-k)
- If m ≤ f ≤ M on [a,b] then m(b-a) ≤ L(f,P) ≤ ∫ₐᵇ f ≤ ∫ₐᵇ f ≤ U(f,P) ≤ M(b-a) for every partition P; in particular every constant function is integrable, with ∫ₐᵇ c = c(b-a)
- The fundamental theorem of calculus and integration by parts
- The real exponential function and the number e by a power series
- The exponential function is strictly increasing
- 1+x≤exp(x) for every real x, hence (1-p)ᵐ≤exp(-mp)
- The principle of mathematical induction
- For |r| < 1, ∑_k ≥ 0 rᵏ = 1/(1-r), and for |r| ≥ 1 the series diverges
- The natural numbers ℕ (von Neumann)
- Finite probability spaces, outcome weights, events, and event probabilities
- Real random variables on finite probability spaces and their finite distributions
- Expectation of a real random variable on a finite probability space
- The indicator random variable of an event
- Indicators turn event probabilities, intersections, and finite counts into expectations and products
- Expectation is linear for every finite family of random variables, without any independence hypothesis
- Markov's inequality on a finite probability space
- The Erdős-Rényi finite random graph G(n,p)
- A finite simple graph is a finite vertex set together with a set of two-element vertex subsets
- A Bernoulli(p) variable has mean p and variance p(1-p); a binomial(n,p) variable has mean np and variance np(1-p)
- Variance, standard deviation, and covariance on a finite probability space
- Variance and covariance identities for random variables
- Independence forces covariance to vanish
- Standard normal and normal laws
- Independent random elements
- The Axiom of Choice
- Expectation of a nonnegative or integrable random variable
- Change of variables for expectation
- Integrating against a density agrees with integrating the product
- If u,v are differentiable on [a,b] with u',v' integrable, then ∫ₐᵇ u v' = u(b)v(b)-u(a)v(a) - ∫ₐᵇ u'v
- The exponential function is smooth and (exp)'=exp
- The exponential dominates every fixed nonnegative integer power at +∞
- Monotone convergence for the integral
- A bounded Riemann integrable function on a closed bounded interval is Lebesgue measurable and has the same integral
- Expectations factor over finite products of independent random variables
- The chain rule, in one line from Carathéodory: if g is differentiable at c and f is differentiable at g(c), then f ∘ g is differentiable at c with (f ∘ g)'(c) = f'(g(c)) g'(c)
- Sums, scalar multiples, products, absolute values, maxima, minima and quotients with nonvanishing denominator of continuous functions are continuous, as are constants, the identity and every polynomial function
- A composite of continuous functions is continuous, with no side hypothesis of the kind the composition of limits needs
- Continuous functions on Euclidean spaces are Borel measurable
- A continuous function on [a,b] is Riemann integrable, by Heine-Cantor and Riemann's criterion
- Monotone change of variable for Riemann-integrable functions
- Laws of integer exponents
- A nondecreasing sequence bounded above converges to the supremum of its range, and a nonincreasing sequence bounded below to the infimum
- Sequences of reals: bounded, eventually, frequently, tails, subsequences
- Integer powers aᵐ
- Finite sums and finite products, by recursion
- Series, partial sums, convergence and the sum, divergence, and the tail series
- A norm on a real vector space, the induced metric, and the dictionary with the metric axioms
- The adjacency matrix of a finite simple graph
- Layer-cake formulas for random variables
- The nonnegative Lebesgue integral
- The Lebesgue integral is linear on L¹(μ)
- Monotonicity and nonnegative homogeneity of the nonnegative integral
- The Gaussian integral ∫_-∞^∞e^-x² dx=√π
- Substitution: if φ is differentiable on [c,d] with φ' integrable and f is continuous on an interval containing φ([c,d]), then ∫_φ(c)^φ(d) f = ∫_cᵈ (f∘φ) φ'
- The Bartle-Sherbert bounds 2.828 < pi < 3.185
- Integrable real and complex functions, and their integrals
- Closure properties of measurable functions used by the integral
- The integral of a nonnegative simple function
- The nonnegative integral agrees with the simple integral on simple functions
- Signs, monotonicity intervals, and ranges of sine and cosine
- Quarter-turn values and shifts by pi/2 and pi
- Principal inverse sine and inverse cosine
- For -1<y<1, (arcsin y)ᵖʳⁱᵐᵉ=1/√1-y² and (arccos y)ᵖʳⁱᵐᵉ=-1/√1-y²
- The mean value theorem, as the case g(x) = x of Cauchy's: for f continuous on [a,b] with a < b and differentiable on (a,b) there is c ∈ (a,b) with f(b) - f(a) = f'(c)(b-a)
- Square roots exist: a unique √a ≥ 0 with (√a)² = a; the positives are {x² : x ≠ 0}
- First-order systems, initial value problems, and solutions on intervals
- Picard-Lindelöf local existence and uniqueness for first-order systems
- Local Lipschitz continuity in the state variable, locally uniform in time and parameters
- The left and right limits of f at c, as limits of the restrictions of f to A ∩ (-∞, c) and A ∩ (c, ∞)
- The derivative and the Riemann integral of a vector-valued function: an intrinsic derivative and a componentwise integral
- The Hessian matrix and critical points of a scalar field
- Fermat's theorem: an interior differentiable local extremum has zero gradient
- Second-order Taylor expansion f(a+h)=f(a)+∇ f(a)· h+1/2h^TH_f(a)h+o(‖h‖²)
- Clairaut--Schwarz theorem for continuous second partial derivatives
- The finite product of finite probability spaces
- Sigma-algebras are closed under countable intersections, differences, symmetric differences, and set limits
- A product of two at most countable sets is at most countable
- For every ε > 0 in a complete ordered field there is a natural n ≥ 1 with 1/n < ε
- Empirical measure of observed units
- The empirical measure is a probability measure
- The Dirac set function at a point
- Weak convergence of borel probability measures
- The indefinite integral of a nonnegative measurable function is a measure
- Measure spaces
- Extended-real-valued measurable functions
- A measurable function between measurable spaces
- Nonnegative simple measurable functions
- A continuous map has Borel preimages of Borel sets
- Integral over a measurable subset
- The Riemannian gradient is the metric dual of the differential
- Integral curves of a vector field
- Every smooth vector field on a compact manifold is complete
- C¹ Euclidean maximal flows, variational dependence and the finite C² upgrade
- Real analytic germs in several variables
- The intrinsic Hessian of a smooth function at a critical point
- The binomial theorem in ℝ: (x+y)ⁿ = ∑_k<n+1 iotaC(n, k) xᵏ y^ n-k
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
DefinitionDependency level 0
Finite phase tori, local lifts and the flat metric
DefinitionDependency level 0
Normalized phase volume and almost-everywhere assertions
DefinitionDependency level 0
Phasors, trigonometric identities and periodic derivatives
DefinitionDependency level 0
Root and sign certificates for parameter-dependent polynomials
DefinitionDependency level 0
Sphere configurations and tangent variations
DefinitionDependency level 0
Point stability, attraction and stability of a set
DefinitionDependency level 0
Independent-entry coordinates, convexity and Lipschitz constants
DefinitionDependency level 0
Signed vector-spin objectives and low-rank Gram factors
DefinitionDependency level 0
Finite vertex Hilbert space in the article convention
ExperimentDependency level 0
Historical report of Huygens’s two clocks
LemmaDependency level 0
Logarithmic tangent bound and a quadratic entropy envelope
LemmaDependency level 0
Vanishing numerical condition for fixed positive Laplacian margins
PropositionDependency level 0
Elementary binomial probability envelopes
PropositionDependency level 0
Binomial mass ratios and geometric tail domination
PropositionDependency level 0
Indicator expectations and exceptional-vertex probabilities
PropositionDependency level 0
Single-vertex degree laws and the shared-edge dependence
PropositionDependency level 0
Gaussian entry moments and triangular independence
PropositionDependency level 0
Geometric growth, truncation and total decrement budgets
PropositionDependency level 0
Norm symmetrization at the actual matrix mean
PropositionDependency level 0
Tail integration and a median-to-mean bridge
PropositionDependency level 0
Angular order and elementary inverse-trigonometric bounds
PropositionDependency level 0
Explicit scalar quadratic descent and its forward domain
PropositionDependency level 0
Local minima and nonnegative second variation
PropositionDependency level 0
Finite Bernoulli difference laws and parity of moments
DefinitionDependency level 1
Point basins, equilibrium-set basins and aggregate volume
DefinitionDependency level 1
Empirical phase measures and Fourier order statistics
DefinitionDependency level 1
Gradient trajectories and complete compact phase dynamics
DefinitionDependency level 1