Alphabeta MathArticle items · 07 — Diagonal phase rotations and relative-phase coordinates
Presents 28 items from the article and its immediate prerequisites, covering dependency level 1.
1 · Prerequisites
- Finite phase tori, local lifts and the flat metric
- Phasors, trigonometric identities and periodic derivatives
- 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
- Homeomorphism, open map, closed map, embedding, and what it means for a property to be topological
- The Erdős-Rényi finite random graph G(n,p)
- Asymptotic quantifiers and uniform parameter ranges
- Finite vertex Hilbert space in the article convention
- Gaussian entry moments and triangular independence
- Standard normal and normal laws
- Independent random elements
- For sigma-finite factors, the product measure exists, has the rectangle formula, is sigma-finite, and is unique
- The Axiom of Choice
- Random elements and real random variables
- The product sigma-algebra and its finite iterates
- Finite symmetric networks and loop removal
- Complex spectral theorem: a normal endomorphism of a finite-dimensional complex inner product space has an orthonormal eigenbasis, and conversely
- The unordered pair {x,y} and the singleton {x} = {x,x}
- Algebraic multiplicity as the exponent of x-λ in χ_T, and geometric multiplicity as dim E_λ(T)
- An endomorphism is diagonalisable if and only if its characteristic polynomial splits and every eigenvalue's geometric multiplicity equals its algebraic multiplicity
- Quarter-turn values and shifts by pi/2 and pi
- Absolute value in an ordered field
- Pi as twice the smallest positive zero of cosine
- Parity and the Pythagorean identity for sine and cosine
- Angular order and elementary inverse-trigonometric bounds
- Integer part: for every real x there is exactly one integer m with m ≤ x < m + 1
- Explicit scalar quadratic descent and its forward domain
- The Hessian matrix and critical points of a scalar field
- Sum, product, quotient, and chain rules from difference quotients
- The Axiom of Countable Choice (AC_ω)
- A box in ℝⁿ with parameters aᵢ≤ bᵢ is Lebesgue measurable of measure ∏_i<n(bᵢ-aᵢ), whichever of its faces are included
- First-order systems, initial value problems, and solutions on intervals
- Historical report of Huygens’s two clocks
- Early pendulum clock by Salomon Coster, c.1657 · Object description, opening historical account
- Huygens’s clocks · Bennett, Schatz, Rockwood and Wiesenfeld, Proc. R. Soc. Lond. A 458 (2002), Sections 1 and 2(a), printed pp. 563–566
- Logarithmic tangent bound and a quadratic entropy envelope
- 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
- 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
- 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)
- 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
- Square roots exist: a unique √a ≥ 0 with (√a)² = a; the positives are {x² : x ≠ 0}
- The left and right limits of f at c, as limits of the restrictions of f to A ∩ (-∞, c) and A ∩ (c, ∞)
- Limits at +∞ and -∞, and infinite limits at a point
- 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
- Nondecreasing, increasing (strictly increasing), nonincreasing, decreasing, monotone and strictly monotone real functions on a subset of ℝ, with the dictionary to monotone sequences
- 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
- Independent-entry coordinates, convexity and Lipschitz constants
- The operator norm is zero on the zero domain and otherwise is max_‖ v‖=1‖ Tv‖
- Cauchy–Schwarz: |⟨ u,v⟩|≤‖ u‖‖ v‖, with equality exactly for linearly dependent vectors
- Closure properties of measurable functions used by the integral
- Monotonicity and nonnegative homogeneity of the nonnegative integral
- The Lebesgue integral is linear on L¹(μ)
- The operator norm as the least bound and as the unit-sphere or unit-ball supremum
- The operator norm is a norm on the space of bounded linear operators
- Continuous functions on Euclidean spaces are Borel measurable
- Composition with a Borel measurable outer map preserves measurability
- A map into ℝⁿ is measurable exactly when its coordinates are measurable
- Geometric growth, truncation and total decrement budgets
- Elementary binomial probability envelopes
- 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 real exponential function and the number e by a power series
- 1+x≤exp(x) for every real x, hence (1-p)ᵐ≤exp(-mp)
- From Kuramoto to Crawford: exploring the onset of synchronization in populations of coupled oscillators · Section 2, p. 2; Section 3.1, p. 3
- Conjugation is an involutive real-field automorphism, zz̄=|z|², and modulus is definite, multiplicative, and subadditive
- The orthogonal projection P_Wv is the W-component in V=W⊕ W^⊥
- The difference a ∖ b, the symmetric difference a △ b, and the complement X ∖ a relative to a set X
- A finite sum in a commutative monoid indexed by an arbitrary finite set
- Quadratic form and kernel of a weighted graph Laplacian
- Connected graphs and connected components defined by the existence of vertex paths
- Eigenvalues, eigenvectors, eigenspaces E_λ(T)=ker(T-λ I), and the spectrum σ_F(T) of an endomorphism
- Self-adjoint and normal endomorphisms of a finite-dimensional real or complex inner product space
- The chain rule for total derivatives: D(g∘ f)(a)=Dg(f(a))∘ Df(a)
- Directional derivatives and partial derivatives of a map U⊆ℝᵐ→ℝⁿ
- Real and complex inner product spaces, with the inner product linear in the first argument
- The norm ‖ v‖=√⟨ v,v⟩ induced by a real or complex inner product
- Linear independence: a finite list v : n → V is independent when ∑_i<n λᵢ vᵢ = 0_V forces every λᵢ = 0_F, and a subset S ⊆ V is independent when every injective finite list into S is independent
- ℂ=ℝ[x]/(x²+1) is a field, every element is uniquely a+bi, and every nonzero element has inverse (a-bi)/(a²+b²)
- Real and complex modulus product and triangle inequalities
- The zero sets of sine and cosine and the least positive common period 2 pi
- 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
- Division with remainder in ℤ: for a ∈ ℤ and b > 0 there are unique q, r ∈ ℤ with a = qb + r and 0 ≤ r < b
- t↦(cos t,sin t) is a bijection from [0,2π) onto the real unit circle
- Real and imaginary parts, complex conjugation, and modulus
- The standard formulas ⟨ x,y⟩=∑_k<nxₖ yₖ on ℝⁿ and ∑_k<nxₖyₖ on ℂⁿ are inner products
- Rectangular matrix multiplication and the identity matrix Iₙ, including zero-sized shapes
- The trace tr(A) as the sum of the diagonal entries
- Real spectral theorem: a self-adjoint endomorphism of a finite-dimensional real inner product space has an orthonormal eigenbasis
- Expectations factor over finite products of independent random variables
- Change of variables for expectation
- Expectation of a nonnegative or integrable random variable
- Similar matrices have the same trace
- Law or distribution of a random element
- Arithmetic and lattice operations preserve measurability whenever they are defined
- Integrable real and complex functions, and their integrals
- Probability measures and probability spaces
- The integral of a nonnegative simple function
- The nonnegative integral agrees with the simple integral on simple functions
- A symmetric matrix operator norm is its largest absolute Rayleigh quotient
- Regular multigraph and normalized adjacency
- Spectral edge and vertex expansion
- Constant vector is a top eigenvector
- Expander mixing lemma
- Phase chaos in coupled oscillators · Equation (1), pp. 065201-1–065201-2 and Figure 1
- Chaotic Attractor in the Kuramoto Model · Publisher abstract
- Signed vector-spin objectives and low-rank Gram factors
- Sphere configurations and tangent variations
- The intrinsic Hessian of a smooth function at a critical point
- Nonconvex landscapes for Z2 synchronization and graph clustering are benign near exact recovery thresholds · Abstract
- Benign landscape for Burer-Monteiro factorizations of MaxCut-type semidefinite programs · Abstract
- Benign landscapes for synchronization on spheres via normalized Laplacian matrices · Abstract
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 1
Diagonal phase rotations and relative-phase coordinates
DefinitionDependency level 1
Daido order parameters
DefinitionDependency level 1
Erdős–Rényi random graph and high probability
DefinitionDependency level 1
Gaussian comparison matrix in the moment argument
DefinitionDependency level 1
Adjacency, degrees, graph Laplacian, and spectral notation
DefinitionDependency level 1
Truncated sine-square function
DefinitionDependency level 1
Kernel for stability amplification
DefinitionDependency level 1
Phase tail sets for amplification
ExampleDependency level 1
Positive-measure attraction to a degenerate saddle
ExperimentDependency level 1
Historical observation of coupled pendulum clocks
LemmaDependency level 1
Entropy endpoint limits and the two monotone branches
LemmaDependency level 1
The sharp coordinate constant already occurs inside the Bernoulli cube
LemmaDependency level 1
Triangular Gaussian coordinates: metric and moment control
LemmaDependency level 1
Elementary angle and logarithm estimates used in amplification
LemmaDependency level 1
Elementary estimates used for degree expectations
PostulateDependency level 1
Scope of the phase-only sinusoidal oscillator idealization
PropositionDependency level 1
A bilinear centered-adjacency estimate without regularity
PropositionDependency level 1
Real symmetric quadratic forms tested on phasors
PropositionDependency level 1
The constant projection and centering decomposition
PropositionDependency level 1
Ordered edge counts and centered cut vectors
PropositionDependency level 1
Support connectivity and the unnormalized Laplacian kernel
PropositionDependency level 1
Differentiation of symmetric periodic pair potentials
PropositionDependency level 1
Rigidity and rotation of a unit-phasor average
PropositionDependency level 1
Trace powers, closed walks and independent edge moments
PropositionDependency level 1
Regular adjacency spectra and normalized expansion
PropositionDependency level 1
Self-adjoint order, norm bounds and diagonal tests
RemarkDependency level 1
An independently attributed heterogeneous-model chaos report
RemarkDependency level 1