Alphabeta MathArticle items · 10 — Łojasiewicz inequality and analytic point stability
Presents 29 items from the article and its immediate prerequisites, covering dependency levels 3–5.
1 · Prerequisites
- Real analytic germs in several variables
- Point stability, attraction and stability of a set
- The Euclidean inner product ⟨ x,y⟩ = ∑_k<n xₖ yₖ on ℝⁿ
- A local minimum with a gradient inequality is stable
- Nearby critical values force nonminimum escape
- The exponent, product, quotient, and iterated-power laws for positive real bases and real exponents
- Holomorphic functions of several variables are smooth and their complex derivatives are holomorphic
- Equilibria, Lyapunov stability and asymptotic stability of an autonomous first-order system
- The Jacobian matrix of partial derivatives and the gradient in the scalar-valued case
- Cᵏ maps and multi-index derivative notation in Euclidean space
- Real powers for positive bases, with the zero-base positive-exponent convention
- Order, continuity, range, and the product, quotient, and reciprocal laws for the natural logarithm
- The exponential function is strictly increasing
- P.-A. Absil and K. Kurdyka, On the stable equilibrium points of gradient systems (accepted 2006 preprint) · Section 3, Lemma 4, printed/PDF p. 4 (inequality); Theorem 3, p. 2, proof pp. 4–5 (stability application)
- Adjacency spectral expander — Definition 1.3
- Adjacency, degrees, graph Laplacian, and spectral notation
- Regular adjacency spectra and normalized expansion
- Centered graph matrices and reference degree
- Loewner order, Hermitian inner product, and operator norm
- Finite symmetric networks and loop removal
- Finite vertex Hilbert space in the article convention
- Covariance under a common positive rescaling of weights
- Loop-dependent weighted degree envelope
- A centered adjacency column retains a high-degree lower bound
- Explicit parameters for the small-gamma logarithmic density
- Independent cross-degrees produce a large degree at logarithmic density
- The real exponential function and the number e by a power series
- The exponential is positive and satisfies exp(-x)=1/exp(x)
- The Erdős-Rényi finite random graph G(n,p)
- The natural logarithm as the inverse of the exponential function
- The operator norm is zero on the zero domain and otherwise is max_‖ v‖=1‖ Tv‖
- Finite probability spaces, outcome weights, events, and event probabilities
- Adjacency, incidence, open and closed neighbourhoods, vertex degree, minimum degree and maximum degree
- Triangular Gaussian coordinates: metric and moment control
- Expected extreme eigenvalues from finite Gaussian comparison
- Cauchy-Schwarz |⟨ x,y⟩| ≤ ‖ x‖₂‖ y‖₂ with its equality case, the triangle inequality for ‖·‖₂, the parallelogram law and polarisation
- Closure properties of measurable functions used by the integral
- Monotonicity and nonnegative homogeneity of the nonnegative integral
- The Axiom of Choice
- Additivity of the nonnegative Lebesgue integral
- Positive real powers preserve measurability
- Probability measures and probability spaces
- The integral of a nonnegative simple function
- The nonnegative integral agrees with the simple integral on simple functions
- The operator norm as the least bound and as the unit-sphere or unit-ball supremum
- Independent random elements
- Standard real and normalized complex Gaussian matrix laws
- Expectation of a nonnegative or integrable random variable
- Moments, variance, and covariance on a probability space
- Smooth manifolds and their smooth charts
- Riemannian metric and riemannian manifold
- The Riemannian gradient is the metric dual of the differential
- Local comparison of a riemannian metric with the euclidean metric
- Cauchy–Schwarz: |⟨ u,v⟩|≤‖ u‖‖ v‖, with equality exactly for linearly dependent vectors
- The differential of a smooth real-valued function
- Coordinate derivations at a point
- Coordinate derivations form a basis of the tangent space
- Oriented edge count between sets
- The sharp coordinate constant already occurs inside the Bernoulli cube
- A large degree with explicitly bounded probability at fixed sparse density
- A high-degree vertex forces a large centered adjacency norm
- Convex and strictly convex functions on Euclidean convex sets
- Lipschitz map, α-Hölder map for rational 0 < α ≤ 1, and contraction
- Daido order parameters
- The constant projection and centering decomposition
- Phasors, trigonometric identities and periodic derivatives
- Even closed walks and traversal multiplicities
- Symmetrization and moments of the adjacency difference
- Gaussian comparison matrix in the moment argument
- Trace powers, closed walks and independent edge moments
- Gaussian entry moments and triangular independence
- Finite Bernoulli difference laws and parity of moments
- The adjacency matrix of a finite simple graph
- Finite phase tori, local lifts and the flat metric
- Energy dissipation along a downward gradient curve
- The Axiom of Countable Choice (AC_ω)
- In any metric space compactness implies countable compactness and limit point compactness, and each of countable compactness and limit point compactness implies sequential compactness; every implication here is proved without a choice principle
- A continuous real-valued function on a nonempty compact metric space is bounded and attains a greatest and a least value
- Continuity and derivatives of positive-base real powers
- The fundamental theorem of calculus and integration by parts
- Riemannian speed and length
- Mean value theorem on a closed real interval
- Sum, product, quotient, and chain rules from difference quotients
- Normalized phase volume and almost-everywhere assertions
- Gradient trajectories and complete compact phase dynamics
- Local center-stable trapping disc for a time map
- C¹ Euclidean maximal flows, variational dependence and the finite C² upgrade
- Finite matrix exponential series and its identities
- Assuming countable choice, every second countable space is Lindelöf
- The image of a lower-dimensional C¹ manifold is null
- An equidimensional C¹ map sends null sets to null sets
- Countable unions and subsets of manifold null sets are null
- A subset of ℝᵐ has Lebesgue outer measure zero if and only if it is null in the sense of countable closed-cube covers
- Null subsets of a smooth manifold
- Global rotation equivalence of phase configurations
- Diagonal phase rotations and relative-phase coordinates
- Rigidity and rotation of a unit-phasor average
- Graph Kuramoto differential equations
- Homogeneous Kuramoto energy
- Gradient and Hessian formulas
- Single-vertex degree laws and the shared-edge dependence
- Asymptotic quantifiers and uniform parameter ranges
- Loop coupling preserves the dynamics and centered Laplacian
- Simultaneous degree deviations on the squared logarithmic scale
- The unnormalized Laplacian needs its own degree deviation
- An elementary sufficient budget for the synchronization criterion
- Every fixed positive squared logarithmic density satisfies the budget
- Analytic potentials on a finite phase torus
- Differentiation of symmetric periodic pair potentials
- The finite union bound
- On the Landscape of Synchronization Networks: A Perspective from Nonconvex Optimization · Energy equation (1.1), PDF p. 1; periodic phase domain and synchronized configuration, PDF p. 2; Theorem 3.3, PDF p. 10; Proposition 5.1, PDF pp. 12–13
- A global synchronization theorem for oscillators on a random graph · Published Chaos 32, 093119 (2022): rotating frame and equation (1), p. 093119-2 (PDF p. 3); equation (2), same page; Theorem 11, p. 093119-8 (PDF p. 9); Theorem 13, p. 093119-9 (PDF p. 10)
- Concentration and regularization of random graphs · Section 1.1 equation (1.2), PDF p. 2; Theorem 1.1 and Section 1.4 example 1, PDF p. 3
- Empirical phase measures and Fourier order statistics
- A scoped continuum self-consistency input
- Width scaling of the continuum branch threshold
- The reals form a totally ordered field
- Lebesgue measurable sets, the family L(ℝⁿ), and the restricted set function λₙ
- Pairings, simple-graph conditioning and model transfer
- Feasible orders for labelled simple regular graphs
- Hamiltonian cycles as finite cyclic-order events
- Vertex relabeling preserves the uniform regular graph law
- Intersecting finitely many high-probability events
- From Hamiltonicity and an invariant event to a prescribed cycle
- R. W. Robinson and N. C. Wormald, Almost All Regular Graphs are Hamiltonian · Section 1, first paragraph and Theorem 1, printed/PDF page 2
- Feasible labelled simple regular graph orders
- Ordered eigenvalues and the centered regular norm
- Disconnected and bipartite spectral witnesses
- Exact normalized spectral threshold transfer
- Simple-model spectral events as conditional pairing counts
- Charles Bordenave, A new proof of Friedman’s second eigenvalue Theorem and its extension to random lifts · arXiv:1502.04482v4 (5 March 2019), Section 1, simple ensemble definition and Theorem 1, PDF/printed p. 2; eigenvalue ordering on p. 1
- Ceilings give the least positive integer meeting a real threshold
- The factorial n! and the falling factorial n^k̲, defined by recursion in ℕ
- Uniform small centered adjacency above logarithmic mean degree
- A positive minimum degree envelope at every fixed supercritical ratio
- Self-adjoint order, norm bounds and diagonal tests
- The integral logarithm is unbounded above and below
- Erdős–Rényi random graph and high probability
- The integral logarithm is continuous and strictly increasing on (0,∞)
- The integral logarithm L is the published natural logarithm
- Normalize the logarithmic random graph for nonbacktracking estimates
- A maximum-degree tail with an explicit logarithmic exponent
- Transfer two nonbacktracking good events to the required norm bound
- A nonbacktracking tail smaller than the required polynomial
- Uniform centered normalization across the full Bernoulli density range
- Uniform squared-row control including dense graphs
- A parameter-uniform nonbacktracking spectral tail
- The inner-product norm is definite, homogeneous, and satisfies the triangle inequality
- Every complete ordered field is Archimedean
- Florent Benaych-Georges, Charles Bordenave and Antti Knowles, Spectral radii of sparse random matrices (19 September 2019) · Section 2, PDF p. 3 Definition 2.1 and (2.2); PDF p. 4 Theorem 2.2
- Cut test-vector identities for Laplacian mixing
- Simple spinning graphs and cluster-constant equilibria
- An explicit eight-phase weighted stability certificate
- Transverse positivity for the special spinning family
- A positive transverse Hessian gives an open nonsynchronous basin
- Eduardo A. Canale, From weighted to unweighted graphs in Synchronizing Graph Theory, arXiv:2209.06362v3 · Section II.A, printed p. 4 (strong density); Section V, printed pp. 14–15 (degree, order, parameters and Theorem 2)
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
TheoremDependency level 3
Łojasiewicz inequality and analytic point stability
DefinitionDependency level 4
Ramanujan graph convention used in the article
DefinitionDependency level 4
Two-sided spectral expander — Definition 1.9
LemmaDependency level 4
A complete weighted network distinguishes a loop cap from spectral control
LemmaDependency level 4
The proposed small-gamma centered-adjacency tail has the opposite limit
LemmaDependency level 4
Extreme-value Lipschitz bounds and centered norm reduction
LemmaDependency level 4
Transfer of the analytic gradient inequality to a smooth metric
LemmaDependency level 4
Internal-edge mixing bound — Lemma 3.3
LemmaDependency level 4
Convex adjacency norm: sharp coordinate constant and median counterexample
LemmaDependency level 4
Spectral estimates for phase Fourier sums
LemmaDependency level 4
Trace-moment comparison with Gaussian matrices
PropositionDependency level 4
Point convergence for compact analytic periodic gradient flows
PropositionDependency level 4
A null basin for all strict saddles of a smooth compact gradient flow
PropositionDependency level 4
Bounds, saturation, and rotation of order parameters
PropositionDependency level 4
Rotating-frame gradient formulation
PropositionDependency level 4
Regular spectral characterization and notation convention
RemarkDependency level 4
Earlier thresholds with explicit source quantifiers and a fixed-scale deduction
RemarkDependency level 4
Background continuum phase transition
RemarkDependency level 4
Self-loops do not affect Kuramoto dynamics
TheoremDependency level 4
Hamiltonicity of a uniform fixed-degree regular graph
TheoremDependency level 4
Friedman spectral expansion in the uniform simple model
CorollaryDependency level 5
Counterexamples to both centered-adjacency tail bounds in Corollary 6.6
DefinitionDependency level 5
Equilibria, stable states, synchronized states, and spurious minima
LemmaDependency level 5
Uniform two-sided expansion sufficient for global synchrony
LemmaDependency level 5
The restricted spectral estimate needed above the connectivity threshold
LemmaDependency level 5
Uniform constant-two adjacency control above a large logarithmic density
LemmaDependency level 5
Degree bounds from two-sided expansion — Lemma 3.1
LemmaDependency level 5
Cut-edge mixing bound — Lemma 3.4
PropositionDependency level 5