Alphabeta MathArticle items · 09 — Bipartite Ramanujan existence and the centered spectrum
Presents 28 items from the article and its immediate prerequisites, covering dependency levels 2–3.
1 · Prerequisites
- Finite symmetric networks and loop removal
- Regular adjacency spectra and normalized expansion
- Interlacing families I: Bipartite Ramanujan graphs of all degrees · Section 2.1, printed p. 308; Theorem 5.5 and its proof, printed p. 316
- The Erdős-Rényi finite random graph G(n,p)
- Connected graphs and connected components defined by the existence of vertex paths
- Asymptotic quantifiers and uniform parameter ranges
- Factorial moments of isolated vertices at the connectivity window
- Finite sieve determines the probability of a zero count
- Components between two vertices and half the graph disappear in the window
- Finite coupling makes connectivity probability monotone
- Finite probability spaces, outcome weights, events, and event probabilities
- Order, continuity, range, and the product, quotient, and reciprocal laws for the natural logarithm
- The logarithm grows more slowly than every positive real power
- The exponential tends to +∞ at +∞ and to 0 at -∞
- The exponential is a continuous bijection from ℝ onto (0,∞)
- Algebra of limits: sums, scalar multiples, products and quotients
- The natural logarithm as the inverse of the exponential function
- Divergence to +∞ and to -∞
- Limits at +∞ and -∞, and infinite limits at a point
- Convex Lipschitz concentration with its coordinate metric
- Independent-entry coordinates, convexity and Lipschitz constants
- Independent random elements
- Random elements and real random variables
- Probability measures and probability spaces
- Adjacency, degrees, graph Laplacian, and spectral notation
- Centered graph matrices and reference degree
- Loewner order, Hermitian inner product, and operator norm
- Oriented edge count between sets
- The cardinality | A| of a finite set
- lvertP(A)| = 2^| A| for finite A
- A subset of a finite set is finite, with | B| ≤ | A|, and equality holds if and only if B = A
- Every nonempty finite set of reals has a maximum and a minimum
- Maximum and minimum of a set
- Graph Kuramoto differential equations
- Finite phase tori, local lifts and the flat metric
- Diagonal phase rotations and relative-phase coordinates
- Phasors, trigonometric identities and periodic derivatives
- The Axiom of Countable Choice (AC_ω)
- Homogeneous Kuramoto energy
- Integer part: for every real x there is exactly one integer m with m ≤ x < m + 1
- The Hessian matrix and critical points of a scalar field
- Sum, product, quotient, and chain rules from difference quotients
- Weighted spectral bounds control the degree without loops
- Finite vertex Hilbert space in the article convention
- Self-adjoint order, norm bounds and diagonal tests
- Loop removal in the weighted Kuramoto calculus
- Erdős–Rényi random graph and high probability
- Expectation of a real random variable on a finite probability space
- Bernoulli random variables and binomial random variables as sums of independent Bernoulli trials
- The operator norm is zero on the zero domain and otherwise is max_‖ v‖=1‖ Tv‖
- Cauchy-Schwarz |⟨ x,y⟩| ≤ ‖ x‖₂‖ y‖₂ with its equality case, the triangle inequality for ‖·‖₂, the parallelogram law and polarisation
- Quadratic Gaussian process and its increment metric
- Triangular Gaussian coordinates: metric and moment control
- The Axiom of Choice
- From finite Gaussian comparisons to a sphere supremum
- Cauchy-Schwarz for random variables
- A finite stack of independent Gaussian vectors is jointly Gaussian
- Multivariate normal law, including singular covariance
- Cramer wold device
- A map into ℝⁿ is measurable exactly when its coordinates are measurable
- Continuous functions on Euclidean spaces are Borel measurable
- Composition with a Borel measurable outer map preserves measurability
- Gaussian entry moments and triangular independence
- The Lebesgue integral is linear on L¹(μ)
- Monotonicity and nonnegative homogeneity of the nonnegative integral
- Law or distribution of a random element
- Change of variables for expectation
- Ordered edge counts and centered cut vectors
- The constant projection and centering decomposition
- Counts of vertices of each degree
- Single-vertex degree laws and the shared-edge dependence
- Indicator expectations and exceptional-vertex probabilities
- Binomial mass ratios and geometric tail domination
- Expectation is linear for every finite family of random variables, without any independence hypothesis
- Indicators turn event probabilities, intersections, and finite counts into expectations and products
- Independent-copy symmetrization matrix
- Norm symmetrization at the actual matrix mean
- Finite Bernoulli difference laws and parity of moments
- Trace powers, closed walks and independent edge moments
- Finite Jensen inequality for a convex function and nonnegative weights summing to one
- Scope of the phase-only sinusoidal oscillator idealization
- Empirical phase measures and Fourier order statistics
- Expectation of a nonnegative or integrable random variable
- 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 scoped continuum self-consistency input
- The reals form a totally ordered field
- A C¹ diffeomorphism satisfies the change-of-variables formula for L¹ functions
- The class L¹(μ) of integrable functions
- Integrable real and complex functions, and their integrals
- Differentiation of symmetric periodic pair potentials
- Real symmetric quadratic forms tested on phasors
- The chain rule for total derivatives: D(g∘ f)(a)=Dg(f(a))∘ Df(a)
- Pairwise and mutual independence of finite-valued random variables
- 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 continuous map has Borel preimages of Borel sets
- A complete-graph second-order landscape application
- 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)
- Markov's inequality on a finite probability space
- Real random variables on finite probability spaces and their finite distributions
- Variance, standard deviation, and covariance on a finite probability space
- An independently attributed heterogeneous-model chaos report
- Chaotic Attractor in the Kuramoto Model — Maistrenko, Popovych and Tass (2005) · Publisher-indexed abstract; bibliographic identity confirmed in JuSER record 14750
- Tail integration and a median-to-mean bridge
- The integral of a nonnegative simple function
- The nonnegative integral agrees with the simple integral on simple functions
- Dyck words and elementary Catalan growth
- Extracting a nonconstant eigenvalue from closed-walk moments
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 2
Bipartite Ramanujan existence and the centered spectrum
TheoremDependency level 2
Connectivity in the logarithmic random-graph window
TheoremDependency level 2
Convex Lipschitz concentration — Theorem 6.5
DefinitionDependency level 3
Adjacency spectral expander — Definition 1.3
DefinitionDependency level 3
Edge isoperimetric constant convention
DefinitionDependency level 3
Phase-locked long-term behavior
DefinitionDependency level 3
Global rotation equivalence of phase configurations
DefinitionDependency level 3
Strict saddle critical point
LemmaDependency level 3
Loop-dependent weighted degree envelope
LemmaDependency level 3
Covariance under a common positive rescaling of weights
LemmaDependency level 3
Loop-free Bernoulli adjacency and its two centerings
LemmaDependency level 3
A high-degree vertex forces a large centered adjacency norm
LemmaDependency level 3
Expected extreme eigenvalues from finite Gaussian comparison
LemmaDependency level 3
Nested-cut polarization identity
LemmaDependency level 3
Cut test-vector identities for Laplacian mixing
LemmaDependency level 3
Lower degree-tail count estimate — Lemma 6.7
LemmaDependency level 3
Upper degree-tail count estimate — Lemma 6.8
LemmaDependency level 3
Symmetrization and moments of the adjacency difference
PostulateDependency level 3
Phase oscillator interpretation and homogeneous modeling assumption
PropositionDependency level 3
Width scaling of the continuum branch threshold
PropositionDependency level 3
Gradient and Hessian formulas
RemarkDependency level 3
Symmetry and moments in the random-matrix discussion
RemarkDependency level 3
Complete-graph positive-semidefinite equilibria
RemarkDependency level 3
Classical Erdős–Rényi connectivity threshold
RemarkDependency level 3
Binomial approximation for vertex degree counts
RemarkDependency level 3
Possible chaos with nonidentical intrinsic frequencies
RemarkDependency level 3
Median–mean gap for convex Lipschitz concentration
TheoremDependency level 3