Alphabeta MathArticle items · 08 — A scoped interacting-particle transformer connection
Presents 28 items from the article and its immediate prerequisites, covering dependency levels 1–2.
1 · Prerequisites
- Sphere configurations and tangent variations
- A mathematical perspective on Transformers · Abstract; Sections 1, 2.2, 7.1 and 7.2, especially equations (SA), (2.6), (7.1) and (7.4)
- Independent-entry coordinates, convexity and Lipschitz constants
- Independent random elements
- Probability measures and probability spaces
- A convex subset of ℝᵐ contains every line segment between two of its points
- Random elements and real random variables
- Concentration Inequalities: A Nonasymptotic Theory of Independence · Section 7.5, Theorem 7.12, printed p. 222, PDF p. 231
- Asymptotic quantifiers and uniform parameter ranges
- Finite vertex Hilbert space in the article convention
- Moments, variance, and covariance on a probability space
- Law or distribution of a random element
- Complex spectral theorem: a normal endomorphism of a finite-dimensional complex inner product space has an orthonormal eigenbasis, and conversely
- The operator norm is zero on the zero domain and otherwise is max_‖ v‖=1‖ Tv‖
- Wigner random matrices with non-symmetrically distributed entries · Section 1, equation (1), Theorem 1.1 and Remark 1.1, PDF pages 1–2
- Adjacency, degrees, graph Laplacian, and spectral notation
- Erdős–Rényi random graph and high probability
- The constant projection and centering decomposition
- Expectation of a real random variable on a finite probability space
- Expectation is linear for every finite family of random variables, without any independence hypothesis
- Bernoulli random variables and binomial random variables as sums of independent Bernoulli trials
- The indicator random variable of an event
- Finite symmetric networks and loop removal
- Ordered edge counts and centered cut vectors
- A finite sum in a commutative monoid indexed by an arbitrary finite set
- The difference a ∖ b, the symmetric difference a △ b, and the complement X ∖ a relative to a set X
- The intersection ⋂ x of a nonempty set, the binary intersection a ∩ b := ⋂{a,b}, and disjointness
- The Cartesian product A × B := { z ∈ P(P(A ∪ B)) : ∃ a ∈ A ∃ b ∈ B z = (a,b) }
- Finite phase tori, local lifts and the flat metric
- Phasors, trigonometric identities and periodic derivatives
- Trace powers, closed walks and independent edge moments
- Walks, closed walks, trails, paths and cycles, with length equal to the number of traversed edges
- A finite simple graph is a finite vertex set together with a set of two-element vertex subsets
- The Axiom of Countable Choice (AC_ω)
- Every smooth vector field on a compact manifold is complete
- C¹ Euclidean maximal flows, variational dependence and the finite C² upgrade
- A smooth vector field is a smooth section of the tangent bundle
- Integral curves of a vector field
- Complete vector fields
- Self-adjoint order, norm bounds and diagonal tests
- The operator norm is 0 on the zero domain and otherwise equals the largest singular value, attained at a right-singular vector
- Partial order and partially ordered set
- Uniform edge orderings, graph-process times and independent labels
- The uniform probability space on a nonempty finite set
- Finite Bernoulli difference laws and parity of moments
- Measures on sigma-algebras
- Finite probability spaces, outcome weights, events, and event probabilities
- Entropy endpoint limits and the two monotone branches
- The natural logarithm has derivative 1/x and equals the integral from 1 to x of 1/t
- Choice-free smooth inverse function theorem in Euclidean space
- Cᵏ Euclidean maps and diffeomorphisms
- 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
- 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
- A function differentiable at c is continuous at c
- The principle of mathematical induction
- Order, continuity, range, and the product, quotient, and reciprocal laws for the natural logarithm
- Rational sine enclosures from a finite factorial sum
- Rational inverse-sine brackets on the principal branch
- Rational logarithm and pi enclosures after range reduction
- A uniform inverse-sine majorant and geometric angular tail
- Differentiation of symmetric periodic pair potentials
- Gradient trajectories and complete compact phase dynamics
- Diagonal phase rotations and relative-phase coordinates
- Normalized phase volume and almost-everywhere assertions
- The Hessian matrix and critical points of a scalar field
- Clairaut--Schwarz theorem for continuous second partial derivatives
- Real spectral theorem: a self-adjoint endomorphism of a finite-dimensional real inner product space has an orthonormal eigenbasis
- If all partial derivatives exist on a neighbourhood and are continuous at a point, then the map is totally differentiable there with Jacobian derivative
- The chain rule for total derivatives: D(g∘ f)(a)=Dg(f(a))∘ Df(a)
- Sum, product, quotient, and chain rules from difference quotients
- The fundamental theorem of calculus and integration by parts
- If f ≤ g on [a,b] and both are integrable then ∫ₐᵇ f ≤ ∫ₐᵇ g; and m(b-a) ≤ ∫ₐᵇ f ≤ M(b-a)
- The exponential function is smooth and (exp)'=exp
- The exponential is positive and satisfies exp(-x)=1/exp(x)
- The exponential tends to +∞ at +∞ and to 0 at -∞
- Eduardo A. Canale, From weighted to unweighted graphs in Synchronizing Graph Theory, arXiv:2209.06362v3 · Section II.B, printed p. 6 (PDF page 6), paragraph beginning "One way to see"
- Independent degrees into a fixed half of a random graph
- An explicit lower bound on a sparse binomial atom
- The Erdős-Rényi finite random graph G(n,p)
- Adjacency, incidence, open and closed neighbourhoods, vertex degree, minimum degree and maximum degree
- Triangular Gaussian coordinates: metric and moment control
- Closure properties of measurable functions used by the integral
- Monotonicity and nonnegative homogeneity of the nonnegative integral
- The Axiom of Choice
- Extended-real-valued measurable functions
- Standard normal and normal laws
- Expectation of a nonnegative or integrable random variable
- Regular adjacency spectra and normalized expansion
- Pairings, simple-graph conditioning and model transfer
- The adjacency matrix of a finite simple graph
- Chernoff bounds with the exact degree mean
- The integral logarithm is unbounded above and below
- The integral logarithm L is the published natural logarithm
- Rigidity and rotation of a unit-phasor average
- Real symmetric quadratic forms tested on phasors
- Support connectivity and the unnormalized Laplacian kernel
- Signs, monotonicity intervals, and ranges of sine and cosine
- Quarter-turn values and shifts by pi/2 and pi
- Cauchy-Schwarz |⟨ x,y⟩| ≤ ‖ x‖₂‖ y‖₂ with its equality case, the triangle inequality for ‖·‖₂, the parallelogram law and polarisation
- Second-order Taylor expansion f(a+h)=f(a)+∇ f(a)· h+1/2h^TH_f(a)h+o(‖h‖²)
- Local minima and nonnegative second variation
- Connected graphs and connected components defined by the existence of vertex paths
- 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
- Truncated sine-square function
- Angular order and elementary inverse-trigonometric bounds
- Complex sine, cosine, hyperbolic sine, and hyperbolic cosine from the complex exponential
- Empirical phase measures and Fourier order statistics
- From Kuramoto to Crawford: exploring the onset of synchronization in populations of coupled oscillators · §§3.1–3.2 and 4, equations (3.1)–(3.3) and (4.1)–(4.6), printed pp. 3–8
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
RemarkDependency level 1
A scoped interacting-particle transformer connection
TheoremDependency level 1
Convex Lipschitz concentration with its coordinate metric
TheoremDependency level 1
A fixed-law Wigner spectral-edge input
DefinitionDependency level 2
Centered graph matrices and reference degree
DefinitionDependency level 2
Counts of vertices of each degree
DefinitionDependency level 2
Oriented edge count between sets
DefinitionDependency level 2
Homogeneous Kuramoto energy
DefinitionDependency level 2
Even closed walks and traversal multiplicities
DefinitionDependency level 2
Graph Kuramoto differential equations
DefinitionDependency level 2
Loewner order, Hermitian inner product, and operator norm
DefinitionDependency level 2
Random graph process and connectivity hitting time
DefinitionDependency level 2
Uniform random regular graph convention
DefinitionDependency level 2
Independent-copy symmetrization matrix
LemmaDependency level 2
Differentiable entropy inverse branches and shifted contracts
LemmaDependency level 2
Exact rational replay of the regular amplification constants
LemmaDependency level 2
Loop removal in the weighted Kuramoto calculus
LemmaDependency level 2
Weighted spectral bounds control the degree without loops
LemmaDependency level 2
A negative adjacency eigenvalue survives all-ones centering
LemmaDependency level 2
A positive transverse Hessian gives an open nonsynchronous basin
LemmaDependency level 2
A large degree with explicitly bounded probability at fixed sparse density
LemmaDependency level 2
Even trace moments transfer from the operator norm
LemmaDependency level 2
Ordered eigenvalues and the centered regular norm
LemmaDependency level 2
Uniform degree slack and the vanishing synchrony condition
PropositionDependency level 2
A complete-graph second-order landscape application
PropositionDependency level 2
Twisted cycle equilibria and transverse positivity
PropositionDependency level 2
One-vertex composition of phase-difference landscapes
PropositionDependency level 2
Elementary properties of the function H
RemarkDependency level 2