Alphabeta MathSpectral estimates and phase landscapes
Supply graph and matrix conventions, centered cut estimates, angular and numerical controls, and graph-specific and vector-spin landscape applications.
1 · Prerequisites
- Finite real interval operations enclose their exact values
- Principal inverse sine and inverse cosine
- Order, continuity, range, and the product, quotient, and reciprocal laws for the natural logarithm
- The derivatives of sine and cosine are cosine and minus sine
- 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
- Exact rational replay of the regular amplification constants
- Real intervals, complex rectangles and ball enclosures
- Outward-rounded interval, rectangle and ball operations
- Dyadic floating-point format and directed rounding
- Directed dyadic rounding encloses with explicit underflow and overflow bounds
- Parity and the Pythagorean identity for sine and cosine
- A function differentiable at c is continuous at c
- Mean value theorem on a closed real interval
- Simple spinning graphs and cluster-constant equilibria
- Self-adjoint order, norm bounds and diagonal tests
- Real spectral theorem: a self-adjoint endomorphism of a finite-dimensional real inner product space has an orthonormal eigenbasis
- Phasors, trigonometric identities and periodic derivatives
- Angular order and elementary inverse-trigonometric bounds
- Quarter-turn values and shifts by pi/2 and pi
- Eduardo A. Canale, From weighted to unweighted graphs in Synchronizing Graph Theory, arXiv:2209.06362v3 · Section II.B, equations (1)–(3), PDF pp. 5–7; Section V, Figure 2 and Lemma 1, PDF pp. 11–14
- A positive transverse Hessian gives an open nonsynchronous basin
- Finite symmetric networks and loop removal
- Normalized phase volume and almost-everywhere assertions
- The Axiom of Countable Choice (AC_ω)
- 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)
- Dedekind completeness: the least-upper-bound property
- Kassabov, Strogatz and Townsend, Sufficiently dense Kuramoto networks are globally synchronizing, arXiv:2105.11406v1 · Section I, p. 1, Eq. (1) and definition of global synchronization; Section II, p. 2, Eq. (2); Section V, p. 5, Theorem 5 and the paragraph immediately following its proof
- 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 (regular degree, 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
Definition
Certificates for finite transcendental inequalities
Lemma
An explicit eight-phase weighted stability certificate
Lemma
Transverse positivity for the special spinning family
Proposition
A simple nonsynchronizing family approaching eleven sixteenths
Remark