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
- Rational sine enclosures from a finite factorial sum
- Rational logarithm and pi enclosures after range reduction
- Principal inverse sine and inverse cosine
- Series, partial sums, convergence and the sum, divergence, and the tail series
- A series of nonnegative terms converges iff its partial sums are bounded, and then the sum is their supremum
- The rationals embed densely in the reals
- Finite phase tori, local lifts and the flat metric
- Phasors, trigonometric identities and periodic derivatives
- Rigidity and rotation of a unit-phasor average
- Differentiation of symmetric periodic pair potentials
- Real symmetric quadratic forms tested on phasors
- Diagonal phase rotations and relative-phase coordinates
- Local minima and nonnegative second variation
- Finite symmetric networks and loop removal
- Connected graphs and connected components defined by the existence of vertex paths
- The chain rule for total derivatives: D(g∘ f)(a)=Dg(f(a))∘ Df(a)
- 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
- 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)
- Rational inverse-sine brackets on the principal branch
- The congruence class [a]ₙ and the quotient set ℤ/n
- Addition and multiplication on ℤ/n by [a]ₙ+[b]ₙ=[a+b]ₙ and [a]ₙ[b]ₙ=[ab]ₙ
- For n≥ 1, every class in ℤ/n has one representative r with 0≤ r<n, so |ℤ/n|=n; while ℤ/0 is in bijection with ℤ
- Parity and the Pythagorean identity for sine and cosine
- The zero sets of sine and cosine and the least positive common period 2 pi
- The derivatives of sine and cosine are cosine and minus sine
- Sum, product, quotient, and chain rules from difference quotients
- Eduardo A. Canale, From weighted to unweighted graphs in Synchronizing Graph Theory, arXiv:2209.06362v3 · Section II.A, p. 4 (spinning construction); Section II.B, pp. 5–7 (equation, potential and Hessian); Section III, Proposition 1, p. 7 (equilibrium lifting)
- Gradient trajectories and complete compact phase dynamics
- Normalized phase volume and almost-everywhere assertions
- The Axiom of Countable Choice (AC_ω)
- 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 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"
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
Lemma
A uniform inverse-sine majorant and geometric angular tail
Proposition
A complete-graph second-order landscape application
Proposition
One-vertex composition of phase-difference landscapes
Proposition
Twisted cycle equilibria and transverse positivity
Lemma
Exact rational replay of the regular amplification constants
Definition
Simple spinning graphs and cluster-constant equilibria
Lemma