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
- A finite simple graph is a finite vertex set together with a set of two-element vertex subsets
- The adjacency matrix of a finite simple graph
- Finite weighted graph Laplacian
- Finite rectangular matrices over a commutative ring, their entries, rows and columns
- A finite sum in a commutative monoid indexed by an arbitrary finite set
- Real and complex inner product spaces, with the inner product linear in the first argument
- The norm ‖ v‖=√⟨ v,v⟩ induced by a real or complex inner product
- The induced length is a norm
- Every finite-dimensional normed space is Banach
- Hilbert space
- The adjoint T^*:W→ V is characterised by ⟨ Tv,w⟩_W=⟨ v,T^*w⟩_V
- Linear map between vector spaces over the same field
- Basis of a vector space: a linearly independent spanning subset; and ordered basis: an injective finite list whose image is a basis
- Eigenvalues, eigenvectors, eigenspaces E_λ(T)=ker(T-λ I), and the spectrum σ_F(T) of an endomorphism
- Vector space over a field
- Real and imaginary parts, complex conjugation, and modulus
- Conjugation is an involutive real-field automorphism, zz̄=|z|², and modulus is definite, multiplicative, and subadditive
- Signs, monotonicity intervals, and ranges of sine and cosine
- The zero sets of sine and cosine and the least positive common period 2 pi
- Quarter-turn values and shifts by pi/2 and pi
- Principal inverse sine and inverse cosine
- For -1<y<1, (arcsin y)ᵖʳⁱᵐᵉ=1/√1-y² and (arccos y)ᵖʳⁱᵐᵉ=-1/√1-y²
- The natural logarithm as the inverse of the exponential function
- Order, continuity, range, and the product, quotient, and reciprocal laws for the natural logarithm
- The natural logarithm has derivative 1/x and equals the integral from 1 to x of 1/t
- The mean value theorem, as the case g(x) = x of Cauchy's: for f continuous on [a,b] with a < b and differentiable on (a,b) there is c ∈ (a,b) with f(b) - f(a) = f'(c)(b-a)
- Sum, product, quotient, and chain rules from difference quotients
- Square roots exist: a unique √a ≥ 0 with (√a)² = a; the positives are {x² : x ≠ 0}
- For |r| < 1, ∑_k ≥ 0 rᵏ = 1/(1-r), and for |r| ≥ 1 the series diverges
- Integer part: for every real x there is exactly one integer m with m ≤ x < m + 1
- The principle of mathematical induction
- Laws of integer exponents
- A nondecreasing sequence bounded above converges to the supremum of its range, and a nonincreasing sequence bounded below to the infimum
- Sequences of reals: bounded, eventually, frequently, tails, subsequences
- Integer powers aᵐ
- Finite sums and finite products, by recursion
- Series, partial sums, convergence and the sum, divergence, and the tail series
- The polynomial ring R[xᵢ:i∈ I] as finitely supported coefficient families on monomials
- Intermediate value theorem, by bisection with a canonical left-half rule: a continuous function on [a,b] takes every value between f(a) and f(b)
- On an interval I, for f continuous on I and differentiable at every interior point: f' ≥ 0 throughout gives f nondecreasing, f' > 0 gives f increasing, f' ≤ 0 and f' < 0 give the two decreasing forms; conversely a nondecreasing f has f' ≥ 0 and a nonincreasing f has f' ≤ 0 wherever it is differentiable, and no strict converse is claimed
- Finite real interval operations enclose their exact values
- Universal property of a polynomial ring on an arbitrary family of indeterminates
- Intervals of ℝ: the nine order-convex forms, nondegeneracy, and length
- Continuity of f : A → ℝ at a point of A and on A: the ε-δ condition, its agreement with lim_x → c f(x) = f(c) at a limit point, and continuity at an isolated point
- Sign rules for products and monotonicity of multiplication
- The derivative f'(c) = lim_x → c f(x) - f(c)/x - c of f : A → ℝ at a point c ∈ A that is a limit point of A, and differentiability on a set
- Finite probability spaces, outcome weights, events, and event probabilities
- Conditional probability ℙ(A∣ B) for ℙ(B)>0
- The factorial n! and the falling factorial n^k̲, defined by recursion in ℕ
- The uniform probability space on a nonempty finite set
- Matchings, saturated vertices, maximal and maximum matchings, perfect matchings and ν(G)
- Empty and complete graphs, complete bipartite graphs, and the convention that Pₙ and Cₙ have n vertices
- The Gram matrix G(v₀,…,vᵣ₋₁)=(⟨ vᵢ,vⱼ⟩)_i,j<r and Gram determinant, with empty value 1
- Row space, column space, nullspace, row rank, column rank and matrix rank
- The Euclidean inner product ⟨ x,y⟩ = ∑_k<n xₖ yₖ on ℝⁿ
- The trace tr(A) as the sum of the diagonal entries
- Rectangular matrix multiplication and the identity matrix Iₙ, including zero-sized shapes
- Laws of finite sums and finite products
- ∑_i ∈ S∑_j ∈ T aᵢⱼ = ∑_(i,j) ∈ S × T aᵢⱼ = ∑_j ∈ T∑_i ∈ S aᵢⱼ for finite index sets S and T
- Row rank equals column rank, and both equal the number of pivots
- If dim_F V = n and U is a linear subspace of V, then U is finite-dimensional, dim_F U ≤ n, and dim_F U = n if and only if U = V
- The standard list e : n → Fⁿ with eᵢ(i) = 1_F and eᵢ(j) = 0_F for j ≠ i is an ordered basis of Fⁿ; hence dim_F Fⁿ = n, and F⁰ is the zero space with basis ∅ and dimension 0
- The transpose A^T of a matrix
- The metric topology: a set is open when every one of its points has a ball around it inside the set; closed means open complement
- Subspace topology: the traces of the open sets, its closed sets and its bases, the continuity of the inclusion, and the characteristic property of a map into a subspace
- Embedded submanifolds and slice charts
- A regular level set is an embedded submanifold
- Slice-chart restrictions form a smooth atlas
- The tangent space to a regular level set
- Cʳ and smooth maps between smooth manifolds
- Cᵏ maps and multi-index derivative notation in Euclidean space
- If all partial derivatives exist on a neighbourhood and are continuous at a point, then the map is totally differentiable there with Jacobian derivative
- Identity maps and composites of smooth maps are smooth
- Critical points and critical values of a smooth function
- The differential of a smooth real-valued function
- The tangent space of a regular level set is the kernel
- The tangent space of Euclidean space
- Coordinate formula for the differential
- Sine and cosine defined by their real power series
- The sine and cosine power series converge absolutely for every real argument
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
Finite symmetric networks and loop removal
Definition
Finite vertex Hilbert space in the article convention
Proposition
Angular order and elementary inverse-trigonometric bounds
Proposition
Geometric growth, truncation and total decrement budgets
Definition
Root and sign certificates for parameter-dependent polynomials
Definition
Pairings, simple-graph conditioning and model transfer
Definition
Signed vector-spin objectives and low-rank Gram factors
Lemma