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Theorems

Relate all of these to multiplication.

FT = Fundamental Theorem

Multiplication has systemic unit 1. Compare with 0 and blank.

• Intrinsic - algorithm construction
• Extension - induction
• Correspondence - bijection
• Unique map - permutation-construction
• Interior and extrema - examination of cases
• Factors - substitution (binomial theorem, lambda calculus, variables)

Yoneda Lemma

Correspondence (division) Bijection

FT Noncommutative algebra. Proper subalgebra of linear transformations of a finite dimensional complex vector space accords with a nontrivial invariant subspace. (Also related to Algebra in that a proper invariant subspace exists.)

FT Finite distributive lattice. Elements can be represented as sets, relations as unions or intersections.

FT Galois theory. Matches intermediate fields of a field extension and the subgroups of its Galois group.

Extension (of domain of addition formula for exponents) Induction

FT Topos theory. The slice E/X of a topos E over its object X is likewise a topos.

FT Algebraic K-theory. How to change the ring.

FT Ultraproducts. Conditions such that a first order formula is true for an ultraproduct.

Non-correspondence

Cantor's theorem. A set has a power set with strictly greater cardinality.

Pigeon hole principle. What happens if pigeons outnumber holes. Non-surjectivity.

Intrinsicness - Algorithm Construction

FT Surface theory. Gaussian curvature is independent of the embedding.

FT Central limit theorem. Sum of independent, identically distributed random variables tends toward normal distribution.

Composition (primes) Substitution

factors

Binomial theorem.

I FT Algebra. Polynomial is composed of factors. Existence of solutions to polynomial equations.

FT Tessarine algebra. Polynomial of degree n has {$n^2$} roots.

FT Arithmetic. Integers are composed of prime factors.

FT Cyclic groups. Determines the subgroups of a cyclic group, one for each divisor of the order.

FT Finitely generated abelian groups. Characterizes their structure as direct sums.

FT Finitely generated modules over a principal ideal domain. Characterizes their structure as direct sums.

FT Ideal theory in number fields. Unique factorization of proper ideals in terms of prime ideals.

FT Symmetric polynomials. Elementary symmetric polynomials form a basis for symmetric polynomials.

FT Projective geometry. Homography is the composition of a finite number of perspectivities.

Classification of closed surfaces. Connected sums of tori and one or two real projective planes.

components

FT Vector calculus. Vector field broken down into curl-free and divergence-free components.

I FT Curves. Shape of a 3-dimensional curve is determined by its curvature and torsion.

Extrema and interior (increasing) Examination of cases

FT Linear programming. Extreme values occur at points on the boundary.

FT Calculus. Integration of a function is matched with differentiation of its antiderivative.

FT Lebesgue integral calculus. Absolute continuity iff is an integral of a derivative.

Stokes theorem.

FT Geometric calculus. Integral of a derivative over a volume equals the integral of the function over the boundary.

Cauchy's integral formula. Complex analytic function, holomorphic function, on a disc is determined by its values on the boundary.

Banach fixed point theorem. Contraction mapping has a unique fixed point.

Universality - Unique map (uniqueness) Construction

FT Homomorphisms. When a subgroup K is mapped to the identity element, then the map can be uniquely interpreted in terms of the cosets of K.

Orbit stabilizer theorem. Bijection between an orbit and the set of cosets of the stabilizer subgroup.

FT Projective geometry. Unique homography.

FT Riemannian geometry. Unique torsion-free metric connection.

• Short exact sequence of chain complexes implies long exact sequence of their homology groups
• Changing the order of integration
• The existence of a primitive root in (Z/pZ)×
• Halting problem
• Existence of nonmeasurable Lebesgue sets
• Row-rank equals column-rank (f.t. of linear algebra?)
• Convergence criteria for geometric series (r<1)
• Convergence of monotone sequences: Any Lp-integrable function can be Lp-approximated by step functions.
• Fourier transform and Fourier series
• Laplace transform
• flowbox theorem - Frobenius theorem