A Reflection Principle As a Reverse-mathematical Fixed Point over the Base Theory ZFC

The Lefschetz Principle , Fixed Point Theory , and Index Theory

This is a rough historical account of some uses of the Lefschetz Principle in fixed point theory and index theory. The Lefschetz Principle states that the alternating sum of the traces on cohomology (a global and rigid invariant) is equal to the alternating sum of the traces on the underlying cochain complex (a local and far less rigid invariant). The original Lefschetz Theorem for compact poly...

The renormalization fixed point as a mathematical object

Stable Homotopy Theory over a Fixed Base Space

A spectral sequence which may be regarded as an 'Adams spectral sequence over a fixed space 2?' has been constructed by J. F. McClendon [6] and J.-P. Meyer [8]. This note describes a generalization in which there is no need for any orientability assumptions. The construction is carried out in a suitable stable category, which may be of independent interest. An application to the enumeration of ...

Inductive Inference on the Base of Fixed Point Theory

Inductive reasoning is an important way to generate knowledge from the propositions reflecting facts or directly from data We intend to extract new knowledge in the form of definitions given as fixed point equations. An appropriate fixed point theory is outlined in favour of our aim. This theory suggests the so called generative fixed point equation system as the form of hypothesis that we are ...

A Discrete Fixed Point Theorem of Eilenberg as a Particular Case of the Contraction Principle

Theorem 1.1 (Eilenberg). Let X be an abstract set and let (Rn)n=0 be a sequence of equivalence relations in X such that (i) X ×X = R0 ⊇ R1 ⊇ ··· ; (ii) ⋂∞ n=0Rn = ∆, the diagonal in X ×X ; (iii) given a sequence (xn)n=0 such that (xn,xn+1) ∈ Rn for all n ∈ N0, there is an x ∈ X such that (xn,x) ∈ Rn for all n∈ N0. If F is a self-map of X such that given n∈ N0 and x, y ∈ X , (x, y) ∈ Rn =⇒ (Fx,F...