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When one considers a rigid body undergoing a plane motion, which is essentially a blend of translational and rotational movement, the application of Newton's second law gives the formula for the translational movement of such a body. If this equation is multiplied by a time interval, dt, and then integrated over the limits of integration, it results in an equation that embodies the principle of linear impulse.
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Metric fixed point theory and partial impredicativity.

D Fernández-Duque1,2, P Shafer3, H Towsner4

  • 1Department of Mathematics WE16, Ghent University, Ghent, Belgium.

Philosophical Transactions. Series A, Mathematical, Physical, and Engineering Sciences
|April 9, 2023
PubMed
Summary

The Priess-Crampe & Ribenboim fixed point theorem is provable in [Formula: see text]. Caristi's fixed point theorem is equivalent to a principle strictly between [Formula: see text] and [Formula: see text].

Keywords:
computability theoryfixed-point theoremsreverse mathematicssecond-order arithmeticvariational principles

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Area of Science:

  • Mathematical analysis
  • Set theory
  • Foundations of mathematics

Background:

  • Fixed-point theorems are fundamental in various mathematical fields.
  • Understanding the axiomatic strength of these theorems is crucial for mathematical logic and proof theory.
  • The relationship between different fixed-point theorems and foundational principles remains an active area of research.

Purpose of the Study:

  • To determine the provability of the Priess-Crampe & Ribenboim fixed point theorem within a specific axiomatic system.
  • To establish the precise logical relationship between Caristi's fixed point theorem and foundational principles.
  • To explore weakenings of Caristi's theorem and their equivalences to specific axioms.

Main Methods:

  • Axiomatic proof-theoretic analysis.
  • Logical equivalences and strict inequalities between mathematical statements.
  • Investigating fixed-point theorems for Baire and Borel functions.

Main Results:

  • The Priess-Crampe & Ribenboim fixed point theorem is shown to be provable in [Formula: see text].
  • Caristi's fixed point theorem for Baire and Borel functions is proven equivalent to the transfinite leftmost path principle.
  • This principle is shown to lie strictly between [Formula: see text] and [Formula: see text].
  • Several weakenings of Caristi's theorem are identified and shown to be equivalent to [Formula: see text] and [Formula: see text].

Conclusions:

  • The study precisely positions the axiomatic strength of key fixed-point theorems within the mathematical landscape.
  • It clarifies the logical dependencies between these theorems and foundational principles like the transfinite leftmost path principle.
  • The findings contribute to a deeper understanding of proof theory and the structure of mathematical reasoning.