Subword complexity and finite characteristic numbers

AbstractWe discuss a problem of Arnold, whether every function is stably equivalent to one which is non-degenerate for its Newton diagram. We argue that the answer is negative. We describe a method to make functions non-degenerate after stabilisation and give examples of singularities where this method does not work. We conjecture that they are in fact stably degenerate, that is not stably equivalent to non-degenerate functions.We review the various non-degeneracy concepts in the literature. For finite characteristic, we conjecture that there are no wild vanishing cycles for non-degenerate singularities. This implies that the simplest example of singularities with finite Milnor number, $$x^p+x^q$$ x p + x q in characteristic p, is not stably equivalent to a non-degenerate function. We argue that irreducible plane curves with an arbitrary number of Puiseux pairs (in characteristic zero) are stably non-degenerate. As the stabilisation involves many variables, it becomes very difficult to determine the Newton diagram in general, but the form of the equations indicates that the defining functions are non-degenerate.

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Subword complexity of the Fibonacci-Thue–Morse sequence: The proof of Dekking’s conjecture

Indagationes Mathematicae ◽

10.1016/j.indag.2021.03.004 ◽

2021 ◽

Author(s):

Jeffrey Shallit

Keyword(s):

Subword Complexity

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Characteristic numbers ofG-manifolds. I

Inventiones mathematicae ◽

10.1007/bf01404631 ◽

1971 ◽

Vol 13 (3) ◽

pp. 213-224 ◽

Cited By ~ 17

Author(s):

Tammo Tom Dieck

Keyword(s):

Characteristic Numbers

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AN INTENSIONAL LEIBNIZ SEMANTICS FOR ARISTOTELIAN LOGIC

The Review of Symbolic Logic ◽

10.1017/s1755020309990396 ◽

2010 ◽

Vol 3 (2) ◽

pp. 262-272 ◽

Cited By ~ 4

Author(s):

KLAUS GLASHOFF

Keyword(s):

Natural Deduction ◽

Predicate Logic ◽

Formal System ◽

Aristotelian Logic ◽

Categorical Logic ◽

Characteristic Numbers ◽

Formal Syntax ◽

Philosophical Standpoint ◽

Classical Philosophy

Since Frege’s predicate logical transcription of Aristotelian categorical logic, the standard semantics of Aristotelian logic considers terms as standing for sets of individuals. From a philosophical standpoint, this extensional model poses problems: There exist serious doubts that Aristotle’s terms were meant to refer always to sets, that is, entities composed of individuals. Classical philosophy up to Leibniz and Kant had a different view on this question—they looked at terms as standing for concepts (“Begriffe”). In 1972, Corcoran presented a formal system for Aristotelian logic containing a calculus of natural deduction, while, with respect to semantics, he still made use of an extensional interpretation. In this paper we deal with a simple intensional semantics for Corcoran’s syntax—intensional in the sense that no individuals are needed for the construction of a complete Tarski model of Aristotelian syntax. Instead, we view concepts as containing or excluding other, “higher” concepts—corresponding to the idea which Leibniz used in the construction of his characteristic numbers. Thus, this paper is an addendum to Corcoran’s work, furnishing his formal syntax with an adequate semantics which is free from presuppositions which have entered into modern interpretations of Aristotle’s theory via predicate logic.

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