Fixed Points on the Real Numbers without the Equality Test

The purpose of this paper is to survey our recent research in computability and definability over continuous data types such as the real numbers, real-valued functions and functionals. We investigate the expressive power and algorithmic properties of the language of Sigma-formulas intended to represent computability over the real numbers. In order to adequately represent computability we extend the reals by the structure of hereditarily finite sets. In this setting it is crucial to consider the real numbers without equality since the equality test is undecidable over the reals. We prove Engeler's Lemma for Sigma-definability over the reals without the equality test which relates Sigma-definability with definability in the constructive infinitary language L_{omega_1 omega}. Thus, a relation over the real numbers is Sigma-definable if and only if it is definable by a disjunction of a recursively enumerable set of quantifier free formulas. This result reveals computational aspects of Sigma-definability and also gives topological characterisation of Sigma-definable relations over the reals without the equality test. We also illustrate how computability over the real numbers can be expressed in the language of Sigma-formulas.

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Dynamics of a family of orbitally continuous mappings

Filomat ◽

10.2298/fil1711507p ◽

2017 ◽

Vol 31 (11) ◽

pp. 3507-3517

Author(s):

Abhijit Pant ◽

R.P. Pant ◽

Kuldeep Prakash

Keyword(s):

Fixed Points ◽

Julia Sets ◽

Real Numbers ◽

Continuous Mappings ◽

Non Linear

The aim of the present paper is to study the dynamics of a class of orbitally continuous non-linear mappings defined on the set of real numbers and to apply the results on dynamics of functions to obtain tests of divisibility. We show that this class of mappings contains chaotic mappings. We also draw Julia sets of certain iterations related to multiple lowering mappings and employ the variations in the complexity of Julia sets to illustrate the results on the quotient and remainder. The notion of orbital continuity was introduced by Lj. B. Ciric and is an important tool in establishing existence of fixed points.

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Shuffling of Linear Orders

Canadian Mathematical Bulletin ◽

10.4153/cmb-1995-032-1 ◽

1995 ◽

Vol 38 (2) ◽

pp. 223-229

Author(s):

John Lindsay Orr

Keyword(s):

Ordered Set ◽

Linear Orders ◽

Real Numbers ◽

The Real ◽

Linearly Ordered Set

AbstractA linearly ordered set A is said to shuffle into another linearly ordered set B if there is an order preserving surjection A —> B such that the preimage of each member of a cofinite subset of B has an arbitrary pre-defined finite cardinality. We show that every countable linearly ordered set shuffles into itself. This leads to consequences on transformations of subsets of the real numbers by order preserving maps.

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A question of van den Dries and a theorem of Lipshitz and Robinson; Not everything is standard

Journal of Symbolic Logic ◽

10.2178/jsl/1174668387 ◽

2007 ◽

Vol 72 (1) ◽

pp. 119-122 ◽

Cited By ~ 4

Author(s):

Ehud Hrushovski ◽

Ya'acov Peterzil

Keyword(s):

Real Numbers ◽

The Real ◽

First Order ◽

Real Closed Fields ◽

Minimal Structure ◽

Closed Fields ◽

New Construction ◽

The Relationship

AbstractWe use a new construction of an o-minimal structure, due to Lipshitz and Robinson, to answer a question of van den Dries regarding the relationship between arbitrary o-minimal expansions of real closed fields and structures over the real numbers. We write a first order sentence which is true in the Lipshitz-Robinson structure but fails in any possible interpretation over the field of real numbers.

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On expressible sets and p-adic numbers

Proceedings of the Edinburgh Mathematical Society ◽

10.1017/s0013091509000091 ◽

2011 ◽

Vol 54 (2) ◽

pp. 411-422

Author(s):

Jaroslav Hančl ◽

Radhakrishnan Nair ◽

Simona Pulcerova ◽

Jan Šustek

Keyword(s):

Haar Measure ◽

Real Numbers ◽

The Real ◽

Expressible Set

AbstractContinuing earlier studies over the real numbers, we study the expressible set of a sequence A = (an)n≥1 of p-adic numbers, which we define to be the set EpA = {∑n≥1ancn: cn ∈ ℕ}. We show that in certain circumstances we can calculate the Haar measure of EpA exactly. It turns out that our results extend to sequences of matrices with p-adic entries, so this is the setting in which we work.

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