Definability and decidability issues in extensions of the integers with the divisibility predicate

1996 ◽  
Vol 61 (2) ◽  
pp. 515-540 ◽  
Author(s):  
Patrick Cegielski ◽  
Yuri Matiyasevich ◽  
Denis Richard
Keyword(s):  
Computational Complexity ◽  
Alternative Proof ◽  
Order Structure ◽  
Prime Divisors ◽  
Additive Theory ◽  
First Order ◽  
Decision Method ◽  
Definable Relations ◽  
Positive Integers ◽  
Number Of Prime Divisors

AbstractLet be a first-order structure; we denote by DEF() the set of all first-order definable relations and functions within . Let π be any one-to-one function from ℕ into the set of prime integers.Let ∣ and • be respectively the divisibility relation and multiplication as function. We show that the sets DEF(ℕ, π, ∣) and DEF(ℕ, π, •) are equal. However there exists function π such that the set DEF(ℕ, +, ∣), or, equivalently, DEF(ℕ, π, •) is not equal to DEF(ℕ, +, •). Nevertheless, in all cases there is an {π, •}-definable and hence also {π, |}-definable structure over π which is isomorphic to 〈ℕ, +, •〉. Hence theories TH(ℕ, π, ∣) and TH(ℕ, π, •) are undecidable.The binary relation of equipotence between two positive integers saying that they have equal number of prime divisors is not definable within the divisibility lattice over positive integers. We prove it first by comparing the lower bound of the computational complexity of the additive theory of positive integers and of the upper bound of the computational complexity of the theory of the mentioned lattice.The last section provides a self-contained alternative proof of this latter result based on a decision method linked to an elimination of quantifiers via specific tables.

scholarly journals Extensions of some formulae of A. Selberg

1985 ◽  
Vol 8 (2) ◽  
pp. 283-302 ◽  
Author(s):  
Claudia A. Spiro
Keyword(s):  
Arithmetic Progression ◽  
Arbitrary Number ◽  
Error Term ◽  
Fixed Number ◽  
Prime Divisors ◽  
Positive Integers ◽  
Number Of Prime Divisors

This paper is concerned with estimating the number of positive integers up to some bound (which tends to infinity), such that they have a fixed number of prime divisors, and lie in a given arithmetic progression. We obtain estimates which are uniform in the number of prime divisors, and at the same time, in the modulus of the arithmetic progression. These estimates take the form of a fixed but arbitrary number of main terms, followed by an error term.

scholarly journals A NEW DP-MINIMAL EXPANSION OF THE INTEGERS

2019 ◽  
Vol 84 (02) ◽  
pp. 632-663 ◽  
Author(s):  
ERAN ALOUF ◽  
CHRISTIAN D’ELBÉE
Keyword(s):  
Analogous Result ◽  
Abelian Groups ◽  
Quantifier Elimination ◽  
Alternative Proof ◽  
Order Structure ◽  
First Order ◽  
Adic Valuation ◽  
Image Position ◽  
Natural Expansion

AbstractWe consider the structure $({\Bbb Z}, + ,0,|_{p_1 } , \ldots ,|_{p_n } )$, where $x|_p y$ means $v_p \left( x \right) \leqslant v_p \left( y \right)$ and vp is the p-adic valuation. We prove that this structure has quantifier elimination in a natural expansion of the language of abelian groups, and that it has dp-rank n. In addition, we prove that a first order structure with universe ${\Bbb Z}$ which is an expansion of $({\Bbb Z}, + ,0)$ and a reduct of $({\Bbb Z}, + ,0,|_p )$ must be interdefinable with one of them. We also give an alternative proof for Conant’s analogous result about $({\Bbb Z}, + ,0, < )$.

scholarly journals Prime divisors of some shifted products

2005 ◽  
Vol 2005 (19) ◽  
pp. 3057-3073
Author(s):  
Eric Levieil ◽  
Florian Luca ◽  
Igor E. Shparlinski
Keyword(s):  
Prime Divisors ◽  
Positive Integers

We study prime divisors of various sequences of positive integersA(n)+1,n=1,…,N, such that the ratiosa(n)=A(n)/A(n−1)have some number-theoretic or combinatorial meaning. In the casea(n)=n, we obviously haveA(n)=n!, for which several new results about prime divisors ofn!+1have recently been obtained.

scholarly journals On the number of prime divisors of the order of elliptic curves modulo p

2005 ◽  
Vol 117 (4) ◽  
pp. 341-352 ◽  
Author(s):  
Jörn Steuding ◽  
Annegret Weng
Keyword(s):  
Elliptic Curves ◽  
Prime Divisors ◽  
Modulo P ◽  
Number Of Prime Divisors

scholarly journals Some questions concerning minimal structures

2007 ◽  
pp. 79-83
Author(s):  
Predrag Tanovic
Keyword(s):  
Order Structure ◽  
First Order ◽  
Definable Subset

An infinite first-order structure is minimal if its each definable subset is either finite or co-finite. We formulate three questions concerning order properties of minimal structures which are motivated by Pillay?s Conjecture (stating that a countable first-order structure must have infinitely many countable, pairwise non-isomorphic elementary extensions).

scholarly journals The distribution of the number of prime divisors of sums a + b

1988 ◽  
Vol 29 (1) ◽  
pp. 94-99 ◽  
Author(s):  
P.D.T.A Elliott ◽  
A Sárközy
Keyword(s):  
Prime Divisors ◽  
Number Of Prime Divisors

scholarly journals Impact of higher order network structure on emergent cortical activity

2020 ◽  
Vol 4 (1) ◽  
pp. 292-314 ◽  
Author(s):  
Max Nolte ◽  
Eyal Gal ◽  
Henry Markram ◽  
Michael W. Reimann
Keyword(s):  
Network Structure ◽  
Morphological Diversity ◽  
Cortical Activity ◽  
Higher Order ◽  
Neuronal Firing ◽  
Network Activity ◽  
Synaptic Connectivity ◽  
Order Structure ◽  
First Order ◽  
Neuronal Types

Synaptic connectivity between neocortical neurons is highly structured. The network structure of synaptic connectivity includes first-order properties that can be described by pairwise statistics, such as strengths of connections between different neuron types and distance-dependent connectivity, and higher order properties, such as an abundance of cliques of all-to-all connected neurons. The relative impact of first- and higher order structure on emergent cortical network activity is unknown. Here, we compare network structure and emergent activity in two neocortical microcircuit models with different synaptic connectivity. Both models have a similar first-order structure, but only one model includes higher order structure arising from morphological diversity within neuronal types. We find that such morphological diversity leads to more heterogeneous degree distributions, increases the number of cliques, and contributes to a small-world topology. The increase in higher order network structure is accompanied by more nuanced changes in neuronal firing patterns, such as an increased dependence of pairwise correlations on the positions of neurons in cliques. Our study shows that circuit models with very similar first-order structure of synaptic connectivity can have a drastically different higher order network structure, and suggests that the higher order structure imposed by morphological diversity within neuronal types has an impact on emergent cortical activity.

Sign in / Sign up

Export Citation Format

Share Document