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 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.

An axiomatization of the logic with the rough quantifier

1991 ◽  
Vol 56 (2) ◽  
pp. 608-617 ◽  
Author(s):  
Michał Krynicki ◽  
Hans-Peter Tuschik
Keyword(s):  
Equivalence Relation ◽  
Equivalence Relations ◽  
Order Structure ◽  
Partition Model ◽  
Weak Model ◽  
First Order ◽  
Basic Properties ◽  
Generalized Quantifier ◽  
Order Language ◽  
The Right

We consider the language L(Q), where L is a countable first-order language and Q is an additional generalized quantifier. A weak model for L(Q) is a pair 〈, q〉 where is a first-order structure for L and q is a family of subsets of its universe. In case that q is the set of classes of some equivalence relation the weak model 〈, q〉 is called a partition model. The interpretation of Q in partition models was studied by Szczerba [3], who was inspired by Pawlak's paper [2]. The corresponding set of tautologies in L(Q) is called rough logic. In the following we will give a set of axioms of rough logic and prove its completeness. Rough logic is designed for creating partition models.The partition models are the weak models arising from equivalence relations. For the basic properties of the logic of weak models the reader is referred to Keisler's paper [1]. In a weak model 〈, q〉 the formulas of L(Q) are interpreted as usual with the additional clause for the quantifier Q: 〈, q〉 ⊨ Qx φ(x) iff there is some X ∊ q such that 〈, q〉 ⊨ φ(a) for all a ∊ X.In case X satisfies the right side of the above equivalence we say that X is contained in φ(x) or, equivalently, φ(x) contains X.

Recursive in a generic real

2000 ◽  
Vol 65 (1) ◽  
pp. 164-172 ◽  
Author(s):  
Juichi Shinoda ◽  
Theodore A. Slaman
Keyword(s):  
Real Number ◽  
Order Structure ◽  
First Order

AbstractThere is a comeager set contained in the set of 1-generic reals and a first order structure such that for any real number X. there is an element of which is recursive in X if and only if there is a presentation of which is recursive in X.

Algebraic theories with definable Skolem functions

1984 ◽  
Vol 49 (2) ◽  
pp. 625-629 ◽  
Author(s):  
Lou van den Dries
Keyword(s):  
Nonempty Subset ◽  
Peano Arithmetic ◽  
Real Closed Field ◽  
Closed Field ◽  
First Order ◽  
Definable Subset ◽  
Skolem Functions ◽  
Algebraic Theories ◽  
Closed Fields ◽  
Definitional Extension

(1.1) A well-known example of a theory with built-in Skolem functions is (first-order) Peano arithmetic (or rather a certain definitional extension of it). See [C-K, pp. 143, 162] for the notion of a theory with built-in Skolem functions, and for a treatment of the example just mentioned. This property of Peano arithmetic obviously comes from the fact that in each nonempty definable subset of a model we can definably choose an element, namely, its least member.(1.2) Consider now a real closed field R and a nonempty subset D of R which is definable (with parameters) in R. Again we can definably choose an element of D, as follows: D is a union of finitely many singletons and intervals (a, b) where – ∞ ≤ a < b ≤ + ∞; if D has a least element we choose that element; if not, D contains an interval (a, b) for which a ∈ R ∪ { − ∞}is minimal; for this a we choose b ∈ R ∪ {∞} maximal such that (a, b) ⊂ D. Four cases have to be distinguished:(i) a = − ∞ and b = + ∞; then we choose 0;(ii) a = − ∞ and b ∈ R; then we choose b − 1;(iii) a ∈ R and b ∈ = + ∞; then we choose a + 1;(iv) a ∈ R and b ∈ R; then we choose the midpoint (a + b)/2.It follows as in the case of Peano arithmetic that the theory RCF of real closed fields has a definitional extension with built-in Skolem functions.

New Method for First-Order Structure Design of Continuous Zoom Lens System

2015 ◽  
Vol 35 (7) ◽  
pp. 0722002
Author(s):  
翟婷婷 Zhai Tingting ◽  
朱健强 Zhu Jianqiang
Keyword(s):  
Structure Design ◽  
New Method ◽  
Zoom Lens ◽  
Order Structure ◽  
Lens System ◽  
First Order

Non-standard lattices and o-minimal groups

2013 ◽  
Vol 19 (1) ◽  
pp. 56-76 ◽  
Author(s):  
Pantelis E. Eleftheriou
Keyword(s):  
Local Level ◽  
Order Structure ◽  
First Order ◽  
Central Notion ◽  
Structure Theorems ◽  
Definable Groups ◽  
Minimal Groups ◽  
Definition Of ◽  
Generic Elements

AbstractWe describe a recent program from the study of definable groups in certain o-minimal structures. A central notion of this program is that of a (geometric) lattice. We propose a definition of a lattice in an arbitrary first-order structure. We then use it to describe, uniformly, various structure theorems for o-minimal groups, each time recovering a lattice that captures some significant invariant of the group at hand. The analysis first goes through a local level, where a pertinent notion of pregeometry and generic elements is each time introduced.

scholarly journals Fields with few types

2013 ◽  
Vol 78 (1) ◽  
pp. 72-84
Author(s):  
Cédric Milliet
Keyword(s):  
Division Ring ◽  
Positive Characteristic ◽  
Field Extension ◽  
Order Structure ◽  
Small Field ◽  
Finite Degree ◽  
Locally Finite ◽  
First Order ◽  
Finite Dimensional ◽  
Characteristic 2

AbstractAccording to Belegradek, a first order structure is weakly small if there are countably many 1-types over any of its finite subset. We show the following results. A field extension of finite degree of an infinite weakly small field has no Artin-Schreier extension. A weakly small field of characteristic 2 is finite or algebraically closed. A weakly small division ring of positive characteristic is locally finite dimensional over its centre. A weakly small division ring of characteristic 2 is a field.

scholarly journals Prefrontal Cortex Activation Reflects Efficient Exploitation of Higher-order Statistical Structure

2016 ◽  
Vol 28 (12) ◽  
pp. 1909-1922 ◽  
Author(s):  
Christiane Ahlheim ◽  
Anne-Marike Schiffer ◽  
Ricarda I. Schubotz
Keyword(s):  
Action Observation ◽  
Second Order ◽  
Order Structure ◽  
Order Information ◽  
First Order ◽  
Action Observation Network ◽  
Observation Network ◽  
Statistical Regularities ◽  
Posterior Parietal ◽  
Action Step

Because everyday actions are statistically structured, knowing which action a person has just completed allows predicting the most likely next action step. Taking even more than the preceding action into account improves this predictability but also causes higher processing costs. Using fMRI, we investigated whether observers exploit second-order statistical regularities preferentially if information on possible upcoming actions provided by first-order regularities is insufficient. We hypothesized that anterior pFC balances whether or not second-order information should be exploited. Participants watched videos of actions that were structured by first- and second-order conditional probabilities. Information provided by the first and by the second order was manipulated independently. BOLD activity in the action observation network was more attenuated the more information on upcoming actions was provided by first-order structure, reflecting expectation suppression for more predictable actions. Activation in posterior parietal sites decreased further with second-order information but increased in temporal areas. As expected, second-order information was integrated more when less first-order information was provided, and this interaction was mediated by anterior pFC (BA 10). Observers spontaneously used both the present and the preceding action to predict the upcoming action, and integration of the preceding action was enhanced when the present action was uninformative.

scholarly journals A notion of functional completeness for first-order structure

2005 ◽  
Vol 2005 (14) ◽  
pp. 2207-2215
Author(s):  
Etienne R. Alomo Temgoua ◽  
Marcel Tonga
Keyword(s):  
Order Structure ◽  
Functional Completeness ◽  
First Order ◽  
Term Functions ◽  
Order Structures

Using☆-congruences and implications, Weaver (1993) introduced the concepts of prevariety and quasivariety of first-order structures as generalizations of the corresponding concepts for algebras. The notion of functional completeness on algebras has been defined and characterized by Burris and Sankappanavar (1981), Kaarli and Pixley (2001), Pixley (1996), and Quackenbush (1981). We study the notion of functional completeness with respect to☆-congruences. We extend some results on functionally complete algebras to first-order structuresA=(A;FA;RA)and find conditions for these structures to have a compatible Pixley function which is interpolated by term functions on suitable subsets of the base setA.

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