scholarly journals AN AXIOMATIC APPROACH TO FREE AMALGAMATION

2017 ◽  
Vol 82 (2) ◽  
pp. 648-671 ◽  
Author(s):  
GABRIEL CONANT
Keyword(s):  
Axiomatic Approach ◽  
Order Property ◽  
Combinatorial Characterization ◽  
First Order ◽  
Homogeneous Structures ◽  
Image Position ◽  
Rosy Theories

AbstractWe use axioms of abstract ternary relations to define the notion of a free amalgamation theory. These form a subclass of first-order theories, without the strict order property, encompassing many prominent examples of countable structures in relational languages, in which the class of algebraically closed substructures is closed under free amalgamation. We show that any free amalgamation theory has elimination of hyperimaginaries and weak elimination of imaginaries. With this result, we use several families of well-known homogeneous structures to give new examples of rosy theories. We then prove that, for free amalgamation theories, simplicity coincides with NTP2 and, assuming modularity, with NSOP3 as well. We also show that any simple free amalgamation theory is 1-based. Finally, we prove a combinatorial characterization of simplicity for Fraïssé limits with free amalgamation, which provides new context for the fact that the generic Kn-free graphs are SOP3, while the higher arity generic $K_n^r$-free r-hypergraphs are simple.

scholarly journals EXISTENCE OF MODELING LIMITS FOR SEQUENCES OF SPARSE STRUCTURES

2019 ◽  
Vol 84 (02) ◽  
pp. 452-472 ◽  
Author(s):  
JAROSLAV NEŠETŘIL ◽  
PATRICE OSSONA DE MENDEZ
Keyword(s):  
Relative Size ◽  
Convergent Sequence ◽  
List Type ◽  
Sparse Graphs ◽  
First Order ◽  
Graph Modeling ◽  
Image Position ◽  
Standard Probability ◽  
Convergent Sequences

AbstractA sequence of graphs is FO-convergent if the probability of satisfaction of every first-order formula converges. A graph modeling is a graph, whose domain is a standard probability space, with the property that every definable set is Borel. It was known that FO-convergent sequence of graphs do not always admit a modeling limit, but it was conjectured that FO-convergent sequences of sufficiently sparse graphs have a modeling limits. Precisely, two conjectures were proposed:1.If a FO-convergent sequence of graphs is residual, that is if for every integer d the maximum relative size of a ball of radius d in the graphs of the sequence tends to zero, then the sequence has a modeling limit.2.A monotone class of graphs ${\cal C}$ has the property that every FO-convergent sequence of graphs from ${\cal C}$ has a modeling limit if and only if ${\cal C}$ is nowhere dense, that is if and only if for each integer p there is $N\left( p \right)$ such that no graph in ${\cal C}$ contains the pth subdivision of a complete graph on $N\left( p \right)$ vertices as a subgraph.In this article we prove both conjectures. This solves some of the main problems in the area and among others provides an analytic characterization of the nowhere dense–somewhere dense dichotomy.

Zero-one laws with variable probability

1993 ◽  
Vol 58 (1) ◽  
pp. 1-14 ◽  
Author(s):  
Joel Spencer
Keyword(s):  
Probability Space ◽  
Equal Weight ◽  
Order Property ◽  
First Order ◽  
Random Structures ◽  
Monadic Logic ◽  
Edge Probability ◽  
Image Position ◽  
Labelled Graphs ◽  
Variable Probability

One of this author's favorite theorems has long been the Zero-One law discovered independently by Glebskii et al. [12] and Ron Fagin [10]. Let A be any first-order property of graphs and let µn(A) be the proportion of labelled graphs on n vertices for which A holds. ThenThis result has inspired much work by logicians, generally in the direction of showing (1) for some powerful languages. Thus it is known [5] that (1) holds when A is a sentence in fixed point logic and it is known [13] that (1) does not always hold when A is a sentence in second-order monadic logic. Here, however, we explore recent work in a totally different direction. Let G(n, p) denote the random graph on n vertices with edge probability p. (In §2 we define the random structures we will deal with.) A property A is an event in the probability space and Pr[G(n, p) ╞ A] is well defined. When p = 1/2, each labelled graph on n vertices has equal weight so that (1) may be rewrittenFagin's proof actually gives that (2) holds for any constant 0 < p < 1.

Quasiunitriangular groups

1993 ◽  
Vol 58 (1) ◽  
pp. 205-218 ◽  
Author(s):  
O. V. Belegradek
Keyword(s):  
Direct Summand ◽  
Algebraic Characterization ◽  
First Order ◽  
Expanded Group ◽  
The Matrix ◽  
Image Position

For a ring with unit R, which need not be associative, denote the group of upper unitriangular 3 × 3 matrices over R by UT3(R). Let e1 = (1,0,0), e2 = (0,1,0), where (α, β, γ) denotes the matrixDenote the expanded group (UT3(R), e1, e2) by (R). A. 1. Mal′cev [M] gave an algebraic characterization of the expanded groups of the form (R) as follows. Let h1, h2 be elements of a group H; then (H, h1, h2) is isomorphic to (R), for some R, if and only if(i) H is 2-step nilpotent;(ii) CH(hi) are abelian, i = 1,2;(iii) CH(h1) ∩ CH(h2) = Z(H);(iv) [CH(h1),h2] = [h1, CH(h2)] = Z(H);(v) Z(H) is a direct summand in both CH(hi).(In [M] condition (v) is a bit stronger; the version above is presented in [B2].)A pair (h1, h2) of elements of a group H is said to be a base if (H, h1, h2) satisfies the conditions (i)–(iv). A. I. Mal′cev [M] found a uniform way of first order interpreting a ring Ring(H, h1, h2) in any group with a base (H, h1, h2); in particular, Ring((R)) ≃ R.

scholarly journals NIP FOR THE ASYMPTOTIC COUPLE OF THE FIELD OF LOGARITHMIC TRANSSERIES

2017 ◽  
Vol 82 (1) ◽  
pp. 35-61 ◽  
Author(s):  
ALLEN GEHRET
Keyword(s):  
Order Theory ◽  
Exchange Property ◽  
Independence Property ◽  
First Order ◽  
Valued Field ◽  
First Order Theory ◽  
Order Language ◽  
Complete Survey ◽  
Image Position

AbstractThe derivation on the differential-valued field Tlog of logarithmic transseries induces on its value group ${{\rm{\Gamma }}_{{\rm{log}}}}$ a certain map ψ. The structure ${\rm{\Gamma }} = \left( {{{\rm{\Gamma }}_{{\rm{log}}}},\psi } \right)$ is a divisible asymptotic couple. In [7] we began a study of the first-order theory of $\left( {{{\rm{\Gamma }}_{{\rm{log}}}},\psi } \right)$ where, among other things, we proved that the theory $T_{{\rm{log}}} = Th\left( {{\rm{\Gamma }}_{{\rm{log}}} ,\psi } \right)$ has a universal axiomatization, is model complete and admits elimination of quantifiers (QE) in a natural first-order language. In that paper we posed the question whether Tlog has NIP (i.e., the Non-Independence Property). In this paper, we answer that question in the affirmative: Tlog does have NIP. Our method of proof relies on a complete survey of the 1-types of Tlog, which, in the presence of QE, is equivalent to a characterization of all simple extensions ${\rm{\Gamma }}\left\langle \alpha \right\rangle$ of ${\rm{\Gamma }}$. We also show that Tlog does not have the Steinitz exchange property and we weigh in on the relationship between models of Tlog and the so-called precontraction groups of [9].

scholarly journals ON CUTS IN ULTRAPRODUCTS OF LINEAR ORDERS II

2018 ◽  
Vol 83 (1) ◽  
pp. 29-39
Author(s):  
MOHAMMAD GOLSHANI ◽  
SAHARON SHELAH
Keyword(s):  
Linear Order ◽  
Linear Orders ◽  
Combinatorial Characterization ◽  
Image Position

AbstractWe continue our study of the class ${\cal C}\left( D \right)$, where D is a uniform ultrafilter on a cardinal κ and ${\cal C}\left( D \right)$ is the class of all pairs $\left( {{\theta _1},{\theta _2}} \right)$, where $\left( {{\theta _1},{\theta _2}} \right)$ is the cofinality of a cut in ${J^\kappa }/D$ and J is some ${\left( {{\theta _1} + {\theta _2}} \right)^ + }$-saturated dense linear order. We give a combinatorial characterization of the class ${\cal C}\left( D \right)$. We also show that if $\left( {{\theta _1},{\theta _2}} \right) \in {\cal C}\left( D \right)$ and D is ${\aleph _1}$-complete or ${\theta _1} + {\theta _2} > {2^\kappa }$, then ${\theta _1} = {\theta _2}$.

Metaontology

2018 ◽  
Author(s):  
Uriah Kriegel
Keyword(s):  
Second Order ◽  
Order Property ◽  
First Order ◽  
First Approximation

Brentano’s theory of judgment serves as a springboard for his conception of reality, indeed for his ontology. It does so, indirectly, by inspiring a very specific metaontology. To a first approximation, ontology is concerned with what exists, metaontology with what it means to say that something exists. So understood, metaontology has been dominated by three views: (i) existence as a substantive first-order property that some things have and some do not, (ii) existence as a formal first-order property that everything has, and (iii) existence as a second-order property of existents’ distinctive properties. Brentano offers a fourth and completely different approach to existence talk, however, one which falls naturally out of his theory of judgment. The purpose of this chapter is to present and motivate Brentano’s approach.

scholarly journals Characterization of Dissipative Structures for First-Order Symmetric Hyperbolic System with General Relaxation

Mathematics ◽  
2021 ◽  
Vol 9 (7) ◽  
pp. 728
Author(s):  
Yasunori Maekawa ◽  
Yoshihiro Ueda
Keyword(s):  
Hyperbolic System ◽  
Dissipative Structure ◽  
Dissipative Structures ◽  
Algebraic Characterization ◽  
Symmetric Hyperbolic System ◽  
Full Generality ◽  
Order Linear ◽  
First Order

In this paper, we study the dissipative structure of first-order linear symmetric hyperbolic system with general relaxation and provide the algebraic characterization for the uniform dissipativity up to order 1. Our result extends the classical Shizuta–Kawashima condition for the case of symmetric relaxation, with a full generality and optimality.

scholarly journals On Consensus Index of Triplex Star-Like Networks: A Graph Spectra Approach

Symmetry ◽  
2021 ◽  
Vol 13 (7) ◽  
pp. 1248
Author(s):  
Da Huang ◽  
Jian Zhu ◽  
Zhiyong Yu ◽  
Haijun Jiang
Keyword(s):  
Algebraic Connectivity ◽  
Multi Agent Systems ◽  
Graph Spectra ◽  
Asymptotic Behaviors ◽  
Agent Systems ◽  
First Order ◽  
Laplacian Spectra ◽  
Multi Agent ◽  
Consensus Index

In this article, the consensus-related performances of the triplex multi-agent systems with star-related structures, which can be measured by the algebraic connectivity and network coherence, have been studied by the characterization of Laplacian spectra. Some notions of graph operations are introduced to construct several triplex networks with star substructures. The methods of graph spectra are applied to derive the network coherence, and some asymptotic behaviors of the indices have been derived. It is found that the operations of adhering star topologies will make the first-order coherence increase a constant value under the triplex structures as parameters tend to infinity, and the second-order coherence have some equality relations as the node related parameters tend to infinity. Finally, the consensus related indices of the triplex systems with the same number of nodes but non-isomorphic graph structures have been compared and simulated to verify the results.

scholarly journals Numerical Characterization of Cohesive and Non-Cohesive ‘Sediments’ under Different Consolidation States Using 3D DEM Triaxial Experiments

Processes ◽  
2020 ◽  
Vol 8 (10) ◽  
pp. 1252
Author(s):  
Hadar Elyashiv ◽  
Revital Bookman ◽  
Lennart Siemann ◽  
Uri ten Brink ◽  
Katrin Huhn
Keyword(s):  
Mechanical Response ◽  
Burial Depth ◽  
Cohesive Sediments ◽  
Third Order ◽  
First Order ◽  
Numerical Characterization ◽  
Response To Stress ◽  
Bond Strengths ◽  
Individual Bond

The Discrete Element Method has been widely used to simulate geo-materials due to time and scale limitations met in the field and laboratories. While cohesionless geo-materials were the focus of many previous studies, the deformation of cohesive geo-materials in 3D remained poorly characterized. Here, we aimed to generate a range of numerical ‘sediments’, assess their mechanical response to stress and compare their response with laboratory tests, focusing on differences between the micro- and macro-material properties. We simulated two endmembers—clay (cohesive) and sand (cohesionless). The materials were tested in a 3D triaxial numerical setup, under different simulated burial stresses and consolidation states. Variations in particle contact or individual bond strengths generate first order influence on the stress–strain response, i.e., a different deformation style of the numerical sand or clay. Increased burial depth generates a second order influence, elevating peak shear strength. Loose and dense consolidation states generate a third order influence of the endmember level. The results replicate a range of sediment compositions, empirical behaviors and conditions. We propose a procedure to characterize sediments numerically. The numerical ‘sediments’ can be applied to simulate processes in sediments exhibiting variations in strength due to post-seismic consolidation, bioturbation or variations in sedimentation rates.

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