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Journal articles on the topic 'Stability theory[Forking theory]'

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Published: 4 June 2021

Last updated: 1 February 2022

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1

Kim, Byunghan. "Simplicity, and stability in there." Journal of Symbolic Logic 66, no. 2 (2001): 822–36. http://dx.doi.org/10.2307/2695047.

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AbstractFirstly, in this paper, we prove that the equivalence of simplicity and the symmetry of forking. Secondly, we attempt to recover definability part of stability theory to simplicity theory. In particular, using elimination of hyperimaginaries we prove that for any supersimple T. canonical base of an amalgamation class is the union of names of ψ-definitions of , ψ ranging over stationary L-formulas in . Also, we prove that the same is true with stable formulas for an 1-based theory having elimination of hyperimaginaries. For such a theory, the stable forking property holds, too.

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2

Ng, Siu-Ah. "A generalization of forking." Journal of Symbolic Logic 56, no. 3 (1991): 813–22. http://dx.doi.org/10.2178/jsl/1183743730.

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Given a subset A of a fixed saturated model , we let denote the algebra of definable subsets of the domain of M of with parameters from A. Then a complete type p over A can be regarded as a measure on , assigning the value 1 to members of p and 0 to nonmembers. In [5] and [6], Keisler developed a theory of forking concerning probability measures. Therefore it generalizes the ordinary theory. On a different track, we can view the complement of the type p, or the collection of null sets of any measure on , as ideals on . Moreover, ideals and the pseudometric of a measure form examples of the so-

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3

ARGOTY, CAMILO. "FORKING AND STABILITY IN THE REPRESENTATIONS OF A C*-ALGEBRA." Journal of Symbolic Logic 80, no. 3 (2015): 785–96. http://dx.doi.org/10.1017/jsl.2015.23.

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AbstractWe show that the theory of a nondegenerate representation of a C*-algebra ${\cal A}$ over a Hilbert space H is superstable. Also, we characterize forking, orthogonality and domination of types.

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4

Hart, Bradd, Byunghan Kim, and Anand Pillay. "Coordinatisation and canonical bases in simple theories." Journal of Symbolic Logic 65, no. 1 (2000): 293–309. http://dx.doi.org/10.2307/2586538.

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In this paper we discuss several generalization of theorems from stability theory to simple theories. Cherlin and Hrushovski, in [2] develop a substitute for canonical bases in finite rank, ω-categorical supersimple theories. Motivated by methods there, we prove the existence of canonical bases (in a suitable sense) for types in any simple theory. This is done in Section 2. In general these canonical bases will (as far as we know) exist only as “hyperimaginaries”, namely objects of the form a/E where a is a possibly infinite tuple and E a type-definable equivalence relation. (In the supersimpl

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5

d’Elbée, Christian. "Expansions and Neostability in Model Theory." Bulletin of Symbolic Logic 27, no. 2 (2021): 216–17. http://dx.doi.org/10.1017/bsl.2021.26.

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AbstractThis thesis is concerned with the expansions of algebraic structures and their fit in Shelah’s classification landscape.The first part deals with the expansion of a theory by a random predicate for a substructure model of a reduct of the theory. Let T be a theory in a language $\mathcal {L}$ . Let $T_0$ be a reduct of T. Let $\mathcal {L}_S = \mathcal {L}\cup \{S\}$ , for S a new unary predicate symbol, and $T_S$ be the $\mathcal {L}_S$ -theory that axiomatises the following structures: $(\mathscr {M},\mathscr {M}_0)$ consist of a model $\mathscr {M}$ of T and S is a predicate for a mo

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6

Ben-Yaacov, Itay, Ivan Tomašić, and Frank O. Wagner. "The Group Configuration in Simple Theories and its Applications." Bulletin of Symbolic Logic 8, no. 2 (2002): 283–98. http://dx.doi.org/10.2178/bsl/1182353874.

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AbstractIn recent work, the authors have established the group configuration theorem for simple theories, as well as some of its main applications from geometric stability theory, such as the binding group theorem, or, in the ω-categorical case, the characterization of the forking geometry of a finitely based non-trivial locally modular regular type as projective geometry over a finite field and the equivalence of pseudolinearity and local modularity.The proof necessitated an extension of the model-theoretic framework to include almost hyperimaginaries, and the study of polygroups.

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7

García, Darío, Dugald Macpherson, and Charles Steinhorn. "Pseudofinite structures and simplicity." Journal of Mathematical Logic 15, no. 01 (2015): 1550002. http://dx.doi.org/10.1142/s0219061315500026.

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We explore a notion of pseudofinite dimension, introduced by Hrushovski and Wagner, on an infinite ultraproduct of finite structures. Certain conditions on pseudofinite dimension are identified that guarantee simplicity or supersimplicity of the underlying theory, and that a drop in pseudofinite dimension is equivalent to forking. Under a suitable assumption, a measure-theoretic condition is shown to be equivalent to local stability. Many examples are explored, including vector spaces over finite fields viewed as 2-sorted finite structures, and homocyclic groups. Connections are made to produc

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8

Tsuboi, Akito. "On the number of independent partitions." Journal of Symbolic Logic 50, no. 3 (1985): 809–14. http://dx.doi.org/10.2307/2274333.

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In [3], Shelah defined the cardinals κn(T) and , for each theory T and n < ω. κn(T) is the least cardinal κ without a sequence (pi)i<κ of complete n-types such that pi is a forking extension of pj for all i < j < κ. It is essential in computing the stability spectrum of a stable theory. On the other hand is called the number of independent partitions of T. (See Definition 1.2 below.) Unfortunately this invariant has not been investigated deeply. In the author's opinion, this unfortunate situation of is partially due to the fact that its definition is complicated in expression. In t

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9

Iovino, José. "On the maximality of logics with approximations." Journal of Symbolic Logic 66, no. 4 (2001): 1909–18. http://dx.doi.org/10.2307/2694984.

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In this paper we analyze some aspects of the question of using methods from model theory to study structures of functional analysis.By a well known result of P. Lindström, one cannot extend the expressive power of first order logic and yet preserve its most outstanding model theoretic characteristics (e.g., compactness and the Löwenheim-Skolem theorem). However, one may consider extending the scope of first order in a different sense, specifically, by expanding the class of structures that are regarded as models (e.g., including Banach algebras or other structures of functional analysis), and

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10

Zhang, Xue Peng. "Stability Bearing Capacity of Concrete Filled Thin-Walled Circular Steel Tubular Column under Axial Compression." Advanced Materials Research 690-693 (May 2013): 720–23. http://dx.doi.org/10.4028/www.scientific.net/amr.690-693.720.

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The bearing capacity formula of concrete filled thin-walled steel tubular (CFTST) short column was established based on limit equilibrium method, and the reasonable value fork1in Richart Strength Model was regression introduced adapted with concrete stress-strain relations. According to the elasticity modulus theory, the calculation formulas of stability bearing capacity of CFTST slender column were deduced, reasonable considering the interactions between steel tube and concrete. And the calculation process is relatively simple which avoiding complicated iterations used in the conventional cal

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11

ADLER, HANS. "THORN-FORKING AS LOCAL FORKING." Journal of Mathematical Logic 09, no. 01 (2009): 21–38. http://dx.doi.org/10.1142/s0219061309000823.

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We introduce the notion of a preindependence relation between subsets of the big model of a complete first-order theory, an abstraction of the properties which numerous concrete notions such as forking, dividing, thorn-forking, thorn-dividing, splitting or finite satisfiability share in all complete theories. We examine the relation between four additional axioms (extension, local character, full existence and symmetry) that one expects of a good notion of independence. We show that thorn-forking can be described in terms of local forking if we localize the number k in Kim's notion of "dividin

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12

ADLER, HANS. "A GEOMETRIC INTRODUCTION TO FORKING AND THORN-FORKING." Journal of Mathematical Logic 09, no. 01 (2009): 1–20. http://dx.doi.org/10.1142/s0219061309000811.

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A ternary relation [Formula: see text] between subsets of the big model of a complete first-order theory T is called an independence relation if it satisfies a certain set of axioms. The primary example is forking in a simple theory, but o-minimal theories are also known to have an interesting independence relation. Our approach in this paper is to treat independence relations as mathematical objects worth studying. The main application is a better understanding of thorn-forking, which turns out to be closely related to modular pairs in the lattice of algebraically closed sets.

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13

Kim, Byunghan, and A. Pillay. "Around stable forking." Fundamenta Mathematicae 170, no. 1 (2001): 107–18. http://dx.doi.org/10.4064/fm170-1-6.

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14

Pourmahdian, M. "The Stable Forking Conjecture in Homogeneous Model Theory." Logic Journal of IGPL 12, no. 3 (2004): 171–80. http://dx.doi.org/10.1093/jigpal/12.3.171.

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15

Onshuus, Alf. "Properties and consequences of Thorn-independence." Journal of Symbolic Logic 71, no. 1 (2006): 1–21. http://dx.doi.org/10.2178/jsl/1140641160.

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AbstractWe develop a new notion of independence (ϸ-independence, read “thorn”-independence) that arises from a family of ranks suggested by Scanlon (ϸ-ranks). We prove that in a large class of theories (including simple theories and o-minimal theories) this notion has many of the properties needed for an adequate geometric structure.We prove that ϸ-independence agrees with the usual independence notions in stable, supersimple and o-minimal theories. Furthermore, we give some evidence that the equivalence between forking and ϸ-forking in simple theories might be closely related to one of the ma

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16

Branigan, Edward. "Nearly True: Forking Plots, Forking Interpretations: A Response to David Bordwell's "Film Futures"." SubStance 31, no. 1 (2002): 105. http://dx.doi.org/10.2307/3685811.

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17

Ealy, Clifton, and Isaac Goldbring. "Thorn-forking in continuous logic." Journal of Symbolic Logic 77, no. 1 (2012): 63–93. http://dx.doi.org/10.2178/jsl/1327068692.

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AbstractWe study thorn forking and rosiness in the context of continuous logic. We prove that the Urysohn sphere is rosy (with respect to finitary imaginaries), providing the first example of an essentially continuous unstable theory with a nice notion of independence. In the process, we show that a real rosy theory which has weak elimination of finitary imaginaries is rosy with respect to finitary imaginaries, a fact which is new even for discrete first-order real rosy theories.

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18

Shami, Ziv. "On the forking topology of a reduct of a simple theory." Archive for Mathematical Logic 59, no. 3-4 (2019): 313–24. http://dx.doi.org/10.1007/s00153-019-00691-w.

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19

YAACOV, ITAÏ BEN, and ARTEM CHERNIKOV. "AN INDEPENDENCE THEOREM FOR NTP2 THEORIES." Journal of Symbolic Logic 79, no. 01 (2014): 135–53. http://dx.doi.org/10.1017/jsl.2013.22.

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Abstract We establish several results regarding dividing and forking in NTP2 theories. We show that dividing is the same as array-dividing. Combining it with existence of strictly invariant sequences we deduce that forking satisfies the chain condition over extension bases (namely, the forking ideal is S1, in Hrushovski’s terminology). Using it we prove an independence theorem over extension bases (which, in the case of simple theories, specializes to the ordinary independence theorem). As an application we show that Lascar strong type and compact strong type coincide over extension bases in a

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20

Dolich, Alfred. "Forking and independence in o-minimal theories." Journal of Symbolic Logic 69, no. 1 (2004): 215–40. http://dx.doi.org/10.2178/jsl/1080938838.

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In the following we try to answer a simple question, “what does forking look like in an o-minimal theory”, or more generally, “what kinds of notions of independence with what kinds of properties are admissible in an o-minimal theory?” The motivation of these question begin with the study of simple theories and generalizations of simple theories. In [3] Kim and Pillay prove that the class of simple theories may be described exactly as those theories bearing a notion of independence satisfying various axioms. Thus it is natural to ask, if we weaken the assumptions as to which axioms must hold, w

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21

McGrail, Tracey. "The model theory of differential fields with finitely many commuting derivations." Journal of Symbolic Logic 65, no. 2 (2000): 885–913. http://dx.doi.org/10.2307/2586576.

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AbstractIn this paper we set out the basic model theory of differential fields of characteristic 0, which have finitely many commuting derivations. We give axioms for the theory of differentially closed differential fields with m derivations and show that this theory is ω-stable, model complete, and quantifier-eliminable, and that it admits elimination of imaginaries. We give a characterization of forking and compute the rank of this theory to be ωm + 1.

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22

Jurkevich, Gayana. "Double Voices and Forking Paths: Baroja'sCamino de Perfección." Symposium: A Quarterly Journal in Modern Literatures 46, no. 3 (1992): 209–24. http://dx.doi.org/10.1080/00397709.1992.10733776.

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23

Chatzidakis, Zoé. "Properties of forking in ω-free pseudo-algebraically closed fields". Journal of Symbolic Logic 67, № 3 (2002): 957–96. http://dx.doi.org/10.2178/jsl/1190150143.

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The study of pseudo-algebraically closed fields (henceforth called PAC) started with the work of J. Ax on finite and pseudo-finite fields [1]. He showed that the infinite models of the theory of finite fields are exactly the perfect PAC fields with absolute Galois group isomorphic to , and gave elementary invariants for their first order theory, thereby proving the decidability of the theory of finite fields. Ax's results were then extended to a larger class of PAC fields by M. Jarden and U. Kiehne [21], and Jarden [19]. The final word on theories of PAC fields was given by G. Cherlin, L. van

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24

Schmid, Peter J. "Nonmodal Stability Theory." Annual Review of Fluid Mechanics 39, no. 1 (2007): 129–62. http://dx.doi.org/10.1146/annurev.fluid.38.050304.092139.

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25

Georgescu, A., and J. T. Stuart. "Hydrodynamic Stability Theory." Journal of Applied Mechanics 54, no. 1 (1987): 250. http://dx.doi.org/10.1115/1.3172987.

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26

Ross, Rocky. "Stability and theory." ACM SIGACT News 35, no. 1 (2004): 49–51. http://dx.doi.org/10.1145/970831.970843.

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27

Vasey, Sebastien. "Infinitary stability theory." Archive for Mathematical Logic 55, no. 3-4 (2016): 567–92. http://dx.doi.org/10.1007/s00153-016-0481-z.

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28

Râsvan, Vladimir. "Stability Theory versus Social Stability." IFAC Proceedings Volumes 42, no. 25 (2009): 45–47. http://dx.doi.org/10.3182/20091028-3-ro-4007.00011.

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29

Pillay, Anand. "The geometry of forking and groups of finite Morley rank." Journal of Symbolic Logic 60, no. 4 (1995): 1251–59. http://dx.doi.org/10.2307/2275886.

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AbstractThe notion of CM-triviality was introduced by Hrushovski, who showed that his new strongly minimal sets have this property. Recently Baudisch has shown that his new ω1-categorical group has this property. Here we show that any group of finite Morley rank definable in a CM-trivial theory is nilpotent-by-finite, or equivalently no simple group of finite Morley rank can be definable in a CM-trivial theory.

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30

Staid, Mairead Small. "What Space is For: The forking paths of memory and return." Yale Review 109, no. 2 (2021): 127–37. http://dx.doi.org/10.1353/tyr.2021.0040.

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31

Benevento, Joseph J. "What Borges Learned from Whitman: The Open Road and Its Forking Paths." Walt Whitman Quarterly Review 2, no. 4 (1985): 21–30. http://dx.doi.org/10.13008/2153-3695.1089.

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32

Shami, Ziv. "Definability in low simple theories." Journal of Symbolic Logic 65, no. 4 (2000): 1481–90. http://dx.doi.org/10.2307/2695059.

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In 1978 Shelah introduced a new class of theories, called simple (see [Shi]) which properly contained the class of stable theories. Shelah generalized part of the theory of forking to the simple context. After approximately 15 years of neglecting the general theory (although there were works by Hrushovski on finite rank with a definability assumption, as well as deep results on specific simple theories by Cherlin, Hrushovski, Chazidakis, Macintyre and Van den dries, see [CH], [CMV], [HP1], [HP2], [ChH]) there was a breakthrough, initiated with the work of Kim ([K1]). Kim proved that almost all

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33

Hodges, Wilfrid. "FUNDAMENTALS OF STABILITY THEORY." Bulletin of the London Mathematical Society 22, no. 5 (1990): 511–12. http://dx.doi.org/10.1112/blms/22.5.511.

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34

Pikul’, V. V. "On shell stability theory." Doklady Physics 52, no. 9 (2007): 513–15. http://dx.doi.org/10.1134/s1028335807090157.

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35

Elliott, D. L. "Stability Theory [Book Reviews]." IEEE Transactions on Automatic Control 41, no. 3 (1996): 473. http://dx.doi.org/10.1109/tac.1996.486655.

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36

Pillay, Anand, and Mike Prest. "Modules and stability theory." Transactions of the American Mathematical Society 300, no. 2 (1987): 641. http://dx.doi.org/10.1090/s0002-9947-1987-0876470-x.

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37

Wellstead, Peter. "Book Review: Stability Theory." International Journal of Electrical Engineering & Education 30, no. 3 (1993): 283. http://dx.doi.org/10.1177/002072099303000326.

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38

Howard, James E. "Unified Hamiltonian stability theory." Celestial Mechanics & Dynamical Astronomy 62, no. 2 (1995): 111–16. http://dx.doi.org/10.1007/bf00692082.

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39

Kerswell, R. R. "Nonlinear Nonmodal Stability Theory." Annual Review of Fluid Mechanics 50, no. 1 (2018): 319–45. http://dx.doi.org/10.1146/annurev-fluid-122316-045042.

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40

Harrington, Alexandra K. "Forking paths and other dramas: Postmodernist features of Anna Achmatova's “Menja, kak reku”." Russian Literature 59, no. 1 (2006): 41–64. http://dx.doi.org/10.1016/j.ruslit.2006.01.003.

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41

Raidl, Eric, and Niels Skovgaard-Olsen. "Bridging Ranking Theory and the Stability Theory of Belief." Journal of Philosophical Logic 46, no. 6 (2016): 577–609. http://dx.doi.org/10.1007/s10992-016-9411-0.

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42

Ra̧czka, R. "A nonperturbative stability theory of quantum field theory models." Annals of Physics 160, no. 2 (1985): 355–405. http://dx.doi.org/10.1016/0003-4916(85)90149-6.

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43

Pillay, Anand. "A note on CM-triviality and the geometry of forking." Journal of Symbolic Logic 65, no. 1 (2000): 474–80. http://dx.doi.org/10.2307/2586549.

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CM-triviality of a stable theory is a notion introduced by Hrushovski [1]. The importance of this property is first that it holds of Hrushovski's new non 1-based strongly minimal sets, and second that it is still quite a restrictive property, and forbids the existence of definable fields or simple groups (see [2]). In [5], Frank Wagner posed some questions about CM-triviality, asking in particular whether a structure of finite rank, which is “coordinatized” by CM-trivial types of rank 1, is itself CM-trivial. (Actually Wagner worked in a slightly more general context, adapting the definitions

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44

Ealy, Clifton, and Alf Onshuus. "Characterizing rosy theories." Journal of Symbolic Logic 72, no. 3 (2007): 919–40. http://dx.doi.org/10.2178/jsl/1191333848.

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AbstractWe examine several conditions, either the existence of a rank or a particular property of þ-forking that suggest the existence of a well-behaved independence relation, and determine the consequences of each of these conditions towards the rosiness of the theory. In particular we show that the existence of an ordinal valued equivalence relation rank is a (necessary and) sufficient condition for rosiness.

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45

Carley, Kathleen. "A Theory of Group Stability." American Sociological Review 56, no. 3 (1991): 331. http://dx.doi.org/10.2307/2096108.

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46

Makkai, Michael, and Anand Pillay. "An Introduction to Stability Theory." Journal of Symbolic Logic 51, no. 2 (1986): 465. http://dx.doi.org/10.2307/2274072.

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47

OHSHIMA, Hiroyuki. "Basic Theory of Colloid Stability." Journal of the Japan Society of Colour Material 77, no. 7 (2004): 328–32. http://dx.doi.org/10.4011/shikizai1937.77.328.

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48

Schwarzenberger, R. L. E., Gerard Looss, and Daniel Joseph. "Elementary Stability and Bifurcation Theory." Mathematical Gazette 75, no. 473 (1991): 391. http://dx.doi.org/10.2307/3619547.

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49

Hastings, Alan. "Food Web Theory and Stability." Ecology 69, no. 6 (1988): 1665–68. http://dx.doi.org/10.2307/1941143.

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50

Seymour, R. M. "Stability in pro-homotopy theory." Proceedings of the Edinburgh Mathematical Society 33, no. 3 (1990): 419–41. http://dx.doi.org/10.1017/s0013091500004843.

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If is a category, an object of pro- is stable if it is isomorphic in pro- to an object of . A local condition on such a pro-object, called strong-movability, is defined, and it is shown in various contexts that this condition is equivalent to stability. Also considered, in the case is a suitable model category, is the stability problem in the homotopy category Ho(pro-), where pro- has the induced closed model category structure defined by Edwards and Hastings [6].

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