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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 (June 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 (September 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-called FN topology. Roughly speaking, an FN topology is one that topologizes the ring structure of a Boolean algebra. Associated with this topology is a special basis, called an FN basis. For , it is natural to identify such a basis with a collection of partial types approximating a certain type.In this paper, we extend the theory of forking and deal with FN bases (hence ideals in particular). By proving a few results here, we hope to indicate that this extension could become as fruitful as Keisler's theory and hence provide an alternative. Our approach here is more algebraic and less analytic. Unlike [5], methods from nonstandard analysis are not used.There are two fundamental reasons why the classical theory should be generalized. One is to extend our investigation to unstable theories. (There are no stability assumptions on the theory in the present work.) The other is to study different and more flexible ways of collecting formulas, such as FN basis.After fixing our notation in §1, we first deal with ideals in §2. We study certain nice extensions. Specifically, we prove the existence of flat nonforking ideals. We then provide a normalization theorem for ideals.

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ARGOTY, CAMILO. "FORKING AND STABILITY IN THE REPRESENTATIONS OF A C*-ALGEBRA." Journal of Symbolic Logic 80, no. 3 (July 22, 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 (March 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 supersimple, ω-categorical case, these reduce to ordinary imaginaries.) So in Section 1 we develop the general theory of hyperimaginaries and show how first order model theory (including the theory of forking) generalises to hyperimaginaries. We go on, in Section 3 to show the existence and ubiquity of regular types in supersimple theories, ω-categorical simple structures and modularity is discussed in Section 4. It is also shown here how the general machinery of simplicity simplifies some of the general theory of smoothly approximable (or Lie-coordinatizable) structures from [2].Throughout this paper we will work in a large, saturated model M of a complete theory T. All types, sets and sequences will have size smaller than the size of M. We will assume that the reader is familiar with the basics of forking in simple theories as laid out in [4] and [6]. For basic stability-theoretic results concerning regular types, orthogonality etc., see [1] or [9].

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5

d’Elbée, Christian. "Expansions and Neostability in Model Theory." Bulletin of Symbolic Logic 27, no. 2 (June 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 model $\mathscr {M}_0$ of $T_0$ which is a substructure of $\mathscr {M}$ . We present a setting for the existence of a model-companion $TS$ of $T_S$ . As a consequence, we obtain the existence of the model-companion of the following theories, for $p>0$ a prime number: • $\mathrm {ACF}_p$ , $\mathrm {SCF}_{e,p}$ , $\mathrm {Psf}_p$ , $\mathrm {ACFA}_p$ , $\mathrm {ACVF}_{p,p}$ in appropriate languages expanded by arbitrarily many predicates for additive subgroups;• $\mathrm {ACF}_p$ , $\mathrm {ACF}_0$ in the language of rings expanded by a single predicate for a multiplicative subgroup;• $\mathrm {PAC}_p$ -fields, in an appropriate language expanded by arbitrarily many predicates for additive subgroups.From an independence relation in T, we define independence relations in $TS$ and identify which properties of are transferred to those new independence relations in $TS$ , and under which conditions. This allows us to exhibit hypotheses under which the expansion from T to $TS$ preserves $\mathrm {NSOP}_{1}$ , simplicity, or stability. In particular, under some technical hypothesis on T, we may draw the following picture (the left column implies the right column): Configuration $T_0\subseteq T$ Generic expansion $TS$ $T_0 = T$ Preserves stability $T_0\subseteq T$ Preserves $\mathrm {NSOP}_{1}$ $T_0 = \emptyset $ Preserves simplicityIn particular, this construction produces new examples of $\mathrm {NSOP}_{1}$ not simple theories, and we study in depth a particular example: the expansion of an algebraically closed field of positive characteristic by a generic additive subgroup. We give a full description of imaginaries, forking, and Kim-forking in this example.The second part studies expansions of the group of integers by p-adic valuations. We prove quantifier elimination in a natural language and compute the dp-rank of these expansions: it equals the number of independent p-adic valuations considered. Thus, the expansion of the integers by one p-adic valuation is a new dp-minimal expansion of the group of integers. Finally, we prove that the latter expansion does not admit intermediate structures: any definable set in the expansion is either definable in the group structure or is able to “reconstruct” the valuation using only the group operation.Abstract prepared by Christian d’Elbée.E-mail: delbee@math.univ-lyon1.frURL: https://choum.net/~chris/page_perso

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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 (June 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 (June 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 products of sets in finite groups, in particular to word maps, and a generalization of Tao's Algebraic Regularity Lemma is noted.

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Tsuboi, Akito. "On the number of independent partitions." Journal of Symbolic Logic 50, no. 3 (September 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 this paper, we shall give equivalents of which can be easily handled.In §1 we shall state the definitions of κn(T) and . Some basic properties of forking will be stated in this section. We shall also show that if = ∞ then T has the independence property.In §2 we shall give some conditions on κ, n, and T which are equivalent to the statement . (See Theorem 2.1 below.) We shall show that does not depend on n. We introduce the cardinal ı(T), which is essential in computing the number of types over a set which is independent over some set, and show that ı(T) is closely related to . (See Theorems 2.5 and 2.6 below.) The author expects the reader will discover the importance of via these theorems.Some of our results are motivated by exercises and questions in [3, Chapter III, §7]. The author wishes to express his heartfelt thanks to the referee for a number of helpful suggestions.

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9

Iovino, José. "On the maximality of logics with approximations." Journal of Symbolic Logic 66, no. 4 (December 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 ask whether the resulting extensions of first order model theory preserve some of its desirable characteristics.A formal framework for the study of structures based on Banach spaces from the perspective of model theory was first introduced by C. W. Henson in [8] and [6]. Notions of syntax and semantics for these structures were defined, and it was shown that using them one obtains a model theoretic apparatus that satisfies many of the fundamental properties of first order model theory. For instance, one has compactness, Löwenheim-Skolem, and omitting types theorems. Further aspects of the theory, namely, the fundamentals of stability and forking, were first introduced in [10] and [9].The classes of mathematical structures formally encompassed by this framework are normed linear spaces, possibly expanded with additional structure, e.g., operations, real-valued relations, and constants. This notion subsumes wide classes of structures from functional analysis. However, the restriction that the universe of a structure be a normed space is not necessary. (This restriction has a historical, rather than technical origin; specifically, the development of the theory was originally motivated by questions in Banach space geometry.) Analogous techniques can be applied if the universe is a metric space. Now, when the underlying metric topology is discrete, the resulting model theory coincides with first order model theory, so this logic extends first order in the sense described above. Furthermore, without any cost in the mathematical complexity, one can also work in multi-sorted contexts, so, for instance, one sort could be an operator algebra while another is. say, a metric space.

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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 calculation method. Finally, the comparative evaluation on calculation formula was done based on 79 groups of test results of concrete filled steel tubular column at home and abroad. Results of calculation formula agree well with the test results in safe range.

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11

ADLER, HANS. "THORN-FORKING AS LOCAL FORKING." Journal of Mathematical Logic 09, no. 01 (June 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 "dividing with respect to k" (using Ben-Yaacov's "k-inconsistency witnesses") rather than the forking formulas. It follows that every theory with an M-symmetric lattice of algebraically closed sets (in Teq) is rosy, with a simple lattice theoretical interpretation of thorn-forking.

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ADLER, HANS. "A GEOMETRIC INTRODUCTION TO FORKING AND THORN-FORKING." Journal of Mathematical Logic 09, no. 01 (June 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 (May 1, 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 (March 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 main open conjectures in simplicity theory, the stable forking conjecture. In particular, we prove that in any simple theory where the stable forking conjecture holds, ϸ-independence and forking independence agree.

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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 (March 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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Shami, Ziv. "On the forking topology of a reduct of a simple theory." Archive for Mathematical Logic 59, no. 3-4 (August 29, 2019): 313–24. http://dx.doi.org/10.1007/s00153-019-00691-w.

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YAACOV, ITAÏ BEN, and ARTEM CHERNIKOV. "AN INDEPENDENCE THEOREM FOR NTP2 THEORIES." Journal of Symbolic Logic 79, no. 01 (March 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 an NTP2 theory. We also define the dividing order of a theory—a generalization of Poizat’s fundamental order from stable theories—and give some equivalent characterizations under the assumption of NTP2. The last section is devoted to a refinement of the class of strong theories and its place in the classification hierarchy.

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20

Dolich, Alfred. "Forking and independence in o-minimal theories." Journal of Symbolic Logic 69, no. 1 (March 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, what kind of theories do we get? Another source of motivation, also stemming from the study of simple theories, comes from the work of Shelah in [8] and [7]. Here Shelah addresses a “classification” type problem for class of models of a theory, showing that a theory will have the appropriate “structure” type property if one can construct a partially ordered set, satisfying various properties, of models of the theory. Using this criterion Shelah shows that the class of simple theories has this “structure” property, yet also that several non-simple examples do as well (though it should be pointed out that o-minimal theories can not be among these since any theory with the strict order property will have the corresponding “non-structure” property [8]). Thus one is lead to ask, what are the non-simple theories meeting this criterion, and one is once again led to study the types of independence relation a theory might bear. Finally, Shelah in [6] provides some possible definitions of what axioms for a notion of independence one should possibly look for in order to hope that theories bearing such a notion of independence should be amenable closer analysis. In studying all of the above mentioned situations it readily becomes clear that dividing and forking play a central role in all of them, even though we are no longer dealing with the simple case where we know that dividing and forking are very well behaved. All of these considerations lead one to look for classes of non-simple theories of which something is known where one can construct interesting notions of independence and consequently also say something about the nature of forking and dividing in these contexts. Given this one is naturally lead to one of the most well behaved classes of non-simple theories, namely the o-minimal theories.

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McGrail, Tracey. "The model theory of differential fields with finitely many commuting derivations." Journal of Symbolic Logic 65, no. 2 (June 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 (September 1992): 209–24. http://dx.doi.org/10.1080/00397709.1992.10733776.

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Chatzidakis, Zoé. "Properties of forking in ω-free pseudo-algebraically closed fields." Journal of Symbolic Logic 67, no. 3 (September 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 den Dries and A. Macintyre [10], see also results by Ju. Ershov [13], [14]. Let K be a PAC field. Then the elementary theory of K is entirely determined by the following data:• The isomorphism type of the field of absolute numbers of K (the subfield of K of elements algebraic over the prime field).• The degree of imperfection of K.• The first-order theory, in a suitable ω-sorted language, of the inverse system of Galois groups al(L/K) where L runs over all finite Galois extensions of K.They also showed that the theory of PAC fields is undecidable, by showing that any graph can be encoded in the absolute Galois group of some PAC field. It turns out that the absolute Galois group controls much of the behaviour of the PAC fields. I will give below some examples illustrating this phenomenon.

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

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Georgescu, A., and J. T. Stuart. "Hydrodynamic Stability Theory." Journal of Applied Mechanics 54, no. 1 (March 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 (March 2004): 49–51. http://dx.doi.org/10.1145/970831.970843.

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Vasey, Sebastien. "Infinitary stability theory." Archive for Mathematical Logic 55, no. 3-4 (March 31, 2016): 567–92. http://dx.doi.org/10.1007/s00153-016-0481-z.

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

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Pillay, Anand. "The geometry of forking and groups of finite Morley rank." Journal of Symbolic Logic 60, no. 4 (December 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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Benevento, Joseph J. "What Borges Learned from Whitman: The Open Road and Its Forking Paths." Walt Whitman Quarterly Review 2, no. 4 (April 1, 1985): 21–30. http://dx.doi.org/10.13008/2153-3695.1089.

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Shami, Ziv. "Definability in low simple theories." Journal of Symbolic Logic 65, no. 4 (December 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 the technical machinery of forking developed in the stable context could be generalized to simple case. However, the theory of multiplicity (i.e., the description of the (bounded) set of non forking extensions of a given complete type) no longer holds in the context of simple theories. Indeed, by contrast to simple theories, stable theories share a strong amalgamation property of types, namely if p and q are two “free” complete extensions over a superset of A, and there is no finite equivalence relation over A which separates them, then the conjunction of p and q is consistent (and even free over A.) In [KP] Kim and Pillay proved a weak version of this property for any simple theory, namely “the Independence Theorem for Lascar strong types”. This was a weaker version both because of the requirement that the sets of parameters of the types be mutually independent, as well as the use of Lascar strong types instead of the usual strong types. A very fundamental and interesting problem is whether the independence theorem can be proved for any simple theory, using only the usual strong types. In 1997 Buechler proved ([Bu]) the strong-type version of the independence theorem for an important subclass of simple theories, namely the class of low theories (which includes the class of stable theories and the class of supersimple theories of finite D-rank.)

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Hodges, Wilfrid. "FUNDAMENTALS OF STABILITY THEORY." Bulletin of the London Mathematical Society 22, no. 5 (September 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 (September 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 (March 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 (February 1, 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 (July 1993): 283. http://dx.doi.org/10.1177/002072099303000326.

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Howard, James E. "Unified Hamiltonian stability theory." Celestial Mechanics & Dynamical Astronomy 62, no. 2 (June 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 (January 5, 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 (January 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 (October 4, 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 (April 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 (March 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 to a certain “local” framework, in which algebraic closure is replaced by P-closure, for P some family of types. We will, however, remain in the standard context, and will just remark here that it is routine to translate our results into Wagner's framework, as well as to generalise to the superstable theory/regular type context.) In any case we answer Wagner's question positively. Also in an attempt to put forward some concrete conjectures about the possible geometries of strongly minimal sets (or stable theories) we tentatively suggest a hierarchy of geometric properties of forking, the first two levels of which correspond to 1-basedness and CM-triviality respectively. We do not know whether this is a strict hierarchy (or even whether these are the “right” notions), but we conjecture that it is, and moreover that a counterexample to Cherlin's conjecture can be found at level three in the hierarchy.

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Ealy, Clifton, and Alf Onshuus. "Characterizing rosy theories." Journal of Symbolic Logic 72, no. 3 (September 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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Carley, Kathleen. "A Theory of Group Stability." American Sociological Review 56, no. 3 (June 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 (June 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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Schwarzenberger, R. L. E., Gerard Looss, and Daniel Joseph. "Elementary Stability and Bifurcation Theory." Mathematical Gazette 75, no. 473 (October 1991): 391. http://dx.doi.org/10.2307/3619547.

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Hastings, Alan. "Food Web Theory and Stability." Ecology 69, no. 6 (December 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 (October 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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