We study model theory of fields with actions of a fixed finite group scheme. We prove the existence and simplicity of a model companion of the theory of such actions, which generalizes our previous results about truncated iterative Hasse–Schmidt derivations [13] and about Galois actions [14]. As an application of our methods, we obtain a new model complete theory of actions of a finite group on fields of finite imperfection degree.

Generalizing and unifying the known theorems for difference and differential fields, it is shown that for every finite free algebra scheme.

In his Ph.D. thesis [7], L. van den Dries studied the model theory of fields with finitely many orderings and valuations where all open sets according to the topology defined by an order or a valuation is globally dense according with all other orderings and valuations. Van den Dries proved that the theory of these fields is companionable and that the theory of the companion is decidable .In this paper we study the case where (...) the fields are expanded with finitely many orderings and an independent derivation. We show that the theory of these fields still admits a model companion in the language Lequation image = {+, –, ·, D, <1, …, model companion by CODFm and give a geometric axiomatization of this theory which uses basic notions of algebraic geometry and some generalized open subsets which appear naturally in this context. This axiomatization allows to recover the one given in [4] for the theory CODF of closed ordered differential fields. Most of the technics we use here are already present in [2] and [4].Finally, we prove that it is possible to describe the completions of CODFm and to obtain quantifier elimination in a slightly enriched language. This generalizes van den Dries' results in the “derivation free” case. (shrink)

In 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 ω m + 1.

We give a survey of results obtained in the model theory of finite and pseudo-finite fields.

Let G be a finite group. We explore the model-theoretic properties of the class of differential fields of characteristic zero in m commuting derivations equipped with a G-action by differential fie...

Vector spaces over unspecified fields can be axiomatized as one-sorted structures, namely, abelian groups with the relation of parallelism. Parallelism is binary linear dependence. When equipped with the n-ary relation of linear dependence for some positive integer n, a vector-space is existentially closed if and only if it is n-dimensional over an algebraically closed field. In the signature with an n-ary predicate for linear dependence for each positive integer n, the theory of infinite-dimensional vector spaces over algebraically closed (...) fields is the model-completion of the theory of vector spaces. (shrink)

he model theory of groups of unitriangular matrices over rings is studied. An important tool in these studies is a new notion of a quasiunitriangular group. The models of the theory of all unitriangular groups are algebraically characterized; it turns out that all they are quasiunitriangular groups. It is proved that if R and S are domains or commutative associative rings then two quasiunitriangular groups over R and S are isomorphic only if R and S are isomorphic (...) or antiisomorphic. This algebraic result is new even for ordinary unitriangular groups. The groups elementarily equivalent to a single unitriangular group UTn are studied. If R is a skew field, they are of the form UTn, for some S ≡ R. In general, the situation is not so nice. Examples are constructed demonstrating that such a group need not be a unitriangular group over some ring; moreover, there are rings P and R such that UTn ≡ UTn, but UTn cannot be represented in the form UTn for S ≡ R. We also study the number of models in a power of the theory of a unitriangular group. In particular, we prove that, for any communicative associative ring R and any infinite power λ, I = I). We construct an associative ring such that I = 3 and I) = 2. We also study models of the theory of UTn in the case of categorical R. (shrink)

We prove some results about the model theory of fields with a derivation of the Frobenius map, especially that the model companion of this theory is axiomatizable by axioms used by Wood in the case of the theory $\operatorname {DCF}_p$ and that it eliminates quantifiers after adding the inverse of the Frobenius map to the language. This strengthens the results from [4]. As a by-product, we get a new geometric axiomatization of this model (...) companion. Along the way we also prove a quantifier elimination result, which holds in a much more general context and we suggest a way of giving “one-dimensional” axiomatizations for model companions of some theories of fields with operators. (shrink)

Fields of characteristic zero with several commuting derivations can be treated as fields equipped with a space of derivations that is closed under the Lie bracket. The existentially closed instances of such structures can then be given a coordinate-free characterization in terms of differential forms. The main tool for doing this is a generalization of the Frobenius Theorem of differential geometry.

Using methods of geometric stability , we determine the structure of Abelian groups definable in ACFA, the model companion of fields with an automorphism. We also give general bounds on sets definable in ACFA. We show that these tools can be used to study torsion points on Abelian varieties; among other results, we deduce a fairly general case of a conjecture of Tate and Voloch on p-adic distances of torsion points from subvarieties.

We study the model theory of fields k carrying a henselian valuation with real closed residue field. We give a criteria for elementary equivalence and elementary inclusion of such fields involving the value group of a not necessarily definable valuation. This allows us to translate theories of such fields to theories of ordered abelian groups, and we study the properties of this translation. We also characterize the first-order definable convex subgroups of a given ordered abelian (...) group and prove that the definable real valuation rings of k are in correspondence with the definable convex subgroups of the value group of a certain real valuation of k. (shrink)

Model theorists have been studying analytic functions since the late 1970s. Highlights include the seminal work of Denef and van den Dries on the theory of the p-adics with restricted analytic functions, Wilkie's proof of o-minimality of the theory of the reals with the exponential function, and the formulation of Zilber's conjecture for the complex exponential. My goal in this talk is to survey these main developments and to reflect on today's open problems, in particular for theories (...) of valued fields. (shrink)

The notion of a D-ring, generalizing that of a differential or a difference ring, is introduced. Quantifier elimination and a version of the Ax-Kochen-Eršov principle is proven for a theory of valued D-fields of residual characteristic zero.

We define a Lie differential field as a field of characteristic 0 with an action, as derivations on , of some given Lie algebra . We assume that is a finite-dimensional vector space over some sub-field given in advance. As an example take the field of rational functions on a smooth algebraic variety, with .For every simple extension of Lie differential fields we find a finite system of differential equations that characterizes it. We then define, using first-order conditions, a (...) collection of allowed systems of differential equations s.t. the above characteristic systems are allowed. We prove that for every allowed system there exists a generic solution in some extension, and this solution is unique .We construct the model completion of the theory of Lie differential fields by adding axioms stating that every allowed system has almost generic solutions. The construction is a generalization of Blum's axioms for DCF0. We also show that this model completion is ω-stable. (shrink)

The notion of a D-ring, generalizing that of a differential or a difference ring, is introduced. Quantifier elimination and a version of the Ax-Kochen-Ersov principle is proven for a theory of valued D-fields of residual characteristic zero.

Major shifts in the field of model theory in the twentieth century have seen the development of new tools, methods, and motivations for mathematicians and philosophers. In this book, John T. Baldwin places the revolution in its historical context from the ancient Greeks to the last century, argues for local rather than global foundations for mathematics, and provides philosophical viewpoints on the importance of modern model theory for both understanding and undertaking mathematical practice. The volume also (...) addresses the impact of model theory on contemporary algebraic geometry, number theory, combinatorics, and differential equations. This comprehensive and detailed book will interest logicians and mathematicians as well as those working on the history and philosophy of mathematics. (shrink)

The mean-field theory with the use of the slave-boson functional method has been generalized to take account of the spin- and/or orbital-ordered state in the doubly degenerate Hubbard model. Numerical calculations are presented of the antiferromagnetic orbital-ordered state in the half-filled simple-cubic model. The orbital order in the present theory is much reduced compared with that in the Hartree–Fock approximation because of the large orbital fluctuations. From a comparison of the ground-state energy, the antiferromagnetic orbital state (...) is shown to be unstable against the antiferromagnetic spin state, although the situation becomes reversed when the exchange interaction is negative. (shrink)

We give foundational results for the model theory of AfinK, the ring of finite adeles over a number field, construed as a restricted product of local fields. In contrast to Weispfenning we work in the language of ring theory, and various sortings interpretable therein. In particular we give a systematic treatment of the product valuation and the valuation monoid. Deeper results are given for the adelic version of Krasner's hyperfields, relating them to the Basarab–Kuhlmann formalism.

We consider the reduct to the module language of certain theories of fields with a non surjective endomorphism. We show in some cases the existence of a model companion. We apply our results for axiomatizing the reduct to the theory of modules of non principal ultraproducts of separably closed fields of fixed but non zero imperfection degree.

We study when the property that a field is dense in its real and p-adic closures is elementary in the language of rings and deduce that all models of the theory of algebraic fields have this property.

In this paper we give a differential lifting principle which provides a general method to geometrically axiomatize the model companion (if it exists) of some theories of differential topological fields. The topological fields we consider here are in fact topological systems in the sense of van den Dries, and the lifting principle we develop is a generalization of the geometric axiomatization of the theory DCF₀ given by Pierce and Pillay. Moreover, it provides a geometric alternative to (...) the axiomatizations obtained by Tressl and Guzy/Point in separate papers where the authors also build general schemes of axioms for some model complete theories of differential fields. We first characterize the existentially closed models of a given theory of differential topological fields and then, under an additional hypothesis of largeness, we show how to modify this characterization to get a general scheme of first-order axioms for the model companion of any large theory of differential topological fields. We conclude with an application of this lifting principle proving that, in existentially closed models of a large theory of differential topological fields, the jet-spaces are dense in their ambient topological space. (shrink)

For the classA of uncountable Archimedian real closed fields we show that the statement “TheL <ω-theory ofA is complete” is independent of ZFC. In particular we have the following results:Assuming the Continuum-Hypothesis (CH) is incomplete. Conversely it is possible to build a model of set theory in which is complete and decidable. The latter can also be deduced from the Proper Forcing Axiom (PFA). In this case turns out to be equivalent to the elementary theory (...) of the real numbers ℝ (by a quantifier-elimination procedure).Formally: is incomplete. is complete and decidable. (shrink)

The phase field microelasticity theory of a three-dimensional, elastically anisotropic system of voids and cracks is proposed. The theory is based on the equation for the strain energy of the continuous elastically homogeneous body presented as a functional of the phase field, which is the effective stress-free strain. It is proved that the stress-free strain minimizing the strain energy of this homogeneous modulus body fully determines the elastic strain and displacement of the body with voids and/or cracks. The (...) proposed phase field integral equation describing the elasticity of an arbitrary system of voids and cracks is exact. The geometry and evolution of multiple voids and/or cracks are described by the phase field, which is the solution of the time-dependent Ginzburg-Landau equation. Other defects, such as dislocations and precipitates, are trivially integrated into this theory. The proposed model does not impose a priori constraints on possible void and crack configurations or their evolution paths. Examples of computations of elastic equilibrium of systems with voids and/or cracks and the evolution of cracks under applied stress are considered. (shrink)

This book contains the material for a first course in pure model theory with applications to differentially closed fields.

We find the model completion of the theory modules over ������, where ������ is a finitely generated commutative algebra over a field K. This is done in a context where the field K and the module are represented by sorts in the theory, so that constructible sets associated with a module can be interpreted in this language. The language is expanded by additional sorts for the Grassmanians of all powers of $K^n $ , which are necessary to (...) achieve quantifier elimination. The result turns out to be that the model completion is the theory of a certain class of "big" injective modules. In particular, it is shown that the class of injective modules is itself elementary. We also obtain an explicit description of the types in this theory. (shrink)

ABSTRACTCritical realism's insight into depth ontology creates the possibility for re-imagining sociology as a science of the social world. However, critical realism has yet to gain a strong foothold in sociological analysis. Challenging the available criticism of critical realism, I argue that its main flaw is its inability to draw an appropriate epistemological strategy from its insights into depth ontology. I propose that this limitation can be overcome when we anchor the depth ontology of critical realism to the two-step epistemological (...) strategies of Bourdieu's field theory: first, objectifying common sense to develop a scientific model of social structure; and second, objectifying the scientific point of view to re-introduce common sense in the scientific model of social structure. I argue that by anchoring the ontological insights of critical realism to the epistemological strategies of field theory, we can create the long-sought core around which sociology can progress as a science of the social world. (shrink)

We introduce the subject of modal model theory, where one studies a mathematical structure within a class of similar structures under an extension concept that gives rise to mathematically natural notions of possibility and necessity. A statement φ is possible in a structure (written φ) if φ is true in some extension of that structure, and φ is necessary (written φ) if it is true in all extensions of the structure. A principal case for us will be the (...) class Mod(T) of all models of a given theory T—all graphs, all groups, all fields, or what have you—considered under the substructure relation. In this article, we aim to develop the resulting modal model theory. The class of all graphs is a particularly insightful case illustrating the remarkable power of the modal vocabulary, for the modal language of graph theory can express connectedness, k-colorability, finiteness, countability, size continuum, size ℵ1, ℵ2, ℵω, ℶω, first ℶ-fixed point, first ℶ-hyper-fixed point, and much more. A graph obeys the maximality principle φ(a)→φ(a) with parameters if and only if it satisfies the theory of the countable random graph, and it satisfies the maximality principle for sentences if and only if it is universal for finite graphs. (shrink)

The purpose of this paper is to study an analogue of Hilbert's seventeenth problem for functions over a valued field which are integral definite on some definable set; that is, that map the given set into the valuation ring. We use model theory to exhibit a uniform method, on various theories of valued fields, for deriving an algebraic characterization of such functions. As part of this method we refine the concept of a function being integral at a (...) point, and make it dependent on the relevant class of valued fields. We apply our framework to algebraically closed valued fields, model complete theories of difference and differential valued fields, and real closed valued fields. (shrink)

A number of philosophers :171–187, 2011; Arnold 2011, in Ethics Politics XV:101–138, 2013) have argued that agent-based, evolutionary game theory models of the evolution of cooperation fail to provide satisfying explanations of cooperation because they are too disconnected from actual biology. I show how these criticisms can be answered by employing modeling approaches from the situated cognition research program that allow for more biologically detailed models. Using cases drawn from recent situated cognition modeling research, I show how agent-based models (...) of the evolution of cooperation can become more empirically-informed and relevant to explaining the evolution of cooperation in real populations. I argue that because situated cognition models allow for more detailed representations of not just agents’ brains but bodies and environments as well, they can be informed by a much wider breadth of empirical research than traditional agent-based models. Moreover, because the simulated bodies of these agents exhibit particular behaviors, they can be more readily compared to the behaviors of actual organisms. I conclude that employing situated cognition approaches to modeling the evolution of cooperation is a more promising way to investigate the evolution of cooperation than more traditional agent-based evolutionary game theory models. (shrink)

§1. Introduction. With Hrushovski's proof of the function field Mordell-Lang conjecture [16] the relevance of geometric stability theory to diophantine geometry first came to light. A gulf between logicians and number theorists allowed for contradictory reactions. It has been asserted that Hrushovski's proof was simply an algebraic argument masked in the language of model theory. Another camp held that this theorem was merely a clever one-off. Still others regarded the argument as magical and asked whether such sorcery (...) could unlock the secrets of a wide coterie of number theoretic problems.In the intervening years each of these prejudices has been revealed as false though such attitudes are still common. The methods pioneered in [16] have been extended and applied to a number of other problems. At their best, these methods have been integrated into the general methods for solving diophantine problems. Moreover, the newer work suggests limits to the application of model theory to diophantine geometry. For example, all such known applications are connected with commutative algebraic groups. This need not be an intrinsic restriction, but its removal requires serious advances in the model theory of fields. (shrink)

The study of consciousness has today moved beyond neurobiology and cognitive models. In the past few years, there has been a surge of research into various newer areas. The present article looks at the non-neurobiological and non-cognitive theories regarding this complex phenomenon, especially ones that self-psychology, self-theory, artificial intelligence, quantum physics, visual cognitive science and philosophy have to offer. Self-psychology has proposed the need to understand the self and its development, and the ramifications of the self for morality and (...) empathy, which will help us understand consciousness better. There have been inroads made from the fields of computer science, machine technology and artificial intelligence, including robotics, into understanding the consciousness of these machines and their implications for human consciousness. These areas are explored. Visual cortex and emotional theories along with their implications are discussed. The phylogeny and evolution of the phenomenon of consciousness is also highlighted, with theories on the emergence of consciousness in fetal and neonatal life. Quantum physics and its insights into the mind, along with the implications of consciousness and physics and their interface are debated. The role of neurophilosophy to understand human consciousness, the functions of such a concept, embodiment, the dark side of consciousness, future research needs and limitations of a scientific theory of consciousness complete the review. The importance and salient features of each theory are discussed along with certain pitfalls, if present. A need for the integration of various theories to understand consciousness from a holistic perspective is stressed. (shrink)

The study of consciousness has today moved beyond neurobiology and cognitive models. In the past few years, there has been a surge of research into various newer areas. The present article looks at the non-neurobiological and non-cognitive theories regarding this complex phenomenon, especially ones that self-psychology, self-theory, artificial intelligence, quantum physics, visual cognitive science and philosophy have to offer. Self-psychology has proposed the need to understand the self and its development, and the ramifications of the self for morality and (...) empathy, which will help us understand consciousness better. There have been inroads made from the fields of computer science, machine technology and artificial intelligence, including robotics, into understanding the consciousness of these machines and their implications for human consciousness. These areas are explored. Visual cortex and emotional theories along with their implications are discussed. The phylogeny and evolution of the phenomenon of consciousness is also highlighted, with theories on the emergence of consciousness in fetal and neonatal life. Quantum physics and its insights into the mind, along with the implications of consciousness and physics and their interface are debated. The role of neurophilosophy to understand human consciousness, the functions of such a concept, embodiment, the dark side of consciousness, future research needs and limitations of a scientific theory of consciousness complete the review. The importance and salient features of each theory are discussed along with certain pitfalls, if present. A need for the integration of various theories to understand consciousness from a holistic perspective is stressed. (shrink)

Summary With regard to the dynamics of human experience and behavior, Gestalt theoretical psychotherapy (GTP) relies mainly on Kurt Lewin’s dynamic field theory of personality. GTP is carried out by including a re-interpretation of Lewin’s theory in some aspects of psychotherapeutic practice in relation to critical realism. Human experience and behavior are understood to be functions of the person and the environment (including the other individuals therein) in a psychic field (life space), which encompasses both of these mutually (...) dependent factors. The anthropological model of this approach is, therefore, not mono-personal but, a priori, structural and relational in nature. It does not one-sidedly focus on the “inner components” of a person, but on the interrelation of the individual and a given environment, which affects experience and behavior. After a brief introduction of these basic concepts, this lecture will focus especially on Lewin’s concept of tension-systems, which may be considered as the Gestalt theoretical counterpart of Freud’s drive theory. Further, we define the basic assumptions which underlie GTP and explain how the person moves through her/his life experience in terms of Gestalt psychology. (shrink)

There is debate as to whether quantum field theory is, at bottom, a quantum theory of fields or particles. One can take a field approach to the theory, using wave functionals over field configurations, or a particle approach, using wave functions over particle configurations. This article argues for a field approach, presenting three advantages over a particle approach: particle wave functions are not available for photons, a classical field model of the electron gives a superior (...) account of both spin and self-interaction as compared to a classical particle model, and the space of field wave functionals appears to be larger than the space of particle wave functions. The article also describes two important tasks facing proponents of a field approach: legitimize or excise the use of Grassmann numbers for fermionic field values and in wave functional amplitudes, and describe how quantum fields give rise to particle-like behavior. (shrink)

Specially selected articles from the Journal of Advanced Nursing have been updated where appropriate by the original author. Models, Theories and Concepts brings together international authorities in their specialist fields to consider the gaps occurring between theory and practice, as well as the evaluation of a selection of models and emerging theories.