We consider first-order theories of topological fields admitting a model-completion and their expansion to differential fields . We give a criterion under which the expansion still admits a model-completion which we axiomatize. It generalizes previous results due to M. Singer for ordered differential fields and of C. Michaux for valued differential fields. As a corollary, we show a transfer result for the NIP property. We also give a geometrical axiomatization of that model-completion. Then, for certain differential (...) valued fields, we extend the positive answer of Hilbert’s seventeenth problem and we prove an Ax–Kochen–Ershov theorem. Similarly, we consider first-order theories of topological fields admitting a model-companion and their expansion to differential fields, and under a similar criterion as before, we show that the expansion still admits a model-companion. This last result can be compared with those of M. Tressl: on one hand we are only dealing with a single derivation whereas he is dealing with several, on the other hand we are not restricting ourselves to definable expansions of the ring language, taking advantage of our topological context. We apply our results to fields endowed with several valuations. (shrink)

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...

We construct a fibered dimension function in some topological differential fields.

In this paper we study predimension inequalities in differential fields and define what it means for such an inequality to be adequate. Adequacy was informally introduced by Zilber, and here we give a precise definition in a quite general context. We also discuss the connection of this problem to definability of derivations in the reducts of differentially closed fields. The Ax-Schanuel inequality for the exponential differential equation (proved by Ax) and its analogue for the differential equation of (...) the j-function (established by Pila and Tsimerman) are our main examples of predimensions. We carry out a Hrushovski construction with the latter predimension and obtain a natural candidate for the first-order theory of the differential equation of the j-function. It is analogous to Kirby's axiomatisation of the theory of the exponential differential equation (which in turn is based on the axioms of Zilber's pseudo-exponentiation), although there are many significant differences. In joint work with Sebastian Eterović and Jonathan Kirby we have recently proven that the axiomatisation obtained in this paper is indeed an axiomatisation of the theory of the differential equation of the j-function, that is, the Ax-Schanuel inequality for the j-function is adequate. (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.

The following strong form of density of definable types is introduced for theoriesTadmitting a fibered dimension functiond: given a modelMofTand a definable setX⊆Mn, there is a definable typepinX, definable over a code forXand of the samed-dimension asX. Both o-minimal theories and the theory of closed ordered differential fields are shown to have this property. As an application, we derive a new proof of elimination of imaginaries for CODF.

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)

Hrushovski proved that the theory of difference-differential fields of characteristic zero has a model-companion, which we shall denote DCFA. Previously, the author proved that this theory is supersimple. In supersimple theories there is a notion of rank defined in analogy with Lascar U-rank for superstable theories. It is also possible to define a notion of dimension for types in DCFA based on transcendence degree of realization of the types. In this paper we compute the rank of a model of (...) DCFA, give some properties regarding rank and dimension, and give an example of a definable set with finite rank but infinite dimension. Finally we prove that for the case of definable subgroup of the additive group being finite-dimensional and having finite rank are equivalent. (shrink)

In this paper we develop a differential analogue of o-minimal cell decomposition for the theory CODF of closed ordered differential fields. Thanks to this differential cell decomposition we define a well-behaving dimension function on the class of definable sets in CODF. We conclude this paper by proving that this dimension is closely related to both the usual differential transcendence degree and the topological dimension associated, in this case, with a natural differential topology on ordered (...) class='Hi'>differential fields. (shrink)

When the history of life on earth is viewed as a history of cell division, all of life becomes a single cell lineage. The growth and differentiation of this lineage in reciprocal interaction with its environment can be viewed as a developmental process; hence the evolution of life on earth can also be seen as the development of life on earth. Here, in reviewing this field, some potentially fruitful research directions suggested by this change in perspective are highlighted. Variation (...) and selection become, for example, bidirectional information flows between scales, while the notions of “cooperation” and “competition” become scale relative. The language of communication, inference, and information processing becomes more useful than the language of causation to describe the interactions of both homogeneous and heterogeneous living systems at any scale. Emerging scale‐free theoretical frameworks such as predictive coding and active inference provide conceptual tools for reconceptualizing biology as the study of a unified, multiscale dynamical system. (shrink)

In [T. Brihaye, C. Michaux, C. Rivière, Cell decomposition and dimension function in the theory of closed ordered differential fields, Ann. Pure Appl. Logic .] the authors proved a cell decomposition theorem for the theory of closed ordered differential fields which generalizes the usual Cell Decomposition Theorem for o-minimal structures. As a consequence of this result, a well-behaving dimension function on definable sets in CODF was introduced. Here we continue the study of this cell decomposition in CODF by (...) proving three additional results. We first discuss the relation between the δ-cells introduced in the above-mentioned reference and the notion of Kolchin polynomial in differential algebra. We then prove two generalizations of classical decomposition theorems in o-minimal structures. More exactly we give a theorem of decomposition into definably d-connected components and a differential cell decomposition theorem for a particular class of definable functions in CODF. (shrink)

In this paper, we develop the beginning of Lie-differential algebra, in the sense of Kolchin by using tools introduced by Hubert in [E. Hubert, Differential algebra for derivations with nontrivial commutation rules, J. Pure Appl. Algebra 200 163–190]. In particular it allows us to adapt the results of Tressl 3933–3951]) by showing the existence of a theory of Lie-differential fields of characteristic zero. This theory will serve as a model companion for every theory of large and Lie- (...) class='Hi'>differential fields extending a model complete theory of pure fields. As an application, we introduce the Lie counterpart of classical theories of differential fields in several commuting derivations. (shrink)

Hrushovski proved that the theory of difference-differential fields of characteristic zero has a model-companion, which we shall denote DCFA. Previously, the author proved that this theory is supersimple. In supersimple theories there is a notion of rank defined in analogy with Lascar U -rank for superstable theories. It is also possible to define a notion of dimension for types in DCFA based on transcendence degree of realization of the types. In this paper we compute the rank of a model (...) of DCFA, give some properties regarding rank and dimension, and give an example of a definable set with finite rank but infinite dimension. Finally we prove that for the case of definable subgroup of the additive group being finite-dimensional and having finite rank are equivalent. (shrink)

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, …, 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)

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.

Irigaray turns to Merleau-Ponty's intuitions about the perception of color to develop her own insights into the creative emergence of sexuate identity. As a quality of the flesh, color cannot be reduced to formal codes. The privileging of word and text inherent to Western culture suppresses the coming into being of the embodied subject in his or her own situated context. Color, tied as it is to a corporeal creativity could provide an important link since it facilitates reflection, and a (...) re-enfleshing through color of a differentiated sexuate identity tied to the imagination as well as to genetic identity. (shrink)

We outline the Hrushovsk-Sokolović proof of Vaught's Conjecture for differentially closed fields, focusing on the use of dimensions to code graphs.

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)

We examine the connections between several automorphism groups associated with a saturated differentially closed field U of characteristic zero. These groups are: Γ, the automorphism group of U; the automorphism group of Γ; , the automorphism group of the differential combinatorial geometry of U and , the group of field automorphisms of U that respect differential closure.Our main results are:• If U is of cardinality λ+=2λ for some infinite regular cardinal λ, then the set of subgroups (...) of Γ consisting of all pointwise stabilizers of Dcl's of finite subsets of U is invariant under Aut .• If U is of arbitrary infinite cardinality, then each automorphism of the differential combinatorial geometry is induced by a field automorphism of U that respects differential closure .• If U is of cardinality λ+=2λ for some infinite regular cardinal λ, then each automorphism of Γ is induced by an element of acting on Γ by conjugation .• If U is of cardinality λ+=2λ for some infinite regular cardinal λ, then the outer automorphism group of Γ is isomorphic to the multiplicative group of the rationals. (shrink)

Analysability of finiteU-rank types are explored both in general and in the theory${\rm{DC}}{{\rm{F}}_0}$. The well-known fact that the equation$\delta \left = 0$is analysable in but not almost internal to the constants is generalized to show that$\underbrace {{\rm{log}}\,\delta \cdots {\rm{log}}\,\delta }_nx = 0$is not analysable in the constants in$\left$-steps. The notion of acanonical analysisis introduced–-namely an analysis that is of minimal length and interalgebraic with every other analysis of that length. Not every analysable type admits a canonical analysis. Using properties of (...) reductions and coreductions in theories with the canonical base property, it is constructed, for any sequence of positive integers$\left$, a type in${\rm{DC}}{{\rm{F}}_0}$that admits a canonical analysis with the property that theith step hasU-rank${n_i}$. (shrink)

A new approach to developing formulisms of physics based solely on laws of mathematics is presented. From simple, classical statistical definitions for the observed space-time position and proper velocity of a particle having a discrete spectrum of internal states we derive u generalized Schrödinger equation on the space-time manifold. This governs the evolution of an N component wave function with each component square integrable over this manifold and is structured like that for a charged particle in an electromagnetic field (...) but also includes SU(N) gauge field couplings. This construction reveals a new hasis for gauge invariance and new insight into the appearance of spin and other such properties in relativistic quantum mechanics and suggests a new charged particle model. (shrink)

In this paper we deal with the model theory of differentially closed fields of characteristic zero with finitely many commuting derivations. First we observe that the only known lower bound for the Lascar rank of types in differentially closed fields, announced in a paper of McGrail, is false. This gives us a new class of regular types which are orthogonal to fields. Then we classify the subgroups of the additive group of Lascar rank omega with differential-type 1 which are (...) nonorthogonal to fields. The last parts consist of an analaysis of the quotients of the heat variety. We show that the generic type of such a quotient is locally modular. Finally, we answer a question of Phylliss Cassidy about the existence of certain Jordan-Holder type series in the negative. (shrink)

We revisit Kolchin's results on definability of differential Galois groups of strongly normal extensions, in the case where the field of constants is not necessarily algebraically closed. In certain classes of differential topological fields, which encompasses ordered or p‐valued differential fields, we find a partial Galois correspondence and we show one cannot expect more in general. In the class of ordered differential fields, using elimination of imaginaries in, we establish a relative Galois correspondence for relatively (...) definable subgroups of the group of differential order automorphisms. (shrink)

Given a model-complete theory of topological fields, we considered its generic differential expansions and under a certain hypothesis of largeness, we axiomatised the class of existentially closed ones. Here we show that a density result for definable types over definably closed subsets in such differential topological fields. Then we show two transfer results, one on the VC-density and the other one, on the combinatorial property NTP2.

We show that in the Maxwell–Lorentz theory of classical electrodynamics most initial values for fields and particles lead to an ill-defined dynamics, as they exhibit singularities or discontinuities along light-cones. This phenomenon suggests that the Maxwell equations and the Lorentz force law ought rather to be read as a system of delay differential equations, that is, differential equations that relate a function and its derivatives at different times. This mathematical reformulation, however, leads to physical and philosophical consequences for (...) the ontological status of the electromagnetic field. In particular, fields cannot be taken as independent degrees of freedom, which suggests that one should not add them to the ontology. (shrink)

Morley rank and Lascar rank are equal on generic types of definable groups in differentially closed fields with finitely many commuting derivations.

Field dependence-independence (FDI) is a widely studied dimension of cognitive styles designed to measure an individual’s ability to identify embedded parts of an organized visual field as entities separate from that given field. The research aims to determine whether the brain activity features that are considered to be perceptual switching indicators could serve as robust features, differentiating Field-Dependent (FD) from Field-Independent (FI) participants. Previous research suggests that various features derived from event related potentials (ERP) and (...) frequency features are associated with the perceptual reversal occurring during the observation of a bistable image. In this study, we combined these features in the context of a different experimental scheme using ambiguous and unambiguous stimuli during participants’ perceptual observations. We assessed the participants’ FD-I classification with the use of the Hidden Figures Test (HFT). Results show that the peak amplitude of the frontoparietal positivity, the late positive deflection in frontal and parietal areas, is higher for the FD group at specific locations of the left lobe, whereas it occurs later for the FD group at the central and occipital electrodes. Additionally, the FD group exhibits higher levels of gamma power before stimulus onset at channel TP10 and higher gamma power during reversal at the right centroparietal electrodes (T8, CP6 and TP10). The peak amplitude of the reversal positivity, the positive deflection during the reversal, is higher for the FD group at the rear right lobe (P4). (shrink)

First, it is pointed out how the author's new differential Galois theory contributes to the understanding of the differential closure of an arbitrary differential field . Secondly, it is shown that a superstable differential field has no proper differential Galois extensions.

This paper introduces a natural extension of Kolchin's differential Galois theory to positive characteristic iterative differential fields, generalizing to the non-linear case the iterative Picard—Vessiot theory recently developed by Matzat and van der Put. We use the methods and framework provided by the model theory of iterative differential fields. We offer a definition of strongly normal extension of iterative differential fields, and then prove that these extensions have good Galois theory and that a G-primitive element theorem (...) holds. In addition, making use of the basic theory of arc spaces of algebraic groups, we define iterative logarithmic equations, finally proving that our strongly normal extensions are Galois extensions for these equations. (shrink)

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

For every natural numberm, the existentially closed models of the theory of fields withmcommuting derivations can be given a first-order geometric characterization in several ways. In particular, the theory of these differential fields has a model-companion. The axioms are that certain differential varieties determined by certain ordinary varieties are nonempty. There is no restriction on the characteristic of the underlying field.

As a continuation of the work of the third author in [5], we make further observations on the features of Galois cohomology in the general model theoretic context. We make explicit the connection between forms of definable groups and first cohomology sets with coefficients in a suitable automorphism group. We then use a method of twisting cohomology (inspired by Serre’s algebraic twisting) to describe arbitrary fibres in cohomology sequences—yielding a useful “finiteness” result on cohomology sets.Applied to the special case of (...) differential fields and Kolchin’s constrained cohomology, we complete results from [3] by proving that the first constrained cohomology set of a differential algebraic group over a bounded, differentially large, field is countable. (shrink)

Using the Lascar inequalities, we show that any finite rank δ-closed subset of a quasiprojective variety is definably isomorphic to an affine δ-closed set. Moreover, we show that if X is a finite rank subset of the projective space P n and a is a generic point of P n , then the projection from a is injective on X. Finally we prove that if RM = RC in DCF 0 , then RM = RU.

Those enlightened philosophers of physics acknowledging some manner of descent from Kant’s ‘Copernican Revolution’ have long found encouragement and inspiration in the writings of Roberto Torretti. In this tribute, I focus on his “perspective on Kant’s perspective on objectivity” (2008), a short but highly stimulating attempt to extract the essential core of the Kantian doctrine that ‘objects of knowledge’ are constituted, not given, or with Roberto’s inimitable pungency, that “objectivity is an achievement, not a gift.” That essential core Roberto locates (...) in the Kantian notion of apperception, or self-activity, manifested in cognition in the idea of combination (Verbindung) or composition, which, Kant tells us, “among all ideas … is the one that is not given through objects, but can only be performed by the subject itself, because it is an act of self-activity” (B 130). I first rehearse Roberto’s proposal for how an imaginative interplay between sensibility and understanding can be fashioned via the productive imagination or power of reflective judgment (of the third Critique). In this way, the notion of composition in general, unfettered from needless period constraints issuing in “pure forms of sensibility” and “pure concepts of the understanding”, can be seen as the intellectual motor for the “free creation” of concepts celebrated by Einstein and others, furnishing structural scaffolding required to articulate and display physical objects and processes, a conceptual panoply that “cannot be fished out of the stream of impressions”. Roberto emphasizes that historical case studies are needed to evaluate his proposal, suggesting one himself, the continuous conceptual development inaugurated by Riemann’s Habilitätionsschrift (1854) resulting, some hundred years later, in the fiber bundle formalism of modern differential geometry and topology. I sketch a related suggestion, that the gauge groups of modern particle physics are the outcome of a similar line of conceptual advance, a structural scaffolding saving the phenomena of high-energy experiment within the framework of ‘effective field theory.’. (shrink)