Maps on Ultrametric Spaces,Hensel's Lemma,and Differential Equations Over Valued Fields

We give a criterion for maps on ultrametric spaces to be surjective and to preserve spherical completeness. We show how Hensel's Lemma and the multidimensional Hensel's Lemma follow from our result. We give an easy proof that the latter holds in every henselian field. We also prove a basic infinite-dimensional Implicit Function Theorem. Further, we apply the criterion to deduce various versions of Hensel's Lemma for polynomials in several additive operators, and to give a criterion for the existence of integration and solutions of certain differential equations on spherically complete valued differential fields, for both valued D-fields in the sense of Scanlon, and differentially valued fields in the sense of Rosenlicht. We modify the approach so that it also covers logarithmic-exponential power series fields. Finally, we give a criterion for a sum of spherically complete subgroups of a valued abelian group to be spherically complete. This in turn can be used to determine elementary properties of power series fields in positive characteristic.     

Metrically and topologically projective ideals of Banach algebras

In the present paper, necessary conditions for the metric and topological projectivity of closed ideals of Banach algebras are given. In the case of commutative Banach algebras, a criterion for the metric and topological projectivity of ideals admitting a bounded approximate identity is obtained. The main result of the paper is as follows: a closed ideal of an arbitrary C*-algebra is metrically or topologically projective if and only if it admits a self-adjoint right identity.     

Special transformations in algebraically closed valued fields

We present two of the three major steps in the construction of motivic integration, that is, a homomorphism between Grothendieck semigroups that are associated with a first-order theory of algebraically closed valued fields, in the fundamental work of Hrushovski and Kazhdan (2006) [8]. We limit our attention to a simple major subclass of V-minimal theories of the form ACV F_S(0,0), that is, the theory of algebraically closed valued fields of pure characteristic 0 expanded by a (V F,Γ)-generated substructure S in the language L_RV. The main advantage of this subclass is the presence of syntax. It enables us to simplify the arguments with many different technical details while following the major steps of the Hrushovski-Kazhdan theory.     

Model-companions and definability in existentially complete structures

The theory of model-companions and existentially complete structures is both reviewed and developed further. The review begins with A. Robinson’s work in the fifties and continues through the definability of second order structures in existentially complete groups. New results include necessary and sufficient conditions for the existence of a model-companion in terms of the definability of general elementary properties. The main theorem of the paper gives necessary and sufficient conditions for the existence of a model-companion for universal theories with finite presentations and the amalgamation property. This result generalizes the result of P. Eklof and G. Sabbagh that the theory ofR-modules has a model-completion if and only ifR is coherent. The author’s survey presentation at the Robinson Memorial Conference included the material in the first and fourth sections of this paper and Theorem 1 and its corollaries and examples in the second section. The other results in Section 2 and the results in Section 3 were obtained after the conference.     

Note sur les corps différentiellement clos valués

In the paper by Guzy and Point, Differential topological fields, the model-completion ${(\mathit{OVF})}_D^1$ of the theory of ordered valued differential fields ${\mathit{OVF}}_D$ is established. Models of this theory are closed ordered differential fields (the theory CODF was studied by Singer) which have a non-trivial convex (for the order) subring as valuation ring. Here we prove the valued analogue of a result of Singer: if K is a model of ${(\mathit{OVF})}_D^1$ then $K(\mathrm{i})$ (${\mathrm{i}}^2=?1$) is a model of the theory of differentially closed valued fields which is the model-completion of the theory of non-trivially valued differential fields of characteristic zero. To cite this article: N. Guzy, C. R. Acad. Sci. Paris, Ser. I 341 (2005).     

Separably closed valued fields: Immediate expansions

It is shown in this paper that every separably closed field with a non-trivial valuation, treated as a first-order structure in the language of valued fields, is an immediate expansion of the underlying first-order field structure; that is to say that there is no proper intermediate first-order structure between these two structures.     

Places of algebraic function fields in arbitrary characteristic

We consider the Zariski space of all places of an algebraic function field F|K of arbitrary characteristic and investigate its structure by means of its patch topology. We show that certain sets of places with nice properties (e.g., prime divisors, places of maximal rank, zero-dimensional discrete places) lie dense in this topology. Further, we give several equivalent characterizations of fields that are large, in the sense of F. Pop's Annals paper Embedding problems over large fields. We also study the question whether a field K is existentially closed in an extension field L if L admits a K-rational place. In the appendix, we prove the fact that the Zariski space with the Zariski topology is quasi-compact and that it is a spectral space.     

Bispaces admitting only bicomplete or only totally bounded quasi-metrics

We characterize quasi-metrizable bispaces that admit only bicomplete quasimetrics by means of doubly primitive sequences, and deduce that if (X, S, T) is a quasi-metrizable bispace admitting only bicomplete quasi-metrics and either (X, S) or (X, T) is hereditarily Lindelöf, then (X, S ∨ T) is compact. We also give an example which shows that hereditary Lindelöfness cannot be omitted in the above result. Finally, we show that a quasi-pseudometrizable bispace (X, S, T) admits only totally bounded quasi-pseudometrics if and only if (X, S ∨ T) is compact, and deduce that a quasi-pseudometrizable topological space admits only totally bounded quasi-pseudometrics if and only if it is hereditarily compact and quasi-sober (equivalently, if and only if it admits a unique quasi-uniformity).     

Complete systems, elementary submodels and the tightness of upper hyperspaces

We introduce and study some completeness properties for systems of open coverings of a given topological space. A Hausdorff space admitting a system of cardinality κ satisfying one of these properties is of type Gκ. Hence, we define several new variants of the ?ech number and use elementary submodels to determine further results. We introduce M-hulls and M-networks, for M elementary submodel. As an application, we give estimates for both the tightness and the Lindelöf number of a generic upper hyperspace. Two recent results of Costantini, Holá and Vitolo on the tightness of co-compact hyperspaces follow from ours as corollaries.     

Retractions onto series-parallel posets

The poset retraction problem for a poset P is whether a given poset Q containing P as a subposet admits a retraction onto P, that is, whether there is a homomorphism from Q onto P which fixes every element of P. We study this problem for finite series-parallel posets P. We present equivalent combinatorial, algebraic, and topological charaterisations of posets for which the problem is tractable, and, for such a poset P, we describe posets admitting a retraction onto P.     

On the cohomology of flows of stochastic and random differential equations

We consider the flow of a stochastic differential equation on d-dimensional Euclidean space. We show that if the Lie algebra generated by its diffusion vector fields is finite dimensional and solvable, then the flow is conjugate to the flow of a non-autonomous random differential equation, i.e. one can be transformed into the other via a random diffeomorphism of d-dimensional Euclidean space. Viewing a stochastic differential equation in this form which appears closer to the setting of ergodic theory, can be an advantage when dealing with asymptotic properties of the system. To illustrate this, we give sufficient criteria for the existence of global random attractors in terms of the random differential equation, which are applied in the case of the Duffing-van der Pol oscillator with two independent sources of noise. Received: 25 May 1999 / Revised version: 19 October 2000 / Published online: 26 April 2001     

On projective symmetries on Finsler spaces

There are two definitions of Einstein-Finsler spaces introduced by Akbar-Zadeh, which we will show is equal along the integral curves of I-invariant projective vector fields. The sub-algebra of the C-projective vector fields, leaving the H-curvature invariant, has been studied extensively. Here we show on a closed Finsler space with negative definite Ricci curvature reduces to that of Killing vector fields. Moreover, if an Einstein-Finsler space admits such a projective vector field then the flag curvature is constant. Finally, a classification of compact isotropic mean Landsberg manifolds admitting certain projective vector fields is obtained with respect to the sign of Ricci curvature.     

Quantifier elimination on some pseudo-algebraically closed valued fields

In this article, some theories of pseudo-algebraically closed non-trivially valued fields are shown to admit quantifier elimination in the language obtained by adjoining to the language of rings the function symbols for splitting coefficients, the function symbols for relative p-coordinate functions, and the division predicate for a valuation.     

Heterogeneous wires made of martensitic materials

In this paper we present a direct derivation of a theory of heterogeneous wires starting from three-dimensional nonlinear hyperelasticity augmented by an interfacial energy term. The derivation involves no a priori choice of asymptotic expansion or ansatz. It yields a wire theory with two Cosserat vector fields. The theory is applied to multiwell energy functions appropriate for martensitic materials. A formal derivation of higher theories of homogeneous wires is given, which yields three additional Cosserat vector fields and an explicit form for the bending and torsion energy. To cite this article: H. Le Dret, N. Meunier, C. R. Acad. Sci. Paris, Ser. I 337 (2003).     

A countably compact free Abelian group of size continuum that admits a non-trivial convergent sequence

We show that it is consistent with ZFC that the free Abelian group of cardinality c admits a topological group topology that makes it countably compact with a non-trivial convergent sequence.     

Dualities for quasi-varieties of bands

We complete the characterization of finite bands admitting a natural duality, by showing that every finite normal band admits a natural duality. In particular we show that every finite normal band is finitely related.     

On differential Galois groups of strongly normal extensions

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.     

Polynomial normal forms with exponentially small remainder for analytic vector fields

A key tool in the study of the dynamics of vector fields near an equilibrium point is the theory of normal forms, invented by Poincaré, which gives simple forms to which a vector field can be reduced close to the equilibrium. In the class of formal vector valued vector fields the problem can be easily solved, whereas in the class of analytic vector fields divergence of the power series giving the normalizing transformation generally occurs. Nevertheless the study of the dynamics in a neighborhood of the origin can very often be carried out via a normalization up to finite order. This paper is devoted to the problem of optimal truncation of normal forms for analytic vector fields in R^m. More precisely we prove that for any vector field in R^m admitting the origin as a fixed point with a semi-simple linearization, the order of the normal form can be optimized so that the remainder is exponentially small. We also give several examples of non-semi-simple linearization for which this result is still true.     

Existence of derivations on near-rings

We obtain conditions on (R,+) which force that the zero map is the only derivation on a zero-symmetric near-ring R. Throughout the paper we construct several new examples of near-rings which are not rings admitting non-zero derivations, non-zero (σ, σ)-derivations and non-zero (1, σ)-derivations.