Related papers: Differentially Large Fields
This paper concerns pairs of models of the theory of the differential field of logarithmic-exponential transseries that are tame as a pair of real closed fields. That is, the smaller model is bounded inside the larger model and there exists…
We give a classification of non-orthogonality classes of trivial order 1 strongly minimal sets in differentially closed fields. A central idea is the introduction of $\tau$-forms, functions on the prolongation of a variety which are…
We develop here the algebra of the differential field of transseries and of related valued differential fields. This book contains in particular our recently obtained decisive positive results on the model theory of these structures.
We define nodal finite dimensional algebras and describe their structure over an algebraically closed field. For a special class of such algebras (type A) we find a criterion of tameness.
Higher derivative scalar field theories have received considerable attention for the potentially explanations of the initial state of the universe or the current cosmic acceleration which they might offer. They have also attracted many…
Finite differences have been widely used in mathematical theory as well as in scientific and engineering computations. These concepts are constantly mentioned in calculus. Most frequently-used difference formulas provide excellent…
It is shown how to extend the formal variational calculus in order to incorporate integrals of divergences into it. Such a generalization permits to study nontrivial boundary problems in field theory on the base of canonical formalism.
Given an algebra $A$ over a differential field $K$, we study derivations on $A$ that are compatible with the derivation on $K$. There is a universal object, which is a twisted version of the usual module of differentials, and we establish…
We investigate the large values of class numbers of cubic fields, showing that one can find arbitrary long sequences of "close" abelian cubic number fields with class numbers as large as possible. We also give a first step toward an…
We use the differential algebra of polytopes to explain the known remarkable relation of the combinatorics of the associahedra and permutohedra with the universal compositional and multiplicative inversion formulas for the formal power…
We study in detail the valuation theory of deeply ramified fields and introduce and investigate several other related classes of valued fields. Further, a classification of defect extensions of prime degree of valued fields that was earlier…
We characterize completey (give a necessary and suffcient condition using special neat embeddings)for a relation algebra to belong to the amalgamation, strong amalgamation, and superamalgamation base of the class of representable algebras.…
We study differential splitting fields of quaternion algebras with derivations. A quaternion algebra over a field $k$ is always split by a quadratic extension of $k$. However, a differential quaternion algebra need not be split over any…
The article is devoted to approximate, global and along curves differentiability of functions over non-archimedean infinite fields with non-trivial valuations. Fields with zero and non-zero characteristics are considered. Spaces of…
We consider cubic number fields ordered by their discriminants, and show that there exist arbitrarily long sequences that contain only fields with class numbers greater than a given bound.
Skew-symmetric forms possess unique capabilities. The properties of closed exterior and dual forms, namely, invariance, covariance, conjugacy and duality, either explicitly or implicitly appear in all invariant mathematical formalisms. This…
Complexes and cohomology, traditionally central to topology, have emerged as fundamental tools across applied mathematics and the sciences. This survey explores their roles in diverse areas, from partial differential equations and continuum…
In this paper, we give a geometrization and a generalization of a lemma of differential Galois theory. This geometrization, in addition of giving a nice insight on this result, offers us the occasion to investigate several points of…
An abstract theory of ultradifferentiable sheafs is developed. Moreover, various applications to the theory of linear partial differential equations, differential geometry and, in particular, CR geometry are discussed.
Ordinary differential equations have an arithmetic analogue in which functions are replaced by numbers and the derivation operator is replaced by a Fermat quotient operator. In this survey we explain the main motivations, constructions,…