Related papers: On generalized series fields and exponential-logar…
Invariants of generalized tensor fields on a line are classified using special polynomials P_mk^(-1/lambda) introduced here for this purpose. For the case of positive characteristic, a new invariant of formal power series, a width, is…
A simple theory of the covariant derivatives, deformed derivatives and relative covariant derivatives of extensor fields is present using algebraic and analytical tools developed in previous papers. Several important formulas are derived.
Projective spaces for finite-dimensional vector spaces over general fields are considered. The geometry of these spaces and the theory of line bundles over these spaces is presented. Particularly, the space of global regular sections of…
We study in detail the general solution for a scalar field cosmology with an exponential potential, correcting some imprecisions, encountered previously in the literature. In addition, we generalize this solution for a piecewise exponential…
These notes contain a survey of some aspects of the theory of differential modules and complexes as well as of their generalization, that is, the theory of $N$-differential modules and $N$-complexes. Several applications and examples coming…
We study in this paper logarithmic derivatives associated to derivations on graded complete Lie algebra, as well as the existence of inverses. These logarithmic derivatives, when invertible, generalize the exp-log correspondence between a…
The differential field of transseries extends the field of real Laurent series, and occurs in various context: asymptotic expansions, analytic vector fields, o-minimal structures, to name a few. We give an overview of the algebraic and…
We generalize the concept of a number derivative, and examine one particular instance of a deformed number derivative for finite field elements. We find that the derivative is linear when the deformation is a Frobenius map and go on to…
We define the universal exponential extension of an algebraically closed differential field and investigate its properties in the presence of a nice valuation and in connection with linear differential equations. Next we prove normalization…
We derive certain systems of differential equations for matrix elements of products and iterates of logarithmic intertwining operators among strongly graded generalized modules for a strongly graded conformal vertex algebra under suitable…
Over a global field any finite number of central simple algebras of exponent dividing $m$ is split by a common cyclic field extension of degree $m$. We show that the same property holds for function fields of two-dimensional excellent…
We develop a notion of (principal) differential rank for differential-valued fields, in analog of the exponential rank and of the difference rank. We give several characterizations of this rank. We then give a method to define a derivation…
We investigate when a computable automorphism of a computable field can be effectively extended to a computable automorphism of its (computable) algebraic closure. We then apply our results and techniques to study effective embeddings of…
In the following article, we give a description of the distingushed irreducible principal series representations of the general linear group over a p-adic field in terms of inducing datum. This provides a counter-example to a conjecture of…
We study subfields of surreal numbers, called hyperseries fields, that are suited to be equipped with derivations and composition laws. We show how to define embeddings on hyperseries fields that commute with transfinite sums and all…
We define generalized vector fields, and contraction and Lie derivatives with respect to them. Generalized commutators are also defined.
The quantum correlations of scalar fields are examined as a power series in derivatives. Recursive algebraic equations are derived and determine the amplitudes; all loop integrations are performed. This recursion contains the same…
We study the logarithmic conformal field theories in which conformal weights are continuous subset of real numbers. A general relation between the correlators consisting of logarithmic fields and those consisting of ordinary conformal…
This monograph, written for educational purposes, serves as an introduction to the concept of integrability as it applies to systems of differential equations (both ordinary and partial) as well as to vector-valued fields. The general cases…
We axiomatize a class of existentially closed exponential fields equipped with an $E$-derivation. We apply our results to the field of real numbers endowed with $exp(x)$ the classical exponential function defined by its power series…