Related papers: Galois groupoid and confluence of difference equat…

In positive characteristic, nearly all Picard-Vessiot extensions are inseparable over some intermediate iterative differential extensions. In the Galois correspondence, these intermediate fields correspond to nonreduced subgroup schemes of…

In this article we discuss some general results on the covariant Picard groupoid in the context of differential geometry and interpret the problem of lifting Lie algebra actions to line bundles in the Picard groupoid approach.

We show that the triviality of the differential Galois cohomologies over a partial differential field K of a linear differential algebraic group is equivalent to K being algebraically, Picard-Vessiot, and linearly differentially closed.…

Here we define a Caputo like discrete fractional difference and we compare it to the earlier defined Riemann-Liouville fractional discrete analog. Then we produce discrete fractional Taylor formulae for the first time, and we estimate their…

In this paper, we develop a theory of Galois descent for equivariant line bundles on partial flag schemes. In particular, we study computational aspects of the classification of descent data of equivariant line bundles attached to…

Let $C$ be a complex algebraic curve uniformised by a Fuchsian group $\Gamma$. In the first part of this paper we identify the automorphism group of the solenoid associated with $\Gamma$ with the Belyaev completion of its commensurator…

By solving a control problem and using Malliavin calculus, explicit derivative formula is derived for the semigroup $P_t$ generated by the Gruschin type operator on $\R^{m}\times \R^{d}:$ $$L (x,y)=\ff 1 2 \bigg\{\sum_{i=1}^m \pp_{x_i}^2…

In this paper, we construct, for some $2$-groups $G$, explicit Galois extensions $E/\mathbb{Q}(T)$ of group $G$ with $E\cap\overline{\mathbb{Q}}=\mathbb{Q}$. We also provide explicit arithmetic progressions of integers $t_0$ such that the…

We express discrete Painlev\'e equations as discrete Hamiltonian systems. The discrete Hamiltonian systems here mean the canonical transformations defined by generating functions. Our construction relies on the classification of the…

We study the Rayleigh-Stokes problem for a generalized second-grade fluid which involves a Riemann-Liouville fractional derivative in time, and present an analysis of the problem in the continuous, space semidiscrete and fully discrete…

Discrete Painlev\'e equations are nonlinear, nonautonomous difference equations of second-order. They have coefficients that are explicit functions of the independent variable $n$ and there are three different types of equations according…

In this paper we present a Galois-Grothendieck-type correspondence for groupoid actions. As an application a Galois-type correspondence is also given.

In the article we discuss the notion of the generalized invariant manifold introduced in our previous study. In the literature the method of the differential constraints is well known as a tool for constructing particular solutions for the…

Inspired by Kummer theory on abelian varieties, we give similar looking descriptions of the Galois groups occuring in the differential Galois theories of Picard-Vessiot, Kolchin and Pillay, and mention some arithmetic applications.

A major open problem in current Galois theory is to characterize those profinite groups which appear as absolute Galois groups of various fields. Obtaining detailed knowledge of the structure of quotients and subgroup filtrations of Galois…

The discrete gradient approach is generalized to yield integral preserving methods for differential equations in Lie groups.

Choose $q\in {\mathbb C}$ with 0<|q|<1. The main theme of this paper is the study of linear q-difference equations over the field K of germs of meromorphic functions at 0. It turns out that a difference module M over K induces in a…

Let K/F be a cyclic field extension of odd prime degree. We consider Galois embedding problems involving Galois groups with common quotient Gal(K/F) such that corresponding normal subgroups are indecomposable Fp[Gal(K/F)]-modules. For these…

In this survey we present the parameterized Galois theory of difference equations, as introduced by Hardouin-Singer. The purpose of this theory is to give a systematic approach to differential transcendence, also called hypertranscendence.…

We introduce a new graph invariant of finite groups that provides a complete characterization of the splitting types of unramified prime ideals in normal number field extensions entirely in terms of the Galois group. In particular, each…