Related papers: Van der Corput's Difference Theorem: some modern d…
This expository paper presents some applications of the parameterized Picard-Vessiot theory for ordinary differential equations, mainly related to monodromy.
In one dimension, the theory of the $G$-normal distribution is well-developed, and many results from the classical setting have a nonlinear counterpart. Significant challenges remain in multiple dimensions, and some of what has already been…
Some important concepts in the nonstandard analysis theory of turbulence are presented in this article. The structure of point, on which differential equations are defined, is analyzed. The distinction between the uniform point and the…
We define vector fields, leaves and trajectories for schemes. With these tools, we are able to give a geometrical interpretation and to generalize several results of differential Galois theory and constructions on differential schemes. We…
We examine some aspects of the recent results by K. Binder. The equilibrium formation/dissolution of droplets in finite systems is discussed in the context of the canonical and the grand canonical distributions.
In this paper we present a multidimensional version of the van der Corput lemma where the decay of the oscillatory integral is gained with respect to all space variables, connecting the standard one-dimensional van der Corput lemma with the…
This article aims to explain essential elements of perturbation theory and their conceptual underpinnings. It is not meant as a summary of popular perturbation methods, though some illustrative examples are given to underline the main…
We prove that some of the basic differential functions appearing in the (unramified) theory of arithmetic differential equations, especially some of the basic differential modular forms in that theory, arise from a "ramified situation".…
I will sketchily illustrate how the theory of symmetry helps in determining solutions of (deterministic) differential equations, both ODEs and PDEs, staying within the classical theory. I will then present a quick discussion of some more…
We study the discrete version of the $p$-Laplacian. Based on its variational properties we discuss some features of the associated parabolic problem. Our approach allows us in turn to obtain interesting information about positivity and…
We obtain global and local theorems on the existence of invariant manifolds for perturbations of non autonomous linear difference equations assuming a very general form of dichotomic behavior for the linear equation. The results obtained…
We give a general method of deriving statistical limit theorems, such as the central limit theorem and its functional version, in the setting of ergodic measure preserving transformations. This method is applicable in situations where the…
This paper deals with the comparison of two common types of equivalence groups of differential equations, and this gives rise to a number of results presented in the form of theorems. It is shown in particular that one type can be…
It is a basic introduction to differential graded Lie algebras, Maurer-Cartan equation and associated deformation functors.
We introduce a new, elementary method for studying random differences in arithmetic progressions and convergence phenomena along random sequences of integers. We apply our method to obtain significant improvements on previously known…
It is demonstrated that today's quantum fluctuation theorems are component part of old quantum fluctuation-dissipation relations [Sov.Phys.-JETP 45, 125 (1977)], and typical misunderstandings in this area are pointed out.
A brief introduction to Topological Quantum Field Theory as well as a description of recent progress made in the field is presented. I concentrate mainly on the connection between Chern-Simons gauge theory and Vassiliev invariants, and…
Recent advances in computational techniques for $K$-theory allow us to describe the $K$-theory of toric varieties in terms of the $K$-theory of fields and simple cohomological data.
We obtain large deviations theorems for nonconventional sums with underlying process being a Markov process satisfying the Doeblin condition or a dynamical system such as subshift of finite type or hyperbolic or expanding transformation.
Consideration of the Noether variational problem for any theory whose action is invariant under global and/or local gauge transformations leads to three distinct theorems. These include the familiar Noether theorem, but also two equally…