Related papers: Iterated Differential Forms IV: C-Spectral Sequenc…
Hopf algebra structure on the differential algebra of the extended $q$-plane is defined. An algebra of forms which is obtained from the generators of the extended $q$-plane is introduced and its Hopf algebra structure is given.
In this paper, we use the language of noncommutative differential geometry to formalise discrete differential calculus. We begin with a brief review of inverse limit of posets as an approximation of topological spaces. We then show how to…
In this paper we construct the jet geometrical extensions of the KCC-invariants, which characterize a given second-order system of differential equations on the 1-jet space $J^1(R,M)$. A generalized theorem of characterization of our jet…
We introduce a notion of elliptic differential graded Lie algebra. The class of elliptic algebras contains such examples as the algebra of differential forms with values in endomorphisms of a flat vector bundle over a compact manifold, etc.…
We present an overview of some recent developments in the theory of generalized formal series, grounded in diffeological geometric framework. These constructions aim to offer new tools for understanding infinite-dimensional phenomena in…
We describe various properties of continued fraction expansions of complex numbers in terms of Gaussian integers. Numerous distinct such expansions are possible for a complex number. They can be arrived at through various algorithms, as…
This paper brings together C*-algebras and algebraic topology in terms of viewing a C*-algebraic invariant in terms of a topological spectrum. E-theory, E(A,B), is a bivariant functor in the sense that is a cohomology functor in the first…
Differential geometry may be generalized to allow infinitesimals to any order. The purpose of the present contribution is to show that the theory so developed expands received geometrical ideas in an interesting way, rich in potential for…
This is the second in a series of papers laying the foundations for a differential graded approach to derived differential geometry (and other geometries in characteristic zero). In this paper, we extend the classical notion of a dg-algebra…
Higher order automorphic forms have recently been introduced to study important questions in number theory and mathematical physics. We investigate the connection between these functions and Chen's iterated integrals. Then using Chen's…
Let $A$ be a finite-dimensional algebra over a field $k$. We define $A$ to be $\mathbf{C}$-dichotomic if it has the dichotomy property of the representation type on complexes of projective $A$-modules. $\mathbf{C}$-dichotomy implies the…
Differential equations are derived for a continuous limit of iterated Schwarzian reflection of analytic curves, and solutions are interpreted as geodesics in an infinite-dimensional symmetric space geometry.
We provide a general theoretical framework allowing us to extend the classical Lie theory for partial differential equations to the case of equations of fractional order. We propose a general prolongation formula for the study of Lie…
We give an abstract formulation of the formal theory partial differential equations (PDEs) in synthetic differential geometry, one that would seamlessly generalize the traditional theory to a range of enhanced contexts, such as…
We define an abstract framework called {\it discrete finite differences embedding} which can be used to obtain discrete analogue of formal functional relations in the spirit of category theory. For ordinary differential equations we exhibit…
Iterated Segre mappings of real analytic generic submanifolds in complex space have been an essential tool in the study of holomorphic, formal, and CR mappings between such manifolds. In this paper we present a theory of iterated Segre…
A delayed term in a differential equation reflects the fact that information takes significant time to travel from one place to another within a process being studied. Despite de apparent similarity with ordinary differential equations,…
In this article we construct what we call a higher spectral sequence for any chain complex (or topological space) that is filtered in $n$ compatible ways. For this we extend the previous spectral system construction of the author, and we…
This paper is divided into two parts. The first is a review, through categorical lenses, of the classical theory of regular-singular differential systems over $C((x))$ and $\mathbb P^1_C\smallsetminus\{0,\infty\}$, where $C$ is…
We present an overview of a theory of complex dimensions of self-similar fractal strings, and compare this theory to the theory of varieties over a finite field from the geometric and the dynamical point of view. Then we combine the several…