Related papers: Iterated Differential Forms III: Integral Calculus
Boolean calculus has been studied extensively in the past in the context of switching circuits, error-correcting codes etc. This work generalizes several approaches to defining a differential calculus for Boolean functions. A unified theory…
In this work nul-filiform and filiform Zinbiel algebras are described up to isomorphism. Moreover, the classification of complex Zinbiel algebras is extended from dimensions $\leq 3$ up to the dimension $4.$
In these expository notes we draw together and develop the ideas behind some recent progress in two directions: the treatment of finite type partial differential operators by prolongation, and a class of differential complexes known as…
$\lambda$-Scale is an enrichment of lambda calculus which is adapted to emergent algebras. It can be used therefore in metric spaces with dilations.
In this paper, a new higher Hochschild Complex is defined with an Iterated Integral map to locally model differential forms on the space of bigons on $M$. In particular, given the local data for a gerbe with structure 2-group given by a…
We formulate a new class of fractional difference and sum operators, study their fundamental properties, and find their discrete Laplace transforms. The method depends on iterating the fractional sum operators corresponding to fractional…
An umbral type formalism is used to derive integrals involving products of Laguerre polynomials and other special functions.
A long-term research proposal on the algebraic structure, the representations and the possible applications of paraparticle algebras is structured in three modules: The first part stems from an attempt to classify the inequivalent gradings…
We investigate linear relations among a class of iterated integrals on the Riemann sphere minus four points $0,1,z$ and $\infty$. Generalization of the duality formula and the sum formula for multiple zeta values to the iterated integrals…
In this paper, we study the consequences of the fundamental theorem of calculus from an algebraic point of view. For functions with singularities, this leads to a generalized notion of evaluation. We investigate properties of such…
An automated treatment of iterated integrals based on letters induced by real-valued quadratic forms and Kummer--Poincar\'e letters is presented. These quantities emerge in analytic single and multi--scale Feynman diagram calculations. To…
We study integrals of the form $\int_{\Omega}f\left( d\omega\right)$, where $1\leq k\leq n$, $f:\Lambda^{k}\rightarrow\mathbb{R}$ is continuous and $\omega$ is a $\left(k-1\right)$-form. We introduce the appropriate notions of convexity,…
The theory of fractional calculus has developed in a number of directions over the years, including: the formulation of multiple different definitions of fractional differintegration; the extension of various properties of standard calculus…
A survey of some results and open questions related to the following algebraic invariants of compact complex manifolds, that can be obtained from differential forms: cohomology groups, Chern classes, rational homotopy groups, and higher…
We study the arithmetic aspects of the finite group of extensions of abelian varieties defined over a number field. In particular, we establish relations with special values of L-functions and congruences between modular forms.
For an abelian category $\mathcal{A}$, we establish the relation between its derived and extension dimensions. Then for an artin algebra $\Lambda$, we give the upper bounds of the extension dimension of $\Lambda$ in terms of the radical…
We introduce a class of iterated integrals, defined through a set of linearly independent integration kernels on elliptic curves. As a direct generalisation of multiple polylogarithms, we construct our set of integration kernels ensuring…
The development of high-degree interpolation polynomials which use the values of the function and its subsequent derivatives is reformulated. Also, we present a variant of new formula in barycentric form.
Braided non-commutative differential geometry is studied. In particular we investigate the theory of (bicovariant) differential calculi in braided abelian categories. Previous results on crossed modules and Hopf bimodules in braided…
In this paper we generalize notions of iterated integral with regard to an unpredictable process. We establish a formula of integration by parts, the existence of a continuous modification and give an expression of the increasing process.