Related papers: Differential Calculus and Integration of Generaliz…
In La G\'eom\'etrie, Descartes proposed a balance between geometric constructions and symbolic manipulation with the introduction of suitable ideal machines. In modern terms, that is a balance between analog and symbolic computation.…
Let $\mathcal{A}$ denote a real, $n$-dimensional, unital, associative algebra.This paper provides an introductory exposition of calculus over $\mathcal{A}$. An $\mathcal{A}$-differentiable function is one for which the differential is…
Using recent advances in integration theory, we give a proof of the fundamental theorem of geometric calculus. We assume only that the tangential derivative $\nabla_VF$ exists and is Lebesgue integrable. We also give sufficient conditions…
We construct a parallel transport on higher loop spaces of a manifold in term of a higher dimensional generalization of iterated path integrals. Under mild assumptions, we define a de Rham complex on higher loop spaces and we recover a…
We first generalize the operation of formal exterior differential in the case of finite dimensional fibered manifolds and then we extend it to certain bundles of smooth maps. In order to characterize the operator order of some morphisms…
General concept of ternary algebras is introduced in this article, along with several examples of its realization. Universal envelope of such algebras is defined, as well as the concept of tri-modules over ternary algebras. The universal…
Euler calculus is based on integrating simple functions with respect to the Euler characteristic. This paper makes the case for extending Euler calculus to continuous integrands by integrating with respect to (Gaussian) curvature. This…
Given a smooth, closed Riemannian manifold $(M,g)$ equipped with a linear connection $\nabla$ (not necessarily metric), we develop the holomorphic functional calculus for operators belonging to the global pseudo-differential classes…
The Euler-Gauss linear transformation formula for the hypergeometric function was extended by Goursat for the case of logarithmic singularities. By replacing the perturbed Bessel differential equation by a monodromic functional equation,…
Differential lambda-calculus was first introduced by Thomas Ehrhard and Laurent Regnier in 2003. Despite more than 15 years of history, little work has been done on a differential calculus with integration. In this paper, we shall propose a…
This paper presents a systematic study of the calculus of interval-valued functions and its application to interval differential equations. To this end, first, we introduce new interval arithmetic operations. Under new operations, the space…
Given a smooth manifold $M$ (with or without boundary), in this paper we establish a global functional calculus (without the standard assumption that the operators are classical pseudo-differential operators) and the G\r{a}rding inequality…
Within the framework of the previous paper [8]. we develop a generalized stochastic calculus for processes associated to higher order diffusion operators. Applications to the study of a Cauchy problem, a Feynman-Kac formula and a…
This paper is devoted to the proof Gauss' divergence theorem in the framework of "ultrafunctions". They are a new kind of generalized functions, which have been introduced recently [2] and developed in [4], [5] and [6]. Their peculiarity is…
This is an introduction to calculus, and its applications to basic questions from physics. We first discuss the theory of functions $f:\mathbb R\to\mathbb R$, with the notion of continuity, and the construction of the derivative $f'(x)$ and…
A deformed differential calculus is developed based on an associative star-product. In two dimensions the Hamiltonian vector fields model the algebra of pseudo-differential operator, as used in the theory of integrable systems. Thus one…
In our previous publications we introduced differential calculus on the enveloping algebras U(gl(m)) similar to the usual calculus on the commutative algebra Sym(gl(m)). The main ingredient of our calculus are quantum partial derivatives…
After an introduction to some aspects of bidifferential calculus on associative algebras, we focus on the notion of a "symmetry" of a generalized zero curvature equation and derive Backlund and (forward, backward and binary) Darboux…
A previously established correspondence between definite-parity real functions and inner analytic functions is generalized to real functions without definite parity properties. The set of inner analytic functions that corresponds to the set…
Differential operators commuting with integral operators were discovered in the work of C. Tracy and H. Widom [37, 38] and used to derive asymptotic expansions of the Fredholm determinants of integral operators arising in random matrix…