相关论文: Towards a geometric theory of integration

Finite dimensional solutions to a class of stochastic partial differential equations are obtained extending the differential constraints method for deterministic PDE to the stochastic framework. A geometrical reformulation of the stochastic…

Fractional calculus is the calculus of differentiation and integration of non-integer orders. In a recently paper (Annals of Physics 323 (2008) 2756-2778), the Fundamental Theorem of Fractional Calculus is highlighted. Based on this…

This article is about the formalization of synthetic differential geometry with the Lean proof assistant and the mathematical library mathlib. The main result we prove and formalize is a Taylor theorem for functions of several variables,…

A generic method for combinatorial constructions of intrinsic geometrical spaces is presented. It is based on the well known inverse sequences of finite graphs that determine (in the limit) topological spaces. If a pattern of the…

We survey recent contributions to finite element exterior calculus on manifolds and surfaces within a comprehensive formalism for the error analysis of vector-valued partial differential equations on manifolds. Our primary focus is on…

It is an established fact that a finite difference operator approximates a derivative with a fixed algebraic rate of convergence. Nevertheless, we exhibit a new finite difference operator and prove it has spectral accuracy. Its rate of…

The goal of this paper consists of developing a new (more physical and numerical in comparison with standard and non-standard analysis approaches) point of view on Calculus with functions assuming infinite and infinitesimal values. It uses…

Any procedure applied to data, and any quantity derived from data, is required to respect the nature and symmetries of the data. This axiom applies to refinement procedures and multiresolution transforms as well as to more basic operations…

This is the first paper in a series on intrinsic Donaldson-Thomas theory, where we develop a new framework for enumerative geometry that allows the generalization of constructions and results from linear moduli stacks to general non-linear…

Infinitesimal contraction analysis, wherein global asymptotic convergence results are obtained from local dynamical properties, has proven to be a powerful tool for applications in biological, mechanical, and transportation systems. The…

Synthetic algebraic geometry uses homotopy type theory extended with three axioms to develop algebraic geometry internal to a higher version of the Zariski topos. In this article we make no essential use of the higher structure and use…

We expose (without proofs) a unified computational approach to integrable structures (including recursion, Hamiltonian, and symplectic operators) based on geometrical theory of partial differential equations. We adopt a coordinate based…

INTRODUCTION This papers deals with partial differential equations of second order, linear, with constant and not constant coefficients, in two variables, which admit real characteristics. I face the study of PDEs with the mentality of the…

A generalization of exterior calculus is considered by allowing the partial derivatives in the exterior derivative to assume fractional orders. That is, a fractional exterior derivative is defined. This is found to generate new vector…

The paper is devoted to developing subdifferential theory for set-valued mappings taking values in ordered infinite-dimensional spaces. This study is motivated by applications to problems of vector and set optimization with various…

We construct the Hodge dual for supermanifolds by means of the Grassmannian Fourier transform of superforms. In the case of supermanifolds it is known that the superforms are not sufficient to construct a consistent integration theory and…

In the context of Synthetic Differential Geometry, we describe a notion of higher connection with values in a cubical groupoid. We do this by exploiting a certain structure of cubical complex derived from the first neighbourhood of the…

We generalize the differential space concept as a tool for developing differential geometry, and enrich this geometry with infinitesimals that allow us to penetrate into the superfine structure of space. This is achieved by Yoneda embedding…

A family of general Master theorems for analytic integration over the real (or imaginary) axis with various reciprocal hyperbolic (trig) kernels ($\sinh and/or \cosh$) with varying arguments is developed. Several examples involving…

We presented a novel geometric interpretation of the Riemann-Liouville fractional integral. We found that a Riemann-Liouville integral can be thought of as the area obtained by summing together the area of an infinite number of…