相关论文: Differential complexes and exterior calculus

This document introduces a generalization of calculus that treats both continuous and discrete variables on an equal footing. This generalization of calculus was developed independently of the "Calculus on Time Scales" literature but may be…

This paper contains a set of lecture notes on manifolds with boundary and corners, with particular attention to the space of quantum states. A geometrically inspired way of dealing with these kind of manifolds is presented,and explicit…

This survey revisits classical results in vector calculus and analysis by exploring a generalised perspective on the exterior derivative, interpreting it as a measure of "infinitesimal flux". This viewpoint leads to a higher-dimensional…

Weinberg-Salam theory and $SU(5)$ grand unified theory are reconstructed using the generalized differential calculus extended on the discrete space $M_4\times Z_{\mathop{}_{N}}$. Our starting point is the generalized gauge field expressed…

Based on the concept of manifold valued generalized functions we initiate a study of nonlinear ordinary differential equations with singular (in particular: distributional) right hand sides in a global setting. After establishing several…

This paper shows that a large class of fading memory state-space systems driven by discrete-time observations of dynamical systems defined on compact manifolds always yields continuously differentiable synchronizations. This general result…

We study covariant differential calculus on the quantum spheres S_q^2N-1. Two classification results for covariant first order differential calculi are proved. As an important step towards a description of the noncommutative geometry of the…

We look at computational physics from an electrical engineering perspective and suggest that several concepts of mathematics, not so well-established in computational physics literature, present themselves as opportunities in the field. We…

We introduce a class of diffeological spaces, called elastic, on which the left Kan extension of the tangent functor of smooth manifolds defines an abstract tangent functor in the sense of Rosicky. On elastic spaces there is a natural…

For differential calculi over certain right coideal subalgebras of quantum groups the notion of quantum tangent space is introduced. In generalization of a result by Woronowicz a one to one correspondence between quantum tangent spaces and…

The notion of a coherent space is a nonlinear version of the notion of a complex Euclidean space: The vector space axioms are dropped while the notion of inner product is kept. Coherent spaces provide a setting for the study of geometry in…

We introduce a novel formulation for geometry on discrete points. It is based on a universal differential calculus, which gives a geometric description of a discrete set by the algebra of functions. We expand this mathematical framework so…

We consider the algebra of N x N matrices as a reduced quantum plane on which a finite-dimensional quantum group H acts. This quantum group is a quotient of U_q(sl(2,C)), q being an N-th root of unity. Most of the time we shall take N=3; in…

In this paper we show that, besides the usual calculus involving K\"ahler differentials, it is also possible to define conical calculus on schemes and perfectoid spaces; this can be done via a stratification process. Following some ideas…

Skew-symmetric differential forms play an unique role in mathematics and mathematical physics. This relates to the fact that closed exterior skew-symmetric differential forms are invariants. The concept of "Exterior differential forms" was…

In differential topology two smooth submanifolds $S_1$ and $S_2$ of euclidean space are said to be transverse if the tangent spaces at each common point together form a spanning set. The purpose of this article is to explore a much more…

A mathematical smooth function means that the function has continuous derivatives to a certain degree C(k). We call it a k-smooth function or a smooth function if k can grow infinitively. Based on quantum physics, there is no such smooth…

Skew-symmetric forms possess unique capabilities. The properties of closed exterior and dual forms, namely, invariance, covariance, conjugacy and duality, either explicitly or implicitly appear in all invariant mathematical formalisms. This…

In exterior calculus on smooth manifolds, the exterior derivative and wedge product are natural with respect to smooth maps between manifolds, that is, these operations commute with pullback. In discrete exterior calculus (DEC), simplicial…

We prove various results in infinite-dimensional differential calculus which relate differentiability properties of functions and associated operator-valued functions (e.g., differentials). The results are applied in two areas: 1. in the…