383 citations · 395 across the 26 of their papers we have counts for
26 papers
Noncommutative Partial Derivative
Keqin Liu
We introduce the axiomatic definition of the point-derivative for noncommutative algebras and present the counterparts of the ordinary multi-variable chain rule and Clairaut's Theo…
Automatic Integration
Keqin Liu
The purpose of this paper is to introduce the concept of the automatic integration and present a new way of approximating definite integrals using the automatic integration based o…
How to Define Automatic Differentiation
Keqin Liu
Based on a class of associative algebras with zero-divisors which are called real-like algebras by us, we introduce a way of defining automatic differentiation and present differen…
Type Stokes' Theorem for Chains
Keqin Liu
We present type Stokes' theorem for type -chains which extends the fundamental theorem of calculus in higher dimensions.
New Characterizations of Algebraic Regularity
Keqin Liu
In this paper, we give new characterizations of algebraic regularity by using differential forms and difference quotients.
Algebraic Regularity over Quaternions and Regular Four-Manifolds
Keqin Liu
Based on a new generalization of Cauchy-Riemann system presented in this paper, we introduce a class of quaternion-valued functions of a quaternionic variable, which are called alg…