Norms of Dual Complex Vectors and Dual Complex Matrices

In this paper,we investigate some properties of dual complex numbers,dual complex vectors,and dual complex matrices.First,based on the magnitude of the dual complex number,we study the Young inequality,the Hölder inequality,and the Minkowski inequality in the setting of dual complex numbers.Second,we define the p-norm of a dual complex vector,which is a nonnegative dual number,and show some related properties.Third,we study the properties of eigenvalues of unitary matrices and unitary triangulation of arbitrary dual complex matrices.In particular,we introduce the operator norm of dual complex matrices induced by the p-norm of dual complex vectors,and give expressions of three important operator norms of dual complex matrices.

Dual Quaternions and Dual Quaternion Vectors

We introduce a total order and an absolute value function for dual numbers.The absolute value function of dual numbers takes dual number values,and has properties similar to those of the absolute value function of real numbers.We define the magnitude of a dual quaternion,as a dual number.Based upon these,we extend 1-norm,co-norm,and 2-norm to dual quaternion vectors.

Vanishing Properties of Dual Bass Numbers

Let R be a Noetherian ring, M an Artinian R-module, and p ∈ CosRM. Thencograden. Homn（Rp, M） =-inf{i ｜πi（p, M） 〉 0} and πi（p, M） 〉 0 =〉 cogradeRpHomR（Rp, M） ≤ i ≤ fdRpHomR（Rp, M）,where πi （p,M） is the i-th dual Bass number of M with respect to p, cogradeRpHomR （Rp ,M） is the common length of any maxima[ HomR（Rp, M）-quasi co-regular sequence contained in pRp, and fdRp HomR（Rp, M） is the flat dimension of the Rp-module HomR（Rp, M）. We also study the relations among cograde, co-dimension and flat dimension of co-localization modules.

Dual-holomorphic Functions and Problems of Lifts

The main purpose of this paper is to study the differential geometrical objects on tangent bundle corresponding to dual-holomorphic objects of dual-holomorphic manifold.As a result of this approach,the authors find a new class of lifts(deformed complete lifts)in the tangent bundle.