Geometric structures of a manifold of positive definite Hermite matrices are considered from the viewpoint of information geometry.A Riemannian metric is defined and dual α-connections are introduced.Then the fact that the manifold is ±l-flat is shown.Moreover,the divergence of two points on the manifold is given through dual potential functions.Furthermore,the optimal approximation of a point onto the submanifold is gotten.Finally,some simulations are given to illustrate our results.
In this paper,we study the periodic solutions to a type of differential delay equations with 2 k-1 lags.The 4 k-periodic solutions are obtained by using the variational method and the method of Kaplan-Yorke coupling system.This is a new type of differential delay equations compared with all the previous researches.And this paper provides a theoretical basis for the study of differential delay equations.An example is given to demonstrate our main results.
Suppose that A and B are two positive-definite matrices,then,the limit of(A^p/2B^pA^p/2)1/p as p tends to 0 can be obtained by the well known Lie-Trotter formula.In this article,we generalize the usual product of matrices to the Hadamard product denoted as*which is commutative,and obtain the explicit formula of the limit(A^p*B^p)^1/p as p tends to 0.Furthermore,the existence of the limit of(A^p*B^p)^1/p as p tends to+∞is proved.
The investigation of novel signal processing tools is one of the hottest research topics in modern signal processing community. Among them, the algebraic and geometric signal processing methods are the most powerful tools for the representation of the classical signal processing method. In this paper, we provide an overview of recent contributions to the algebraic and geometric signal processing. Specifically, the paper focuses on the mathematical structures behind the signal processing by emphasizing the algebraic and geometric structure of signal processing. The two major topics are discussed. First, the classical signal processing concepts are related to the algebraic structures, and the recent results associated with the algebraic signal processing theory are introduced. Second, the recent progress of the geometric signal and information processing representations associated with the geometric structure are discussed. From these discussions, it is concluded that the research on the algebraic and geometric structure of signal processing can help the researchers to understand the signal processing tools deeply, and also help us to find novel signal processing methods in signal processing community. Its practical applications are expected to grow significantly in years to come, given that the algebraic and geometric structure of signal processing offer many advantages over the traditional signal processing.
