Study On Some Matrix Equations Involving The Weighted Geometric Mean and Their Application

In this paper we consider two matrix equations that involve the weighted geometric mean. We use the fixed point theorem in the cone of positive definite matrices to prove the existence of a unique positive definite solution. In addition, we study the multi-step stationary iterative method for those equations and prove the corresponding convergence. A fidelity measure for quantum states based on the matrix geometric mean is introduced as an application of matrix equation.

Common Positive Solution of Two Nonlinear Matrix Equations Using Fixed Point Results

We discuss a pair of nonlinear matrix equations (NMEs) of the form X=R1+∑i=1kAi*F(X)Ai, X=R2+∑i=1kBi*G(X)Bi, where R1,R2∈P(n), Ai,Bi∈M(n), i=1,⋯,k, and the operators F,G:P(n)→P(n) are continuous in the trace norm. We go through the necessary criteria for a common positive definite solution of the given NME to exist. We develop the concept of a joint Suzuki-implicit type pair of mappings to meet the requirement and achieve certain existence findings under weaker assumptions. Some concrete instances are provided to show the validity of our findings. An example is provided that contains a randomly generated matrix as well as convergence and error analysis. Furthermore, we offer graphical representations of average CPU time analysis for various initializations.

Latest Inversion-Free Iterative Scheme for Solving a Pair of Nonlinear Matrix Equations

In this work, the following system of nonlinear matrix equations is considered, X 1 + A ∗ X 1 − 1 A + B ∗ X 2 − 1 B = I and X 2 + C ∗ X 2 − 1 C + D ∗ X 1 − 1 D = I , where A , B , C , and D are arbitrary n × n matrices and I is the identity matrix of order n . Some conditions for the existence of a positive-definite solution as well as the convergence analysis of the newly developed algorithm for finding the maximal positive-definite solution and its convergence rate are discussed. Four examples are also provided herein to support our results.

On the mechanism of movement of the palatine plates

Professor of Embryology, Histology and Anatomy at the Imperial University of Tokyo.C 3 fig.(Abstract).The question of the movement of the palatine plates to this day is still the subject of lively discussion, which, however, has not yet led to a definite solution. The reason for this was a preparation of a pig embryo, 3 ml long, described by Dursu, in which one palatal plate was located next to the tongue, while the other lay on top of the tongue. Dursy thought this embryo was apparently pathological. In 1901. a second such case was published by His'om, suggesting that this embryo was in a normal physiological movement. His'a observation prompted many researchers to study this issue.

Emotional preparation for the unification of Korea: Through the embracement, forgiveness and love shown in the Gospel of Matthew

The greatest wish of the Baeda l people, or South Koreans, living in the Korean Peninsula is the unification of Korea. However, even when it has been 70 years since the outbreak of the Korean War, the two Koreas that used to be one nation are still in conflict. There have been many discourses on unification over the past 70 years, but these discourses still fail to create clear rules and a framework for unification. Discourses from the perspective of biblical theology offer insight regarding the problems at stake as well as alternatives, but they do not offer perspectives on a definite solution. This article, in line with its statement on biblical theology, does not pretend to contribute to a clear-cut solution to meet the challenge of Korea’s unification. Rather, this article intends to contribute to the formation of an emotional consensus that can contribute to the unification of Korea by looking at the teachings about peace in the Gospel of Matthew. The article explores whether the notion of embracement, forgiveness and love written during the conflict between the Matthean community and Formative Judaism can be applied to help create an emotional bond between the two Koreas. The terms embracement, forgiveness and love found in the Gospel of Matthew are upheld as symbolising peace derived from opposition and conflict. Jesus taught us to overcome situations of opposition and conflict through embracement, forgiveness and love.Contribution: The purpose and contribution of this article was to discuss if the notions of embracement, forgiveness and love found in the Gospel of Matthew can be helpful for the emotional preparation of the peaceful unification of the Korean Peninsula by comparing the situations reflected in the Gospel of Matthew and that of the Korean Peninsula.

The Reproduction of money material in Marx’s Capital II (by way of a critique of Sandemose’s ‘gold digging’)

The question of the reproduction of money material is a crucial feature in the investigation of the cycle of the total social capital that Marx tackled in the context of simple reproduction analysis in Part Three of Capital II. However, his inquiry was left unfinished, so the problem remained ultimately unsolved. This unsettled character was first identified by Luxemburg and later by Grossman. Sandemose attempted to reconstruct the ‘missing fragment’ of Marx’s investigation, where the analysis of the reproduction of the constant capital of gold producers should have been accomplished, alleging that its absence contributed to keep out of sight a central problem addressed there by Marx: that is, ‘the problem of the excess money necessary for the passage from simple reproduction to accumulation’. Sandemose claims also to have accurately reconstructed the passage, attaining a definite solution to the problem that Marx left unanswered. This article shows that both allegations are completely unwarranted. First, it demonstrates that under simple reproduction assumptions there is actually no (net) hoard formation (‘excess money’), in spite of Sandemose’s claim. This also entails a critique of Marx’s conclusions. As a corollary, Sandemose’s thesis that the central problem Marx addressed there was that of the ‘excess money’ needed for accumulation proves to be untenable. Second, the article also demonstrates that Sandemose’s reconstruction of Marx’s ‘missing fragment’ is fundamentally flawed, offering at the same time a consistent alternative. By this means, the investigation initiated by Marx is finally completed. As a result, and beyond its outward polemic character, this article actually renders an ideal (i.e. in thought) simple reproduction of the real process of reproduction of the total social capital, where replacement of the money material is fully taken into account.