scholarly journals New Results for Arithmetic –Geometric Mean Inequalities and Singular Values in Matrices

2021 ◽  
Vol 20 ◽  
pp. 625-629
Author(s):  
Ahmad Abu Rahma ◽  
Aliaa Burqan ◽  
Özen Özer
Keyword(s):  
Computer Science ◽  
Open Problem ◽  
Matrix Theory ◽  
Geometric Mean ◽  
Singular Values ◽  
Direct Sums ◽  
Block Matrices ◽  
Theoretical Results

Matrix theory is very popular in different kind of sciences such as engineering, architecture, physics, chemistry, computer science, IT, so on as well as mathematics many theoretical results dealing with the structure of the matrices even this topic seems easy to work. That is why many scientists still consider some open problem in matrix theory. In this paper, generalizations of the arithmetic-geometric mean inequality is presented for singular values related to block matrices. Singular values are also given for sums, products and direct sums of the matrices.

scholarly journals Asymptotic Stability of Caputo Type Fractional Neutral Dynamical Systems with Multiple Discrete Delays

2014 ◽  
Vol 2014 ◽  
pp. 1-10 ◽  
Author(s):  
Hai Zhang ◽  
Daiyong Wu ◽  
Jinde Cao
Keyword(s):  
Asymptotic Stability ◽  
Matrix Theory ◽  
Sufficient Conditions ◽  
Algebraic Approach ◽  
Stability Criteria ◽  
Differential Systems ◽  
Discrete Delays ◽  
Transcendental Equations ◽  
The Stability ◽  
Theoretical Results

We discuss the delay-independent asymptotic stability of Caputo type fractional-order neutral differential systems with multiple discrete delays. Based on the algebraic approach and matrix theory, the sufficient conditions are derived to ensure the asymptotic stability for all time-delay parameters. By applying the stability criteria, one can avoid solving the roots of transcendental equations. The results obtained are computationally flexible and convenient. Moreover, an example is provided to illustrate the effectiveness and applicability of the proposed theoretical results.

scholarly journals Electromagnetic wave scattering on imperfect cloaking devices

2008 ◽  
Vol 40 (3) ◽  
pp. 245-250
Author(s):  
G. Isic ◽  
A. Beltaos ◽  
R. Gajic ◽  
K. Hingerl
Keyword(s):  
Electromagnetic Wave ◽  
Scattering Theory ◽  
Wave Scattering ◽  
Singular Values ◽  
Shell Material ◽  
Electromagnetic Wave Scattering ◽  
Material Parameters ◽  
Full Wave ◽  
Overall Performance ◽  
Theoretical Results

Cloaking devices based on the coordinate transform approach enable, in principle, a perfect concealment of a region in space provided that the material composing the cloaking shell meets certain criteria. To achieve ideal cloaking it is necessary that the shell material parameters have singular values on the surface bounding the cloaked region which is unphysical. In this paper we assume finite values of cloak parameters and apply the scattering theory formalism to give an estimate of the overall performance of an 'imperfect' cloak. We perform full-wave numerical calculations and use our theoretical results to discuss them.

scholarly journals The Expected Characteristic and Permanental Polynomials of the Random Gram Matrix

10.37236/3709 ◽  
2014 ◽  
Vol 21 (3) ◽  
Author(s):  
Jacob G. Martin ◽  
E. Rodney Canfield
Keyword(s):  
Random Matrix ◽  
Generating Functions ◽  
Numerical Algorithms ◽  
Singular Values ◽  
Gram Matrix ◽  
Efficient Computation ◽  
Underlying Distribution ◽  
Characteristic Roots ◽  
Speed Up ◽  
Theoretical Results

A $t \times n$ random matrix $A$ can be formed by sampling $n$ independent random column vectors, each containing $t$ components. The random Gram matrix of size $n$, $G_{n}=A^{T}A$, contains the dot products between all pairs of column vectors in the randomly generated matrix $A$, and has characteristic roots coinciding with the singular values of $A$. Furthermore, the sequences $\det{(G_{i})}$ and $\text{perm}(G_{i})$ (for $i = 0, 1, \dots, n$) are factors that comprise the expected coefficients of the characteristic and permanental polynomials of $G_{n}$. We prove theorems that relate the generating functions and recursions for the traces of matrix powers, expected characteristic coefficients, expected determinants $E(\det{(G_{n})})$, and expected permanents $E(\text{perm}(G_{n}))$ in terms of each other. Using the derived recursions, we exhibit the efficient computation of the expected determinant and expected permanent of a random Gram matrix $G_{n}$, formed according to any underlying distribution. These theoretical results may be used both to speed up numerical algorithms and to investigate the numerical properties of the expected characteristic and permanental coefficients of any matrix comprised of independently sampled columns.

Formation-containment control of sampled-data second-order multi-agent systems with sampling delay

2018 ◽  
Vol 40 (16) ◽  
pp. 4369-4381 ◽  
Author(s):  
Baojie Zheng ◽  
Xiaowu Mu
Keyword(s):  
Matrix Theory ◽  
Sufficient Conditions ◽  
Second Order ◽  
Control Problems ◽  
Sampled Data ◽  
Multi Agent Systems ◽  
Containment Control ◽  
Agent Systems ◽  
Multi Agent ◽  
Theoretical Results

The formation-containment control problems of sampled-data second-order multi-agent systems with sampling delay are studied. In this paper, we assume that there exist interactions among leaders and that the leader’s neighbours are only leaders. Firstly, two different control protocols with sampling delay are presented for followers and leaders, respectively. Then, by utilizing the algebraic graph theory and matrix theory, several sufficient conditions are obtained to ensure that the leaders achieve a desired formation and that the states of the followers converge to the convex hull formed by the states of the leaders, i.e. the multi-agent systems achieve formation containment. Furthermore, an explicit expression of the formation position function is derived for each leader. An algorithm is provided to design the gain parameters in the protocols. Finally, a numerical example is given to illustrate the effectiveness of the obtained theoretical results.

scholarly journals A New Asymptotic Notation: Weak Theta

2015 ◽  
Vol 2015 ◽  
pp. 1-15
Author(s):  
Andrei-Horia Mogoş ◽  
Bianca Mogoş ◽  
Adina Magda Florea
Keyword(s):  
Artificial Intelligence ◽  
Computer Science ◽  
Main Tool ◽  
Complexity Of Algorithms ◽  
Complexity Function ◽  
Theoretical Results

Algorithms represent one of the fundamental issues in computer science, while asymptotic notations are widely accepted as the main tool for estimating the complexity of algorithms. Over the years a certain number of asymptotic notations have been proposed. Each of these notations is based on the comparison of various complexity functions with a given complexity function. In this paper, we define a new asymptotic notation, called “Weak Theta,” that uses the comparison of various complexity functions with two given complexity functions. Weak Theta notation is especially useful in characterizing complexity functions whose behaviour is hard to be approximated using a single complexity function. In addition, in order to highlight the main particularities of Weak Theta, we propose and prove several theoretical results: properties of Weak Theta, criteria for comparing two complexity functions, and properties of a new set of complexity functions (also defined in the paper) based on Weak Theta. Furthermore, to illustrate the usefulness of our notation, we discuss an application of Weak Theta in artificial intelligence.

scholarly journals Exponential Stability of Stochastic Nonlinear Dynamical Price System with Delay

2013 ◽  
Vol 2013 ◽  
pp. 1-9 ◽  
Author(s):  
Wenli Zhu ◽  
Xinfeng Ruan ◽  
Ye Qin ◽  
Jie Zhuang
Keyword(s):  
Time Delay ◽  
Exponential Stability ◽  
Matrix Theory ◽  
Sufficient Conditions ◽  
Lyapunov Stability Theory ◽  
Price System ◽  
Nonlinear Dynamical ◽  
System With Delay ◽  
Exponentially Stable ◽  
Theoretical Results

Based on Lyapunov stability theory, Itô formula, stochastic analysis, and matrix theory, we study the exponential stability of the stochastic nonlinear dynamical price system. Using Taylor's theorem, the stochastic nonlinear system with delay is reduced to ann-dimensional semilinear stochastic differential equation with delay. Some sufficient conditions of exponential stability and corollaries for such price system are established by virtue of Lyapunov function. The time delay upper limit is solved by using our theoretical results when the system is exponentially stable. Our theoretical results show that if the classical price Rayleigh equation is exponentially stable, so is its perturbed system with delay provided that both the time delay and the intensity of perturbations are small enough. Two examples are presented to illustrate our results.

Multiconsensus of first-order multiagent systems with directed topologies

2020 ◽  
Vol 34 (23) ◽  
pp. 2050240
Author(s):  
Xiao-Wen Zhao ◽  
Guangsong Han ◽  
Qiang Lai ◽  
Dandan Yue
Keyword(s):  
Multiagent Systems ◽  
Matrix Theory ◽  
Sufficient Conditions ◽  
Consensus Problem ◽  
Multiple Agents ◽  
First Order ◽  
The Matrix ◽  
Directed Topologies ◽  
Necessary And Sufficient ◽  
Theoretical Results

The multiconsensus problem of first-order multiagent systems with directed topologies is studied. A novel consensus problem is introduced in multiagent systems — multiconsensus. The states of multiple agents in each subnetwork asymptotically converge to an individual consistent value in the presence of information exchanges among subnetworks. Linear multiconsensus protocols are proposed to solve the multiconsensus problem, and the matrix corresponding to the protocol is designed. Necessary and sufficient conditions are derived based on matrix theory, under which the stationary multiconsensus and dynamic multiconsensus can be reached. Simulations are provided to demonstrate the effectiveness of the theoretical results.

scholarly journals Group Consensus with a Dynamic Leader for Multiagent Systems via Sampled-Data Control

2014 ◽  
Vol 2014 ◽  
pp. 1-9
Author(s):  
Hong Xia ◽  
Ting-Zhu Huang ◽  
Jin-Liang Shao ◽  
Jun-Yan Yu
Keyword(s):  
Multiagent Systems ◽  
Matrix Theory ◽  
Sampling Period ◽  
Group Consensus ◽  
Sampled Data ◽  
Ultimate Boundedness ◽  
Tracking Errors ◽  
Data Control ◽  
Sampled Data Control ◽  
Theoretical Results

This paper considers a group consensus problem with a dynamic leader for multiagent systems in a sampled-data setting. With the leader’s state available to only a fraction of the followers, a distributed linear protocol based on sampled-data control is proposed for group consensus under fixed directed topology. On basis ofM-matrix theory, we derive a sufficient condition on the sampling period and the control parameter for ultimate boundedness of the tracking errors. Furthermore, simulation examples are provided to demonstrate the effectiveness of the theoretical results.

scholarly journals Complete Periodic Synchronization of Memristor-Based Neural Networks with Time-Varying Delays

2013 ◽  
Vol 2013 ◽  
pp. 1-12 ◽  
Author(s):  
Huaiqin Wu ◽  
Luying Zhang ◽  
Sanbo Ding ◽  
Xueqing Guo ◽  
Lingling Wang
Keyword(s):  
Neural Networks ◽  
Periodic Solution ◽  
Matrix Theory ◽  
Adaptive Controller ◽  
Network System ◽  
Time Varying ◽  
State Dependent ◽  
Coincidence Theorem ◽  
Error System ◽  
Theoretical Results

This paper investigates the complete periodic synchronization of memristor-based neural networks with time-varying delays. Firstly, under the framework of Filippov solutions, by usingM-matrix theory and the Mawhin-like coincidence theorem in set-valued analysis, the existence of the periodic solution for the network system is proved. Secondly, complete periodic synchronization is considered for memristor-based neural networks. According to the state-dependent switching feature of the memristor, the error system is divided into four cases. Adaptive controller is designed such that the considered model can realize global asymptotical synchronization. Finally, an illustrative example is given to demonstrate the validity of the theoretical results.

scholarly journals Reaching a Group-Agreement in Finite Time Under Acyclic Interaction Topology

2021 ◽  
Author(s):  
Xinru Ma ◽  
Hengyu Li ◽  
Tiehui Zhang ◽  
Jun Liu ◽  
Shaorong Xie ◽  
...  
Keyword(s):  
Finite Time ◽  
Multiagent System ◽  
Matrix Theory ◽  
Structural Characteristics ◽  
Mathematical Induction ◽  
Time Required ◽  
Interaction Topology ◽  
Theoretical Results ◽  
Group Agreement ◽  
Agreement Problem

Abstract This paper discusses the finite time agreement problem of networks with acyclic partition topology. In view of the structural characteristics of such network topology, mathematical induction is particularly suitable to prove the main conclusions in the paper. In addition, for the consideration of the finite time consensus problem, in addition to using basic matrix theory to verify the solution of the problem, this brief also has a more detailed analysis of the time required to reach consensus. Based on these two points, it is observed that the solution of this problem is due to the features of acyclic partition interactions and the continuity of the related finite time protocol and contributes to the research on the grouping consensus of multiagent system. Furthermore, simulation examples are presented to verify the theoretical results.

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