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...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.展开更多
It is well established that Nash equilibrium exists within the framework of mixed strategies in strategic-form non-cooperative games. However, finding the Nash equilibrium generally belongs to the class of problems kn...It is well established that Nash equilibrium exists within the framework of mixed strategies in strategic-form non-cooperative games. However, finding the Nash equilibrium generally belongs to the class of problems known as PPAD (Polynomial Parity Argument on Directed graphs), for which no polynomial-time solution methods are known, even for two-player games. This paper demonstrates that in fixed-sum two-player games (including zero-sum games), the Nash equilibrium forms a convex set, and has a unique expected payoff. Furthermore, these equilibria are Pareto optimal. Additionally, it is shown that the Nash equilibrium of fixed-sum two-player games can theoretically be found in polynomial time using the principal-dual interior point method, a solution method of linear programming.展开更多
Let the triangle matrix A^(ru)be a generalization of the Cesàro matrix and U∈{c_(0),c,l_(∞)}.In this study,we essentially deal with the space U(A^(ru))defined by the domain of A^(ru)in the space U and give the ...Let the triangle matrix A^(ru)be a generalization of the Cesàro matrix and U∈{c_(0),c,l_(∞)}.In this study,we essentially deal with the space U(A^(ru))defined by the domain of A^(ru)in the space U and give the bases,and determine the Kothe-Toeplitz,generalized K?theToeplitz and bounded-duals of the space U(A^(ru)).We characterize the classes(l_(∞)(A^(ru)):l_(∞)),(l_(∞)(A^(ru)):c),(c(A^(ru)):c),and(U:V(A^(ru)))of infinite matrices,where V denotes any given sequence space.Additionally,we also present a Steinhaus type theorem.As an another result of this study,we investigate the l_(p)-norm of the matrix A^(ru)and as a result obtaining a generalized version of Hardy's inequality,and some inclusion relations.Moreover,we compute the norm of well-known operators on the matrix domain l_(p)(A^(ru)).展开更多
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 rea...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.展开更多
The expanding role of the Internet in consumer purchasing activities has created substantial new opportunities accessing to end-consumers. More and more manufacturers are beginning to sell products to potential consum...The expanding role of the Internet in consumer purchasing activities has created substantial new opportunities accessing to end-consumers. More and more manufacturers are beginning to sell products to potential consumers directly online while continuing to sell through the traditional brick-and-mortar retailers, a phenomenon leading to intense channel competition and conflicts. Using game theory, this research examines the effect of market segments, consumer choice and the acceptance of direct online channels on firm performance and the whole system’s profit. The analysis indicates that the addition of direct online channel does not necessarily harm the incumbent retailers. A win-win zone is proposed, in which both the manufacturer and the retailer benefit from the encroachment.展开更多
In heterogeneous network with hybrid energy supplies including green energy and on-grid energy, it is imperative to increase the utilization of green energy as well as to improve the utilities of users and networks. A...In heterogeneous network with hybrid energy supplies including green energy and on-grid energy, it is imperative to increase the utilization of green energy as well as to improve the utilities of users and networks. As the difference of hybrid energy source in stability and economy, thus, this paper focuses on the network with hybrid energy source, and design the utility of each user in the hybrid energy source system from the perspective of stability, economy and environment pollution. A dual power allocation algorithm based on Stackelberg game to maximize the utilities of users and networks is proposed. In addition, an iteration method is proposed which enables all players to reach the Stackelberg equilibrium(SE). Simulation results validate that players can reach the SE and the utilities of users and networks can be maximization, and the green energy can be efficiently used.展开更多
In this paper,we consider nonlinear infinity-norm minimization problems.We device a reliable Lagrangian dual approach for solving this kind of problems and based on this method we propose an algorithm for the mixed li...In this paper,we consider nonlinear infinity-norm minimization problems.We device a reliable Lagrangian dual approach for solving this kind of problems and based on this method we propose an algorithm for the mixed linear and nonlinear infinity- norm minimization problems.Numerical results are presented.展开更多
基金the National Natural Science Foundation of China(Grant No.11871051).
文摘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.
文摘It is well established that Nash equilibrium exists within the framework of mixed strategies in strategic-form non-cooperative games. However, finding the Nash equilibrium generally belongs to the class of problems known as PPAD (Polynomial Parity Argument on Directed graphs), for which no polynomial-time solution methods are known, even for two-player games. This paper demonstrates that in fixed-sum two-player games (including zero-sum games), the Nash equilibrium forms a convex set, and has a unique expected payoff. Furthermore, these equilibria are Pareto optimal. Additionally, it is shown that the Nash equilibrium of fixed-sum two-player games can theoretically be found in polynomial time using the principal-dual interior point method, a solution method of linear programming.
文摘Let the triangle matrix A^(ru)be a generalization of the Cesàro matrix and U∈{c_(0),c,l_(∞)}.In this study,we essentially deal with the space U(A^(ru))defined by the domain of A^(ru)in the space U and give the bases,and determine the Kothe-Toeplitz,generalized K?theToeplitz and bounded-duals of the space U(A^(ru)).We characterize the classes(l_(∞)(A^(ru)):l_(∞)),(l_(∞)(A^(ru)):c),(c(A^(ru)):c),and(U:V(A^(ru)))of infinite matrices,where V denotes any given sequence space.Additionally,we also present a Steinhaus type theorem.As an another result of this study,we investigate the l_(p)-norm of the matrix A^(ru)and as a result obtaining a generalized version of Hardy's inequality,and some inclusion relations.Moreover,we compute the norm of well-known operators on the matrix domain l_(p)(A^(ru)).
基金supported by Hong Kong Innovation and Technology Commission(InnoHK Project CIMDA)supported by the National Natural Science Foundation of China(No.11971138)+3 种基金the Natural Science Foundation of Zhejiang Province of China(Nos.LY19A010019,LD19A010002)supported by Hong Kong Research Grants Council(Project 11204821)Hong Kong Innovation and Technology Commission(InnoHK Project CIMDA)City University of Hong Kong(Project 9610034).
文摘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.
文摘The expanding role of the Internet in consumer purchasing activities has created substantial new opportunities accessing to end-consumers. More and more manufacturers are beginning to sell products to potential consumers directly online while continuing to sell through the traditional brick-and-mortar retailers, a phenomenon leading to intense channel competition and conflicts. Using game theory, this research examines the effect of market segments, consumer choice and the acceptance of direct online channels on firm performance and the whole system’s profit. The analysis indicates that the addition of direct online channel does not necessarily harm the incumbent retailers. A win-win zone is proposed, in which both the manufacturer and the retailer benefit from the encroachment.
基金supported by the Beijing Natural Science Foundation (4142049)863 project No. 2014AA01A701the Fundamental Research Funds for Central Universities of China No. 2015XS07
文摘In heterogeneous network with hybrid energy supplies including green energy and on-grid energy, it is imperative to increase the utilization of green energy as well as to improve the utilities of users and networks. As the difference of hybrid energy source in stability and economy, thus, this paper focuses on the network with hybrid energy source, and design the utility of each user in the hybrid energy source system from the perspective of stability, economy and environment pollution. A dual power allocation algorithm based on Stackelberg game to maximize the utilities of users and networks is proposed. In addition, an iteration method is proposed which enables all players to reach the Stackelberg equilibrium(SE). Simulation results validate that players can reach the SE and the utilities of users and networks can be maximization, and the green energy can be efficiently used.
文摘In this paper,we consider nonlinear infinity-norm minimization problems.We device a reliable Lagrangian dual approach for solving this kind of problems and based on this method we propose an algorithm for the mixed linear and nonlinear infinity- norm minimization problems.Numerical results are presented.