Nowadays the semi-tensor product(STP)approach to finite games has become a promising new direction.This paper provides a comprehensive survey on this prosperous field.After a brief introduction for STP and finite(netw...Nowadays the semi-tensor product(STP)approach to finite games has become a promising new direction.This paper provides a comprehensive survey on this prosperous field.After a brief introduction for STP and finite(networked)games,a description for the principle and fundamental technique of STP approach to finite games is presented.Then several problems and recent results about theory and applications of finite games via STP are presented.A brief comment about the potential use of STP to artificial intelligence is also proposed.展开更多
Let φ be an Orlicz function that has a complementary function φ＊ and let lφ be an Orlicz sequence space. We prove a similar version of Rearrangement Inequality and Chebyshev＇s Sum Inequality in lφ FX, the Freml...Let φ be an Orlicz function that has a complementary function φ＊ and let lφ be an Orlicz sequence space. We prove a similar version of Rearrangement Inequality and Chebyshev＇s Sum Inequality in lφ FX, the Fremlin projective tensor product of lφ with a Banach lattice X, and in lφ iX, the Wittstock injective tensor product of lφ with a Banach lattice X.展开更多
A new matrix product, called the second semi-tensor product (STP-Ⅱ) of matrices is proposed. It is similar to the classical semi-tensor product (STP-I). First, its fundamental properties are presented. Then, the equi...A new matrix product, called the second semi-tensor product (STP-Ⅱ) of matrices is proposed. It is similar to the classical semi-tensor product (STP-I). First, its fundamental properties are presented. Then, the equivalence relation caused by STP-II is obtained. Using this equivalence, a quotient space is also obtained. Finally, the vector space structure, the metric and the metric topology, the projection and subspaces, etc. of the quotient space are investigated in detail.展开更多
The kernel function method in support vector machine(SVM)is an excellent tool for nonlinear classification.How to design a kernel function is difficult for an SVM nonlinear classification problem,even for the polynomi...The kernel function method in support vector machine(SVM)is an excellent tool for nonlinear classification.How to design a kernel function is difficult for an SVM nonlinear classification problem,even for the polynomial kernel function.In this paper,we propose a new kind of polynomial kernel functions,called semi-tensor product kernel(STP-kernel),for an SVM nonlinear classification problem by semi-tensor product of matrix(STP)theory.We have shown the existence of the STP-kernel function and verified that it is just a polynomial kernel.In addition,we have shown the existence of the reproducing kernel Hilbert space(RKHS)associated with the STP-kernel function.Compared to the existing methods,it is much easier to construct the nonlinear feature mapping for an SVM nonlinear classification problem via an STP operator.展开更多
The cross-dimensional dynamical systems have received increasing research attention in recent years.This paper characterizes the structure features of the cross-dimensional vector space.Specifically,it is proved that ...The cross-dimensional dynamical systems have received increasing research attention in recent years.This paper characterizes the structure features of the cross-dimensional vector space.Specifically,it is proved that the completion of cross-dimensional vector space is an infinite-dimensional separable Hilbert space.Hence,it means that one can isometrically and linearly embed the crossdimensional vector space into theℓ^(2),which is known as the space of square summable sequences.This result will be helpful in the modeling and analyzing the dynamics of cross-dimensional dynamical systems.展开更多
River discharge data offer a rich source of information for reservoir management and flood control, if modelling can separate out the effects of rainfall, land use, soil type, relief, and weather conditions. In this p...River discharge data offer a rich source of information for reservoir management and flood control, if modelling can separate out the effects of rainfall, land use, soil type, relief, and weather conditions. In this paper, we model river discharge data from the Black Volta River, using Generalised Additive Mixed Models (GAMMs) with a space-time interaction represented via a tensor product of continuous time and discrete space. River discharge data from January 2000 to December 2009 for the four gauge stations along the Black Volta River namely, Lawra, Chache, Bui and Bamboi were obtained from the hydrological services department of Ghana and used for model fitting. Four GAMMs were explored, two with space-time interactions and two without space-time interactions. The comparison of the performance of the models with space-time interactions and those without space-time interactions based on Akaike Information Criterion (AIC) and Bayesian Information Criterion (BIC) suggests that in this application, the former is better overall and in particular for modelling local variations. Further, a model with space and time main effects performed better compared with one without space and time main effects. After model selection, checking and validation, there is evidence for increasing river discharge from the most upstream gauge station to the most downstream gauge station for the study period.展开更多
In 1934, Hardy, Littlewood and Polya introduced a rearrangement inequality：∑i=1,aib（m＋1-i）≤∑i=1maibp（i）≤∑i=1,aibi,in which the real sequences {ai}i and {bi}i are in increasing order, and p（i） indicates a ...In 1934, Hardy, Littlewood and Polya introduced a rearrangement inequality：∑i=1,aib（m＋1-i）≤∑i=1maibp（i）≤∑i=1,aibi,in which the real sequences {ai}i and {bi}i are in increasing order, and p（i） indicates a random permutation. We now consider a sequence in lp with 1 〈 p 〈 ∞, and a sequence in a Banach lattice X. Instead of normal multiplication, we consider the tensor product of lp and X. We show that in Wittstock injective tensor product, lp iX, and Fremlin projective tensor product, lp FX, the rearrangement inequality still exists.展开更多
Properties for tensor products of semigroups are considered and the solutions of the equationAC - CB = Q are discussed. Results obtained in this paper considerably generalize thoseobtained in [9].
This paper investigates the basis and pure Nash equilibrium of finite pure harmonic games(FPHGs) based on the vector space structure. First, a new criterion is proposed for the construction of pure harmonic subspace, ...This paper investigates the basis and pure Nash equilibrium of finite pure harmonic games(FPHGs) based on the vector space structure. First, a new criterion is proposed for the construction of pure harmonic subspace, based on which, a more concise basis is constructed for the pure harmonic subspace. Second, based on the new basis of FPHGs and auxiliary harmonic vector, a more easily verifiable criterion is presented for the existence of pure Nash equilibrium in basis FPHGs. Third,by constructing a pure Nash equilibrium cubic matrix, the verification of pure Nash equilibrium in three-player FPHGs is given.展开更多
利用张量积构造正则空间,用于场函数逼近,得到MLS(moving least squares)场函数插值形式的一个特例.在规则域上,通过基空间扩张,构造满足齐次边界条件的场函数,使得MLS插值形式变得更简单有效,扩大了基函数空间的选择范围,克服了一般ML...利用张量积构造正则空间,用于场函数逼近,得到MLS(moving least squares)场函数插值形式的一个特例.在规则域上,通过基空间扩张,构造满足齐次边界条件的场函数,使得MLS插值形式变得更简单有效,扩大了基函数空间的选择范围,克服了一般MLS逼近所特有的大量求逆、计算效率低下、判据复杂的缺点.数值实例表明,该方法数学概念简单,加密点阵重复计算方便,计算精度高,计算结果连续性好,总体计算量少,计算过程收敛快.展开更多
In this paper we define the tensor products of spaces of exponential type vectors of closed unbounded operators in Banach spaces. Using the real method of interpolation （K-functional） we prove the interpolation theo...In this paper we define the tensor products of spaces of exponential type vectors of closed unbounded operators in Banach spaces. Using the real method of interpolation （K-functional） we prove the interpolation theorems that permit to characterize of tensor products of spaces of exponential type vectors, We show an application of abstract results to the theory of regular elliptic operators on bounded domains. For such operators the exponential type vectors are root vectors. Thus we describe the tensor products of root vectors of regular elliptic operators on bounded domains.展开更多
Motivated by the inconvenience or even inability to explain the mathematics of the state space optimization of finite state machines(FSMs)in most existing results,we consider the problem by viewing FSMs as logical dyn...Motivated by the inconvenience or even inability to explain the mathematics of the state space optimization of finite state machines(FSMs)in most existing results,we consider the problem by viewing FSMs as logical dynamic systems.Borrowing ideas from the concept of equilibrium points of dynamic systems in control theory,the concepts of t-equivalent states and t-source equivalent states are introduced.Based on the state transition dynamic equations of FSMs proposed in recent years,several mathematical formulations of t-equivalent states and t-source equivalent states are proposed.These can be analogized to the necessary and sufficient conditions of equilibrium points of dynamic systems in control theory and thus give a mathematical explanation of the optimization problem.Using these mathematical formulations,two methods are designed to find all the t-equivalent states and t-source equivalent states of FSMs.Further,two ways of reducing the state space of FSMs are found.These can be implemented without computers but with only pen and paper in a mathematical manner.In addition,an open question is raised which can further improve these methods into unattended ones.Finally,the correctness and effectiveness of the proposed methods are verified by a practical language model.展开更多
基金the National Natural Science Foundation of China(NSFC)under Grant Nos.62073315,61074114,and 61273013。
文摘Nowadays the semi-tensor product(STP)approach to finite games has become a promising new direction.This paper provides a comprehensive survey on this prosperous field.After a brief introduction for STP and finite(networked)games,a description for the principle and fundamental technique of STP approach to finite games is presented.Then several problems and recent results about theory and applications of finite games via STP are presented.A brief comment about the potential use of STP to artificial intelligence is also proposed.
文摘Let φ be an Orlicz function that has a complementary function φ＊ and let lφ be an Orlicz sequence space. We prove a similar version of Rearrangement Inequality and Chebyshev＇s Sum Inequality in lφ FX, the Fremlin projective tensor product of lφ with a Banach lattice X, and in lφ iX, the Wittstock injective tensor product of lφ with a Banach lattice X.
基金National Natural Science Foundation of China (Nos. 61733018, 61333001, 61773371).
文摘A new matrix product, called the second semi-tensor product (STP-Ⅱ) of matrices is proposed. It is similar to the classical semi-tensor product (STP-I). First, its fundamental properties are presented. Then, the equivalence relation caused by STP-II is obtained. Using this equivalence, a quotient space is also obtained. Finally, the vector space structure, the metric and the metric topology, the projection and subspaces, etc. of the quotient space are investigated in detail.
基金supported by the National Natural Science Foundation of China(61573288)the Key Programs in Shaanxi Province of China(2021JZ-12)and the Yulin Science and Technology Bureau project(2019-89-2).
文摘The kernel function method in support vector machine(SVM)is an excellent tool for nonlinear classification.How to design a kernel function is difficult for an SVM nonlinear classification problem,even for the polynomial kernel function.In this paper,we propose a new kind of polynomial kernel functions,called semi-tensor product kernel(STP-kernel),for an SVM nonlinear classification problem by semi-tensor product of matrix(STP)theory.We have shown the existence of the STP-kernel function and verified that it is just a polynomial kernel.In addition,we have shown the existence of the reproducing kernel Hilbert space(RKHS)associated with the STP-kernel function.Compared to the existing methods,it is much easier to construct the nonlinear feature mapping for an SVM nonlinear classification problem via an STP operator.
基金supported by the National Natural Science Foundation of China under Grant No.61673129the Key Programs in Shaanxi Province of China under Grant No.2021JZ-12Science and the Technology Bureau Project of Yulin under Grant Nos.2019-89-2 and 2019-89-4。
文摘The cross-dimensional dynamical systems have received increasing research attention in recent years.This paper characterizes the structure features of the cross-dimensional vector space.Specifically,it is proved that the completion of cross-dimensional vector space is an infinite-dimensional separable Hilbert space.Hence,it means that one can isometrically and linearly embed the crossdimensional vector space into theℓ^(2),which is known as the space of square summable sequences.This result will be helpful in the modeling and analyzing the dynamics of cross-dimensional dynamical systems.
文摘River discharge data offer a rich source of information for reservoir management and flood control, if modelling can separate out the effects of rainfall, land use, soil type, relief, and weather conditions. In this paper, we model river discharge data from the Black Volta River, using Generalised Additive Mixed Models (GAMMs) with a space-time interaction represented via a tensor product of continuous time and discrete space. River discharge data from January 2000 to December 2009 for the four gauge stations along the Black Volta River namely, Lawra, Chache, Bui and Bamboi were obtained from the hydrological services department of Ghana and used for model fitting. Four GAMMs were explored, two with space-time interactions and two without space-time interactions. The comparison of the performance of the models with space-time interactions and those without space-time interactions based on Akaike Information Criterion (AIC) and Bayesian Information Criterion (BIC) suggests that in this application, the former is better overall and in particular for modelling local variations. Further, a model with space and time main effects performed better compared with one without space and time main effects. After model selection, checking and validation, there is evidence for increasing river discharge from the most upstream gauge station to the most downstream gauge station for the study period.
文摘In 1934, Hardy, Littlewood and Polya introduced a rearrangement inequality：∑i=1,aib（m＋1-i）≤∑i=1maibp（i）≤∑i=1,aibi,in which the real sequences {ai}i and {bi}i are in increasing order, and p（i） indicates a random permutation. We now consider a sequence in lp with 1 〈 p 〈 ∞, and a sequence in a Banach lattice X. Instead of normal multiplication, we consider the tensor product of lp and X. We show that in Wittstock injective tensor product, lp iX, and Fremlin projective tensor product, lp FX, the rearrangement inequality still exists.
文摘Properties for tensor products of semigroups are considered and the solutions of the equationAC - CB = Q are discussed. Results obtained in this paper considerably generalize thoseobtained in [9].
基金supported by the National Natural Science Foundation of China under Grant No.62073202the Young Experts of Taishan Scholar Project under Grant No.tsqn201909076。
文摘This paper investigates the basis and pure Nash equilibrium of finite pure harmonic games(FPHGs) based on the vector space structure. First, a new criterion is proposed for the construction of pure harmonic subspace, based on which, a more concise basis is constructed for the pure harmonic subspace. Second, based on the new basis of FPHGs and auxiliary harmonic vector, a more easily verifiable criterion is presented for the existence of pure Nash equilibrium in basis FPHGs. Third,by constructing a pure Nash equilibrium cubic matrix, the verification of pure Nash equilibrium in three-player FPHGs is given.
文摘利用张量积构造正则空间,用于场函数逼近,得到MLS(moving least squares)场函数插值形式的一个特例.在规则域上,通过基空间扩张,构造满足齐次边界条件的场函数,使得MLS插值形式变得更简单有效,扩大了基函数空间的选择范围,克服了一般MLS逼近所特有的大量求逆、计算效率低下、判据复杂的缺点.数值实例表明,该方法数学概念简单,加密点阵重复计算方便,计算精度高,计算结果连续性好,总体计算量少,计算过程收敛快.
文摘In this paper we define the tensor products of spaces of exponential type vectors of closed unbounded operators in Banach spaces. Using the real method of interpolation （K-functional） we prove the interpolation theorems that permit to characterize of tensor products of spaces of exponential type vectors, We show an application of abstract results to the theory of regular elliptic operators on bounded domains. For such operators the exponential type vectors are root vectors. Thus we describe the tensor products of root vectors of regular elliptic operators on bounded domains.
基金Project supported by the National Natural Science Foundation of China(Nos.U1804150,62073124,and 61973175)。
文摘Motivated by the inconvenience or even inability to explain the mathematics of the state space optimization of finite state machines(FSMs)in most existing results,we consider the problem by viewing FSMs as logical dynamic systems.Borrowing ideas from the concept of equilibrium points of dynamic systems in control theory,the concepts of t-equivalent states and t-source equivalent states are introduced.Based on the state transition dynamic equations of FSMs proposed in recent years,several mathematical formulations of t-equivalent states and t-source equivalent states are proposed.These can be analogized to the necessary and sufficient conditions of equilibrium points of dynamic systems in control theory and thus give a mathematical explanation of the optimization problem.Using these mathematical formulations,two methods are designed to find all the t-equivalent states and t-source equivalent states of FSMs.Further,two ways of reducing the state space of FSMs are found.These can be implemented without computers but with only pen and paper in a mathematical manner.In addition,an open question is raised which can further improve these methods into unattended ones.Finally,the correctness and effectiveness of the proposed methods are verified by a practical language model.
