Theory Of Fuzzy Tensors And Its Applications

This dissertation mainly aims to study fuzzy tensor based on multi-dimensional data,which proves to be crucial in tackling many problems.Firstly the dissertation gives the definition of fuzzy tensors,followed by the discussion over fuzzy tensors from the perspective of operation properties and order structure aspect,and then illustrates the application of fuzzy tensors with multi-dimensional data example.The research results and contributions of this dissertation mainly are rendered in the following four aspects:1.The definition of fuzzy tensors is introduced and the algebraic structure of fuzzy tensors is constructed.Since the idea of fuzzy mathematics is applied to investigate tensors,the dissertation puts forward the concept of fuzzy tensors.Targeting at the problem of algebraic structure,the dissertation studies fuzzy tensors in theory,using multiple linear algebra principle and fuzzy logic operations;hence,some basic operation relations between two fuzzy tensors of the same size are given respectively,such as union operation,intersection operation,Order and semi ordered structure(size comparison relation),equality relation and(remaining)complementary operation and so on.We also present the properties of union,intersection and complement operation of fuzzy tensor,so as to construct the algebraic structure of fuzzy tensors.2.The decomposition theorem of fuzzy tensors is proposed and proved.Herein the application of fuzzy tensors decomposition theorem I in Congou black tea is discussed.Using cutting tensors and scalar product,we propose and prove decomposition theorem I of fuzzy tensors.By decomposing a fuzzy tensor that multiple scalar product union form of the same dimension of the same order of cutting tensors and fuzzy number,and the feasibility and correctness of fuzzy tensors decomposition theorem I are verified through numerical examples.Based on anti-fuzzy cutting tensors and anti-fuzzy scalar product,we present and prove fuzzy tensors decomposition theorem II.A fuzzy tensor is decomposed that anti-fuzzy scalar product union form of the same dimension with the same order anti-fuzzy cutting tensors and fuzzy number.By using the fuzzy tensors decomposition theorem I,the dissertation tries to solve grade comparison practical problems of Congou black tea through the water extracts of Congou black tea,that made from fresh leaves of old tree tea from Pu'an County,Southwest of Guizhou province.The application case shows that the fuzzy tensors decomposition theorem I is feasible and correct.3.The convergence of fuzzy tensors is studied.We introduce the notions of directed paths and fuzzy directed path systems.Their convergent theorems are proposed by using matrices principle.By using the monotonicity of fuzzy tensors and the characteristics of the elements contained in fuzzy tensors,two algorithms for judging the convergence of fuzzy tensors are given.Numerical examples are given to illustrate the effectiveness and feasibility of the presented methods.4.The oscillation period and index of fuzzy tensors are explored.The oscillation period and index of fuzzy tensors are obtained on the basis of Power Method with max-min operation.Secondly,by adopting graph theory principle and relying on Minimal Strong Component we find the oscillation period of fuzzy tensors.Furthermore,numerical results demonstrate that the two proposed algorithms are effective and feasible.