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E-raamat: Introduction To Semi-tensor Product Of Matrices And Its Applications, An

(Chinese Academy Of Sciences, China), (Chinese Academy Of Sciences, China), (Chinese Academy Of Sciences, China)
  • Formaat: 612 pages
  • Ilmumisaeg: 31-May-2012
  • Kirjastus: World Scientific Publishing Co Pte Ltd
  • Keel: eng
  • ISBN-13: 9789814458016
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Introduction To Semi-tensor Product Of Matrices And Its Applications, AnSuurem pilt
  • Formaat: 612 pages
  • Ilmumisaeg: 31-May-2012
  • Kirjastus: World Scientific Publishing Co Pte Ltd
  • Keel: eng
  • ISBN-13: 9789814458016
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A generalization of Conventional Matrix Product (CMP), called the Semi-Tensor Product (STP), is proposed. It extends the CMP to two arbitrary matrices and maintains all fundamental properties of CMP. In addition, it has a pseudo-commutative property, which makes it more superior to CMP. The STP was proposed by the authors to deal with higher-dimensional data as well as multilinear mappings. After over a decade of development, STP has been proven to be a powerful tool in dealing with nonlinear and logical calculations.This book is a comprehensive introduction to the theory of STP and its various applications, including logical function, fuzzy control, Boolean networks, analysis and control of nonlinear systems, amongst others.
Preface v
Notations xiii
1 Multi-Dimensional Data
1(22)
1.1 Multi-Dimensional Data
1(3)
1.2 Arrangement of Data
4(3)
1.3 Matrix Products
7(4)
1.3.1 Kronecker Product of Matrices
8(1)
1.3.2 Hadamard Product
9(1)
1.3.3 Khatri-Rao Product
10(1)
1.4 Tensor
11(3)
1.5 Nash Equilibrium
14(2)
1.6 Symmetric Group
16(2)
1.7 Swap Matrix
18(5)
Exercises
20(3)
2 Semi-Tensor Product of Matrices
23(28)
2.1 Multilinear Function
23(4)
2.2 Left Semi-Tensor Product of Matrices
27(5)
2.3 Fundamental Properties
32(6)
2.4 Pseudo-Commutativity via Swap Matrix
38(6)
2.5 Semi-Tensor Product as Bilinear Mapping
44(7)
Exercises
47(4)
3 Multilinear Mappings among Vector Spaces
51(34)
3.1 Cross Product on R3
51(4)
3.2 General Linear Algebra
55(3)
3.3 Mappings over Matrices
58(11)
3.4 Converting Matrix Expressions
69(6)
3.5 Two Applications
75(10)
3.5.1 General Linear Group and Its Algebra
75(3)
3.5.2 Hautus and Sylvester Equations
78(2)
Exercises
80(5)
4 Right and General Semi-Tensor Products
85(16)
4.1 Right STP
85(7)
4.2 Semi-Tensor Product of Arbitrary Matrices
92(9)
Exercises
96(5)
5 Rank, Pseudo-Inverse, and Positivity of STP
101(12)
5.1 Rank of Products
101(1)
5.2 Pseudo-Inverse of STP
102(6)
5.2.1 Moore-Penrose Inverse
103(4)
5.2.2 Drazin Inverse
107(1)
5.3 Positivity of Products
108(5)
Exercises
111(2)
6 Matrix Expression of Logic
113(32)
6.1 Logic and Its Expression
113(8)
6.2 General Structure of Logical Operators
121(3)
6.3 Fundamental Properties of Logical Operators
124(6)
6.4 Logical System and Logical Inference
130(6)
6.5 Multi-Valued Logic
136(9)
Exercises
141(4)
7 Mix-Valued Logic
145(18)
7.1 Normal Form of Logical Operators
145(4)
7.2 Mix-Valued Logic
149(3)
7.3 General Logical Mappings
152(5)
7.4 Two Practical Examples
157(6)
7.4.1 Mix-Valued Logical Form of Rules in Fuzzy Control
157(3)
7.4.2 Expression of Strategies of Dynamic Games
160(1)
Exercises
161(2)
8 Logical Matrix, Fuzzy Set and Fuzzy Logic
163(22)
8.1 Matrices of General Logical Variables
163(3)
8.2 Logical Operators for k-Valued Matrices
166(3)
8.3 Fuzzy Sets
169(5)
8.4 Mappings over Fuzzy Sets
174(4)
8.5 Fuzzy Logic and Its Computation
178(7)
Exercises
180(5)
9 Fuzzy Relational Equation
185(16)
9.1 k-Valued Matrix and Fuzzy Relational Equations
185(3)
9.2 Structure of the Set of Solutions
188(3)
9.3 Solving Fuzzy Relational Equation
191(2)
9.4 Numerical Examples
193(8)
Exercises
199(2)
10 Fuzzy Control with Coupled Fuzzy Relations
201(24)
10.1 Multiple Fuzzy Relations
201(7)
10.1.1 Matrix Expression
201(2)
10.1.2 Multiple Fuzzy Inference
203(1)
10.1.3 Compounded Multiple Fuzzy Relations
204(4)
10.2 Fuzzy Control of Coupled Multiple Fuzzy Relations
208(8)
10.2.1 Fuzzification via Dual Fuzzy Structure
208(2)
10.2.2 Design of Fuzzy Controller
210(3)
10.2.3 Defuzzification
213(3)
10.3 Numerical Solution for Fuzzy Control Design
216(9)
Exercises
220(5)
11 Representation of Boolean Functions
225(28)
11.1 Boolean Functions in Galois Field Z2
225(3)
11.2 Polynomial Form of Boolean Functions
228(5)
11.3 Walsh Transformation
233(7)
11.4 Linear Structure
240(4)
11.5 Nonlinearity
244(3)
11.6 Symmetry of Boolean Function
247(6)
Exercises
251(2)
12 Decomposition of Logical Functions
253(22)
12.1 Disjoint Bi-Decomposition
253(7)
12.2 Non-Disjoint Bi-Decomposition
260(5)
12.3 Decomposition of Multi-Valued Logical Functions
265(4)
12.4 Decomposition of Mix-Valued Logical Functions
269(6)
Exercises
272(3)
13 Boolean Calculus
275(22)
13.1 Boolean Derivatives
275(6)
13.2 Boolean Differential Equations
281(3)
13.3 Boolean Integral
284(13)
13.3.1 Primitive Function
285(2)
13.3.2 Indefinite Integral
287(5)
13.3.3 Definite Integral
292(1)
Exercises
293(4)
14 Lattice, Graph, and Universal Algebra
297(40)
14.1 Lattice
297(4)
14.2 Isomorphic Lattices and Sublattices
301(2)
14.3 Matrix Expression of Finite Lattice
303(6)
14.4 Distributive and Modular Lattices
309(1)
14.5 Graph and Its Adjacency Matrix
310(4)
14.6 Vector Space Structure of Graph
314(4)
14.7 Planar Graph and Coloring Problem
318(9)
14.8 Universal Algebra
327(5)
14.9 Lattice-Based Logics
332(5)
Exercises
334(3)
15 Boolean Network
337(28)
15.1 An Introduction
337(3)
15.2 Fixed Points and Cycles
340(3)
15.3 Invariant Subspace and Input-State Description
343(10)
15.3.1 State Space and Subspaces
343(7)
15.3.2 Input-State Description
350(3)
15.4 Higher-Order Boolean Networks
353(4)
15.4.1 First Algebraic Form of Higher-Order Boolean Networks
354(2)
15.4.2 Second Algebraic Form of Higher-Order Boolean Networks
356(1)
15.5 Dynamic-Static Boolean Networks
357(8)
Exercises
360(5)
16 Boolean Control System
365(36)
16.1 Dynamics of Boolean Control Networks
365(5)
16.2 Controllability
370(3)
16.3 Observability
373(3)
16.4 Disturbance Decoupling
376(9)
16.5 Some Other Control Problems
385(16)
16.5.1 Stability and Stabilization
385(6)
16.5.2 Optimal Control
391(2)
16.5.3 Identification
393(3)
Exercises
396(5)
17 Game Theory
401(18)
17.1 An Introduction to Game Theory
401(6)
17.2 Infinitely Repeated Games
407(3)
17.3 Local Optimization of Strategies and Local Nash/Sub-Nash Equilibrium
410(9)
Exercises
415(4)
18 Multi-Variable Polynomials
419(38)
18.1 Matrix Expression of Multi-Variable Polynomials
419(11)
18.2 Differential Form of Functional Matrices
430(9)
18.3 Conversion of Generators
439(4)
18.4 Taylor Expansion of Multi-Variable Functions
443(4)
18.5 Fundamental Formula of Differential
447(2)
18.6 Lie Derivative
449(8)
Exercises
453(4)
19 Some Applications to Differential Geometry and Algebra
457(42)
19.1 Calculation of Connection
457(6)
19.2 Contraction of Tensor Field
463(4)
19.3 Structure Matrix of Finite-Dimensional Algebra
467(6)
19.4 Two-Dimensional Algebras
473(3)
19.5 Three-Dimensional Algebras
476(5)
19.6 Lower-Dimensional Lie Algebra and Invertible Algebra
481(11)
19.7 Tensor Product Algebra
492(7)
Exercises
495(4)
20 Morgan's Problem
499(16)
20.1 Input-Output Decomposition
499(3)
20.2 Problem Formulation
502(3)
20.3 Numerical Expression of Solvability
505(10)
Exercises
512(3)
21 Linearization of Nonlinear Control Systems
515(28)
21.1 Carleman Linearization
515(4)
21.2 First Integral
519(6)
21.3 Invariance of Polynomial System
525(2)
21.4 Feedback Linearization of Nonlinear Control System
527(2)
21.5 Single Input Feedback Linearization
529(3)
21.6 Algorithm for Non-Regular Feedback Linearization
532(11)
Exercises
537(6)
22 Stability Region of Dynamic Systems
543(24)
22.1 Stability Region
543(2)
22.2 Stable Submanifold
545(3)
22.3 Quadratic Approximation
548(4)
22.4 Higher Order Approximation
552(8)
22.5 Differential-Algebraic System
560(7)
Exercises
563(4)
Appendix A Numerical Algorithms
567(6)
A.1 Basic Functions
567(3)
A.2 Some Examples
570(3)
Bibliography 573(10)
Index 583
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