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E-raamat: From Dimension-Free Matrix Theory to Cross-Dimensional Dynamic Systems

(Institute of Systems Science, AMSS, Chinese Academy of Sciences, China)
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From Dimension-Free Matrix Theory to Cross-Dimensional Dynamic SystemsSuurem pilt
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From Dimension-Free Matrix Theory to Cross-Dimensional Dynamic Systems illuminates the underlying mathematics of semi-tensor product, a generalized matrix product that extends the conventional matrix product to two matrices of arbitrary dimensions. As dimension-varying systems are everywhere, this newly developed theory can revolutionize large data systems, such as genomics and bio-systems, deep learning, IT and information-based engineering applications.

  • Provides, for the first time, cross-dimensional system theory that is useful for modeling dimension-varying systems
  • Includes a brand new dimension-free matrix theory and cross-dimensional dynamic system theory
  • Investigates the underlying mathematics of semi-tensor product, including the equivalence and lattice structure of matrices and monoid (semi-group with identity) of matrices with arbitrary dimensions

Arvustused

"Dimension-free matrix theory and the cross-dimensional dynamic system theory presented in this book may be useful tools to handle and even model dimension-varying systems. The intended application areas are (i) genetic regulatory networks, where cells may die or be born at any time; (ii) the internet or some other service-based networks, where some users may join in or withdraw from time to time; (iii) some engineering applications, the modeling of spacecraft systems and vehicle clutch systems; etc. The results in the book are new. This book starts a new direction for further study." --Mathematical Reviews Clippings

Preface xi
References xiv
Notations xvii
1 Semi-tensor product of matrices
1.1 Two kinds of semi-tensor product of matrices
1(3)
1.1.1 Left semi-tensor product
1(2)
1.1.2 Right semi-tensor product
3(1)
1.2 Fundamental properties
4(2)
1.3 Expression of Kronecker product via STP
6(2)
1.4 Swap and permutation matrices
8(5)
References
12(1)
2 Boolean networks
2.1 Algebraic form of logical operators
13(2)
2.2 Boolean networks
15(3)
2.3 Boolean control network
18(1)
2.4 Fixed points and cycles of Boolean networks
19(1)
2.5 Controllability of Boolean control networks
20(2)
2.6 Set controllability of Boolean control networks
22(2)
2.7 Controllability via mix-type controls
24(4)
2.8 Observability of Boolean control networks
28(5)
References
32(1)
3 Finite games
3.1 General logical functions
33(2)
3.2 Finite games and its vector space structure
35(1)
3.3 Networked evolutionary games
35(4)
3.4 Potential game
39(5)
3.5 Decomposition of finite games
44(7)
References
50(1)
4 Equivalence and lattice structures
4.1 Equivalence and matrix lattice
51(9)
4.1.1 M-1 equivalence of matrices
51(3)
4.1.2 Lattice structure within an equivalence class
54(2)
4.1.3 Properties of equivalence class
56(2)
4.1.4 Sublattice and lattice homomorphism
58(2)
4.2 Semi-group structure of matrices
60(6)
4.2.1 Monoid of matrices
60(3)
4.2.2 Vector space structure on subset of matrices
63(3)
4.2.3 Group structure on subset of matrices
66(1)
4.3 Lattice of matrix subspace
66(5)
4.3.1 Lattice structure on subset of matrices
66(3)
4.3.2 Relationship between lattices of subspaces and matrices
69(2)
4.3.3 Right lattice relations
71(1)
4.4 Quotient space with vector space structure
71(8)
4.4.1 Quotient space as a monoid
71(3)
4.4.2 M-1 addition on subspace of matrices
74(1)
4.4.3 Vector space structure on subset of matrices
75(2)
References
77(2)
5 Topological structure on quotient space
5.1 Topology on matrices
79(11)
5.1.1 Product topology on quotient subset
79(4)
5.1.2 Bundle structure on matrices
83(3)
5.1.3 Coordinate frame on quotient space
86(4)
5.2 Metrics
90(11)
5.2.1 Inner product
90(5)
5.2.2 Metric and metric topology on quotient space
95(3)
5.2.3 Subspaces of quotient space
98(2)
References
100(1)
6 Differential geometry on set of matrices
6.1 Manifold structure and functions on quotient space
101(12)
6.1.1 Bundled manifold
101(4)
6.1.2 Smooth functions on quotient space
105(4)
6.1.3 Generalized inner products
109(4)
6.2 Differential geometry on quotient space
113(9)
6.2.1 Vector fields
113(3)
6.2.2 Integral curves
116(2)
6.2.3 Forms
118(2)
6.2.4 Tensor fields
120(2)
6.3 Ring structure on M-1 equivalent square matrices
122(7)
6.3.1 Ring of quotient square matrices
122(3)
6.3.2 Polynomials on ring of quotient matrices
125(1)
6.3.3 Analytic functions on ring of quotient square matrics
126(1)
References
127(2)
7 Cross-dimensional Lie algebra and Lie group
7.1 Lie algebra on quotient square matrices
129(9)
7.1.1 Bundled Lie algebra
129(2)
7.1.2 Bundled Lie sub-algebra
131(3)
7.1.3 Further properties of Lie algebra on quotient matrices
134(4)
7.2 Lie group on quotient square matrices
138(7)
7.2.1 Bundled Lie group
138(1)
7.2.2 Relationship with its Lie algebra
139(2)
7.2.3 Lie subgroups
141(1)
7.2.4 Symmetric group
142(2)
References
144(1)
8 Second matrix-matrix semi-tensor product
8.1 MM-2 STP and its fundamental properties
145(2)
8.2 M-2 equivalence and quotient space
147(1)
8.3 Lattice structure on equivalence class
148(1)
8.4 Vector space structure
149(2)
8.5 M-II metric on matrices and their quotients
151(4)
8.6 M-II metric topology on quotient space
155(1)
8.7 Subspaces of quotient space
156(3)
References
157(2)
9 Structure on set of vectors
9.1 Dimension-free vector space
159(8)
9.1.1 Vector space structure
159(2)
9.1.2 V-equivalence of vectors
161(2)
9.1.3 Lattice structure of vectors
163(1)
9.1.4 Metric on vector space
164(1)
9.1.5 Right equivalence
165(2)
9.2 Quotient vector space
167(6)
9.2.1 Quotient vector space
167(2)
9.2.2 Topological structure on quotient vector space
169(2)
9.2.3 Subspace and bundle structure
171(2)
9.3 Cross-dimensional projection
173(2)
9.3.1 Projection between spaces of different dimensions
173(2)
9.4 Least square approximated linear system
175(13)
9.4.1 Projected system of dimension-varying system
178(3)
9.4.2 Transient dynamics of dimension-varying process
181(4)
References
185(3)
10 Dimension-varying linear system
10.1 Cross-dimensional linear system
188(10)
10.1.1 S-system
188(1)
10.1.2 Pseudo linear system
189(4)
10.1.3 Linearity of S-system
193(3)
10.1.4 Operator norm of matrices
196(2)
10.1.5 S-system on quotient space
198(1)
10.2 Invariant subspace
198(14)
10.2.1 Fixed dimension invariant subspace
198(6)
10.2.2 Cross-dimensional invariant subspace
204(3)
10.2.3 Higher order linear mapping
207(3)
10.2.4 Invariant subspace on quotient vector space
210(2)
10.3 Formal polynomial of matrices
212(8)
10.3.1 Joint pseudo vector space
212(2)
10.3.2 Direct sum of vector spaces
214(2)
10.3.3 Formal polynomials of matrices
216(4)
10.4 Cross-dimensional linear system
220(13)
10.4.1 Discrete-time linear pseudo dynamic system
220(2)
10.4.2 Time invariant linear system
222(3)
10.4.3 Trajectory of discrete time linear systems
225(2)
10.4.4 Action of matrices on vectors
227(1)
10.4.5 Trajectory of continuous time linear systems
228(3)
References
231(2)
11 Dimension-varying linear control system
11.1 Modeling and analysis of dimensional-varying linear control system
233(7)
11.1.1 Discrete time cross-dimensional linear control system
233(3)
11.1.2 Continuous time linear control systems
236(4)
11.2 Dynamic and control systems on quotient space
240(13)
11.2.1 Quotient vector space and quotient matrix space
240(2)
11.2.2 Dynamic system on quotient space
242(2)
11.2.3 Formal quotient polynomials
244(1)
11.2.4 Lie algebra of formal quotient polynomials
245(1)
11.2.5 Linear (control) system on quotient space
246(3)
11.2.6 Stationary realization on quotient space
249(2)
11.2.7 Other quotient spaces
251(2)
11.3 Finite dimensional projective realization
253(6)
11.3.1 Projective realization of discrete time systems
253(2)
11.3.2 Projective realization of discrete time control systems
255(1)
11.3.3 Continuous time systems
256(1)
References
257(2)
12 Generalized dynamic systems
12.1 Constructing general STP
259(8)
12.1.1 Constructing general STPs
259(5)
12.1.2 Different types of semi-tensor product
264(3)
12.2 General equivalcence
267(9)
12.2.1 Equivalence by matrix multipler
267(2)
12.2.2 Quotient matrix space
269(1)
12.2.3 Topology on quotient matrix space
270(2)
12.2.4 Equivalence on vector space
272(1)
12.2.5 Inner product
273(2)
12.2.6 Quotient vector space
275(1)
12.3 S-system
276(5)
12.3.1 Linear S-system
276(4)
12.3.2 Linear S-system on quotient spaces
280(1)
12.4 Formal polynomials
281(12)
12.4.1 Direct sum of matrices
281(4)
12.4.2 Connected topology on formal polynomial space
285(2)
12.4.3 Quotient formal polynomials
287(3)
12.4.4 Polynomial based structures
290(1)
References
291(2)
13 Dimension-varying nonlinear dynamic systems
13.1 Discrete-time nonlinear systems
293(12)
13.1.1 Cross-dimensional nonlinear mapping
293(4)
13.1.2 Discrete-time nonlinear dynamic systems
297(4)
13.1.3 Discrete-time nonlinear control systems
301(4)
13.2 Continuous-time nonlinear systems
305(1)
13.2.1 Weak solution
305(1)
13.2.2 Continuous-time cross-dimensional nonlinear systems
306(2)
13.2.3 Continuous-time cross-dimensional nonlinear control systems
308(2)
References
310(1)
A Mathematical preliminaries
A.1 Topology
311(284)
A.1.1 Topological space
311(2)
A.1.2 Metric space
313(1)
A.1.3 Subspace, product space, and quotient space
314(1)
A.2 Fiber bundle
315(2)
A.2.1 Bundle and cross section
315(1)
A.2.2 Bundle morphism
316(1)
A.3 Algebra
317(6)
A.3.1 Vector space
317(2)
A.3.2 Group
319(3)
A.3.3 Ring
322(1)
A.4 Lattice
323(7)
A.4.1 Two definitions of lattice
324(1)
A.4.2 Lattice isomorphism
325(2)
A.4.3 Congruence of lattice
327(2)
A.4.4 Distributive and modular lattices
329(1)
A.5 Differential geometry
330(4)
A.5.1 Differentiable manifold
330(1)
A.5.2 Vector field
331(2)
A.5.3 Co-vector field and Lie derivative
333(1)
A.6 Lie groups and Lie algebras
334(5)
A.6.1 Lie group
334(2)
A.6.2 Lie algebra
336(1)
A.6.3 Lie algebra of Lie group
337(1)
References
338(1)
Index 339
Daizhan Cheng is the creator of the novel and highly-useful product of matrices called the semi-tensor product (STP) or Cheng product. He holds a PhD from Washington University, St. Louis, and since 1990, he has served as a professor with the Institute of Systems Science, AMSS, Chinese Academy of Sciences. He is the author / co-author of 14 books, over 250 journal papers, and more than 150 conference papers. He received the Second National Natural Science Award in both 2008 and 2014, the Outstanding Science and Technology Achievement Price of CAS in 2015, and the Automatica Best Paper Award (20082010), bestowed by the IFAC.
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