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＜電子ブック＞
Multidimensional Analysis : Algebras and Systems for Science and Engineering / by George W. Hart

版	1st ed. 1995.
出版者	(New York, NY : Springer New York : Imprint: Springer)
出版年	1995
大きさ	XII, 236 p : online resource
著者標目	*Hart, George W author SpringerLink (Online service)
件　名	LCSH:Mathematics LCSH:Physics LCSH:Astronomy LCSH:Numerical analysis LCSH:Engineering mathematics LCSH:Engineering—Data processing FREE:Applications of Mathematics FREE:Physics and Astronomy FREE:Numerical Analysis FREE:Mathematical and Computational Engineering Applications
一般注記	0. Introductory -- 0.1 Physical Dimensions -- 0.2 Mathematical Dimensions -- 0.3 Overview -- Exercises -- 1. The Mathematical Foundations of Science and Engineering -- 1.1 The Inadequacy of Real Numbers -- 1.2 The Mathematics of Dimensioned Quantities -- 1.3 Conclusions -- Exercises -- 2. Dimensioned Linear Algebra -- 2.1 Vector Spaces and Linear Transformations -- 2.2 Terminology and Dimensional Inversion -- 2.3 Dimensioned Scalars -- 2.4 Dimensioned Vectors -- 2.5 Dimensioned Matrices -- Exercises -- 3. The Theory of Dimensioned Matrices -- 3.1 The Dimensional Freedom of Multipliable Matrices -- 3.2 Endomorphic Matrices and the Matrix Exponential -- 3.3 Square Matrices, Inverses, and the Determinant -- 3.4 Squarable Matrices and Eigenstructure -- 3.5 Dimensionally Symmetric Multipliable Matrices -- 3.6 Dimensionally Hankel and Toeplitz Matrices -- 3.7 Uniform, Half Uniform, and Dimensionless Matrices -- 3.8 Conclusions -- 3.A Appendix: The n + m ? 1 Theorem -- Exercises -- 4. Norms, Adjoints, and Singular Value Decomposition -- 4.1 Norms for Dimensioned Spaces -- 4.2 Dimensioned Singular Value Decomposition (DSVD) -- 4.3 Adjoints -- 4.4 Norms for Nonuniform Matrices -- 4.5 A Control Application -- 4.6 Factorization of Symmetric Matrices -- Exercises -- 5. Aspects of the Theory of Systems -- 5.1 Differential and Difference Equations -- 5.2 State-Space Forms -- 5.3 Canonical Forms -- 5.4 Transfer Functions and Impulse Responses -- 5.5 Duals and Adjoints -- 5.6 Stability -- 5.7 Controllability, Observability, and Grammians -- 5.8 Expectations and Probability Densities -- Exercises -- 6. Multidimensional Computational Methods -- 6.1 Computers and Engineering -- 6.2 Representing and Manipulating Dimensioned Scalars -- 6.3 Dimensioned Vectors -- 6.4 Representing Dimensioned Matrices -- 6.5 Operations on Dimensioned Matrices -- 6.6 Conclusions -- Exercises -- 7. Forms of Multidimensional Relationships -- 7.1 Goals -- 7.2 Operations -- 7.3 Procedure -- Exercises -- 8. Concluding Remarks -- 9. Solutions to Odd-Numbered Exercises -- References This book deals with the mathematical properties of dimensioned quantities, such as length, mass, voltage, and viscosity. Beginning with a careful examination of how one expresses the numerical results of a measurement and uses these results in subsequent manipulations, the author rigorously constructs the notion of dimensioned numbers and discusses their algebraic structure. The result is a unification of linear algebra and traditional dimensional analysis that can be extended from the scalars to which the traditional analysis is perforce restricted to multidimensional vectors of the sort frequently encountered in engineering, systems theory, economics, and other applications HTTP:URL=https://doi.org/10.1007/978-1-4612-4208-6

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