Modern Mathematical Methods In Technology

Editorial NoteIntroductionChapter 1 Linear Algebra 101. Matrix Theory 101.01. Linear mappings 101.02. Matrices 101.03. Basic matrix operations 101.04. Hypermatrices 101.05. Linearly independent vectors 101.06. Orthogonal and biorthogonal systems of vectors 101.07. The inverse of a matrix 101.08. The dyadic decomposition of matrices 101.09. The rank of a vector system 101.10. The rank of a matrix 101.11. The minimal decomposition of a matrix 101.12. A few theorems on products of matrices 101.13. The dyadic decomposition of certain important matrices 101.14. Eigenvalues and eigenvectors of matrices 101.15. Symmetric and hermitian matrices 101.16. Matrix polynomials 101.17. The characteristic polynomial of a matrix. The Cayley-Hamilton theorem 101.18. The minimum polynomial of a matrix 101.19. The biorthogonal minimal decomposition of a square matrix 102. Matrix Analysis 102.01. Sequences, series, continuity, differentiation and integration of matrices 102.02. Power series of matrices 102.03. Analytical matrix functions 102.04. Decomposition of rational matrices 103. A Few Applications of Matrix Calculus 103.01. The theory of systems of linear equations 103.02. Linear integral equations 103.03. Linear systems of differential equations 103.04. The motion of a particle 103.05. The stability of linear systems 103.06. Bending of a supported beam 103.07. Application of matrix techniques to linear electrical networks 103.08. The application of matrices to the theory of four-pole devices Chapter 2 Optimization Theory 201. Linear Optimization 201.01. The problem 201.02 Geometrical approaches 201.03. Minimum vectors for a linear programming problem 201.04. Solution of the linear programming problem 201.05. Dual linear programming problems 201.06. Transportation problems and their solution by the Hungarian method 202. Convex Optimization 202.01. The problem 202.02. Definitions and lemmas 202.03. The Kuhn-Tucker theorem 202.04. Convex optimization with differentiable functionsChapter 3 Elements of the Theory of Graphs 301.01. Introduction 301.02. The idea of a graph 301.03. Sub-graphs and complete graphs; complementary graphs 301.04. Chains, paths and cycles 301.05. Components and blocks of a graph 301.06. Trees and spanning trees of a graph 301.061. An application 301.07. Fundamental systems of cycles and sheaves 301.08. Graphs on surfaces 301.09. Duality 301.10. Boolean algebra 301.101. Incidence matrices 301.102. Cycle matrices 301.103. Sheaf matrices 301.104. Vectors spaces generated by graphs 301.11. Directed graphs 301.111. Matrices associated with directed graphs 301.12. The application of graph theory to the theory of electric networks 301.13. The Ford-Fulkerson theoremBibliographyIndex