By Kazuo Murota
ISBN-10: 3642039936
ISBN-13: 9783642039935
A matroid is an summary mathematical constitution that captures combinatorial homes of matrices. This e-book deals a different advent to matroid thought, emphasizing motivations from matrix conception and purposes to structures analysis.
This publication serves additionally as a finished presentation of the idea and alertness of combined matrices, built basically by means of the current writer within the final decade. A combined matrix is a handy mathematical instrument for platforms research, appropriate with the actual remark that "fixed constants" and "system parameters" are to be extraordinary within the description of engineering systems.
This ebook may be tremendous valuable to graduate scholars and researchers in engineering, arithmetic and computing device science.
From the reviews:
"…The ebook has been ready very conscientiously, encompasses a lot of attention-grabbing effects and is very prompt for graduate and postgraduate students."
András Recski, Mathematical studies Clippings 2000m:93006
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Extra info for Matrices and Matroids for Systems Analysis (Algorithms and Combinatorics)
Example text
4) shows how to combine the structural information from Q-part and T -part to obtain the information about A. 22 1. 2. For a nonsingular mixed polynomial matrix A(s) = Q(s) + T (s), degs det A = max {degs det Q[I, J] + degs det T [R \ I, C \ J]}. 34) is only an upper bound on degs det A. 34) involves a maximization over all pairs (I, J), the number of which is almost as large as 2|R|+|C| , too large for an exhaustive search for maximization. Fortunately, however, it is possible to design an efficient algorithm to compute this maximum on the basis of the facts that each of the functions fQ (I, J) = degs det Q[I, J], fT (I, J) = degs det T [I, J] can be evaluated easily, and that the maximization (“combination of Q-part and T -part”) can be done efficiently, as follows.
Block-triangular decompositions of the above kind are one of the major topics studied in this book; among which are the Dulmage–Mendelsohn decomposition in Chap. 2 and the combinatorial canonical form (CCF) in Chap. 4. Notes. 3 is due to Murota [200]. This chapter is an improved version of a presentation (Murota [223]) at ICIAM 95. 2. Matrix, Graph, and Matroid This chapter lays the mathematical foundation for combinatorial methods of systems analysis. Combinatorial properties of numerical matrices can be stated and analyzed with the aid of matroid theory, whereas those of polynomial matrices are formulated in the language of valuated matroids in Chap.
Ij = 0}. A˜ = {(xj , ei ) | A¯ij = 0 or F¯ij = 0} ∪ {(uj , ei ) | B It is sometimes convenient to assign weight 1 to arc (xj , ei ) with F¯ij = 0 and weight 0 to the other arcs. The above distinction between standard form and the descriptor form implies, in particular, that the finest decomposition of Aˆ is obtained through the strong component decomposition, whereas that of A¯ is through the Dulmage– Mendelsohn decomposition. 20) another graph representation is sometimes useful. For k ≥ 1 the dynamic graph of time-span k is defined to be Gk0 = ) with (X0k ∪ U0k−1 , Ak−1 0 k X0k = X t = {xti | i = 1, · · · , n} X t, (t = 0, 1, · · · , k), t=0 k−1 U0k−1 = U t, U t = {utj | j = 1, · · · , m} (t = 0, 1, · · · , k − 1), t=0 Ak−1 = {(xtj , xt+1 ) | Aˆij = 0; t = 0, 1, · · · , k − 1} 0 i ˆij = 0; t = 0, 1, · · · , k − 1}.
Matrices and Matroids for Systems Analysis (Algorithms and Combinatorics) by Kazuo Murota
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