Download e-book for kindle: A First Course in Graph Theory and Combinatorics by Sebastian M. Cioaba, M. Ram Murty

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By Sebastian M. Cioaba, M. Ram Murty

The idea that of a graph is key in arithmetic because it with ease encodes various kin and enables combinatorial research of many advanced counting difficulties. during this publication, the authors have traced the origins of graph idea from its humble beginnings of leisure arithmetic to its sleek atmosphere for modeling conversation networks as is evidenced by means of the realm broad internet graph utilized by many web se's. This publication is an creation to graph conception and combinatorial research. it really is in keeping with classes given via the second one writer at Queen's collage at Kingston, Ontario, Canada among 2002 and 2008. The classes have been aimed toward scholars of their ultimate yr in their undergraduate program.

Errate: http://www.math.udel.edu/~cioaba/book_errata.pdf

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Example text

However in some cases it is possible to solve the classification problem. 1. 2-dimensional systems Let R be a PID and Σ = (A, B) a reachable m-input 2-dimensional linear system over R. Since U1 (B) = R there exist invertible matrices P and Q such that P BQ = 1 0 0 d ... 0 ... 0 where d is a generator of U2 (B). Considering the relevant actions of the feedback group we obtain the following result. P ROPOSITION 97. Let Σ = (A, B) be a 2-dimensional reachable linear system over a principal ideal domain R.

An−1 b)P −1 = c(A). Taking an adequate feedback matrix one has the following result. P ROPOSITION 82. Let Σ = (A, b) be a reachable single-input n-dimensional system. Then Σ is feedback equivalent to the system ⎛ ⎛ 0 1 ⎜ ⎜0 0 ⎜ ⎜. ⎜. Σ =⎜ ⎜A = ⎜ . ⎝ ⎝0 0 0 0 0 1 .. ··· ··· .. 0 0 ··· ··· ⎛ ⎞⎞ ⎞ 0 0 ⎜ 0 ⎟⎟ 0⎟ ⎜ ⎟⎟ .. ⎟ ˆ ⎜ .. ⎟ ⎟ , b = ⎜ . ⎟⎟ . ⎝ 0 ⎠⎠ 1⎠ 0 1 In particular, there exists a unique class of reachable single-input n-dimensional systems. The next objective is to study the multi-input case (m > 1).

Let R be a commutative ring and M an R-module. (i) M is projective if M is a direct summand of a free R-module or equivalently the functor Hom(M, − ) is exact. (ii) M is flat if the functor M ⊗R − is exact. See [5, Chapters I and II] and [82] for the main properties of these modules. When M is finitely generated one has the following characterization. T HEOREM 19. Let M be a finitely generated R-module. Then: (i) M is flat if and only if Mp is a free Rp -module for all prime ideals p of R. Linear algebra over commutative rings 13 (ii) M is projective if and only if M is flat and the rank function rk : Spec(R) → Z p → rankRp Mp is continuous.

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