By R. C. Penner

Measured geodesic laminations are a typical generalization of straightforward closed curves in surfaces, and so they play a decisive function in quite a few advancements in two-and third-dimensional topology, geometry, and dynamical structures. This ebook offers a self-contained and complete remedy of the wealthy combinatorial constitution of the distance of measured geodesic laminations in a set floor. households of measured geodesic laminations are defined through specifying a educate music within the floor, and the gap of measured geodesic laminations is analyzed via learning homes of educate tracks within the floor. the fabric is built from first rules, the strategies hired are basically combinatorial, and just a minimum heritage is needed at the a part of the reader. particularly, familiarity with effortless differential topology and hyperbolic geometry is thought. the 1st bankruptcy treats the fundamental concept of educate tracks as came across via W. P. Thurston, together with recurrence, transverse recurrence, and the categorical development of a measured geodesic lamination from a measured educate music. the following chapters advance convinced fabric from R. C. Penner's thesis, together with a average equivalence relation on measured educate tracks and conventional types for the equivalence sessions (which are used to investigate the topology and geometry of the gap of measured geodesic laminations), a duality among transverse and tangential buildings on a teach music, and the categorical computation of the motion of the mapping category workforce at the house of measured geodesic laminations within the floor.

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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.