Walking through the corridors of a Fire and Rescue Service (FRS) headquarters in the north-east of England, you encounter an array of posters showing charts, graphs, and tables containing a variety ...of different information pertaining to fire emergencies. Affixed to walls, multi-coloured scattergraphs indicate the age of those most vulnerable to fire. Adjacent, a bar chart shows which fire stations have attended the most fire incidents on a monthby-month basis. A few further steps along, a map purports to show the distribution of fire incidents year on year. These posters boldly sit on the walls of the FRS headquarters as
In this chapter we shall treat yet another case of processes obtained by inductive functions; this time however the results obtained will have a direct bearing on our problem of statistical ...predictability. As in the last chapter we consider inductive functions Z of X where X is a finite state Markoff process. The range A of Z will be a subset of a projective space and the operation ψ(π) associated with the variables of X will be projective transformations of A. The results of the preceding chapter, in particular Theorem 29.2, will not cover this situation because the transformations ψ(π)
Let F be a closed orientable surface of genus g. The fundamental group of F can be expressed in terms of 2g generators and a single relator as given below, and is called asurface group.
\{{\pi ...}_{1}}(\text{F})=<{{\text{x}}_{1}},{{\text{y}}_{1}},\cdots ,{{\text{x}}_{\text{g}}},{{\text{y}}_{\text{g}}}|{{\text{x}}_{1}},{{\text{y}}_{1}}{{\text{x}}_{2}},{{\text{y}}_{2}}\cdots {{\text{x}}_{\text{g}}},{{\text{y}}_{\text{g}}}>.\
An observation of Nielsen allows one to study the automorphisms of surface groups geometrically. In particular:
Lemma (Nielsen 9).Ifg ≥ 1then every element ofOut(π
1(F))is represented by a unique isotopy class of self-homeomorphisms ofF.
There is no difference in considering surface automorphisms up to isotopy or up to homotopy, as homotopic automorphisms of a closed
We will describe an ergodicity phenomenon for the action of an arbitrary discrete group of hyperbolic isometries on the tangent spaces of the sphere at ∞. One application (Section VII) is a maximal ...extension of Mostow’s rigidity theorem “rough isometry → isometry” from finite volume hyperbolic manifolds to manifolds whose volume grows slower than that of hyperbolic space. Another application (Section V) allows a complete description of the finitely generated Kleinian groups in one quasiconformal conjugacy class in terms of a nice complex manifold Teichmüller space. Along the way to the main theorem we characterize ergodicity of the action of
Let Ω be an open set in the Euclidean n-dimensional space Rn. For the sake of simplicity let us assume that Ω be a bounded and connected open set with smooth boundary; moreover, assume that n ≥ 2. We ...shall denote by x = (x1, …, xn) points in Rnand by dx = dx1, …, dxnor$\mathrm{d\mathfrak{L}^{n}}$the Lebesgue volume element in Rn.¹)
Let u(x) = (u1(x), …, uN(x)) be a vector valued function defined in Ω with value in RN, N ≥ 1. We shall denote by Du or$\mathrm{\triangledown u}$the gradient of u, i.e. the set
Concepts such as topological index, order of mappings, looping coefficients, etc. are generally discussed in topology see P. Alexandroff and H. Hopf, I, Chapter XII; S. Lefshetz, I, Chapter IV; M. H. ...A. Newman, I, Chapter VII. Here only the particular case of the "order of a point p∊E2with respect to a closed plane curve C ⊂ E2" (topological index) is needed. This concept admits of a well known and very simple metrical approach which is given below (8.1-2) Cf. P. Alexandroff and Hopf, I, Chapter XII, §1, No. 6, p. 462; T. Radó, II, Chapter II, 4.34, p.
MINIMAL RESOLUTIONS Laufer, Henry B
Normal Two-Dimensional Singularities. (AM-71),
03/2016, Letnik:
71
Book Chapter
Resolutions of normal singularities are not unique. We may always perform a quadratic transformation at a point of π-¹(p). In this section we shall show that there is a unique minimal resolution. All ...other resolutions may be obtained from the minimal resolution by quadratic transformations.
We first need some machinery of a general nature about sheaves.
Proposition 5.1. (Mayer-Vietoris)LetAandBbe open sets in the topological spaceXandSa sheaf overX.Then the following sequence is exact
$0 \to \Gamma \left( {AUB,S} \right) \to \Gamma \left( {A,S} \right) \oplus \Gamma \left( {B,S} \right)\Gamma \left( {A \cap B,S} \right)$
${H^1}\left( {AUB,S} \right) \to ... \to {H^i}\left( {A \cup B,S} \right) \to {H^I}\left( {A,S} \right) \oplus {H^i}\left( {B,S} \right)$
$\rho \to {H^i}\left( {A \cup B,S} \right) \to \delta ...$
ιis induced by
$\iota \left( a \right) = a \oplus a$
andρis induced by
$\rho \left( {\beta \oplus \gamma } \right) = \beta - \gamma $.
Proof: Let
$0 \to S \to {e^ \circ } \to {e^1} \to {e^2} \to $
CLASSIFICATION DAVID EISENBUD; WALTER NEUMANN
Three-Dimensional Link Theory and Invariants of Plane Curve Singularities. (AM-110),
03/2016
Book Chapter
Recall that a Seifert link is a link$\text{L}=(\Sigma ,\text{K})=(\Sigma ^{'},{{\text{S}}_{1}}\cup \cdots \cup {{\text{S}}_{\text{n}}})$whose exterior Σ0= Σ′ — int N(K) admits a Seifert fibration.
...Lemma 7.1.An irreducible linkL = (Σ,K)is a Seifert link if and only ifKis an invariant set for some effectiveS1-action onΣ.Moreover, this action onΣcan be chosen fixed point free unlessLis the following link inS3:
Proof. Suppose L is a Seifert link and letπ:Σ →F be the Seifert fibration. If F were non-orientable, its orientation cover would induce a nontrivial double covering of Σ0, trivial along ∂Σ0,
INVARIANTS DAVID EISENBUD; WALTER NEUMANN
Three-Dimensional Link Theory and Invariants of Plane Curve Singularities. (AM-110),
03/2016
Book Chapter
Let Γ be a splice diagram with no multiplicity weights, representing a graph link${\text{L}}(\Gamma)=\Sigma,{\text{S}}_{1}\cup \cdots \cup {\text{S}}_{\text{n}}$and let$\Gamma ({\underline {\text ...m}})=\Gamma({\text {m}}_{1},\cdots ,{\text {m}}_{\text {n}})$be the splice diagram for$(\Sigma ,{{\text{m}}_{1}}{{\text{S}}_{1}}\cup \cdots \cup {{\text{m}}_{\text{n}}}{{\text{S}}_{\text{n}}})$. We fix this notation throughout this section.
Moreover, we shall number the vertices of Γ as\{{\text{v}}_{1}},\cdots ,{{\text{v}}_{\text{n}}},\ {{\text{v}}_{\text{n}+1}},\cdots ,{{\text{v}}_{\text{k}}},\, with${{\text{v}}_{1}},\cdots ,{{\text{v}}_{\text{n}}},$being the arrowheads and${{\text{v}}_{\text{n}+1}},\cdots ,{{\text{v}}_{\text{k}}}$being the remaining vertices. For${\text {i}}={\text{n}}+1,\cdots,{\text{k}}$, let Sibe a nonsingular fiber of the Seifert component corresponding to vi, if viis a node, and let Sibe the (possibly) singular fiber corresponding to viif viis a boundary vertex. We always orient Sias in section 7.
PLATS AND LINKS Birman, Joan S
Braids, Links, and Mapping Class Groups. (AM-82),
03/2016, Letnik:
82
Book Chapter
Our emphasis in Chapters 1-4 has been on the relationship between closed braids and links. It would, however, be amiss to end this monograph without mention of another, and quite different, ...algebraic-topological connection between Artin’s braid group and links in S³, the concept of a “plat.” This latter notion has only recently begun to receive attention, yet it appears to be of considerable importance in connection with the classification of 3-manifolds see Viro, 1972; Birman and Hilden, 1973a and 1974; Montesinos, 1974, and it may also offer the possibility of a new approach to the study of knots and links
