Topology in Molecular Biology
212 R. Brooks x= 0 x=1 y= y y=y 1 0 Fig. 12.3.What a cusp looks like Theorem 1.Let Sbe a Riemann surface of finite area, and let ...
12 The Spectral Geometry of Riemann Surfaces 213 Both worries will be taken care of by showing that “not much is happening far o ...
214 R. Brooks and it is easily seen that ∫∞ y 1 a^20 1 y^2 dy= (const)(y 1 )−^2 s. Hence ∫∞ y 1 a^20 1 y^2 dy= ( y 0 y 1 )∫∞ y 0 ...
12 The Spectral Geometry of Riemann Surfaces 215 After a M ̈obius transformation, we may assume that the three points are 0, 1, ...
216 R. Brooks 2π/3 Fig. 12.4.The fundamental domain forPSL(2,Z) (1 +i √ 3)/2, respectively. Removing the inverse images of these ...
12 The Spectral Geometry of Riemann Surfaces 217 ThenSis a Belyi surface. Our proof is based on the proof in [4]. The converse t ...
218 R. Brooks 12.5 The Basic Construction Let Γ be a three-regular graph. An orientationOon Γ is an assignment, for each vertexv ...
12 The Spectral Geometry of Riemann Surfaces 219 Fig. 12.6.The graph on two vertices SO(Γ,O) as a covering space ofH^2 /P SL(2,Z ...
220 R. Brooks Fig. 12.7.Building on the tetrahedron Fig. 12.8.The tetrahedron with one orientation reversed may take the standar ...
12 The Spectral Geometry of Riemann Surfaces 221 again the geometric center, and, while it is not the case that the conformal ce ...
222 R. Brooks Fig. 12.10.Various options for orientations on the cube got surfaces of genus two, because there are two LHT paths ...
12 The Spectral Geometry of Riemann Surfaces 223 Proof.Consider the function g(z)= {f(z) z ,z=0 f′(0),z=0. On|z|=r<1, we hav ...
224 R. Brooks Proof.: Let us write the “pullback metric” of ds^22 by f∗ds^22 =g^2 (z)ds^21 , whereg(z) is a real function, which ...
12 The Spectral Geometry of Riemann Surfaces 225 The proof will be a combination of the two arguments above. We again look at th ...
226 R. Brooks Suppose that there exist constantsC 1 andC 2 such that C 1 sup(κ 1 ) inf(κ 2 )≤sup(κ 2 )≤C 2 inf(κ 1 )≤C 2 sup(κ 1 ...
12 The Spectral Geometry of Riemann Surfaces 227 Fig. 12.11.A simple example We remark that part of the statement of the theorem ...
228 R. Brooks wherefwill be a function that is equal to one outside of cusp neighborhoods. We will want to choosefso that: The ...
12 The Spectral Geometry of Riemann Surfaces 229 The coefficient of d/theta^2 is [( 2 1 −tanh^2 (r/2) (tanh(r/2)) )] 2 = [ 2sinh ...
230 R. Brooks there are other features of the geometry of Riemann surfaces which can be read off from this picture. The key poin ...
12 The Spectral Geometry of Riemann Surfaces 231 Theorem 10.The cycle decomposition of a randomly picked element ofS(n) has the ...
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