Introduction Fundamental Concepts Results Bibliography Fuchsian group generators associated with the C 2 , 8 channel quantization Anderson Jos´ e de Oliveira Reginaldo Palazzo Jr. Federal University of Alfenas - UNIFAL-MG State University of Campinas - UNICAMP July 25th to 27th, 2018 anderson.oliveira@unifal-mg.edu.br/ palazzo@dt.fee.unicamp.br LAWCI 2018
Introduction Fundamental Concepts Results Bibliography Motivation Figure: Research Problem anderson.oliveira@unifal-mg.edu.br/ palazzo@dt.fee.unicamp.br LAWCI 2018
Introduction Fundamental Concepts Results Bibliography Outline 1 Introduction 2 Fundamental Concepts Concepts related to graphs and surfaces Hyperbolic Geometry Fuchsian Differential Equations 3 Results 4 References anderson.oliveira@unifal-mg.edu.br/ palazzo@dt.fee.unicamp.br LAWCI 2018
Introduction Fundamental Concepts Results Bibliography Introduction We consider the steps to be followed by a designer of a communication system regarding the channel output quantization problem under the topological space approach. 1 Determine the minimum and the maximum genus associated with the embedding of the given DMC channel. 2 Establish the algebraic curve for each genus from Step 1. 3 Solve the linear second order differential equation for each associated algebraic curve from Step 2. 4 Determine the algebraic structure (Fuchsian group generators) associated with the fundamental region of each surface/algebraic curve. anderson.oliveira@unifal-mg.edu.br/ palazzo@dt.fee.unicamp.br LAWCI 2018
Introduction Fundamental Concepts Results Bibliography Definitions and concepts related to graphs and surfaces Definition [4] A graph G ′ is called an embedding in a surface Ω when no two of its edges meet except at a vertex. The complement of G ′ in Ω is called region . A region which is homeomorphic (topological equivalence) to an open disk is called 2-cell ; if the entire region is a 2-cell, the embedding is said to be a 2-cell embedding . anderson.oliveira@unifal-mg.edu.br/ palazzo@dt.fee.unicamp.br LAWCI 2018
Introduction Fundamental Concepts Results Bibliography Definitions and concepts related to graphs and surfaces A complete bipartite graph with m and n vertices, denoted by K m , n , is a graph consisting of two disjoint vertex sets with m and n vertices, where each vertex of a set is connected by an edge to every vertex of the other set. An important topological invariant of graphs and surfaces is the E¨ uler characteristic. anderson.oliveira@unifal-mg.edu.br/ palazzo@dt.fee.unicamp.br LAWCI 2018
Introduction Fundamental Concepts Results Bibliography Definitions and concepts related to graphs and surfaces Theorem [5] For m , n ≥ 2 , the E¨ uler characteristic of the complete bipartite graph K m , n is given by χ ( K m , n ) = 2[( m + n − mn / 2) / 2] , (1) where [ a ] denotes the greatest integer less than or equal to the real number a. anderson.oliveira@unifal-mg.edu.br/ palazzo@dt.fee.unicamp.br LAWCI 2018
Introduction Fundamental Concepts Results Bibliography Definitions and concepts related to graphs and surfaces When considering 2-cell embedding of complete bipartite graphs K m , n , the minimum and the maximum genus of the corresponding surfaces may be determined. These values are given by: ( i ) the minimum genus of an oriented compact surface is, [1]: g m ( K m , n ) = { ( m − 2) ( n − 2) / 4 } , for m , n ≥ 2 , (2) where { a } denotes the least integer greater than or equal to the real number a . ( ii ) the maximum genus of an oriented compact surface is, [8]: g M ( K m , n ) = [( m − 1) ( n − 1) / 2] , for m , n ≥ 1 , (3) where [ a ] denotes the greatest integer less than or equal to the real number a . anderson.oliveira@unifal-mg.edu.br/ palazzo@dt.fee.unicamp.br LAWCI 2018
Introduction Fundamental Concepts Results Bibliography Elements of hyperbolic geometry Definition [2][3] The transformations identified in PSL (2 , Z ) are classified into three types as to the value of the associated matrix trace module. Let T ( z ) = az + b cz + d , a , b , c , d ∈ Z , with ad − bc = 1. In this way, T is an elliptic transformation, if Tr ( T ) = | a + d | < 2, a parabolic transformation, if Tr ( T ) = | a + d | = 2, and a hyperbolic transformation, if Tr ( T ) = | a + d | > 2. anderson.oliveira@unifal-mg.edu.br/ palazzo@dt.fee.unicamp.br LAWCI 2018
Introduction Fundamental Concepts Results Bibliography Elements of hyperbolic geometry Proposition [2][3] The Mobius transformations are isometries, that is, a subgroup of the isometry group of the upper-half plane Isom ( H 2 ) . Definition [2][3] A regular tessellation of the hyperbolic plane is a partition consisting of polygons, all congruent, subject to the constraint of intercepting only at edges and vertices, so as to have the same number of polygons sharing a same vertex, independent of the vertex. Therefore, there are infinite regular tessellations in H 2 . anderson.oliveira@unifal-mg.edu.br/ palazzo@dt.fee.unicamp.br LAWCI 2018
Introduction Fundamental Concepts Results Bibliography Fuchsian differential equations Definition [3], [4] An equation of the type: ′ ( z )+ p n ( z ) y ( z ) = 0 , (4) y ( n ) ( z )+ p 1 ( z ) y ( n − 1) ( z )+ ... + p n − 1 ( z ) y is an equation of the Fuchsian type if every singular point in the extended complex plane is regular . anderson.oliveira@unifal-mg.edu.br/ palazzo@dt.fee.unicamp.br LAWCI 2018
Introduction Fundamental Concepts Results Bibliography Fuchsian differential equations For the second order equation case: ′′ ( z ) + p 1 ( z ) y ′ ( z ) + p 2 ( z ) y ( z ) = 0 , y (5) a singular point is said to be regular if the singularity in p 1 ( z ) is a simple pole and in p 2 ( z ) is at most one pole of order 2. A second-order ODE with n regular singular points is of the form: ′′ ( z ) + p ( z ) y ′ ( z ) + q ( z ) y ( z ) = 0 , y (6) anderson.oliveira@unifal-mg.edu.br/ palazzo@dt.fee.unicamp.br LAWCI 2018
Introduction Fundamental Concepts Results Bibliography Fuchsian differential equations with: A 1 A n p ( z ) = z − ǫ 1 + · · · + z − ǫ n + K 1 , B 1 C 1 B n C n q ( z ) = ( z − ǫ 1 ) 2 + z − ǫ 1 + · · · + ( z − ǫ n ) 2 + z − ǫ n + K 2 , where: A 1 + · · · + A n = 2 , C 1 + · · · + C n = 0 , ( B 1 + · · · + B n ) + ( ǫ 1 C 1 + · · · + ǫ n C n = 0 , (2 ǫ 1 B 1 + · · · + 2 ǫ n B n ) + ( ǫ 2 1 C 1 + · · · + ǫ 2 n C n ) = 0 . anderson.oliveira@unifal-mg.edu.br/ palazzo@dt.fee.unicamp.br LAWCI 2018
Introduction Fundamental Concepts Results Bibliography Results Since we are interested in 2-cell embedding of the complete bipartite graph K 2 , 8 , it follows that the minimum and the maximum genus of the corresponding surface are given by: anderson.oliveira@unifal-mg.edu.br/ palazzo@dt.fee.unicamp.br LAWCI 2018
Introduction Fundamental Concepts Results Bibliography Results ( i ) the minimum genus of an oriented compact surface is g min ( K 2 , 8 ) = { ( m − 2) ( n − 2) / 4 } = { 0 } = 0 (7) where { a } denotes the least integer greater than or equal to the real number a . ( ii ) the maximum genus of an oriented compact surface is g max ( K 2 , 8 ) = [( m − 1) ( n − 1) / 2] = [3 . 5] = 3 (8) where [ a ] denotes the greatest integer less than or equal to the real number a . anderson.oliveira@unifal-mg.edu.br/ palazzo@dt.fee.unicamp.br LAWCI 2018
Introduction Fundamental Concepts Results Bibliography Results Thus, g min = 0 and g max = 3 implying 0 ≤ g ≤ 3. We consider only the case where g = 2 due to its specificity. The corresponding planar algebraic curve is y 2 = z 5 − 1. Note that these five singularities may be viewed as the vertices of a regular pentagon as shown in Fig. 2. anderson.oliveira@unifal-mg.edu.br/ palazzo@dt.fee.unicamp.br LAWCI 2018
Introduction Fundamental Concepts Results Bibliography Results Figure: Singularities of y 2 = z 5 − 1 anderson.oliveira@unifal-mg.edu.br/ palazzo@dt.fee.unicamp.br LAWCI 2018
Introduction Fundamental Concepts Results Bibliography Results Note that there is an one-to-one correspondence between the set of solutions of z 5 − 1 and the values − 2, − 1, 0, 1 and 2, [7]. From this fact, the corresponding second order Fuchsian differential equation is given by � � 2 �� ( z 5 − 5 z 3 + 4 z ) · ( z 5 − 5 z 3 +4 z ) y ′′ + y ′ +[( z 5 − 5 z 3 +4 z ) · k 2 ] y = 0 , z + 1 + k 1 (9) k 1 , k 2 ∈ C . anderson.oliveira@unifal-mg.edu.br/ palazzo@dt.fee.unicamp.br LAWCI 2018
Introduction Fundamental Concepts Results Bibliography Results The linearly independent solutions of (9) result in elliptic transformations of the form: S i ( t ) = a i t + b i with | a i + d i | = 0 , for each 1 ≤ i ≤ 5 . (10) c i t + d i , anderson.oliveira@unifal-mg.edu.br/ palazzo@dt.fee.unicamp.br LAWCI 2018
Introduction Fundamental Concepts Results Bibliography Results We call attention to the fact that these elliptic transformations are the generators of the Fuchsian group associated with the second order Fuchsian differential equation, (9). Note that the Euler characteristic of the pentagon as shown in Figure 2 is given by: χ ( S ) = V − E + F = 2 , χ ( S ) = 2 − 2 g , g = 0 . Therefore, the surface is a sphere. anderson.oliveira@unifal-mg.edu.br/ palazzo@dt.fee.unicamp.br LAWCI 2018
Recommend
More recommend
