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Fuchsian group generators associated with the C2,8 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

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Motivation

Figure: Research Problem

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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

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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.

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Definitions and concepts related to graphs and surfaces

Definition [4] A graph G ′ is called an embedding in a surface Ω when no two

• f 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.

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Definitions and concepts related to graphs and surfaces

A complete bipartite graph with m and n vertices, denoted by Km,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.

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Definitions and concepts related to graphs and surfaces

Theorem [5] For m, n ≥ 2, the E¨ uler characteristic of the complete bipartite graph Km,n is given by χ (Km,n) = 2[(m + n − mn/2) /2], (1) where [a] denotes the greatest integer less than or equal to the real number a.

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Definitions and concepts related to graphs and surfaces

When considering 2-cell embedding of complete bipartite graphs Km,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]: gm (Km,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]: gM (Km,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.

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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.

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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(H2). 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 H2.

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Fuchsian differential equations

Definition [3], [4] An equation of the type: y(n)(z)+p1(z)y(n−1)(z)+...+pn−1(z)y

′(z)+pn(z)y(z) = 0, (4)

is an equation of the Fuchsian type if every singular point in the extended complex plane is regular.

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Fuchsian differential equations

For the second order equation case: y

′′(z) + p1(z)y ′(z) + p2(z)y(z) = 0,

(5) a singular point is said to be regular if the singularity in p1(z) is a simple pole and in p2(z) is at most one pole of order 2. A second-order ODE with n regular singular points is of the form: y

′′(z) + p(z)y ′(z) + q(z)y(z) = 0,

(6)

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Fuchsian differential equations

with: p(z) = A1 z − ǫ1 + · · · + An z − ǫn + K1, q(z) = B1 (z − ǫ1)2 + C1 z − ǫ1 + · · · + Bn (z − ǫn)2 + Cn z − ǫn + K2, where: A1 + · · · + An = 2, C1 + · · · + Cn = 0, (B1 + · · · + Bn) + (ǫ1C1 + · · · + ǫnCn = 0, (2ǫ1B1 + · · · + 2ǫnBn) + (ǫ2

1C1 + · · · + ǫ2 nCn)

= 0.

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Results

Since we are interested in 2-cell embedding of the complete bipartite graph K2,8, it follows that the minimum and the maximum genus of the corresponding surface are given by:

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Results

(i) the minimum genus of an oriented compact surface is gmin (K2,8) = {(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 gmax (K2,8) = [(m − 1) (n − 1) /2] = [3.5] = 3 (8) where [a] denotes the greatest integer less than or equal to the real number a.

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Results

Thus, gmin = 0 and gmax = 3 implying 0 ≤ g ≤ 3. We consider

• nly the case where g = 2 due to its specificity. The corresponding

planar algebraic curve is y2 = z5 − 1. Note that these five singularities may be viewed as the vertices of a regular pentagon as shown in Fig. 2.

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Results

Figure: Singularities of y 2 = z5 − 1

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Results

Note that there is an one-to-one correspondence between the set

• f solutions of z5 − 1 and the values −2, −1, 0, 1 and 2, [7]. From

this fact, the corresponding second order Fuchsian differential equation is given by

(z5−5z3+4z)y ′′+

• (z5 − 5z3 + 4z) ·
• 2

z + 1 + k1

• y ′+[(z5−5z3+4z)·k2]y = 0,

(9) k1, k2 ∈ C.

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Results

The linearly independent solutions of (9) result in elliptic transformations of the form:

Si(t) = ait + bi cit + di , with |ai + di| = 0, for each 1 ≤ i ≤ 5. (10)

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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 − 2g, g = 0. Therefore, the surface is a sphere.

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Results

In order to find the subgroup associated with the fundamental region which will uniformize the corresponding algebraic curve, we have to fix one of these transformations and multiplying it by the remaining ones, results in four sided pairing hyperbolic transformations (bitorus) as shown in Figs. 2 and 3, where ei for each 1 ≤ i ≤ 5 represent the singularities and e′

i for each

1 ≤ i ≤ 5 represent the midpoints of the polygon sides.

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Results

Figure: Mobius Transformations

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Results

S1S2 = 1, 414214t − (0, 9511 + 0, 3090i) (0, 9511 − 0, 3090i)t − 1, 414214 · 1, 414214t − i −it − 1, 414214 S1S2 = 1, 6909831 + 0, 9510565i 1, 3449971 − 0, 9771976i 1, 3449971 + 0, 9771976i 1, 6909831 − 0, 9510565i

• .

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S1S3 = 1, 414214t − (0, 9511 + 0, 3090i) (0, 9511 − 0, 3090i)t − 1, 414214 · 1, 414214t − (−0, 9511 + 0, 3090i) (−0, 9511 − 0, 3090i)t − 1, 414214 S1S3 = 2, 8090171 + 0, 5877853i 2, 6899941 − 1, 665D − 16i 2, 6899941 + 1, 665D − 16i 2, 8090171 − 0, 5877853i

• .

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S1S4 = 1, 414214t − (0, 9511 + 0, 3090i) (0, 9511 − 0, 3090i)t − 1, 414214 · 1, 414214t − (−0, 5878 − 0, 8091i) (−0, 5878 + 0, 8091i)t − 1, 414214 S1S4 = 2, 8090171 − 0, 5877853i 2, 176251 + 1, 5811389i 2, 176251 − 1, 5811389i 2, 8090171 + 0, 5877853i

• .

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S1S5 = 1, 414214t − (0, 9511 + 0, 3090i) (0, 9511 − 0, 3090i)t − 1, 414214 · 1, 414214t − (0, 5878 − 0, 8091i) (0, 5878 + 0, 8091i)t − 1, 414214 S1S5 = 1, 6909831 − 0, 9510565i 0, 5137432 + 1, 5811389i 0, 5137432 − 1, 5811389i 1, 6909831 + 0, 9510565i

• .

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Results

The trace of each one of the previous transformations is Tr(S1S2) = 3, 3819662, Tr(S1S3) = 5, 6180342, Tr(S1S4) = 5, 6180342 and Tr(S1S5) = 3, 3819662. Note that every trace is greater than 2 (hyperbolic transformations), thus identifying the maximum genus surface associated with the eight-sided polygon.

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Results

These four transformations are the generators of the fundamental region of the bitorus, an eight sided regular hyperbolic polygon.

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Bibliography

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Massey, J.L. “Coding and modulation in digital communication,” 1974 Zurich Seminar, pp.E2(1)-E2(4), 1974. Vaz, J., Methods of Applied Mathematics I, Class Notes, IMECC, UNICAMP, 2012. White, A.T., “Orientable embeddings of Cayley graphs,” Bulletin of the American Mathematical Society,

• vol. 69, pp.272-275.

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• nig, D., Theirue der Endlichen und Unendlichen Graphen, Leipzig, 1936, reprinted, Chelsea, New York,

1950. White, A.T., Graphs, Groups and Surfaces, North-Holland Mathematics Studies 8, North-Holland Publishing Co., 1973. Firby, P.A. and Gardiner, C.F., Surface Topology, Ellis Horwood Limited, England, 1991. Ringeisen, R.D., “Determining all compact orientable 2-manifolds upon which Km,n has 2-cell embeddings,” Journal Combinatorial Theory, vol. 12, pp.101-104, 1972. anderson.oliveira@unifal-mg.edu.br/ palazzo@dt.fee.unicamp.br LAWCI 2018

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Bibliography

Ringel, G., “Das Geschlecht des Vollst¨ andigen paaren Graphen,” Abh. Math. Sem. Univ. Hamburg, vol. 28, pp.139-150, 1965. Katok, S., Fuchsian Groups, The University of Chicago Press, 1992. Beardon, A., The Geometry of Discret Groups, Springer-Verlag, New York, 1983. Kristensson, G., Second Order Differential Equations - Special Functions and Their Classification, Springer, New York, 2010. Whittaker E. T., “On Hyperlemniscate functions, a family of automorphic functions”, em Records of Proceedings at Meetings, pp 274-278, 1929. Sotomayor, J., Lessons on Ordinary Differential Equations, Rio de Janeiro: IMPA, 1979. Oliveira, A. J.; Palazzo Jr., R., ”Geometric and algebraic structures associated with the channel quantization problem,” Computational and Applied Mathematics, 2017. anderson.oliveira@unifal-mg.edu.br/ palazzo@dt.fee.unicamp.br LAWCI 2018

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Thank you!!

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