Citation

Material Information

Title:
Drawing "good" graph diagrams
Creator:
Kume, Noriyuki
Place of Publication:
Denver, CO
Publisher:
Publication Date:
Language:
English
Physical Description:
iv, 64 leaves : illustrations ; 29 cm

Thesis/Dissertation Information

Degree:
Master's ( Master of Science)
Degree Grantor:
Degree Divisions:
Department of Mathematical and Statistical Sciences, CU Denver
Degree Disciplines:
Applied Mathematics
Committee Chair:
Fisher, David C.
Committee Members:
Lodwick, Weldon
Billups, Stephen

Subjects

Subjects / Keywords:
Geometrical drawing -- Computer simulation ( lcsh )
Graphic methods ( lcsh )
Genre:
bibliography ( marcgt )
theses ( marcgt )
non-fiction ( marcgt )

Notes

Bibliography:
Includes bibliographical references (leaf 64).
Thesis:
Submitted in partial fulfillment of the requirements for the degree, Master of Science, Applied Mathematics
General Note:
Department of Mathematical and Statistical Sciences
Statement of Responsibility:
by Noriyuki Kume.

Record Information

Source Institution:
Holding Location:
|Auraria Library
Rights Management:
All applicable rights reserved by the source institution and holding location.
Resource Identifier:
37887131 ( OCLC )
ocm37887131
Classification:
LD1190.L622 1997m .K86 ( lcc )

Full Text
DRAWING GOOD GRAPH DIAGRAMS
by
Noriyuki Kume
B.S., University of Colorado at Denver, 1993
A thesis submitted to the
in partial fulfillment
of the requirements for the degree of
Master of Science
Applied Mathematics
1997

This thesis for the Master of Science
degree by
Noriyuki Kume
has been approved
by
David C. Fisher
Weldon Lodwick
Date
to

Kume, Noriyuki (M.S., Applied Mathematics)
DRAWING GOOD GRAPH DIAGRAMS
Thesis directed by Associate Professor David C. Fisher
ABSTRACT
Producing graph diagrams is a non-trivial problem. We propose a
method to automate this task. Criteria for defining and evaluating graph
diagrams will be presented, as well as a numerical algorithm for producing
diagrams according to these criteria. Results from a computer program im-
plementing this method will be shown.
This abstract accurately represents the content of the candidates
thesis. I recommend its publicatior
Signed I
David C. Fisher
m

CONTENTS
Chapter
1. Introduction......................................... 1
1.1 Motivation........................................... 1
1.2 What is a Good Graph?.............................. 2
1.3 Energy Function...................................... 3
2. Observation of Force................................. 5
2.1 Connected Components................................. 5
2.2 Configuration........................................ 5
2.3 Interaction between nodes............................ 7
2.4 Various Energy Configuration......................... 9
2.5 An Exception........................................ 14
3 An Automation with Newtons Method.................. 16
3.1 Algorithm........................................... 16
3.2 Animation of Algorithm.............................. 18
4 An Application...................................... 22
5 Rotation............................................ 28
5.1 Automatic Algorithm................................. 28
5.2 Rotation and Slope.................................. 33
Appendix
A: Graph Diagrams............................... 36
B: Animation of Algorithm....................... 38
C: An Implementation in Pascal.................. 41
References ............................................. 64
IV

Dedication
I would like to dedicate this thesis to my wife Jenny and my parents
Katsuo and Eiko Kume for their love and unconditional support.

Acknowledgement
In preparation and completion of my thesis, I must thank Dr. David
C. Fisher for his inspiration and my thesis committee for their support. Es-
pecially, Dr. Stephen Billups and Dr. David C. Fisher for giving me valuable
advise at the final edit of this thesis. I also would like to thank the graduate
school for giving me an opportunity to study and learn. There are so many
friends who supported me throughout the project. However if I were to list
and thank everybody, eventually the acknowledgment section would become
longer than the thesis itself. But, I cannot end the section without extending
my special thanks to Dr. Zenas R. Hartvigson and Leanne D. Holder for their
understanding and assistance. Finally, my special thanks to Michael H. Kawai
for helping me throughout the project. Mike, thank you very much for helping
me with the preparation and validation of the presentation materials for my
oral defense, with contribution of interesting test cases for the algorithm, with
the LaTex, and with making me realize what I was trying to do on the 11th
hour.

1. INTRODUCTION
1.1 MOTIVATION
The study of graph theory naturally involves the visual representation
of graphs. Unfortunately, the use of computers in processing graphs has been
confined primarily to the mathematical rather than the visual aspects of a
graph. For example, computer algorithms exist that find chromatic numbers,
cliques, and minimal/maximal dominations for a given graph. In contrast,
very little has been done to automatically produce good diagrams of graphs
from their adjacency matrices. In this thesis we will focus on this problem by
proposing an algorithm to produce good drawings of graphs.
Definition 1.1.1 Graphs
A graph G may be defined as a pair (v, e), where v is a set of nodes (vertices)
and e is a set of edges (unordered pairs of elements from v). We denote this
with the symbol G (v, e).
Definition 1.1.2 Graph diagrams
A diagram of a graph maps the nodes to points in the plane and each edge to
the line segments between its end points.
A graph is not a graph diagram. Though we sometimes use the
term graph when we really mean graph diagram, we hope the actual
1

meaning is clear from context. Strictly speaking, a graph diagram is a 2-
dimensional representation of the graph. Thus, we will see that each graph can
be represented by a number of graph diagrams that are only drawn differently.
1.2 What is a Good Graph?
As Supreme Court Justice Felix Frankfurter said of the challenge of
defining pornography, I may not be able to describe it, but I know it when I
see it. Defining goodness is more of a philosophical question than something
we can describe mathematically. For an example, let us draw a 5-cycle in two
different ways.
Fig. 1.2.1 Pentagonal form of C5
Fig. 1.2.2 Another C5
We often associate certain canonical diagrams with a particular graph.
This does not imply that everyone will consider the same canonical form as
the best diagram. In the example above, we might consider the pentagon
diagram simpler than the other representation of the 5-cycle shown in Fig.
1.2.1., because none of the edges are too close to each other. But, someone
else may decide Fig. 1.2.2 is more aesthetically pleasing.
Nevertheless, when we draw a graph diagram, we want each edge
to be distinguishable, and in general, as short as possible. Additionally, we
2

want the nodes to be sufficiently separated. In order to achieve these goals we
attempt to minimize the energy function described below.
1.3 Energy Function
Whether a pair of nodes is connected by an edge or not, we do not
want the nodes to be too close together. We must devise a node property
which will encourage them to separate themselves from one another. We can
model the situation by thinking of Coulombs Law. We model the nodes as
being particles with a like charge on the 2 dimensional Cartesian coordinate
plane, and thus we have mutual repulsion between any pair of nodes.
r
Axiom 1.3.1 Energy due to repulsing force for a pair of nodes
The energy due to the repulsing force between a pair of nodes located at Zi =
(xi, Vi) and Zj = {xj,yf) is where d (zh zj) = xj)2 + (y* yf)2
is the distance between the pair of nodes.
However, unless we introduce an attracting force, the nodes would
tend to drift as far away from each other as possible, making the distance
between any two of them equal to +oo as energy approaches 0. In general, we
do not care if non-adjacent nodes are far apart. So, we devise a node property
which will keep adjacent nodes from drifting too far apart. We can model
the situation by thinking of certain pairs as being connected by a spring.
3

Axiom 1.3.2 Energy due to the attracting force between a pair of
The energy due to the attracting force between a pair of adjacent nodes located
at Zi and Zj is d (Zi} Zj).
This node property prevents edges from becoming too long. We use
the attraction property of our model selectively. We combine the two energies
and obtain the total energy for the pairs.
Axiom 1.3.3 Energy function
Energy of a graph diagram at a position of nodes z = (Zj, Z2, Z3, ...Zn) can
be defined as E{z) = i(z^z ) + 12 d {Z%, Zf).
4

2. OBSERVATION OF FORCE
So far we claim that by minimizing the energy exerted by the repuls-
ing forces and the attracting forces, we should be able to place the nodes in a
good configuration. Let us observe how this works by the energy function
in action.
2.1 Connected Components
For non-adjacent pairs there is no direct force of attraction. Thus,
we represent the energy of these pairs of nodes with Axiom 1.3.1. Therefore,
isolated nodes or disconnected components will diverge from other connected
components. Hence, our model will not handle more than one connected com-
ponent at a time. For example, suppose that we have a graph consisting
only of two non-adjacent nodes (i.e., two isolated nodes or two disconnected
components). Only the repulsing force exists between them, and the distance
which minimizes the system energy is +oo. Therefore, disconnected compo-
nents must be handled separately because our model will only accommodate
connected components.
2.2 Configuration
We demonstrate our model with a 4-cycle. Our algorithm will pro-
duce the canonical diagram on the left in Fig. 2.2.1.
5

Fig. 2.2.1 Two possible diagrams for C4
If we fix the position of nodes a, c, and d, and send node b outward
as in the diagram on the right in Fig. 2.2.1, the energy level increases. Fig.
2.2.2 shows node interaction in terms of the energy function. Previously, we
said that we wanted non-adjacent nodes to be sufficiently far apart from one
another.
E(x) = x + 5
Fig. 2.2.2 Node interactions in C4
On the right in Fig. 2.2.1, node b is now too far apart from node a
6

and c. Since node b is connected to both node a and c, these two nodes will
try to attract node b closer. Hence, node b will be positioned at the coordinate
where it is relatively close to a and c, but because of repulsive forces, will not
too close.
2.3 Interaction Between Nodes
Let us analyze two more graph diagrams to show how adjacent and
non-adjacent nodes interact with each other. Both graphs illustrate the fol-
lowing principle.
(1) Non-adjacent nodes repulse each other
Here is the adjacency matrix for a triangle with an extra adjacent
node:
0 110
10 10
110 1
0 0 10
7

Energy is : 8.94201
Fig. 2.3.1 Triangle with an extra adjacent node
The edges of the triangle are not equal because of the interaction
between the topmost nodes and the bottom two nodes.
Here is the adjacency matrix for a 4-cycle with an extra adjacent
node:
0 10 10
10 10 0
0 10 10
10 10 1
0 0 0 1 0
8

Energy is : 12.54175
Fig. 2.3.2 4-cycle with an extra adjacent node
The four cycle is not square because of the interaction between the
topmost node and the non-adjacent nodes.
2.4 Various Energy Configurations
The question still remains why our algorithm chooses a square to
represent the 4-cycle. Since we are minimizing the energy function, we know
that the model will favor edges with length near 1.
Consider the base case for a pair of adjacent nodes. Given the adja-
cency matrix,
0 1
1 0
we can calculate energy.
9

Let the distance between nodes Z\ and Z^ be x. The energy function
is
E(x) = x + -.
x
We find the optimal distance between the nodes by differentiating the function
and setting it equal to zero.
dE(x)
dx
xz
xt
x =
x -
i-4
xz
-1
1
1
1.
= 0
Since we will not allow negative distances, we choose x = 1 which
implies that E( 1) = 2 is minimal.
In the case of a 4-cycle, the algorithm will place the nodes in a
square configuration and will choose the outer rim node pairs as edges over
the diagonal node pairs. (By the Pythagorean theorem, the diagonal edges
are longer than the rim edges.)
Energy is : 9.30739 Energy is : 9.97676
I
Fig. 2.4.1 Two possible configurations for C4
10

The canonical (square) form consists of:
4 edges of distance x
2 non-adjacent edges of length x\/2.
The energy function is:
Â£!(*)= 4(* + |)+2(^).
Differentiation yields = q
x = yj2^ + 1 => E + l) = = 9.30738, which agrees with the
output produced by our algorithm.
Our algorithm approximated the energy of the configuration depicted
on the right in Fig. 2.4.1 to 9.97676 unit energy. Thus, the configuration
depicted on the left in Fig. 2.4.1 has a lower energy.
We also analyze the complete graph on five vertices, K5.
Canonical (Pentagonal) K5:
5 edges of distance x
5 (4x2 + 2x2 y/5 + 3 + V5)
(l + y/5) x
By setting E' (x) = 0, we get x = y/^2 1 ^ -786151. Then the value of E is
approximately 20.5817.
11

Energy is : 20.67962
4 edges of distance x
4 edges of distance y/2x
2 edges of distance 2x
The energy function is:
B(x) = 4(z + i)+4(v5x+-i-)+2(2z + -L)
8a:2 + 4y^a?2 + 5 + 2^2
x
By setting E' (a:) = 0, we get x = \]\~ ^ = .757115. The value of E is
approximately 20.6796.
4 edges of distance x
3 edges of distance 2x
2 edges of distance 3a;
1 edge of distance 4x
12

The energy function is:
E(x)
4 ix+*)+3 (2x+h)+213*+s)+ (4x+s)
240a;2 + 77
12x '
By setting E' (x) = 0, we get x = ^q55 The value of E is approximately
22.6569.
Thus, the canonical K5 has the lowest energy of the three observed
configurations. The program produces the following output which agrees with
the previous analysis.
x-coordinate y-coordinate
-0.638 0.201
-0.006 0.669
-0.388 -0.544
0.634 0.212
0.398 -0.538
Table 2.4.1 Program output for K5
Fig. 2.4.2 Best K5 diagram
13

2.5 An Exception
The model does not handle all cases satisfactorily. For example,
0 1 0 11 0 1 1
1 0 1 0 0 0 1
0 1 0 1 0 0 1
1 0 1 0 1 0 1
0 0 0 1 0 1 1
1 0 0 0 1 0 1
1 1 1 1 1 1 0
The model produces Fig. 2.5.1.
Fig. 2.5.1 An exception
There is actually an edge which connects node a and node d. Since
there is nothing in the model to discourage edges from coinciding, the edge
14

connecting a to d overlaps the edges which connect d to g, and g to a.
If we draw a diagram of this graph manually, we can avoid the over-
lapping edges, but our graph would contain a curved edge (connecting a and
d) which may not be considered aesthetically pleasing.
This exception illustrates the fact that there are many other features
of good graph diagrams which we elect not to model.
15

3. AN AUTOMATION WITH NEWTONS
METHOD
To automate the process of finding the minimum energy for a system
of charged particles, we use Newtons Method. We can make a few modifica-
tions to the algorithm from Burden and Faires[2] to accommodate the model.
3.1 Algorithm
Input: We input the number of nodes (n) and the adjacency matrix from a
data file.
Output: We output an approximate solution z = {(xi,Ui) (X2, IJ2), , {xn,Vn)}
which represents the position of the nodes in a good graph diagram.
Step 1: Read and construct the adjacency matrix from a data file.
Step 2: Repeat Step 3 through Step 12 for a pre-determined number of iter-
ations, N.
Step 3: Generate the initial coordinates of the nodes by using a standard
random number generator.
Step 4: Set k = 1.
While (k < N) and (tol < TOL) do Step 5 through Step 12, where TOL is the
constant threshold of the difference in norm of gradient G(z) defined at the
next step.
Step 5: Construct G(z) and H(z), where G(z) = VÂ£/(z) and H(z) = V2E(z)
16

Step 6: Attempt to solve the nxn linear system H(z) m = G(z) by direct
method.
Step 7: IÂ£ the system is inconsistent, approximate the solution by using Gauss-
Seidel or some other iterative method.
Step 8: Calculate the energy level according to Axiom 1.3.3, the energy func-
tion. In an attempt to minimize energy, it is important that the energy level
decreases at each iteration.
Step 9a: If the energy level did not decrease from the previous iteration, then
cut the step size m; Set z' = z + for l = 1 ..d, until E(z') -< E(z),where
l and d are integers and z' indicates the current position of the nodes. This
step calculates the current energy level until either there is an improvement or
the pre-determined number of iterations d is reached. If we can not find lower
energy level within the pre-determined number of iterations d, then we move
to step 9b and take a full step anyway
Step 9b: Else if the energy level did decrease from the previous iteration or
we could not find lower energy state by cutting the step size m. Then, set
z' = z + m, where z' is the position of the nodes at the current iteration or a
solution calculated at step 7 and m is a full step.
Step 10: Calculate centroids, then make an adjustment: Subtract an average
of the x coordinates and y coordinates from the (x, y) coordinates of each node
respectively, thereby moving the centroid of each graph diagram to the origin.
Step 11: Set k = k + 1.
Step 12: For our implementation, we store the coordinates in an array to
produce a graph in Latex format. However, the output can be used to produce
17

graph diagrams in any format.
3.2 Animation of the algorithm
We can show how our algorithm works by printing the graph configu-
ration at each iteration of Newtons Method. (We posted a few more of these
animations in the Appendix.) For example, consider the following adjacency
matrix:
0 0 0 1
0 0 0 1
0 0 0 1
1110
Our initial (random) guess looks like this:
Energy is : 11.79678
We can follow the diagrams development.
18

Energy is : 9.07786
Energy is : 8.38551
19

Energy is : 7.58100
Energy is : 7.53873
20

Energy is : 7.53556
21

4. AN APPLICATION
As an application of this research, we chose a graph theory paper in
which drawing the graph diagrams was left to the reader. The paper by Edwin
W. Clark and Larry A. Dunning is Sharp upper bounds for the domination
numbers of graphs with given order and minimum degree [3]. The authors
used the following notation to denote a graph:
G (# of nodes, Minimum degree for each node, Domination)
We display our output diagrams for the various graphs described in that
paper.
22

G (9,4,3) =
0 110
10 10
110 1
0 0 10
0 10 1
10 0 1
0 0 10
0 10 0
10 0 1
0 10 0 1
10 0 10
0 0 10 0
110 0 1
0 10 10
10 10 0
0 10 11
10 10 1
0 0 110
Energy is : 50.33236
Fig. 4.1 Our diagram of G (9,4,3)
23

0 1 1 0 0 1 0 0 0 0
1 0 1 0 1 0 0 0 0 0
1 1 0 1 0 0 0 0 0 0
0 0 1 0 1 0 0 1 0 0
0 1 0 1 0 1 0 0 0 0
1 0 0 0 1 0 1 0 0 0
0 0 0 0 0 1 0 0 1 1
0 0 0 1 0 0 0 0 1 1
0 0 0 0 0 0 1 1 0 0
0 0 0 0 0 0 1 1 0 0
0 0 0 0 1 0 0 0 1 1
Energy is : 52.34533
Fig. 4.2 Our diagram of G(ll,3,4)
0
0
0
0
1
0
0
0
1
1
0
24

0(11,6,3)
0 1 1 0 1 0 0 1 0 1 1
1 0 1 1 0 1 0 1 0 0 1
1 1 0 1 1 0 1 0 1 0 0
0 1 1 0 1 1 0 0 1 1 0
1 0 1 1 0 1 1 0 0 1 0
0 1 0 1 1 0 1 1 0 0 1
0 0 1 0 1 1 0 1 1 0 1
1 1 0 0 0 1 1 0 1 1 0
0 0 1 1 0 0 1 1 0 1 1
1 0 0 1 1 0 0 1 1 0 1
1 1 0 0 0 1 1 0 1 1 0
Energy is : 86.98557
Fig. 4.3 Our diagram of G(ll, 6,3)
25

G(13,7,3)
0 1 1 1 0 0 1 1 1 0 0 0 1
1 0 1 1 1 0 1 0 0 0 1 0 1
1 1 0 1 1 1 0 0 0 0 1 1 0
1 1 1 0 1 1 0 0 1 1 0 0 0
0 1 1 1 0 1 1 1 0 1 0 0 0
0 0 1 1 1 0 1 1 0 0 0 1 1
1 1 0 0 1 1 0 1 1 0 0 1 1
1 0 0 0 1 1 1 0 1 1 1 0 0
1 0 0 1 0 0 1 1 0 1 1 1 0
0 0 0 1 1 0 0 1 1 0 1 1 1
0 1 1 0 0 0 0 1 1 1 0 1 1
0 0 1 0 0 1 1 0 1 1 1 0 1
1 1 0 0 0 1 1 0 0 1 1 1 0
26

Energy is : 122.58273
Fig. 4.4 Our diagram of G(13,7,3)
27

5. ROTATION
So far, we can produce graphs with good spacing with respect
to the nodes. Now, we would like to rotate the graph about its centroid
and find the orientation which is most aesthetically pleasing. We present an
automatic algorithm which gives aesthetic preference to graphs with certain
characteristics. As an alternative, the User can manually rotate the graph if
his aesthetic preferences do not match the ones in the automatic algorithm.
5.1 Automatic Algorithm
We formulate rules for the automatic algorithm by examining some
simple graphs. We start with K2, which is the smallest graph the algorithm
can handle. There are two canonical forms. Each is parallel to one of the
coordinate axes. We arbitrarily decided that the horizontal form (parallel
to the x-axis) would be preferred over the vertical form. (The fact that we
save paper (vertical space) using the horizontal form had some impact on our
decision.)
Energy is : 2.00000
--------------
Fig. 5.1.1 Preferred K2 horizontal form
28

Energy is : 2.00000
Fig. 5.1.2 K2 vertical form
As part of the automatic algorithm, we demand that at least one pair
of nodes (adjacent or non-adjacent) lie on an imaginary horizontal line. This
forces all the linear graphs (L3, L4, etc.) to be output as horizontal forms also.
Next, we consider a 3-cycle or equilateral triangle. We prefer to have
the apex of the triangle point upward. The base of the triangle is horizontal.
Energy is : 6.00000
Fig. 5.1.3 Preferred 3-cycle orientation
Next, we consider a 4 cycle or square. For a square graph, we would
like all of the edges to be parallel either to the x-axis or the y-axis.
29

Energy is : 9.30739
Fig. 5.1.4 Preferred 4-cycle orientation
The next basic graph we consider is a house graph, which is a
mixture of a 3 cycle and K4. We would like the house to stand up (the
pointy edge points upward and the walls and a floor are parallel to the y-axis
and the x-axis respectively).
Energy is : 17.18001
Fig. 5.1.5 Preferred orientation for the house
The table summarizes some of the preferred characteristics for our
best graphs.
30

Number of edges Number of edges
parallel to x-axis parallel to y-axis
l2 1 0
Triangle 1 0
Square 2 2
House 2 2
Table 5.1.1 Parallel edge table
Thus, we derived two rules in order to optimize the rotation of the
graph automatically.
Rule #1
We want the graph diagram to have the greatest possible number of node pairs
Recall that the original algorithm fails to give us a good diagram for
certain graph diagrams such as figure 2.5.1. The algorithm cannot deal with
overlapping edges. Although we cannot visualize the overlapping edges here,
(since the rotation algorithm only deals with node spacing), Rule #1 takes
the number of edges while rotating a graph diagram. However, if we are to
use only this criterion, the rotation algorithm could produce graphs that still
do not meet our requirements.
31

Energy is : 6.00000
Fig. 5.1.6 Upside-down 3-cycle
Fig. 5.1.7 Upside-down house
The diagrams in Fig. 5.1.6 and Fig. 5.1.7 indeed meet the criterion
of the first rule. Thus, we need to come up with a scheme in order to deal
with these cases.
Rule #2
We break ties from Rule by choosing the graph diagram which maximizes
the sum of Max(0,yi) over all nodes i with respect to its centroid.
In other words, if there is a tie from Rule #1, we will choose the
tallest graph.
32

5.2 Rotation and Slope
We describe how the rotation portion of the automatic algorithm
works.
Fig. 5.2.1 An unrotated diagram
Given a graph, we identify in Cartesian coordinates the location of
each node relative to the centroid. For this calculation, we assign the cen-
troid to the origin of our new coordinate system. For the sake of clarity, we
have internally transformed each Cartesian coordinate to its polar coordinate
counterpart. The value of 9 (measured in standard position) for each point is
denoted in the figure. In Fig. 5.2.1, we see that the value of 6 for node b is
59.
For each pair of nodes, the algorithm calculates the slope between
those nodes. For example, if node a has Cartesian coordinates (xj,yi) and
node b has Cartesian coordinates (x2,1/2) then the slope would be ^3^7- Thus,
33

if the two nodes lie on the same imaginary horizontal line, the slope will be
zero. If the two nodes lie on the same imaginary vertical line, the slope will
be a very large positive or negative number.
During each iteration, the algorithm rotates the nodes clockwise by
0.1. If any pair of nodes has a slope close to zero (within a constant thresh-
old), we calculate all the slopes between every possible pair of nodes. We score
this configuration by the total number of slopes which are either horizontal or
vertical. (We consider a slope to be vertical if its magnitude exceeds a large
constant threshold.) Thus, the two 3-cycles in Fig. 5.2.2 and Fig. 5.2.3 have
the highest score of or the same number of horizontal/vertical slopes.
a: 270 degrees b: 30 degrees c: 150 degrees
\ / a
Fig. 5.2.2 Upside-down 3-cycle
34

y
Fig. 5.2.3 Right-side up 3-cycle
The second rule chooses the diagram in Fig. 5.2.3 since the apex of
the 3-cycle, node c, has the largest y-axis value with respect to its centroid.
35

Appendix A: Graphs
Here are some selected results from our program.
Energy is : 12.72020 Energy is : 23.44287
Energy is : 27.14550
Fig. A-3 Nine cycle
Energy is : 43.89305
Fig. A-2 Eight cycle
Energy is : 31.12898
Energy is : 59.03932
Fig. A-6 K8
36

Energy is : 117.97083
Fig. A-7 K12
Fig. A-8 Petersons graph
We note that Fig.
A-8 is not the typical
diagram we see in most books.
37

Appendix B: Animation of the algorithm
Here is another animation of the algorithm. We chose a simple linear graph.
Energy is : 5.13249
Fig. A-9 Our initial (random) guess
Energy is : 4.52890
(
Fig. A-10 Iteration #1
Energy is : 4.47317
Fig. A-12 Iteration #3 Fig. A-13 Iteration #4
Energy is : 4.49156
Energy is : 4.47214
38

We animate our algorithm for the graph K3i3 or commonly known as
Gas-Water-Electricity Graph.
Energy is : 28.84328
Energy is : 24.27690
Energy is : 24.52922
Energy is : 23.74992
39

Energy is : 23.28304
Energy is : 23.08933
Energy is : 22.94107
Energy is : 23.23460
Energy is : 22.95004
Energy is : 22.94073
40

Appendix C: An implementation of algorithm in Pascal
PROGRAM Thesis(INPUT,OUTPUT);
CONST
{Defines the maximum size of matrices}
Max.Size = 30;
{This is always twice the size of Max.Size}
J_Size = 60;
{This number needs to be larger than C(Max.size,2)}
Number_of_edges = 450;
TYPE
Sizes = l..Max_Size;
Sizes2 = l..J_Size;
Sizes3 = 1..Number_of.edges;
Matrix = ARRAY[Sizes2,Sizes2] OF REAL;
Vector = Array[Sizes] OF REAL;
Vector2 = Array[Sizes2] OF REAL;
Vector3 = Array[Sizes3] OF REAL;
VAR
matrix.size : INTEGER;
jacobi : Matrix; {Jacobi matrix used for Newtons method}
b : Vector2; {Vector used for gauss sidel method}
xs : Vector; {These are x and y coordinates of nodes}
ys : Vector;
prevx, {Temporary variable used to store }
prevy : Vector; {coordinates}
slope : Vector3; {Slope used to rotate the graph}
r, {Used to store coordinates in polar}
theta : VECTOR;
slope.index : INTEGER;
{----------------------------------------------------------------}
PROCEDURE GAUSS.SIDEL(VAR x,xo : Vector2);
{
This is an implementation of Gauss-Sidel algorithm to
solve system of equation iteratively.
}
var
k,i,n,j,h,l,m :integer;
suml,sum2,
tol,toll,tol2 : REAL;
begin
for k: = 1 to matrix_size*2 do
begin
xo [k] : = 0;
end;
n := matrix_size*2;
suml := 0;{initialize sum}
sum2 := 0;{initialize sum}
k := 1;
tol := 10;
toll := 0;
tol2 := 0;
41

while((k<=500) and (tol>=0.0001) and (abs(tol)<100))do
begin
toll := 0;
tol2 := 0;
for i := 1 to n do
begin
for h := 1 to i-1 do
begin
suml := suml + j acobi[i,h]*x[h];
end;{k}
for j := i+1 to n do
begin
sum2 := sum2 + jacobi [i, j] *X0 [j] ;
end;{j}
x[i] := (-suml-simi2 + b[i] )/jacobi [i,i] ;
suml := 0;{re-initialize sum>
sum2 := 0;{re-initialize sum}
end;{i>
k := k + 1;
for i := 1 to n do begin toll := toll + x[i]; end;
for i := 1 to n do begin tol2 := tol2 + XD[ij; end;
tol := abs(toll tol2);
{to the next iteration}
for i := 1 to n do begin X0[i] := x[i]; end;
end;{while}
end;{Gausss_Sidel}
PROCEDURE Get_Matrix;
{
This method, upon called, will process adjacency matrix
that is in text file format.
}
VAR
file_name
f
i,
STRING;
TEXT;
j : INTEGER;
BEGIN
ASSIGN(f,file_name);
RESET(f);
FOR i := 1 TO matrix_size DO
BEGIN
FOR j := 1 TO matrix_size DO
BEGIN
END;
END;
CLOSE(f);
END;

42

{--------------------------------------------------------------->
PROCEDURE GUESSES;
{
This method, upon called, will obtain initial guess for
the Newton's method.
>
VAR
i : INTEGER;
BEGIN
RANDOMIZE;
FOR i := 1 TD matrix_size DO
BEGIN
xs[i] := SQRT(matrix_size)*((RANDOM(10000)/5000)-1);
ys[i] := SQRT(matrix_size)*((RANDOM(10000)/5000)-1);
END;
END; {Guesses}-
{---------------------------------------------------------------}
{ Equations used in the Newton's method BEGIN }
{--------------------------------------------------------------->
FUNCTION Dist(xi,xj,yi,yj:REAL):REAL;
BEGIN
Dist := sqrt(sqr(xi-xj) + sqr(yi-yj));
END;{Dist>
{--------------------------------------------------------------->
FUNCTION Dist2(xi,xj,yi,yj:REAL):REAL;
BEGIN
Dist2 := EXP(3*LN(Dist(xi,xj,yi,yj)));
END;{Dist2}
{--------------------------------------------------------------->
FUNCTION Dist3(xi,xj,yi,yj:REAL):REAL;
BEGIN
Dist3 := EXP(5*LN(Dist(xi,xj,yi,yj)));
END;{Dist3>
{---------------------------------L----------------------------->
FUNCTION _xixj(xi,xj,yi,yj:REAL):REAL;
BEGIN
_xixj := -Csqr(yi-yj))/dist2(xi,xj,yi,yj);
END;
{--------------------------------------------------------------->
FUNCTION _yiyj(xi,xj,yi,yj:REAL):REAL;
BEGIN
_yiyj := -(sqr(xi-xj))/dist2(xi,xj,yi,yj);
END;
{--------------------------------------------------------------->
FUNCTION _xiyj(xi,xj,yi,yj:REAL):REAL;
BEGIN
-Xiyj := ((xi-xj)*(yi_yj))/dist2(xi,xj ,yi,yj) ;
END;
{--------------------------------------------------------------->
FUNCTION _xjyi(xi,xj,yi,yj:REAL):REAL;
BEGIN
-xjyi := ((xi-xj)*(yi-yj))/dist2(xi,xj,yi,yj);
END;
{--------------------------------------------------------------->
FUNCTION _xixi(xi,xj,yi,yj:REAL):REAL;
43

BEGIN
_xixi := (sqr(yi-yj))/dist2(xi,xj,yi,yj);
END;
{---------------------------------------------------------------->
FUNCTION _xjx j(xi,xj,yi,yj:REAL):REAL;
BEGIN
xjxj := (sqr(yi-yj))/dist2(xi,xj,yi,yj);
END;
{---------------------------------------------------------------->
FUNCTION _yiyi(xi,xj,yi,yj:REAL):REAL;
BEGIN
_yiyi := (sqr(xi-xj))/dist2(xi,xj,yi,yj);
END;
{---------------------------------------------------------------->
FUNCTION _yjyj(xi,xj,yi,yj:REAL):REAL;
BEGIN
_yjyj := (sqr(xi-xj))/dist2(xi,xj,yi,yj);
END;
{---------------------------------------------------------------->
FUNCTION _xiyi(xi,xj,yi,yj:REAL):REAL;
BEGIN
_xiyi := ((xi-xj)*(yj-yi))/dist2(xi,xj,yi,yj);
END;
{---------------------------------------------------------------->
FUNCTION _xj yj(xi,xj,yi,yj:REAL):REAL;
BEGIN
_xjyj := ((xi-xj)*(yj-yi))/dist2(xi,xj,yi,yj);
END;
{---------------------------------------------------------------->
FUNCTION xixj(xi,xj,yi,yj : REAL):REAL;
BEGIN
xixj := -(2*sqr(xi-xj) sqr(yi-yj))/dist3(xi,xj,yi,yj);
END;{xixj}
{---------------------------------------------------------------->
FUNCTION yiyj(xi,xj,yi,yj : REAL):REAL;
BEGIN
yiyj :=(sqr(xi-xj) 2*sqr(yi-yj))/dist3(xi,xj,yi,yj);
END;{yiyj}
{---------------------------------------------------------------->
FUNCTION xiyj(xi.xj,yi,yj : REAL):REAL;
BEGIN
xiyj := (3*(yj-yi)*(xi-xj))/dist3(xi,xj,yi,yj);
END;{xiyj>
{---------------------------------------------------------------->
FUNCTION xjyi(xi,xj,yi,yj : REAL):REAL;
BEGIN
xjyi := (3*(yj-yi)*(xi-xj))/dist3(xi,xj,yi,yj);
END;{xjyi}
{---------------------------------------------------------------->
FUNCTION xixi(xi,xj,yi,yj : REAL):REAL;{main diag upper}
BEGIN {this one nededs +/- 2}
xixi: = (2*SG)R(xi-xj ) -SQR(yi-yj ) ) /dist3 (xi, xj yi, yj ) ;
END;{xixi}
44

>
{--------------------------------------------------------
FUNCTION xjxj (xi,xj ,yi,yj : REAL):REAL;{main diag upper}
BEGIN {this one needs +/- 2}
xjxj:=(2*SQR(xi-xj)-SQR(yi-yj))/dist3(xi,xj,yi,yj);
END;{xjxj}
{--------------------------------------------------------
FUNCTION yiyi(xi,xj,yi,yj : REAL):REAL;{main diag lower}
BEGIN {this one needs +/- 2}
yiyi:=-(SQR(xi-xj)-2*SQR(yi-yj))/dist3(xi,xj,yi,yj);
END;{yiyi}
{--------------------------------------------------------
FUNCTION yjyj Cxi,xj ,yi,yj : REAL)-.REAL;{main diag lower}
BEGIN {this one needs +/- 2}
yj yj:=-(SQR(xi-xj)-2*SQR(yi-yj))/dist3Cxi, xj yi, yj ) ;
END; {yjyj}
{--------------------------------------------------------
FUNCTION xiyi(xi,xj,yi,yj : REAL):REAL;{main diag lower}
BEGIN
xiyi:=(3*(xi-xj)*(yi-yj))/dist3(xi,xj,yi,yj);
END;{xiyi}
{--------------------------------------------------------
FUNCTION xjyj(xi,xj,yi,yj : REAL):REAL;{main diag lower}
BEGIN
xj yj: = (3*(xi-xj)*(yi-yj))/dist3(xi,xj,yi,yj);
END;{xjyj}
{--------------------------------------------------------
FUNCTION _xi(xi,xj,yi,yj : REAL):REAL;
BEGIN
_xi := (xi-xj)/dist(xi,xj,yi,yj);
END;
{--------------------------------------------------------
FUNCTION _xj(xi,xj,yi,yj : REAL):REAL;
BEGIN
xj := (xj-xi)/dist(xi,xj,yi,yj);
END;
{--------------------------------------------------------
FUNCTION _yi(xi,xj,yi,yj : REAL):REAL;
BEGIN
_yi := (yi-yj)/dist(xi,xj,yi,yj);
END;
{--------------------------------------------------------
FUNCTION _yj(xi,xj,yi,yj : REAL):REAL;
BEGIN
-yj := (yj-yi)/dist(xi,xj,yi,yj);
END;
{--------------------------------------------------------
FUNCTION x_i(xi,xj,yi,yj : REAL):REAL;
BEGIN
x_i := (xj-xi)/dist2(xi,xj,yi,yj);
END;
{--------------------------------------------------------
FUNCTION x_j(xi,xj,yi,yj : REAL):REAL;
BEGIN
}
}
}
}
}
}
}
}
}
}
45

x_j := (xi-xj)/dist2(xi,xj,yi,yj);
END;
{------------------------------------------------------
FUNCTION y_i(xi,xj,yi,yj : REAL):REAL;
BEGIN
y_i := Cyj-yi)/dist2(xi,xj,yi,yj);
END;
{------------------------------------------------------
FUNCTION y_j(xi,xj,yi,yj : REAL):REAL;
BEGIN
y_j := Cyi-yj)/dist2(xi,xj,yi,yj);
END;
{------------------------------------------------------
{ Equations used in the Newtons method END
{------------------------------------------------------
{------------------------------------------------------
{ Hessian matrix BEGIN
{------------------------------------------------------
PROCEDURE Upper_Left;
>
VAR
i,
j
temp
temp2
Processes upper left block of hessian matrix.
INTEGER;
REAL;
real;
BEGIN
FOR i : =
BEGIN
1 TO matrix_size DO
FOR j:= i+1 TO matrix_size DO
END
BEGIN
temp := xixj (xs [i] ,xs [j] ,ys [i] ,ys [j] ) ;
IF (Alii,j] = 1) THEN
BEGIN
temp2 := _xixj (xs [i] ,xs [j] ,ys [i] ,ys [j] ) ;
jacobi[i,j] := temp2 + temp;
END
ELSE
BEGIN
jacobi[i,j]
END;
:= {-2 }temp;
END;
END;{Upper_Left}
{-------------------------------------------------------
PROCEDURE Lower_Right;
C
Processes Lower right block of hessian matrix.
>
VAR
i,
j : INTEGER;
temp : REAL;
t emp2,t emp3 : REAL;
>
>
>
>
>
>
>
>
}
46

BEGIN
FOR i := 1 TO matrix_size DO
BEGIN
FOR j := i+1 TO matrix_size DO
BEGIN
temp := 0;
temp2 := 0;
temp := yiyj (xs [i] ,xs [j] ,ys [i] ,ys [j] ) ;
IF (A[i,j] = 1) THEN{if connected}
BEGIN
temp2 := _yiyj (xs [i] ,xs [j] ,ys [i] ,ys [j]);
temp3 := temp2 + temp;
j acobi[i+matrix_size,j +matrix_size]
:= temp3;
END
ELSE {elseif disconnected}
BEGIN
jacobi[i+matrix_size,j+matrix_size] := temp;
END;
END;{j}
END;{i}
END;{Lower_Right}
{------------------------------------------------------------
PROCEDURE Up_Up_Right;
Processes upper triangle of upper right block of hessian
matrix.
}
VAR
i,
j : INTEGER;
interm : REAL;
BEGIN
FOR i := 1 TO matrix_size DO
BEGIN
FOR j := i+1 TO matrix_size DO
BEGIN
interm := xiyj (xs [i] ,xs[j] ,ys[i] ,ys [j]);
IF (A[i,j] = 1) THEN
BEGIN
j acobi [i] [matrix_size+j]
:= _xiyj (xs [i] ,xs[j] ,ys[i] ,ys [j] )+interm;
END
ELSE
BEGIN
jacobi[i] [matrix_size+j]
END;
interm;
END;
END;
END;{Up_Up_Right}
{------------------------------------------------------
PROCEDURE Up_Low_Right;
-C
Processes lower triangle of upper right block of
hessianmatrix.
}
}
}
47

VAR
i,
j : INTEGER;
interm : REAL;
BEGIN
FDR i := 2 TO matrix_size DD
BEGIN
j := l;
WHILE (j < i) DD
BEGIN
interm := xjyi(xs [i] ,xs [j] ys [i] ,ys [j] ) ;
IF (A[i,j] = 1) THEN
BEGIN
jacobi[i] [j+matrix_size]
: = _xjyi(xs[i] ,xs[j] ,ys[i] ,ys[j]) + interm;
END
ELSE
BEGIN
jacobi[i] [j+matrix_size] := interm;
END;
j := j + 1;
END;{while j>
END;{for..do i}
END;{Up_Low_Right}
{-----------------------------------------------------------------:
PROCEDURE Up_Diag_Right;
{
Processes diagnal entries of upper right block of hessian
matrix.
>
VAR
i,
j.
k : INTEGER;
sum,
temp,
temp2 : REAL;
BEGIN
FDR i:= 1 TO matrix_size DO
BEGIN
sum := 0;
k := 1;
WHILE (k < i) DO
BEGIN
temp : = xjyj (xs [k] xs [i] ys [k] ys [i] ) ;
IF (A[k,i] = 1) THEN
BEGIN
temp2 := 0;
temp2 : = _xj yj (xs [k] xs [i] ys [k] ys [i] ) +temp;
sum := sum + temp2;
END
ELSE
BEGIN
sum := sum + temp;
END;
k : = k + 1
END;{While k
48

j := i+i;
WHILE (j <= matrix_size) DO
BEGIN
temp := xiyi(xs [i] ,xs [j] ,ys [i] ,ys [j] ) ;
IF (A[i,j] = 1) THEN
BEGIN
temp2 := 0;
temp2 := _xiyi(xs [i] ,xs [j] ,ys [i] ,ys [j] )+temp;
siim := sum + temp2;
END
ELSE
BEGIN
sum := sum + temp;
END;
j := j + l;
END;{WHILE j < matrix_size}
Jacobi[i,i+matrix_size] := sum;
temp := 0;
END;{i>
END; {Up_Diag_Right}
{----------------------------------------------------------------}
PROCEDURE Main_Diag_Upper;
{
Processes upper main diagonal entries of hessian matrix.
>
VAR
i,
j.
k : INTEGER;
interm,
sum,
temp : REAL;
BEGIN
FOR i:= 1 TO matrix_size DO
BEGIN
sum := 0;
k := 1;
WHILE (k < i) DO
BEGIN
temp : = xjxj (xs [k] xs [i] ys [k] ys [i] ) ;
IF (A[k,i] = 1) THEN
BEGIN
interm := _xjxj(xs[k],xs[i],ys[k],ys[i])+temp;
END
ELSE
BEGIN
interm := temp;
END;
sum := sum + interm;
k := k + 1;
END;{While k
j := i+1;
WHILE (j <= matrix_size) DO
BEGIN
temp := xixi(xs [i] ,xs [j] ,ys [i] ,ys [j] ) ;
IF (A[i,j] = 1) THEN
49

BEGIN
interm := _xixi(xs [i] ,xs [j] ,ys [i] ,ys [j] )+temp;
END
ELSE
BEGIN
interm := temp;
END;
sum := sum + interm;
j := j + 1;
END;{WHILE j < matrix_size}
Jacobi[i,i] := sum;
sum := 0;
END;{i}
END;{Main_Diag}
{---------------------------------------------------------------->
PROCEDURE Main_Diag_Lower;
{
Processes lower main diagonal entries of hessian matrix.
>
VAR
i,
k : INTEGER;
interm,
sum,
temp : REAL;
BEGIN
FOR i:= 1 TO matrix_size DO
BEGIN
sum :- 0;
temp := 0;
interm := 0;
k := 1;
WHILE (k < i) DO
BEGIN
temp : = yjyj (xs [k] xs [i] ys [k] ys [i] ) ;
IF CA[k,i] = 1) THEN
BEGIN
interm : = _yjyj (xs [k] xs [i] ys [k] ys [i] ) +temp;
END
ELSE
BEGIN
interm := {2 -]- temp;
END;
sum := sum + interm;
k := k + 1;
END;{While k
j := i+1;
WHILE (j <= matrix_size) DO
BEGIN
temp := yiyiCxs [i] ,xs [j] ,ys [i] ,ys [j]);
IF (A [i,j] = 1) THEN
BEGIN
interm := _yiyi(xs[i],xs[j],ys[i],ys[j])+temp;
END
ELSE
BEGIN
50

interm := {2 -} temp;
END;
sum := sum + interm;
j := j + l;
END;{WHILE j < matrix_size}
Jacobi [i+matrix_size,i+matrix_size] := sum;
END;{i}
END;{Main_Diag>
PROCEDURE Copy_Jacobi;
{
Copies upper triangle of hessian to lower triangle.
>
VAR
i
big_size : INTEGER;
BEGIN
big_size := 2*matrix_size;
FOR i := 1 TO big_size DO
BEGIN
FOR j := 1 TO big.size DO IF (i <> j) THEN
BEGIN
jacobi[j,i] := jacobi[i,j];
END;{j>
END;{i>
END;{Copy_Jacobi}
{----------------------------------------------------------------
PROCEDURE RHS_Upper;
{
Processes upper half of right hand side equation used in
the Newton's method.
>
VAR
i,
j.
k : INTEGER;
sum,
temp : REAL;
BEGIN
temp := 0;
FOR i := 1 TO matrix_size DO
BEGIN
k := 1;
sum := 0;
WHILE (k BEGIN
temp : = _xj (xs [k] ,xs [i] ys [k] ys [i] ) ;
IF (A[k, i] = 1) THEN
BEGIN
temp : = temp + x_j (xs [k] xs [i] ys [k] ys [i] ) ;
END
ELSE
BEGIN
temp := 0;
temp := {temp }x_j(xs[k],xs[i],ys[k],ys[i]);
}
51

END;
sum := sum + temp;
k := k + 1;
END;{k
j := i + 1;
WHILE Cj <= matrix.size) DD
BEGIN
temp := _xi(xs[i] ,xs[j] ,ys[i] ,ys[j]) ;
IF (A[i,j] = 1) THEN
BEGIN
temp := temp + x_i(xs [i] ,xs [j] ,ys [i] ,ys [j] ) ;
END
ELSE
BEGIN
temp := 0;
temp := {temp }x_i(xs [i] ,xs [j] ,ys [i] ,ys [j] ) ;
END;
sum := sum + temp;
j := j + 1;
END;
b[i] := -sum;
temp := 0;
END;{i}
END;{RHS>
PROCEDURE RHS_Lower;
{
Processes lower half of right hand side equation used in the
Newtons method.
>
VAR
i,
J.
k : INTEGER;
sum,
temp : REAL;
BEGIN
temp :=0;
FOR i := 1 TO matrix_size DO
BEGIN
k := 1;
sum := 0;
WHILE (k BEGIN
temp : = _yj (xs [k] xs [i], ys [k], ys [i])
IF (A[k, i] = 1) THEN
BEGIN
temp := temp + y_j(xs[k],xs[i]
END
ELSE
BEGIN
, ys [k] ,ys [i]) ;
temp := 0;
temp := {temp }y_j(xs[k],xs[i],ys[k],ys[i]);
END;
sum := sum + temp;
k := k + 1;
END;{k
52

j := i + 1;
WHILE (j <= matrix_size) DO
BEGIN
temp := _yi(xs [i] ,xs [j] ,ys [i] ,ys [j] ) ;
IF (A[i, j] = 1) THEN
BEGIN
temp := temp + y_i(xs [i] ,xs [j] ,ys [i] ,ys [j] ) ;
END
ELSE
BEGIN
temp := 0;
temp := {temp ->y_i(xs [i] ,xs [j] ,ys [i] ,ys [j] ) ;
END;
sum := sum + temp;
j := j + i;
END;
b[i+matrix_size] := -sum;
temp :=0;
END;{i>
END;{RHS>
{---------------------------------------------------------------->
PROCEDURE CRUNCH;
{
Processes entire hessian matrix
>
VAR
i, j : INTEGER;
BEGIN
FOR i := 1 TO 2*matrix_size DO
BEGIN
FOR j := 1 TO 2*matrix_size DO
BEGIN
jacobi[i,j] := 0;
END;
END;
Upper_Left;
Lower_Right;
Up_Up_Right;
Up_Low_Right;
Up_Diag_Right;
Main_D i ag_Upp er;
Main_Diag_Lower;
Copy_Jacobi;
RHS_Upper;
RHS_Lower;
END;
{----------------------------------------------------------------}
{ Hessian matrix BEGIN 3-
{---------------------------------------------------------------->
{---------------------------------------------------------------->
FUNCTION Energy(x_coord, y_coord : VECTOR):REAL;
{
This method, upon called, will calculate an enrgy value
of given state of given graph.
>
53

: INTEGER;
VAR
i,
j
interm,
temp,
sum : REAL;
BEGIN
sum := 0;
FDR i := 1 TO matrix.size DO
BEGIN
FOR j := i+1 TO matrix_size DO
BEGIN
interm := 0;
interm :=
dist (x_coord[i] ,x_coord[j] ,y_coord[i] ,y_coord[j]);
IF (A[i,j] = 1) THEN
BEGIN
temp := (interm + 1/interm);
END
ELSE
BEGIN
temp := 0;
temp := 1/interm;
END;
sum := sum + temp;
END;{j>
END;{i>
Energy : = sum;
END;{Energy}
PROCEDURE Direct.Method(var x : VECT0R2;VAR singular : BOOLEAN);
{
This is an implementation of solving system of linear
equation directly. (Gauss elimination)
}
VAR
i,

k,
n : INTEGER;
m,
interm : REAL;
BEGIN
singular := FALSE;
n := 2* matrix.size;
FOR i := 1 TO n 1
BEGIN
IF (Jacobi[i,i]
BEGIN
singular :
DO IF NOT(singular) THEN
= 0) THEN
= TRUE;
END;
FOR k := i+1 TO n DO IF NOT(singular) THEN
BEGIN
m := Jacobi[k,i]/Jacobi[i,i];
FOR j := i+1 TO n DO
BEGIN
54

Jacobi[k,j] := Jacobi[k,j] m*Jacobi[i,j];
END;{ j loop}
b [k] : = b [k] m*b [i] ;
END;{k}
END;{i>
{
step 7 If a(n,n) = 0 then OUTPUT (no unique solution) STOP
}
IF NOT(singular) THEN
BEGIN
IF Jacobi[n,n] = 0 THEN
BEGIN
singular := TRUE;
END
ELSE
BEGIN
x[n] := b[n]/Jacobi[n,n];
FOR i := n-1 DOWNTO 1 DO
BEGIN
interm := 0;
FOR j := i+1 TO n DO
BEGIN
interm := interm + Jacobi[i, j]*x[j] ;
END;
x[i] := (b[i] interm)/Jacobi[i,i] ;
END;
END;{else}
END;
END;
{----------------------------------------------------------------}
PROCEDURE Latex_Begin(VAR out_f : TEXT;file_name : STRING);
BEGIN
assign(out_f,file_name) ;
REWRITE(out_f);
WRITELN(out_f,\documentstyle[12pt]{article});
WRITELN(out_f,\begin{document});
END;
{----------------------------------------------------------------}
PROCEDURE Latex_End(VAR close.f : TEXT);
BEGIN
WRITELN(close_f,\end{document});
CLOSE(close_f);
END;
{----------------------------------------------------------------}
FUNCTION FInd_Max(abcd: Vector):REAL;
VAR
i : INTEGER;
larger,
compare : REAL;
BEGIN
larger := -10000;
FOR i := 1 TO matrix_size DO
BEGIN
compare := abcd[i] ;
IF (larger <= compare) THEN
55

BEGIN
larger := compare;
END
END;
Find_Max := larger;
END;
FUNCTION Find_Min(abcd : Vector):REAL;
VAR
i : INTEGER;
smaller,
compare : REAL;
BEGIN
smaller := 10000;
FOR i := 1 TO matrix_size DO
BEGIN
compare := abcd[i];
IF (smaller >= compare) THEN
BEGIN
smaller := compare;
END
END;
Find_Min := smaller;
END;
->
PROCEDURE LaTexCenergy
VAR
i.
j
1
xmax,
xmin,
ymax,
ymin
BEGIN
WRITELN(f,}energy is
WRITELN(f);
WRITELN(f,1\smallskip);
WRITELN(f, ,\^Initlength=lin,);
REAL; VAR f : TEXT);
INTEGER;
REAL;
REAL;
1,energy:10:5);
xmax := Find_Max(xs);
xmin := Find_Min(xs);
ymax := Find_Max(ys);
ymin := Find_Min(ys);
WRITELN(f, 'Xbegin-CpictureX > ,xmax-xmin:7:4, , ,ymax-ymin: 7:4,
)(,xmin:7:4,3, 3,ymin:7:4,'));
FOR i := 1 TO matrix_size DO
BEGIN
FOR j := i+1 TO matrix_size DO
BEGIN
WRITECf,XputC1,xs[i]:6:4,*, >,ys[i] :6:4,0 0;
WRITELN (f,{\circle*{. 155) ;
IF (A[i,j] = 1) THEN
BEGIN
1 := SQRT(dist (xs [i] ,xs [j] ,ys [i] ,ys [j] ));
WRITECf,,\multiput(,,xs[i]:6:4,J,,ys[i]:6:4);
56

WRITE(f,J)(J,(xs[j]-xs[i])/(100*1):6:4,J,);
WRITE(f,(ys[j]-ys[i])/(100*l):6:4,J){,100*1:6:0,>});
WRITELN(f, {\circle*{. 001 ) ;
END;
END;
END;
WRITELN(f,\end{picture});
WRITELN(f);
END;-CEnd LaTex}
{---------------------------------------------------------------->
{
THis function, upon called, will return quadrant #
according to the sign of (x,y).
>
BEGIN
IF (x>0) THEN
BEGIN
IF (y>0) THEN
BEGIN
END
ELSE
BEGIN
END;
END
ELSE
BEGIN
IF (y>0) THEN
BEGIN
END
ELSE
BEGIN
END;
END;
{---------------------------------------------------------------->
PROCEDURE ROTATION(degree : REAL);
{
This method, upon called, will rotate given graph
in polar coordinate by the degree specified by
the variable degree.
>
VAR
angle : REAL;
i : INTEGER;
BEGIN
{polar conversion start}
FOR i := 1 TO matrix_size DO
BEGIN
r[i] := SQRT(SQR(xs[i]) + SQR(ys[i]));
END;
FOR i := 1 TO matrix_size DO
57

BEGIN
angle := ArcTanCprevy[i]/prevx[i])*(180/pi);
1 : theta[i] := angle;
2 : theta [i] := 180 + angle;
3 : theta [i] := 180 + angle;
4 : theta[i] := 360 + angle;
END;
theta[i] := theta[i] + degree; {gives rotation}
END;
{polar conversion end}
END;
{------------------------------------------------------
PROCEDURE Polar_To_Cartesian;
{
This method, upon called, will convert
poloar coordinate into cartesian coordinate.
}
VAR
i : INTEGER;
BEGIN
FOR i := 1 TO matrix.size DO
BEGIN
prevx[i] := r[i]*cos((theta[i])*(pi/180)) ;
prevy [i] := r[i]*sin((theta[i])*(pi/180)) ;
END;
END;
{------------------------------------------------------
PROCEDURE Orig_to_Temp;
{
This method, upon called, will copy values
stored in the xs[] and ys[] into prevx [] and
prevy[] respectively.
INTEGER;
= 1 TO matrix.size DO
: = xs [i] ;
: = ys [i] ;
}
VAR
BEGIN
FOR i
BEGIN
prevx[i]
prevy[i]
END;
END;
{----------------------------------------------------------
PROCEDURE Temp_to_Orig;
{
This method, upon called, will copy values
stored in the prevx [] and prevy[] into xs[] and
ys [] respectively.
}
VAR
BEGIN
FOR
INTEGER;
i := 1 TO matrix.size DO
58

END;
C-
BEGIN
xs [i]
ys [i]
END;
:= prevx[i];
:= prevy[i];
: INTEGER;
: LONGINT;
:= 0;
:= 1 TO slope_index
FUNCTION Count_Score:INTEGER;
This method, upon called, will check number of
edges that are parallel to x and y-axis in the
given graph.
}
VAR
i,
a_score
a_slope
BEGIN
a_score
FOR i := 1 TO slope_index DO
BEGIN
a_slope := TRUNC(slope[i]*1000);
a_slope := ROUND(a_slope/1000);
CASE ABS(a_slope) OF
0..10 : a_score := a_score + 1;
80..100: a_score := a_score + 1;
else
END;
END;
Count_Score := a_score;
END;
{---------------------------------------------------------------
FUNCTION The_Heighest:REAL;
This method, upon called, will check the heighest
point of given graph on the cartesian coordinate.
>
VAR
i : INTEGER;
compare : REAL;
BEGIN
compare := prevy[1];
FOR i := 2 TO matrix_size DO
BEGIN
IF compare < prevy[i] THEN
BEGIN
compare := prevy[i];
END;
END;
The_Heighest := compare;
END;
{---------------------------------------------------------------
PROCEDURE Rotate;
This method, upon called, will rotate the given graph
automatically according to following criteria:
Here, a good graph has:
59

1. more edges that are parallel to x and y-axis
2. A good graph is the tallest graph on top of
above criteria #1.
>
VAR
i,
k,
check_me,
deg,
prev_score,
curr_score : INTEGER;
maxy,
miny,
numerator,
denominator,
prev_height,
curr_height : REAL;
BEGIN
prev_score := 0;{initialization}
prev_height := 0;
FOR deg := 1 TO 360 DO
BEGIN
Orig_to_Temp;
Rotation(deg);
polar_to_cartesian;
slope_index := 0;
FOR i := 1 TO matrix_size 1 DO
BEGIN
FOR j := i+1 TO matrix_size DO
BEGIN
slope_index := slope_index + 1;
numerator := prevy[j]-prevy[i];
IF ABS(numerator) < 0.005 THEN
BEGIN
numerator := 0;
END;{IF}
denominator := prevx[j]-prevx[i] ;
IF ABS(denominator) < 0.005 THEN
BEGIN
slope [slope_index] := 1000;
END{IF>
ELSE
BEGIN
slope[slope_index]enumerator/denominator;
END;{ELSE}
END;{j}
END;{i}
FOR check_me := 1 TO slope_index DO
BEGIN
IF (ABS(slope[check_me]) < 0.05) THEN
BEGIN
FOR k := 1 TO slope_index DO
BEGIN
slope[k] := arctan(slope[k])*(180/pi);
END;
60

curr_score := Count.Score;
curr.height : = Th.e_Heigh.est;
IF ((curr_score >= prev_score)
AND (curr_height >= prev_height )) THEN
BEGIN
prev.score := curr.score;
prev_height := curr.height;
Temp_to_Qrig;
END;{if max height}
END;{if slope = 0}
END;{check_me}
END;{degj-
END;{Rotate}
{-----------------------------------------------------------------
VAR
i,
j,
ii,
energy counter INTEGER;
x, xo Vector2;
temp_x, temp_y VECTOR;
energyl, energy2, x sum, y sum, x_avg, y_avg, half, prev, curr, norml, norm2 REAL;
singular BOOLEAN;
dif real;
f TEXT;
GIN Get.Matrix; curr := 0; prev := 10000;
FOR jj := 1 TO 50 DO
BEGIN
Guesses;
{
Calcurate initial energy with initial
While loop is there to make sure that
energy is none-negative to start.
}
energyi := -1;
WHILE (energyl < 0) DO
BEGIN
energyl := Energy(xs,ys);
END;
norml := 100;
guesses.
the initial
}
61

norm2 := 101;
ii := 0;
WHILE ((iiClOO) AND (ABSCnorml norm2)>0.00005)) DO
BEGIN
IF (norm2 < norml) THEN
BEGIN
norml := norm2;
END;
ii := ii+1;
CRUNCH;
singular := FALSE;
Direct_Method(xo,singular);
{
if the jacobi is singular for the iteration,
then switch to gauss sidel method.
}
IF singular THEN
BEGIN
CRUNCH;
Gauss_sidel(x,xo);
END;
{
Check current energy, if energy is increasing,
cut the step by half.
>
FOR i := 1 TO matrix_size DO
BEGIN
temp_x[i] := xs [i] + xo[i];
temp_y[i] := ys [i] + xo [i+matrix_size] ;
END;
energy2 := Energy(temp_x,temp_y);
dif := energyl energy2;
energy.counter := 0;
WHILE C(energy2 > energyl) AND
(abs(energy2 energyl) >= 0.001) AND
(energy.counter <= 100)) DO
BEGIN
dif := energyl-energy2;
energy.counter := energy.counter + 1;
FOR i := 1 TO 2*matrix_size DO
BEGIN
xo[i] := xo[i]/2;
END;
FOR i := 1 TO matrix.size DO
BEGIN
temp_x[i] := xs[i] + xo[i];
temp_y[i] := ys[i]+xo[i+matrix_size] ;
END;
energy2 := Energy(temp_x,temp_y);
END; {while}
energyl := energy2;
FOR i := 1 TO matrix.size DO
BEGIN
xs[i] := xs[i] + xo[i];
ys[i] := ys[i] + xo[i+matrix_size] ;
62

END;
{calculate centeroid, then make an adjustment}
x_sum := 0;
y_sum := 0;
x_avg := 0;
y_avg := 0;
FDR i := 1 TD matrix_size DD
BEGIN
x_sum := x_sum + xs [i];
y_sum := y_sum + ys[i];
END;
x_avg := x_sum/matrix_size;
y_avg : = y_sum/matrix_size;
FOR i := 1 TO matrix_size DO
BEGIN
xs[i] := xs[i] x_avg;
ys[i] := ys[i] y_avg;
END;
norm2 := 0;
FOR i := 1 TO matrix_size DO
BEGIN
norm2 := norm2 + b[i];
END;
norm2 := ABS(norm2);
END;{while: Multivariable newtons method}
curr : = energyl;
IF (curr <= prev) THEN
BEGIN
prev := curr;
FOR i := 1 TO matrix_size DO
BEGIN
prevx[i] := xs [i];
prevy[i] := ys[i];
END;
END;
WRITELN
(It took ,ii, iterations and energy = ,energy2:10:2);
END;{for jj}
FOR i := 1 TO matrix_size DO
BEGIN
xs[i] := prevx [i];
ys [i] : = prevy [i] ;
END;
Rotate;
WRITELN;
FOR i := 1 TO matrix_size DO
BEGIN
writeln(xs[i]:10:3, ,ys[i]:10:3);
END;
Latex_Begin(f,lowest.tex);
LaTex(prev,f);
Latex_End(f) ;
END.{end of driver}
63

REFERENCES
[1] Arfken, George, Mathematical Methods for Physicists,
Third Edition,Academic Press, San Diego, California.
[2] Burden, Richard L. and Faires, Douglas J.,
Numerical Analysis, Fifth Edition, PWS Publishing,
Boston, Massachusetts, 1993, Chapter 10.
[3] Dunning, Larry A. and Clark, Edwin W. Sharp upper
bounds for the domination numbers of graphs with given
order and minimum degree
[4] Horn, Roger A. and Johnson, Charles R., Matrix Analysis,
Cambridge University Press, New York,
Cambridge University Press, 1985.
[5] Kaplan, Wilfred, Advanced Calculus, Third Edition,
Addison-Wesley, Menlo Park, California, Chapter 5.
[6] Roberts, Fred S., Applied Combinatorics,
Prentice-Hall, Englewood Cliffs, New Jersey, 1984.
64

Full Text

PAGE 1

# +  W B R B # B + O # W B + 5 M ) L h ) k 4 i 5 c N ? i h t | ) L u L L h @ L @ | # i ? i h c Â b b | i t t t M 4 | | i | L | i N ? i h t | ) L u L L h @ L @ | # i ? i h ? T @ h | @ u Â€ * 4 i ? | L u | i h i ^ h i 4 i ? | t u L h | i i } h i i L u @ t | i h L u 5 U i ? U i T T i @ | i 4 @ | U t Â b b

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A t | i t t u L h | i @ t | i h L u 5 U i ? U i i } h i i M ) L h ) k 4 i @ t M i i ? @ T T h L i M ) # @ 6 t i h  i L ? w L U 5 | i T i ? * T t # @ | i

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k 4 i c L h ) E 5 c T T i @ | i 4 @ | U t # +  W B R B # B + O # W B + 5 A i t t h i U | i M ) t t L U @ | i h L u i t t L h # @ 6 t i h 5 A + A h L U ? } } h @ T @ } h @ 4 t t @ ? L ? | h @ T h L M i 4  i T h L T L t i @ 4 i | L | L @ | L 4 @ | i | t | @ t h | i h @ u L h i Â€ ? ? } @ ? i @ @ | ? } } h @ T @ } h @ 4 t * M i T h i t i ? | i c @ t i * @ t @ ? 4 i h U @ @ } L h | 4 u L h T h L U ? } @ } h @ 4 t @ U U L h ? } | L | i t i U h | i h @ + i t | t u h L 4 @ U L 4 T | i h T h L } h @ 4 4 T i 4 i ? | ? } | t 4 i | L * M i t L ? A t @ M t | h @ U | @ U U h @ | i ) h i T h i t i ? | t | i U L ? | i ? | L u | i U @ ? _ @ | i < t | i t t W h i U L 4 4 i ? | t T M U @ | L ? 5 } ? i # @ 6 t i h

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A A 5 @ T | i h Â Â Â Â 2 Â 2 2 Â 2 2 2 2 e 2 D Â 2 e D D Â D 2 W ? | h L U | L ? L | @ | L ?  @ | t @ R B L L B h @ T q ? i h } ) 6 ? U | L ? M t i h @ | L ? L u 6 L h U i L ? ? i U | i L 4 T L ? i ? | t L ? Â€ } h @ | L ? W ? | i h @ U | L ? M i | i i ? ? L i t V @ h L t ? i h } ) L ? Â€ } h @ | L ? ? U i T | L ? ? | L 4 @ | L ? | i | L ? < t i | L } L h | 4 ? 4 @ | L ? L u } L h | 4 ? T T U @ | L ? + L | @ | L ? | L 4 @ | U } L h | 4 + L | @ | L ? @ ? 5 L T i Â Â 2 D D D b Â e Â S Â S Â H 2 2 2 H 2 H T T i ? G G G + i u i h i ? U i t B h @ T # @ } h @ 4 t ? 4 @ | L ? L u } L h | 4 ? W 4 T i 4 i ? | @ | L ? ? @ t U @ S H e Â S e

PAGE 5

# i U @ | L ? W L * i | L i U @ | i | t | i t t | L 4 ) u i a i ? ? ) @ ? 4 ) T @ h i ? | t k @ | t L @ ? L k 4 i u L h | i h L i @ ? ? U L ? | L ? @ t T T L h |

PAGE 6

U ? L i } i 4 i ? | W ? T h i T @ h @ | L ? @ ? U L 4 T i | L ? L u 4 ) | i t t c W 4 t | | @ ? # h # @ 6 t i h u L h t ? t T h @ | L ? @ ? 4 ) | i t t U L 4 4 | | i i u L h | i h t T T L h | t T i U @ * ) c # h 5 | i T i ? * T t @ ? # h # @ 6 t i h u L h } ? } 4 i @ @ M i @ t i @ | | i Â€ ? @ i | L u | t | i t t W @ t L L * i | L | @ ? | i } h @ @ | i t U L L u L h } ? } 4 i @ ? L T T L h | ? | ) | L t | ) @ ? i @ h ? A i h i @ h i t L 4 @ ? ) u h i ? t L t T T L h | i 4 i | h L } L | | i T h L i U | O L i i h u W i h i | L t | @ ? | @ ? i i h ) M L ) c i i ? | @ * ) | i @ U ? L i } 4 i ? | t i U | L ? L M i U L 4 i L ? } i h | @ ? | i | i t t | t i u | c W U @ ? ? L | i ? | i t i U | L ? | L | i | i ? ? } 4 ) t T i U @ | @ ? t | L # h ~ i ? @ t + O @ h | } t L ? @ ? w i @ ? ? i # O L i h u L h | i h ? i h t | @ ? ? } @ ? @ t t t | @ ? U i 6 ? @ * ) c 4 ) t T i U @ | @ ? t | L U @ i O k @ @ u L h i T ? } 4 i | h L } L | | i T h L i U | i c | @ ? ) L i h ) 4 U u L h i T ? } 4 i | | i T h i T @ h @ | L ? @ ? @ @ | L ? L u | i T h i t i ? | @ | L ? 4 @ | i h @ t u L h 4 ) L h @ i u i ? t i c | U L ? | h M | L ? L u ? | i h i t | ? } | i t | U @ t i t u L h | i @ } L h | 4 c | | i w @ A i c @ ? | 4 @ ? } 4 i h i @ 3 i @ | W @ t | h ) ? } | L L L ? | i Â Â | L h

PAGE 7

Â W A + # N A W Â Â A W V A W A i t | ) L u } h @ T | i L h ) ? @ | h @ * ) ? L i t | i t @ h i T h i t i ? | @ | L ? L u } h @ T t N ? u L h | ? @ | i ) c | i t i L u U L 4 T | i h t ? T h L U i t t ? } } h @ T t @ t M i i ? U L ? Â€ ? i T h 4 @ h ) | L | i 4 @ | i 4 @ | U @ h @ | i h | @ ? | i t @ @ t T i U | t L u @ } h @ T 6 L h i @ 4 T i c U L 4 T | i h @ } L h | 4 t i t | | @ | Â€ ? U h L 4 @ | U ? 4 M i h t c U ^ i t c @ ? 4 ? 4 @ % 4 @ 4 @ L 4 ? @ | L ? t u L h @ } i ? } h @ T W ? U L ? | h @ t | c i h ) | | i @ t M i i ? L ? i | L @ | L 4 @ | U @ * ) T h L U i } L L _ @ } h @ 4 t L u } h @ T t u h L 4 | i h @ @ U i ? U ) 4 @ | h U i t W ? | t | i t t i * u L U t L ? | t T h L M i 4 M ) T h L T L t ? } @ ? @ } L h | 4 | L T h L U i R } L L _ h @ ? } t L u } h @ T t # i Â€ ? | L ? Â Â Â B h @ T t } h @ T C 4 @ ) M i i Â€ ? i @ t @ T @ h E c e c i h i t @ t i | L u ? L i t E i h | U i t @ ? e t @ t i | L u i } i t E ? L h i h i T @ h t L u i i 4 i ? | t u h L 4  i i ? L | i | t | | i t ) 4 M L C E c e # i Â€ ? | L ? Â Â 2 B h @ T @ } h @ 4 t @ } h @ 4 L u @ } h @ T 4 @ T t | i ? L i t | L T L ? | t ? | i T @ ? i @ ? i @ U i } i | L | i ? i t i } 4 i ? | t M i | i i ? | t i ? T L ? | t } h @ T t ? L | @ } h @ T @ } h @ 4 A L } i t L 4 i | 4 i t t i | i | i h 4 R } h @ T i ? i h i @ * ) 4 i @ ? R } h @ T @ } h @ 4 c i L T i | i @ U | @ Â

PAGE 8

4 i @ ? ? } t U i @ h u h L 4 U L ? | i | 5 | h U | ) t T i @ ? } c @ } h @ T @ } h @ 4 t @ 2 4 i ? t L ? @ h i T h i t i ? | @ | L ? L u | i } h @ T A t c i * t i i | @ | i @ U } h @ T U @ ? M i h i T h i t i ? | i M ) @ ? 4 M i h L u } h @ T @ } h @ 4 t | @ | @ h i L ? ) h @ ? g i h i ? | ) Â 2  @ | t @ R B L L B h @ T q t 5 T h i 4 i L h | a t | U i 6 i 6 h @ ? u h | i h t @ L u | i U @ * i ? } i L u i Â€ ? ? } T L h ? L } h @ T ) c R W 4 @ ) ? L | M i @ M i | L i t U h M i | c M | W ? L | i ? W t i i | # i Â€ ? ? } } L L ? i t t t 4 L h i L u @ T L t L T U @ ^ i t | L ? | @ ? t L 4 i | ? } i U @ ? i t U h M i 4 @ | i 4 @ | U @ * ) 6 L h @ ? i @ 4 T i c i | t h @ @ D U ) U i ? | L g i h i ? | @ ) t 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 6 } Â 2 Â i ? | @ } L ? @ u L h 4 L u D 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 6 } Â 2 2 ? L | i h D  i L u | i ? @ t t L U @ | i U i h | @ ? U @ ? L ? U @ @ } h @ 4 t | @ T @ h | U @ h } h @ T A t L i t ? L | 4 T ) | @ | i i h ) L ? i * U L ? t i h | i t @ 4 i U @ ? L ? U @ u L h 4 @ t | i R M i t | @ } h @ 4 W ? | i i @ 4 T i @ M L i c i 4 } | U L ? t i h | i T i ? | @ } L ? @ } h @ 4 R t 4 T i h | @ ? | i L | i h h i T h i t i ? | @ | L ? L u | i D U ) U i t L ? ? 6 } Â 2 Â c M i U @ t i ? L ? i L u | i i } i t @ h i | L L U L t i | L i @ U L | i h | c t L 4 i L ? i i t i 4 @ ) i U i 6 } Â 2 2 t 4 L h i @ i t | i | U @ * ) T i @ t ? } i i h | i i t t c i ? i h @ @ } h @ T @ } h @ 4 c i @ ? | i @ U i } i | L M i t | ? } t @ M i c @ ? ? } i ? i h @ c @ t t L h | @ t T L t t M i _ | L ? @ * ) c i 2

PAGE 9

@ ? | | i ? L i t | L M i t U i ? | ) t i T @ h @ | i W ? L h i h | L @ U i i | i t i } L @ t i @ | | i 4 T | | L 4 ? 4 3 i | i i ? i h } ) u ? U | L ? i t U h M i M i L Â ? i h } ) 6 ? U | L ?  i | i h @ T @ h L u ? L i t t U L ? ? i U | i M ) @ ? i } i L h ? L | c i L ? L | @ ? | | i ? L i t | L M i | L L U L t i | L } i | i h  i 4 t | i t i @ ? L i T h L T i h | ) U * i ? U L h @ } i | i 4 | L t i T @ h @ | i | i 4 t i i t u h L 4 L ? i @ ? L | i h  i U @ ? 4 L i | i t | @ | L ? M ) | ? ? } L u L L 4 M < t w @  i 4 L i | i ? L i t @ t M i ? } T @ h | U i t | @ i U @ h } i L ? | i 2 4 i ? t L ? @ @ h | i t @ ? U L L h ? @ | i T @ ? i c @ ? | t i @ i 4 | @ h i T t L ? M i | i i ? @ ? ) T @ h L u ? L i t L 4 Â Â ? i h } ) i | L h i T t ? } u L h U i u L h @ T @ h L u ? L i t A i i ? i h } ) i | L | i h i T t ? } u L h U i M i | i i ? @ T @ h L u ? L i t L U @ | i @ | ~ E % c + @ ? ~ E % c + t Â E ~ c ~ c i h i E ~ c ~ t E % % 2 n E + + 2 t | i t | @ ? U i M i | i i ? | i T @ h L u ? L i t O L i i h c ? i t t i ? | h L U i @ ? @ | | h @ U | ? } u L h U i c | i ? L i t L | i ? | L h u | @ t u @ h @ @ ) u h L 4 i @ U L | i h @ t T L t t M i c 4 @ ? } | i t | @ ? U i M i | i i ? @ ? ) | L L u | i 4 i ^ @ | L n 4 @ t i ? i h } ) @ T T h L @ U i t f W ? } i ? i h @ c i L ? L | U @ h i u ? L ? @ @ U i ? | ? L i t @ h i u @ h @ T @ h | 5 L c i i t i @ ? L i T h L T i h | ) U * i i T @ @ U i ? | ? L i t u h L 4 h u | ? } | L L u @ h @ T @ h |  i U @ ? 4 L i | i t | @ | L ? M ) | ? ? } L u U i h | @ ? T @ h t @ t M i ? } U L ? ? i U | i M ) @ t T h ? }

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L 4 Â 2 ? i h } ) i | L | i @ | | h @ U | ? } u L h U i M i | i i ? @ T @ h L u @ @ U i ? | ? L i t A i i ? i h } ) i | L | i @ | | h @ U | ? } u L h U i M i | i i ? @ T @ h L u @ @ U i ? | ? L i t L U @ | i @ | ~ @ ? ~ t E ~ c ~ A t ? L i T h L T i h | ) T h i i ? | t i } i t u h L 4 M i U L 4 ? } | L L L ? }  i t i | i @ | | h @ U | L ? T h L T i h | ) L u L h 4 L i t i i U | i )  i U L 4 M ? i | i | L i ? i h } i t @ ? L M | @ ? | i | L | @ i ? i h } ) u L h | i T @ h t L 4 Â ? i h } ) u ? U | L ? ? i h } ) L u @ } h @ T @ } h @ 4 @ | @ T L t | L ? L u ? L i t 3 E ~ Â c ~ 2 c ~ c ~ ? U @ ? M i i Â€ ? i @ t E 3 S Â  ? Â E ~ c ~ n S E c M e E ~ c ~ e

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2 5 + V A W 6 6 + 5 L u @ h i U @ 4 | @ | M ) 4 ? 4 3 ? } | i i ? i h } ) i i h | i M ) | i h i T t ? } u L h U i t @ ? | i @ | | h @ U | ? } u L h U i t c i t L M i @ M i | L T @ U i | i ? L i t ? @ R } L L U L ? Â€ } h @ | L ? w i | t L M t i h i L | t L h t M ) | i i ? i h } ) u ? U | L ? ? @ U | L ? 2 Â L ? ? i U | i L 4 T L ? i ? | t 6 L h ? L ? @ @ U i ? | T @ h t | i h i t ? L h i U | u L h U i L u @ | | h @ U | L ? A t c i h i T h i t i ? | | i i ? i h } ) L u | i t i T @ h t L u ? L i t | L 4 Â Â A i h i u L h i c t L @ | i ? L i t L h t U L ? ? i U | i U L 4 T L ? i ? | t * i h } i u h L 4 L | i h U L ? ? i U | i U L 4 T L ? i ? | t O i ? U i c L h 4 L i * ? L | @ ? i 4 L h i | @ ? L ? i U L ? ? i U | i U L 4 T L ? i ? | @ | @ | 4 i 6 L h i @ 4 T i c t T T L t i | @ | i @ i @ } h @ T U L ? t t | ? } L ? ) L u | L ? L ? @ @ U i ? | ? L i t E i c | L t L @ | i ? L i t L h | L t U L ? ? i U | i U L 4 T L ? i ? | t ? ) | i h i T t ? } u L h U i i t | t M i | i i ? | i 4 c @ ? | i t | @ ? U i U 4 ? 4 3 i t | i t ) t | i 4 i ? i h } ) t n 4 A i h i u L h i c t U L ? ? i U | i U L 4 T L ? i ? | t 4 t | M i @ ? i t i T @ h @ | i ) M i U @ t i L h 4 L i * L ? ) @ U U L 4 4 L @ | i U L ? ? i U | i U L 4 T L ? i ? | t 2 2 L ? Â€ } h @ | L ?  i i 4 L ? t | h @ | i L h 4 L i | @ e U ) U i h @ } L h | 4 * T h L U i | i U @ ? L ? U @ @ } h @ 4 L ? | i i u | ? 6 } 2 2 Â D

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6 } 2 2 Â A L T L t t M i @ } h @ 4 t u L h e W u i Â€ | i T L t | L ? L u ? L i t @ c S c @ ? _ c @ ? t i ? ? L i K L | @ h @ t ? | i @ } h @ 4 L ? | i h } | ? 6 } 2 2 Â c | i i ? i h } ) i i ? U h i @ t i t 6 } 2 2 2 t L t ? L i ? | i h @ U | L ? ? | i h 4 t L u | i i ? i h } ) u ? U | L ? h i L t ) c i t @ | @ | i @ ? | i ? L ? @ @ U i ? | ? L i t | L M i t U i ? | ) u @ h @ T @ h | u h L 4 L ? i @ ? L | i h 6 } 2 2 2 L i ? | i h @ U | L ? t ? e ? | i h } | ? 6 } 2 2 Â c ? L i K t ? L | L L u @ h @ T @ h | u h L 4 ? L i @ S

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@ ? S 5 ? U i ? L i K t U L ? ? i U | i | L M L | ? L i @ @ ? S c | i t i | L ? L i t * | h ) | L @ | | h @ U | ? L i K U L t i h O i ? U i c ? L i K * M i T L t | L ? i @ | | i U L L h ? @ | i i h i | t h i @ | i ) U L t i | L @ @ ? S c M | M i U @ t i L u h i T t i u L h U i t c * ? L | | L L U L t i 2 W ? | i h @ U | L ? i | i i ? L i t w i | t @ ? @ ) 3 i | L 4 L h i } h @ T @ } h @ 4 t | L t L L @ @ U i ? | @ ? ? L ? @ @ U i ? | ? L i t ? | i h @ U | | i @ U L | i h L | } h @ T t * t | h @ | i | i u L * L ? } T h ? U T i E Â L ? @ @ U i ? | ? L i t h i T t i i @ U L | i h E 2 @ U i ? | ? L i t | i ? | L M i @ M L | Â ? | t | @ ? U i @ T @ h | O i h i t | i @ @ U i ? U ) 4 @ | h u L h @ | h @ ? } i | @ ? i | h @ @ @ U i ? | ? L i G 5 9 9 9 9 9 9 9 9 9 9 7 f Â Â f Â f Â f Â Â f Â f f Â f 6 : : : : : : : : : : 8

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, ? i h } ) t G H b e 2 f Â 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 6 } 2 Â A h @ ? } i | @ ? i | h @ @ @ U i ? | ? L i A i i } i t L u | i | h @ ? } i @ h i ? L | i ^ @ M i U @ t i L u | i ? | i h @ U | L ? M i | i i ? | i | L T 4 L t | ? L i t @ ? | i M L | | L 4 | L ? L i t O i h i t | i @ @ U i ? U ) 4 @ | h u L h @ e U ) U i | @ ? i | h @ @ @ U i ? | ? L i G 5 9 9 9 9 9 9 9 9 9 9 9 9 9 9 7 f Â f Â f Â f Â f f f Â f Â f Â f Â f Â f f f Â f 6 : : : : : : : : : : : : : : 8 H

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, ? i h } ) t G Â 2 D e Â D 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 6 } 2 2 e U ) U i | @ ? i | h @ @ @ U i ? | ? L i A i u L h U ) U i t ? L | t ^ @ h i M i U @ t i L u | i ? | i h @ U | L ? M i | i i ? | i | L T 4 L t | ? L i @ ? | i ? L ? @ @ U i ? | ? L i t 2 e V @ h L t ? i h } ) L ? Â€ } h @ | L ? t A i ^ i t | L ? t | * h i 4 @ ? t ) L h @ } L h | 4 U L L t i t @ t ^ @ h i | L h i T h i t i ? | | i e U ) U i 5 ? U i i @ h i 4 ? 4 3 ? } | i i ? i h } ) u ? U | L ? c i ? L | @ | | i 4 L i * u @ L h i } i t | i ? } | ? i @ h Â L ? t i h | i M @ t i U @ t i u L h @ T @ h L u @ @ U i ? | ? L i t B i ? | i @ @ U i ? U ) 4 @ | h c 5 9 9 7 f Â Â f 6 : : 8 i U @ ? U @ U @ | i i ? i h } ) b

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w i | | i t | @ ? U i M i | i i ? ? L i t ~ Â @ ? ~ 2 M i % A i i ? i h } ) u ? U | L ? t E % % n Â %  i Â€ ? | i L T | 4 @ t | @ ? U i M i | i i ? | i ? L i t M ) g i h i ? | @ | ? } | i u ? U | L ? @ ? t i | | ? } | i ^ @ | L 3 i h L E % % Â Â % 2 f Â % 2 Â Â % 2 Â % 2 Â % Â 5 ? U i i * ? L | @ * L ? i } @ | i t | @ ? U i t c i U L L t i % Â U 4 T i t | @ | E Â 2 t 4 ? 4 @ W ? | i U @ t i L u @ e U ) U i c | i @ } L h | 4 * T @ U i | i ? L i t ? @ t ^ @ h i U L ? Â€ } h @ | L ? @ ? * U L L t i | i L | i h h 4 ? L i T @ h t @ t i } i t L i h | i @ } L ? @ ? L i T @ h t E ) | i ) | @ } L h i @ ? | i L h i 4 c | i @ } L ? @ i } i t @ h i L ? } i h | @ ? | i h 4 i } i t ? i h } ) t G b f b ? i h } ) t G b b S S 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 6 } 2 e Â A L T L t t M i U L ? Â€ } h @ | L ? t u L h e Â f

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G ? 4 T i 4 i ? | @ | L ? L u @ } L h | 4 ? @ t U @ ( @ ( } ? f p p C S L ? L ? L ? % 3 ? v f q = f p z f 2 > 2 2 p 1 f J q 2 > z e Q f p } > a 3 1 f$ n d % v ? p p > > & p z Q f z f p 1 f J q } > a 3 1 f ] a 3 1 f $O d % v ? p = 2 K f e = f f [ p z J K f > e { f e z > = C } > a p 1 f 0 2 K f e a J q a f [ { f p$ c B d % ? r ) 3 1 f p $} > a 3 1 f % 3 1 f p 0$ ] a 3 1 f % 3 1 f p n $2 K f e a J q a f [ { f p % [ a } > z e$ ( ( r b 3 1 f p 3 1 f p l 4 S ? ) @ ) ( % } > z e $( ( r b 3 1 f p 0 3 1 f p 0 l 4 ( ) s % R f Q z J e$ e e > & b 3 1 f p l 4 ( ) s % R f Q z J e 0 $e e > & b 3 1 f p 0 l 4 ( ) s % R f Q z J e n$ e e > & b 3 1 f p n l 4 ( ) s % R ( 2 > z e a p 1 f D S ? ) @ ) ( % > Q J K D } > z e % v ] > Q J K 2 > z e p f [ q J e f z J = p ; 2 f z J [ K D R f Q z J e 0 % v R f Q z J e p f [ q J e { > p p p [ f 2 f z J [ D [ a } > z e % v ? p p > [ > Q f = Q & 2 > z e p D R f Q z J e % v ? f p f > e f > = [ & Q J J e [ = > z f p J q = J [ f p & p D R f Q z J e % P e f v ? f 2 P J e > e & > e > K f p f [ z J p z J e f P e f & D R f Q z J e % v Q J J e [ = > z f p p J P f D R f Q z J e n % v 3 J P f p f [ z J e J z > z f z f { e > P e v L p f [ z J p z J e f Q J J e [ = > z f p = P J > e z f z > D R ) ? ( % p J P f a = [ f D S ? ) @ ) ( % v ( ) L ( ) @ L 3 3 a 3 S ) s C R ( J D R f Q z J e 0 % v ? p p > = 2 P f 2 f = z > z J = J q @ > p p 3 [ f > { J e z 2 z J p J f p & p z f 2 J q f Z > z J = z f e > z f & > e _ = _ 2 D = z f { f e % p 2 p 2 0 z J z J z J 0 D ( ) s % K f { = q J e D $z J 2 > z e a p 1 f 6 0 [ J K f { = J b l D$ d % f = [ % = D $2 > z e a p 1 f 6 0 % p 2 D$ d % v = z > 1 f p 2 p 2 0 D $d % v = z > 1 f p 2 D$ % z J D $d % z J D$ d % z J 0 D $d % e Â PAGE 48 ' f C C$ B d d > = [ C z J 9 $d d d d > = [ C > K p C z J d d [ J K f { = z J D$ d % z J 0 D $d % q J e D$ z J = [ J K f { = q J e D $z J [ J K f { = p 2 D$ p 2 k > Q J K b l 6 b l % f = [ % v q J e D $k z J = [ J K f { = p 2 0 D$ p 2 0 k > Q J K b l 6 g b l % f = [ % v b l D $C p 2 p 2 0 k K b l > Q J K b l % p 2 D$ d % v e f = z > 1 f p 2 p 2 0 D $d % v e f = z > 1 f p 2 f = [ % v D$ k % q J e D $z J = [ J K f { = z J D$ z J k b l % f = [ % q J e D $z J = [ J K f { = z J 0 D$ z J 0 k g b l % f = [ % z J D $> K p C z J z J 0 % v z J z f = f z z f e > z J = q J e D$ z J = [ J K f { = g b l D $b l % f = [ % f = [ % v f f = [ % v @ > p p p a 3 [ f v ( ) L ( ) @ f z a } > z e % v ? p 2 f z J [ P J = Q > ' f [ ' P e J Q f p p > [ > Q f = Q & 2 > z e z > z p = z f z q f q J e 2 > z R ( q f a = > 2 f D 3 ? ( S @ % q D ? ) g ? % D S ? ) @ ) ( % ) @ S \ ( S ? ) C L ? L ? ; f > p f f = z f e > q f = > 2 f D ; % ( ) s C S L ? q f a = > 2 f % 3 3 S @ C q q f a = > 2 f % ( ) 3 ) ? C q % ( ) s C q 2 > z e a p 1 f % 4 ( D$ ? 2 > z e a p 1 f ) @ S 4 ( D $? 2 > z e a p 1 f ) @ S ( ) C q b l % ) % ( ) s C q % ) % s 3 ) C q % ) % e 2 PAGE 49 v ( ) L ( ) @ L ) 3 3 ) 3 % v ? p 2 f z J [ P J = Q > ' f [ ' J K z > = = z > { f p p q J e z f f z J = ; p 2 f z J [ R ( D S ? ) @ ) ( % ) @ S ( } S | ) % 4 ( D$ ? 2 > z e a p 1 f ) @ S p b l D $3 ( ? C 2 > z e a p 1 f 6 C C ( } C d d d d B d d d % & p b l D$ 3 ( ? C 2 > z e a p 1 f 6 C C ( } C d d d d B d d d % ) % ) % v @ f p p f p v v ) Z > z J = p p f [ = z f f z J = ; p 2 f z J [ ) @ S v 4 L ? S p z C _ & & D ( ) s D ( ) s % ) @ S p z D $p Z e z C p Z e C k p Z e C & & % ) % v p z v 4 L ? S p z 0 C _ & & D ( ) s D ( ) s % ) @ S p z 0 D$ ) g C n 6 s C p z C _ & & % ) % v p z 0 v 4 L ? S p z n C _ & & D ( ) s D ( ) s % ) @ S p z n D $) g C B 6 s C p z C _ & & % ) % v p z n v 4 L ? S a C _ & & D ( ) s D ( ) s % ) @ S a D$ C p Z e C & & [ p z 0 C _ & & % ) % v 4 L ? S a & & C _ & & D ( ) s D ( ) s % ) @ S a & & D $C p Z e C [ p z 0 C _ & & % ) % v 4 L ? S a & C _ & & D ( ) s D ( ) s % ) @ S a & D$ C C 6 C & & [ p z 0 C _ & & % ) % v 4 L ? S a & C _ & & D ( ) s D ( ) s % ) @ S a & D $C C 6 C & & [ p z 0 C _ & & % ) % v 4 L ? S a C _ & & D ( ) s D ( ) s % e PAGE 50 ) @ S a D$ C p Z e C & & [ p z 0 C _ & & % ) % v 4 L ? S a C _ & & D ( ) s D ( ) s % ) @ S a D $C p Z e C & & [ p z 0 C _ & & % ) % v 4 L ? S a & & C _ & & D ( ) s D ( ) s % ) @ S a & & D$ C p Z e C [ p z 0 C _ & & % ) % v 4 L ? S a & & C _ & & D ( ) s D ( ) s % ) @ S a & & D $C p Z e C [ p z 0 C _ & & % ) % v 4 L ? S a & C _ & & D ( ) s D ( ) s % ) @ S a & D$ C C 6 C & & [ p z 0 C _ & & % ) % v 4 L ? S a & C _ & & D ( ) s D ( ) s % ) @ S a & D $C C 6 C & & [ p z 0 C _ & & % ) % v 4 L ? S C _ & & D ( ) s D ( ) s % ) @ S D$ C 0 6 p Z e C p Z e C & & [ p z n C _ & & % ) % v v 4 L ? S & & C _ & & D ( ) s D ( ) s % ) @ S & & D $C p Z e C 0 6 p Z e C & & [ p z n C _ & & % ) % v & & v 4 L ? S & C _ & & D ( ) s D ( ) s % ) @ S & D$ C n 6 C & & 6 C [ p z n C _ & & % ) % v & v 4 L ? S & C _ & & D ( ) s D ( ) s % ) @ S & D $C n 6 C & & 6 C [ p z n C _ & & % ) % v & v 4 L ? S C _ & & D ( ) s D ( ) s % v 2 > = [ > { P P f e ) @ S v z p J = f = f [ f [ p k 0 D$ C 0 6 3 ( C 3 ( C & & [ p z n C _ & & % ) % v e e

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v 4 L ? S C _ & & D ( ) s D ( ) s % v 2 > = [ > { P P f e ) @ S v z p J = f = f f [ p k 0 D $C 0 6 3 ( C 3 ( C & & [ p z n C _ & & % ) % v v 4 L ? S & & C _ & & D ( ) s D ( ) s % v 2 > = [ > { J f e ) @ S v z p J = f = f f [ p k 0 & & D$ C 3 ( C 0 6 3 ( C & & [ p z n C _ & & % ) % v & & v 4 L ? S & & C _ & & D ( ) s D ( ) s % v 2 > = [ > { J f e ) @ S v z p J = f = f f [ p k 0 & & D $C 3 ( C 0 6 3 ( C & & [ p z n C _ & & % ) % v & & v 4 L ? S & C _ & & D ( ) s D ( ) s % v 2 > = [ > { J f e ) @ S & D$ C n 6 C 6 C & & [ p z n C _ & & % ) % v & v 4 L ? S & C _ & & D ( ) s D ( ) s % v 2 > = [ > { J f e ) @ S & D $C n 6 C 6 C & & [ p z n C _ & & % ) % v & v 4 L ? S a C _ & & D ( ) s D ( ) s % ) @ S a D$ C [ p z C _ & & % ) % v 4 L ? S a C _ & & D ( ) s D ( ) s % ) @ S a D $C [ p z C _ & & % ) % v 4 L ? S a & C _ & & D ( ) s D ( ) s % ) @ S a & D$ C & & [ p z C _ & & % ) % v 4 L ? S a & C _ & & D ( ) s D ( ) s % ) @ S a & D $C & & [ p z C _ & & % ) % v 4 L ? S a C _ & & D ( ) s D ( ) s % ) @ S a D$ C [ p z 0 C _ & & % ) % v 4 L ? S a C _ & & D ( ) s D ( ) s % ) @ S e D

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a D $C [ p z 0 C _ & & % ) % v 4 L ? S & a C _ & & D ( ) s D ( ) s % ) @ S & a D$ C & & [ p z 0 C _ & & % ) % v 4 L ? S & a C _ & & D ( ) s D ( ) s % ) @ S & a D $C & & [ p z 0 C _ & & % ) % v v ) Z > z J = p p f [ = z f f z J = ; p 2 f z J [ ) v v v M f p p > = 2 > z e ) @ S v ( ) L ( ) L P P f e a s f q z % v e J Q f p p f p P P f e f q z K J Q J q f p p > = 2 > z e R ( D S ? ) @ ) ( % z f 2 P D ( ) s % z f 2 P 0 D e f > % ) @ S 4 ( D$ ? 2 > z e a p 1 f ) @ S 4 ( D $k ? 2 > z e a p 1 f ) @ S z f 2 P D$ C p b l p b l & p b l & p b l % S 4 C b l $? M ) ) @ S z f 2 P 0 D$ a C p b l p b l & p b l & p b l % > Q J K b l D $z f 2 P 0 k z f 2 P % ) ) s 3 ) ) @ S > Q J K b l D$ v 0 z f 2 P % ) % ) % ) % ) % v L P P f e a s f q z v ( ) L ( ) s J f e a ( { z % v e J Q f p p f p s J f e e { z K J Q J q f p p > = 2 > z e R ( D S ? ) @ ) ( % z f 2 P D ( ) s % z f 2 P 0 z f 2 P n D ( ) s % e S

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) @ S 4 ( D $? 2 > z e a p 1 f ) @ S 4 ( D$ k ? 2 > z e a p 1 f ) @ S z f 2 P D $d % z f 2 P 0 D$ d % z f 2 P D $& & C p b l p b l & p b l & p b l % S 4 C b l$ ? M ) v q Q J = = f Q z f [ ) @ S z f 2 P 0 D $a & & C p b l p b l & p b l & p b l % z f 2 P n D$ z f 2 P 0 k z f 2 P % > Q J K b k 2 > z e a p 1 f k 2 > z e a p 1 f l D $z f 2 P n % ) ) s 3 ) v f p f q [ p Q J = = f Q z f [ ) @ S > Q J K b k 2 > z e a p 1 f k 2 > z e a p 1 f l D$ z f 2 P % ) % ) % v ) % v ) % v s J f e a ( { z v ( ) L ( ) L P a L P a ( { z % v e J Q f p p f p P P f e z e > = { f J q P P f e e { z K J Q J q f p p > = 2 > z e R ( D S ? ) @ ) ( % = z f e 2 D ( ) s % ) @ S 4 ( D $? 2 > z e a p 1 f ) @ S 4 ( D$ k ? 2 > z e a p 1 f ) @ S = z f e 2 D $& C p b l p b l & p b l & p b l % S 4 C b l$ ? M ) ) @ S > Q J K b l b 2 > z e a p 1 f k l D $a & C p b l p b l & p b l & p b l k = z f e 2 % ) ) s 3 ) ) @ S > Q J K b l b 2 > z e a p 1 f k l D$ = z f e 2 % ) % ) % ) % ) % v L P a L P a ( { z v ( ) L ( ) L P a s J a ( { z % v e J Q f p p f p J f e z e > = { f J q P P f e e { z K J Q J q f p p > = 2 > z e e

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R ( D S ? ) @ ) ( % = z f e 2 D ( ) s % ) @ S 4 ( D $0 ? 2 > z e a p 1 f ) @ S D$ % \ M S s ) C ) @ S = z f e 2 D $& C p b l p b l & p b l & p b l % S 4 C b l$ ? M ) ) @ S > Q J K b l b k 2 > z e a p 1 f l D $a & C p b l p b l & p b l & p b l k = z f e 2 % ) ) s 3 ) ) @ S > Q J K b l b k 2 > z e a p 1 f l D$ = z f e 2 % ) % D $k % ) % v f ) % v q J e [ J ) % v L P a s J a ( { z v ( ) L ( ) L P a > { a ( { z % v e J Q f p p f p [ > { = > f = z e f p J q P P f e e { z K J Q J q f p p > = 2 > z e R ( _ D S ? ) @ ) ( % p 2 z f 2 P z f 2 P 0 D ( ) s % ) @ S 4 ( D$ ? 2 > z e a p 1 f ) @ S p 2 D $d % D$ % \ M S s ) C ) @ S z f 2 P D $& C p b l p b l & p b l & p b l % S 4 C b l$ ? M ) ) @ S z f 2 P 0 D $d % z f 2 P 0 D$ a & C p b l p b l & p b l & p b l k z f 2 P % p 2 D $p 2 k z f 2 P 0 % ) ) s 3 ) ) @ S p 2 D$ p 2 k z f 2 P % ) % D $k % ) % v \ f e H PAGE 55 D$ k % \ M S s ) C $2 > z e a p 1 f ) @ S z f 2 P D$ & C p b l p b l & p b l & p b l % S 4 C b l $? M ) ) @ S z f 2 P 0 D$ d % z f 2 P 0 D $a & C p b l p b l & p b l & p b l k z f 2 P % p 2 D$ p 2 k z f 2 P 0 % ) ) s 3 ) ) @ S p 2 D $p 2 k z f 2 P % ) % D$ k % ) % v \ M S s ) 2 > z e a p 1 f ] > Q J K b k 2 > z e a p 1 f l D $p 2 % z f 2 P D$ d % ) % v ) % v L P a > { a ( { z v ( ) L ( ) } > = a > { a L P P f e % v e J Q f p p f p P P f e 2 > = [ > { J = > f = z e f p J q f p p > = 2 > z e R ( _ D S ? ) @ ) ( % = z f e 2 p 2 z f 2 P D ( ) s % ) @ S 4 ( D $? 2 > z e a p 1 f ) @ S p 2 D$ d % D $% \ M S s ) C ) @ S z f 2 P D$ C p b l p b l & p b l & p b l % S 4 C b l $? M ) ) @ S = z f e 2 D$ a C p b l p b l & p b l & p b l k z f 2 P % ) ) s 3 ) ) @ S = z f e 2 D $z f 2 P % ) % p 2 D$ p 2 k = z f e 2 % D $k % ) % v \ f D$ k % \ M S s ) C $2 > z e a p 1 f ) @ S z f 2 P D$ C p b l p b l & p b l & p b l % S 4 C b l $? M ) e b PAGE 56 ) @ S = z f e 2 D$ a C p b l p b l & p b l & p b l k z f 2 P % ) ) s 3 ) ) @ S = z f e 2 D $z f 2 P % ) % p 2 D$ p 2 k = z f e 2 % D $k % ) % v \ M S s ) 2 > z e a p 1 f ] > Q J K b l D$ p 2 % p 2 D $d % ) % v ) % v } > = a > { v ( ) L ( ) } > = a > { a s J f e % v e J Q f p p f p J f e 2 > = [ > { J = > f = z e f p J q f p p > = 2 > z e R ( _ D S ? ) @ ) ( % = z f e 2 p 2 z f 2 P D ( ) s % ) @ S 4 ( D$ ? 2 > z e a p 1 f ) @ S p 2 D $d % z f 2 P D$ d % = z f e 2 D $d % D$ % \ M S s ) C ) @ S z f 2 P D $& & C p b l p b l & p b l & p b l % S 4 C b l$ ? M ) ) @ S = z f e 2 D $a & & C p b l p b l & p b l & p b l k z f 2 P % ) ) s 3 ) ) @ S = z f e 2 D$ v 0 z f 2 P % ) % p 2 D $p 2 k = z f e 2 % D$ k % ) % v \ f D $k % \ M S s ) C$ 2 > z e a p 1 f ) @ S z f 2 P D $& & C p b l p b l & p b l & p b l % S 4 C b l$ ? M ) ) @ S = z f e 2 D $a & & C p b l p b l & p b l & p b l k z f 2 P % ) ) s 3 ) ) @ S D f PAGE 57 = z f e 2 D$ v 0 z f 2 P % ) % p 2 D $p 2 k = z f e 2 % D$ k % ) % v \ M S s ) 2 > z e a p 1 f ] > Q J K b k 2 > z e a p 1 f k 2 > z e a p 1 f l D $p 2 % ) % v ) % v } > = a > { v ( ) L ( ) J P & a ] > Q J K % v J P f p P P f e z e > = { f J q f p p > = z J J f e z e > = { f R ( _ K { a p 1 f D S ? ) @ ) ( % ) @ S K { a p 1 f D$ 0 6 2 > z e a p 1 f % 4 ( D $? K { a p 1 f ) @ S 4 ( D$ ? K { a p 1 f S 4 C 9 ? M ) ) @ S > Q J K b l D $> Q J K b l % ) % v ) % v ) % v J P & a ] > Q J K v ( ) L ( ) ( M 3 a L P P f e % v e J Q f p p f p P P f e > q J q e { z > = [ p [ f f Z > z J = p f [ = z f f z J = ; p 2 f z J [ R ( _ D S ? ) @ ) ( % p 2 z f 2 P D ( ) s % ) @ S z f 2 P D$ d % 4 ( D $? 2 > z e a p 1 f ) @ S D$ % p 2 D $d % \ M S s ) C ) @ S z f 2 P D$ a C p b l p b l & p b l & p b l % S 4 C b l $? M ) ) @ S z f 2 P D$ z f 2 P k a C p b l p b l & p b l & p b l % ) ) s 3 ) ) @ S z f 2 P D $d % z f 2 P D$ v z f 2 P a C p b l p b l & p b l & p b l % D Â

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) % p 2 D $p 2 k z f 2 P % D$ k % ) % v D $k % \ M S s ) C$ 2 > z e a p 1 f ) @ S z f 2 P D $a C p b l p b l & p b l & p b l % S 4 C b l$ ? M ) ) @ S z f 2 P D $z f 2 P k a C p b l p b l & p b l & p b l % ) ) s 3 ) ) @ S z f 2 P D$ d % z f 2 P D $v z f 2 P a C p b l p b l & p b l & p b l % ) % p 2 D$ p 2 k z f 2 P % D $k % ) % K b l D$ p 2 % z f 2 P D $d % ) % v ) % v ( M 3 v ( ) L ( ) ( M 3 a s J f e % v e J Q f p p f p J f e > q J q e { z > = [ p [ f f Z > z J = p f [ = z f f z J = ; p 2 f z J [ R ( _ D S ? ) @ ) ( % p 2 z f 2 P D ( ) s % ) @ S z f 2 P D$ d % 4 ( D $? 2 > z e a p 1 f ) @ S D$ % p 2 D $d % \ M S s ) C ) @ S z f 2 P D$ a & C p b l p b l & p b l & p b l % S 4 C b l $? M ) ) @ S z f 2 P D$ z f 2 P k & a C p b l p b l & p b l & p b l % ) ) s 3 ) ) @ S z f 2 P D $d % z f 2 P D$ v z f 2 P & a C p b l p b l & p b l & p b l % ) % p 2 D $p 2 k z f 2 P % D$ k % ) % v D 2

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D $k % \ M S s ) C$ 2 > z e a p 1 f ) @ S z f 2 P D $a & C p b l p b l & p b l & p b l % S 4 C b l$ ? M ) ) @ S z f 2 P D $z f 2 P k & a C p b l p b l & p b l & p b l % ) ) s 3 ) ) @ S z f 2 P D$ d % z f 2 P D $v z f 2 P & a C p b l p b l & p b l & p b l % ) % p 2 D$ p 2 k z f 2 P % D $k % ) % K b k 2 > z e a p 1 f l D$ p 2 % z f 2 P D $d % ) % v ) % v ( M 3 v ( ) L ( ) ( L M % v e J Q f p p f p f = z e f f p p > = 2 > z e R ( D S ? ) @ ) ( % ) @ S 4 ( D$ ? 0 6 2 > z e a p 1 f ) @ S 4 ( D $? 0 6 2 > z e a p 1 f ) @ S > Q J K b l D$ d % ) % ) % L P P f e a s f q z % s J f e a ( { z % L P a L P a ( { z % L P a s J a ( { z % L P a > { a ( { z % } > = a > { a L P P f e % } > = a > { a s J f e % J P & a ] > Q J K % ( M 3 a L P P f e % ( M 3 a s J f e % ) % v v M f p p > = 2 > z e ) @ S v v 4 L ? S ) = f e { & C a Q J J e [ & a Q J J e [ D R ) ? ( D ( ) s % v ? p 2 f z J [ P J = Q > ' f [ ' Q > Q > z f > = f = e { & > f J q { f = p z > z f J q { f = { e > P D

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R ( D S ? ) @ ) ( % = z f e 2 z f 2 P p 2 D ( ) s % ) @ S p 2 D $d % 4 ( D$ ? 2 > z e a p 1 f ) @ S 4 ( D $k ? 2 > z e a p 1 f ) @ S = z f e 2 D$ d % = z f e 2 D $[ p z C a Q J J e [ b l a Q J J e [ b l & a Q J J e [ b l & a Q J J e [ b l % S 4 C b l$ ? M ) ) @ S z f 2 P D $C = z f e 2 k = z f e 2 % ) ) s 3 ) ) @ S z f 2 P D$ d % z f 2 P D $= z f e 2 % ) % p 2 D$ p 2 k z f 2 P % ) % v ) % v ) = f e { & D $p 2 % ) % v ) = f e { & v ( ) L ( ) e f Q z a } f z J [ C > e D R ) ? ( 0 % R ( p = { > e D s ) % v ? p p > = 2 P f 2 f = z > z J = J q p J = { p & p z f 2 J q = f > e f Z > z J = [ e f Q z & C @ > p p f 2 = > z J = R ( _ = D S ? ) @ ) ( % 2 = z f e 2 D ( ) s % ) @ S p = { > e D$ 4 s 3 ) % = D $0 6 2 > z e a p 1 f % 4 ( D$ ? = S 4 ? C p = { > e ? M ) ) @ S S 4 C ] > Q J K b l $d ? M ) ) @ S p = { > e D$ ? ( L ) % ) % 4 ( D $k ? = S 4 ? C p = { > e ? M ) ) @ S 2 D$ ] > Q J K b l ] > Q J K b l % 4 ( D $k ? = ) @ S D e PAGE 61 ] > Q J K b l D$ ] > Q J K b l 2 6 ] > Q J K b l % ) % v J J P K b l D $K b l 2 6 K b l % ) % v ) % v v p z f P + S q > C = =$ d z f = L ? L ? C ; = J = Z f p J z J = ; 3 ? S 4 ? C p = { > e ? M ) ) @ S S 4 ] > Q J K b = = l $d ? M ) ) @ S p = { > e D$ ? ( L ) % ) ) s 3 ) ) @ S b = l D $K b = l ] > Q J K b = = l % 4 ( D$ = \ ? ) @ S = z f e 2 D $d % 4 ( D$ k ? = ) @ S = z f e 2 D $= z f e 2 k ] > Q J K b l 6 b l % ) % b l D$ C K b l = z f e 2 ] > Q J K b l % ) % ) % v f p f ) % ) % v ( ) L ( ) s > z f a f { = C R ( J z a q D ? ) g ? % q f a = > 2 f D 3 ? ( S @ % ) @ S > p p { = C J z a q q f a = > 2 f % ( ) \ ( S ? ) C J z a q % \ ( S ? ) s C J z a q ; [ J Q 2 f = z p z & f b 0 P z l v > e z Q f ; % \ ( S ? ) s C J z a q ; K f { = v [ J Q 2 f = z ; % ) % v ( ) L ( ) s > z f a ) = [ C R ( Q J p f a q D ? ) g ? % ) @ S \ ( S ? ) s C Q J p f a q ; f = [ v [ J Q 2 f = z ; % s 3 ) C Q J p f a q % ) % v 4 L ? S 4 S = [ a } > C > K Q [ D R f Q z J e D ( ) s % R ( D S ? ) @ ) ( % > e { f e Q J 2 P > e f D ( ) s % ) @ S > e { f e D $d d d d % 4 ( D$ ? 2 > z e a p 1 f ) @ S Q J 2 P > e f D $> K Q [ b l % S 4 C > e { f e$ Q J 2 P > e f ? M ) D D

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) @ S > e { f e D $Q J 2 P > e f % ) ) % 4 = [ a } > D$ > e { f e % ) % v 4 L ? S 4 = [ a } = C > K Q [ D R f Q z J e D ( ) s % R ( D S ? ) @ ) ( % p 2 > ' f e Q J 2 P > e f D ( ) s % ) @ S p 2 > ' f e D $d d d d % 4 ( D$ ? 2 > z e a p 1 f ) @ S Q J 2 P > e f D $> K Q [ b l % S 4 C p 2 > ' f e 9$ Q J 2 P > e f ? M ) ) @ S p 2 > ' f e D $Q J 2 P > e f % ) ) % 4 = [ a } = D$ p 2 > ' f e % ) % v ( ) L ( ) s > ? f C f = f e { & D ( ) s % R ( q D ? ) g ? % R ( D S ? ) @ ) ( % D ( ) s % 2 > 2 = & 2 > & 2 = D ( ) s % ) @ S \ ( S ? ) s C q ; f = f e { & p D ; f = f e { & D d D B % \ ( S ? ) s C q % \ ( S ? ) s C q ; p 2 > ' p P ; % \ ( S ? ) s C q ; = z f = { z $= ; % 2 > D$ 4 = [ a } > C p % 2 = D $4 = [ a } = C p % & 2 > D$ 4 = [ a } > C & p % & 2 = D $4 = [ a } = C & p % \ ( S ? ) s C q ; K f { = v P Q z e f C ; 2 > 2 = D + D c ; ; & 2 > & 2 = D + D c ; C ; 2 = D + D c ; ; & 2 = D + D c ; ; % 4 ( D$ ? 2 > z e a p 1 f ) @ S 4 ( D $k ? 2 > z e a p 1 f ) @ S \ ( S ? ) C q ; P z C ; p b l D O D c ; ; & p b l D O D c ; ; % \ ( S ? ) s C q ; v Q e Q f 6 v B ; % S 4 C b l$ ? M ) ) @ S D $3 ( ? C [ p z C p b l p b l & p b l & p b l % \ ( S ? ) C q ; 2 z P z C ; p b l D O D c ; ; & p b l D O D c % D S PAGE 63 \ ( S ? ) C q ; C ; C p b l p b l C d d 6 D O D c ; ; % \ ( S ? ) C q C & p b l & p b l C d d 6 D O D c ; v ; d d 6 D O D d ; ; % \ ( S ? ) s C q ; v Q e Q f 6 v d d ; % ) % ) % ) % \ ( S ? ) s C q ; f = [ v P Q z e f ; % \ ( S ? ) s C q % ) % v ) = [ s > ? f v 4 L ? S > [ e > = z C & D ( ) s D S ? ) @ ) ( % v ? M p q = Q z J = P J = Q > ' f [ ' e f z e = Z > [ e > = z n > Q Q J e [ = { z J z f p { = J q C & ) @ S S 4 C 9 d ? M ) ) @ S S 4 C & 9 d ? M ) ) @ S > [ e > = z D$ % ) ) s 3 ) ) @ S > [ e > = z D $c % ) % ) ) s 3 ) ) @ S S 4 C & 9 d ? M ) ) @ S > [ e > = z D$ 0 % ) ) s 3 ) ) @ S > [ e > = z D $n % ) % ) % ) % v > [ e > = z v ( ) L ( ) ( ? ? S C [ f { e f f D ( ) s % v ? p 2 f z J [ P J = Q > ' f [ ' e J z > z f { f = { e > P = P J > e Q J J e [ = > z f K & z f [ f { e f f p P f Q q f [ K & z f > e > K f [ f { e f f R ( > = { f D ( ) s % Q a Z > [ e > = z D S ? ) @ ) ( % ) @ S v P J > e Q J = f e p J = p z > e z 4 ( D$ ? 2 > z e a p 1 f ) @ S e b l D $3 ( ? C 3 ( C p b l k 3 ( C & p b l % ) % 4 ( D$ ? 2 > z e a p 1 f D

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) @ S > = { f D $e Q ? > = C P e f & b l P e f b l 6 C E d P % Q a Z > [ e > = z D$ > [ e > = z C P e f b l P e f & b l % 3 ) Q a Z > [ e > = z 4 D z f z > b l D $> = { f % 0 D z f z > b l D$ E d k > = { f % n D z f z > b l D $E d k > = { f % c D z f z > b l D$ n O d k > = { f % ) % z f z > b l D $z f z > b l k [ f { e f f % v { f p e J z > z J = ) % v P J > e Q J = f e p J = f = [ ) % v ( ) L ( ) J > e a ? J a > e z f p > = % v ? p 2 f z J [ P J = Q > ' f [ ' Q J = f e z P J J > e Q J J e [ = > z f = z J Q > e z f p > = Q J J e [ = > z f R ( D S ? ) @ ) ( % ) @ S 4 ( D$ ? 2 > z e a p 1 f ) @ S P e f b l D $e b l 6 Q J p C C z f z > b l 6 C P E d % P e f & b l D$ e b l 6 p = C C z f z > b l 6 C P E d % ) % ) % v ( ) L ( ) e { a z J a ? f 2 P % v ? p 2 f z J [ P J = Q > ' f [ ' Q J P & > f p p z J e f [ = z f p b l > = [ & p b l = z J P e f b l > = [ P e f & b l e f p P f Q z f & R ( D S ? ) @ ) ( % ) @ S 4 ( D $? 2 > z e a p 1 f ) @ S P e f b l D$ p b l % P e f & b l D $& p b l % ) % ) % v ( ) L ( ) ? f 2 P a z J a e { % v ? p 2 f z J [ P J = Q > ' f [ ' Q J P & > f p p z J e f [ = z f P e f b l > = [ P e f & b l = z J p b l > = [ & p b l e f p P f Q z f & R ( D S ? ) @ ) ( % ) @ S 4 ( D$ ? 2 > z e a p 1 f D H

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) @ S p b l D $P e f b l % & p b l D$ P e f & b l % ) % ) % v 4 L ? S J = z a 3 Q J e f D S ? ) @ ) ( % v ? p 2 f z J [ P J = Q > ' f [ ' Q f Q = 2 K f e J q f [ { f p z > z > e f P > e > ' f z J > = [ & > p = z f { f = { e > P R ( > a p Q J e f D S ? ) @ ) ( % > a p J P f D s @ S ? % ) @ S > a p Q J e f D $d % 4 ( D$ ? p J P f a = [ f ) @ S > a p J P f D $? ( L C p J P f b l 6 d d d % > a p J P f D$ ( L C > a p J P f d d d % 3 ) 3 C > a p J P f 4 d d D > a p Q J e f D $> a p Q J e f k % E d d d D > a p Q J e f D$ > a p Q J e f k % f p f ) % ) % J = z a 3 Q J e f D $> a p Q J e f % ) % v 4 L ? S ? f a M f { f p z D ( ) s % v ? p 2 f z J [ P J = Q > ' f [ ' Q f Q z f f { f p z P J = z J q { f = { e > P J = z f Q > e z f p > = Q J J e [ = > z f R ( D S ? ) @ ) ( % Q J 2 P > e f D ( ) s % ) @ S Q J 2 P > e f D$ P e f & b l % 4 ( D $0 ? 2 > z e a p 1 f ) @ S S 4 Q J 2 P > e f P e f & b l ? M ) ) @ S Q J 2 P > e f D$ P e f & b l % ) % ) % ? f a M f { f p z D $Q J 2 P > e f % ) % v ( ) L ( ) ( J z > z f % v ? p 2 f z J [ P J = Q > ' f [ ' e J z > z f z f { f = { e > P > z J 2 > z Q > ' & > Q Q J e [ = { z J q J ' J = { Q e z f e > D M f e f > { J J [ { e > P > p D D b PAGE 66 2 J e f f [ { f p z > z > e f P > e > ' f z J > = [ & > p 0 { J J [ { e > P p z f z > ' f p z { e > P J = z J P J q > K J f Q e z f e > n R ( _ Q f Q a 2 f [ f { P e f a p Q J e f Q e e a p Q J e f D S ? ) @ ) ( % 2 > & 2 = & = 2 f e > z J e [ f = J 2 = > z J e P e f a f { z Q e e a f { z D ( ) s % ) @ S P e f a p Q J e f D$ d % v = z > 1 > z J = P e f a f { z D $d % 4 ( [ f { D$ ? n O d ) @ S e { a z J a ? f 2 P % ( J z > z J = C [ f { % P J > e a z J a Q > e z f p > = % p J P f a = [ f D $d % 4 ( D$ ? 2 > z e a p 1 f ) @ S 4 ( D $k ? 2 > z e a p 1 f ) @ S p J P f a = [ f D$ p J P f a = [ f k % = 2 f e > z J e D $P e f & b l P e f & b l % S 4 3 C = 2 f e > z J e d d d B ? M ) ) @ S = 2 f e > z J e D$ d % ) % v S 4 [ f = J 2 = > z J e D $P e f b l P e f b l % S 4 3 C [ f = J 2 = > z J e d d d B ? M ) ) @ S p J P f b p J P f a = [ f l D$ d d d % ) v S 4 ) s 3 ) ) @ S p J P f b p J P f a = [ f l D $= 2 f e > z J e [ f = J 2 = > z J e % ) % v ) s 3 ) ) % v ) % v 4 ( Q f Q a 2 f D$ ? p J P f a = [ f ) @ S S 4 C 3 C p J P f b Q f Q a 2 f l d d B ? M ) ) @ S 4 ( D $? p J P f a = [ f ) @ S p J P f b l D$ > e Q z > = C p J P f b l 6 C E d P % ) % S f

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Q e e a p Q J e f D $J = z a 3 Q J e f % Q e e a f { z D$ ? f a M f { f p z % S 4 C C Q e e a p Q J e f 9 $P e f a p Q J e f C Q e e a f { z 9$ P e f a f { z ? M ) ) @ S P e f a p Q J e f D $Q e e a p Q J e f % P e f a f { z D$ Q e e a f { z % ? f 2 P a z J a e { % ) % v q 2 > f { z ) % v q p J P f $d ) % v Q f Q a 2 f ) % v [ f { ) % v ( J z > z f v R ( _ _ f = f e { & a Q J = z f e D S ? ) @ ) ( % J D R f Q z J e 0 % z f 2 P a z f 2 P a & D R ) ? ( % f = f e { & f = f e { & 0 a p 2 & a p 2 a > { & a > { > q P e f Q e e = J e 2 = J e 2 0 D ( ) s % p = { > e D s ) % [ q D e f > % q D ? ) g ? % ) @ S @ f z a } > z e % Q e e D$ d % P e f D $d d d d % 4 ( D$ ? B d ) @ S @ f p p f p % v > Q e > z f = z > f = f e { & z = z > { f p p f p \ f J J P p z f e f z J 2 > f p e f z > z z f = z > f = f e { & p = J = f = f { > z f z J p z > e z f = f e { & D $% \ M S s ) C f = f e { & d ) @ S f = f e { & D$ ) = f e { & C p & p % ) % = J e 2 D $d d % S Â PAGE 68 = J e 2 0 D$ d % D $d % \ M S s ) C C d d C 3 C = J e 2 = J e 2 0 9 d d d d d B ) @ S S 4 C = J e 2 0 = J e 2 ? M ) ) @ S = J e 2 D$ = J e 2 0 % ) % D $k % ( L M % p = { > e D$ 4 s 3 ) % e f Q z a } f z J [ C J p = { > e % v q z f > Q J K p p = { > e q J e z f z f e > z J = z f = p z Q z J { > p p p [ f 2 f z J [ S 4 p = { > e ? M ) ) @ S ( L M % @ > p p a p [ f C J % ) % v f Q Q e e f = z f = f e { & q f = f e { & p = Q e f > p = { Q z z f p z f P K & > q 4 ( D $? 2 > z e a p 1 f ) @ S z f 2 P a b l D$ p b l k J b l % z f 2 P a & b l D $& p b l k J b k 2 > z e a p 1 f l % ) % f = f e { & 0 D$ ) = f e { & C z f 2 P a z f 2 P a & % [ q D $f = f e { & f = f e { & 0 % f = f e { & a Q J = z f e D$ d % \ M S s ) C C f = f e { & 0 9 f = f e { & C > K p C f = f e { & 0 f = f e { & 9 $d d d C f = f e { & a Q J = z f e$ d d ) @ S [ q D $f = f e { & f = f e { & 0 % f = f e { & a Q J = z f e D$ f = f e { & a Q J = z f e k % 4 ( D $? 0 6 2 > z e a p 1 f ) @ S J b l D$ J b l 0 % ) % 4 ( D $? 2 > z e a p 1 f ) @ S z f 2 P a b l D$ p b l k J b l % z f 2 P a & b l D $& p b l k J b k 2 > z e a p 1 f l % ) % f = f e { & 0 D$ ) = f e { & C z f 2 P a z f 2 P a & % ) % v f f = f e { & D $f = f e { & 0 % 4 ( D$ ? 2 > z e a p 1 f ) @ S p b l D $p b l k J b l % & p b l D$ & p b l k J b k 2 > z e a p 1 f l % S 2

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) % v Q > Q > z f Q f = z f e J [ z f = 2 > f > = > [ p z 2 f = z a p 2 D $d % & a p 2 D$ d % a > { D $d % & a > { D$ d % 4 ( D $? 2 > z e a p 1 f ) @ S a p 2 D$ a p 2 k p b l % & a p 2 D $& a p 2 k & p b l % ) % a > { D$ a p 2 2 > z e a p 1 f % & a > { D $& a p 2 2 > z e a p 1 f % 4 ( D$ ? 2 > z e a p 1 f ) @ S p b l D $p b l a > { % & p b l D$ & p b l & a > { % ) % = J e 2 0 D $d % 4 ( D$ ? 2 > z e a p 1 f ) @ S = J e 2 0 D $= J e 2 0 k K b l % ) % = J e 2 0 D$ 3 C = J e 2 0 % ) % v f D } z > e > K f = f z J = ; p 2 f z J [ Q e e D $f = f e { & % S 4 C Q e e$ P e f ? M ) ) @ S P e f D $Q e e % 4 ( D$ ? 2 > z e a p 1 f ) @ S P e f b l D $p b l % P e f & b l D$ & p b l % ) % ) % \ ( S ? ) s C ; S z z J J ; _ ; z f e > z J = p > = [ f = f e { & $; f = f e { & 0 D d D 0 % ) % v q J e 4 ( D$ ? 2 > z e a p 1 f ) @ S p b l D $P e f b l % & p b l D$ P e f & b l % ) % ( J z > z f % \ ( S ? ) s % 4 ( D \$ ? 2 > z e a p 1 f ) @ S e z f = C p b l D d D n ; ; & p b l D d D n % ) % s > z f a f { = C q ; J f p z z f ; % s > ? f C P e f q % s > z f a ) = [ C q % ) v f = [ J q [ e f e S

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+ 6 + , 5 d Â o h u i ? c B i L h } i c @ | i 4 @ | U @ i | L t u L h ) t U t | t c A h _ | L ? c U @ i 4 U h i t t c 5 @ ? # i } L c @ u L h ? @ d 2 o h i ? c + U @ h w @ ? 6 @ h i t c # L } @ t a c 4 i h U @ ? @ ) t t c 6 u | | L ? c  5 M t ? } c L t | L ? c @ t t @ U t i | | t c Â b b c @ T | i h Â f d o # ? ? ? } c w @ h h ) @ ? @ h c ?  5 @ h T T T i h M L ? t u L h | i L 4 ? @ | L ? ? 4 M i h t L u } h @ T t | } i ? L h i h @ ? 4 ? 4 4 i } h i i d e o O L h ? c + L } i h @ ? a L ? t L ? c @ h i t + c @ | h ? @ ) t t c @ 4 M h } i N ? i h t | ) h i t t c i v L h c @ 4 M h } i N ? i h t | ) h i t t c Â b H D d D o k @ T @ ? c  u h i c @ ? U i @ U t c A h _ | L ? c _ t L ?  i t i ) c i ? L @ h c @ u L h ? @ c @ T | i h D d S o + L M i h | t c 6 h i 5 c T T i L 4 M ? @ | L h U t c h i ? | U i O @ * c ? } i L L g t c i a i h t i ) c Â b H e S e