UFDC Home myUFDC Home  |   Help
<%BANNER%>

# Continuous Approach to the Lightning Discharge

## Material Information

Title: Continuous Approach to the Lightning Discharge
Physical Description: 1 online resource (57 p.)
Language: english
Creator: Aslan, Beyza Caliskan
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2007

## Subjects

Subjects / Keywords: complete, eigenbasis, eigenproblem, electric, generalized, laplacian, lightning, maxwell, potential
Mathematics -- Dissertations, Academic -- UF
Genre: Mathematics thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

## Notes

Abstract: We develop a continuous model for the lightning discharge. We consider Maxwell's equations in three dimensions and obtain a formula for the limiting potential as conductivity tends to infinity in a three-dimensional subdomain (the lightning channel) of the modeled domain. The limit is expressed in terms of the eigenfunctions for a generalized eigenvalue problem for the Laplacian operator. The potential in the breakdown region can be expressed in terms of a harmonic function which is constant in the breakdown region.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Beyza Caliskan Aslan.
Thesis: Thesis (Ph.D.)--University of Florida, 2007.

## Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2007
System ID: UFE0019659:00001

## Material Information

Title: Continuous Approach to the Lightning Discharge
Physical Description: 1 online resource (57 p.)
Language: english
Creator: Aslan, Beyza Caliskan
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2007

## Subjects

Subjects / Keywords: complete, eigenbasis, eigenproblem, electric, generalized, laplacian, lightning, maxwell, potential
Mathematics -- Dissertations, Academic -- UF
Genre: Mathematics thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

## Notes

Abstract: We develop a continuous model for the lightning discharge. We consider Maxwell's equations in three dimensions and obtain a formula for the limiting potential as conductivity tends to infinity in a three-dimensional subdomain (the lightning channel) of the modeled domain. The limit is expressed in terms of the eigenfunctions for a generalized eigenvalue problem for the Laplacian operator. The potential in the breakdown region can be expressed in terms of a harmonic function which is constant in the breakdown region.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Beyza Caliskan Aslan.
Thesis: Thesis (Ph.D.)--University of Florida, 2007.

## Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2007
System ID: UFE0019659:00001

Full Text

CONTINUOUS APPROACH TO THE LIGHTNING DISCHARGE

By

BEYZA CALISKAN ASLAN

A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA

2007

2007 B.--I Calihkan Asian

To my parents,

Kamile Qallkan and Yusuf Qahskan,

and to my husband,

Omer Deniz Asian

ACKNOWLEDGMENTS

First of all, I would like to express my gratitude to my advisor, Professor William

W. Hager. Without his encouragement, consistent support and guidance, this dissertation

could not have been completed. I am grateful to have had the opportunity to study

under such a caring, intelligent, and energetic advisor. His confidence in me will ahv---

encourage me to move forward on my research.

Second, I would also like to thank Dr. .1i1-,d.-.p Gopalakrishnan, Dr. Shari Moskow,

Dr. Sergei S. Pilyugin, and Dr. Vladimir A. Rakov for serving on my supervisory

committee. Their valuable -Ir.-. -Ii..i, have been very helpful to my research.

Third, thanks go to my officemates (Dr. Hongchao Zhang, Dr. Shu-Jen Huang, and

Sukanya K i-li:i ,i'.--v i), and all colleagues and friends in the Department of Mathematics

at the University of Florida. Their c~ rl'' -: alleviated the stress and frustration of this

time.

Last, but not least, I wish to express my special thanks to my family: to my husband,

Deniz, for his love and his endless support to pursue and complete my degree; to our

daughter, Erin Basak, for being a glorious joy to us; to my parents for their immeasurable

support and love; to my parents-in-law for their wholehearted understanding and

encouragement; and to my brother for his unstopping support and encouragement.

Without their support and encouragement, this dissertation could not have been

completed successfully.

page

ACKNOWLEDGMENTS ...............

LIST OF FIGURES ...................

ABSTRACT .......................

CHAPTER

1 INTRODUCTION .................

2 LITERATURE REVIEW ............

2.1 Lightning Models with Explicit Lightning (C!
2.1.1 Helsdon's Model ............
2.1.2 MacGorman's Model .........
2.1.3 Mansell's Model ............
2.2 Hager's Model: The Discrete Model .....

3 THE DISCRETE MODEL .............

3.1 Governing Equations .............
3.1.1 Maxwell's Equations .........
3.1.2 Derivation of the Governing Equations
3.2 The Model in One-dimension .. ......
3.3 Generalization to Three-dimension .....

4 THE CONTINUOUS MODEL .. ........

4.1 Formulation of the Equations .. ......
4.2 Eigenproblem for A .............

5 GENERALIZED EIGENPROBLEM FOR THE L.

5.1 Introduction .. .. .. ... .. .. .. ..
5.2 Eigenfunctions of Type 1, 2, and 3 .....
5.3 Reformulation of Eigenproblem in 7- Using D
5.4 Eigenvalue Separation and Completeness of E

6 THE LIMIT .....................

6.1 Introduction .. .. .. ... .. .. .. ..
6.2 Reformulation of the Continuous Equation .
6.3 Potential C!i ,. for the Continuous Equatio

7 APPLICATION TO ONE-DIMENSION .....

7.1 Application of the Generalized Eigenproblem

ill., . . .

PLACIAN . .

'ouble-Layer Potential
.igenfunctions ..

n . . .
. . . .

7.2 Application of the Continuous Model ................ .... .. 48

8 CONCLUSIONS ............... ................ ..52

REFERENCES .. . ....... .. ........ ....... 54

BIOGRAPHICAL SKETCH ............... . . ..57

LIST OF FIGURES
Figure page

4-1 A sketch of L and Q for a lightning discharge .................. 20

7-1 Eigenfunctions in -F in one dimension .................. .... 49

Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy

CONTINUOUS APPROACH TO THE LIGHTNING DISCHARGE

By

B.-i- Cahlkan Asian

August 2007

C'!h wi: William W. Hager
Major: Mathematics

We develop a continuous model for the lightning discharge. We consider Maxwell's

equations in three dimensions and obtain a formula for the limiting potential as

conductivity tends to infinity in a three-dimensional subdomain (the lightning channel)

of the modeled domain. The limit is expressed in terms of the eigenfunctions for a

generalized eigenvalue problem for the Laplacian operator. The potential in the breakdown

region can be expressed in terms of a harmonic function which is constant in the

breakdown region.

CHAPTER 1
INTRODUCTION

Lightning is one of the most beautiful di-p-v'1 in nature, however, it is also

frightening. It can destroy buildings and even kill people. It is a costly as well as deadly

natural event that mankind can not avoid.

The fear and respect for lightning attracted many people's attention over the years.

Tod iw, the physical processes involved in lightning are the focus of intensive research

throughout the world. Lightning is a result of charge separation inside a cloud. As the

graupel and ice particles within a cloud grow in size and increase in number, under the

influence of the wind, collisions between them may occur resulting in charge exchanges

between the particles. In general, smaller particles acquire positive charge, while larger

particles acquire negative charge. The charge separation occurs when these particles

separate under the influence of updrafts and gravity, and as a result, upper portion of the

cloud becomes positively charged and the lower portion of the cloud becomes negatively

charged. This results in huge electrical potential difference within the cloud as well as

between the cloud and the ground causing a flash to occur moving charges between

positive and negative regions of a thunderstorm.

Detailed history of early lightning research can be found in Uman [29]. Be-,i i,,ii

Franklin was the first person who performed a scientific study of lightning. In the second

half of eighteenth century, he designed an experiment that proved the lightning was

electrical. It was after photographic and spectroscopic tools became available towards the

end of the nineteenth century that more studies about lightning started being conducted.

Lightning current measurements were first proposed by Pockels [26-28]. He estimated the

amount of current by analyzing the magnetic field induced by lightning currents. Later,

Wilson [32, 33] was the first researcher to use the electric field measurements to estimate

the structure of thunderstorm charges involved in lightning discharges. He won the Nobel

Prize for inventing the Cloud C'i i,: her to track high energy particles and made 1i i ir

contributions to our present understanding of lightning. Lightning research has been

particularly active since about 1970. This increased interest was motivated by

The damage to aircraft or spacecraft due to lightning,

Vulnerability of solid-state electronics used in computers and other devices,

Development of new techniques of data taking and improvement of observational
capabilities.

Most lightning research is done by physicists, chemists, meteorologists, and electrical

engineers. Hager [11, 13, 14] was the first mathematician using Maxwell's equations to

develop a three-dimensional mathematical model to simulate a lightning discharge. His

discharge model [14] was obtained by discretizing Maxwell's equations to obtain a relation

between the potential field and current density due to the motion of charged particles

under the influence of the wind. Spatial derivatives in his equation were approximated

by using volume elements in space, while the temporal derivatives were estimated by

a backward Euler scheme in time. Since conductivity is very large in the region where

the electric field reaches the breakdown threshold, he evaluated the solution limit as the

conductivity tends to infinity in the breakdown region. In his model [14], the output was

the electric field as a function of time, and the inputs were currents generated by the flow

of charged particles within the thundercloud under the influence of the wind.

This dissertation is based on Hager's mathematical model. Some improvements are

made compared to Hager's earlier work. For example, the solution is computed without

discretizing the equations. Consequently, we do not have huge matrix systems to compute

and therefore it is computationally much more efficient and less expensive.

CHAPTER 2
LITERATURE REVIEW

Numerous studies in lightning from different aspects have been reported in the past

few decades. This review is focused on different approaches for the change in charge due to

lightning. The neutralization of charge by lightning in the models with explicit lightning

channels is discussed in Section 2.1. The approach used by Hager et al. is discussed briefly

in Section 2.2, and studied further in ('! Ilpter 3.

2.1 Lightning Models with Explicit Lightning Channels

2.1.1 Helsdon's Model

Helsdon et al. [15-18] estimated both the geometry and charge distribution of an

intercloud lightning flash in a two-dimensional Storm Electrification Model (SEM)which

has been extended to a three-dimensional numerical cloud model later. Adapting ideas

from Kasemir [19], the parameterized lightning propagated bidirectionally (initially

parallel and antiparallel to the electric field) from the point of initial breakdown and

developed segments of opposite charge polarity.

Initiation, propagation direction, and termination of the discharge were computed

using the magnitude and direction of the electric field vector as the determining criteria.

The charge redistribution associated with lightning was approximated by assuming that

the channel remained electrically neutral over its total length.Their discharge followed

the electric-field lines until the termination condition was satisfied. Therefore, their

parametrization produced a single, unbranched channel.

As an initial criteria, a threshold of electric field of 400 kV/m was chosen. The

channel was extended in both directions along the field line until the ambient electric-field

magnitude fell below a certain threshold (150 kV/m) at the locations of the channel-termination

points. They assumed that the linear charge density at a grid point, P, along the channel

was proportional to the difference between the potential at the point where the discharge

originated, and the potential at P. The linear charge density can be given by

Qp -k(p Do),

where Qp was the charge density at P, and 4)p and 4o were the potentials at P and the

initiation point of the discharge, respectively. The value of this proportionality constant k

controlled the amount of charge transferred by the discharge. They extended the channel

by four grid points at each end and adjusted the charge distribution at each end of the

channel in order to maintain charge neutrality over the channel. In this extended region,

they assumed that the charge density decreased like e-,"2, where x is the distance from

the channel.

2.1.2 MacGorman's Model

MacGorman et al. [23] slr.-.- -1 .I a lightning parametrization that was considered

an extension of the parametrization of Helsdon et al. [18] in conjunction with some of

the bulk-lightning parametrization methods presented by Ziegler and MacGorman [34].

MacGorman et al. [23] developed a parametrization to enable cloud models to simulate

the location and structure of individual lightning flashes by using the conceptual model of

MacGorman et al. [22] and Williams et al. [31]. Their parametrization proceeded in two

stages. Using the ideas of Helsdon et al. [18], a flash traced the electric-field line from an

initiation point outward in both parallel and antiparallel directions until the magnitude

of the ambient electric field at each end fell below some certain threshold value. When

one end of the channel reached ground, the parametrization terminated at that end, but

allowed the other end to continue developing.

('C! Irge estimation and neutralization were parameterized by applying the technique

proposed by Ziegler and MacGorman [34], except that Ziegler and MacGorman neutralized

charge at all grid points having Ip(i,j, k) > pi (where p(i,j, k) was the net charge

density at the grid point (i,j, k) and pi was the minimum |p(i, j, k) for all grid points

to be involved in lightning beyond initial propagation) throughout the storm, but their

parametrization neutralized charge only at such grid point within a single localized flash.

2.1.3 Mansell's Model

Mansell et al. [24] proposed a lightning parametrization derived from the dielectric

breakdown model that was developed by Niemeyer et al. [25] and Wiesmann and Zeller

[30] to simulate electric discharges. They extended the dielectric breakdown model to

a three-dimensional domain to represent more realistic electric field and thunderstorm

dynamics.

In their work, the stochastic lightning model (SLM) was an application of the

Wiesmann-Zeller model to simulate bidirectional discharges in the regions of varying net

charge density (e.g., in an electrified thunderstorm). Procedures for simulating lightning

flashes in the thunderstorm model were as follows. A flash occurred when the magnitude

of the electric field exceeded the initiation threshold Eiu anywhere in the model domain.

The lightning initiation point was chosen randomly from among all the points where

the magnitude of the electric field is greater than 0.9Einit. Both decisions for choosing

the initiation threshold and the initiation point were made according to MacGorman et

al. [23]. Positive and negative parts of the flash were propagated independently so that

up to two new channel segments (positive and negative) could be added at each step.

Both ends had default initial propagation thresholds of 0.75Einit. For flash neutrality,

they applied the ideas from Kasemir [19] and assumed that the channel structure would

maintain overall charge neutrality as long as neither end reached the ground. But, for

computational simplicity, their parametrization maintained near-neutrality (within 5' ) by

a technique of adjusting the reference potential to the growth of the lightning structure

2.2 Hager's Model: The Discrete Model

Hager et al. [11, 13, 14] proposed a three-dimensional lightning-discharge model that

produced bidirectional IC and -CG flashes. The model generated the discharge region,

charge transfer, and detailed charge rearrangement associated with the flash.

Their approach to lightning was quite different from those in Section 2.1. Their

breakdown model was based on Maxwell's equations. They assumed that current due to

transport of charge under the influence of wind was known. They obtained an equation

governing the evolution of the electric potential under the assumption that the time

derivative of the magnetic field can be disregarded. After integrating this equation over

boxes and approximating derivatives by finite differences, they obtained an implicit system

of difference equations describing the evolution of the electric field. Their approach to

lightning was to let the conductivity tend to infinity wherever the electric field reached the

breakdown threshold. This approach appeals to our basic conception of nature: When the

electric field reaches breakdown threshold, conductivity becomes very large as a plasma

forms.

When the electric field reaches the breakdown threshold, the electric potential changes

instantaneously everywhere within the thundercloud. The Inverse Matrix Modification

Formula [10] was applied to evaluate this change:

After Ibefore A- U(UTA-1U) -1UTbefore, (2-1)

where before was the electric potential before discharge, after was the electric potential

after discharge, A was the discrete Laplacian, and U was a matrix with a +1 and -1 in

each column corresponding to the arcs associated with the breakdown. There were no

parameters in Equation (2-1) besides the electric potential before discharge. This was

consistent with experimental observations: The charge is controlled predominately by a

single parameter: the local electrostatic field. This was observed in experiments reported

by Williams et al. [31].

CHAPTER 3
THE DISCRETE MODEL

3.1 Governing Equations

3.1.1 Maxwell's Equations

Maxwell's equations are a set of four equations, first written down in complete form

by physicist James Clerk Maxwell, that describe the behavior of both the electric and

magnetic fields. Maxwell's four equations express how electric charges produce electric

fields (Gauss's law), the experimental absence of magnetic monopoles, how currents and

changing electric fields produce magnetic fields (Ampre's law), and how changing magnetic

fields produce electric fields (FEaYd'i's law of induction).

In the absence of magnetic or polarizable media, the differential form of these

equations are:

1. Gauss' law for electricity: V E =

2. Gauss' law for magnetism: V H = 0
9B
3. Faraday's law of induction: V x E =
at
9~E
4. Ampere's law: V x H = Jo + E-

where E is the electric field, H is the magnetic field strength, B is the magnetic flux

density, p is the charge density, E is the permittivity of air, Jo is the current density, V. is

the divergence operator, and Vx is the curl operator.

3.1.2 Derivation of the Governing Equations

By Ampere's law, the curl of the magnetic field strength H is given by

dE
Vx H Jo + a (3-1)
at *

Since Jo is partly due to the movement of charged ice and water particles in the cloud and

partly due to the electrical conductivity of the cloud, we write

Jo Jp + aoE

where or is the conductivity of the atmosphere. In this model, we assume the time

derivative of the magnetic flux density is zero, i.e.,

9B
0.
at

Hence, the curl of E vanishes by Faraday's law and E is the gradient of a potential Q:

E -Vl.

Therefore (3-1) becomes

V x H = Jp aoV0

avo
at

(3-2)

Taking the divergence of (3-2), it follows that

0 -VVxH V-Jp -V-a oV

avt
at

(3-3)

Letting a = ao/ and J = Jp/E, we obtain

V +aV V- V J 0.
at't

(3-4)

In our model we also have the following assumptions:

Let EB be the breakdown field strength. Then the electric field magnitude is alvb--,
less than or equal to the breakdown threshold EB. That is, |E| < EB.

When the electric field reaches the breakdown threshold EB at some point, the
conductivity tends to infinity in a small neighborhood of that point. That is,
a(x) oo wherever |E(x)| = EB.

3.2 The Model in One-dimension

To illustrate the ideas used to obtain (2-1) in Section 2.2, we now focus on the

equation (3-4) in one dimension. In one-dimension, (3-4) reduces to

" + (,7')' = o, 0(0) 0, (H) = V,

(3-5)

where H is the distance to ionosphere and the domain is from the surface of the earth

up to the ionosphere. the domain [0, H] is discretized using N discretization points. Let

h = H/N be the mesh spacing, xi be the ith grid point, 4i be the approximation to

potential at xi + h/2, ai be the conductivity at xi, and Ji be the current density at xi.

Integrating (3-5) over [x', xi+,] gives

((x+i) )'(Xi)) + (i+1'(X+) i4'(X)) (J+ Ji) 0

To approximate 0', [11, 13, 14] used backward differences,

wci te-1
0((Xi) i ,
h

which yield the discrete equation

i hi-1 \ +1 i
h h

This can be written in matrix form as

> A + B4 = hAJ

where

-1

2
So

and

-1 + -2 -72

-72 92 + a -3 3

-a3 U3 + 4 -94

-O4 a4 + 5
S S

(i Oi-1 -
o"- I -- (J4+1 J) 0
&I

Si+1 i
h h

Integrating the matrix equation over the interval [0, At] gives

4(t + At) e-A Bt (t) + h C-A-1B(zt-s)AJ(s)ds

If = Eb, then Hager lets Jk -- Tk + 7 where 7 -+ +oo. It is shown that
h
as 7k k-- k + T, B B + TwwT, where ,' = 1, ',,+1i -1, = 0 otherwise. Next,

they calculated the limit as 7 -- o0 and At -+ 0 and obtained the formula for the potential

after the lightning:

4(t+) (t) A- w(wTA- w)-wT(t).

3.3 Generalization to Three-dimension

Suppose =-l b Eb at k = ko, k,... ki. For each such k, k -- +00,

B -+ B + Y wT WT.
i=0
Here, wi is zero except for +1 and -1 in components ki and ki+l. Taking limits as

-Ti +oo and At 0 yields

4(t+) = )(t) A-1WI(WTA-'1W)- W 4t)

where WI = [wol... wi].

CHAPTER 4
THE CONTINUOUS MODEL

4.1 Formulation of the Equations

By (3-4), the following equations model the evolution of the electric potential in a

domain Q, assuming the time derivative of the magnetic flux density can be neglected:

aA-V (aV) + V J in 2x [0, o), (4-1)
at
0(x, t) 0, (x, t) e 09 x [0, oo), (4-2)

(x, 0) -= o(x), x e (4-3)

where 2 C PR is a bounded domain with boundary 02, a > 0 lies in the space L(Q2)

of essentially bounded functions defined on 2, and the forcing term J lies in L2(2), the

usual space of square integrable functions defined on 2. The divergence V J as well as the

equation (4-1) are interpreted in a weak sense, as explained later. The initial condition

Qo is assumed to lie in H o'(), the Sobolev space consisting of functions which vanish on

0L and with first derivatives in L2(2). The evolution problem (4-1)-(4-3) has a solution

0(x,t) with (., t) E Hol(0) and the partial derivative At(-, t) E H0o'(). Although
Maxwell's equations describe the electromagnetic fields in 3-dimensions, the mathematical

analysis throughout this dissertation is developed in an n-dimensional setting, where n is

an arbitrary positive integer.

In a thunderstorm, a is the conductivity divided by the permittivity of the atmosphere,

2 is any large domain extending from the Earth to the ionosphere which contains the

thundercloud in its interior, and J is due to transport by wind of charged ice and water

particles in the cloud. Although J is a function of time, we focus on the potential change

during lightning, which we consider infinitely fast. Hence, during a lightning event, J is

essentially time invariant. The potential of the ionosphere is not zero, however, we can

make a change of variables to subtract off the "fair-field potential" (the potential of the

atmosphere when the thundercloud is removed) and transform the problem to the form

(4-1)-(4-3) where the potential vanishes on 9Q.

A possible lightning channel is sketched in Figure 4-1. Mathematically, the lightning

Figure 4-1: A sketch of L and 2 for a lightning discharge.

channel L could be any connected, open set contained in 2 with its complement C = 2\

connected. More realistically, we should view L as a connected network of thin open tubes.

The case where L touches 02, as would happen during a cloud-to-ground flash, is treated

as a limit in which L approaches arbitrarily closely to the boundary of 2.

Making the change of variables 0 = (-A)- ~, we rewrite (4-1) as

A, + V J, (4-4)
at

where

A, -(-A)-7(V.(aV))(-A) 2

Here (-A)-2 : L2() H- Ho(2) denotes the inverse of the square root of the Laplacian

([3]).

Let T be the characteristic function for L (T is identically 0 except in L where T is

1). The effect of lightning is to ionize the domain in essence, replacing a in (4-4) by

a + 7r where r is large. If the lightning occurs at t = 0, then in the moments after the

lightning, the electric potential is governed by the equation

S-(A, + TAv) + V. J, (4-5)

where

A, -(- A)-(V.7 (TV))(-A)-2,

in 2 x [0, oc) subject to the boundary conditions (4-2) and (4-3). Here the scalar r reflects

the change in conductivity in the lightning channel.

Let 4,(x, t) denote the solution of (4-5) at time t, and let ,-(x, t) = (-A)-'-,(x,t)

be the corresponding solution of (4-1). If the lightning occurs at time t 0, then the

electric potential right after the flash is given by

lim lim r (x,t). (4-6)
t- + -Too

Using an eigendecomposition for Ap, the limit (4-6) can be evaluated.

4.2 Eigenproblem for A1

In this section, we show ([2]) that the eigenproblem for Ap reduces to the following

generalized eigenproblem: Find u E Hi(0), u / 0, and A E IR such that

(Vu, Vv), A(Vu, Vv)n (4-7)

for all v e Hs(Q), where (, .)n is the L2(Q) inner product

(Vu, Vv)n Vu Vv dx. (4-8)

We view Hi(Q) as a Hilbert space for which the inner product between functions u and

v C Hi(Q2) is given by (4-8).

The weak form of the eigenproblem for A, is to find U e L2(2) such that

(A, U, V) ((V)(-A)- U, V(-A)- V), A(U, V)Q (4-9)

for all V E L2(Q). Let u = (-A)-1/2U and v = (-A)-1/2V denote the corresponding
functions in H0(/). Consequently, (4-9) reduces to the problem of finding u H0(Q/ ) such
that
(Vu, Vv) = A((-A) ~, (-A)'v) (4-10)

for all v c H(Q). If u c Co-(Q), then

((-A)> (-A)v)n -(Au, v)A = (Vu, Vv)n. (4-11)

Since Co"(Q) is dense in H (Q) and the operators (-A)i and V are both bounded in
HO(Q), the identity (4-11) is valid for all u c H1(Q). Hence, (4-10) reduces to (4-7).

CHAPTER 5
GENERALIZED EIGENPROBLEM FOR THE LAPLACIAN

This chapter is based on the paper [2].

5.1 Introduction

Our analysis identifies four classes of eigenfunctions for the generalized eigenproblem

(4-7):

1. The function H which is 1 on L and harmonic on 2 \ L; the eigenvalue is 0.

2. Functions in H (Q) with support in 2 \ L; the eigenvalue is 0.

3. Functions in H/(0) with support in L; the eigenvalue is 1.

4. Excluding H, the harmonic extension of the eigenfunctions of a double li. r potential
on 9. The eigenvalues are contained in the open interval (0, 1). The only possible
accumulation point is A = 1/2.

5.2 Eigenfunctions of Type 1, 2, and 3

In this section, we derive the eigenfunctions of types 1, 2, and 3. By (4-7), we have

A (Vu, Vu) (Vu, Vu) )
A =w(5+1)
(Vu, Vu)n (Vu, Vu)L + (Vu, Vu) '

which implies that 0 < A < 1. Let Ho([) C Ho(Q) denote the subspace consisting of

functions with support in L. Similarly, let Ho([c) C Hol() denote the subspace consisting

of functions with support in Lc.

Proposition 1. A = 1 and u E H (Q) is an eigenpair of (4-7) if and only if the support

of u is contained in L. If u E H (c)0, then u is an .:, ,. function of (4-7) corresponding to

the .: ,': ;in .l, 0. The only other .:, ,function of (4-7) corresponding to the ':, 'o.: ;. : 0,

which is or'l,. ',j.,,.'1 to H([ c), is the solution HI c Ho(') of

(VI, Vv)n 0 for all v E Hol(), H = 1 on L. (5-2)

Proof. If A = 1 and u E Ho() is an eigenpair of (4-7), then by (5-1), we have

(Vu, Vu)c = 0.

Hence, Vu = 0 in c, which implies that u is constant in L since L' is connected. Since

u E H1J(), u 0 in c. Conversely, if u 0 in Lc, then by (4-7), u is an eigenfunction

corresponding to the eigenvalue 1. If u 0 in L, then u is an eigenfunction corresponding

to the eigenvalue 0. The solution H of (5-2) is an eigenfunction of (4-7) corresponding to

the eigenvalue 0 since VH = 0 in .

Let w E H1(Q) be any eigenfunction of (4-7) corresponding to the eigenvalue 0 which

is orthogonal to H1(Lc). By (4-7), we have (Vw, Vw) = 0, which implies that Vw 0

in L, or w is constant in L since L is connected. Without loss of generality, let us assume

that w 1 in L. Since w is orthogonal to the functions v c H[(c), we have

(Vw, Vv)Q 0 for all v H(C).

Combining this with (5-2) gives

(V(w n),Vv)n 0 for all ve HC (.

Since 0I = 9fQ U 9i and since w H vanishes on both fQ and 9, it follows that

w = H. O

5.3 Reformulation of Eigenproblem in -H Using Double-Layer Potential

Proposition 1 describes eigenfunctions of type 1, 2, and 3. In this section, we focus on

type 4 eigenfunctions. Let H- be the space which consists of all u E H (Q) satisfying the

conditions

(Vu, Vv)n 0 for all v H0() and (5-3)

(Vu, Vw)n 0 for all w E H(Lc). (5-4)

H is a subspace of H (Q) consisting of functions harmonic in L and Lc (Au = 0 in L and

Au = 0 in c). Note that II e U. Since H0() and H0(c) are orthogonal with respect to

the Ho(Q) inner product, and since 1H is the orthogonal complement of Ho() E Hl(c) in

H01(), we have the orthogonal decomposition

H'(Q) = E H'() H0'(c).

The following series of lemmas reformulates the generalized eigenvalue problem (4-7) on R
in terms of an integral operator.

Lemma 1. u cE is a solution of the generalized .:i. ',,.'l. 11 ,, (4-7) if and only if

O- A -O on (L, (55)
On [9n\

where
S E H-1/2 ().
on] an On
Here n is the outward unit normal to L and the and + refer to the limits from the
interior and exterior of L -1./,' ;

Proof. First we show a generalized eigenpair also satisfies (5-5). By (4-7) we have

(Vu, Vv) =- A(Vu, Vv)n Q A ((Vu, Vv)Lc + (Vu, Vv)L) (5-6)

for any v c Ho'(Q). Integrating by parts and utilizing the fact that u is harmonic in both
L and Lc gives
f u- ou+ f u-
SV d -jA v d7 + A v- d', (57)
JL On JQ On 11L On
where 7 denotes the boundary measure on ai. Hence, we have

[F u +A dOu+ u
JQC Ln On n
for any v c Hd(Q). Since any v E H1/2(aO) has an H{(Q) extension, (5-5) holds.
Conversely, suppose that u satisfies (5-5). As in (5-6)-(5-7), we have

A(Vv, Vu)Q --A v d7+AJ vJU d7
far On J" On

Applying (5-5) gives
A(Vv,Vu)Q v O- (Vv, Vu)

since u is harmonic in L. Hence, u satisfies (4-7). D

Now let us introduce the Green's function on Q:

AyG(x, y)= 6,(y) in Q, G(x, y) = 0 for ye 9, (5-8)

where 6x is the Dirac delta function located at x. The piecewise harmonic functions u CE 7
can be described in terms of the jump on L of the normal derivative.

Lemma 2. Suppose that 9 and 89 are C2. If u cE -, x cE and x 9i, then

u(x) "a I (y)cG(x, y)d^y. (5-9)

Proof. Since u is harmonic away from 9, it is continuous there, and so for x 9,

u(x) = I u(y)AyG(x, y)dy

= u(y)AG(x,y)dy+ j u(y)AyG(x,y)dy.
SLc

Since u is smooth in each subdomain, we can integrate by parts to obtain

u(x) = Vu VGdy + U dyd Vu -VGdy- u dy.

Since u is smooth on each subdomain and u E H1(Q), the traces u+ and u- e H1/2(aL)
must satisfy u- = u on 9. Since Au = 0 on each subdomain and G = 0 on 89, we have

u(x) = Vu VGdy Vu VGdy
JI Jc
u (y)G(x, y)dy,+ / u (y)G(x, y)dy,
Jwh On l 5 O9n
= [ (y) G (x, y) d- (5-10)

which yields (5-9). D

The following Lemma is well known for free space potentials (see for example
Theorem 3.22 of [8]); we state it here for the case of our Green's function corresponding to
a bounded outer domain.

Lemma 3. Suppose E H1/2([) and both a and OQ are C2. For x E Q, x f 9 let

v(x) be /. F, .1 by

S anC
v~x) = (y) 4){y {(x,y)dy,.

The trace v+ of v onto 9 from the exterior of L and the trace v- of v onto 9 from the
interior L are given by

1 OG
v+(x) =- (x) + (y) (x, y)d
2 J i I Yn,

and

v-(x) = -(x)W + 0(y)() (x,y)dy,.
2 J+ J a ay
Proof. Let N(x, y) be the free space Green's function for the Laplacian,

|X y>2-n
Sn > 2,
N(x, y) (2=) (5-11)
-- log Ix yl n = 2,
27r
where w is the surface area of the unit sphere in R". Recall that

avN(x, y) = 6,(y).

Define

H(x,y):= G(x, y)-N(x,y).

By the definition of v,

v(x)= y) (x, y)dy + 0(y) (x,y)dy,. (512)
JQC any JQC any

For any x E Q, H satisfies

AH(x, y) = 0 for y E ,

H(x, y) = -N(x, y) for y e 9.

Hence, H(x, y) is harmonic for y E Q with smooth boundary data. This implies that the

function w(x) defined by

w(x) = 0(y) OH(x, y) dy

is continuous in a neighborhood of 0i since the kernel has no singularity. For the second
term of (5-12) we can apply the well known result (see [8]) for the limit x -- a0. iFrom

the exterior of we have

O "H ) ON
v+(x) = (y) (x, y)dy,- -b(x) + (y) (x, y)dy,
at ha 2 OG Bu
SOW + (y) n (x, y)dy,.
z'\ Jd any

The proof for the interior limit is similar. D

Using Lemma 1, 2, and 3, we reformulate the generalized eigenproblem (4-7) on

R- in terms of a boundary integral operator. By the trace theorem [1, Thm. 7.53], any
u c R c Ho'() has a trace on 9A in H1/2 (A). Conversely, u c H1/2(/[) has a unique

harmonic extension into both L and [c with u = 0 on a9. Hence, there is a one-to-one

correspondence between elements of and elements of H1/2(/[).

Define

T: L2(a!) L- L2(a2)

by
T(x)= (y)K(x, y)d K(x, y) : (x, Y). (5-13)

By [8, Prop. 3.17], K is a continuous kernel of order n 2 on 9[. It follows from [8, Prop.
3.12] that T is a compact operator from L2( /) to itself.

Proposition 2. If both a and OQ are C2, then (u, A) cE H x R is a generalized .,. ,l"'.:'
for (4-7) if and only if the corresponding u E H1/2(a) is an .u ,. function ofT with
associated .:I ,.'', ; l;,. 1/2 A; that is,

Tu = (1/2 A)u. (5-14)

Proof. First, let us assume that (u, A) E -H x R is a generalized eigenpair for (4 -7). By
Lemmas 1 and 2, we have

Au(x) U (y)G(x,y) dy

for x E Q and x 5. We integrate by parts to obtain

Au(x) = Vu(y) VG(x, y)dy
Jc
u(y) (x, y)d y + u(y)AG(x, y)dy.

If x e [C, then the second term above disappears, and we have

Au(x)= () (x,y)d-

an equation for a double liv-r potential. We let x E [C approach L. According to
Lemma 3,
1 OG
Au(x) = u(x) (y) x, y)d-
2 uYn,
which is equivalent to (5-14).
Conversely, suppose that u E H1/2 ([) satisfies (5-14). We identify u with its
harmonic extension in H-, and we define w(x) by

u (x, y)dy for x e C,
W(X) = (5-15)
u(y) (x, y)dy, + u(x) for x e L.
/1 aGny

In either I' and w is harmonic. By Lemma 3, we have

w(x)= u(x) u(y) (ary)dcy
W 21 JaL (x, y) dy

and

w- () = ux)- () (x, y)dy,
1 OG
-u2() u (y) an (x,y)dy,.

Utilizing (5-14) yields

w+ w- (1/2- T)u = Au on 9. (5-16)

Observe that w vanishes on 9Q due to the symmetry of G(x, y) [8, Lem. 2.33]; that
is, since G(x, y) = 0 when y E 0Q, we have by symmetry G(x, y) = 0 when x E 0Q.
Hence, the normal derivative in (5-15) vanishes when x cE O. Since w is harmonic in each
subdomain and it is equal to Au on both 9 (see (5-16)) and 0Q (they both vanish), it
follows that w = Au in Q. We replace w with Au in (5-15) to obtain

j u(y) (.x,y)dy, for x e C,

a u )(y)G(x,y)d( (VudfG),
\u( (xy if (5x 17)

) f 9 u
Au(x) (y) G(x, y)d (5 8)
JQC 8n~

By Lemma 2,

Au(x) =A "~n (y)G(x,y)d7,. (5-19)

Subtracting (5-19) from (5-18) gives

s(x): j (y)G(x,y)dy= 0 for any x a[,
J8

where

O( -u (y) (?)- [A ] (y).

Hence, s = 0 almost everywhere in Q. If =- 0, then

(9u(- 9u
an Ln

in which case Lemma 1 completes the proof.

To prove that = 0, suppose to the contrary that 0 does not vanish. Let r be any
smooth function defined on aL for which

j (y)r(y)d7y/ 0.

Let r also denote any smooth extension in Q which vanishes on a0. By the symmetry of

G, we have

r(y) I' [Axr (x)]G(x, y)dx.

Forming the L2(Q) inner product between s (which vanishes almost everywhere) and Ar

yields

0 (s, Ar)n- (y) [Axr(x)]G(x, y)d- Jr (, r)aL / 0.

Hence, we have a contradiction and the proof is complete. E

Corollary 1. If both aO and OQ are C2, then the ,. :'ii ,,.i, I of the double '7.;,. potential

operator T in (5-13) are real and contained in the half-open interval (-1/2, 1/2]. The only
possible accumulation point for the spectrum is 0.

Proof. The eigenvalues of the generalized eigenproblem (4-7) are all real due to symmetry

of the inner product. By Proposition 2, the eigenvalues of T are all real. As noted before

Proposition 1, the eigenvalues of (4-7) are contained on the interval [0, 1]. Moreover, by

Proposition 1, the only eigenfunction corresponding to the eigenvalue 1 has support in L.

The trace of this eigenfunction on 9iC is 0. The only element in R- with vanishing trace on

at is the zero function. Consequently, there is no eigenfunction in R- corresponding to the

eigenvalue 1. There is one eigenfunction in R- corresponding to the eigenvalue 0, namely

the function H of Proposition 1. Except for the eigenvalue 0, all the remaining eigenvalues

for the generalized eigenproblem lie in the open interval (0, 1). Since the eigenvalues of

T are 1/2 minus the corresponding eigenvalue of (4-7) in [0, 1), the proof is complete.

Since T is compact on L2(Q) [8, Prop. 3.12], the only possible accumulation point for the

spectrum is 0. O

A lower bound for the separation between the largest and second largest eigenvalues

of T is obtained from Proposition 3.

5.4 Eigenvalue Separation and Completeness of Eigenfunctions

Due to Proposition 1, the generalized eigenproblem (4-7) restricted to R- has a

simple eigenvalue A = 0 corresponding to the eigenfunction HIE while the remaining

eigenvalues are positive. By Proposition 2, the only possible accumulation point for the

spectrum is A = 1/2. Hence, there is an interval (0, p), p > 0, where the generalized

eigenproblem has no eigenvalues. We now give an explicit positive lower bound for p in

terms of three embedding constants:

El. Let Ua denote the constant function on 2 whose value is the average of u E H1(2)
over L:
Ua measure-t) f u(x)dx.
measure(L) I
By [7, Thm. 1, p. 275], there exists a constant 01 > 0 such that

Vfr al ) > 01 HI(U

for all u c H1(Q).

E2. By [1, Thm. 7.53], there exists a constant 02 > 0 such that

|H ||H (>) 2 2 H1/2(ac)

for all u c H1(Q).

E3. There exists a constant 03 > 0 such that

I 1/|2I/ >( 0 H llHi(c (5-20)

for all u E HiJ(Q) which are harmonic in c (in other words, (5-4) holds). The
following proof of E3 was sil-'-, -1'. by .1 v-'- p Gopalakrishnan: For u E H (Q),
let T(u) e H1/2(L) denote the trace of u evaluated on ai. By [9, Thm. 1.5.1.3],
T has a continuous right inverse which we denote T-1. In other words, for each
g e H1/2(a[), we have T-l(g) e H'(Q), TT- (g) and

IT -(g)IHI(Q) < i-1 H1/2(A)*
Define vo u-T-l(g). Since vo vanishes on both 0Q and 9, there exists a constant
c > 0 such that (see [7, Thm. 3, p. 265])

'I,, H1(c ) c< c IV ,,, LL2(L). (5-21)
Moreover, since u is harmonic in [c and i,, vanishes on both 0Q and [, we have

(VT,, V,,)>, (V,, V(u T- (g)))c -(V,, VT- (g))Lc
< lVvo oILc2(C) Ir-(g) IL2(c)
< |V ,,|L2(C.L )iT- (- )||HI(c),
which gives ||V,, lL2(L) < I 1VT-'(g)11H1(Lc). We combine this with (5-21) to obtain

',,I HI(oc) < c 1 T-l (g)llH (o)-
Hence, by the triangle inequality,

I|u|HI( c) < ,,11 HI(c) + 7-1'(g\)\H1Lc)
< (1 +c)llr-l(g)llH- (c)

< ((1+ c)7 -1|) gH/2(),

which yields (5-20).

Proposition 3. If both a and OQ are Lipschitz, then the generalized .:j, ig,'l. 'i,, (4-7)

has no .:j, n' in the interval (0,p) where

p =min{l, 0203}01/2.

Proof. Let p be the smallest positive eigenvalue for the generalized eigenproblem (4-7),

and let u be an associated eigenfunction with normalization (Vu, Vu)Q = 1. If IIE 7- is

the eigenfunction described in (5-2), then we have

> Oiu a H1() (5-22)

0111u- U HI() (5-23)

> 0102 1U FUa 1u2 1 (5-24)

> 0028 i -nUa 2HI(. (5-25)

Above, (5-23) is due to the fact that II 1 on L, while (5-22), (5-24), and (5-25) come

from El, E2, and E3 respectively.

Suppose that the proposition does not hold, in which case p < 01/2 and p < 010203/2.

By (5-23) and (5-25), we have

H(Lu- H ) < 1/2 and Iu Hwll1i|c) < 1/2.

Combining these gives

1U- nu ll||1() < 1. (5-26)

On the other hand, u and II are orthogonal since these eigenfunctions correspond to

distinct eigenvalues. Since Hll is a multiple of H which is orthogonal to u, it follows that

1 < I|V(u InU)|| 2() < |IV( nUf ) ||H(). (5-27)

Comparing (5-26) and (5-27), we have a contradiction. Hence, either p > 01/2 or

P > 010203/2. EO

We continue to develop properties for the eigenfunctions of the generalized eigenproblem
(4-7) by exploiting the connection, given in Proposition 2, between the eigenfunctions of
the generalized eigenproblem (4-7) and those of the double 1v,--r potential T in (5-13).
As noted before Proposition 2, there is a one-to-one correspondence between elements

of R and elements of H1/2(O). If u e H1/2([), then the corresponding E(u) E R
is the harmonic extension of u e H1/2( O) into Q which vanishes on 90. For any
u,v E H1/2(O[), we define the inner product

(u, v) (VE(u), VE(v))n. (5-28)

In other words, harmonically extend u and v in 2 and form the H1J() inner product of

the extended functions. We now show that T is self adjoint and compact relative to this
new inner product.
Lemma 4. The following properties are -,;.:/7. ,1

T1. If 9Q and a are Lipschitz, then the norm (., )1/2 is equivalent to the usual norm for
H1/2(aL). That is, there exist positive constants cl and c2 such that

C (v, v) < I, II1 /2(aC) < C2( v, v)

for all v E H1/2(a).

T2. If 9Q and 9 are C2, then the double '7..;. potential operator T in (5-13) is self-
adjoint relative to the inner product (5-28).

T3. If O9 is C2 and a Z is C2'0, then T is a compact operator from H1/2(a[) into
H1/2(aO).

Proof. We begin by showing that the norm of H1/2([) and the norm (., .)1/2 are
equivalent. First, recall [7, p. 265] that there exists a constant 04 > 0 such that

IIV 11L 2(Q) > 04 I :|i1(Q)

for each v e H- which vanishes on 9Q. Combining this with E2 gives the lower bound

(v, v) (VE(v), VE(v)) > 04 1E(v) H\\(Q)

> e04E(v)i HI(f) > ~H4 /2(). (5-29)

An upper bound for (v, v) is obtained from E3:

(v, v) (VE(v), VE(v))n < ||E(v) ||2H(Q)

< (031 +01) ,l/yC). (5 30)

Here 03 > 0 is analogous to 03 in (5-20) except that it relates L to [L:

1 H111 2(aL) >03|E(v) HW(L)

Relations (5-29) and (5-30) yield T1.
To show that T is self adjoint relative to the inner product (5-28), we must verify the
identity
(Tu, v) (VE(Tu),VE(v))n = (VE(u),VE(Tv))Q = (u,Tv) (5-31)

for all u and v E H1/2(O). We first observe that the extension of Tu has the form

-E(u(x)) + u) (Y )dy for x c C',
E(Tu) 2 (5-32)
--E(u(x)) + U(y) G( dy for x c L.
2 JQ any

By Lemma 3, the trace of the right side of (5-32) is Tu from either side of 9L. Moreover,
the right side is harmonic and it vanishes on 9Q since E(u) vanishes on 9Q and G(x, y) =
0, independent of y cE 2, when x e 92. Since the right side is harmonic and satisfies the
boundary conditions associated with E(Tu), it must equal E(Tu).

Integrating by parts and utilizing (5-32), we obtain

(Tu, v) = (VE(Tu), Vv)n

= (VE(Tu), VE(v)), + (VE(Tu), VE(v)),c

t f E(u)- OE(u)+
a- a +E(v)dy. (5-33)
2 O n On j

The term in E(Tu) associated with the Green's function cancels since the normal

derivative of a double l1-cv-r potential operator is continuous across a0 (for example,

see [6, Thm. 3.1], [5, Thm. 2.21], [21, Thm. 6.13]).

For any p and q e R-, we have the identities

(Vp, Vq)c = q- 7 d7,
JaL On JaL On

and
f ap' + Oq+
(Vp, Vq)LI = q d7 = p- d.
JaL On OQ on
Hence, the normal derivatives in (5 33) can be moved from the u terms to v to obtain

(Tu, (OE(v)- OE, E(u)d7 (u, Tv),

which establishes T2.

We now show that T is compact on H1/2 (a). Consider the corresponding free space

double l-1.,-r potential operator TF defined by

JQC any
TF(x) aN) (x, y)d

where N is the free space Green's function defined in (5-11). For n = 2, TF is compact

by [21, Thm. 8.20]. For n > 3, Theorem 4.2 in [20] gives the boundedness of TF as a map

from L2(a) to H1(a). This result extends to our operator T as follows. The difference,

T TF, is an integral operator on aL with kernel

anH anG any

For x cE 0, H has no singularity since it is harmonic with smooth boundary data (see

the proof of Lemma 3). Consequently, T TF is bounded from L2(t) to H1(9).

Since both TF and T TF are bounded from L2(a) to H'1(), we conclude that T is

bounded from L2(t) to H'1(). This implies that T is compact on H1/2( ) since H1

embeds compactly in H1/2; that is, by [9, Thm. 1.4.3.2] H8 embeds compactly in Ht when

s > t > 0. Hence, T is compact on H1/2(a). E

Theorem 1. If 98 is C2 and at is C2',, for some a E (0, 1) (the exponent of Holder

conl/.,',iI, for the second derivative), then I w.; f c Ho(Q) has an expansion of the form
00
f Z Yi,
i=1

where the Qi are :,. ,functions of (4-7) which are orl,. .,. ,,,al relative to the inner product

(6-5). Here the convergence is with respect to the norm of Hdo(Q).

Proof. As pointed out earlier, we have the orthogonal decomposition

H/(Q) = E H'() E H'(c).

By Proposition 1, any complete orthonormal basis for H'() is an eigenfunction basis

corresponding to the eigenvalue 1. Likewise, any complete orthonormal basis for Ho1(gC) is

a basis whose elements are eigenfunctions of the generalized eigenproblem corresponding to

the eigenvalue 0. To complete the proof, we need to show that any f E R- lies in the span

of the remaining eigenfunctions for (4-7).

By Lemma 4, T is compact and self adjoint relative to the inner product (., .) defined

in (5-28). Hence, every f e H1/2(t9) has a unique expansion in terms of orthogonal

eigenfunctions of T (for example, see [4, Thm. 1.28]). Given f E R-, its restriction to aL

lies in H1/2([C). Therefore, there exist orthogonal eigenfunctions 0i, i > 1, of T such that

00
f = i on L.
i= 1

By the linearity and boundedness of the extension operator, we have

OO
f Z= E(0) on Q.
i= 1

By Proposition 2, E( i) is an eigenfunction for the generalized eigenproblem (4-7). D

CHAPTER 6
THE LIMIT

This chapter is based on the paper [12].

6.1 Introduction

We expand the solution to (4-1) in terms of the eigenfunctions of (4-7) and analyze

limits to compute the change in the electric potential due to lightning discharge. Our main

result is the following:

Theorem 2. If 02 is C2 and Z is C2',, for some ca E (0, 1) (the exponent of Holder

conlii,',il', for the second derivative), then the electric potential + :ii,,/,, .:al. /;/l after the

lightning 1.:-. lI.,r, ,- is given by

(x ) L if x E L,
+(x) (6 -1)
o (x) + (x) if x e c,

where
(Vo, vn),
L V (6 2)
(vn, vn), '
and where II and are harmonic functions in c with boundary conditions as ... i'', ,

below:

AII 0 in c, II =0 on 0n II 1 in L, (6-3)

AO = 0 in = oon Q, L 4o on (6-4)

Here (-, .)Q is the L2(Q) inner product

(Vu, Vv)n Vu Vv dx. (6-5)

Thus 0+ has the constant value OL along the lightning channel L and the change

in the potential due to lightning has been expressed in terms of the potential 4o before

the lightning and the lightning channel L. When L touches 0Q, as it would during a

cloud-to-ground flash, L = 0 and II can be eliminated. That is, as [ approaches the

boundary of Q, H develops a jump singularity since I = 1 on C and H = 0 on a9. Hence,

VII approaches a delta function as approaches a0. Since the delta function is squared

in the denominator of QL while the numerator is finite, QL tends to 0 as approaches aQ.

Thus in a cloud-to-ground flash, the change ( in electric potential due to the lightning is

the solution to (6-4) with L 0.

6.2 Reformulation of the Continuous Equation

Let (0i, Ai), i E N, denote a complete orthonormal set of eigenfunctions for the

generalized eigenproblem (4-7), as given by Theorem 1. We decompose N into the disjoint

union of four sets corresponding to the four classes of eigenfunctions described in the

introduction:

Sn = {i i N: O 11/VIII||L2(Q)},

So = {i N: A, 0, (VQ, Vn) = 0},

S1 {i N: Ai },

S+ = N: 0 < A{ < A 1}.

The set Sn contains precisely one element corresponding to the eigenfunction I given by

(6-3). The set So corresponds to eigenfunctions supported on [c, while S1 corresponds to

eigenfunctions supported on L. The set S+ corresponds to functions in Ho'1() which are

harmonic in both L and [C, and with the eigenvalues uniformly bounded away from 0.

The weak form of (4-5) is to find Q such that

a (VQ, V)n = -(aVQ, V)n T ,(VQ, Vv) + (Jp, VV)> (6-6)

for all v E Ho(Q). We substitute the eigenexpansion

(x, t) aM(t)(x)
iEN

into (6-6). Taking v = j, j = 1, 2,..., and utilizing the orthonormality of the
eigenfunctions yields the linear system

&a -Aa Da + f, a(0) = ao, (6-7)

where the dot denotes time derivative and

(ao)i (V- o, Vo)Q, i N, (6-8)

ail = <(7V4, V )n, (6-9)

dij = (V V, j)V ,

fi = (Jp, Vd)n. (6-10)

Since the Oi are orthonormal eigenfunctions for (4-7), we have

J A\ if i j,
dij = (VA Vj) Ai(0i, 0j) if i
0 otherwise.

Hence, D is a diagonal matrix with the eigenvalues Ai, i E N, on the diagonal. Since the
eigenvalues are nonnegative, D is positive semidefinite. We now consider A and f:

Lemma 5. The matrix A is positive -, i,,../. f,i../ The 2-norms of A and f, /. I,.. in
(6-9) and (6-10) ,'. /.. -/'.; /1,; are both finite, and we have

IIA | < ess sup la(x)| := ||Ia|- and fl = JJp 112( ). (6-11)
xen

Proof. By the definition of A, we have

xTAx a7 xiV ,i, xiV)i >0 (6-12)
i= 1 i= 1 9

for all x e (2 since a > 0. Since A is positive semidefinite, the C li. !:-Schwarz inequality
yields

yTAxx < yTAyvxTAx (6 13)

for all x and y e 2. If |x|| = 1, then by (6-12) we have

XTAx =I< |1| 7||1 xvo, xioV I Io-I \1,o.
i=1 i= 1 9
Hence, (6-13) yields the first inequality in (6-11).

Let u E Ho(Q) be the weak solution to Au = -V Jp:

(Vu, Vv) (Jp, Vv) for all v e HI(Q).

We have

IIVi, l-= (Vu ) = (Jp,Vu) < II||J ||||V ||.

Dividing by |IVull gives

I|VUl < I< llJ. (6-14)

For f defined in (6-10),

llfll2 Z(Jp, V7)2 (Vu7,V )2 (Vu:,VU) < IIJ P 2.
i= 1 i 1
The last equality is due to the fact that the Oi are a complete orthonormal basis relative
to the Ho(Q) inner product, while the last inequality is (6-14). E

6.3 Potential Change for the Continuous Equation

We now prove Theorem 2. Multiply (6-7) by aT and utilize the fact that D and A

are positive semidefinite to obtain

aTd -c aTAa T-TDDa + aTf < aTf.

Hence, we have
1d 1
|dlla 2 aT < jllall l l< (11a 112 + lfll2).
2 dt 2
Multiplying by e-t and rearranging yields

S(e-lal) < e-lfll2.

Integration over the interval [0, t] gives

Ija(t)|2 _< et a(0)||2 + (t- 1) lf2 (6 15)

This shows that I|a(.-) is uniformly bounded over any finite interval.

For the remaining analysis, it is convenient if the eigenvalues are arranged in

decreasing order. Thus without loss of generality, we assume that

A 0
D
0 0

where A is a diagonal matrix with strictly positive diagonal and the 0's represent blocks

whose elements are all zero. The components of a are partitioned in a compatible way

into [p ; z] where p corresponds to the positive eigenvalues Ai and z corresponds to the

zero eigenvalues.

Multiply (6-7) by [pT ; 0] to obtain

pTp = [p; O]T

[p; O]TAa- r[p; 0T Ap + [p; 0]Tf.
0

Taking norms on the right side gives

ld
dtllP 2 _< alla2IAII Tp p+llpll Ifl
2 dt
< la1211AII TAollP+12 + (lll2 +lf12), (6-16)
2

where Ao denotes the smallest positive eigenvalue; a positive lower bound for Ao is

obtained in Proposition 3. Suppose 7 is large enough that TAo > 1. ('!....-- t > 0

and let c be the constant defined by

c -lf|2 + 211All max lla(s)l2,
sE[o,t]

which is finite due to (6-15). By (6-16), we have

dt
llpll2 C- T- 0A0pll2

on the interval [0, t] assuming TAo > 1. Multiplying both sides by e'xot yields

dt (TAOtpll 2) < CAOtC.

Integration over the interval [0, t] gives

1
IlP(t) 2 < e-TA7t (0) 12 + ( 1 e- -TOt)

< e-TAOt||p( (0)||2 +
TAo

Since the right side approaches 0 as 7 tends to oo, we conclude that for any t > 0,

lim p,(t) = 0.
T---00

Here we have inserted a 7 subscript on p to remind us that the p-component of the

solution a to (6-7) depends on 7.

Now consider the bottom half of the equation (6-7):

iz = A2a, f2, (6-17)

where A2 denotes the bottom half of A and f2 is the bottom half of f. Since the bottom

half of D is zero, the D term of (6-7) is not present in (6-17). Since ,(.) is bounded over

any finite interval, independent of 7 by (6-15), it follows from (6-17) that z.(t) approaches

z(0) as t tends to 0, independent of 7. To summarize, we have

lim lim a,() = [0; z(0)],
t-0+ T--oo

where z(0) is the vector of coefficients in the eigenfunction expansion of 0o corresponding

to the eigenvalues A = 0. These coefficients correspond to the index set Sn U So. It follows

that

(x) = lim lim r (x, t) a (0) 4O(x), (6-18)
t-0+ T-Oo
iESnUSo
where

ai(0)= (Vo, Voi)n.

For x E L and i E So, Oi(x) = 0 since Oi, i E So, is supported on Lc. Hence, for x E L,

we have

0+(x) = Y (O)a,( x).
iESn
Since O for i E Sn is the normalized II, ai(0))O(x) is simply the projection of Oo along I:

Y a(0)(x) ( oVn)o, n)Q )(x) On(x). (6-19)
.( (VII, VfII) )
Since H(x) = 1 for x E the top hal' of (6 ) has been established.
Since II(x) = 1 for x E L, the top half of (6-1) has been established.

Now suppose that x e Lc. By the completeness of the ji, we have

0o(X) a(0) ) (x). (6-20)
iEN

Consequently, for x E [c, (6-18) can be rewritten

+(x) o(x) a(0) (x) 0o(x) a(0) (x) (6-21)
iES uS+ iES+

since Oi for i e S1 vanishes on '. Let ( denote the final term in (6-21):

(x) a(0)O(x)
iES+

For i E S+, we have Ai = 0 on LC since the eigenfunctions associated with indices in S+

are harmonic in either L or '. Hence, A =- 0 in c. ( vanishes on fQ since i e Hl(fQ).

To obtain the boundary values for ( on 9, we examine the eigenexpansion (6-20), which

can be rearranged in the form

Y a(0o (x) o(x) 5 a,(0) O(x). (6-22)
iES+ iESnUSoUSi

For x E 9I, Oi(x) = 0 if i E So U Si since Oi for i E So is supported on Lc, while Oi for

i SE S is supported on L. Consequently, for x E a0, it follows from (6-19) and (6-22) that

-(x) =- ai(0)O(x)= o(x) ai(O)dO(x) o(x) OLn(x).
iES+ iESn

Since n (x) = 1 for x e aC, (x) = O 0o(x) on 9C. This completes the proof of Theorem
2.

CHAPTER 7
APPLICATION TO ONE-DIMENSION

In this chapter, we will present the results in one-dimension for both the generalized

eigenproblem for the Laplacian and the continuous model for the lightning discharge.

7.1 Application of the Generalized Eigenproblem

Let us consider the generalized eigenproblem (4-7) in one dimension where 2 is the

interval [0, 1] and L is a subinterval [a, b] C (0, 1). In this case, there are precisely 2

eigenfunctions in -H. The functions which are harmonic on both L and [c are piecewise

linear. The eigenfunction H of Proposition 1, corresponding to the eigenvalue 0, is defined

by its boundary values H(0) = H(1) = 0 and the values H(x) = 1 on L. Let si, s2,

and S3 be the slope on the intervals [0, a], [a, b], and [b, 1] respectively of the remaining

eigenfunction u e H-. The jump condition of Lemma 1 yields

s2 = -A(s S2) and 2 = -A(s3 S2). (71)

Hence, si = 83. Let s denote either s, or 83. The boundary conditions u(0) = u() = 0

imply that

0 j '(x) dx s1a + s2(b- a) + S3(1 b) = (1 + a b) + 2(b- a).

This gives
(b-a-l1
S2 S b a )

With this substitution in (7-1), we have

A= 1 (b- a).

A sketch of these two eigenfunctions appears in Figure 7-1.

7.2 Application of the Continuous Model

In this section, we focus on Theorem 2 in dimension 1 (n = 1) with 2 the open

interval (0, 1) and L a subinterval (a, b) whose closure is contained in (0, 1). In this case,

Figure 7-1: Eigenfunctions in Fi in one dimension.

the equations describing II reduce to

I" 0 in(0, a) U(b, ), II in

[a, b], P1(o) P 1(1) 0 .

The solution is

n H(x)

Six

1

s2(1 -

if xE (0, a),

if xe [a, b],

x) if x E (b, 1),

where

1 1
s8 and S2
a l-b

Hence, we have

(/ 1"')Q sio8(a) + 8200(b)
LL / t
(Tr', 7')Q a + 1-b

Let us define the parameters

1-b
01 -
S-b+a

a
and 02 =
1 -b+a

With these definitions,

OL O 0o(a) + 0200(b),

(1 b)Oo(a) + aOo(b)
1 b+a

\ b
0\ 1

(7-2)

where 01 > 0, 02 > 0, and 01 + 02 = 1. Thus the effect of the lightning is to make the

potential QL on the lightning channel (a, b) a convex combination of the potential Qo(a)

and 0o(b) at the ends of the channel. The coefficients 01 and 02 in the convex combination

depend on the distance between the ends of the channel and the boundary of the domain

Q. It is interesting to note that the potential QL on the lightning channel only depends

on the pre-flash potentials Qo(a) and Qo(b) at the ends of the channel; in other words, the

pre-flash potential at interior points along the channel apparently has no effect on the

potential that is achieved along the lightning channel after the flash. Also, notice that

as one of the channel ends, a approaches the boundary, QL approaches zero since 02

and Qo(a) both approach 0 as a approaches 0 (recall that 0o(0) = 0). A more general

discussion of a cloud-to-ground flash is given after Theorem 2.

Now let us focus on the potential change ( outside the lightning channel L. According

to Theorem 2,

(" =0 on (0,a), ((0) = 0, ((a) =L Qo(a),

1/ 0 on (b, ), (1) 0, ((b) =fL Qo(b).

The solution is
S rx on (0, a),
r2(1 x) on (b, 1),

where
L o(a) a L 0(b)
r.= and r2
a 1-b
Substituting for QL using (7-2), we obtain

(01 1)yo(a) + 02o(b) 02 ) 0(b) yo(a)
r = ( o(b) 1o(a)) -+2.
a a 1-b+a

Hence, by Theorem 2, we have

(X) 0 () + x 1- if x (0, a),

0o(x) ( x) if x (b, 1),

where 6o0 = Oo(b) Oo(a) and I|| b a is the length of the lightning channel. Thus

lightning causes a linear change in the electric potential, where the size of the linear

perturbation is proportional to the pre-flash potential difference across the ends of the

channel.

CHAPTER 8
CONCLUSIONS

In this dissertation, Maxwell's equations are used to establish a continuous lightning

discharge model:

aA -V (aV) + V J, (x,t) eQx [0,oo), (8-1)

0(x, t) 0, (x, t) e 9 x [0, oo),

(x, 0) -= o(x), x Q,

in a bounded domain f C R" with a connected subdomain When the electric field in

a thundercloud reaches the Ii o, i.1lwn threshold," the atmosphere turns into a plasma,

locally, where conductivity is large. When conditions are right, a lightning discharge can

occur. In the lightning domain L the conductivity a is essentially infinite. To evaluate the

change in the electric potential due to lightning, we replace a by a + Tr where T is the

characteristic function for L, and we consider the differential equation

=4 -(A, + TA- ) + V.J. (8-2)

where = (-A)-2'. In C'! lpter 4 we show that the eigenproblem for AP is equivalent to

a generalized eigenproblem for the Laplacian (4-7). We analyze the eigenproblem (4-7) in

C'! lpter 5 and obtain the following results: The elements of Ho (,c) are eigenfunctions

corresponding to the eigenvalue 0, while the elements of Hi(C) are eigenfunctions

corresponding to the eigenvalue 1. The remaining eigenfunctions are elements of the

piecewise harmonic space 9-, consisting of functions in Hlo'() which are harmonic in both

L and Lc. There is a one-to-one correspondence between eigenfunctions of (4-7) in 1H

and eigenfunctions of the double 1 vrir potential T in (5-13). The eigenfunctions of (4-7)

are the harmonic extensions of the eigenfunctions of T, and if p is an eigenvalue of T,

then A = 1/2 p is the corresponding eigenvalue of (4-7). H E G (see Proposition

1) is the only eigenfunction in 1H corresponding to the eigenvalue 0. All the remaining

eigenvalues corresponding to eigenfunctions in H are contained in the open interval

(0, 1) and A = 1/2 is the only possible accumulation point. Since the eigenvalues of the

generalized eigenproblem (4-7) associated with eigenfunctions in H are contained in the

half-open interval [0, 1), the eigenvalues of the double 1 V-,-r potential T in (5-13) are

contained in [-1/2, 1/2). Proposition 3 gives a lower bound for the positive eigenvalues

of the generalized eigenproblem, or equivalently, a lower bound for the gap between the

largest and the second largest eigenvalue of T. Based on the fact that the double l1- -r

potential T is self adjoint and compact relative to the inner product (5-28), as established

in Lemma 4, we conclude that any f c H0(Q) can be expressed as a linear combination

of orthogonal eigenfunctions for (4-7). The potential immediately after the lightning

discharge is computed in C'!i plter 6 by expanding the potential Q using the orthonormal

eigenfunctions of (4-7) and studying the limits as T tends to infinity and t tends to zero to

compute the solution to (8-1). We find that the potential immediately after the lightning

discharge is constant throughout the lightning domain and the constant value depends on

the initial potential and the eigenfunction II of (4-7). Outside the lightning domain, the

change in the potential is the solution to the problem

A =- 0 in L', 0 on 0Q, = IL 0o on aL.

Applications of both the generalized eigenproblem and the continuous model for the

lightning discharge to one dimension are given in ('!, plter 7.

REFERENCES

[2] B. C. ASLAN, W. W. HAGER, AND S. MOSKOW, A generalized eigenproblem for
the Laplacian which arises in lightning, J. Math. Pures Appl., (2007, submitted).

[3] P. AUSCHER AND P. TCHAMITCHIAN, Square roots of elliptic second order diver-
gence operators on strongly Lipchitz domains: L2 I,,,,. J. Anal. Math., 90 (2003),
pp. 1-12.

[4] F. CAKONI AND D. COLTON, Qualitative Methods in Inverse Scattering Th(..,;, An
Introduction, Springer, New York, 2006.

[5] D. COLTON AND R. KRESS, Il,/, g,, Equation Methods in Scattering Th(..,;I John
Wiley-Interscience, New York, 1983.

[6] Inverse Acoustic and Electromagnetic Scattering Ti, .., Springer-V, i1 -
Berlin, 1998.

[7] L. C. EVANS, Partial Differential Equations, American Mathematical Society,
Providence, RI, 1998.

[8] G. B. FOLLAND, An Introduction to Partial Differential Equations, Princeton
University Press, New Jersey, 1976.

[9] P. GRISVARD, Elliptic Problems in Nonsmooth Domains, Pitman, Boston, 1985.

[10] W. W. HAGER, Updating the inverse of a matrix, SIAM Review, 31 (1989),
pp. 221-239.

[11] A discrete model for the lightning 1.:. li,,hg J. Comput. Phys., 144 (1998),
pp. 137-150.

[12] W. W. HAGER AND B. C. ASLAN, The, l.,i,,g, in electric potential due to lightning
1.- ,ig. SIAM J. Appl. Math., (2007, submitted).

[13] W. W. HAGER, J. S. NISBET, AND J. R. KASHA, The evolution and 1.:- I,,,g of
electric fields within a thunderstorm, J. Comput. Phys., 82 (1989), pp. 193-217.

[14] W. W. HAGER, J. S. NISBET, J. R. KASHA, AND W.-C. SHANN, Simulations of
electric fields within a thunderstorm, J. Atmos. Sci., 46 (1989), pp. 3542-3558.

[15] J. H. HELSDON AND R. D. FARLEY, A numerical modeling study of a montana
thunderstorm: 1. model results versus observations involving nonelectrical aspects,
J. Geophys. Res., 92 (1987), pp. 5645-5659.

[16] A numerical modeling study of a montana thunderstorm: 2. model results versus
observations involving electrical aspects, J. Geophys. Res., 92 (1987), pp. 5661-5675.

[17] J. H. HELSDON, R. D. FARLEY, AND G. FU, Lightning parametrization in a storm
electric i..r.I, model, in Proceedings on the Conference on Atmospheric Electricity,
1988, pp. 849-854.

[18] J. H. HELSDON, G. FU, AND R. D. FARLEY, An intercloud lightning parametriza-
tion scheme for a storm electr':f;. li..., model, J. Geophys. Res., 97 (1992),
pp. 5865-5884.

[19] H. W. KASEMIR, A contribution to the electrostatic theory of a lightning 1.:. h.llg
J. Geophys. Res., 65 (1960), pp. 1873-1878.

[20] A. KIRSCH, S, f,., : gradients and coni ,,,i., properties for some '*,.I, gi,,, operators in
classical scattering i',, .,;; Math. Methods Appl. Sci., 11 (1989), pp. 789-804.

[21] R. KRESS, Linear I,.,I gpd Equations, Springer-Verlag, Berlin, 1989.

[22] D. R. MACGORMAN, A. A. FEW, AND T. L. TEER, I.;, ,, 1.l lightning i.:/;.:'; J.
Geophys. Res., 81 (1986), pp. 9900-9910.

[23] D. R. MACGORMAN, J. M. STRAKA, AND C. L. ZIEGLER, A lightning
parametrization for numerical cloud models, J. Appl. Meteorol., 40 (2001),
pp. 459-478.

[24] E. R. MANSELL, D. R. MACGORMAN, C. L. ZIEGLER, AND J. M. STRAKA,
Simulated three-dimensional branched lightning in a numerical thunderstorm model,
J. Geophys. Res., 107 (2002), pp. 4075-4086.

[25] L. NIEMEYER, L. PIETRONERO, AND H. J. WIESMAN, Fractal dimension of
dielectric breakdown, Physical Review Letter, 52 (1984), pp. 1033-1036.

[26] F. POCKELS, Uber das magnetische verhalten einger basaltischer gesteien, Ann. Phys.
C'!. n 63 (1897), pp. 195-201.

[27] Bestimmung maximaler /1,'l;J,,ig,-stromstdrken aus ihrer magnetisirenden
.:,l.,, Ann. Phys. C'!. ii 65 (1898), pp. 458-475.

[28] Uber die '1l..: ,,,'iI,, l/,,.. ,. erreicht stronstdrke, Phys. Z., 2 (1900), pp. 307-307.

[29] M. A. UMAN, The Lightning D.>. lrg, Academic Press, San Diego, CA, 1987.

[30] H. J. WIESMANN AND H. R. ZELLER, A fractal model of dielectric breakdown
and prebreakdown in and around space lIrg, clouds, J. Geophys. Res., 90 (1985),
pp. 6054-6070.

[31] E. R. WILLIAMS, Electrical .:-. hnrl-g propagation in and around space I,,.g, clouds,
J. Geophys. Res., 90 (1985), pp. 6059-6070.

[32] C. T. R. WILSON, On some determinations of the sign and magnitude of electric
1.:. I,,,,. in lightning flashes, Proc. R. Soc., Ser. A, 92 (1916), pp. 555-574.

[33] Investigations on lightning I.:,. Li.,r and on the electric field of thunderstorms,
Phil. Trans. R. Soc., Ser. A, 221 (1920), pp. 73-115.

[34] C. L. ZIEGLER AND D. R. MACGORMAN, Observed lightning me, //,. I ,I;/ relative
to modeled space .,-/,l/.- and electric field distributions in a tornadic storm, J. Atmos.
Sci., 51 (1994), pp. 833-851.

BIOGRAPHICAL SKETCH

B.vi, Cahllkan Asian was born in Kiitahya, Turkey, in 1977. She was awarded a

Bachelor of Science degree in mathematics in 1999 from Middle East Technical University

(\ IlTU), Ankara, Turkey. In 2000, she started her graduate study in mathematics at the

University of Florida, from which she received her M.S. in mathematics in 2003 and her

Ph.D. in mathematics in 2007.

PAGE 1

1

PAGE 2

2

PAGE 3

KamileCalskanandYusufCalskan, andtomyhusband, OmerDenizAslan 3

PAGE 4

PAGE 5

page ACKNOWLEDGMENTS ................................. 4 LISTOFFIGURES .................................... 7 ABSTRACT ........................................ 8 CHAPTER 1INTRODUCTION .................................. 9 2LITERATUREREVIEW .............................. 11 2.1LightningModelswithExplicitLightningChannels ............. 11 2.1.1Helsdon'sModel ............................. 11 2.1.2MacGorman'sModel .......................... 12 2.1.3Mansell'sModel ............................. 13 2.2Hager'sModel:TheDiscreteModel ...................... 14 3THEDISCRETEMODEL .............................. 15 3.1GoverningEquations .............................. 15 3.1.1Maxwell'sEquations .......................... 15 3.1.2DerivationoftheGoverningEquations ................ 15 3.2TheModelinOne-dimension ......................... 16 3.3GeneralizationtoThree-dimension ...................... 18 4THECONTINUOUSMODEL ........................... 19 4.1FormulationoftheEquations ......................... 19 4.2EigenproblemforA 21 5GENERALIZEDEIGENPROBLEMFORTHELAPLACIAN .......... 23 5.1Introduction ................................... 23 5.2EigenfunctionsofType1,2,and3 ...................... 23 5.3ReformulationofEigenprobleminHUsingDouble-LayerPotential .... 24 5.4EigenvalueSeparationandCompletenessofEigenfunctions ......... 32 6THELIMIT ...................................... 40 6.1Introduction ................................... 40 6.2ReformulationoftheContinuousEquation .................. 41 6.3PotentialChangefortheContinuousEquation ................ 43 7APPLICATIONTOONE-DIMENSION ...................... 48 7.1ApplicationoftheGeneralizedEigenproblem ................. 48 5

PAGE 6

..................... 48 8CONCLUSIONS ................................... 52 REFERENCES ....................................... 54 BIOGRAPHICALSKETCH ................................ 57 6

PAGE 7

Figure page 4-1AsketchofLandforalightningdischarge ................... 20 7-1EigenfunctionsinHinonedimension ........................ 49 7

PAGE 8

Wedevelopacontinuousmodelforthelightningdischarge.WeconsiderMaxwell'sequationsinthreedimensionsandobtainaformulaforthelimitingpotentialasconductivitytendstoinnityinathree-dimensionalsubdomain(thelightningchannel)ofthemodeleddomain.ThelimitisexpressedintermsoftheeigenfunctionsforageneralizedeigenvalueproblemfortheLaplacianoperator.Thepotentialinthebreakdownregioncanbeexpressedintermsofaharmonicfunctionwhichisconstantinthebreakdownregion. 8

PAGE 9

PAGE 10

Mostlightningresearchisdonebyphysicists,chemists,meteorologists,andelectricalengineers.Hager[ 11 13 14 ]wastherstmathematicianusingMaxwell'sequationstodevelopathree-dimensionalmathematicalmodeltosimulatealightningdischarge.Hisdischargemodel[ 14 ]wasobtainedbydiscretizingMaxwell'sequationstoobtainarelationbetweenthepotentialeldandcurrentdensityduetothemotionofchargedparticlesundertheinuenceofthewind.Spatialderivativesinhisequationwereapproximatedbyusingvolumeelementsinspace,whilethetemporalderivativeswereestimatedbyabackwardEulerschemeintime.Sinceconductivityisverylargeintheregionwheretheelectriceldreachesthebreakdownthreshold,heevaluatedthesolutionlimitastheconductivitytendstoinnityinthebreakdownregion.Inhismodel[ 14 ],theoutputwastheelectriceldasafunctionoftime,andtheinputswerecurrentsgeneratedbytheowofchargedparticleswithinthethundercloudundertheinuenceofthewind. ThisdissertationisbasedonHager'smathematicalmodel.SomeimprovementsaremadecomparedtoHager'searlierwork.Forexample,thesolutioniscomputedwithoutdiscretizingtheequations.Consequently,wedonothavehugematrixsystemstocomputeandthereforeitiscomputationallymuchmoreecientandlessexpensive. 10

PAGE 11

PAGE 12

23 ]suggestedalightningparametrizationthatwasconsideredanextensionoftheparametrizationofHelsdonetal.[ 18 ]inconjunctionwithsomeofthebulk-lightningparametrizationmethodspresentedbyZieglerandMacGorman[ 34 ].MacGormanetal.[ 23 ]developedaparametrizationtoenablecloudmodelstosimulatethelocationandstructureofindividuallightningashesbyusingtheconceptualmodelofMacGormanetal.[ 22 ]andWilliamsetal.[ 31 ].Theirparametrizationproceededintwostages.UsingtheideasofHelsdonetal.[ 18 ],aashtracedtheelectric-eldlinefromaninitiationpointoutwardinbothparallelandantiparalleldirectionsuntilthemagnitudeoftheambientelectriceldateachendfellbelowsomecertainthresholdvalue.Whenoneendofthechannelreachedground,theparametrizationterminatedatthatend,butallowedtheotherendtocontinuedeveloping. ChargeestimationandneutralizationwereparameterizedbyapplyingthetechniqueproposedbyZieglerandMacGorman[ 34 ],exceptthatZieglerandMacGormanneutralizedchargeatallgridpointshavingj(i;j;k)j1(where(i;j;k)wasthenetchargedensityatthegridpoint(i;j;k)and1wastheminimumj(i;j;k)jforallgridpoints 12

PAGE 13

PAGE 14

11 13 14 ]proposedathree-dimensionallightning-dischargemodelthatproducedbidirectionalICand-CGashes.Themodelgeneratedthedischargeregion,chargetransfer,anddetailedchargerearrangementassociatedwiththeash. TheirapproachtolightningwasquitedierentfromthoseinSection2.1.TheirbreakdownmodelwasbasedonMaxwell'sequations.Theyassumedthatcurrentduetotransportofchargeundertheinuenceofwindwasknown.Theyobtainedanequationgoverningtheevolutionoftheelectricpotentialundertheassumptionthatthetimederivativeofthemagneticeldcanbedisregarded.Afterintegratingthisequationoverboxesandapproximatingderivativesbynitedierences,theyobtainedanimplicitsystemofdierenceequationsdescribingtheevolutionoftheelectriceld.Theirapproachtolightningwastolettheconductivitytendtoinnitywherevertheelectriceldreachedthebreakdownthreshold.Thisapproachappealstoourbasicconceptionofnature:Whentheelectriceldreachesbreakdownthreshold,conductivitybecomesverylargeasaplasmaforms. Whentheelectriceldreachesthebreakdownthreshold,theelectricpotentialchangesinstantaneouslyeverywherewithinthethundercloud.TheInverseMatrixModicationFormula[ 10 ]wasappliedtoevaluatethischange: wherebeforewastheelectricpotentialbeforedischarge,afterwastheelectricpotentialafterdischarge,AwasthediscreteLaplacian,andUwasamatrixwitha+1and-1ineachcolumncorrespondingtothearcsassociatedwiththebreakdown.TherewerenoparametersinEquation( 2{1 )besidestheelectricpotentialbeforedischarge.Thiswasconsistentwithexperimentalobservations:Thechargeiscontrolledpredominatelybyasingleparameter:thelocalelectrostaticeld.ThiswasobservedinexperimentsreportedbyWilliamsetal.[ 31 ]. 14

PAGE 15

3.1.1Maxwell'sEquations Intheabsenceofmagneticorpolarizablemedia,thedierentialformoftheseequationsare: 1. Gauss'lawforelectricity:rE= Gauss'lawformagnetism:rH=0 3. Faraday'slawofinduction:rE=@B Ampere'slaw:rH=J0+"@E SinceJ0ispartlyduetothemovementofchargediceandwaterparticlesinthecloudandpartlyduetotheelectricalconductivityofthecloud,wewriteJ0=Jp+0E

PAGE 16

3{1 )becomes @t:(3{2) Takingthedivergenceof( 3{2 ),itfollowsthat 0=rrH=rJpr0r"r@r @t:(3{3) Letting=0="andJ=Jp=",weobtain @t+rrrJ=0:(3{4) Inourmodelwealsohavethefollowingassumptions: 2{1 )inSection2:2,wenowfocusontheequation( 3{4 )inonedimension.Inone-dimension,( 3{4 )reducesto _00+(0)0J0=0;(0)=0;(H)=V;(3{5) 16

PAGE 17

Integrating( 3{5 )over[xi;xi+1]gives_0(xi+1)_0(xi)+(i+10(xi+1)i0(xi))(Ji+1Ji)=0 Toapproximate0,[ 11 13 14 ]usedbackwarddierences,0(xi)ii1 Thiscanbewritteninmatrixformas)A_+B=hJ

PAGE 18

whereWl=[w0j:::jwl]. 18

PAGE 19

3{4 ),thefollowingequationsmodeltheevolutionoftheelectricpotentialinadomain,assumingthetimederivativeofthemagneticuxdensitycanbeneglected: @t=r(r)+rJin[0;1); whereRnisaboundeddomainwithboundary@,0liesinthespaceL1()ofessentiallyboundedfunctionsdenedon,andtheforcingtermJliesinL2(),theusualspaceofsquareintegrablefunctionsdenedon.ThedivergencerJaswellastheequation( 4{1 )areinterpretedinaweaksense,asexplainedlater.Theinitialcondition0isassumedtolieinH10(),theSobolevspaceconsistingoffunctionswhichvanishon@andwithrstderivativesinL2().Theevolutionproblem( 4{1 ){( 4{3 )hasasolution(x;t)with(;t)2H10()andthepartialderivative@t(;t)2H10().AlthoughMaxwell'sequationsdescribetheelectromagneticeldsin3-dimensions,themathematicalanalysisthroughoutthisdissertationisdevelopedinann-dimensionalsetting,wherenisanarbitrarypositiveinteger. Inathunderstorm,istheconductivitydividedbythepermittivityoftheatmosphere,isanylargedomainextendingfromtheEarthtotheionospherewhichcontainsthethundercloudinitsinterior,andJisduetotransportbywindofchargediceandwaterparticlesinthecloud.AlthoughJisafunctionoftime,wefocusonthepotentialchangeduringlightning,whichweconsiderinnitelyfast.Hence,duringalightningevent,Jisessentiallytimeinvariant.Thepotentialoftheionosphereisnotzero,however,wecanmakeachangeofvariablestosubtractothe\fair-eldpotential"(thepotentialofthe 19

PAGE 20

4{1 ){( 4{3 )wherethepotentialvanisheson@. ApossiblelightningchannelissketchedinFigure 4-1 .Mathematically,thelightning Figure4-1: AsketchofLandforalightningdischarge. channelLcouldbeanyconnected,opensetcontainedinwithitscomplementLc=nLconnected.Morerealistically,weshouldviewLasaconnectednetworkofthinopentubes.ThecasewhereLtouches@,aswouldhappenduringacloud-to-groundash,istreatedasalimitinwhichLapproachesarbitrarilycloselytotheboundaryof. Makingthechangeofvariables=()1 2,werewrite( 4{1 )as whereA=()1 2(r(r))()1 2: 2:L2()!H10()denotestheinverseofthesquarerootoftheLaplacian([ 3 ]). LetbethecharacteristicfunctionforL(isidentically0exceptinLwhereis1).TheeectoflightningistoionizethedomainL,inessence,replacingin( 4{4 )by+whereislarge.Ifthelightningoccursatt=0,theninthemomentsafterthe 20

PAGE 21

whereA=()1 2(r(r))()1 2; 4{2 )and( 4{3 ).Herethescalarreectsthechangeinconductivityinthelightningchannel. Let(x;t)denotethesolutionof( 4{5 )attimet,andlet(x;t)=()1 2(x;t)bethecorrespondingsolutionof( 4{1 ).Ifthelightningoccursattimet=0,thentheelectricpotentialrightaftertheashisgivenby limt!0+lim!1(x;t):(4{6) UsinganeigendecompositionforA,thelimit( 4{6 )canbeevaluated. 2 ])thattheeigenproblemforAreducestothefollowinggeneralizedeigenproblem:Findu2H10(),u6=0,and2Rsuchthat forallv2H10(),whereh;iistheL2()innerproduct WeviewH10()asaHilbertspaceforwhichtheinnerproductbetweenfunctionsuandv2H10()isgivenby( 4{8 ). TheweakformoftheeigenproblemforAistondU2L2()suchthat 2U;r()1 2Vi=hU;Vi(4{9) 21

PAGE 22

4{9 )reducestotheproblemofndingu2H10()suchthat 2u;()1 2vi(4{10) forallv2H10().Ifu2C10(),then 2u;()1 2vi=hu;vi=hru;rvi:(4{11) SinceC10()isdenseinH10()andtheoperators()1 2andrarebothboundedinH10(),theidentity( 4{11 )isvalidforallu2H10().Hence,( 4{10 )reducesto( 4{7 ). 22

PAGE 23

Thischapterisbasedonthepaper[ 2 ]. 4{7 ): 1.Thefunctionwhichis1onLandharmoniconnL;theeigenvalueis0. 2.FunctionsinH10()withsupportinnL;theeigenvalueis0. 3.FunctionsinH10()withsupportinL;theeigenvalueis1. 4.Excluding,theharmonicextensionoftheeigenfunctionsofadoublelayerpotentialon@L.Theeigenvaluesarecontainedintheopeninterval(0;1).Theonlypossibleaccumulationpointis=1=2. 4{7 ),wehave whichimpliesthat01.LetH10(L)H10()denotethesubspaceconsistingoffunctionswithsupportinL.Similarly,letH10(Lc)H10()denotethesubspaceconsistingoffunctionswithsupportinLc. 4{7 )ifandonlyifthesupportofuiscontainedinL.Ifu2H10(Lc),thenuisaneigenfunctionof( 4{7 )correspondingtotheeigenvalue0.Theonlyothereigenfunctionof( 4{7 )correspondingtotheeigenvalue0,whichisorthogonaltoH10(Lc),isthesolution2H10()of 4{7 ),thenby( 5{1 ),wehavehru;ruiLc=0:

PAGE 24

4{7 ),uisaneigenfunctioncorrespondingtotheeigenvalue1.Ifu=0inL,thenuisaneigenfunctioncorrespondingtotheeigenvalue0.Thesolutionof( 5{2 )isaneigenfunctionof( 4{7 )correspondingtotheeigenvalue0sincer=0inL. Letw2H10()beanyeigenfunctionof( 4{7 )correspondingtotheeigenvalue0whichisorthogonaltoH10(Lc).By( 4{7 ),wehavehrw;rwiL=0,whichimpliesthatrw=0inL,orwisconstantinLsinceLisconnected.Withoutlossofgenerality,letusassumethatw=1inL.Sincewisorthogonaltothefunctionsv2H10(Lc),wehavehrw;rvi=0forallv2H10(Lc): 5{2 )giveshr(w);rvi=0forallv2H10(Lc): 1 describeseigenfunctionsoftype1,2,and3.Inthissection,wefocusontype4eigenfunctions.LetHbethespacewhichconsistsofallu2H10()satisfyingtheconditions (5{3) 24

PAGE 25

4{7 )onHintermsofanintegraloperator. 4{7 )ifandonlyif @n=@u @non@L;(5{5) @n=@u @n+@u @n2H1=2(@L): Proof. 5{5 ).By( 4{7 )wehave foranyv2H10().IntegratingbypartsandutilizingthefactthatuisharmonicinbothLandLcgives @nd=Z@Lv@u @n+d+Z@Lv@u @nd;(5{7) wheredenotestheboundarymeasureon@L.Hence,wehaveZ@Lv@u @n+@u @n+@u @nd=0 foranyv2H10().Sinceanyv2H1=2(@L)hasanH10()extension,( 5{5 )holds. Conversely,supposethatusatises( 5{5 ).Asin( 5{6 ){( 5{7 ),wehavehrv;rui=Z@Lv@u @n+d+Z@Lv@u @nd

PAGE 26

5{5 )giveshrv;rui=Z@Lv@u @n=hrv;ruiL 4{7 ). NowletusintroducetheGreen'sfunctionon: yG(x;y)=x(y)in;G(x;y)=0fory2@;(5{8) wherexistheDiracdeltafunctionlocatedatx.Thepiecewiseharmonicfunctionsu2Hcanbedescribedintermsofthejumpon@Lofthenormalderivative. @n(y)G(x;y)dy:(5{9) @nydyZLcruryGdyZ@Lu+@G @nydy: @n(y)G(x;y)dy+Z@L@u @n+(y)G(x;y)dy=Z@L@u @n(y)G(x;y)dy; whichyields( 5{9 ). 26

PAGE 27

8 ]);westateithereforthecaseofourGreen'sfunctioncorrespondingtoaboundedouterdomain. @ny(x;y)dy: 2(x)+Z@L(y)@G @ny(x;y)dy; 2(x)+Z@L(y)@G @ny(x;y)dy: 2logjxyjn=2;(5{11) where!nisthesurfaceareaoftheunitsphereinRn.RecallthatyN(x;y)=x(y): @ny(x;y)dy+Z@L(y)@N @ny(x;y)dy:(5{12) 27

PAGE 28

@ny(x;y)dy 5{12 )wecanapplythewellknownresult(see[ 8 ])forthelimitx!@L.>FromtheexteriorofL,wehave @ny(x;y)dy1 2(x)+Z@L(y)@N @ny(x;y)dy=1 2(x)+Z@L(y)@G @ny(x;y)dy: UsingLemma 1 2 ,and 3 ,wereformulatethegeneralizedeigenproblem( 4{7 )onHintermsofaboundaryintegraloperator.Bythetracetheorem[ 1 ,Thm.7.53],anyu2HH10()hasatraceon@LinH1=2(@L).Conversely,u2H1=2(@L)hasauniqueharmonicextensionintobothLandLcwithu=0on@.Hence,thereisaone-to-onecorrespondencebetweenelementsofHandelementsofH1=2(@L). DeneT:L2(@L)!L2(@L) by @ny(x;y):(5{13) By[ 8 ,Prop.3.17],Kisacontinuouskernelofordern2on@L.Itfollowsfrom[ 8 ,Prop.3.12]thatTisacompactoperatorfromL2(@L)toitself. 28

PAGE 29

4{7 )ifandonlyifthecorrespondingu2H1=2(@L)isaneigenfunctionofTwithassociatedeigenvalue1=2;thatis, 47 ).ByLemmas 1 and 2 ,wehaveu(x)=Z@L@u @n(y)G(x;y)dy @ny(x;y)dy+ZLu(y)yG(x;y)dy: @ny(x;y)dy; 3 ,u(x)=1 2u(x)Z@Lu(y)@G @ny(x;y)dy; 5{14 ). Conversely,supposethatu2H1=2(@L)satises( 5{14 ).WeidentifyuwithitsharmonicextensioninH,andwedenew(x)by @ny(x;y)dyforx2Lc;Z@Lu(y)@G @ny(x;y)dy+u(x)forx2L:(5{15) 29

PAGE 30

3 ,wehavew+(x)=1 2u(x)Z@Lu(y)@G @ny(x;y)dy 2u(x)Z@Lu(y)@G @ny(x;y)dy=1 2u(x)Z@Lu(y)@G @ny(x;y)dy: 5{14 )yields Observethatwvanisheson@duetothesymmetryofG(x;y)[ 8 ,Lem.2.33];thatis,sinceG(x;y)=0wheny2@,wehavebysymmetryG(x;y)=0whenx2@.Hence,thenormalderivativein( 5{15 )vanisheswhenx2@.Sincewisharmonicineachsubdomainanditisequaltouonboth@L(see( 5{16 ))and@(theybothvanish),itfollowsthatw=uin.Wereplacewwithuin( 5{15 )toobtain @ny(x;y)dyforx2Lc;Z@Lu(y)@G @ny(x;y)dy+u(x)forx2L:(5{17) Integratingbypartsandusing( 5{8 )givesZ@L@u @n(y)G(x;y)dy=hru;ryGiL=8>>><>>>:Z@Lu(y)@G @ny(x;y)dyifx2Lc;Z@Lu(y)@G @ny(x;y)dyu(x)ifx2L: 5{17 ),weconcludethat @n(y)G(x;y)dy:(5{18) 30

PAGE 31

2 @n(y)G(x;y)dy:(5{19) Subtracting( 5{19 )from( 5{18 )givess(x):=Z@L(y)G(x;y)dy=0foranyx=2@L; @n(y)@u @n(y): @n=@u @n; 1 completestheproof. Toprovethat=0,supposetothecontrarythatdoesnotvanish.Letrbeanysmoothfunctiondenedon@LforwhichZ@L(y)r(y)dy6=0: 5{13 )arerealandcontainedinthehalf-openinterval(1=2;1=2].Theonlypossibleaccumulationpointforthespectrumis0.

PAGE 32

4{7 )areallrealduetosymmetryoftheinnerproduct.ByProposition 2 ,theeigenvaluesofTareallreal.AsnotedbeforeProposition 1 ,theeigenvaluesof( 4{7 )arecontainedontheinterval[0;1].Moreover,byProposition 1 ,theonlyeigenfunctioncorrespondingtotheeigenvalue1hassupportinL.Thetraceofthiseigenfunctionon@Lis0.TheonlyelementinHwithvanishingtraceon@Listhezerofunction.Consequently,thereisnoeigenfunctioninHcorrespondingtotheeigenvalue1.ThereisoneeigenfunctioninHcorrespondingtotheeigenvalue0,namelythefunctionofProposition 1 .Exceptfortheeigenvalue0,alltheremainingeigenvaluesforthegeneralizedeigenproblemlieintheopeninterval(0;1).SincetheeigenvaluesofTare1/2minusthecorrespondingeigenvalueof( 4{7 )in[0;1),theproofiscomplete.SinceTiscompactonL2()[ 8 ,Prop.3.12],theonlypossibleaccumulationpointforthespectrumis0. AlowerboundfortheseparationbetweenthelargestandsecondlargesteigenvaluesofTisobtainedfromProposition 3 1 ,thegeneralizedeigenproblem( 4{7 )restrictedtoHhasasimpleeigenvalue=0correspondingtotheeigenfunction2Hwhiletheremainingeigenvaluesarepositive.ByProposition 2 ,theonlypossibleaccumulationpointforthespectrumis=1=2.Hence,thereisaninterval(0;),>0,wherethegeneralizedeigenproblemhasnoeigenvalues.Wenowgiveanexplicitpositivelowerboundforintermsofthreeembeddingconstants: E1. Letuadenotetheconstantfunctiononwhosevalueistheaverageofu2H1()overL:ua=1 measure(L)ZLu(x)dx: 7 ,Thm.1,p.275],thereexistsaconstant1>0suchthatkruk2L2(L)1kuuak2H1(L) 32

PAGE 33

By[ 1 ,Thm.7.53],thereexistsaconstant2>0suchthatkuk2H1(L)2kuk2H1=2(@L) E3. Thereexistsaconstant3>0suchthat forallu2H10()whichareharmonicinLc(inotherwords,( 5{4 )holds).ThefollowingproofofE3wassuggestedbyJayadeepGopalakrishnan:Foru2H10(),letT(u)2H1=2(@L)denotethetraceofuevaluatedon@L.By[ 9 ,Thm.1.5.1.3],ThasacontinuousrightinversewhichwedenoteT1.Inotherwords,foreachg2H1=2(@L),wehaveT1(g)2H10(),TT1(g)=g,andkT1(g)kH1()kT1kkgkH1=2(@L): 7 ,Thm.3,p.265]) Moreover,sinceuisharmonicinLcandv0vanishesonboth@and@L,wehavehrv0;rv0iLc=hrv0;r(uT1(g))iLc=hrv0;rT1(g)iLckrv0kL2(Lc)krT1(g)kL2(Lc)krv0kL2(Lc)kT1(g)kH1(Lc); 5{21 )toobtainkv0kH1(Lc)ckrT1(g)kH1(Lc): 5{20 ). 33

PAGE 34

4{7 )hasnoeigenvaluesintheinterval(0;)where=minf1;23g1=2: 4{7 ),andletubeanassociatedeigenfunctionwithnormalizationhru;rui=1.If2Histheeigenfunctiondescribedin( 5{2 ),thenwehave =1kuuak2H1(L) 2(@L) Above,( 5{23 )isduetothefactthat=1onL,while( 5{22 ),( 5{24 ),and( 5{25 )comefromE1,E2,andE3respectively. Supposethatthepropositiondoesnothold,inwhichcase<1=2and<123=2.By( 5{23 )and( 5{25 ),wehavekuuak2H1(L)<1=2andkuuak2H1(Lc)<1=2: Ontheotherhand,uandareorthogonalsincetheseeigenfunctionscorrespondtodistincteigenvalues.Sinceuaisamultipleofwhichisorthogonaltou,itfollowsthat 1kr(uua)k2L2()kr(uua)k2H1():(5{27) 34

PAGE 35

5{26 )and( 5{27 ),wehaveacontradiction.Hence,either1=2or123=2. Wecontinuetodeveloppropertiesfortheeigenfunctionsofthegeneralizedeigenproblem( 4{7 )byexploitingtheconnection,giveninProposition 2 ,betweentheeigenfunctionsofthegeneralizedeigenproblem( 4{7 )andthoseofthedoublelayerpotentialTin( 5{13 ).AsnotedbeforeProposition 2 ,thereisaone-to-onecorrespondencebetweenelementsofHandelementsofH1=2(@L).Ifu2H1=2(@L),thenthecorrespondingE(u)2Histheharmonicextensionofu2H1=2(@L)intowhichvanisheson@.Foranyu;v2H1=2(@L),wedenetheinnerproduct (u;v)=hrE(u);rE(v)i:(5{28) Inotherwords,harmonicallyextenduandvinandformtheH10()innerproductoftheextendedfunctions.WenowshowthatTisselfadjointandcompactrelativetothisnewinnerproduct. 5{13 )isself-adjointrelativetotheinnerproduct( 5{28 ). Proof. 7 ,p.265]thatthereexistsaconstant4>0suchthatkrvk2L2()4kvk2H1()

PAGE 36

(v;v)=hrE(v);rE(v)i4kE(v)k2H1()4kE(v)k2H1(L)24kvk2H1=2(@L): Anupperboundfor(v;v)isobtainedfromE3: (v;v)=hrE(v);rE(v)ikE(v)k2H1()(13+13)kvk2H1=2(@L): Here3>0isanalogousto3in( 5{20 )exceptthatitrelatesLto@L:kvk2H1=2(@L)3kE(v)k2H1(L) 5{29 )and( 5{30 )yieldT1. ToshowthatTisselfadjointrelativetotheinnerproduct( 5{28 ),wemustverifytheidentity (Tu;v)=hrE(Tu);rE(v)i=hrE(u);rE(Tv)i=(u;Tv)(5{31) foralluandv2H1=2(@L).WerstobservethattheextensionofTuhastheform 2E(u(x))+Z@Lu(y)@G(x;y) 2E(u(x))+Z@Lu(y)@G(x;y) ByLemma 3 ,thetraceoftherightsideof( 5{32 )isTufromeithersideof@L.Moreover,therightsideisharmonicanditvanisheson@sinceE(u)vanisheson@andG(x;y)=0,independentofy2,whenx2@.SincetherightsideisharmonicandsatisestheboundaryconditionsassociatedwithE(Tu),itmustequalE(Tu). 36

PAGE 37

5{32 ),weobtain (Tu;v)=hrE(Tu);rvi=hrE(Tu);rE(v)iL+hrE(Tu);rE(v)iLc=1 2Z@L@E(u) TheterminE(Tu)associatedwiththeGreen'sfunctioncancelssincethenormalderivativeofadoublelayerpotentialoperatoriscontinuousacross@L(forexample,see[ 6 ,Thm.3.1],[ 5 ,Thm.2.21],[ 21 ,Thm.6.13]). Foranypandq2H,wehavetheidentitieshrp;rqiL=Z@Lq@p @nd=Z@Lp@q @nd; @n+d=Z@Lp@q @n+d: 5{33 )canbemovedfromtheutermstovtoobtain(Tu;v)=1 2Z@L@E(v) WenowshowthatTiscompactonH1=2(@L).ConsiderthecorrespondingfreespacedoublelayerpotentialoperatorTFdenedbyTF(x)=Z@L(y)@N @ny(x;y)dy; 5{11 ).Forn=2,TFiscompactby[ 21 ,Thm.8.20].Forn3,Theorem4.2in[ 20 ]givestheboundednessofTFasamapfromL2(@L)toH1(@L).ThisresultextendstoouroperatorTasfollows.Thedierence, 37

PAGE 38

@ny(x;y)=@G @ny(x;y)@N @ny(x;y): 3 ).Consequently,TTFisboundedfromL2(@L)toH1(@L).SincebothTFandTTFareboundedfromL2(@L)toH1(@L),weconcludethatTisboundedfromL2(@L)toH1(@L).ThisimpliesthatTiscompactonH1=2(@L)sinceH1embedscompactlyinH1=2;thatis,by[ 9 ,Thm.1.4.3.2]HsembedscompactlyinHtwhens>t0.Hence,TiscompactonH1=2(@L). 4{7 )whichareorthogonalrelativetotheinnerproduct( 6{5 ).HeretheconvergenceiswithrespecttothenormofH10(). Proof. 1 ,anycompleteorthonormalbasisforH10(L)isaneigenfunctionbasiscorrespondingtotheeigenvalue1.Likewise,anycompleteorthonormalbasisforH10(Lc)isabasiswhoseelementsareeigenfunctionsofthegeneralizedeigenproblemcorrespondingtotheeigenvalue0.Tocompletetheproof,weneedtoshowthatanyf2Hliesinthespanoftheremainingeigenfunctionsfor( 4{7 ). ByLemma 4 ,Tiscompactandselfadjointrelativetotheinnerproduct(;)denedin( 5{28 ).Hence,everyf2H1=2(@L)hasauniqueexpansionintermsoforthogonaleigenfunctionsofT(forexample,see[ 4 ,Thm.1.28]).Givenf2H,itsrestrictionto@L

PAGE 39

2 ,E(i)isaneigenfunctionforthegeneralizedeigenproblem( 4{7 ). 39

PAGE 40

Thischapterisbasedonthepaper[ 12 ]. 4{1 )intermsoftheeigenfunctionsof( 4{7 )andanalyzelimitstocomputethechangeintheelectricpotentialduetolightningdischarge.Ourmainresultisthefollowing: =0inLc;=0on@;=L0on@L: Thus+hastheconstantvalueLalongthelightningchannelLandthechangeinthepotentialduetolightninghasbeenexpressedintermsofthepotential0beforethelightningandthelightningchannelL.WhenLtouches@,asitwouldduringacloud-to-groundash,L=0andcanbeeliminated.Thatis,asLapproachesthe 40

PAGE 41

6{4 )withL=0. 4{7 ),asgivenbyTheorem 1 .WedecomposeNintothedisjointunionoffoursetscorrespondingtothefourclassesofeigenfunctionsdescribedintheintroduction:S=fi2N:i==krkL2()g;S0=fi2N:i=0;hri;ri=0g;S1=fi2N:i=1g;S+=fi2N:0
PAGE 42

6{6 ).Takingv=j,j=1;2;:::,andutilizingtheorthonormalityoftheeigenfunctionsyieldsthelinearsystem wherethedotdenotestimederivativeand (0)i=hr0;rii;i2N; Sincetheiareorthonormaleigenfunctionsfor( 4{7 ),wehavedij=hri;rjiL=ihi;ji=8><>:iifi=j;0otherwise. Hence,Disadiagonalmatrixwiththeeigenvaluesi,i2N,onthediagonal.Sincetheeigenvaluesarenonnegative,Dispositivesemidenite.WenowconsiderAandf: 6{9 )and( 6{10 )respectively,arebothnite,andwehave forallx2`2since0.SinceAispositivesemidenite,theCauchy-Schwarzinequalityyields 42

PAGE 43

6{12 )wehavexTAx=kkL1*1Xi=1xiri;1Xi=1xiri+=kkL1: 6{13 )yieldstherstinequalityin( 6{11 ). Letu2H10()betheweaksolutiontou=rJp:hru;rvi=hJp;rviforallv2H10(): Forfdenedin( 6{10 ),kfk2=1Xi=1hJp;rii2=1Xi=1hru;rii2=hru;ruikJpk2: 6{14 ). 2 .Multiply( 6{7 )byTandutilizethefactthatDandAarepositivesemidenitetoobtainT_=TATD+TfTf: 2d dtkk2=T_kkkfk1 2(kk2+kfk2): dtetkk2etkfk2:

PAGE 44

Thisshowsthatk()kisuniformlyboundedoveranyniteinterval. Fortheremaininganalysis,itisconvenientiftheeigenvaluesarearrangedindecreasingorder.Thuswithoutlossofgenerality,weassumethatD=264000375 Multiply( 6{7 )by[pT;0]toobtainpT_p=[p;0]T_=[p;0]TA[p;0]T264p0375+[p;0]Tf: 1 2d dtkpk2kk2kAkpTp+kpkkfkkk2kAk0kpk2+1 2(kpk2+kfk2); where0denotesthesmallestpositiveeigenvalue;apositivelowerboundfor0isobtainedinProposition 3 .Supposeislargeenoughthat01.Chooset>0andletcbetheconstantdenedbyc=kfk2+2kAkmaxs2[0;t]k(s)k2;

PAGE 45

6{15 ).By( 6{16 ),wehaved dtkpk2c0kpk2 dte0tkpk2e0tc: 0: 6{7 )dependson. Nowconsiderthebottomhalfoftheequation( 6{7 ): whereA2denotesthebottomhalfofAandf2isthebottomhalfoff.SincethebottomhalfofDiszero,theDtermof( 6{7 )isnotpresentin( 6{17 ).Since()isboundedoveranyniteinterval,independentofby( 6{15 ),itfollowsfrom( 6{17 )thatz(t)approachesz(0)asttendsto0,independentof.Tosummarize,wehavelimt!0+lim!1(t)=[0;z(0)]; 45

PAGE 46

wherei(0)=hr0;rii: Since(x)=1forx2L,thetophalfof( 6{1 )hasbeenestablished. Nowsupposethatx2Lc.Bythecompletenessofthei,wehave Consequently,forx2Lc,( 6{18 )canberewritten sinceifori2S1vanishesonLc.Letdenotethenaltermin( 6{21 ):(x)=Xi2S+i(0)i(x) Fori2S+,wehavei=0onLcsincetheeigenfunctionsassociatedwithindicesinS+areharmonicineitherLorLc.Hence,=0inLc.vanisheson@sincei2H10().Toobtaintheboundaryvaluesforon@L,weexaminetheeigenexpansion( 6{20 ),whichcanberearrangedintheform 46

PAGE 47

6{19 )and( 6{22 )that(x)=Xi2S+i(0)i(x)=0(x)Xi2Si(0)i(x)=0(x)L(x): 2 47

PAGE 48

Inthischapter,wewillpresenttheresultsinone-dimensionforboththegeneralizedeigenproblemfortheLaplacianandthecontinuousmodelforthelightningdischarge. 4{7 )inonedimensionwhereistheinterval[0;1]andLisasubinterval[a;b](0;1).Inthiscase,thereareprecisely2eigenfunctionsinH.ThefunctionswhichareharmoniconbothLandLcarepiecewiselinear.TheeigenfunctionofProposition 1 ,correspondingtotheeigenvalue0,isdenedbyitsboundaryvalues(0)=(1)=0andthevalues(x)=1onL.Lets1,s2,ands3betheslopeontheintervals[0;a],[a;b],and[b;1]respectivelyoftheremainingeigenfunctionu2H.ThejumpconditionofLemma 1 yields Hence,s1=s3.Letsdenoteeithers1ors3.Theboundaryconditionsu(0)=u(1)=0implythat0=Z10u0(x)dx=s1a+s2(ba)+s3(1b)=s(1+ab)+s2(ba): 7{1 ),wehave=1(ba): 7-1 2 indimension1(n=1)withtheopeninterval(0;1)andLasubinterval(a;b)whoseclosureiscontainedin(0;1).Inthiscase, 48

PAGE 49

EigenfunctionsinHinonedimension. theequationsdescribingreduceto00=0in(0;a)[(b;1);=1in[a;b];(0)=(1)=0: 1b: 1b=(1b)0(a)+a0(b) 1b+a: 49

PAGE 50

2 NowletusfocusonthepotentialchangeoutsidethelightningchannelL.AccordingtoTheorem 2 ,00=0on(0;a);(0)=0;(a)=L0(a);00=0on(b;1);(1)=0;(b)=L0(b): 1b: 7{2 ),weobtainr1=(11)0(a)+20(b) 1b+a=r2:

PAGE 51

2 ,wehave+(x)=8><>:0(x)+x0 51

PAGE 52

Inthisdissertation,Maxwell'sequationsareusedtoestablishacontinuouslightningdischargemodel: @t=r(r)+rJ;(x;t)2[0;1); where=()1 2.InChapter4weshowthattheeigenproblemforAisequivalenttoageneralizedeigenproblemfortheLaplacian( 4{7 ).Weanalyzetheeigenproblem( 4{7 )inChapter5andobtainthefollowingresults:TheelementsofH10(Lc)areeigenfunctionscorrespondingtotheeigenvalue0,whiletheelementsofH10(L)areeigenfunctionscorrespondingtotheeigenvalue1.TheremainingeigenfunctionsareelementsofthepiecewiseharmonicspaceH,consistingoffunctionsinH10()whichareharmonicinbothLandLc.Thereisaone-to-onecorrespondencebetweeneigenfunctionsof( 4{7 )inHandeigenfunctionsofthedoublelayerpotentialTin( 5{13 ).Theeigenfunctionsof( 4{7 )aretheharmonicextensionsoftheeigenfunctionsofT,andifisaneigenvalueofT,then=1=2isthecorrespondingeigenvalueof( 4{7 ).2H(seeProposition 1 )istheonlyeigenfunctioninHcorrespondingtotheeigenvalue0.Alltheremaining 52

PAGE 53

4{7 )associatedwitheigenfunctionsinHarecontainedinthehalf-openinterval[0;1),theeigenvaluesofthedoublelayerpotentialTin( 5{13 )arecontainedin[1=2;1=2).Proposition 3 givesalowerboundforthepositiveeigenvaluesofthegeneralizedeigenproblem,orequivalently,alowerboundforthegapbetweenthelargestandthesecondlargesteigenvalueofT.BasedonthefactthatthedoublelayerpotentialTisselfadjointandcompactrelativetotheinnerproduct( 5{28 ),asestablishedinLemma 4 ,weconcludethatanyf2H10()canbeexpressedasalinearcombinationoforthogonaleigenfunctionsfor( 4{7 ).ThepotentialimmediatelyafterthelightningdischargeiscomputedinChapter6byexpandingthepotentialusingtheorthonormaleigenfunctionsof( 4{7 )andstudyingthelimitsastendstoinnityandttendstozerotocomputethesolutionto( 8{1 ).Wendthatthepotentialimmediatelyafterthelightningdischargeisconstantthroughoutthelightningdomainandtheconstantvaluedependsontheinitialpotentialandtheeigenfunctionof( 4{7 ).Outsidethelightningdomain,thechangeinthepotentialisthesolutiontotheproblem=0inLc;=0on@;=L0on@L: 53

PAGE 54

[1] [2] [3] [4] [5] [6] ,InverseAcousticandElectromagneticScatteringTheory,Springer-Verlag,Berlin,1998. [7] [8] [9] [10] [11] ,Adiscretemodelforthelightningdischarge,J.Comput.Phys.,144(1998),pp.137{150. [12] [13] [14] [15] [16] ,Anumericalmodelingstudyofamontanathunderstorm:2.modelresultsversusobservationsinvolvingelectricalaspects,J.Geophys.Res.,92(1987),pp.5661{5675. 54

PAGE 55

[18] [19] [20] [21] [22] [23] [24] [25] [26] [27] ,Bestimmungmaximalerentladungs-stromstarkenausihrermagnetisirendenwirkung,Ann.Phys.Chem.,65(1898),pp.458{475. [28] ,Uberdieblizentlandungenerreichtstronstarke,Phys.Z.,2(1900),pp.307{307. [29] [30] [31] [32] 55

PAGE 56

,Investigationsonlightningdischargesandontheelectriceldofthunderstorms,Phil.Trans.R.Soc.,Ser.A,221(1920),pp.73{115. [34] 56

PAGE 57

xml version 1.0 encoding UTF-8
REPORT xmlns http:www.fcla.edudlsmddaitss xmlns:xsi http:www.w3.org2001XMLSchema-instance xsi:schemaLocation http:www.fcla.edudlsmddaitssdaitssReport.xsd
INGEST IEID E20101207_AAAACL INGEST_TIME 2010-12-07T16:26:27Z PACKAGE UFE0019659_00001
AGREEMENT_INFO ACCOUNT UF PROJECT UFDC
FILES
FILE SIZE 1634 DFID F20101207_AABRGV ORIGIN DEPOSITOR PATH aslan_b_Page_56thm.jpg GLOBAL false PRESERVATION BIT MESSAGE_DIGEST ALGORITHM MD5
263f1503f671b477fca7f029e576a773
SHA-1
225c25bd02240456f6eb82cfde24be801ba91143
25526 F20101207_AABQZX aslan_b_Page_27.pro
6fd4fe876a023fd6715bc5b9d52496fc
6422c99d17cccb0f2250c6392d8511d0b990cec2
1657 F20101207_AABRBY aslan_b_Page_28.txt
513bffbf218634bf003c00361430761a
e93be227d2776342db6159397531211d04a77c4c
63155 F20101207_AABQSC aslan_b_Page_16.jpg
70722b241bd0e5954f3b3bd12b9dce7a
299c78a199d734f13b6de970c18eca69c8a9d00c
2031 F20101207_AABRGW aslan_b_Page_57thm.jpg
312330641f84d186d0fa5ff2c1dcae88
8ef5d4b323948a034dd3f6f0612f25eaa2047a89
26239 F20101207_AABQXA aslan_b_Page_57.jp2
a91c6ecebddd89098cf3795fa6f26a97
9c5775d39df91ee1a26739a7a380e6d9f4691c1c
34013 F20101207_AABQZY aslan_b_Page_28.pro
10e34445ace030c94e4b43bc766c88ba
ea6d044d11593756c149b712136e268b2a8bd521
1557 F20101207_AABRBZ aslan_b_Page_29.txt
2c8e27bed1c12cf9607224dfb1dd6fff
b4a2cc60dff988916c1214a982fdb07b04ae9536
32511 F20101207_AABQSD aslan_b_Page_12.QC.jpg
5170ef958fa99e4b580db539bed54d24c7648ef7
7909 F20101207_AABRGX aslan_b_Page_57.QC.jpg
8bb95d2d6aa7b41c322c7804c2149efb
1053954 F20101207_AABQXB aslan_b_Page_01.tif
d4ee5d46175f0dea13911aae027b2a0c
0deb7207eceeff148ca9733a805f5eba505346d9
28728 F20101207_AABQZZ aslan_b_Page_29.pro
1e6f23ebaa6dc64bf415354cd9839df0c0a6b3ba
25271604 F20101207_AABQSE aslan_b_Page_42.tif
25bd73ffc89a6388b9327012f382238b
f9a92f98353db28faf15d9116ae83a25b51fe0f6
67865 F20101207_AABRGY UFE0019659_00001.mets FULL
383b7a57396125259f505c755d2a7ff9
97c7b61aa4ded05f6ddebb156e372d0f4f70a639
F20101207_AABQXC aslan_b_Page_02.tif
2232d31113a46364fc370035cf3dc798
125454 F20101207_AABQSF aslan_b_Page_54.jp2
b1982c7d220bbf80cd812c3e9848279a
7948 F20101207_AABREA aslan_b_Page_14thm.jpg
64d0728b8047199b1cf4622f5d6725b6
1bcb3aaa79429f2e4f2184d65a25a4dafe6dcdc3
F20101207_AABQXD aslan_b_Page_03.tif
9bf2d3dededdbffbec9962b663266307
26328 F20101207_AABQSG aslan_b_Page_53.QC.jpg
7ef62c561e4b6c18aac6028a985fba14
b807b03d6146aa98a04c1ef23b8f806c196356fd
6118 F20101207_AABREB aslan_b_Page_15thm.jpg
68720dd240d718c873f5a5c394df76a4
be3c9645e317a30f456a613b52a0d3c04ec00df1
F20101207_AABQXE aslan_b_Page_04.tif
ca8b773246cbfb45771a2653a1f24e6d
172172 F20101207_AABQSH aslan_b_Page_07.jp2
fe0d3fa77e7d0705fa1e8d93690606f0
092764576f366a271288d2871025bc988af6ec5a
24652 F20101207_AABREC aslan_b_Page_15.QC.jpg
396eeb05386c2aa639bb9c2d411acdc7
b9c1e577b561a3a1a1cc16554a11beb6eba8e1e7
F20101207_AABQXF aslan_b_Page_05.tif
72b19cec5a5f9cc5ac5dacae6f7be410
8336 F20101207_AABQSI aslan_b_Page_55thm.jpg
26a7004e848356b3fa36ecf76eb3509b
2507800639624b5be5fdf285823458957a0946a4
5721 F20101207_AABRED aslan_b_Page_16thm.jpg
23d53a3c7b17a99bc13c386e886014ae
25d51f88962a0bd285eae98160106584d74efa9b
F20101207_AABQXG aslan_b_Page_06.tif
dcba73b6e9b0c1f4222ceaebcd093c06
53401 F20101207_AABQSJ aslan_b_Page_14.pro
841b4cfe25d3895c5fccff9fd20cbf15
bf7315fda7d6269f5a585dee2c04e901c8f0af3d
19126 F20101207_AABREE aslan_b_Page_16.QC.jpg
c4cb3a7d71f83c29410981874cf30639
d134c628317b5e910d19bcbd4674b787cd24ccbd
F20101207_AABQXH aslan_b_Page_07.tif
584ffd2048be35b4048c8fe7d0f9d4d0
64bbb37a772215510fb01e6208e87ea0b93d87be
102855 F20101207_AABQSK aslan_b_Page_32.jpg
31d54f1155755bcd2834cb4e3f7ba58a
f890ef01c0c04e153ab32180a92dcbf78fb2d9e6
4985 F20101207_AABREF aslan_b_Page_17thm.jpg
e6c84cd4164525e629acfdbbaed9b372
24a94bbe50f1bd55842e10cbef0de5c332e00082
F20101207_AABQXI aslan_b_Page_08.tif
2d7950feff6b31cf20a25d363f09498f
8ef492eec40c8311b264d4faac302ab68d890037
31318 F20101207_AABQSL aslan_b_Page_05.QC.jpg
788a0eedda30935597eb1c31a5f1fcd8
b7788422d4a8046f83a926047afa9b0210413e53
16548 F20101207_AABREG aslan_b_Page_17.QC.jpg
e973eb4eda7634ea7e3ce86e23a1d691
F20101207_AABQXJ aslan_b_Page_09.tif
ea89f21b777d52e0e61004c91d75dc80
7c52c2ac51b827517e50f087e943dc47cd28a1e4
306732 F20101207_AABQSM aslan_b_Page_22.jp2
0d7c022ed29fa590cd9922a8002c822f
8eca8481cc0ff4883d19ea366968e014ba46f14d
4219 F20101207_AABREH aslan_b_Page_18thm.jpg
05425d6b368c34abca513d450cecc387
73df575d9687ebb3824bf92941f6a91faf685481
F20101207_AABQXK aslan_b_Page_10.tif
03ba8375b370e3495e02ec7dcb389f29
dd2a5f5d8a4fc2bab04579c11ee747e8c3e1fd90
1597 F20101207_AABQSN aslan_b_Page_21.txt
14656 F20101207_AABREI aslan_b_Page_18.QC.jpg
5cbd2bb360589c3d700d098db8d11e5a
bc531e0665bd874014d9b70037ed48b8b59704b3
F20101207_AABQXL aslan_b_Page_11.tif
4a0d29586280863f44f5f0c3db408a6e
d495453b766e88f7f84d7798330636a9178d5035
5572 F20101207_AABQSO aslan_b_Page_43thm.jpg
208ae1da757e7da62e63693121fec00a
9721ab51242bd8943607958bbd7b2e052d9451e9
7174 F20101207_AABREJ aslan_b_Page_19thm.jpg
b49d1b9765b0380c44c84d3849292712
c2c2d52b3522c210ea80d2d57292a19d6afb460f
F20101207_AABQXM aslan_b_Page_12.tif
916120305b0299f566c3a2b4559e6c16517c08d2
30403 F20101207_AABREK aslan_b_Page_19.QC.jpg
6d9409eb25beb401ba47399ca831951d
204f10ed64e87d93a1939fe88445e21a1c98961b
F20101207_AABQXN aslan_b_Page_13.tif
1e8785b85005503a7e4bee67b17dded5
66848c616781894a2c32da12ab1f1d9af9439a19
66010 F20101207_AABQSP aslan_b_Page_26.jpg
42108e4f0211a3a1b015bda4f5b416a3
fdc0f1cce233c360ecce3336846e38f3666c251b
5999 F20101207_AABREL aslan_b_Page_20thm.jpg
0b95b84277b88ca852e67e0267d4ce11
7cc192b9e318025b9023f79697633c1ebdc622c6
F20101207_AABQXO aslan_b_Page_14.tif
94f03e9dce9d69b487055e6b5f547241
45099b94823e2fc39b268a1e1447f97b7b2766a7
393 F20101207_AABQSQ aslan_b_Page_56.txt
9c0df9ee32b7c18b95d7d06a784e3416
f2f1494f1410ca0219a825eaae09c3bd9e93f772
22211 F20101207_AABREM aslan_b_Page_20.QC.jpg
a21ecff4e059c7529d4576e4b2496810
F20101207_AABQXP aslan_b_Page_16.tif
8d4ee1353628225a131338b77dff6e82
87fc330ec6a42f38d7b8af3375184d4985758dd5
33546 F20101207_AABQSR aslan_b_Page_14.QC.jpg
b372b6a2a611be2e7f185a6b0a77524f
f21a8fcd5fb6209169fc1822554b405614aec371
5956 F20101207_AABREN aslan_b_Page_21thm.jpg
1ab1b6edc0da48d1bbdd39786ca8d71b
c0741547e0ca69800e01c347de07898fbb898a5f
F20101207_AABQXQ aslan_b_Page_17.tif
e7fd547fe1a17c4f0fc4a8a092e81fed
95a1f004a86cd0bb8eb031c8a721176b56620f81
43535 F20101207_AABQSS aslan_b_Page_18.jp2
74a969c1a5a97ce37a2510a59f7d339e
96de16ace579ddaee8c79d0efda6b775799233cd
21571 F20101207_AABREO aslan_b_Page_21.QC.jpg
1f3502554551af5c6739417c162b3ed1
F20101207_AABQXR aslan_b_Page_18.tif
17a6cb155308b41e8bcde872633e36f2
9d270c42872b747bc3512de7c54f7160a2eac8ed
47509 F20101207_AABQST aslan_b_Page_17.jpg
93768e2a61411104920ee9bd188ae6b9
770f16df55c95bb996f63ab30f33b74fc1dddd32
2916 F20101207_AABREP aslan_b_Page_22thm.jpg
5a570273c1e478a553453dbd0c203acd
F20101207_AABQXS aslan_b_Page_19.tif
15435e9b44a3c645fc28ba241f22bed4
60f0c7e499a1596b360f9cb6bbf1056293553c2c
18212 F20101207_AABQSU aslan_b_Page_43.QC.jpg
2ce21d8fa8d44abbf12bab1f0be9bbe1
f23080b89fd1147d7bfb5f43573088cae8d4a3f0
10448 F20101207_AABREQ aslan_b_Page_22.QC.jpg
818cccababdd4e49044b3a8e60f20969
06ab046023e2139f74fd887023ed9959fd45e8d4
F20101207_AABQXT aslan_b_Page_20.tif
c196158b08d6a54206f9747cbb78a7f2
cb2da5b987221fb9929ce531ae66fbae4dfe5e26
29944 F20101207_AABQSV aslan_b_Page_52.QC.jpg
c3b0dd350c91e1163cb5bee336e70095
0e0bc6e06514b637c9c108f70e6a51873362bea4
7053 F20101207_AABRER aslan_b_Page_23thm.jpg
7391f64f27c30963c342e6dee70f2f0d
94e5feec37089a0d8c65af45717fc36100a9a762
21608 F20101207_AABQSW aslan_b_Page_36.QC.jpg
069f3a7831572006479e003b3446c00a
020e60d92f000721a411caa590b6a2197cba700d
F20101207_AABQXU aslan_b_Page_21.tif
eed125936797016d2f3fe3ffd927fdc8
26c493064ca3ec57b5b2bb0bf38998f9923bcd0b
32126 F20101207_AABQSX aslan_b_Page_32.QC.jpg
6d2c1195bf292baeee53e263f8cee4d6
bbf5ea229fe9220f2ea92d15330b82df5b917c21
27532 F20101207_AABRES aslan_b_Page_23.QC.jpg
1802fb5fa5203e8619d5c1caba539c23
ee66838fdaba0b9ed42539c1b5b15bcd956bdfc6
F20101207_AABQXV aslan_b_Page_22.tif
1caf2bdfdda5e249ebcf9d167b77fdb5
033d15362ffff100d56151f157232de525cd7caf
F20101207_AABQSY aslan_b_Page_15.tif
f82a444c6183328d5b319b1c54f49678
9f2929c8cd572e54afff8ac44a03e411a615c172
7201 F20101207_AABRET aslan_b_Page_24thm.jpg
f31f1bf8e4826f5f55a7026c1d7ddc27
405fffbf0db4a3af03eb3483dcf61df3e95f449e
F20101207_AABQXW aslan_b_Page_23.tif
b1c750412cde705ffd012e933a96810e4277066d
704329 F20101207_AABQSZ aslan_b_Page_21.jp2
17d102c8371d87343c1fec07a42383ee
8f310392018ebaf0158514fb0c5c490200ca491e
26843 F20101207_AABREU aslan_b_Page_24.QC.jpg
2c26044ec82af2e1921a86a740611ae4
F20101207_AABQXX aslan_b_Page_24.tif
8bd46e44c47322e324a14df4b5db2727
2c312f1e50640028b6623a4d5b705ca53da2e4bf
21522 F20101207_AABREV aslan_b_Page_25.QC.jpg
8873986926a55b0d4963e6aa0786f52e
F20101207_AABQXY aslan_b_Page_26.tif
ee52aca4e6d658fb7f237c467335edde
91a28f4dd7ac19c243023a5a9a410943288a4c6d
6052 F20101207_AABREW aslan_b_Page_26thm.jpg
ee63a400601ab279244ffc33285576fa
aa4a6b27cd3309d8a802b3c8695cafc5be7cdf09
119437 F20101207_AABQVA aslan_b_Page_55.jpg
9ca443c1474ff3eea876a4e7dde17695
71f46f5b20c6b4b5c3480d805fa1a5c3c6d399d4
F20101207_AABQXZ aslan_b_Page_28.tif
affe07b41661847588c7b2a68402cc6a
9c3df4a178fe0374d3a7d2b3defa1b5f58a3e177
21546 F20101207_AABREX aslan_b_Page_26.QC.jpg
1a49155795daf80c85a67e32a4330d88d3b080fc
20710 F20101207_AABQVB aslan_b_Page_56.jpg
6ea6f3d475d12ecb150856b243520d97
3597a53b87ace8deeb6e2cd9e47997ec1a930542
5097 F20101207_AABREY aslan_b_Page_27thm.jpg
24809 F20101207_AABQVC aslan_b_Page_57.jpg
85e5112ce8c363923a1c539554d0e136
1256 F20101207_AABRCA aslan_b_Page_30.txt
26afa207a0652c918a17e006db4db325
5cf4fa7ea94d194a015fee68a77f42012b06ccac
17808 F20101207_AABREZ aslan_b_Page_27.QC.jpg
8583d42691f5ece794e092f735e9ddf6
44550f0676b8f3ca8c9cebe4dca4d6e29c3259d1
22711 F20101207_AABQVD aslan_b_Page_01.jp2
2a8ccdc86449be24fee72ece797c0774
5046a5a60c688be538f62183dd7a1dab792cbf2f
1376 F20101207_AABRCB aslan_b_Page_31.txt
965ebcf3d774d0e468f0c499500fcf06
46befb57f4e90c98672bcceca7d2c498b869d67f
5781 F20101207_AABQVE aslan_b_Page_02.jp2
7d5f42b558b136e7c0f607d853d8218c
50bbeb4d2be29a117eeb9ceea050f34b8a7036cf
2188 F20101207_AABRCC aslan_b_Page_32.txt
8b9836d9f3311e30fb893d2a9679306d
66b4cb4b7352b2f24985c0637413b9034300086b
9020 F20101207_AABQVF aslan_b_Page_03.jp2
e9c519383cfbe39d7e8f39a2d47d4312
5ecf57140413680d5216177a7b802c03ed53341d
1703 F20101207_AABRCD aslan_b_Page_33.txt
79205bd3e425602294fddce86600ba74
14f2d1c051b8f5fe3dce56fc9977e5bb7e2d0399
82417 F20101207_AABQVG aslan_b_Page_04.jp2
61dec404a06010efd9f4f8244ec4a937f2921419
1536 F20101207_AABRCE aslan_b_Page_34.txt
bd9df6f00f8d59a300d3c8530af86775
edb3da02e03c5c1eb81c5e7ce341ece3aa1cd88d
1051961 F20101207_AABQVH aslan_b_Page_05.jp2
0e3e735fb5980ed1552c1ac46e35f6db
df508591f2f823b259c3c20a4e13bcdc8626cc18
1865 F20101207_AABRCF aslan_b_Page_35.txt
867b8358dbd793473bcab9520c55d386
db3d9e438f781f5958b6dd484833101cd88f7c4e
205492 F20101207_AABQVI aslan_b_Page_06.jp2
e92dd8d6852f6d0a57b2d9a91b34d920
6c0b8a16d7ecc8e5c446fa2b4534bf1685ed3a13
1549 F20101207_AABRCG aslan_b_Page_36.txt
e60e1b302af0848dbaf5c17b5d1d20a7
457258b34eb5c6cc0105bf7eb8416502d43a317e
49693 F20101207_AABQVJ aslan_b_Page_08.jp2
8c9c64d4d3f71733aa43c5e8966bb4e0
e67fcd21971b71e6c0ea6e8197b3e57afb8dbf6a
1686 F20101207_AABRCH aslan_b_Page_37.txt
980878ed58f97102da70a8417ab2ef79
1051956 F20101207_AABQVK aslan_b_Page_09.jp2
2e908510cc563e0ed4c85fc2d7fff561
b3ce99230e0376d158db9718f5ee30bf09b0fb45
1971 F20101207_AABRCI aslan_b_Page_38.txt
ece998b4113040124e03ac09bd5fcfe6
1045466 F20101207_AABQVL aslan_b_Page_10.jp2
602d240266f7a0b087ed5eda20666aae
db6e414806e91d8f40df48d27d8a840b1b246f7c
564 F20101207_AABRCJ aslan_b_Page_39.txt
3845acbffb505bc1e2666701238c8ee5
163e2a8ce4dce283d7abce209aea5c401e2d6f07
1051969 F20101207_AABQVM aslan_b_Page_11.jp2
b076e6fce0381d00f0e3bf62c55554c9462d7101
1823 F20101207_AABRCK aslan_b_Page_40.txt
51c48f6ae5df566d2c38eb3edb085237
db536d1e28febb6b72934b98dcc9b98e6dc1b716
1051968 F20101207_AABQVN aslan_b_Page_12.jp2
658973e69095c32bccf857142880f02e
1668 F20101207_AABRCL aslan_b_Page_41.txt
7181c1ffe75ee9559fbc67035142c660
7074fba7bf2a9f60065399351ffd828b0bc7774d
1051914 F20101207_AABQVO aslan_b_Page_13.jp2
cfec76520a440f9e912a63b1e8b432fa
1696 F20101207_AABRCM aslan_b_Page_42.txt
b833ff26b3523fac97048e4367991166ebdc8fc0
80088 F20101207_AABQVP aslan_b_Page_15.jp2
020cd243a7044a4196685ee4d4aa6a63
1c3cf44b171f528d0aff8f2f3f1393d36580d356
1447 F20101207_AABRCN aslan_b_Page_43.txt
ef6c7cd23522366af41de0c60fc99b2db65e1348
660296 F20101207_AABQVQ aslan_b_Page_16.jp2
ed2f032ddce859cc0614d0878ebed8b7
0395da2be1fa0c7cbdcb0bd81b905157baa7d7c9
1656 F20101207_AABRCO aslan_b_Page_44.txt
4f99ae50a1a54dfdf14dd5a881faa159f888e7d2
485097 F20101207_AABQVR aslan_b_Page_17.jp2
d7728b0d2b867885f6a3d3278958df4b
d06ae59c87d939a3b1c4712879bfaecbe0ce2d75
1702 F20101207_AABRCP aslan_b_Page_45.txt
2cdc675490396dbb6058c833eb5f1325
bba87f9de9434370a9cbee3a52ba0cce91316ce9
1930 F20101207_AABRCQ aslan_b_Page_46.txt
063cb28086a76dc2186049be13e14473
F20101207_AABQVS aslan_b_Page_19.jp2
863693882d53642a2d26f9ee9aa018f7
739ac6db66c05267283983aa3f8f4fd00e271bc6
429 F20101207_AABRCR aslan_b_Page_47.txt
8a9eaa363d762306bce0bfd0b2405c2c
9d42c4cf8db1edb44a4cdf94d5a8f904e308baaf
704161 F20101207_AABQVT aslan_b_Page_20.jp2
5ccc1a747c24cfc5c33f62a4a3c5b14e
645fd2692e012ce63a732b83453de8b84c999d9b
1774 F20101207_AABRCS aslan_b_Page_48.txt
848ac63efbc9d2e540b8eff2d86ed865
baa5f3f2a364acb67102bc5df17bcb194cdcfc46
921519 F20101207_AABQVU aslan_b_Page_23.jp2
b560c4b9aa797b8c25c4c3b5a32a12c1
d51cd4c7d4562f52252ba51123cc318545154976
835 F20101207_AABRCT aslan_b_Page_49.txt
2be741df5d88706aec18714109e7bdba
639c7766e70dbed22e976dbdde1cbf6bfe7e3af5
913853 F20101207_AABQVV aslan_b_Page_24.jp2
a27e047742045e32d49c4484ca9aafe1
cb285a9519f079e1e9597debb410c307217ec257
1832 F20101207_AABRCU aslan_b_Page_50.txt
4d0393b3f40e28937fc61387eede6a4b
843fea28bd65b3a52a647dcd51d7043b08d3aae7
714409 F20101207_AABQVW aslan_b_Page_25.jp2
1355fb915e1e579cf8bf743ee052efbd
25571d923ca4d22da923b0a070d364b33226903c
477 F20101207_AABRCV aslan_b_Page_51.txt
5e81942cb4dc6cdd320ffe3e70d9d6b9
a5287f2715d9ccdd10820c973c1d947d80dc461a
714285 F20101207_AABQVX aslan_b_Page_26.jp2
2042 F20101207_AABRCW aslan_b_Page_52.txt
702405c97ab3dfb6d893206b35663faa
b58234eed59810fc250b080970c37d9a9c348a70
87368 F20101207_AABQTA UFE0019659_00001.xml
c1de493ea4cc10a3f37d470be8b41f38c25f34d9
559525 F20101207_AABQVY aslan_b_Page_27.jp2
e577f63ab1f5f4cca1d227997a0b2c03
4522edaf1a0195e33a15d3a5637257960fab714e
1680 F20101207_AABRCX aslan_b_Page_53.txt
de22f02fc89e71866b96f078c6b1a832
6a4c7fdfd1bd3d74903920a1099be1fa1d0e35e2
761983 F20101207_AABQVZ aslan_b_Page_28.jp2
7ee5f9777fa48bcf6cf26ddda6ab3cde
83d5b11f603115b5927085ca83f0df85f037bf8e
2227 F20101207_AABRCY aslan_b_Page_54.txt
388856716f38647a18d34aa2f7095fee
c44e345c1d1edb46b6c885a4510b7ea9d0f938dd
22706 F20101207_AABRAA aslan_b_Page_30.pro
be0e4c4be17a6b68403695de2ddd2527
c8ceee6eedcdbf3966ecb3d279c2f01d7bece036
2442 F20101207_AABRCZ aslan_b_Page_55.txt
295aa74c347cf09c14556187a63fac19
ac85a95432522a078d146abe1e5ed1433925c6fd
24586 F20101207_AABQTD aslan_b_Page_01.jpg
16971ca0b678c6d17419286a73c42089
8052dde50291569282bbd72d33405d24c1053fa6
F20101207_AABQYA aslan_b_Page_29.tif
f5b4b2f8daef5f98eb829a35d70703d4
4287 F20101207_AABQTE aslan_b_Page_02.jpg
df83f1f8e2c660dcafe88f112b1e67fe
79681fd257042d925219bd09c8b741c76ce6fa1a
F20101207_AABQYB aslan_b_Page_30.tif
479e89d1c587009e29ea41a593042e10c5748355
27816 F20101207_AABRAB aslan_b_Page_31.pro
ddb9a91c6e6f3248bedc62f691b64061
444c86975f3c205b194557630c038416d3853d75
8117 F20101207_AABQTF aslan_b_Page_03.jpg
06f1893f9ff1777c337985909cf2fcd9
acc11992557bd7ab4f6e5354ea7e262125a7e236
6162 F20101207_AABRFA aslan_b_Page_28thm.jpg
5c1c4f2a25e327c1f3051e7bba8fd6d1b61f216e
F20101207_AABQYC aslan_b_Page_31.tif
19bff8c24936a1db17693005e1aa10cd
bf861ef1824429d6868a0234ff9f8a9dc332367b
33162 F20101207_AABRAC aslan_b_Page_33.pro
53ee83736b283728e4e713b7a144e47a
8df2e14dc8cbba4484e0288eebe45cdbe9db8c37
75847 F20101207_AABQTG aslan_b_Page_04.jpg
9d8d18e59c743d6b3e87fd6a2a7c330c
139248f7d449271602ba23990f1d6180df688a0b
F20101207_AABQYD aslan_b_Page_32.tif
912cdd93e28e6454a6de8d88bdafdc2f
01bb7e6f11ebbe530b0277ae4c0596af
df59cb87801ccf732a4d78906328662520199418
109632 F20101207_AABQTH aslan_b_Page_05.jpg
021c00aa88221317016583072b152862
d56c32620cd71b64318a0eea3e910e2e5ec3ac06
23431 F20101207_AABRFB aslan_b_Page_28.QC.jpg
bacc839b7da2e031de745cba1dab4bbb
f6e1f61a2cb4ef29e611b7dbf07ca94eecb5ded1
F20101207_AABQYE aslan_b_Page_33.tif
1c25a4cf1f6fa2c25cce426050f23ba4
ede0c4bed12c158f008d715589d4c3794a268578
43351 F20101207_AABRAE aslan_b_Page_35.pro
267d36f2e8de0fe65bc88685f826e9b840672916
18167 F20101207_AABQTI aslan_b_Page_06.jpg
a643a96906b56d5e0272e83a0a18687e
c3782277ac52bc881d3013470f42935a8aacc5af
5741 F20101207_AABRFC aslan_b_Page_29thm.jpg
c6d9a7aaa92a67fe5ec25db77ba7e0bbc64c6378
F20101207_AABQYF aslan_b_Page_34.tif
eebf72b14edefbf0ac71141cffc234e9
3b78ab94c5cd87a7132a61c82e3e8883fa65cf22
31389 F20101207_AABRAF aslan_b_Page_36.pro
c1e050f33e911d2635987510a460772c
232df422ef9f49fbd376ebe50156fbc5e7ec8f17
13627 F20101207_AABQTJ aslan_b_Page_07.jpg
717a12ab30a23f44f12cc7953b004dc0
19656 F20101207_AABRFD aslan_b_Page_29.QC.jpg
9b38a3efd3e9905a551b7de3513a3d3a
63f5e9cfcb76ba69b4daba06edba593d5d5b1ced
F20101207_AABQYG aslan_b_Page_35.tif
4c7843ebe59e603228a9579ca53fe02a
cc2c6179286706e3ae1bf72cb1439f10dabe8d8b
31931 F20101207_AABRAG aslan_b_Page_37.pro
4f7e91a7923680c5ec60971d96d9598d
81eb75110b1e5d6b6e874cdd83a6d52e707623eb
110498 F20101207_AABQTK aslan_b_Page_09.jpg
515c1ccc4ecb6bce006348fde7a5f779
5621 F20101207_AABRFE aslan_b_Page_30thm.jpg
9cf2e2861c98bee61f9ec15aec7166f5
f5eeb69080bacbdf0aaa82be7d298ecae5940618
F20101207_AABQYH aslan_b_Page_37.tif
91647c0798b6d6145dc00ff6e24e3028
f3c52c047f6e7e3d3affdd865fcb4d1d955b8f47
44728 F20101207_AABRAH aslan_b_Page_38.pro
4bcc18c7c714e92bc16e96907cd036c2
4f0ba25df947990ae1f9eeb9af1fb4cb5a920557
92478 F20101207_AABQTL aslan_b_Page_10.jpg
e74585e8c3acc24895246716fbe79f5c
0639c2da8efdeff1826f0d79e28019eb0787a048
19476 F20101207_AABRFF aslan_b_Page_30.QC.jpg
1bec79bdc279bf973fbcaf6a1a9b6161
32b3fe0738a9048050a8638d2ddd64eb0be3d8c9
F20101207_AABQYI aslan_b_Page_38.tif
005d19749a86f70da3969e93f4fe78a1
f71d794f8e0f62f7fae09a9f55b89c158dbfe78c
8151 F20101207_AABRAI aslan_b_Page_39.pro
7bc3e058bccc3e4dd24b7d9825a836a9
48be456d6e6fb33dda99e3234c0c9fdb98755ce3
106829 F20101207_AABQTM aslan_b_Page_11.jpg
669c2ac1d43ac497a7f396fc8e109f90
36f43b27fa72cba7b310dc7c222d5c22d04c29b1
5647 F20101207_AABRFG aslan_b_Page_31thm.jpg
256f30d868696d27824309db4b9b1310
5ece74b1242c9d77a8bb830c4c4a5a5ab6267da3
F20101207_AABQYJ aslan_b_Page_39.tif
33697413c0da48f8179b7a5a07b5f38f
973732044b53c6e810c6b12de9d1f9accb9371fc
34524 F20101207_AABRAJ aslan_b_Page_40.pro
9fb2542c91cf05cefb6be5e5deb41140
5059f2242bbcc50fb6ffdd3a4c61977869f83de7
105130 F20101207_AABQTN aslan_b_Page_12.jpg
7f220ff1fb38ea0c46d20520f1b98291
7ba56429010e982bf9f591b5b4ab9332bcdb8e66
20289 F20101207_AABRFH aslan_b_Page_31.QC.jpg
1a63ce68c4b6e39a1183ce07f3e213d4
c63dc733a97311d06a8d25c2e6061339f9b28b80
F20101207_AABQYK aslan_b_Page_40.tif
26c68385c0d3279abd6d30c3726ae566a63a9038
30582 F20101207_AABRAK aslan_b_Page_42.pro
5d8c9146862e28099c0a35f2469a329e833ccd13
103277 F20101207_AABQTO aslan_b_Page_13.jpg
fdbc40be34cf26566a3c0970bd166d285cb6b5cf
7614 F20101207_AABRFI aslan_b_Page_32thm.jpg
a87402efc639ba32d6294849d17015d2
b161c22feff18989c3db9f38d2ebc0c414389cdb
F20101207_AABQYL aslan_b_Page_41.tif
1067478d24f52aaedae3e0e491c9c135
184d67035e248d799fa7141cb21c00692ef41094
27052 F20101207_AABRAL aslan_b_Page_43.pro
7d838c5cd72ce99223392170dc4379ea
1d482df8007f4f5a4ddd540986dbb9693275d0d8
107348 F20101207_AABQTP aslan_b_Page_14.jpg
1ef1aa93e9675f515da8ed92e437f2fb
dd10901ba9ef3d0e9bb55893f39725b995482d88
20978 F20101207_AABRFJ aslan_b_Page_33.QC.jpg
6b92fa77781c2bc6e16b47f4ca42956b
52dc56c78aa7851413df1c5125e71cebd37c0983
F20101207_AABQYM aslan_b_Page_43.tif
fb77d20bee808ab2f02515a22e07f8a42d84242a
29345 F20101207_AABRAM aslan_b_Page_44.pro
d83d2f8549580c6fb9820c24419d04ef
55693a525cfc5fcb11b88ae04f6a50375818c292
5918 F20101207_AABRFK aslan_b_Page_34thm.jpg
68027d9aee00675c257716792d30c7db
b3b20cac92570f5cb444f3f8b708c72d96d2a78f
F20101207_AABQYN aslan_b_Page_44.tif
53edbb3ff081cb7e89666ca058d514aa
a3ae31c42e42250d25f4c697494ee37067decb02
31404 F20101207_AABRAN aslan_b_Page_45.pro
63ed0ff6b187e87202fd7c44f5a6c1e6
ccef54560017f896c3108a14796bdc79ba9602a9
74047 F20101207_AABQTQ aslan_b_Page_15.jpg
697fa1ffb822edab6be0346cf638ea33
16fd8afdf6db91d1e1ae1219827d199cbab11f86
20152 F20101207_AABRFL aslan_b_Page_34.QC.jpg
4ff31ea3b2b28e442a0e685e2c66f0b3
19bf94785625e54279cee98549226a9422fbab38
F20101207_AABQYO aslan_b_Page_45.tif
c2c88ce17cf105c02abe3f9b4583ea22
e2e4d3a50af5d445bd583abbfca6c3708cea89ed
33715 F20101207_AABRAO aslan_b_Page_46.pro
dd3921d7a1c9d925d2b4609bb321ed29
c1f89fc6c73cb4e67e240335a7a45cd1aab4e379
41093 F20101207_AABQTR aslan_b_Page_18.jpg
98fcc9e346bd3f494374b90920a4c87048ece8c9
6996 F20101207_AABRFM aslan_b_Page_35thm.jpg
75973abbe4282448fc859f09368a4116
fb5d28a8c0eb6ca342d02b907216214721232bcc
F20101207_AABQYP aslan_b_Page_46.tif
68df8742b4b5c225ea7ef8776c43ae27
9c14b08b1c939fd5ca58b3a52ee22956ccc35982
9454 F20101207_AABRAP aslan_b_Page_47.pro
a423ffe5059d465fd19a50308f78e01c
fc99dc53bb1525c7ff94034b0749b1c9ac8ea746
96232 F20101207_AABQTS aslan_b_Page_19.jpg
ed648726772d0e2407c16c081f0309c1
4aa015a24909e945bb9d6d4270dd976fa1ccbfda
27006 F20101207_AABRFN aslan_b_Page_35.QC.jpg
f0715a9365610651662ce30747125b75
F20101207_AABQYQ aslan_b_Page_47.tif
81c6f6961cec694d636449676ec3e9f2
35299c6e9d8435ccfa41c98b1b3503f987e88a8b
38247 F20101207_AABRAQ aslan_b_Page_48.pro
90285ac5044c34c9212639167f20d150
4fbdf82300bbb647d1d3151b0ffc2befd1da59b7
65735 F20101207_AABQTT aslan_b_Page_20.jpg
4dc671afce29c7ce25e1e1466a7ca368
5874 F20101207_AABRFO aslan_b_Page_36thm.jpg
b6ae9b674e9ec0bfc79e1ef6b046518c
10eab775a0c112918514f3c51d44a43b3f56f8d3
F20101207_AABQYR aslan_b_Page_48.tif
15247 F20101207_AABRAR aslan_b_Page_49.pro
119d8ca3e918a99b199bbf01543ac6f1
b4a5b9a34e794df4cf21cd1aa78827e4be6a4efa
67237 F20101207_AABQTU aslan_b_Page_21.jpg
8cd0279b12cec717e70398be18b5baa0
5d17302397443f6d13bcd0c1d61927275a6492f3
5979 F20101207_AABRFP aslan_b_Page_37thm.jpg
faaeb684e59b0f5390e82684b6c0e53f
00eaf2ac063f048e49594f64d141232dd4f3580a
F20101207_AABQYS aslan_b_Page_49.tif
7c2a07b0c508015f459b6554d6132e9d
77ba15889756509d9df3d7410ff9f0a1d869056e
37862 F20101207_AABRAS aslan_b_Page_50.pro
69d0b235834095377c6a92aaa6d5fbd5
81e3099017498b88b27bbb91285d19d8e372bbc2
30432 F20101207_AABQTV aslan_b_Page_22.jpg
dd8ed701da360613556055822ac25596
71d39f1c8b32f1b606e46a5fd437364158dfc3c5
21725 F20101207_AABRFQ aslan_b_Page_37.QC.jpg
f80cac71127084f1c9ae3d8056838010
F20101207_AABQYT aslan_b_Page_51.tif
451ce90f0536c68ff9cf0d73e67a7bd2
c817541584a5a348628e56eb2139bbe8562cc8a7
10235 F20101207_AABRAT aslan_b_Page_51.pro
480674b0c5d7ae09fffbd5273fbb9df2
74da89a04cec2d3c344b97bebd49725ef0318c90
7319 F20101207_AABRFR aslan_b_Page_38thm.jpg
f44d40b9e1fc55ba6d1d6f530397864a
2e05e9ab0d9aaf466f0e21dda6db0bd206d58baa
F20101207_AABQYU aslan_b_Page_52.tif
3657b0137c5744ec43f51f3cbcedc352398edf7c
47465 F20101207_AABRAU aslan_b_Page_52.pro
30e5526b21b0ce302355b56e88748023
104b4a20aabff9eedd80099d511d13a9902ce942
86992 F20101207_AABQTW aslan_b_Page_23.jpg
3720c18d9428089217c8423aee6985ee
29341 F20101207_AABRFS aslan_b_Page_38.QC.jpg
f78ba4c54e4bb8b1e272aca407cb6603
7dfe0e82051bf187e854cd1dceb9fb59a4d1784a
41689 F20101207_AABRAV aslan_b_Page_53.pro
f2cd6c7c4926ac08ae5511294b457f28
a24cd66a1a5bf018b200fae26668572b2763dedb
86724 F20101207_AABQTX aslan_b_Page_24.jpg
ff71dfbd0bf25a49352db81361cc91b6
c3e50bdf51207f61ba59b9e6ce9ebd6a6024633b
2118 F20101207_AABRFT aslan_b_Page_39thm.jpg
111cb55a7d9bedab44b5bd62a3d5d833
09f5fc8b35f582b543e6b611628ea97732c269c6
F20101207_AABQYV aslan_b_Page_53.tif
d085946b15b4b27db11f9e3ec1872f86
a6273ed8ffdc4d301c5c56149cc36d34c56fc030
55526 F20101207_AABRAW aslan_b_Page_54.pro
c3c835e04fd0c4eb9d080161a2302bdf
068bdaf2d1a5fb073a2ec2a703a0d39b879c42da
65614 F20101207_AABQTY aslan_b_Page_25.jpg
6544 F20101207_AABRFU aslan_b_Page_39.QC.jpg
282961a85477492d8ebb8f9b9419560c
e102ec283fafd05331ae6ed3898a2a070219c06b
F20101207_AABQYW aslan_b_Page_54.tif
7320bf1511bfc0472cd76e49531a8c71
6d05f5c9d818d62dc91cb66a570b75b36db6f6ec
61022 F20101207_AABRAX aslan_b_Page_55.pro
34e2333b15e73c76356248959204c5f7
73c0e9a3c61cac1775c6cdbd0cc956cb6d35191a
51639 F20101207_AABQTZ aslan_b_Page_27.jpg
bf1667d99816483495181f270eb1b8c1
137d4f848de94558327df3f817e55da7882d4cd4
6083 F20101207_AABRFV aslan_b_Page_40thm.jpg
ea9abdc38b172336cc5808947513459b86487a53
F20101207_AABQYX aslan_b_Page_55.tif
858a59f5ab76fdb92086033af9a379be
41b91cb7da7e0fac856e1a29bd44a61fff869720
9615 F20101207_AABRAY aslan_b_Page_56.pro
975a2ac4c8cd294940cb3c53e9b4e8f0
21354 F20101207_AABRFW aslan_b_Page_40.QC.jpg
3e5bafd6bf427782fb6f018fdc33c2ec49d8516a
661534 F20101207_AABQWA aslan_b_Page_29.jp2
698d978e4142ce2bd5842bbd84453e05
ecab4b4fa0d0d6b92efc8a1144069a8865effe41
F20101207_AABQYY aslan_b_Page_56.tif
b5c4b2bac6bc2db0a8f393e8e05960e6
4562035e4c19ebf61dd370a1f7671bd969230553
10500 F20101207_AABRAZ aslan_b_Page_57.pro
24636 F20101207_AABRFX aslan_b_Page_41.QC.jpg
c5f57de48ac3e5e83c70b7a929655199
e0ff73fd86577f6b4dbc5464cd7ae8e55fb70564
656348 F20101207_AABQWB aslan_b_Page_30.jp2
f9df82c599e9c19b63f3fbe7ac6fbf32
F20101207_AABQYZ aslan_b_Page_57.tif
4940e1a4903e33368d36ae9b3f534dee
5815 F20101207_AABRFY aslan_b_Page_42thm.jpg
a67bf953a2daf947c4c047cffcd02b51
9fd2a8c0881e78b11ca0defa9fa2607a5a0ce586
648991 F20101207_AABQWC aslan_b_Page_31.jp2
ca632164a8e7de28deae91c12c359df3
ae74ab5fe3ce00c219f7f4e1061376620b2c123b
19770 F20101207_AABRFZ aslan_b_Page_42.QC.jpg
a247a1bc2b038dbb65bd0cec5cac803b
6f0b2bf30bbf62fbb3e386cbc2318974fccf08bc
1051930 F20101207_AABQWD aslan_b_Page_32.jp2
7f3abc84326318ddbb06eb66a3a485ac
6e5e140b1bceae2ec4f30ef3e0296705acd49a83
454 F20101207_AABRDA aslan_b_Page_57.txt
481997bef44cbbd3cd7df3eaa42220f239d271bf
721196 F20101207_AABQWE aslan_b_Page_33.jp2
20bd1c417f17d7b71aa5022d4707f965
3fb4dea242e7cc76fbd70b9fc23a160293d780f9
344516 F20101207_AABRDB aslan_b.pdf
9d9229a662dc74cbb6216eee96002216
684663 F20101207_AABQWF aslan_b_Page_34.jp2
4c16ed95e79627743268669a688acd49
7e33c835d8e8582e413e66586f85c54eb71f7392
1819 F20101207_AABRDC aslan_b_Page_01thm.jpg
7ffb00bfc252588ece676fd4fe77b4e6
e235c5637253c5dfd21c77407f267ed8b9f1f264
719156 F20101207_AABQWG aslan_b_Page_36.jp2
17488cf6e96bc7abee8cfbc9baafa20d
a57a4a06521f9ee6e81bbd83124d46bc1589d33e
7485 F20101207_AABRDD aslan_b_Page_01.QC.jpg
5bb2901e62e68b7ac33ee9438a9c825b
e585fde7755732c736bc8b9f74833f4144cf7ab0
724193 F20101207_AABQWH aslan_b_Page_37.jp2
3c8fe06320ff69576352f2d772a1a0f1
d81a94a8be258db6565f274c4be9582c573ae9eb
512 F20101207_AABRDE aslan_b_Page_02thm.jpg
77be8d692a21904757224148268d22cd
1021378 F20101207_AABQWI aslan_b_Page_38.jp2
1a808b81a04023fccda693f43ce02f4e
25253b83fc4a648e5e2ee1febcdd9066753d649e
968142 F20101207_AABQRL aslan_b_Page_35.jp2
31c9962ac97c7bf9de903f51970e5481
43aca737627bbacda24498ab33d0a4752444cd17
1537 F20101207_AABRDF aslan_b_Page_02.QC.jpg
7f615b92bff6713ec478fd05f99a7e86
3c2d42bdd595647bb1f8619468a55c2c82a79ca0
189497 F20101207_AABQWJ aslan_b_Page_39.jp2
4cdb0a59f964cf29ddac49f98df6f9cac9df4703
36098 F20101207_AABQRM aslan_b_Page_41.pro
e1806ab83f5b5a05f815c2eb7f2cc8cfb350f290
816 F20101207_AABRDG aslan_b_Page_03thm.jpg
fd03cd7e03ebb80f07f820027b8b1dff
1fd668e92bfea14778d4fe0821ce6809090debb4
724483 F20101207_AABQWK aslan_b_Page_40.jp2
f7a398930189744e39b73b4b12da5d8c7869d965
51275 F20101207_AABQRN aslan_b_Page_32.pro
a75b7b40fafae7ba5db7b2996172ff15
aa16feacd8a6fcd6df8278f1fdeca0834d1412b3
2680 F20101207_AABRDH aslan_b_Page_03.QC.jpg
c0628ed6d8dda40b2ab69b0a1039544b
9d4c5c0dcaae48356e1be51e74bb3acf4919d565
812443 F20101207_AABQWL aslan_b_Page_41.jp2
331c7568d7493622e17155177eca77a9
59f1a814877f745b4d43ed2dd58711f27fac4ec5
5753 F20101207_AABRDI aslan_b_Page_04thm.jpg
bab714e41b5f6b288db1a197eb2e4182
316a21946bbaf1312d3bbe3bc501f3dfa207763f
633177 F20101207_AABQWM aslan_b_Page_42.jp2
5ff64b2f99493e409d1a9ae1e773bbcf
c52a5501256236f781086536553dac1cb43d39c0
F20101207_AABQRO aslan_b_Page_50.tif
1b5247386a736510d5471a59f6aec7ce
2c92d9c148281a2c2d50a2619fd0d0dde8f9a802
24343 F20101207_AABRDJ aslan_b_Page_04.QC.jpg
7751bc34b7e948ba1212dedd18287e0f
588975 F20101207_AABQWN aslan_b_Page_43.jp2
422a62daf9e3d043254d1db9510085e6
c0dc2b1b5a9dc393b69c355891bdf76bba8532ca
F20101207_AABQRP aslan_b_Page_27.tif
25682b4c2046c7289eeb3c74fa19a010
567718c02b169868d1f2e2e425f5df0fc193fe8e
6590 F20101207_AABRDK aslan_b_Page_05thm.jpg
531f09cc358f7d4437c3fce470b8325e
200a7ebb1e1b574bd2d99a3c82e43d5e78224d21
622188 F20101207_AABQWO aslan_b_Page_44.jp2
d44dc4ffc6276aa9ef8ecfd89193f790
50930 F20101207_AABQRQ aslan_b_Page_13.pro
89f45fa0718e61e5bd5cc037453c24f3
0c14aa098640a6d32c50a2e6079513ee044ee096
1469 F20101207_AABRDL aslan_b_Page_06thm.jpg
fa24652ba3d0cd91c33341658303899d
2eb8e2cab1b3a89158a679fe714569cda52167a1
684427 F20101207_AABQWP aslan_b_Page_45.jp2
71cde58b5489b9797928206967871db4
ec8a41568001252a6f29f47e02775042247d573c
2155 F20101207_AABQRR aslan_b_Page_14.txt
ab7fa1841767e7a11a371a586e59fa46
d1b9e45b087ce4521480dc50f56aaa0392cafe26
6092 F20101207_AABRDM aslan_b_Page_06.QC.jpg
1baa8286be66f2687ae5401fc16f01ab
dbe3be853744a7c994ef5078d51dcc7611ba93c1
694887 F20101207_AABQWQ aslan_b_Page_46.jp2
db0824c1f0a9acffbe3fdd8ebb4c00a0
d2b35922afe46205fbda96cf2e2291e7bfe0c229
5504 F20101207_AABQRS aslan_b_Page_33thm.jpg
60d78daff46e8910202f03b9a5b0352b
16b1a65aee4b7b536fde9c48fc7328f144306924
1526 F20101207_AABRDN aslan_b_Page_07thm.jpg
628be7d223aac924d1b1542a85684ba7
6b740674b435d02b7f1719e9c4d090b3b0d0051e
222297 F20101207_AABQWR aslan_b_Page_47.jp2
6136 F20101207_AABQRT aslan_b_Page_25thm.jpg
c703e928f327a6828551cddf53cd1b36
561b940d93661bb2b86c824f7c4c66269864ac19
4566 F20101207_AABRDO aslan_b_Page_07.QC.jpg
6dafbc9b635c7457c4b6803edb0e5495
874f8ed50be38cdd633158ebe2b391cd4816ddde
830388 F20101207_AABQWS aslan_b_Page_48.jp2
f29ccc3d153e50c6f7273f3372bf4a67
8f4b11bb42c60cf7b9b8080c54c63574a18c49dd
47130 F20101207_AABQRU aslan_b_Page_08.jpg
787b6eee74222be95d77c411a0601066
3e9100e4503b10266868fdb59703a993134f6627
3549 F20101207_AABRDP aslan_b_Page_08thm.jpg
a1ef1dc83d602f693c893fd059ea6155
01088628e1a36044abdb40db972304a7b069b06b
29895 F20101207_AABQRV aslan_b_Page_20.pro
d0d659c27751d13b4bd095dc6efa370d
c8998f15018cb19118ca465de523d5e789a19e8f
14191 F20101207_AABRDQ aslan_b_Page_08.QC.jpg
58fe86785425d910de0fb44a050d4de9
9ea4b55bb038e8c7568a9e46086e506fbec02ba2
1051974 F20101207_AABQRW aslan_b_Page_14.jp2
f0ff23553371d13a5fce87b0a986f37d
b263ea695b76dd3060eb1d9777960c8bcf122b3a
8056 F20101207_AABRDR aslan_b_Page_09thm.jpg
4006ccce3e1d09d7ab5ba61c024fbc57
bf87739e50b77197256e07007ec26ee2bdecf584
33259 F20101207_AABQWT aslan_b_Page_49.jp2
0d79c1e33d78faae917fc707326d7a39
e9440938945e7a41736130713a361320882fa996
F20101207_AABQRX aslan_b_Page_25.tif
9c2f253d836794b142428503bd4076a5
58337d79c7f2241df02605f3dddbdf46baaac5d9
34451 F20101207_AABRDS aslan_b_Page_09.QC.jpg
fc4f48320f3d6896834a8f9f712e075a
796834 F20101207_AABQWU aslan_b_Page_50.jp2
16e90212f9fae312e021db5e2486756a
4ab302859c53ea9755922821f2388d675346b484
2312 F20101207_AABQRY aslan_b_Page_09.txt
c615be4e50d742574ec78427518bc8bc
e79b36bfa4cfc7e1c544bf7347298c80ab8f4b19
6708 F20101207_AABRDT aslan_b_Page_10thm.jpg
b09c58cb9e6672cce4934cf3b1bd7433ce6f8b9c
226347 F20101207_AABQWV aslan_b_Page_51.jp2
d1c4e4626954198cc8fbf1952eb932f3
27f385fd73a949bf81f921e8558670450c56c581
5990 F20101207_AABQRZ aslan_b_Page_56.QC.jpg
a1a4e84997d9e7ab8b261d825a105f7c
5777f39b2fe521a78b7329210f5d69ee8fe2526c
29175 F20101207_AABRDU aslan_b_Page_10.QC.jpg
f0d2dac454986fd66f87971b6506cbe5
afcb6b4ebb775461f64228a1aa39205328de6e1a
1039947 F20101207_AABQWW aslan_b_Page_52.jp2
0f8ed024e82764f063b06554fa6db873
8df45d9954784d378a8c42bf49efff5e2eed841b
7604 F20101207_AABRDV aslan_b_Page_11thm.jpg
648e6884bc3d256ca9dcafdc19cffc3f24c6cc98
938368 F20101207_AABQWX aslan_b_Page_53.jp2
d25f699c6fa227d5a86261dd8b60d308
4f96b3c007648247297e10fcef0c626fc0b7b23e
32643 F20101207_AABRDW aslan_b_Page_11.QC.jpg
e854da9720215f6defd72b907dfb26cb
72382 F20101207_AABQUA aslan_b_Page_28.jpg
7a0b6444267ba488651f14999f7971f435b02c9d
135730 F20101207_AABQWY aslan_b_Page_55.jp2
a08221ce4d358d9ed60a9fb3d6a01768
ab7a2e3e31060758f7bb63684ba9dabfe5631cb4
7617 F20101207_AABRDX aslan_b_Page_12thm.jpg
0c4c55928ef2f6308f7dd8755db59f99
60605 F20101207_AABQUB aslan_b_Page_29.jpg
56f95f582a9bda369ec788e2d26a98dc
969c620de6251f49450e8e4ef2c62c83076f34ec
24186 F20101207_AABQWZ aslan_b_Page_56.jp2
778c5d7a7ee2c1a3e9a58f7f3c2ea070
3ac90f7cc654d0b9d0086cbf031f0a7d2600e9e2
7312 F20101207_AABRDY aslan_b_Page_13thm.jpg
f7e138512cc2e7c6384e2eb20b6499c6
c3bf23bf1d7b3873bc2bcb1458b70a37a14613e1
58944 F20101207_AABQUC aslan_b_Page_30.jpg
d40c9f4e0307d11c0b640596dd1ae8c5
6956a6c18884b024223ddfe2044f3b4b12c86ae9
7497 F20101207_AABQZA aslan_b_Page_01.pro
3ef47e7e5caf72935a25d54a8900714f
6161ff33bf96075c55d1cacd9a3da2ca053ffe0d
419 F20101207_AABRBA aslan_b_Page_01.txt
fddba2257284a0e92043d78e5a6dde12
c6201593a237bc5ed8d85e7fa8dc38df484d16be
31808 F20101207_AABRDZ aslan_b_Page_13.QC.jpg
3bc9da04619d32033831c2f4c174ae96
71f2356c736f938f7cdb31d14a52f9631b7fa29f
60362 F20101207_AABQUD aslan_b_Page_31.jpg
5726bc319213c5c2942ec2988ac0909bfafe8784
1002 F20101207_AABQZB aslan_b_Page_02.pro
13d887b35ff4d8f3de771d4937e225fe
e31043ed8104ab82f747f39aa44861a977520eaa
95 F20101207_AABRBB aslan_b_Page_02.txt
926477b65bc26f26aec58bc0ec060d68
41fd32e0c15760c18361707dfbb4a359a592e17c
68097 F20101207_AABQUE aslan_b_Page_33.jpg
5c61db379b93909745c9b732908d5d3a
109e974f7d6a0b4511ecae4d293cd9707feedb67
2514 F20101207_AABQZC aslan_b_Page_03.pro
e3f1a4ded513b1d1820a3bb11321748b
852b03dd0958aeec0a653f440e2527022de7d23a
167 F20101207_AABRBC aslan_b_Page_03.txt
6a918eb58d8b93cdef0fceea786e99dd
d325a377803ffc81cc1eea81d3810feffd35e798
61699 F20101207_AABQUF aslan_b_Page_34.jpg
9ef253fb969de84d04f1d438f898c2a4
a44288322c584700903fa35d3f356e671df9e796
5639 F20101207_AABRGA aslan_b_Page_44thm.jpg
17137ebb384626d10fd792b5c650251f
6bb11d2c535a120249f39469989b533050f61d38
38167 F20101207_AABQZD aslan_b_Page_04.pro
9edf9b8c7f45f5b1b7d09b2744910b48
1566 F20101207_AABRBD aslan_b_Page_04.txt
95239b3e1a855885e446e420a407face
7461ecf747cf8baf9d2fc6a5860dc925b6a1d806
85378 F20101207_AABQUG aslan_b_Page_35.jpg
95154c6f59c1d87a5aab3fa8d70f04f8
8830867ab4d1d18cd21b7f2094efccc837b99b0b
19951 F20101207_AABRGB aslan_b_Page_44.QC.jpg
d073647c9a16dd59518fbaa19c5cce8e
55dea78c28b8a11c81bb00f9a6b25b3f25bd0657
37970 F20101207_AABQZE aslan_b_Page_05.pro
d5918802bbd79a4af5643f8f8a8bb58c
b850cc3ca360871cf98174cdd9546cf4038275be
1681 F20101207_AABRBE aslan_b_Page_05.txt
be2c58cdf2e7297601cf52151e2dfbed
c96105ef5214bf7844fe15b602146d9aa2f066bd
64625 F20101207_AABQUH aslan_b_Page_36.jpg
23061999528acf20997d615a5a3ed0e8
9a569d49d5f71b63a667949dd96af590ce8db8ac
6842 F20101207_AABQZF aslan_b_Page_06.pro
1be8949c9c9e3bd8c06639f08f180d74
972ec0e2347b79596ce5dcec08eb1198bd2c22b2
285 F20101207_AABRBF aslan_b_Page_06.txt
2c4d750f7c91b3100e0cce8257527b72
65145 F20101207_AABQUI aslan_b_Page_37.jpg
ac849d85cf376a30430a086deaa2cd6e
6e0b5703ed5745e714390c113af39636e590e00b
5640 F20101207_AABRGC aslan_b_Page_45thm.jpg
7a3fa5a9bca0891228c1bf0a48d96d11
ea7c67f2de568829bb56cc22ce69225f003ea23a
6002 F20101207_AABQZG aslan_b_Page_07.pro
34108f8a351df3f69f7145200baab51d
315 F20101207_AABRBG aslan_b_Page_07.txt
8d38329f9a83c71d88dd677dcc6e6665
764b39d51d50f4be1207566ee53832317eed3e5b
91516 F20101207_AABQUJ aslan_b_Page_38.jpg
a0d7c9a74097262118a2b42225991b2c
2d5760210d19128c1405576e2216ab935a0566e8
20488 F20101207_AABRGD aslan_b_Page_45.QC.jpg
e5f65a410f8d125f489be079df4b41cf
40fb0c9f7a61e757c0ec253b8eb8e43ddaf6cec5
21753 F20101207_AABQZH aslan_b_Page_08.pro
8c1e73d28b0fb0bdec01b76318440279
f3314849aaaf7e30f3b6ded5d9e5d8c54844138f
1042 F20101207_AABRBH aslan_b_Page_08.txt
20473 F20101207_AABQUK aslan_b_Page_39.jpg
ed5db4880503153c5253efe02c378a50
fe9da5126a0f532b867e93cd8707ebec522225c2
5819 F20101207_AABRGE aslan_b_Page_46thm.jpg
0c72d48a9b3913bebf8bb2477fb7a9f4
2f6e26cbbf5cd7c3aecc11e84578069e5b00656e
56900 F20101207_AABQZI aslan_b_Page_09.pro
11e9c8f467f2e1a03bb26e2a35f211df
1874 F20101207_AABRBI aslan_b_Page_10.txt
8c7e63a07f80cc3b893516ebcdbea39c
e5a29b79117c3bc840951e64f25871803268fb21
69257 F20101207_AABQUL aslan_b_Page_40.jpg
6052e9ee6650db495e82f2c3a76b92d0
15b3bf70a5b40e0c2e2393769b6c2db5e9e1a451
20826 F20101207_AABRGF aslan_b_Page_46.QC.jpg
9f784753fe615bf86890a620369a7c7b
340f4d52e65e18779a03e083c7cf9a22517f77c0
46651 F20101207_AABQZJ aslan_b_Page_10.pro
6ae953db2e3a1d73067f34aabf9d7815
dc99164f77dc43b122bf2ba98d960ce99e0c25df
2124 F20101207_AABRBJ aslan_b_Page_11.txt
26dc345b8580e318ff2dfa99ccdf4a06
306543f612681378d13e09197e39d08fabae1247
77768 F20101207_AABQUM aslan_b_Page_41.jpg
66f3afba7f50026c1b3f26cb36d68060
527daf7054fe5ec8614a0e789900bdbbc32d9283
2210 F20101207_AABRGG aslan_b_Page_47thm.jpg
d8ffb512bf11847eed5527478377bfcd
2c9a0503d04f44d722c9128c1d74a08a6f14730e
51552 F20101207_AABQZK aslan_b_Page_11.pro
4ce1cd610e73cab1a0a3d089d2ea6436
2071 F20101207_AABRBK aslan_b_Page_12.txt
501808c2392686320e561d5435489252
3dcd7a6aaab7d0d99f86592d363413c9c98e8a66
59070 F20101207_AABQUN aslan_b_Page_42.jpg
998b1c0d6e3ebe51b1072be9b0ca3f7b
d7bb16a67ca0530abc263ff188fb15a9478dae1c
7624 F20101207_AABRGH aslan_b_Page_47.QC.jpg
b7c9b19c56e2ef94378e02c11ebafd86
34e278cd3e9b49ef1254af429765f89f24ea2270
51668 F20101207_AABQZL aslan_b_Page_12.pro
42e09977bc840fbf3c4a5dfabb96a748
f8be9051dedc8d2d91e8155230718e8102339064
2017 F20101207_AABRBL aslan_b_Page_13.txt
579944c452e8dacab7775be203580bd6
7fb8e0f8f54debbfa877da18cee56338cc349dfa
56090 F20101207_AABQUO aslan_b_Page_43.jpg
6ccd0a8d2c92bf0804ac1be5b25ae48c
64b9e6aae95bf02347f6bf984c910dd1428f5c2c
6690 F20101207_AABRGI aslan_b_Page_48thm.jpg
f39a17eda73505597502240d5c251ca4
6949272566d5d604b9a84428c6b2d320f93de202
37520 F20101207_AABQZM aslan_b_Page_15.pro
69677923c61b0a728e32c97971cc7c50
58dff65c8f7326504c8d71f00e65e5b34cd72ebb
1921 F20101207_AABRBM aslan_b_Page_15.txt
a2223885b4495f46d823a621f4c0b8702ea46223
63702 F20101207_AABQUP aslan_b_Page_44.jpg
baf16b9ce919282d20ff7ced2d0cfdf9
e6d7baa0ab7590b7dedd0855d88c94b52500742c
25829 F20101207_AABRGJ aslan_b_Page_48.QC.jpg
17661f7c913d741e70bc1f3540af72d3
8649c5b75a37c894392664e0e3f746946d826ab0
29984 F20101207_AABQZN aslan_b_Page_16.pro
7e30382658fedfe5f70e01ce47cdf77b
1378 F20101207_AABRBN aslan_b_Page_16.txt
d390076002d411cbd5bae2542563cae0
0517e170a35f69db134702b7386e1bd9cdb8a48d
63485 F20101207_AABQUQ aslan_b_Page_45.jpg
a3a746b5bf5dcec19d670bd97860ef9bf9b25883
3217 F20101207_AABRGK aslan_b_Page_49thm.jpg
e96fae3b3a306c70666375ae6fb0061a
abbb5284f33cccb1783f50a47a1becbce93b331c
22164 F20101207_AABQZO aslan_b_Page_17.pro
4a732303d3726e9cd6d9df211a8db128
36a975907d9b0d41bd2167230e84de8495e85d2c
1191 F20101207_AABRBO aslan_b_Page_17.txt
6183064e04cb237539704c3dff475243
d713fc9957b6dd0fe5bea5144ac458768f631a95
10721 F20101207_AABRGL aslan_b_Page_49.QC.jpg
9bbe0e3d490d9045c8fbe94c93defea5
42158d5b8b7970401441db5c7f7b7baaf85fa5b8
964 F20101207_AABRBP aslan_b_Page_18.txt
4c7a352ed755e33853ab1bfca418b95e
46693ba4354de9924512c7c7aa2d683fc1a9efc6
65027 F20101207_AABQUR aslan_b_Page_46.jpg
f4b36c77fd393f3a99aeb6edf8cfedfc
f37756cfeb6451113a13aa633392d5b9c9530d24
6152 F20101207_AABRGM aslan_b_Page_50thm.jpg
8cf73f0939d0eae0530c77521253f021
a6fd4ff953359cd162351424c45f8f1a88323b22
19295 F20101207_AABQZP aslan_b_Page_18.pro
bb9c47d1c75bcb09b80f1039e029cc76
e36e7f6ea9d958f884c7247095424840f6004c67
2161 F20101207_AABRBQ aslan_b_Page_19.txt
2f5b2bf23af5d4f3598555fe466c34766861eba1
22936 F20101207_AABQUS aslan_b_Page_47.jpg
c83fbfd39cab609f7a68f381828e95f7
27ed84b1e99d6882fb5eca6427f9e6138e5dfc0d
24781 F20101207_AABRGN aslan_b_Page_50.QC.jpg
5cd0376bf07e2a55d6c35215e293ebb2
c52cd59df0a46679cbf4280d696849ce88d3edce
49402 F20101207_AABQZQ aslan_b_Page_19.pro
cb8fa54d53f5bdeb963679ef702b0dbf
c9f53e444b0534384c1e4344e4bf68d15d727a48
1338 F20101207_AABRBR aslan_b_Page_20.txt
c52814c93caaa6f1f649a3c70b882374
78754 F20101207_AABQUT aslan_b_Page_48.jpg
2057 F20101207_AABRGO aslan_b_Page_51thm.jpg
7567c657e7089389cd377a108c01d4f6a7064d55
32162 F20101207_AABQZR aslan_b_Page_21.pro
dd8f1f725ae5db1af4826216c1268b0134a2bc28
576 F20101207_AABRBS aslan_b_Page_22.txt
965f6441d7582589ffd675b8a71fd039
3f4a4452255862fc72c9d63ed50fa1361c46b85e
31568 F20101207_AABQUU aslan_b_Page_49.jpg
759a8d33dc69eb1acf2cb246181eb2cc
8b2d5cfd18b4720c82b18d8bc63599ac8380ed78
7883 F20101207_AABRGP aslan_b_Page_51.QC.jpg
3fdab435b36d8b4e5dc4681f2c18e8c8
12956 F20101207_AABQZS aslan_b_Page_22.pro
1909dc73f44e5c09c36bb08509f369a1
b5705a8b05525de0f1f08eda63d01a6078a384a1
1913 F20101207_AABRBT aslan_b_Page_23.txt
de33e170b81f89a519d2e5cd890e4fc1
9235a7078308947a6fce24f34a74916554f44759
75835 F20101207_AABQUV aslan_b_Page_50.jpg
c47bc7fc3657666486b2498ab9bf6030557f40d1
7246 F20101207_AABRGQ aslan_b_Page_52thm.jpg
d3a0a94640d56f13e0cc2a96591aa49f
6e731b24f8ee59a34db9e1afca820f42ba53b1d8
41306 F20101207_AABQZT aslan_b_Page_23.pro
2f24909fe01ffa30dfef91a1261a6193
2f71452128e85a61a7d7a6f7202b1694f97f6f95
1861 F20101207_AABRBU aslan_b_Page_24.txt
a7f9db87c4476f3955a4cd117a4ef965
48be53f7b5d33ab687ed81dfb1b7bed919602fa7
22581 F20101207_AABQUW aslan_b_Page_51.jpg
054eb875e6c9924bb08e7479dbc74260
69da3549bb4b099c152a3ec9abde7c1ba4a0a4bd
F20101207_AABRGR aslan_b_Page_53thm.jpg
634914fbbd6010d612d31456679b13ff
a8f9bd6cc1a906cd6513c4ccd74dbfc42291de34
42078 F20101207_AABQZU aslan_b_Page_24.pro
da2e3c08ef05796b881640d5b62aa77e
02fa432ce4ed8c43f6bc0e45f23a37f7ffe06c27
1663 F20101207_AABRBV aslan_b_Page_25.txt
317c599f2122dc677e216e6922ca16cd
739881ff19d970bd4dca708a730c75e766daaab4
93818 F20101207_AABQUX aslan_b_Page_52.jpg
bdfc454714d16d1d293920664681aa3f
3ddca5797fbb4fccbe43ccde7c7d41ac2047a0ba
8096 F20101207_AABRGS aslan_b_Page_54thm.jpg
5ba9074e622ab28bff7555178566c9b1
F20101207_AABQZV aslan_b_Page_25.pro
a1be3b548a525fc9a1b021c658449c26
0a8a70f60a8a46b43dc161e30b239be1e16b3300
F20101207_AABQSA aslan_b_Page_36.tif
bed3871ef1cccbe59e8d622399d9a8fd
ae33d6e81106379ce2892daf2418866f4fff85d7
84517 F20101207_AABQUY aslan_b_Page_53.jpg
2fc843fbf9e0ea382dcc4bb5af6e62cc
e869b5b65b84b496c5e47e388a3d7061241c168a
32332 F20101207_AABRGT aslan_b_Page_54.QC.jpg
3ed5b3c673c52ff1f61b0bf78ee1f073
65c1c0603ed91c96c00a18b7fdb3515a5ec0bd2d
1602 F20101207_AABRBW aslan_b_Page_26.txt
a41a6af14aac63c247e226e15ae39676
f731b0378d6c05a73651beb5d1468a44edc3742b
6178 F20101207_AABQSB aslan_b_Page_41thm.jpg
de0db50bb753e2ba5836188e4f8afd6c
19be268bcd7237a74fb64c78f11a94512468dd00
111732 F20101207_AABQUZ aslan_b_Page_54.jpg
ea3fb237e67fc8b9c7b0dc51374e007e
dc70cea084695ddffc6d74ca7a71a4e625856ee5
34713 F20101207_AABRGU aslan_b_Page_55.QC.jpg
b33e8cf09433852a65f44b9af1132332
bf22b9006ac3cb9e2c13e542c84fea6150c2df49
31145 F20101207_AABQZW aslan_b_Page_26.pro
e1f23796fe2e03e9806b4ee7c12f9fe3
2bf3c02fa2c6c7e83eaf7dce48ac9301775939ca
F20101207_AABRBX aslan_b_Page_27.txt
1ef39e6a8f7511b5b433a79ca9186bef
7b36bac0de2e10637d06af50358bf8dd75a8a8e9