Citation
Fluid dynamics of circumstellar material associated with Be stars

Material Information

Title:
Fluid dynamics of circumstellar material associated with Be stars
Creator:
Morgan, Thomas Harlow, 1945-
Publication Date:
Language:
English
Physical Description:
xvii, 174 leaves. : ; 28 cm.

Subjects

Subjects / Keywords:
Approximation ( jstor )
Be stars ( jstor )
Coordinate systems ( jstor )
Emission spectra ( jstor )
Equations of motion ( jstor )
Mathematical variables ( jstor )
Stellar rotation ( jstor )
Stellar spectra ( jstor )
Time dependence ( jstor )
Velocity ( jstor )
B stars ( lcsh )
Stars -- Atmospheres ( lcsh )
Stars -- Spectra ( lcsh )
Genre:
bibliography ( marcgt )
non-fiction ( marcgt )

Notes

Bibliography:
Bibliography: leaves 171-173.
General Note:
Typescript.
General Note:
Vita.

Record Information

Source Institution:
University of Florida
Holding Location:
University of Florida
Rights Management:
Copyright [name of dissertation author]. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
Resource Identifier:
13989425 ( OCLC )
ocm13989425
00577545 ( ALEPH )

Downloads

This item has the following downloads:


Full Text











FLUID DYNAMICS OF CIRCUMSTELLAR MATERIAL
ASSOCIATED WITH Bb STARS











By

Thomas Harlow Morgan


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

UNIVERSITY OF FLORIDA
1972




FLUID DYNAMICS OF CIRCUMSTELLAR MATERIAL
ASSOCIATED WITH Bfe STARS
By
Thomas Harlow Morgan
A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF
THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
1972


To KYC and HENboth of whom pushed.


ACKNOWLEDGEMENTS
I am debtor both to the Greeks, and to the barbarians;
both to the wise, and the unwise.
Romans, i!4
I would like to thank my parents for their
encouragement throughout my education.
I am indebted to Dr. D. C. Swanson for first interesting
me in physics as well as to Drs. S. S. Ballard, C. F. Hooper,
Jr., and F. E. Dunnam for help and advice in the summer of
1970. Dr. F. B. Wood introduced me to variable stars, and
Dr. K-Y Chen suggested the topic of this dissertation.
Mr. W. W. Richardson drew the figures that appear
here. The members of my committee, particularly Drs. K-Y
Chen and A. G. Smith, read and criticised the rough draft.
Dr. R. S. Leacock also read sections of the rough draft,
and in addition, offered many practical suggestions. Mr.
H. E. Nutter was of great help. I am indebted to them all.
iii


TABLE OF CONTENTS
Page
ACKNOWLEDGEMENTS iii
LIST OF TABLES ix
LIST OF FIGURES X
KEY TO SYMBOLS xii
ABSTRACT xv
CHAPTER
I INTRODUCTION 1
Discovery and Taxonomy of Emission-Line
B Stars. . 1
A Brief History 1
Nomenclature 2
Characteristics of Be Stars 3
Spectral and Luminosity Class.... 3
Galactic Distribution, Membership
in Multiple Star Systems and
Incidence Among Early-Type Stars 3
Rotational Velocity 4
Spectral Variations 5
The Struve Model 6
Origin 6
Line Profiles 10
Hydrodynamical Models for the Flow 10
Spectral Variations and their
Possible Causes 11
Modifications of the Struve Model. 12
Major Topics of this Work 13
II HYDRODYNAMICAL APPROACHES AND STEADY-STATE
SOLUTIONS 15
Background 15
iv


CHAPTER
Page
The Hydrodynamical Equations and Notations. . 15
Static Solutions 18
The More General Solutions 20
Form of the Circular Velocity Law 23
Final Comments 27
III APPLICATION OF HYDRODYNAMICS TO THE ENVELOPES
OF Be STARS 29
Disk Dimensions 29
Theoretical Foundations of Hydrodynamics. ... 31
Validity of the Boltzmann Transport
Equation 31
Relaxation Times 32
The Hydrodynamical Prescription 35
The Hydrodynamical Equations 35
Navier-Stokes Equations and Similarity
Numbers 37
Importance of Viscous Term in the Equation
Of Motion 41
IV MATHEMATICAL ANALYSIS OF THE HYDRODYNAMICAL
EQUATIONS 44
Preliminary Remarks 44
Linearization of the Equations for Temporal
and Angular Dependence 45
Introduction of Time and Angle
Dependent Terms 45
Selection of a Subscript-Zero
Solution 50
Considerations Concerning Approximations
Near the Plane 53
Additional Relations 56
Integration Over z 58
Further Restrictions on the Flow. ... 58
Relations Among the Flow Variables. . 64
v


CHAPTER
Page
A Reprise. 68
V COMPUTATIONS AND RESULTS 70
Introduction to the Computations .... 70
The Equation for k 71
The Consequences of Imaginary Roots. 71
Initial Values for the Parameters. 72
Machine Computation of the Roots . 74
The Case n=0 75
Distinctive Aspects of the n = 0
Case 75
Variation with o)*r and A 75
The Case n = l 78
Preliminary Remarks 78
The Effect of c, 79
The Effect of A 93
Extreme Cases 93
The Case n > 1 107
General Features 107
Large n and the Effects of c^ and r. 107
Sample Flow Variable Solutions 114
Comparison with Observations 115
Actual Stellar Values for Vs . . 115
Qualitative Aspects of the Predicted
Spectral Variations 116
Comparison with Observations Reported
in the Literature 118
VI EQUATION FOR ENERGY TRANSPORT 121
Form of the Equation of Energy Transport 121
The Distinctive Nature of the Energy
Equation 121
An Approximate Form 123
Relative Importance of the Terms of
the Equations 125
vi


CHAPTER
Page
A Linearized Equation 129
Separation of the Time and Angle
Dependence 129
An Equation for the Subscript-One
Terms 131
Relationship to Previous Work 134
Reduction to Comparable Form .... 134
The Question of Consistency 137
A Reconsideration of the Approximate
Energy Equation 138
Non-Static Solutions and the
Adequacy of the Equations 138
Final Comments 140
VII FINAL COMMENTS 142
Stability of the Steady-State Flows. . 142
Conclusions 144
APPENDIX A
Exchange of a Differential and an
Integral Operator
APPENDIX B
Separation of the Equations and
the k Relation
APPENDIX C
Evaluation of the Determinant formed
from the Equations Relating
"nk'
and S
nk
APPENDIX D
The Program CALSOL
APPENDIX E
Pertinent Integrals
148
148
151
151
155
155
157
157
165
165
vii


Page
APPENDIX F ...... 168
An Equation for r ^ 168
BIBLIOGRAPHY 171
BIOGRAPHICAL SKETCH 174


LIST OF TABLES
Figure Page
1 Characteristic Central Star and Disk
Values 30
2 The values of the Coefficients Appearing in
the Energy Equation 128
3 The Program Names for Important Quantities . 159
4 Main Program 160
5 Subroutine ALPl 163
6 Subroutine BET 1 164
ix


LIST OF FIGURES
Figure Page
1 A Be star viewed equator-on and pole-on ... 8
2 The coordinate system 17
3 The radial dependence of |PER| for n = 0
and for large c^ 77
4 The radial dependence of |PER^ | and
|PER2 I for small values of c^ 81
5 The radial dependence of |PER.jJand
|PER2|for larger values of c^ 83
6 The radial dependence of |PER.,|for four
values of 85
7 The radial dependence |PER.| and |PER2|
for small values of T 88
8 The radial dependence of |PER,| and
|PER2|for large values of T 90
9 The radial dependence of |PER.,| for
small values of r 92
10 The radial dependence of |PER-| for three
values of r 95
11 The radial dependence of all three periods
for large c^ and r 97
12 The radial dependence of |PER, | and|PER2|
for small values of c^ and r 99
13 The radial dependence of the three periods
for three different values of n 102
14 The quantity |PER^|j.,^. as a function of n . 104
15 The effect of large parameter values on
the periods when n is large 106
x


Figure
Page
16
The radial dependence of u .(0) and
U'., (0) for various times. .
111
17
The radial dependence of ,, (90) -
u,. (270) for various times'.
113
41
xi


KEY TO SYMBOLS
In order to avoid an excessive number of symbols,
extensive use was made of subscripting. The subscripts fall
mainly into four classes. First, subscripts are used to
denote a component of a vector. For the cylindrical
coordinate system in use here (see Figure 2), these subscripts
are w,, and z. Second, the subscript c indicates a
dimensioned quantity. Third, a capital letter subscripted o
is a characteristic dimensioning quantity. Finally, a subscript
o appearing with a lower case letter indicates a steady-state
cylindrically symmetric term, and the subscript 1 indicates
a term with temporal and angular dependence.
The abbreviations for units follow the usage of
the Manual of Style for the Astrophysical Journal. Symbols
whose usage is confined to a page or a section will not be
listed here.
Important Symbols
p density
v velocity
P, p' Pressure, dimensionless pressure
ux integrated velocity
xii


^1 integrated density
n integer controlling the angular
dependence
k quantity controling the temporal
dependence
A More General Listing
The numbers below are the pages on which the symbols
are introduced.
A, 127
B, 127
Cnk' 64 Cl' 61 C2' 61
DET, 68
FN, 40 f, 56
G, 16
, unit vector i, imaginary number
k, 63
L, 61
M 30 m, proton mass
s
Ne, 32 Nhe, 34
|PER|, 70 |PER,|, 86 |PERj|, 86 |PER,|, 86 PN, 127
p (as a parameter in the cross-sectional flow), 22 p^, 137
Q, 50 qQ, 44
R 29 RN, 40 r, 17
s
Snk' 64 SN/ 40
xiii


Te, .29 Tg, .29 t, time
Unk' 64
V r 29 v (as ve), 58 and 133 ve, 59 and 139
s
X, 67
Y, 67
z, 17 zq/ 59
r, 59 T, 73
n / 37
k, 123
A, 60 A, 73 136 AJ 136
u, 30
Pd' 29
o', 124
17
51
), 17
xiv


Abstract of Dissertation Presented to the Graduate Council of
the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy
FLUID DYNAMICS OF CIRCUMSTELLAR MATERIAL
ASSOCIATED WITH Be STARS
By
Thomas Harlow Morgan
December, 1972
Chairman: Alex G. Smith
Co-Chairman: Kwan-Yu Chen
Major Department: Physics and Astronomy
Temporal and angular variations in the motion and
distribution of circumstellar material about typical Be stars
are studied by means of a simple hydrodynamical approach. The
validity of any hydrodynamical study of the dynamics of the
envelope is first examined, and the problem is found to lie
within the realm of hydrodynamics for typical disk conditions.
The problem is, however, close to the limit of validity of
that discipline. Further, Euler's equation may be used as
the equation of motion throughout most of the gaseous disk
surrounding the central star.
Each flow variable is written as the sum of a known
steady-state axially-symmetric term and an unknown term
containing the temporal and angular dependence. The squares
of these latter quantities are assumed small. Using this,
the equations of motion and continuity are reduced to a set
xv v


of linear equations for the terms with temporal and angular
dependence. The known steady-state axially-symmetric terms
are taken from the literature (Limber, D. N. 1964, Ap. J.,
140, 1391). This solution is a static isothermal one. The
geometry of the problem suggests several approximations which
may be used in the linearized equations? these approximations
introduce two parameters. Solutions can be found to this
somewhat approximated version of the linearized equations.
Since the boundary conditions have not been treated, a fully
determined set of solutions is not possible. The dependence
on both time and angle is, however, explicit.
The angular dependence enters the solutions in an
extremely simple form involving an integer n, but understanding
the temporal dependence requires computation of the periods
characteristic of the temporal variation at a given location
in the gaseous envelope and for a given choice of the two
parameters discussed above. The IBM 360/65 at the University
of Florida was used to perform these computations. Under
most conditions the characteristic periods are found to be
3/2
real. Further, their dependence on n and w is w /n where
(i) is the distance from the center of the star in the equatorial
plane. These calculations predict temporal variation down to
fractions of an hour, and in qualitative agreement with
relevant observations.
The third equation of hydrodynamics, the energy
equation,is examined insofar as it can be in the absence of
an effective treatment of the radiation-fluid interaction.
xvi


This approach leads to inconsistencies, with the results
of the study of the equations of motion and continuity.
It is concluded that in the absence of some accommodation
of radiation effects, no equation for energy transport can
improve the existing solutions.
Finally, the stability of steady-state flows is
examined. The steady-state flows are not absolutely
unstable, but the most general solutions appear to be
periodic in character.
xvii


CHAPTER I
INTRODUCTION
Discovery and Taxonomy of Emission-Line B Stars
A Brief History
In August 1866 Father Secchi observed H0.in emission
in y Cas. Over the next half-century Campbell, Frost, Miss
Cannon, and others observed hydrogen in emission in other
stars of what would now be called spectral type B as well
as in members of the remaining early spectral types. Miss
Cannon appears to have been the first to observe variations
in the emission features in one of these stars (Cannon, 1898) .
Pickering (1912) published the first catalogue of early-type
emission-line stars; at that time there were 94.
Further work was in two directions; more emission
line objects were discovered and individual stars were
studied in detail. The catalogue of Merrill and Burwell
(1933, 1943, 1949) contained over 200 B-type emission-line
stars. Detailed spectral studies led Struve (1931, 1942)
to interpret the spectral features characteristic of Be
stars in terms of emission and absorption in a gaseous
disk-shaped envelope surrounding a rapidly rotating central
star; this interpretation has not changed substantially
since.
In recent years both long term variations (Limber, 1969)
1


2
and shorter term variations (Lacoarret, 1965; Slettebak,
1969; Hutchings, 1969 and 1970; Hutchings et al, 1971) in
the characteristic emission-absorption features have received
attention. The most recent listing of Be stars (Jaschek
et al., 1971) contains 1900 objects.
Nomenclature
The basis of modern systems of spectral classification
is the system adopted for the Henry Draper catalogue (Cannon
and Pickering, 1918-24). The original Harvard arrangement
of the spectra was in order of decreasing ratios of intensities
of the hydrogen Balmer lines to the intensities of a number
of other absorption lines. The Harvard observers soon
realized that a rearrangement of the original alphabetical
order into the present spectral" sequence0, B, A, F, G,
K, Mrepresented an ordering in temperature with decreasing
temperatures from O to M.
Among the HD objects typed B, those possessing
hydrogen-emission lines were marked with an additional
lower case p for peculiarfollowing the B. Modern
classification uses the notation Be for such stars, that
is, stars with B-type spectral features plus emission in
one or more of the Balmer lines. The suffix p is now
used only to indicate the presence of annomalously strong
or weak lines. If a Be star shows sharp metallic absorption
lines superposed on the Be spectrum, the star is called a
shell star.


3
Characteristics of Be Stars
Spectral and Luminosity Class
In this work the term "Be star" applied to hydrogen
emission-line stars whose underlying spectral types are
early of mid B (B0-B7) and whose luminosity classifications
are either II, IV, or V. This is somewhat at variance with
much of the literature, which considers Be stars to be
basically main-sequence (luminosity class V) objects. But
Mendoza's (1958) comprehensive luminosity class survey of
Be stars in the Merrill and Burwell catalogue simply placed
the stars in two groupsvery luminous objects (I, II) and
"less" luminous objects (III, IV, V)on the basis of a
pronounced dichotomy in his observational material. In
fact, strong doubts (Underhill, 1966) have been expressed
as to the existence of a systematic distinction between MK
luminosity classes IV and V in this early spectral region.
Absolute magnitudes of Be stars determined on the basis of
cluster or association membership appear to fall about one
magnitude above the zero age main sequence. Finally the
appellation shell will be similarly restricted although
objects with A and even early F spectral classes are some
times called shell objects.
Galactic Distribution, Membership in Multiple Star Systems,
and Incidence Among Early-Type Stars
Mendoza (1958) took for his sample all the stars in
the Merrill and Burwell catalogue whose spectral types fell


4
in the range from BOe to B2..5e and whose luminosity classes
were III, IV, or V. He found that the Be stars populate in
a roughly uniform manner the spiral arms as delineated by
the 0-B associations. Be stars are found in 0-B associations
but are not strongly concentrated in them. They are to be
found both in galactic clusters (x Persei, Coma) and also
in binary systems (a Tau,

Rotational Velocity
Be stars are the most rapidly rotating class of
stars. The most recent catalogue of observed rotational
velocities (Bernacca and Perimotto, 1971) lists 189 objects
in this spectral class whose rotational velocities have
been measured. The mean rotational velocity of the group
is 262 km/s, where the observed rotational velocity is
the true rotational velocity times sin i, i being the angle
between the axis of rotation and the observer's line of
sight. On the assumption that the axes of stellar rotation
are directed randomly in space one finds the true mean
rotational velocity (Huang and Struve, 1960) to be 344 km/s.
Traditional methods of line profile analysis used to
determine the observed rotational velocities are now believed
(Collins, 1970; Hardrop and Strittmatter, 1968) to underestimate
the actual values by as much as 40 percent. If this belief
is true the Be stars are rotating at large fractions of
their equatorial break-up velocities. Balmer emissions


5
occur in approximately 15 percent of all B-.type spectra
(Underhill, 1966).
Spectral Variations
Spectral variations occur in most Be and shell stars.
Following the terminology of McLaughlin (1961) the spectral
variations fall into three major classes:
(1) appearance or disappearance of a shell
absorption spectrum,
(2) changes in the ratio of the intensity of an
emission line to the neighboring continuum (E/C),
(3) changes in the relative intensities of the red
and violet components of the Balmer emission
lines (V/R) .
The classic example of the first variation is
Pleione (28 Tau), which has been observed spectrographically
since 1888. Pleione was first classified as an emission
line star, but by 1905 it showed no emission features and
possessed a normal B spectrum (with large rotational velocity).
In 1938, emission lines were present; a short time later
the star displayed a shell spectrum. By 1954 the emission
features were lost. Recent observations (Sharov and Lyutiv,
1972) indicate that Pleione is entering a new cycle of
activity.
E/C and V/R variations, though strictly periodic
only when they reflect orbital motions of stars in binary


6
systems, show much quasi-periodic behavior. The time scales
for these variations may be on the order of a few years or
as short as days or fractions of a day (Slettebak, 1969;
Hutchings, 1969 and 1970; Peters, 1972).
The Struve Model
Origin
In 1931 Struve suggested that the emission-absorption
features that are the definitive characteristic of Be stars
are formed in gaseous circum-equatorial rings surrounding
(and ejected from) B-type central stars that are rotating
at their break-up velocites. Later, Struve (1942) was able
to present the model on the basis of more detailed spectral
considerations. He also proposed a circular velocity law
for the equatorial disk,
V. (w
4>c oc
(w /)
c oc
)
)
where w /tu is the radial distance from the central axis
c oc
to any point in the disk divided by the stellar radius and
Vc^uoc^ the rotat;*-onal velocity at the stellar equator.
Figure 1, which draws its inspiration from Hack (1970),
illustrates the model's ability to explain the major features
of Be spectra. The first case is a Be star whose axis of
rotation is perpendicular to the line of sight; the second,
one whose axis of rotation is parallel to the observer's line
O


Figure 1
Rotating Shell. Shows a Be star viewed equator-on.
The envelope is assumed for simplicity to have on
rotational motion. Note the central absorption
from region 1 and the emission lobes from the
areas labeled 2 and 3
Shell Star Pole-on. Shows a Be star viewed
pole-on.


Rotating Shell


9
of sight.. A schematic profile of the expected emission at
one of the Balmer lines is also shown for each case. The
imposition of a small expansion velocity on the large
circular velocity gives a V/R asymmetry. That the disk is
indeed cixcum -.equatorial was given credence by the work of
the Burbidges. They analyzed (Burbidge and Burbidge, 1953)
high dispersion spectra of six stars whose emission features
were like the second case above, often called "pole-on."
They found that, although a very thin layer of gas did
appear to exist above the polar regions of the photosphere,
the bulk of the gas was in or near the equatorial plane.
Another aspect of the Struve model is that it explains the
differences in Be and shell spectra as due to differences
in the density and extent of the envelope.
Although the Struve model provides a qualitative
explanation for the spectral features found in Be stars
and shell stars, detailed calculations of line profiles
require
(1) a model for the radiation field of the central
star,
(2) a model which gives the physical state,
density, and velocity of the gas at all
points in the disk,
(3) the changes in the radiation field resulting
from its passage through the gaseous disk.
The work of Mihalas (1965) and of others now
provides the first requirement. The second and third


10
requirements, have not been so satisfactorily achieved and
are not even truly separable.
The Struve hypothesis says nothing about the
mechanism(s) that cause the formation and destruction of
the diskthat is, changes from B to Be to shell star and
vice versanor does it say anything about the mechanisms
that produce other spectral variations.
Line Profiles
The hydrogen-emission spectrum is a recombination
spectrum formed in a disk the various portions of which
Can move relative to one another. A great deal of work
on hydrogen emission in a stationary atmosphere has been
done (Kogure, 1959a, 1959b, 1961). Sobolev (1960) and
Rothenberg (1952) have studied the formation of the hydrogen
lines in spherically symmetric envelopes expanding with
constant velocity. Marlborough (Marlborough, 1969 and 1970;
Marlborough and Roy, 1971) has calculated Ha profiles using
a 6-level hydrogen atom. He used a model for the distribu
tion and motion of the material due to Limber (1964) which
will be discussed below.
Hydrodynamical Models for the Flow
Determination of the density and velocity field
throughout the disk is a hydrodynamical problem. In the
first such study (Limber, 1964),a steady-state axially-
symmetric disk was examined for a parameterized form of


11
circular velocity, law and in the absence of any radial
velocity. Limber treated isothermal envelopes, polytropes,
and envelopes which have a specified temperature law. For
each case he was able to calculate the density at any
point in the envelope. Later, Limber (1967) studied the
more general case in which a radial velocity component was
present.
Spectral Variations and Their Possible Causes
A rough division of spectral changes in Be stars
will be made here to facilitate the discussion; the divisions
are:
(1) long term changes (time scale longer than
10 years) which characterize the appearance
and disappearance of shell or emission spectra
as in Pleione,
(2) medium term changes (1 to 10 years) particularly
V/R changes with time scales in this time
range,
(3) short term changes ( a year or less).
There is a certain inevitability to the long term
changes; Crampin and Hoyle (1960) argued that any initial
magnetic fields present at the formation of the disk would
render it unstable in 100 years or less. Limber has
suggested that there may be a non-uniform transfer of
momentum from rapidly rotating inner regions of the central


12
star out to the photospheric regions resulting in turn in
non-uniform mass loss. Limber (1969) analyzed the shell
phase of Pleione in terms of a time-dependent mass flux
from the central star's equatorial regions to the disk. He
found good agreement with the observations for a flux whose
time dependence showed a slow increase to maximum followed
by a steep drop to zero.
McLaughlin (1961) reviewed what are called here
medium term variations. He observed that the most difficult
problem concerned V/R changes. McLaughlin (1962) discussed
the major attempts to explain the quasi-periodic V/R changes
of the sort seen in ir Aqr, and he concluded that only a
suggestion credited to Otto Struve was consistent with the
observations. Struve's suggestion had been that the ring
or disk was elliptical in shape and that the V/R variations
result from a line of apsides rotation of the disk. There
has been no recent examination of this idea.
The most comprehensive work on short term variations
is Huang's (1972a). Huang showed that an asymmetry in the
disk will produce spectral variations observable after time
spans as short as a fraction of a day; these variations can
persist for as much as a year before the disk's differential
rotation destroys the asymmetry. Hutchings (1969 1970.)
suggested that photospheric fluctuations or "dumpiness"
in the disk may be responsible for short time changes.
Modifications of the Struve Model
It now appears that several aspects of Struve's


13
original model must be modified. First, Struve's model
requires that the equatorial regions of the photosphere
be rotating at the break-up velocity. Observations suggest
that Be stars' rotational velocities lie below the break-up
limit. Huang (1972b) recently argued that the ejection of
matter is due to both rapid rotation and a "temperature-
dependent instability." Huang feels that the "temperature-
dependent instability" may be the mechanism studied by Lucy
and Solomon (1970) in their work on stellar winds in early-
type giant atmospheres; this work argued for a mass outflow
due to radiation pressure from mid-UV resonances in silicon,
carbon, nitrogen, and sulfur, these being in high ionization
states.
Limber and Marlborough (1968) as part of a general
discussion of the physical processes at work in Be stellar
envelopes, reanalyzed the work of Struve and Wurm (1938) in
order to show that the velocity law,
Vi|ic^oc^
<1/2
was in better accord with the data than the circular
velocity law originally proposed by Struve.
Major Topics of this Work
A principal contention of this dissertation is that
the existing steady-state, axially-symmetric hydrodynamical


14
solutions explain the key elements of the Be star phenomenon.
More exact and comprehensive explanation of the body of
observational material will require solutions to the hydro-
dynamical equations which possess time and angle dependence.
The approach here will be to view the expression for each
flow variable as the superposition of a detailed but in some
sense second-order term onto the time-independent axially-
symmetric solution.
Chapter II, a literature review, discusses the
previous hydrodynamical work on Be stars. Chapter III
studies the extent to which the motion and distribution of
circumstellar matter can be treated as a hydrodynamical
problem. Chapter IV introduces the dichotomous view of the
flow discussed above into the hydrodynamical equations;
this chapter then presents the mathematical reduction of
the original set of equations for the time and angle
dependence to a set of solvable equations. The next chapter
(V) discusses calculations based on the solution to these
equations. In Chapter VI the energy equation is considered,
and Chapter VII, after a discussion of the relation of the
results of this work to turbulence and stability arguments,
summarizes the work.


CHAPTER II
HYDRODYNAMICAL APPROACHES AND STEADY-STATE SOLUTIONS
Background
The steady-state axially-symmetric solutions of the
hydrodynamical equations offer an acceptable description for
the overall phenomena presented by Be stars and shell stars.
These published solutions are the starting point for the work
which is the principal subject of this dissertation. For
this reason, and because the discussion will outline the
hydrodynamical approach to the motion of circumstellar material
in Be stars, the literature on steady-state axially-symmetric
solutions will be reviewed in some detail. After a brief
discussion of notation, the derivation and properties of
static envelopes will be discussed; then the permissible
solutions with non-zero radial velocity will be considered;
finally the evidence available for choosing among the
various possible circular velocity laws will reviewed. There
will be a short summary at the last. The order of presenta
tion of the three main sections represents the order in which
these ideas were developed by Limber and Marlborough (Limber,
1964, 1967; Limber and Marlborough, 1968).
The Hydrodynamical Equations and Notation
The hydrodynamical equations used here are the simplest
15


16
ones that could be chosen; they are Euler's equation and the
continuity equation. In dimensioned coordinates the former
is
3v
pc (wr
+ v
and the latter takes the form
3t
+ V
(p v ) =
Kc c
The justification of the use of Euler's equation will
be a principal concern of the next chapter.
The coordinate system used here is shown in Figure
2. The three coordinates (w, , z) may appear with or
without the subscript c. The presence of the subscript
signifies that the coordinate in question is dimensioned in
cgs units; without the subscript, it is a dimensionless
variable in the system of dimensionless coordinates which is
developed in the next chapter. A similar procedure is
-y
followed for the flow variables and time: velocity v,
density p, pressure P, and time t. The equations already
presented in the first chapter are consistent with this
practice. In cases where non-cgs units are used, these units
will be explicitly stated.
In the body of this work symbols will on occasion
undergo some change in definition; for instance, a variable
symbol may be required to absorb an integrating factor. This


17
Figure 2
2
The cylindrical coordinate system in
use here.


18
is done only when necessary to prevent an undue proliferation
of closely related variable names. When questions arise
about the meaning of a symbol, the best recourse is a check
of £he Key to Symbols. Each change in usage is noted there
as is the first page on which the new usage occurs; .
Static Solutions
Limber (1964) attempted the first hydrodynamical
study of the material outside rapidly rotating stars. He
took the central star to be rotating at the break-up
velocity and made the following assumptions:
1) viscous and magnetic terms in the equation of
motion can be ignored;
2) radiation forces are either negligible or
includable through the use of an appropriately
reduced stellar mass in the gravitational
potential;
3) disk self-gravitational effects are negligible;
4) steady-state conditions prevail;
5) only axially-symmetric flows are included;
6) the Z component of the fluid velocity (perpendicular
to the equatorial plane) is zero;
7) the radial component of the fluid velocity is
zero;
the circular velocity has the parameterized form
V %c(Mo>
to
8)


19
where the new symbol a is a parameter whose
value lies between 1/2 and 1;
9) the density is an explicit function of the
pressure alone.
The first three conditions determine the nature
of the equation of motion which can be usedEuler's equation.
The third approximation, neglect of self-gravitational terms,
is quite reasonable. The validity of the first and second
conditions will be discussed in later sections. The use of
Euler's equation as the equation of motion is common to all
hydrodynamical studies of Be star envelopes. The justification
for the use of Euler's equation or, indeed, any hydrodynamical
equation under disk physical conditions will be the subject
of the next chapter.
The next group of approximations, (4) through (7),
reduces the equation of motion to the form
where v^ is the circular velocity (4> component of the veloc
ity) and iu is the unit vector for the radial direction in
cylindrical coordinates. Conditions (4) through (7) also
result in solutions which will identically satisfy the
continuity equation, the second of the three equations
needed to specify completely the problem. Since violation
of (5) will show up observationally as variations with time,
both (4) and (5) are good approximations to the extent that


20
they describe phenomena with either no time dependence or
long term time dependence. Near the equatorial plane
condition (6) is a good approximation, while condition (7)
is clearly an idealization which will be partially removed
in subsequent work.
The radial dependence of the circular velocity
law can be at least roughly determined from dilution factors
for shell absorption lines, from the widths of absorption
lines, and from the widths of emission features. It is,
then, possible to choose a parameterized form on the basis
of the observations. Calculations were done for three values
of a : 1/2, 3/4, and 1. In all cases the circular velocitis
were required to match the stellar rotational velocity at the
stellar surface.
The last condition, that the density is a function
of the pressure alone, circumvents the failure to treat
the last of the three hydrodynamical equations, the equation
for energy transport, in the disk. The condition imposed
may be alternately expressed as requiring that surfaces of
constant pressure and of constant density be identical.
Limber is then able to calculate the density
throughout the disk for isothermal envelopes, for polytropes,
and for envelopes in which the temperature though constant
along any one surface of constant pressure may vary from
surface to surface.
The More General Solutions
The static solutionssolutions in which the radial


21
velocity is zeroare, strictly speaking, nonphysical for the
same reason that static solutions are nonphysical for the
solar wind (Parker, 1958). Limber extended his analysis to
non-static solutions (Limber, 1967). Three assumptions of
the approach presented in the preceding section are modified;
they are (6), (7), and (9). These three are replaced by the
following less restrictive conditions:
1) only flow near the plane is considered;
2) the radial cross section of flow must be specified
(in parameterized form);
3) the temperature distribution throughout the disk
is known.
As in the static case the ideal gas law is used. Euler's
equation takes the form
where the new symbol v^ is the radial component of the
velocity. The equation of continuity appears in the inte
grated form
vw(u)p(aj) A(w) = %(0)p(0) A(w0)
where A(to) is the radial component of the cross-sectional
flow. This form of the continuity equation is similar to
that seen in solar wind theory. The quantity A()/A(toQ) is
analogous to the term
F
s


22
in the elementary solar wind theory (Parker, 1963) where b is
a parameter (usually given a value near 2) Limber chose,
instead, a one-parameter family of cross-sectional flow
terms
)
U>
A = S
( (¡J p )
oc
oc
where p is a parameter. The meridional projections of the
stream lines are straight lines running from a point
c = puoc
in the equatorial plane and sloping slowly away from the
equatorial plane.
Limber went on to establish the types of solutions
that are now permissible. Valid solutions must be subsonic
below the star's surface and approach interstellar values
at great distances from the star. The only solutions which
satisfy these two boundary conditions are those whose radial
velocity
1) first decreases with increasing w,
2) reaches a minimum as w continues to grow,
3) then increases as w continues its increase,
4) becomes supersonic at sufficient distance from
the central star.
Limber examined the range of validity of his earlier
work, the calculations for static envelopes, in the context


23
of his new results. He found that the density calculations
from the earlier approach agreed quite well with the new
results over most of the near-equatorial regions of the disk.
At large distances from the star the old solutions no longer
agreed with the new calculations; the distance at which the
breakdown of the old approach occurred depended on the exact
physical conditions, and geometric dimensions chosen for
the disk.
Form of the Circular Velocity Law
The solutions developed by Limber in his 1967 study
contain two parameterized functions, the cross-sectional flow
expression and the circular velocity. Of these two the circular
velocity is the more crucial; the solutions are fairly insen
sitive to changes in p, the parameter which appears in the
cross-sectional term. Limber and Marlborough (1968) examined
the physical conditions and observational evidence for the
circular velocity's dependence on co throughout the disk.
The equatorial break-up velocity is only 1/^2 times
the escape velocity. This means that if a small element of
matter were perturbed with a small radial velocity so that
it began to move outward, and if the element were not further
disturbed, then the element would eventually fall back onto
the star. The available evidence indicates that shell material,
rather than falling back onto the star, is continually escaping
the system; there is mass loss. Limber and Marlborough call
this the problem of "support." The physical processes which


24
act to provide the "support" profoundly influence the radial
dependence of the circular velocity law. Limber and Marlborough
divide the mechanisms for support into two classes: "direct"
and "centrifugal." The first type, "direct" mechanisms, are
radially directed forces which act on any fluid element to
overcome the difference between the gravitational force on that
element and the centrifugal force. These direct forces would
appear as additional terms in the equation of motion. The
"centrifugal" classification refers to mechanisms which act
to transfer angular momentum out into the disk so that the
centrifugal force always balances the gravitational force due
to the central star.
Direct mechanisms do not act to affect the angular
momentum of a small fluid element which is moving from the
stellar equator out into the disk; the angular momentum of
the fluid element is conserved. In this case
v Quasi-static motion of a small fluid element away from the
star with centrifugal forces present requires that
GM
s
at any point in the disk; hence
V ^72


25
There are four direct-support mechanisms available;
they are:
1) thermal support,
2) turbulent support,
3) radiative support,
4) magnetic support.
Limber was able to show that the first possibility, thermal
support, requires temperatures and densities throughout the
disk that are totally at variance with present knowledge
of the values of these quantities. Turbulent support requires
a highly turbulent flow whose effects on spectral features
should be observable but have not been reported.
Limber and Marlborough considered two possible
radiative support mechanisms, electron scattering and photo
ionization of neutral hydrogen. The net outward-directed
radiation force on a small fluid element due to Thomson
scattering was calculated for typical disk electron densities
and was found to be negligible in comparison to the gravitational
force acting on that element. They noted that for a given
neutral hydrogen density there was an upper limit to the radial
force that could be generated by photo-ionization of neutral
hydrogen. Maximum momentum transfer in the radial direction
from photo-ionization occurs if the recombination following
the photo-ionization emits a photon backwards toward the
source of the original photon. To establish an upper limit,
Limber and Marlborough assumed that all the flux from the
central star shortward of Lyman a and incident on the disk


26
was absorbed and back-emitted. On the basis of both this
assumption and other considerations, these authors calculated
an upper limit for support due to photo-ionization. This value
was far less than the value needed to provide radiative
support. While both these calculations are dependent on
estimates of typical disk values for the physical parameters,
they are so much smaller than the value needed for direct
support that they may be omitted from further discussion
until such time as the best estimates of these disk quantities
change dramatically.
Finally radial support from magnetic forces was
considered. The same investigators used elementary considera
tions to show that the field strength, H, required was
H > 75 gauss.
They then showed that fields of this sort in the disk
would disrupt it on a time scale of days. The observational
results of the disruption of the disk, loss of emission lines
or shell absorption lines, show much longer time scales.
Thus they concluded that magnetic fields of the required order
are not present in the disk.
Analogously there are four mechanisms which can
act to transfer angular momentum:
1) thermal viscosity,
2) turbulent viscosity,
3) radiative viscosity,
4) magnetic viscosity.


27
Limber and Marlborough used the term "viscous" to denote
any phenomenon which acts to transfer angular momentum. They
found that two of these possibilitis, the first and third,
were insignificant under disk conditions. However, either
small scale turbulence or small (< 5 gauss) magnetic fields
could provide the needed angular momentum transfer. They
concluded that the observational evidence was consistent
with either interpretation. It is important here to note
that the amount of turbulence or the magnetic field strength
required for angular momentum transfer is far too small to
provide any direct support.
Limber and Marlborough noted that the work of Hynek
and Struve (1938) represented the only attempt to draw
quantitative inferences from observations about the form of
the circular velocity law and that this attempt had considered
only the 1/to behavior. The data were reanalyzed to see if
the l//w form gave a more consistent interpretation of the
data. The conclusion of this reanalysis of the data was
that the form of the circular velocity law associated with
centrifugal support was in better accord with the data than
the old Struve form.
Final Comments
Steady-stte solutions have been applied to the
shell phase of Pleione (Limber, 1969) by allowing the
envelope to be at any moment very near steady-state. This
will, of course, work only for long term phenomena. Limber


28
suggested that the end of the shell phase represented the
result of increased matter outflow not matched by sufficient
energy flow into the disk; hence,a collapse.
Two results of Limber's studies are of sufficient
importance to the remainder of this work to bear restating.
First, over most of the disk, static isothermal envelope
calculations agree well with the more exact theory including
a radial velocity. Second, the circular velocity law is that
for centrifugal effects. The static isothermal envelopes
with a circular velocity law can be presented in closed form
and do not require the specification of a cross-sectional
flow parameter.


CHAPTER III
APPLICATION OF HYDRODYNAMICS TO THE
ENVELOPES OF Be STARS
Disk Dimensions
Table 1 gives the values used here for the physical
and geometrical quantities which characterize the star and
its envelope. The mass (M ) and surface temperature (T )
S b
of the star are those of BO dwarf. The radius is larger
than one might expect for such a star and represents acknowl
edging the evidence that the star is somewhat evolved and
rotationally distorted. These quantities are those
commonly found in the literature. The equatorial rotational
velocity (V ) represents a slight departure in that the
rotational velocity was set at just slightly more than half
the equatorial break-up velocity. The disk temperature,
which is taken to be the electron temperature (Te>, the
mean molecular weight (y) of the disk gas, and the average
disk density (p^) are typical of values found in the
journals.
From the emission and absorption features at
hydrogen and helium resonance lines one can conclude that
in a typical Be star envelope a large percentage of the
hydrogen is ionized while most of the helium is not.
29


30
TABLE I
Characteristic Central Star and Disk Values
Quantity
Symbol
Value
Stellar Mass
Ms
10 ^0
Stellar Equatorial Radius
R
s
10R G
Surface Temperature
T
s
25,000K
Equatorial Rotational
Velocity
V
s
250 km/s
Disk Temperature
T
e
10,000K
Molecular Weight
y
0.68
Average Disk Density
pd
IQ"12


31
Theoretical Foundations of Hydrodynamics
Validity of the Boltzmann Transport Equation
The most satisfying foundation for the hydrodynamical
equations is the Boltmann transport equation. The hydro-
dynamical equations are the result of integrals of the
form
Y D f dv
c op C
Â¥c is a conserved quantity, D0p* the Boltzmann operator,
and f, the distribution function which is a solution of the
integro-differential equation
f
0 .
Two assumptions are essential to the derivation of
the Boltzmann transport equation. The first, the assumption
of molecular chaos, is mentioned only for completeness;
the second, the restriction to binary forces, is open to
question in the presence of Coulomb forces. The so-called
collision integral of the Boltzmann equation includes only
the effects of binary collisions. More fundamentally, the
Bolbanann. Weltansicht is that the individual particles
comprising the gas spend most of their time in free flight,
this condition being interrupted only occasionally and
briefly by collisions.


32
The long-range Coulomb forces present, when, charged
particles are constituents of the gas, generally result in
an infinite contribution from the collision term. The
underlying view of the gas motion is suspect as several
distant collisions may simultaneously interact with one
another. There is, however, one set of circumstances under
which the Boltzmann approach is still valid (Zel'dovich
and Raizer, 1966). The conditions are:
(1) the average Coulomb potential energy at the
mean separation distance is much less than
the average thermal energy;
(2) the Debye length is much greater than the
mean separation distance.
Assume that the disk gas is made of equal numbers of
electrons and protons. The ratio of the Coulomb potential
energy at mean separation to average thermal energy for
such a gas under disk physical conditions gives
(Ze) 2
1/3
1.2 x 10
where Ne, e, k, and Z are the electron number density,
the electron charge, the Boltzmann constant, and the average
ionic charge. The ratio of the Debye length to average
separation is
6.90(T /N )1/2

^¡73
e
8.0


33
The first condition is nicely satisfied; the second, less
well so but within the limit of toleration. The conclusion
to be- made is that the Boltzmann approach and, hence, the
hydrodynamical description are justified, but are near the
limit of validity. The additional species present in a
more realistic representative disk gas will not alter this
result.
Relaxation Times
A fundamental consideration for a gaseous mixture
is the extent to which that mixture can be treated as a
single fluid, particularly to what extent a single temperature
can be used to describe the flow. The relaxation time for
collisions between two species is a measure of the time
required after an initial disturbance in one of the species
to establish equilibrium among the translational degrees
of freedom in both. For a system of the type under study
here, there is some time which is characteristic of the
time scale on which hydrodynamic variations occur. If
the ratio of the relaxation time between any two major
species of the gas and the characteristic time of the
system is equal to or greater than one, the single-fluid
hydrodynamical description is invalid.
For a gas under disk physical conditions there
are two relevant relaxation timethat between ions and
electrons and that between electrons and neutral atoms.


34
For purposes of studying the nature of the disk gas, that
gas will be approximatedby a three-species gas containing
electron, ionized hydrogen, and neutral helium; further,
the characteristic time (x ) for the system will be taken
V/
to be 2 ttR /V .
s s
The ratio of the electron-proton relaxation time
(x ) to the characteristic time (x ) is
6p c
x 3.5 x 10+8V T3/2
e-p s e
Tc 2ttR N AnA,
s e d
where is the reduced Debye length (Spitzer, 1962). The
ratio of the electron-helium relaxation time (x ) to
e-ne
the characteristic time is approximately
Te-he
x
c
V
s
2RsNhe Te/ he
(Zel'dovich and Raizer, 1966) where Q^e is the helium-
electron collision cross-section for typical disk conditions
+4
and N^e is the helium number density. If Tfi is 10 K
16 2
Q^e is approximately 5.7 x 10 cm .
For the typical physical state of the disk, the
ratio of the electron-proton relaxation time (Te_p) to the


35
-4
characteristic time is 4.6 x 10 The ratio which compares
electron-helium relaxation time to the characteristic time
_5
is approximately 1.79 x 10 Finally one can compare
e--he- a 0.4 x 10-1
Te-P
The first two of these three numbers indicate that the
temperature should be the same for all three components;
electrons, protons, and neutral helium.
These two conclusions should hold in the more complex
disk gas. The principal disparity between the three-element
picture and the actual fluid is the number of neutral
species. The quantity Te_jie is being used as a measure
of the relaxation time for electron neutral collisions. The
relaxation time for collisions of electrons with all neutral
atoms, in the presence of additional neutral species besides
helium, can only be smaller than that for helium alone.
Therefore, the quantity Te_jie overestimates the actual
electron-neutral atom relaxation time. Substitution of
a smaller number for x in this section will only
e-he
strengthen the conclusions.
The Hydrodynamical Prescription
The Hydrodynamical Equations
The three conserved quantities used to generate the
hydrodynamical equations are mass, momentum, and energy; these


36
give an equation of continuity, an equation of motion,
and an energy equation, in that order. However, quantities
such as the heat flux and the deformation tensor which
appear in the equations are defined by integrals that
contain the distribution function. Indeed, the quantities
that are identified as flow variables (the pressure, density
and fluid velocity) are also so defined.
If the transport equation is solved by a successive
approximation technique in which the distribution function
is written as a series of terms generated by increasingly
higher degrees of approximation, a set of hydrodynamical
equations may be formed at each stage of the approximation.
The "zeroth" approximation, in which only the initial term
for the distribution function appears,is a locally Maxwell-
Boltmann distribution; the resultant hydrodynamical
equations are those for an ideal inviscid fluid. The
equation of motion in this case, Euler's equation, has the
form
c
Vv
c
1
P
c
VP
c
c
where t is the force per unit mass. The continuity
equation is
0


37
The next approximation leads to the hydrodynamical
equations for a viscous gas. In this case the equation of
continuity is unchanged but a new equation of motion, called
the Navier-Stokes equation, is formed. The Navier-Stokes
equation will be examined below to ascertain the extent to
which Euler's equation is a good approximation for a fluid
under disk physical conditions.
Navier-Stokes Equations and Similarity Numbers
The Navier-Stokes equation in the presence of a
gravitational potential is
+ v
U + n/3)
V(V v ) GM V (
-1
The quantities n and £ are the first and second vicosity
coefficients, respectively. The quantities n and B, are both
always positive. The second viscosity coefficient
represents effects that occur at high density or when
species with slowly excited degrees of freedom are present.
It is included only for completeness. Even if these effects
were present, as long as n > £, all the arguments presented
below are valid.


38
The second term on the left-hand side in the Navier-
Stokes equation is called the inertia term; the second
and third terms on the right are the viscous terms. The
last term contains gravitational effects.
Let R be a dimension characteristic of the
o
boundary surface, Vq a typical value of the fluid velocity,
no a representative density. These characteristic values
are chosen such that the associated physical quantities
(r v p ) vary from a large fraction to several times
the characteristic values, that is, each of the values
represents the order of magnitude of the quantity of which
it is a characteristic. Characteristic values for the
pressure and time which result from these choices are
and
respectively. Note that the characteristic numbers are
dimensional.
The characteristic numbers can be used to set up
a system of dimensionless variables defined by the following
relations:
(1) r
c
f
(2) v
c
-+
V V
o
f


39
(3) Pc = V '
(4) P^ = P P
c o
(5) t = T t
c o
Note that the subscript zero quantities are dimensioned
numbers. These relations can be used to rewrite the
Navier-Stokes equations in dimensionless coordinates. The
result is shown below:
o 3v
R0 9t
V2
o
R
v Vv =
- v2
o p
GM
s
? (- b )
nVo 72v + (5 -f n/3) v 7(7 V)
P n td2 O p
n r
o o
o o
Division by VZ/R^
* O o
yields
9v
3t
+
v
GM
s
R V2
o o
+
R v n
o o o
a + n/3)
Rvn
. o o o
V(V v)
P
Since the characteristic parameters represent
the> order of magnitude of their respective variables, the
order of magnitude of the ratio of any two terms is the
ratio of their coefficients. This statement is, however,


40
open to question near the flow boundary. The flow variables
may change rapidly over short distances near the boundary
surface thereby causing terms involving their gradients
to be quite large. These coefficient: ratios taken together
qualitatively describe the flow. Historically they have
been called similarity numbers and given the following
names:
1.Reynolds Number. This dimensionless number,
labeled RN here, is the inverse of the ratio
of the viscous terms to the inertial terms,
1_ = 0
RN R V n
o o o
2.Froud Number. This value, referred to
symbolically as FN, is
FN
Ro(GM/r2)
Its inverse is the ratio of the gravitational
term to the inertial term.
3.Strouhal Number. Called SN, it is given by
the expression
V T
o o
The inverse of this quantity is the ratio of the
time derivative of the velocity to the inertial
term


41
Importance of Viscous Term in the
Equation of Motion
The relative importance of the term in the Navier-
Stokes equation can be determined by calculating the
similarity numbers. To calculate these numbers, the
viscosity must be estimated and the characteristic values
(R V n ) chosen,
o o o
Suppose the gas to be a ternary mixture composed
of hydrogen ions, electrons, and neutral helium. Because
the ratio of the electron mass to that of either of the
other two species is so small, the viscosity of the mixture
is essentially determined by the viscosities of the
hydrogen ions and neutral helium (Chapman and Cowling, 1970).
For hydrogen ions under physical disk condition the
expression given by Spitzer (1962) may be used. It is
2.2 x IQ"15 T5/2
i.nA,
which for disk temperature (Te) gives
H = 2.2 x 10
-6
Chapman and Cowling (1970) give an expression that can be
used to calculate helium viscosity.
t) =
5(]cmheTe/7r)
8a2 W
1/2


42
where a is the molecular diameter and W is a tabulated
function ( ir is the mathematical number) Under disk
conditions the two quantities a and W are
a = 2.7 x 10"8 cm2
W = 6.00
This gives
-4
n = 2.4 x 10
The following characteristic values are used;
(1) R = R ,
o s
(2) V = V /2tt ,
O s
(3) nQ = pa .
For viscous effects due to hydrogen the similarity numbers
are:
(1) RN = 1.2 x 10
+12
(2) FH = 8 x 10
(3) SN = 1
-3
The helium viscosity results in a Reynolds number which
is two orders of magnitude smaller than that for hydrogen,
but this is of little import. Nor would a more accurate
calculation change the basic result that the Reynolds


43
number for the disk fluid is very large. The ratio of the
viscous terms to the inertial terms is so small that the
latter completely dominate the former. Removal of the
viscous terms reduces the Navier-Stokes equation to
3v
3t
+ v
GM
s
R V2
o o
Euler's equation


CHAPTER IV
MATHEMATICAL ANALYSIS OF THE
HYDRODYNAMICAL EQUATIONS
Preliminary Remarks
In the work which follows magnetic effects will be
ignored and any radiation effects will be assumed either
entirely negligible or such that they may be included through
a small adjustment in the gravitational term. On the basis
of the discussion in the last chapter, the viscous terms in
the Navier-Stokes equation can be discarded. The momentum
equation is
3v
8t
+ v
Vv
VP
P
GM
s
R V
o o
The coordinates, time, and the flow variables are dimensionless
and related to the normal dimensioned quantities through the
relation of the preceding chapter. In the rest of this work
the symbol
GM
s
RoVo
GM (2ir)
s
R V2
s s
will be used. The equation of continuity,
rf + V (p V) = 0 ,
44


45
provides an additional relation between pand v.
The full hydrodynamical prescription requires the
inclusion of a third equation, that for energy transport.
No effort will be made in this chapter to introduce such
an equation. Failure to study the energy equation will
necessitate, at some point, the specification of an additional
condition linking two of the flow variables. Introduction of
this new relation reduced the number of unknowns from five to
four. Euler's equation, which contains no viscous terms, is
not valid in the boundary layer between the bulk of the
circumstellar matter and the stellar photosphere. No analysis
of the flow in this region has been made, nor will any be
attempted here.
Linearization of the Equations for Temporal
and Angular Dependence
Introduction of Time and Angle Dependent Terms
As the steady-state axially-syrametric solutions to
the hydrodynamical equations appear to explain much of the
Be star phenomenon, each flow variable will be each divided
into two parts: a steady-state axially-symmetric part and
a term containing both angular and temporal dependence.
The notation is shown below:
(1) v = VQ(a),z) + ^ z,t)
(2) p = po(w,z) + p1 (to, ,z,t)


46
3) P = po(t,z). + p1(w/(J>/Z./t)
The subscript-zero identifies the steady-state axially-
syinmetric terras; the subscript-one# the terms containing
the temporal and angular dependence. In order to avoid
confusion with the dimensioning parameter PQ used earlier#
a lower case p is used with subscript in the case of
pressure.
These pairs of functions are substituted into the
equation of motion to give
*+
v
o
7vo + V1
VV + v
o o
)VPl- qoV(-
and
3 p 3 p 1 ^ ^
3? + "3t + Vpo + V Vpo + ^o VP1 + ^1 Vpl + poV ^o
+ P^V v1 + p V v + pnV v1 = 0
o K1
The denominator in the term
7(po + pi>
has been approximated by first expanding in the series
HlM


47
/
where x is P^/PQ r and then excluding terms in the third
and in higher powers of x. Subsequent conditions placed on
the flow will limit the problem to cases where
With the exception of this approximation for the denominator,
these equations are exact.
Since the subscript-zero terms are themselves steady-
state axially-symmetric solutions of the problem, they satisfy
the equations
o
1
VPD qoV (- i )
and
(p v ) = 0
'Ko o
The equations for the subscript-one terms become


48
and
V1 + pl7 vo + pl
+ v
The underlying physical picture presented thus far
has been that the steady-state axially-symmetric solutions,
the subscript-zero terms, represent the gross nature of the
flow. The subscript-one terms are taken to represent a
refinement to this gross nature. These refinement terms
would then be expected to be smallthough not negligible.
The assumption will be made here that subscript-one terms are
sufficiently smaller than the subscript-zero quantities that
elements of the equations which include the products of
subscript-one quantities can be omitted. The resulting
linearized equations are
Vv. + v
1
1
P
o
1
and
+
v
o
+ V
0


49
These must be rewritten as scalar equations
3v
wl
9.v.
+ v
0)0
3t 0)1 3o)
3v v,, 3v(1_ v, 3v .
, o)l V0)0 3o) o) 3 0) 3 + V
3v
a + v
zl 3Z zo 3z
3v 2v, v,,
oil (fro 0)
i +
p 3o) 2 3p
po
Sv*l
+ v ?> +
3v,,
41 .
3v
(fro
v
+ .
3t
1 v0)l Su +
V0)O
3o)
0)
3 (fr
0)
vzl
_i v + v
3z (fro zo
_3. .
3z
%1 +
V0)lV(frO +
0)
v V ,
0)0 (frl
0)
1
8Pi + pi
9po
P 0)
IT 2
T p 0)
o
3 3v
zl
3 v.
+ V,
zo
3v
v., 3v
3v.
+ v zl _4>1 zo + (fro zl
3t '0)1 3a) wo 3o) o) 3 3v_
+ v
zo
3v
+ V
zl
zl 3z zo 3z
_1 ffl £l 2o
Po 3z P2 dz '
and
3p, p, v p v 3p, 3v 3p
K1 K1 Q)Q o 0)1 H1 p 0)0 Ko
3t co w vwo 3o) 1 3o) vo)l 3o)
. 3vMl 1 r:. i i
po 3w w iV(fro 3 3p_ 3v p, 3v
+ p. -#) zo
3z


50
9pl + r 9po + 9Vzl
Vzo 3z ... Vzl 3z Po 3z
0
Selection of a Subscript-zero Solution
The subscript-zero solution which will be used is
that for a static isothermal envelope (Limber/ 1964). This
solution is chosen for the following reasons:
(1) over most of the disk, the isothermal envelopes
are in agreement with calculations based on
more advanced treatments (See Chapter II);
(2) the solution is in closed form;
(3) the simplicity of this solution results, in
turn, in a comparatively simple set of expressions
for the subscript-one terms.
This subscript-zero solution (in dimensionless coordinates)
is
v
wo
0
9
i i r 1
eo = exp f Q.fc *
P0 = {- § (| -
o,2 7 7771 ] 1'
(d)2 + zV'2 ] ] '
The quantity Q is defined
kR T
o = ^ e
ymM G '
s


51
while ¥ is given by
. (.2tt) kt
Y =
ym Vr
The ratio of these two, Q/V, is the Froude number (FN).
Substitution of these expressions into the linearized
equations for the subscript-one quantities gives
3vu>l .. 2tt aVul 4tt _ 4U f 1 f 1 _1 vli 8P1
Tt" + ^372 ~W ¡372 l y iexPLQ l a) r JJl gw
+ 4'p1 {exp[ § ( ^ p )}}^ {exp [-1 ( I I) ] } ,
3v
4>1
at
TT
V +
2ir
u
372 ul ^3/2 3
i = I {expf i (i i
34) (D texp*- Q r j JJ34,
+ P1H' {exp L§ F ] ] } -gf exp[ CT. w I ) ] '
3v av ,
zl j_. 2tt zl
at ^372 34)
3Pi
feitp [§£-?)]> -57
+ l fei£p[ § (s ? )] t {-5 ( S I )]) '
and
3pl vwl r r 1 fl 1 >, 1 1
Tt + TT exp L Q fe r J J }
+ v
col 3o)
t exp[ - ?) ]1


52
+ ( exp [ -§ ( i | ) ] } jf1
O
While the formal division of the flow into two
elements is quite general, the division has physical significance
only if what are called here the subscript-zero terms adequately
represent the steady-state axially-symmetric part of the flow.
The static isothermal solutions do not adequately represent
such flow at large distance from the equatorial plane. The
solutions to the above equations are useful only for regions
near the plane. It will be the practice in the rest of the
chapter to limit the discussion to the region |z| /w £ 0.1
(quite a conservative value). Such a limitation may seem a
drastic approximation; in fact, it is not. This "thin"
region is a solar diameter in thickness near the stellar
surface and larger farther out. Additionally, examination
of the z-component of Euler's equation reveals that only the
pressure gradient balances the z-component of the gravitational
force, but the ratio of the former to the latter gives
o
1
8.3 x 10
-3


53
The dominance of the z-component of the gravitational term
suggests that the disk material is highly concentrated
toward the plane.
Considerations Concerning Approximations Near the Plane
3
For z < a) the term 1/r becomes
9
similarly, the 1/r term is
All derivatives of known functions should be
performed before approximations such as those described
above are included. This has been the practice here. On
. 3
substitution of the appropriate expressions for 1/r and
1/r, the set of four equations becomes
3v n
,0)1
3t
.+
0)
2rr
3/2
3v
tul
W~
(0
4tt
3/2 V(j)l
= [ exp ( ZT3 ) ]
9Pi
2QW
3 a)
3WP]
2Q
( h ) [ exp( 3 ) ]
2Qo)'


54
3v
J.1
w
IT
3/2 vwl T ~Y72 3
2ir
u
£L = 1
0)
to
j_exp (
2Qto'
)] 3Pl
d 3vzl
"TF"
to
2tt
3/2
3v
zl
3
= [exp (-^-3 ) ]
L 2Qw J
3p
_1
3z
fp,Z r 2
T" [exp (j)}
' n L 2Qto J
to Q
and
3pi V-U1 r ,
TE + ST Lexp ("
2Qu
)]
3z
2Qto
4 Vtol
[ exp ( ZT3 ^ ]
2Qto'
+ t eXP (~ 2Qm3 1 ^ + ^72 7T- + [ exp 2Qm3 ^
3Pi
,3v
*1
3 4>
zv
Qto
zl
y
3v
[exp (" ZT5 ) ] + [exp (- 5 )] = -
2Qto
The form of the last and most complicated of the set
of four equations suggests a change of variable which will
simplify the equation. Define the variable v^ by the
expression
vii = vii exp ( -
2Qto'
) *
the i standing for u),



55
Comparison of this result with the terms in the last of the
set of four equations above shows that the last equation can
be written
2 TT
~T/2
CO '
+
V
->e
V1
0 .
Study of the first three members of the set, which were
derived from Euler's equation, indicates:
(1) that no component of v1 appears in a partial
derivative with respect to either id or z,
(2) that the exponentials, when they appear in any
term, come before all operators.
2 3
On multiplication by exp (~z /2Qco ) and with the introduction
of the variable v^ the first three equations of the set
become
3v
col
9v'
3t
+ f
col
a
- 2fvix=
3V1.,- fvi = 1
at 2 3




56
and
3.V
zl
3.v
3t
+ f
zl
3(J>
Tzpi
3
Qto
Note that the function f used above is
f
The exponential in the definition of ^ makes the function
strongly peaked toward the equatorial plane .
Additional Relations
There are four equations, but five unknowns. As
mentioned in the beginning of this chapter the missing
equation is that for energy transport. This equation
requires more physical information about the nature of the
flow than is needed for either the equation of motion or
the continuity equation and it is mathematically more
complex than either. The traditional approach has been to
circumvent the need for such an equation by specifying
in some manner the relationship between any two of the five
unknowns; this will also be done here. The density and
pressure will be assumed to be related by a condition of
the form
Pi = rPl


57
where T is a parameter. As an illustration, suppose the
relationship between the subscript-one density and the
subscript-one pressure to be isentropic with the isentropic
constant determined from the subscript-zero quantities,
then
. ... 2. ~
r = y a 102

2
where c is the isothermal speed of sound determined from
the subscript-zero terms.
The boundary conditions for viscous hydrodynamics
require that both the tangential and normal components of
the velocity on either side of the boundary between flow
and bounding surface be equal. In the steady-state axially-
symmetric solutions, the interior boundary surface is
rotating and has no radial velocity. The steady-state
solutions do not represent a true use of boundary conditions,
for the velocity dependence is in effect assumed. Further,
the flow is taken to be Eulerian up to the boundary. An
adequate treatment of the boundary-value problem requires:
(1) treating the flow in the boundary layer,
(2) describing the behavior of both the tangential
and normal components of the velocity
over the boundary surface, particularly
dependence on time and angle.
There is no adequate treatment of the boundary layer flow,
and the observations provide little information about the


58
temporal or angular dependence of the stellar surface
velocity. For these two reasons boundary conditions are
of little use in the study of the nature of the subscript-one
solutions.
To this point the four equations which together
comprise Euler's equation and continuity are
9v
o>l
9v
3t
+ f
ul
94)
- 2f =
- r
9 Pi
9o)
3Â¥z2
2Qo)
3V1 fV03l + f8Vl = L i
at + 2 9 o) 34> f
3v
zl
+ f
9v -
zl _p
-TT
8pl
VzP]
and
3p, 9p. v 9v , 3v,t 9v ,
_1 1 0)1 0)1 1 4>1 zl
9t 94> 0) 3o) 0) 34 9z
Note that the e superscripts have been dropped. The symbol
v-j^ now contains the exponential term, and will for the rest
of this chapter.
Integration Over z
Further Restrictions on the Flow
The subscript-one flow will be assumed to possess
symmetry about the equatorial plane; that is, the flow below


59
the plane is the mirror image of the flow above the plane.
This condition requires that the flow variables, with the
exception of vz^, must be odd functions of z.
The set of four equations which describe the flow
will be integrated with respect to z over the interval
[-z + z ] where z is a function of w alone. New variables
o o o
must be introduced; they are:
(1) ux = / v^dZj^ ,
Zo
+z
(2) a, J p.dz .
-z A
o
Under the condition requiring symmetry about the equatorial
plane, the z-component of the integrated velocity (ulz) is
identically zero. The integration with respect to z and the
introduction of new variables require that operator reversals
of the type
+ ZQ(0)) +ZQ(0))
!-z M d;9('2) dz -z (U)9('z) dz'
o o
where g(u,z) is a well-defined function, be performed. The
justification for such changes in the order of operators
appears in Appendix A.
These two steps result in the three equations
_3
3t
Ua)l+ f
_3
3(j)
u
wl
2fu 3a^
* 3 )
3T
2Q u
+z
/
-z_
z Pxdz ,


60
3u
at
+
f
1
u
tol
+ f
a
<) u
*1
r
0)
a*
and
3a, u t
1 + 0)1
&t a)
cTa> Uwl + w 3l
.3.a,
rU,, + f
d 2vzl(z
V
= 0
The last term of the first equation above will be
rewritten
+z^ +z
/ zZp^ dz=A I dz
-z -z
o o
where the quantity A may be considered defined by this rela
tion. A is, strictly speaking, a function, but it will be
treated as a constant parameter. The validity of this
approximation can be tested; the solutions should change
only slowly with changes in A.
The last term of the fourth equation, 2vzi(z0)/
represents mass flow perpendicular to the equatorial plane;
a quantity generally believed small. It is a reasonable
approximation to set this term to zero. Such a step, while
not crucial, does simplify the calculation.
The third term in the continuity equation, which
contains a partial derivative with respect to w, should
be dominated by the neighboring terms which contain

partial derivatives. This term will therefore' be dropped


61
from further consideration,as is a similar term in the first
equation of the set.
The result of all these steps is shown below:
(1> It i + f It i 2f V
-m
2Q u
4 1 *
(2)
and
3U f - 3U .
at + 2 wi + f a* *i
. rffi
0) 3
3a.
3a, u)_ 3u,,
1 + t + i -i =
(3) at + f 3* T io T lo ~5?
Solutions
The following symbols are introduced to simplify the
notation:
(1)
(2) c2 = -r ,
<3> L'at+fl? '
The equations of the previous section become
Clal
L uol 2iu*l= Hr '
f f2: 3crl
L U4>1 + 2 Ul u 3?
9


62
and
u V 3.,.
LCTi + +
1-w o) 3
0 ..
This set of simultaneous differential equations must
now be reduced to a set in which there is a separate equation
for each independent variable. This involves much differentia
tion and manipulation. These reductions are shown in Appendix
B. The results of these manipulation are that all three
equations share the common form
{ o>L
3
2o)'
)f2 ) L + (
2c f c.f a
2 L_ 1JL } x
0) 2w4 h* 1 x
(variable) = 0
where "variable" may be either u^^/m u^ or .
Each of these three equations has the form
oj) y = 0
Let
y = exp [a(o)) where a(u) and 3(a) are as yet unspecified functions. Since
3
jjjj- exp Ia(w) + 3 (w) t]


63
and
exp [ct(w) + 3(w)t] = 3(w) exp [ct (oj) 4> + 3(oj)t]
one has that
F (|j g|; /w) y = F(a(u), 3(w),to)y
A solution (y) exists only for a(ai) and 3(w) such that
F'Xaiw) 3 () rw) = 0
Unfortunately there are infinitely many such (a(w), 3(w))pairs.
The temporal and angular dependence of the flow
variables takes the form exp[ikt + in] where the substitutions
a = ilc
3 = in
have been made. The quantity k is, in effect, a frequency.
For comparison with observations the associated quantity
2irt/k which is the period associated with such a frequency
is more useful. Since + 2tt) are physically the
same point, n must be an integer. The equation which k and
n must satisfy is derived in Appendix B. This equation is
tok2 + 3 unfit2 + ( 2 w f2 + 3uf2n2


For a given n and to there will be in general, three values
for
Relations Among the Flow Variables
The temporal and angular dependence of each flow
variable takes the form
exp [in The quantity n is an integer, and the lc(a),n) are roots of a
polynomial equation whose coefficients are functions of to
and n; The general solution for the problem may be written
exp[in + ik(to,n)t],
exp[in<(> + ik( exp[in + ik(to,n)t]
u
= LUnk<>
Bl nk ak
u
n,k
and
= LSnE<>
n,k
The relationships among the three quantities, U^, C ^ ,
Snkf determine the relationship among the flow variables,
and


65
Substitution of the general solution into the
original set of three equations yields
l_ [i(k + nf)
n,k
u : 2f c r
nk nk
ID,
nk
]{ exp[in + ik(w,n)t]}
0/
£_[ + + i(k + nf) ]{ exp[in + ik(w,n)t]} = 0/
n, k 2 w
j + i (k + nf)
n,k
inc -i _
Sn^{exp[in<|, + ik(w,n) t]} = 0*
These three equations all require summations over n and k.
What is desired is a set of relations among the three
quantities (Un£, Snk' Cnk} for a given n and k.
Each equation is multiplied by the quantity
exp[in' + ik'(w,n')t] .
Next, double integration of the form
Lim
T
1
4irT
+T
/ dt
-T
+TT
/ d exp[-in -IT
is performed on each equation (and the order of summation and
integration reversed). One finds
i(k* + n'f) un'k' 2f Cn'k' 4 Snk' 0 '
0)


66
f
2 wn'k'
an Vc
£ U_,r, + i(k + n'f) Cn,p -
2
co
Sn'k' 0
1
co
Un'k'
in'
co
Cn 'Kf'
+ i (k' + n'f)
Sn'K'
In the following pactes the primes will be surpressed.
These expressions form a system of homogeneous linear
equations. There is no a priori reason to assume that any
non-trivial solutions exist. If solutions do exist, they will
constitute at best a single infinity of solutions. Such a
case can only provide unique values for the ratios of two
of the unknowns with respect to the third.
If non-trivial solutions exist, the determinant
formed by the coefficients of the variables in the system
above,
i (k + nf)
-'2f
-C][/w4
f/2
i (k + nf)
-in c2/w
1/w
in/co
i (k + nf)
should be zero. The details of the calculation of the
determinant are left to Appendix C but the results are
pleasantly familiar. The condition that the determinant
be zero is a polynomial in k which must be zero. This
polynomial is the same as that used to calculate k. Therefore


67
the determinant is identically zero.
There will be a single infinity of solutions if
the rank of the matrix from which the determinant above was
formed is two; then the equations can be solved for the
ratios of two of the unknowns with respect to the third.
The rank of this matrix is, indeed, two (See Appendix C).
The ratios which will be determined are
and
They satisfy the system of equations
cl
- (k + nf) x 2fy = ^
)
These two ratios are
x
c^.(k + nf)
2fnc
01
DET


68
nc0(k .+. nf)
Y
DET
where
DET = [-(k + nf)2 + f2] .
A Reprise
The analysis of this chapter is valid only to the
extent that the underlying physical picture, discussed in
the second section of this chapter, is a reasonable representa
tion of Be stars. This requires the dichotomous view of
the flow presented there to be a real one. The steady-state
axially-symmetric effects must dominate the flow. The
elements of the flow which depend on time and angle must
be secondary.
Further,solutions were obtained only when:
(1) known functions appearing in the differential
equations were approximated by formulae valid
only for z < w;
(2) the flow variables were assumed to possess
symmetry about the equatorial plane;
(3) two parameters, T and A, were introduced;
(4) the equations were integrated with respect
to z (and new dependent variables introduced);


69
(5) two terms in the continuity, equation and one
in the equation of motion which appear to be
of secondary importance were dropped.
The solutions to this set of equations are proportional
to exp[in<}> + ik(wrn)t], where n is any integer and the values
of k for any n and w are the roots of a polynomial equation.
The general solution for the flow can be determined to
within the infinite set of multiplicative factors Sn^ which
appear in the integrated density,
indeterminance awaits a boundary va
Removal of the S t-


CHAPTER V
COMPUTATIONS AND RESULTS
Introduction to the Computations
Both the angular and the temporal dependence of the
$
solutions in the previous chapter are controlled by the
integers n and the quantities k, respectively. The calculation
for k, which depends on n, w, A, and r, emerges as the
principal difficulty. Accordingly, the three principal
concerns of this chapter will be:
1) the nature of the roots of the equation for k,
2) the means used to calculate them,
3) the results of the computations for various
choices on which the k equation depends.
As the solutions cannot be fully evaluated in the
absence of a proper boundary treatment, these results will
constitute much of what can be learned from the analysis.
In most of the discussion the period
2ttT
| PER | =
1*1
(generally in hours) will be used.
The first major section of this chapter is devoted
to the first two topics above, the nature of the roots and
70


71
the procedure used to compute them. The results of these
computations, the third major concern of this chapter,
naturally group themselves into three divisions:
(1) calculation of the pertinent quantities
in the absence of any angular dependence
(n = 0) ,
(2) temporal behavior in the presence of the
simplest angular dependence (n = 1),
(3) the trends of the results for more complex
angular behavior (n < 1).
The three central sections of the chapter reflect this
three-fold division. In each the dependence of the
results on w, A, and T is discussed. The penultimate
section presents sample calculations of the flow
variables which illustrate the qualitative aspects of the
solutions. The last section of the chapter compares
the results with relevant observations.
The Equation for k
The Consequences of Imaginary Roots
The equation for k, derived in Chapter IV is
wk? + 3wnfk^ + (
+ 3limV % ) ir
0)
(D


72
+ (
c2n3f
0)
2c nf o 7 c,nf
- wf n + un'£~' 7) = 0
a) 2(u
Being an equation of the third degree, this equation has
either three real roots or one real and two complex roots.
If there are complex roots, one is the complex conjugate
of the other. The appearance of complex roots had immediate
and serious consequences for the methodology of Chapter IV.
One of these roots must of necessity have a negative imaginary
part which will appear in the solutions as an increasing
exponential term, exp (-+-1 11) where | | is the absolute
value of the imaginary part. Flow variables which increase
exponentially in time will, at some time, violate the condition
under which linearization of the flow variables is valid.
It is possible to choose the three multiplicative factors
^Snk' Cnk' Unk^ *n sucl1 manner that these increasing terms
do not contribute to the final solution, but such choices
are suspect. Only for conditions in which all the roots
are real can the linearization procedure, and hence all
thereafter, be considered valid.
Initial Values for the Parameters
The quantities T and A, which appear in the
coefficients of the equation for 1c (in c, and c2 respectively)
enter the problem through the relations
pi = rpi
#


73
and
+z +z
/ tt p1 dz = A / dz ,
respectively. The calculations, whose results are described
in later sections,will be performed over a whole range of
values for each of these quantities; however, an initial
or representative value for each quantity must be determined.
The representative value for Tcalled Tis the
value for T in the case that p1 and p1 are related
isentropically. The isentropic coefficient is determined
from the subscript-zero quantities; it is
(2tt) 2 kT
r= () -
v2 ym
s
+4
For an electron temperature (Te) of 10 K,
T = 7.67 x 10"2 .
A representative value for Acalled Acan be
calculated be setting p1 equal to a constant; for zq equal
to 0.1000,
- zo 3
A = = 3.334 x 10~ .
3


74
Machine Computation of the Roots
Although the mathematics required to determine the
values of k at any point in the disk is not difficult, it
is time consuming. This calculation must be performed at
many points and for a number of different values for the
quantities n, A, and T; machine computation is neccessary
to perform as many calculations as are needed. A program
was devised which will perform the following:
1. Input. The program reads in the values which
are to be assumed by the physical and geometrical quantities
appearing in the coefficients of the k equation. The program
must also be given initial values for A and r.
2. Roots of the k equation. On the basis of internal
control statements, the program establishes a sequence of
A, T, n, and w combinations. For each of these combinations,
the program calculates first the coefficient and then the
roots of the equation for k. Each root is searched for an
imaginary part; if the root is real, the quantity 2Tr/k (in
hours) is calculated.
3. Solutions for the flow variables. When given
as a function of in Chapter IV to calculate both and Un£* The quantities
Re{Un^()) exp[in<}> + ik(w,n) t]}
Re{Cn£(w) exp[in<{> + ik(aj,n)t]> ,


75
and
Re{Snk() exP[in are calculated by the program throughout the disk for
various choices of angles and times.
4. Readout. The program prints the results of the
calculations in tabular form.
The actual program steps are contained in Appendix D.
The Case n = 0
Distinctive Aspects of the n = 0 Case
The k values here are for solutions with no angular
dependence. In the absence of dependence, the k equation
takes the form
Q
wk3 (wf2 + -j) k = 0 .
0)
the three roots are
2 cl X/2
k = 0, + (f + -£)
J
The first value corresponds to a trivial solution. The
last expression is real for all reasonable values of c^.
Variation with o), r and A
The expression for the non-trivial roots depends on
w and c^ but not on c2*
Now


Figure 3
The dependence of [PER| on
a) is shown for the case n =
and c^ = 36.122.


77


78
(2tt )
= 0.152
where
5. = 2M
1 2Q
Setting one finds that
k =
2tt
372
0)
(1 + ( i,) i +...)
2 (2ir) z w
2n
1/2
0)
The absolute value of the corresponding period is
[PER|= T
Ikl
or
|PER|= 48.5 10
3/2
For large the behavior of |per| near the stellar surface
3/2
(w = 1) departs slightly from w behavior, as would be
expected from the full expression for |Jc| but the outer
3/2
regions of the disk still show the oj behavior. These
features are shown in Figure 3.
The Case n = 1
Preliminary Remarks
The n = 0 case cannot be used for a general discussion.


79
Exploratory calculations: indicated that the n = 1 case
illustrates most of the characteristic features of the
general non-zero n case, yet possesses certain unique
features themselves worthy of investigation. Accordingly,
the most extensive series of computations were performed
for n = 1.
On the basis of both these preliminary calculations
and physical considerations the range of A was chosen
to be [0.000, 0.200] and the range of T was taken to be
[0.001, 1.000], Calculations were made for each of the 150
T-A pairs; these pairs uniformly spanned the combined ranges.
In a later series of calculations r was allowed to take values
as high 10.00. In subsequent discussion, the quantity c^,
through which A enters the equation for k, will be used
rather than A; however, T continues in use since c2 = -T .
Initial studies also indicated that an evenly spaced 100-point
grid spanning the range [1.0, 11.0] in the radial coordinate
was sufficient to study the variation of the calculated
quantities throughout the disk.
The dependence of the solutions to the k equation
on c^, T, and w will be the subject of the next three
subsections. One important result should be mentioned at
the outsetthe roots of the k equation under almost all
circumstances studied are real.
The Effect of c^^
Rather than k, the absolute value of the period


Figure 4
The quantities |PER^| and IPER2|
are plotted against1 radial
coordinate for two different values of
c^. Curves a and b are ¡PER2| and
I PER,I respectively, when c, = 0.361;
c and d, when c, = 2.709. In'1 all
cases T = 0.0757


81


Figure 5
The dependence of |PER.| and
| PERjl I on w is shown for two
different values of c,. Curves
a and b are | PER2 | and-1" | PER^ | ,
respectively/ when Cn = 18. 061
c and d, when c^ = 36.122. In
all cases r = 0.075.


Full Text

PAGE 1

)/8,' '<1$0,&6 2) &,5&8067(//$5 0$7(5,$/ $662&,$7(' :,7+ %IH 67$56 %\ 7KRPDV +DUORZ 0RUJDQ $ ',66(57$7,21 35(6(17(' 72 7+( *5$'8$7( &281&,/ 2) 7+( 81,9(56,7< 2) )/25,'$ ,1 3$57,$/ )8/),//0(17 2) 7+( 5(48,5(0(176 )25 7+( '(*5(( 2) '2&725 2) 3+,/2623+< 81,9(56,7< 2) )/25,'$

PAGE 2

7R .<& DQG +(1f§ERWK RI ZKRP SXVKHG

PAGE 3

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

PAGE 4

7$%/( 2) &217(176 3DJH $&.12:/('*(0(176 LLL /,67 2) 7$%/(6 L[ /,67 2) ),*85(6 ; .(< 72 6<0%2/6 [LL $%675$&7 [Y &+$37(5 ,1752'8&7,21 'LVFRYHU\ DQG 7D[RQRP\ RI (PLVVLRQ/LQH % 6WDUV $ %ULHI +LVWRU\ 1RPHQFODWXUH &KDUDFWHULVWLFV RI %H 6WDUV 6SHFWUDO DQG /XPLQRVLW\ &ODVV *DODFWLF 'LVWULEXWLRQ 0HPEHUVKLS LQ 0XOWLSOH 6WDU 6\VWHPV DQG ,QFLGHQFH $PRQJ (DUO\7\SH 6WDUV 5RWDWLRQDO 9HORFLW\ 6SHFWUDO 9DULDWLRQV 7KH 6WUXYH 0RGHO 2ULJLQ /LQH 3URILOHV +\GURG\QDPLFDO 0RGHOV IRU WKH )ORZ 6SHFWUDO 9DULDWLRQV DQG WKHLU 3RVVLEOH &DXVHV 0RGLILFDWLRQV RI WKH 6WUXYH 0RGHO 0DMRU 7RSLFV RI WKLV :RUN ,, +<'52'<1$0,&$/ $3352$&+(6 $1' 67($'<67$7( 62/87,216 %DFNJURXQG LY

PAGE 5

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

PAGE 6

&+$37(5 3DJH $ 5HSULVH 9 &20387$7,216 $1' 5(68/76 ,QWURGXFWLRQ WR WKH &RPSXWDWLRQV 7KH (TXDWLRQ IRU N 7KH &RQVHTXHQFHV RI ,PDJLQDU\ 5RRWV ,QLWLDO 9DOXHV IRU WKH 3DUDPHWHUV 0DFKLQH &RPSXWDWLRQ RI WKH 5RRWV 7KH &DVH Q 'LVWLQFWLYH $VSHFWV RI WKH Q &DVH 9DULDWLRQ ZLWK Rfr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

PAGE 7

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n DQG 6 QN $33(1',; 7KH 3URJUDP &$/62/ $33(1',; ( 3HUWLQHQW ,QWHJUDOV YLL

PAGE 8

3DJH $33(1',; ) $Q (TXDWLRQ IRU U A %,%/,2*5$3+< %,2*5$3+,&$/ 6.(7&+

PAGE 9

/,67 2) 7$%/(6 )LJXUH 3DJH &KDUDFWHULVWLF &HQWUDO 6WDU DQG 'LVN 9DOXHV 7KH YDOXHV RI WKH &RHIILFLHQWV $SSHDULQJ LQ WKH (QHUJ\ (TXDWLRQ 7KH 3URJUDP 1DPHV IRU ,PSRUWDQW 4XDQWLWLHV 0DLQ 3URJUDP 6XEURXWLQH $/3O 6XEURXWLQH %(7 L[

PAGE 10

/,67 2) ),*85(6 )LJXUH 3DJH $ %H VWDU YLHZHG HTXDWRURQ DQG SROHRQ 7KH FRRUGLQDWH V\VWHP 7KH UDGLDO GHSHQGHQFH RI _3(5_ IRU Q DQG IRU ODUJH FA 7KH UDGLDO GHSHQGHQFH RI _3(5A DQG _3(5 IRU VPDOO YDOXHV RI FA 7KH UDGLDO GHSHQGHQFH RI _3(5M-DQG _3(5_IRU ODUJHU YDOXHV RI FA 7KH UDGLDO GHSHQGHQFH RI _3(5_IRU IRXU YDOXHV RI 7KH UDGLDO GHSHQGHQFH _3(5_ DQG _3(5_ IRU VPDOO YDOXHV RI 7 7KH UDGLDO GHSHQGHQFH RI _3(5_ DQG _3(5_IRU ODUJH YDOXHV RI 7 7KH UDGLDO GHSHQGHQFH RI _3(5_ IRU VPDOO YDOXHV RI U 7KH UDGLDO GHSHQGHQFH RI _3(5_ IRU WKUHH YDOXHV RI U 7KH UDGLDO GHSHQGHQFH RI DOO WKUHH SHULRGV IRU ODUJH FA DQG U 7KH UDGLDO GHSHQGHQFH RI _3(5 DQG_3(5_ IRU VPDOO YDOXHV RI FA DQG U 7KH UDGLDO GHSHQGHQFH RI WKH WKUHH SHULRGV IRU WKUHH GLIIHUHQW YDOXHV RI Q 7KH TXDQWLW\ _3(5A_MA DV D IXQFWLRQ RI Q 7KH HIIHFW RI ODUJH SDUDPHWHU YDOXHV RQ WKH SHULRGV ZKHQ Q LV ODUJH [

PAGE 11

)LJXUH 3DJH 7KH UDGLDO GHSHQGHQFH RI X rf DQG 8n rf IRU YDULRXV WLPHV SO 7KH UDGLDO GHSHQGHQFH RI rf X rf IRU YDULRXV WLPHVn } [L

PAGE 12

.(< 72 6<0%2/6 ,Q RUGHU WR DYRLG DQ H[FHVVLYH QXPEHU RI V\PEROV H[WHQVLYH XVH ZDV PDGH RI VXEVFULSWLQJ 7KH VXEVFULSWV IDOO PDLQO\ LQWR IRXU FODVVHV )LUVW VXEVFULSWV DUH XVHG WR GHQRWH D FRPSRQHQW RI D YHFWRU )RU WKH F\OLQGULFDO FRRUGLQDWH V\VWHP LQ XVH KHUH VHH )LJXUH f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n 3UHVVXUH GLPHQVLRQOHVV SUHVVXUH X[ LQWHJUDWHG YHORFLW\ [LL

PAGE 13

A LQWHJUDWHG GHQVLW\ Q LQWHJHU FRQWUROOLQJ WKH DQJXODU GHSHQGHQFH N TXDQWLW\ FRQWUROLQJ WKH WHPSRUDO GHSHQGHQFH $ 0RUH *HQHUDO /LVWLQJ 7KH QXPEHUV EHORZ DUH WKH SDJHV RQ ZKLFK WKH V\PEROV DUH LQWURGXFHG $ % &QNn &On &n '(7 )1 I  XQLW YHFWRU L LPDJLQDU\ QXPEHU N / 0 P SURWRQ PDVV V 1H 1KH _3(5_ _3(5_ _3(5M_ _3(5_ 31 S DV D SDUDPHWHU LQ WKH FURVVVHFWLRQDO IORZf SA 4 T4 5 51 U V 6QNn 61 [LLL

PAGE 14

7H 7J W WLPH 8QNn 9 U Y DV YHf DQG YH DQG V ; < ] ]T U 7 Q N $ $ ƒ $X 3Gn Rn f [LY

PAGE 15

$EVWUDFW RI 'LVVHUWDWLRQ 3UHVHQWHG WR WKH *UDGXDWH &RXQFLO RI WKH 8QLYHUVLW\ RI )ORULGD LQ 3DUWLDO )XOILOOPHQW RI WKH 5HTXLUHPHQWV IRU WKH 'HJUHH RI 'RFWRU RI 3KLORVRSK\ )/8,' '<1$0,&6 2) &,5&8067(//$5 0$7(5,$/ $662&,$7(' :,7+ %H 67$56 %\ 7KRPDV +DUORZ 0RUJDQ 'HFHPEHU &KDLUPDQ $OH[ 6PLWK &R&KDLUPDQ .ZDQ
PAGE 16

RI OLQHDU HTXDWLRQV IRU WKH WHUPV ZLWK WHPSRUDO DQG DQJXODU GHSHQGHQFH 7KH NQRZQ VWHDG\VWDWH D[LDOO\V\PPHWULF WHUPV DUH WDNHQ IURP WKH OLWHUDWXUH /LPEHU 1 $S f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n Q ZKHUH Lf LV WKH GLVWDQFH IURP WKH FHQWHU RI WKH VWDU LQ WKH HTXDWRULDO SODQH 7KHVH FDOFXODWLRQV SUHGLFW WHPSRUDO YDULDWLRQ GRZQ WR IUDFWLRQV RI DQ KRXU DQG LQ TXDOLWDWLYH DJUHHPHQW ZLWK UHOHYDQW REVHUYDWLRQV 7KH WKLUG HTXDWLRQ RI K\GURG\QDPLFV WKH HQHUJ\ HTXDWLRQLV H[DPLQHG LQVRIDU DV LW FDQ EH LQ WKH DEVHQFH RI DQ HIIHFWLYH WUHDWPHQW RI WKH UDGLDWLRQIOXLG LQWHUDFWLRQ [YL

PAGE 17

7KLV DSSURDFK OHDGV WR LQFRQVLVWHQFLHV ZLWK WKH UHVXOWV RI WKH VWXG\ RI WKH HTXDWLRQV RI PRWLRQ DQG FRQWLQXLW\ ,W LV FRQFOXGHG WKDW LQ WKH DEVHQFH RI VRPH DFFRPPRGDWLRQ RI UDGLDWLRQ HIIHFWV QR HTXDWLRQ IRU HQHUJ\ WUDQVSRUW FDQ LPSURYH WKH H[LVWLQJ VROXWLRQV )LQDOO\ WKH VWDELOLW\ RI VWHDG\VWDWH IORZV LV H[DPLQHG 7KH VWHDG\VWDWH IORZV DUH QRW DEVROXWHO\ XQVWDEOH EXW WKH PRVW JHQHUDO VROXWLRQV DSSHDU WR EH SHULRGLF LQ FKDUDFWHU [YLL

PAGE 18

&+$37(5 ,1752'8&7,21 ‘'LVFRYHU\ DQG 7D[RQRP\ RI (PLVVLRQ/LQH % 6WDUV $ %ULHI +LVWRU\ ,Q $XJXVW )DWKHU 6HFFKL REVHUYHG +LQ HPLVVLRQ LQ \ &DV 2YHU WKH QH[W KDOIFHQWXU\ &DPSEHOO )URVW 0LVV &DQQRQ DQG RWKHUV REVHUYHG K\GURJHQ LQ HPLVVLRQ LQ RWKHU VWDUV RI ZKDW ZRXOG QRZ EH FDOOHG VSHFWUDO W\SH % DV ZHOO DV LQ PHPEHUV RI WKH UHPDLQLQJ HDUO\ VSHFWUDO W\SHV 0LVV &DQQRQ DSSHDUV WR KDYH EHHQ WKH ILUVW WR REVHUYH YDULDWLRQV LQ WKH HPLVVLRQ IHDWXUHV LQ RQH RI WKHVH VWDUV &DQQRQ f 3LFNHULQJ f SXEOLVKHG WKH ILUVW FDWDORJXH RI HDUO\W\SH HPLVVLRQOLQH VWDUV DW WKDW WLPH WKHUH ZHUH )XUWKHU ZRUN ZDV LQ WZR GLUHFWLRQV PRUH HPLVVLRQn OLQH REMHFWV ZHUH GLVFRYHUHG DQG LQGLYLGXDO VWDUV ZHUH VWXGLHG LQ GHWDLO 7KH FDWDORJXH RI 0HUULOO DQG %XUZHOO f FRQWDLQHG RYHU %W\SH HPLVVLRQOLQH VWDUV 'HWDLOHG VSHFWUDO VWXGLHV OHG 6WUXYH f WR LQWHUSUHW WKH VSHFWUDO IHDWXUHV FKDUDFWHULVWLF RI %H VWDUV LQ WHUPV RI HPLVVLRQ DQG DEVRUSWLRQ LQ D JDVHRXV GLVNVKDSHG HQYHORSH VXUURXQGLQJ D UDSLGO\ URWDWLQJ FHQWUDO VWDU WKLV LQWHUSUHWDWLRQ KDV QRW FKDQJHG VXEVWDQWLDOO\ VLQFH ,Q UHFHQW \HDUV ERWK ORQJ WHUP YDULDWLRQV /LPEHU f

PAGE 19

DQG VKRUWHU WHUP YDULDWLRQV /DFRDUUHW 6OHWWHEDN +XWFKLQJV DQG +XWFKLQJV HW DOA f LQ WKH FKDUDFWHULVWLF HPLVVLRQDEVRUSWLRQ IHDWXUHV KDYH UHFHLYHG DWWHQWLRQ 7KH PRVW UHFHQW OLVWLQJ RI %H VWDUV -DVFKHN HW DO f FRQWDLQV REMHFWV 1RPHQFODWXUH 7KH EDVLV RI PRGHUQ V\VWHPV RI VSHFWUDO FODVVLILFDWLRQ LV WKH V\VWHP DGRSWHG IRU WKH +HQU\ 'UDSHU FDWDORJXH &DQQRQ DQG 3LFNHULQJ f 7KH RULJLQDO +DUYDUG DUUDQJHPHQW RI WKH VSHFWUD ZDV LQ RUGHU RI GHFUHDVLQJ UDWLRV RI LQWHQVLWLHV RI WKH K\GURJHQ %DOPHU OLQHV WR WKH LQWHQVLWLHV RI D QXPEHU RI RWKHU DEVRUSWLRQ OLQHV 7KH +DUYDUG REVHUYHUV VRRQ UHDOL]HG WKDW D UHDUUDQJHPHQW RI WKH RULJLQDO DOSKDEHWLFDO RUGHU LQWR WKH SUHVHQW VSHFWUDO VHTXHQFHf§ % $ ) 0f§UHSUHVHQWHG DQ RUGHULQJ LQ WHPSHUDWXUH ZLWK GHFUHDVLQJ WHPSHUDWXUHV IURP 2 WR 0 $PRQJ WKH +' REMHFWV W\SHG % WKRVH SRVVHVVLQJ K\GURJHQHPLVVLRQ OLQHV ZHUH PDUNHG ZLWK DQ DGGLWLRQDO ORZHU FDVH Sf§ IRU SHFXOLDUf§IROORZLQJ WKH % 0RGHUQ FODVVLILFDWLRQ XVHV WKH QRWDWLRQ %H IRU VXFK VWDUV WKDW LV VWDUV ZLWK %W\SH VSHFWUDO IHDWXUHV SOXV HPLVVLRQ LQ RQH RU PRUH RI WKH %DOPHU OLQHV 7KH VXIIL[ S LV QRZ XVHG RQO\ WR LQGLFDWH WKH SUHVHQFH RI DQQRPDORXVO\ VWURQJ RU ZHDN OLQHV ,I D %H VWDU VKRZV VKDUS PHWDOOLF DEVRUSWLRQ OLQHV VXSHUSRVHG RQ WKH %H VSHFWUXP WKH VWDU LV FDOOHG D VKHOO VWDU

PAGE 20

&KDUDFWHULVWLFV RI %H 6WDUV 6SHFWUDO DQG /XPLQRVLW\ &ODVV ,Q WKLV ZRUN WKH WHUP %H VWDU DSSOLHG WR K\GURJHQ HPLVVLRQOLQH VWDUV ZKRVH XQGHUO\LQJ VSHFWUDO W\SHV DUH HDUO\ RI PLG % %%f DQG ZKRVH OXPLQRVLW\ FODVVLILFDWLRQV DUH HLWKHU ,, ,9 RU 9 7KLV LV VRPHZKDW DW YDULDQFH ZLWK PXFK RI WKH OLWHUDWXUH ZKLFK FRQVLGHUV %H VWDUV WR EH EDVLFDOO\ PDLQVHTXHQFH OXPLQRVLW\ FODVV 9f REMHFWV %XW 0HQGR]DnV f FRPSUHKHQVLYH OXPLQRVLW\ FODVV VXUYH\ RI %H VWDUV LQ WKH 0HUULOO DQG %XUZHOO FDWDORJXH VLPSO\ SODFHG WKH VWDUV LQ WZR JURXSVf§YHU\ OXPLQRXV REMHFWV ,,f DQG OHVV OXPLQRXV REMHFWV ,,, ,9 9ff§RQ WKH EDVLV RI D SURQRXQFHG GLFKRWRP\ LQ KLV REVHUYDWLRQDO PDWHULDO ,Q IDFW VWURQJ GRXEWV 8QGHUKLOO f KDYH EHHQ H[SUHVVHG DV WR WKH H[LVWHQFH RI D V\VWHPDWLF GLVWLQFWLRQ EHWZHHQ 0. OXPLQRVLW\ FODVVHV ,9 DQG 9 LQ WKLV HDUO\ VSHFWUDO UHJLRQ $EVROXWH PDJQLWXGHV RI %H VWDUV GHWHUPLQHG RQ WKH EDVLV RI FOXVWHU RU DVVRFLDWLRQ PHPEHUVKLS DSSHDU WR IDOO DERXW RQH PDJQLWXGH DERYH WKH ]HUR DJH PDLQ VHTXHQFH )LQDOO\ WKH DSSHOODWLRQ VKHOO ZLOO EH VLPLODUO\ UHVWULFWHG DOWKRXJK REMHFWV ZLWK $ DQG HYHQ HDUO\ ) VSHFWUDO FODVVHV DUH VRPHn WLPHV FDOOHG VKHOO REMHFWV *DODFWLF 'LVWULEXWLRQ 0HPEHUVKLS LQ 0XOWLSOH 6WDU 6\VWHPV DQG ,QFLGHQFH $PRQJ (DUO\7\SH 6WDUV 0HQGR]D f WRRN IRU KLV VDPSOH DOO WKH VWDUV LQ WKH 0HUULOO DQG %XUZHOO FDWDORJXH ZKRVH VSHFWUDO W\SHV IHOO

PAGE 21

LQ WKH UDQJH IURP %2H WR %H DQG ZKRVH OXPLQRVLW\ FODVVHV ZHUH ,,, ,9 RU 9 +H IRXQG WKDW WKH %H VWDUV SRSXODWH LQ D URXJKO\ XQLIRUP PDQQHU WKH VSLUDO DUPV DV GHOLQHDWHG E\ WKH % DVVRFLDWLRQV %H VWDUV DUH IRXQG LQ % DVVRFLDWLRQV EXW DUH QRW VWURQJO\ FRQFHQWUDWHG LQ WKHP 7KH\ DUH WR EH IRXQG ERWK LQ JDODFWLF FOXVWHUV [ 3HUVHL &RPDf DQG DOVR LQ ELQDU\ V\VWHPV D 7DX S 3HU 9 &\Jf 5RWDWLRQDO 9HORFLW\ %H VWDUV DUH WKH PRVW UDSLGO\ URWDWLQJ FODVV RI VWDUV 7KH PRVW UHFHQW FDWDORJXH RI REVHUYHG URWDWLRQDO YHORFLWLHV %HUQDFFD DQG 3HULPRWWR f OLVWV REMHFWV LQ WKLV VSHFWUDO FODVV ZKRVH URWDWLRQDO YHORFLWLHV KDYH EHHQ PHDVXUHG 7KH PHDQ URWDWLRQDO YHORFLW\ RI WKH JURXS LV NPV ZKHUH WKH REVHUYHG URWDWLRQDO YHORFLW\ LV WKH WUXH URWDWLRQDO YHORFLW\ WLPHV VLQ L L EHLQJ WKH DQJOH EHWZHHQ WKH D[LV RI URWDWLRQ DQG WKH REVHUYHUnV OLQH RI VLJKW 2Q WKH DVVXPSWLRQ WKDW WKH D[HV RI VWHOODU URWDWLRQ DUH GLUHFWHG UDQGRPO\ LQ VSDFH RQH ILQGV WKH WUXH PHDQ URWDWLRQDO YHORFLW\ +XDQJ DQG 6WUXYH f WR EH NPV 7UDGLWLRQDO PHWKRGV RI OLQH SURILOH DQDO\VLV XVHG WR GHWHUPLQH WKH REVHUYHG URWDWLRQDO YHORFLWLHV DUH QRZ EHOLHYHG &ROOLQV +DUGURS DQG 6WULWWPDWWHU f WR XQGHUHVWLPDWH WKH DFWXDO YDOXHV E\ DV PXFK DV SHUFHQW ,I WKLV EHOLHI LV WUXH WKH %H VWDUV DUH URWDWLQJ DW ODUJH IUDFWLRQV RI WKHLU HTXDWRULDO EUHDNXS YHORFLWLHV %DOPHU HPLVVLRQV

PAGE 22

RFFXU LQ DSSUR[LPDWHO\ SHUFHQW RI DOO %W\SH VSHFWUD 8QGHUKLOO f 6SHFWUDO 9DULDWLRQV 6SHFWUDO YDULDWLRQV RFFXU LQ PRVW %H DQG VKHOO VWDUV )ROORZLQJ WKH WHUPLQRORJ\ RI 0F/DXJKOLQ f WKH VSHFWUDO YDULDWLRQV IDOO LQWR WKUHH PDMRU FODVVHV f DSSHDUDQFH RU GLVDSSHDUDQFH RI D VKHOO DEVRUSWLRQ VSHFWUXP f FKDQJHV LQ WKH UDWLR RI WKH LQWHQVLW\ RI DQ HPLVVLRQ OLQH WR WKH QHLJKERULQJ FRQWLQXXP (&f f FKDQJHV LQ WKH UHODWLYH LQWHQVLWLHV RI WKH UHG DQG YLROHW FRPSRQHQWV RI WKH %DOPHU HPLVVLRQ OLQHV 95f 7KH FODVVLF H[DPSOH RI WKH ILUVW YDULDWLRQ LV 3OHLRQH 7DXf ZKLFK KDV EHHQ REVHUYHG VSHFWURJUDSKLFDOO\ VLQFH 3OHLRQH ZDV ILUVW FODVVLILHG DV DQ HPLVVLRQn OLQH VWDU EXW E\ LW VKRZHG QR HPLVVLRQ IHDWXUHV DQG SRVVHVVHG D QRUPDO % VSHFWUXP ZLWK ODUJH URWDWLRQDO YHORFLW\f ,Q HPLVVLRQ OLQHV ZHUH SUHVHQW D VKRUW WLPH ODWHU WKH VWDU GLVSOD\HG D VKHOO VSHFWUXP %\ WKH HPLVVLRQ IHDWXUHV ZHUH ORVW 5HFHQW REVHUYDWLRQV 6KDURY DQG /\XWLY f LQGLFDWH WKDW 3OHLRQH LV HQWHULQJ D QHZ F\FOH RI DFWLYLW\ (& DQG 95 YDULDWLRQV WKRXJK VWULFWO\ SHULRGLF RQO\ ZKHQ WKH\ UHIOHFW RUELWDO PRWLRQV RI VWDUV LQ ELQDU\

PAGE 23

V\VWHPV VKRZ PXFK TXDVLSHULRGLF EHKDYLRU 7KH WLPH VFDOHV IRU WKHVH YDULDWLRQV PD\ EH RQ WKH RUGHU RI D IHZ \HDUV RU DV VKRUW DV GD\V RU IUDFWLRQV RI D GD\ 6OHWWHEDN +XWFKLQJV DQG 3HWHUV f 7KH 6WUXYH 0RGHO 2ULJLQ ,Q 6WUXYH VXJJHVWHG WKDW WKH HPLVVLRQDEVRUSWLRQ IHDWXUHV WKDW DUH WKH GHILQLWLYH FKDUDFWHULVWLF RI %H VWDUV DUH IRUPHG LQ JDVHRXV FLUFXPHTXDWRULDO ULQJV VXUURXQGLQJ DQG HMHFWHG IURPf %W\SH FHQWUDO VWDUV WKDW DUH URWDWLQJ DW WKHLU EUHDNXS YHORFLWHV /DWHU 6WUXYH f ZDV DEOH WR SUHVHQW WKH PRGHO RQ WKH EDVLV RI PRUH GHWDLOHG VSHFWUDO FRQVLGHUDWLRQV +H DOVR SURSRVHG D FLUFXODU YHORFLW\ ODZ IRU WKH HTXDWRULDO GLVN 9 Z !F RF Z f F RF f f ZKHUH Z WX LV WKH UDGLDO GLVWDQFH IURP WKH FHQWUDO D[LV F RF WR DQ\ SRLQW LQ WKH GLVN GLYLGHG E\ WKH VWHOODU UDGLXV DQG 9M!FAXRFA WKH URWDWrRQDO YHORFLW\ DW WKH VWHOODU HTXDWRU )LJXUH ZKLFK GUDZV LWV LQVSLUDWLRQ IURP +DFN f LOOXVWUDWHV WKH PRGHOnV DELOLW\ WR H[SODLQ WKH PDMRU IHDWXUHV RI %H VSHFWUD 7KH ILUVW FDVH LV D %H VWDU ZKRVH D[LV RI URWDWLRQ LV SHUSHQGLFXODU WR WKH OLQH RI VLJKW WKH VHFRQG RQH ZKRVH D[LV RI URWDWLRQ LV SDUDOOHO WR WKH REVHUYHUnV OLQH 2

PAGE 24

)LJXUH 5RWDWLQJ 6KHOO 6KRZV D %H VWDU YLHZHG HTXDWRURQ 7KH HQYHORSH LV DVVXPHG IRU VLPSOLFLW\ WR KDYH RQ URWDWLRQDO PRWLRQ 1RWH WKH FHQWUDO DEVRUSWLRQ IURP UHJLRQ DQG WKH HPLVVLRQ OREHV IURP WKH DUHDV ODEHOHG DQG 6KHOO 6WDU 3ROHRQ 6KRZV D %H VWDU YLHZHG SROHRQ

PAGE 25

5RWDWLQJ 6KHOO

PAGE 26

RI VLJKW $ VFKHPDWLF SURILOH RI WKH H[SHFWHG HPLVVLRQ DW RQH RI WKH %DOPHU OLQHV LV DOVR VKRZQ IRU HDFK FDVH 7KH LPSRVLWLRQ RI D VPDOO H[SDQVLRQ YHORFLW\ RQ WKH ODUJH FLUFXODU YHORFLW\ JLYHV D 95 DV\PPHWU\ 7KDW WKH GLVN LV LQGHHG FL[FXP HTXDWRULDO ZDV JLYHQ FUHGHQFH E\ WKH ZRUN RI WKH %XUELGJHV 7KH\ DQDO\]HG %XUELGJH DQG %XUELGJH f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f D PRGHO IRU WKH UDGLDWLRQ ILHOG RI WKH FHQWUDO VWDU f D PRGHO ZKLFK JLYHV WKH SK\VLFDO VWDWH GHQVLW\ DQG YHORFLW\ RI WKH JDV DW DOO SRLQWV LQ WKH GLVN f WKH FKDQJHV LQ WKH UDGLDWLRQ ILHOG UHVXOWLQJ IURP LWV SDVVDJH WKURXJK WKH JDVHRXV GLVN 7KH ZRUN RI 0LKDODV f DQG RI RWKHUV QRZ SURYLGHV WKH ILUVW UHTXLUHPHQW 7KH VHFRQG DQG WKLUG

PAGE 27

UHTXLUHPHQWV KDYH QRW EHHQ VR VDWLVIDFWRULO\ DFKLHYHG DQG DUH QRW HYHQ WUXO\ VHSDUDEOH 7KH 6WUXYH K\SRWKHVLV VD\V QRWKLQJ DERXW WKH PHFKDQLVPVf WKDW FDXVH WKH IRUPDWLRQ DQG GHVWUXFWLRQ RI WKH GLVNf§WKDW LV FKDQJHV IURP % WR %H WR VKHOO VWDU DQG YLFH YHUVDf§QRU GRHV LW VD\ DQ\WKLQJ DERXW WKH PHFKDQLVPV WKDW SURGXFH RWKHU VSHFWUDO YDULDWLRQV /LQH 3URILOHV 7KH K\GURJHQHPLVVLRQ VSHFWUXP LV D UHFRPELQDWLRQ VSHFWUXP IRUPHG LQ D GLVN WKH YDULRXV SRUWLRQV RI ZKLFK &DQ PRYH UHODWLYH WR RQH DQRWKHU $ JUHDW GHDO RI ZRUN RQ K\GURJHQ HPLVVLRQ LQ D VWDWLRQDU\ DWPRVSKHUH KDV EHHQ GRQH .RJXUH D E f 6REROHY f DQG 5RWKHQEHUJ f KDYH VWXGLHG WKH IRUPDWLRQ RI WKH K\GURJHQ OLQHV LQ VSKHULFDOO\ V\PPHWULF HQYHORSHV H[SDQGLQJ ZLWK FRQVWDQW YHORFLW\ 0DUOERURXJK 0DUOERURXJK DQG 0DUOERURXJK DQG 5R\ f KDV FDOFXODWHG +D SURILOHV XVLQJ D OHYHO K\GURJHQ DWRP +H XVHG D PRGHO IRU WKH GLVWULEXn WLRQ DQG PRWLRQ RI WKH PDWHULDO GXH WR /LPEHU f ZKLFK ZLOO EH GLVFXVVHG EHORZ +\GURG\QDPLFDO 0RGHOV IRU WKH )ORZ 'HWHUPLQDWLRQ RI WKH GHQVLW\ DQG YHORFLW\ ILHOG WKURXJKRXW WKH GLVN LV D K\GURG\QDPLFDO SUREOHP ,Q WKH ILUVW VXFK VWXG\ /LPEHU fD VWHDG\VWDWH D[LDOO\ V\PPHWULF GLVN ZDV H[DPLQHG IRU D SDUDPHWHUL]HG IRUP RI

PAGE 28

FLUFXODU YHORFLW\ ODZ DQG LQ WKH DEVHQFH RI DQ\ UDGLDO YHORFLW\ /LPEHU WUHDWHG LVRWKHUPDO HQYHORSHV SRO\WURSHV DQG HQYHORSHV ZKLFK KDYH D VSHFLILHG WHPSHUDWXUH ODZ )RU HDFK FDVH KH ZDV DEOH WR FDOFXODWH WKH GHQVLW\ DW DQ\ SRLQW LQ WKH HQYHORSH /DWHU /LPEHU f VWXGLHG WKH PRUH JHQHUDO FDVH LQ ZKLFK D UDGLDO YHORFLW\ FRPSRQHQW ZDV SUHVHQW 6SHFWUDO 9DULDWLRQV DQG 7KHLU 3RVVLEOH &DXVHV $ URXJK GLYLVLRQ RI VSHFWUDO FKDQJHV LQ %H VWDUV ZLOO EH PDGH KHUH WR IDFLOLWDWH WKH GLVFXVVLRQ WKH GLYLVLRQV DUH f ORQJ WHUP FKDQJHV WLPH VFDOH ORQJHU WKDQ \HDUVf ZKLFK FKDUDFWHUL]H WKH DSSHDUDQFH DQG GLVDSSHDUDQFH RI VKHOO RU HPLVVLRQ VSHFWUD DV LQ 3OHLRQH f PHGLXP WHUP FKDQJHV WR \HDUVf SDUWLFXODUO\ 95 FKDQJHV ZLWK WLPH VFDOHV LQ WKLV WLPH UDQJH f VKRUW WHUP FKDQJHV D \HDU RU OHVVf 7KHUH LV D FHUWDLQ LQHYLWDELOLW\ WR WKH ORQJ WHUP FKDQJHV &UDPSLQ DQG +R\OH f DUJXHG WKDW DQ\ LQLWLDO PDJQHWLF ILHOGV SUHVHQW DW WKH IRUPDWLRQ RI WKH GLVN ZRXOG UHQGHU LW XQVWDEOH LQ \HDUV RU OHVV /LPEHU KDV VXJJHVWHG WKDW WKHUH PD\ EH D QRQXQLIRUP WUDQVIHU RI PRPHQWXP IURP UDSLGO\ URWDWLQJ LQQHU UHJLRQV RI WKH FHQWUDO

PAGE 29

VWDU RXW WR WKH SKRWRVSKHULF UHJLRQV UHVXOWLQJ LQ WXUQ LQ QRQXQLIRUP PDVV ORVV /LPEHU f DQDO\]HG WKH VKHOO SKDVH RI 3OHLRQH LQ WHUPV RI D WLPHGHSHQGHQW PDVV IOX[ IURP WKH FHQWUDO VWDUnV HTXDWRULDO UHJLRQV WR WKH GLVN +H IRXQG JRRG DJUHHPHQW ZLWK WKH REVHUYDWLRQV IRU D IOX[ ZKRVH WLPH GHSHQGHQFH VKRZHG D VORZ LQFUHDVH WR PD[LPXP IROORZHG E\ D VWHHS GURS WR ]HUR 0F/DXJKOLQ f UHYLHZHG ZKDW DUH FDOOHG KHUH PHGLXP WHUP YDULDWLRQV +H REVHUYHG WKDW WKH PRVW GLIILFXOW SUREOHP FRQFHUQHG 95 FKDQJHV 0F/DXJKOLQ f GLVFXVVHG WKH PDMRU DWWHPSWV WR H[SODLQ WKH TXDVLSHULRGLF 95 FKDQJHV RI WKH VRUW VHHQ LQ LU $TU DQG KH FRQFOXGHG WKDW RQO\ D VXJJHVWLRQ FUHGLWHG WR 2WWR 6WUXYH ZDV FRQVLVWHQW ZLWK WKH REVHUYDWLRQV 6WUXYHnV VXJJHVWLRQ KDG EHHQ WKDW WKH ULQJ RU GLVN ZDV HOOLSWLFDO LQ VKDSH DQG WKDW WKH 95 YDULDWLRQV UHVXOW IURP D OLQH RI DSVLGHV URWDWLRQ RI WKH GLVN 7KHUH KDV EHHQ QR UHFHQW H[DPLQDWLRQ RI WKLV LGHD 7KH PRVW FRPSUHKHQVLYH ZRUN RQ VKRUW WHUP YDULDWLRQV LV +XDQJnV Df +XDQJ VKRZHG WKDW DQ DV\PPHWU\ LQ WKH GLVN ZLOO SURGXFH VSHFWUDO YDULDWLRQV REVHUYDEOH DIWHU WLPH VSDQV DV VKRUW DV D IUDFWLRQ RI D GD\ WKHVH YDULDWLRQV FDQ SHUVLVW IRU DV PXFK DV D \HDU EHIRUH WKH GLVNnV GLIIHUHQWLDO URWDWLRQ GHVWUR\V WKH DV\PPHWU\ +XWFKLQJV f VXJJHVWHG WKDW SKRWRVSKHULF IOXFWXDWLRQV RU GXPSLQHVV LQ WKH GLVN PD\ EH UHVSRQVLEOH IRU VKRUW WLPH FKDQJHV 0RGLILFDWLRQV RI WKH 6WUXYH 0RGHO ,W QRZ DSSHDUV WKDW VHYHUDO DVSHFWV RI 6WUXYHnV

PAGE 30

RULJLQDO PRGHO PXVW EH PRGLILHG )LUVW 6WUXYHnV PRGHO UHTXLUHV WKDW WKH HTXDWRULDO UHJLRQV RI WKH SKRWRVSKHUH EH URWDWLQJ DW WKH EUHDNXS YHORFLW\ 2EVHUYDWLRQV VXJJHVW WKDW %H VWDUVn URWDWLRQDO YHORFLWLHV OLH EHORZ WKH EUHDNXS OLPLW +XDQJ Ef UHFHQWO\ DUJXHG WKDW WKH HMHFWLRQ RI PDWWHU LV GXH WR ERWK UDSLG URWDWLRQ DQG D WHPSHUDWXUH GHSHQGHQW LQVWDELOLW\ +XDQJ IHHOV WKDW WKH WHPSHUDWXUH GHSHQGHQW LQVWDELOLW\ PD\ EH WKH PHFKDQLVP VWXGLHG E\ /XF\ DQG 6RORPRQ f LQ WKHLU ZRUN RQ VWHOODU ZLQGV LQ HDUO\ W\SH JLDQW DWPRVSKHUHV WKLV ZRUN DUJXHG IRU D PDVV RXWIORZ GXH WR UDGLDWLRQ SUHVVXUH IURP PLG89 UHVRQDQFHV LQ VLOLFRQ FDUERQ QLWURJHQ DQG VXOIXU WKHVH EHLQJ LQ KLJK LRQL]DWLRQ VWDWHV /LPEHU DQG 0DUOERURXJK f DV SDUW RI D JHQHUDO GLVFXVVLRQ RI WKH SK\VLFDO SURFHVVHV DW ZRUN LQ %H VWHOODU HQYHORSHV UHDQDO\]HG WKH ZRUN RI 6WUXYH DQG :XUP f LQ RUGHU WR VKRZ WKDW WKH YHORFLW\ ODZ 9L_LFARFA ffRF! ZDV LQ EHWWHU DFFRUG ZLWK WKH GDWD WKDQ WKH FLUFXODU YHORFLW\ ODZ RULJLQDOO\ SURSRVHG E\ 6WUXYH 0DMRU 7RSLFV RI WKLV :RUN $ SULQFLSDO FRQWHQWLRQ RI WKLV GLVVHUWDWLRQ LV WKDW WKH H[LVWLQJ VWHDG\VWDWH D[LDOO\V\PPHWULF K\GURG\QDPLFDO

PAGE 31

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f GLVFXVVHV FDOFXODWLRQV EDVHG RQ WKH VROXWLRQ WR WKHVH HTXDWLRQV ,Q &KDSWHU 9, WKH HQHUJ\ HTXDWLRQ LV FRQVLGHUHG DQG &KDSWHU 9,, DIWHU D GLVFXVVLRQ RI WKH UHODWLRQ RI WKH UHVXOWV RI WKLV ZRUN WR WXUEXOHQFH DQG VWDELOLW\ DUJXPHQWV VXPPDUL]HV WKH ZRUN

PAGE 32

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n WLRQ RI WKH WKUHH PDLQ VHFWLRQV UHSUHVHQWV WKH RUGHU LQ ZKLFK WKHVH LGHDV ZHUH GHYHORSHG E\ /LPEHU DQG 0DUOERURXJK /LPEHU /LPEHU DQG 0DUOERURXJK f 7KH +\GURG\QDPLFDO (TXDWLRQV DQG 1RWDWLRQ 7KH K\GURG\QDPLFDO HTXDWLRQV XVHG KHUH DUH WKH VLPSOHVW

PAGE 33

RQHV WKDW FRXOG EH FKRVHQ WKH\ DUH (XOHUnV HTXDWLRQ DQG WKH FRQWLQXLW\ HTXDWLRQ ,Q GLPHQVLRQHG FRRUGLQDWHV WKH IRUPHU LV Y SF ZU Y DQG WKH ODWWHU WDNHV WKH IRUP W 9 S Y f .F F 7KH MXVWLILFDWLRQ RI WKH XVH RI (XOHUnV HTXDWLRQ ZLOO EH D SULQFLSDO FRQFHUQ RI WKH QH[W FKDSWHU 7KH FRRUGLQDWH V\VWHP XVHG KHUH LV VKRZQ LQ )LJXUH 7KH WKUHH FRRUGLQDWHV Z -! ]f PD\ DSSHDU ZLWK RU ZLWKRXW WKH VXEVFULSW F 7KH SUHVHQFH RI WKH VXEVFULSW VLJQLILHV WKDW WKH FRRUGLQDWH LQ TXHVWLRQ LV GLPHQVLRQHG LQ FJV XQLWV ZLWKRXW WKH VXEVFULSW LW LV D GLPHQVLRQOHVV YDULDEOH LQ WKH V\VWHP RI GLPHQVLRQOHVV FRRUGLQDWHV ZKLFK LV GHYHORSHG LQ WKH QH[W FKDSWHU $ VLPLODU SURFHGXUH LV \ IROORZHG IRU WKH IORZ YDULDEOHV DQG WLPH YHORFLW\ Y GHQVLW\ S SUHVVXUH 3 DQG WLPH W 7KH HTXDWLRQV DOUHDG\ SUHVHQWHG LQ WKH ILUVW FKDSWHU DUH FRQVLVWHQW ZLWK WKLV SUDFWLFH ,Q FDVHV ZKHUH QRQFJV XQLWV DUH XVHG WKHVH XQLWV ZLOO EH H[SOLFLWO\ VWDWHG ,Q WKH ERG\ RI WKLV ZRUN V\PEROV ZLOO RQ RFFDVLRQ XQGHUJR VRPH FKDQJH LQ GHILQLWLRQ IRU LQVWDQFH D YDULDEOH V\PERO PD\ EH UHTXLUHG WR DEVRUE DQ LQWHJUDWLQJ IDFWRU 7KLV

PAGE 34

)LJXUH 7KH F\OLQGULFDO FRRUGLQDWH V\VWHP LQ XVH KHUH

PAGE 35

LV GRQH RQO\ ZKHQ QHFHVVDU\ WR SUHYHQW DQ XQGXH SUROLIHUDWLRQ RI FORVHO\ UHODWHG YDULDEOH QDPHV :KHQ TXHVWLRQV DULVH DERXW WKH PHDQLQJ RI D V\PERO WKH EHVW UHFRXUVH LV D FKHFN RI eKH .H\ WR 6\PEROV (DFK FKDQJH LQ XVDJH LV QRWHG WKHUH DV LV WKH ILUVW SDJH RQ ZKLFK WKH QHZ XVDJH RFFXUV 6WDWLF 6ROXWLRQV /LPEHU f DWWHPSWHG WKH ILUVW K\GURG\QDPLFDO VWXG\ RI WKH PDWHULDO RXWVLGH UDSLGO\ URWDWLQJ VWDUV +H WRRN WKH FHQWUDO VWDU WR EH URWDWLQJ DW WKH EUHDNXS YHORFLW\ DQG PDGH WKH IROORZLQJ DVVXPSWLRQV f YLVFRXV DQG PDJQHWLF WHUPV LQ WKH HTXDWLRQ RI PRWLRQ FDQ EH LJQRUHG f UDGLDWLRQ IRUFHV DUH HLWKHU QHJOLJLEOH RU LQFOXGDEOH WKURXJK WKH XVH RI DQ DSSURSULDWHO\ UHGXFHG VWHOODU PDVV LQ WKH JUDYLWDWLRQDO SRWHQWLDO f GLVN VHOIJUDYLWDWLRQDO HIIHFWV DUH QHJOLJLEOH f VWHDG\VWDWH FRQGLWLRQV SUHYDLO f RQO\ D[LDOO\V\PPHWULF IORZV DUH LQFOXGHG f WKH = FRPSRQHQW RI WKH IOXLG YHORFLW\ SHUSHQGLFXODU WR WKH HTXDWRULDO SODQHf LV ]HUR f WKH UDGLDO FRPSRQHQW RI WKH IOXLG YHORFLW\ LV ]HUR WKH FLUFXODU YHORFLW\ KDV WKH SDUDPHWHUL]HG IRUP 9 bF0R! SF D WR f

PAGE 36

ZKHUH WKH QHZ V\PERO D LV D SDUDPHWHU ZKRVH YDOXH OLHV EHWZHHQ DQG f WKH GHQVLW\ LV DQ H[SOLFLW IXQFWLRQ RI WKH SUHVVXUH DORQH 7KH ILUVW WKUHH FRQGLWLRQV GHWHUPLQH WKH QDWXUH RI WKH HTXDWLRQ RI PRWLRQ ZKLFK FDQ EH XVHGf§(XOHUnV HTXDWLRQ 7KH WKLUG DSSUR[LPDWLRQ QHJOHFW RI VHOIJUDYLWDWLRQDO WHUPV LV TXLWH UHDVRQDEOH 7KH YDOLGLW\ RI WKH ILUVW DQG VHFRQG FRQGLWLRQV ZLOO EH GLVFXVVHG LQ ODWHU VHFWLRQV 7KH XVH RI (XOHUnV HTXDWLRQ DV WKH HTXDWLRQ RI PRWLRQ LV FRPPRQ WR DOO K\GURG\QDPLFDO VWXGLHV RI %H VWDU HQYHORSHV 7KH MXVWLILFDWLRQ IRU WKH XVH RI (XOHUnV HTXDWLRQ RU LQGHHG DQ\ K\GURG\QDPLFDO HTXDWLRQ XQGHU GLVN SK\VLFDO FRQGLWLRQV ZLOO EH WKH VXEMHFW RI WKH QH[W FKDSWHU 7KH QH[W JURXS RI DSSUR[LPDWLRQV f WKURXJK f UHGXFHV WKH HTXDWLRQ RI PRWLRQ WR WKH IRUP ZKHUH YA LV WKH FLUFXODU YHORFLW\ FRPSRQHQW RI WKH YHORFn LW\f DQG LX LV WKH XQLW YHFWRU IRU WKH UDGLDO GLUHFWLRQ LQ F\OLQGULFDO FRRUGLQDWHV &RQGLWLRQV f WKURXJK f DOVR UHVXOW LQ VROXWLRQV ZKLFK ZLOO LGHQWLFDOO\ VDWLVI\ WKH FRQWLQXLW\ HTXDWLRQ WKH VHFRQG RI WKH WKUHH HTXDWLRQV QHHGHG WR VSHFLI\ FRPSOHWHO\ WKH SUREOHP 6LQFH YLRODWLRQ RI f ZLOO VKRZ XS REVHUYDWLRQDOO\ DV YDULDWLRQV ZLWK WLPH ERWK f DQG f DUH JRRG DSSUR[LPDWLRQV WR WKH H[WHQW WKDW

PAGE 37

WKH\ GHVFULEH SKHQRPHQD ZLWK HLWKHU QR WLPH GHSHQGHQFH RU ORQJ WHUP WLPH GHSHQGHQFH 1HDU WKH HTXDWRULDO SODQH FRQGLWLRQ f LV D JRRG DSSUR[LPDWLRQ ZKLOH FRQGLWLRQ f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f§VROXWLRQV LQ ZKLFK WKH UDGLDO

PAGE 38

YHORFLW\ LV ]HURf§DUH VWULFWO\ VSHDNLQJ QRQSK\VLFDO IRU WKH VDPH UHDVRQ WKDW VWDWLF VROXWLRQV DUH QRQSK\VLFDO IRU WKH VRODU ZLQG 3DUNHU f /LPEHU H[WHQGHG KLV DQDO\VLV WR QRQVWDWLF VROXWLRQV /LPEHU f 7KUHH DVVXPSWLRQV RI WKH DSSURDFK SUHVHQWHG LQ WKH SUHFHGLQJ VHFWLRQ DUH PRGLILHG WKH\ DUH f f DQG f 7KHVH WKUHH DUH UHSODFHG E\ WKH IROORZLQJ OHVV UHVWULFWLYH FRQGLWLRQV f RQO\ IORZ QHDU WKH SODQH LV FRQVLGHUHG f WKH UDGLDO FURVV VHFWLRQ RI IORZ PXVW EH VSHFLILHG LQ SDUDPHWHUL]HG IRUPf f WKH WHPSHUDWXUH GLVWULEXWLRQ WKURXJKRXW WKH GLVN LV NQRZQ $V LQ WKH VWDWLF FDVH WKH LGHDO JDV ODZ LV XVHG (XOHUnV HTXDWLRQ WDNHV WKH IRUP ZKHUH WKH QHZ V\PERO YA LV WKH UDGLDO FRPSRQHQW RI WKH YHORFLW\ 7KH HTXDWLRQ RI FRQWLQXLW\ DSSHDUV LQ WKH LQWHn JUDWHG IRUP YZXfSDMf $Zf bffSff $Zf ZKHUH $WRf LV WKH UDGLDO FRPSRQHQW RI WKH FURVVVHFWLRQDO IORZ 7KLV IRUP RI WKH FRQWLQXLW\ HTXDWLRQ LV VLPLODU WR WKDW VHHQ LQ VRODU ZLQG WKHRU\ 7KH TXDQWLW\ $mf$WR4f LV DQDORJRXV WR WKH WHUP )UF! V

PAGE 39

LQ WKH HOHPHQWDU\ VRODU ZLQG WKHRU\ 3DUNHU f ZKHUH E LV D SDUDPHWHU XVXDOO\ JLYHQ D YDOXH QHDU f /LPEHU FKRVH LQVWHDG D RQHSDUDPHWHU IDPLO\ RI FURVVVHFWLRQDO IORZ WHUPV f 8! $fF! 6 cS f RF RF ZKHUH S LV D SDUDPHWHU 7KH PHULGLRQDO SURMHFWLRQV RI WKH VWUHDP OLQHV DUH VWUDLJKW OLQHV UXQQLQJ IURP D SRLQW fF SXRF LQ WKH HTXDWRULDO SODQH DQG VORSLQJ VORZO\ DZD\ IURP WKH HTXDWRULDO SODQH /LPEHU ZHQW RQ WR HVWDEOLVK WKH W\SHV RI VROXWLRQV WKDW DUH QRZ SHUPLVVLEOH 9DOLG VROXWLRQV PXVW EH VXEVRQLF EHORZ WKH VWDUnV VXUIDFH DQG DSSURDFK LQWHUVWHOODU YDOXHV DW JUHDW GLVWDQFHV IURP WKH VWDU 7KH RQO\ VROXWLRQV ZKLFK VDWLVI\ WKHVH WZR ERXQGDU\ FRQGLWLRQV DUH WKRVH ZKRVH UDGLDO YHORFLW\ f ILUVW GHFUHDVHV ZLWK LQFUHDVLQJ Z f UHDFKHV D PLQLPXP DV Z FRQWLQXHV WR JURZ f WKHQ LQFUHDVHV DV Z FRQWLQXHV LWV LQFUHDVH f EHFRPHV VXSHUVRQLF DW VXIILFLHQW GLVWDQFH IURP WKH FHQWUDO VWDU /LPEHU H[DPLQHG WKH UDQJH RI YDOLGLW\ RI KLV HDUOLHU ZRUN WKH FDOFXODWLRQV IRU VWDWLF HQYHORSHV LQ WKH FRQWH[W

PAGE 40

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n VLWLYH WR FKDQJHV LQ S WKH SDUDPHWHU ZKLFK DSSHDUV LQ WKH FURVVVHFWLRQDO WHUP /LPEHU DQG 0DUOERURXJK f H[DPLQHG WKH SK\VLFDO FRQGLWLRQV DQG REVHUYDWLRQDO HYLGHQFH IRU WKH FLUFXODU YHORFLW\n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

PAGE 41

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f 4XDVLVWDWLF PRWLRQ RI D VPDOO IOXLG HOHPHQW DZD\ IURP WKH VWDU ZLWK FHQWULIXJDO IRUFHV SUHVHQW UHTXLUHV WKDW *0 V DW DQ\ SRLQW LQ WKH GLVN KHQFH 9f A f

PAGE 42

7KHUH DUH IRXU GLUHFWVXSSRUW PHFKDQLVPV DYDLODEOH WKH\ DUH f WKHUPDO VXSSRUW f WXUEXOHQW VXSSRUW f UDGLDWLYH VXSSRUW f PDJQHWLF VXSSRUW /LPEHU ZDV DEOH WR VKRZ WKDW WKH ILUVW SRVVLELOLW\ WKHUPDO VXSSRUW UHTXLUHV WHPSHUDWXUHV DQG GHQVLWLHV WKURXJKRXW WKH GLVN WKDW DUH WRWDOO\ DW YDULDQFH ZLWK SUHVHQW NQRZOHGJH RI WKH YDOXHV RI WKHVH TXDQWLWLHV 7XUEXOHQW VXSSRUW UHTXLUHV D KLJKO\ WXUEXOHQW IORZ ZKRVH HIIHFWV RQ VSHFWUDO IHDWXUHV VKRXOG EH REVHUYDEOH EXW KDYH QRW EHHQ UHSRUWHG /LPEHU DQG 0DUOERURXJK FRQVLGHUHG WZR SRVVLEOH UDGLDWLYH VXSSRUW PHFKDQLVPV HOHFWURQ VFDWWHULQJ DQG SKRWRn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

PAGE 43

ZDV DEVRUEHG DQG EDFNHPLWWHG 2Q WKH EDVLV RI ERWK WKLV DVVXPSWLRQ DQG RWKHU FRQVLGHUDWLRQV WKHVH DXWKRUV FDOFXODWHG DQ XSSHU OLPLW IRU VXSSRUW GXH WR SKRWRLRQL]DWLRQ 7KLV YDOXH ZDV IDU OHVV WKDQ WKH YDOXH QHHGHG WR SURYLGH UDGLDWLYH VXSSRUW :KLOH ERWK WKHVH FDOFXODWLRQV DUH GHSHQGHQW RQ HVWLPDWHV RI W\SLFDO GLVN YDOXHV IRU WKH SK\VLFDO SDUDPHWHUV WKH\ DUH VR PXFK VPDOOHU WKDQ WKH YDOXH QHHGHG IRU GLUHFW VXSSRUW WKDW WKH\ PD\ EH RPLWWHG IURP IXUWKHU GLVFXVVLRQ XQWLO VXFK WLPH DV WKH EHVW HVWLPDWHV RI WKHVH GLVN TXDQWLWLHV FKDQJH GUDPDWLFDOO\ )LQDOO\ UDGLDO VXSSRUW IURP PDJQHWLF IRUFHV ZDV FRQVLGHUHG 7KH VDPH LQYHVWLJDWRUV XVHG HOHPHQWDU\ FRQVLGHUDn WLRQV WR VKRZ WKDW WKH ILHOG VWUHQJWK + UHTXLUHG ZDV + JDXVV 7KH\ WKHQ VKRZHG WKDW ILHOGV RI WKLV VRUW LQ WKH GLVN ZRXOG GLVUXSW LW RQ D WLPH VFDOH RI GD\V 7KH REVHUYDWLRQDO UHVXOWV RI WKH GLVUXSWLRQ RI WKH GLVN ORVV RI HPLVVLRQ OLQHV RU VKHOO DEVRUSWLRQ OLQHV VKRZ PXFK ORQJHU WLPH VFDOHV 7KXV WKH\ FRQFOXGHG WKDW PDJQHWLF ILHOGV RI WKH UHTXLUHG RUGHU DUH QRW SUHVHQW LQ WKH GLVN $QDORJRXVO\ WKHUH DUH IRXU PHFKDQLVPV ZKLFK FDQ DFW WR WUDQVIHU DQJXODU PRPHQWXP f WKHUPDO YLVFRVLW\ f WXUEXOHQW YLVFRVLW\ f UDGLDWLYH YLVFRVLW\ f PDJQHWLF YLVFRVLW\

PAGE 44

/LPEHU DQG 0DUOERURXJK XVHG WKH WHUP YLVFRXV WR GHQRWH DQ\ SKHQRPHQRQ ZKLFK DFWV WR WUDQVIHU DQJXODU PRPHQWXP 7KH\ IRXQG WKDW WZR RI WKHVH SRVVLELOLWLV WKH ILUVW DQG WKLUG ZHUH LQVLJQLILFDQW XQGHU GLVN FRQGLWLRQV +RZHYHU HLWKHU VPDOO VFDOH WXUEXOHQFH RU VPDOO JDXVVf PDJQHWLF ILHOGV FRXOG SURYLGH WKH QHHGHG DQJXODU PRPHQWXP WUDQVIHU 7KH\ FRQFOXGHG WKDW WKH REVHUYDWLRQDO HYLGHQFH ZDV FRQVLVWHQW ZLWK HLWKHU LQWHUSUHWDWLRQ ,W LV LPSRUWDQW KHUH WR QRWH WKDW WKH DPRXQW RI WXUEXOHQFH RU WKH PDJQHWLF ILHOG VWUHQJWK UHTXLUHG IRU DQJXODU PRPHQWXP WUDQVIHU LV IDU WRR VPDOO WR SURYLGH DQ\ GLUHFW VXSSRUW /LPEHU DQG 0DUOERURXJK QRWHG WKDW WKH ZRUN RI +\QHN DQG 6WUXYH f UHSUHVHQWHG WKH RQO\ DWWHPSW WR GUDZ TXDQWLWDWLYH LQIHUHQFHV IURP REVHUYDWLRQV DERXW WKH IRUP RI WKH FLUFXODU YHORFLW\ ODZ DQG WKDW WKLV DWWHPSW KDG FRQVLGHUHG RQO\ WKH WR EHKDYLRU 7KH GDWD ZHUH UHDQDO\]HG WR VHH LI WKH OZ IRUP JDYH D PRUH FRQVLVWHQW LQWHUSUHWDWLRQ RI WKH GDWD 7KH FRQFOXVLRQ RI WKLV UHDQDO\VLV RI WKH GDWD ZDV WKDW WKH IRUP RI WKH FLUFXODU YHORFLW\ ODZ DVVRFLDWHG ZLWK FHQWULIXJDO VXSSRUW ZDV LQ EHWWHU DFFRUG ZLWK WKH GDWD WKDQ WKH ROG 6WUXYH IRUP )LQDO &RPPHQWV 6WHDG\VW£WH VROXWLRQV KDYH EHHQ DSSOLHG WR WKH VKHOO SKDVH RI 3OHLRQH /LPEHU f E\ DOORZLQJ WKH HQYHORSH WR EH DW DQ\ PRPHQW YHU\ QHDU VWHDG\VWDWH 7KLV ZLOO RI FRXUVH ZRUN RQO\ IRU ORQJ WHUP SKHQRPHQD /LPEHU

PAGE 45

VXJJHVWHG WKDW WKH HQG RI WKH VKHOO SKDVH UHSUHVHQWHG WKH UHVXOW RI LQFUHDVHG PDWWHU RXWIORZ QRW PDWFKHG E\ VXIILFLHQW HQHUJ\ IORZ LQWR WKH GLVN KHQFHD FROODSVH 7ZR UHVXOWV RI /LPEHUnV VWXGLHV DUH RI VXIILFLHQW LPSRUWDQFH WR WKH UHPDLQGHU RI WKLV ZRUN WR EHDU UHVWDWLQJ )LUVW RYHU PRVW RI WKH GLVN VWDWLF LVRWKHUPDO HQYHORSH FDOFXODWLRQV DJUHH ZHOO ZLWK WKH PRUH H[DFW WKHRU\ LQFOXGLQJ D UDGLDO YHORFLW\ 6HFRQG WKH FLUFXODU YHORFLW\ ODZ LV WKDW IRU FHQWULIXJDO HIIHFWV 7KH VWDWLF LVRWKHUPDO HQYHORSHV ZLWK D FLUFXODU YHORFLW\ ODZ FDQ EH SUHVHQWHG LQ FORVHG IRUP DQG GR QRW UHTXLUH WKH VSHFLILFDWLRQ RI D FURVVVHFWLRQDO IORZ SDUDPHWHU

PAGE 46

&+$37(5 ,,, $33/,&$7,21 2) +<'52'<1$0,&6 72 7+( (19(/23(6 2) %H 67$56 'LVN 'LPHQVLRQV 7DEOH JLYHV WKH YDOXHV XVHG KHUH IRU WKH SK\VLFDO DQG JHRPHWULFDO TXDQWLWLHV ZKLFK FKDUDFWHUL]H WKH VWDU DQG LWV HQYHORSH 7KH PDVV 0 f DQG VXUIDFH WHPSHUDWXUH 7 f 6 E RI WKH VWDU DUH WKRVH RI %2 GZDUI 7KH UDGLXV LV ODUJHU WKDQ RQH PLJKW H[SHFW IRU VXFK D VWDU DQG UHSUHVHQWV DFNQRZOn HGJLQJ WKH HYLGHQFH WKDW WKH VWDU LV VRPHZKDW HYROYHG DQG URWDWLRQDOO\ GLVWRUWHG 7KHVH TXDQWLWLHV DUH WKRVH FRPPRQO\ IRXQG LQ WKH OLWHUDWXUH 7KH HTXDWRULDO URWDWLRQDO YHORFLW\ 9 f UHSUHVHQWV D VOLJKW GHSDUWXUH LQ WKDW WKH URWDWLRQDO YHORFLW\ ZDV VHW DW MXVW VOLJKWO\ PRUH WKDQ KDOI WKH HTXDWRULDO EUHDNXS YHORFLW\ 7KH GLVN WHPSHUDWXUH ZKLFK LV WDNHQ WR EH WKH HOHFWURQ WHPSHUDWXUH 7H! WKH PHDQ PROHFXODU ZHLJKW \f RI WKH GLVN JDV DQG WKH DYHUDJH GLVN GHQVLW\ SAf DUH W\SLFDO RI YDOXHV IRXQG LQ WKH MRXUQDOV )URP WKH HPLVVLRQ DQG DEVRUSWLRQ IHDWXUHV DW K\GURJHQ DQG KHOLXP UHVRQDQFH OLQHV RQH FDQ FRQFOXGH WKDW LQ D W\SLFDO %H VWDU HQYHORSH D ODUJH SHUFHQWDJH RI WKH K\GURJHQ LV LRQL]HG ZKLOH PRVW RI WKH KHOLXP LV QRW

PAGE 47

7$%/( &KDUDFWHULVWLF &HQWUDO 6WDU DQG 'LVN 9DOXHV 4XDQWLW\ 6\PERO 9DOXH 6WHOODU 0DVV 0V A 6WHOODU (TXDWRULDO 5DGLXV 5 V 5 6XUIDFH 7HPSHUDWXUH 7 V r. (TXDWRULDO 5RWDWLRQDO 9HORFLW\ 9 V NPV 'LVN 7HPSHUDWXUH 7 H r. 0ROHFXODU :HLJKW \ $YHUDJH 'LVN 'HQVLW\ SG ,4

PAGE 48

7KHRUHWLFDO )RXQGDWLRQV RI +\GURG\QDPLFV 9DOLGLW\ RI WKH %ROW]PDQQ 7UDQVSRUW (TXDWLRQ 7KH PRVW VDWLVI\LQJ IRXQGDWLRQ IRU WKH K\GURG\QDPLFDO HTXDWLRQV LV WKH %ROWPDQQ WUDQVSRUW HTXDWLRQ 7KH K\GUR G\QDPLFDO HTXDWLRQV DUH WKH UHVXOW RI LQWHJUDOV RI WKH IRUP < I GY F RS & gF LV D FRQVHUYHG TXDQWLW\ 'Sr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

PAGE 49

7KH ORQJUDQJH &RXORPE IRUFHV SUHVHQW ZKHQ FKDUJHG SDUWLFOHV DUH FRQVWLWXHQWV RI WKH JDV JHQHUDOO\ UHVXOW LQ DQ LQILQLWH FRQWULEXWLRQ IURP WKH FROOLVLRQ WHUP 7KH XQGHUO\LQJ YLHZ RI WKH JDV PRWLRQ LV VXVSHFW DV VHYHUDO GLVWDQW FROOLVLRQV PD\ VLPXOWDQHRXVO\ LQWHUDFW ZLWK RQH DQRWKHU 7KHUH LV KRZHYHU RQH VHW RI FLUFXPVWDQFHV XQGHU ZKLFK WKH %ROW]PDQQ DSSURDFK LV VWLOO YDOLG =HOnGRYLFK DQG 5DL]HU f 7KH FRQGLWLRQV DUH f WKH DYHUDJH &RXORPE SRWHQWLDO HQHUJ\ DW WKH PHDQ VHSDUDWLRQ GLVWDQFH LV PXFK OHVV WKDQ WKH DYHUDJH WKHUPDO HQHUJ\ f WKH 'HE\H OHQJWK LV PXFK JUHDWHU WKDQ WKH PHDQ VHSDUDWLRQ GLVWDQFH $VVXPH WKDW WKH GLVN JDV LV PDGH RI HTXDO QXPEHUV RI HOHFWURQV DQG SURWRQV 7KH UDWLR RI WKH &RXORPE SRWHQWLDO HQHUJ\ DW PHDQ VHSDUDWLRQ WR DYHUDJH WKHUPDO HQHUJ\ IRU VXFK D JDV XQGHU GLVN SK\VLFDO FRQGLWLRQV JLYHV =Hf [ ZKHUH 1H H N DQG = DUH WKH HOHFWURQ QXPEHU GHQVLW\ WKH HOHFWURQ FKDUJH WKH %ROW]PDQQ FRQVWDQW DQG WKH DYHUDJH LRQLF FKDUJH 7KH UDWLR RI WKH 'HE\H OHQJWK WR DYHUDJH VHSDUDWLRQ LV 7 1 f k k f§Ac H

PAGE 50

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f§WKDW EHWZHHQ LRQV DQG HOHFWURQV DQG WKDW EHWZHHQ HOHFWURQV DQG QHXWUDO DWRPV

PAGE 51

)RU SXUSRVHV RI VWXG\LQJ WKH QDWXUH RI WKH GLVN JDV WKDW JDV ZLOO EH DSSUR[LPDWHGE\ D WKUHHVSHFLHV JDV FRQWDLQLQJ HOHFWURQ LRQL]HG K\GURJHQ DQG QHXWUDO KHOLXP IXUWKHU WKH FKDUDFWHULVWLF WLPH [ f IRU WKH V\VWHP ZLOO EH WDNHQ 9 WR EH WW5 9 V V 7KH UDWLR RI WKH HOHFWURQSURWRQ UHOD[DWLRQ WLPH [ f WR WKH FKDUDFWHULVWLF WLPH [ f LV f§S F [ [ 9 7 HS B V H 7F WW5 1 $Q$ V H G ZKHUH LV WKH UHGXFHG 'HE\H OHQJWK 6SLW]HU f 7KH UDWLR RI WKH HOHFWURQKHOLXP UHOD[DWLRQ WLPH [ f WR HQH WKH FKDUDFWHULVWLF WLPH LV DSSUR[LPDWHO\ 7HKH [ F 9 V f5V1KH 7H£ mKH =HOnGRYLFK DQG 5DL]HU f ZKHUH 4AH LV WKH KHOLXP HOHFWURQ FROOLVLRQ FURVVVHFWLRQ IRU W\SLFDO GLVN FRQGLWLRQV DQG 1AH LV WKH KHOLXP QXPEHU GHQVLW\ ,I 7IL LV r. f§ 4AH LV DSSUR[LPDWHO\ [ FP )RU WKH W\SLFDO SK\VLFDO VWDWH RI WKH GLVN WKH UDWLR RI WKH HOHFWURQSURWRQ UHOD[DWLRQ WLPH 7HBSf WR WKH

PAGE 52

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

PAGE 53

JLYH DQ HTXDWLRQ RI FRQWLQXLW\ DQ HTXDWLRQ RI PRWLRQ DQG DQ HQHUJ\ HTXDWLRQ LQ WKDW RUGHU +RZHYHU TXDQWLWLHV VXFK DV WKH KHDW IOX[ DQG WKH GHIRUPDWLRQ WHQVRU ZKLFK DSSHDU LQ WKH HTXDWLRQV DUH GHILQHG E\ LQWHJUDOV WKDW FRQWDLQ WKH GLVWULEXWLRQ IXQFWLRQ ,QGHHG WKH TXDQWLWLHV WKDW DUH LGHQWLILHG DV IORZ YDULDEOHV WKH SUHVVXUH GHQVLW\ DQG IOXLG YHORFLW\f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nV HTXDWLRQ KDV WKH IRUP F f 9Y F 3 F 93 F F ZKHUH W LV WKH IRUFH SHU XQLW PDVV 7KH FRQWLQXLW\ HTXDWLRQ LV

PAGE 54

7KH QH[W DSSUR[LPDWLRQ OHDGV WR WKH K\GURG\QDPLFDO HTXDWLRQV IRU D YLVFRXV JDV ,Q WKLV FDVH WKH HTXDWLRQ RI FRQWLQXLW\ LV XQFKDQJHG EXW D QHZ HTXDWLRQ RI PRWLRQ FDOOHG WKH 1DYLHU6WRNHV HTXDWLRQ LV IRUPHG 7KH 1DYLHU6WRNHV HTXDWLRQ ZLOO EH H[DPLQHG EHORZ WR DVFHUWDLQ WKH H[WHQW WR ZKLFK (XOHUnV HTXDWLRQ LV D JRRG DSSUR[LPDWLRQ IRU D IOXLG XQGHU GLVN SK\VLFDO FRQGLWLRQV 1DYLHU6WRNHV (TXDWLRQV DQG 6LPLODULW\ 1XPEHUV 7KH 1DYLHU6WRNHV HTXDWLRQ LQ WKH SUHVHQFH RI D JUDYLWDWLRQDO SRWHQWLDO LV Y 8 Qf 99 f Y f *0 9 7KH TXDQWLWLHV Q DQG e DUH WKH ILUVW DQG VHFRQG YLFRVLW\ FRHIILFLHQWV UHVSHFWLYHO\ 7KH TXDQWLWLHV Q DQG % DUH ERWK DOZD\V SRVLWLYH 7KH VHFRQG YLVFRVLW\ FRHIILFLHQW UHSUHVHQWV HIIHFWV WKDW RFFXU DW KLJK GHQVLW\ RU ZKHQ VSHFLHV ZLWK VORZO\ H[FLWHG GHJUHHV RI IUHHGRP DUH SUHVHQW ,W LV LQFOXGHG RQO\ IRU FRPSOHWHQHVV (YHQ LI WKHVH HIIHFWV ZHUH SUHVHQW DV ORQJ DV Q e DOO WKH DUJXPHQWV SUHVHQWHG EHORZ DUH YDOLG

PAGE 55

7KH VHFRQG WHUP RQ WKH OHIWKDQG VLGH LQ WKH 1DYLHU 6WRNHV HTXDWLRQ LV FDOOHG WKH LQHUWLD WHUP WKH VHFRQG DQG WKLUG WHUPV RQ WKH ULJKW DUH WKH YLVFRXV WHUPV 7KH ODVW WHUP FRQWDLQV JUDYLWDWLRQDO HIIHFWV /HW 5 EH D GLPHQVLRQ FKDUDFWHULVWLF RI WKH R ERXQGDU\ VXUIDFH 9T D W\SLFDO YDOXH RI WKH IOXLG YHORFLW\ QR D UHSUHVHQWDWLYH GHQVLW\ 7KHVH FKDUDFWHULVWLF YDOXHV DUH FKRVHQ VXFK WKDW WKH DVVRFLDWHG SK\VLFDO TXDQWLWLHV U Y S f YDU\ IURP D ODUJH IUDFWLRQ WR VHYHUDO WLPHV WKH FKDUDFWHULVWLF YDOXHV WKDW LV HDFK RI WKH YDOXHV UHSUHVHQWV WKH RUGHU RI PDJQLWXGH RI WKH TXDQWLW\ RI ZKLFK LW LV D FKDUDFWHULVWLF &KDUDFWHULVWLF YDOXHV IRU WKH SUHVVXUH DQG WLPH ZKLFK UHVXOW IURP WKHVH FKRLFHV DUH DQG UHVSHFWLYHO\ 1RWH WKDW WKH FKDUDFWHULVWLF QXPEHUV DUH GLPHQVLRQDO 7KH FKDUDFWHULVWLF QXPEHUV FDQ EH XVHG WR VHW XS D V\VWHP RI GLPHQVLRQOHVV YDULDEOHV GHILQHG E\ WKH IROORZLQJ UHODWLRQV f U F I f Y F 9 9 R I

PAGE 56

f 3F 9 n f 3A 3 3 F R f W 7 W F R 1RWH WKDW WKH VXEVFULSW ]HUR TXDQWLWLHV DUH GLPHQVLRQHG QXPEHUV 7KHVH UHODWLRQV FDQ EH XVHG WR UHZULWH WKH 1DYLHU6WRNHV HTXDWLRQV LQ GLPHQVLRQOHVV FRRUGLQDWHV 7KH UHVXOW LV VKRZQ EHORZ R Y 5 W 9 R 5 Y f 9Y Y r1 R S *0 V E f Q9R Y I Qf Y f 9f 3 Q WG 2 S Q U R R R R 'LYLVLRQ E\ 9=5A ‘r 2 R \LHOGV Y W Y *0 V 5 9 R R 5 Y Q R R R D Qf 5YQ R R R 99 ‘ Yf 3 6LQFH WKH FKDUDFWHULVWLF SDUDPHWHUV UHSUHVHQW WKH! RUGHU RI PDJQLWXGH RI WKHLU UHVSHFWLYH YDULDEOHV WKH RUGHU RI PDJQLWXGH RI WKH UDWLR RI DQ\ WZR WHUPV LV WKH UDWLR RI WKHLU FRHIILFLHQWV 7KLV VWDWHPHQW LV KRZHYHU

PAGE 57

RSHQ WR TXHVWLRQ QHDU WKH IORZ ERXQGDU\ 7KH IORZ YDULDEOHV PD\ FKDQJH UDSLGO\ RYHU VKRUW GLVWDQFHV QHDU WKH ERXQGDU\ VXUIDFH WKHUHE\ FDXVLQJ WHUPV LQYROYLQJ WKHLU JUDGLHQWV WR EH TXLWH ODUJH 7KHVH FRHIILFLHQW UDWLRV WDNHQ WRJHWKHU TXDOLWDWLYHO\ GHVFULEH WKH IORZ +LVWRULFDOO\ WKH\ KDYH EHHQ FDOOHG VLPLODULW\ QXPEHUV DQG JLYHQ WKH IROORZLQJ QDPHV 5H\QROGV 1XPEHU 7KLV GLPHQVLRQOHVV QXPEHU ODEHOHG 51 KHUH LV WKH LQYHUVH RI WKH UDWLR RI WKH YLVFRXV WHUPV WR WKH LQHUWLDO WHUPV B 51 5 9 Q R R R )URXG 1XPEHU 7KLV YDOXH UHIHUUHG WR V\PEROLFDOO\ DV )1 LV )1 5R*0Uf ,WV LQYHUVH LV WKH UDWLR RI WKH JUDYLWDWLRQDO WHUP WR WKH LQHUWLDO WHUP 6WURXKDO 1XPEHU &DOOHG 61 LW LV JLYHQ E\ WKH H[SUHVVLRQ 9 7 R R 7KH LQYHUVH RI WKLV TXDQWLW\ LV WKH UDWLR RI WKH WLPH GHULYDWLYH RI WKH YHORFLW\ WR WKH LQHUWLDO WHUP

PAGE 58

,PSRUWDQFH RI 9LVFRXV 7HUP LQ WKH (TXDWLRQ RI 0RWLRQ 7KH UHODWLYH LPSRUWDQFH RI WKH WHUP LQ WKH 1DYLHU 6WRNHV HTXDWLRQ FDQ EH GHWHUPLQHG E\ FDOFXODWLQJ WKH VLPLODULW\ QXPEHUV 7R FDOFXODWH WKHVH QXPEHUV WKH YLVFRVLW\ PXVW EH HVWLPDWHG DQG WKH FKDUDFWHULVWLF YDOXHV 5 9 Q f FKRVHQ R R R 6XSSRVH WKH JDV WR EH D WHUQDU\ PL[WXUH FRPSRVHG RI K\GURJHQ LRQV HOHFWURQV DQG QHXWUDO KHOLXP %HFDXVH WKH UDWLR RI WKH HOHFWURQ PDVV WR WKDW RI HLWKHU RI WKH RWKHU WZR VSHFLHV LV VR VPDOO WKH YLVFRVLW\ RI WKH PL[WXUH LV HVVHQWLDOO\ GHWHUPLQHG E\ WKH YLVFRVLWLHV RI WKH K\GURJHQ LRQV DQG QHXWUDO KHOLXP &KDSPDQ DQG &RZOLQJ f )RU K\GURJHQ LRQV XQGHU SK\VLFDO GLVN FRQGLWLRQ WKH H[SUHVVLRQ JLYHQ E\ 6SLW]HU f PD\ EH XVHG ,W LV [ ,4 7 LQ$ ZKLFK IRU GLVN WHPSHUDWXUH 7Hf JLYHV + [ &KDSPDQ DQG &RZOLQJ f JLYH DQ H[SUHVVLRQ WKDW FDQ EH XVHG WR FDOFXODWH KHOLXP YLVFRVLW\ Wf @FPKH7HUf D :

PAGE 59

ZKHUH D LV WKH PROHFXODU GLDPHWHU DQG : LV D WDEXODWHG IXQFWLRQ LU LV WKH PDWKHPDWLFDO QXPEHUf 8QGHU GLVN FRQGLWLRQV WKH WZR TXDQWLWLHV D DQG : DUH D [ FP : 7KLV JLYHV Q [ 7KH IROORZLQJ FKDUDFWHULVWLF YDOXHV DUH XVHG f 5 5 R V f 9 9 WW 2 V f Q4 SD )RU YLVFRXV HIIHFWV GXH WR K\GURJHQ WKH VLPLODULW\ QXPEHUV DUH f 51 [ f )+ [ f 61 7KH KHOLXP YLVFRVLW\ UHVXOWV LQ D 5H\QROGV QXPEHU ZKLFK LV WZR RUGHUV RI PDJQLWXGH VPDOOHU WKDQ WKDW IRU K\GURJHQ EXW WKLV LV RI OLWWOH LPSRUW 1RU ZRXOG D PRUH DFFXUDWH FDOFXODWLRQ FKDQJH WKH EDVLF UHVXOW WKDW WKH 5H\QROGV

PAGE 60

QXPEHU IRU WKH GLVN IOXLG LV YHU\ ODUJH 7KH UDWLR RI WKH YLVFRXV WHUPV WR WKH LQHUWLDO WHUPV LV VR VPDOO WKDW WKH ODWWHU FRPSOHWHO\ GRPLQDWH WKH IRUPHU 5HPRYDO RI WKH YLVFRXV WHUPV UHGXFHV WKH 1DYLHU6WRNHV HTXDWLRQ WR Y W Y *0 V 5 9 R R (XOHUnV HTXDWLRQ

PAGE 61

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f V 5 9 V V ZLOO EH XVHG 7KH HTXDWLRQ RI FRQWLQXLW\ UI 9 r S 9f

PAGE 62

SURYLGHV DQ DGGLWLRQDO UHODWLRQ EHWZHHQ SDQG Y 7KH IXOO K\GURG\QDPLFDO SUHVFULSWLRQ UHTXLUHV WKH LQFOXVLRQ RI D WKLUG HTXDWLRQ WKDW IRU HQHUJ\ WUDQVSRUW 1R HIIRUW ZLOO EH PDGH LQ WKLV FKDSWHU WR LQWURGXFH VXFK DQ HTXDWLRQ )DLOXUH WR VWXG\ WKH HQHUJ\ HTXDWLRQ ZLOO QHFHVVLWDWH DW VRPH SRLQW WKH VSHFLILFDWLRQ RI DQ DGGLWLRQDO FRQGLWLRQ OLQNLQJ WZR RI WKH IORZ YDULDEOHV ,QWURGXFWLRQ RI WKLV QHZ UHODWLRQ UHGXFHG WKH QXPEHU RI XQNQRZQV IURP ILYH WR IRXU (XOHUn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f Y 94Df]f A ]Wf f S SRZ]f S WR M!]Wf

PAGE 63

f 3 SRW]f SZ-!=Wf 7KH VXEVFULSW]HUR LGHQWLILHV WKH VWHDG\VWDWH D[LDOO\ V\LQPHWULF WHUUDV WKH VXEVFULSWRQH WKH WHUPV FRQWDLQLQJ WKH WHPSRUDO DQG DQJXODU GHSHQGHQFH ,Q RUGHU WR DYRLG FRQIXVLRQ ZLWK WKH GLPHQVLRQLQJ SDUDPHWHU 34 XVHG HDUOLHU D ORZHU FDVH S LV XVHG ZLWK VXEVFULSW LQ WKH FDVH RI SUHVVXUH 7KHVH SDLUV RI IXQFWLRQV DUH VXEVWLWXWHG LQWR WKH HTXDWLRQ RI PRWLRQ WR JLYH rf Y R YR 9 99f Yf R R f93O TR9 DQG S S A A f" W r 9SR 9 9SR AR r 93 A n 9SO SR9 r AR 3A9 r Y S 9 f Yf SQ9 r Y R 7KH GHQRPLQDWRU LQ WKH WHUP SR SL! KDV EHHQ DSSUR[LPDWHG E\ ILUVW H[SDQGLQJ LQ WKH VHULHV +O0

PAGE 64

f f f ZKHUH [ LV 3A34 U DQG WKHQ H[FOXGLQJ WHUPV LQ WKH WKLUG DQG LQ KLJKHU SRZHUV RI [ 6XEVHTXHQW FRQGLWLRQV SODFHG RQ WKH IORZ ZLOO OLPLW WKH SUREOHP WR FDVHV ZKHUH :LWK WKH H[FHSWLRQ RI WKLV DSSUR[LPDWLRQ IRU WKH GHQRPLQDWRU WKHVH HTXDWLRQV DUH H[DFW 6LQFH WKH VXEVFULSW]HUR WHUPV DUH WKHPVHOYHV VWHDG\ VWDWH D[LDOO\V\PPHWULF VROXWLRQV RI WKH SUREOHP WKH\ VDWLVI\ WKH HTXDWLRQV R 93' TR9 L f DQG S Y f n.R R 7KH HTXDWLRQV IRU WKH VXEVFULSWRQH WHUPV EHFRPH

PAGE 65

DQG 9 SO f YR SO Y 7KH XQGHUO\LQJ SK\VLFDO SLFWXUH SUHVHQWHG WKXV IDU KDV EHHQ WKDW WKH VWHDG\VWDWH D[LDOO\V\PPHWULF VROXWLRQV WKH VXEVFULSW]HUR WHUPV UHSUHVHQW WKH JURVV QDWXUH RI WKH IORZ 7KH VXEVFULSWRQH WHUPV DUH WDNHQ WR UHSUHVHQW D UHILQHPHQW WR WKLV JURVV QDWXUH 7KHVH UHILQHPHQW WHUPV ZRXOG WKHQ EH H[SHFWHG WR EH VPDOOf§WKRXJK QRW QHJOLJLEOH 7KH DVVXPSWLRQ ZLOO EH PDGH KHUH WKDW VXEVFULSWRQH WHUPV DUH VXIILFLHQWO\ VPDOOHU WKDQ WKH VXEVFULSW]HUR TXDQWLWLHV WKDW HOHPHQWV RI WKH HTXDWLRQV ZKLFK LQFOXGH WKH SURGXFWV RI VXEVFULSWRQH TXDQWLWLHV FDQ EH RPLWWHG 7KH UHVXOWLQJ OLQHDUL]HG HTXDWLRQV DUH 9Y Y 3 R DQG ‘ Y R 9

PAGE 66

7KHVH PXVW EH UHZULWWHQ DV VFDODU HTXDWLRQV Y ZO Y Y f W f Rf Y n Y YB Y Y RfO IUO ZR `fR f 9f Rf Rf -! f Mf 9 Y fD Y ]O = ]R ] Y Y Y RLO B IUR IUO B f L S Rf f S r SR 6YrO Y "! Y } Y IUR Y W YfO 6X 9f2 Rf f IU f Y]O BL Y Y ] IUR ]R B ] b 9fO9IU2 f Y 9 f IUO f 3L SL SR 3 f ,7 7 S f R IU Y ]O Y 9 ]R Y Y Y Y Y ]O B! ]R IUR ]O W n nf Df ZR Rf Rf IU FR IU YB Y ]R Y 9 ]O ]O ] ]R ] B IIO eO R 3R ] 3 G] n DQG S S Y S Y S Y S . 4f4 R f + S f .R W FR Z YZR Rf Rf YRfO Rf Y0O U L L f f SR Z Z L9IUR IU 3 SB Y S Y r S f ]R IUO IU .R IU ]

PAGE 67

SO U SR 9]O 9]R ] 9]O ] 3R ] 6HOHFWLRQ RI D 6XEVFULSW]HUR 6ROXWLRQ 7KH VXEVFULSW]HUR VROXWLRQ ZKLFK ZLOO EH XVHG LV WKDW IRU D VWDWLF LVRWKHUPDO HQYHORSH /LPEHU f 7KLV VROXWLRQ LV FKRVHQ IRU WKH IROORZLQJ UHDVRQV f RYHU PRVW RI WKH GLVN WKH LVRWKHUPDO HQYHORSHV DUH LQ DJUHHPHQW ZLWK FDOFXODWLRQV EDVHG RQ PRUH DGYDQFHG WUHDWPHQWV 6HH &KDSWHU ,,f f WKH VROXWLRQ LV LQ FORVHG IRUP f WKH VLPSOLFLW\ RI WKLV VROXWLRQ UHVXOWV LQ WXUQ LQ D FRPSDUDWLYHO\ VLPSOH VHW RI H[SUHVVLRQV IRU WKH VXEVFULSWRQH WHUPV 7KLV VXEVFULSW]HUR VROXWLRQ LQ GLPHQVLRQOHVV FRRUGLQDWHVf LV Y ZR L L U HR H[S I f 4IF r 3 ^ i R @ n Gf ]9n @ @ n 7KH TXDQWLW\ 4 LV GHILQHG N5 7 R A H \P0 n V

PAGE 68

ZKLOH g LV JLYHQ E\ WWf NW < \P 9U 7KH UDWLR RI WKHVH WZR 49 LV WKH )URXGH QXPEHU )1f 6XEVWLWXWLRQ RI WKHVH H[SUHVVLRQV LQWR WKH OLQHDUL]HG HTXDWLRQV IRU WKH VXEVFULSWRQH TXDQWLWLHV JLYHV YX!O WW D9XO B WW f B B 8 I I B YOL 3 7W A a: c M!O \ LH[3/4 O Df U --O JZ nS ^H[S> i A S f``A ^H[S > ,f @ ` Y DW 77 9 LU X XO A L ^H[SI L L L f WH[Sr 4 U M -3+n ^H[S /i f ) @ @ ` JI H[S> f &7 Z f @ r n Y B DY ]O MB WW ]O DW A f 3L ‘ IHLWS >ie"f@! }O r IHLeS> i V f@ W ^ 6 f@f n DQG SO YZO U U IO 7W 77 H[S /f 4 IH U ` Y FRO Rf W H[S> "f @

PAGE 69

H[S > i L f @ ` f§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e TXLWH D FRQVHUYDWLYH YDOXHf 6XFK D OLPLWDWLRQ PD\ VHHP D GUDVWLF DSSUR[LPDWLRQ LQ IDFW LW LV QRW 7KLV WKLQ UHJLRQ LV D VRODU GLDPHWHU LQ WKLFNQHVV QHDU WKH VWHOODU VXUIDFH DQG ODUJHU IDUWKHU RXW $GGLWLRQDOO\ H[DPLQDWLRQ RI WKH ]FRPSRQHQW RI (XOHUnV HTXDWLRQ UHYHDOV WKDW RQO\ WKH SUHVVXUH JUDGLHQW EDODQFHV WKH ]FRPSRQHQW RI WKH JUDYLWDWLRQDO IRUFH EXW WKH UDWLR RI WKH IRUPHU WR WKH ODWWHU JLYHV R [

PAGE 70

7KH GRPLQDQFH RI WKH ]FRPSRQHQW RI WKH JUDYLWDWLRQDO WHUP VXJJHVWV WKDW WKH GLVN PDWHULDO LV KLJKO\ FRQFHQWUDWHG WRZDUG WKH SODQH &RQVLGHUDWLRQV &RQFHUQLQJ $SSUR[LPDWLRQV 1HDU WKH 3ODQH )RU ] Df WKH WHUP U EHFRPHV VLPLODUO\ WKH U WHUP LV $OO GHULYDWLYHV RI NQRZQ IXQFWLRQV VKRXOG EH SHUIRUPHG EHIRUH DSSUR[LPDWLRQV VXFK DV WKRVH GHVFULEHG DERYH DUH LQFOXGHG 7KLV KDV EHHQ WKH SUDFWLFH KHUH 2Q VXEVWLWXWLRQ RI WKH DSSURSULDWH H[SUHVVLRQV IRU U DQG U WKH VHW RI IRXU HTXDWLRQV EHFRPHV Y Q f W f UU Y WXO :a WW 9MfO > H[S =7 f @ 3L 4: Df :3@ 4 K f > H[S  f @ 4Rfn

PAGE 71

Y mZ ,7 YZO 7 a< M! LU X e/ B f WR MBH[S 4WRn f@ 3O GS Y]O 7) WR WW Y ]O I! >H[S A f @ / 4Z S B ] IS= U 7 >H[S f§Mf` n Q / 4WR WR 4 DQG SL 98 U 7( f67 /H[S 4Xf f@ ] 4WR 9WRO > H[S f =7 A @ 4WRn W H;3 a 4P A A 7  > H[S 4P A 3L Y r ]Y 4WR ]O \ Yf >H[S =7 f @ >H[S  f@ r 4WR 7KH IRUP RI WKH ODVW DQG PRVW FRPSOLFDWHG RI WKH VHW RI IRXU HTXDWLRQV VXJJHVWV D FKDQJH RI YDULDEOH ZKLFK ZLOO VLPSOLI\ WKH HTXDWLRQ 'HILQH WKH YDULDEOH YA E\ WKH H[SUHVVLRQ YLL YLL H[S 4WRn f r WKH L VWDQGLQJ IRU XfS} RU ] 1RZ

PAGE 72

&RPSDULVRQ RI WKLV UHVXOW ZLWK WKH WHUPV LQ WKH ODVW RI WKH VHW RI IRXU HTXDWLRQV DERYH VKRZV WKDW WKH ODVW HTXDWLRQ FDQ EH ZULWWHQ 77 a7 &2 n 9 !H 9 6WXG\ RI WKH ILUVW WKUHH PHPEHUV RI WKH VHW ZKLFK ZHUH GHULYHG IURP (XOHUnV HTXDWLRQ LQGLFDWHV f WKDW QR FRPSRQHQW RI Y DSSHDUV LQ D SDUWLDO GHULYDWLYH ZLWK UHVSHFW WR HLWKHU LG RU ] f WKDW WKH H[SRQHQWLDOV ZKHQ WKH\ DSSHDU LQ DQ\ WHUP FRPH EHIRUH DOO RSHUDWRUV 2Q PXOWLSOLFDWLRQ E\ H[S a] 4FR f DQG ZLWK WKH LQWURGXFWLRQ RI WKH YDULDEOH YA WKH ILUVW WKUHH HTXDWLRQV RI WKH VHW EHFRPH Y FRO Yn W I FRO DW! IYL[ 9} IYrL DW S FR DM!

PAGE 73

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

PAGE 74

ZKHUH 7 LV D SDUDPHWHU $V DQ LOOXVWUDWLRQ VXSSRVH WKH UHODWLRQVKLS EHWZHHQ WKH VXEVFULSWRQH GHQVLW\ DQG WKH VXEVFULSWRQH SUHVVXUH WR EH LVHQWURSLF ZLWK WKH LVHQWURSLF FRQVWDQW GHWHUPLQHG IURP WKH VXEVFULSW]HUR TXDQWLWLHV WKHQ a U \ D f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f WUHDWLQJ WKH IORZ LQ WKH ERXQGDU\ OD\HU f GHVFULELQJ WKH EHKDYLRU RI ERWK WKH WDQJHQWLDO DQG QRUPDO FRPSRQHQWV RI WKH YHORFLW\ RYHU WKH ERXQGDU\ VXUIDFH SDUWLFXODUO\ GHSHQGHQFH RQ WLPH DQG DQJOH 7KHUH LV QR DGHTXDWH WUHDWPHQW RI WKH ERXQGDU\ OD\HU IORZ DQG WKH REVHUYDWLRQV SURYLGH OLWWOH LQIRUPDWLRQ DERXW WKH

PAGE 75

WHPSRUDO RU DQJXODU GHSHQGHQFH RI WKH VWHOODU VXUIDFH YHORFLW\ )RU WKHVH WZR UHDVRQV ERXQGDU\ FRQGLWLRQV DUH RI OLWWOH XVH LQ WKH VWXG\ RI WKH QDWXUH RI WKH VXEVFULSWRQH VROXWLRQV 7R WKLV SRLQW WKH IRXU HTXDWLRQV ZKLFK WRJHWKHU FRPSULVH (XOHUnV HTXDWLRQ DQG FRQWLQXLW\ DUH Y R!O Y W I XO f I U 3L Rf g] 4Rf 9} I9O I9}O / L DW W! Rf I Y ]O I Y ]O B BS 77 SO 9]3@ DQG S S Y Y YW Y B f f ]O W f Rf f } ] 1RWH WKDW WKH H VXSHUVFULSWV KDYH EHHQ GURSSHG 7KH V\PERO YMA QRZ FRQWDLQV WKH H[SRQHQWLDO WHUP DQG ZLOO IRU WKH UHVW RI WKLV FKDSWHU ,QWHJUDWLRQ 2YHU ] )XUWKHU 5HVWULFWLRQV RQ WKH )ORZ 7KH VXEVFULSWRQH IORZ ZLOO EH DVVXPHG WR SRVVHVV V\PPHWU\ DERXW WKH HTXDWRULDO SODQH WKDW LV WKH IORZ EHORZ

PAGE 76

WKH SODQH LV WKH PLUURU LPDJH RI WKH IORZ DERYH WKH SODQH 7KLV FRQGLWLRQ UHTXLUHV WKDW WKH IORZ YDULDEOHV ZLWK WKH H[FHSWLRQ RI Y]A PXVW EH RGG IXQFWLRQV RI ] 7KH VHW RI IRXU HTXDWLRQV ZKLFK GHVFULEH WKH IORZ ZLOO EH LQWHJUDWHG ZLWK UHVSHFW WR ] RYHU WKH LQWHUYDO >] ] @ ZKHUH ] LV D IXQFWLRQ RI Z DORQH 1HZ YDULDEOHV R R R PXVW EH LQWURGXFHG WKH\ DUH f X[ r YAG=MA =R ] f D SG] ] $ R 8QGHU WKH FRQGLWLRQ UHTXLULQJ V\PPHWU\ DERXW WKH HTXDWRULDO SODQH WKH ]FRPSRQHQW RI WKH LQWHJUDWHG YHORFLW\ XO]f LV LGHQWLFDOO\ ]HUR 7KH LQWHJUDWLRQ ZLWK UHVSHFW WR ] DQG WKH LQWURGXFWLRQ RI QHZ YDULDEOHV UHTXLUH WKDW RSHUDWRU UHYHUVDOV RI WKH W\SH =4ff =4ff ] 0 Gfnf G] ] 8ffn]f G]n R R ZKHUH JX]f LV D ZHOOGHILQHG IXQFWLRQ EH SHUIRUPHG 7KH MXVWLILFDWLRQ IRU VXFK FKDQJHV LQ WKH RUGHU RI RSHUDWRUV DSSHDUV LQ $SSHQGL[ $ 7KHVH WZR VWHSV UHVXOW LQ WKH WKUHH HTXDWLRQV B W 8DfO I B Mf X ZO IXSL DA r f 7 4 X ] r ]B ] 3[G]

PAGE 77

X DW I X WRO I D f3 X r U f Dr DQG D X W f ‘ tW Df f F7D! 8ZO Z If 8F-!O D U8 I G3 Y]O] 9 7KH ODVW WHUP RI WKH ILUVW HTXDWLRQ DERYH ZLOO EH UHZULWWHQ ]A B ] r ]=SA G] $ r G] ] ] R R ZKHUH WKH TXDQWLW\ $ PD\ EH FRQVLGHUHG GHILQHG E\ WKLV UHODn WLRQ $ LV VWULFWO\ VSHDNLQJ D IXQFWLRQ EXW LW ZLOO EH WUHDWHG DV D FRQVWDQW SDUDPHWHU 7KH YDOLGLW\ RI WKLV DSSUR[LPDWLRQ FDQ EH WHVWHG WKH VROXWLRQV VKRXOG FKDQJH RQO\ VORZO\ ZLWK FKDQJHV LQ $ 7KH ODVW WHUP RI WKH IRXUWK HTXDWLRQ Y]L]f UHSUHVHQWV PDVV IORZ SHUSHQGLFXODU WR WKH HTXDWRULDO SODQH D TXDQWLW\ JHQHUDOO\ EHOLHYHG VPDOO ,W LV D UHDVRQDEOH DSSUR[LPDWLRQ WR VHW WKLV WHUP WR ]HUR 6XFK D VWHS ZKLOH QRW FUXFLDO GRHV VLPSOLI\ WKH FDOFXODWLRQ 7KH WKLUG WHUP LQ WKH FRQWLQXLW\ HTXDWLRQ ZKLFK FRQWDLQV D SDUWLDO GHULYDWLYH ZLWK UHVSHFW WR Z VKRXOG EH GRPLQDWHG E\ WKH QHLJKERULQJ WHUPV ZKLFK FRQWDLQ S DQG W SDUWLDO GHULYDWLYHV 7KLV WHUP ZLOO WKHUHIRUHn EH GURSSHG

PAGE 78

IURP IXUWKHU FRQVLGHUDWLRQDV LV D VLPLODU WHUP LQ WKH ILUVW HTXDWLRQ RI WKH VHW 7KH UHVXOW RI DOO WKHVH VWHSV LV VKRZQ EHORZ ,W fL I ,W fL I 9 f P 4 X r r f DQG 8 I ‘ 8 DW ZL I Dr rL UIIL f -! D D XfB X f§W L L f DW I r 7 LR 7 OR a" 6ROXWLRQV 7KH IROORZLQJ V\PEROV DUH LQWURGXFHG WR VLPSOLI\ WKH QRWDWLRQ f f F U /nDWIO" n 7KH HTXDWLRQV RI WKH SUHYLRXV VHFWLRQ EHFRPH &ODO / XRO f LXrO f+U n I I FUO / 8! 8O X f"

PAGE 79

DQG X 9 /&7L f Z Rf I! 7KLV VHW RI VLPXOWDQHRXV GLIIHUHQWLDO HTXDWLRQV PXVW QRZ EH UHGXFHG WR D VHW LQ ZKLFK WKHUH LV D VHSDUDWH HTXDWLRQ IRU HDFK LQGHSHQGHQW YDULDEOH 7KLV LQYROYHV PXFK GLIIHUHQWLDn WLRQ DQG PDQLSXODWLRQ 7KHVH UHGXFWLRQV DUH VKRZQ LQ $SSHQGL[ % 7KH UHVXOWV RI WKHVH PDQLSXODWLRQ DUH WKDW DOO WKUHH HTXDWLRQV VKDUH WKH FRPPRQ IRUP ^ R!/ Rfn fI f / F I FI D /B -/ ` [ f Z Kr [ YDULDEOHf ZKHUH YDULDEOH PD\ EH HLWKHU XAAP XA RU (DFK RI WKHVH WKUHH HTXDWLRQV KDV WKH IRUP RMf \ /HW \ H[S >DRff-f DMfW@ ZKHUH DXf DQG Df DUH DV \HW XQVSHFLILHG IXQFWLRQV 6LQFH MMMM H[S ,DZf-f XfW@ DRMf H[S >DRffI! Zf W@

PAGE 80

DQG H[S >FWZfL! ZfW@ Zf H[S >FW RMf RMfW@ RQH KDV WKDW ) _M J_ Zf \ )DXf ZfWRf\ $ VROXWLRQ \f H[LVWV RQO\ IRU DDLf DQG Zf VXFK WKDW )n;DLZf ff UZf 8QIRUWXQDWHO\ WKHUH DUH LQILQLWHO\ PDQ\ VXFK DZf ZffSDLUV 7KH WHPSRUDO DQG DQJXODU GHSHQGHQFH RI WKH IORZ YDULDEOHV WDNHV WKH IRUP H[S>LNW LQL!@ ZKHUH WKH VXEVWLWXWLRQV D LOF LQ KDYH EHHQ PDGH 7KH TXDQWLW\ N LV LQ HIIHFW D IUHTXHQF\ )RU FRPSDULVRQ ZLWK REVHUYDWLRQV WKH DVVRFLDWHG TXDQWLW\ LUWN ZKLFK LV WKH SHULRG DVVRFLDWHG ZLWK VXFK D IUHTXHQF\ LV PRUH XVHIXO 6LQFH I} DQG M! WWf DUH SK\VLFDOO\ WKH VDPH SRLQW Q PXVW EH DQ LQWHJHU 7KH HTXDWLRQ ZKLFK N DQG Q PXVW VDWLVI\ LV GHULYHG LQ $SSHQGL[ % 7KLV HTXDWLRQ LV WRN XQILW f§ Z I XIQ

PAGE 81

)RU D JLYHQ Q DQG WR WKHUH ZLOO EH LQ JHQHUDO WKUHH YDOXHV IRU 5HODWLRQV $PRQJ WKH )ORZ 9DULDEOHV 7KH WHPSRUDO DQG DQJXODU GHSHQGHQFH RI HDFK IORZ YDULDEOH WDNHV WKH IRUP H[S >LQMM L-7FRQfW@ 7KH TXDQWLW\ Q LV DQ LQWHJHU DQG WKH OFDfQf DUH URRWV RI D SRO\QRPLDO HTXDWLRQ ZKRVH FRHIILFLHQWV DUH IXQFWLRQV RI WR DQG Q 7KH JHQHUDO VROXWLRQ IRU WKH SUREOHP PD\ EH ZULWWHQ H[S>LQI! LNWRQfW@ H[S>LQ! LNRQfW@ H[S>LQ-! LNWRQfW@ X /8QNf! %O QfN DN X MO QN DQG /6Q(f! QN 7KH UHODWLRQVKLSV DPRQJ WKH WKUHH TXDQWLWLHV 8A & A 6QNI GHWHUPLQH WKH UHODWLRQVKLS DPRQJ WKH IORZ YDULDEOHV DQG

PAGE 82

6XEVWLWXWLRQ RI WKH JHQHUDO VROXWLRQ LQWR WKH RULJLQDO VHW RI WKUHH HTXDWLRQV \LHOGV OB >LN QIf QN X f§ I F U QN QN ,' QN ‘@^ H[S>LQ-! LNZQfW@` eB> f§ LN QIf @^ H[S>LQ-! LNZQfW@` Q N Z f§Mf§ L N QIf QN  LQFf L B f§f§ 6QA^H[S>LQ_ LNZQf W@` r 7KHVH WKUHH HTXDWLRQV DOO UHTXLUH VXPPDWLRQV RYHU Q DQG N :KDW LV GHVLUHG LV D VHW RI UHODWLRQV DPRQJ WKH WKUHH TXDQWLWLHV 8Qe 6QNn &QN` IRU D JLYHQ Q DQG N (DFK HTXDWLRQ LV PXOWLSOLHG E\ WKH TXDQWLW\ H[S>LQnL! LNnZQnfW@ 1H[W GRXEOH LQWHJUDWLRQ RI WKH IRUP /LP 7 LU7 7 GW 7 77 GM! H[S>LQMf LNW@ ,7 LV SHUIRUPHG RQ HDFK HTXDWLRQ DQG WKH RUGHU RI VXPPDWLRQ DQG LQWHJUDWLRQ UHYHUVHGf 2QH ILQGV LNr QnIf XQnNn f I &QnNn 6QfNn n f

PAGE 83

I ZQnNn DQ 9F e 8BU LN QnIf &QS FR 6QnNn FR 8QnNn LQn FR &Q n.In L Nn QnIf 6Qn.n ,Q WKH IROORZLQJ SDFWHV WKH SULPHV ZLOO EH VXUSUHVVHG 7KHVH H[SUHVVLRQV IRUP D V\VWHP RI KRPRJHQHRXV OLQHDU HTXDWLRQV 7KHUH LV QR D SULRUL UHDVRQ WR DVVXPH WKDW DQ\ QRQWULYLDO VROXWLRQV H[LVW ,I VROXWLRQV GR H[LVW WKH\ ZLOO FRQVWLWXWH DW EHVW D VLQJOH LQILQLW\ RI VROXWLRQV 6XFK D FDVH FDQ RQO\ SURYLGH XQLTXH YDOXHV IRU WKH UDWLRV RI WZR RI WKH XQNQRZQV ZLWK UHVSHFW WR WKH WKLUG ,I QRQWULYLDO VROXWLRQV H[LVW WKH GHWHUPLQDQW IRUPHG E\ WKH FRHIILFLHQWV RI WKH YDULDEOHV LQ WKH V\VWHP DERYH L N QIf nI &@>Z I L N QIf LQ FZ Z LQFR L N QIf VKRXOG EH ]HUR 7KH GHWDLOV RI WKH FDOFXODWLRQ RI WKH GHWHUPLQDQW DUH OHIW WR $SSHQGL[ & EXW WKH UHVXOWV DUH SOHDVDQWO\ IDPLOLDU 7KH FRQGLWLRQ WKDW WKH GHWHUPLQDQW EH ]HUR LV D SRO\QRPLDO LQ N ZKLFK PXVW EH ]HUR 7KLV SRO\QRPLDO LV WKH VDPH DV WKDW XVHG WR FDOFXODWH N 7KHUHIRUH

PAGE 84

WKH GHWHUPLQDQW LV LGHQWLFDOO\ ]HUR 7KHUH ZLOO EH D VLQJOH LQILQLW\ RI VROXWLRQV LI WKH UDQN RI WKH PDWUL[ IURP ZKLFK WKH GHWHUPLQDQW DERYH ZDV IRUPHG LV WZR WKHQ WKH HTXDWLRQV FDQ EH VROYHG IRU WKH UDWLRV RI WZR RI WKH XQNQRZQV ZLWK UHVSHFW WR WKH WKLUG 7KH UDQN RI WKLV PDWUL[ LV LQGHHG WZR 6HH $SSHQGL[ &f 7KH UDWLRV ZKLFK ZLOO EH GHWHUPLQHG DUH DQG 7KH\ VDWLVI\ WKH V\VWHP RI HTXDWLRQV FO N QIf [ I\ f§A f 7KHVH WZR UDWLRV DUH [ fFAN QIf IQF '(7

PAGE 85

QFN QIf < '(7 ZKHUH '(7 >N QIf I@ $ 5HSULVH 7KH DQDO\VLV RI WKLV FKDSWHU LV YDOLG RQO\ WR WKH H[WHQW WKDW WKH XQGHUO\LQJ SK\VLFDO SLFWXUH GLVFXVVHG LQ WKH VHFRQG VHFWLRQ RI WKLV FKDSWHU LV D UHDVRQDEOH UHSUHVHQWDn WLRQ RI %H VWDUV 7KLV UHTXLUHV WKH GLFKRWRPRXV YLHZ RI WKH IORZ SUHVHQWHG WKHUH WR EH D UHDO RQH 7KH VWHDG\VWDWH D[LDOO\V\PPHWULF HIIHFWV PXVW GRPLQDWH WKH IORZ 7KH HOHPHQWV RI WKH IORZ ZKLFK GHSHQG RQ WLPH DQG DQJOH PXVW EH VHFRQGDU\ )XUWKHUVROXWLRQV ZHUH REWDLQHG RQO\ ZKHQ f NQRZQ IXQFWLRQV DSSHDULQJ LQ WKH GLIIHUHQWLDO HTXDWLRQV ZHUH DSSUR[LPDWHG E\ IRUPXODH YDOLG RQO\ IRU ] Z f WKH IORZ YDULDEOHV ZHUH DVVXPHG WR SRVVHVV V\PPHWU\ DERXW WKH HTXDWRULDO SODQH f WZR SDUDPHWHUV 7 DQG $ ZHUH LQWURGXFHG f WKH HTXDWLRQV ZHUH LQWHJUDWHG ZLWK UHVSHFW WR ] DQG QHZ GHSHQGHQW YDULDEOHV LQWURGXFHGf

PAGE 86

f WZR WHUPV LQ WKH FRQWLQXLW\ HTXDWLRQ DQG RQH LQ WKH HTXDWLRQ RI PRWLRQ ZKLFK DSSHDU WR EH RI VHFRQGDU\ LPSRUWDQFH ZHUH GURSSHG 7KH VROXWLRQV WR WKLV VHW RI HTXDWLRQV DUH SURSRUWLRQDO WR H[S>LQ`! LNZUQfW@ ZKHUH Q LV DQ\ LQWHJHU DQG WKH YDOXHV RI N IRU DQ\ Q DQG Z DUH WKH URRWV RI D SRO\QRPLDO HTXDWLRQ 7KH JHQHUDO VROXWLRQ IRU WKH IORZ FDQ EH GHWHUPLQHG WR ZLWKLQ WKH LQILQLWH VHW RI PXOWLSOLFDWLYH IDFWRUV 6QA ZKLFK DSSHDU LQ WKH LQWHJUDWHG GHQVLW\ LQGHWHUPLQDQFH DZDLWV D ERXQGDU\ YD 5HPRYDO RI WKH 6 W

PAGE 87

&+$37(5 9 f&20387$7,216 $1' 5(68/76 ,QWURGXFWLRQ WR WKH &RPSXWDWLRQV %RWK WKH DQJXODU DQG WKH WHPSRUDO GHSHQGHQFH RI WKH VROXWLRQV LQ WKH SUHYLRXV FKDSWHU DUH FRQWUROOHG E\ WKH LQWHJHUV Q DQG WKH TXDQWLWLHV N UHVSHFWLYHO\ 7KH FDOFXODWLRQ IRU N ZKLFK GHSHQGV RQ Q Z $ DQG U HPHUJHV DV WKH SULQFLSDO GLIILFXOW\ $FFRUGLQJO\ WKH WKUHH SULQFLSDO FRQFHUQV RI WKLV FKDSWHU ZLOO EH f WKH QDWXUH RI WKH URRWV RI WKH HTXDWLRQ IRU N f WKH PHDQV XVHG WR FDOFXODWH WKHP f WKH UHVXOWV RI WKH FRPSXWDWLRQV IRU YDULRXV FKRLFHV RQ ZKLFK WKH N HTXDWLRQ GHSHQGV $V WKH VROXWLRQV FDQQRW EH IXOO\ HYDOXDWHG LQ WKH DEVHQFH RI D SURSHU ERXQGDU\ WUHDWPHQW WKHVH UHVXOWV ZLOO FRQVWLWXWH PXFK RI ZKDW FDQ EH OHDUQHG IURP WKH DQDO\VLV ,Q PRVW RI WKH GLVFXVVLRQ WKH SHULRG WW7 3(5 r JHQHUDOO\ LQ KRXUVf ZLOO EH XVHG 7KH ILUVW PDMRU VHFWLRQ RI WKLV FKDSWHU LV GHYRWHG WR WKH ILUVW WZR WRSLFV DERYH WKH QDWXUH RI WKH URRWV DQG

PAGE 88

WKH SURFHGXUH XVHG WR FRPSXWH WKHP 7KH UHVXOWV RI WKHVH FRPSXWDWLRQV WKH WKLUG PDMRU FRQFHUQ RI WKLV FKDSWHU QDWXUDOO\ JURXS WKHPVHOYHV LQWR WKUHH GLYLVLRQV f FDOFXODWLRQ RI WKH SHUWLQHQW TXDQWLWLHV LQ WKH DEVHQFH RI DQ\ DQJXODU GHSHQGHQFH Q f f WHPSRUDO EHKDYLRU LQ WKH SUHVHQFH RI WKH VLPSOHVW DQJXODU GHSHQGHQFH Q f f WKH WUHQGV RI WKH UHVXOWV IRU PRUH FRPSOH[ DQJXODU EHKDYLRU Q f 7KH WKUHH FHQWUDO VHFWLRQV RI WKH FKDSWHU UHIOHFW WKLV WKUHHIROG GLYLVLRQ ,Q HDFK WKH GHSHQGHQFH RI WKH UHVXOWV RQ Z $ DQG 7 LV GLVFXVVHG 7KH SHQXOWLPDWH VHFWLRQ SUHVHQWV VDPSOH FDOFXODWLRQV RI WKH IORZ YDULDEOHV ZKLFK LOOXVWUDWH WKH TXDOLWDWLYH DVSHFWV RI WKH VROXWLRQV 7KH ODVW VHFWLRQ RI WKH FKDSWHU FRPSDUHV WKH UHVXOWV ZLWK UHOHYDQW REVHUYDWLRQV 7KH (TXDWLRQ IRU N 7KH &RQVHTXHQFHV RI ,PDJLQDU\ 5RRWV 7KH HTXDWLRQ IRU N GHULYHG LQ &KDSWHU ,9 LV ZN" ZQINA f OLP9 b f LU f '

PAGE 89

FQI f F QI R FQI f§f§ ZI Q XQnean f§f§f Df X %HLQJ DQ HTXDWLRQ RI WKH WKLUG GHJUHH WKLV HTXDWLRQ KDV HLWKHU WKUHH UHDO URRWV RU RQH UHDO DQG WZR FRPSOH[ URRWV ,I WKHUH DUH FRPSOH[ URRWV RQH LV WKH FRPSOH[ FRQMXJDWH RI WKH RWKHU 7KH DSSHDUDQFH RI FRPSOH[ URRWV KDG LPPHGLDWH DQG VHULRXV FRQVHTXHQFHV IRU WKH PHWKRGRORJ\ RI &KDSWHU ,9 2QH RI WKHVH URRWV PXVW RI QHFHVVLW\ KDYH D QHJDWLYH LPDJLQDU\ SDUW ZKLFK ZLOO DSSHDU LQ WKH VROXWLRQV DV DQ LQFUHDVLQJ H[SRQHQWLDO WHUP H[S f ZKHUH _ LV WKH DEVROXWH YDOXH RI WKH LPDJLQDU\ SDUW )ORZ YDULDEOHV ZKLFK LQFUHDVH H[SRQHQWLDOO\ LQ WLPH ZLOO DW VRPH WLPH YLRODWH WKH FRQGLWLRQ XQGHU ZKLFK OLQHDUL]DWLRQ RI WKH IORZ YDULDEOHV LV YDOLG ,W LV SRVVLEOH WR FKRRVH WKH WKUHH PXOWLSOLFDWLYH IDFWRUV A6QNn &QNn 8QNA frfQ VXFO PDQQHU WKDW WKHVH LQFUHDVLQJ WHUPV GR QRW FRQWULEXWH WR WKH ILQDO VROXWLRQ EXW VXFK FKRLFHV DUH VXVSHFW 2QO\ IRU FRQGLWLRQV LQ ZKLFK DOO WKH URRWV DUH UHDO FDQ WKH OLQHDUL]DWLRQ SURFHGXUH DQG KHQFH DOO WKHUHDIWHU EH FRQVLGHUHG YDOLG ,QLWLDO 9DOXHV IRU WKH 3DUDPHWHUV 7KH TXDQWLWLHV 7 DQG $ ZKLFK DSSHDU LQ WKH FRHIILFLHQWV RI WKH HTXDWLRQ IRU F LQ F DQG F UHVSHFWLYHO\f HQWHU WKH SUREOHP WKURXJK WKH UHODWLRQV SL USL

PAGE 90

DQG ] ] r WW S G] $ r G] UHVSHFWLYHO\ 7KH FDOFXODWLRQV ZKRVH UHVXOWV DUH GHVFULEHG LQ ODWHU VHFWLRQVZLOO EH SHUIRUPHG RYHU D ZKROH UDQJH RI YDOXHV IRU HDFK RI WKHVH TXDQWLWLHV KRZHYHU DQ LQLWLDO RU UHSUHVHQWDWLYH YDOXH IRU HDFK TXDQWLW\ PXVW EH GHWHUPLQHG 7KH UHSUHVHQWDWLYH YDOXH IRU 7f§FDOOHG 7f§LV WKH YDOXH IRU 7 LQ WKH FDVH WKDW S DQG S DUH UHODWHG LVHQWURSLFDOO\ 7KH LVHQWURSLF FRHIILFLHQW LV GHWHUPLQHG IURP WKH VXEVFULSW]HUR TXDQWLWLHV LW LV WWf N7 U f§ f§f Y \P V )RU DQ HOHFWURQ WHPSHUDWXUH 7Hf RI r. 7 [ $ UHSUHVHQWDWLYH YDOXH IRU $f§FDOOHG $f§FDQ EH FDOFXODWHG EH VHWWLQJ S HTXDO WR D FRQVWDQW IRU ]T HTXDO WR ]R $ f§ [ a

PAGE 91

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f LV FDOFXODWHG 6ROXWLRQV IRU WKH IORZ YDULDEOHV :KHQ JLYHQ DV D IXQFWLRQ RI R WKH SURJUDP XVHV WKH IRUPDOLVP GHYHORSHG LQ &KDSWHU ,9 WR FDOFXODWH ERWK DQG 8Qer 7KH TXDQWLWLHV 5H^8QAff H[S>LQ`! LNZQf W@` 5H^&QeZf H[S>LQ^! LNDMQfW@!

PAGE 92

DQG 5H^6QNff H[3>LQIf LNZQfW@` DUH FDOFXODWHG E\ WKH SURJUDP WKURXJKRXW WKH GLVN IRU YDULRXV FKRLFHV RI DQJOHV DQG WLPHV 5HDGRXW 7KH SURJUDP SULQWV WKH UHVXOWV RI WKH FDOFXODWLRQV LQ WDEXODU IRUP 7KH DFWXDO SURJUDP VWHSV DUH FRQWDLQHG LQ $SSHQGL[ 7KH &DVH Q 'LVWLQFWLYH $VSHFWV RI WKH Q &DVH 7KH N YDOXHV KHUH DUH IRU VROXWLRQV ZLWK QR DQJXODU GHSHQGHQFH ,Q WKH DEVHQFH RI GHSHQGHQFH WKH N HTXDWLRQ WDNHV WKH IRUP 4 ZN ZI Mf N f WKH WKUHH URRWV DUH FO ; N I ef 7KH ILUVW YDOXH FRUUHVSRQGV WR D WULYLDO VROXWLRQ 7KH ODVW H[SUHVVLRQ LV UHDO IRU DOO UHDVRQDEOH YDOXHV RI FA 9DULDWLRQ ZLWK Rf U DQG $ 7KH H[SUHVVLRQ IRU WKH QRQWULYLDO URRWV GHSHQGV RQ Z DQG FA EXW QRW RQ Fr 1RZ

PAGE 93

)LJXUH 7KH GHSHQGHQFH RI >3(5_ RQ Df LV VKRZQ IRU WKH FDVH Q DQG FA

PAGE 95

WW f ZKHUH 0 4 6HWWLQJ RQH ILQGV WKDW N WW f f§Lf L f LUf ] Z Q f 7KH DEVROXWH YDOXH RI WKH FRUUHVSRQGLQJ SHULRG LV >3(5_ f§ 7 ,NO r RU _3(5_ )RU ODUJH WKH EHKDYLRU RI _SHU_ QHDU WKH VWHOODU VXUIDFH Z f GHSDUWV VOLJKWO\ IURP Z n EHKDYLRU DV ZRXOG EH H[SHFWHG IURP WKH IXOO H[SUHVVLRQ IRU _-F_ EXW WKH RXWHU UHJLRQV RI WKH GLVN VWLOO VKRZ WKH RM EHKDYLRU 7KHVH IHDWXUHV DUH VKRZQ LQ )LJXUH 7KH &DVH Q 3UHOLPLQDU\ 5HPDUNV 7KH Q FDVH FDQQRW EH XVHG IRU D JHQHUDO GLVFXVVLRQ

PAGE 96

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f§WKH URRWV RI WKH N HTXDWLRQ XQGHU DOPRVW DOO FLUFXPVWDQFHV VWXGLHG DUH UHDO 7KH (IIHFW RI FAA 5DWKHU WKDQ N WKH DEVROXWH YDOXH RI WKH SHULRG

PAGE 97

)LJXUH 7KH TXDQWLWLHV _3(5A_ DQG ,3(5_ DUH SORWWHG DJDLQVW UDGLDO FRRUGLQDWH IRU WZR GLIIHUHQW YDOXHV RI FA &XUYHV D DQG E DUH c3(5_ DQG 3(5, UHVSHFWLYHO\ ZKHQ F F DQG G ZKHQ F ,Qn DOO FDVHV 7

PAGE 99

)LJXUH 7KH GHSHQGHQFH RI _3(5_ DQG 3(5MO RQ Z LV VKRZQ IRU WZR GLIIHUHQW YDOXHV RI F &XUYHV D DQG E DUH 3(5 DQG 3(5A UHVSHFWLYHO\ ZKHQ &Q F DQG G ZKHQ FA ,Q DOO FDVHV U

PAGE 101

)LJXUH 7KH GHSHQGHQFH RI _3(5_ RQ Z LV VKRZQ IRU IRXU YDOXHV F ,Q DOO FDVHV 7

PAGE 103

ZLOO EH WKH EDVLV RI WKH IROORZLQJ GLVFXVVLRQ $W HDFK SRLQW LQ WKH GLVN WKHUH DUH WKUHH N YDOXHV DQG KHQFH WKUHH SHULRGV 7KHVH ZLOO EH RUGHUHG DQG QXPEHUHG DFFRUGLQJ WR WKHLU DEVROXWH YDOXHf§WKDW LV 3(5A _3(5 B 3(5 f 7AH N V ZLOO WDNH WKHLU QXPEHULQJ IURP WKH SHULRGV IRU LQVWDQFH NA LV WKH YDOXH IRU N ZKLFK JLYHV _3(5A_ ,Q )LJXUHV DQG WKH YDULDWLRQ RI _3(5A_ DQG _3(5_WKURXJKRXW WKH GLVN LV SORWWHG IRU IRXU GLIIHUHQW YDOXHV RI FA 7KH FRQWULEXWLRQ RI FA WR WKH FRHIILFLHQWV RI WKH N HTXDWLRQ DUH VPDOO IRU VPDOO YDOXHV RI FA EXW WKH\ DUH QRW VPDOO IRU ODUJH YDOXHV RI FA
PAGE 104

)LJXUH 3(5 FXUYHV FXUYHV D DQG 7KH GHSHQGHQFH RI E DQG Gf DQG _3(5c Ff RQ WR IRU WZR YDOXHV RI < &XUYHV D DQG E DUH IRU WKH FDVH 7 FXUYHV F DQG G IRU WKH FDVH 7 ,Q ERWK FDVHV FA

PAGE 106

)LJXUH 7KH GHSHQGHQFH RI _3(5A_FXUYHV E DQG Gf DQG ,3(5, FXUYHV D DQG Ff RQ D! IRU WZR YDOXHV RI 7 &XUYHV D DQG E DUH IRU WKH FDVH 7 DQG FXUYHV F DQG G IRU WKH FDVH 7 ,Q ERWK FDVHV FA

PAGE 108

)LJXUH 7KH GHSHQGHQFH RI c3(5A, RQ R LV VKRZQ IRU WZR YDOXHV RI 7 7 DQG U ,Q ERWK FDVHV FA

PAGE 110

GHSHQGHQW RQ FA %H\RQG WKLV LQILQLW\ IXUWKHU RXW LQ WKH GLVN _3(5_ IDOOV RII ILUVW UDSLGO\ DQG WKHQ VORZO\ ZLWK LQFUHDVLQJ Z ,W DSSURDFKHV D YDOXH RI DERXW [ K QHDU WKH RXWHU UHJLRQ RI WKH GLVN 7KH (IIHFW RI $ 7KHUH DUH PDQ\ VLPLODULWLHV ZLWK WKH SUHFHGLQJ FDVH 7KH GHSHQGHQFH RI _3(5A_ DQG ,3(5, RQ WKH UDGLDO FRRUGLQDWH IRU IRXU GLIIHUHQW 7 YDOXHV LV VKRZQ LQ )LJXUHV DQG 7KH FXUYHV FKDQJH RQO\ D OLWWOH ZLWK ODUJH FKDQJHV LQ 7 2QO\ IRU U FXUYH F RI )LJXUH f GR DQ\ RI WKH FXUYHVf§DQG WKHQ RQO\ ,3(5,f§ VKRZ D QRWLFHDEOH GHSDUWXUH IURP Z EHKDYLRU 7KH TXDQWLW\ _3(5_LV PXFK PRUH GHSHQGHQW RQ 7 ([DPLQDWLRQ RI )LJXUHV DQG VKRZV WKDW DV WKH YDOXH RI 7 LQFUHDVHV WKH VLQJXODULW\ LQ _3(5_PRYHV LQZDUG E\ 7 FXUYH F LQ )LJXUH f LW LV LQWHULRU WR WKH ILUVW JULG SRLQW )XUWKHU WKHUH LV D JHQHUDO GHFUHDVH RI _3(5_f§ DZD\ IURP WKH VLQJXODULW\f§DV 7 LQFUHDVHV )RU 7 FXUYH H LQ )LJXUH f _3(5_ UHDFKHV LWV PD[LPXP GD\V DW WR DQG GHFUHDVHV RXWZDUG ([WUHPH &DVHV ,Q )LJXUH WKH UDGLDO GHSHQGHQFH RI WKH WKUHH SHULRGV ZKHQ ERWK FA DQG U DUH ODUJH LV GLVSOD\HG ,Q WKH RXWHU SDUW RI WKH GLVN ,3(5, VKRZV WKH VDPH GHSDUWXUH IURP

PAGE 111

)LJXUH 7KH GHSHQGHQFH RI _3(5RQ RM LV VKRZQ IRU WKUHH YDOXHVn RI 7 U Ff U Gf DQG 7 Hf $JDLQ FA LQ DOO FDVHV

PAGE 113

)LJXUH 7KH GHSHQGHQFH RI WKH RQ WR LV VKRZQ ZKHQ &Q 7 _3(5;_ LV DQGM3(5JO LV F WKUHH SHULRGV DQG L _3(5_ LV E

PAGE 115

)LJXUH 7KH GHSHQGHQFH RI _3(5_ Df _3(5_ Ef DQG _3(5B_;Ff RQ R ZKHQ F DQG7 LV VKRZQ

PAGE 117

EHKDYLRU WKDW LV IRXQG IRU _3(5, LQ FXUYH F LQ )LJXUH 2WKHUZLVH _3(5A_ DQG _3(5_ VKRZ WKH VDPH LQVHQVLn WLYLW\ WR WKH H[DFW YDOXH RI FA RU ) WKDW ZDV VKRZQ LQ HDUOLHU FDVHV 7KH ORFDWLRQ RI WKH VLQJXODULW\ LQ M 3(5 LV WKH UHVXOW RI FRPSHWLQJ WUHQGV IRU ODUJH FA DQG ODUJH 7 /DUJH YDOXHV RI FA PRYH WKH VLQJXODULW\ RXWZDUG ZKLOH ODUJH YDOXHV RI 7 PRYH LW LQZDUG WRZDUG WKH VWDU 7KH ORFDWLRQ RI WKH VLQJXODULW\ LV DSSUR[LPDWHO\ PLGZD\ EHWZHHQ LWV ORFDWLRQ IRU ODUJH FA DORQH FA LQ )LJXUH f DQG WKH SRVLWLRQ RI WKH VLQJXODULW\ ZKHQ RQO\ 7 LV ODUJH 7 FXUYH H LQ )LJXUH f ,Q WKH RXWHU UHJLRQV RI WKH GLVN RQH VHHV HIIHFWV YHU\ PXFK OLNH WKRVH LQ )LJXUH 7KH UDGLDO GHSHQGHQFH RI WKHVH TXDQWLWLHV IRU VPDOO SDUDPHWHU YDOXHV LV VKRZQ LQ )LJXUH _3(5A_ DQG _3(5_ D DQG E LQ WKH ILJXUHf FRQILUP WKH SDWWHUQ RI UHODWLYH LQVHQVLWLYLW\ RI WKH WZR TXDQWLWLHV WR FKDQJHV LQ WKH SDUDPHWHUV _3(5_ FXUYH Ff LV TXLWH ODUJH VKRZV QR VLQJXODULW\ DQG LQFUHDVHV LQ WKH GLVN DV Z n 7KH SDUDPHWHU 7 GHWHUPLQHV WKH SUHVHQFH RU DEVHQFH RI LPDJLQDU\ URRWV &DOFXODWLRQV IRU ODUJH YDOXHV VKRZ WKDW LPDJLQDU\ URRWV DSSHDU LQ WKH RXWHU UHJLRQV RI WKH GLVN DW DERXW 9 $V 7 LQFUHDVHV EH\RQG WKLV YDOXH ODUJHU DQG ODUJHU UHJLRQV RI WKH GLVN H[KLELW LPDJLQDU\ URRWV 7KH RQVHW RI LPDJLQDU\ URRWV LV LQVHQVLWLYH WR FA

PAGE 118

)LJXUH 7KH GHSHQGHQFH RI WKH WKUHH SHULRGV RQ Df LV VKRZQ IRU WKUHH YDOXHV RI Qf§ Df Ef DQG Ff ,Q HDFK FDVH FA DQG 7

PAGE 120

)LJXUH 7KH TXDQWLW\ M3(5MDW G LV SORWWHG DJDLQVW Q )RU DOO Q DQG 7 7KH VROLG FXUYH LV Q

PAGE 122

)LJXUH 7KHVH FXUYHV LOOXVWUDWH WKH HIIHFW RI H[WUHPH YDOXHV RI F DQG U IRU Q &XUYH D FRUUHVSRQGV WR _3(5_ ZKHQ F DQG 7 7KH RWKHU WZR SHULRGV ZKLFK IDOO TXLWH FORVH WR WKLV FXUYH DUH QRW VKRZQ 7KH FXUYHV E _3(5,f F _3(5 _f DQG G _3(5 DUH IRU WKH FDVH LQ ZKLFK FA DQG 7

PAGE 124

*HQHUDO )HDWXUHV 7KH &DVH Q 7KH DSSHDUDQFH RI D VLQJXODULW\ LQ _3(5_ LV XQLTXH WR Q )LJXUH VKRZV DOO WKUHH SHULRGV SORWWHG DV D IXQFWLRQ RI WKH UDGLDO FRRUGLQDWH IRU WKUHH YDOXHV RI Qf§ DQG FXUYHV D E DQG F UHVSHFWLYHO\f $JDLQ RQH VHHV WKH RR GHSHQGHQFH LQ 3(5A DQG c 3(5 DQG QRZ LQ _3(5A, DV ZHOO 1RWH WKDW WKH VHSDUDWLRQ DPRQJ WKH WKUHH SHULRGV GHFUHDVHV ZLWK LQFUHDVLQJ Q 0RUH LPSRUWDQWO\ WKH DEVROXWH YDOXHV RI WKH SHULRGV IDOO RII PDUNHGO\ IURP Q WR Q DQG OHVV PDUNHGO\ IURP Q WR Q 7KLV EHKDYLRU D GHFUHDVH ZLWK LQFUHDVLQJ Q LV FRQILUPHG LQ )LJXUH +HUH WKH YDOXH RI _3(5A_ DW X KDV EHHQ SORWWHG DJDLQVW Q 7KH VROLG OLQH LQ WKH ILJXUH LV [ L Q WKH ILW IRU Q JUHDWHU WKDQ LV UHPDUNDEOH 7KH RWKHU WZR SHULRGV VKRZ VLPLODU EHKDYLRU /DUJH Q DQG WKH (IIHFW RI FA DQG 7 )LJXUH VKRZV WKH HIIHFW RI H[WUHPH YDOXHV RI WKH FRQVWDQW DQG U RQ WKH WKUHH SHULRGV LQ WKH FDVH Q ,Q JHQHUDO WKHVH DQG RWKHU FDOFXODWLRQ VKRZ VRPH GHSHQGHQFH RQ 7 DQG DOPRVW QRQH RQ FA )XUWKHU IRU WKRVH Q YDOXHV

PAGE 125

WHVWHG QR URRWV RI WKH N HTXDWLRQ DUH LPDJLQDU\ HYHQ IRU U ,W ZRXOG EH SURKLELWLYHO\ H[SHQVLYH WR SHUIRUP WKH H[WHQVLYH FDOFXODWLRQV GRQH IRU WKH Q FDVH RQ YHU\ PDQ\ Q YDOXHV %XW RQH FDQ ILQG DW OHDVW TXDOLWDWLYH VXSSRUW IRU WKHVH ILQGLQJV LQ WKH -F HTXDWLRQ :ULWLQJ N Q$ DQG VXEVWLWXWLQJ WKLV LQWR WKH HTXDWLRQ RQH ILQGV WKDW $ I $ F &O I Uf f Q Z Q FI FBI FQI I I B B IB @ n X Df Q f RL Q Q 7KH LQWHJHU Q DSSHDUV LQ ILYH WHUPV DV Q )RU ODUJH Q WKHVH WHUPV DUH QHJOLJLEOH DQG FA RFFXUV RQO\ LQ WKHVH WHUPV )RU ODUJH Q DW OHDVW WKH VROXWLRQV RI WKH N HTXDWLRQ VKRXOG QRW GHSHQG RQ FA )RU ODUJH Q WKH HTXDWLRQ IRU $ LV $ I$ If $ a If f X )RU 7 LQ WKH UDQJH > @ DQG X LQ > @ =EL 7KH WHUPV ZKLFK FRQWDLQ U DUH QHYHU GRPLQDQW JHQHUDOO\ WKH\

PAGE 126

PD\ WR D ILUVW DSSUR[LPDWLRQ EH QHJOHFWHG ,I WKH\ DUH RQH KDV WKDW $ I$ I$ I r &OHDUO\ 7KLV JLYHV 3(5 WW7 7 X R B R QI Q Z Q LQ KRXUV $WX 3(5 Z OO Q 7KLV UHVXOW FRPSDUHV IDYRUDEO\ ZLWK WKH FXUYH LQ )LJXUH 7KH DSSUR[LPDWH QDWXUH RI WKLV UHVXOW VKRXOG EH HPSKDVL]HG &DOFXODWLRQV VKRZ WKDW LQ DOO FDVHV WKHUH DUH WKUHH GLIIHUHQW URRWV :KHQ U LV ODUJH RQH RI WKHVH URRWV PD\ EH H[SHFWHG WR GHSDUW IURP DQ WR n EHFRPLQJ PRUH OLNH Z UDGLDO GHSHQGHQFH

PAGE 127

)LJXUH 7KH YDULDEOHV XXLrf DQG XZOrf DUH SORWWHG DJDLQVW Z IRU WLPHV KRXUV Df KRXUV E DQG Gf DQG KRXUV F DQG Hf 7KHVH FRUUHVSRQG WR WKH FDVH AN N ff GLVFXVVHG LQ WKH WH[W $W KRXUV X Arf LV ]HUR

PAGE 128

5DGLXV 6WHOODU 5DGLLf $PSOLWXGH 'LPHQVLRQOHVV 8QLWVf ,77

PAGE 129

)LJXUH 7KH TXDQWLW\ XArf XA rf IRU WKH FDVH GLVFXVVHG LQ WKH WH[W LV SORWWHG DJDLQVW WKH UDGLDO FRRUGLQDWH DW KRXUV DQG DW KRXUV IURP DQ LQLWLDO WLPH

PAGE 130

5DGLXV 6WHOODU 5DGLLf $PSOLWXGH 'LPHQVLRQOHVV 8QLWVf HQ

PAGE 131

6DPSOH )ORZ 9DULDEOH 6ROXWLRQV $V ZDV GLVFXVVHG LQ &KDSWHU ,9 WKHUH FDQ EH QR IXOO\ GHWHUPLQHG VROXWLRQV IURP WKLV DQDO\VLV 7KHVH VROXWLRQV FDQ EH GHWHUPLQHG RQO\ ZLWKLQ D IDFWRU 6QfZfW ZKLFK PXVW EH JLYHQ 7KH TXDOLWDWLYH QDWXUH RI WKH VROXWLRQV FDQ EH VWXGLHG E\ FKRRVLQJ VRPH VLPSOH 6 A DQG FDOFXODWLQJ WKH VROXWLRQV DV IXQFWLRQV RI XM 3 DQG W $ VLPSOH FKRLFH LV 6QNf! a mQL NN ,2f ZKHUH QA DQG A DUH .URQHFNHU GHOWDV )LJXUH VKRZV XA DQG XA DV IXQFWLRQV RI LG QHDU WKH VWDU > @ DQG IRU WKUHH GLIIHUHQW WLPHV KRXUV KRXUV DQG KRXUV )LJXUH VKRZV WKH TXDQWLW\ >XA rf XA rf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
PAGE 132

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f WR WKH EUHDNn XS YHORFLW\ 7KLV UDQJH FDQ JLYH DW PRVW D IDFWRU RI GHFUHDVH LQ FA DQG 7 7KLV YDOXH LV VPDOO LQ FRPSDULVRQ ZLWK WKH UDQJH RI YDOXHV RI FA DQG $ RYHU ZKLFK FDOFXODWLRQV ZHUH SHUIRUPHG &RQVLGHULQJ WKH UHVXOWV RI WKLV FKDSWHU

PAGE 133

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f ORQJ WHUP \HDUVf f PHGLXP WHUP WR \HDUVf f VKRUW WHUP \HDUf &OHDUO\ WKH SKHQRPHQD ZKLFK PD\ UHVXOW IURP WKHVH VROXWLRQV IDOO LQWR WKH ODVW FDWHJRU\ VKRUW WHUP YDULDWLRQV WKH RQO\ H[FHSWLRQ EHLQJ WKH FDVH IRU Q DQG N NAf 'HSHQGLQJ RQ WKH YDOXH RI Q WKH SHULRGV LQ WKH LQQHU SDUW RI WKH GLVN UDQJH IURP VHYHUDO GD\V WR IUDFWLRQV RI D GD\ DQG IRU ODUJHU Q YDOXHV IUDFWLRQV RI DQ KRXU )RU D JLYHQ Q WKH YDOXHV RI WKH SHULRGV FKDQJH

PAGE 134

WKURXJKRXW WKH GLVN DQG WKH SUHVHQFH RI VHYHUDO Q YDOXHV DW RQFH FDQQRW EH SUHFOXGHG 7KH PXOWLSOLFLW\ RI SHULRGV VXJJHVWV WKDW SKHQRPHQD GHSHQGHQW RQ WKHVH VROXWLRQV VKRXOG VKRZ D FRPSOH[ YDULDWLRQ ZLWK WLPH $OWKRXJK RQH SHULRG PD\ GRPLQDWH WKH REVHUYDWLRQV LQ JHQHUDO QR VLPSOH SHULRGLFLWLHV VKRXOG EH H[SHFWHG )ROORZLQJ WKH WD[RQRP\ RI 0F/DXJKOLQ f WKH WZR PDMRU W\SHV RI VSHFWUDO FKDQJH ZKLFK FRXOG EH H[SHFWHG WR UHVXOW IURP VXEVFULSWRQH VROXWLRQV DUH 95 YDULDWLRQV DQG (& YDULDWLRQV 7KH WRWDO HPLVVLRQ LQ D OLQH LQ WKH FLUFXPVWHOODU HQYHORSH VKRXOG EH PRUH GHSHQGHQW RQ WKH GHQVLW\ WKDQ RQ WKH IOXLG YHORFLW\ 7KH UDWLR RI WKH LQWHQVLW\ RI WKH YLROHW HPLVVLRQ OREH WR WKDW RI WKH UHG OREH LV VWURQJO\ GHSHQGHQW RQ WKH IOXLG YHORFLW\f§SDUWLFXODUO\ WKH UDGLDO FRPSRQHQW ,Q WKH VDPSOH VROXWLRQV ZKLFK ZHUH FDOFXODWHG WKH DPSOLWXGH RI WKH GHQVLW\ YDULDWLRQ LV PXFK VPDOOHU WKDQ WKH DPSOLWXGH RI WKH YHORFLW\ YDULDWLRQ 7KH SUHVHQFH RI WKHVH VROXWLRQV VKRXOG EH PRVW HYLGHQW LQ WKH 95 YDULDWLRQV 7KH ODUJHVW SHULRG IRU WKH FDVH Q LV DQ DQRPDORXV RQH )RU W\SLFDO SDUDPHWHU YDOXHV WKLV SHULRG LV JUHDWHU WKDQ \HDUV RYHU PRVW RI WKH GLVN /RQJ WHUP FKDQJHV REVHUYHG LQ %H VWDUV KDYH JHQHUDOO\ EHHQ ODUJH FKDQJHVf§QRW RQHV ZKLFK RQH ZRXOG H[SHFW IRU VROXWLRQV WR WKH OLQHDUL]HG HTXDWLRQV )RU FHUWDLQ FRPELQDWLRQV RI WKH SDUDPHWHUV _3(5, IRU Q LV IRU PRVW RI WKH GLVN RQ WKH RUGHU RI \HDUV DQGf§IRU ODUJH HQRXJK If§IUDFWLRQV RI D \HDU )XUWKHU

PAGE 135

IRU ODUJH YDOXHV RI U ,SHUA, FKDQJHV RQO\ VORZO\ ZLWK Z VHH FXUYH H RI )LJXUH f 7KLV VXJJHVWV WKDW DQ\ REVHUYDEOH HIIHFW RI _3(5_ IRU ODUJH 7 ZLOO SRVVHVV D VLPSOH SHULRGLFLW\ ,W LV LQWHUHVWLQJ WR FRPSDUH WKLV ZLWK WKH GD\ SHULRG IRXQG LQ +' E\ 0UV 3HWHUV f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f§WKLV ODVW UHTXLUHV VWDWH RI WKH DUW REVHUYLQJ WHFKQLTXHV +XWFKLQJV HWBDO f 7KH SULQFLSDO VWXGLHV LQ WKLV WLPH UHJLRQ VHYHUDO GD\V DQG VKRUWHUf ZLOO EH GLVFXVVHG EHORZ 7KHUH

PAGE 136

DUH WZR UHSRUWV LQ WKH OLWHUDWXUH RI VWULFWO\ SHULRGLF YDULDWLRQV ERWK DUH 95 YDULDWLRQV /DFRDUUHW f IRXQG D VHYHQGD\ SHULRGLFLW\ LQ +' +XWFKLQJV f IRXQG D GD\ SHULRGLFLW\ LQ \ &DV ,Q WKH VHFRQG FDVH \ &DVf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f DQG +XWFKLQJV f DQG RI +' DV ZHOO DV 'UD +XWFKLQJV HW DO f 7KHVH REVHUYDWLRQV DUH FKDUDFWHUL]HG E\ WKH IROORZLQJ f KLJKO\ FRPSOH[ WLPH GHSHQGHQFH f UDSLG WHPSRUDO YDULDWLRQV f 95 YDULDWLRQV LQ D FRPSOH[ HPLVVLRQDEVRUSWLRQ VWUXFWXUH ,Q WKH REVHUYDWLRQV RI +' DQG L" 'UD WHPSRUDO UHVROXWLRQ ZDV GRZQ WR PLQXWHV YDULDWLRQV ZHUH REVHUYHG WR WKLV OLPLW 7LPHV RI YDULDWLRQ DUH QRW SHULRGV EXW WKH WLPH RYHU ZKLFK

PAGE 137

D FKDQJH ZDV QRWHG 7KHVH FRPSDUH QRW WR SHULRGV EXW UDWKHU WR D QXPEHU OLNH >SHU_ RU _3(5_ ,I ODUJH Q YDOXHV DSSUR[LPDWHO\ WR f DUH LQFOXGHG WKH SUHGLFWLRQV RI WKH DQDO\VLV DQG WKH REVHUYDWLRQV DJUHH RQ DOO WKUHH FKDUDFWHULVWLFV

PAGE 138

&+$37(5 9, (48$7,21 )25 (1(5*< 75$163257 )RUP RI WKH (TXDWLRQ RI (QHUJ\ 7UDQVSRUW 7KH 'LVWLQFWLYH 1DWXUH RI WKH (QHUJ\ (TXDWLRQ ,Q RQH UHVSHFW WKH HTXDWLRQ IRU HQHUJ\ WUDQVSRUW VWDQGV LQ VKDUS GLVWLQFWLRQ WR WKH RWKHU WZR K\GURQ\QDPLFDO HTXDWLRQV WKH YHU\ IRUPXODWLRQ RI D JHQHUDO HTXDWLRQ IRU HQHUJ\ WUDQVSRUW UHTXLUHV DQ LQWLPDWH NQRZOHGJH RI WKH LQWHUDFWLRQ EHWZHHQ WKH UDGLDWLRQ ILHOG DQG WKH GLVN IOXLG $EVRUSWLRQ DQG HPLVVLRQ LQ WKH GLVN GHWHUPLQH WKH H[WHQW RI LRQL]DWLRQ WKURXJKRXW WKH GLVN DQG VLQFH WKHVH SURFHVVHV DFW DV KHDWLQJ DQG FRROLQJ PHFKDQLVPV WKH WHPSHUDWXUH
PAGE 139

UDGLDWLRQIOXLG LQWHUDFWLRQ D PLQLPXP FRQGLWLRQ FDQ EH LPSRVHG 2QH FDQ GHPDQG IRUPDO VLPLODULW\ WR WKH DQDO\VLV RI &KDSWHU ,,, DQG ,9 $ VLPSOH FULWHULRQ SURYLGHV WKLV *LYHQ D FDQGLGDWH HTXDWLRQ WKH IROORZLQJ SURFHGXUHV DUH IROORZHG f W\SLFDO GLVN FRQGLWLRQV DUH LPSRVHG f HDFK IORZ SDUDPHWHU LV H[SUHVVHG DV WKH 6LDP RI WZR WHUPVf§D VWHDG\VWDWH D[LDOO\V\PPHWULF SDUW VXEVFULSW]HUR FRPSRQHQWf DQG D SDUW FRQWDLQLQJ WKH DQJXODU DQG WHPSRUDO GHSHQGHQFH VXEVFULSWRQH FRPSRQHQWf f WKH HTXDWLRQ LV OLQHDUL]HG DQG D UHODWLRQ ZKLFK PXVW EH VDWLVILHG E\ WKH VXEVFULSW]HUR WHQQV LV FRQVWUXFWHG 7KH FULWHULRQ LV WKDW WKH VWDWLF LVRWKHUPDO VROXWLRQV WR WKH HTXDWLRQV RI FRQWLQXLW\ DQG PRWLRQ DOVR EH VROXWLRQV RI WKH UHODWLRQ IRXQG LQ VWHS f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

PAGE 140

7KH ILQDO VHFWLRQV RI WKLV FKDSWHU H[DPLQH ILUVW WKH DJUHHPHQW ZLWK WKH ZRUN RI &KDSWHUV ,,, ,9 DQG 9 DQG WKHQ WKH DGHTXDF\ RI WKH DSSUR[LPDWH IRUP RI WKH HQHUJ\ HTXDWLRQ $Q $SSUR[LPDWH )RUP $Q\ UDGLDWLRQ HIIHFWV HQWHU RQO\ LQGLUHFWO\ WKURXJK WKH LQWHUQDO HQHUJ\ DQG SRVVLEO\ WKURXJK D VRPHZKDW VWHOODU PDVV WHUP RULJLQDWLQJ LQ WKH HTXDWLRQ RI PRWLRQ DQG LQFOXGHG KHUH RQO\ IRU FRPSOHWHQHVVf $ VLPSOH WKUHHHOHPHQW IOXLG FRQVLVWLQJ RI SURWRQV HOHFWURQV DQG QHXWUDO KHOLXP ZLOO EH XVHG 7KH HTXDWLRQ IRU HQHUJ\ WUDQVSRUW FRXOG SURFHHG LQ WKH VDPH PDQQHU DV WKH HTXDWLRQ RI PRWLRQ WKDW LV WKH HTXDWLRQ IRU HQHUJ\ WUDQVSRUW FRXOG EH GHULYHG IURP DQ LQWHJUDO LQYROYLQJ D FRQVHUYHG TXDQWLW\f§WKH WKHUPDO HQHUJ\ 7KH HTXDWLRQ JLYHQ EHORZ LV DQ HTXDWLRQ IRU HQHUJ\ WUDQVSRUW LQ WKH SUHVHQFH RI YLVFRVLW\ ,W LV IURP WKH PDWKHPDWLFDO VWDQGSRLQW WKH VDPH RUGHU RI DSSUR[LPDWLRQ DV WKH 1DYLHU 6WRNHV HTXDWLRQ ,Q SUDFWLFH WKH HTXDWLRQ LV GHULYHG KHXULVWLFDOO\ /DQGDX DQG /LIVKLW] f 7KH HTXDWLRQ LV DW F

PAGE 141

,Q WKLV HTXDWLRQ WKH V\PEROV QRW XVHG EHIRUH DUH f H IRU LQWHUQDO HQHUJ\ GHQVLW\ F f Z IRU HQWKDOS\ F f N IRU WKHUPDO FRQGXFWLYLW\ F f FWn IRU WKH YLVFRVLW\ VWUHVV WHQVRU F f IRU WKH JUDYLWDWLRQDO SRWHQWLDO F 7KH SURGXFW f DA KDV HOHPHQWV YLD FL M ZKHUH Dn F L Q $ [ GY= A Ye f§ f "f§ [f [f DQG UHSHDWHG LQGLFHV LQGLFDWH D VXP 1RWH WKDW WKH VXEVFULSW F ZKLFK ZRXOG QRUPDOO\ DSSHDU ZLWK Y DQG [ DERYH KDV EHHQ VXSUHVVHG LQ WKH H[SUHVVLRQ WR DYRLG FRQIXVLRQ ZLWK WKH LQGLFHV RI VXPPDWLRQ 7KH LQWHUQDO HQHUJ\ GHQVLW\ DQG HQWKDOS\ IRU WKH WKUHHHOHPHQW JDV DUH H F N7 F `LP ;, SP DQG Z F N7 F \P ;, \P

PAGE 142

UHVSHFWLYHO\ ZKHUH ; LV WKH PDVV IUDFWLRQ RI K\GURJHQ DQG LV WKH LRQL]DWLRQ SRWHQWLDO RI K\GURJHQ 5HODWLYH ,PSRUWDQFH RI WKH 7HUPV RI WKH (TXDWLRQV $ QHZ TXDQWLW\ WKH WKHUPDO FRQGXFWLYLW\ DSSHDUV LQ WKH HQHUJ\ HTXDWLRQ 6LQFH WKH PDVV RI WKH HOHFWURQ LV VR PXFK VPDOOHU WKHP WKDW RI HLWKHU WKH SURWRQ RU WKH KHOLXP DWRP WKH WKHUPDO FRQGXFWLYLW\ LQ WKH WKUHHVSHFLHV PL[WXUH LV HVVHQWLDOO\ GHWHUPLQHG E\ WKH HOHFWURQ 7KH WKHUPDO FRQGXFWLYLW\ IRU WKH HOHFWURQ FRPSRQHQW RI WKH JDV LV & [ f 7 F ORJ $Gf H ZKHUH A DQG A DUH GLPHQVLRQOHVV QXPEHUV RI RUGHU XQLW\ JLYHQ E\ 6SLW]HU f 7KH LGHDO JDV ODZ ZLOO EH XVHG WR H[SUHVV WKH ODVW WHUP f LQ WKH HTXDWLRQ DV D IXQFWLRQ RI SUHVVXUH DQG GHQVLW\ 7KH GLPHQVLRQOHVV YDULDEOHV GLVFXVVHG LQ &KDSWHU ,,, DUH LQWURGXFHG LQWR WKH HTXDWLRQ IRU HQHUJ\ WUDQVSRUW JLYLQJ /B DW / ;,S 9WP9 R 0 V 9a5 R R

PAGE 143

;,39 fP9R 0 V 95 R R 9e B U K 9 5 R R R fYLMfM LL U R U +8 N f ZKHUH DQG [ O2A f§ ( O4 $G %RWK VLGHV RI WKLV HTXDWLRQ KDYH EHHQ GLYLGHG E\ WKH TXDQWLW\ -,R9A7Rf 7KH TXDQWLW\ LV RI RUGHU XQLW\ 7KH FRHIILFLHQWV VKRXOG EH PHDVXUHV RI WKH UHODWLYH LPSRUWDQFH RI WKHLU UHVSHFWLYH WHUPV 7R HYDOXDWH WKHP WKH EKDUDFWHULVWLF YDOXHV IROORZLQJ &KDSWHU ,,,f DUH WDNHQ WR EH f Q4 SG f 64 5V 9 9f f

PAGE 144

f 7 5 9 R R f S Q Y R R R 7DEOH FRQWDLQV WKH YDOXHV XVHG IRU WKH TXDQWLWLHV 5 9 V V DQG SA 7KH HTXDWLRQ LV UHZULWWHQ œ DO $ 3 9 $ ) f n >% SY Y %3Y E SY % += % Y D f %IL -f 9 f @ U LL@ n‘Sn .S7KH YDOXHV IRU WKHVH FRHIILFLHQWV DUH JLYHQ LQ 7DEOH 7ZR WHUPV PHULW VSHFLDO FRPPHQW 7KHVH DUH WKH ODVW WZR WHUPV ZKLFK DUH PXFK VPDOOHU WKDQ WKH RWKHU HOHPHQWV RI WKH HTXDWLRQf§VR VPDOO WKDW WKH\ PD\ EH GURSSHG ,Q WKH FDVH RI WKH ODVW WHUP %Jf ZKLFK LV FOHDUO\ QRQn OLQHDU WKLV LV IRUWXQDWH (YHQ WKRXJK ERWK DUH VPDOO WKH GLVSDULW\ EHWZHHQ WKH WZR YDOXHV % DQG %J LV DW ILUVW WKRXJKW GLVWXUELQJ 7KH FRHIILFLHQW %A LV WKH LQYHUVH RI WKH 5H\QROGV QXPEHU WKH WKHUPDO FRQGXFWLYLW\ WHUP LV DSSUR[LPDWHO\ Q 5 R R U Q Y R R R Q U Y F R R R S 31 ZKHUH &S LV WKH VSHFLILF KHDW DW FRQVWDQW SUHVVXUH DQG 31 LV

PAGE 145

7$%/( 7KH 9DOXHV RI WKH &RHIILFLHQWV $SSHDULQJ LQ WKH (QHUJ\ (TXDWLRQ &RHIILFLHQW $ % % % % % % 9DOXH [ [ [ O SURWRQVf RU [ A KHOLXPf [ r

PAGE 146

WKH 3HFOHW QXPEHU 1RZ WKH UDWLR \HW WKH FRQYHQWLRQDO ZLVGRP RI K\GURG\QDPLFV LV WKDW WKHVH WZR QXPEHUV VKRXOG EH URXJKO\ HTXDO 7KLV GLVFUHSDQF\ DSSHDUV WR EH QRW DQ HUURU LQ WKH DQDO\VLV EXW UDWKHU D SHFXOLDULW\ RI WKH WHUQDU\ PL[WXUH FKRVHQ IRU VWXG\ KHUH 7KH SULQFLSDO DJHQW IRU WKHUPDO FRQGXFWLRQ LV WKH HOHFWURQ FRPSRQHQW RI WKH JDV &KDSPDQ DQG &RZOLQJ f ZKLOH SURWRQV DQG QHXWUDO KHOLXP SURYLGH PRVW RI WKH YLVFRVLW\ $ /LQHDUL]HG (TXDWLRQ 6HSDUDWLRQ RI WKH 7LPH DQG $QJOH 'HSHQGHQFH 7KH HTXDWLRQ IRU HQHUJ\ FRQVHUYDWLRQ LV 7KH IORZ YDULDEOHV DUH VHSDUDWHG LQWR VXEVFULSWRQH DQG VXEVFULSW]HUR TXDQWLWLHV DV EHIRUH 7KHVH DUH WKHQ VXEVWLWXWHG LQWR WKH HQHUJ\ HTXDWLRQ ZKLFK EHFRPHV

PAGE 147

Y 9Lf @ AYY 3RAO 9 9R` % 9 SR R %SR9O S 9 UR S Y r R )1U YR SR 3 A SRf 7KH WHUPV ZKLFK LQYROYH VXEVFULSW]HUR TXDQWLWLHV DORQH DUH FROOHFWHG RQ HDFK VLGH RI WKH HTXDWLRQ DQG DUH VHW HTXDO WR HDFK RWKHU 7KH HTXDWLRQ ZKLFK PXVW EH VDWLVILHG E\ WKH VXEVFULSW ]HUR VROXWLRQV LV %SR 9 %S Y R R 9 R )1U S 9 .R RM 6LQFH WKH VXEVFULSW]HUR TXDQWLWLHV DUH ILUVW VWHDG\VWDWH DQG D[LDOO\V\PPHWULF WKHUH LV QRW WLPH RU DQJOH GHSHQGHQFHf WKH HTXDWLRQ IRU WKH VXEVFULSW]HUR WHUPV EHFRPHV > XS 9 0R XR Y R 8%S 9 R XR XS Y rR XR XS 9 .R XR @ X )1U

PAGE 148

U SR9!BR S A6/3 B/ SY -] / R ]R )1U 0RY]R ZKHUH 9 9 9 9 2 =2 S2 7KH VXEVFULSW]HUR VROXWLRQV XVHG KHUH DUH DOVR VWDWLF DQG LVRWKHUPDO VROXWLRQV LQ ZKLFK YR f YRXf 7KHVH VROXWLRQV VDWLVI\ WKH HTXDWLRQ DERYH LGHQWLFDOO\ 7KLV IRUP RI WKH HQHUJ\ HTXDWLRQ VDWLVILHV WKH FULWHULRQ SURSRVHG LQ WKH ILUVW VHFWLRQ RI WKLV FKDSWHU $Q (TXDWLRQ IRU WKH 6XEVFULSW2QH 7HUPV 7KH VWDWLF LVRWKHUPDO VROXWLRQV GR VDWLVI\ WKH UHODWLRQ IRU VXEVFULSW]HUR TXDQWLWLHV ZKLFK UHVXOWV IURP WKH HQHUJ\ HTXDWLRQ $FFRUGLQJO\ WKH VWDWLF LVRWKHUPDO VROXWLRQV XVHG IRU VXEVFULSW]HUR TXDQWLWLHV LQ &KDSWHUV ,9 DQG 9 ZLOO EH XVHG KHUH 7KH\ DUH O! YFRR 9 WW 7 U

PAGE 149

f Y]4 2 f SR H[S >L / 4 f 34 7H[S ON f Y W s 4 f Lf@ f@f 7KH HTXDWLRQ IRU WKH VXEVFULSWRQH WHUPV LV A 3 f} 7W  SRYL!RYW!O YrR 3 %SO )“ f L LX AU %3 L ,-A )1U .RFRO 3 %SR 77 E, f Ym 7 )) YrR %YrRf 3 YcWRSO n A )1U 7-!R ‘ a I!RSRf %SR SR f Y]O @  >rerr}rfr "ef Y0O@ ZSUW XS 6 8 W IY/SR %S 3T B3T f )1U n f r )1U bR A 3 9-!RSO @ U U SRYG! B rR SR Y ] /  a %SR )1U 9]O 3

PAGE 150

7KH VXEVFULSW]UR TXDQWLWLHV DUH NQRZQ IXQFWLRQV DQG FRQVHTXHQWO\ DOO GHULYDWLYHV RI WKHVH NQRZQ TXDQWLWLHV PD\ EH REWDLQHG )XUWKHU WKH PDWKHPDWLFDO IRUP RI WKHVH IXQFWLRQV DQG WKHLU GHULYDWLYHV VXJJHVWV WKDW WKH VXEVWLWXWLRQ RI H[SUHVVLRQV YDOLG IRU ]Xf ZLOO VLPSOLI\ WKH FDOFXODn WLRQV DW QR JUHDW FRVW LQ XVHIXOQHVV 7KH QHZ GHSHQGHQW YDULDEOH 9" 9 L L H[S ] 4Zn LV LQWURGXFHG LQWR WKH HTXDWLRQ 7KH UHVXOW RI DOO WKHVH VWHSV LV fI L DX SL SL SL [ SL W W DW fDW )“U fDW I % A H < FMI < YfO I 9 A A 8f n 9GfO f§ % L L Gf DZ 31U f )1U rVrO \ Y X Z )1UZ Dr B DS %I )1U n W! I IIL BD I If )1 ] ]O A E 6_ /IO ,Q WKH UHVW RI WKLV FKDSWHU WKH VXSHUVFULSW H ZLOO EH GURSSHG WKH YHORFLW\ ZLOO FRQWDLQ DQ DEVRUEHG H[SRQHQWLDO XQOHVV RWKHUn ZLVH QRWHG

PAGE 151

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

PAGE 152

'RLQJ WKLV ZLOO UHTXLUH WKH IROORZLQJ LQWHJUDWLRQV ] Q S f (f G] n ] = V L KL 8f9 f X Z ] R f G] ] 9 U r f f§ rU[a G] UDf BB R = A3L mm U -7 G= f f O=r g f f R 7KH UDWKHU WHGLRXV PDWKHPDWLFV UHTXLUHG WR FRQVWUXFW VXLWDEOH H[SUHVVLRQV IRU WKH LQWHJUDOV LV FRQWDLQHG LQ $SSHQGL[ ( WKH UHVXOWLQJ H[SUHVVLRQV IRU WKHVH LQWHJUDOV DUH f

PAGE 153

D m!‘ $ f 7 X ^ 7a f Y]O]R! 7KH QHZ SDUDPHWHUV $nDQG $n DUH GHILQHG E\ WKH H[SUHVVLRQV U]r ]=YfLG] ‘ Dn]R G] = ] f R R DQG ] YG] T!L G] ) UHVSHFWLYHO\ %RWK RI WKHVH SDUDPHWHUV DUH FRQVLGHUHG FRQVWDQW 2QH GHSDUWXUH IURP WKH SDWWHUQ RI DQDO\VLV IURXQG LQ &KDSWHU ,9 LV PDGH KHUH 7KH TXDQWLW\ K &r SL G] 2 LV GHILQHG 7KLV UHPRYHV WKH QHHG IRU DQ DVVXPHG UHODWLRQ EHWZHHQ DQG SA LW LV WKLV UHODWLRQ ZKLFK GHILQHV WKH SDUDPHWHU 7 2WKHUZLVH WKH VWHSV RXWOLQHG LQ WKH ODVW VHFWLRQ RI &KDSWHU ,9 DUH IROORZHG KHUH 7KH UHVXOW RI DOO WKLV LV

PAGE 154

W W L KDr n D A Z f 7 )1WR )1&7KH 4XHVWLRQ RI &RQVLVWHQF\ 7KH DSSUR[LPDWHG IRUP RI WKH HQHUJ\ HTXDWLRQ DQG WKH UHVXOWV RI &KDSWHU ,9 DQG 9 DUH FRQVLVWHQW LI WKH VROXWLRQV ZRUNHG RXW LQ WKHVH FKDSWHUV VDWLVI\ WKDW HTXDWLRQ 7R WHVW WKLV WKH UHODWLRQ D"O UHII LRO LV VXEVWLWXWHG LQWR WKH HTXDWLRQ GHULYHG DW WKH HQG RI WKH ODVW VHFWLRQ 7KH H[SUHVVLRQ IRU WKH IORZ YDULDEOHV XA XZO DQG 2Mf GHULYHG LQ &KDSWHU ,9 DUH XVHG WR HYDOXDWH WHUPV LQ WKH HQHUJ\ HTXDWLRQ ZKLFK FRQWDLQ WKHVH YDULDEOHV RU WKHLU GHULYDWLYHV 2QH WKHQ KDV DQ DOJHEUDLF H[SUHVVLRQ ZKLFK FDQ EH XVHG 6HH $SSHQGL[ )f WR GHYHORS DQ H[SUHVVLRQ HII I IOF' Q$,M$nf

PAGE 155

7KH QRWDWLRQ N7f VLJQLILHG WKDW 7 HQWHUV RQO\ WKURXJK N 7KLV HTXDWLRQ GHSHQGV GLUHFWO\ RQ Q$7n DQG $n DQG LQGLUHFWO\ RQ 7 +RZHYHU WKH TXDQWLWLHV $n DQG $ KDYH OLWWOH LQIOXHQFH RQ WKH HTXDWLRQ $ VLPSOH SURFHGXUH VXJJHVWV LWVHOI 9DOXHV DUH FKRVHQ IRU HDFK RI WKH TXDQWLWLHV Q$$n$n DQG 7 $V LQ &KDSWHU 9 WKH QXPEHUV N DUH GHWHUPLQHG 7KH H[SUHVVLRQ IRU 7Hee PD\ WKHQ EH HYDOXDWHG ,I WKH VROXWLRQV GHULYHG HDUOLHU VDWLVI\ WKH HQHUJ\ HTXDWLRQ WKHQ U U HII 7KH DGGLWLRQ RI D VKRUW VXEURXWLQH WR WKH H[LVWLQJ SURJUDPV DOORZV RQH WR SHUIRUP WKHVH VWHSV VHH $SSHQGL[ 'f 7KH UDQJH RI 7 ZDV VHW IURP WR DQG UHSUHVHQWDWLYH YDOXHV RI WKH RWKHU TXDQWLWLHV XVHG RYHU WKLV UDQJH LW LV IRXQG WKDW U } L f U 7KXV WKH DSSUR[LPDWHG IRUP RI WKH HQHUJ\ HTXDWLRQ DQG WKH UHVXOWV RI &KDSWHU ,9 DQG 9 DUH LQFRQVLVWHQW $ 5HFRQVLGHUDWLRQ RI WKH $SSUR[LPDWH (QHUJ\ (TXDWLRQ 1RQ6WDWLF 6ROXWLRQV DQG WKH $GHTXDF\ RI WKH (TXDWLRQV ,Q YLHZ RI WKH LQFRQVLVWHQF\ EHWZHHQ WKH VROXWLRQV

PAGE 156

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nV QRQVWDWLF VROXWLRQV VHH &KDSWHU ,,f GR QRW H[LVW LQ FORVHG IRUP EXW IRU DQ\ RI KLV VROXWLRQV WW a 3f Z f Sf

PAGE 157

ZKHUH S LV WKH SDUDPHWHU GLVFXVVHGLQ &KDSWHU ;, )XUWKHU WKH FLUFXODU YHORFLW\ LV NQRZQ 7KH VWUDWHJ\ KHUH ZLOO EH WR H[DPLQH WKH QH[W PRVW VLPSOH FDVH DIWHU WKH VWDWLF LVRWKHUPDO HQYHORSHf§WKH QRQn VWDWLF LVRWKHUPDO HQYHORSH 2QO\ WKH UHJLRQ QHDU WKH HTXDWRULDO SODQH ZLOO EH FRQVLGHUHG 7KH DERYH HTXDWLRQ XQGHU WKHVH FRQGLWLRQV OHDGV LPPHGLDWHO\ WR WKH UHVXOW WKDW WKH TXDQWLW\ %/ 6,` f Z )1U Z Sf VKRXOG EH FRQVWDQW ZLWKLQ D VPDOO VHFRQGRUGHU WHUP -2 AWWf Sf 9 Mf f S +RZHYHU WKH TXDQWLW\ DERYH LV FOHDUO\ QRW D FRQVWDQW 7KHUHIRUH WKH HTXDWLRQ LV QRW VDWLVILHG LQ WKLV VLPSOH FDVH 7KH DSSUR[LPDWH IRUP RI WKH HQHUJ\ HTXDWLRQ PXVW EH KHOG LQ GRXEW )LQDO &RPPHQW ,Q WKLV FKDSWHU WKH GLVWLQFWLYH QDWXUH RI WKH HQHUJ\ HTXDWLRQ ZDV GLVFXVVHG $ VLPSOH FULWHULRQ IRU

PAGE 158

FKRRVLQJ WKH DSSUR[LPDWH IRUP RI WKH HQHUJ\ HTXDWLRQ ZKLFK UHSUHVHQWV WKH VDPH IRUPDO PDWKHPDWLFDO RUGHU RI DSSUR[LPDWLRQ DV WKH DQDO\VLV RI WKH SUHYLRXV WZR FKDSWHUV ZDV SURSRVHG 6XFK DQ HTXDWLRQ ZDV IRXQG ,Q WKH HQG KRZHYHU WKLV HTXDWLRQ ZDV LQFRQVLVWHQW ERWK ZLWK WKH DQDO\VLV RI &KDSWHUV ,9 DQG 9 DQG ZLWK WKH QRQVWDWLF VWHDG\VWDWH D[LDOO\ V\PPHWULF VROXWLRQV 7KH VROXWLRQV GHYHORSHG LQ ERWK WKH SUHYLRXV VWHDG\ VWDWH VWXGLHV DQG LQ WKLV ZRUN GHSHQG RQ HTXDWLRQV FRQWLQXLW\ DQG PRWLRQf ZKLFK FRQWDLQ QR UDGLDWLRQ GHSHQGHQW WHUPV VDYH IRU D VOLJKWO\ UHGXFHG VWHOODU PDVV LQ WKH JUDYLWDWLRQDO WHUPf 7KLV QHJOHFW KDV EHHQ DGHTXDWHO\ MXVWLILHG /LPEHU DQG 0DUOn ERURXJK f 7KDW WKH VROXWLRQV EDVHG RQ WKHVH HTXDWLRQV GR QRW VDWLVI\ DQ HQHUJ\ WUDQVSRUW HTXDWLRQ ZLWKRXW UDGLDWLRQ WHUPV LV QRW DQ LQGLFDWLRQ WKDW WKH VROXWLRQV DUH LQYDOLG EXW UDWKHU WKDW UDGLDWLRQ PXVW SOD\ DQ LPSRUWDQW UROH LQ WKH HQHUJ\ HTXDWLRQ 7KLV LV VR HYHQ WKRXJK UDGLDWLRQ HIIHFWV DUH RI QR FRQVHTXHQFH LQ WKH RWKHU WZR HTXDWLRQV $Q HQHUJ\ HTXDWLRQ ZKLFK FDQ EH XVHG WR LPSURYH WKH H[LVWLQJ VROXWLRQV DZDLWV IXUWKHU VWXG\ RI WKH UDGLDWLRQ ILHOG LQ WKH GLVN

PAGE 159

&+$37(5 9,, f),1$/ &200(176 6WDELOLW\ RI WKH 6WHDG\6WDWH )ORZV 7KH H[LVWLQJ VWHDG\VWDWH VROXWLRQV WR WKH K\GUR G\QDPLFDO IORZ LQ %H VWDUV DUH DSSUR[LPDWH VROXWLRQV WR DSSUR[LPDWH IRUPV RI WKH HTXDWLRQV RI FRQWLQXLW\ DQG PRWLRQ \HW LQ SULQFLSOH WKHUH LV DQ H[DFW VWHDG\VWDWH VROXWLRQ WR WKH H[DFW K\GURG\QDPLFDO HTXDWLRQV IRU DQ\ SUREOHP :LWK UHJDUG WR VWDUV WKLV PD\ EH RI OLWWOH YDOXH IRU RQH QHHG ORRN QR IXUWKHU WKDQ RXU 6XQ WR VHH WKDW D VWHOODU VXUIDFH LV QRW D WUXH VWHDG\VWDWH ERXQGDU\
PAGE 160

DQG 9 PD\ EH XVHG LQ DQ LQYHVWLJDWLRQ RI WKH VWDELOLW\ RI VWHDG\VWDWH IORZV 7KH VXEVFULSWRQH WHUUD PD\ EH YLHZHG QRZ DV D SHUWXUEDWLRQ 7KH JHQHUDO VWDELOLW\ SUREOHP IRU VWHDG\ IORZV DERXW D ILQLWH ERG\ KDV QHYHU EHHQ VROYHG 2Q WKH EDVLV RI WKH VSHFLDO FDVHV ZKLFK KDYH EHHQ VXFFHVVIXOO\ WUHDWHG D SDWWHUQ HPHUJHV 7KH VROXWLRQV WR WKH SHUWXUEDWLRQ HTXDWLRQV DUH WKRVH ZLWK D WLPH GHSHQGHQFH LQ WKH IRUP H[S LXWf ZKHUH X LV D IUHTXHQF\ ZKLFK FDQ EH GHWHUPLQHG RQO\ E\ VROYLQJ WKH SHUWXUEDWLRQ HTXDWLRQV ZLWK UHVSHFW WR WKH SUHVFULEHG FRQGLWLRQV 7KLV PXFK LV VLPLODU WR WKH DQDO\VLV RI WKH DSSUR[LPDWHG YHUVLRQ RI WKH OLQHDUL]HG HTXDWLRQV GLVFXVVHG LQ &KDSWHU ,9 +RZHYHU X EHLQJ D UHDO QXPEHU LV JHQHUDOO\ WKH H[FHSWLRQ UDWKHU WKDQ WKH UXOH 'HSHQGLQJ XSRQ WKH SUREOHP X PD\ KDYH LPDJLQDU\ SDUWV ZKLFK OHDG WR GDPSHG SHUWXUEDn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f§QRU LV WKHUH DQ\ UHDVRQ WR VXSSRVH WKDW WKH\ ZLOO

PAGE 161

7KH JHQHUDO VROXWLRQ IRU HDFK IORZ YDULDEOH LV VWLOO WKH VXP RI WKH VWHDG\VWDWH WHUP DQG WKH SHUWXUEDWLRQ 2QH KDV SDVVHG LQWR WKH UHJLRQ RI SHULRGLF IORZV LQ WKH VHQVH WKDW WKH VXEVFULSWRQH IORZV DUH SHULRGLFfIDQG WKH VWHDG\VWDWH IORZV DUH DW EHVW DSSUR[LPDWLRQV WKRXJK WKH\ PD\ EH JRRG RQHVf /RJLFDOO\ RQH PXVW WKHQ SURFHHG WR WKH TXHVWLRQ RI VWDELOLW\ RI HYHQ SHULRGLF IORZV ,Q SULQFLSOH RQH FRXOG SHUIRUP D VWDELOLW\ DQDO\VLV RQ WKH VXEVFULSWRQH WHUPV LQ PXFK WKH VDPH PDQQHU WKDW ZDV IROORZHG LQ WKH DQDO\VLV RI WKH VWDELOLW\ RI VWHDG\VWDWH IORZV 6XFK DQ DQDO\VLV LV FKDUDFWHUL]HG E\ H[FHSWLRQDO PDWKHPDWLFDO GLIILFXOW\ DQG LV VHOGRP DWWHPSWHG IRU HYHQ WKH VLPSOHVW IORZV 1RQH ZLOO EH DWWHPSWHG KHUH ,I WKH SHULRGLF IORZ ZHUH XQVWDEOH WKH IORZ ZRXOG EH WXUEXOHQW $ WUHDWPHQW RI WXUEXOHQFH LV EH\RQG WKH VFRSH RI WKLV ZRUN &RQFOXVLRQV ,W ZDV FRQFOXGHG WKDW WKH PRWLRQ DQG GLVWULEXWLRQ RI WKH FLUFXPVWHOODU PDWWHU DERXW W\SLFDO %H VWDUV FRXOG EH WUHDWHG K\GURG\QDPLFDOO\ 7\SLFDO GLVN FRQGLWLRQV SODFHG WKH IORZ ZLWKLQ WKH UHDOP RI K\GURG\QDPLFV EXW FORVH WR WKH OLPLWV RI YDOLGLW\ RI WKDW GLVFLSOLQH ,W ZDV DOVR VKRZQ WKDW RYHU PRVW RI WKH GLVN (XOHUnV HTXDWLRQ FRXOG EH XVHG IRU WKH HTXDWLRQ RI PRWLRQ (DFK IORZ YDULDEOH ZDV ZULWWHQ DV WKH VXP RI D NQRZQ

PAGE 162

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f ZHUH XQGHU PRVW FRQGLWLRQV UHDO )XUWKHU WKHLU GHSHQGHQFH RQ Q DQG Z ZDV JHQHUDOO\ f Q 7KHVH FDOFXODWLRQV ZHUH LQ TXDOLWDWLYH DJUHHPHQW ZLWK UHOHYDQW REVHUYDWLRQV 7KH WKLUG HTXDWLRQ RI K\GURG\QDPLFV WKH HQHUJ\ HTXDWLRQ ZDV H[DPLQHG LQVRIDU DV FRXOG EH LQ WKH DEVHQFH

PAGE 163

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

PAGE 164

$33(1',&(6

PAGE 165

$33(1',; $ ([FKDQJH RI D 'LIIHUHQWLDO DQG DQ ,QWHJUDO 2SHUDWRU /HW IAZf DQG AAfA EH IXQFWLRQV RI X VXFK WKDW IXf IZf IXUWKHU OHW JZ-!]f EH D ZHOO GHILQHG IXQFWLRQ LQ WKH LQWHUYDO IURP ] IXf WR ] Iff :KDW LV WR EH VKRZQ LV WKDW IZ! D I Xf JZIL]f G] I  IAZf 'HILQH WKH TXDQWLW\ ]n + Z I! ] f JZM!]f G] FRQVWDQW 1RZ I/ RMf JRfI!]f G] IMALZf

PAGE 166

+Z-!If + !IAf KHQHH D f IWf D77 JDcW!]f G] f [0 >+Pref +FRrI-f@ ,Q WKH SDUWLDO GLIIHUHQWLDWLRQ WKH Z GHSHQGHQFH WKDW HQWHUV WKURXJK WKH IXQFWLRQV IAZf DQG ILf FDQ EH IRUPDOO\ VHSDUDWHG IURP WKH JHQHUDO Z GHSHQGHQFH WKLV LV VKRZQ EHORZ 6L W+fnrnIf UDI A emfnrnI! IO! n ZKHUH WKH EDU RYHU IA DQG I V\PEROL]HV WKDW WKH SDUWLDO GLIIHUHQWLDWLRQ WUHDWV WKHVH DV XQUHODWHG YDULDEOHV %XW _MM JRLL]f G] DQG VLQFH WKH SDUWLDO GLIIHUHQWLDWLRQ GRHV QRW VHH IA DQG I a JZ-!]f G] ‘ JDf-f]f G]

PAGE 167

,W UHPDLQV WKHQ RQO\ WR VKRZ WKDW I I Lmfn e! W, DVU 6LQFH I/ Zf I WRf LW IROORZV WKDW mf DQG A IA IIA Ir &OHDUO\ + IO B B+ I IA I Z KHQFH + ee B B+ e A IA Gf DQG D Iff I Rf JU J'Lf]f G] JDf I! ]f G] IMLZf I OGfr

PAGE 168

$33(1',; % 6HSDUDWLRQ RI WKH (TXDWLRQV DQG WKH N 5HODWLRQ 7KH WKUHH HTXDWLRQV ZKLFK FRQWDLQ WKH GHSHQGHQW YDULDEOHV DUH OZO eY ,, r, n 8f I &B DA 8XfO f X X /D f Lf -! ,Q RUGHU WR DSSO\ WUDGLWLRQDO PDWKHPDWLFDO SURFHGXUHV WR WKLV SUREOHP LW LV XVHIXO WR FRQVWUXFW VHSDUDWH HTXDWLRQV IRU HDFK IORZ YDULDEOH 8VLQJ WKH WKLUG HTXDWLRQ RQH FDQ ZULWH WKDW 9O f>/RL Y@ f 7KLV PD\ EH XVHG LQ WKH UHPDLQLQJ WZR HTXDWLRQV JLYLQJ > / I @ XQ O! c @r

PAGE 169

DQG U O f e 9 IFR/ & ‘/ / / I! B B FR S 7KH RSHUDWRUV ZKLFK SUHFHGH DA LQ HDFK HTXDWLRQ DUH DOORZHG WR RSHUDWH RQ WKH RWKHU HTXDWLRQ 6LQFH WKHVH WZR RSHUDWRUV FRPPXWH WKH ULJKWKDQG VLGHV RI WKH UHVXOWLQJ HTXDWLRQV DUH HTXDO 6XEWUDFWLQJ RQH IURP WKH RWKHU \LHOGV F/ FI U B  D B  R B A r B K? K r WR f X I/nn DW f 7 M I R/ R L 7 X/ 7 Dr 7 / f FOI nD mR A XfO RU ^ LR/n F/ Ef AU A -! ZI f / FfI F I f f6f§ AXQ /HW 8-! 8QN H[S >LQWr! LNZQfW@ n DQG VXEVWLWXWH WKLV LQWR WKH HTXDWLRQ DERYH 7KH UHVXOW RI WKHVH VWHSV LV fD FQ F R L ^ FR ,F QIf f§f§ N QIf f§M XIA f N QIf FQI FI Laf§U Q `

PAGE 170

+HQFH U I &Q RN DPIN F RIA WRQ I f N WR WR FBQ I FB IQ FQQI DIQ XQI $ f f WR 0XFK RI WKLV DQDO\VLV PD\ EH XVHG IRU WKH UHGXFWLRQ WR DQ HTXDWLRQ IRU JA 7KH RSHUDWRUV ZKLFK SUHFHGH X A LQ HDFK RI WKH WZR HTXDWLRQV / K I ` A Z/ f D WR DQG X I!O IWR/ OL B WR f n DUH DOORZHG WR RSHUDWH RQ WKH RWKHU HTXDWLRQ 6LQFH WKH RSHUDWRUV FRPPXWH WKH OHIWKDQG VLGHV DUH QRZ HTXDO 6XEWUDF WLRQ DQG VRPH UHDUUDQJHPHQW JLYHV FB/ 2I f / WR FI &OI L L n D f ` 7KH VWHSV WR WKH N HTXDWLRQ QHHG QRW EH UHSHDWHG 7KH WKLUG RI WKH WKUHH RULJLQDO HTXDWLRQV FDQ EH /FU XP L D f 8! r UHZULWWHQ

PAGE 171

6XEVWLWXWLRQ LQWR WKH UHPDLQLQJ HTXDWLRQV JLYHV DQG L n F f RLO I/ f U B -f X !O I BL WN n f I! n XO U I BO R LW Of X A 7KH PXWXDOO\FRPPXWLQJ RSHUDWRUV SUHFHGLQJ XA LQ HDFK HTXDWLRQ DUH DOORZHG WR RSHUDWH RQ WKH RWKHU HXTDWLRQ 2Q VXEWUDFWLRQ DQG UHDUUDQJHPHQW RQH KDV ^Df/f§L LI f / G3 FI RLn FM/ R -! ` Y RL f

PAGE 172

$33(1',; & (YDOXDWLRQ RI WKH 'HWHUPLQDQW )RUPHG IURP WKH (TXDWLRQV 5HODWLQJ 8A &A 7KH GHWHUPLQDQW ZKLFK PXVW EH HYDOXDWHG LV LN QIf I X I L N QIf LQRf LQFR} LN QIf 2Q H[SDQVLRQ LQ WHUPV RI LN QIf [ GHWHUPLQDQW RQH KDV LN QIf LQFR! LQX LN QIf I I LQFRf & f I LN QIf f LN QIf &2 f LQRf 7KLV JLYHV B A Q e QFB LN QIf N QIf A A f LI N QIf f f

PAGE 173

IL f U IQ B N nn KIf 9 Z X n Z RU L ^ e QI-F I IQ U f e f & FQ I FBIQ A FQQI 9 IQ eQ f ` ‘ f Z I 7KH TXDQWLW\ LQ EUDFNHWV LV WKH HTXDWLRQ IRU N 7KH GHWHUPLQDQW LV LGHQWLFDOO\ ]HUR ,I D QRQ]HUR [ GHWHUPLQDQW FDQ EH IRUPHG IURP PDWUL[ HOHPHQWV DERYH WKH PDWULF LV RI UDQN 7KHUH DUH WKUHH [ GHWHUPLQDQWV ZKLFK DSSHDU LQ WLLH H[SDQVLRQ RI WKH [ GHWHUPLQDQW DERYH 7KH FRQGLWLRQ WKDW DOO WKUHH EH ]HUR LV WKDW 2YHU PRVW RI WKH GLVN VXFK D YDOXH UHSUHVHQWV D ODUJHUYDOXH RI UFf WKDQ WKRVH LQ XVH KHUH )RU WKH FRPSXWDWLRQV GHVFULEHG LQ &KDSWHU 9 WKH PDWUL[ LV RI UDQN

PAGE 174

$33(1',; 7KH 3URJUDP &$/62/ 7KH PDMRU YDULDEOH QDPHV IRU WKH SURJUDP DUH UHODWHG WR WKH V\PEROV XVHG LQ WKLV ZRUN LQ 7DEOH 7KH SURJUDP KDV IRXU PDMRU GLYLVLRQV f 0DLQ SURJUDP f 6XEURXWLQH 32/57 f 6XEURXWLQH $/3O f 6XEURXWLQH %(7 7KH VHFRQG GLYLVLRQ 32/57 LV HVVHQWLDOO\ XQFKDQJHG IURP WKH GLVFXVVLRQ JLYHQ LQ WKH ,%0 SXEOLFDWLRQ 6\VWHP 6FLHQWLILF 6XEURXWLQH 3DFNDJH 9HUVLRQ ,,, 3URJUDPPHUnV 0DQXDO 7KH PDLQ SURJUDPf§PLQXV &20021 DQG ',0(16,21 VWDWHPHQWVf§LV JLYHQ LQ 7DEOH 7KHUH DUH LQWHUQDO FRQWURO VWDWHPHQWV LQ WKH SURJUDP ZKLFK FKDQJH IURP UXQ WR UXQ :KHQ WKHVH RFFXU LQ WKH SURJUDP WKH H[SUHVVLRQ FRQWURO n[n DSSHDUV ZKHUH n [n LV D QXPEHU 7KH ORFDWLRQV RI WKHVH VWDWHPHQWV LV PDUNHG E\ D GRXEOH DVWHULVN rrf WR WKH OHIW RI WKH VWDWHPHQW &21752/ DQG &21752/ WRJHWKHU FRQWURO WKH YDOXHV RU VHWV RI YDOXHV ZKLFK FA DQG DUH WR DVVXPH &21752/ WHOOV WKH SURJUDP DW ZKDW YDOXHV RI Q WKH FDOFXODWLRQ LV WR EH SHUIRUPHG DQG &21752/ FKRRVHV WKH SRLQWV LQ WKH GLVN IRU ZKLFK WKH FRPSXWDWLRQV DUH WR EH SHUIRUPHG

PAGE 175

6XEURXWLQH $/3 7KLV SURJUDP FDOFXODWHV WKH TXDQWLW\ U A LQ &KDSWHU 9,, 7KH SURJUDP LV VKRZQ LQ 7DEOH GLVFXVVHG 6XEURXWLQH %(7, 7KLV VXEURXWLQH FDOFXODWHV WKH GHQVLW\ WKH FLUFXODU YHORFLW\ DQG WKH UDGLDO YHORFLW\ LQ WKDW RUGHU ZKHQ JLYHQ 6Qe 7KLV SURJUDP LV VKRZQ LQ 7DEOH $JDLQ DOO VHFRQGDU\ VWDWHPHQWV KDYH EHHQ GHOHWHG

PAGE 176

7$%/( 7KH 3URJUDP 1DPHV IRU ,PSRUWDQW 4XDQWLWHV 60$6 0V & 6 F 65$' 5V 20(* f 69(/ 9 V 1 Q 508 6 9 52<75 5HDO SDUW RI 57(03 7 H 52<7, 3DUW RI N 5/$0 $ &$/*$0 UHII 5*$0 6 U '(11 D 4 4 62/1 X}O 36, r 5$'1 XFRO &O & 6. 6QN

PAGE 177

7$%/( 0DLQ 3URJUDP 5($'f60$665$'69(/I50857(03 )250$7)f 5($'f5/$0Of5*$0f )250$7)f 5($'f)$&725 )250$7)f '2 ,1 )$&75,1 f )$&725,1 f &217,18( :5,7(f )250$7;n0$66 5$',86 685)$&( 9 02/ :7 7(03(5$785(nf :5,7(f60$665$'69(/50857(03 :5,7(f )250$7n 7+( 7:2 3$5$0(7(56 $5(nf :5,7(f5/$0f5*$0f ( 5. /( 5$7:7 ( 4 5.r65$'r(frfr57(03f508r5$7:7r60$6r*r(f 36, rfrrr5.r57(03rf508r5$7:7r69(/rrfr( ff :5,7(f )250$7;n4 $1' 36, $5( *,9(1 %(/2: nf :5,7(f436, )250$7)f &219/O r36,r5/$0Offr*f &219/ 5*$0f &21752/ &21752/ &21,,1f &219/Or)$&725,1f &21,1f &219/ r)$&75,1 f :5,7( f )250$7;n9$/8( )25 &2167$176 $5( *,9(1 %(/2:nf

PAGE 178

7$%/( FRQWLQXHG :5,7(f&21,,1f&21,1f )250$7)f &21752/ :5,7( f )250$7 n $1* 1 5$',86 &$/ 1 3(5,2'nn 21 )2//2:,1* /,1(6 '(16,7< &,5 9(/ $1' 5$' 9(/ 7,0( 9$5,(6 0267 5$3,'/< nf &21752/ 20(* O-f 9& r20(*rrOf &O &21O,1f & &21,1f <&2)f &rrrfr9&f20(*r&r9&r,f20(*20(*r9&rrfr 0(*rrrfr9&rrf&r,r9&fr20(*rrff <&2)f &rrrff20(*20(*r9&rrfr20(*rrrfr9&rrf &0(*rrf <&2)f r20(*r,r9& <&2)f 20(* &$// 32/57<&2)<2)52<7552<7,,',f '2 1 ,)52<7,1ff &217,18( 9$/$% $%652<751f 9$/&3 ( ,)9$/$%/79$/&3f *R 7R 3(5 rrfrfr65$'69(/r52<751frff &$// $/3 &$// %(7 ,'(7 ,);&.f &$/*$0 ,'(7 &217,18( :5,7(f20(*13(5'(111/.f62/11/.f5$'11/.f. f/ f

PAGE 179

7$%/( FRQWLQXHG )250$7;;);,,;()ff *2 72 &217,18( :5,7(f )250$7 n ,6 $3352$&+,1* =(52 +(5( nf &217,1-( &217,18( &217,18( 6723 (1'

PAGE 180

7$%/( 6XEURXWLQH $/3 %)*01 & &Of & &Of 3$5 52<751f,r9&ff '(7 3$5rrf9&rrfff ;&. ,)$%6'(7ff &217,18( *2 72 &217,18( ;&. *2 72 &217,18( ;%$5 &/r3$520(*rrffr9&r,r&20(*ff'(7 < ,r&r3$520(7f&r9&r20(*rrffff'(7 )&1f 20(*r9& )&1f 20(*rrfr9&rr#ff%)*0136,*f20(*f&Of20(*rr fff )&1f %)*01f20(*fr36,fr20(*ff&20(*rrfff )&1f r20(*r9&rrfff%)*0120(*fr36,fr20(*ff36 ,4f20(*rrff&r20(*rrffff )&1 f 20(*rrf r 9&rrf f%)*0120(*f 9&r 36,4f 20(*f O&Or9&fr20(*rrffff &$/*$0 )&1 f r72<75&f f )&1 f r,f )&1f r;%$5f )&1 f r,r
PAGE 181

7$%/( 6XEURXWLQH %(7O $5* rf3(5 $5* fr 6.-1f Mff '2 ,3+& '2 ,7,& ,7,0 ,7/f ,3+, ,3+2O '(111,3+2,7,2f 6.-1fr&26$5*Or,7,0f$5*r,3+,fff 62/11,3+,7,f 6.-1fr
PAGE 182

$33(1',; ( 3HUWLQHQW ,QWHJUDOV 7KH LQWHJUDO ] ] 3; G] ,W 67 LI f S[G] UH ] L MB W D Z f 2O ZKHUH ] Z DQG KLJKHU SRZHUV RI ]Z KDYH EHHQ QHJOHFWHG )RU WKH UHJLRQ RI LQWHUHVW _]_R f WKHVH PD\ EH QHJOHFWHG HYHQ XQGHU WKH LQWHJUDO 8VLQJ WKH GHILQLWLRQV RI &KDSWHU ,9 ] ] A U G] I s $ f rff f LI D B 7KH LQWHJUDO ] ] f W B L M B U rILL AYZO MB RM UYr!OnG] Z ‘r AW f G U YZOn G] f§ ] B ] f ] R ]B f= YPO G= D ] f YRfO Xf G]

PAGE 183

f fcf\ YXLD] i c rY}LG] ] ]B R R $JDLQ WKH WHUPV LQ ] Z DUH VR VPDOO WKDW WKH\ FDQ EH VDIHO\ QHJOHFWHG 7KLV H[SUHVVLRQ UHGXFHV WR ]R L U ,6 } @ GV $ $ X WR n X nfU 9GfOfn nfZ D! A ZOr ] Zn 7KH LQWHJUDO ] ] I LLY UX I! G] f§ B GWS ] I f§ Y G] BB UX I!O ]B ] W R Y I!O 7f G] 7 RU WR WHUPV LQ ]AZA DQG KLJKHU 7KLV H[SUHVVLRQ JLYHV W 9! MB B XFIfO UX I! G] / r A ZA I! 7KH LQWHJUDO ] ]B ] U O U Z UO = A U If SOG] If B] 3 / XAG= WR WKH XVXDO DSSUR[LPDWLRQ 7KLV FDQ EH ZULWWHQ

PAGE 184

SO U If G] 7KH LQWHJUDO UHGXFHV WR f LPPHGLDWHO\

PAGE 185

$33(1',; ) $Q (TXDWLRQ IRU 7KH TXDQWLWLHV DQG 8% ‘ 8QN H[S >LQ+ LNW@ Q N XrL OB &QN H[S >LQ-fLNW@ QN 4 ,, &nnO 6QN H[S >LQ_!LNW@ QN IURP &KDSWHU ,9 DUH VXEVWLWXWHG LQWR ZI 8-! f I % $ 7L RO A )18 n W 6L8 X A V B : f )1 GL X f§ )1Rf )1Z ! 3 %I I $I VLU "L rr G_! )1: Z)1 $Q RSHUDWLRQ RI WKH W\SH /LP 7 77 aa B AW GI! H[S >LQI!LNW@ 7 f‘ f LU7 7 LW

PAGE 186

LV SHUIRUPHG RQ WKH HTXDWLRQ DQG LQWHJUDOV DQG VXPV DUH LQWHUFKDQJHG )LQDOO\ WKH HTXDWLRQ LV GLYLGHG E\ 6QNn DQG WKH UDWLRV [ L[ 8 6 QN QN LQWURGXFHG 7KH UHVXOW RI WKHVH VWHSV LV ZIN\ f I % )1RM $ /B /f N )1Xf i i - L $/f Q\ m )1Rfn D f S1Z )1A rr UfI B-B $I BBIB7f f )1f R)1 7KH TXDQWLW\ U ZKHUH LW DSSHDUV H[SOLFLWO\ LQ WKLV HTXDWLRQ LV FDOOHG 7 6ROYLQJ IRU UJII RQH ILQGV HII ^ 0 I % X N 4Z Rf UGLI % I8} &OI L B U% LO F af§ ZR6 7nQ W Z ; Rf Rf I }G % cW-BOL7O Q\ f f 4X c-

PAGE 187

fr % II 4Z f fN IQ $ r r ZKHUH Q ZIN\ ` DQG

PAGE 188

%,%/,2*5$3+< %HUQDFFD 3 / DQG 3HULPRWWR 0 &RQWU 2VnVn $VWU $VLDJR 1R %XUELGJH 5 DQG %XUELGJH ( 0 $S &DQQRQ $QQLH $QQ $VWU RQ 2EnVnn +DUYDUG &ROOHJH &DQQRQ $QQLH DQG 3LFNHULQJ ( & $QQ $VWURQ 2EnVnn +DUYDUG &ROOHJH &KDSPDQ 6 DQG &RZOLQJ 7 ( 7KH 0DWKHPDWLFDO 7KHRU\ RI 1RQ8QLIRUP *DVHV &DPEULGJH 7KH 8QLYHUVLW\ 3UHVVf &ROOLQV : ,, 6WHOODU 5RWDWLRQ HG $6OHWWHEDN 1HZ
PAGE 189

6WHOODU 5RWDWLRQ HG $ 6OHWWHEDN 1HZ
PAGE 190

0HUULOO 3 ( DQG %XUZHOO & $S LELG B LELG 0LKDODV $S 6XSSO B 3DUNHU ( 1 $S ,QWHUSODQHWDU\ '\QDPLFDO 3URFHVVHV 1HZ
PAGE 191

%,2*5$3+,&$/ 6.(7&+ 7KRPDV +DUORZ 0RUJDQ ZDV ERUQ 0DUFK LQ -DFNVRQYLOOH )ORULGD DWWHQGHG WKH SXEOLF VFKRROV WKHUH DQG ZDV JUDGXDWHG IURP /DQGRQ +LJK 6FKRRO LQ ,Q -XQH RI WKDW \HDU KH HQWHUHG WKH 8QLYHUVLW\ RI )ORULGD +H UHFHLYHG WKH %DFKHORU RI 6FLHQFH GHJUHH LQ 'HFHPEHU DQG HQWHUHG WKH 8QLYHUVLW\ RI )ORULGD *UDGXDWH 6FKRRO LQ SK\VLFV WKH IROORZLQJ \HDU 8SRQ JUDGXDWLRQ KH ZLOO EHFRPH D SRVWGRFWRUDO DVVRFLDWH DW WKH 0DQQHG 6SDFHFUDIW &HQWHU LQ +RXVWRQ +H LV D PHPEHU RI (,,( WKH SK\VLFV KRQRUDU\ VRFLHW\ WKH LG! KRQRU VRFLHW\ WKH $PHULFDQ 3K\VLFDO 6RFLHW\ DQG WKH $VWURQRPLFDO 6RFLHW\ RI WKH 3DFLILF

PAGE 192

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

PAGE 193

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

PAGE 194

81,9(56,7< 2) )/25,'$


UNIVERSITY OF FLORIDA
3 1262 08554 5332