Citation
Algebraic properties of noncommensurate systems and their applications in robotics

Material Information

Title:
Algebraic properties of noncommensurate systems and their applications in robotics
Creator:
Schwartz, Eric Michael, 1959- ( Dissertant )
Doty, Keith L. ( Thesis advisor )
Bullock, Thomas E. ( Thesis advisor )
Staudhammer, John ( Reviewer )
Crane, Carl D. ( Reviewer )
Yeralan, Sencer ( Reviewer )
Phillips, Winfred M. ( Degree grantor )
Holbrook, Karen A. ( Degree grantor )
Place of Publication:
Gainesville, Fla.
Publisher:
University of Florida
Publication Date:
Copyright Date:
1995
Language:
English
Physical Description:
xii, 124 leaves : ill. ; 29 cm.

Subjects

Subjects / Keywords:
Coordinate systems ( jstor )
Coordinate transformations ( jstor )
Distance functions ( jstor )
Eigenvalues ( jstor )
Ellipsoids ( jstor )
Jacobians ( jstor )
Mathematical vectors ( jstor )
Matrices ( jstor )
Robotics ( jstor )
Wrenches ( jstor )
Dissertations, Academic -- Electrical Engineering -- UF
Electrical Engineering thesis, Ph. D
Robots -- Control systems ( lcsh )
City of Boca Raton ( local )
Genre:
bibliography ( marcgt )
non-fiction ( marcgt )

Notes

Abstract:
Several algebraic properties for systems in which either or both the input and output vectors have elements with different physical units. The condition son linear transformation A for a physically consistent noncommensurate system, u=Ax, are given. Linear noncommensurate systems do no generally have eigenvalues and eigenvectors. The requirements for noncommensurate linear systems do not have a physically consistent singular value decomposition. The manipulator Jacobian maps possibly noncommnesurate robot joint-rate vectores into noncommensurate twist vectores. The inverse velocity problem is often solved through the use of the pseudo-inverse of the Jacobian. This solution is generally scale and frame dependent. The pseudo-inverse solution is physically inconsistent, in general, requiring the addition of elements of unlike physical units. For some manipulators there may exist points—called decouple points—at which the pseudo-inverse of the Jacobian is physically consistent for all frames at these points. In decouple frames, the pseudo-inverse is shown to be equivalent to the weighted generalized-inverse with identity metrics. An entire class of nonidentity metrics used with the weighted generalized-inverse are shown to give identical solutions to the pseudo-inverse solution at decouple points. At decouple points, the twist and wrench spaces can be decomposed into two metric-independent subspaces. This decomposition is accomplished with kinestatic filtering projection matrices. The Mason/Raibert hybrid control theory of robotics is shown to be useful only for frames located at decouple points and is not optimal in any objective sense. The current manipulability theory, which depends on the eigensystem of various functions of the Jacobian, is shown to be invalid. Two new classes of manipulators are introduced, self-reciprocal manipulators and decoupled manipulators. The twists of freedom of a self-reciprocal manipulator are reciprocal. The class of self-reciprocal manipulators consists of planar manipulators, spherical manipulators, and prismatic-jointed manipulators. Decoupled manipulators are show to decouple at every point. The manipulators of this class are planar manipulators, prismatic-jointed manipulators, and SCARA-type manipulators. Results that are generalized from decoupled manipulators often prove to b invalid for manipulators that do not decouple at every point.
Thesis:
Thesis (Ph. D.)--University of Florida, 1995.
Bibliography:
Includes bibliographical references (leaves 119-123).
General Note:
Typescript.
General Note:
Vita.
Statement of Responsibility:
by Eric M. Schwartz.

Record Information

Source Institution:
University of Florida
Holding Location:
University of Florida
Rights Management:
Copyright Eric Michael Schwartz. 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:
002046269 ( ALEPH )
33417401 ( OCLC )
AKN4201 ( NOTIS )

Downloads

This item has the following downloads:


Full Text










ALGEBRAIC PROPERTIES OF NONCOMMENSURATE SYSTEMS
AND THEIR APPLICATIONS IN ROBOTICS














By

ERIC M. SCHWARTZ


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

UNIVERSITY OF FLORIDA


1995






























To My Wife Gabriella
&
To My Parents Marilyn and Seymour














ACKNOWLEDGMENTS


I will always be grateful for the opportunity I have had to work with Professor Keith L. Doty. Our discussions on everything from robotics, to politics and religion have made my work especially enjoyable. This dissertation would not have been possible without his direction and support. He has encouraged and enhanced my growth as an engineer and as a member of the species. I will never forget the summer in Italy that he made possible and that through him I met my wife.

I would like to thank Professor Thomas E. Bullock for making the considerable time we have spent working together in the controls area both informative and interesting. I have learned a great deal from him and have very much enjoyed doing SO.

My thanks are offered to the Electrical Engineering Department for supporting me through the first half of my graduate studies and to the Electronics Communication Laboratory for supporting me through most of the second half.
I would like to express my gratitude to my friends in the Machine Intelligence Laboratory, both past and present, who have allowed me to debate, listen, learn and party with them. In particular, I, thank Kimberly Cephus for her technical advise, moral support, and friendship.

Finally, I would like to thank my family, especially my parents Seymour and Marilyn, for being proud of me and for doing whatever it was they did to make me me. And my very special thanks go to my wife Gabriella for putting up with my long times away from her, for pushing me along, and for choosing me for her husband.














TABLE OF CONTENTS




ACKNOWLEDGMENTS . ii

LIST OF TABLES . vi

LIST OF FIGURES . viii

KEY TO SYMBOLS . ix

ABSTRACT . . . . . . . . . . . . . . xi

CHAPTERS

1 INTRODUCTION . 1

1.1 Noncommensurate Vector Spaces . 3
1.2 The Pseudo- and Generalized-Inverses . 14
1.2.1 The Moore-Penrose Pseudo-Inverse . 15 1.2.2 The Weighted Generalized-Inverse . 16
1.3 Eigenvalues, Eigenvectors and SVD . 18 2 LINEAR NONCOMMENSURATE SYSTEMS . 19

2.1 Eigensystem In Noncommensurate Systems . 20 2.2 Conditions for Physically Consistent Eigensystems . 21 3 PHYSICAL CONSISTENCY OF JACOBIAN FUNCTIONS . 24

3.1 Inappropriate Uses of the Euclidean Norm in Robotics . 24 3.2 Physical Consistency of JrJ and JJ . . 28
3.2.1 Consistency of Iu = Axi . 32 3.2.2 Invalid use of Eigensystem and SVD of JJT. . . 36 4 INVERSE VELOCITY KINEMATICS . 40

4.1 Physical Consistency of Jt . . 42
4.1.1 Rotations and Consistency of Jt . . .42 4.1.2 Translations and Consistency of jt . . .44 4.1.3 Consistency of Jt in All Frames . 45
4.2 Invariance of Jt to Scaling . 47 4.3 Equivalent Generalized Inverses . 54









5 MANIPULATOR MANIPULABILITY .

6 DECOMPOSITION OF SPACES .

6.1 Projections and Kinestatic Filters.
6.2 Twist Decomposition .
6.3 Wrench Decomposition .
6.4 Hybrid Control .
6.5 Decomposition with Ray Coordinate Twist 6.6 Space Decomposition at Decouple Point .
6.7 Self-Reciprocal Manipulators .


Space . . .


7 SUMMARY AND CONCLUSIONS . APPENDIX

A D-H PARAMETERS FOR VARIOUS MANIPULATORS. REFERENCES .
BIOGRAPHICAL SKETCH .














LIST OF TABLES




D-H parameters for GE P50 manipulator . 29

Physical units of Det[JTJ] for various non-redundant manipulators. 30 Physical units of Det[JJ] for various redundant manipulators. 32


D-H parameters for D-H parameters for D-H parameters for


PR virtual manipulator. the SCARA manipulator . the PRP Small Assembly Robot


. . . . . . . . . . 44

. . . . . . . . . . 46

(SAR) . 55


6.1 D-H parameters for a non-planar RRR manipulator .


D-H parameters for D-H parameters for D-H parameters for D-H parameters for D-H parameters for D-H parameters for D-H parameters for D-H parameters for D-H parameters for D-H parameters for D-H parameters for D-H parameters for D-H parameters for


PR virtual manipulator . 106 an RR manipulator . 107 a general RRR manipulator . 107 the Planar RRR manipulator . 108 the Spherical RRR manipulator . 108 the Non-planar RRR manipulator . .109 the PPP orthogonal manipulator . .109 the PRP Small Assembly Robot (SAR) . .110 the RPR manipulator . 110 the RRRP-1 SCARA manipulator . 111 the RRRP-2 manipulator . 111 the RRRP-3 manipulator . 112 5R GE-P50 manipulator . 113


A.1 A.2 A.3 A.4 A.5 A.6 A.7 A.8 A.9 A.10 A.11 A.12 A.13









A.14 D-H parameters for A.15 D-H parameters for A.16 D-H parameters for A.17 D-H parameters for A.18 D-H parameters for A.19 D-H parameters for


the 7R Redundant Anthropomorphic Arm. .114 the 7R CESAR Research Manipulator . .115 the 7R K-1207 Robot Research Arm . .115 the 7R PUMA-260+1 Spherical Wrist Manipulator. 116 the 3P-4R Redundant Spherical Wrist Robot. . . 117 the 2R-P-4R GP66+1 Manipulator . .117














LIST OF FIGURES




4.1 Peg-in-the-hole with PR virtual manipulator . 45 4.2 SCARA manipulator . 46 4.3 Small Assembly Robot (SAR) . 56 6.1 Decomposition of the twist space in frame i into decoupled subspaces. 81














KEY TO SYMBOLS


Symbol or
Variable Definition
t for Xt, the Moore-Penrose pseudo-inverse of X
# for X#, the weighted generalized-inverse of X
I I for lxi where x is a vector, IxI = Vfx0
I" IM for IXIMX where x is a vector, Ixi = V/x � Mx
x for x x y, the vector cross product of vectors x and y
G for x G y, the inner (or dot) product of vectors x and y
0 for X o Y, the klein (or reciprocal) product of screws X and Y
E) for X E Y, the direct sum of the subspaces X and Y
M M
E) for X ED Y, the direct sum of the M-orthogonal subspaces X and y
? possibly equal, often physically inconsistent
def
_ defined as
N numerically equal to
(.)(ij) for matrix X(ij), element of X in i-th row, j-th column
.). for matrix X(.,j), the j-th column of X
(, for matrix X(i,.), the i-th row of X
[']r,c for [X]r,can r x c matrix with all units identical to the units of L
[01r,c r x c matrix of zeros
[Ib for [X]6, matrix where the column vectors constitute a basis for X
[.]'r the transpose operator
0," zero vector of dimension n
a angle between successive joint axes projected on plane with common
normal used in D-H parameterization
A orthogonal 6 x 6 matrix that converts between ray and axis
coordinates
0 angle about a joint axes used in D-H parameterization
v cos(a )
0i sin(ai)
T the generalized-force vector containing n joint forces and/or joint
torques corresponding to prismatic and/or revolute joints
w angular velocity 3-vector
A wrench coordinate transformation matrix
a perpendicular distance between successive joint axes used in D-H
parameterization









Symbol or
Variable Definition
B skew-symmetric 3 x 3 translation matrix of b
b translation 3-vector
ei+j cos(Oi + 0j)
D9 defect manifold
d distance along joint axis used in D-H paramtetrization
E. matrix such that XE, is the column-reduced echelon form of X
f force 3-vecor
G twist coordinate transformation matrix
b body's inertia tensor at the center-of-mass expressed
in principal corrdinates-a diagonal matrix
Ij j x j identity matrix
J manipulator Jacobian that transforms joint rates into twists, V =J
ill, first three rows of J, such that v = Jv4
JI rows four through six of J, such that w = J,,
[L]r,c r x c units matrix with all units of length
n number of joints in manipulator
n moment of force 3-vector
Null[A] null space of matrix A, i.e., all x such that Ax = 0 Q joint-rates vector space
R 3x3 rotation matrix
1Z radical subspace
R" commensurate m-space over reals
Range[A] range space of matrix A, i.e., all y such that y = Ax S or Si rotation vector of screw i So or Soi translation vector of screw i
Sq change of units scaling matrix for joint rates
S,, change of units scaling matrix for twists
si+j sin(0i + Oj)
T generalized (joint) forces vector space
[U] -,C r x c unitless units matrix
units[.] the physical dimensions of the matrix inside the brackets V twists in Pldcker ray coordinates, V = [VT, W T]T
V twists screw space
V1 twists of freedom subspace
Vnfi twists of nonfreedom manifold
v linear velocity 3-vector
W wrench in Plicker axis coordinates, W = [fT, nT]T
W wrenches screw space
Y'vl wrenches of constraint subspace
IV. wrenches of nonconstraint manifold
z unit vector in z direction ([0, 0, 1]')














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

ALGEBRAIC PROPERTIES OF NONCOMMENSURATE SYSTEMS
AND THEIR APPLICATIONS IN ROBOTICS By

ERIC M. SCHWARTZ

May 1995

Chairman: Keith L. Doty
Major Department: Electrical Engineering

Several algebraic properties are given for systems in which either or both the input and output vectors have elements with different physical units. The conditions on linear transformation A for a physically consistent noncommensurate system, u = Ax, are given. Linear noncommensurate systems do not generally have eigenvalues and eigenvectors. The requirements for a noncommensurate system to possess a physically consistent eigensystem are presented. It is also shown that noncommensurate linear systems do not have a physically consistent singular value decomposition.

The manipulator Jacobian maps possibly noncommensurate robot joint-rate vectors into noncommensurate twist vectors. The inverse velocity problem is often solved through the use of the pseudo-inverse of the Jacobian. This solution is generally scale and frame dependent. The pseudo-inverse solution is physically inconsistent, in general, requiring the addition of elements of unlike physical units. For some manipulators there may exist points-called decouple points-at which the pseudo-inverse of the Jacobian is physically consistent for all frames at these points.









In decouple frames, the pseudo-inverse is shown to be equivalent to the weighted generalized-inverse with identity metrics. An entire class of nonidentity metrics used with the weighted generalized-inverse are shown to give identical solutions to the pseudo-inverse solution at decouple points.

At decouple points, the twist and wrench spaces can be decomposed into two metric-independent subspaces. This decomposition is accomplished with kinestatic filtering projection matrices.

The Mason/Raibert hybrid control theory of robotics is shown to be useful only for frames located at decouple points and is not optimal in any objective sense.

The current manipulability theory, which depends on the eigensystem of various functions of the Jacobian, is shown to be invalid.

Two new classes of manipulators are introduced, self-reciprocal manipulators and decoupled manipulators. The twists of freedom of a self-reciprocal manipulator are reciprocal. The class of self-reciprocal manipulators consists of planar manipulators, spherical manipulators, and prismatic-jointed manipulators. Decoupled manipulators are shown to decouple at every point. The manipulators of this class are planar manipulators, prismatic-jointed manipulators, and SCARA-type manipulators. Results that are generalized from decoupled manipulators often prove to be invalid for manipulators that do not decouple at every point.














CHAPTER 1
INTRODUCTION

Optimum, according to Webster [58], means "best; most favorable." In real physical systems, to say a solution is optimum or optimal one must specify the criteria for optimality.

The theory of hybrid control of manipulators developed by Mason in 1978 [41, 40] and then tested and expanded by Raibert in 1981 [51] has been shown by Lipkin and Duffy [37, 36] and others [1, 19] to be erroneous. Lipkin and Duffy explain that the failure of Mason and Raibert's hybrid control theory (MRHCT) is in their use of orthogonality. In MRHCT, the orthogonality of two vectors with terms of unlike units is used when it is easily seen that the inner product of these vectors in not invariant to scaling. Because so many authors continued to use MRHCT, Duffy [22] found it necessary to write an editorial debunking this theory.

The problem with MRHCT, in this author's view, is that the terms of their optimal solution were not sufficiently defined. An exploration of the meaning of their optimal solution would have shown that the solution is based on minimizing the Euclidean norms of two non-Euclidean vectors.

In 1989, Doty noticed and eventually published research [14, 19] that the MoorePenrose pseudo-inverse solution in the robotics inverse velocity problem gives results that are dependent on the frames of reference. Doty's algebraic viewpoint, together with Duffy and Lipkin's geometric results using screw theory, suggested a further investigation of the possible non-invariance of solution techniques in several areas of robotics and applied mathematics in general.









This dissertation is based in part on correcting the inappropriate use of the pseudoinverse in the field of robotics. Researchers such as Doty [18], Duffy [22], Lipkin and Duffy [37, 36], Lipkin [35], Griffis [26], and Schwartz [54, 53] have shown the fallacy of incorrectly applying optimization techniques to robotics problems without a judicious investigation of the underlying metrics incorporated. This dissertation intends to formalize and explain these problems and offer consistent solutions and interpretations of these solutions.

Each of these problems involves solving a set of linear equations which by some manipulation can be put in the form u = Ax, where A is nonsquare or singular. More often than not, a multitude of robotics researchers including [12, 23, 29, 32, 38, 39, 43, 44, 45] have solved these problems by using the pseudo-inverse. The inconsistent results generated through the use of the pseudo-inverse (without a metric or metrics) are explained in this dissertation.
The robotics literature [10, 31, 46, 57, 59, 60] also makes use of the eigenvalues, eigenvectors, or singular values of matrices whose eigenvalues and singular values are not invariant to changes in scale or coordinate transformations, and are therefore not true "eigensolutions". The eigensolution problem is also discussed in this dissertation.

The basic mathematics and terminology of robotics and screw theory necessary for an understanding of the issues discussed will be introduced in this chapter. There is no original work in this chapter other than some basic definitions with regard to noncommensurate systems. Since a general understanding of the Euclidean vector norm, the pseudo-inverse, the weighted generalized-inverse, eigenvalues, eigenvectors, and singular value decomposition are paramount to understanding this dissertation, these topics will also be presented and examples (with references) of their use in robotic systems will be given in this chapter.









1.1 Noncommensurate Vector Spaces

Systems involving elements of different physical units are defined here as noncommensurate systems. Robotics systems are noncommensurate when they deal with both position and orientation or have both revolute and prismatic joints. A vector of elements of unlike physical units is defined as a noncommensurate vector.! (The noncommensurate vector is also called a compound vector [14, 53] and non-homogeneous vector [151.)

In robotics, the equation that relates joint velocities to twists (1.1) describes a noncommensurate system,

V=Jq � (1.1)

The manipulator joint-rate vector is

q = [41, 42, ., ql] , (1.2)

where n represents the total number of revolute and prismatic joints of the manipulator. The manipulator's instantaneous twist vector, V = [V', wTlr , (1.3)

is composed of the linear velocity v [v, vY , v] and the angular velocity w [w,, wi, w2]T. The Jacobian J is a 6 x n matrix, where 6 is the number of coordinates necessary to describe the position and orientation of a body in space.

The twist represents a noncommensurate vector since the units of v and w differ. When the manipulator has both revolute and prismatic joints, the joint-rate vector is also noncommensurate and the manipulator is called a noncommensurate manipulator.

The vector i'm Vk represents the twist of a point p, fixed to frame k, and expressed in frame i coordinates with respect to a fixed frame m. Since the Jacobian i'mJp,k has columns that are also twists, the superscript i and m and the subscripts p and k








have the same interpretations as in i'mVp,k. When the subscripts p and k and the superscript m are omitted in 'V and "J, it is understood that k is the end-effector frame n of an n-jointed manipulator, m is the base frame (frame 0), and point p is at the origin of frame i, the frame of expression (2V = ',�Vi,).
To transform twists or Jacobians to representations in different frames, the twist coordinate transformation matrix G is used, =Gp'q [iRj iBpq 'R ] (1.4)
- 1= 0]3,3 iRj '

where [013,3 is a 3 x 3 matrix of zeros and 'Rj is a rotation transformation which rotates a vector from frame j into frame i. Since rotation matrices are orthogonal, the inverse is equal to the transpose, i.e.,

'R11 = 'R; = . (1.5)

(By convention, the term "orthogonal matrix" refers to matrices with orthonormal columns [56].) The matrix 'Bp,q = ['bp,q x] is a skew symmetric matrix that represents translation from point p to q expressed in frame i. The B matrix is the matrix-form of the vector cross-product, i.e., Bc = b x c, where b and c are arbitrary 3-vectors and B is defined as 0 -bz iby
lBPbq = 2 0 -1b2. (1.6)
- by ibx 0

The vector ibp,q = [ibx, iby, ib.]T is a position vector from point p to point q expressed in frame i coordinates.
Since B is skew symmetric, it has the following properties:

Bqp = -Bp,q , (1.7)

(Bp,q)" = -Bp,q , and (1.8)
RjJBpq = 'Bp,q1Rj. (1.9)









With the above equations it is easily shown that (iGP'q)- = . (1.10)

Note that (iPQ)T "# Gq"P


The expressions for the frame transformations of twists and Jacobians are

iVp,k = (iGp'q) jVq,k , and (1.11)

iJp,k = (iGp'q) 3Jq,k (1.12)

The shorthand notation tGj is used when the transformation has no translation and the notation iGp,q is used when the transformation has no rotation.
The twists that a manipulator can accomplish with joint-rate control in a given configuration are know as the twists of freedom [5, 8, 22],

iV! = Range[tJ] , (1.13)

where V represents a twist manifold and i is the frame of expression. The twist of freedom manifold is a subspace.

It is important when writing vectors, matrices, and manifolds to make the frame of expression clear. In this dissertation, the expression frame, if not explicitly written as a leading superscript, will be otherwise described in the context of the discussion.

Note that throughout this dissertation, a calligraphic symbol (such as V) represents a manifold (or set) of vectors or screws. Therefore, X = {Xi} is the manifold of vectors or screws Xi, for various i. The column vectors of the matrix [X]b constitute a basis for X. The matrix E., converts the basis set, [X]b, to a matrix in column-reduced echelon form [56], [X]bE,.

The application of a wrench W at the end-effector of a static serial manipulator will induce a balancing generalized-force vector rT,, TW = J'W , (1.14)









where a wrench, W = [f', n']', is the noncommensurate 6-vector composed of the two 3-vectors of forces f and moments n. A generalized-force vector, 7, is the n-vector of joint torques (for revolute joints) and/or joint forces (for prismatic joints).

The matrix 'Wa,,p = [if, 'n T represent a wrench at point p expressed in frame i, with the moments taken about point a. When the subscript a is omitted it is understood that the point a is at the point p, so that 'Wp = 'Wpp. When both subscripts are omitted the origin of the frame is the point at which moments are taken, i.e., iW =

Wrenches transform via the wrench coordinate transformation matrix A,

zWP = (iAypq) JWq , (1.15)


where
r R, [9]3,3fpq 2R (1.16)
3 v, q i3 'Rj

Equations (1.7)-(1.9) can also be used to show that

='" = _' , and (1.17)

(iG?'Py = j . (1.18)

The wrenches applied at the end effector that require no joint forces for balancing are know as the wrenches of constraint, 'W,, and form a subspace, W, = Null['JT] . (1.19)

These wrenches will cause no joint motion when applied to a static manipulator. Manipulators (of at least 6 joints) in configurations with Jacobian of rank 6 have no constraint wrenches, i.e., some nonzero joint forces are required to balance every possible wrench.

Notice that the above twists and wrenches are screws (defined below) expressed in axis coordinates and ray coordinates [27, 30], respectively. The designations of Plicker









ray coordinates and Plicker axis coordinates are based on the original formulation of screw theory by Ball in 1900 [5]. Ball defined lines in two ways, each independently leading to coordinate system definitions: the join of two points lead to ray coordinates and the meet of two planes lead to axis coordinates. These sets of identical but reordered coordinates are know as the homogeneous Plicker line coordinates. The distinction is only necessary when lines or screws in different Plicker coordinates are used simultaneously, as is the case with the traditional algebraic descriptions of twists and wrenches previously defined.

A screw $ is defined as a line with an associated pitch h. For example, the motion defined by a physical screw being advanced into a pre-threaded hole can be characterized by the following screw (in axis coordinates), $ai [S So [A (in axis coords), (1.20)

where the line passes through the coordinate system origin. (A more general description is given in (1.24) below.) The vector S is a commensurate 3-vector in the direction of linear motion and the rotation is about this axis using the right-hand-rule. For every 0 radians of rotation, the screw advances by hO in the S direction.

A screw may also be defined as a linear combination of unlimited lines [5, 25]. An unlimited line L is defined with two vectors: a unit vector S in the direction of the line and a vector r from the coordinate system origin to any point on the line,

Laxs= [] (in axis coords). (1.21)

Lines also have the property that S G So = S 0 (r x S) = 0. The ray coordinate version of this same line is

LraY= [ ] = [ ] (in ray cordss. (1.22)









A linear combination of two lines in axis (ray) coordinates creates a screw in axis (ray) coordinates,

10 ais xiss 7(rx S1) + y12(r2 X S2) (1r.231
7r$fl = iLfls� y2Llc =[ 1 S1 + 72 S2 J- r I Sor (.3

For screws, Sr G S0r = hr, where h, the pitch of the resultant screw. Therefore screws are not lines except in the special case when the pitch is zero. The resultant screw can be written as

[ Sr+ ](1.24)

The differences in equations (1.20) and (1.24) are due to different coordinate system definitions. If r, = 0, i.e., the coordinate system origin is on the line of rotation, the two equations are identical. A general screw can always be converted to a "pure screw" as in (1.20) by a twist coordinate transformation for axis coordinate screws or a wrench coordinate transformation for ray coordinate screws. For example, a twist coordinate transformation will transform the pure axis coordinate screw into a general axis coordinate screw,

Ghj ] [ R4 BR][h][ h ~w +BRw] (1.25)


Note that coordinate translations (B) do not affect the angular velocity vectorthe bottom component in the right-hand-side of equation (1.25). Although rotations affect both parts of the screw, if there is no translation, the rotation will not affect the apparent purity of a screw viewed in each of the frames.

As stated above, the pitch of a screw can be found simply by So�S
h = SI2 , IS[ =0. (1.26)

If S is the zero vector, the screw is said to have infinite pitch and (1.25) is replaced by
G 03 [0]3,3 R 03 03









Note that the translation B has no affect on the resulting screw representation. If the pitch is zero, So is zero and the screw represents a pure rotation.

The translation that will move a general axis screw to a pure axis screw is SoxS
b = l-1- ISI $ 0. (1.28)


where B can be found from b with (1.6).

All rigid body motion is instantaneously equivalent to a screw motion twist [9]. The twist defined previously, V = [v0, w], is equal to a linear velocity v0 (referenced to some origin 0) and an angular velocity w, a free vector [25] and is here defined as a screw in Plicker axis coordinates [48, 49]. A twist can also be represented in Plicker ray coordinates, Vray = [W, vo].

Similarly, a wrench is instantaneously equivalent to a force and moment on a rigid body. The Plicker ray coordinates of a wrench, W = [f, no], is equal to a force f in the direction of the wrench and a moment n referenced about origin 0. A wrench can also be represented in Pldcker axis coordinates, Waxs = [rn0, f].

Unless otherwise noted, twists will be expressed in Plicker axis coordinates and wrenches will be expressed in Plicker ray coordinates.
The matrix A [36] transforms a screw or line in axis coordinates to a screw or line in ray coordinates and a screw or line in ray coordinates to a screw or line in axis coordinates,

$ray = A$ls (1.29)

$axis A Asray (1.30)
A :[[013,3 13 ] (1.31)
/3 [013,3

The matrix A is an unitary matrix (and therefore also an orthogonal matrix) with the properties
A = A-' A = AT AA=I (1.32)


y









The matrix A is an example of a more general transformation, defined as a correlation [27] that maps an axis screw to a ray screw (or a ray screw to an axis screw). A collineation maps a ray screw to a ray screw (or an axis screw to an axis screw).

The reciprocal or Klein product [5, 22] of any two screws in identical axis or ray coordinates-twists V1 and V2, for example-is defined as


V o V2 = V1 � AV2 = VAV2 (1.33)

= V1�w2 "-2 �22 , (1.34)

where G represents the Euclidean inner or dot product.

The Klein product of a screw in axis coordinates, and a screw in ray coordinates V and W is

VoW = VG W = V'W = vGf +wo n . (1.35)

Notice that no A matrix is needed in the expansion of the Klein product of a twist and a wrench, whereas the A matrix is needed in the expansion of the Klein product of two twists or two wrenches. The Klein product of a twist and wrench of the endeffector of a serial manipulator gives the instantaneous virtual power (work) [36] that the manipulator end-effector contributes to the environment.

A well known important characteristic of the reciprocal product is that it is invariant to coordinate transformations. This is shown in the following theorem and proof [5]. The proof is given to provide the reader an understanding of the notation and mathematics involved.

Theorem 1 The reciprocal product of a manipulator twist and wrench expressed at the same point and in the same coordinate system is invariant to coordinate transformations.








Proof

'Vp,q = (iG'q) 1 Pq (1.36)

- (iGpp) jGp V,q , as shown in (1.10) (1.37)

iV o " (') G ,q 0' Wp (1.38)

. p,q i (q,p) Pq i W (1.39)
- iV p o ( P ,, as shown in (1.18) (1.40)

- JVq,k 0 JWq , as shown in (1.11) and (1.15). (1.41)


The twist or screw motion created by a single revolute joint i is a pure rotation, i-itrvr�= [0O, O, 00001 [
ilrv=IJ = [ 31, (1.42)

where the above equation is expressed in the frame of the previous joint i - 1, and
is the vector [0, 0, 1]r. The twist coordinate transformation matrix enables this screw to be expressed in different coordinate frames-as shown in (1.25). To express this screw in various coordinate frames, the twist coordinate transformation matrix may be employed as shown in (1.25). When the frame is translated to a frame j that is located by vector b from the frame i - 1, the screw motion is j Vrev [ x (1.43)

[Be] (1.44)

[by, O J (1.45)

where B is the screw symmetric matrix of (1.6) corresponding to the translation vector b = [b., by, b]r.
When the frame is rotated to a frame k with no translation from frame i - 1, the screw motion is

kVrev k 3 (1.46)









The twist or screw motion created by a single prismatic joint i is a pure translation, i-lV$Prs [0f, 0, L, 0, 0, 0, ]T = [i 1147 i 03 (1.47)


Again the twist coordinate transformation matrix can be employed to express this screw in various coordinate systems. An arbitrary coordinate transformation kGP'qi rotates the twist to frame k while the translation has no affect for any p and q, kVpris = k~pq, i-lvpris = [ ]Ri-l(1.48
i-1 0 1 -(1.48)


That translation has no affect on this twist was verified by symbolically performing the multiplication kGP,1 i-VPris in (1.27).

Screws can be added to form new screws. In this manner the motion of the endeffector (or any other point) of a serial manipulator may be found by a summation of the screws of each of the joints,

V = V1+V2 + V.V, (1.49)

= 41$1 + 42$2 +''" + 4n$n (1.50)

- [$1, $2, "', $nlq (1.51)

J4 (1.52)

where 4 is the vector of manipulator joint rates of (1.2) and (1.52) is identical to (1.1). To perform the addition of screws, it is first necessary to reference them to the same coordinate frame and point via the appropriate screw coordinate transformation matrices, e.g., the summation of the screws in (1.50) is actually accomplished with the equation
iv = E4'G'is,, . (1.53)
j=1
Any twist can be constructed by six or less independent screws each representing either a prismatic or a revolute motion. Therefore a virtual manipulator can always be constructed to instantaneously accomplish any twist. Griffis [25] defines a virtual









manipulator as any imaginary serial manipulator "whose joint displacements and speeds uniquely describe any permissible twist (Vf) and any permissible position and orientation of its end-effector." The permissible end-effector wrenches (W) together with the twists completely describe the instantaneous kinematics of a real or virtual manipulator end-effector.

Theorem 2 below, given in [5], shows that the reciprocal product of any twist of freedom and any wrench of constraint must be zero. The proof is shown to give the reader an insight to the concept of reciprocity. Theorem 2 The Klein or reciprocal product of Vf and W,, is zero, i.e., VfoW,=O ,VV EVf andVWEW, (1.54)


Proof
V = J4 , VV E V1 and some (1.55)

(V) T = JV (1.56)

(V) W = -JW. (1.57)

Now let W be a constraint wrench W, E W,, so that

(v),rW = 4TJTWC (1.58)

But JTW, = 0 by definition in (1.19), so

(V1)TWC = 0. (1.59)

But by the definition of the Klein product in (1.35),

(Vf)TWo = V1 o WC, (1.60)

so that V o W = 0.
E









This means that the manipulator can do no work with any wrench of constraint or, alternatively, can not move with the screw motion of any ray coordinate constraint wrench interpreted as an axis coordinate twist.

The reciprocity relationship between Vf and WY has been inadvertently (and inappropriately) used by researchers to characterize the entire space through the use of the direct sum decomposition of the 6-space of position and orientation.

The fundamental theorem of linear algebra [56] states that


R'R = Range[A] E Null[A'] , (1.61)

where m is the number of rows of A. The symbol D represents the direct sum and implies that Range[A] n Null[A'] = {0} and ' = Range[A] U Null[AT]. Applying this theorem to robotics by letting A be the Jacobian can be misleading,

6 I Range[J] � Null[JT] (1.62)

Since J has physical meaning, with terms not all of the same units, the implication of this theorem applied to robotics is that the total space is a combination of axis coordinate (twists) and ray coordinate (wrenches). The subspaces Range[J] and Null[JT] are noncommensurate. What does it mean to decompose a vector (or screw) into the sum of an axis coordinate vector and ray coordinate vector? This problem will be addressed in Chapter 6.

1.2 The Pseudo- and Generalized-Inverses

The Moore-Penrose pseudo-inverse and the weighted generalized-inverse can both be used to solve linear equations. Of course each of the solutions is based on different optimality conditions for their solutions.








1.2.1 The Moore-Penrose Pseudo-Inverse

The Moore-Penrose pseudo-inverse gives a unique minimum norm least-squares solution to a linear equation, u = Ax, (1.63)

for example. The pseudo-inverse of A (A C R(,Xn)), is denoted At and has the following properties [6, 34]: AAtA = A, (1.64)

AtAAt = At, (1.65)

(AAt)T = AAt, (1.66)

(AtA) = AtA . (1.67)

The pseudo-inverse can be found through a full-rank factorization of A, A = FC, where F E R(,xr) has full column rank r and C E R(rx ) has full row rank r. The pseudo-inverse of A can be expressed as At = Cr(FrACT) -FT (1.68)

= Cr(CCT)-1(F-F)-iF" (1.69)

= CtFt. (1.70)

The unique minimum norm least-squares solution to (1.63) is therefore

x, = Atu . (1.71)

The solution x', is a least-squares solution in that the residual (if any), Iu - Axi, is minimized, where I" I is the Euclidean vector norm (see equation (1.72)). The solution x, is minimum norm since any other solutions x, to Ax = u has Euclidean norm Ixi > Ixsl.
A least-squares solution is obtained if (1.64) and (1.66) are true and the solution is minimum norm if (1.64) and (1.67) are true [6].









It is a fortunate fact that the least-squares solution and the minimum norm solution are identical for linear systems and equal to the pseudo-inverse solution.

The Euclidean norm of a vector x E Rn (also known as the square root of the Euclidean inner-product of x with itself) is defined as lXi = +\[1X12
n
I12 = < X,. >=X(DX =X = X? (1.72)
i=l

If matrix A has full row rank or full column rank, (1.68)-(1.70) has the simplified solutions

At = A(AA)-1 , A full row rank, and (1.73)

At = (ArA)-Ar , A full column rank. (1.74)

These equations are derived directly from (1.69), substituting F = Ir when A has full column rank and C = Ir when A has full row rank. (Matrix Ir is the r x r identity matrix.)

1.2.2 The Weighted Generalized-Inverse

The weighted generalized-inverse gives a unique minimum Ms-norm least M,,squares solution to a linear equation. The weighted generalized-inverse of A (called the generalized-inverse throughout the rest of this dissertation), is denoted A# and has the following properties [6, 19]: AA#A = A, (1.75)

A#AA# = A#, (1.76)

(M AA#)Tr = MUAA#, (1.77)

(MA#A)T = MA#A (1.78)

The matrices M, and M, are metrics. A metric is a symmetric positive definite matrix.









The generalized-inverse of A [6, 7, 19], with the same full-rank factorization A = FC discussed previously, is

A# = MxjCT(FTMuAMxlCT)-FM (1.79)

= [Mx-1Cr(CM.j1C'ry-1] [(FTMuF)-FTMI (1.80)

= C#F# , (1.81)

where F# and C# are defined by (1.81) and the bracketed expressions in (1.80).
The unique minimum Mx-norm least M,-squares solution to (1.63) is therefore xS = A#u . (1.82)

The solution xs, is a least Mn-squares solution in that the residual (if any), Iu - AxIM., is minimized, where I['M is defined below in (1.83). The solution x, is minimum Mx-norm since any other solutions x, to Ax = u has Ms-norm IX1IM. > IXIM,.
The M-norm of vector a,

lalM = + la 1M,
n
Iai2M = =aOMa= Zai(Ma)i � (1.83)

In addition to the positive definite requirement for a metric, a metric must also make the corresponding square of the M-norm physically consistent [15], e.g., a � Ma must be physically consistent for any a if M is to be considered a valid metric.
A least-M, squares solution is obtained if (1.75) and (1.77) are true and the solution is minimum Mx-norm if (1.75) and (1.78) are true [7].

It is a fortunate fact that the least Mu-squares solution and the minimum Mxnorm solution are identical for linear systems and equal to the generalized-inverse solution given in (1.79)-(1.81).

In order for solutions to be invariant to coordinate transformations [19] in both the spaces defined by u and x, the metrics must transform via a specific congruence









transformation [56],
M' = GMG (1.84)

If u' = G-ju, then the metric for u' must be M,,, = GM G,. This will insure that the Mu-norm is invariant, Iu 12, = Ju12. The metric M must also transform via a congruence transformation, M, = GxMxGX, where x' = G.1x.

1.3 Eigenvalues, Eigenvectors and SVD

Eigenvalues and eigenvectors of an n x n matrix A are defined [34] by the equation, Ae(i) = A(')e(i) , (1.85)

where the i eigenvalues and eigenvectors are represented by 0() and e('), respectively.

The singular value decomposition (SVD) of an m x n matrix A of rank r is defined [34, 56] by the equation
A = UEV (1.86)

where E is an m x n matrix with the singular values of A (cri) on the main diagonal, U is an m x m orthogonal matrix, and V is an n x n orthogonal matrix.

The columns of U are the eigenvectors of AAr and the columns of V are the eigenvectors of A TA. The r singular values are the nonnegative square roots of the nonzero eigenvalues of both AAT and ATA, i.e., the eigenvalues of AAT and A TA are equal to the square of the singular values of A.

The eigenvalues are preserved for similarity transformations, B = S-1AS, and the eigenvectors of B are S- e(?). Eigenvalues are not preserved under congruence transformations, B = STAS (unless S is a rotation, in which case ST = S-1 and B is also a similarity transformation).














CHAPTER 2
LINEAR NONCOMMENSURATE SYSTEMS

For linear noncommensurate system, u = Ax, the requirements on the structure of A are determined in this section, where A is an n x m matrix, x is a noncommensurate m-vector, and u is a noncommensurate n-vector. Upon expanding u = Ax, it is found that

u= aijxj , (2.1)
j=l,m
so that

units[aij]units[xj] = units[uj] (2.2)

Using two terms in the sum of (2.1) for two elements of u, we get Zi = aijxj + aikxk (2.3)

z, = aljxj + alkxk , (2.4)

for all i, j, k, 1, where units[zi] = units[uj]. Solve (2.3) for xk and substitute the result into (2.4) to get
alk (akai3 (2.5)
z- -Zi = (akt - ai1atk) xj aik aik
Therefore, for physically consistency, units[aik] units[aj] = units [aij]units [alk] , (2.6)

or

units[aik] = units[ak] . (2.7)
aij aij

In other words, if m - 2 columns and n - 2 rows are eliminated, the determinant of the remaining 2 x 2 matrix must be physically consistent for the system to be noncommensurate.









Using three terms in the sum of (2.1) for three elements of u, we get three equations similar to (2.3) and (2.4). Solving these equations leads to a condition similar to that shown in (2.6), i.e., if m - 3 columns and n - 3 rows are eliminated, the determinant of the remaining 3 x 3 matrix must be physically consistent for the system to be noncommensurate.

By induction, the above technique shows that for all i > 2, and i < m < n or i < n < m? if m - i columns and n - i rows are eliminated, the determinant of the remaining i x i matrix must be physically consistent for the system to be noncommensurate.

Another requirement on matrix A is found by viewing A as a matrix of column vectors A(.,0,

u= xiA(.,i) (2.8)
i=l,m
It is evident that the units of any two columns of A must be proportional. This is an alternate way to express the results of (2.7) and a simple way to visually deduce whether or not a matrix is a candidate noncommensurate linear system matrix.
All linear systems are either commensurate, noncommensurate, or physically inconsistent. Commensurate and noncommensurate systems are physically consistent systems. For commensurate systems, all elements of the A matrix have identical units.

2.1 Eigensystem In Noncommensurate Systems

As was mentioned at the start of Section 1.1, many researchers make use of the eigenvalues, eigenvectors, or singular values of matrices whose eigenvalues and singular values are not invariant to changes in scale or coordinate transformations. These are therefore not true "eigensolutions" in the sense that they may only subjectively characterize a manipulator configuration based on a particular observer (with a choice of scale and coordinate frame of reference) as opposed to a more relevant objective characterization of a manipulator configuration.









2.2 Conditions for Physically Consistent Eigensystems

When does a matrix A have physically consistent eigenvalues and eigenvectors? Let A be an n x n matrix,
all a12 �. aln
A a21 a22 a2n (2.9)

-an an2 �. ann
and let the domain of A be X'2, where Xn is a space with physical units. The Xn-space can be characterized as follows. Let P3 be an n-vector of possibly distinct physical units

3= [3#1 32 " 3]T (2.10)

Any x E Xn is equivalent to an item-wise multiplication of 3 and y, y E ' , i.e.,

* = 0 (D y=[ lY #2Y2 . fln]T (2.11)

x E Xn (2.12)

y E , (2.13)

so that + A Xn and each eigenvector of A from (1.85), e(i), is an element of Xnspace.
Substituting (2.9) into (1.85) and performing the matrix multiplication results in the following equations:
allel(') + ax2e2(i) + - + alnen() = A()el() a21ea(i) + a22e2() -- + a2nen(i) = A(i)e2(i)
: : (2.14)

ailel(0 + an2e2(i) + - + anen(i = A((e)(=
Recognizing that only quantities with identical physical units may be added leads to the following theorem.

Theorem 3 The equation Ax = Ax is physically consistent if and only if units[akjlunits[xj] = units[A]units[xk], for all j and k.









Proof

By hypothesis, 1%~ akjxj = AXk for all k. Recognizing that only identical physical units can be added together, we immediately conclude that units[akjjunits[x] = units[A]units[xk], for all j and k.

Now, assume units[akjlunits[xj] = units[A]units[xk] for all j and k. Clearly, the equation E' =j akjx = Axk is physically consistent for all k, i.e., Ax = Ax is physically consistent.

0

Observe that units[akjlunits[xj] = units[A]units[xk] implies that units[A] = units[aii], for all i. Hence, any matrix with a physically consistent eigenvalue equation must have diagonal elements with the same physical units and all its eigenvalues must have those same units.

A simple test for a physically consistent eigensystem is the validity of the below equation for each element in matrix A,

units[akj]units[ajk] = units[a?.] (2.15)

Since the singular values of A are the nonnegative square roots of the nonzero eigenvalues of both AAT and ATA, a test on these matrix products (similar to the tests discussed above for the eigensystem of A) will determine if the SVD of A is physically consistent. The conditions for the physical consistency of the SVD of A are stated in Corollary 1 below.


Corollary 1 The singular value decomposition of A, A = UEVT, is physically consistent if and only if units[bkj]units[xj] = units[A]units[xk], for all j and k, where B = AAr or B = ATA, and Bx = Ax.









Proof

This follows directly from Theorem 3 and the properties of SVD, i.e., the eigensystem tests on the matrices AAT and ATA determine the singular values and orthogonal matrices U and V7. Therefore the test of Theorem 3 and (2.15) can be directly applied to AAT and ArA to determine if the SVD of A is physically consistent.

M

Let j in Corollary 1 be equal to k. Then units[bkk] = units[A], and all diagonal elements of B must have the same physical units. If A is an n x m matrix, then each diagonal term of B is

E I akj ,for B =AAr
bkk = or ,for all k. (2.16)
E =i ak ,for B = ArA

Therefore, all the elements in the k-th row or k-column A must have identical units, for B = AAr or B = A*TA, respectively. But since units[bkkl = units[bjj] for all j and k, all the elements of A must have the same units. Therefore, singular value decomposition is only valid for commensurate systems, i.e., Theorem 4 Noncommensurate system never have a physically consistent singular value decomposition.


The major results of this chapter are summarized below. The requirements on A for all physically consistent linear noncommensurate systems, u = Ax, were given in (2.6) and (2.7). The requirements for A to have a physically consistent eigensystem were given in (2.15). And, finally, it was shown that physically consistent linear noncommensurate systems do not have a physically consistent singular value decomposition. Only commensurate systems have a physically consistent SVD.














CHAPTER 3
PHYSICAL CONSISTENCY OF JACOBIAN FUNCTIONS

The manipulator Jacobian is used by many researchers in ways which result in physically inconsistent results. Several of these will be discussed in this chapter.

3.1 Inappropriate Uses of the Euclidean Norm in Robotics

A multitude of researchers [3, 31, 45, 59, 60] have characterized a robot configuration or condition in terms of the scalar quantity of the Euclidean norm. This will be shown to be invalid, in general. One or more non-Euclidean metrics are often necessary [14, 17, 19, 53, 54] to find a physically consistent (non-Euclidean) norm. Although this may not seem obvious at first glance, consider the following examples.

The twist vector V-defined in (1.3)-is composed of the translational velocity vector v and the angular velocity vector w. The square of the Euclidean norm is often inappropriately applied to the twist vector, IV12=VV=V V. (3.1)

But the expression V G V is physically inconsistent since IV2?vOv+wGw , (3.2)

and v has units of [length/time] while w has units of [angle/time]. This is like adding apples to oranges, generally inappropriate without a metric on the worth of an apple compared to an orange ([length/time] compared to [angle/time]).

For example, if vT = [1 1 1]cm, and w' = [11 Ilrd, then tfl 1 6. Changing the scale from cm to mm will change the result to 1i2 ? 303 5 6!

If we define a metric for twists, M,, we can use the Me-norm -defined in (1.83)to get a measure of twists, IV12 = VOMV. The metric M, must be positive definite 24









and make IVI' physically consistent. A metric M, can be selected such that this norm describes the kinetic energy, K, of a rigid body,

1 1
K = -V G MYV = -VMV (3.3)
2 2

The kinetic energy metric expressed at the center-of-mass with axes aligned with the body's principal axes is the principal mass-inertia matrix of a rigid body, MKE [mbI3 [013,3] (3.4)
S[013,3 Ib J
where mb is the body's mass and lb is the body's inertia tensor at the center-of-mass expressed in principal coordinates-a diagonal matrix. We must express this metric in the same frame as the twists- see (1.84)-(or express the twists at the body's centerof-mass aligned with the body's principal axis). Transforming the metric MKE to the frame of expression of the twist results in the metric
M, 'ME =[ mbI3 mbR B R (35
V 'MbR7Brl? Rr(Ib + mbBrB)R
Mv= G MKEGV = [mbI3 R( mbB R ,(3.5)

where G, is defined in (1.4), with R = iRp, B = 'Bp,i, i is the expression frame for the twist V = iV, and p is the frame of the principal axes of the body. The lower right 3 x 3 matrix in (3.5) is the inertia matrix of the body in the twist frame. If there is no rotation between the twist frame and the principal frame, the inertia matrix is Ib = (lb + mbB"B). This inertia matrix could have also been determined using the parallel axis theorem [42].

The metric of (3.5) is the twist inertia matrix of a rigid body composed of the zero-order mass-moment (mass), the first-order mass-moment (momentum), and the second-order mass-moment (inertia).

For a second example of the problem of using Euclidean norms in robotic applications, let us look at the generalized-force vector r of the manipulator joints. The square of the Euclidean norm of T is


I TI2 ( 622 +) +Tn2


(3.6)









If all the joints of the manipulator are revolute or all are prismatic, (3.6) is physically consistent (but this measure of the sum of the square of joint torques is probably of little value since the driving component of some joints is generally quite different from other joints). But, if the manipulator has both revolute and prismatic joints i.e., the manipulator joints angles or velocities form a noncommensurate vector-this equation sums physically inconsistent force-squared and moment-squared terms.

Let us view the Euclidean norms of V and r from a different perspective - namely, by looking at the manipulator Jacobian defined in (1.1). Of course if the manipulator has 6 joints and J has full rank, then J-1 can be found and the solution to (1.1) is q, = J-1 V .(3.7)

To solve for 4 when J is not a square matrix, many researchers use the pseudo-inverse Jr-see (1.68)-(1.70)-and the equation

, y? Vt . (3.8)

For a full row rank matrix J, the pseudo inverse is

jt = Jr(JJr)- , J full row rank. (3.9)


Equation (3.9) is often used with redundant manipulators (manipulators with more than 6 joints). For manipulators with less than 6 joints, the pseudo-inverse for a full column rank matrix is often used,

jt = (Jrj)-J' , J full column rank. (3.10)


Note that the pseudo-inverse in one case involves the term JJ and in the other case involves the term JJ. There is often a units problem (physical inconsistency) with both of these terms. One of these terms also appears in each of the norm of a twist V and the norm of the generalized-force vector r-, as will be shown below.









From equations (3.1) and (1.1), we get the Euclidean norm of twist V of

IV12 ? Vw V = (j 4)(J 4) = 4- (J-J) 4 . (3.11)

A similar technique will be used to find an alternate form of the Euclidean norm of the generalized-force vector.

The static wrench defined in (1.14) is repeated here for convenience, r,,, = JW, where Tr, is the n-vector of generalized-forces-joint torques (for revolute joints) and/or joint forces (for prismatic joints)-induced by an end-effector wrench W, and J is the manipulator Jacobian. A wrench W = [f7, n*]* is composed of the 3-vectors of force f and moment n.

The term JJT again appears in the Euclidean norm of r,. Equation (3.6) can be rewritten using (1.14) as

17-12 ? T 7-W = w1 (JJT) w . (3.12)

Let us now look at the physical consistency of these Euclidean norms by performing a units analysis on JJ and JJr.

The units of a manipulator Jacobian matrix is found simply by noting that the units of the range of J is equal to the units of V and is not dependent on the structure of the manipulator. Therefore the units of elements in a Jacobian column have one of the following two forms [13, 16, 53]:

* If manipulator joint i is revolute, the i-th column of the Jacobian has the units units[J(.,i)] = [[L]3,1 for revolute joints (3.13)
[U]3,1]

� If manipulator joint i is prismatic, the i-th column of the Jacobian has the units units[J(.,] =[[U]3,1 for prismatic joints (3.14)
[013,1

The [']j,k in the above equations corresponds to a j x k matrix whose elements have units of L for units of length or U for unitless. The [0]j,k term identifies a matrix whose elements are equal to zero (and says nothing about the elements' units).









3.2 Physical Consistency of JrJ and JJr

If all n joints manipulator are revolute, the units of JrJ is

units[JTJ] =? [ [L2 + U],,,' ] , for n revolute joints, (3.15)

i.e., each term sums a length-squared term with a unitless term. Since the Euclidean norm of V in (3.11) requires the product (J*J), the Euclidean norm of V is obviously physically inconsistent, as shown in (3.2).

For noncommensurate manipulators, if the i-th and j-th joints of a manipulator are revolute, then the (i,j)-th element of the matrix JrJ is physically inconsistent with units of

units[(J J)(,)] ? L2 + U , for i-th and j-th joints revolute. (3.16) If the i-th joint is revolute and the j-th joint is prismatic, then the (i, j)-th element of the matrix JrJ is physically consistent with units of

units[(J'J)(ij)] = L , for i-th joint revolute, j-th joints prismatic. (3.17) If the i-th and j-th joints are both prismatic, then the (i,j)-th element of the matrix JTJ is physically consistent with units of

units[(JJ)(ij)] = U , for i-th and j-th joints prismatic. (3.18)

Similarly, the Euclidean norm of 4 is also physically inconsistent for noncommensurate manipulators, i.e.,

1411 ? 2 + 2 +. + 2 ,(3.19) making a noncommensurate vector of joint rates with units of (L2 + U)/T2, where T represents time units.

The Euclidean norm of V and the matrix JrJ are physically consistent for an all prismatic-jointed manipulator since the entire JrJ matrix is unitless and V = [vT , 0, 0, O]r, i.e., the angular velocity is zero.









Table 3.1. D-H parameters for GE P50 manipulator.
Joint Type d a 0 a
R 0 0 01 7r/2
R a2 0 02 0
R a3 0 03 0
R 0 0 04 7r/2
R 0 0 05 0


The General Electric P50 manipulator (with 5 revolute joints) has Denavit-Hartenberg parameters given in Table 3.1 and a frame 2 Jacobian

0 0 0 a333 0
0 a2 0 -a3C3 0
2= -a2c2 0 0 0 -a3c4 (3.20)
52 0 0 0 83+4
C2 0 0 0 -C3+4
0 1 1 1 0

The matrix 2J 2J = 2(JXJ) has elements with inconsistent physical units such as the (4,4) term whose calculated value is 1 + a3.

The determinant of 2(Jrj) for the P50 manipulator has terms that sum elements with units of L4 with L6. The determinant of JrJ for a variety of manipulators was calculated in various frames and generally found to be physically inconsistent. A summary appears in Table 3.2. This table also shows the units of the determinant for each of the manipulators in various frames. (Refer to Appendix A for the D-H parameters for each manipulator in this table. This appendix also has the Jacobian and the determinant of JTJ in a particular frame or frames for each of the manipulators.) The frame "general" corresponds to any nonzero translation. Pure rotations have no affect on the value of JrJ since

(J')J' = (GJ)-(GJ) (3.21)
= JGT*GJ (3.22)

G' = G-1 , for rotations (no translation), (3.23) = (J)TXJ' = JJ , for rotations (no translation). (3.24)









Table 3.2. Physical units of Det[JTJ] for various non-redundant manipulators.
Manipulator Coordinate Units of
Description Frame Det[JTJ]
PR Virtual 0,1,2 U
PR Virtual general U + L2
Planar RRR All L4
Non-planar RRR 0,1,2,general U + L2 + L4
General RRR 0,1,2,general U + L2 + La
PPP Orthogonal All U
SAR (PRP) 0,1,2 U
SAR (PRP) 3,general U + L2
RPR 0,1,2,3 U-L2
RPR general U + L2 + L4
SCARA(RRRP) Any L4
RRRP-2 0 L2
RRRP-2 1,2,3,4,general L2 + L4
RRRP-3 0,1 U+L2
RRRP-3 2,3,4,general U + L2 + L4
P50 (5R) 0,1,2,3,4,5 L 4+ L6
P50 (5R) t L4
6-jointed, Det[J] _ 0 _Any frame IL_6-2p

Although the physical consistency of JrJ assures the physical consistency of the

determinant of Jrj, the inverse of this statement is not always true. For instance,

the RRRP-2 manipulator in frame 0 has physically inconsistent terms in O(JrJ), but

Det[0(J-J)] = a2S3 is physically consistent.
It will be shown in Section 6.6 that the physical consistency of the determinant

of JTJ assures that J is physically consistent.

Frames in which jt is physically consistent are called decoupled frames. The
reason these frames are called decoupled frames will be made clear in Chapter 6.

Definition 1 A frame is called a decouple frame of a manipulator if the pseudo-inverse

of the manipulator Jacobian in this frame is physically consistent.

The determinant of J for a manipulator with six joints can always be calculated
since J is 6 x 6 for these robots. The physical dimensions of Det[J] (always physically

consistent) is L3-P, where p is the number of prismatic joints up to three. (Any more









than three prismatic joints will mean the manipulator always has Det[J] = 0.) The determinants of JTJ and JJr therefore have physical dimensions L2(3-P) and are equal since
Det[A]Det[B] = Det[AB] (3.25)

for all square matrices A and B with identical matrix dimensions. Equation (3.25) also guarantees the equality Det[JJ] = Det[JJ] = (Det[J])2.
The determinant of JTJ is zero for manipulators with more than six, joints since JTJ can have at most rank 6, the maximum rank of J (not rank n). So instead we look at the matrix JJ for redundant manipulators.

The units of JJT for an all revolute joint manipulator is
units[JJ*] = [L213,3 [L]3,3 ] for all revolutejoints. (3.26)
[L]3,3 [U ',3

The units of this matrix are physically consistent, as is the case for an all prismaticjointed manipulator where

units[JJT =[[U3,3 [013,3 ] for all prismatic joints. (3.27) [013,3 [013,3

For a noncomnensurate manipulator, the JJ units matrix of

unitsiJJT} ? [ [L2[ZU]3,3 13,3 for noncommensurate manipulator, (3.28) is physically inconsistent.

The determinant of JJr is frame independent (i.e., invariant to both rotations and translations) since for J' = GJ,

Det[J'(J')] = Det[GJ(GJ)] = Det[GJJG] (3.29)

= Det[G] Det[JJ] Det[G] (3.30)

- Det[JJ-] , (3.31)

and the determinant of the twist coordinate transformation matrix G is one.









Table 3.3. Physical units of Det[JJT] for various redundant manipulators.
Manipulator Coordinate Units of
Description Frame Det[JJ-]
6-jointed, Det[J] :A 0 Any frame L6-2p
Anthropomorphic Arm (7R) Any L6
Puma-260 +1 (7R) Any L6
CESAR (7R) Any L6
K-1207 (7R) Any L6
3P-4R Any U
GP66 +1 (2R-P-4R) Any L4 + L6


(The determinant of JJ for manipulators with less than six joints is of course zero since the rank of J and thus the rank of JJ is less than six for these robots.)

The determinant of JJT for several redundant manipulators was calculated and the physical consistency of the determinants corresponded to the physical consistency discussed above for the matrix JJr in all cases. Table 3.3 shows the units of the determinant for each of the manipulators. See Appendix A for the Denavit-Hartenberg parameters of each of these manipulators, the Jacobian in a particular midframe, and the determinant of JJ in this frame.

3.2.1 Consistency of lu = AxI

A generalization of some of the above results for the physical consistency of the Euclidean norm will be shown in this section. For a linear set of equations u = Ax, Theorem 5 and Corollary 2 (both below) show that the physical consistency (or inconsistency) of the Euclidean norm of u can be determined by the physical consistency (or inconsistency) of ArA.


Theorem 5 If u = Ax, where A is an m x n matrix (m _ n), then the for the following statements S1 through S3, Si implies S2 and S2 implies S3, so that S1 implies S3.

S1 The equation Jul2 = u 0 u = u7u is physically consistent (inconsistent).









S2 The nonzero elements in a given column of A have identical units (not all identical units), i.e.,

If aik = 0 and aJk 0, then units[aik] = units[aik],

for k E {1,2,.,n} and ij E {1,2,.,m}. (3.32)


S3 The matrix A'A is physically consistent (inconsistent).


In other words, Theorem 5 tells us that the physical consistency of the Euclidean norm of u implies that all elements in a given column of A have identical units (or are equal to zero) and that ATA is physically consistent. Proof
This proof is split up into two parts: the first proof shows that S1 implies S2; the second proof shows that S2 implies S3. Then by transitivity, S1 implies S3.
The following hold throughout these proofs: i,j E {1, 2,. ,m} and k, h E {1,2,. ,n}.

e Assume S1 to prove S2.

- Since u'u is physically consistent, units[uil = units[uj] = units[u].

- Since u = Ax, ui = E' 1 aikxk.

- Since ui is physically consistent, units[aikxk] = units[aihXh].

- Since units[ui] = units[uj], units[E'=1 aikxkI = units[E'=1 ajkxk].

- But units[E'=1 aikxk] = units[aikxk], so that units[aikxkl = units[ajkxk].

- Therefore, units[aik] = units[ajk] and all terms in column k of A have
identical units. This proves S2 given S1.









* Assume S2 to prove S3.

- Given that units[aik] = units[ajk].

- Let B = ATA, so that bhk = Z=I aihaik.

- Since all elements in a column k of A are identical (units[aik] = units[ajk]),

units[bhk] = units[aihaik] so that each element bhk of B = ArA is physically

consistent. This proves S3 given S2.



Corollary 2 below follows directly from the above theorem when the Euclidean norm of x is physically consistent. Corollary 2 If u = Ax, where A is an m x n matrix (m > n), and the Euclidean norm of x is physically consistent, then the three statements in Theorem 5 are equivalent and are equivalent to the statement

S4 All elements of A must have (must not have) identical units.


Proof

To prove the corollary, it is only necessary to show that with the added condition of a physically consistent Ixi, statement S3 of Theorem 5 implies S4 of the corollary and S4 implies S1 of the theorem.

Throughout this corollary, let i,j E {1,2,. . ,m} and k,h E {1,2,. ,n}.

e Assume xTx, and ATA are physically consistent.

* Since u = Ax, u = 1 aikxk.

* Since xTx is physically consistent, units[xk] = units[xhl = units[x].

* Then units[uj] = units[aikx] = units[aihx], and units[aik] = units[aih]. This

means that all elements in the i-th row of A have identical units.









e The diagonal elements of B = ATA are bkk = l aikaik. Since B is physically
consistent, units[aik] = units[aik]. This means that all elements in the k-th

column of A have identical units.

e Since all elements in any row or any column of A have identical units, then all

elements of A have identical units. This proves statement S4.

* Finally, I will show that statement S4 implies SI. Since the elements of x have

identical units and S4 tells us that the elements of A have identical units, then the equation u = Ax forces the elements of u to have identical units. Therefore,

u has a physically consistent Euclidean norm. This proves statement S1.



A theorem similar to Theorem 6 (offered without proof) can be constructed with the following conditions relating the physical consistency of 1x12, the units of all elements in each row of A, and the physical consistency of AA.


Theorem 6 If u = Ax, where A is an m x n matrix (m n), then the for the following statements S1 through S3, S1 implies S2 and S2 implies S3, so that S1 implies S3.


S1 The equation 1x12 = x 0 x = xTx is physically consistent (inconsistent).

S2 The nonzero elements in a given row of A have identical units (not all identical
units), i.e.,

If aki $ 0 and akj 5 0, then units[aki] = units[akj],

for k E {1,2,.,m} and i,j E {1,2,.,n}. (3.33)


S3 The matrix AA' is physically consistent (inconsistent).









In other words, Theorem 6 tells us that the physical consistency of the Euclidean norm of x implies that all elements in a given row of A have identical units (or are equal to zero) and that AAT is physically consistent.

Corollary 3 follows directly from the above theorem when the Euclidean norm of u is physically consistent (and is also offered without proof).


Corollary 3 If u = Ax, where A is an m x n matrix (m _ n), and the Euclidean norm of u is physically consistent, then the three statements in Theorem 6 are equivalent and are equivalent to the statement

S4 All elements of A must have (must not have) identical units.


The implications of these two theorems and two corollaries are that noncommensurate systems generally need be dealt with in a more considered manner than commensurate systems which has often not been the case in robotics. Since the matrices ATrA and AAr are used in the pseudo-inverse solution x, = Atu, for full column rank A or full row rank A, respectively, the above theorems can be used to determine the general validity of these results. (The validity is not absolutely determined by the physical consistencies of these matrix products as was evidenced in the fact that the RRRP-2 has a physically inconsistent o(JTJ) but a physically consistent Det[O(JTJ)] and OJ.)

In the robotics inverse velocity problem, solving V = J4 for 4, given V, through use of the pseudo-inverse gives physically inconsistent results due to the non-Euclidean nature of the twist and (sometimes) joint spaces. This physical inconsistency is apparent in the physical inconsistency of JrJ or JJT.

3.2.2 Invalid use of Eigensystem and SVD of JJ7Since the pseudo-inverse for redundant manipulators of equation (3.9) contains the matrix JJr, many researchers have used this factor in solving (1.1) for the joint









rates or to characterize a manipulator configuration [2, 3, 12, 23, 29, 32, 39, 44, 45, 52, 57, 59, 60]. Yoshikawa [59, 60], for example, was the first of many to use

Det(JJ-) as a manipulability measure for a manipulator in a given configuration. Further, Yoshikawa (and others including [31, 46]) defined a manipulability ellipsoid with principal axes in the direction of the eigenvectors of JJT. Each ellipsoid axis was given the length of r/A(i), where (i) is an "eigenvalue" of JJP.

Recall that Theorem 3 in Section 2.2 gives the requirements for meaningful eigenvalues and eigenvectors. Even though JJ' is physically consistent for an all revolute joint manipulator (see the units matrix of 3.26), this matrix does not have an invariant eigensystem since (2.15) requires that the units of each term on the main diagonal of the matrix must be identical where in fact they are [L2, L2, L2, U, U, U].

The matrix JJT for most noncommensurate manipulators also does not have meaningful eigensystems since the matrix is itself physically inconsistent. An exception to the general physical inconsistency of JJr for noncommensurate manipulators occurs with the 3P-4R Redundant Spherical Wrist Robot with D-H parameters given in Table A.18 when expressed in a particular set of frames.

The matrix JJT for the 3P-4R manipulator is generally physically inconsistent as expected. But in any frame with origin located at the center of the spherical wrist (the origin of frames 4, 5, 6, and 7), the matrix JJr is physically consistent and unitless. The eigenvalues and eigenvectors of JJ are therefore well defined by the rules given in Theorem 3 and (2.15) and are dimensionless. The eigenvalues are [1, 1, 1, 2,0.873,1.912] and are invariant to rotation of the frame (with this origin). The invariance of eigenvalues to rotations can be deduced from the well known theorem that similarity transformations preserve eigenvalues, i.e., if Ae = Ae, then SAS-'e' = Ae' for full rank S. The twist coordinate transformation matrix G acts like S in the similarity transformation derived below:











JJ'e = Ae

GJJ'e = AGe

e = GYe'

GJJG*Te' = AGG e' (3.34)

GT = G-1 , for rotations (no translation), GJJTG-le = AGG-'e' (3.35)

J' = GJ
J'(J')T'I = Ae'

=. A invariant to rotations. (3.36)

Notice that if translations are allowed, the congruence transformations of (3.34) results. Since GGr $ 16, translations (and congruence transformations) do not preserve eigenvalues.

Even though JJ for the 3P-4R manipulator in frames located at the intersection of the spherical joint axes appears to have physically meaningful eigenvalues and eigenvectors, the interpretation of this manipulability ellipsoid is problematic since the eigenvectors appears to be unitless (not the necessary wrenches that should be expected for the wrench manipulability ellipsoid discussed in Chapter 5). Moreover, as was stated in Theorem 4, noncommensurate systems never have a physically consistent SVD.

The matrix JJT for an all prismatic-jointed manipulator (with at most three degrees-of-freedom and no orientation capabilities) also has a meaningful eigensystem but these limited manipulators will not be discussed.

Therefore, since JJ- does not have eigenvalues or eigenvectors (except for the special cases mentioned above), the above configuration characterization theory is






39

invalid. (Several of the commonly used manipulability ellipsoids are shown in [17] to be physically inconsistent.)

It will be shown later, in Section 5, that the use of metrics on the appropriate noncomnensurate twist and joint spaces (discussed in the next chapter) does not change the fact that the manipulability ellipsoid theory violates the eigensystem and SVD theorems of Section 2.2.














CHAPTER 4
INVERSE VELOCITY KINEMATICS Several authors [14, 19, 35, 53, 54] have discussed the inappropriateness of using the pseudo-inverse in solving for the joint rates given a desired twist vector since this inverse utilizes the Euclidean norms of both the joint-rate vector and the twist vector. But the twist is not a Euclidean space (and neither is the joint-rate vector when the manipulator is composed of both revolute and prismatic joints). This problem has been addressed in these above papers and extensively in [19] by using the (weighted) generalized-inverse along with metrics on both the twist (M,) and joint rates (Mq).
From (1.68)-(1.70) and (1.79)-(1.80), the pseudo-inverse and generalized-inverse of the manipulator Jacobian [19] are jt ? C(F-JC-)-F (4.1)
? C-(CC-)-1(F-F)-1F (4.2)

Ctt, (4.3)

and

J# = M IC (FrMjMqICr) -'FrM (4.4)

= [M1CT(CMq-1cT)-1] [(F-MF)-1F-M.] (4.5)

= C#F#, (4.6)

respectively. A full-rank factorization of J, J = FC, is used in the above equations, where F E R(6xr) has full column rank r , C E 3?(rx) has full row rank r, and n is the number of joints in the manipulator.









Two special cases of the generalized-inverse of a Jacobian are obtained when J is either full row rank or full column rank, i.e.,

i# = MglJr(JMJl)-1 , J full row rank (4.7)

J# = (JTMVJ)-IJTM, , J full column rank, (4.8)


where (4.7) is found by letting F = 16 and (4.8) is found by letting C = I,, in (4.5).

As stated earlier, the metrics must be positive definite, and for invariance to coordinate transformations and scaling, the metrics must transform according to (1.84), i.e.,


M,, = G'M,, G for V' = G,V, (4.9)

Mq = GMqGq for j' Gq. (4.10)

If the desired twist is in the range of the Jacobian, then no metric on the twists is necessary since the residual V - J4, is zero, i.e.,

J# = [M,-Cr(CM,-1Cr)-] [(F-F)-IFr] , V E Range[J] (4.11)


This equation is found by substituting M, = 16 in (4.5).

If the Jacobian has full column rank, then no metric on joint rates is necessary and (4.8) may be used.

If the conditions of both (4.11) and (4.8) are valid-i.e., V is in the range of J and J has full column rank-then neither metric is needed and the generalized-inverse is equal to the pseudo-inverse,

j# = jt , V E Range[J] and J full column rank. (4.12)


But, since all manipulators (including redundant manipulators) have singular configurations [41, and at singular configurations there exist V's not in the range of J, every manipulator has configurations in which a twist metric is needed.









For redundant manipulators, where J has full row rank except in singular configurations, the generalized-inverse is independent of the twist metric and (4.7) may be used. Furthermore, if all joints are revolute (or all are prismatic) the metric on the joint space is not needed for physical consistency-and the pseudo-inverse can be used-but the metric is needed for invariance to coordinate transformations and scaling.

For noncommensurate manipulators with J full row rank, the pseudo-inverse will generally be physically inconsistent (and not invariant to coordinate transformations and scaling) since the minimum norm 141 is physically inconsistent.

Sections 4.1-4.2 will discuss the situations in which the pseudo-inverse solution is physically consistent, invariant to scaling, and invariant to rigid body transformations.

4.1 Physical Consistency of jt

Although the pseudo-inverse of the manipulator Jacobian may be physically consistent in a given frame, there may be other frames in which jt is not physically consistent. (This was suggested by equations (3.9), (3.10), (3.11), all of which have the terms JrJ or JJr embedded in them, and Section 3.2 which discussed the possible physical inconsistencies of these matrices.)

4.1.1 Rotations and Consistency of jt

Theorem 7 shows that if the pseudo-inverse is physically consistent in a given frame then it will remain physically consistent under any rigid body rotation.

Theorem 7 If the pseudo-inverse of J in frame i (ijt) is physically consistent, then for every rigid body rotation from frame i to frame j the pseudo-inverse of J in frame j (ijt) is physically consistent.








Proof
Let 'V and iV be twists such that frame j is a rotation of frame i (no translation), iV = iGj 'V.
Assume that the pseudo-inverse of iJ is physically consistent. The pseudo-inverse of the Jacobian in frame i is

UJt = C,(CC)-1(F-F)-Fr , (4.13)

where J = FC is a full-rank factorization, F full column rank and C full row rank. The pseudo-inverse of the Jacobian in frame j is jt = (JGi'J)t = [(jGiF)C]t (4.14)

= Cr(CCT)-i(Fr JGj~ G F)-F'r jT (4.15)

= C-(CC-)-1(FTF)-F'Gj (4.16)
=ij tGj , (4.17)

where (4.16) follows from (4.15) since ja = (JGi)-' = 'Gj for the case under discussion of JG a rotation (with no translation). It is now only necessary to prove that iJt iG is physically consistent.

Partition the pseudo-inverses in frames i and j into two n x 3 matrices, W and X, and Y and Z, respectively, ijt = [WX] (4.18)

f jr= [Y Z] = [WR XR] , (4.19)

where R = iRj. Since ijT operates on 'V = [v', w']', then each component in a row of W (or a row of X) must have like units or have zero value. Since R is dimensionless, the units of the elements in a row of Y (or Z) are identical to the units of the elements









Table 4.1. D-H parameters for PR virtual manipulator.
Joint Type d a 0 a P di 0 0 0
R d2 0 02 0

in a row of W (or X) and are therefore of consistent physical dimension. Therefore Jtj is physically consistent.

E

Decouple frames are therefore actually decouple points, points at which the pseudoinverse of the manipulator Jacobian (with respect to any frame at the decouple point) is physically consistent. The reason this point is called a decouple point will be made clear in Chapter 6.

Definition 2 A point is called a decouple point of a manipulator if the pseudo-inverse

of the manipulator Jacobian in any frame located at this point is physically

consistent.

4.1.2 Translations and Consistency of jt

A rigid body translation may cause a physically consistent J to become physically inconsistent. An example will demonstrate this fact.
The virtual manipulator [25] associated with the peg-in-the-hole problem [19, 37] after insertion has begun is shown in Figure 4.1. This PR manipulator has the Denavit-Hartenberg parameters given in Table 4.1.

The Jacobian in frame 2 is

2jr0 0 1 0 0 0] (4.20)
100000001
and the pseudo-inverse in this frame, 2jt = 2Jr, is physically consistent.

In an arbitrarily translated frame (no rotation) the Jacobian is tJ = (2Gt,2) 2j, where
2 [ [0 13 [I (4.21)


























Figure 4.1. Peg-in-the-hole with PR virtual manipulator.

and p = [p., py ,1pz]. The Jacobian in this arbitrarily translated frame is j=[ 0 0 1 0 0 0]O(422

Py p 0 0 0 1 (4.22)

and the pseudo-inverse is
[ ? 0 0 1 0 0 0
tjt- -- -P 0 0 0 1 (4.23)
1+p2 +p2 X+p20p0 1+p2+p2 Note the physical inconsistency in the denominator of the terms in t.j The physical inconsistency of this virtual manipulator model of the peg-in-the-hole problem is an alternative demonstration for the non-validity of the Mason-Raibert hybrid control techniques stated in published research [19, 22, 24].

4.1.3 Consistency of Jt in All Frames

The SCARA manipulator (Selective Compliant Articulated Robot for Assembly) [11] in Figure 4.2, with Denavit-Hartenberg parameters in Table 4.2 has a frame 2









Table 4.2. D-H parameters for the SCARA manipulator.
Joint Type d a 0 a
R 0 a, 01 0
R 0 a2 02 0
R 0 0 03 0
P d4 0 0 0


Figure 4.2. SCARA manipulator.


Jacobian of
als2 0 0 0 a2+alc2 a2 0 0 2j 0 0 0 1
0 0 0 0 (4.24)
0 0 0 0
1 1 1 0
Translating the frame of expression of the manipulator by an arbitrary vector p,

results in a Jacobian, tj = (2G',2) 2J, whose pseudo-inverse is

1 0 0 0 0
als2 als2
a2+alc2 1 0 0 0 a2py+alc2py+ajpxs2 tlatS2 a2 0 -p1+a2 32--2px
0= a11202 a 0 0 a1282 (4.25)
a2s5 a2 a2s 2
0 0 1 0 0
Since this pseudo-inverse is physically consistent, the pseudo-inverse in any translated or rotated frame (see Theorem 7) will be physically consistent for the SCARA

manipulator.









The planar RRR manipulator, with its three revolute joints identical to the first three joints of the SCARA, also has a physically consistent pseudo-inverse in any frame. These two manipulators are often used as example manipulators to demonstrate new algorithms [2, 60]. Perhaps this is not appropriate, given their aforementioned special properties.

4.2 Invariance of jt to Scaling

When the pseudo-inverse of the manipulator Jacobian is physically inconsistent, terms of unlike physical units are summed. If the parameters in this manipulator were re-scaled, perhaps from British to SI units, the physically inconsistent terms will cause the resulting pseudo-inverse to give a different result.

It has been argued that the problem of physical inconsistencies can be "factored out" by scaling the problem. The fallacy of this statement will presently be shown.

A change of units scaling matrix is a diagonal matrix that converts a physically consistent vector with physical units into a vector with similar physical units or no units. For example, if V = [vT,wr]r, units[vx] = units[vw] = units[v,] = m/s, and units[wx] = units[w.] = units[wI = rad/s, then S, is a change of units scaling matrix if

s= [av13 01] (4.26)
S,= 0 a,, 13 '

where, for example, ao = (100cm/m)(60s/min) and a, = (60s/min). The scaled twist, V' = [avv, a,wr], has similar units to V, i.e., each element of v and av has units of L/T and each element of w and aow has units of 1/T.

A manipulator joint-rate vector 4 should have the change of units joint-rate scaling matrix

,= Diag[e, C2, .en] where ei {a, if joint i is revolute
qe w a,, if joint i is prismatic '

where the scalar physical unit transformations a, and a, are the identical to those used in (4.26).









Any scaling of a physical unit for a single element of a noncommensurate vector must be identically scaled in all other elements of the noncommensurate vector. For instance, in the example discussed above, the time units were necessarily converted from seconds to minutes in both a, and a,,.

The change of units scaling matrix S, is also normalizing if only the units-not the numerical value-of the noncommensurate vector is changed, i.e., for the twist example above a, = (s/m) and a, = s. A normalizing units scaling matrix is numerically equal to the identity matrix, e.g., S, N 16.

Scaling will now be applied to the inverse velocity problem. The twist vectors are scaled with the change of units diagonal scaling matrix S, and the joint-rate vectors are scaled with the change of units diagonal scaling matrix Sq [15] such that V = S" V (4.28)

4s = SqI . (4.29)

The scaled version of the mapping of joint rates to twist of (1.1) is

V = SV = S JS'SqA = Jj , (4.30)

where the scaled Jacobian is

4 = S'JSqT (4.31)

To obtain the pseudo-inverse of J,, first get the full rank factorization J = FC so that J = FsCs = (S F)(CS '). Equation (4.2) is then used replacing all F's with F's and all C's with C's so that

(j)t ? S.lCr(CS,2Cr)-l(FTS2F)-FrS (4.32)

The scaled joint-rate solution is thus
qrs = J)V (4.33)

(J)tS"V (4.34)

? S-1CT(CS-2CT)-1 (F-S F)lFrSV , (4.35)









and the unscaled joint-rate solution is = Sq ? S V (4.36)


S -- 2 -F) F (4.37)


Compare (4.37) with the generalized-inverse solution of 4. = J#V obtained using (4.5), i.e.,
= M.lCT(CMq 1CT)-(FrM F)-FMV (4.38)

It is evident that the two scaling matrices act as metrics where S2 and Sq2 in (4.37) correspond to the metrics M, and Mq in (4.38), respectively. Since S2 and Sq are both positive definite and symmetric, they need only meet the additional requirements that V G S2V and 4 D Sq24 are physically consistent in order for the ' symbol in (4.37) to become an equal sign.

When the desired twist V is in the range of J, the solution 4, = JtV is always physically consistent. If jt is physically inconsistent, the inconsistencies are canceled out when Jt is multiplied by V.

The RRRP-2 manipulator has a physically consistent pseudo-inverse in frame 0 and physically inconsistent pseudo-inverse in frame 2,

0 0 0 0 0 1
[ 12+3 SIC2+3 s2+3 sl(a1s2+3+a2s3) -cl(a1s2+3+a2S3) 0
Ojt a2s3 a2S3 a2S3 a2S3 a2S3.
C1C2+3 -SC2+3 -2+3 -a18182+3 alcS2+3 0_(.
a2S3 a2S3 a2S3 a2S3 a2s3
C1 C2z A1C2_ aisisq -aict s? 0

0 0 -(al+a2C2) 0. 0
2t? C 1_ 0 0 0 0
2jt . a2S3 a2 (4.40)
. =1a - _ 0 0 0 1
a2S3 a2
L 0 0 0 0 0
S3
where/3 +a 2 + a2c2 + 2aa2c2 is physically inconsistent. When the desired twist is in the range of J, the solution in each of the frames are identical and physically consistent. For instance, the twist for an arbitrary joint-rate vector, q = [41, q2, 43, 4]',








in each of frames 0 and 2 are
cl(a243s2 + 44s2+3) 4453
si(a24352 + 44S2+3) a242 - c344
ov= -al(42 + q3) - a2c2q3 - c2+344 2v -4q(al + a2c2) (4.41)
si(2 + 43) ' 241
-CJ(W2 + 3) C241
L1 L 2 + 43

where 2V = 2Go�V. Substituting �V and (4.39) into 4, = JtV, and substituting 2V and (4.40) into 4. = JtV, both the solutions are 4 = 4 = [41, 42, 43, 4]*.T In frame 2, the physically inconsistent terms in 2J cancel when multiplied by any V E Range[J].
For any twist not in the range of J, the solution is frame dependent. In frame 0 the solution is independent of scaling; in frame 2 the solution is not independent of scaling. For example, let the configuration be defined by

01= 0.1rad , 02 = 0.2rad , 03 = 0.3rad , d4= 4m (4.42)

and let
a, = 0.3m , a2 =lm . (4.43)

Now consider the equivalent desired twists

2. 4m 1.329m
0.2m1 0.4640!I! -1.240m
Vd = , 2Vd = 2G0�V = 0.1967rad (4.44)
5 5
-6rad0. 9805rad 1ra~.6.030rad

not in the range of J. The solution for �Vd is
5 ]

-sa = �J �v0 = 11.000rad, ra4.759d 1.271 rad 4.496 (4.45) The resulting actual twist obtained by substituting this joint-rate vector into �V. = 0J41 is

m rad m rad rad1
V = 2.396 0.2404-, -7.000- 0.6020-, -6.000 ,1.000 ,(4.46)
S5 S S 5 S









which in frame 2 coordinates is 2Va = 2Go�V,
2VmMm rad rad radio
2Va = .329, 0.4640m, -1.280-, 0.1987-, 0.9801-, 6.030 (4.47)
s S S S S

The solutions found in frame 0 will now be compared with those found in frame

2. The solution for 2Vd is
0.9686(o.6301m2+1m4)
o.6103m2 s+m s
4 2jt2V ? 4.759/s
qb -1.271/s ,in frame 2, (4.48)
4.496m/s
4tbN [0.9805, 4.759, 1.271, 4.496]r , using units of m and s. (4.49) The joint-rate solution 4,b in (4.48) is physically inconsistent. The resulting actual twist obtained by using qb in 2Vb = 2J4,b is

2Vb N [1.329, 0.4641, -1.255, 0.1948, 0.9610, 6.030]T (4.50)


which transformed to frame 0 is

ov b = 0G2 Vb N [2.396,0.2404, -7.000, 0.6020, -6.000, 0.9805]r (4.51) These twists are different from the desired twists in (4.44).

If the twists are scaled according to (4.26) and the joint rates are scaled according to (4.27), where a, = 100cm/rnm and a, = 1, then the numerical solution in frame 2 equals


SC" [0.9686, 4.759, 1.271, 449.6]' , using units of cm and s. (4.52) The resulting actual twist obtained by using 4s, in 2s, = 2j& is

2 [132.8, 46.48, -124.0, 0.1924, 0.9493, 6.030]- , (4.53)


which transformed to frame 0 is

0V_ = N [239.6, 24.04, -700.0, 0.6020, -6.000, 0.9686]T (4.54)









Notice that the results of (4.49) and (4.52) differ, i.e., qb : 4,,. The first joint-rate components differ by more than 10%, the second and third joint rates are numerically identical, and the fourth joint-rate component (corresponding to the prismatic joint) in (4.52) is (as expected) 100 times the fourth component in (4.49). Since only terms in the first row of 2j in (4.40) are physically inconsistent, then only the first component of the joint-rate solution is adversely affected by scaling; the other components are scaled appropriately.

The solutions 4,b and &~ are as nearlyl as they are only because the specified twist vector is "nearly" in the range of J, i.e., the desired twist of (4.44) is "almost the same" (whatever that means!) as

m m rad rad radio
V 2.501m, 0.2510-, -7.951-, 0.4992-, -4.975-, 1.000- (4.55)
S S S S S S

2= Mm m rad rad radio
V 1.182m, -1.821-, -1.280m, 0.1987-, 0.9801-, 5.000- , (4.56)
1 S S S S S S
which are in the range of J.

The resulting actual twists Vb and V,, are not equal, are both different from the desired twist Vd, and are both also different form the physically consistent result found in V,.

For the special cases of unitless J, j is physically consistent.

Theorem 8 If J in some frame is unitless, then jt in this frame is physically consistent.


Proof

Since the pseudo-inverse does not introduce any units not already in J, then jt can have only the units of J and the inverse of the units of J or any combination of the two. Therefore, if J is unitless, then jt is unitless.

E









For example, the Jacobians expressed in frames 1 and 2 for the SAR (PRP) manipulator,
0 0 s2 0 0 0
0 0 -c2 1 0 0
11 1 0 0 2j. 0 0 1 (4.57)
0 0 0 0 0 0
0 0 0 0 1 0

01 0 0 0 0
are unitless and the pseudo-inverses, V1 rJ and 2J = 2Jr, are physically consistent.

Of course the inverse of Theorem 8-i.e., if J in some frame is not unitless, then j in this frame is not physically consistent-is not true. For example, the RRRP-2 manipulator has a physically consistent inverse in frame 0, yet the frame 0 Jacobian is not unitless.

Assume that the q, = JtV is scaleable. Then rewriting (4.36), the scaled inverse velocity equation,

? (S X (JX)tSI) V , (4.58)

it is apparent that (S,'(J)tS,) acts like jt in the unscaled equation ci JtV. When
(J)t is physically consistent, the =? can be replaced by an = since scaleability means that q 4 " q. In this case,

jt = Sj'(J,)tS, , when (J,)t physically consistent. (4.59)


Theorem 9 below must be used to verify this equation.


Theorem 9 If D and E are physically consistent invertable diagonal matrices, then A is physically consistent if and only if DAE is physically consistent.


Proof

Let B = DAE, where dii and ejj are the diagonal elements of the diagonal matrices D and E, respectively. Then bij = dijaijejj. Since there is no addition in the equation for bij and no dii or ejj is zero, then bij is physically consistent if and only









if aij is physically consistent. Therefore, B = DAE is physically consistent if A is physically consistent. The other direction of the proof follows directly from the fact that D-1 and E-' are diagonal matrices and A = D-1BE-1 has the same form as B = DAE.



Theorem 9 and (4.59) tell us that if (J3)t is physically consistent, then Pt is physically consistent. Conversely, solve (4.59) for (J,)t,

SqJtS,' = (J,)t , when jt physically consistent, (4.60)

to show that if jt is physically consistent, so is any scaling (J8)t of j. These results lead us directly to the fact that

Fact 1 If Jt is physically consistent, the solution - JtV is independent of scaling

for all V.


If jt is not physically consistent, then (4.60) is not valid, and the pseudo-inverse solution to the inverse velocity problem is not scaleable.

The results of this section can be summarized as follows. A real physical system is always scaleable, e.g., V = J4 can always be scaled. The inverse velocity solution, 4 1 JtV, is scaleable for all twists if and only if jt is physically consistent; in this case q, = JtV. If jt is physically consistent, i.e., the frame of expression has its origin at a decouple point, then scaling will not affect the resulting joint rates and the solution , is independent of scaling.

4.3 Equivalent Generalized Inverses

If an identity metric is assumed in a particular frame, the pseudo-inverse is equal to the generalized inverse. But in addition, there are other metrics that also give the same result.









Table 4.3. D-H parameters for the PRP Small Assembly Robot (SAR).
Joint Type d a 0 a
P di 0 0 0
R 0 0 02 7r/2
P d3 0 0 0


Using Theorem 10 below, all metrics which result in identical joint velocities can be found. Theorem 10 stems from Theorem 2.2 in [19] and the facts that JJ# = FF# and J#J = C#C. The proof of these Theorems is given in [19].


Theorem 10 All statements in the left column are equivalent statements and all statements in the right column are equivalent statements [19]: JJ# = (JJ#)T J#J = (J#J)T (4.61)

M JJ# = JJ#M' MqJ#J = J#JM (4.62)
jt = j# jt = j# (4.63)

MJJt = JJtMV MqJtJ = JtJMq . (4.64)


If we assume (4.63), that the pseudo-inverse is equal to the generalized inverse, then the left equation of (4.64) may be used to solve for all equivalent twist metrics, M'JJtJJtM =0 . (4.65)

For example, the PRP Small Assembly Robot (SAR) shown in Figure 4.3, with Denavit-Hartenberg parameters given in Table 4.3, has a pseudo-inverse in frame 2 of
of [0 1 0 00 0]
[jt0 0 0 0 1 0 (4.66)

0 0 1 0 0 0






















Figure 4.3. Small Assembly Robot (SAR). Any metric of the form in (4.67) that is also positive definite will cause the generalized-inverse to equal the pseudo-inverse, i.e., ill 0 0 in14 0 M16
0 M22 m23 0 M25 0
M = 0 m23 M33 0 M35 0 (4.67)
M14 0 0 M44 0 M46
0 n25 M35 0 n55 0 iM16 0 0 M46 0 M66

The important result of this section is that if a pseudo-inverse is physically consistent, then there are a set of metrics which give identical results when using the generalized-inverse, i.e., for every decouple point of a manipulator, a class of metrics exist for which the pseudo-inverse and generalized-inverse of the Jacobian are equal.














CHAPTER 5
MANIPULATOR MANIPULABILITY

As was discussed in Section 3.2.2, the matrices JJ* and JrJ do not have physically consistent eigenvalues, eigenvectors, or a SVD. A few authors [17, 20, 31, 46, 601 have used other Jacobian functions-some Jacobian functions incorporating metrics-in manipulability definitions. In this section several of these manipulability ellipsoids will be introduced and their eigensystems will be explored.

There are three basic types of manipulability ellipsoids. Each of these arise from setting the square of the a Euclidean or non-Euclidean norm to less then or equal to 1. The manipulability ellipsoid discussed previously is called the wrench manipulability ellipsoid (or force manipulability ellipsoid) since this ellipsoid is defined as 1r12 = Wr(JJr)W < 1 . (5.1)

The "eigenvalues," Aj, and "eigenvectors," ej, of JJ7 are used to create the ellipsoid with each principal axis in the direction of an ei and axis length equal to A singular value decomposition of J can be used to deduce these same quantities (see Section 1.3).
As discussed previously in Section 3.2.2, this analysis is faulty due to the failure of JJr to have a physically meaningful eigensystem (see Theorem 3).

It was proposed in [20] that incorporating a metric to replace the Euclidean norm of T might correct this problem. The resulting equation if a metric is used to determine the Mr-norm of Tw is

IT12d = WT(JMMjT)W - 1 (5.2)

It will now be shown that the ellipsoid defined by the eigenvalues and eigenvectors of JM.J does not meet the requirements for a physically consistent eigensystem.

57








The physical units of M , found by forcing -r M,-T to be physically consistent, are

units[M] = - T C , where C = [cij] and (5.3)
F2L2
U , joints i and j revolute
cij = L , either joint i or j revolute, the other prismatic (5.4)
1L2 , joints i and j prismatic. The units variable -t, is equal to the desired units of ITI.
With the above units for M , the resulting units matrix for JMJr is

____ [L213,3 [L]3,31
units[JMJ] = F2L2 [[L]3'3 [U]3,3] (5.5)
F2L2 I[L]3,3 1U]3,3I

The units matrix for JMJ is a scalar multiple of the units matrix of JJ for manipulators with all revolute joints-given in (3.26). Therefore, by Theorem 3, the wrench manipulability ellipsoid with metric M, is also based on a physically inconsistent eigensystem.
It should be pointed out that no metric is needed for a physically consistent ITn if all the joints are of identical type, therefore the above result could have been immediately deduced.
A units analysis of the Mq metric used to make 142q physically consistent leads to the units matrix

units[Mq] = q -2C , where C = [cij] and (5.6)
L ,joints i and j revolute
cij = L ,joint i or j revolute, other prismatic (5.7)
U ,joints i and j prismatic,
where the units variable q is equal to the desired units of 1412qM. The units matrix M -I is therefore

units[MT] = 'SqC , where C = [cij] and (5.8)
U ,joints i and j revolute
cij = L ,joint i or j revolute, other prismatic (5.9)
L2 ,joints i and j prismatic.








This units matrix differs by a scalar constant from the units matrix of M,. Therefore, metrics derived for joint rates can be inverted and then used for joint torques, i.e., Mr = Mq.
The twist manipulability ellipsoid was defined originally [59] as

= Vr ((Jt)jt) V < 1 . (5.10)

The twist manipulability ellipsoid can alternatively be defined with a generalizedinverse and/or with a joint-rate metric as

14s12 = VT ((J#)J#) V < 1 (5.11)

IqsM = V ((Jt)-MqJt) V 1 or (5.12)

I1Mq - V ((J#)TMqJ#) V < 1 . (5.13)

Since noncommensurate manipulators generally have physically inconsistent Jt and thus can not have physically consistent eigensystems, only all revolute-jointed manipulators will be analyzed for the definitions in (5.10) and (5.12). The units analysis below for revolute joints using jt and (5.12) is equivalent to the units analysis of any manipulator using J# and (5.13).
Each of the n rows of jt has the units

units[Jt](.) = [ L' L, U, U, U] ,for all revolute joints. (5.14) (Notice that the rows of this jt are ray coordinate screws as opposed to the axis coordinate screws of the columns of J.) Therefore, the units of (Jt)TJt for an all revolute-jointed manipulator are
units[(Jt)7Jt] - I [1U]3,3 [L]3,3 ] for all revolute joints. (5.15) LntZ- [L]3,3 [L2]3,3 '

And since for an all revolute joint manipulator the metric Mq is entirely composed of identical units, the units of (Jt)TMqJt are proportional to the units of (jt)T jt. By Theorem 3, the matrix (Jt) Jt does not have a physically meaningful eigensystem.






60

Replacing the pseudo-inverse of J with the weighted generalized inverse of J does not change the fact that the matrix (J#)7MqJ# does not have a physically meaningful eigensystem. (The physical units of J# are a scalar multiple of the units of jt when j is physically consistent.) But the matrix (J#)T*J# is physically consistent even for noncommensurate manipulators.

The dynamic-manipulability ellipsoid [17, 20, 60] is derived from the manipulator dynamics equation
r = M(q)j + h(q, 4) + g(q) , (5.16)

where r represents the generalized-force vector at the joints, M(q) is a positive definite mass matrix, 4 is the joint acceleration, h(q, 4) represents the Coriolis and centrifugal forces, and g(q) represents the gravitational forces. Solving for 4 results in

4 = M-1 [r - h(q, i) - g(q)] , (5.17)

where the dependency in M(q) on q has been dropped for simplicity of notation.

The development here follows from [20] and is given here to demonstrate the method with which manipulability matrices have been derived. Differentiating V = J4j with respect to time results in v=J + .(5.18)


Again to simplify the notation, define A as the frame acceleration, A=Jq=V-Jq , (5.19)

and ? as
r= -h(q, )-g(q) (5.20)

Substituting (5.20) into (5.17) and the result into (5.19) yields A = jM-I . (5.21)








Solving for "r we get
(JM-')tA , (5.22)

or
f= (JM-')#A . (5.23)

The Mr-norm of f, (using only the generalized inverse since the pseudo-inverse may be physically inconsistent) is

AM. = X ([(JM-1)#]'M.(JM-1)#) A (5.24)
= 4" (J[(JM-1)#]T'M.(JM-1)#J) ij . (5.25)

If J has full column rank, then

(JM-')# = MJ# , J full column rank (5.26)

f, = MJ#A , J full column rank. (5.27)

and

IsIM = (MJ#A)rM.(MJ#A) , J full column rank (5.28)
= Ar ((J#)TMTMTMJ#) A , J full column rank (5.29)

The dynamic-manipulability ellipsoid is found using (5.29) so that

i [ Mr = AT ((J#)rMTMTMJ#) A < 1 , J full column rank, (5.30) and the ellipsoid is found from eigensystem of (J#)'TMTM-MJ#. As discussed previously, a metric can be used for M,. If Mq = M so that 1I1q is the kinetic energy of the manipulator, then (5.30) reduces to

I-'2M, = A((J#)rMJ#) A <
J full column rank and Mr = M-1. (5.31)

The ellipsoid found from the eigensystem J#TMJ# (J full column rank) is physically consistent but does not meet the criteria of a valid eigensystem in (2.15), since the









units of this matrix are proportional to the units of (5.15). (Notice that the matrix defining the dynamic manipulability ellipsoid is identical to the matrix defining the twist manipulability ellipsoid.)

Let us look a little further J#TMJ#, the definition for the dynamic manipulability ellipsoid as originally developed in [59]. Expanding (5.31) by substituting (5.21) for A yields

M = (Jr(j#)TMj#j) 4 , J full column rank, M, = M-1. (5.32)

But for full column rank J, J#J = I, and (5.32) to the trivial equation


--= 4'Mq , J full column rank, Mr = , (5.33)

and the ellipsoid is dependent only on the metric. But since M has the units of Mq and M does not satisfy the conditions necessary for a valid eigensystem for noncommensurate manipulators, again the dynamic manipulability ellipsoid is shown to have an invalid eigensystem. Note that although Mq is unitless for commensurate manipulators and thus M, has a valid eigensystem, the dynamic manipulability ellipsoid does not have a valid eigensystem even for commensurate manipulators.

For the case when J does not have full column rank, (5.24) is used to define the ellipsoid [601. But again, a units analysis of the matrices shows that the eigensystem requirements are violated. This is also true for the expanded version of this ellipsoid determined by (5.25) when the manipulator is noncommensurate; but, if the manipulator is commensurate, each term of the matrix determining the ellipsoid has identical units and the eigensystem is physically meaningful.

To summarize, none of the manipulability ellipsoids possess geometric invariance. The wrench manipulability ellipsoid defined by the eigensystem of matrix JMrJr is not valid for any manipulator. The twist manipulability ellipsoid originally defined by the eigensystem of (Jt)rJt and subsequently modified to (Jt)rMqJt and then to
(J#)rMqJ#, is not valid for any manipulators. The dynamic-manipulability ellipsoid,









defined by the eigensystem of matrix [(JM-)#*]TMr(JM-)#, is not a physically consistent eigensystem even for the case when J has full column rank. If J has full column rank and M, = M-', this matrix product reduces to (J#)-MqJ# which also does not have a valid eigensystem. An expansion of the dynamic-manipulability equation leads to jr ((J#)TMqJ#) J = M,, which has a valid eigensystem if the manipulator is commensurate.

Although the existing manipulability theory has been shown to be invalid in all cases for manipulators with six or more joints, for manipulators with six or fewer joints, the scalar manipulability measure, Det[JrJ], is physically meaningful at decouple points. At decouple points, the manipulability measure is physically consistent (see equation (6.105)). Thus, when a decoupled coordinate frame is used, the manipulability of these manipulators in one configuration can be meaningfully compared to the manipulability at other configurations.














CHAPTER 6
DECOMPOSITION OF SPACES

Griffis recently introduced a special six dimensional spring for use as a wrist placed on a 6-jointed manipulator [26]. He thus created a wrench space via small displacements (or twists) creating a K-orthogonal complement to the twists of freedom, which he called the twists of compliance. With this technique Griffis and Duffy [28] showed that independent position and force control can be accomplished for a two-dimensional example and that the twists of compliance are in fact K-orthogonal complements to the twists of freedom. Without adding such a wrist, this chapter explores several techniques for twist and wrench space decomposition.

Let us assume that a twist space referenced to a particular coordinate system is decomposed into two manifolds, and one of these manifolds is the twists of freedom subspace, V1 = Range[J], as previously defined in (1.13). The other manifold is the twists of nonfreedom, V,1, introduced by Lipkin and Duffy [36] in their important article on the nature of twists and wrenches as screws.
The twists of nonfreedom are the twists that are not possible to accomplish in a given configuration. Lipkin and Duffy [36] define this as a "subspace which is the orthogonal complement of" the twists of freedom, although Duffy later repudiates this notion in [22]. But since V1 is a noncommensurate space, the orthogonal complement of Vf is not an appropriate manifold to introduce since it does not have the physical dimensions of a twist manifold. This manifold would have the strange property of dependence on the units of expression of V1. The wrenches of constraint subspace, VV,, when viewed as a unitless vector space in R6, is recognized as the orthogonal complement of an assumed unitless version of V1. But WVc only in special cases









appear to have the physical units of twist vectors, which is necessary for the manifold V,,1 to be meaningful. (To be fair, [36] defines twists of nonfreedom in the context of an example that appears to have a unitless basis for W,, which could therefore be viewed as an appropriate twist subspace. This dissertation defines wrenches of constraint in a manner consistent to the definition given in [36].)

6.1 Projections and Kinestatic Filters

In commensurate systems, the pseudo-inverse and generalized-inverse can be used to separate various spaces into two disjoint spaces [34, 56]. In noncommensurate systems, care must be taken when using the pseudo-inverse. If the pseudo-inverse is physically inconsistent, projections using this inverse are also generally physically inconsistent.

All types of projections for the various manipulator spaces are derived below using the generalized-inverse, although in cases of a physically consistent pseudo-inverse, the generalized-inverse may be replaced by the pseudo-inverse.

The twist space projection is found through the following series of equations:

V = J4 (6.1)

4. = J#V (6.2)

V, = J4, (6.3)

V = JJ#Vd, (6.4)

where the s subscript is for "solution", the "d" subscript is for "desired', and the "r" subscript is for "resulting."

The joint-rate space projection, obtained by substituting (6.1) into (6.2), is

4, = J#Jld � (6.5)

The wrench space projection is found through the following series of equations: r = J W (6.6)






66

W = J#*'Td (6.7)

W = J#Jd = (JJ#)Td . (6.8)

The generalized-force space projection, obtained by substituting (6.7) into (6.6), is
r = JTJ#Trd = (J#J)Td (6.9)

The various projection matrices are the four kinestatic filters [19],

P, = JJ# , Pq = J#J , P = (JJ#) , P, = (J#J) (6.10)

The various spaces can now be decomposed into disjoint spaces using the above projection matrices and (6.4), (6.5), (6.8), and (6.9), V = Null[JJ#1 E) Range[JJ#] (6.11)

Q = Null[J#J] (D Range[J#J] (6.12)

W = Null[(DJJ#)T] M Range[(JJ#)r] (6.13)

T = Null[(J#J)] MD Range[(J#J)r] (6.14)
M,
where the symbol E means that the two subspaces on either side of this symbol are Me-orthogonal. The normal direct sum (E) means that the two spaces are orthogonal (in the Euclidean sense). Notice that the above decompositions do not follow from the fundamental theorem of linear algebra, R- = Null[Ar] E Range[A], where the range and null operators operate on a matrix and its transpose. For the metric-dependent decompositions, the range and null operators operate on the same matrix.
The above decomposition equations can be simplified by applying some facts about the full rank decomposition of the Jacobian, J = FC and J# = C#F# of (4.5), JJ# = FCC#F# = FF# (6.15)

J#J = C#F#FC = C#C , (6.16)








and some facts about the null and range space operators, Null[JJ#] = Null[FF#] = Null[F#] (6.17)

Range[JJ#] = Range[FF#] = Range[F] (6.18)

Null[J#J] = Null[C#C] = Null[C] (6.19)

Range[J#J] = Range[C#C] = Range[C#] (6.20)

Each of the statements in (6.17)-(6.20) can be proven in a manner similar to that shown below for (6.17).
Let FF#x = 0. Multiply both sides by F# to give F#FF#x = 0. But by the property of the generalized-inverse given in (1.76), F#FF# = F#, so that F#x = 0. Therefore, Null[FF#] = Null[F#].
These simplifications lead to the below simplified decomposition equations:

V = Null[J#] ED Range[J] (6.21)
Mq
Q = Null[J] (D Range[J#] (6.22)

W = Null[J] ED Range[(J#)] (6.23)
MVT
T = Null[(J#)] E) Range[J] , (6.24)

and the even simpler decomposition equations: V = Null[F#] e' Range[F] (6.25)
Mq
Q = Null[C] E Range[C#] (6.26)
M;-1
W = Null[F'] M& Range[(F#)*r] (6.27)

T = Null[(C#)T] MD Range[C] (6.28)

Each metric will give a different decomposition. If the metric has the required property (that it transforms via a congruence transformation, (1.84)), then the frame of expression has no bearing on the decomposition.








The below two facts allow us, in some cases, to apply the above metric dependent decompositions, which use the generalized-inverse, to a metric independent decomposition, which uses the pseudo-inverse. Fact 2 If J = J# for some metric M, and some metric Mq, then Jt is physically

consistent.

Fact 3 If jt is physically consistent, then jt = J# for some metric M, and some

metric Mq.

If the pseudo-inverse is used instead of the generalized-inverse by choosing change of unit identity scaling metrics for M, and Mq, the decomposition is frame dependent and only valid if the pseudo-inverse is physically consistent. The decomposition for physically consistent jt is

V = Null[JJt] 0 Range[JJt] (6.29)

Q = Null[JtJ] E Range[JtJ] (6.30)

1WV = Null[(JJt)] E Range[(JJt)T] (6.31)

T = Null[(JtJ)r] e Range[(JtJ)r] . (6.32)

From Theorem 10 and the fact that jt = J# for some metric (since jt is assumed physically consistent), Jjt = (jjt)T and JtJ = (JtJ)-. Therefore the above decompositions simplify to

V = W = Null[JJt] E Range[JJt] (6.33)

Q = T = Null[JtJ] 0 Range[JtJ] , (6.34)

when jt is physically consistent. The spaces V and W are decomposed identically as are the spaces Q and T.








The above decomposition can be further simplified by using the below equations:


jjt jtj

Null[JJt]



Range[JJt] Null[JtJ] Range[JtJ]


= FFt

= CtC

= Null[FFt] = Null[Ft] = Null[(F"F)-F ] = Null[F'] = Null[C'F'] = Null[JT] = Range[J] = Null[J]

= Range[CtC] = Range[Ct] = Range[C'(CCT)-1] = Range[C'] = Range[C'F'] = Range[J'] .


The space decompositions for frames in which jt is physically consistent are therefore

V = 1 = Null[J'] D Range[J] (6.43)

Q = T = Null[J] G Range[J] . (6.44)

Equations (6.43) and (6.44) appear to be direct applications of the fundamental theorem of linear algebra; this is a deceptive notion. The reader should remember the limited scope of these equations-i.e., they are only valid in frames in which jt is physically consistent-and their rather involved derivations.
This decomposition will be explored further in the subsequent sections.

6.2 Twist Decomposition

In order to demonstrate the problem with defining a twist of nonfreedom manifold as a subspace, two examples will be shown. One example will show when these twists constitute a subspace and the other will show when they do not form a subspace.
First consider the SCARA manipulator of Figure 4.2. The SCARA Jacobian expressed in frame 2 coordinates was given in (4.24). The column-reduced echelon


(6.35) (6.36) (6.37) (6.38) (6.39) (6.40) (6.41) (6.42)









form of the wrench of constraint subspace in this frame, 'V, = Null[2JT], is [2W E 0 0 0 1 0 (6.45)


where E, is the matrix that converts [WYJb to column-reduced echelon form. Note that these wrenches might also be interpreted as twists of nonfreedom with no discrepancy with units,
[2v ] E [0 0 0 10 (6.46)
E 0 0 0 1 0 0
The SAR (PRP) manipulator of Figure 4.3 has the Jacobian and wrench of constraint subspace basis vectors expressed in frame 3 coordinates of

0 d3 0 0 -_ 0
1 0 0 0 0 0
3j 0 0 0 [3W0]bE= 0 0 1 (6.47)

0 1 0 0 1 0
L0 0 0 JL1 0 0J

Note that these basis wrenches cannot be interpreted as twists of nonfreedom since the second basis vector does not have the units of a twist (an axis coordinate screw). Therefore, for this manipulator expressed in frame 3 coordinates, the concept of twists of nonfreedom as described previously (as a subspace) is untenable.

A slightly modified definition of twists of nonfreedom is therefore necessary and is given below.

Definition 3 Twists of nonfreedom are twists that the manipulator cannot fully generate in a given configuration,

"nf = V - Vf . (6.48)

The meaning of the above equation might need explanation. The manifold Vf include all the twists of V except those twists in Vf. This is not the orthogonal complement of Vf, which (as stated previously) is physically inconsistent for screws.









The nonfreedom twist manifold might also be defined as

V,, = {V,,: Vq E V and Vq i Vj} (6.49)

In general, the manifold Vq is not a subspace. Typically, two twists of nonfreedom might sum to a twist of freedom or a nonfreedom twist.

For example, two nonfreedom twists for the SAR manipulator expressed in frame

3 coordinates are
0, 0
0 0
is M - 1rad (6.50)
1an 1rad_5 s
0 0
0 0

The sum of these two nonfreedom twists is the twist of freedom [0, 0, 1, 0, 0, OT. The difference of these two nonfreedom twists is the nonfreedom twist

[0, 0, 1M, s 0, 0]-.

Since Vn1 is not, in general, a subspace, a direct sum decomposition of twists of freedom and twists of nonfreedom is not typically possible, i.e., V 7� vf E) vs (6.51)

In the special cases when W, can be interpreted entirely as twists, the twist space can be decomposed as the direct sum decomposition, V = V1f Vi, where Vi are the subspace of inaccessible twists defined below. Definition 4 Inaccessible twists constitute the screw subspace of twists such that "i C "q ,(6.52)


and the inner product Vi G V - vi D v1 + wi ) WI (which is generally physically inconsistent) is physically consistent for any Vi E Vi and any Vf E Vf. The
subspace Vi may not exist.









If Vi = V,1, then the twist space is uniquely decomposed by the direct sum decomposition

V=Vf eD V, ifVi =Vf. (6.53)

6.3 Wrench Decomposition

Assume that a wrench space referenced to a particular coordinate system is decomposed into two manifolds. One of these manifolds equals the wrenches of constraint, W, = Null[Jr], as previously defined in (1.19). Since W, is the null space of a matrix, it must be a subspace. The other manifold is the wrenches of nonconstraint manifold [36], Wno.

The wrenches of nonconstraint, when applied at the end effector of a manipulator, require some nonzero joint forces for static balancing or will cause some motion of the manipulator. Lipkin and Duffy [36] define wrenches of nonconstraint in an analogous fashion to the twists of nonfreedom, i.e., according to [36], the wrench of nonconstraint manifold is the orthogonal complement of the wrench of constraint subspace. But the orthogonal complement of W, has physical dimensions of a twist manifold. Furthermore, when W, is viewed as a unitless vector space in R6, the orthogonal complement is a unitless version of Vf. But the axis screw vectors of Vf, only in special cases appear to have the physical units of wrench vectors, a necessary requirement for the manifold W, to be meaningful.

For example, the orthogonal complement of 3yV for the SAR manipulator is the Jacobian, 3J, given in (6.47). The second basis vector of 3j in (6.47) is obviously not a wrench (a ray coordinate screw), so this subspace cannot describe wrenches of nonconstraint.

To avoid the above problems, a slight modification of the definition of wrenches of nonconstraint is given below.









Definition 5 Nonconstraint wrenches are wrenches that will produce a nonzero power

with some twist of freedom [25],

IN== W - W. (6.54)


The manifold of nonconstraint wrenches are all the wrenches of W except those wrenches in W,. This is not, in general, the orthogonal complement of W,, which (as stated previously) is physically inconsistent for screws.

Wrenches of nonconstraint might also be defined as

I, = {Wn,: Wc E W and W, IV} (6.55)

Note that W, is a manifold that, in general, is not a subspace, so that no direct sum decomposition of wrenches of constraint and wrenches of nonconstraint is generally possible, i.e.,

W 0 Wc EWn . (6.56)

In the special cases when the twists of Null[W/f] possess a meaningful interpretation as W, , the wrench space can be decomposed via the direct sum decomposition, W = Wc E Wd, where Wd are the subspace of driving wrenches defined below. Definition 6 Driving wrenches constitute the screw subspace of wrenches such that Wd 9 W. , (6.57)


and the inner product Wd 0 Wrtc I fd G f,, + nd 0 nc (which is generally physically inconsistent) is physically consistent for any Wd E Wd and any Wc E

W,,. The subspace Wd may not exist.


If Wd = W,,, then the wrench space is uniquely decomposed by the direct sum decomposition


W = Wd e Wc , if Wd = Wnc.


(6.58)









If both twists of nonfreedom and wrenches of nonconstraint are subspaces (and thus are identically the inaccessible twists and the driving wrenches, respectively), then a hybrid control is accomplished by decomposing the desired twist into twists of freedom and twists of nonfreedom and the desired wrench into wrenches of constraint and wrenches of nonconstraint, and then filtering out the inaccessible twists and constraint wrenches. This assures that the control inputs will be entirely composed of twists of freedom and driving wrenches.

6.4 Hybrid Control

The hybrid control algorithms of Mason [40, 41] and Raibert [51] inherently assume a decomposition essentially equivalent to

- Range[J] D Null[Jr] = V1 E) W, (6.59)


This theory splits the hybrid control problem into "natural" and "artificial" constraints at what is now commonly know as the "center of compliance" or "compliance center" [2, 25, 60] (called a constraint frame in [11, 51]). A center of compliance is defined as a point through which pure forces produce only pure translations and pure couples produce only pure rotations about that point. This point may or may not exist, or may exist at more than one point.

When the coordinate reference frame origin is located at the center of compliance, the MRHCT (Mason and Raibert's hybrid control theory) states that the diagonal selection matrices [11, 51] are used to determine the appropriate action for each loop of the hybrid position and force control., i.e., each joint is used to control either a position component (twist) or a force component (wrench).

The MRHCT calls these two subspaces orthogonal complements, which these subspaces appear to be if the screw spaces were instead commensurate six dimensional vector subspaces as in (1.61). But they are not orthogonal complement screw subspaces.








An example will now demonstrate the MRHCT [1, 2, 19]. The task at hand is to place a peg into a hole as shown in Figure 4.1. (In this example, the virtual PR manipulator of the figure is not involved.) The "natural" and "artificial" constraints, taken together (since the distinction between the two is sometimes open to interpretation), with respect to frame 2 are v, = v, = 0, f, = 0, and n, = 0. Both the twist and wrench selection matrices are diagonal matrices, both with elements of either 0 or 1. This leads to the twist selection matrix, 2P,, and the wrench selection matrix 2pm, i.e.,

0 00 0 00 1 00 00 0
0 00 0 00 0 10 00 0
2p 0 0 1 0 0 0 2p 0 0 0 0 0 0
0 0 00 00 0 0 0 1 0 0 (6.60)
0 0 0 0 0 0 0 0 0 0 1 0
0 0 0 0 0 1 0 0 0 0 0 0

The selection matrices are always related by the equation

P. = 16- P . (6.61)

The hybrid control then filters the specified twist, V,, and wrench, W,, with the selection matrices as follows:

2v = 2 ,2V 2w = 2P, 2Ws (6.62)


This guarantees that the twist 2V E 2Vf and 2W E 2W, in frame 2.
It is apparent that the selection matrices, P, and Pw, act as filters on twists and wrenches. In fact, P, and Pw are projection matrices,

B _ 2B [2B(2B B)-2B-] (6.63)
2p 2C2Ct = 2c [2C(2CT2C)-12C-] , (6.64)









where B represents a basis for the twists of freedom and C represents a basis for the wrenches of constraint,

0 0 1 0 0 0
0 0 0 1 0 0
1 0 2c= 0 0 0 0 (6.65)
0B 00 0 0010

00 0001
01 0000

In frame 2 the MRHCT seems to work. But in a frame t (see Figure 4.1), arbitrarily translated from frame 2, the MRHCT fails. In this frame the projection matrices, P, = Jt and PL, = tW C , W, are physically inconsistent, i.e.,

2 +2
-Ppv 0 0 0 0 0 ._e
-?JL P, 0 0 0 - - 0 0 0 -TV 0 0 100 0 ,p ? 0 0 000 0
0 0 000 0 0 0 010 0
0 0 000 0 0 0 001 0
PE 2 0 00 & P. 0 0 0 '
(6.66)

where - 1+ p2 + p2, a physically inconsistent quantity.

6.5 Decomposition with Ray Coordinate Twist Space

Recently several authors [1, 24] have expanded a discussion on isotropic subspaces begun in [52] and greatly enhanced in [37]. These articles have attempted a different decomposition using four manifolds, two of which are the twists of freedom and wrenches of constraint. Manipulate the twists space via the A matrix so that the twists of freedom and wrenches of constraint subspaces are both defined using ray coordinate screws, i.e.,

V y = AVJ = Range(AJ) . (6.67)

The radical manifold, 1?, is the screw manifold of the common elements in V'ray and wM)


RiV?= ry (6w6)


(6.68)









The defect manifold, E), is the manifold not covered by V7 Y and Wc, (V7Y u WC) u D = $6 , (6.69)

where $6 is the full 6-dimensional ray coordinate screw space.
Let us investigate how each of these manifolds relate to the others. As shown in Theorem 1, V and W, are reciprocal subspaces. Since v;aY is the ray coordinate version of Vj, then V ry and W, are also reciprocal subspaces. This theorem leads to the corollary below which states that the radical manifold is a self-reciprocal subspace. The proof for the theorem below is based in part on Theorem 1 which states that coordinate transformations do not affect the reciprocal product. Corollary 4 The radical screw subspace 1?. is self-reciprocal, riorj=O , Vri,rjEIZ . (6.70)



Proof

Since r E R, r E V7aY, and r E Wc, and all TVaY E Vray and W, E W, are reciprocal (Vray o W, = 0) by Theorem 2, then ri o rj = 0 for all i and j.



Since the screw subspace R. is self-reciprocal, the screws in this subspace are selfreciprocal and mutually reciprocal. The theorem below also shows that each column of a manipulator Jacobian is self-reciprocal. Theorem 11 For revolute and/or prismatic jointed manipulators, each column of a manipulator Jacobian is self-reciprocal.









Proof

If the i-th joint in a manipulator is revolute, the i-th column of the manipulator Jacobian in frame i - 1 is [0, 0, 0, 0, 0, 1]*T. If the i-th joint in a manipulator is prismatic, the i-th column of the manipulator Jacobian in frame i - 1 is [0, 0, 1, 0, 0, 01. Since both these screws are self-reciprocal and reciprocity is invariant to coordinate transformations, then regardless of the frame, the i-th column of the Jacobian is selfreciprocal.

N

The radical is always a subspace since it is the intersection of two subspaces. But Vray U W, is generally not a subspace as is shown in the below example.

The P50 manipulator with 02 = 03 = r/2 and 04 = 0 has

0 0 0 0 0 0
-1 0 0 0 -1 0
0 1 1 1 0 0
[VaY]b=' 0 'a2' 0 0 '0 ,[c]b= 1
0 a3 a3 0 0 0
a3 0 0 0 0 0
(6.71)

Summing the fifth screw of Vfay and - times the only screw of V, results in the vector [0, -1, 0, 7, 0, 0]", for all -, where units[7] = L. This screw is not in V} Y U W, for any nonzero Y. Therefore VaY U W, is not a screw subspace.

Similarly, the defect manifold is generally not a screw subspace, since D2 = $6 (V'Y U We), although [24, 37] both claim that the defect is a subspace. For example, the SAR manipulator in frame 2 has twist of freedom and wrench of constraint basis sets of
0 0 0"-' 0 0 1
0 1 0 0 0 0
_al 2 0 0 0 _ [ 0 0 0
[2VfaY]b=A2J= 0 ' 0 ' 0 [Wrb- 0 0 1 0
1 0 0 0 0 0
0 0 1 L 1 0 0
(6.72)






79

so that the radical basis set is

[2Z]b = {[0, 0, 0, 0, 0, 11] (6.73)

The defect manifold contains all screws [2D]b = {I[3[, fl, 'Y, 6., 6y, 62]} (6.74)

with nonzero -. This is not a subspace, although [24] claims that a basis can be selected for the defect, [2 D]b = {[0, 0, -y, 0, 0, 0]'}.

In frame 3, the SAR manipulator has twist of freedom and wrench of constraint basis sets of
0 0 0 0 -' 0
0 1 0 0 0 0
[3oy= 0 0 0 [3W]b 1 0 0
S0' d3' 0 [ ]0 0 ' 1
1 0 0 0 1 0
L 0 0 1 0 0 0
(6.75)

so that the radical basis set is empty, i.e., [31Z]b = 0. The defect manifold is also empty for the SAR manipulator in frame 3.

It is apparent now that the decomposition theory of [24, 37] is not unique and the claims made are generally invalid. Therefore a new technique for screw and wrench space decomposition is presented in the next section and the results of the previous sections of this chapter are tied together.

6.6 Space Decomposition at Decouple Point

In Section 6.2, it was shown that in some cases the twist space can be decomposed uniquely via a (Euclidean) direct sum decomposition, (see (6.53)) and in other cases not. In this section, the conditions for which this decomposition is possible are found.

When the wrenches of constraint are put in column-reduced echelon form, [Wj bE , some of the columns may appear unitless. Since wrenches are screws, unitless columns will only exist in columns that have zeros in the force or moment positions. Each unitless column of [W,]bE represents one of the following two types of









wrenches: the wrench is a pure force, i.e.,

Wforce = [fX, fy, fZ, 0, 0, O , (6.76)

or the wrench is a pure moment with respect to a frame on the wrench (screw) axis, i.e.,

Woment = [0, 0, 0, nx, nl, n,] (6.77)

Group these apparently unitless columns into [W ]bEo, the wrenches of constraint with either zero force or zero moment. The columns of [W,]bEo that are not unitless are called the nonzero force and nonzero moment wrenches of constraint, [WVz]bEw.
If [W ]bE = [WZbE , then the manipulator twist space decouples as shown in Theorem 12 below.


Theorem 12






Proof

First prove that, in a given frame, there exists a direct sum decomposition of V if Wc = Wcz; and then prove that, in a given frame, if there is a direct sum decomposition of V, then Wc = Wz.
If WC = Wz, the column-reduced echelon form basis vectors of [WV]bE have no units and can therefore be used for a basis of Vi. But since the dimension of Wc plus the dimension of Vf is six and Wc = Wz, then Vnf = V14. Therefore [Vf]bEv = [Wz]bE . This proves one half of the theorem.
The second half of the theorem is proven as follows. If the decomposition V = Vf E) V?,f is assumed, then the projection involved is Euclidean, i.e.,


Vf = Range[JJt] = Range[J] ,


(6.78)
















Figure 6.1. Decomposition of the twist space in frame i into decoupled subspaces. and


V,,f = Null[JJt] = Null[Jt] = Null[Jr] ,


(6.79)


where J must be physically consistent from the assumption. But W, = Null[J'] by definition. Since Null[Jr] can be interpreted as both a twist (of nonfreedom) and a wrench (of constraint), then W, = Wf.

U

The twist space decomposition, when possible, is shown schematically in Figure 6.1. Conditions for this decomposition are given in Theorem 12 above and Theorem 13 below.

The above proof leads to a corollary that a subspace, V,, of V containing the twists of freedom, V, D Vf, always has a direct sum decomposition V, = V1 E) Vi, i.e.,


[VibE = [Wf]bE,


(6.80)


where Vi does not exist (is empty) if there are no wrenches of constraint with zero force or zero moment in the chosen frame. Corollary 5
WS2 = WV e t (6.81)

where iV) C V. C V.









Proof

If 'Wz = 'W,, then the proof of this corollary is identical to the proof of Theorem 12 and 'V, = iV. Otherwise, if 'IW C iWc, then the proof again follows the reasoning of the proof of Theorem 12, although the dimensions of the space iV, is reduced from 6 (the dimensions of iV) to Dim[Vf] + Dim['Wz].

U

To continue this discussion of twist space decomposition, separate the twists of freedom into linear velocities of freedom and angular velocities of freedom, and separate the wrenches of constraint into forces of constraint and moments of constraint,


V f] , w [ ] (6.82)

In a given frame i, if all if, are orthogonal to all 'vf and all in, are orthogonal to all iwf, then the manipulator decouples and the twist space can be uniquely decomposed into twists of freedom and twists of nonfreedom subspaces. Theorem 13
� if� ] = 0, V fc, ]v
i2V = zVf EiVnf inc 0 wf-=- 0, Vinc,iwf



Proof

Assume iv = "V$ e iV, f and remember from (1.35) that iV o W, = 0. From Theorem 12, 'W, = iWz, which implies that either ifc = 0 or in, = 0 for each wrench in 'Wc. In the case = 0,

Vo iC = 'wf �n, = 0 ,(6.83)


and the right-hand side of the theorem is proven. In the case inc = 0, then


iVf o %WC = v 0 oC = 0


(6.84)








completing the proof that the right-hand side of the theorem follows from the lefthand side.
The other direction of the proof proceeds as follows. The right-hand side of the theorem implies that iVV, = i1z, and then the proof of Theorem 12 will suffice.

N

For example, the wrenches of constraint in column-reduced echelon form of the PR virtual manipulator of Figure 4.1, expressed in frame 2 is 0001
0010
02 ] 021 0
[W]E = [�Nuill[2J] E,,J 0 1 0 0 (6.85)
0 1 0 0
1 0 0 0
0 0 0 0J
where (4.20) gives the Jacobian of this manipulator. The conditions on the right hand side of Theorem 12 are satisfied since the above matrix is also [2wz]bE,,. The conditions on the right hand side of Theorem 13 are also met since 2f, G) 2vf = 0 and 2n, 0 2Uwf = 0 for all f", 2Vf, 2 n, and 2Wf . Therefore, both Theorem 12 and Theorem 13 tell us that the decomposition of the twist space into unique disjoint subspaces is valid in this frame.
The Jacobian of the PR manipulator expressed in the translated frame t was given in (4.22). The wrenches of constraint in column-reduced echelon form are

-1 0 0 P.
Py Py
0 0 0 1
0 0 0 0(6.86) I'W1 bE' 0 0 1 0 0 1 0 0
L1 0 0 0J
where Py :A 0. If Py = 0, the first column of [tkW]bE,, is replaced by [0, 1/p,, O, O, O, 1] and the last column by [ 1, 0, 0, 0, 0, 0] .









The requirement of the right hand side of Theorem 12 is violated by the above [twC~bEW. Also, both conditions on the right hand side of Theorem 13 are violated by the wrench in the first column of (6.86).

Theorem 12 and Theorem 13 lead to a similar unique decomposition of the wrench space. The wrench space can sometimes be split into two disjoint subspaces, the wrenches of constraint and the wrenches of nonconstraint, . But first define a subspace Vf in a manner similar to the definition of Wz, i.e., [VI]bE, are the twists of freedom with either zero linear velocity or zero angular velocity. This leads to Theorem 14 below.


Theorem 14

2W = WcQ E Wc <= Vf = V



Proof

First prove that there exists a direct sum decomposition of W if Vf = Vfz; and then prove that if there is a direct sum decomposition of W, then V1 = Vf.

If V1 = Vf, the column-reduced echelon form basis vectors of [Vf]bE, have no units and can therefore be used for a basis of Wd. But since the dimension of V1 plus the dimension of W, is six and V1 = Vf, then W, = Wd. Therefore [W"JbE = [V]]bEv. This proves one half of the theorem.

The second half of the theorem is proven as follows. If the decomposition VV = WV, D W,, is assumed, then the projection involved is Euclidean, i.e., Wc = Null[JJt] = Null[Jd] , (6.87)

and

Vc = Range[JJt] = Range[J] , (6.88)

where jt must be physically consistent from the assumption. But V1 = Range[J] by






85

definition. Since Range[J] can be interpreted as both a wrench (of nonconstraint) and a twist (of freedom), then Vf = Vz.



Finally, Theorem 15 below shows the equivalence of the decomposition of the twist and wrench spaces when jt is physically consistent i.e., the unique Euclidean decomposition of the twists space results in the unique Euclidean decomposition of the wrench space, and vice-versa. Theorem 15 If jt is physically consistent, the following are equivalent statements:

2Vf = tV; (6.89)

iwC = (6.90)

iW = iWcED'Wc (6.91)
iV = iVf e VC (6.92)

iv = W = Tf D iW = Range[J] D Null[J'] (6.93)



Proof

If Vf = V.j W, = Wz, Theorem 12 and Theorem 14 can be used to prove the equivalence of the rest of the statements. From Theorem 12, W, = Wz * V = Vf ED Vf = Vf E W, (6.94)

From Theorem 14,

Vf = V; 4* W = W, E W, = W ED .f (6.95)

Since the right-hand-side decomposition of these two equations are identical, (6.89) and (6.90) are equivalent statements.

0









If the twist and wrench screw spaces are uniquely decomposable in a chosen frame, then a rotation of the frame of expression on the disjoint subspaces will preserve disjointedness since 'Gj = 'Aj. But a translation of the frame of expression will not preserve the decomposition of the subspaces. In fact, only special manipulators have the two unique subspaces (twists of constraint and wrenches of freedom) for twist and wrench space decompositions in all configurations. (These manipulators will be discussed in Section 6.7.) Generally, the set of twists that a manipulator cannot achieve, V,f, is not a subspace of twists so no unique V,,f can be found; and generally, the set of wrenches that a manipulator can apply, W,,,, is not a subspace of wrenches so no unique W,,, can be found.

The SCARA and the planar RRR manipulator discussed earlier are special manipulators that decouple the twist and wrench spaces into two disjoint subspaces in all frames of expression. For the SCARA manipulator in a frame arbitrarily translated from frame 2, the column-reduced echelon form twists of freedom and the columnreduced echelon form wrenches of constraint are
0 0 0 1 0 0
0 0 1 0 0 0
t ,E, 0 10 0 0 0
[0vf/bEv 0 0 0 0 [ E 0 1 (6.96)
0000 10
1000 00

Since each of the column-reduced echelon form twists of freedom have zero linear velocity or zero angular velocity, the manipulator decouples. It also decouples since each of the column-reduced echelon form wrenches of constraint have zero force. Since both of the constraint wrenches have zero force, this manipulator can apply a force to the environment in any direction as long as the manipulator is not in a singular configuration.

The terms decouple frame and decouple point are defined in Section 3.2 and Section 4.1.1, respectively. The pseudo-inverse of the manipulator Jacobian in a









frame located at a decouple point (a decouple frame) is physically consistent. Some new meaning of decouple points can now be presented.

Theorem 15 is based on the condition that jt is physically consistent, i.e., the frame of expression is located at a decouple point. All of the statements in this theorem are therefore the requirements necessary for a manipulator space, with respect to a particular frame, to decouple. If the frame of expression is at a decouple point, the twist and wrench spaces decouple identically as shown in (6.43) and (6.93).

Raibert and Craig [51] define a "constraint frame" as a frame in which the natural and "orthogonal" artificial constraints can be independently specified. A constraint frame or a compliant frame [2] is a frame in which the twist and wrench spaces decouple entirely into subspaces, and therefore twists and wrenches may be uniquely decomposed into constraint and freedom components. For the SCARA and the planar RRR manipulators, all frames are compliant frames.

The author of this paper prefers the term decouple point to describe a point at which a frame can be placed that will allow the twist and wrench spaces to be uniquely decomposed. This is also a point at which the pseudo-inverse is physically consistent. In fact, at a decouple point, the fundamental theorem of algebra for commensurate systems is meaningful for this noncommensurate system. As was shown in the Chapter 4, any rotations of the frame at this point will not affect the decoupled nature of the spaces.

When the frame of expression is not located at a decouple point, the twist and wrench spaces cannot be uniquely decomposed by a direct sum. But, a part of the twist or wrench spaces may be uniquely decomposable so that

Subspace[VI = eVf E 'Vi = iVf E 'w/z $ iv (6.97)

Subspace['W] = W,: id =1 i Vf # . (6.98)

For any frame i, a wrench coordinate transformation iAt,i exist that will convert any single wrench of constraint with nonzero force and nonzero moment to a wrench









with a zero moment and the same force. This particular wrench coordinate transformation consists of a translation vector of n x f (6.99)


and no rotation. Note that this transformation will also generally convert other wrenches that had zero moments to wrenches with nonzero moments.

Therefore, for all manipulators with Jacobian of rank less then six (i.e., a nonempty wrench of constraint subspace), there exists a frame that makes at least one of the constraint wrenches into an element of W,, and thus W : 0 in some frame.

For example, a P50 manipulator in frame 3 coordinates has the Jacobian and column-reduced echelon form wrench of constraint basis of

0 a233 0 00 0
0 a3 + a2c3 a3 0 0 0
82+3+4
3j= -a2c2 - a3C2+3 0 0 0 0 [3Wc]bEw 3 84(a2c2"+a3c2+3)
-52+3 0 0 0 S4 a
84
C2+3 0 0 0 -C4 1
0 1 1 1 0 0
(6.100)

Note that frame 3 is not a decouple frame. But (6.99) can be used with (1.16) to find a frame where the manipulator does decouple,

= (a2c2 + a3c2+3)s4 (a2c2 + a3c2+3)c4 0 (6.101)
S2+3+4 S2+3+4
r 82+3+4
tWV, = 3At,3[3 Vc~bEw = ,0 234_ 612
A,[ IE2 0, 0 2 + a3c2+3 0, 0, (6.102)

The physically consistent determinant of JTJ in frame t is Det[tJr t J] = (a2a3s3s2+3+4)2 . (6.103)

A non-planar RRR manipulator with Denavit-Hartenberg parameters given in Table 6.1 has a frame, 2 Jacobian and column-reduced echelon form wrenches of




Full Text

PAGE 1

$/*(%5$,& 3523(57,(6 2) 121&200(1685$7( 6<67(06 $1' 7+(,5 $33/,&$7,216 ,1 52%27,&6 %\ (5,& 0 6&+:$57= $ ',66(57$7,21 35(6(17(' 72 7+( *5$'8$7( 6&+22/ 2) 7+( 81,9(56,7< 2) )/25,'$ ,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 0\ :LIH *DEULHOOD t 7R 0\ 3DUHQWV 0DULO\Q DQG 6H\

PAGE 3

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n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

PAGE 4

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m8 7UDQVODWLRQV DQG &RQVLVWHQF\ RI m8 &RQVLVWHQF\ RI -W LQ $OO )UDPHV ,QYDULDQFH RI -W WR 6FDOLQJ (TXLYDOHQW *HQHUDOL]HG ,QYHUVHV ,9

PAGE 5

0$1,38/$725 0$1,38/$%,/,7< '(&20326,7,21 2) 63$&(6 3URMHFWLRQV DQG .LQHVWDWLF )LOWHUV 7ZLVW 'HFRPSRVLWLRQ :UHQFK 'HFRPSRVLWLRQ +\EULG &RQWURO 'HFRPSRVLWLRQ ZLWK 5D\ &RRUGLQDWH 7ZLVW 6SDFH 6SDFH 'HFRPSRVLWLRQ DW 'HFRXSOH 3RLQW 6HOI5HFLSURFDO 0DQLSXODWRUV 6800$5< $1' &21&/86,216 $33(1',; $ '+ 3$5$0(7(56 )25 9$5,286 0$1,38/$7256 5()(5(1&(6 %,2*5$3+,&$/ 6.(7&+ Y

PAGE 6

/,67 2) 7$%/(6 '+ SDUDPHWHUV IRU *( 3 PDQLSXODWRU 3K\VLFDO XQLWV RI 'HW>-7m-@ IRU YDULRXV QRQUHGXQGDQW PDQLSXODWRUV 3K\VLFDO XQLWV RI 'HW>--7@ IRU YDULRXV UHGXQGDQW PDQLSXODWRUV '+ SDUDPHWHUV IRU 35 YLUWXDO PDQLSXODWRU '+ SDUDPHWHUV IRU WKH 6&$5$ PDQLSXODWRU '+ SDUDPHWHUV IRU WKH 353 6PDOO $VVHPEO\ 5RERW 6$5f '+ SDUDPHWHUV IRU D QRQSODQDU 555 PDQLSXODWRU $O '+ SDUDPHWHUV IRU 35 YLUWXDO PDQLSXODWRU $ '+ SDUDPHWHUV IRU DQ 55 PDQLSXODWRU $ '+ SDUDPHWHUV IRU D JHQHUDO 555 PDQLSXODWRU $ '+ SDUDPHWHUV IRU WKH 3ODQDU 555 PDQLSXODWRU $ '+ SDUDPHWHUV IRU WKH 6SKHULFDO 555 PDQLSXODWRU $ '+ SDUDPHWHUV IRU WKH 1RQSODQDU 555 PDQLSXODWRU $ '+ SDUDPHWHUV IRU WKH 333 RUWKRJRQDO PDQLSXODWRU $ '+ SDUDPHWHUV IRU WKH 353 6PDOO $VVHPEO\ 5RERW 6$5f $ '+ SDUDPHWHUV IRU WKH 535 PDQLSXODWRU $ '+ SDUDPHWHUV IRU WKH 5553 6&$5$ PDQLSXODWRU ,OO $ '+ SDUDPHWHUV IRU WKH 5553 PDQLSXODWRU ,OO $ '+ SDUDPHWHUV IRU WKH 5553 PDQLSXODWRU $ '+ SDUDPHWHUV IRU 5 *(3 PDQLSXODWRU YL

PAGE 7

$ '+ SDUDPHWHUV IRU WKH 5 5HGXQGDQW $QWKURSRPRUSKLF $UP $ '+ SDUDPHWHUV IRU WKH 5 &(6$5 5HVHDUFK 0DQLSXODWRU $ '+ SDUDPHWHUV IRU WKH 5 . 5RERW 5HVHDUFK $UP $ '+ SDUDPHWHUV IRU WKH 5 380$ 6SKHULFDO :ULVW 0DQLSXODWRU $ '+ SDUDPHWHUV IRU WKH 35 5HGXQGDQW 6SKHULFDO :ULVW 5RERW $ '+ SDUDPHWHUV IRU WKH 535 *3 0DQLSXODWRU YLL

PAGE 8

/,67 2) ),*85(6 3HJLQWKHKROH ZLWK 35 YLUWXDO PDQLSXODWRU 6& $5$ PDQLSXODWRU 6PDOO $VVHPEO\ 5RERW 6$5f 'HFRPSRVLWLRQ RI WKH WZLVW VSDFH LQ IUDPH L LQWR GHFRXSOHG VXEVSDFHV YLLL

PAGE 9

.(< 72 6<0%2/6 6\PERO RU 9DULDEOH 7 [ k R k 0 k GHM 1 nfmn-f 2¯ Mf fµffµ} >f¬@UF >@UF >fµ@} +7 Rf± D $ m‘ 7 8$ D 'HILQLWLRQ IRU ;? WKH 0RRUH3HQURVH SVHXGRLQYHUVH RI ; IRU ; WKH ZHLJKWHG JHQHUDOL]HGLQYHUVH RI ; IRU _FF_ ZKHUH D LV D YHFWRU _[_ \M[ [ IRU ?[?P[ ZKHUH D LV D YHFWRU _[_ ?-[ k 0[[ IRU [ [ \ WKH YHFWRU FURVV SURGXFW RI YHFWRUV [ DQG \ IRU [ 4\ WKH LQQHU RU GRWf SURGXFW RI YHFWRUV [ DQG \ IRU ; R < WKH NOHLQ RU UHFLSURFDOf SURGXFW RI VFUHZV ; DQG < IRU ; k \ WKH GLUHFW VXP RI WKH VXEVSDFHV ; DQG A 0 IRU ; k \ WKH GLUHFW VXP RI WKH 0RUWKRJRQDO VXEVSDFHV ; DQG \ SRVVLEO\ HTXDO RIWHQ SK\VLFDOO\ LQFRQVLVWHQW GHILQHG DV QXPHULFDOO\ HTXDO WR IRU PDWUL[ HOHPHQW RI ; LQ WK URZ MWK FROXPQ IRU PDWUL[ <@UW&DQ U [ F PDWUL[ ZLWK DOO XQLWV LGHQWLFDO WR WKH XQLWV RI / U [ F PDWUL[ RI ]HURV IRU >9@ PDWUL[ ZKHUH WKH FROXPQ YHFWRUV FRQVWLWXWH D EDVLV IRU ; WKH WUDQVSRVH RSHUDWRU ]HUR YHFWRU RI GLPHQVLRQ Q DQJOH EHWZHHQ VXFFHVVLYH MRLQW D[HV SURMHFWHG RQ SODQH ZLWK FRPPRQ QRUPDO XVHG LQ '+ SDUDPHWHUL]DWLRQ RUWKRJRQDO [ PDWUL[ WKDW FRQYHUWV EHWZHHQ UD\ DQG D[LV FRRUGLQDWHV DQJOH DERXW D MRLQW D[HV XVHG LQ '+ SDUDPHWHUL]DWLRQ FRV Df VLQDM WKH JHQHUDOL]HGIRUFH YHFWRU FRQWDLQLQJ Q MRLQW IRUFHV DQGRU MRLQW WRUTXHV FRUUHVSRQGLQJ WR SULVPDWLF DQGRU UHYROXWH MRLQWV DQJXODU YHORFLW\ YHFWRU ZUHQFK FRRUGLQDWH WUDQVIRUPDWLRQ PDWUL[ SHUSHQGLFXODU GLVWDQFH EHWZHHQ VXFFHVVLYH MRLQW D[HV XVHG LQ '+ SDUDPHWHUL]DWLRQ ,;

PAGE 10

6\PERO RU 9DULDEOH 'HILQLWLRQ % E VNHZV\PPHWULF [ WUDQVODWLRQ PDWUL[ RI E WUDQVODWLRQ YHFWRU FLM 9 G ([ I * K FRV Mf GHIHFW PDQLIROG GLVWDQFH DORQJ MRLQW D[LV XVHG LQ '+ SDUDPWHWUL]DWLRQ PDWUL[ VXFK WKDW ;([ LV WKH FROXPQUHGXFHG HFKHORQ IRUP RI ; IRUFH YHFRU WZLVW FRRUGLQDWH WUDQVIRUPDWLRQ PDWUL[ ERG\f¬V LQHUWLD WHQVRU DW WKH FHQWHURIPDVV H[SUHVVHG LQ SULQFLSDO FRUUGLQDWHVf§D GLDJRQDO PDWUL[ K M -XL -\ >/?UF Q M [ M LGHQWLW\ PDWUL[ PDQLSXODWRU -DFRELDQ WKDW WUDQVIRUPV MRLQW UDWHV LQWR WZLVWV 9 -T ILUVW WKUHH URZV RI VXFK WKDW Y -YT URZV IRXU WKURXJK VL[ RI VXFK WKDW X! f§ -ZT U [ F XQLWV PDWUL[ ZLWK DOO XQLWV RI OHQJWK QXPEHU RI MRLQWV LQ PDQLSXODWRU Q 1XOS@ 4 5 Q 5DQJH>$@ 6 RU 6L 6R RU 6T¯ VT VY VLM 7 >88 XQLWV>@ 9 9 9 9QI 9 Z Z ZF ZQF ] PRPHQW RI IRUFH YHFWRU QXOO VSDFH RI PDWUL[ $ LH DOO [ VXFK WKDW $[ MRLQWUDWHV YHFWRU VSDFH [ URWDWLRQ PDWUL[ UDGLFDO VXEVSDFH FRPPHQVXUDWH PVSDFH RYHU UHDOV UDQJH VSDFH RI PDWUL[ $ LH DOO \ VXFK WKDW \ f§ $[ URWDWLRQ YHFWRU RI VFUHZ L WUDQVODWLRQ YHFWRU RI VFUHZ L FKDQJH RI XQLWV VFDOLQJ PDWUL[ IRU MRLQW UDWHV FKDQJH RI XQLWV VFDOLQJ PDWUL[ IRU WZLVWV VLQ Mf JHQHUDOL]HG MRLQWf IRUFHV YHFWRU VSDFH U ; F XQLWOHVV XQLWV PDWUL[ WKH SK\VLFDO GLPHQVLRQV RI WKH PDWUL[ LQVLGH WKH EUDFNHWV WZLVWV LQ 3O¾FNHU UD\ FRRUGLQDWHV 9 >X7 X!7@7 WZLVWV VFUHZ VSDFH WZLVWV RI IUHHGRP VXEVSDFH WZLVWV RI QRQIUHHGRP PDQLIROG OLQHDU YHORFLW\ YHFWRU ZUHQFK LQ 3O¾FNHU D[LV FRRUGLQDWHV : f§ >7 Q7@7 ZUHQFKHV VFUHZ VSDFH ZUHQFKHV RI FRQVWUDLQW VXEVSDFH ZUHQFKHV RI QRQFRQVWUDLQW PDQLIROG XQLW YHFWRU LQ ] GLUHFWLRQ > @7f ;

PAGE 11

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f§ $[ DUH JLYHQ /LQHDU QRQFRPPHQVXUDWH V\VWHPV GR QRW JHQHUDOO\ KDYH HLJHQYDOXHV DQG HLJHQYHFWRUV 7KH UHTXLUHPHQWV IRU D QRQFRPPHQVXUDWH V\VWHP WR SRVVHVV D SK\VLFDOO\ FRQVLVWHQW HLJHQV\VWHP DUH SUHVHQWHG ,W LV DOVR VKRZQ WKDW QRQFRPPHQVXUDWH OLQHDU V\VWHPV GR QRW KDYH D SK\VLFDOO\ FRQVLVWHQW VLQJXODU YDOXH GHFRPSRVLWLRQ 7KH PDQLSXODWRU -DFRELDQ PDSV SRVVLEO\ QRQFRPPHQVXUDWH URERW MRLQWUDWH YHFn WRUV LQWR QRQFRPPHQVXUDWH WZLVW YHFWRUV 7KH LQYHUVH YHORFLW\ SUREOHP LV RIWHQ VROYHG WKURXJK WKH XVH RI WKH SVHXGRLQYHUVH RI WKH -DFRELDQ 7KLV VROXWLRQ LV JHQHUDOO\ VFDOH DQG IUDPH GHSHQGHQW 7KH SVHXGRLQYHUVH VROXWLRQ LV SK\VLFDOO\ LQFRQVLVWHQW LQ JHQHUDO UHTXLULQJ WKH DGGLWLRQ RI HOHPHQWV RI XQOLNH SK\VLFDO XQLWV )RU VRPH PDQLSn XODWRUV WKHUH PD\ H[LVW SRLQWVf§FDOOHG GHFRXSOH SRLQWVf§DW ZKLFK WKH SVHXGRLQYHUVH RI WKH -DFRELDQ LV SK\VLFDOO\ FRQVLVWHQW IRU DOO IUDPHV DW WKHVH SRLQWV [L

PAGE 12

,Q GHFRXSOH IUDPHV WKH SVHXGRLQYHUVH LV VKRZQ WR EH HTXLYDOHQW WR WKH ZHLJKWHG JHQHUDOL]HGLQYHUVH ZLWK LGHQWLW\ PHWULFV $Q HQWLUH FODVV RI QRQLGHQWLW\ PHWULFV XVHG ZLWK WKH ZHLJKWHG JHQHUDOL]HGLQYHUVH DUH VKRZQ WR JLYH LGHQWLFDO VROXWLRQV WR WKH SVHXGRLQYHUVH VROXWLRQ DW GHFRXSOH SRLQWV $W GHFRXSOH SRLQWV WKH WZLVW DQG ZUHQFK VSDFHV FDQ EH GHFRPSRVHG LQWR WZR PHWULFLQGHSHQGHQW VXEVSDFHV 7KLV GHFRPSRVLWLRQ LV DFFRPSOLVKHG ZLWK NLQHVWDWLF ILOWHULQJ SURMHFWLRQ PDWULFHV 7KH 0DVRQ5DLEHUW K\EULG FRQWURO WKHRU\ RI URERWLFV LV VKRZQ WR EH XVHIXO RQO\ IRU IUDPHV ORFDWHG DW GHFRXSOH SRLQWV DQG LV QRW RSWLPDO LQ DQ\ REMHFWLYH VHQVH 7KH FXUUHQW PDQLSXODELOLW\ WKHRU\ ZKLFK GHSHQGV RQ WKH HLJHQV\VWHP RI YDULRXV IXQFWLRQV RI WKH -DFRELDQ LV VKRZQ WR EH LQYDOLG 7ZR QHZ FODVVHV RI PDQLSXODWRUV DUH LQWURGXFHG VHOIUHFLSURFDO PDQLSXODWRUV DQG GHFRXSOHG PDQLSXODWRUV 7KH WZLVWV RI IUHHGRP RI D VHOIUHFLSURFDO PDQLSXODWRU DUH UHFLSURFDO 7KH FODVV RI VHOIUHFLSURFDO PDQLSXODWRUV FRQVLVWV RI SODQDU PDQLSXODWRUV VSKHULFDO PDQLSXODWRUV DQG SULVPDWLFMRLQWHG PDQLSXODWRUV 'HFRXSOHG PDQLSXODn WRUV DUH VKRZQ WR GHFRXSOH DW HYHU\ SRLQW 7KH PDQLSXODWRUV RI WKLV FODVV DUH SODQDU PDQLSXODWRUV SULVPDWLFMRLQWHG PDQLSXODWRUV DQG 6&$5$W\SH PDQLSXODWRUV 5Hn VXOWV WKDW DUH JHQHUDOL]HG IURP GHFRXSOHG PDQLSXODWRUV RIWHQ SURYH WR EH LQYDOLG IRU PDQLSXODWRUV WKDW GR QRW GHFRXSOH DW HYHU\ SRLQW ;OO

PAGE 13

&+$37(5 ,1752'8&7,21 2SWLPXP DFFRUGLQJ WR :HEVWHU >@ PHDQV f¯EHVW PRVW IDYRUDEOHf° ,Q UHDO SK\Vn LFDO V\VWHPV WR VD\ D VROXWLRQ LV RSWLPXP RU RSWLPDO RQH PXVW VSHFLI\ WKH FULWHULD IRU RSWLPDOLW\ 7KH WKHRU\ RI K\EULG FRQWURO RI PDQLSXODWRUV GHYHORSHG E\ 0DVRQ LQ > @ DQG WKHQ WHVWHG DQG H[SDQGHG E\ 5DLEHUW LQ >@ KDV EHHQ VKRZQ E\ /LSNLQ DQG 'XII\ > @ DQG RWKHUV > @ WR EH HUURQHRXV /LSNLQ DQG 'XII\ H[SODLQ WKDW WKH IDLOXUH RI 0DVRQ DQG 5DLEHUWf¬V K\EULG FRQWURO WKHRU\ 05+&7f LV LQ WKHLU XVH RI RUWKRJRQDOLW\ ,Q 05+&7 WKH RUWKRJRQDOLW\ RI WZR YHFWRUV ZLWK WHUPV RI XQOLNH XQLWV LV XVHG ZKHQ LW LV HDVLO\ VHHQ WKDW WKH LQQHU SURGXFW RI WKHVH YHFWRUV LQ QRW LQYDULDQW WR VFDOLQJ %HFDXVH VR PDQ\ DXWKRUV FRQWLQXHG WR XVH 05+&7 'XII\ >@ IRXQG LW QHFHVVDU\ WR ZULWH DQ HGLWRULDO GHEXQNLQJ WKLV WKHRU\ 7KH SUREOHP ZLWK 05+&7 LQ WKLV DXWKRUf¬V YLHZ LV WKDW WKH WHUPV RI WKHLU RSWLPDO VROXWLRQ ZHUH QRW VXIILFLHQWO\ GHILQHG $Q H[SORUDWLRQ RI WKH PHDQLQJ RI WKHLU RSWLPDO VROXWLRQ ZRXOG KDYH VKRZQ WKDW WKH VROXWLRQ LV EDVHG RQ PLQLPL]LQJ WKH (XFOLGHDQ QRUPV RI WZR QRQ(XFOLGHDQ YHFWRUV ,Q 'RW\ QRWLFHG DQG HYHQWXDOO\ SXEOLVKHG UHVHDUFK > @ WKDW WKH 0RRUH 3HQURVH SVHXGRLQYHUVH VROXWLRQ LQ WKH URERWLFV LQYHUVH YHORFLW\ SUREOHP JLYHV UHVXOWV WKDW DUH GHSHQGHQW RQ WKH IUDPHV RI UHIHUHQFH 'RW\f¬V DOJHEUDLF YLHZSRLQW WRJHWKHU ZLWK 'XII\ DQG /LSNLQf¬V JHRPHWULF UHVXOWV XVLQJ VFUHZ WKHRU\ VXJJHVWHG D IXUWKHU LQYHVWLJDWLRQ RI WKH SRVVLEOH QRQLQYDULDQFH RI VROXWLRQ WHFKQLTXHV LQ VHYHUDO DUHDV RI URERWLFV DQG DSSOLHG PDWKHPDWLFV LQ JHQHUDO

PAGE 14

7KLV GLVVHUWDWLRQ LV EDVHG LQ SDUW RQ FRUUHFWLQJ WKH LQDSSURSULDWH XVH RI WKH SVHXGRn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f§ $[ ZKHUH $ LV QRQVTXDUH RU VLQJXODU 0RUH RIWHQ WKDQ QRW D PXOWLWXGH RI URERWLFV UHVHDUFKHUV LQFOXGLQJ > @ KDYH VROYHG WKHVH SUREOHPV E\ XVLQJ WKH SVHXGRLQYHUVH 7KH LQFRQVLVWHQW UHVXOWV JHQHUDWHG WKURXJK WKH XVH RI WKH SVHXGRLQYHUVH ZLWKRXW D PHWULF RU PHWULFVf DUH H[SODLQHG LQ WKLV GLVVHUWDWLRQ 7KH URERWLFV OLWHUDWXUH > @ DOVR PDNHV XVH RI WKH HLJHQYDOXHV HLJHQYHFWRUV RU VLQJXODU YDOXHV RI PDWULFHV ZKRVH HLJHQYDOXHV DQG VLQJXODU YDOXHV DUH QRW LQYDULDQW WR FKDQJHV LQ VFDOH RU FRRUGLQDWH WUDQVIRUPDWLRQV DQG DUH WKHUHIRUH QRW WUXH f¯HLJHQVROXWLRQVf° 7KH HLJHQVROXWLRQ SUREOHP LV DOVR GLVFXVVHG LQ WKLV GLVVHUWDWLRQ 7KH EDVLF PDWKHPDWLFV DQG WHUPLQRORJ\ RI URERWLFV DQG VFUHZ WKHRU\ QHFHVVDU\ IRU DQ XQGHUVWDQGLQJ RI WKH LVVXHV GLVFXVVHG ZLOO EH LQWURGXFHG LQ WKLV FKDSWHU 7KHUH LV QR RULJLQDO ZRUN LQ WKLV FKDSWHU RWKHU WKDQ VRPH EDVLF GHILQLWLRQV ZLWK UHJDUG WR QRQFRPPHQVXUDWH V\VWHPV 6LQFH D JHQHUDO XQGHUVWDQGLQJ RI WKH (XFOLGHDQ YHFWRU QRUP WKH SVHXGRLQYHUVH WKH ZHLJKWHG JHQHUDOL]HGLQYHUVH HLJHQYDOXHV HLJHQYHFWRUV DQG VLQJXODU YDOXH GHFRPSRVLWLRQ DUH SDUDPRXQW WR XQGHUVWDQGLQJ WKLV GLVVHUWDWLRQ WKHVH WRSLFV ZLOO DOVR EH SUHVHQWHG DQG H[DPSOHV ZLWK UHIHUHQFHVf RI WKHLU XVH LQ URERWLF V\VWHPV ZLOO EH JLYHQ LQ WKLV FKDSWHU

PAGE 15

1RQFRPPHQVXUDWH 9HFWRU 6SDFHV 6\VWHPV LQYROYLQJ HOHPHQWV RI GLIIHUHQW SK\VLFDO XQLWV DUH GHILQHG K«UH DV QRQn FRPPHQVXUDWH V\VWHPV 5RERWLFV V\VWHPV DUH QRQFRPPHQVXUDWH ZKHQ WKH\ GHDO ZLWK ERWK SRVLWLRQ DQG RULHQWDWLRQ RU KDYH ERWK UHYROXWH DQG SULVPDWLF MRLQWV ƒ YHFWRU RI HOHPHQWV RI XQOLNH SK\VLFDO XQLWV LV GHILQHG DV D QRQFRPPHQVXUDWH YHFWRU 7KH QRQn FRPPHQVXUDWH YHFWRU LV DOVR FDOOHG D FRPSRXQG YHFWRU > @ DQG QRQKRPRJHQHRXV YHFWRU >@f ,Q URERWLFV WKH HTXDWLRQ WKDW UHODWHV MRLQW YHORFLWLHV WR WZLVWV f GHVFULEHV D QRQFRPPHQVXUDWH V\VWHP \ -T f 7KH PDQLSXODWRU MRLQWUDWH YHFWRU LV >L fµfµfµ TQ@7 f , ZKHUH Q UHSUHVHQWV WKH WRWDO QXPEHU RI UHYROXWH DQG SULVPDWLF MRLQWV RI WKH PDQLSXn ODWRU 7KH PDQLSXODWRUf¬V LQVWDQWDQHRXV WZLVW YHFWRU 9 >Y7 X!7@f¬ f LV FRPSRVHG RI WKH OLQHDU YHORFLW\ Y >Y[ Y\ Y]@7 DQG WKH DQJXODU YHORFLW\ X >ZA RM\ X`]@7 7KH -DFRELDQ LV D [ Q PDWUL[ ZKHUH LV WKH QXPEHU RI FRRUGLQDWHV QHFHVVDU\ WR GHVFULEH WKH SRVLWLRQ DQG RULHQWDWLRQ RI D ERG\ LQ VSDFH 7KH WZLVW UHSUHVHQWV D QRQFRPPHQVXUDWH YHFWRU VLQFH WKH XQLWV RI Y DQG X! GLIn IHU :KHQ WKH PDQLSXODWRU KDV ERWK UHYROXWH DQG SULVPDWLF MRLQWV WKH MRLQWUDWH M YHFWRU LV DOVR QRQFRPPHQVXUDWH DQG WKH PDQLSXODWRU LV FDOOHG D QRQFRPPHQVXUDWH PDQLSXODWRU 7KH YHFWRU KP93WN UHSUHVHQWV WKH WZLVW RI D SRLQW S IL[HG WR IUDPH N DQG H[SUHVVHG M LQ IUDPH L FRRUGLQDWHV ZLWK UHVSHFW WR D IL[HG IUDPH P 6LQFH WKH -DFRELDQ KP-3WN KDV FROXPQV WKDW DUH DOVR WZLVWV WKH VXSHUVFULSW L DQG P DQG WKH VXEVFULSWV S DQG N

PAGE 16

KDYH WKH VDPH LQWHUSUHWDWLRQV DV LQ nP93WN :KHQ WKH VXEVFULSWV S DQG N DQG WKH VXSHUVFULSW P DUH RPLWWHG LQ ;9 DQG rLW LV XQGHUVWRRG WKDW N LV WKH HQGHIIHFWRU IUDPH Q RI DQ QMRLQWHG PDQLSXODWRU P LV WKH EDVH IUDPH IUDPH f DQG SRLQW S LV DW WKH RULJLQ RI IUDPH L WKH IUDPH RI H[SUHVVLRQ f¬9 .!f 7R WUDQVIRUP WZLVWV RU -DFRELDQV WR UHSUHVHQWDWLRQV LQ GLIIHUHQW IUDPHV WKH WZLVW FRRUGLQDWH WUDQVIRUPDWLRQ PDWUL[ * LV XVHG L43f¬O f§ >@ O5 b5 8SWT OOM L5 f ZKHUH >@ LV D [ PDWUL[ RI ]HURV DQG n5M LV D URWDWLRQ WUDQVIRUPDWLRQ ZKLFK URWDWHV D YHFWRU IURP IUDPH M LQWR IUDPH L 6LQFH URWDWLRQ PDWULFHV DUH RUWKRJRQDO WKH LQYHUVH LV HTXDO WR WKH WUDQVSRVH LH 5M O5@ b f %\ FRQYHQWLRQ WKH WHUP f¯RUWKRJRQDO PDWUL[f° UHIHUV WR PDWULFHV ZLWK RUWKRQRUPDO FROXPQV >@f 7KH PDWUL[ %Sf [@ LV D VNHZ V\PPHWULF PDWUL[ WKDW UHSUHVHQWV WUDQVODWLRQ IURP SRLQW S WR T H[SUHVVHG LQ IUDPH L 7KH % PDWUL[ LV WKH PDWUL[IRUP RI WKH YHFWRU FURVVSURGXFW LH %H E [ F ZKHUH E DQG F DUH DUELWUDU\ YHFWRUV DQG % LV GHILQHG DV LS f§ R b b E\ OE[ ] b f 7KH YHFWRU OES!T >OE[ OE\ WE]?U LV D SRVLWLRQ YHFWRU IURP SRLQW S WR SRLQW T H[SUHVVHG LQ IUDPH L FRRUGLQDWHV 6LQFH % LV VNHZ V\PPHWULF LW KDV WKH IROORZLQJ SURSHUWLHV %TS f§ f DQG f LS LS e!S\T IWM fµ f

PAGE 17

:LWK WKH DERYH HTXDWLRQV LW LV HDVLO\ VKRZQ WKDW M*IS f 1RWH WKDW *ST\ s A*IS 7KH H[SUHVVLRQV IRU WKH IUDPH WUDQVIRUPDWLRQV RI WZLVWV DQG -DFRELDQV DUH bL DQG f f«RUf br ‘ LLf 7KH VKRUWKDQG QRWDWLRQ b*M LV XVHG ZKHQ WKH WUDQVIRUPDWLRQ KDV QR WUDQVODWLRQ DQG WKH QRWDWLRQ W*ST LV XVHG ZKHQ WKH WUDQVIRUPDWLRQ KDV QR URWDWLRQ 7KH WZLVWV WKDW D PDQLSXODWRU FDQ DFFRPSOLVK ZLWK MRLQWUDWH FRQWURO LQ D JLYHQ FRQILJXUDWLRQ DUH NQRZ DV WKH WZLVWV RI IUHHGRP > @ n9 5DQJH>r-@ f ZKHUH 9 UHSUHVHQWV D WZLVW PDQLIROG DQG L LV WKH IUDPH RI H[SUHVVLRQ 7KH WZLVW RI IUHHGRP PDQLIROG LV D VXEVSDFH ,W LV LPSRUWDQW ZKHQ ZULWLQJ YHFWRUV PDWULFHV DQG PDQLIROGV WR PDNH WKH IUDPH RI H[SUHVVLRQ FOHDU ,Q WKLV GLVVHUWDWLRQ WKH H[SUHVVLRQ IUDPH LI QRW H[SOLFLWO\ ZULWWHQ DV D OHDGLQJ VXSHUVFULSW ZLOO EH RWKHUZLVH GHVFULEHG LQ WKH FRQWH[W RI WKH GLVFXVVLRQ 1RWH WKDW WKURXJKRXW WKLV GLVVHUWDWLRQ D FDOOLJUDSKLF V\PERO VXFK DV 9f UHSUHn VHQWV D PDQLIROG RU VHWf RI YHFWRUV RU VFUHZV 7KHUHIRUH ; ^:` LV WKH PDQLIROG RI YHFWRUV RU VFUHZV ;L IRU YDULRXV L 7KH FROXPQ YHFWRUV RI WKH PDWUL[ >;?L FRQn VWLWXWH D EDVLV IRU ; 7KH PDWUL[ ([ FRQYHUWV WKH EDVLV VHW >;@E WR D PDWUL[ LQ FROXPQUHGXFHG HFKHORQ IRUP >@ >;@E([ 7KH DSSOLFDWLRQ RI D ZUHQFK : DW WKH HQGHIIHFWRU RI D VWDWLF VHULDO PDQLSXODWRU ZLOO LQGXFH D EDODQFLQJ JHQHUDOL]HGIRUFH YHFWRU WZ 7Z UZ f

PAGE 18

ZKHUH D ZUHQFK : f§ >I7 QU@7 LV WKH QRQFRPPHQVXUDWH YHFWRU FRPSRVHG RI WKH WZR YHFWRUV RI IRUFHV DQG PRPHQWV Q $ JHQHUDOL]HGIRUFH YHFWRU W LV WKH QYHFWRU RI MRLQW WRUTXHV IRU UHYROXWH MRLQWVf DQGRU MRLQW IRUFHV IRU SULVPDWLF MRLQWVf 7KH PDWUL[ O:DI3 f§ >n UHSUHVHQW D ZUHQFK DW SRLQW S H[SUHVVHG LQ IUDPH L ZLWK WKH PRPHQWV WDNHQ DERXW SRLQW D :KHQ WKH VXEVFULSW D LV RPLWWHG LW LV XQGHUVWRRG WKDW WKH SRLQW D LV DW WKH SRLQW S VR WKDW r:3 O:3W3 :KHQ ERWK VXEVFULSWV DUH RPLWWHG WKH RULJLQ RI WKH IUDPH LV WKH SRLQW DW ZKLFK PRPHQWV DUH WDNHQ LH O: r :UHQFKHV WUDQVIRUP YLD WKH ZUHQFK FRRUGLQDWH WUDQVIRUPDWLRQ PDWUL[ $ ^:3 If M:T f ZKHUH (TXDWLRQV ff FDQ DOVR EH XVHG WR VKRZ WKDW 9"f¬fB M$IS DQG f M*I3< ^$SMf¬ f 7KH ZUHQFKHV DSSOLHG DW WKH HQG HIIHFWRU WKDW UHTXLUH QR MRLQW IRUFHV IRU EDODQFLQJ DUH NQRZ DV WKH ZUHQFKHV RI FRQVWUDLQW r:& DQG IRUP D VXEVSDFH f«f«!9& 1X8>97@ f 7KHVH ZUHQFKHV ZLOO FDXVH QR MRLQW PRWLRQ ZKHQ DSSOLHG WR D VWDWLF PDQLSXODWRU 0DQLSXODWRUV RI DW OHDVW MRLQWVf LQ FRQILJXUDWLRQV ZLWK -DFRELDQ RI UDQN KDYH QR FRQVWUDLQW ZUHQFKHV LH VRPH QRQ]HUR MRLQW IRUFHV DUH UHTXLUHG WR EDODQFH HYHU\ SRVVLEOH ZUHQFK 1RWLFH WKDW WKH DERYH WZLVWV DQG ZUHQFKHV DUH VFUHZV GHILQHG EHORZf H[SUHVVHG LQ D[LV FRRUGLQDWHV DQG UD\ FRRUGLQDWHV > @ UHVSHFWLYHO\ 7KH GHVLJQDWLRQV RI 3OLLFNHU O5L O5 O3 8S\T $OM >A b f

PAGE 19

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f LQ D[LV FRRUGVf f ZKHUH WKH OLQH SDVVHV WKURXJK WKH FRRUGLQDWH V\VWHP RULJLQ $ PRUH JHQHUDO GHn VFULSWLRQ LV JLYHQ LQ f EHORZf 7KH YHFWRU 6n LV D FRPPHQVXUDWH YHFWRU LQ WKH GLUHFWLRQ RI OLQHDU PRWLRQ DQG WKH URWDWLRQ LV DERXW WKLV D[LV XVLQJ WKH ULJKWKDQGUXOH )RU HYHU\ UDGLDQV RI URWDWLRQ WKH VFUHZ DGYDQFHV E\ K LQ WKH 6 GLUHFWLRQ $ VFUHZ PD\ DOVR EH GHILQHG DV D OLQHDU FRPELQDWLRQ RI XQOLPLWHG OLQHV > @ $Q XQOLPLWHG OLQH / LV GHILQHG ZLWK WZR YHFWRUV D XQLW YHFWRU 6 LQ WKH GLUHFWLRQ RI WKH OLQH DQG D YHFWRU U IURP WKH FRRUGLQDWH V\VWHP RULJLQ WR DQ\ SRLQW RQ WKH OLQH AD[LV U [ 6 n6Rn 6 V LQ D[LV FRRUGVf f /LQHV DOVR KDYH WKH SURSHUW\ WKDW 6 4 6 6 4 U [ 6f 7KH UD\ FRRUGLQDWH YHUVLRQ RI WKLV VDPH OLQH LV 6 U [ 6 V 6R XD\ LQ UD\ FRRUGVf f

PAGE 20

$ OLQHDU FRPELQDWLRQ RI WZR OLQHV LQ D[LV UD\f FRRUGLQDWHV FUHDWHV D VFUHZ LQ D[LV UD\f FRRUGLQDWHV UU;6 /I6 LUL [ 6Lf U [ 6f n6RUn OO6O A f§ U 6r f )RU VFUHZV 6U46RU . ZKHUH KU WKH SLWFK RI WKH UHVXOWDQW VFUHZ 7KHUHIRUH VFUHZV DUH QRW OLQHV H[FHSW LQ WKH VSHFLDO FDVH ZKHQ WKH SLWFK LV ]HUR 7KH UHVXOWDQW VFUHZ FDQ EH ZULWWHQ DV UU [ 6Uf KU6U UUD;6 U f 7KH GLIIHUHQFHV LQ HTXDWLRQV f DQG f DUH GXH WR GLIIHUHQW FRRUGLQDWH V\VWHP GHILQLWLRQV ,I U7 LH WKH FRRUGLQDWH V\VWHP RULJLQ LV RQ WKH OLQH RI URWDWLRQ WKH WZR HTXDWLRQV DUH LGHQWLFDO $ JHQHUDO VFUHZ FDQ DOZD\V EH FRQYHUWHG WR D f¯SXUH VFUHZf° DV LQ f E\ D WZLVW FRRUGLQDWH WUDQVIRUPDWLRQ IRU D[LV FRRUGLQDWH VFUHZV RU D ZUHQFK FRRUGLQDWH WUDQVIRUPDWLRQ IRU UD\ FRRUGLQDWH VFUHZV )RU H[DPSOH D WZLVW FRRUGLQDWH WUDQVIRUPDWLRQ ZLOO WUDQVIRUP WKH SXUH D[LV FRRUGLQDWH VFUHZ LQWR D JHQHUDO D[LV FRRUGLQDWH VFUHZ WXR 5 %5 f« KXL K5XM %5MM /2 >2OD 5 8f 5LR f 1RWH WKDW FRRUGLQDWH WUDQVODWLRQV %f GR QRW DIIHFW WKH DQJXODU YHORFLW\ YHFWRUf§ WKH ERWWRP FRPSRQHQW LQ WKH ULJKWKDQGVLGH RI HTXDWLRQ f $OWKRXJK URWDWLRQV DIIHFW ERWK SDUWV RI WKH VFUHZ LI WKHUH LV QR WUDQVODWLRQ WKH URWDWLRQ ZLOO QRW DIIHFW WKH DSSDUHQW SXULW\ RI D VFUHZ YLHZHG LQ HDFK RI WKH IUDPHV $V VWDWHG DERYH WKH SLWFK RI D VFUHZ FDQ EH IRXQG VLPSO\ E\ 6R46 ,6 _6_A f ,I 6 LV WKH ]HUR YHFWRU WKH VFUHZ LV VDLG WR KDYH LQILQLWH SLWFK DQG f LV UHSODFHG E\ 9 5 %5 9 5Y r >@f± 5 2 f

PAGE 21

1RWH WKDW WKH WUDQVODWLRQ % KDV QR DIIHFW RQ WKH UHVXOWLQJ VFUHZ UHSUHVHQWDWLRQ ,I WKH SLWFK LV ]HUR 6R LV ]HUR DQG WKH VFUHZ UHSUHVHQWV D SXUH URWDWLRQ 7KH WUDQVODWLRQ WKDW ZLOO PRYH D JHQHUDO D[LV VFUHZ WR D SXUH D[LV VFUHZ LV E ) f¬_V_ r r /f ZKHUH % FDQ EH IRXQG IURP E ZLWK f $OO ULJLG ERG\ PRWLRQ LV LQVWDQWDQHRXVO\ HTXLYDOHQW WR D VFUHZ PRWLRQ WZLVW >@ 7KH WZLVW GHILQHG SUHYLRXVO\ 9 >X R!@ LV HTXDO WR D OLQHDU YHORFLW\ Y UHIHUHQFHG WR VRPH RULJLQ f DQG DQ DQJXODU YHORFLW\ Z D IUHH YHFWRU >@ DQG LV KHUH GHILQHG DV D VFUHZ LQ 3OILFNHU D[LV FRRUGLQDWHV > @ $ WZLVW FDQ DOVR EH UHSUHVHQWHG LQ 3OLLFNHU UD\ FRRUGLQDWHV 97D\ f§ >X! QR@ 6LPLODUO\ D ZUHQFK LV LQVWDQWDQHRXVO\ HTXLYDOHQW WR D IRUFH DQG PRPHQW RQ D ULJLG ERG\ 7KH 3OLLFNHU UD\ FRRUGLQDWHV RI D ZUHQFK : f§ > Q@ LV HTXDO WR D IRUFH LQ WKH GLUHFWLRQ RI WKH ZUHQFK DQG D PRPHQW Q UHIHUHQFHG DERXW RULJLQ $ ZUHQFK FDQ DOVR EH UHSUHVHQWHG LQ 3OLLFNHU D[LV FRRUGLQDWHV ,8D[LV >PR @ 8QOHVV RWKHUZLVH QRWHG WZLVWV ZLOO EH H[SUHVVHG LQ 3OLLFNHU D[LV FRRUGLQDWHV DQG ZUHQFKHV ZLOO EH H[SUHVVHG LQ 3OLLFNHU UD\ FRRUGLQDWHV 7KH PDWUL[ $ >@ WUDQVIRUPV D VFUHZ RU OLQH LQ D[LV FRRUGLQDWHV WR D VFUHZ RU OLQH LQ UD\ FRRUGLQDWHV DQG D VFUHZ RU OLQH LQ UD\ FRRUGLQDWHV WR D VFUHZ RU OLQH LQ D[LV FRRUGLQDWHV UD\ B $D[LV f D[LV $UD\ $ f f 7KH PDWUL[ $ LV DQ XQLWDU\ PDWUL[ DQG WKHUHIRUH DOVR DQ RUWKRJRQDO PDWUL[f ZLWK WKH SURSHUWLHV $ $ $ $7 $$ f

PAGE 22

7KH PDWUL[ $ LV DQ H[DPSOH RI D PRUH JHQHUDO WUDQVIRUPDWLRQ GHILQHG DV D FRUUHODWLRQ >@ WKDW PDSV DQ D[LV VFUHZ WR D UD\ VFUHZ RU D UD\ VFUHZ WR DQ D[LV VFUHZf $ FROOLQHDWLRQ PDSV D UD\ VFUHZ WR D UD\ VFUHZ RU DQ D[LV VFUHZ WR DQ D[LV VFUHZf 7KH UHFLSURFDO RU .OHLQ SURGXFW > @ RI DQ\ WZR VFUHZV LQ LGHQWLFDO D[LV RU UD\ FRRUGLQDWHVf§WZLVWV 9? DQG 9 IRU H[DPSOHf§LV GHILQHG DV \O\ 9L k $9 9"$9 f 9? k 8" k f ZKHUH k UHSUHVHQWV WKH (XFOLGHDQ LQQHU RU GRW SURGXFW 7KH .OHLQ SURGXFW RI D VFUHZ LQ D[LV FRRUGLQDWHV DQG D VFUHZ LQ UD\ FRRUGLQDWHV 9 DQG : LV 9R: 94: 97: Y4I XpQ f 1RWLFH WKDW QR $ PDWUL[ LV QHHGHG LQ WKH H[SDQVLRQ RI WKH .OHLQ SURGXFW RI D WZLVW DQG D ZUHQFK ZKHUHDV WKH $ PDWUL[ LV QHHGHG LQ WKH H[SDQVLRQ RI WKH .OHLQ SURGXFW RI WZR WZLVWV RU WZR ZUHQFKHV 7KH .OHLQ SURGXFW RI D WZLVW DQG ZUHQFK RI WKH HQG HIIHFWRU RI D VHULDO PDQLSXODWRU JLYHV WKH LQVWDQWDQHRXV YLUWXDO SRZHU ZRUNf >@ WKDW WKH PDQLSXODWRU HQGHIIHFWRU FRQWULEXWHV WR WKH HQYLURQPHQW $ ZHOO NQRZQ LPSRUWDQW FKDUDFWHULVWLF RI WKH UHFLSURFDO SURGXFW LV WKDW LW LV LQn YDULDQW WR FRRUGLQDWH WUDQVIRUPDWLRQV 7KLV LV VKRZQ LQ WKH IROORZLQJ WKHRUHP DQG SURRI >@ 7KH SURRI LV JLYHQ WR SURYLGH WKH UHDGHU DQ XQGHUVWDQGLQJ RI WKH QRWDWLRQ DQG PDWKHPDWLFV LQYROYHG 7KHRUHP 7KH UHFLSURFDO SURGXFW RI D PDQLSXODWRU WZLVW DQG ZUHQFK H[SUHVVHG DW WKH VDPH SRLQW DQG LQ WKH VDPH FRRUGLQDWH V\VWHP LV LQYDULDQW WR FRRUGLQDWH WUDQVIRUPDn WLRQV

PAGE 23

3URRI -"If OM*"f¬TbT f -*ST O9S!T DV VKRZQ LQ f f r90Rr; ^*ISf c* O@U 7KH WZLVW FRRUGLQDWH WUDQVIRUPDWLRQ PDWUL[ HQDEOHV WKLV VFUHZ WR EH H[SUHVVHG LQ GLIIHUHQW FRRUGLQDWH IUDPHVf§DV VKRZQ LQ f 7R H[SUHVV WKLV VFUHZ LQ YDULRXV FRRUGLQDWH IUDPHV WKH WZLVW FRRUGLQDWH WUDQVIRUPDWLRQ PDWUL[ PD\ EH HPSOR\HG DV VKRZQ LQ f :KHQ WKH IUDPH LV WUDQVODWHG WR D IUDPH M WKDW LV ORFDWHG E\ YHFWRU E IURP WKH IUDPH L f§ WKH VFUHZ PRWLRQ LV M\7HY K efµ L= %] 4L= >E\ E[ @ L= f f f ZKHUH % LV WKH VFUHZ V\PPHWULF PDWUL[ RI f FRUUHVSRQGLQJ WR WKH WUDQVODWLRQ YHFWRU E>E[ E\ E]@7 :KHQ WKH IUDPH LV URWDWHG WR D IUDPH N ZLWK QR WUDQVODWLRQ IURP IUDPH L f§ WKH VFUHZ PRWLRQ LV N\UHY R R R R R c 7 R LN5L] B f

PAGE 24

7KH WZLVW RU VFUHZ PRWLRQ FUHDWHG E\ D VLQJOH SULVPDWLF MRLQW L LV D SXUH WUDQVODWLRQ rf§O\SULV GL 7 VX f $JDLQ WKH WZLVW FRRUGLQDWH WUDQVIRUPDWLRQ PDWUL[ FDQ EH HPSOR\HG WR H[SUHVV WKLV VFUHZ LQ YDULRXV FRRUGLQDWH V\VWHPV $Q DUELWUDU\ FRRUGLQDWH WUDQVIRUPDWLRQ N*O; URWDWHV WKH WZLVW WR IUDPH N ZKLOH WKH WUDQVODWLRQ KDV QR DIIHFW IRU DQ\ S DQG T N SSULV NALST Lf§OSUSULV GLK5L , rrn f§ b 2Q f 7KDW WUDQVODWLRQ KDV QR DIIHFW RQ WKLV WZLVW ZDV YHULILHG E\ V\PEROLFDOO\ SHUIRUPLQJ WKH PXOWLSOLFDWLRQ N*r1L \SQV LQ f 6FUHZV FDQ EH DGGHG WR IRUP QHZ VFUHZV ,Q WKLV PDQQHU WKH PRWLRQ RI WKH HQG HIIHFWRU RU DQ\ RWKHU SRLQWf RI D VHULDO PDQLSXODWRU PD\ EH IRXQG E\ D VXPPDWLRQ RI WKH VFUHZV RI HDFK RI WKH MRLQWV 9L Y fµ fµfµ . f OO n fµ fµ UDQ f > fµ fµfµ f -T f ZKHUH T LV WKH YHFWRU RI PDQLSXODWRU MRLQW UDWHV RI f DQG f LV LGHQWLFDO WR f 7R SHUIRUP WKH DGGLWLRQ RI VFUHZV LW LV ILUVW QHFHVVDU\ WR UHIHUHQFH WKHP WR WKH VDPH FRRUGLQDWH IUDPH DQG SRLQW YLD WKH DSSURSULDWH VFUHZ FRRUGLQDWH WUDQVIRUPDWLRQ PDWULFHV HJ WKH VXPPDWLRQ RI WKH VFUHZV LQ f LV DFWXDOO\ DFFRPSOLVKHG ZLWK WKH HTXDWLRQ L9 nWLLL*7b ‘ f f L $Q\ WZLVW FDQ EH FRQVWUXFWHG E\ VL[ RU OHVV LQGHSHQGHQW VFUHZV HDFK UHSUHVHQWLQJ HLWKHU D SULVPDWLF RU D UHYROXWH PRWLRQ 7KHUHIRUH D YLUWXDO PDQLSXODWRU FDQ DOZD\V EH FRQVWUXFWHG WR LQVWDQWDQHRXVO\ DFFRPSOLVK DQ\ WZLVW *ULIILV >@ GHILQHV D YLUWXDO

PAGE 25

PDQLSXODWRU DV DQ\ LPDJLQDU\ VHULDO PDQLSXODWRU f¯ZKRVH MRLQW GLVSODFHPHQWV DQG VSHHGV XQLTXHO\ GHVFULEH DQ\ SHUPLVVLEOH WZLVW 9If DQG DQ\ SHUPLVVLEOH SRVLWLRQ DQG RULHQWDWLRQ RI LWV HQGHIIHFWRUf° 7KH SHUPLVVLEOH HQGHIIHFWRU ZUHQFKHV :Ff WRJHWKHU ZLWK WKH WZLVWV FRPSOHWHO\ GHVFULEH WKH LQVWDQWDQHRXV NLQHPDWLFV RI D UHDO RU YLUWXDO PDQLSXODWRU HQGHIIHFWRU 7KHRUHP EHORZ JLYHQ LQ >@ VKRZV WKDW WKH UHFLSURFDO SURGXFW RI DQ\ WZLVW RI IUHHGRP DQG DQ\ ZUHQFK RI FRQVWUDLQW PXVW EH ]HUR 7KH SURRI LV VKRZQ WR JLYH WKH UHDGHU DQ LQVLJKW WR WKH FRQFHSW RI UHFLSURFLW\ 7KHRUHP 7KH .OHLQ RU UHFLSURFDO SURGXFW RI 9M DQG :F LV ]HUR LH 9IR:F ?9I * 9I DQG 9:& fü :F f 3URRI 9I -T :I 9I DQG VRPH ^T` f Y\ \U f ^9If7: T7-7: f 1RZ OHW : EH D FRQVWUDLQW ZUHQFK :F * :F VR WKDW n9If7:F T7-7:& f %XW -7:& f§ E\ GHILQLWLRQ LQ f VR Y\ZF R f %XW E\ WKH GHILQLWLRQ RI WKH .OHLQ SURGXFW LQ f YI\ZF 9IRZF LRf VR WKDW 9I R :F

PAGE 26

7KLV PHDQV WKDW WKH PDQLSXODWRU FDQ GR QR ZRUN ZLWK DQ\ ZUHQFK RI FRQVWUDLQW RU DOWHUQDWLYHO\ FDQ QRW PRYH ZLWK WKH VFUHZ PRWLRQ RI DQ\ UD\ FRRUGLQDWH FRQVWUDLQW ZUHQFK LQWHUSUHWHG DV DQ D[LV FRRUGLQDWH WZLVW 7KH UHFLSURFLW\ UHODWLRQVKLS EHWZHHQ 9 DQG :F KDV EHHQ LQDGYHUWHQWO\ DQG LQn DSSURSULDWHO\f XVHG E\ UHVHDUFKHUV WR FKDUDFWHUL]H WKH HQWLUH VSDFH WKURXJK WKH XVH RI WKH GLUHFW VXP GHFRPSRVLWLRQ RI WKH VSDFH RI SRVLWLRQ DQG RULHQWDWLRQ 7KH IXQGDPHQWDO WKHRUHP RI OLQHDU DOJHEUD >@ VWDWHV WKDW 5DQJH>$@ k 1XOO>$7@ f ZKHUH P LV WKH QXPEHU RI URZV RI $ 7KH V\PERO p UHSUHVHQWV WKH GLUHFW VXP DQG LPSOLHV WKDW 5DQJH >$@ IO 1XOO>$7@ ^` DQG 5DQJH>$@ 8 1XOO>$7@ $SSO\LQJ WKLV WKHRUHP WR URERWLFV E\ OHWWLQJ $ EH WKH -DFRELDQ FDQ EH PLVOHDGLQJ 6IF 5DQJH>-@ p 1XOO>-7@ f 6LQFH KDV SK\VLFDO PHDQLQJ ZLWK WHUPV QRW DOO RI WKH VDPH XQLWV WKH LPSOLFDWLRQ RI WKLV WKHRUHP DSSOLHG WR URERWLFV LV WKDW WKH WRWDO VSDFH LV D FRPELQDWLRQ RI D[LV FRRUGLQDWH WZLVWVf DQG UD\ FRRUGLQDWH ZUHQFKHVf 7KH VXEVSDFHV 5DQJH>-@ DQG 1XOO>-7@ DUH QRQFRPPHQVXUDWH :KDW GRHV LW PHDQ WR GHFRPSRVH D YHFWRU RU VFUHZf LQWR WKH VXP RI DQ D[LV FRRUGLQDWH YHFWRU DQG UD\ FRRUGLQDWH YHFWRU" 7KLV SUREOHP ZLOO EH DGGUHVVHG LQ &KDSWHU 7KH 3VHXGR DQG *HQHUDOL]HG,QYHUVHV 7KH 0RRUH3HQURVH SVHXGRLQYHUVH DQG WKH ZHLJKWHG JHQHUDOL]HGLQYHUVH FDQ ERWK EH XVHG WR VROYH OLQHDU HTXDWLRQV 2I FRXUVH HDFK RI WKH VROXWLRQV LV EDVHG RQ GLIIHUHQW RSWLPDOLW\ FRQGLWLRQV IRU WKHLU VROXWLRQV

PAGE 27

7KH 0RRUH3HQURVH 3VHXGR,QYHUVH 7KH 0RRUH3HQURVH SVHXGRLQYHUVH JLYHV D XQLTXH PLQLPXP QRUP OHDVWVTXDUHV VROXWLRQ WR D OLQHDU HTXDWLRQ X $[ f IRU H[DPSOH 7KH SVHXGRLQYHUVH RI $ $ ( eP;Qff LV GHQRWHG $r DQG KDV WKH IROORZLQJ SURSHUWLHV > @ $$n$ $n$$r ^$$n< $8f7 $ $$r $A$ f f f f 7KH SVHXGRLQYHUVH FDQ EH IRXQG WKURXJK D IXOOUDQN IDFWRUL]DWLRQ RI $ $ f§ )& ZKHUH ) fü "P;Uf KDV IXOO FROXPQ UDQN U DQG & ( KDV IXOO URZ UDQN U 7KH SVHXGRLQYHUVH RI $ FDQ EH H[SUHVVHG DV $W &7)7$&7fa)7 f &7&&7f)7)fa)U f &)W f 7KH XQLTXH PLQLPXP QRUP OHDVWVTXDUHV VROXWLRQ WR f LV WKHUHIRUH [V $AX f 7KH VROXWLRQ [V LV D OHDVWVTXDUHV VROXWLRQ LQ WKDW WKH UHVLGXDO LI DQ\f _X f§ $[_ LV PLQLPL]HG ZKHUH _ fµ _ LV WKH (XFOLGHDQ YHFWRU QRUP VHH HTXDWLRQ ff 7KH VROXWLRQ [V LV PLQLPXP QRUP VLQFH DQ\ RWKHU VROXWLRQV DT WR $[ X KDV (XFOLGHDQ QRUP _DT_ ! _DV_ $ OHDVWVTXDUHV VROXWLRQ LV REWDLQHG LI f DQG f DUH WUXH DQG WKH VROXWLRQ LV PLQLPXP QRUP LI f DQG f DUH WUXH >@

PAGE 28

,W LV D IRUWXQDWH IDFW WKDW WKH OHDVWVTXDUHV VROXWLRQ DQG WKH PLQLPXP QRUP VROXn WLRQ DUH LGHQWLFDO IRU OLQHDU V\VWHPV DQG HTXDO WR WKH SVHXGRLQYHUVH VROXWLRQ 7KH (XFOLGHDQ QRUP RI D YHFWRU [ * DOVR NQRZQ DV WKH VTXDUH URRW RI WKH (XFOLGHDQ LQQHUSURGXFW RI [ ZLWK LWVHOIf LV GHILQHG DV r ?cr) ! Q _UFM [ [ ! [ [ [7[ f§ A [I f  L ,I PDWUL[ $ KDV IXOO URZ UDQN RU IXOO FROXPQ UDQN ff KDV WKH VLPSOLILHG VROXWLRQV $r $7$$7f $ IXOO URZ UDQN DQG f $r $7MfB\O7 $ IXOO FROXPQ UDQN f 7KHVH HTXDWLRQV DUH GHULYHG GLUHFWO\ IURP f VXEVWLWXWLQJ ) f§ ,U ZKHQ $ KDV IXOO FROXPQ UDQN DQG & f§ ,U ZKHQ $ KDV IXOO URZ UDQN 0DWUL[ ,U LV WKH U [ U LGHQWLW\ PDWUL[f 7KH :HLJKWHG *HQHUDOL]HG,QYHUVH 7KH ZHLJKWHG JHQHUDOL]HGLQYHUVH JLYHV D XQLTXH PLQLPXP 0AQRUP OHDVW 0X VTXDUHV VROXWLRQ WR D OLQHDU HTXDWLRQ 7KH ZHLJKWHG JHQHUDOL]HGLQYHUVH RI $ FDOOHG WKH JHQHUDOL]HGLQYHUVH WKURXJKRXW WKH UHVW RI WKLV GLVVHUWDWLRQf LV GHQRWHG $r DQG KDV WKH IROORZLQJ SURSHUWLHV > @ $ $r $ $ f $r $ $r $r f ^0X$$r\ 08$$r f 0;$r$f7 0;$r$ f 7KH PDWULFHV 0[ DQG 0X DUH PHWULFV $ PHWULF LV D V\PPHWULF SRVLWLYH GHILQLWH PDWUL[

PAGE 29

7KH JHQHUDOL]HGLQYHUVH RI $ > @ ZLWK WKH VDPH IXOOUDQN IDFWRUL]DWLRQ $ )& GLVFXVVHG SUHYLRXVO\ LV $r 0&7)70X$0&7fa)70Q 0a&7&0a&7faA >^)709)fa)709 &r)r f f f ZKHUH )t DQG & DUH GHILQHG E\ f DQG WKH EUDFNHWHG H[SUHVVLRQV LQ f 7KH XQLTXH PLQLPXP 0[QRUP OHDVW 0XVTXDUHV VROXWLRQ WR f LV WKHUHIRUH [V $X f 7KH VROXWLRQ [V LV D OHDVW 0XVTXDUHV VROXWLRQ LQ WKDW WKH UHVLGXDO LI DQ\f _X f§ $[?PX LV PLQLPL]HG ZKHUH __P LV GHILQHG EHORZ LQ f 7KH VROXWLRQ [V LV PLQn LPXP 0AQRUP VLQFH DQ\ RWKHU VROXWLRQV ;? WR $[ X KDV 0[QRUP _AL_P[ ! 7KH 0QRUP RI YHFWRU D ,T?P “K ! Q _D_P D 0D ! D 4 0D D0Df f  L ,Q DGGLWLRQ WR WKH SRVLWLYH GHILQLWH UHTXLUHPHQW IRU D PHWULF D PHWULF PXVW DOVR PDNH WKH FRUUHVSRQGLQJ VTXDUH RI WKH 0QRUP SK\VLFDOO\ FRQVLVWHQW >@ HJ D40D PXVW EH SK\VLFDOO\ FRQVLVWHQW IRU DQ\ D LI 0 LV WR EH FRQVLGHUHG D YDOLG PHWULF $ OHDVW0f± VTXDUHV VROXWLRQ LV REWDLQHG LI f DQG f DUH WUXH DQG WKH VROXWLRQ LV PLQLPXP 0[QRUP LI f DQG f DUH WUXH >@ ,W LV D IRUWXQDWH IDFW WKDW WKH OHDVW 0XVTXDUHV VROXWLRQ DQG WKH PLQLPXP 0[ QRUP VROXWLRQ DUH LGHQWLFDO IRU OLQHDU V\VWHPV DQG HTXDO WR WKH JHQHUDOL]HGLQYHUVH VROXWLRQ JLYHQ LQ ff ,Q RUGHU IRU VROXWLRQV WR EH LQYDULDQW WR FRRUGLQDWH WUDQVIRUPDWLRQV >@ LQ ERWK WKH VSDFHV GHILQHG E\ X DQG [ WKH PHWULFV PXVW WUDQVIRUP YLD D VSHFLILF FRQJUXHQFH

PAGE 30

WUDQVIRUPDWLRQ >@ 0n *70* f ,I Xn f§ *AX WKHQ WKH PHWULF IRU Xn PXVW EH 0X! f§ *7808*8 7KLV ZLOO LQVXUH WKDW WKH 0XQRUP LV LQYDULDQW ?Xn?0 W f§ _X_DX 7KH PHWULF 0[ PXVW DOVR WUDQVIRUP YLD D FRQJUXHQFH WUDQVIRUPDWLRQ 0[! *7;0;*; ZKHUH [n f§ *a[[ (LJHQYDOXHV (LJHQYHFWRUV DQG 69' (LJHQYDOXHV DQG HLJHQYHFWRUV RI DQ Q [ Q PDWUL[ $ DUH GHILQHG >@ E\ WKH HTXDWLRQ f ZKHUH WKH L HLJHQYDOXHV DQG HLJHQYHFWRUV DUH UHSUHVHQWHG E\ $Ef DQG HEf UHVSHFWLYHO\ 7KH VLQJXODU YDOXH GHFRPSRVLWLRQ 69'f RI DQ P [ Q PDWUL[ $ RI UDQN U LV GHILQHG > @ E\ WKH HTXDWLRQ $ 8+97 f ZKHUH ( LV DQ P [ Q PDWUL[ ZLWK WKH VLQJXODU YDOXHV RI $ Df RQ WKH PDLQ GLDJRQDO 8 LV DQ P [ P RUWKRJRQDO PDWUL[ DQG 9 LV DQ Q [ Q RUWKRJRQDO PDWUL[ 7KH FROXPQV RI 8 DUH WKH HLJHQYHFWRUV RI $$7 DQG WKH FROXPQV RI 9 DUH WKH HLJHQYHFWRUV RI $7$ 7KH U VLQJXODU YDOXHV DUH WKH QRQQHJDWLYH VTXDUH URRWV RI WKH QRQ]HUR HLJHQYDOXHV RI ERWK $$7 DQG $7$ LH WKH HLJHQYDOXHV RI $$7 DQG $U $ DUH HTXDO WR WKH VTXDUH RI WKH VLQJXODU YDOXHV RI $ 7KH HLJHQYDOXHV DUH SUHVHUYHG IRU VLPLODULW\ WUDQVIRUPDWLRQV % f§ 6aO$6 DQG WKH HLJHQYHFWRUV RI % DUH 6nBHEf (LJHQYDOXHV DUH QRW SUHVHUYHG XQGHU FRQJUXHQFH WUDQVIRUPDWLRQV % f§ 67$6 XQOHVV 6 LV D URWDWLRQ LQ ZKLFK FDVH 67 6n DQG % LV DOVR D VLPLODULW\ WUDQVIRUPDWLRQf

PAGE 31

&+$37(5 /,1($5 121&200(1685$7( 6<67(06 )RU OLQHDU QRQFRPPHQVXUDWH V\VWHP X f§ $[ WKH UHTXLUHPHQWV RQ WKH VWUXFWXUH RI $ DUH GHWHUPLQHG LQ WKLV VHFWLRQ ZKHUH $ LV DQ Q [ P PDWUL[ [ LV D QRQFRPPHQVXUDWH PYHFWRU DQG X LV D QRQFRPPHQVXUDWH QYHFWRU 8SRQ H[SDQGLQJ X f§ $[ LW LV IRXQG WKDW X f§ A @ ¾MM;M f M OP VR WKDW XQLWV>DM@XQLWV>DM@ XQLWV>X@ f 8VLQJ WZR WHUPV LQ WKH VXP RI f IRU WZR HOHPHQWV RI X ZH JHW =b f§ =^M;M f¯I 4nLNn(N f => DLM;M DLN;N f IRU DOO L M N ZKHUH XQLWVI]@ XQLWV>X@ 6ROYH f IRU ;N DQG VXEVWLWXWH WKH UHVXOW LQWR f WR JHW ]c =^ f§ GLN&WLM f GLN GLN 7KHUHIRUH IRU SK\VLFDOO\ FRQVLVWHQF\ XQLWV>DIF@XQLWV>D\@ XQLWV>DM@XQLWV>DIF@ f RU XQLWV>f§@ XQLWV>f§@ DOM DLM f ,Q RWKHU ZRUGV LI P f§ FROXPQV DQG Q f§ URZV DUH HOLPLQDWHG WKH GHWHUPLQDQW RI WKH UHPDLQLQJ [ PDWUL[ PXVW EH SK\VLFDOO\ FRQVLVWHQW IRU WKH V\VWHP WR EH QRQFRPPHQVXUDWH

PAGE 32

8VLQJ WKUHH WHUPV LQ WKH VXP RI f IRU WKUHH HOHPHQWV RI X ZH JHW WKUHH HTXDWLRQV VLPLODU WR f DQG f 6ROYLQJ WKHVH HTXDWLRQV OHDGV WR D FRQGLWLRQ VLPLODU WR WKDW VKRZQ LQ f LH LI P f§ FROXPQV DQG Q f§ URZV DUH HOLPLQDWHG WKH GHWHUPLQDQW RI WKH UHPDLQLQJ [ PDWUL[ PXVW EH SK\VLFDOO\ FRQVLVWHQW IRU WKH V\VWHP WR EH QRQFRPPHQVXUDWH %\ LQGXFWLRQ WKH DERYH WHFKQLTXH VKRZV WKDW IRU DOO r ! DQG L P Q RU L Q P LI P f§ L FROXPQV DQG Q f§ L URZV DUH HOLPLQDWHG WKH GHWHUPLQDQW RI WKH UHPDLQLQJ L [ L PDWUL[ PXVW EH SK\VLFDOO\ FRQVLVWHQW IRU WKH V\VWHP WR EH QRQFRPPHQVXUDWH $QRWKHU UHTXLUHPHQW RQ PDWUL[ $ LV IRXQG E\ YLHZLQJ $ DV D PDWUL[ RI FROXPQ YHFWRUV $LWf m e [L$Lf fµ f L ?P ,W LV HYLGHQW WKDW WKH XQLWV RI DQ\ WZR FROXPQV RI $ PXVW EH SURSRUWLRQDO 7KLV LV DQ DOWHUQDWH ZD\ WR H[SUHVV WKH UHVXOWV RI f DQG D VLPSOH ZD\ WR YLVXDOO\ GHGXFH ZKHWKHU RU QRW D PDWUL[ LV D FDQGLGDWH QRQFRPPHQVXUDWH OLQHDU V\VWHP PDWUL[ $OO OLQHDU V\VWHPV DUH HLWKHU FRPPHQVXUDWH QRQFRPPHQVXUDWH RU SK\VLFDOO\ LQn FRQVLVWHQW &RPPHQVXUDWH DQG QRQFRPPHQVXUDWH V\VWHPV DUH SK\VLFDOO\ FRQVLVWHQW V\VWHPV )RU FRPPHQVXUDWH V\VWHPV DOO HOHPHQWV RI WKH $ PDWUL[ KDYH LGHQWLFDO XQLWV (LJHQVYVWHP ,Q 1RQFRPPHQVXUDWH 6\VWHPV $V ZDV PHQWLRQHG DW WKH VWDUW RI 6HFWLRQ PDQ\ UHVHDUFKHUV PDNH XVH RI WKH HLJHQYDOXHV HLJHQYHFWRUV RU VLQJXODU YDOXHV RI PDWULFHV ZKRVH HLJHQYDOXHV DQG VLQJXODU YDOXHV DUH QRW LQYDULDQW WR FKDQJHV LQ VFDOH RU FRRUGLQDWH WUDQVIRUPDWLRQV 7KHVH DUH WKHUHIRUH QRW WUXH f¯HLJHQVROXWLRQVf° LQ WKH VHQVH WKDW WKH\ PD\ RQO\ VXEMHFWLYHO\ FKDUDFWHUL]H D PDQLSXODWRU FRQILJXUDWLRQ EDVHG RQ D SDUWLFXODU REVHUYHU ZLWK D FKRLFH RI VFDOH DQG FRRUGLQDWH IUDPH RI UHIHUHQFHf DV RSSRVHG WR D PRUH UHOHYDQW REMHFWLYH FKDUDFWHUL]DWLRQ RI D PDQLSXODWRU FRQILJXUDWLRQ

PAGE 33

&RQGLWLRQV IRU 3K\VLFDOO\ &RQVLVWHQW (LJHQVYVWHPV :KHQ GRHV D PDWUL[ $ KDYH SK\VLFDOO\ FRQVLVWHQW HLJHQYDOXHV DQG HLJHQYHFWRUV" /HW $ EH DQ Q [ Q PDWUL[ D fµ fµ fµ m Q m r n n n rQ $ QO Q DQG OHW WKH GRPDLQ RI $ EH ;Q ZKHUH ;Q LV D VSDFH ZLWK SK\VLFDO XQLWV 7KH $AVSDFH FDQ EH FKDUDFWHUL]HG DV IROORZV /HW EH DQ QYHFWRU RI SRVVLEO\ GLVWLQFW SK\VLFDO XQLWV 3 f§ 3L 3Q f $Q\ [ fü ;Q LV HTXLYDOHQW WR DQ LWHPZLVH PXOWLSOLFDWLRQ RI DQG \ \  LH [ p\ @L 3L\[ \ [ H ;Q \ fü 7 3Q9Q f f f VR WKDW LQ Lf§! ;Q DQG HDFK HLJHQYHFWRU RI $ IURP f HKf LV DQ HOHPHQW RI ;Q VSDFH 6XEVWLWXWLQJ f LQWR f DQG SHUIRUPLQJ WKH PDWUL[ PXOWLSOLFDWLRQ UHVXOWV LQ WKH IROORZLQJ HTXDWLRQV HOA AA fµ fµ ‘ RLQHQA rLHLA G DHA$ fµ fµ ‘ DQHQAO 2QOAOA AA ‘ n fµ D!QQfüQA f§ 5HFRJQL]LQJ WKDW RQO\ TXDQWLWLHV ZLWK LGHQWLFDO SK\VLFDO XQLWV PD\ EH DGGHG OHDGV WR WKH IROORZLQJ WKHRUHP $mHLm $:H: $mHQm f 7KHRUHP 7KH HTXDWLRQ $[ f§ ;[ LV SK\VLFDOO\ FRQVLVWHQW LI DQG RQO\ LI XQLWVIDMWMMXQLWVI[M@ XQLWV>$@XQLWV>DIF@ IRU DOO M DQG N

PAGE 34

3URRI %\ K\SRWKHVLV DIFM@XQLWV>[M@ XQLWV>$@XQLWV>[IF@ IRU DOO M DQG N 1RZ DVVXPH XQLWV >DM\@ XQLWV >[\@ XQLWV>$@ XQLWV >[A@ IRU DOO M DQG N &OHDUO\ WKH HTXDWLRQ DMWM@XQLWV>[M@ XQLWV>$@XQLWV>[MW@ LPSOLHV WKDW XQLWV>$@ XQLWV>D@ IRU DOO L +HQFH DQ\ PDWUL[ ZLWK D SK\VLFDOO\ FRQVLVWHQW HLJHQYDOXH HTXDWLRQ PXVW KDYH GLDJRQDO HOHPHQWV ZLWK WKH VDPH SK\VLFDO XQLWV DQG DOO LWV HLJHQYDOXHV PXVW KDYH WKRVH VDPH XQLWV $ VLPSOH WHVW IRU D SK\VLFDOO\ FRQVLVWHQW HLJHQV\VWHP LV WKH YDOLGLW\ RI WKH EHORZ HTXDWLRQ IRU HDFK HOHPHQW LQ PDWUL[ $ XQLWV>DIFM@XQLWV>DMIF@ XQLWV>DW@ f 6LQFH WKH VLQJXODU YDOXHV RI $ DUH WKH QRQQHJDWLYH VTXDUH URRWV RI WKH QRQ]HUR HLJHQYDOXHV RI ERWK $$U DQG $7$ D WHVW RQ WKHVH PDWUL[ SURGXFWV VLPLODU WR WKH WHVWV GLVFXVVHG DERYH IRU WKH HLJHQV\VWHP RI $f ZLOO GHWHUPLQH LI WKH 69' RI $ LV SK\VLFDOO\ FRQVLVWHQW 7KH FRQGLWLRQV IRU WKH SK\VLFDO FRQVLVWHQF\ RI WKH 69' RI $ DUH VWDWHG LQ &RUROODU\ EHORZ &RUROODU\ 7KH VLQJXODU YDOXH GHFRPSRVLWLRQ RI $ $ f§ 8e9U LV SK\VLFDOO\ FRQn VLVWHQW LI DQG RQO\ LI XQLWV>M@XQLWV>[M@ XQLWV>$@XQLWV>[IF@ IRU DOO M DQG N ZKHUH % f§ $$7 RU % $7$ DQG %[ $[

PAGE 35

3URRI 7KLV IROORZV GLUHFWO\ IURP 7KHRUHP DQG WKH SURSHUWLHV RI 69' LH WKH HLJHQV\V WHP WHVWV RQ WKH PDWULFHV $$7 DQG $7$ GHWHUPLQH WKH VLQJXODU YDOXHV DQG RUWKRJRQDO PDWULFHV 8 DQG 97 7KHUHIRUH WKH WHVW RI 7KHRUHP DQG f FDQ EH GLUHFWO\ DSn SOLHG WR $$7 DQG $7$ WR GHWHUPLQH LI WKH 69' RI $ LV SK\VLFDOO\ FRQVLVWHQW /HW M LQ &RUROODU\ EH HTXDO WR N 7KHQ XQLWV>IFIF@ XQLWV>$@ DQG DOO GLDJRQDO HOHPHQWV RI % PXVW KDYH WKH VDPH SK\VLFDO XQLWV ,I $ LV DQ Q [ P PDWUL[ WKHQ HDFK GLDJRQDO WHUP RI % LV  (M/L DM\ IRU % $$7 ENN RU IRU DOO N f e rr IRU % $7$ 7KHUHIRUH DOO WKH HOHPHQWV LQ WKH IFWK URZ RU IFFROXPQ $ PXVW KDYH LGHQWLFDO XQLWV IRU % $$U RU % $7$ UHVSHFWLYHO\ %XW VLQFH XQLWV>rIF@ XQLWVIRU DOO M DQG N DOO WKH HOHPHQWV RI $ PXVW KDYH WKH VDPH XQLWV 7KHUHIRUH VLQJXODU YDOXH GHFRPSRVLWLRQ LV RQO\ YDOLG IRU FRPPHQVXUDWH V\VWHPV LH 7KHRUHP  1RQFRPPHQVXUDWH V\VWHP QHYHU KDYH D SK\VLFDOO\ FRQVLVWHQW VLQJXODU YDOXH GHFRPSRVLWLRQ 7KH PDMRU UHVXOWV RI WKLV FKDSWHU DUH VXPPDUL]HG EHORZ 7KH UHTXLUHPHQWV RQ $ IRU DOO SK\VLFDOO\ FRQVLVWHQW OLQHDU QRQFRPPHQVXUDWH V\VWHPV X $[ ZHUH JLYHQ LQ f DQG f 7KH UHTXLUHPHQWV IRU $ WR KDYH D SK\VLFDOO\ FRQVLVWHQW HLJHQ V\VWHP ZHUH JLYHQ LQ f $QG ILQDOO\ LW ZDV VKRZQ WKDW SK\VLFDOO\ FRQVLVWHQW OLQHDU QRQFRPPHQVXUDWH V\VWHPV GR QRW KDYH D SK\VLFDOO\ FRQVLVWHQW VLQJXODU YDOXH GHFRPSRVLWLRQ 2QO\ FRPPHQVXUDWH V\VWHPV KDYH D SK\VLFDOO\ FRQVLVWHQW 69'

PAGE 36

&+$37(5 3+<6,&$/ &216,67(1&< 2) -$&2%,$1 )81&7,216 7KH PDQLSXODWRU -DFRELDQ LV XVHG E\ PDQ\ UHVHDUFKHUV LQ ZD\V ZKLFK UHVXOW LQ SK\VLFDOO\ LQFRQVLVWHQW UHVXOWV 6HYHUDO RI WKHVH ZLOO EH GLVFXVVHG LQ WKLV FKDSWHU ,QDSSURSULDWH 8VHV RI WKH (XFOLGHDQ 1RUP LQ 5RERWLFV $ PXOWLWXGH RI UHVHDUFKHUV > @ KDYH FKDUDFWHUL]HG D URERW FRQILJXn UDWLRQ RU FRQGLWLRQ LQ WHUPV RI WKH VFDODU TXDQWLW\ RI WKH (XFOLGHDQ QRUP 7KLV ZLOO EH VKRZQ WR EH LQYDOLG LQ JHQHUDO 2QH RU PRUH QRQ(XFOLGHDQ PHWULFV DUH RIWHQ QHFHVVDU\ > @ WR ILQG D SK\VLFDOO\ FRQVLVWHQW QRQ(XFOLGHDQf QRUP $OWKRXJK WKLV PD\ QRW VHHP REYLRXV DW ILUVW JODQFH FRQVLGHU WKH IROORZLQJ H[DPSOHV 7KH WZLVW YHFWRU 9f§GHILQHG LQ ff§LV FRPSRVHG RI WKH WUDQVODWLRQDO YHORFLW\ YHFWRU Y DQG WKH DQJXODU YHORFLW\ YHFWRU X! 7KH VTXDUH RI WKH (XFOLGHDQ QRUP LV RIWHQ LQDSSURSULDWHO\ DSSOLHG WR WKH WZLVW YHFWRU ?9? 949 9U 9 f %XW WKH H[SUHVVLRQ 9 k 9 LV SK\VLFDOO\ LQFRQVLVWHQW VLQFH _8_ Y4Y ZZ f DQG Y KDV XQLWV RI >OHQJWKWLPH@ ZKLOH XL KDV XQLWV RI >DQJOHWLPH@ 7KLV LV OLNH DGGLQJ DSSOHV WR RUDQJHV JHQHUDOO\ LQDSSURSULDWH ZLWKRXW D PHWULF RQ WKH ZRUWK RI DQ DSSOH FRPSDUHG WR DQ RUDQJH >OHQJWKWLPH@ FRPSDUHG WR >DQJOHWLPH@f )RU H[DPSOH LI YU f§ > O@AMS DQG Z7 > @AA WKHQ _8_ &KDQJLQJ WKH VFDOH IURP FP WR PP ZLOO FKDQJH WKH UHVXOW WR _9_ A ,I ZH GHILQH D PHWULF IRU WZLVWV 0Y ZH FDQ XVH WKH 0XQRUP f§GHILQHG LQ ff§ WR JHW D PHDVXUH RI WZLVWV ?9?0Y f§ 940Y9 7KH PHWULF 0Y PXVW EH SRVLWLYH GHILQLWH

PAGE 37

DQG PDNH AA SK\VLFDOO\ FRQVLVWHQW $ PHWULF 0Y FDQ EH VHOHFWHG VXFK WKDW WKLV QRUP GHVFULEHV WKH NLQHWLF HQHUJ\ . RI D ULJLG ERG\ . O940Y9 L970Y9 f 7KH NLQHWLF HQHUJ\ PHWULF H[SUHVVHG DW WKH FHQWHURIPDVV ZLWK D[HV DOLJQHG ZLWK WKH ERG\f¬V SULQFLSDO D[HV LV WKH SULQFLSDO PDVVLQHUWLD PDWUL[ RI D ULJLG ERG\ 0NH PEO] >f >@ f ZKHUH UDt LV WKH ERG\f¬V PDVV DQG t LV WKH ERG\f¬V LQHUWLD WHQVRU DW WKH FHQWHURIPDVV H[SUHVVHG LQ SULQFLSDO FRRUGLQDWHVf§D GLDJRQDO PDWUL[ :H PXVW H[SUHVV WKLV PHWULF LQ WKH VDPH IUDPH DV WKH WZLVWVf§ VHH ff§RU H[SUHVV WKH WZLVWV DW WKH ERG\f¬V FHQWHU RIPDVV DOLJQHG ZLWK WKH ERG\f¬V SULQFLSDO D[LVf 7UDQVIRUPLQJ WKH PHWULF 0NH WR WKH IUDPH RI H[SUHVVLRQ RI WKH WZLVW UHVXOWV LQ WKH PHWULF PLK PW.U%5 UUQ!57%75 57,E PE%7%f5 f¬ ZKHUH *Y LV GHILQHG LQ f ZLWK 5 n5S % f§ n%3WL L LV WKH H[SUHVVLRQ IUDPH IRU WKH WZLVW 9 f§ r9 DQG S LV WKH IUDPH RI WKH SULQFLSDO D[HV RI WKH ERG\ 7KH ORZHU ULJKW [ PDWUL[ LQ f LV WKH LQHUWLD PDWUL[ RI WKH ERG\ LQ WKH WZLVW IUDPH ,I WKHUH LV QR URWDWLRQ EHWZHHQ WKH WZLVW IUDPH DQG WKH SULQFLSDO IUDPH WKH LQHUWLD PDWUL[ LV )E K PE%7%f 7KLV LQHUWLD PDWUL[ FRXOG KDYH DOVR EHHQ GHWHUPLQHG XVLQJ WKH SDUDOOHO D[LV WKHRUHP >@ 7KH PHWULF RI f LV WKH WZLVW LQHUWLD PDWUL[ RI D ULJLG ERG\ FRPSRVHG RI WKH ]HURRUGHU PDVVPRPHQW PDVVf WKH ILUVWRUGHU PDVVPRPHQW PRPHQWXPf DQG WKH VHFRQGRUGHU PDVVPRPHQW LQHUWLDf )RU D VHFRQG H[DPSOH RI WKH SUREOHP RI XVLQJ (XFOLGHDQ QRUPV LQ URERWLF DSSOLn FDWLRQV OHW XV ORRN DW WKH JHQHUDOL]HGIRUFH YHFWRU U RI WKH PDQLSXODWRU MRLQWV 7KH VTXDUH RI WKH (XFOLGHDQ QRUP RI U LV 0Y *WY0.(*Y _ " f§ L L 7 f§ 7? 7 f« f« 7Q f

PAGE 38

,I DOO WKH MRLQWV RI WKH PDQLSXODWRU DUH UHYROXWH RU DOO DUH SULVPDWLF f LV SK\VLFDOO\ FRQVLVWHQW EXW WKLV PHDVXUH RI WKH VXP RI WKH VTXDUH RI MRLQW WRUTXHV LV SUREDEO\ RI OLWWOH YDOXH VLQFH WKH GULYLQJ FRPSRQHQW RI VRPH MRLQWV LV JHQHUDOO\ TXLWH GLIIHUHQW IURP RWKHU MRLQWVf %XW LI WKH PDQLSXODWRU KDV ERWK UHYROXWH DQG SULVPDWLF MRLQWV f§ LH WKH PDQLSXODWRU MRLQWV DQJOHV RU YHORFLWLHV IRUP D QRQFRPPHQVXUDWH YHFWRUf§WKLV HTXDWLRQ VXPV SK\VLFDOO\ LQFRQVLVWHQW IRUFHVTXDUHG DQG PRPHQWVTXDUHG WHUPV /HW XV YLHZ WKH (XFOLGHDQ QRUPV RI 9 DQG U IURP D GLIIHUHQW SHUVSHFWLYH f§ QDPHO\ E\ ORRNLQJ DW WKH PDQLSXODWRU -DFRELDQ GHILQHG LQ f 2I FRXUVH LI WKH PDQLSXODWRU KDV MRLQWV DQG KDV IXOO UDQN WKHQ -a[ FDQ EH IRXQG DQG WKH VROXWLRQ WR f LV T f§ -a; 9 f 7R VROYH IRU T ZKHQ LV QRW D VTXDUH PDWUL[ PDQ\ UHVHDUFKHUV XVH WKH SVHXGRLQYHUVH -Of§VHH fff§DQG WKH HTXDWLRQ TV 9 fµ f )RU D IXOO URZ UDQN PDWUL[ WKH SVHXGR LQYHUVH LV -r -7--7fB IXOO URZ UDQN f (TXDWLRQ f LV RIWHQ XVHG ZLWK UHGXQGDQW PDQLSXODWRUV PDQLSXODWRUV ZLWK PRUH WKDQ MRLQWVf )RU PDQLSXODWRUV ZLWK OHVV WKDQ MRLQWV WKH SVHXGRLQYHUVH IRU D IXOO FROXPQ UDQN PDWUL[ LV RIWHQ XVHG -r -7-fa-7 IXOO FROXPQ UDQN f 1RWH WKDW WKH SVHXGRLQYHUVH LQ RQH FDVH LQYROYHV WKH WHUP --7 DQG LQ WKH RWKHU FDVH LQYROYHV WKH WHUP -77KHUH LV RIWHQ D XQLWV SUREOHP SK\VLFDO LQFRQVLVWHQF\f ZLWK ERWK RI WKHVH WHUPV 2QH RI WKHVH WHUPV DOVR DSSHDUV LQ HDFK RI WKH QRUP RI D WZLVW 9 DQG WKH QRUP RI WKH JHQHUDOL]HGIRUFH YHFWRU WZ DV ZLOO EH VKRZQ EHORZ

PAGE 39

)URP HTXDWLRQV f DQG f ZH JHW WKH (XFOLGHDQ QRUP RI WZLVW 9 RI ?9?r 97 9 f§ TI7 Q7@7 LV FRPSRVHG RI WKH YHFWRUV RI IRUFH DQG PRPHQW Q 7KH WHUP --7 DJDLQ DSSHDUV LQ WKH (XFOLGHDQ QRUP RI WZ (TXDWLRQ f FDQ EH UHZULWWHQ XVLQJ f DV : UZ7 WZ :7 --7f : f /HW XV QRZ ORRN DW WKH SK\VLFDO FRQVLVWHQF\ RI WKHVH (XFOLGHDQ QRUPV E\ SHUIRUPn LQJ D XQLWV DQDO\VLV RQ -7DQG --7 7KH XQLWV RI D PDQLSXODWRU -DFRELDQ PDWUL[ LV IRXQG VLPSO\ E\ QRWLQJ WKDW WKH XQLWV RI WKH UDQJH RI LV HTXDO WR WKH XQLWV RI 9 DQG LV QRW GHSHQGHQW RQ WKH VWUXFWXUH RI WKH PDQLSXODWRU 7KHUHIRUH WKH XQLWV RI HOHPHQWV LQ D -DFRELDQ FROXPQ KDYH RQH RI WKH IROORZLQJ WZR IRUPV > @ fµ ,I PDQLSXODWRU MRLQW L LV UHYROXWH WKH LWK FROXPQ RI WKH -DFRELDQ KDV WKH XQLWV IRU UHYROXWH MRLQWV f fµ ,I PDQLSXODWRU MRLQW L LV SULVPDWLF WKH ]nWK FROXPQ RI WKH -DFRELDQ KDV WKH XQLWV IRU SULVPDWLF MRLQWV f XQLWV>-cf@ > >R@0 XQLWV>-cf@ >e@L 0P 7KH LQ WKH DERYH HTXDWLRQV FRUUHVSRQGV WR D M [ N PDWUL[ ZKRVH HOHPHQWV KDYH XQLWV RI / IRU XQLWV RI OHQJWK RU 8 IRU XQLWOHVV 7KH >@IF WHUP LGHQWLILHV D PDWUL[ ZKRVH HOHPHQWV DUH HTXDO WR ]HUR DQG VD\V QRWKLQJ DERXW WKH HOHPHQWVf¬ XQLWVf

PAGE 40

3K\VLFDO &RQVLVWHQF\ RI -7DQG --7 ,I DOO Q MRLQWV PDQLSXODWRU DUH UHYROXWH WKH XQLWV RI -7LV XQLWV>-7-@ >e 9 @f± IRU Q UHYROXWH MRLQWV f LH HDFK WHUP VXPV D OHQJWKVTXDUHG WHUP ZLWK D XQLWOHVV WHUP 6LQFH WKH (XFOLGHDQ QRUP RI 9 LQ f UHTXLUHV WKH SURGXFW -7-f WKH (XFOLGHDQ QRUP RI 9 LV REYLRXVO\ SK\VLFDOO\ LQFRQVLVWHQW DV VKRZQ LQ f )RU QRQFRPPHQVXUDWH PDQLSXODWRUV LI WKH WK DQG MWK MRLQWV RI D PDQLSXODWRU DUH UHYROXWH WKHQ WKH LMfWK HOHPHQW RI WKH PDWUL[ -7LV SK\VLFDOO\ LQFRQVLVWHQW ZLWK XQLWV RI XQLWV> -7 / 8 IRU WK DQG MWK MRLQWV UHYROXWH f ,I WKH WK MRLQW LV UHYROXWH DQG WKH MWK MRLQW LV SULVPDWLF WKHQ WKH  MfWK HOHPHQW RI WKH PDWUL[ -7LV SK\VLFDOO\ FRQVLVWHQW ZLWK XQLWV RI XQLWV> -7-fMf@ f§ / IRU WK MRLQW UHYROXWH MWK MRLQWV SULVPDWLF f ,I WKH WK DQG MWK MRLQWV DUH ERWK SULVPDWLF WKHQ WKH  MfWK HOHPHQW RI WKH PDWUL[ -7LV SK\VLFDOO\ FRQVLVWHQW ZLWK XQLWV RI XQLWV> -7-fLMf@ 8 IRU WK DQG MWK MRLQWV SULVPDWLF f 6LPLODUO\ WKH (XFOLGHDQ QRUP RI T LV DOVR SK\VLFDOO\ LQFRQVLVWHQW IRU QRQFRPPHQn VXUDWH PDQLSXODWRUV H ?T? œc Tc TO f PDNLQJ D QRQFRPPHQVXUDWH YHFWRU RI MRLQW UDWHV ZLWK XQLWV RI / 8f7 ZKHUH 7 UHSUHVHQWV WLPH XQLWV 7KH (XFOLGHDQ QRUP RI 9 DQG WKH PDWUL[ -7DUH SK\VLFDOO\ FRQVLVWHQW IRU DQ DOO SULVPDWLFMRLQWHG PDQLSXODWRU VLQFH WKH HQWLUH -7PDWUL[ LV XQLWOHVV DQG 9 >‘Y7 @7 LH WKH DQJXODU YHORFLW\ LV ]HUR

PAGE 41

7DEOH '+ SDUDPHWHUV IRU *( 3 PDQLSXODWRU -RLQW 7\SH G D H D 5 W 5 5 5 L W 5 6 7KH *HQHUDO (OHFWULF 3 PDQLSXODWRU ZLWK UHYROXWH MRLQWVf KDV 'HQDYLW+DUW HQEHUJ SDUDPHWHUV JLYHQ LQ 7DEOH DQG D IUDPH -DFRELDQ DV D f§ DF M B f§F f§DF 6 6_ F F 7KH PDWUL[ -7 G -7-f KDV HOHPHQWV ZLWK LQFRQVLVWHQW SK\VLFDO XQLWV VXFK DV WKH f WHUP ZKRVH FDOFXODWHG YDOXH LV D? 7KH GHWHUPLQDQW RI -7-f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n ODWRUVf 7KH IUDPH f¯JHQHUDOf° FRUUHVSRQGV WR DQ\ QRQ]HUR WUDQVODWLRQ 3XUH URWDWLRQV KDYH QR DIIHFW RQ WKH YDOXH RI -7VLQFH -nf7-n *-f7*-f f -7*7*f *7 *a[ IRU URWDWLRQV QR WUDQVODWLRQf f }-nf7-n -7 IRU URWDWLRQV QR WUDQVODWLRQf f f

PAGE 42

7DEOH 3K\VLFDO XQLWV RI 'HW>-7-@ IRU YDULRXV QRQUHGXQGDQW PDQLSXODWRUV 0DQLSXODWRU 'HVFULSWLRQ &RRUGLQDWH )UDPH 8QLWV RI 'HW>-7-@ 35 9LUWXDO 8 35 9LUWXDO JHQHUDO 8 / 3ODQDU 555 $OO / 1RQSODQDU 555 JHQHUDO 8 / / *HQHUDO 555 JHQHUDO 8 / / 333 2UWKRJRQDO $OO 8 6$5 353f X 6$5 353f JHQHUDO 8 / 535 8 / 535 JHQHUDO 8 / / 6&$5$ 5553f $Q\ / 5553 / 5553 JHQHUDO / / 5553 8 / 5553 JHQHUDO 8 // 3 5f / /t 3 5f W / MRLQWHG 'HW>-@ $Q\ IUDPH /aS $OWKRXJK WKH SK\VLFDO FRQVLVWHQF\ RI -7DVVXUHV WKH SK\VLFDO FRQVLVWHQF\ RI WKH GHWHUPLQDQW RI -7 WKH LQYHUVH RI WKLV VWDWHPHQW LV QRW DOZD\V WUXH )RU LQVWDQFH WKH 5553 PDQLSXODWRU LQ IUDPH KDV SK\VLFDOO\ LQFRQVLVWHQW WHUPV LQ r-7-f EXW 'HW>r-7-f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f LV /aS ZKHUH S LV WKH QXPEHU RI SULVPDWLF MRLQWV XS WR WKUHH $Q\ PRUH

PAGE 43

WKDQ WKUHH SULVPDWLF MRLQWV ZLOO PHDQ WKH PDQLSXODWRU DOZD\V KDV 'HW>-@ f 7KH GHWHUPLQDQWV RI -7DQG --7 WKHUHIRUH KDYH SK\VLFDO GLPHQVLRQV LSf DQG DUH HTXDO VLQFH 'HW>L@'HW>IO@ 'HW>$%@ f IRU DOO VTXDUH PDWULFHV $ DQG % ZLWK LGHQWLFDO PDWUL[ GLPHQVLRQV (TXDWLRQ f DOVR JXDUDQWHHV WKH HTXDOLW\ 'HW>-7-@ 'HW>--7@ 'HW>-@f 7KH GHWHUPLQDQW RI -7 LV ]HUR IRU PDQLSXODWRUV ZLWK PRUH WKDQ VL[ MRLQWV VLQFH -7FDQ KDYH DW PRVW UDQN WKH PD[LPXP UDQN RI QRW UDQN Qf 6R LQVWHDG ZH ORRN DW WKH PDWUL[ --7 IRU UHGXQGDQW PDQLSXODWRUV 7KH XQLWV RI --7 IRU DQ DOO UHYROXWH MRLQW PDQLSXODWRU LV XQLWV >--7@ f§ 7KH XQLWV RI WKLV PDWUL[ DUH SK\VLFDOO\ FRQVLVWHQW DV LV WKH FDVH IRU DQ DOO SULVPDWLF MRLQW HG PDQLSXODWRU ZKHUH >e@f± >@ A >&nLVV IRU DOO UHYROXWH MRLQWV f XQLWV>--7@ 0V >@Y 0 0 IRU DOO SULVPDWLF MRLQWV f )RU D QRQFRPPHQVXUDWH PDQLSXODWRU WKH --7 XQLWV PDWUL[ RI XQLWV>--7@ >/ 8@L >;@I : A IRU QRQFRPPHQVXUDWH PDQLSXODWRU f LV SK\VLFDOO\ LQFRQVLVWHQW 7KH GHWHUPLQDQW RI --7 LV IUDPH LQGHSHQGHQW LH LQYDULDQW WR ERWK URWDWLRQV DQG WUDQVODWLRQVf VLQFH IRU -n *'HW>-nM\@ 'HW>*-*-f7@ 'HW>*--7*7@ 'HW>*@ 'HW>--7@ 'HW>*7@ 'HW>--U@ f f f DQG WKH GHWHUPLQDQW RI WKH WZLVW FRRUGLQDWH WUDQVIRUPDWLRQ PDWUL[ * LV RQH

PAGE 44

7DEOH 3K\VLFDO XQLWV RI 'HW>--7@ IRU YDULRXV UHGXQGDQW PDQLSXODWRUV 0DQLSXODWRU 'HVFULSWLRQ &RRUGLQDWH )UDPH 8QLWV RI 'HW>--7@ MRLQWHG 'HW>-@ $Q\ IUDPH /tS $QWKURSRPRUSKLF $UP (f $Q\ /t 3XPD 5f $Q\ / &(6$5 5f $Q\ / . 5f $Q\ / 35 $Q\ 8 *3 535f $Q\ /r /t 7KH GHWHUPLQDQW RI --7 IRU PDQLSXODWRUV ZLWK OHVV WKDQ VL[ MRLQWV LV RI FRXUVH ]HUR VLQFH WKH UDQN RI DQG WKXV WKH UDQN RI --7 LV OHVV WKDQ VL[ IRU WKHVH URERWVf 7KH GHWHUPLQDQW RI --7 IRU VHYHUDO UHGXQGDQW PDQLSXODWRUV ZDV FDOFXODWHG DQG WKH SK\VLFDO FRQVLVWHQF\ RI WKH GHWHUPLQDQWV FRUUHVSRQGHG WR WKH SK\VLFDO FRQVLVWHQF\ GLVFXVVHG DERYH IRU WKH PDWUL[ --7 LQ DOO FDVHV 7DEOH VKRZV WKH XQLWV RI WKH GHn WHUPLQDQW IRU HDFK RI WKH PDQLSXODWRUV 6HH $SSHQGL[ $ IRU WKH 'HQDYLW+DUWHQEHUJ SDUDPHWHUV RI HDFK RI WKHVH PDQLSXODWRUV WKH -DFRELDQ LQ D SDUWLFXODU PLGIUDPH DQG WKH GHWHUPLQDQW RI --7 LQ WKLV IUDPH &RQVLVWHQF\ RI _X $[? $ JHQHUDOL]DWLRQ RI VRPH RI WKH DERYH UHVXOWV IRU WKH SK\VLFDO FRQVLVWHQF\ RI WKH (XFOLGHDQ QRUP ZLOO EH VKRZQ LQ WKLV VHFWLRQ )RU D OLQHDU VHW RI HTXDWLRQV X f§ $[ 7KHRUHP DQG &RUROODU\ ERWK EHORZf VKRZ WKDW WKH SK\VLFDO FRQVLVWHQF\ RU LQFRQn VLVWHQF\f RI WKH (XFOLGHDQ QRUP RI X FDQ EH GHWHUPLQHG E\ WKH SK\VLFDO FRQVLVWHQF\ RU LQFRQVLVWHQF\f RI $7$ 7KHRUHP ,IX f§ $[ ZKHUH $ LV DQ P[Q PDWUL[ P!Qf WKHQ WKH IRU WKH IROORZLQJ VWDWHPHQWV 6, WKURXJK 6 6L LPSOLHV 6 DQG 6 LPSOLHV 6 VR WKDW 6, LPSOLHV 6 6L 7KH HTXDWLRQ _X_ X4X X7X LV SK\VLFDOO\ FRQVLVWHQW LQFRQVLVWHQWf

PAGE 45

6 7KH QRQ]HUR HOHPHQWV LQ D JLYHQ FROXPQ RI $ KDYH LGHQWLFDO XQLWV QRW DOO LGHQn WLFDO XQLWVf LH ,I DLN DQG DMN A WKHQ XQLWV>DIF@ XQLWVID\IF@ IRU N fü ^ Q` DQG LM ^P` f 6 7KH PDWUL[ $7$ LV SK\VLFDOO\ FRQVLVWHQW LQFRQVLVWHQWf ,Q RWKHU ZRUGV 7KHRUHP WHOOV XV WKDW WKH SK\VLFDO FRQVLVWHQF\ RI WKH (XFOLGHDQ QRUP RI X LPSOLHV WKDW DOO HOHPHQWV LQ D JLYHQ FROXPQ RI $ KDYH LGHQWLFDO XQLWV RU DUH HTXDO WR ]HURf DQG WKDW $7$ LV SK\VLFDOO\ FRQVLVWHQW 3URRI 7KLV SURRI LV VSOLW XS LQWR WZR SDUWV WKH ILUVW SURRI VKRZV WKDW 6, LPSOLHV 6 WKH VHFRQG SURRI VKRZV WKDW 6 LPSOLHV 6 7KHQ E\ WUDQVLWLYLW\ 6, LPSOLHV 6 7KH IROORZLQJ KROG WKURXJKRXW WKHVH SURRIV LM * ^ P` DQG NK * ^Q` fµ $VVXPH 6L WR SURYH 6 f§ 6LQFH X7X LV SK\VLFDOO\ FRQVLVWHQW XQLWV>X@ XQLWV>X@ XQLWV>X@ f§ 6LQFH X $[ 8^ f§ DLN;N f§ 6LQFH 8L LV SK\VLFDOO\ FRQVLVWHQW XQLWV>DIF;@ XQLWVIDMA[A@ 6LQFH XQLWV>X@ XQLWV>X@ XQLWV>efe DA;N@ f§ XQLWV>;A DMN;r@ %XW XQLWV>A DLN;N@ XQLWV>DIF;IF@ VR WKDW XQLWV>D[@ XQLWV>DMN;N@ f§ 7KHUHIRUH XQLWV>DIF@ XQLWV>DA@ DQG DOO WHUPV LQ FROXPQ N RI $ KDYH LGHQWLFDO XQLWV 7KLV SURYHV 6 JLYHQ 6L

PAGE 46

fµ $VVXPH 6 WR SURYH 6 f§ *LYHQ WKDW XQLWV>DcW@ XQLWV>DIF@ /HW % f§ $7$ VR WKDW EKN f§ DIF@ XQLWV>DMIF@f XQLWVIAIF@ XQLWVIDM7MDA@ VR WKDW HDFK HOHPHQW EKN RI % $7$ LV SK\VLFDOO\ FRQVLVWHQW 7KLV SURYHV 6 JLYHQ 6 &RUROODU\ EHORZ IROORZV GLUHFWO\ IURP WKH DERYH WKHRUHP ZKHQ WKH (XFOLGHDQ QRUP RI [ LV SK\VLFDOO\ FRQVLVWHQW &RUROODU\ ,IX f§ $[ ZKHUH $ LV DQ P[Q PDWUL[ P ! Qf DQG WKH (XFOLGHDQ QRUP RI [ LV SK\VLFDOO\ FRQVLVWHQW WKHQ WKH WKUHH VWDWHPHQWV LQ 7KHRUHP DUH HTXLYDOHQW DQG DUH HTXLYDOHQW WR WKH VWDWHPHQW 6 $OO HOHPHQWV RI $ PXVW KDYH PXVW QRW KDYHf LGHQWLFDO XQLWV 3URRI 7R SURYH WKH FRUROODU\ LW LV RQO\ QHFHVVDU\ WR VKRZ WKDW ZLWK WKH DGGHG FRQGLWLRQ RI D SK\VLFDOO\ FRQVLVWHQW _[_ VWDWHPHQW 6 RI 7KHRUHP LPSOLHV 6 RI WKH FRUROODU\ DQG 6 LPSOLHV 6, RI WKH WKHRUHP 7KURXJKRXW WKLV FRUROODU\ OHW LM ^P` DQG NK ^UH` fµ $VVXPH [7[ DQG $7$ DUH SK\VLFDOO\ FRQVLVWHQW fµ 6LQFH X $[ X X@ XQLWV>DIFD@ XQLWV >DA[@ DQG XQLWV>DMW@ XQLWVIDA@ 7KLV PHDQV WKDW DOO HOHPHQWV LQ WKH WK URZ RI $ KDYH LGHQWLFDO XQLWV

PAGE 47

fµ 7KH GLDJRQDO HOHPHQWV RI % $-$ DUH ENN DIF@ XQLWV>DM@ 7KLV PHDQV WKDW DOO HOHPHQWV LQ WKH IFWK FROXPQ RI $ KDYH LGHQWLFDO XQLWV fµ 6LQFH DOO HOHPHQWV LQ DQ\ URZ RU DQ\ FROXPQ RI $ KDYH LGHQWLFDO XQLWV WKHQ DOO HOHPHQWV RI $ KDYH LGHQWLFDO XQLWV 7KLV SURYHV VWDWHPHQW 6 fµ )LQDOO\ , ZLOO VKRZ WKDW VWDWHPHQW 6 LPSOLHV 6, 6LQFH WKH HOHPHQWV RI [ KDYH LGHQWLFDO XQLWV DQG 6 WHOOV XV WKDW WKH HOHPHQWV RI $ KDYH LGHQWLFDO XQLWV WKHQ WKH HTXDWLRQ X f§ $[ IRUFHV WKH HOHPHQWV RI X WR KDYH LGHQWLFDO XQLWV 7KHUHIRUH X KDV D SK\VLFDOO\ FRQVLVWHQW (XFOLGHDQ QRUP 7KLV SURYHV VWDWHPHQW 6L $ WKHRUHP VLPLODU WR 7KHRUHP RIIHUHG ZLWKRXW SURRIf FDQ EH FRQVWUXFWHG ZLWK WKH IROORZLQJ FRQGLWLRQV UHODWLQJ WKH SK\VLFDO FRQVLVWHQF\ RI _[_ WKH XQLWV RI DOO HOHPHQWV LQ HDFK URZ RI $ DQG WKH SK\VLFDO FRQVLVWHQF\ RI $$7KHRUHP ,IX f§ $[ ZKHUH $ LV DQ P[Q PDWUL[ P Qf WKHQ WKH IRU WKH IROORZLQJ VWDWHPHQWV 6, WKURXJK 6 6, LPSOLHV 6 DQG 6 LPSOLHV 6 VR WKDW 6, LPSOLHV 6 7KH HTXDWLRQ _D_ f§ [4 [ f§ [7[ LV SK\VLFDOO\ FRQVLVWHQW LQFRQVLVWHQWf 7KH QRQ]HUR HOHPHQWV LQ D JLYHQ URZ RI $ KDYH LGHQWLFDO XQLWV QRW DOO LGHQWLFDO XQLWVf LH ,I D ccL A DQG DNM WKHQ XQLWVMDIH@ XQLWVIDA@ IRU N * ^ P` DQG LM e ^ Q` f 67KH PDWUL[ $$7 LV SK\VLFDOO\ FRQVLVWHQW LQFRQVLVWHQWf

PAGE 48

,Q RWKHU ZRUGV 7KHRUHP WHOOV XV WKDW WKH SK\VLFDO FRQVLVWHQF\ RI WKH (XFOLGHDQ QRUP RI [ LPSOLHV WKDW DOO HOHPHQWV LQ D JLYHQ URZ RI $ KDYH LGHQWLFDO XQLWV RU DUH HTXDO WR ]HURf DQG WKDW $$7 LV SK\VLFDOO\ FRQVLVWHQW &RUROODU\ IROORZV GLUHFWO\ IURP WKH DERYH WKHRUHP ZKHQ WKH (XFOLGHDQ QRUP RI X LV SK\VLFDOO\ FRQVLVWHQW DQG LV DOVR RIIHUHG ZLWKRXW SURRIf &RUROODU\ ,IX $[ ZKHUH $ LV DQP[Q PDWUL[ P Qf DQG WKH (XFOLGHDQ QRUP RI X LV SK\VLFDOO\ FRQVLVWHQW WKHQ WKH WKUHH VWDWHPHQWV LQ 7KHRUHP DUH HTXLYDOHQW DQG DUH HTXLYDOHQW WR WKH VWDWHPHQW 6 $OO HOHPHQWV RI $ PXVW KDYH PXVW QRW KDYHf LGHQWLFDO XQLWV 7KH LPSOLFDWLRQV RI WKHVH WZR WKHRUHPV DQG WZR FRUROODULHV DUH WKDW QRQFRP PHQVXUDWH V\VWHPV JHQHUDOO\ QHHG EH GHDOW ZLWK LQ D PRUH FRQVLGHUHG PDQQHU WKDQ FRPPHQVXUDWH V\VWHPV ZKLFK KDV RIWHQ QRW EHHQ WKH FDVH LQ URERWLFV 6LQFH WKH PDn WULFHV $7$ DQG $$7 DUH XVHG LQ WKH SVHXGRLQYHUVH VROXWLRQ [V f§ $AX IRU IXOO FROXPQ UDQN $ RU IXOO URZ UDQN $ UHVSHFWLYHO\ WKH DERYH WKHRUHPV FDQ EH XVHG WR GHWHUPLQH WKH JHQHUDO YDOLGLW\ RI WKHVH UHVXOWV 7KH YDOLGLW\ LV QRW DEVROXWHO\ GHWHUPLQHG E\ WKH SK\VLFDO FRQVLVWHQFLHV RI WKHVH PDWUL[ SURGXFWV DV ZDV HYLGHQFHG LQ WKH IDFW WKDW WKH 5553 KDV D SK\VLFDOO\ LQFRQVLVWHQW r-7 -f EXW D SK\VLFDOO\ FRQVLVWHQW 'HW>r -7 -f@ DQG r-Wf ,Q WKH URERWLFV LQYHUVH YHORFLW\ SUREOHP VROYLQJ 9 -T IRU T JLYHQ 9 WKURXJK XVH RI WKH SVHXGRLQYHUVH JLYHV SK\VLFDOO\ LQFRQVLVWHQW UHVXOWV GXH WR WKH QRQ(XFOLGHDQ QDWXUH RI WKH WZLVW DQG VRPHWLPHVf MRLQW VSDFHV 7KLV SK\VLFDO LQFRQVLVWHQF\ LV DSSDUHQW LQ WKH SK\VLFDO LQFRQVLVWHQF\ RI -7RU --7 ,QYDOLG XVH RI (LJHQV\VWHP DQG 69' RI --7 6LQFH WKH SVHXGRLQYHUVH IRU UHGXQGDQW PDQLSXODWRUV RI HTXDWLRQ f FRQWDLQV WKH PDWUL[ --7 PDQ\ UHVHDUFKHUV KDYH XVHG WKLV IDFWRU LQ VROYLQJ f IRU WKH MRLQW

PAGE 49

UDWHV RU WR FKDUDFWHUL]H D PDQLSXODWRU FRQILJXUDWLRQ > @ @ IRU H[DPSOH ZDV WKH ILUVW RI PDQ\ WR XVH f DV D PDQLSXODELOLW\ PHDVXUH IRU D PDQLSXODWRU LQ D JLYHQ FRQILJXUDWLRQ )XUWKHU @f GHILQHG D PDQLSXODELOLW\ HOOLSVRLG ZLWK SULQFLSDO D[HV LQ WKH GLUHFWLRQ RI WKH HLJHQYHFWRUV RI --7 (DFK HOOLSVRLG D[LV ZDV JLYHQ WKH OHQJWK RI \M O$: ZKHUH $Af LV DQ f¯HLJHQYDOXHf° RI --7 5HFDOO WKDW 7KHRUHP LQ 6HFWLRQ JLYHV WKH UHTXLUHPHQWV IRU PHDQLQJIXO HLJHQn YDOXHV DQG HLJHQYHFWRUV (YHQ WKRXJK --7 LV SK\VLFDOO\ FRQVLVWHQW IRU DQ DOO UHYROXWH MRLQW PDQLSXODWRU VHH WKH XQLWV PDWUL[ RI f WKLV PDWUL[ GRHV QRW KDYH DQ LQYDULn DQW HLJHQV\VWHP VLQFH f UHTXLUHV WKDW WKH XQLWV RI HDFK WHUP RQ WKH PDLQ GLDJRQDO RI WKH PDWUL[ PXVW EH LGHQWLFDO ZKHUH LQ IDFW WKH\ DUH >/ // 8 8 8@ 7KH PDWUL[ --7 IRU PRVW QRQFRPPHQVXUDWH PDQLSXODWRUV DOVR GRHV QRW KDYH PHDQLQJIXO HLJHQV\VWHPV VLQFH WKH PDWUL[ LV LWVHOI SK\VLFDOO\ LQFRQVLVWHQW $Q H[FHSn WLRQ WR WKH JHQHUDO SK\VLFDO LQFRQVLVWHQF\ RI --7 IRU QRQFRPPHQVXUDWH PDQLSXODWRUV RFFXUV ZLWK WKH 35 5HGXQGDQW 6SKHULFDO :ULVW 5RERW ZLWK '+ SDUDPHWHUV JLYHQ LQ 7DEOH $ ZKHQ H[SUHVVHG LQ D SDUWLFXODU VHW RI IUDPHV 7KH PDWUL[ -7 IRU WKH 35 PDQLSXODWRU LV JHQHUDOO\ SK\VLFDOO\ LQFRQVLVWHQW DV H[SHFWHG %XW LQ DQ\ IUDPH ZLWK RULJLQ ORFDWHG DW WKH FHQWHU RI WKH VSKHULFDO ZULVW WKH RULJLQ RI IUDPHV DQG f WKH PDWUL[ --U LV SK\VLFDOO\ FRQVLVWHQW DQG XQLWOHVV 7KH HLJHQYDOXHV DQG HLJHQYHFWRUV RI --7 DUH WKHUHIRUH ZHOO GHILQHG E\ WKH UXOHV JLYHQ LQ 7KHRUHP DQG f DQG DUH GLPHQVLRQOHVV 7KH HLJHQYDOXHV DUH >@ DQG DUH LQYDULDQW WR URWDWLRQ RI WKH IUDPH ZLWK WKLV RULJLQf 7KH LQYDULDQFH RI HLJHQYDOXHV WR URWDWLRQV FDQ EH GHGXFHG IURP WKH ZHOO NQRZQ WKHRUHP WKDW VLPLODULW\ WUDQVIRUPDWLRQV SUHVHUYH HLJHQYDOXHV LH LI $H f§ $H WKHQ 6$6AHn f§ $Hn IRU IXOO UDQN 6 7KH WZLVW FRRUGLQDWH WUDQVIRUPDWLRQ PDWUL[ * DFWV OLNH 6 LQ WKH VLPLODULW\ WUDQVIRUPDWLRQ GHULYHG EHORZ

PAGE 50

--7H $H *--7H f§ $ *H H f§ *7Hn *--7*7Hn f§ $ **7Hn f *7 *a[ IRU URWDWLRQV QR WUDQVODWLRQf *-7*nHn $ **nHn f -n f§ *-n^-nf7Hn $Hn r $ LQYDULDQW WR URWDWLRQV f 1RWLFH WKDW LI WUDQVODWLRQV DUH DOORZHG WKH FRQJUXHQFH WUDQVIRUPDWLRQV RI f UHn VXOWV 6LQFH **7 A WUDQVODWLRQV DQG FRQJUXHQFH WUDQVIRUPDWLRQVf GR QRW SUHVHUYH HLJHQYDOXHV (YHQ WKRXJK --7 IRU WKH 35 PDQLSXODWRU LQ IUDPHV ORFDWHG DW WKH LQWHUVHFn WLRQ RI WKH VSKHULFDO MRLQW D[HV DSSHDUV WR KDYH SK\VLFDOO\ PHDQLQJIXO HLJHQYDOXHV DQG HLJHQYHFWRUV WKH LQWHUSUHWDWLRQ RI WKLV PDQLSXODELOLW\ HOOLSVRLG LV SUREOHPDWLF VLQFH WKH HLJHQYHFWRUV DSSHDUV WR EH XQLWOHVV QRW WKH QHFHVVDU\ ZUHQFKHV WKDW VKRXOG EH H[SHFWHG IRU WKH ZUHQFK PDQLSXODELOLW\ HOOLSVRLG GLVFXVVHG LQ &KDSWHU f 0RUHRYHU DV ZDV VWDWHG LQ 7KHRUHP QRQFRPPHQVXUDWH V\VWHPV QHYHU KDYH D SK\VLFDOO\ FRQn VLVWHQW 69' 7KH PDWUL[ --7 IRU DQ DOO SULVPDWLFMRLQWHG PDQLSXODWRU ZLWK DW PRVW WKUHH GHJUHHVRIIUHHGRP DQG QR RULHQWDWLRQ FDSDELOLWLHVf DOVR KDV D PHDQLQJIXO HLJHQV\VWHP EXW WKHVH OLPLWHG PDQLSXODWRUV ZLOO QRW EH GLVFXVVHG 7KHUHIRUH VLQFH --7 GRHV QRW KDYH HLJHQYDOXHV RU HLJHQYHFWRUV H[FHSW IRU WKH VSHFLDO FDVHV PHQWLRQHG DERYHf WKH DERYH FRQILJXUDWLRQ FKDUDFWHUL]DWLRQ WKHRU\ LV

PAGE 51

LQYDOLG 6HYHUDO RI WKH FRPPRQO\ XVHG PDQLSXODELOLW\ HOOLSVRLGV DUH VKRZQ LQ >@ WR EH SK\VLFDOO\ LQFRQVLVWHQWf ,W ZLOO EH VKRZQ ODWHU LQ 6HFWLRQ WKDW WKH XVH RI PHWULFV RQ WKH DSSURSULDWH QRQFRPPHQVXUDWH WZLVW DQG MRLQW VSDFHV GLVFXVVHG LQ WKH QH[W FKDSWHUf GRHV QRW FKDQJH WKH IDFW WKDW WKH PDQLSXODELOLW\ HOOLSVRLG WKHRU\ YLRODWHV WKH HLJHQV\VWHP DQG 69' WKHRUHPV RI 6HFWLRQ

PAGE 52

&+$37(5 ,19(56( 9(/2&,7< .,1(0$7,&6 6HYHUDO DXWKRUV > @ KDYH GLVFXVVHG WKH LQDSSURSULDWHQHVV RI XVLQJ WKH SVHXGRLQYHUVH LQ VROYLQJ IRU WKH MRLQW UDWHV JLYHQ D GHVLUHG WZLVW YHFWRU VLQFH WKLV LQYHUVH XWLOL]HV WKH (XFOLGHDQ QRUPV RI ERWK WKH MRLQWUDWH YHFWRU DQG WKH WZLVW YHFWRU %XW WKH WZLVW LV QRW D (XFOLGHDQ VSDFH DQG QHLWKHU LV WKH MRLQWUDWH YHFWRU ZKHQ WKH PDQLSXODWRU LV FRPSRVHG RI ERWK UHYROXWH DQG SULVPDWLF MRLQWVf 7KLV SUREOHP KDV EHHQ DGGUHVVHG LQ WKHVH DERYH SDSHUV DQG H[WHQVLYHO\ LQ >@ E\ XVLQJ WKH ZHLJKWHGf JHQHUDOL]HGLQYHUVH DORQJ ZLWK PHWULFV RQ ERWK WKH WZLVW 0Yf DQG MRLQW UDWHV 0Tf )URP ff DQG ff WKH SVHXGRLQYHUVH DQG JHQHUDOL]HGLQYHUVH RI WKH PDQLSXODWRU -DFRELDQ >@ DUH DQG MW O &7)7-&7fa)7 f O &7&&7fa^)7)f)7 f s &r)r f -r 0a&7)709-0a&7fa)709 f 0a&U^&0a&Ufa[ )7 09)fa; )U 0Y f &r)r f UHVSHFWLYHO\ $ IXOOUDQN IDFWRUL]DWLRQ RI )& LV XVHG LQ WKH DERYH HTXDWLRQV ZKHUH ) * "[Uf KDV IXOO FROXPQ UDQN U & fü KDV IXOO URZ UDQN U DQG Q LV WKH QXPEHU RI MRLQWV LQ WKH PDQLSXODWRU

PAGE 53

7ZR VSHFLDO FDVHV RI WKH JHQHUDOL]HGLQYHUVH RI D -DFRELDQ DUH REWDLQHG ZKHQ LV HLWKHU IXOO URZ UDQN RU IXOO FROXPQ UDQN LH -r 0-7-0-7f IXOO URZ UDQN f -r -709-fa-709 IXOO FROXPQ UDQN f ZKHUH f LV IRXQG E\ OHWWLQJ ) f§ ,t DQG f LV IRXQG E\ OHWWLQJ & f§ ,Q LQ f $V VWDWHG HDUOLHU WKH PHWULFV PXVW EH SRVLWLYH GHILQLWH DQG IRU LQYDULDQFH WR FRRUn GLQDWH WUDQVIRUPDWLRQV DQG VFDOLQJ WKH PHWULFV PXVW WUDQVIRUP DFFRUGLQJ WR f LH 0Y *7Y0Y*Y IRU 9 *Y9 f 0T *7T0T*T IRU Tn *TT f ,I WKH GHVLUHG WZLVW LV LQ WKH UDQJH RI WKH -DFRELDQ WKHQ QR PHWULF RQ WKH WZLVWV LV QHFHVVDU\ VLQFH WKH UHVLGXDO 9 f§ -TV LV ]HUR LH &7&0a&7fa? >)7)fa)7 9 fü 5DQJH>-@ f 7KLV HTXDWLRQ LV IRXQG E\ VXEVWLWXWLQJ 0Y LQ f ,I WKH -DFRELDQ KDV IXOO FROXPQ UDQN WKHQ QR PHWULF RQ MRLQW UDWHV LV QHFHVVDU\ DQG f PD\ EH XVHG ,I WKH FRQGLWLRQV RI ERWK f DQG f DUH YDOLGf§LH 9 LV LQ WKH UDQJH RI DQG KDV IXOO FROXPQ UDQNf§WKHQ QHLWKHU PHWULF LV QHHGHG DQG WKH JHQHUDOL]HGLQYHUVH LV HTXDO WR WKH SVHXGRLQYHUVH 9 5DQJH>-@ DQG IXOO FROXPQ UDQN f %XW VLQFH DOO PDQLSXODWRUV LQFOXGLQJ UHGXQGDQW PDQLSXODWRUVf KDYH VLQJXODU FRQILJn XUDWLRQV >@ DQG DW VLQJXODU FRQILJXUDWLRQV WKHUH H[LVW 9f¬V QRW LQ WKH UDQJH RI HYHU\ PDQLSXODWRU KDV FRQILJXUDWLRQV LQ ZKLFK D WZLVW PHWULF LV QHHGHG

PAGE 54

)RU UHGXQGDQW PDQLSXODWRUV ZKHUH KDV IXOO URZ UDQN H[FHSW LQ VLQJXODU FRQn ILJXUDWLRQV WKH JHQHUDOL]HGLQYHUVH LV LQGHSHQGHQW RI WKH WZLVW PHWULF DQG f PD\ EH XVHG )XUWKHUPRUH LI DOO MRLQWV DUH UHYROXWH RU DOO DUH SULVPDWLFf WKH PHWULF RQ WKH MRLQW VSDFH LV QRW QHHGHG IRU SK\VLFDO FRQVLVWHQF\f§DQG WKH SVHXGRLQYHUVH FDQ EH XVHGf§EXW WKH PHWULF LV QHHGHG IRU LQYDULDQFH WR FRRUGLQDWH WUDQVIRUPDWLRQV DQG VFDOLQJ )RU QRQFRPPHQVXUDWH PDQLSXODWRUV ZLWK IXOO URZ UDQN WKH SVHXGRLQYHUVH ZLOO JHQHUDOO\ EH SK\VLFDOO\ LQFRQVLVWHQW DQG QRW LQYDULDQW WR FRRUGLQDWH WUDQVIRUPDWLRQV DQG VFDOLQJf VLQFH WKH PLQLPXP QRUP __ LV SK\VLFDOO\ LQFRQVLVWHQW 6HFWLRQV ZLOO GLVFXVV WKH VLWXDWLRQV LQ ZKLFK WKH SVHXGRLQYHUVH VROXWLRQ LV SK\VLFDOO\ FRQVLVWHQW LQYDULDQW WR VFDOLQJ DQG LQYDULDQW WR ULJLG ERG\ WUDQVIRUPDWLRQV 3K\VLFDO &RQVLVWHQF\ RI -W $OWKRXJK WKH SVHXGRLQYHUVH RI WKH PDQLSXODWRU -DFRELDQ PD\ EH SK\VLFDOO\ FRQn VLVWHQW LQ D JLYHQ IUDPH WKHUH PD\ EH RWKHU IUDPHV LQ ZKLFK -W LV QRW SK\VLFDOO\ FRQVLVWHQW 7KLV ZDV VXJJHVWHG E\ HTXDWLRQV f f f DOO RI ZKLFK KDYH WKH WHUPV -7RU --7 HPEHGGHG LQ WKHP DQG 6HFWLRQ ZKLFK GLVFXVVHG WKH SRVVLEOH SK\VLFDO LQFRQVLVWHQFLHV RI WKHVH PDWULFHVf 5RWDWLRQV DQG &RQVLVWHQF\ RI -W 7KHRUHP VKRZV WKDW LI WKH SVHXGRLQYHUVH LV SK\VLFDOO\ FRQVLVWHQW LQ D JLYHQ IUDPH WKHQ LW ZLOO UHPDLQ SK\VLFDOO\ FRQVLVWHQW XQGHU DQ\ ULJLG ERG\ URWDWLRQ 7KHRUHP ,I WKH SVHXGRLQYHUVH RI LQ IUDPH L L-c>f LV SK\VLFDOO\ FRQVLVWHQW WKHQ IRU HYHU\ ULJLG ERG\ URWDWLRQ IURP IUDPH L WR IUDPH M WKH SVHXGRLQYHUVH RI LQ IUDPH M $rf LV SK\VLFDOO\ FRQVLVWHQW

PAGE 55

3URRI /HW n9 DQG -9 EH WZLVWV VXFK WKDW IUDPH M LV D URWDWLRQ RI IUDPH L QR WUDQVODWLRQf M9 M*L r9 $VVXPH WKDW WKH SVHXGRLQYHUVH RI ?LV SK\VLFDOO\ FRQVLVWHQW 7KH SVHXGRLQYHUVH RI WKH -DFRELDQ LQ IUDPH L LV I &7&&7f^)7)fa)7 f ZKHUH )& LV D IXOOUDQN IDFWRUL]DWLRQ ) IXOO FROXPQ UDQN DQG & IXOO URZ UDQN 7KH SVHXGRLQYHUVH RI WKH -DFRELDQ LQ IUDPH M LV M*W?M\ >M*L)fFI f &7&&7fnL)7 -*L )fB)7 M*f &7&&7fO)7)f)7L*M f :n*M f ZKHUH f IROORZV IURP f VLQFH L*M f§ -f O*M IRU WKH FDVH XQGHU GLVFXVn VLRQ RI -" D URWDWLRQ ZLWK QR WUDQVODWLRQf ,W LV QRZ RQO\ QHFHVVDU\ WR SURYH WKDW *M LV SK\VLFDOO\ FRQVLVWHQW 3DUWLWLRQ WKH SVHXGRLQYHUVHV LQ IUDPHV L DQG M LQWR WZR Q [ PDWULFHV : DQG ; DQG < DQG = UHVSHFWLYHO\ fµ-W >: ;@ f >< =@ f§ >:5 ;5@ f ZKHUH 5 f§ r5M 6LQFH  RSHUDWHV RQ O9 >X7X7@7 WKHQ HDFK FRPSRQHQW LQ D URZ RI : RU D URZ RI ;f PXVW KDYH OLNH XQLWV RU KDYH ]HUR YDOXH 6LQFH 5 LV GLPHQVLRQOHVV WKH XQLWV RI WKH HOHPHQWV LQ D URZ RI < RU =f DUH LGHQWLFDO WR WKH XQLWV RI WKH HOHPHQWV

PAGE 56

7DEOH '+ SDUDPHWHUV IRU 35 YLUWXDO PDQLSXODWRU -RLQW 7\SH G D D 3 GL 5 G LQ D URZ RI : RU ;f DQG DUH WKHUHIRUH RI FRQVLVWHQW SK\VLFDO GLPHQVLRQ 7KHUHIRUH : LV SK\VLFDOO\ FRQVLVWHQW 'HFRXSOH IUDPHV DUH WKHUHIRUH DFWXDOO\ GHFRXSOH SRLQWV SRLQWV DW ZKLFK WKH SVHXGRn LQYHUVH RI WKH PDQLSXODWRU -DFRELDQ ZLWK UHVSHFW WR DQ\ IUDPH DW WKH GHFRXSOH SRLQWf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f DQG WKH SVHXGRLQYHUVH LQ WKLV IUDPH -W -U LV SK\VLFDOO\ FRQVLVWHQW ,Q DQ DUELWUDULO\ WUDQVODWHG IUDPH QR URWDWLRQf WKH -DFRELDQ LV Of§ *Lf¬f ZKHUH K >S[@ 4W >@ f

PAGE 57

= r= [W 3HJ P ; %ORFN +ROA ] AA 9LUWXDO 3, 0DQLSXODG !U =O G L ;O ; )LJXUH 3HJLQWKHKROH ZLWK 35 YLUWXDO PDQLSXODWRU DQG S >Sr S\ S]@7 7KH -DFRELDQ LQ WKLV DUELWUDULO\ WUDQVODWHG IUDPH LV O3\ a3[ f DQG WKH SVHXGRLQYHUVH LV : O 3X U 3UU f± f± f± fµ n f 3 3\ 3[3\ 3O3O 1RWH WKH SK\VLFDO LQFRQVLVWHQF\ LQ WKH GHQRPLQDWRU RI WKH WHUPV LQ 7KH SK\VLFDO LQFRQVLVWHQF\ RI WKLV YLUWXDO PDQLSXODWRU PRGHO RI WKH SHJLQWKHKROH SUREOHP LV DQ DOWHUQDWLYH GHPRQVWUDWLRQ IRU WKH QRQYDOLGLW\ RI WKH 0DVRQ5DLEHUW K\EULG FRQWURO WHFKQLTXHV VWDWHG LQ SXEOLVKHG UHVHDUFK > @ &RQVLVWHQF\ RI -W LQ $OO )UDPHV 7KH 6&$5$ PDQLSXODWRU 6HOHFWLYH &RPSOLDQW $UWLFXODWHG 5RERW IRU $VVHPEO\f >@ LQ )LJXUH ZLWK 'HQDYLW+DUWHQEHUJ SDUDPHWHUV LQ 7DEOH KDV D IUDPH

PAGE 58

7DEOH '+ SDUDPHWHUV IRU WKH 6&$5$ PDQLSXODWRU -RLQW 7\SH G D D 5 2O L 5 5 3  LOO ;; )LJXUH 6&$5$ PDQLSXODWRU -DFRELDQ RI DLV D & f 7UDQVODWLQJ WKH IUDPH RI H[SUHVVLRQ RI WKH PDQLSXODWRU E\ DQ DUELWUDU\ YHFWRU S UHVXOWV LQ D -DFRELDQ Of§ *Wnf ZKRVH SVHXGRLQYHUVH LV f 6LQFH WKLV SVHXGRLQYHUVH LV SK\VLFDOO\ FRQVLVWHQW WKH SVHXGRLQYHUVH LQ DQ\ WUDQVn ODWHG RU URWDWHG IUDPH VHH 7KHRUHP f ZLOO EH SK\VLFDOO\ FRQVLVWHQW IRU WKH 6&$5$ PDQLSXODWRU 99 DL6 6 D"DL & G3Y-UDL&nS\WD?S[6 DL6 f¯ DDV &B &3Vn663[ 6 m 6

PAGE 59

7KH SODQDU 555 PDQLSXODWRU ZLWK LWV WKUHH UHYROXWH MRLQWV LGHQWLFDO WR WKH ILUVW WKUHH MRLQWV RI WKH 6&$5$ DOVR KDV D SK\VLFDOO\ FRQVLVWHQW SVHXGRLQYHUVH LQ DQ\ IUDPH 7KHVH WZR PDQLSXODWRUV DUH RIWHQ XVHG DV H[DPSOH PDQLSXODWRUV WR GHPRQn VWUDWH QHZ DOJRULWKPV > @ 3HUKDSV WKLV LV QRW DSSURSULDWH JLYHQ WKHLU DIRUHPHQn WLRQHG VSHFLDO SURSHUWLHV ,QYDULDQFH RI -W WR 6FDOLQJ :KHQ WKH SVHXGRLQYHUVH RI WKH PDQLSXODWRU -DFRELDQ LV SK\VLFDOO\ LQFRQVLVWHQW WHUPV RI XQOLNH SK\VLFDO XQLWV DUH VXPPHG ,I WKH SDUDPHWHUV LQ WKLV PDQLSXODWRU ZHUH UHVFDOHG SHUKDSV IURP %ULWLVK WR 6, XQLWV WKH SK\VLFDOO\ LQFRQVLVWHQW WHUPV ZLOO FDXVH WKH UHVXOWLQJ SVHXGRLQYHUVH WR JLYH D GLIIHUHQW UHVXOW ,W KDV EHHQ DUJXHG WKDW WKH SUREOHP RI SK\VLFDO LQFRQVLVWHQFLHV FDQ EH f¯IDFWRUHG RXWf° E\ VFDOLQJ WKH SUREOHP 7KH IDOODF\ RI WKLV VWDWHPHQW ZLOO SUHVHQWO\ EH VKRZQ $ FKDQJH RI XQLWV VFDOLQJ PDWUL[ LV D GLDJRQDO PDWUL[ WKDW FRQYHUWV D SK\VLFDOO\ FRQVLVWHQW YHFWRU ZLWK SK\VLFDO XQLWV LQWR D YHFWRU ZLWK VLPLODU SK\VLFDO XQLWV RU QR XQLWV )RU H[DPSOH LI 9 f§ >X7Z7@7 XQLWVA@ XQLWVIX\@ XQLWV>Xr@ PV DQG XQLWV>FMU@ XQLWV>Z\@ XQLWV>RY@ UDGV WKHQ 6Y LV D FKDQJH RI XQLWV VFDOLQJ PDWUL[ LI D , f¬ > DM] ZKHUH IRU H[DPSOH DY FPPfVPLQf DQG DZ f§ VPLQf 7KH VFDOHG WZLVW 9 f§ >DYY7 DZX!7@7 KDV VLPLODU XQLWV WR 9 LH HDFK HOHPHQW RI Y DQG DYY KDV XQLWV RI /7 DQG HDFK HOHPHQW RI X! DQG RFZX KDV XQLWV RI 7 $ PDQLSXODWRU MRLQWUDWH YHFWRU T VKRXOG KDYH WKH FKDQJH RI XQLWV MRLQWUDWH VFDOLQJ PDWUL[ f 6J 'LDJMHL H HQ@ ZKHUH H^ DZ LI MRLQW L LV UHYROXWH D LI MRLQW L LV SULVPDWLF f ZKHUH WKH VFDODU SK\VLFDO XQLW WUDQVIRUPDWLRQV DY DQG DZ DUH WKH LGHQWLFDO WR WKRVH XVHG LQ f

PAGE 60

$Q\ VFDOLQJ RI D SK\VLFDO XQLW IRU D VLQJOH HOHPHQW RI D QRQFRPPHQVXUDWH YHFWRU PXVW EH LGHQWLFDOO\ VFDOHG LQ DOO RWKHU HOHPHQWV RI WKH QRQFRPPHQVXUDWH YHFWRU )RU LQVWDQFH LQ WKH H[DPSOH GLVFXVVHG DERYH WKH WLPH XQLWV ZHUH QHFHVVDULO\ FRQYHUWHG IURP VHFRQGV WR PLQXWHV LQ ERWK DY DQG DZ 7KH FKDQJH RI XQLWV VFDOLQJ PDWUL[ 6Y LV DOVR QRUPDOL]LQJ LI RQO\ WKH XQLWVf§QRW WKH QXPHULFDO YDOXHf§RI WKH QRQFRPPHQVXUDWH YHFWRU LV FKDQJHG LH IRU WKH WZLVW H[DPSOH DERYH DY f§ VPf DQG DZ V $ QRUPDOL]LQJ XQLWV VFDOLQJ PDWUL[ LV QXPHULFDOO\ HTXDO WR WKH LGHQWLW\ PDWUL[ HJ 6Y , 6FDOLQJ ZLOO QRZ EH DSSOLHG WR WKH LQYHUVH YHORFLW\ SUREOHP 7KH WZLVW YHFWRUV DUH VFDOHG ZLWK WKH FKDQJH RI XQLWV GLDJRQDO VFDOLQJ PDWUL[ 6Y DQG WKH MRLQWUDWH YHFWRUV DUH VFDOHG ZLWK WKH FKDQJH RI XQLWV GLDJRQDO VFDOLQJ PDWUL[ 6T >@ VXFK WKDW 6Y9 f TV 6TT f 7KH VFDOHG YHUVLRQ RI WKH PDSSLQJ RI MRLQW UDWHV WR WZLVW RI f LV 9V 6Y9 6Y-66TT -VJV f ZKHUH WKH VFDOHG -DFRELDQ LV -A6Y-6 f 7R REWDLQ WKH SVHXGRLQYHUVH RI -V ILUVW JHW WKH IXOO UDQN IDFWRUL]DWLRQ )& VR WKDW -V f§ )6&6 69)f&6rf (TXDWLRQ f LV WKHQ XVHG UHSODFLQJ DOO )f¬V ZLWK M)Vf¬V DQG DOO &f¬V ZLWK &Vf¬V VR WKDW -VfW O 6O&U^&6&7
PAGE 61

DQG WKH XQVHDOHG MRLQWUDWH VROXWLRQ LV O 6? , 6n-fn6f±9 O $nW&U&6n&nUf)U6)f)76nc9 f f &RPSDUH f ZLWK WKH JHQHUDOL]HGLQYHUVH VROXWLRQ RI TV f§ -9 REWDLQHG XVLQJ f LH TV 0&7^&0&7f^)70Y)faO)70Y9 f ,W LV HYLGHQW WKDW WKH WZR VFDOLQJ PDWULFHV DFW DV PHWULFV ZKHUH 6L DQG 6b LQ f FRUUHVSRQG WR WKH PHWULFV 0Y DQG 0T LQ f UHVSHFWLYHO\ 6LQFH 6, DQG 6b DUH ERWK SRVLWLYH GHILQLWH DQG V\PPHWULF WKH\ QHHG RQO\ PHHW WKH DGGLWLRQDO UHTXLUHPHQWV WKDW 9 2 6O9 DQG T k 6AT DUH SK\VLFDOO\ FRQVLVWHQW LQ RUGHU IRU WKH V\PERO LQ f WR EHFRPH DQ HTXDO VLJQ :KHQ WKH GHVLUHG WZLVW 9 LV LQ WKH UDQJH RI WKH VROXWLRQ TV -: LV DOZD\V SK\VLFDOO\ FRQVLVWHQW ,I -W LV SK\VLFDOO\ LQFRQVLVWHQW WKH LQFRQVLVWHQFLHV DUH FDQFHOHG RXW ZKHQ -W LV PXOWLSOLHG E\ 9 7KH 5553 PDQLSXODWRU KDV D SK\VLFDOO\ FRQVLVWHQW SVHXGRLQYHUVH LQ IUDPH DQG SK\VLFDOO\ LQFRQVLVWHQW SVHXGRLQYHUVH LQ IUDPH && 6& 6ORL66f &Or2 rf m f¯r V 6 r f§&O & f§ m& f§r rr f¯&6 m A && 6?& DLVMVJ f§ 2W&W 6 m m m 6 r f f§DLJ Hf 3 e Q S 8 f ZKHUH c f§ DI DLD& LV SK\VLFDOO\ LQFRQVLVWHQW :KHQ WKH GHVLUHG WZLVW LV LQ WKH UDQJH RI WKH VROXWLRQ LQ HDFK RI WKH IUDPHV DUH LGHQWLFDO DQG SK\VLFDOO\ FRQn VLVWHQW )RU LQVWDQFH WKH WZLVW IRU DQ DUELWUDU\ MRLQWUDWH YHFWRU T f§ >ML M IH "@U

PAGE 62

LQ HDFK RI IUDPHV DQG DUH &O f OR Wf & f§2O f f§ 2& f§ & 9 f§LDL DFf O f M Y f§ fµ6 a&O f FL O f ZKHUH 9 f§ *r9 6XEVWLWXWLQJ r9 DQG f LQWR TV -A9 DQG VXEVWLWXWLQJ 9 DQG f LQWR TV -A9 ERWK WKH VROXWLRQV DUH TV T > T TA7 ,Q IUDPH WKH SK\VLFDOO\ LQFRQVLVWHQW WHUPV LQ -f«> FDQFHO ZKHQ PXOWLSOLHG E\ DQ\ 9 * 5DQJH >-@ )RU DQ\ WZLVW QRW LQ WKH UDQJH RI WKH VROXWLRQ LV IUDPH GHSHQGHQW ,Q IUDPH WKH VROXWLRQ LV LQGHSHQGHQW RI VFDOLQJ LQ IUDPH WKH VROXWLRQ LV QRW LQGHSHQGHQW RI VFDOLQJ )RU H[DPSOH OHW WKH FRQILJXUDWLRQ EH GHILQHG E\ L 2OUDG UDG f§ UDG  P f DQG OHW mL P D OP f 1RZ FRQVLGHU WKH HTXLYDOHQW GHVLUHG WZLVWV n p n p p I BP r V p e_G 9G *r9 p BeDG pA UDD V pA QRW LQ WKH UDQJH RI 7KH VROXWLRQ IRU r9G LV T 9 r9M A A f§ f§ 6 6 6 6 f f 7KH UHVXOWLQJ DFWXDO WZLVW REWDLQHG E\ VXEVWLWXWLQJ WKLV MRLQWUDWH YHFWRU LQWR rKVD r-TVD LV r9 f§ \ VD f§ f§ f§ f§ V V V V V V f

PAGE 63

ZKLFK LQ IUDPH FRRUGLQDWHV LV 9VD f§ *Rr9VD O9 \VD f§ f± f±P f± f±f±f±UDG UDG UDG f§ f§ f§ f§ f§ f§ V V V V V V f 7KH VROXWLRQV IRXQG LQ IUDPH ZLOO QRZ EH FRPSDUHG ZLWK WKRVH IRXQG LQ IUDPH 7KH VROXWLRQ IRU 9 LV N \9GO TVE > @7 XVLQJ XQLWV RI P DQG V f LPLPf P VLPV V V PV LQ IUDPH f 7KH MRLQWUDWH VROXWLRQ TVE LQ f LV SK\VLFDOO\ LQFRQVLVWHQW 7KH UHVXOWLQJ DFWXDO WZLVW REWDLQHG E\ XVLQJ TVE LQ 9VE -TVE LV 9VE > @7 f ZKLFK WUDQVIRUPHG WR IUDPH LV r9VE r*9VE > @7 f 7KHVH WZLVWV DUH GLIIHUHQW IURP WKH GHVLUHG WZLVWV LQ f ,I WKH WZLVWV DUH VFDOHG DFFRUGLQJ WR f DQG WKH MRLQW UDWHV DUH VFDOHG DFFRUGLQJ WR f ZKHUH DY FPP DQG DZ WKHQ WKH QXPHULFDO VROXWLRQ LQ IUDPH HTXDOV TVF > @7 XVLQJ XQLWV RI FP DQG V f 7KH UHVXOWLQJ DFWXDO WZLVW REWDLQHG E\ XVLQJ TVF LQ 96& f§ -TVF LV 96& > @7 f ZKLFK WUDQVIRUPHG WR IUDPH LV r\VF r*9VF > @7 f

PAGE 64

1RWLFH WKDW WKH UHVXOWV RI f DQG f GLIIHU LH TVE A TVF 7KH ILUVW MRLQWUDWH FRPSRQHQWV GLIIHU E\ PRUH WKDQ b WKH VHFRQG DQG WKLUG MRLQW UDWHV DUH QXPHULFDOO\ LGHQWLFDO DQG WKH IRXUWK MRLQWUDWH FRPSRQHQW FRUUHVSRQGLQJ WR WKH SULVPDWLF MRLQWf LQ f LV DV H[SHFWHGf WLPHV WKH IRXUWK FRPSRQHQW LQ f 6LQFH RQO\ WHUPV LQ WKH ILUVW URZ RI -W LQ f DUH SK\VLFDOO\ LQFRQVLVWHQW WKHQ RQO\ WKH ILUVW FRPSRQHQW RI WKH MRLQWUDWH VROXWLRQ LV DGYHUVHO\ DIIHFWHG E\ VFDOLQJ WKH RWKHU FRPSRQHQWV DUH VFDOHG DSSURSULDWHO\ 7KH VROXWLRQV TVE DQG TVF DUH DV f¯QHDUf° DV WKH\ DUH RQO\ EHFDXVH WKH VSHFLILHG WZLVW YHFWRU LV f¯QHDUO\f° LQ WKH UDQJH RI LH WKH GHVLUHG WZLVW RI f LV f¯DOPRVW WKH VDPHf° ZKDWHYHU WKDW PHDQVf DV r9 9 P LQ LQ UDG B B B UDG UDG f§ f§ f§f§ f§ f§f§ V V V V V V f§f§ f§ f§ f§ V V V V V V f f ZKLFK DUH LQ WKH UDQJH RI 7KH UHVXOWLQJ DFWXDO WZLVWV 9VE DQG 9VF DUH QRW HTXDO DUH ERWK GLIIHUHQW IURP WKH GHVLUHG WZLVW 9 DQG DUH ERWK DOVR GLIIHUHQW IRUP WKH SK\VLFDOO\ FRQVLVWHQW UHVXOW IRXQG LQ 9VD )RU WKH VSHFLDO FDVHV RI XQLWOHVV -W LV SK\VLFDOO\ FRQVLVWHQW 7KHRUHP ,I LQ VRPH IUDPH LV XQLWOHVV WKHQ -W LQ WKLV IUDPH LV SK\VLFDOO\ FRQVLVn WHQW 3URRI 6LQFH WKH SVHXGRLQYHUVH GRHV QRW LQWURGXFH DQ\ XQLWV QRW DOUHDG\ LQ WKHQ -W FDQ KDYH RQO\ WKH XQLWV RI DQG WKH LQYHUVH RI WKH XQLWV RI RU DQ\ FRPELQDWLRQ RI WKH WZR 7KHUHIRUH LI LV XQLWOHVV WKHQ -W LV XQLWOHVV

PAGE 65

)RU H[DPSOH WKH -DFRELDQV H[SUHVVHG LQ IUDPHV DQG IRU WKH 6$5 353f PDQLSXODWRU n V n n F f DUH XQLWOHVV DQG WKH SVHXGRLQYHUVHV f§ -7 DQG -W -7 DUH SK\VLFDOO\ FRQVLVWHQW 2I FRXUVH WKH LQYHUVH RI 7KHRUHP f§LH LI LQ VRPH IUDPH LV QRW XQLWOHVV WKHQ -W LQ WKLV IUDPH LV QRW SK\VLFDOO\ FRQVLVWHQWf§LV QRW WUXH )RU H[DPSOH WKH 5553 PDQLSXODWRU KDV D SK\VLFDOO\ FRQVLVWHQW LQYHUVH LQ IUDPH \HW WKH IUDPH -DFRELDQ LV QRW XQLWOHVV $VVXPH WKDW WKH T7 -A9 LV VFDOHDEOH 7KHQ UHZULWLQJ f WKH VFDOHG LQYHUVH YHORFLW\ HTXDWLRQ L VAM\Vf Y f LW LV DSSDUHQW WKDW nf¯ -V\6Yf DFWV OLNH m, LQ WKH XQVHDOHG HTXDWLRQ TU -A9 :KHQ -VfO LV SK\VLFDOO\ FRQVLVWHQW WKH FDQ EH UHSODFHG E\ DQ VLQFH VFDOHDELOLW\ PHDQV WKDW T7 TnV ,Q WKLV FDVH 6T?-V\6Y ZKHQ -fW SK\VLFDOO\ FRQVLVWHQW f 7KHRUHP EHORZ PXVW EH XVHG WR YHULI\ WKLV HTXDWLRQ 7KHRUHP ,I ' DQG ( DUH SK\VLFDOO\ FRQVLVWHQW LQYHUWDEOH GLDJRQDO PDWULFHV WKHQ $ LV SK\VLFDOO\ FRQVLVWHQW LI DQG RQO\ LI '$( LV SK\VLFDOO\ FRQVLVWHQW 3URRI /HW % '$( ZKHUH GD DQG HMM DUH WKH GLDJRQDO HOHPHQWV RI WKH GLDJRQDO PDn WULFHV ' DQG ( UHVSHFWLYHO\ 7KHQ EcM GDDLMHMM 6LQFH WKHUH LV QR DGGLWLRQ LQ WKH HTXDWLRQ IRU EM DQG QR GD RU HMM LV ]HUR WKHQ E^M LV SK\VLFDOO\ FRQVLVWHQW LI DQG RQO\

PAGE 66

LI DLM LV SK\VLFDOO\ FRQVLVWHQW 7KHUHIRUH % '$( LV SK\VLFDOO\ FRQVLVWHQW LI $ LV SK\VLFDOO\ FRQVLVWHQW 7KH RWKHU GLUHFWLRQ RI WKH SURRI IROORZV GLUHFWO\ IURP WKH IDFW WKDW ' DQG (B DUH GLDJRQDO PDWULFHV DQG $ 'aO%(a[ KDV WKH VDPH IRUP DV % '$( 7KHRUHP DQG f WHOO XV WKDW LI -VfW LV SK\VLFDOO\ FRQVLVWHQW WKHQ ' LV SK\VLFDOO\ FRQVLVWHQW &RQYHUVHO\ VROYH f IRU -VfW 6T-A6aO -Vfr ZKHQ -W SK\VLFDOO\ FRQVLVWHQW f WR VKRZ WKDW LI -W LV SK\VLFDOO\ FRQVLVWHQW VR LV DQ\ VFDOLQJ -VfW RI -K 7KHVH UHVXOWV OHDG XV GLUHFWO\ WR WKH IDFW WKDW )DFW ,I LV SK\VLFDOO\ FRQVLVWHQW WKH VROXWLRQ TV -W\ LV LQGHSHQGHQW RI VFDOLQJ IRU DOO 9 ,I -W LV QRW SK\VLFDOO\ FRQVLVWHQW WKHQ f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

PAGE 67

7DEOH '+ SDUDPHWHUV IRU WKH 353 6PDOO $VVHPEO\ 5RERW 6$5f -RLQW 7\SH G D H D 3 G? 5 WW 3 G] 8VLQJ 7KHRUHP EHORZ DOO PHWULFV ZKLFK UHVXOW LQ LGHQWLFDO MRLQW YHORFLWLHV FDQ EH IRXQG 7KHRUHP VWHPV IURP 7KHRUHP LQ >@ DQG WKH IDFWV WKDW --t )) DQG -r&r& 7KH SURRI RI WKHVH 7KHRUHPV LV JLYHQ LQ >@ 7KHRUHP $OO VWDWHPHQWV LQ WKH OHIW FROXPQ DUH HTXLYDOHQW VWDWHPHQWV DQG DOO VWDWHn PHQWV LQ WKH ULJKW FROXPQ DUH HTXLYDOHQW VWDWHPHQWV >@ MMr MMrfW -r-rM\ f 09--r --r09 0T-r-r-0T f ,, r b ,, A f 0Y--I --n09 0TM8 -I-0T f ,I ZH DVVXPH f WKDW WKH SVHXGRLQYHUVH LV HTXDO WR WKH JHQHUDOL]HG LQYHUVH WKHQ WKH OHIW HTXDWLRQ RI f PD\ EH XVHG WR VROYH IRU DOO HTXLYDOHQW WZLVW PHWULFV 09f )RU H[DPSOH WKH 353 6PDOO $VVHPEO\ 5RERW 6$5f VKRZQ LQ )LJXUH ZLWK 'HQDYLW+DUWHQEHUJ SDUDPHWHUV JLYHQ LQ 7DEOH KDV D SVHXGRLQ YHUVH LQ IUDPH RI -I -7 f

PAGE 68

=R ]L\ -G G \L  GU \R )LJXUH 6PDOO $VVHPEO\ 5RERW 6$5f $Q\ PHWULF RI WKH IRUP LQ f WKDW LV DOVR SRVLWLYH GHILQLWH ZLOO FDXVH WKH JHQHUDOL]HGLQYHUVH WR HTXDO WKH SVHXGRLQYHUVH LH 0Y PQ P PLH P P P P P P PL P P P P P V PLH P P f 7KH LPSRUWDQW UHVXOW RI WKLV VHFWLRQ LV WKDW LI D SVHXGRLQYHUVH LV SK\VLFDOO\ FRQn VLVWHQW WKHQ WKHUH DUH D VHW RI PHWULFV ZKLFK JLYH LGHQWLFDO UHVXOWV ZKHQ XVLQJ WKH JHQHUDOL]HGLQYHUVH LH IRU HYHU\ GHFRXSOH SRLQW RI D PDQLSXODWRU D FODVV RI PHWULFV H[LVW IRU ZKLFK WKH SVHXGRLQYHUVH DQG JHQHUDOL]HGLQYHUVH RI WKH -DFRELDQ DUH HTXDO

PAGE 69

&+$37(5 0$1,38/$725 0$1,38/$%,/,7< $V ZDV GLVFXVVHG LQ 6HFWLRQ WKH PDWULFHV --7 DQG -7GR QRW KDYH SK\VLFDOO\ FRQVLVWHQW HLJHQYDOXHV HLJHQYHFWRUV RU D 69' $ IHZ DXWKRUV > @ KDYH XVHG RWKHU -DFRELDQ IXQFWLRQVf§VRPH -DFRELDQ IXQFWLRQV LQFRUSRUDWLQJ PHWULFVf§LQ PDQLSXODELOLW\ GHILQLWLRQV ,Q WKLV VHFWLRQ VHYHUDO RI WKHVH PDQLSXODELOLW\ HOOLSVRLGV ZLOO EH LQWURGXFHG DQG WKHLU HLJHQV\VWHPV ZLOO EH H[SORUHG 7KHUH DUH WKUHH EDVLF W\SHV RI PDQLSXODELOLW\ HOOLSVRLGV (DFK RI WKHVH DULVH IURP VHWWLQJ WKH VTXDUH RI WKH D (XFOLGHDQ RU QRQ(XFOLGHDQ QRUP WR OHVV WKHQ RU HTXDO WR 7KH PDQLSXODELOLW\ HOOLSVRLG GLVFXVVHG SUHYLRXVO\ LV FDOOHG WKH ZUHQFK PDQLSXODELOLW\ HOOLSVRLG RU IRUFH PDQLSXODELOLW\ HOOLSVRLGf VLQFH WKLV HOOLSVRLG LV GHILQHG DV _U_ :U--7f: f 7KH f¯HLJHQYDOXHVf° $ DQG f¯HLJHQYHFWRUVf° H RI --7 DUH XVHG WR FUHDWH WKH HOOLSVRLG ZLWK HDFK SULQFLSDO D[LV LQ WKH GLUHFWLRQ RI DQ H DQG D[LV OHQJWK HTXDO WR \nM $ VLQJXODU YDOXH GHFRPSRVLWLRQ RI FDQ EH XVHG WR GHGXFH WKHVH VDPH TXDQWLWLHV VHH 6HFWLRQ f $V GLVFXVVHG SUHYLRXVO\ LQ 6HFWLRQ WKLV DQDO\VLV LV IDXOW\ GXH WR WKH IDLOXUH RI --7 WR KDYH D SK\VLFDOO\ PHDQLQJIXO HLJHQV\VWHP VHH 7KHRUHP f ,W ZDV SURSRVHG LQ >@ WKDW LQFRUSRUDWLQJ D PHWULF WR UHSODFH WKH (XFOLGHDQ QRUP RI U PLJKW FRUUHFW WKLV SUREOHP 7KH UHVXOWLQJ HTXDWLRQ LI D PHWULF LV XVHG WR GHWHUPLQH WKH 0UQRUP RI WZ LV 0PW :7-07Uf: f ,W ZLOO QRZ EH VKRZQ WKDW WKH HOOLSVRLG GHILQHG E\ WKH HLJHQYDOXHV DQG HLJHQYHFWRUV RI -0WGRHV QRW PHHW WKH UHTXLUHPHQWV IRU D SK\VLFDOO\ FRQVLVWHQW HLJHQV\VWHP

PAGE 70

7KH SK\VLFDO XQLWV RI 07 IRXQG E\ IRUFLQJ U k 0WW WR EH SK\VLFDOO\ FRQVLVWHQW DUH XQLWV >07@ 7 & ZKHUH & >F@ DQG FLM )/ 8 MRLQWV L DQG M UHYROXWH / HLWKHU MRLQW L RU M UHYROXWH WKH RWKHU SULVPDWLF / MRLQWV L DQG M SULVPDWLF 7KH XQLWV YDULDEOH 7 LV HTXDO WR WKH GHVLUHG XQLWV RI _U_A :LWK WKH DERYH XQLWV IRU 07 WKH UHVXOWLQJ XQLWV PDWUL[ IRU -0U-7 LV f f XQLWV >07 -7@ U )/ P 0V >9?Y f 7KH XQLWV PDWUL[ IRU -07-7 LV D VFDODU PXOWLSOH RI WKH XQLWV PDWUL[ RI --7 IRU PDQLSXODWRUV ZLWK DOO UHYROXWH MRLQWVf§JLYHQ LQ f 7KHUHIRUH E\ 7KHRUHP WKH ZUHQFK PDQLSXODELOLW\ HOOLSVRLG ZLWK PHWULF 07 LV DOVR EDVHG RQ D SK\VLFDOO\ LQFRQVLVWHQW HLJHQV\VWHP ,W VKRXOG EH SRLQWHG RXW WKDW QR PHWULF LV QHHGHG IRU D SK\VLFDOO\ FRQVLVWHQW _U_ LI DOO WKH MRLQWV DUH RI LGHQWLFDO W\SH WKHUHIRUH WKH DERYH UHVXOW FRXOG KDYH EHHQ LPPHGLDWHO\ GHGXFHG $ XQLWV DQDO\VLV RI WKH 0T PHWULF XVHG WR PDNH ?T??MT SK\VLFDOO\ FRQVLVWHQW OHDGV WR WKH XQLWV PDWUL[ XQLWV>0J@ &LM ML f§& ZKHUH & >F\@ DQG / MRLQWV L DQG M UHYROXWH / MRLQW L RU M UHYROXWH RWKHU SULVPDWLF 8 MRLQWV L DQG M SULVPDWLF f f ZKHUH WKH XQLWV YDULDEOH LV HTXDO WR WKH GHVLUHG XQLWV RI ?T??MT 7KH XQLWV PDWUL[ 0f° LV WKHUHIRUH XQLWV>0" @ ZKHUH & >F-@ DQG 8 MRLQWV L DQG M UHYROXWH / MRLQW L RU M UHYROXWH RWKHU SULVPDWLF / MRLQWV L DQG M SULVPDWLF f f

PAGE 71

7KLV XQLWV PDWUL[ GLIIHUV E\ D VFDODU FRQVWDQW IURP WKH XQLWV PDWUL[ RI 0U 7KHUHIRUH PHWULFV GHULYHG IRU MRLQW UDWHV FDQ EH LQYHUWHG DQG WKHQ XVHG IRU MRLQW WRUTXHV LH 07 0 7KH WZLVW PDQLSXODELOLW\ HOOLSVRLG ZDV GHILQHG RULJLQDOO\ >@ DV _"V_ -WfnMWf 9 f 7KH WZLVW PDQLSXODELOLW\ HOOLSVRLG FDQ DOWHUQDWLYHO\ EH GHILQHG ZLWK D JHQHUDOL]HG LQYHUVH DQGRU ZLWK D MRLQWUDWH PHWULF DV YW Mr\Mrf Y L f WµI LL MV ,$ R f ,"ON ,, V r ,$ K f 6LQFH QRQFRPPHQVXUDWH PDQLSXODWRUV JHQHUDOO\ KDYH SK\VLFDOO\ LQFRQVLVWHQW -W DQG WKXV FDQ QRW KDYH SK\VLFDOO\ FRQVLVWHQW HLJHQV\VWHPV RQO\ DOO UHYROXWHMRLQWHG PDQLSn XODWRUV ZLOO EH DQDO\]HG IRU WKH GHILQLWLRQV LQ f DQG f 7KH XQLWV DQDO\VLV EHORZ IRU UHYROXWH MRLQWV XVLQJ -W DQG f LV HTXLYDOHQW WR WKH XQLWV DQDO\VLV RI DQ\ PDQLSXODWRU XVLQJ -t DQG f (DFK RI WKH Q URZV RI KDV WKH XQLWV XQLWV>mr@f >f§ \ \ 8 8 8@ IRU DOO UHYROXWH MRLQWV f -M -M /L 1RWLFH WKDW WKH URZV RI WKLV -W DUH UD\ FRRUGLQDWH VFUHZV DV RSSRVHG WR WKH D[LV FRRUGLQDWH VFUHZV RI WKH FROXPQV RI -f 7KHUHIRUH WKH XQLWV RI -If7-W IRU DQ DOO UHYROXWHMRLQWHG PDQLSXODWRU DUH XQLWV>MWf7 -W@ _A@f¬ ¯Lf¬ IRU DOO UHYROXWH MRLQWV f / > ONM $QG VLQFH IRU DQ DOO UHYROXWH MRLQW PDQLSXODWRU WKH PHWULF 0T LV HQWLUHO\ FRPSRVHG RI LGHQWLFDO XQLWV WKH XQLWV RI -Af70T-A DUH SURSRUWLRQDO WR WKH XQLWV RI MWf7-K %\ 7KHRUHP WKH PDWUL[ -If7-W GRHV QRW KDYH D SK\VLFDOO\ PHDQLQJIXO HLJHQV\VWHP

PAGE 72

5HSODFLQJ WKH SVHXGRLQYHUVH RI ZLWK WKH ZHLJKWHG JHQHUDOL]HG LQYHUVH RI GRHV QRW FKDQJH WKH IDFW WKDW WKH PDWUL[ ^-rf7 0T-r GRHV QRW KDYH D SK\VLFDOO\ PHDQLQJIXO HLJHQV\VWHP 7KH SK\VLFDO XQLWV RI -r DUH D VFDODU PXOWLSOH RI WKH XQLWV RI 3 ZKHQ -W LV SK\VLFDOO\ FRQVLVWHQWf %XW WKH PDWUL[ -f7LV SK\VLFDOO\ FRQVLVWHQW HYHQ IRU QRQFRPPHQVXUDWH PDQLSXODWRUV 7KH G\QDPLFPDQLSXODELOLW\ HOOLSVRLG > @ LV GHULYHG IURP WKH PDQLSXODWRU G\QDPLFV HTXDWLRQ 7 0TfT KTTfJTf f ZKHUH U UHSUHVHQWV WKH JHQHUDOL]HGIRUFH YHFWRU DW WKH MRLQWV 0Tf LV D SRVLWLYH GHILQLWH PDVV PDWUL[ T LV WKH MRLQW DFFHOHUDWLRQ KTTf UHSUHVHQWV WKH &RULROLV DQG FHQWULIXJDO IRUFHV DQG JTf UHSUHVHQWV WKH JUDYLWDWLRQDO IRUFHV 6ROYLQJ IRU T UHVXOWV LQ T 0a[ >U KT Tf JTf@ f ZKHUH WKH GHSHQGHQF\ LQ 0Tf RQ T KDV EHHQ GURSSHG IRU VLPSOLFLW\ RI QRWDWLRQ 7KH GHYHORSPHQW KHUH IROORZV IURP >@ DQG LV JLYHQ KHUH WR GHPRQVWUDWH WKH PHWKRG ZLWK ZKLFK PDQLSXODELOLW\ PDWULFHV KDYH EHHQ GHULYHG 'LIIHUHQWLDWLQJ 9 f§ -T ZLWK UHVSHFW WR WLPH UHVXOWV LQ 9 f§ -T -T f $JDLQ WR VLPSOLI\ WKH QRWDWLRQ GHILQH $ DV WKH IUDPH DFFHOHUDWLRQ $ -T f§ 9 f§ MT f DQG I DV I 7 KTTfJTf f 6XEVWLWXWLQJ f LQWR f DQG WKH UHVXOW LQWR f \LHOGV $ f§ -0a[W f

PAGE 73

6ROYLQJ IRU I ZH JHW 76 ^-0a\$ f RU IV f 7KH 07QRUP RI IV XVLQJ RQO\ WKH JHQHUDOL]HG LQYHUVH VLQFH WKH SVHXGRLQYHUVH PD\ EH SK\VLFDOO\ LQFRQVLVWHQWf LV ?I?0U $7 >-0f@W0W-0frf $ f T7 -7>-0f@707-0f-f T f ,I KDV IXOO FROXPQ UDQN WKHQ -0fr IV DQG _UV_A 0-ƒf707^0$f IXOO FROXPQ UDQN f $U -rf7070U0-rf $ IXOO FROXPQ UDQN f 7KH G\QDPLFPDQLSXODELOLW\ HOOLSVRLG LV IRXQG XVLQJ f VR WKDW ?IV?0U $7 rf $ IXOO FROXPQ UDQN f DQG WKH HOOLSVRLG LV IRXQG IURP HLJHQV\VWHP RI -f707070$V GLVFXVVHG SUHn YLRXVO\ D PHWULF 0 FDQ EH XVHG IRU 07 ,I 0T 0 VR WKDW ?T?0T LV WKH NLQHWLF HQHUJ\ RI WKH PDQLSXODWRU WKHQ f UHGXFHV WR -r<0-rf $ IXOO FROXPQ UDQN DQG 07 0a[ f 7KH HOOLSVRLG IRXQG IURP WKH HLJHQV\VWHP -70IXOO FROXPQ UDQNf LV SK\VLFDOO\ FRQVLVWHQW EXW GRHV QRW PHHW WKH FULWHULD RI D YDOLG HLJHQV\VWHP LQ f VLQFH WKH 0-r IXOO FROXPQ UDQN f 0-$ IXOO FROXPQ UDQN f

PAGE 74

XQLWV RI WKLV PDWUL[ DUH SURSRUWLRQDO WR WKH XQLWV RI f 1RWLFH WKDW WKH PDWUL[ GHILQLQJ WKH G\QDPLF PDQLSXODELOLW\ HOOLSVRLG LV LGHQWLFDO WR WKH PDWUL[ GHILQLQJ WKH WZLVW PDQLSXODELOLW\ HOOLSVRLGf /HW XV ORRN D OLWWOH IXUWKHU -70WKH GHILQLWLRQ IRU WKH G\QDPLF PDQLSXODELOLW\ HOOLSVRLG DV RULJLQDOO\ GHYHORSHG LQ >@ ([SDQGLQJ f E\ VXEVWLWXWLQJ f IRU $ \LHOGV ?WV?PW T7 ^M7-rf70-rmf T IXOO FROXPQ UDQN 07 0 f %XW IRU IXOO FROXPQ UDQN -r ,Q DQG f WR WKH WULYLDO HTXDWLRQ ?WV?0W T70T IXOO FROXPQ UDQN 0U f§ 0B f DQG WKH HOOLSVRLG LV GHSHQGHQW RQO\ RQ WKH PHWULF %XW VLQFH 0 KDV WKH XQLWV RI 0T DQG 0T GRHV QRW VDWLVI\ WKH FRQGLWLRQV QHFHVVDU\ IRU D YDOLG HLJHQV\VWHP IRU QRQFRP PHQVXUDWH PDQLSXODWRUV DJDLQ WKH G\QDPLF PDQLSXODELOLW\ HOOLSVRLG LV VKRZQ WR KDYH DQ LQYDOLG HLJHQV\VWHP 1RWH WKDW DOWKRXJK 0T LV XQLWOHVV IRU FRPPHQVXUDWH PDQLSn XODWRUV DQG WKXV 0T KDV D YDOLG HLJHQV\VWHP WKH G\QDPLF PDQLSXODELOLW\ HOOLSVRLG GRHV QRW KDYH D YDOLG HLJHQV\VWHP HYHQ IRU FRPPHQVXUDWH PDQLSXODWRUV )RU WKH FDVH ZKHQ GRHV QRW KDYH IXOO FROXPQ UDQN f LV XVHG WR GHILQH WKH HOOLSVRLG >@ %XW DJDLQ D XQLWV DQDO\VLV RI WKH PDWULFHV VKRZV WKDW WKH HLJHQV\VWHP UHTXLUHPHQWV DUH YLRODWHG 7KLV LV DOVR WUXH IRU WKH H[SDQGHG YHUVLRQ RI WKLV HOOLSVRLG GHWHUPLQHG E\ f ZKHQ WKH PDQLSXODWRU LV QRQFRPPHQVXUDWH EXW LI WKH PDQLSXn ODWRU LV FRPPHQVXUDWH HDFK WHUP RI WKH PDWUL[ GHWHUPLQLQJ WKH HOOLSVRLG KDV LGHQWLFDO XQLWV DQG WKH HLJHQV\VWHP LV SK\VLFDOO\ PHDQLQJIXO 7R VXPPDUL]H QRQH RI WKH PDQLSXODELOLW\ HOOLSVRLGV SRVVHVV JHRPHWULF LQYDULDQFH 7KH ZUHQFK PDQLSXODELOLW\ HOOLSVRLG GHILQHG E\ WKH HLJHQV\VWHP RI PDWUL[ -07-7 LV QRW YDOLG IRU DQ\ PDQLSXODWRU 7KH WZLVW PDQLSXODELOLW\ HOOLSVRLG RULJLQDOO\ GHILQHG E\ WKH HLJHQV\VWHP RI -Wf7-W DQG VXEVHTXHQWO\ PRGLILHG WR -Af70T-A DQG WKHQ WR -rf70TLV QRW YDOLG IRU DQ\ PDQLSXODWRUV 7KH G\QDPLFPDQLSXODELOLW\ HOOLSVRLG

PAGE 75

GHILQHG E\ WKH HLJHQV\VWHP RI PDWUL[ >-0BfA@707-0f LV QRW D SK\VLFDOO\ FRQVLVWHQW HLJHQV\VWHP HYHQ IRU WKH FDVH ZKHQ KDV IXOO FROXPQ UDQN ,I KDV IXOO FROXPQ UDQN DQG 07 0B WKLV PDWUL[ SURGXFW UHGXFHV WR -f70JZKLFK DOVR GRHV QRW KDYH D YDOLG HLJHQV\VWHP $Q H[SDQVLRQ RI WKH G\QDPLFPDQLSXODELOLW\ HTXDWLRQ OHDGV WR -7 -rf70T-f§ 0T ZKLFK KDV D YDOLG HLJHQV\VWHP LI WKH PDQLSXODWRU LV FRPPHQVXUDWH $OWKRXJK WKH H[LVWLQJ PDQLSXODELOLW\ WKHRU\ KDV EHHQ VKRZQ WR EH LQYDOLG LQ DOO FDVHV IRU PDQLSXODWRUV ZLWK VL[ RU PRUH MRLQWV IRU PDQLSXODWRUV ZLWK VL[ RU IHZHU MRLQWV WKH VFDODU PDQLSXODELOLW\ PHDVXUH 'HW>-7-@ LV SK\VLFDOO\ PHDQLQJIXO DW GHn FRXSOH SRLQWV $W GHFRXSOH SRLQWV WKH PDQLSXODELOLW\ PHDVXUH LV SK\VLFDOO\ FRQVLVWHQW VHH HTXDWLRQ ff 7KXV ZKHQ D GHFRXSOHG FRRUGLQDWH IUDPH LV XVHG WKH PDQLSn XODELOLW\ RI WKHVH PDQLSXODWRUV LQ RQH FRQILJXUDWLRQ FDQ EH PHDQLQJIXOO\ FRPSDUHG WR WKH PDQLSXODELOLW\ DW RWKHU FRQILJXUDWLRQV

PAGE 76

&+$37(5 '(&20326,7,21 2) 63$&(6 *ULIILV UHFHQWO\ LQWURGXFHG D VSHFLDO VL[ GLPHQVLRQDO VSULQJ IRU XVH DV D ZULVW SODFHG RQ D MRLQWHG PDQLSXODWRU >@ +H WKXV FUHDWHG D ZUHQFK VSDFH YLD VPDOO GLVSODFHPHQWV RU WZLVWVf FUHDWLQJ D $RUWKRJRQDO FRPSOHPHQW WR WKH WZLVWV RI IUHHn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f 7KH RWKHU PDQLIROG LV WKH WZLVWV RI QRQIUHHGRP 9f± LQWURGXFHG E\ /LSNLQ DQG 'XII\ >@ LQ WKHLU LPSRUWDQW DUWLFOH RQ WKH QDWXUH RI WZLVWV DQG ZUHQFKHV DV VFUHZV 7KH WZLVWV RI QRQIUHHGRP DUH WKH WZLVWV WKDW DUH QRW SRVVLEOH WR DFFRPSOLVK LQ D JLYHQ FRQILJXUDWLRQ /LSNLQ DQG 'XII\ >@ GHILQH WKLV DV D f¯VXEVSDFH ZKLFK LV WKH RUWKRJRQDO FRPSOHPHQW RI f° WKH WZLVWV RI IUHHGRP DOWKRXJK 'XII\ ODWHU UHSXGLDWHV WKLV QRWLRQ LQ >@ %XW VLQFH 9 LV D QRQFRPPHQVXUDWH VSDFH WKH RUWKRJRQDO FRPSOHPHQW RI 9 LV QRW DQ DSSURSULDWH PDQLIROG WR LQWURGXFH VLQFH LW GRHV QRW KDYH WKH SK\VLFDO GLPHQVLRQV RI D WZLVW PDQLIROG 7KLV PDQLIROG ZRXOG KDYH WKH VWUDQJH SURSHUW\ RI GHSHQGHQFH RQ WKH XQLWV RI H[SUHVVLRQ RI 9 7KH ZUHQFKHV RI FRQVWUDLQW VXEVSDFH :F ZKHQ YLHZHG DV D XQLWOHVV YHFWRU VSDFH LQ  LV UHFRJQL]HG DV WKH RUWKRJRQDO FRPSOHPHQW RI DQ DVVXPHG XQLWOHVV YHUVLRQ RI 9 %XW :F RQO\ LQ VSHFLDO FDVHV

PAGE 77

DSSHDU WR KDYH WKH SK\VLFDO XQLWV RI WZLVW YHFWRUV ZKLFK LV QHFHVVDU\ IRU WKH PDQLIROG 9QI WR EH PHDQLQJIXO 7R EH IDLU >@ GHILQHV WZLVWV RI QRQIUHHGRP LQ WKH FRQWH[W RI DQ H[DPSOH WKDW DSSHDUV WR KDYH D XQLWOHVV EDVLV IRU :F ZKLFK FRXOG WKHUHIRUH EH YLHZHG DV DQ DSSURSULDWH WZLVW VXEVSDFH 7KLV GLVVHUWDWLRQ GHILQHV ZUHQFKHV RI FRQVWUDLQW LQ D PDQQHU FRQVLVWHQW WR WKH GHILQLWLRQ JLYHQ LQ >@f 3URMHFWLRQV DQG .LQHVWDWLF )LOWHUV ,Q FRPPHQVXUDWH V\VWHPV WKH SVHXGRLQYHUVH DQG JHQHUDOL]HGLQYHUVH FDQ EH XVHG WR VHSDUDWH YDULRXV VSDFHV LQWR WZR GLVMRLQW VSDFHV > @ ,Q QRQFRPPHQVXUDWH V\VWHPV FDUH PXVW EH WDNHQ ZKHQ XVLQJ WKH SVHXGRLQYHUVH ,I WKH SVHXGRLQYHUVH LV SK\VLFDOO\ LQFRQVLVWHQW SURMHFWLRQV XVLQJ WKLV LQYHUVH DUH DOVR JHQHUDOO\ SK\VLFDOO\ LQFRQVLVWHQW $OO W\SHV RI SURMHFWLRQV IRU WKH YDULRXV PDQLSXODWRU VSDFHV DUH GHULYHG EHORZ XVLQJ WKH JHQHUDOL]HGLQYHUVH DOWKRXJK LQ FDVHV RI D SK\VLFDOO\ FRQVLVWHQW SVHXGRLQYHUVH WKH JHQHUDOL]HGLQYHUVH PD\ EH UHSODFHG E\ WKH SVHXGRLQYHUVH 7KH WZLVW VSDFH SURMHFWLRQ LV IRXQG WKURXJK WKH IROORZLQJ VHULHV RI HTXDWLRQV 9 -T f TV -r9G f 9U f§ &>V f 9U --r9G f ZKHUH WKH VXEVFULSW LV IRU f¯VROXWLRQf° WKH f¯Gf° VXEVFULSW LV IRU f¯GHVLUHGf¬ DQG WKH f¯Uf° VXEVFULSW LV IRU f¯UHVXOWLQJf° 7KH MRLQWUDWH VSDFH SURMHFWLRQ REWDLQHG E\ VXEVWLWXWLQJ f LQWR f LV TV -r-TG ‘ f 7KH ZUHQFK VSDFH SURMHFWLRQ LV IRXQG WKURXJK WKH IROORZLQJ VHULHV RI HTXDWLRQV U -7: f

PAGE 78

: -r77G f :V -rU-U7G --rf7UG f 7KH JHQHUDOL]HGIRUFH VSDFH SURMHFWLRQ REWDLQHG E\ VXEVWLWXWLQJ f LQWR f LV WY -7-r7UG -r-f7UG f 7KH YDULRXV SURMHFWLRQ PDWULFHV DUH WKH IRXU NLQHVWDWLF ILOWHUV >@ 3Y --r 3T -r3Z --r< 3U --f7 fµ f 7KH YDULRXV VSDFHV FDQ QRZ EH GHFRPSRVHG LQWR GLVMRLQW VSDFHV XVLQJ WKH DERYH SURMHFWLRQ PDWULFHV DQG f f f DQG f 1XOO>--@ 5DQJH>--@ f 1XOO>--@ k 5DQJH>--@ f $ 1XOO>--f7@ k 5DQJH>--f7@ f 0 1XOO>--fU@ k 5DQJH>--f7@ f ZKHUH WKH V\PERO k PHDQV WKDW WKH WZR VXEVSDFHV RQ HLWKHU VLGH RI WKLV V\PERO DUH 0f±RUWKRJRQDO 7KH QRUPDO GLUHFW VXP pf PHDQV WKDW WKH WZR VSDFHV DUH RUWKRJRQDO LQ WKH (XFOLGHDQ VHQVHf 1RWLFH WKDW WKH DERYH GHFRPSRVLWLRQV GR QRW IROORZ IURP WKH IXQGDPHQWDO WKHRUHP RI OLQHDU DOJHEUD 1XOO>$7@ k 5DQJH >$@ ZKHUH WKH UDQJH DQG QXOO RSHUDWRUV RSHUDWH RQ D PDWUL[ DQG LWV WUDQVSRVH )RU WKH PHWULFGHSHQGHQW GHFRPSRVLWLRQV WKH UDQJH DQG QXOO RSHUDWRUV RSHUDWH RQ WKH VDPH PDWUL[ 7KH DERYH GHFRPSRVLWLRQ HTXDWLRQV FDQ EH VLPSOLILHG E\ DSSO\LQJ VRPH IDFWV DERXW WKH IXOO UDQN GHFRPSRVLWLRQ RI WKH -DFRELDQ )& DQG RI f --r )&&r)r f§ ))r -r&r)r)& &r& f f

PAGE 79

DQG VRPH IDFWV DERXW WKH QXOO DQG UDQJH VSDFH RSHUDWRUV 1XOO>--@ 1XOO>))@ 1XOO>)@ f 5DQJH>--A@ 5DQJH>))A@ 5DQJH>)@ f 1XOO>--@ 1XOO>&@ 1XOO>&@ f 5DQJH>--@ 5DQJH>&&@ 5DQJH>@ f (DFK RI WKH VWDWHPHQWV LQ ff FDQ EH SURYHQ LQ D PDQQHU VLPLODU WR WKDW VKRZQ EHORZ IRU f /HW ))[ 0XOWLSO\ ERWK VLGHV E\ ) WR JLYH )A))r[ f§ %XW E\ WKH SURSHUW\ RI WKH JHQHUDOL]HGLQYHUVH JLYHQ LQ f )r))r f§ )t VR WKDW )[ 7KHUHIRUH 1XOO>))@ 1XOO>)@ 7KHVH VLPSOLILFDWLRQV OHDG WR WKH EHORZ VLPSOLILHG GHFRPSRVLWLRQ HTXDWLRQV 9 1XOO>-@ k 5DQJH>-@ f 0T 4 1XOO>-@ k 5DQJH>-@ f 0a : 1XOO>-7@ k 5DQJH>-f7@ f 0aO 7 1XOO>-f7@ k 5DQJH>-7@ f DQG WKH HYHQ VLPSOHU GHFRPSRVLWLRQ HTXDWLRQV 9 1XOO>)@ k 5DQJH>)@ f 4 1XOO>&@ k 5DQJHI&@ f 0a : 1XOO>)7@ k 5DQJH>)f7@ f f± 0 7 1XOO>&f7@ k 5DQJH>&n7@ f (DFK PHWULF ZLOO JLYH D GLIIHUHQW GHFRPSRVLWLRQ ,I WKH PHWULF KDV WKH UHTXLUHG SURSHUW\ WKDW LW WUDQVIRUPV YLD D FRQJUXHQFH WUDQVIRUPDWLRQ ff WKHQ WKH IUDPH RI H[SUHVVLRQ KDV QR EHDULQJ RQ WKH GHFRPSRVLWLRQ

PAGE 80

7KH EHORZ WZR IDFWV DOORZ XV LQ VRPH FDVHV WR DSSO\ WKH DERYH PHWULF GHSHQGHQW GHFRPSRVLWLRQV ZKLFK XVH WKH JHQHUDOL]HGLQYHUVH WR D PHWULF LQGHSHQGHQW GHFRPn SRVLWLRQ ZKLFK XVHV WKH SVHXGRLQYHUVH )DFW ,I -W -r IRU VRPH PHWULF 0Y DQG VRPH PHWULF 0T WKHQ -O LV SK\VLFDOO\ FRQVLVWHQW )DFW ,I LV SK\VLFDOO\ FRQVLVWHQW WKHQ -W IRU VRPH PHWULF 0Y DQG VRPH PHWULF 0T ,I WKH SVHXGRLQYHUVH LV XVHG LQVWHDG RI WKH JHQHUDOL]HGLQYHUVH E\ FKRRVLQJ FKDQJH RI XQLW LGHQWLW\ VFDOLQJ PHWULFV IRU 0Y DQG 0T WKH GHFRPSRVLWLRQ LV IUDPH GHSHQGHQW DQG RQO\ YDOLG LI WKH SVHXGRLQYHUVH LV SK\VLFDOO\ FRQVLVWHQW 7KH GHFRPSRVLWLRQ IRU SK\VLFDOO\ FRQVLVWHQW LV 9 1XOO>-MW@ k 5DQJH >--A@ f 4 1XOO>-I -@ k 5DQJH>-I-@ f : 1XOO>--Wf7@ k 5DQJH>--Wf7@ f 7 1XOO> -I -f7@ k 5DQJH> -f7@ f )URP 7KHRUHP DQG WKH IDFW WKDW -W -r IRU VRPH PHWULF VLQFH mr LV DVVXPHG SK\VLFDOO\ FRQVLVWHQWf --O --Wf7 DQG -O-W-f7 7KHUHIRUH WKH DERYH GHFRPn SRVLWLRQV VLPSOLI\ WR 9 : 1XOO>--W@ p 5DQJHIMMW@ f 47 1XOO>M9@ k 5DQJH>-I -@ f ZKHQ -W LV SK\VLFDOO\ FRQVLVWHQW 7KH VSDFHV 9 DQG : DUH GHFRPSRVHG LGHQWLFDOO\ DV DUH WKH VSDFHV 4 DQG 7

PAGE 81

7KH DERYH GHFRPSRVLWLRQ FDQ EH IXUWKHU VLPSOLILHG E\ XVLQJ WKH EHORZ HTXDWLRQV ))A f -WM t& f 1X8>-MW@ 1XOOI)LA@ 1XOOI)r@ 1XOO>)7)fB)7@ f 1XOO>)7@ 1XOO>7)7@ 1XOO>-7@ f 5DQJHI--@ 5DQJH>-@ f 1XOOIM9@ 1XOO>-@ f 5DQJ H>-r-@ 5DQJHA&@ 5DQJHG@ 5DQJH>&n7&n&7f@ f 5DQJH>&7@ 5DQJHI&A)@ 5DQJH>-7@ f 7KH VSDFH GHFRPSRVLWLRQV IRU IUDPHV LQ ZKLFK m) LV SK\VLFDOO\ FRQVLVWHQW DUH WKHUHn IRUH 9 : 1XOO>-7@ k 5DQJH>-@ f 47 1XOO>-@ k 5DQJH>-7@ f (TXDWLRQV f DQG f DSSHDU WR EH GLUHFW DSSOLFDWLRQV RI WKH IXQGDPHQWDO WKHRUHP RI OLQHDU DOJHEUD WKLV LV D GHFHSWLYH QRWLRQ 7KH UHDGHU VKRXOG UHPHPEHU WKH OLPLWHG VFRSH RI WKHVH HTXDWLRQVf§LH WKH\ DUH RQO\ YDOLG LQ IUDPHV LQ ZKLFK -W LV SK\VLFDOO\ FRQVLVWHQWf§DQG WKHLU UDWKHU LQYROYHG GHULYDWLRQV 7KLV GHFRPSRVLWLRQ ZLOO EH H[SORUHG IXUWKHU LQ WKH VXEVHTXHQW VHFWLRQV 7ZLVW 'HFRPSRVLWLRQ ,Q RUGHU WR GHPRQVWUDWH WKH SUREOHP ZLWK GHILQLQJ D WZLVW RI QRQIUHHGRP PDQLIROG DV D VXEVSDFH WZR H[DPSOHV ZLOO EH VKRZQ 2QH H[DPSOH ZLOO VKRZ ZKHQ WKHVH WZLVWV FRQVWLWXWH D VXEVSDFH DQG WKH RWKHU ZLOO VKRZ ZKHQ WKH\ GR QRW IRUP D VXEVSDFH )LUVW FRQVLGHU WKH 6&$5$ PDQLSXODWRU RI )LJXUH 7KH 6&$5$ -DFRELDQ H[SUHVVHG LQ IUDPH FRRUGLQDWHV ZDV JLYHQ LQ f 7KH FROXPQUHGXFHG HFKHORQ

PAGE 82

IRUP RI WKH ZUHQFK RI FRQVWUDLQW VXEVSDFH LQ WKLV IUDPH :F 1XOO>-7@ LV :& (f± f§ n n U U F E : f ZKHUH (Z LV WKH PDWUL[ WKDW FRQYHUWV >:F@^ WR FROXPQUHGXFHG HFKHORQ IRUP 1RWH WKDW WKHVH ZUHQFKHV PLJKW DOVR EH LQWHUSUHWHG DV WZLVWV RI QRQIUHHGRP ZLWK QR GLVFUHSDQF\ ZLWK XQLWV f 7KH 6$5 353f PDQLSXODWRU RI )LJXUH KDV WKH -DFRELDQ DQG ZUHQFK RI FRQn VWUDLQW VXEVSDFH EDVLV YHFWRUV H[SUHVVHG LQ IUDPH FRRUGLQDWHV RI ‘ G] ‘ n B -B G] n :& (Z f 1RWH WKDW WKHVH EDVLV ZUHQFKHV FDQQRW EH LQWHUSUHWHG DV WZLVWV RI QRQIUHHGRP VLQFH WKH VHFRQG EDVLV YHFWRU GRHV QRW KDYH WKH XQLWV RI D WZLVW DQ D[LV FRRUGLQDWH VFUHZf 7KHUHIRUH IRU WKLV PDQLSXODWRU H[SUHVVHG LQ IUDPH FRRUGLQDWHV WKH FRQFHSW RI WZLVWV RI QRQIUHHGRP DV GHVFULEHG SUHYLRXVO\ DV D VXEVSDFHf LV XQWHQDEOH $ VOLJKWO\ PRGLILHG GHILQLWLRQ RI WZLVWV RI QRQIUHHGRP LV WKHUHIRUH QHFHVVDU\ DQG LV JLYHQ EHORZ 'HILQLWLRQ 7ZLVWV RI QRQIUHHGRP DUH WZLVWV WKDW WKH PDQLSXODWRU FDQQRW IXOO\ JHQn HUDWH LQ D JLYHQ FRQILJXUDWLRQ 9} 99 f 7KH PHDQLQJ RI WKH DERYH HTXDWLRQ PLJKW QHHG H[SODQDWLRQ 7KH PDQLIROG 9Qc LQFOXGH DOO WKH WZLVWV RI 9 H[FHSW WKRVH WZLVWV LQ 9 7KLV LV QRW WKH RUWKRJRQDO FRPSOHPHQW RI 9 ZKLFK DV VWDWHG SUHYLRXVO\f LV SK\VLFDOO\ LQFRQVLVWHQW IRU VFUHZV

PAGE 83

7KH QRQIUHHGRP WZLVW PDQLIROG PLJKW DOVR EH GHILQHG DV 9f± ^. .fü9DQG.A9` f ,Q JHQHUDO WKH PDQLIROG 9QM LV QRW D VXEVSDFH 7\SLFDOO\ WZR WZLVWV RI QRQIUHHGRP PLJKW VXP WR D WZLVW RI IUHHGRP RU D QRQIUHHGRP WZLVW )RU H[DPSOH WZR QRQIUHHGRP WZLVWV IRU WKH 6$5 PDQLSXODWRU H[SUHVVHG LQ IUDPH FRRUGLQDWHV DUH p V L UDG V ,, L UDG V f 7KH VXP RI WKHVH WZR QRQIUHHGRP WZLVWV LV WKH WZLVW RI IUHHGRP > LI @7 7KH GLIIHUHQFH RI WKHVH WZR QRQIUHHGRP WZLVWV LV WKH QRQIUHHGRP WZLVW >R R LI R R\ 6LQFH 9QI LV QRW LQ JHQHUDO D VXEVSDFH D GLUHFW VXP GHFRPSRVLWLRQ RI WZLVWV RI IUHHGRP DQG WZLVWV RI QRQIUHHGRP LV QRW W\SLFDOO\ SRVVLEOH LH 9 A 9 k 9QI f ,Q WKH VSHFLDO FDVHV ZKHQ :F FDQ EH LQWHUSUHWHG HQWLUHO\ DV WZLVWV WKH WZLVW VSDFH FDQ EH GHFRPSRVHG DV WKH GLUHFW VXP GHFRPSRVLWLRQ 9 9 k 9 ZKHUH 9 DUH WKH VXEVSDFH RI LQDFFHVVLEOH WZLVWV GHILQHG EHORZ 'HILQLWLRQ ,QDFFHVVLEOH WZLVWV FRQVWLWXWH WKH VFUHZ VXEVSDFH RI WZLVWV VXFK WKDW 9L F 9% f UI DQG WKH LQQHU SURGXFW 9L49I 9L4YI Z kZ ZKLFK LV JHQHUDOO\ SK\VLFDOO\ LQFRQVLVWHQWf LV SK\VLFDOO\ FRQVLVWHQW IRU DQ\ 9L 9 DQG DQ\ 9f fü 9 7KH VXEVSDFH 9 PD\ QRW H[LVW

PAGE 84

,I 9 9QI WKHQ WKH WZLVW VSDFH LV XQLTXHO\ GHFRPSRVHG E\ WKH GLUHFW VXP GHFRPn SRVLWLRQ 9 9k9L LI 9W 9QI f :UHQFK 'HFRPSRVLWLRQ $VVXPH WKDW D ZUHQFK VSDFH UHIHUHQFHG WR D SDUWLFXODU FRRUGLQDWH V\VWHP LV GHFRPn SRVHG LQWR WZR PDQLIROGV 2QH RI WKHVH PDQLIROGV HTXDOV WKH ZUHQFKHV RI FRQVWUDLQW :F 1XOO>-7@ DV SUHYLRXVO\ GHILQHG LQ f 6LQFH :F LV WKH QXOO VSDFH RI D PDWUL[ LW PXVW EH D VXEVSDFH 7KH RWKHU PDQLIROG LV WKH ZUHQFKHV RI QRQFRQVWUDLQW PDQLIROG >@ :QF 7KH ZUHQFKHV RI QRQFRQVWUDLQW ZKHQ DSSOLHG DW WKH HQG HIIHFWRU RI D PDQLSXODWRU UHTXLUH VRPH QRQ]HUR MRLQW IRUFHV IRU VWDWLF EDODQFLQJ RU ZLOO FDXVH VRPH PRWLRQ RI WKH PDQLSXODWRU /LSNLQ DQG 'XII\ >@ GHILQH ZUHQFKHV RI QRQFRQVWUDLQW LQ DQ DQDORJRXV IDVKLRQ WR WKH WZLVWV RI QRQIUHHGRP LH DFFRUGLQJ WR >@ WKH ZUHQFK RI QRQFRQn VWUDLQW PDQLIROG LV WKH RUWKRJRQDO FRPSOHPHQW RI WKH ZUHQFK RI FRQVWUDLQW VXEVSDFH %XW WKH RUWKRJRQDO FRPSOHPHQW RI :F KDV SK\VLFDO GLPHQVLRQV RI D WZLVW PDQLIROG )XUWKHUPRUH ZKHQ :F LV YLHZHG DV D XQLWOHVV YHFWRU VSDFH LQ 65 WKH RUWKRJRQDO FRPn SOHPHQW LV D XQLWOHVV YHUVLRQ RI 9 %XW WKH D[LV VFUHZ YHFWRUV RI 9 RQO\ LQ VSHFLDO FDVHV DSSHDU WR KDYH WKH SK\VLFDO XQLWV RI ZUHQFK YHFWRUV D QHFHVVDU\ UHTXLUHPHQW IRU WKH PDQLIROG :QF WR EH PHDQLQJIXO )RU H[DPSOH WKH RUWKRJRQDO FRPSOHPHQW RI :& IRU WKH 6$5 PDQLSXODWRU LV WKH -DFRELDQ JLYHQ LQ f 7KH VHFRQG EDVLV YHFWRU RI LQ f LV REYLRXVO\ QRW D ZUHQFK D UD\ FRRUGLQDWH VFUHZf VR WKLV VXEVSDFH FDQQRW GHVFULEH ZUHQFKHV RI QRQ FRQVWUDLQW 7R DYRLG WKH DERYH SUREOHPV D VOLJKW PRGLILFDWLRQ RI WKH GHILQLWLRQ RI ZUHQFKHV RI QRQFRQVWUDLQW LV JLYHQ EHORZ

PAGE 85

'HILQLWLRQ 1RQFRQVWUDLQW ZUHQFKHV DUH ZUHQFKHV WKDW ZLOO SURGXFH D QRQ]HUR SRZHU ZLWK VRPH WZLVW RI IUHHGRP >@ :QF : :F f 7KH PDQLIROG RI QRQFRQVWUDLQW ZUHQFKHV DUH DOO WKH ZUHQFKHV RI : H[FHSW WKRVH ZUHQFKHV LQ :F 7KLV LV QRW LQ JHQHUDO WKH RUWKRJRQDO FRPSOHPHQW RI :F ZKLFK DV VWDWHG SUHYLRXVO\f LV SK\VLFDOO\ LQFRQVLVWHQW IRU VFUHZV :UHQFKHV RI QRQFRQVWUDLQW PLJKW DOVR EH GHILQHG DV :f±F ^ZQF :QF H : DQG :QF e :F` f 1RWH WKDW :QF LV D PDQLIROG WKDW LQ JHQHUDO LV QRW D VXEVSDFH VR WKDW QR GLn UHFW VXP GHFRPSRVLWLRQ RI ZUHQFKHV RI FRQVWUDLQW DQG ZUHQFKHV RI QRQFRQVWUDLQW LV JHQHUDOO\ SRVVLEOH LH ::Fk:QF f ,Q WKH VSHFLDO FDVHV ZKHQ WKH WZLVWV RI 1XOOI:Mn@ SRVVHVV D PHDQLQJIXO LQWHUSUHWDWLRQ DV :QF WKH ZUHQFK VSDFH FDQ EH GHFRPSRVHG YLD WKH GLUHFW VXP GHFRPSRVLWLRQ : :F k :G ZKHUH :G DUH WKH VXEVSDFH RI GULYLQJ ZUHQFKHV GHILQHG EHORZ 'HILQLWLRQ 'ULYLQJ ZUHQFKHV FRQVWLWXWH WKH VFUHZ VXEVSDFH RI ZUHQFKHV VXFK WKDW :G & :f±F f DQG WKH LQQHU SURGXFW : k :QF IG k QF Q k QQF ZKLFK LV JHQHUDOO\ SK\VLFDOO\ LQFRQVLVWHQWf LV SK\VLFDOO\ FRQVLVWHQW IRU DQ\ fü : DQG DQ\ :QF * :QF 7KH VXEVSDFH :G PD\ QRW H[LVW ,I :G :QF WKHQ WKH ZUHQFK VSDFH LV XQLTXHO\ GHFRPSRVHG E\ WKH GLUHFW VXP GHFRPSRVLWLRQ : :G k :F LI :G :QF f

PAGE 86

,I ERWK WZLVWV RI QRQIUHHGRP DQG ZUHQFKHV RI QRQFRQVWUDLQW DUH VXEVSDFHV DQG WKXV DUH LGHQWLFDOO\ WKH LQDFFHVVLEOH WZLVWV DQG WKH GULYLQJ ZUHQFKHV UHVSHFWLYHO\f WKHQ D K\EULG FRQWURO LV DFFRPSOLVKHG E\ GHFRPSRVLQJ WKH GHVLUHG WZLVW LQWR WZLVWV RI IUHHGRP DQG WZLVWV RI QRQIUHHGRP DQG WKH GHVLUHG ZUHQFK LQWR ZUHQFKHV RI FRQVWUDLQW DQG ZUHQFKHV RI QRQFRQVWUDLQW DQG WKHQ ILOWHULQJ RXW WKH LQDFFHVVLEOH WZLVWV DQG FRQVWUDLQW ZUHQFKHV 7KLV DVVXUHV WKDW WKH FRQWURO LQSXWV ZLOO EH HQWLUHO\ FRPSRVHG RI WZLVWV RI IUHHGRP DQG GULYLQJ ZUHQFKHV +\EULG &RQWURO 7KH K\EULG FRQWURO DOJRULWKPV RI 0DVRQ > @ DQG 5DLEHUW >@ LQKHUHQWO\ DVVXPH D GHFRPSRVLWLRQ HVVHQWLDOO\ HTXLYDOHQW WR s 5DQJH>-@ k 1XOO>-U@ 9 :F f 7KLV WKHRU\ VSOLWV WKH K\EULG FRQWURO SUREOHP LQWR f¯QDWXUDOf° DQG f¯DUWLILFLDOf° FRQn VWUDLQWV DW ZKDW LV QRZ FRPPRQO\ NQRZ DV WKH f¯FHQWHU RI FRPSOLDQFHf° RU f¯FRPSOLDQFH FHQWHUf° > @ FDOOHG D FRQVWUDLQW IUDPH LQ > @f $ FHQWHU RI FRPSOLDQFH LV GHILQHG DV D SRLQW WKURXJK ZKLFK SXUH IRUFHV SURGXFH RQO\ SXUH WUDQVODWLRQV DQG SXUH FRXSOHV SURGXFH RQO\ SXUH URWDWLRQV DERXW WKDW SRLQW 7KLV SRLQW PD\ RU PD\ QRW H[LVW RU PD\ H[LVW DW PRUH WKDQ RQH SRLQW :KHQ WKH FRRUGLQDWH UHIHUHQFH IUDPH RULJLQ LV ORFDWHG DW WKH FHQWHU RI FRPSOLDQFH WKH 05+&7 0DVRQ DQG 5DLEHUWf¬V K\EULG FRQWURO WKHRU\f VWDWHV WKDW WKH GLDJRQDO VHOHFWLRQ PDWULFHV > @ DUH XVHG WR GHWHUPLQH WKH DSSURSULDWH DFWLRQ IRU HDFK ORRS RI WKH K\EULG SRVLWLRQ DQG IRUFH FRQWURO LH HDFK MRLQW LV XVHG WR FRQWURO HLWKHU D SRVLWLRQ FRPSRQHQW WZLVWf RU D IRUFH FRPSRQHQW ZUHQFKf 7KH 05+&7 FDOOV WKHVH WZR VXEVSDFHV RUWKRJRQDO FRPSOHPHQWV ZKLFK WKHVH VXEn VSDFHV DSSHDU WR EH LI WKH VFUHZ VSDFHV ZHUH LQVWHDG FRPPHQVXUDWH VL[ GLPHQVLRQDO YHFWRU VXEVSDFHV DV LQ f %XW WKH\ DUH QRW RUWKRJRQDO FRPSOHPHQW VFUHZ VXEn VSDFHV

PAGE 87

$Q H[DPSOH ZLOO QRZ GHPRQVWUDWH WKH 05+&7 > @ 7KH WDVN DW KDQG LV WR SODFH D SHJ LQWR D KROH DV VKRZQ LQ )LJXUH ,Q WKLV H[DPSOH WKH YLUWXDO 35 PDQLSXODWRU RI WKH ILJXUH LV QRW LQYROYHGf 7KH f¯QDWXUDOf° DQG f¯DUWLILFLDOf° FRQVWUDLQWV WDNHQ WRJHWKHU VLQFH WKH GLVWLQFWLRQ EHWZHHQ WKH WZR LV VRPHWLPHV RSHQ WR LQWHUSUHn WDWLRQf ZLWK UHVSHFW WR IUDPH DUH Y[ Y\ f§ I] DQG Q] %RWK WKH WZLVW DQG ZUHQFK VHOHFWLRQ PDWULFHV DUH GLDJRQDO PDWULFHV ERWK ZLWK HOHPHQWV RI HLWKHU RU 7KLV OHDGV WR WKH WZLVW VHOHFWLRQ PDWUL[ 39 DQG WKH ZUHQFK VHOHFWLRQ PDWUL[ 3Z¯ n n n ‘ S BB " : f§ B f 7KH VHOHFWLRQ PDWULFHV DUH DOZD\V UHODWHG E\ WKH HTXDWLRQ 3ZK3Y fµ f 7KH K\EULG FRQWURO WKHQ ILOWHUV WKH VSHFLILHG WZLVW 9V DQG ZUHQFK :V ZLWK WKH VHOHFWLRQ PDWULFHV DV IROORZV 9 396 : 3Z:V f 7KLV JXDUDQWHHV WKDW WKH WZLVW 9 fü 9 DQG : H :& LQ IUDPH ,W LV DSSDUHQW WKDW WKH VHOHFWLRQ PDWULFHV 3Y DQG 3Z DFW DV ILOWHUV RQ WZLVWV DQG ZUHQFKHV ,Q IDFW 3Y DQG 3Z DUH SURMHFWLRQ PDWULFHV 39 %%@ 3: &&W % >%%W%fa%W & >&^&W&faX& f f

PAGE 88

ZKHUH % UHSUHVHQWV D EDVLV IRU WKH WZLVWV RI IUHHGRP DQG & UHSUHVHQWV D EDVLV IRU WKH ZUHQFKHV RI FRQVWUDLQW n ‘ n n F A f ,Q IUDPH WKH 05+&7 VHHPV WR ZRUN %XW LQ D IUDPH W VHH )LJXUH f DUELWUDULO\ WUDQVODWHG IURP IUDPH WKH 05+&7 IDLOV ,Q WKLV IUDPH WKH SURMHFWLRQ PDWULFHV 3Y DQG 3Z W:FW:I DUH SK\VLFDOO\ LQFRQVLVWHQW LH $ f§S[S\ (MO f§3[3\ S_ f§ 3[ (MO f§ 3[ 3L (L6[ ](M/ 3[3\ SY f§ ]6MO (V SrSY f ZKHUH 3\ D SK\VLFDOO\ LQFRQVLVWHQW TXDQWLW\ 'HFRPSRVLWLRQ ZLWK 5D\ &RRUGLQDWH 7ZLVW 6SDFH 5HFHQWO\ VHYHUDO DXWKRUV > @ KDYH H[SDQGHG D GLVFXVVLRQ RQ LVRWURSLF VXEVSDFHV EHJXQ LQ >@ DQG JUHDWO\ HQKDQFHG LQ >@ 7KHVH DUWLFOHV KDYH DWWHPSWHG D GLIIHUn HQW GHFRPSRVLWLRQ XVLQJ IRXU PDQLIROGV WZR RI ZKLFK DUH WKH WZLVWV RI IUHHGRP DQG ZUHQFKHV RI FRQVWUDLQW 0DQLSXODWH WKH WZLVWV VSDFH YLD WKH $ PDWUL[ VR WKDW WKH WZLVWV RI IUHHGRP DQG ZUHQFKHV RI FRQVWUDLQW VXEVSDFHV DUH ERWK GHILQHG XVLQJ UD\ FRRUGLQDWH VFUHZV LH 9I\ $9M 5DQJH$-f f 7KH UDGLFDO PDQLIROG LV WKH VFUHZ PDQLIROG RI WKH FRPPRQ HOHPHQWV LQ 9fD\ DQG

PAGE 89

7KH GHIHFW PDQLIROG 9 LV WKH PDQLIROG QRW FRYHUHG E\ 979 DQG :F 9I\ 8 :Ff X ' f ZKHUH LV WKH IXOO GLPHQVLRQDO UD\ FRRUGLQDWH VFUHZ VSDFH /HW XV LQYHVWLJDWH KRZ HDFK RI WKHVH PDQLIROGV UHODWH WR WKH RWKHUV $V VKRZQ LQ 7KHRUHP 9M DQG :F DUH UHFLSURFDO VXEVSDFHV 6LQFH 9fD\ LV WKH UD\ FRRUGLQDWH YHUVLRQ RI 9 WKHQ 9AD\ DQG :F DUH DOVR UHFLSURFDO VXEVSDFHV 7KLV WKHRUHP OHDGV WR WKH FRUROODU\ EHORZ ZKLFK VWDWHV WKDW WKH UDGLFDO PDQLIROG LV D VHOIUHFLSURFDO VXEVSDFH 7KH SURRI IRU WKH WKHRUHP EHORZ LV EDVHG LQ SDUW RQ 7KHRUHP ZKLFK VWDWHV WKDW FRRUGLQDWH WUDQVIRUPDWLRQV GR QRW DIIHFW WKH UHFLSURFDO SURGXFW &RUROODU\  7KH UDGLFDO VFUHZ VXEVSDFH -= LV VHOIUHFLSURFDO Q R UM 9 ULUM ( = f 3URRI 6LQFH U ( 7= U ( 9fD\ DQG U :F DQG DOO 9S\ ( 9fD\ DQG :F ( :F DUH UHFLSURFDO 9-D\ R :F f E\ 7KHRUHP WKHQ U R UM IRU DOO L DQG M 6LQFH WKH VFUHZ VXEVSDFH = LV VHOIUHFLSURFDO WKH VFUHZV LQ WKLV VXEVSDFH DUH VHOIn UHFLSURFDO DQG PXWXDOO\ UHFLSURFDO 7KH WKHRUHP EHORZ DOVR VKRZV WKDW HDFK FROXPQ RI D PDQLSXODWRU -DFRELDQ LV VHOIUHFLSURFDO 7KHRUHP )RU UHYROXWH DQGRU SULVPDWLF MRLQWHG PDQLSXODWRUV HDFK FROXPQ RI D PDQLSXODWRU -DFRELDQ LV VHOIUHFLSURFDO

PAGE 90

3URRI ,I WKH WK MRLQW LQ D PDQLSXODWRU LV UHYROXWH WKH WK FROXPQ RI WKH PDQLSXODWRU -DFRELDQ LQ IUDPH  f§ LV >@7 ,I WKH LWK MRLQW LQ D PDQLSXODWRU LV SULVn PDWLF WKH LWK FROXPQ RI WKH PDQLSXODWRU -DFRELDQ LQ IUDPH r f§ LV >@7 6LQFH ERWK WKHVH VFUHZV DUH VHOIUHFLSURFDO DQG UHFLSURFLW\ LV LQYDULDQW WR FRRUGLQDWH WUDQVIRUPDWLRQV WKHQ UHJDUGOHVV RI WKH IUDPH WKH WK FROXPQ RI WKH -DFRELDQ LV VHOIn UHFLSURFDO 7KH UDGLFDO LV DOZD\V D VXEVSDFH VLQFH LW LV WKH LQWHUVHFWLRQ RI WZR VXEVSDFHV %XW 9\D\ 8 :F LV JHQHUDOO\ QRW D VXEVSDFH DV LV VKRZQ LQ WKH EHORZ H[DPSOH 7KH 3 PDQLSXODWRU ZLWK f§ WW DQG KDV U U U L U U L  ! " ! 2 R DL ! f 6XPPLQJ WKH ILIWK VFUHZ RI 9AD\ DQG WLPHV WKH RQO\ VFUHZ RI :F UHVXOWV LQ WKH YHFWRU > f§ @7 IRU DOO ZKHUH XQLWV>@ / 7KLV VFUHZ LV QRW LQ \r1\ M 99& IRU DQ\ QRQ]HUR 7KHUHIRUH 9AD\ 8 :F LV QRW D VFUHZ VXEVSDFH 6LPLODUO\ WKH GHIHFW PDQLIROG LV JHQHUDOO\ QRW D VFUHZ VXEVSDFH VLQFH 7! f§ 9MD\8 :Ff DOWKRXJK > @ ERWK FODLP WKDW WKH GHIHFW LV D VXEVSDFH )RU H[DPSOH WKH 6$5 PDQLSXODWRU LQ IUDPH KDV WZLVW RI IUHHGRP DQG ZUHQFK RI FRQVWUDLQW EDVLV VHWV RI >9I\@c $L f§ n n n 2 Wf§+ WR ( ,, " « } L M Ur+ r f

PAGE 91

VR WKDW WKH UDGLFDO EDVLV VHW LV _5@ ^> @n` f 7KH GHIHFW PDQLIROG FRQWDLQV DOO VFUHZV ^>IW IW IW IW IWS` f ZLWK QRQ]HUR 7KLV LV QRW D VXEVSDFH DOWKRXJK >@ FODLPV WKDW D EDVLV FDQ EH VHOHFWHG IRU WKH GHIHFW >!@L ^> @7` ,Q IUDPH WKH 6$5 PDQLSXODWRU KDV WZLVW RI IUHHGRP DQG ZUHQFK RI FRQVWUDLQW EDVLV VHWV RI LYUQE n n ‘ f« f« n n U e n n n >:F@E G] « r r W f VR WKDW WKH UDGLFDO EDVLV VHW LV HPSW\ LH >"@t 7KH GHIHFW PDQLIROG LV DOVR HPSW\ IRU WKH 6$5 PDQLSXODWRU LQ IUDPH ,W LV DSSDUHQW QRZ WKDW WKH GHFRPSRVLWLRQ WKHRU\ RI > @ LV QRW XQLTXH DQG WKH FODLPV PDGH DUH JHQHUDOO\ LQYDOLG 7KHUHIRUH D QHZ WHFKQLTXH IRU VFUHZ DQG ZUHQFK VSDFH GHFRPSRVLWLRQ LV SUHVHQWHG LQ WKH QH[W VHFWLRQ DQG WKH UHVXOWV RI WKH SUHYLRXV VHFWLRQV RI WKLV FKDSWHU DUH WLHG WRJHWKHU 6SDFH 'HFRPSRVLWLRQ DW 'HFRXSOH 3RLQW ,Q 6HFWLRQ LW ZDV VKRZQ WKDW LQ VRPH FDVHV WKH WZLVW VSDFH FDQ EH GHFRPSRVHG XQLTXHO\ YLD D (XFOLGHDQf GLUHFW VXP GHFRPSRVLWLRQ VHH ff DQG LQ RWKHU FDVHV QRW ,Q WKLV VHFWLRQ WKH FRQGLWLRQV IRU ZKLFK WKLV GHFRPSRVLWLRQ LV SRVVLEOH DUH IRXQG :KHQ WKH ZUHQFKHV RI FRQVWUDLQW DUH SXW LQ FROXPQUHGXFHG HFKHORQ IRUP >:F@E(Z VRPH RI WKH FROXPQV PD\ DSSHDU XQLWOHVV 6LQFH ZUHQFKHV DUH VFUHZV XQLWn OHVV FROXPQV ZLOO RQO\ H[LVW LQ FROXPQV WKDW KDYH ]HURV LQ WKH IRUFH RU PRPHQW SRVLn WLRQV (DFK XQLWOHVV FROXPQ RI >:AW(X UHSUHVHQWV RQH RI WKH IROORZLQJ WZR W\SHV RI

PAGE 92

ZUHQFKHV WKH ZUHQFK LV D SXUH IRUFH LH :IUFH >I[ I\ I] @7 f RU WKH ZUHQFK LV D SXUH PRPHQW ZLWK UHVSHFW WR D IUDPH RQ WKH ZUHQFK VFUHZf D[LV LH IAPRPHQ¯ >2 7O[ IO\ 7O]? fµ f *URXS WKHVH DSSDUHQWO\ XQLWOHVV FROXPQV LQWR >:]@E(Z WKH ZUHQFKHV RI FRQVWUDLQW ZLWK HLWKHU ]HUR IRUFH RU ]HUR PRPHQW 7KH FROXPQV RI ?:I?E(Z WKDW DUH QRW XQLWOHVV DUH FDOOHG WKH QRQ]HUR IRUFH DQG QRQ]HUR PRPHQW ZUHQFKHV RI FRQVWUDLQW ?:A]@E(Z ,I >:]@E(Z >:F@E(Z WKHQ WKH PDQLSXODWRU WZLVW VSDFH GHFRXSOHV DV VKRZQ LQ 7KHRUHP EHORZ 7KHRUHP m9 f«9 k n9% f«:F n:r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k 9QI LV DVVXPHG WKHQ WKH SURMHFWLRQ LQYROYHG LV (XFOLGHDQ LH 9I 5DQJH>A@ 5DQJH>-@ f

PAGE 93

r9 )LJXUH 'HFRPSRVLWLRQ RI WKH WZLVW VSDFH LQ IUDPH L LQWR GHFRXSOHG VXEVSDFHV DQG 9QI 1XOO>--I@ 1XOO>-W@ 1XOO>-U@ f ZKHUH -W PXVW EH SK\VLFDOO\ FRQVLVWHQW IURP WKH DVVXPSWLRQ %XW :F 1XOO>-7@ E\ GHILQLWLRQ 6LQFH 1XOO>-7@ FDQ EH LQWHUSUHWHG DV ERWK D WZLVW RI QRQIUHHGRPf DQG D ZUHQFK RI FRQVWUDLQWf WKHQ :F :r 7KH WZLVW VSDFH GHFRPSRVLWLRQ ZKHQ SRVVLEOH LV VKRZQ VFKHPDWLFDOO\ LQ )LJn XUH &RQGLWLRQV IRU WKLV GHFRPSRVLWLRQ DUH JLYHQ LQ 7KHRUHP DERYH DQG 7KHn RUHP EHORZ 7KH DERYH SURRI OHDGV WR D FRUROODU\ WKDW D VXEVSDFH 9V RI 9 FRQWDLQLQJ WKH WZLVWV RI IUHHGRP 9V ' 9 DOZD\V KDV D GLUHFW VXP GHFRPSRVLWLRQ 9 9 9 H >9L@E(Y >:]F@E(Z f ZKHUH 9 GRHV QRW H[LVW LV HPSW\f LI WKHUH DUH QR ZUHQFKHV RI FRQVWUDLQW ZLWK ]HUR IRUFH RU ]HUR PRPHQW LQ WKH FKRVHQ IUDPH &RUROODU\ b On9 k b f ZKHUH -9 & r96 & r9

PAGE 94

3URRI ,I r:r f¬:F WKHQ WKH SURRI RI WKLV FRUROODU\ LV LGHQWLFDO WR WKH SURRI RI 7KHn RUHP DQG f«9V 9 2WKHUZLVH LI n:r & :& WKHQ WKH SURRI DJDLQ IROORZV WKH UHDVRQLQJ RI WKH SURRI RI 7KHRUHP DOWKRXJK WKH GLPHQVLRQV RI WKH VSDFH n9V LV UHGXFHG IURP WKH GLPHQVLRQV RI n9f WR 'LP>9@ 'LPS99I@ 7R FRQWLQXH WKLV GLVFXVVLRQ RI WZLVW VSDFH GHFRPSRVLWLRQ VHSDUDWH WKH WZLVWV RI IUHHGRP LQWR OLQHDU YHORFLWLHV RI IUHHGRP DQG DQJXODU YHORFLWLHV RI IUHHGRP DQG VHSDn UDWH WKH ZUHQFKHV RI FRQVWUDLQW LQWR IRUFHV RI FRQVWUDLQW DQG PRPHQWV RI FRQVWUDLQW 9 YM 8I -)& 6F QU f ,Q D JLYHQ IUDPH L LI DOO b DUH RUWKRJRQDO WR DOO OYM DQG DOO f«QF DUH RUWKRJRQDO WR DOO OX!I WKHQ WKH PDQLSXODWRU GHFRXSOHV DQG WKH WZLVW VSDFH FDQ EH XQLTXHO\ GHFRPSRVHG LQWR WZLVWV RI IUHHGRP DQG WZLVWV RI QRQIUHHGRP VXEVSDFHV 7KHRUHP f¬9 rf¬9 k n9f± L r b k [9M 9nFn\ OQF k OL2M 9 nQF 3URRI $VVXPH f¬9 O9M k r9Q DQG UHPHPEHU IURP f WKDW b9M R :& )URP 7KHRUHP :& r:J ZKLFK LPSOLHV WKDW HLWKHU b RU OQF IRU HDFK ZUHQFK LQ r:& ,Q WKH FDVH b f§ n9M R :F OXM k LQF f DQG WKH ULJKWKDQG VLGH RI WKH WKHRUHP LV SURYHQ ,Q WKH FDVH OQF WKHQ n9 R r:& ^Yc nF f

PAGE 95

FRPSOHWLQJ WKH SURRI WKDW WKH ULJKWKDQG VLGH RI WKH WKHRUHP IROORZV IURP WKH OHIW KDQG VLGH 7KH RWKHU GLUHFWLRQ RI WKH SURRI SURFHHGV DV IROORZV 7KH ULJKWKDQG VLGH RI WKH WKHRUHP LPSOLHV WKDW f¬:F f«:_ DQG WKHQ WKH SURRI RI 7KHRUHP ZLOO VXIILFH )RU H[DPSOH WKH ZUHQFKHV RI FRQVWUDLQW LQ FROXPQUHGXFHG HFKHORQ IRUP RI WKH 35 YLUWXDO PDQLSXODWRU RI )LJXUH H[SUHVVHG LQ IUDPH LV >1XOO>-7@@(Z nn f ZKHUH f JLYHV WKH -DFRELDQ RI WKLV PDQLSXODWRU 7KH FRQGLWLRQV RQ WKH ULJKW KDQG VLGH RI 7KHRUHP DUH VDWLVILHG VLQFH WKH DERYH PDWUL[ LV DOVR >:r@L"Z 7KH FRQGLWLRQV RQ WKH ULJKW KDQG VLGH RI 7KHRUHP DUH DOVR PHW VLQFH F k X DQG QF k X!M f§ IRU DOO F YM QF DQG D! 7KHUHIRUH ERWK 7KHRUHP DQG 7KHRUHP WHOO XV WKDW WKH GHFRPSRVLWLRQ RI WKH WZLVW VSDFH LQWR XQLTXH GLVMRLQW VXEVSDFHV LV YDOLG LQ WKLV IUDPH 7KH -DFRELDQ RI WKH 35 PDQLSXODWRU H[SUHVVHG LQ WKH WUDQVODWHG IUDPH W ZDV JLYHQ LQ f 7KH ZUHQFKHV RI FRQVWUDLQW LQ FROXPQUHGXFHG HFKHORQ IRUP DUH W :F (f± f§ 3\ 3\ f ZKHUH S\ A ,I S\ WKH ILUVW FROXPQ RI A:AE(Z LV UHSODFHG E\ > OS[ @7 DQG WKH ODVW FROXPQ E\ > @7

PAGE 96

7KH UHTXLUHPHQW RI WKH ULJKW KDQG VLGH RI 7KHRUHP LV YLRODWHG E\ WKH DERYH >W:F@E(Z $OVR ERWK FRQGLWLRQV RQ WKH ULJKW KDQG VLGH RI 7KHRUHP DUH YLRODWHG E\ WKH ZUHQFK LQ WKH ILUVW FROXPQ RI f 7KHRUHP DQG 7KHRUHP OHDG WR D VLPLODU XQLTXH GHFRPSRVLWLRQ RI WKH ZUHQFK VSDFH 7KH ZUHQFK VSDFH FDQ VRPHWLPHV EH VSOLW LQWR WZR GLVMRLQW VXEVSDFHV WKH ZUHQFKHV RI FRQVWUDLQW DQG WKH ZUHQFKHV RI QRQFRQVWUDLQW O:QF %XW ILUVW GHILQH D VXEVSDFH 9M LQ D PDQQHU VLPLODU WR WKH GHILQLWLRQ RI 99" LW >9M@E(Y DUH WKH WZLVWV RI IUHHGRP ZLWK HLWKHU ]HUR OLQHDU YHORFLW\ RU ]HUR DQJXODU YHORFLW\ 7KLV OHDGV WR 7KHRUHP EHORZ 7KHRUHP ,I n: r:QF k :& !9I 9M 3URRI )LUVW SURYH WKDW WKHUH H[LVWV D GLUHFW VXP GHFRPSRVLWLRQ RI : LI 9 9M? DQG WKHQ SURYH WKDW LI WKHUH LV D GLUHFW VXP GHFRPSRVLWLRQ RI : WKHQ 9 9M ,I 9 9M WKH FROXPQUHGXFHG HFKHORQ IRUP EDVLV YHFWRUV RI >9M@E(Z KDYH QR XQLWV DQG FDQ WKHUHIRUH EH XVHG IRU D EDVLV RI : %XW VLQFH WKH GLPHQVLRQ RI 9 SOXV WKH GLPHQVLRQ RI :F LV VL[ DQG 9 9M WKHQ :QF :M 7KHUHIRUH >:QF@E(Z f§ >9M@E(Y 7KLV SURYHV RQH KDOI RI WKH WKHRUHP 7KH VHFRQG KDOI RI WKH WKHRUHP LV SURYHQ DV IROORZV ,I WKH GHFRPSRVLWLRQ : :F k :QF LV DVVXPHG WKHQ WKH SURMHFWLRQ LQYROYHG LV (XFOLGHDQ LH :F 1XOOI--W@ 1XOO>-7@ f :QF f§ 5DQJH>--r@ 5DQJH>-@ f ZKHUH -W PXVW EH SK\VLFDOO\ FRQVLVWHQW IURP WKH DVVXPSWLRQ %XW 9 5DQJHI-@ E\

PAGE 97

GHILQLWLRQ 6LQFH 5DQJH>-@ FDQ EH LQWHUSUHWHG DV ERWK D ZUHQFK RI QRQFRQVWUDLQWf DQG D WZLVW RI IUHHGRPf WKHQ 9 9I )LQDOO\ 7KHRUHP EHORZ VKRZV WKH HTXLYDOHQFH RI WKH GHFRPSRVLWLRQ RI WKH WZLVW DQG ZUHQFK VSDFHV ZKHQ -W LV SK\VLFDOO\ FRQVLVWHQW LH WKH XQLTXH (XFOLGHDQ GHFRPSRVLWLRQ RI WKH WZLVWV VSDFH UHVXOWV LQ WKH XQLTXH (XFOLGHDQ GHFRPSRVLWLRQ RI WKH ZUHQFK VSDFH DQG YLFHYHUVD 7KHRUHP ,I -W LV SK\VLFDOO\ FRQVLVWHQW WKH IROORZLQJ DUH HTXLYDOHQW VWDWHPHQWV b ,, f ZF f«f«:F f f¬: f«:QFkf¬A f fµnY O9I p b f 9 r: n9I k r:& 5DQJH>-@ k 1XOO>-7@ f 3URRI ,I 9I 9M :F :r 7KHRUHP DQG 7KHRUHP FDQ EH XVHG WR SURYH WKH HTXLYDOHQFH RI WKH UHVW RI WKH VWDWHPHQWV )URP 7KHRUHP :F :r}9 9k9Q 9k:F f )URP 7KHRUHP 9I 9M :&p :QF :F k 9 f 6LQFH WKH ULJKWKDQGVLGH GHFRPSRVLWLRQ RI WKHVH WZR HTXDWLRQV DUH LGHQWLFDO f DQG f DUH HTXLYDOHQW VWDWHPHQWV

PAGE 98

,I WKH WZLVW DQG ZUHQFK VFUHZ VSDFHV DUH XQLTXHO\ GHFRPSRVDEOH LQ D FKRVHQ IUDPH WKHQ D URWDWLRQ RI WKH IUDPH RI H[SUHVVLRQ RQ WKH GLVMRLQW VXEVSDFHV ZLOO SUHVHUYH GLVMRLQWHGQHVV VLQFH O*M n$M %XW D WUDQVODWLRQ RI WKH IUDPH RI H[SUHVVLRQ ZLOO QRW SUHVHUYH WKH GHFRPSRVLWLRQ RI WKH VXEVSDFHV ,Q IDFW RQO\ VSHFLDO PDQLSXODWRUV KDYH WKH WZR XQLTXH VXEVSDFHV WZLVWV RI FRQVWUDLQW DQG ZUHQFKHV RI IUHHGRPf IRU WZLVW DQG ZUHQFK VSDFH GHFRPSRVLWLRQV LQ DOO FRQILJXUDWLRQV 7KHVH PDQLSXODWRUV ZLOO EH GLVFXVVHG LQ 6HFWLRQ f *HQHUDOO\ WKH VHW RI WZLVWV WKDW D PDQLSXODWRU FDQQRW DFKLHYH 9Q LV QRW D VXEVSDFH RI WZLVWV VR QR XQLTXH 9QI FDQ EH IRXQG DQG JHQHUDOO\ WKH VHW RI ZUHQFKHV WKDW D PDQLSXODWRU FDQ DSSO\ :QF LV QRW D VXEVSDFH RI ZUHQFKHV VR QR XQLTXH :QF FDQ EH IRXQG 7KH 6& $5$ DQG WKH SODQDU 555 PDQLSXODWRU GLVFXVVHG HDUOLHU DUH VSHFLDO PDQLSn XODWRUV WKDW GHFRXSOH WKH WZLVW DQG ZUHQFK VSDFHV LQWR WZR GLVMRLQW VXEVSDFHV LQ DOO IUDPHV RI H[SUHVVLRQ )RU WKH 6&$5$ PDQLSXODWRU LQ D IUDPH DUELWUDULO\ WUDQVODWHG IURP IUDPH WKH FROXPQUHGXFHG HFKHORQ IRUP WZLVWV RI IUHHGRP DQG WKH FROXPQ UHGXFHG HFKHORQ IRUP ZUHQFKHV RI FRQVWUDLQW DUH n f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

PAGE 99

IUDPH ORFDWHG DW D GHFRXSOH SRLQW D GHFRXSOH IUDPHf LV SK\VLFDOO\ FRQVLVWHQW 6RPH QHZ PHDQLQJ RI GHFRXSOH SRLQWV FDQ QRZ EH SUHVHQWHG 7KHRUHP LV EDVHG RQ WKH FRQGLWLRQ WKDW -W LV SK\VLFDOO\ FRQVLVWHQW LH WKH IUDPH RI H[SUHVVLRQ LV ORFDWHG DW D GHFRXSOH SRLQW $OO RI WKH VWDWHPHQWV LQ WKLV WKHn RUHP DUH WKHUHIRUH WKH UHTXLUHPHQWV QHFHVVDU\ IRU D PDQLSXODWRU VSDFH ZLWK UHVSHFW WR D SDUWLFXODU IUDPH WR GHFRXSOH ,I WKH IUDPH RI H[SUHVVLRQ LV DW D GHFRXSOH SRLQW WKH WZLVW DQG ZUHQFK VSDFHV GHFRXSOH LGHQWLFDOO\ DV VKRZQ LQ f DQG f 5DLEHUW DQG &UDLJ >@ GHILQH D f¯FRQVWUDLQW IUDPHf° DV D IUDPH LQ ZKLFK WKH QDWXUDO DQG f¯RUWKRJRQDOf°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n WHU DQ\ URWDWLRQV RI WKH IUDPH DW WKLV SRLQW ZLOO QRW DIIHFW WKH GHFRXSOHG QDWXUH RI WKH VSDFHV :KHQ WKH IUDPH RI H[SUHVVLRQ LV QRW ORFDWHG DW D GHFRXSOH SRLQW WKH WZLVW DQG ZUHQFK VSDFHV FDQQRW EH XQLTXHO\ GHFRPSRVHG E\ D GLUHFW VXP %XW D SDUW RI WKH WZLVW RU ZUHQFK VSDFHV PD\ EH XQLTXHO\ GHFRPSRVDEOH VR WKDW 6XEVSDFHS9@ b k b b k :&rA f 6XEVSDFHSn:@ :& p s : f )RU DQ\ IUDPH L D ZUHQFK FRRUGLQDWH WUDQVIRUPDWLRQ L$WW H[LVW WKDW ZLOO FRQYHUW DQ\ VLQJOH ZUHQFK RI FRQVWUDLQW ZLWK QRQ]HUR IRUFH DQG QRQ]HUR PRPHQW WR D ZUHQFK

PAGE 100

ZLWK D ]HUR PRPHQW DQG WKH VDPH IRUFH 7KLV SDUWLFXODU ZUHQFK FRRUGLQDWH WUDQVIRUn PDWLRQ FRQVLVWV RI D WUDQVODWLRQ YHFWRU RI 3 Q [ a: f DQG QR URWDWLRQ 1RWH WKDW WKLV WUDQVIRUPDWLRQ ZLOO DOVR JHQHUDOO\ FRQYHUW RWKHU ZUHQFKHV WKDW KDG ]HUR PRPHQWV WR ZUHQFKHV ZLWK QRQ]HUR PRPHQWV 7KHUHIRUH IRU DOO PDQLSXODWRUV ZLWK -DFRELDQ RI UDQN OHVV WKHQ VL[ LHD QRQn HPSW\ ZUHQFK RI FRQVWUDLQW VXEVSDFHf WKHUH H[LVWV D IUDPH WKDW PDNHV DW OHDVW RQH RI WKH FRQVWUDLQW ZUHQFKHV LQWR DQ HOHPHQW RI :r DQG WKXV :_ LQ VRPH IUDPH )RU H[DPSOH D 3 PDQLSXODWRU LQ IUDPH FRRUGLQDWHV KDV WKH -DFRELDQ DQG FROXPQUHGXFHG HFKHORQ IRUP ZUHQFK RI FRQVWUDLQW EDVLV RI U ,6 ‘ m A * f§DF f§ 2& fµ6 6L >:F@E(Z f§ 6D&&f e/ 6L & F f 1RWH WKDW IUDPH LV QRW D GHFRXSOH IUDPH %XW f FDQ EH XVHG ZLWK f WR ILQG D IUDPH ZKHUH WKH PDQLSXODWRU GRHV GHFRXSOH 3 r:& G& Ff r& D f fµ6 fµ6 m& m&fA 7 f f 7KH SK\VLFDOO\ FRQVLVWHQW GHWHUPLQDQW RI -7LQ IUDPH W LV 'HWSW-@ f§ XrnVnVf fµ f $ QRQSODQDU 555 PDQLSXODWRU ZLWK 'HQDYLW+DUWHQEHUJ SDUDPHWHUV JLYHQ LQ 7DEOH KDV D IUDPH -DFRELDQ DQG FROXPQUHGXFHG HFKHORQ IRUP ZUHQFKHV RI

PAGE 101

7DEOH ' + SDUDPHWHUV IRU D QRQ MODQDU 555 PDQLSXODWRU -RLQW 7\SH G D D 5 DL L W 5  r WW 5 FRQVWUDLQW EDVLV RI & n U 6 6 DL& DF & G& f§ 2O f§ & 6 6 ZF (Z f§ [ F f 6LQFH QR VLQJOH ZUHQFK FRRUGLQDWH WUDQVIRUPDWLRQ ZLOO FRQYHUW ERWK RI WKH ILUVW WZR FROXPQV RI >:F(Z?E WR D IRUP WKDW VDWLVILHV WKH ULJKW KDQG VLGH RI 7KHRUHP WKLV PDQLSXODWRU KDV QR IUDPH DW ZKLFK WKH WZLVW DQG ZUHQFK VSDFHV GHFRXSOH :UHQFK FRRUGLQDWH WUDQVIRUPDWLRQV ZLOO QRW DIIHFW FROXPQ D SXUH IRUFH IURP UHPDLQLQJ D SXUH IRUFH (LWKHU FROXPQ RU FROXPQ FDQ EH PDGH LQWR D ]HUR IRUFH ZUHQFK RI FRQVWUDLQW JLYHQ DQ DSSURSULDWH ZUHQFK FRRUGLQDWH WUDQVIRUPDWLRQ ZLWK WUDQVODWLRQ FDOFXODWHG IURP f ,Q WKLV FDVH :r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a7 (79-7 (Y(aO@ 'HW>-(Yf7-(Yf? 'HW >Be"f±@ f

PAGE 102

VLQFH 'HW>(M@ 'HW>(O@ 7KH XQLWV RI HDFK HOHPHQW LQ D JLYHQ URZ RI (Y DUH LGHQWLn FDO DQG WKHUHIRUH WKH GHWHUPLQDQW RI (Y LV SK\VLFDOO\ FRQVLVWHQW 7KHQ IURP f 'HW>-7-@ LV SK\VLFDOO\ FRQVLVWHQW LI DQG RQO\ LI 'HW>-(9f7^-(9f? LV SK\VLFDOO\ FRQVLVn WHQW 7KH PDWUL[ >9I?E(Y -(9 KDV QR SK\VLFDO XQLWV LI -W LV SK\VLFDOO\ FRQVLVWHQW DQG WKHUHIRUH 'HW>-(YfU-(Yf? LV SK\VLFDOO\ FRQVLVWHQW LI -W LV SK\VLFDOO\ FRQVLVWHQW +HQFH WKH IUDPH RI H[SUHVVLRQ LV D GHFRXSOH SRLQW DQG 'HW>-7 -@ LV SK\VLFDOO\ FRQVLVn WHQW LI DQG RQO\ LI -W LV SK\VLFDOO\ FRQVLVWHQW )RU H[DPSOH WKH 6&$5$ PDQLSXODWRU LQ IUDPH FRRUGLQDWHV KDV -(9 n (Y D L mr 33& r 3L 3L r f 7KH GHWHUPLQDQWV LQ IUDPH FRRUGLQDWHV 'HW>-7-f@ DDV? DQG 'HW>-(9f7-(Yf` DUH ERWK SK\VLFDOO\ FRQVLVWHQW DQG WKXV IUDPH LV D GHFRXSOH IUDPH IRU WKH 6&$5$ PDQLSXODWRU 7KH GHWHUPLQDQW RI (Y LV f§ODLDVf )RU DQRWKHU H[DPSOH WKH 5553 PDQLSXODWRU H[SUHVVHG LQ IUDPH FRRUGLQDWHV KDV R F R R ‘ U R DL f§D\F F &B & i ,, 3r WR 3r 7KH GHWHUPLQDQWV LQ IUDPH FRRUGLQDWHV f 'HW> -7 -f@ DVfO D DA D[DFf f DQG 'HWO-(Yf7-(Yf? DM DF DDFf f F DUH ERWK SK\VLFDOO\ LQFRQVLVWHQW DQG WKXV IUDPH LV QRW D GHFRXSOH IUDPH IRU WKH 5553 PDQLSXODWRU 7KH GHWHUPLQDQW RI (Y LV f§OD&6f

PAGE 103

7HVWV FDQ QRZ EH FOHDUO\ VWDWHG WR GHWHUPLQH WKH FRQGLWLRQV IRU GHFRXSOH SRLQWV $ WHVW WR GHWHUPLQH LI D PDQLSXODWRU GHFRXSOHV IRU DOO FRQILJXUDWLRQV ZLWK RULJLQV DW WKH RULJLQ RI IUDPH L LV SK\VLFDOO\ FRQVLVWHQW 'HW>n7 r-f@ IRU DOO FRQILJXUDWLRQV f $ WHVW WR GHWHUPLQH LI D PDQLSXODWRU GHFRXSOHV IRU DOO FRQILJXUDWLRQV DW HYHU\ SRLQW LV SK\VLFDOO\ FRQVLVWHQW 'HW O*WW ?SK\VLFDOO\ FRQVLVWHQW 'HW ^-7 O*Wnb ?IRU DOO FRQILJXUDWLRQV RU IRU DOO FRQILJXUDWLRQV f f ZKHUH O*WW LV WKH JHQHUDO WUDQVODWLRQ PDWUL[ JLYHQ E\ f ZLWK QR URWDWLRQ 5 f DQG L LV VRPH FRQYHQLHQW IUDPH 0LGIUDPH -DFRELDQV DUH DOZD\V VLPSOHU V\PEROLn FDOO\ WKDQ HQGIUDPH RU EDVHIUDPH -DFRELDQV >@f 1RWH WKDW WKH PDWUL[ SURGXFW W*Wf7 r*rf¬r LQ f LV SK\VLFDOO\ LQFRQVLVWHQW 2I FRXUVH WKH DERYH WHVW RI f DQG f RU DQ\ RI WKH RWKHU WHVWV IRU GHFRXSOH SRLQWV FDQ EH XVHG ZLWK D JHQHUDO WUDQVODWLRQ PDWUL[ WR GHWHUPLQH WKH YDULRXV FRQGLWLRQV IRU GHFRXSOH SRLQWV 3HUKDSV WKH VLPSOHVW WHVW WR ILQG D PDQLSXODWRUf¬V GHFRXSOH SRLQWV LV WR GHWHUPLQH WKH FRQGLWLRQV LI DQ\f IRU ZKLFK O*WU ^-(p9_ ZKHUH (ILV WKH PDWUL[ WKDW SXWV L*WnW L-(ALQ FROXPQUHGXFHG HFKHORQ IRUP 7KH ILQDO SDUW RI WKH WZLVW VSDFH LV WKH WZLVW GHIHFW PDQLIROG 9P 9GP 99I 9L f 6LPLODUO\ WKH ILQDO SDUW RI WKH ZUHQFK VSDFH LV WKH ZUHQFK GHIHFW PDQLIROG :72 :G72 ::F:G f $ QRQHPSW\ WZLVW RU ZUHQFK GHIHFW PDQLIROG LV QHYHU D VXEVSDFH %RWK WKH WZLVW DQG ZUHQFK GHIHFW PDQLIROGV DUH HPSW\ LI WKH IUDPH RI H[SUHVVLRQ LV D GHFRXSOH IUDPH

PAGE 104

7R VXPPDUL]H VHYHUDO WHVWV WR GHWHUPLQH LI D SRLQW DW ZKLFK WKH FRRUGLQDWHV RI D PDQLSXODWRU DUH H[SUHVVHGf LV D GHFRXSOH SRLQW DUH DV IROORZV fµ IRU DOO FRQILJXUDWLRQV WKH PDWUL[ f¬-O LV SK\VLFDOO\ FRQVLVWHQW fµ IRU DOO FRQILJXUDWLRQV WKH 'HW>r-7 -@ LV SK\VLFDOO\ FRQVLVWHQW fµ IRU DOO FRQILJXUDWLRQV >n9M@E(Y f§ >W9\@IFef¬O DQG fµ IRU DOO FRQILJXUDWLRQV >n:F@E(Z f§ A:I?E(Z 6HOI5HFLSURFDO 0DQLSXODWRUV $ FODVV RI PDQLSXODWRUV DUH QRZ LQWURGXFHG IRU ZKLFK WKH WZLVWV RI IUHHGRP DUH VHOIUHFLSURFDO LH -7$>@QQ f ([SDQGLQJ f WKH VHOIUHFLSURFDO WHVW EHFRPHV -WDM UYMf± UMY >R@%Q +f ZKHUH 7 -Y M f§ MM DQG Z ‘ mQVf 'HILQLWLRQ $ PDQLSXODWRU LV VHOIUHFLSURFDO IRU FRQILJXUDWLRQ T LQ IUDPH L LI DQG RQO\ LI L-Tf7$L-^Tf >@Q% f 7KHRUHP EHORZ VKRZV WKDW L-7$L>@QQ LV D IUDPH LQGHSHQGHQW FKDUDFWHULVWLF

PAGE 105

7KHRUHP ,I D PDQLSXODWRU LV VHOIUHFLSURFDO IRU FRQILJXUDWLRQ T LQ IUDPH L WKHQ LW LV DOVR VHOIUHFLSURFDO IRU FRQILJXUDWLRQ T LQ DQ\ RWKHU IUDPH M 3URRI $VVXPH O-7$-f f§ >@f±!Q ZKHUH f§ -Tf LH WKH FRQILJXUDWLRQ GHSHQGHQF\ LV DVVXPHG )URP f DQG f 9 M*In On 5 %5 n LM 2 \ 5O-Y %5% 5 L 5b f 7KHQ -7$8 ?UY-Y -7-9f L-8-57%7 %f5-f±f f %XW E\ f %7 % >@L VR WKDW XVLQJ f UHVXOWV LQ M-7$M?-79-: -0 97$r>@QQ fµ f 7KLV WKHRUHP OHDGV WR WKH GHILQLWLRQ RI VHOIUHFLSURFDO PDQLSXODWRUV EHORZ 'HILQLWLRQ $ PDQLSXODWRU LV VHOIUHFLSURFDO LI DQG RQO\ LI -Tf7$-Tf >@QQ IRU DOO FRQILJXUDWLRQV T $ VXEVHW RI WKH VHOIUHFLSURFDO PDQLSXODWRUV DUH WKH PDQLSXODWRUV IRU ZKLFK ERWK WHUPV LQ WKH VXPPDWLRQ RI f DUH ]HUR LH UYMZ UMY >R@QQ f 7KHUH DUH WZR W\SHV RI PDQLSXODWRUV WKDW VDWLVI\ f DOO SULVPDWLFMRLQWHG PDn QLSXODWRUV DQG SODQDU PDQLSXODWRUV 3ODQDU PDQLSXODWRUV DUH GHILQHG EHORZ 'HILQLWLRQ $ SODQDU PDQLSXODWRU ZLOO FUHDWH RQO\ OLQHDU PRWLRQ LQ D SODQH DQG DQJXODU PRWLRQ SHUSHQGLFXODU WR WKDW SODQH IRU DOO FRQILJXUDWLRQV

PAGE 106

&RQVHTXHQFHV RI WKLV GHILQLWLRQ IRU URERWV FRPSRVHG HQWLUHO\ RI UHYROXWH DQG SULVn PDWLF MRLQWV DUH JLYHQ LQ )DFW EHORZ )DFW 3ODQDU PDQLSXODWRUV KDYH WKH IROORZLQJ FKDUDFWHULVWLFV fµ 7KH FURVV SURGXFWV RI DOO SULVPDWLF MRLQW D[HV DUH SDUDOOHO fµ 5HYROXWH MRLQW D[HV DUH SDUDOOHO fµ $OO UHYROXWH MRLQW D[HV DUH RUWKRJRQDO WR DOO SULVPDWLF MRLQW D[HV fµ 7KH FURVV SURGXFWV RI DOO SULVPDWLF MRLQW D[HV DUH SDUDOOHO WR DOO UHYROXWH MRLQW D[HV 6SKHULFDO PDQLSXODWRUV DUH VHOIUHFLSURFDO EXW GR QRW VDWLVI\ f 7KH GHILQLn WLRQ RI VSKHULFDO PDQLSXODWRUV LV JLYHQ LQ WKH GHILQLWLRQ EHORZ 'HILQLWLRQ $ VSKHULFDO PDQLSXODWRU ZLOO FUHDWH OLQHDU PRWLRQ RI DQ\ IL[HG SRLQW LQ WKH WRROIUDPH RQO\ LQ GLUHFWLRQV WDQJHQW WR D VSKHUHf¬V VXUIDFH 7KH VSKHUH LV IL[HG IRU HYHU\ WRRO VKDSH 7KHUH DUH QR FRQVWUDLQWV RQ DQJXODU PRWLRQ IRU VSKHULFDO PDQLSXODWRUV &RQVHTXHQFHV RI WKLV GHILQLWLRQ IRU URERWV FRPSRVHG HQWLUHO\ RI UHYROXWH DQG SULVn PDWLF MRLQWV DUH JLYHQ LQ )DFW EHORZ )DFW 6SKHULFDO PDQLSXODWRUV KDYH WKH IROORZLQJ FKDUDFWHULVWLFV fµ 7KH\ DUH FRPSRVHG HQWLUHO\ RI UHYROXWH MRLQWV LH KDYH QR SULVPDWLF MRLQWV fµ $OO UHYROXWH MRLQW D[HV LQWHUVHFW DW D VLQJOH SRLQW 7KH VHOIUHFLSURFDO FRQGLWLRQ -7$>@f±Q LV YDOLG IRU DOO VSKHULFDO PDQLSXODWRUV EXW -f±-: f§ >@%Q RQO\ LQ D IUDPH ORFDWHG DW WKH LQWHUVHFWLRQ SRLQW RI HDFK VSKHULFDO PDQLSXODWRUf¬V MRLQW D[HV 7KLV LV LOOXVWUDWHG E\ WKH EHORZ H[DPSOH

PAGE 107

7KH 55 PDQLSXODWRU LV D VSKHULFDO PDQLSXODWRU ZKRVH -DFRELDQ LQ D IUDPH DUELWUDULO\ WUDQVODWHG IURP IUDPH HTXDOV O*n 3\ A 3]& t 3\ 3[t 7 3]t A 3[ 3[:O 3\9?6 O6 FDL .L f 7KLV UHVXOWV LQ -7$>@Q!Q EXW -7 B BW77 B "? ^3[&3\6f UL^S[F a3\6f f 7KH 55 PDQLSXODWRU LV VHOIUHFLSURFDO EXW -A-: >f@f±f± IRU DOO FRQILJXUDWLRQV RQO\ DW WKH D[HV LQWHUVHFWLRQ KH S[ f§ S\ 6LPLODUO\ WKH 555 VSKHULFDO PDQLSXODWRU LV VHOIUHFLSURFDO EXW WO7 f§ f§r7 f§ &S[ S]6 a3\6 A 3[ 3]t 3[ 3\6 3; f ,I S[ SY S] f§ LQ f WKHQ WKH IUDPH LV DW WKH LQWHUVHFWLRQ RI WKH MRLQW D[HV DQG -O-Z f§ ,Q RUGHU WR ILQG WKH FODVV RI DOO 555 PDQLSXODWRUV WKDW DUH VHOIUHFLSURFDO OHW XV ORRN DW -7$LQ IUDPH IRU WKH JHQHUDO 555 PDQLSXODWRU -7$-f 4L?M f§DL7L f§ DD f§DFU f ZKHUH f§ f§DLQFUL f§ D.F7L f§ D.?FU f§ D?.?FM G7?UV f 7KH FRQGLWLRQV IRU ZKLFK -7$>f@f±f± LQ DOO FRQILJXUDWLRQV IRU WKH JHQHUDO 555 PDQLSXODWRU DUH DL DQG D f§ DQG 7? f§ RU FU RU G f f

PAGE 108

RU 7L DQG (DFK RI WKHVH FDVHV UHVXOW LQ D PDQLSXODWRU WKDW LV HLWKHU SODQDU RU RI WKH VHW RI FRQGLWLRQV RI f LV YDOLG WKHQ WKH PDQLSXODWRU LV FRQGLWLRQV RI f DUH YDOLG WKHQ WKH PDQLSXODWRU LV SODQDU 7KH UHVXOWV DUH VXPPDUL]HG LQ WKH IDFWV EHORZ )DFW 6HOIUHFLSURFDO PDQLSXODWRUV DUH fµ 2QO\ SULVPDWLFMRLQWHG fµ 3ODQDU RU fµ 6SKHULFDO )DFW (QWLUHO\ SULVPDWLFMRLQWHG PDQLSXODWRUV DQG SODQDU PDQLSXODWRUV GHFRXSOH DQG KDYH --: f§ >@f±LQ LQ DOO IUDPHV )DFW 6SKHULFDO PDQLSXODWRUV DUH VHOIUHFLSURFDO )RU WKHVH PDQLSXODWRUV -A-: f§-A-\ $W WKH LQWHUVHFWLRQ RI WKH UHYROXWH D[HV >@f±!Q ,I D PDQLSXODWRU LV VHOIUHFLSURFDO WKHQ 5DQJH>$-@ & :F f IRU VRPH VFDODU 7KLV LV GHULYHG IURP 7KHRUHP DQG WKH UHVXOWLQJ IDFW WKDW -7:& >@Qf§U IrU DQ\ :F DQG -7$-f >@QLf± IRU DOO FRQILJXUDWLRQV ZKHUH U LV WKH FROXPQ UDQN RI ,I WKH FROXPQ UDQN RI LV WKHQ 5DQJH>$-@ :F f VLQFH WKH UDQN RI SOXV WKH UDQN RI :F LV DOZD\V HTXDO WR VL[ 7KHUHIRUH WKH PD[LPXP QXPEHU RI LQGHSHQGHQW MRLQWV IRU D UHFLSURFDO PDQLSXODWRU LV WKUHH 0DQLSXODWRUV ZLWK f VSKHULFDO ,I DQ\ VSKHULFDO ,I WKH

PAGE 109

PRUH WKDQ WKUHH MRLQWV PD\ EH VHOIUHFLSURFDO RQO\ LI WKH\ DUH UHGXQGDQW LH WKH UDQN RI LV OHVV WKDQ RU HTXDO WR 7KH  fWK WHUP RI -7$LV DOZD\V ]HUR IRU DQ\ PDQLSXODWRU VLQFH E\ 7KHRUHP DOO FROXPQV RI D PDQLSXODWRU -DFRELDQ DUH VHOIUHFLSURFDO LH -7$-fL -f $-f f 2 -Zf f VR WKDW k AfR f /HW WKH WK FROXPQ RI -Y EH UHSUHVHQWHG E\ Yc DQG WKH WK FROXPQ RI EH UHSUHVHQWHG E\ Z 7KH WZLVW GXH WR WKH MRLQW LV WKHUHIRUH 9c >X Z@7 DQG f EHFRPHV 9L -L YMX!L f UHJDUGOHVV RI WKH IUDPH RI H[SUHVVLRQ RU WKH FRQILJXUDWLRQ RI WKH PDQLSXODWRU )RU SODQDU PDQLSXODWRUV WKH FRQGLWLRQV LQ )DFW PDNH WKH IXUWKHU UHTXLUHPHQWV WKDW 9L 4L-M 9 L M 8L ; /œ9 L M 9Nr9M EW[YLL RU 9 L M N O 7KLV OHDGV WR 7KHRUHP EHORZ 9L ; 9M N [ YM? f f f 7KHRUHP 3ODQDU PDQLSXODWRUV GHFRXSOH LQ HYHU\ IUDPH 3URRI %\ WKH GHILQLWLRQ RI SODQDU PRWLRQ WKHUH H[LVWV D IUDPH VXFK WKDW WKH OLQHDU PRWLRQ LV LQ WKH [\SODQH DQG WKH DQJXODU PRWLRQ LV DERXW WKH ]D[LV IRU DOO FRQILJXUDWLRQV

PAGE 110

,I WKH UDQN RI WKH -DFRELDQ LV WKHQ LQ WKLV IUDPH WKH WZLVW RI IUHHGRP DUH 9I 5DQJH>5@ %I f 7KHQ WUDQVODWLRQ LQ WKH ] GLUHFWLRQ DQG URWDWLRQ DERXW WKH D DQG \ D[LV DUH LQDFFHVVLEOH WZLVWV 9L f§ 5DQJH>,"@ %^ f 7KHVH WZLVWV FRQVWLWXWH D VXEVSDFH 6LQFH HDFK RI WKH LQDFFHVVLEOH WZLVWV KDYH HLWKHU Y f§ RU Xf WKH\ PD\ DSSURSULDWHO\ EH LQWHUSUHWHG DV ZUHQFKHV RI FRQVWUDLQW 7KXV E\ 7KHRUHP DOO SODQDU PDQLSXODWRUV ZLWK -DFRELDQ RI UDQN GHFRXSOH LQ WKLV IUDPH ,I WKH PDQLSXODWRU KDV OHVV WKDQ WKUHH MRLQWV RU WKH UDQN RI LV OHVV WKDQ WKUHHf WKHQ RQH RU PRUH RI WKH IUHHGRP WZLVWV DERYH ZLOO EHFRPH LQDFFHVVLEOH WZLVWV 7KH QHZ LQDFFHVVLEOH WZLVW RU WZLVWV PD\ DOVR EH LQWHUSUHWHG DV ZUHQFKHV RI FRQVWUDLQW VLQFH HDFK RI WKH SRVVLEOH WZLVW RI IUHHGRP LQ f DOVR PHHW WKH UHTXLUHPHQW RI HLWKHU Y RU XM 7KXV DOO SODQDU PDQLSXODWRU GHFRXSOH LQ WKLV IUDPH $ WUDQVODWLRQ RI WKH IUDPH GRHV QRW DIIHFW WKH LQDFFHVVLEOH WZLVWV IRU SODQDU PDn QLSXODWRUV ,W KDV EHHQ SUHYLRXVO\ VKRZQ WKDW IUDPH URWDWLRQV KDYH QR DIIHFW RQ ZKHWKHU D PDQLSXODWRU GHFRXSOHV RU QRW 7KHUHIRUH SODQDU PDQLSXODWRUV GHFRXSOH LQ DOO IUDPHV 6LQFH SODQDU PDQLSXODWRUV GHFRXSOH DW HYHU\ SRLQW WKHQ -W LV SK\VLFDOO\ FRQVLVWHQW UHJDUGOHVV RI WKH IUDPH RI H[SUHVVLRQ :KHQ XVLQJ WKH SVHXGRLQYHUVH VROXWLRQ RI WKH LQYHUVH YHORFLW\ SUREOHP VRPH UHVHDUFKHUV KDYH WKHUHIRUH WHUPHG D VROXWLRQ DV

PAGE 111

f¯RSWLPDOf° ZKHQ LQ IDFW WKH VROXWLRQ LV VWLOO RQO\ f¯RSWLPDOf° ZLWK UHVSHFW WR D SK\VLFDOO\ LQFRQVLVWHQW (XFOLGHDQ QRUP 7KLV f¯RSWLPDOf° UHVXOW LV D JHQHUDOL]HGLQYHUVH VROXWLRQ XVLQJ LGHQWLW\ PHWULFV LQ WKH f¯RSWLPDOf° IUDPH $SSO\LQJ WKH SVHXGRLQYHUVH VROXWLRQ LQ DQRWKHU IUDPH ZLOO JLYH D GLIIHUHQW f¯RSWLPDOf° VROXWLRQ FRUUHVSRQGLQJ WR DQ LGHQWLW\ PHWULF LQ WKLV QHZ IUDPH )RU H[DPSOH WKH SODQDU 555 PDQLSXODWRU LV VROYHG EHORZ LQ IUDPH FRRUGLQDWHV DQG LQ D IUDPH DUELWUDULO\ WUDQVODWHG IURP IUDPH 7KH GHVLUHG WZLVW LV 9 U Lf§ L V ‘ L P L n [ 6 n V V RP ] V ,,, S[ L 6 V U V A R P L S[ 3Y UDG rY *Wn9 r 6 6 6 D UDG A V A V 7t V P / V / V f 7KH IROORZLQJ SDUDPHWHU YDOXHV KDYH EHHQ VHOHFWHG D[ D OP L UUDG f§ LUUDG LUUDG f 7KH SVHXGRLQYHUVH VROXWLRQV LQ HDFK RI WKHVH IUDPHV DUH RUDG r MW 9 UDG ¯ VW : r9 RUDG B S] ] V PV W SJ 2= f± I PV UDr S] f )RU DQ\ QRQ]HUR S] WKHVH VROXWLRQ DUH GLIIHUHQW 7KH UHVXOWLQJ DFWXDO WZLVWV REWDLQHG E\ VXEVWLWXWLQJ WKHVH MRLQWUDWH YHFWRUV LQWR 9 f§ -T H[SUHVVHG LQ IUDPH FRRUGLQDWHV DUH WR Kr OOBe [ 6 6 RP Sr  6 A 6 9 -T r9VW W*f¬WW-TVW UDG / V mg f 7KH TXHVWLRQ VKRXOG WKHQ EH DVNHG f¯ZK\ LV VROXWLRQ TV EHWWHU WKDQ DQ\ RI WKH SRVVLEOH VROXWLRQV RI ,I RQH LV WR FODLP WKDW WKHUH LV f¯VRPHWKLQJ VSHFLDOf° DERXW IUDPH

PAGE 112

IRU LQVWDQFH VXFK WKDW _9M X X Z KDV VRPH XVHIXO PHDQLQJ WKHQ WR JHW WKH VDPH DQVZHU LQ DQRWKHU IUDPH WKH DVVXPHG LGHQWLW\ PHWULF XVHG WR DUULYH DW D SK\VLFDOO\ FRQVLVWHQW VROXWLRQ PXVW EH DSSURSULDWHO\ WUDQVIRUPHG LQ WKH VROXWLRQ IRU WKH WUDQVODWHG IUDPH 7KH LGHQWLW\ PHWULF XVHG LQ VROYLQJ IRU TV LV 0Y 6n RI f ZLWK DYDZ XQLWV>R@XQLWV>X@ VR WKDW 9 k 0Y9 LV SK\VLFDOO\ FRQVLVWHQW 7KH PHWULF LQ WKH WUDQVODWHG IUDPH LV IRXQG IURP f WR EH 0nY 09*f¬ f ,I 0nY LV XVHG LQ WKH JHQHUDOL]HGLQYHUVH HTXDWLRQ f WKHQ TnVW O9 TV LH WKH JHQHUDOL]HGLQYHUVH VROXWLRQ LV LQYDULDQW WR WUDQVODWLRQV DQG JLYHV WKH VDPH VROXWLRQ DV WKDW REWDLQHG XVLQJ WKH SVHXGRLQYHUVH LQ IUDPH 1R PHWULF 0T LV QHHGHG VLQFH KDV IXOO FROXPQ UDQN IRU WKH JLYHQ FRQILJXUDWLRQf 6XPPDUL]LQJ WKH UHVXOWV RI WKH DERYH H[DPSOH QHLWKHU WKH MRLQWUDWH VROXWLRQV QRU WKH LQGXFHG WZLVWV DUH JHQHUDOO\ HTXDO ZKHQ WKH SVHXGRLQYHUVH VROXWLRQ WHFKQLTXH LV XVHG LQ WZR IUDPHV WKDW DUH WUDQVODWHG IURP HDFK RWKHU $V IRU DOO PDQLSXODWRUV QRW MXVW SODQDU PDQLSXODWRUV f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

PAGE 113

:F :J DQG D IUDPH FDQ DOZD\V EH IRXQG VXFK WKDW >:F@E(Z f« n f 3ODQDU DQG 6&$5$W\SH PDQLSXODWRUV ZKLFK KDYH DV RI QRZ EHHQ LGHQWLILHG DV WKH RQO\ PDQLSXODWRUV W\SHV WKDW GHFRXSOH LQ HYHU\ IUDPH DUH RIWHQ XVHG E\ UHVHDUFKHUV WR GHPRQVWUDWH YDULRXV FRQWURO DOJRULWKPV 7KLV PD\ VLPSOLI\ WKH VROXWLRQV EXW PD\ OHDG WR LQYDOLG JHQHUDOL]DWLRQV

PAGE 114

&+$37(5 6800$5< $1' &21&/86,216 6HYHUDO DOJHEUDLF WHFKQLTXHV QRW JHQHUDOO\ DSSURSULDWH IRU QRQFRPPHQVXUDWH V\Vn WHPV KDYH EHHQ QRWHG DV EHLQJ ZLGHO\ DSSOLHG LQ WKH OLWHUDWXUH WR SUREOHPV LQ WKH QRQFRPPHQVXUDWH V\VWHP RI URERWLFV 3ULPDU\ DPRQJ WKHVH ZDV WKH SVHXGRLQYHUVH DQG WKH HLJHQVWUXFWXUH RI WKHVH V\VWHPV ,Q OLQHDU QRQFRPPHQVXUDWH V\VWHPV X f§ $[ FRQVWUDLQWV RQ WKH SRVVLEOH SK\VLFDO XQLWV RI $ ZHUH JLYHQ LQ 6HFWLRQ ,I WKHVH UHTXLUHPHQWV DUH YLRODWHG WKH V\VWHP LV SK\VLFDOO\ LQFRQVLVWHQW $OO OLQHDU V\VWHPV FDQ WKHUHIRUH EH FODVVLILHG LQWR HLWKHU SK\Vn LFDOO\ FRQVLVWHQW RU SK\VLFDOO\ LQFRQVLVWHQW V\VWHPV )RU FRPPHQVXUDWH V\VWHPV WKH SK\VLFDO XQLWV RI DOO WKH HOHPHQWV LQ $ DUH LGHQWLFDO 7KH UHTXLUHPHQWV RQ WKH SK\VLn FDO XQLWV RI $ JLYHQ IRU QRQFRPPHQVXUDWH V\VWHPV DUH LQ IDFW YDOLG IRU DOO SK\VLFDOO\ FRQVLVWHQW V\VWHPV 7KLV GLVVHUWDWLRQ SXWV WR UHVW WKH FXUUHQW PDQLSXODELOLW\ WKHRU\ /LQHDU QRQFRPn PHQVXUDWH V\VWHPV GR QRW JHQHUDOO\ KDYH LQYDULDQW RU SK\VLFDOO\ FRQVLVWHQW HLJHQYDOXHV DQG HLJHQYHFWRUV 7KH UHTXLUHPHQWV IRU D QRQFRPPHQVXUDWH V\VWHP WR SRVVHVV DQ LQn YDULDQW HLJHQV\VWHP ZDV SUHVHQWHG LQ 6HFWLRQ ,Q URERWLFV WKH ZLGHO\ DFFHSWHG WKHRU\ RI PDQLSXODWRU PDQLSXODELOLW\ EDVHG RQ WKH HLJHQVWUXFWXUH RI YDULRXV IXQFWLRQV RI WKH PDQLSXODWRU -DFRELDQ ZDV VKRZQ LQ &KDSWHU WR EH LQYDOLG LQ DOO FDVHV 7KH PDQLSXODELOLW\ PHDVXUH 'HW>-7-@ LV YDOLG DW GHFRXSOH SRLQWV VLQFH WKLV PHDn VXUH LV SK\VLFDOO\ FRQVLVWHQW 7KXV WKH PDQLSXODELOLW\ DW D VLQJOH GHFRXSOH SRLQW LQ RQH FRQILJXUDWLRQ FDQ EH PHDQLQJIXOO\ FRPSDUHG WR WKH PDQLSXODELOLW\ DW RWKHU FRQn ILJXUDWLRQV %XW VLQFH 'HW>-7-@ LV QRW LQYDULDQW WR WUDQVODWLRQV WKH PDQLSXODELOLW\ PHDVXUH DW GLIIHUHQW GHFRXSOH SRLQWV FDQ QRW EH PHDQLQJIXOO\ FRPSDUHG

PAGE 115

,W ZDV DOVR VKRZQ WKDW SK\VLFDOO\ FRQVLVWHQW OLQHDU QRQFRPPHQVXUDWH V\VWHPV GR QRW KDYH SK\VLFDOO\ FRQVLVWHQW VLQJXODU YDOXH GHFRPSRVLWLRQV 2QO\ FRPPHQVXUDWH OLQHDU V\VWHPV KDYH SK\VLFDOO\ FRQVLVWHQW VLQJXODU YDOXH GHFRPSRVLWLRQV 7KH PDQLSXODWRU -DFRELDQ PDSV SRVVLEO\ QRQFRPPHQVXUDWH URERW MRLQWUDWH YHFn WRUV LQWR QRQFRPPHQVXUDWH WZLVW YHFWRUV 7KH LQYHUVH YHORFLW\ SUREOHP LQ URERWLFV LV RIWHQ VROYHG WKURXJK WKH XVH RI WKH SVHXGRLQYHUVH RI WKH -DFRELDQ 7KLV VROXWLRQ LV JHQHUDOO\ DUELWUDU\ DQG IUDPH GHSHQGHQW ,W KDV EHHQ VKRZQ WKDW WKH SVHXGRLQYHUVH VROXWLRQ LV SK\VLFDOO\ LQFRQVLVWHQW LQ JHQHUDO UHTXLULQJ WKH DGGLWLRQ RI HOHPHQWV RI XQOLNH SK\VLFDO XQLWV 7KLV LV GXH WR WKH IDFW WKDW WKH SVHXGRLQYHUVH VROXWLRQ UHVXOWV LQ D PLQLPXP (XFOLGHDQQRUP (XFOLGHDQ OHDVWVTXDUHV VROXWLRQ RQ WZR JHQHUDOO\ QRQ (XFOLGHDQ QRQFRPPHQVXUDWHf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f±QRUP RI WZLVWV DQG WKH 0QRUP RI MRLQW UDWHV WR EH SK\VLFDOO\ FRQVLVWHQW ,Q GHFRXSOH IUDPHV WKH SVHXGRLQYHUVH LV VKRZQ WR EH HTXLYDOHQW WR WKH JHQHUDOL]HGLQYHUVH ZLWK LGHQWLW\ PHWULFV $ ZKROH FODVV RI

PAGE 116

QRQLGHQWLW\ PHWULFV XVHG ZLWK WKH JHQHUDOL]HGLQYHUVH DUH VKRZQ WR JLYH LGHQWLFDO VROXWLRQV WR WKH SVHXGRLQYHUVH VROXWLRQ 7KLV GLVVHUWDWLRQ SXWV WR UHVW WKH DUJXPHQWV DERXW WKH YDOLGLW\ RI WKH 0D VRQ5DLEHUW K\EULG FRQWURO WKHRU\ RI URERWLFV DQG WKH VHDUFK IRU D f¯QDWXUDO GHFRPn SRVLWLRQf°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n FRPSRVLWLRQ LV DFFRPSOLVKHG ZLWK NLQHVWDWLF ILOWHULQJ SURMHFWLRQ PDWULFHV VKRZQ LQ 6HFWLRQ $ PHWULFGHSHQGHQW JHQHUDOL]HGLQYHUVHf K\EULG FRQWURO LV XQLTXH ZLWK UHVSHFW WR WKH FKRVHQ PHWULFV DQG LV IUDPH LQGHSHQGHQW ,Q WKLV GLVVHUWDWLRQ D FODVV RI PDQLSXODWRUV FDOOHG VHOIUHFLSURFDO ZHUH LQWURGXFHG 3ODQDU PDQLSXODWRUV RQH W\SH RI VHOIUHFLSURFDO PDQLSXODWRU KDYH WKH SHFXOLDU SURSn HUW\ WKDW WKH\ GHFRXSOH DW DOO SRLQWV 6&$5$W\SH PDQLSXODWRUV DOVR GHFRXSOH DW HYHU\ SRLQW )RU WKLV UHDVRQ FDUH PXVW EH H[HUFLVHG LQ JHQHUDOL]LQJ DOJRULWKPV DQG FKDUDFWHULVWLF SURSHUWLHV RI PDQLSXODWRUV EDVHG RQ SODQDU DQG 6&$5$W\SH PDQLSXODWRUV 5HVXOWV WKDW DUH JHQHUDOL]HG IURP WKHVH VSHFLDO PDQLSXODWRUV PD\ SURYH WR EH LQYDOLG IRU PDQLSXODWRUV WKDW GR QRW GHFRXSOH DW HYHU\ SRLQW

PAGE 117

6HYHUDO WHVWV IRU GHWHUPLQLQJ LI D PDQLSXODWRU KDV GHFRXSOH SRLQWV DQG LI VR ZKHUH WKH\ DUH ORFDWHG ZHUH LGHQWLILHG 7KH HTXLYDOHQFH RI WKHVH WHVWV ZDV GLVFXVVHG LQ 6HFWLRQ 7KUHH FODVVHV RI PDQLSXODWRUV KDYH EHHQ LGHQWLILHG LQ WKLV GLVVHUWDWLRQ ZLWK UHn VSHFW WR GHFRXSOH SRLQWV PDQLSXODWRUV WKDW GHFRXSOH DW HYHU\ SRLQW DOO SODQDU DQG 6&$5$W\SH PDQLSXODWRUVf DW D SODQH RI SRLQWV HP HJ WKH 6$5 PDQLSXODWRU DW D OLQH RI SRLQWV ^HJ WKH *( 3 PDQLSXODWRUf DW D VLQJOH SRLQW DOO VSKHULFDO PDQLSXODWRUVf DQG DW QR SRLQWV ^HJ WKH 535 PDQLSXODWRUf )XWXUH UHODWHG ZRUN LQFOXGHV WKH GHYHORSPHQW RI DQ DOJHEUD WKDW LQFRUSRUDWHV SK\VLFDO XQLWV ZKHUH HDFK HOHPHQWV LV FRPSRVHG RI D QXPHULFDO YDOXH DQG D SK\VLFDO XQLW 2WKHU DOJHEUDLF SURSHUWLHV RI SK\VLFDOO\ FRQVLVWHQW QRQFRPPHQVXUDWH OLQHDU V\VWHPV ZLOO DOVR EH H[SORUHG $QG ILQDOO\ IXUWKHU LQYHVWLJDWLRQ RI PDQLSXODWRUV WKDW GHFRXSOH DW HYHU\ SRLQW ZLOO EH SXUVXHG

PAGE 118

$33(1',; $ '+ 3$5$0(7(56 )25 9$5,286 0$1,38/$7256 'HQDYLW+DUWHQEHUJ SDUDPHWHUV IRU DOO WKH YDULRXV PDQLSXODWRUV LQWURGXFHG LQ WKH ERG\ RI WKLV GLVVHUWDWLRQ DUH IRXQG LQ WKH IROORZLQJ WDEOHV DORQJ ZLWK WKH PDQLSXODWRU -DFRELDQV H[SUHVVHG LQ YDULRXV IUDPHV )RU HDFK PDQLSXODWRU ZLWK OHVV WKDQ VL[ MRLQWV WKH GHWHUPLQDQWV RI -7LQ YDULRXV IUDPHV DUH JLYHQ )RU HDFK PDQLSXODWRU ZLWK PRUH WKDQ VL[ MRLQWV WKH GHWHUPLQDQW RI --7 LV JLYHQ 6RPH -DFRELDQV IRU WKH 35 YLUWXDO PDQLSXODWRU RI 7DEOH $O DUH 3\ a3[ $f DQG 'HW>-7-f@ 'HW>¯-7mf@ S[ S\ $f 7KH GHWHUPLQDQW LQ IUDPH FRRUGLQDWHV LV SK\VLFDOO\ FRQVLVWHQW DQG WKH GHWHUPLQDQW LQ IUDPH W FRRUGLQDWHV LV QRW SK\VLFDOO\ FRQVLVWHQW 6RPH -DFRELDQV IRU WKH 55 PDQLSXODWRU RI 7DEOH $ DUH DQG n n 3\.;S]&7? 3\ 3[rO 3]&7O6 3[ ,, WR WR ,, 3[& t 3\tO FULV " 6 &" FFUL .L &O 'HW>-7-f@ FUc $f $f 7DEOH $O ' + SDUDPHWHUV IRU 35 YLUWXDO PDQLSXODWRU -RLQW 7\SH G D H D 3 G[ 5 G

PAGE 119

7DEOH $ '+ SDUDPHWHUV IRU DQ 55 PDQLSXODWRU -RLQW 7\SH G D D 5 L mL 5 'HW>L-7-f@ D?^O S[ S\ S]f f± f±[ 3O3\a A3[3\&t SL a SJfFb V_f $f 7KH GHWHUPLQDQW LQ IUDPH FRRUGLQDWHV LV SK\VLFDOO\ FRQVLVWHQW DQG WKH GHWHUPLQDQW LQ IUDPH W FRRUGLQDWHV LV QRW SK\VLFDOO\ FRQVLVWHQW 7DEOH $ '+ SDUDPHWHUV IRU D JHQHUDO 555 PDQLSXODWRU -RLQW 7\SH G D H D 5 G? 2[ L RW? 5 G D D 5 G] m $ PLGIUDPH -DFRELDQ IRU WKH JHQHUDO 555 PDQLSXODWRU RI 7DEOH $ LV GF7L DLFLV n T.L. D?.? NF f§ T?-?W f§ DFM? f§ G.;?V FON M f§G?.7? f§ D.&-? f§D.?7 f§&L?.?&I G-?-6 f§ a D[V NFWL N[M D .L. f§ &?* . $f DQG ZLWK mL _ mL DQG _ 'HW>-U-f@ R RLR R  DG Rr R[RI D2 [A DG fµ $f 7KLV GHWHUPLQDQW LQ IUDPH FRRUGLQDWHV LV QRW SK\VLFDOO\ FRQVLVWHQW

PAGE 120

SDUDPHWHUV IRU W UH 3 DQDU 5, -RLQW 7\SH G D D 5 2L L 5 5 6RPH -DFRELDQV IRU WKH 3ODQDU 555 PDQLSXODWRU RI 7DEOH $ DUH DLA n 3\ 26 3\ 3\ m *& f§ 3[ 2& 3[ a3[ Of§ *W $f DQG 'HW>-9f@ 'HWS-7-f@ D?D?V? $f 7KH GHWHUPLQDQWV LQ IUDPH DQG IUDPH W FRRUGLQDWHV DUH SK\VLFDOO\ FRQVLVWHQW 7DEOH $ '+ SDUDPHWHUV IRU WKH 6SKHULFDO 555 PDQLSXODWRU -RLQW 7\SH G D D 5 L W 5 W 5 6RPH -DFRELDQV IRU WKH 6SKHULFDO 555 PDQLSXODWRU RI 7DEOH $ DUH n n 3\F a3] 3\ 3[& 3]6 3[ ,, 7R F ,, a3\V 3[ 6 F 'HWS27-f@ A 9HO??UMf? O9O3O3O" ‘ $f $f 7KH GHWHUPLQDQW LQ IUDPH FRRUGLQDWHV LV SK\VLFDOO\ FRQVLVWHQW DQG WKH GHWHUPLQDQW LQ IUDPH W FRRUGLQDWHV LV QRW SK\VLFDOO\ FRQVLVWHQW

PAGE 121

7DEOH $ '+ SDUDPHWHUV IRU WKH 1RQSODQDU 555 PDQLSXODWRU -RLQW 7\SH G D D 5 2O L WW 5 G m W 5 6RPH -DFRELDQV IRU WKH 1RQSODQDU 555 PDQLSXODWRU RI 7DEOH $ DUH Y DQG FG GF n GV f§8L f§ RF f§DL f§D V G V D F F &&G f§ 2O6 f§ &,&6 f§DL& f§ DFF f§ FGV GV f§D &6 V f§66 & f§F 'HW>-7-f¬f@ f§  _ R I R +r  rE DLDF I RA 'HW>-7-f@ 'HW>-7-f@ V D D? G DDF D $f $f $f D DLDF FD Gbf $f 7KH GHWHUPLQDQWV LQ IUDPH DQG FRRUGLQDWHV DUH QRW SK\VLFDOO\ FRQVLVWHQW 7DEOH $ '+ SDUDPHWHUV nRU W UH 333 RUWKRJRQ -RLQW 7\SH G D D 3 GL WW 3 G! W WW 3 G! 6RPH -DFRELDQV IRU WKH 333 2UWKRJRQDO PDQLSXODWRU RI 7DEOH $ DUH rWr*WR r2 ‘ n f§ ,, ,, $Lf

PAGE 122

DQG 'HW>V-U-f@ $f IRU DQ\ IUDPH J 7KH GHWHUPLQDQW LQ DQ\ IUDPH FRRUGLQDWHV LV SK\VLFDOO\ FRQVLVWHQW 7DEOH $ '+ SDUDPHWHUV IRU WKH 353 6PDOO $VVHPEO\ 5RERW 6$5f -RLQW 7\SH G D D 3 GL 5 WW 3 G] 6RPH -DFRELDQV IRU WKH 6$5 PDQLSXODWRU RI 7DEOH $ DUH DQG R R R A f§ R R R ,, R R R 'HW>-7-f@ 'HW> -7-f? G $ f $f 7KH GHWHUPLQDQW LQ IUDPH FRRUGLQDWHV LV SK\VLFDOO\ FRQVLVWHQW DQG WKH GHWHUPLQDQW LQ IUDPH FRRUGLQDWHV LV QRW SK\VLFDOO\ FRQVLVWHQW 7DEOH $ '+ SDUDPHWHUV IRU WKH 535 PDQLSXODWRU -RLQW 7\SH G D D 5 L WW 3 G WW 7 5 ] 6RPH -DFRELDQV IRU WKH 535 PDQLSXODWRU RI 7DEOH $ DUH n n n 3\ 3] 3[  R R WM aL M f§ 8W f§ GL 3\ R R $f

PAGE 123

,OO 7DEOH $ '+ SDUDPHWHUV IRU WKH 5553 6& $5$ PDQLSXODWRU -RLQW 7\SH G D H D 5 mL L 5 5 3  DQG 'HW>-7-f@ 'HWI27-f@ 3cf a W! IW 3cf ‘ $f 7KH GHWHUPLQDQW LQ IUDPH FRRUGLQDWHV LV SK\VLFDOO\ FRQVLVWHQW DQG WKH GHWHUPLQDQW LQ IUDPH W FRRUGLQDWHV LV QRW SK\VLFDOO\ FRQVLVWHQW 6RPH -DFRELDQV IRU WKH 6&$5$ PDQLSXODWRU RI 7DEOH $ DUH DL6 !O& f¬ *W L D?6 DDL& D3YDLF3YDL3[6 [ 6" D DODV & &3\D663[ 6 $f $f DQG 'HW>-7-f@ 'HWA-9f@ RR6f $f 7KH GHWHUPLQDQWV LQ IUDPH DQG IUDPH W FRRUGLQDWHV DUH SK\VLFDOO\ FRQVLVWHQW 7DEOH $OO '+ SDUDPHWHUV IRU WKH 5553 PDQLSXODWRU -RLQW 7\SH G D D 5 2L WW 5 5 7W 3 

PAGE 124

6RPH -DFRELDQV IRU WKH 5553 PDQLSXODWRU RI 7DEOH $ DUH r‘ 2&O &6 266 66 & 2L fµ6L f§&O L f§ 8& fµ6L f§& f§&/? f§ GF &O f¯&O F DQG f¬9 V ! 'HW>r-7-fM DV 'HW> -7 -f? D?V? D?DVO DLD&VO DFVO $f $f $f 7KH GHWHUPLQDQW LQ IUDPH FRRUGLQDWHV LV SK\VLFDOO\ FRQVLVWHQW DQG WKH GHWHUPLQDQW LQ IUDPH FRRUGLQDWHV LV QRW SK\VLFDOO\ FRQVLVWHQW 7DEOH $ '+ SDUDPHWHUV IRU WKH 5553 PDQLSXODWRU -RLQW 7\SH G D H D 5 L U 5 W 5 WW 3  6RPH -DFRELDQV IRU WKH 5553 PDQLSXODWRU RI 7DEOH $ DUH Y ‘ & n ‘ RF F & 9 F aF 'HW>[ -9f@ & F? A I A 6 nr $f DQG 'HW>-7-f@ V D?V? DFV DFV 7KH GHWHUPLQDQWV LQ IUDPH DQG FRRUGLQDWHV DUH QRW SK\VLFDOO\ FRQVLVWHQW $f $f

PAGE 125

'+ SDUDPHWHUV IRU 5* (3 -RLQW 7\SH G D D 5 WW 5 R 5 R 5 WW 5 R PDQLSXODWRU 6RPH -DFRELDQV IRU WKH *(3 PDQLSXODWRU RI 7DEOH $ DUH f§DF V F D& f§ 2& & f§& Dm D& & & V F $f $f DQG 'HW>-7-f@ DODOV.DMFO A RF f $f 'HW>-7-f@ DD6 DF mA & f§ 6f $f ,Q D IUDPH W WUDQVODWHG IURP IUDPH E\ 3 RF f f fµ6 R& f J $f WKH -DFRELDQ f«*W 9 LV FDFDFf &&f& ,&D&fF m m m DF t& fm D&f¯mfV m 6&&fm m 6 Ofµ6 & $f

PAGE 126

DQG 'HW>f«-9f@ WcDLVOVcZ $f 7KH GHWHUPLQDQWV LQ IUDPH DQG FRRUGLQDWHV DUH QRW SK\VLFDOO\ FRQVLVWHQW DQG WKH GHWHUPLQDQW LQ IUDPH W FRRUGLQDWHV LV SK\VLFDOO\ FRQVLVWHQW 7KH 5 5HGXQGDQW $QWKURSRPRUSKLF $UP PDQLSXODWRU RI 7DEOH $ KDV WKUHH LQWHUVHFWLQJ VKRXOGHU MRLQWV DQ HOERZ MRLQW DQG WKUHH LQWHUVHFWLQJ ZULVW MRLQWV $ PLGIUDPH -DFRELDQ IRU WKLV PDQLSXODWRU LV 66 f§ž&6 *& *& eA &T ,, f§,&66 DFF f§ &&6 f§ &6 &6 V A &6 t6 & F 66 && &66 V & & :KHQ ? OUDG t f§ UDG ,, &2 UDG H UDG UDG $f UDG DQG UDG 'HW>--7@ RA DA D_De $f 7KLV GHWHUPLQDQW LV SK\VLFDOO\ FRQVLVWHQW

PAGE 127

7DEOH $ '+ SDUDPHWHUV IRU WKH 5 &(6$5 5HVHDUFK 0DQLSXODWRU -RLQW 7\SH G D D 5 2[ WW 5 G WW 5 f§ WW 5  5 f§WW 5 WW 5  $ PLGIUDPH -DFRELDQ IRU WKH 5 &(6$5 5HVHDUFK 0DQLSXODWRU >@ RI 7DEOH $ LV FFG G]V6 F]G 4&T6 f§GV DVV & f§DF f§DF&4 f§DF f§ FGV FGV GV & 6 &6 fµ6 f§6 &6 F F ‘66 VV F & $f :KHQ %? OUDG UDG f§ UDG f§ UDG UDG UDG 2 f§ UDG R " DQG G G f§ 'HW>--7@ " A " " "9 $f 7KLV GHWHUPLQDQW LV SK\VLFDOO\ FRQVLVWHQW 7DEOH $ '+ SDUDPHWHUV IRU WKH 5 . 5RERW 5HVHDUFK $UP -RLQW 7\SH G D D 5 D[ WW 5 D W 5 D WW 5 WW 5 WW 5 N 5 7KH 5 . 5RERW 5HVHDUFK $UP >@ RI 7DEOH $ LV IXQFWLRQDOO\ HTXLYDOHQW ZLWK FHUWDLQ Df¬V DQG FIV VHW WR ]HURf WR WKH PDQLSXODWRU GHVFULEHG LQ >@ DQG WR WKH

PAGE 128

8-,%27 LQ >@ $ PLGIUDPH -DFRELDQ IRU WKLV PDQLSXODWRU LV D6 af¯&,A&A&T6 DVV f§DF 8& & DF f§D t4!4! FV fµ6 ‘6 &6 f§&T6¯ &&6 f§F 66 && &66 VV aF & ViV UDG G FVL ,, UDG UDG UDG ,, H " DQG G G $f 'HW>--7@ " " A9 " " A $f 7KLV GHWHUPLQDQW LV SK\VLFDOO\ FRQVLVWHQW $ PLGIUDPH -DFRELDQ IRU WKH 5 380$ 6SKHULFDO :ULVW 0DQLSXODWRU RI 7DEOH $ LV FFG f§ D&6L f§ 6 &&" D&A6 & G n A f§DF r&& FLAnV &G6s 6  &6 fµ6 m V 64 f§& 66 ‘66 & 2IL $f 'HW>--7@ D?F?G?^DF GA]f O V V FFO VLV V?FO FVf $f

PAGE 129

7DEOH $ '+ SDUDPHWHUV IRU WKH 35 5HGXQGDQW 6SKHULFDO :ULVW 5RERW -RLQW 7\SH G D H D 3 G? WW 3 G U W 3 G] 5 WW 5 W 5 W 5 7KLV GHWHUPLQDQW LV SK\VLFDOO\ FRQVLVWHQW $ PLGIUDPH -DFRELDQ IRU WKH 35 5HGXQGDQW 6SKHULFDO :ULVW 5RERW RI 7Dn EOH $ LV F V F fµ6 &6 & fµ66 & 'HW>--7@ O VM VJ &J&J VMVO V?b FMVIM $f $f 7KLV GHWHUPLQDQW LV SK\VLFDOO\ FRQVLVWHQW 7DEOH $ '+ SDUDPHWHUV IRU WKH 535 *3 0DQLSXODWRU -RLQW 7\SH G D FF 5 L WW 5 D W 3 G] 5 WW 5 Gi WW 5 WW 5 $ PLGIUDPH -DFRELDQ IRU WKH 535f *3 0DQLSXODWRU RI 7DEOH $ LV f§DF6 f§ GV6 FG FG f§GVV6H f§m &6 FAGV64 M &-&& E AAA &6 V V FV F VV f§& f§& &6 F 66 $f

PAGE 130

:KHQ L OUDG UDG UDG UDG f§ UDG 2T f§ UDG f§ UDG DQG f§ " 'HW>--7@ " "G G : G " "G "A $f 7KLV GHWHUPLQDQW LV QRW SK\VLFDOO\ FRQVLVWHQW

PAGE 131

5()(5(1&(6 >@ $EEDWL0DUHVFRWWL $ DQG %RQLYHQWR & DQG 0HOFKLRUUL & f¯2Q WKH ,QYDULDQFH RI WKH +\EULG 3RVLWLRQ)RUFH &RQWUROf° -RXUQDO RI ,QWHOOLJHQW DQG 5RERWLF 6\VWHPV >@ $VDGD +DUXKLNR DQG 6ORWLQH -( 5RERW $QDO\VLV DQG &RQWURO -RKQ :LOH\ DQG 6RQV ,QF 1HZ @ %DLOOLHXO -RKQ f¯$ &RQVWUDLQW 2ULHQWHG $SSURDFK WR ,QYHUVH 3UREOHPV IRU .LQHPDWLFDOO\ 5HGXQGDQW 0DQLSXODWRUVf° ,((( &RQIHUHQFH RQ 5RERWLFV DQG $XWRPDWLRQ 5DOHLJK 1RUWK &DUROLQD SS >@ %DNHU 'DQLHO 5 DQG :DPSOHU ,, &KDUOHV : f¯2Q WKH ,QYHUVH .LQHPDn WLFV RI 5HGXQGDQW 0DQLSXODWRUVf° ,QWHUQDWLRQDO -RXUQDO RI 5RERWLFV 5HVHDUFK f >@ %DOO 5REHUW 6WDZHOO6LU $ 7UHDWLVH RQ WKH 7KHRU\ RI 6FUHZV &DPEULGJH 8QLYHUVLW\ 3UHVV &DPEULGJH >@ %HQ,VUDHO $GL DQG *UHYLOOH 7KRPDV 1( *HQHUDOL]HG ,QYHUVHV 7KHRU\ DQG $SSOLFDWLRQV -RKQ :LOH\ DQG 6RQV ,QF 1HZ @ &DPSEHOO 6WHSKHQ / DQG 0H\HU &DUO ' *HQHUDOL]HG ,QYHUVHV RI /LQHDU 7UDQVIRUPDWLRQV 3LWPDQ 3XEOLVKLQJ /LPLWHG /RQGRQ (QJODQG >@ &HSKXV .LPEHUO\ * DQG 'RW\ .HLWK / f¯,VVXHV LQ 3DUDOOHO 0HFKDQLVP .LQHVWDWLFVf° )LIWK &RQIHUHQFH RQ 5HFHQW $GYDQFHV LQ 5RERWLFV )ORULGD $WODQWLF 8QLYHUVLW\ %RFD 5DWRQ )/ -XQH SS >@ &KDVOHV 0 f¯1RWH VXU OHV SURSUL«W«V JHQ«UDOHV GX V\VW«PH GH GHX[ FRUSV f° %XOOHWLQ 6FL 0DWK )HUUXVDF >@ &KLDFFKLR 3DVTXDOH DQG &KLDYHULQL 6WHIDQR DQG 6FLDYLFFR /RUHQ]R DQG 6LFLOn LDQR %UXQR f¯*OREDO 7DVN 6SDFH 0DQLSXODELOLW\ (OOLSVRLGV IRU 0XOWLSOH $UP 6\VWHPVf° ,((( -RXUQDO RI 5RERWLFV DQG $XWRPDWLRQ f >@ &UDLJ -RKQ ,QWURGXFWLRQ WR 5RERWLFV 0HFKDQLFV DQG &RQWURO $GGLVRQ :HVOH\ 3XEOLVKLQJ &RPSDQ\ 5HDGLQJ 0DVVDFKXVHWWV >@ 'H /XFD $ DQG 0DQHV & DQG 1LFROR ) f¯+\EULG )RUFH9HORFLW\ &RQWURO 8VLQJ 5HGXQGDQW 0DQLSXODWRUVf° 1$72 $GYDQFHG 5HVHDUFK :RUNVKRS 5RERWV ZLWK 5HGXQGDQF\ 'HVLJQ 6HQVLQJ DQG &RQWURO 6DOµ /DJR GL *DUGD ,WDO\ -XQH -XO\ >@ 'RW\ .HLWK / f¯7DEXODWLRQ RI WKH 6\PEROLF 0LGIUDPH -DFRELDQ RI D 5RERWLF 0DQLSXODWRUf° ,QWHUQDWLRQDO -RXUQDO RI 5RERWLFV 5HVHDUFK f

PAGE 132

>@ 'RW\ .HLWK / f¯$Q (VVD\ RQ WKH $SSOLFDWLRQ RI :HLJKWHG *HQHUDOL]HG ,QYHUVHV LQ 5RERWLFVf° )LIWK &RQIHUHQFH RQ 5HFHQW $GYDQFHV LQ 5RERWLFV )ORULGD $WODQWLF 8QLYHUVLW\ %RFD 5DWRQ )/ -XQH SS >@ 'RW\ .HLWK / f¯,QYHUWLQJ .LQHVWDWLF (TXDWLRQV 7KH :HLJKWHG *HQHUDOn L]HG ,QYHUVH 9HUVXV WKH 0RRUH3HQURVH ,QYHUVHf° ,((( ,QWHUQDWLRQDO &RQn IHUHQFH RQ 5RERWLFV DQG $XWRPDWLRQf§ 7XWRULDO 6f§)RUFH DQG &RQWDFW &RQWURO LQ 5RERWLF 6\VWHP $WODQWD *HRUJLD 0D\ SS >@ 'RW\ .HLWK / DQG 0HOFKLRUUL &ODXGLR DQG %RQLYHQWR &ODXGLR f¯:KDW LV :URQJ ZLWK 5RERW .LQHPDWLF DQG '\QDPLF 0DQLSXODELOLW\ 7KHn RU\"f° 0,/./' 0DFKLQH ,QWHOOLJHQFH /DERUDWRU\ 8QLYHUVLW\ RI )ORULGD *DLQHVYLOOH )ORULGD >@ 'RW\ .HLWK / DQG 0HOFKLRUUL &ODXGLR DQG %RQLYHQWR &ODXGLR D f¯$ &ULWLFDO 5HYLHZ RI WKH &XUUHQW .LQHPDWLF DQG '\QDPLF 0DQLSXODELOLW\ 7KHn RU\f° 7KLUG ,QWHUQDWLRQDO :RUNVKRS RQ $GYDQFHV LQ 5RERW .LQHPDWLFV 6HSW )HUUDUD ,WDO\ SS >@ 'RW\ .HLWK / DQG 0HOFKLRUUL &ODXGLR DQG %RQLYHQWR &ODXGLR E f¯5RERW .LQHVWDWLF DQG '\QDPLF 0DQLSXODELOLW\f° 6XEPLWWHG 6HSW 5HYLVHG 6HSW ,((( -RXUQDO RI 5RERWLFV DQG $XWRPDWLRQ 0,/./' 0DFKLQH ,QWHOOLJHQFH /DERUDWRU\ 8QLYHUVLW\ RI )ORULGD *DLQHVYLOOH )ORULGD >@ 'RW\ .HLWK / DQG 0HOFKLRUUL &ODXGLR DQG %RQLYHQWR &ODXGLR f¯$ 7KHRU\ RI *HQHUDOL]HG ,QYHUVHV $SSOLHG WR 5RERWLFVf° ,QWHUQDWLRQDO -RXUQDO RI 5RERWLFV 5HVHDUFK f >@ 'RW\ .HLWK / DQG 0HOFKLRUUL &ODXGLR DQG 6FKZDUW] (ULF 0 DQG %RQLYHQWR &ODXGLR ,Q SUHVV f¯5RERW 0DQLSXODELOLW\f° ,((( -RXUQDO RI 5RERWLFV DQG $Xn WRPDWLRQ >@ 'XEH\ 5DMLY 9 f¯5HDO7LPH ,PSOHPHQWDWLRQ RI DQ 2SWLPL]DWLRQ 6FKHPH IRU 6HYHQ'HJUHHRI)UHHGRP 5HGXQGDQW 0DQLSXODWRUVf° ,((( -RXUQDO RI 5RERWLFV DQG $XWRPDWLRQ f >@ 'XII\ -RVHSK f¯7KH )DOODF\ RI 0RGHUQ +\EULG &RQWURO 7KHRU\ WKDW LV %DVHG RQ f«2UWKRJRQDO &RPSOHPHQWVf¬ RI 7ZLVW DQG :UHQFK 6SDFHVf° -RXUQDO RI 5RERWLF 6\VWHPV f >@ (XOHU $ DQG 'XEH\ 5 9 DQG %DEFRFN 6 0 f¯6HOI 0RWLRQ 'Hn WHUPLQDWLRQ %DVHG RQ $FWXDWRU 9HORFLW\ %RXQGV IRU 5HGXQGDQW 0DQLSXODWRUVf° -RXUQDO RI 5RERWLF 6\VWHPV f >@ *ROGHQEHUJ $ $ f¯,QGHILQLWH ,QQHU 3URGXFW%DVHG 'HFRPSRVLWLRQ DQG $SSOLFDWLRQ WR &RQWURO RI )RUFH DQG 3RVLWLRQ RI 5RERW 0DQLSXODWRUVf° 7XWRn ULDO f¯)RUFH DQG &RQWDFW &RQWURO LQ 5RERWLF 6\VWHPV $ +LVWRULFDO 3HUVSHFWLYH DQG &XUUHQW 7HFKQRORJLHVf° ,((( 5RERWLFV DQG $XWRPDWLRQ &RQIHUHQFH $WODQWD *HRUJLD 0D\ SS >@ *ULIILV 0LFKDHO :LOOLDP .LQHWLF &RQVLGHUDWLRQV LQ WKH +\EULG &RQWURO RI 5RERWLF 0DQLSXODWRUV 0DVWHUf¬V 7KHVLV 8QLYHUVLW\ RI )ORULGD >@ *ULIILV 0LFKDHO :LOOLDP .LQHVWDWLF &RQWURO $ 1RYHO 7KHRU\ IRU 6LPXOn WDQHRXVO\ 5HJXODWLQJ )RUFH DQG 'LVSODFHPHQW 3K' 'LVVHUWDWLRQ 8QLYHUVLW\ RI )ORULGD

PAGE 133

>@ *ULIILV 0LFKDHO DQG 'XII\ -RVHSK f¯.LQHVWDWLF &RQWURO $ 1RYHO 7KHn RU\ IRU 6LPXOWDQHRXVO\ 5HJXODWLQJ )RUFH DQG 'LVSODFHPHQWf° 3URFHHGLQJV VW $60( 0HFKDQLVPV &RQIHUHQFH 6HSW &KLFDJR ,/ SS >@ *ULIILV 0LFKDHO :LOOLDP DQG 'XII\ -RVHSK f¯1RWHV RQ .LQHPDWLF &RQWURO 8VLQJ 'LVSODFHPHQWV WR 1XOO )RUFHVf° ,((( ,QWHUQDWLRQDO &RQIHUHQFH RQ 5RERWLFV DQG $XWRPDWLRQf§ 7XWRULDO 6f§)RUFH DQG &RQWDFW &RQWURO LQ 5RERWLF 6\VWHP $WODQWD *HRUJLD 0D\ SS >@ +VX 3LQJ DQG +DXVHU -RKQ DQG 6DVWU\ 6KDQNDU f¯'\QDPLF &RQWURO RI 5HGXQGDQW 0DQLSXODWRUVf° -RXUQDO RI 5RERWLF 6\VWHPV f >@ +XQW .HQQHWK + f¯6SHFLDO &RQILJXUDWLRQV RI 5RERW$UPV 9LD 6FUHZ 7KHn RU\f° ,QWHUQDWLRQDO -RXUQDO RI ,QIRUPDWLRQ (GXFDWLRQ DQG 5HVHDUFK LQ 5RERWLFV DQG $UWLILFLDO ,QWHOOLJHQFH >@ .OHLQ &KDUOHV $ DQG %ODKR %UXFH ( f¯'H[WHULW\ 0HDVXUHV IRU WKH 'HVLJQ DQG &RQWURO RI .LQHPDWLFDOO\ 5HGXQGDQW 0DQLSXODWRUVf° ,QWHUQDWLRQDO -RXUQDO RI 5RERWLFV 5HVHDUFK f >@ .OHLQ &KDUOHV $ DQG +XDQJ &KLQJ+VLDQJ f¯5HYLHZ RI 3VHXGRLQYHUVH &RQWURO IRU 8VH ZLWK .LQHPDWLFDOO\ 5HGXQGDQW 0DQLSXODWRUVf° ,((( 7UDQVDFn WLRQV RQ 6\VWHPV 0DQ DQG &\EHUQHWLFV 60&f >@ .UHXW]'HOJDGR . DQG /RQJ 0 DQG 6HUDML + f¯.LQHPDWLF $QDO\VLV RI '2) $QWKURSRPRUSKLF $UPVf° 3URFHHGLQJV RI WKH ,((( ,QWHUQDWLRQDO &RQIHUHQFH RQ 5RERWLFV DQG $XWRPDWLRQ &LQFLQQDWL 2KLR >@ /HRQ 6WHYHQ /LQHDU $OJHEUD ZLWK $SSOLFDWLRQV 0DFPLOODQ 3XEOLVKLQJ &R ,QF 1HZ @ /LSNLQ +DUYH\ f¯,QYDULDQW 3URSHUWLHV RI WKH 3VHXGRLQYHUVH LQ 5RERWLFVf° 3URFHHGLQJV RI 16) 'HVLJQ DQG 0DQXIDFWXULQJ 6\VWHPV &RQIHUHQFH 7HPSH $ULn ]RQD -DQXDU\ SS >@ /LSNLQ +DUYH\ DQG 'XII\ -RVHSK f¯7KH (OOLSWLF 3RODULW\ RI 6FUHZVf° $60( -RXUQDO RI 0HFKDQLVPV 7UDQVPLVVLRQV DQG $XWRPDWLRQ LQ 'HVLJQ >@ /LSNLQ +DUYH\ DQG 'XII\ -RVHSK f¯+\EULG 7ZLVW DQG :UHQFK &RQWURO IRU D 5RERWLF 0DQLSXODWRUf° $60( -RXUQDO RI 0HFKDQLVPV 7UDQVPLVVLRQV DQG $XWRPDWLRQ LQ 'HVLJQ -XQH >@ /X =LUHQ DQG 6KLPRJD .DUXQDNDU % DQG *ROGHQEHUJ $QGUHZ $ f¯([n SHULPHQWDO 'HWHUPLQDWLRQ RI '\QDPLF 3DUDPHWHUV RI 5RERWLF $UPVf° -RXUQDO RI 5RERWLFV 6\VWHPV f >@ 0DUWLQ ' 3 %DLOOLHXO DQG +ROOHUEDFK 0 f¯5HVROXWLRQ RI .LQHPDWLF 5HGXQGDQF\ 8VLQJ 2SWLPL]DWLRQ 7HFKQLTXHVf° ,((( -RXUQDO RI 5RERWLFV DQG $XWRPDWLRQ f >@ 0DVRQ 0DWWKHZ 7 &RPSOLDQFH DQG )RUFH &RQWURO IRU &RPSXWHU &RQn WUROOHG 0DQLSXODWRUV 0DVWHUf¬V 7KHVLV 0DVVDFKXVHWWV ,QVWLWXWH RI 7HFKQRORJ\ >@ 0DVRQ 0DWWKHZ 7 f¯&RPSOLDQFH DQG )RUFH &RQWURO IRU &RPSXWHU &RQn WUROOHG 0DQLSXODWRUVf° ,((( 7UDQVDFWLRQV RQ 6\VWHPV 0DQ DQG &\EHUQHWLFV 60&OOf

PAGE 134

>@ 0HULDP -DPHV / (QJLQHHULQJ 0HFKDQLFV 9ROXPH 6WDWLFV -RKQ :LOH\ DQG 6RQV ,QF 1HZ @ 1DNDPXUD @ 1DNDPXUD @ 1HQFKHY 'UDJRPLU 1 f¯5HGXQGDQF\ 5HVROXWLRQ WKURXJK /RFDO 2SWLPL]Dn WLRQ $ 5HYLHZf° -RXUQDO RI 5RERWLF 6\VWHPV f >@ 3DUN @ 3DWWHUVRQ 7LPRWK\ )UHGULFN 5HGXFWLRQ RI &RPSOLDQFH LQ 6SDFH%DVHG 5Hn GXQGDQW 'HJUHHRI)UHHGRP 0DQLSXODWRUV 3K' 'LVVHUWDWLRQ *HRUJLD ,QVWLWXWH RI 7HFKQRORJ\ >@ 3O¾FNHU f¯2Q D 1HZ *HRPHWU\ RI 6SDFHf° 3KLO 7UDQV 5R\DO 6RFLHW\ RI /RQGRQ >@ 3O¾FNHU f¯)XQGDPHQWDO 9LHZV 5HJDUGLQJ 0HFKDQLFVf° 3KLO 7UDQV 5R\DO 6RFLHW\ RI /RQGRQ >@ 3RGKRURGHVNL 5 3 DQG *ROGHQEHUJ $ $ DQG )HQWRQ 5 * f¯5HVROYLQJ 5HGXQGDQW 0DQLSXODWRU -RLQW 5DWHV DQG ,GHQWLI\LQJ 6SHFLDO $UP &RQILJXUDWLRQV 8VLQJ -DFRELDQ 1XOO6SDFH %DVHVf° ,((( -RXUQDO RI 5RERWLFV DQG $XWRPDWLRQ f >@ 5DLEHUW 0DUF + DQG &UDLJ -RKQ f¯+\EULG 3RVLWLRQ)RUFH &RQWURO RI 0DQLSXODWRUVf° $60( '\QDPLF 6\V 0HDV &RQWU >@ 6DOLVEXU\ .HQQHWK DQG &UDLJ -RKQ f¯$UWLFXODWHG +DQGV )RUFH &RQn WURO DQG .LQHPDWLF ,VVXHVf° ,QWHUQDWLRQDO -RXUQDO RI 5RERWLFV 5HVHDUFK f >@ 6FKZDUW] (ULF 0 DQG 'RW\ .HLWK / f¯$SSOLFDWLRQ RI WKH :HLJKWHG *HQn HUDOL]HG,QYHUVHV LQ 5RERWLFV DQG 'LVFUHWH7LPH &RQWUROVf° )LIWK &RQIHUHQFH RQ 5HFHQW $GYDQFHV LQ 5RERWLFV )ORULGD $WODQWLF 8QLYHUVLW\ %RFD 5DWRQ )/ -XQH SS >@ 6FKZDUW] (ULF 0 DQG 'RW\ .HLWK / f¯7KH :HLJKWHG *HQHUDOL]HG,QYHUVH $SSOLHG WR 0HFKDQLVP &RQWUROODELOLW\f° 6L[WK $QQXDO &RQIHUHQFH RQ 5HFHQW $Gn YDQFHV LQ 5RERWLFV 8QLYHUVLW\ RI )ORULGD *DLQHVYLOOH )ORULGD $SULO SS O%OO >@ 6KLHOGV 3DXO & (OHPHQWDU\ /LQHDU $OJHEUD :RUWK 3XEOLVKHUV ,QF 1HZ @ 6WUDQJ *LOEHUW /LQHDU $OJHEUD DQG LWV $SSOLFDWLRQV +DUFRXUW %UDFH -R YDQRYLFK 6DQ 'LHJR &$

PAGE 135

>@ :DPSOHU ,, &KDUOHV : f¯0DQLSXODWRU ,QYHUVH .LQHPDWLF 6ROXWLRQV %DVHG RQ 9HFWRU )RUPXODWLRQV DQG 'DPSHG /HDVW6TXDUHV 0HWKRGVf° ,((( 7UDQVDFn WLRQV RQ 6\VWHPV 0DQ DQG &\EHUQHWLFV 60&f >@ :HEVWHUf¬V 1HZ 7ZHQWLHWK &HQWXU\ 'LFWLRQDU\ RI WKH (QJOLVK /DQJXDJH 8QDEULGJHG 6HFRQG (GLWLRQ 1HZ :RUOG 'LFWLRQDULHV6LPRQ DQG 6FKXVn WHU 1HZ @ @
PAGE 136

%,2*5$3+,&$/ 6.(7&+ (ULF 0LFKDHO 6FKZDUW] ZDV ERUQ LQ 0LDPL )ORULGD RQ $SULO ,Q KH JUDGXDWHG IURP 6RXWKZHVW 0LDPL 6HQLRU +LJK 6FKRRO LQ WKH WRS RQH SHUFHQW RI KLV FODVV +H UHFHLYHG WZR %DFKHORU RI 6FLHQFH GHJUHHV IURP WKH 8QLYHUVLW\ RI )ORULGD LQ $SULO ERWK ZLWK KLJK KRQRUVff§RQH LQ HOHFWULFDO HQJLQHHULQJ DQG WKH RWKHU LQ PHFKDQLFDO HQJLQHHULQJ +H UHFHLYHG D 0DVWHU RI (QJLQHHULQJ GHJUHH IURP WKH 8QLYHUVLW\ RI )ORULGD LQ $XJXVW ,QWHUVSHUVHG ZLWK KLV XQGHUJUDGXDWH DQG JUDGXDWH VFKRRO FDUHHU KH KDV ZRUNHG DW YDULRXV MREV LQ WKH HQJLQHHULQJ LQGXVWU\ KROGLQJ SRVLWLRQV ZLWK %HQGL[ $YLRQLFV 8QLn YHUVDO 6HFXULWLHV ,QVWUXPHQWV ,%0 WKH $OOHQ%UDGOH\ &RPSDQ\ DQG WKH (OHFWURQLFV &RPPXQLFDWLRQV /DERUDWRU\ DW WKH 8QLYHUVLW\ RI )ORULGD +H KDV DOVR ZRUNHG DW WKH 8QLYHUVLW\ RI )ORULGD DV D WHDFKLQJ DVVLVWDQW DQG D UHVHDUFK DVVLVWDQW DQG ZDV DQ LQn VWUXFWRU IRU WZR HOHFWULFDO HQJLQHHULQJ FRXUVHV +H WDXJKW D VKRUW URERWLFV ODERUDWRU\ FRXUVH DW 7HFKQRSROLV ,QVWLWXWH LQ %DUL ,WDO\ +H KDV EHHQ D PHPEHU RI 3URIHVVRU .HLWK / 'RW\f¬V 0DFKLQH ,QWHOOLJHQFH /DERUDWRU\ DW WKH 8QLYHUVLW\ RI )ORULGD VLQFH +H ZLOO UHFHLYH KLV 'RFWRU RI 3KLORVRSK\ IURP WKH 8QLYHUVLW\ RI )ORULGD LQ 0D\ +LV LPPHGLDWH JRDOV DUH WR ILQG D IDFXOW\ SRVLWLRQ DW D XQLYHUVLW\ VRPHZKHUH LQ WKH 86$ DQG WR OHDUQ WR VSHDN DQG ZULWH IOXHQWO\ LQ LWDOLDQ

PAGE 137

, FHUWLI\ WKDW , KDYH UHDG WKLV VWXG\ DQG WKDW LQ P\ RSLQLRQ LW FRQIRUPV WR DFFHSWn DEOH VWDQGDUGV RI VFKRODUO\ SUHVHQWDWLRQ DQG LV IXOO\ DGHTXDWH LQ VFRSH DQG TXDOLW\ DV D GLVVHUWDWLRQ IRU WKH GHJUHH RI 'RFWRU RI 3KLORVRSK\ / 'RW\ &KDLU RI (OHFWULFDO (QJLQHHULQJ , FHUWLI\ WKDW , KDYH UHDG WKLV VWXG\ DQG WKDW LQ P\ RSLQLRQ LW FRQIRUPV WR DFFHSWn DEOH VWDQGD[GV RI VFKRODUO\ SUHVHQWDWLRQ DQG LV IXOO\ DGHTXDWH LQ VFRSH DQG TXDOLW\ DV D GLVVHUWDWLRQ IRU WKH GHJUHH RI 'RFWRU RI 3KLORVRSK\ 7KRPDV ( %XOORFN &RFKDLU 3URIHVVRU RI (OHFWULFDO (QJLQHHULQJ , FHUWLI\ WKDW , KDYH UHDG WKLV VWXG\ DQG WKDW LQ P\ RSLQLRQ LW FRQIRUPV WR DFFHSWn DEOH VWDQGDUGV RI VFKRODUO\ SUHVHQWDWLRQ DQG LV IXOO\ DGHTXDWH LQ VFRSH DQG TXDOLW\ DV D GLVVHUWDWLRQ IRU WKH GHJUHH RI 'RFWRU RI 3KLORVRSK\ P 6WDXGKDPPHU 3URIHVVRU RI (OHFWULFDO (QJLQHHULQJ , FHUWLI\ WKDW , KDYH UHDG WKLV VWXG\ DQG WKDW LQ P\ RSLQLRQ LW FRQIRUPV WR DFFHSWn DEOH VWDQGDUGV RI VFKRODUO\ SUHVHQWDWLRQ DQG LV IXOO\ DGHTXDWH LQ VFRSH DQG TXDOLW\ DV D GLVVHUWDWLRQ IRU WKH GHJUHH RI 'RFWRU RI 3KLORVRSK\ &DUO ' &UDQH ,,, $VVRFLDWH 3URIHVVRU RI 0HFKDQLFDO (QJLQHHULQJ , FHUWLI\ WKDW , KDYH UHDG WKLV VWXG\ DQG WKDW LQ P\ RSLQLRQ LW FRQIRUPV WR DFFHSWn DEOH VWDQGDUGV RI VFKRODUO\ SUHVHQWDWLRQ DQG LV IXOO\ DGHTXDWH LQ VFRSH DQG TXDOLW\ DV D GLVVHUWDWLRQ IRU WKH GHJUHH RI 'RFWRU RI 3KLORVRSK\ 6HQFHU
PAGE 138

7KLV GLVVHUWDWLRQ ZDV VXEPLWWHG WR WKH *UDGXDWH )DFXOW\ RI WKH &ROOHJH RI (Qn JLQHHULQJ DQG WR WKH *UDGXDWH 6FKRRO DQG ZDV DFFHSWHG DV SDUWLDO IXOILOOPHQW RI WKH UHTXLUHPHQWV IRU WKH GHJUHH RI 'RFWRU RI 3KLORVRSK\ 0D\ :LQIUHG 0 3KLOOLSV 'HDQ &ROOHJH RI (QJLQHHULQJ .DUHQ $ +ROEURRN 'HDQ *UDGXDWH 6FKRRO


xml record header identifier oai:www.uflib.ufl.edu.ufdc:UF0008236300001datestamp 2009-02-16setSpec [UFDC_OAI_SET]metadata oai_dc:dc xmlns:oai_dc http:www.openarchives.orgOAI2.0oai_dc xmlns:dc http:purl.orgdcelements1.1 xmlns:xsi http:www.w3.org2001XMLSchema-instance xsi:schemaLocation http:www.openarchives.orgOAI2.0oai_dc.xsd dc:title Algebraic properties of noncommensurate systems and their applications in roboticsdc:creator Schwartz, Eric Michaeldc:publisher Eric M. Schwartzdc:date 1995dc:type Bookdc:identifier http://www.uflib.ufl.edu/ufdc/?b=UF00082363&v=0000133417401 (oclc)002046269 (alephbibnum)dc:source University of Floridadc:language English