Citation |

- Permanent Link:
- http://ufdc.ufl.edu/AA00003652/00001
## Material Information- Title:
- Sister trajectories in string theory
- Creator:
- Carbon, Steven L., 1958-
- Publication Date:
- 1993
- Language:
- English
- Physical Description:
- iv, 85 leaves : ill. ; 29 cm.
## Subjects- Subjects / Keywords:
- Algebra ( jstor )
Critical points ( jstor ) Diagrams ( jstor ) Mathematical vectors ( jstor ) Mathematics ( jstor ) Matrices ( jstor ) Oscillators ( jstor ) String theory ( jstor ) Trajectories ( jstor ) Vertices ( jstor ) Dissertations, Academic -- Physics -- UF ( lcsh ) Physics thesis Ph. D ( lcsh ) String models ( lcsh ) - Genre:
- bibliography ( marcgt )
non-fiction ( marcgt )
## Notes- Thesis:
- Thesis (Ph. D.)--University of Florida, 1993.
- Bibliography:
- Includes bibliographical references (leaves 82-84).
- General Note:
- Typescript.
- General Note:
- Vita.
- Statement of Responsibility:
- by Steven L. Carbon.
## Record Information- Source Institution:
- University of Florida
- Holding Location:
- University of Florida
- Rights Management:
- Copyright [name of dissertation author]. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
- Resource Identifier:
- 001897880 ( ALEPH )
29809574 ( OCLC ) AJX3177 ( NOTIS )
## UFDC Membership |

Downloads |

## This item has the following downloads: |

Full Text |

SISTER TRAJECTORIES IN STRING THEORY By STEVEN L. CARBON 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 1993 ACKNOWLEDGEMENTS I owe many thanks to Professor Charles Thorn, my thesis advisor, for many enlightening discussions, and for his encouragement. I would also like to thank the University of Florida particle theory group for presenting a stimulating atmosphere in which to work. TABLE OF CONTENTS Page ACKNOWLEDGEMENTS . ii ABSTRACT . iv 1 INTRODUCTION ....................... 1 2 SISTER TRAJECTORIES ................... 5 Tree Level Six-Particle Scattering . 9 Tree Level Eight-Particle Scattering 14 3 DOUBLE-LOOP FOUR-TACHYON SCATTERING 18 4 OPEN STRING SISTERS IN CLOSED STRING SCATTERING 29 5 OSCILLATOR REPRESENTATION OF SISTER TRAJECTORIES 37 6 CONCLUDING REMARKS ................. 50 APPENDIX: WEIGHT DIAGRAMS AND LAX OPERATORS 52 Review of Weight Diagrams . 53 Standard Construction of Lax Operators 58 Diagrammatic Construction of Lax Operators 61 Proof of Diagrammatic Scheme. .. 70 Supersymmetric Lax Operators . 75 Discussion . 80 REFERENCES .. .82 BIOGRAPHICAL SKETCH .................... 85 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 SISTER TRAJECTORIES IN STRING THEORY By STEVEN L. CARBON May 1993 Chairman: Charles B. Thorn Major Department: Physics It is shown that by using higher-order corrections that include sister trajec- tories, it may be possible to restore the Cerulus-Martiii bound in string theory, which would resolve an objection to locality. In the single-Regge limit, we review the existence of the first sister trajectory in the six-point function and then exhibit the second sister in the eight-point function. New work includes demonstrating that the first sister enters the four-point function at two loops, and that it can be seen across the intermediate open string propagator which appears as a higher order correction to closed string four-point scattering. We then introduce a procedure for determining the state representation of the sister trajectories, am(t), for space-like momentum transfer squared t. These sister states are obtained by analytically continuing from the physical states, and involve reciprocal oscillators signalling the sisters unplihyical nature. We consider both tree level and higher order scattering. CHAPTER 1 INTRODUCTION A major hope of string theory has been that it would describe physics at the Planck scale. This entails understanding its short distance behavior and considering the related issue of locality. The latter is important because if the theory is nonlocal at a fundamental level then acausality may result, which is probably unacceptable. The high energy behavior at tree level seems to suggest that it is not local, although Gross and Mende1 claim that it still obeys causal- ity because the interaction of strings is local. The tree level objection to locality in string theory is that in the limit of high center of mass energy, s --+ 00, for fixed scattering angle sin2(0/2) a -_, it does not behave as do theories for nonextended objects. Cerulus and Martin2(CM) found that for general theo- ries, under certain assumptions which include locality, the scattering amplitude obeys the rigorous lower bound | A(s, t) I> e-vlIn sc(9). String theory, on the other hand, has the tree level fixed-angle behavior I A(s, t) I-- e-sf(0), which was pointed out even in Veneziano's original paper.3 Gross and Mende1 and later Mende and Ooguri,4 attempted to determine if stringy perturbative cor- rections could restore the CM bound, but were unable to control the higher order corrections to reach a decisive conclusion. The CM bound was derived using the assumptions of unitarity, existence of a finite mass gap, and polynomial boundedness. Off-shell, covariant string theory must introduce Fadeev-Popov ghosts to be unitary. However, unitarity can be established on-shell in the critical dimension, for instance, by breaking 2 manifest Lorentz invariance and fixing to the light-cone gauge. String theory also does not have a finite mass gap since it has massless particles. But this probably does not lead to the violation of the CM bound. The final assump- tion of uniform polynomial boundedness states that, for fixed t, the amplitude I A(s, t) I is bounded by sN where N does not depend on s or t. String theory does not obey this condition either for t > N since its fixed t behavior goes as st. Nor does this behavior obey even the weaker condition where N ~ 0(1 t 12) which Martin showed also gives the CM bound.5 Furthermore, in quantum field theory, polynomial boundedness is a consequence of locality. As a result, it is generally thought this power behavior of s leads to the CM bound violation.1'6 Although restoring the CM bound is necessary for a local string theory, it is not sufficient. Nevertheless, resolving this issue may be important for future development of string theory. In this thesis we will show how stringy corrections can slow the exponen- tial fall-off of the scattering amplitude for t -- -oo. However, unlike Gross and Mende who examined the high energy behavior that dominates over the entire moduli space, we will focus on particular processes that dominate just a small region. Our analysis will expose an overall subdominant behavior that is consistent with the lower limit of the CM bound. This suggests that the more dominant behavior considered by Gross and Mende may actually exceed the lower bound. However, we have yet to sum the perturbative series to ver- ify if in fact the bound is obeyed. By considering subdominant behavior we hope that in performing the perturbative sum it may be possible to avoid the uncontrollable corrections which obstructed Gross and Mende. The plan is as follows. In Chap. 2 we review the emergence of the lin- ear Regge trajectories, called sisters, which have more gradual slopes than the 3 standard a(t) Regge trajectories at large negative transfer momentum squared t. Rather than working directly in the fixed angle limit, we find it more conve- nient to take s --* oc with t held fixed. We show that the complete set of sister trajectories is consistent with the CM bound. We then present a slightly new approach for obtaining the sister contributions and discuss six-point and eight- point tree level scattering. The generalization to N-point scattering should then be apparent. Since the sister trajectories occur in the bosonic sector, our results equally apply to the Superstring and Heterotic string. In Chap. 3 we will show that sisters first appear in the open string four- point scattering amplitude at the double-loop level. We focus on this process because four-point scattering is the simplest case which must be shown to obey the CM bound. An interesting result suggested from our analysis is that the sisters also occur in the non-planar case defined when twists are placed on both loops while they are separated by the intermediate sister propagator. Thus, in Chap. 4 we are led to consider the case of four interacting closed strings with an intermediate open string propagator. We will find that the open string propagator supports the sister trajectory, which is degenerate with the dilaton trajectory that may appear on the connecting closed string propagators. In Chap. 5 the focus is shifted to the the state interpretation of the sisters. We will determine the oscillator representation of the sisters by isolating the appropriate propagator, and then saturating it with string oscillators. We then analytically continue to the unphysical sister state. This procedure also affords us the opportunity to confirm the amplitudes derived in the high energy analysis. In Chap. 6 we give some concluding remarks and discuss a possible physical interpretation of the sisters. 4 In the appendix we discuss the weight diagram construction of Lax oper- ators, which have no connection with sister trajectories. Lax operators have gained recent popularity in their application to the theory of matrix models, which have been shown to be related to low dimensional string theories.7,8,9 Each Lax operator can be associated with a particular representation of an affine Lie algebra, and generates a corresponding KdV equation. These KdV equations, in turn, generate integrable systems which can reproduce matrix model results. Our notation is as follows. The standard Regge trajectory is given by a(t) = a't + ao, where we choose the open string slope a' = 1 and intercept a0 = 1. This leads to a tachyon mass of m2 = -1. In the same units, for the closed string we have a(t) = t + 2. The trajectory a(ti) is associated with the momentum transfer squared ti across the propagator zi. Finally, the trajectory t(sij) is defined with respect to the energy sij = -(pi + Pi+1 + + Pj)2. CHAPTER 2 SISTER TRAJECTORIES A central feature of the dual resonance theory3 was that in the high energy limit s --+ co the scattering amplitude scales as A(s, t) oc sa(t), for fixed t. Until the mid-70's, it was thought that the Regge trajectories a(t) were linear and parallel, i.e., a(t) = a't+a0, differing only in their intercept ao. Then, in 1976, Hoyer, T6rnqvist and Webber10 discovered that the theory also predicted a new "sister" trajectory with a slope half that of the leading Regge trajectory. They were led to this result by a careful examination of the six particle scattering tree amplitude of Fig. 1. Hoyer et al. argued as follows. In the limit s --+* o, the six-point amplitude factorizes as follows: A6 D(aa)V(aa, ab)D(ab)V(ab, ac)D(ac). (2.1) Here, the propagators D(a) have zeros for a = -1,-2,.... On the other hand, the vertices V(aa, ab) have unphysical poles for aa, ab = -1,-2,..., which have the undesirable properties of negative spin(nonsense) and wrong- signature. Now, for ab there is only one zero coming from the central propaga- tor, while there are two poles coming from the adjacent vertices. This leaves an unphysical pole, which does not appear in the exact expression for the scatter- ing amplitude. For the theory to be consistent there must be some mechanism to cancel this unwanted pole. 2 3 4 5 X 1zl z2 z3 1 1 6 S Figure 1. The first sister f(t) requires twists on both zj and z3. At that time, most people worked with four-dimensional models. Although string theories are simplest in critical space-time dimensions of 10 and 26, we can still adopt this restriction in the high energy limit by permitting momentum to grow large in only four dimensions. In this case, for six particle scattering, there are eight kinematic degrees of freedom yet nine free parameters. Thus, one must apply a four-dimensionality constraint.11 For the higher dimension string theories, one would have to consider a higher N-point function with corresponding dimensionality condition. The following discussion holds in any case. To examine the high energy behavior of the amplitudes in a way that makes sense requires that one first analytically continue the energy into the complex plane, e.g., s -- ioc.12 What Hoyer et al. observed was that previous analyses had imposed the dimensionality condition only after the energy had been analytically continued back to the physical plane. The effect of not main- taining the constraint throughout the calculation is that some critical point remains hidden. Fixing this oversight and imposing the constraint before the back continuation, allows a factorization to occur in the amplitude, which then 7 exposes the critical point. Integration about this point then leads to the be- havior I A(s,t) 1--+ (t) where /3(t) = -a(t) 1 is the first sister trajectory, for 03(t) = -1,0, 1,... or equivalently a(t) = -1, 1,3,.... Furthermore, at a(t) = -1, it was explicitly shown that the pole due to the sister trajectory 03(t) precisely cancels the remaining unphysical pole coming from the a(t) tra- jectory. In addition, the sister trajectory has associated daughters which cancel the poles at a(t) = -2, -3,.... Ref. 10 also noted that the sister trajectory had been elusive in the past because at each vertex it can not couple to more than one on-shell state. This decoupling can easily be understood by considering the factorization (2.1). In the high energy limit, the end vertices that couple an intermediate propagator to two on-shell states are represented by factors of unity, which obviously do not have poles. Shortly after the discovery of the first sister in the six-point amplitude, working in the helicity-pole limit, Hoyer13 showed that the eight-point tree amplitude predicts a second sister trajectory -y(t) = a(t) 1. The purpose of this second sister is to cancel unphysical poles occurring on the /(t) sister trajectory for 03(t) = -2, -3,.... He then proposed the generalization, 1 1 am(t) = -a(t) -(m -1), (2.2) m 2 where the mth sister first appears at tree level for 2m + 4 interacting particles. Hoyer et al. then showed that the first sister 3(t) could be obtained using the more general single-Regge limit,14 as opposed to the helicity-pole limit used in Ref. 10. Other work followed which examined the sister trajectories under the, even less general, multi-Regge limit.15'16 Sisters were subsequently found in the Neveu-Schwarz sector of the NSR superstring,17 and related phenomenological implications were discussed.18 0(t) 0(t) Figure 2. Plot showing the leading a(t) Regge trajectory and the first two sisters, 3(t) and 7(t). Quir6s showed that the sister 3(t) appears in the single-loop six-point diagram, and that it renormalizes the corresponding tree level sister.19 Further, several papers also considered the closed bosonic string which found that, as in the open string, sisters appear at tree level when there are at least six interacting particles.20,21,22 The first two sister trajectories are shown in Fig. 2 along with the leading Regge trajectory. Due to successively more gradual slopes, the net behavior of the sisters is clearly not linear as t -+ -oo. We can find the asymptotic behavior by considering the intersection of two neighboring curves am(t) and am+i(t) and then letting m -- oo. Equating these using (2.2), we easily find that to lowest order a(t) -m2. Comparing this with (2.2) gives am(t) -m = -V-2a(t) (2-.)t. (2.3) 9 For fixed angle scattering we conclude sam(t) -* s-v', which is the CM bound. This short calculation also demonstrates why locality may be responsible for the CM bound violation. In string theory, the fundamental length scale is defined by I = VaV. Further, the limit m -* oo is completely equivalent to a' -- 0. Thus, the length scale I associated with each sister approaches zero as the order of the sister increase. This means that each successive sister appears more local than the previous one, and in the asymptotic limit we reach point- like behavior. The possibility for restoring the CM bound in four-point scattering exists if we can show that the entire set of sisters, am(t), is present. Since, at each vertex, sisters do not couple to more then one on-shell state, we must consider higher order corrections. We make an initial step in this direction by showing that the first sister 0(t) couples at the double-loop level. Because t is to be held fixed, and in order to work under the most general conditions, we apply the single-Regge limit s --* oo. However, since the original approach found in Appendix B of Ref. 14 requires a priori knowledge of any twists, we modify the calculation to remove the need for their explicit presence in the initial expression of the amplitude. This has the advantage of allowing us to consider many cases simultaneously, which significantly reduces the amount of work evaluating higher order functions. Note that in the multi-Regge limit, one can only determine the need for twists by first inserting them, and then computing the final result to see if sisters appear. Tree Level Six-Particle Scattering The sister trajectory /(t) is seen in the high energy limit of the six-point function only if twists are placed on both of the adjoining propagators as shown 10 in Fig 1. We begin, however, with the corresponding untwisted amplitude. This is easily calculated in the Fubini-Veneziano formalism from A6 = (0,P1 I V(p2)AV(p3)AV(p4)AV(p5) I 0,P6). (2.4) In general, passing the vertex operators through each other produces factors of the form exp [-2pi pZ (1 z)2piPj, (2.5) n=1 where z is the product of coordinates zi which are associated with the propa- gators connecting the vertices. Final expressions for the amplitude are usually written in terms of the right-hand factors. The left-hand form is more conve- nient, however, for locating critical points in the high energy limit s -+ ioo. Consequently, we use the left-hand side of (2.5) if one of the connecting propa- gators sees the energy s, and the right-hand side for non-overlapping quantities. In the particular case of Fig. 1, the complete exponential factor is then easily found to be oo n exp [2 z(-P2z P3) (P4 + P5z) (2.6) n=1 Substituting in the momentum scalar products, the full amplitude becomes A 1 dz dz2dzz --a(t) z-1-a(t2) -1-a(t3) A6 ,- dzadz2az3z1 z2 23 x (1 zl)-1-a(s23)(1 z3)-1-a(s45) (2.7) x exp z nz(s24 34) + zzs +s 34 + z( (335 34) ) n=l where we have defined s= 34 + S61 s24 s35. (2.8) In writing (2.7), we have also dropped terms in the exponential which can be safely neglected in the high energy limit. 11 We are now in a position to impose the four-dimensionality constraint, which, in the high energy limit, reduces to s35s24 1. (2.9) s34s61 Applying this constraint to (2.7) allows the argument of the exponential to be factorized, giving A6 01dzldz2dz3z-1-a(ti) -1-a(t)3 -1-a(t3)(1 l)-1-a(23) 0 0 n )] (2.10) x (1 z3)-1-a(45) exp[ E 2 1 -- Xl)(Z2 - n=1 where 1 = l )- 1, X3 -_ 1 (2.11) s35 S24 To discuss the high energy limit we must let s -+ ocei6 where the real part of s is held fixed, and 6 is such that the real part of s is in the strip of convergence.14 The result is a Fourier integral whose asymptotic behavior is dominated by its critical points.12 For (xl,x3) to be a useful critical point it must fall within the integration region, 0 < z1, z3 < 1. Critical points taken at the boundaries do not produce sisters. Since the boundary of the integration region is not included, the factors in (2.7), other than the exponential, can be ignored during integration. To recover the proper limit s -+ ooe i, we obtain a double critical point by choosing the phases s34, 61 -* ooei, (2.12) s24, 35 -+ -cei This is completely equivalent to twisting the propagators corresponding to zi and z3 since energies that overlap an odd number of twisted propagators change sign. In other words, the role of the twists here is to place the critical point inside the integration region. 12 To obtain the leading sister trajectory by evaluating (2.7) about the critical points, we keep only the lowest order terms in the exponential and integrate about Iz x1 5e, I z3 x3 1< e, (2.13) for e small. Choosing higher powers of z1 and z3 would lead to daughter trajectories. After shifting z1 and z3, we obtain A6 -~ l 1-a(ti) x1-a(t6)(1 l)-1-'a(s2)(1 x3)-1-0a(s45)6, (2.14) where 16= dz2z2 )ez dz1dz3 exp [z2z1z31, (2.15) and 1 c = zX3(x1 1)(x3 1). (2.16) Setting y = -iez2z3s, gives 6 = i()-1 dz22-a(t2)e f d 0 dy exp yz), (2.17) where yo = -iE2z2s. Integrating over z1 we easily find 6 -1 1 -2-(t2) zjc y exp(iy) exp(-iy) I6 = -1 dz2 z2 e P )f0dy (2.18) o Y-0 y where the y integral is symmetric. If z3 were not critical, taking the limit e -- 0 now would give L6 = 0 (use dy ~ e). This demonstrates the need for a double critical point. Now, define z = -sz2c, which gives I6 = _(-c)a(t)+-1 1 dzz--a(t2)e-z 0Y dexp(iy) -exp(-iy) (2.19) For us to consistently write z = y2e-4g-1c we must have 1 > e > g-4 to reach the lower limit z -+ 0 for fixed y. Consquently, in the high energy limit 13 s -t ioo, yo -+ 00 and so the integration over y gives iwr. Next, the z integral gives r(-1 a(t2)) which is valid only for a(t2) < -1. Thus, the complete amplitude is .1+ )1 X 1 1 \ -1-a(ti) -1-a(ta) A6 -c 2 1 3 (2.20) x (1 x1)-1-a(s23)(1 x3)-1-a(45). We now analytically continue the energy back by making the replacement -A- e-i'. Defining f3(t2) = a(t2) 1, which corresponds to the first sister trajectory, and simplifying, we finally arrive at A6 ~Zire-i7r(t2)2-l(t2)-10(t2)F(-/3(t2) 1)X(t2)-0a(1) 4(t2)-a(t3) 3 (2.21) x (1 x1),(t2)-a(S23)(l x3)(t2)-a(s4). Since each of the energies comprising 9 overlaps with s34, the Regge behavior )(Wt2) shows that the central propagator in Fig. 1 sees the sister. Using the four-dimensionality constraint (2.9) we can write s = 834 + 61 s24 35 = s61(s35 861)(s24 s61), (2.22) and easily recover Eq. B.19 of Ref. 14. Examining the F function in (2.21), we see that the poles of the sister trajectory are for 3(t2) = -1,0,1,.... Our approach makes it particularly easy to determine the signature r of these poles. Twisting the sister propagator t2 changes the sign of all overlapping energies. Although both the numerator and denominator of x1 and x3 change sign in Eq. (2.11), the signs of the energy ratios remain unchanged. Thus, the twisted and untwisted diagrams can be added together giving an overall factor r + 1. Therefore, the poles of 0(t2) have pure positive signature. Since these poles correspond to odd values of spin, i.e., a(t2) = -1,1,3,..., they have unphysical wrong-signature. For the existence of the sister it was necessary that the argument of the exponential factorize, producing a 2-tuple critical point. Integrating over both 2 3 4 5 6 7 \P(t) y(t) 0(t) / z1 z2 z3 z4 z5 1 8 Figure 3. At tree level, the second sister 7(t) first appears in eight-point scattering. Concurrently, z2 and z4 see #3(t). coordinates, in effect, removed the linear power of the propagator variable 22 from the exponential. In general, integrals of the form I= 1 -a-1-n exp(-czrn) (2.23) in the limit c -+ oo integrate to _I (- -- )- -~ for a < -n. (2.24) m mm Thus, sisters do not appear in 4- or 5-pt scattering since both retain the linear power of z. Furthermore, to produce the second sister "7(t), both the linear and quadratic powers of z must be integrated away, leaving the cubic power. This occurs when the critical point is a 4-tuple, which first arises in the eight-point scattering amplitude. Tree Level Eight-Particle Scattering In this section we will expose the second sister, -"(t), in the open string tree diagram of Fig. 3, where the sister appears across the propagator with z3.16 In the corresponding amplitude, we isolate the relevant terms by including, in the exponential only quantities which overlap the central propagator. We gather the other terms into a function f(zi, z2, z4, zs), whose exact form can 15 be ignored since, as shown in the last section, the sisters depend only on the exponential factor. The advantage of using the single-Regge limit over past approaches becomes more apparent in this example. From Fig. 3(without twists), we immediately write down 1 5 AS = 0 j dzif(zl,z2,Z4,z5)z3at3) 0i= 0 (2.25) x exp [2 n(-P2zz p3z2 p4) (P5 + P6z + P7z z ). n=1 Substituting in the high energy limit values of the momentum scalar products gives the eight-tachyon amplitude 1 5 A8 f- 0 dzif(z,'z2,z4,z5)z31 i=1 00 n x exp[ znz(s25 -- s35) + z1zz4 (s26 s25 + s35 s36) n=1 (2.26) + 45 + 2z z2 z4 + 5 2(s35 845) + z4 (s46 S45) + z (s47 s46) + 4zz (836 s35 + S45 s46) + z2z4z5(s37 s36 + s46 S47)) where, now, = (s81 826 + s36 s37). (2.27) Applying the four-dimensionality constraints s81836 8135 -= 1, 81-46 = 1, 81845 = 1, (2.28) s26s37 s25s37 s26847 s25847 factorizes the argument of the exponential yielding 1 51-a(t3) AS 0 dzif(zl,z2, z4, z5)z3 ( i= l (2.29) x exp [ (z X)(Z x2)(z x4)(z x5)], n=1 where s36 s37 + s47 s46 s25 35 l -4= s s26 s25 + s35 s36 (2.30) S47 s46 25 s26 + s36 s35 X2 = 5 5 s37 s36 + s46 47' For the critical point (xl, X2, X4, x5) to be inside the integration region, we must place twist on each of the associated propagators, and apply an additional four-dimensionality constraint: 3645 1. (2.31) s35s46 Consequently, due to the twists we have the sign changes 826, 35, 37, s46 --+ -ooe. (2.32) To remove the first two powers of z3 in the exponential in (2.29), and to obtain a leading trajectory, we will integrate around z1 x1 1e, z2- x2 15 E, (2.33) z4 4 |< E, I25 Vx5 1 E- Clearly, this is just one of many critical points that we could have chosen. By writing z x4 = (z4 V/4)(z4 + v'4) ~ 2/f-4 (z4 ViT-), etc., and shifting the z's, we find A8 ~ f(x1, x2, vx4, /i) dz3z31-O(t3)esz3c x dzl dz2 exp z3zlz24(v/4 -4)(v5 x5) (2.34) x dz4 dz5 exp[2z2z4z5s. (x2 X1)(12 x2)], where 13 3 3 c = 3( l)(X2 x2)(x4 x4)( x5). (2.35) The last four integrals in (2.34) can be done in pairs, resulting in A8 ~ 47r2(2.i2x,(x2 l1)(2 x2)(1 V )(1 i5))-1 (2.36) x f(x1, x2, V/-5, v/i dz3-4-Q))ea3 c 0 Using (2.24) then gives A8 ~ f(x1,x 2,v)2(-sc) (- ) 1) 3-1 (2.37) x (22x4x5(x al)(a2 x2)(1 v/4)(1 V/- )) Again, we analytically continue back by replacing -9 with e-"9. Thus, we find the Regge behavior A8 oc 9 a(ta)-1 g(t3), which corresponds to the second sister trajectory. The first pole at q-(t3) = -2 cancels the pole of the first sister trajectory at /(t3) = -2. The daughters of 7(t3) cancel the other poles at 3(t3) = -3, -4, .. When the central propagator in Fig. 3 carries the second sister 'Y(t), the adjacent propagators, z2 and z4, see the first sister 3(t). Each of these sisters can easily be computed by constructing the exponential term in (2.25) from the appropriate overlap quantities, taking the corresponding high energy limit and then integrating over a 2-tuple. Finally, if we had chosen to initially integrate over a 2-tuple critical point for z3, then we would have found the first sister trajectory, /(t). CHAPTER 3 DOUBLE-LOOP FOUR-TACHYON SCATTERING We now adopt our procedure to handle loop corrections. As in the tree level case, we must first isolate terms in the corresponding amplitude which overlap the appropriate propagator. In particular, to search for sisters in the double- loop four-point amplitude, we consider the limiting situation where the two loops are sufficiently separated such that they and the connecting propagator can be treated as individual objects. Two such topologies are shown in Fig. 4. Both may be constructed by sewing together two single-loop diagrams. For this, we use the formalism from appendix D of Di Vecchia et al. 23 where the open string N-point multi-loop vertex has the form V(N;g) oc dDp exp [i7Tp r- 1 + p B + C], (3.1) and where 7r is the period matrix. Completing the square and integrating over the loop momentum p gives V(N;g) c(det D/2 exp [-rB 7-1 B + Cl. (3.2) The factorized four-tachyon double-loop amplitude is then 1 4 1 2 A(4;2) = dz dzi f d(0,p,p2 exp 2 + CL i1 (3.3) L(<'-2 < r 1 (BR')2 + t x -2 exp +[ ]0,p3,P4), where the subscripts L and R refer to the left and right loop, resp., and the superscript on Lc) labels the leg connecting the loops. The period matrix 18 2 3 \ / 2 3 S (t) j3(t) 1 4 14 (a) (b) Figure 4. Two distinct topologies for producing the 3/(t) sister in double- loop four-point scattering. has been reduced to the single-loop case r = 2ri ln k, where k will be defined below. The details of the measure dtz, which is a function of k1 and k2, may be suppressed in the analysis below as long as we avoid the boundaries of the integration region. In the multi-loop case the coefficient BY in (3.1) is given by(with a' = 1) --/ --.am) qm( n Tt (z)) ^ Ta (zo) B 'a= \/2 I) In (3.4) i=1 m=O P Ta(Vi(z)) P Ta(zo)) z=O' (3.4) where zo, irp, and p are fixed points, and a product of Schottky group elements is defined by Ta=S SS S r=1,2,...,g; nieZ/{O}; pi Ipi+1, (3.5) where g is the genus number. Also, (P) Ec, means that the sum is over all elements of the Schottky group except that the leftmost element in Ta can not be S,. In the single-loop case Ta = S' and S'(y) = kny, where k is the multiplier and related to the radius of the loop. Here, however, the sum restriction leaves just the identity. Finally, for one loop o1 -- oo, i71 -+ 0. Thus, dropping the loop index, B = V2 1 m In (3.6) i=1 m=O MzO 20 where the projective transformation is explicitly given by T(z) = zi-l(zi zi+l)z + zi(zi+l zi-1) alz + a2 (zi zi+l)z + (zi+1 zi-1_l) a3z +4 (3.7) To reduce (3.3) we will need the commutator OO [B. a()] = 12 ( lnVc(z)) (3.8) [ c -(m 1)! z= ( m=l Partial derivatives of the projective transformation can easily be taken giving ala4 a2a3 (zi zi+)(zi-1 zi) (a3z + a4)2 z=0 (zi+1 zi_) (3.9) or, more generally, m!(-a3)mn-(ala4 a2a3) (a3z + a4)m+l z= =_ m-lm! (zi zi+l)m(zi-1 zi) (Zi+1 zi-1) The single-loop three-point diagram is constructed by sewing together two legs of a five-point diagram, and then fixing three of the projective coordinates. For that case, following Di Vecchia et al. 24 we sew together legs 3 and 4 and then choose z3 = k, z4 = oo, and z5 = 1. In the present case, we will associate the coordinate z5 with the connecting leg coordinate zc. This gives 9zVc(z = 0) = 21 1, along with Oz'Vc(z = 0) = 0 for m > 2. Thus, the commutator (3.8) becomes \n+l(\rl(z)\m [B, = c) (-)m E (z))m z= = 2 E(1 zl)m. (3.11) m= 1 m= 1 Next, the coefficient C in (3.1) is given by 3 00 C i) (i) 1 ln[i'(z)]z=0 i=1 m=0 3 o0 (i) (j) + 2 E > n! rm! ln[(y) Vj(z)]y= z=0 (3.12) i i,j=n O Vm=O) y=z=O where the prime form is defined by wi \z T/ (w) w T (z) E(z,w) = (z w) z T() w T,(w) (3.13) Sz To,(z) w Tc,(w)' and the indicates the the identity is not included. For a single loop the prime form reduces to 00 nw w kz E(z,w) = (z w) I z k w k (3.14) z knz w knw" n=l Below, we will need the commutator 00 (0 [Cc)] O,pi) = ( 1)! mln[Ve(z)]zo m=l 1 Oo +2 (Pi mln[zi Vc(z)]z=O i c m= 1 +2E Pi 9 zi knVc(z) Vc(z) knz = Sm=1 = (m 1)!z n zi knzi Vc(z) k Vc(z) z=0' (3.15) where pc = p. Due to momentum conservation we can neglect the second denominator in the last term. Further, in the high energy limit s -- oo, we have Pl P3 -+ s/2, Pl P4 --* -s/2, P2 P3 -- -s/2 and P2 P4 -+ s/2. These imply, P Pi = (P1 +P2)Pi --+ 0. (3.16) Consequently, some of the terms in (3.15) do not survive the high energy limit in (3.3). This permits us to drop the entire first term, and the i = c term in the last sum. Rearrangement then yields 00 (0 I[C, aMIc)] O,p) =2 E (m -)! i c m=1 00 00 x [in [zi knV (z)] [Vc(z) krz] z= n=O r=l (3.17) Taking the derivatives gives 00 (0 [C, c)] 0,pi) =2 pi[ ic m=1 n=O -kmn(Vc(z))m (zi knVc(z))m (3.18) + (_)+l(Vi(z))m S(V(z) knz1)m =0' m,n=l which simplifies to (0 l [C, a,)] lO, Pi) kmn(1 z)m + (k --Y E m,n=l (- Z1)m (-knzi)mJ (3.19) We also need the single-loop result exp 1-2 + C 0, P10,P2) = ,' 2 2 I 0, p,p2), (3.20) where f arises in planar loop amplitudes and can be expressed in terms of the Jacobi theta function. Substituting (3.11) and (3.19) into (3.3) then gives A(4;2)" 1j dzz-1-a(t) Jd 1j dazi122p P2V,2'p4 i= 1 ep2 zm(1 zl)m(1 z3)m Pi .j In zi In zj x exp 2 n m=1 ij 1 k kmn I:P (k 2 zj)m n=0 2 1- kmn in Z1 [E _I PInk2 L (k ) -zi)m n=0 1 i j5c -EEPi i#c j + E pi Pj kmr X 2 = (kr z)m r -- 2 O n= 1 1 S (1- kfzi)m n=1 00 1 +E(1 knz)m) n=1 00 1 Z=1(1 kyJ)m)) (3.21) where i and j correspond to the different loops, and we have dropped a mo- mentum independent factor which can be ignored in the high energy limit. 0C =-2Ep [E ic m=l1 n=0 (0 k"mn = (k" zi)m n=0 1 I 23 Replacing the momentum scalar products by their high energy limits allows us to factorize the argument of the exponential to get A(4;2) dzz-1-(t) dp dz( ','12'34)-1-ra(t) JO zm(1 zi)m( z x exp s z zl)( z3)mgm(zl, z2, kl)gm(z3, z4, k2) m=1 (3.22) where we have defined 9in y, k) = (k-m (1 )m -< x --+ y >. (3.23) n=O n=1 The function gm(x, y, k) is for orientable planar loops and is essentially the mth derivative of in Thus, we can immediately write down the expression in the non-orientable case: o In x (-k)mn 00 1 = Ink ((-k)n )m- (1- (-k)nx)m < > n=0 n=1 (3.24) and for the non-planar case: SIn (_ )mkmn 00 1 gmn(x, y, k) = ( (1+ knx)m- n=0 n=1 Now, we search for critical points which do not reside on the boundary of the integration region. Unfortunately, due to its complicated form, one must numerically search for zeros in gm(x, y, k). It is found that gm(x, y, k), for all m, does indeed possess zeros that are exclusively within the integration range. These zeros generate the critical-point curve x = P(y, k), for some function P(y, k) which satisfies gm(P(y, k), y, k) = 0. In addition, numerically analysis indicates that both non-orientable and non-planar cases also possess critical- point curves. In all these cases the zeros do not seem to be confined to any particular region of integration space. 24 This case differs from the tree calculation in two respects. First, to factorize Eq. (3.21) it was not necessary to impose a dimensionality constraint. Clearly, this is due to the fact that there are only four interacting particles, and not due to the loops. Second, unlike the tree amplitudes, the presence of twists is not significant. In the former case, the twists were necessary to change the sign of some of the energies to place critical points inside the integration region. In the loop amplitudes, the signs change as a result of the periodicity of the Jacobi theta function. Continuing with the calculation, in the limit s -* ooei6 (3.22) becomes A(4 2) ~ dzz-1-a() d dzi(0 12 34)1-(t)esZh22 ) J O i=1 (3.26) x exp [sz(1 z1)(1 z3)g1(zl, z2, kl)gl(z33, z4, k2)] where h2 = (1 21)2(1 z3)2g2(zl, z2, kl)92(z3, z4, k2) (3.27) 2 We will evaluate about the critical curve I 21 P(z2, k) 1 e, z3 P(z4, k2) 1< (3.28) Expanding the gl's about this curve, and then shifting z21 and 23, gives A(4;2) ~ 1 dzz-1-a(t) / 1 dz2dz4(12 034)-1-a(t)esz h2 Jo J Jo (3.29) x dz1dz3 exp[szzlz3h ], where hl =(1 P(z2, kl))2(1 P(z4, k2))2 (3.30) x gj(zl = P(z2, kl), z2, kl)g(z3 = P(z4,k2),z4, k2), and h2, '12, and 034 are now evaluated on the critical curve. The integration of zi and z3 proceeds as before, giving A(4;2) irs-1 dzz-2-a(t) f d (3.31) O (3.31) x j dz2dz4(12 34)-l-a(t)h-lesz2h2. J0 25 Similarly, the z integration is also easily done giving A(4;2) ~ iTre-ir(t) (t) n(-f(t) 1) I dy 11 (3.32) x dz2dz4(12034)-1-(t)h1h 0(t)+1, which exhibits the first sister trajectory /(t). Since the integrands involve derivatives of the Jacobi theta functions, we are unable to complete the cal- culation showing explicitly that the sister does not decouple. For the planar diagram, however, in the special case 0(t) = -1, it can easily be shown that the signs of each of the integrand factors are the same over the entire integra- tion region. On the other hand, to show that decoupling does not occur in the non-orientable and non-planar cases is more difficult, although the results of the next chapter indicate that the sister survives the latter case. The existence of the the second sister requires that two of the g's share the same critical point. Using (1 + x)-(r+l) (1 + X)-r = -xe, (3.33) it follows that 00 gr(x, y, k) gr+1(x, y, k) = (knxe-knz knye-kY) n=O O (3.34) SE -x/k /kn n0 e- /" kne n=0 Since the difference is independent of the index r, for any given critical point either one gr vanishes, resulting in a single sister, or they all vanish simultane- ously. In the latter case, (3.22) results in the form A(4;2) j1dzz-1-a(t) dl dz2dz4(012'34)-1-a(t) Sf (3.35) x dzdz3 exp sziz3 Z zmem]. rm=1 2 3 1 4 (a) 2 3 1 4 (b) Figure 5. General 2m-loop four-point diagrams for generating the mth sister am(t). Integrating over z1 and z3, we obtain A(4;2) -i7rs-1 1 dzz-1-a(t) fdp 1 0 1 (3.36) x dz2dz4(012034)-1-a(t) zmem) )J m=1 The right factor gives a z-1 in leading order. Consequently, the z integral generates a leading pole at a(t) = -1, whereas the second sister requires a(t) = -2. Presumably, the 7*(t) trajectory is present if there are at least two loops on both sides of the propagator. We suspect that, in this case, there would be a factorization of the form Gmn(x, y, k, k2) = gm(x,y, kl)gm(x,y, k2) (3.37) where k1 and k2 correspond to same-side loops. In Fig. 5. we display two distinct possible multi-loop topologies for producing the higher order sisters. s s (a) (b) Figure 6. The Regge cut behavior is across the dotted lines. In both cases the central propagator may allow up to the mth sister if there are at least m loops on either side. However, evaluating Fig. 5a is not practical since the Schottky representation of the prime form (3.13) is much too formal when two or more unfactorized loops are present. On the other hand, since Fig. 5b completely factorizes the loops it requires no more than the techniques presented in this chapter. The sister trajectories may also appear across propagators which are em- bedded in an irreducible diagram. An example is the double-loop diagram displayed in Fig. 6. The sister here may be across one of the horizontal propa- gators. Such diagrams are, however, dominated by the behavior of Regge cuts. In the present case, the cut in Fig. 6a gives '(t In s e2 A (~ (3.38) (ln s)P ' for some p at fixed t. The cut has the same Regge slope as the first sister, yet its a(t)-intercept is higher. In general, the nth cut occurs at the same order as that of the n4h sister, but with a trajectory lying above the sister. This implies that the collective behavior of the cuts would actually exceed the 28 CM bound. A high energy analysis of the entire moduli space, such as that of Gross and Mende, would be dominated by the cuts. This is supported, in part, by their proposal that the fixed t behavior have the form (t Ins A ~ +1- (3.39) (In S)12g where g is the genus number. The single-loop amplitude, computed first in the fixed angle limit, was shown explicitly to reduce to (3.39), for p = 1, in the fixed t limit 0 -- 0. CHAPTER 4 OPEN STRING SISTERS IN CLOSED STRING SCATTERING An unexpected result of the last chapter is uncovered by considering the non-planar diagram in Fig. 7. The central propagator that carries the sister f(t) is that of the open string, while the non-planar loops on either side contain closed string poles. This raises the interesting possibility of open string sisters coupling to closed string propagators as in the diagram shown in Fig. 8. Below we show that this is in fact the case. In the case of the Heterotic string, however, the diagram in Fig. 8 decouples since the open string propagator can not accommodate the achiral boundary conditions required by the closed string propagators. The amplitude for four-tachyon closed string scattering with an intermedi- ate open string propagator, takes the form AG2 =(2--) d2zld2z2cl(O,p4 V't(p3,z l,2-11) x o(O I T(At,a) I0)c (41r), 1J|<1 x Ao c(0 I T(A, a) I 0)o x V(p2, z2,2) I 0, p)c. (4.1) Among the many expressions appearing in the literature for the transition op- erator T between the open and closed string state, we will use that of Shapiro and Thorn.25 We will ignore here the ghosts terms given in their explicit expres- sion for T. These give a non-trivial contribution only if loops are present. Even then, the ghosts can be ignored since they have no bearing on the calculation which focuses on the exponential contributions away from the integration Figure 7. Non-planar double-loop four-point diagram. The loops contain closed string poles. 0o(t) Figure 8. Four-point string propagator. closed string interaction with an intermediate open boundary region. The transition operator is then given by n,m=0 E C)As As - 2 nm n Am n,m=0O (n,m)?(0,0) (4.2) where, (1) (_)n+m nm -2m 2n + 1 and 721 I_2I -2 - n m 00 + (2) + nm-2n-I -2m-1 n,m=0 1 0-0 a-2c m=1 (2) 1 (-)n+m 1 11 , nm4n+m+l[n m (3) (-)n+m _ n+m n m (4.3) (4.4) (4.5) 31 The sine and cosine oscillators of the closed string are given by 1 i Ac = -(Ar + Ar), As = -(Ar Ar), (4.6) where Ar and Ar correspond to the left and right movers, respectively. In terms of the sine and cosine modes, the closed string vertex operator is written V(P2,z,z) 0, Pl)cl =ei(p+P)x z 1-2-a( exp[p (-- n(zn + -) n=1 -As (zn )n 0)c,, n /i (4.7) where the first two factors are the zero modes. In Eq. (4.7) we have written the closed string trajectory a(t) acl(t). To eliminate confusing notation, we will write the open string trajectory also in terms of a(t), i.e., ao(t) = 2a(t) 3. Then, we have zLo-2 0)o = zp2-2 0)o = z-t-2 0)o = z2-2a(t) 0)o. (4.8) Pushing T to the right, Tt to the left, moving the propagator to the right, and then using momentum conservation to eliminate part of the zero modes, Eq. (4.2) becomes AG2 =()2 dzz2-2a(t) < d2z1d2z2 zlz2 -2-a(t <47-r ) o 1 |zj|<1 1 0 + i x o(0 I exp [/- P3 (-a2n( 1) n=1 00 + i Cn a2m+l( - m=0 0O x exp [ C $ma2n+1 '2m+1l (4.9) x exp [ (a-2n-l a -2m-z2(n+m+1) n,m=0 x exp [P2 (E-- 2nz2n (z + 42 ) V'_ n=1 o nV2 -Z-(1) 2m+l n _z2 n \) + j CnmaQ-2m-1z2m1(4 2 0)o. m=0 We can easily move the even oscillators through to the vacuum states since they only appear at the far left and far right. This produces the factor exp [-P2 P3 z 2nzl + 2i)(' + (.) n=1 (4.10) = (1 z2zlz2)(1 z2zl12) p2-P3, which can be pulled outside the vacuum states. Next, pushing the quadratic terms past each other produces the factor 00 exp 4(2(k + )C (Ca42n-a 2m+2(n+k+l)], (4.11) enk km -2n-lOa2m+1 k,n,m=0 and an oscillator independent exponential which we can neglect since it will not survive in the high energy limit. Moving the quadratic terms to the vac- uum states will produce no other permanent effect as they pass by the vertex 33 operators. Pushing the factor (4.11) past the right vertex then results in AG2 )2 j1 dzz2-2a(t) jz 1 d d2z22 zlz2-2-a(t) 2 2 2p2IP3 x (1 z2zlz2)(1-_ z2z12) 2p2 p3 00 x o(0 exp[-iVzp3 Cnma2m+1i(z - n=1 m=0 00 x exp[ivP2 E C( -2m-2z2m+1(z2 i) n=1 m=0 (x + 4(2m + 1) (2k + I)a-2j-1 .C2)Cz2(j+k+1) 0)o. j,k=0 (4.12) Again, the quadratic term will not leave any permanent imprint after moving to the left-hand side. Finally, AG2 )211 dzz2-2a(t) j d2z1d2z2 ZlZ2 -2-a(t) x (1 z2zlz2)(1 z2zl2) 2p2.p3 00 xexp [2p2 P3 E(2m + 1)z2m+1kmnm(zk kl)(z -2 k,n=l m=0 (4.13) To perform the sums in the second exponential, we go to the limit s -- oceib and keep only the term linear in z. The sums can now be done by noting 2 2.7 (4.14) n --1 2n (4.14) Thus, 00 00 1 L znO E 1- 2n = (1 z)I 1. (4.15) n=1 n=1 Eq. (4.13) then becomes AG2 (1)2 1 dzz2-2a(t)z<1 d2zd2z2 I zlz2-2-a(t) x (1 z2zlz2)(1 Z2Z12) -s-8 (4.16) x exp [-sz((1 z 1) (1 2i) (1 z2)5 -(1- z2))] As in the double-loop case, since there are only four interacting particles, the argument of the exponential appears in a factorized form without resorting to a dimensionality constraint. Examining the second exponential term in (4.16), we see that there is a critical point when z1 = -\ or z2 = 52. Writing z = pei', implies 0 = 0 or 7r. To integrate (4.16) about these points, we return to the Taylor series expansion in (4.13) and (4.10), i.e., AG2 ~ dzz2-2a(t) dpdP2(P1P2)-1-a(t) 2 d01 d02 Jo Jo Jo 47r 47r x exp [2sz2P1p2 cos 01 cos 02 (4.17) 00 x exp [4sz E CkkC)p ^sin(k91) sin(n02)] k,n=l We expand by setting sin(nO) w nO for 0 = 0, and sin(nO) ; (-)nnO for 0 = 7r. The p sums for 0 = 0 can easily be carried out as follows: 00 00 (_pn 1 1 , P n-'nCO( P 2n = --p(1 p)-. (4.18) n=l n=l1 With a similar expression for 0 = r, eq. (4.17) becomes AG2 ~2 dzz2-2a(t) dpdp2(P1P2)-1-(t) fd0 d02 O JO J- 47r 47r x (exp [sz90192P1P2(1 PI)(1 P2) + 2sp1p2z2 (4.19) + exp -sz9Ol2PlP2(1 + Pl) (1 + P2) 2splP2z2), where the first exponential is for 91 and 02 expanded around the same value, and the second for the converse case. Integration over 01 yields AG2 8 dzzl-2a(t) dpidP2(PIP2 )-2-a(t) dy [eY e-' x ((1 pi)(1 p2)2 exp [2splP2 21 + (1 + pl)1(1 + p2)1 exp -2spiP2z2]), (4.20) where the exact expression for yo is not needed. The integration over y gives 2i7r. Unlike the previous examples, the sister is not necessarily the dominant behavior. This requires that we extend our considerations to higher orders. By Taylor expanding (1 p)2 and (1 + p)2, we obtain many terms which may indicate the presence of the open string sister, o. To make a firm determina- tion requires some care since the O3o trajectory is degenerate with the dilaton trajectories that may appear across the adjacent closed string propagators. There is no doubt, however, when (4.20) generates a triple pole. The form of the required solution is suggested by the partial wave analysis term /-+ sat In2 s, (4.21) C (ao(t) t)(acl() t)(acl(t) t) sa ln2 21) for the case ao(t) = acl(t). Eq. (4.20) yields this result if we select the p2 expansion terms for both pl and P2. This gives, AG2 2T 1 dzzl-2(t) dpidp2(plP2)-a(t) 28s7r 0Jo 0 (4.22) x (exp [2splp2z2 + exp 2sp1P2z21. Let w = P1P2. Then AG2 -2- 1 dzzl-2a'(t) dww-a(t) e2swz+-2swz2) dP2P 1 ,2- 7 1Q Jo (p2P (4.23) 36 The P2 integral easily gives In w. Next, defining w = z-2y, we integrate over z to obtain S [1 (22sy e2sy) AG2 210s 1 dyy-a(t) In2 y e2sy -2sy i d2 dyy-a(t) e 2sy +e -2sy Finally, the end result is AG2 ir-1 (1 e-irflo(t) )2-9+ (t)I(-1 /o(t))so(t) In2 s, (4.25) where the open string sister 03o(t) = a(t) 2. One possible concern that may arise in the above calculation is that in writing Eq. (4.17) we have discarded the term [(1 z2zlz2)(1 z2zlz2)(1 z2 lz2)(1 z2z12)] 2 (4.26) When tachyons are present this may diverge at the critical points in the neigh- borhood z = z = I z2 = 1. Fortunately, the sister trajectory emerges from the other end of the integration region where these quantities approach zero. In place of Fig. 8b, we could also represent the open string propagator as a disk that is cut out of a plane which parametrizes the world-sheet.26 In principle, we can recover the situation discussed in this chapter if we impose Neumann boundary conditions on the hole, and then factorize by restricting the locations of the vertices. An alternative case is when the hole obeys Dirichlet boundary conditions. In this case, the open string propagator is physical only when there is zero momentum across it. Since the sister trajectories occur in the limit of large s momentum transfer squared, we can rule out their existence in the Dirichlet theory. CHAPTER 5 OSCILLATOR REPRESENTATION OF SISTER TRAJECTORIES In this chapter we will determine the state representation of the sister tra- jectories. The basic nature of sister states will differ from the states associated with the standard Regge trajectory a(t) since the corresponding sister poles are not physical. In the space-like t region, the poles have the manifest unphysical characteristic of nonsense, i.e., negative spin J. The time-like resonances are not physical either. In exact expressions, the residues associated with physi- cal resonances can always be written as polynomials in the energies.10 At tree level, we can see from the final expressions (2.21) and (2.37) that the energies overlapping the sister propagator are not in this form. However, this is not the case in the double-loop expression (3.32) where only a single energy appears. Here, repeating the argument given for the six-point case in Chap. 2, we rely on the fact that the sister has wrong-signature. Constructing the sister states will give us another means for uniformly displaying the unphysical nature of the poles in all regions of t. There exists, yet, a second motivation for being interested in the state representations of the sisters. At about the same time sisters were found in the high energy analysis, they were noticed in an entirely different context by Goldstone27 who was investigating the problem of counting physical states at each mass level.28,29 This is a nontrivial problem because in D dimensions the physical states transform under the group O(D 1), whereas, due to gauge invariance, string states fill multiplets of the transverse group O(D 2). For 38 the case of four space-time dimensions, the counting problem was solved in 1976 when Goldstone presented the generating function 00 00 x(x, J) = [E(1 n2] XrJ+r(r-l)/2()r(-l(l xr)2. (5.1) n=1 r=1 This has since been generalized to higher dimensions, and for the Superstring and Heterotic string.29'30'31 Expanding out (5.1), the coefficient of x counts the number of 0(3) representations of spin J, while the exponent is the corre- sponding mass level. The connection to sisters can be made if in (5.1) one sets the x exponent in the second factor equal to M2, i.e., M2 1 Jr(M) = (r 1). (5.2) r 2 Since poles in the a(t) plane are labeled by (M2, J), we can identify (5.2) with (2.2). The state analysis of Goldstone is in the time-like t region, while the high energy analyses exposes dominant behavior in the space-like region. Determin- ing the state representation of the sister trajectories will provide a more direct link between these two approaches. The unifying feature of pole cancellation can be seen in Fig. 9 which, for a(t) > 0, displays the lowest mass levels obtained from Goldstone's formula (5.1). The figure shows how the various trajectories conspire to form the physical states(solid dots) and remove some of the pure gauge states(crosses), and that the 3(t) trajectory in both regimes enters with the opposite sign to the a(t) and *y(t) trajectories. The state representations of the physical states, defined at the poles of the standard Regge a(t) trajectories and its daughters, are well known. The first three states of the leading trajectory displayed in Fig. 9 are given by the tachyon | 0), the "photon" a_ | 0), and the massive spin two symmetric state a1[i-ai 0), where the transverse index i = 0, D 2. Suppressing the space-time index, the general leading state is given by an 0), n > 0. It is J a(t) Cx cc X S(t) a"02a-2 (t) Figure 9. Lower mass states in the open bosonic string. Dots denote phys- ical states, and crosses denote pure gauge states. The coefficients indicate con- tributions from the various leading and daughter trajectories. Nonsense poles are from high energy scattering analysis, and sense poles are from Goldstone's formula. important to note that the mode number of the states along the a(t) trajectory differ by one. This implies that by varying nr in the general open string state -na n-- a nr--- 0), (5.3) we move along a path in Fig. 9 that parallels the rth sister trajectory. Al- though the poles of the rA sister trajectory do differ by mode number r, the corresponding sister states can not be represented by the physical states (5.3). Instead, by analogy with the high energy analysis, we must analytically continue away from the states defined by (5.3). To proceed, we will work directly with the factorized scattering amplitude. This isolates the appropriate propagator which allows us project onto it all possible classes of physical states. We are then free to select the states which lead to the sisters. For the six-point 40 diagram of Fig. 1, again ignoring the twists, we project the physical states onto the central propagator. To preserve unitarity, we insert the corresponding identity operator on adjacent sides of the propagator, i.e., A6 = dzld30,p I V(p2,z-l)V(p3,)Io 1IV(p4,1)V(p5,z3) 0,P6), JQ zlz3 L 1 (5.4) where n00= 00 I = E-- ansla) 0) E 2, -1 2 '( n=0 .n2=0 The normalizations in (5.5) are fixed by the projector condition 12 = I, and the commutation relations [a4a, 4] = mrnm+nrl7'. (5.6) When we project I onto the central propagator, we easily obtain 1 00 00 1 1 ,I I I- n, J0) ( anr r (57) S 1 rn nr! 1 n ljy a(t), (5.7) r=l nr=0 where t t2. Substituting this into (5.4), using the four-dimensionality con- straint (2.9), and then taking the high energy limit s34 -+ oo, yields Ag j dz1dz3zz 1-a(tl) z 1-(t3)(1 zl)-1-a(s23)( z3)-1-a(s4s) r=1 n,= where x 1, x3 and 9 are as before. Note, by using the integral representation jnj_ -a(t) jdzzr1=,-(1)-I (5.9) we can replace the sums in (5.8) by exponential functions to get A6 1 dzdz3dzz1-(t)Z 1-a()(l zl)-l-a(s23)(1 z)-l-Q(S4) x z-1-a(t)exs X ~ exp I X 3- x3 r=51 (5.10) 41 Thus, we have completely recovered Eq. (2.10). In fact, at virtually each step of our computations below, there is a parallel step using the exponentiated form. This provides a useful check on our results, and allows us to be brief in much of the derivation. As in Chap. 2, to obtain the leading /3(t) trajectory we require that (xl, x3) be a critical point. In this case, not only do we twist the propagators zl and z3, but the index n1 must be analytically continued to a negative value. The standard procedure is to replace the infinite sum by a Sommerfeld-Watson contour integral, and then push back the contour exposing the poles on the negative real axis. To generate the necessary pole in n1 we first evaluate (5.8) at the critical point. About (xl, x3), the amplitude (5.8) is approximately A6 ~ x -c(txl)X3 1-a(t3)(1 Xl)-l+a(s23)(1 x3)-+a(s45)I6, (5.11) where, after shifting z1 and z3, the integral becomes r00 00 r Or=2 n=O (5.12) x 1 1 dzldz3(gzlz3)nl nl=0 n1 j a(t) The double integral is easily performed, giving 16 =2N o1 11 1 [K X) (r x1]3 r r=2nr=O (5.13) x y-+ 1 1 (-1)n+1 ( n= nl! = l =1jnj a(t) (nl + 1)2 where yo = e2.. As we will see shortly, it is crucial for the sister that n1 appear as a double pole. As a prelude to replacing the discrete variable ni by a continuous one, we must replace n1! by its gamma function representation F(nI + 1). To analyt- ically continue to the pole at nI = -1 we must be careful since Eq. (5.13) 42 vanishes for odd n1 because of the numerator in the last factor. Using the Sommerfeld-Watson transformation to convert the sum over n into a contour integral, we get 1 1/E \llr r=2 n.=0 (5.14) S dx yz+1 1 1 (-1)x+1 x sinxr(x+l)x+E=2jnj-a(t) ( + 1)2 ' where we have separated out, and displayed, the odd n contributions. To continue back to the pole, we must first signaturize the last factor by setting -1 = e". Pushing back the contour then exposes the double pole with residue 16 =29-1 J00 00 -n X1)Xr -X3)] r=2 nr=0 (5.15) d[ 1 y+ 1 eir(x+l) dx sinrxr(x+1)x + jnj -a(t) x=-1 Since this expression vanishes when we set x = -1 in the last factor, we only need to differentiate this term. The result is -ir. The reason why we require a double critical point should now be clear. If only z\ or z3 were critical, a single nl pole would result whose residue vanishes. Next, in the limit x -+ -1, sin x7rx(x + 1) -- 1. (5.16) Thus, 00 00 1 r X Xr n I6 =- 2i7r-1 11 E rnrnr! ) 3 r r=2 n,=0 (5.17) 1 1 + Ej-=2 IJnj a (t) In the special case nr = 0 for r = 2,3,..., we have the nonsense pole at a(t) = -1, i.e.., 6 = -2i 1 a(t) (5.18) 43 Since the analytical continuation was along the curve described by an"1 0), it is clears that this pole is generated by the leading Regge trajectory a(t). The analysis above shows that this pole is given by "(a_1)-1 | 0)". The inverse oscillator indicates the unphysical nature of this state. Now, we must explicitly show that the pole (5.18) is canceled by a cor- responding pole on the leading first sister trajectory, /3(t). To analytically continue to this pole along /3(t), we must convert the n2 sum in (5.13) to a contour integral 1 -10 n+ 1 (-l)n+1 16 = g E c rZ =O nl! (ni + 1)2 n=O "(5.19) f dx 1 1+2 2 A _ C sin rx 2r(x + 1) 13 n + 2x a(t)' where we have set nr = 0 for r = 3,4,.... Picking up the pole at x = 2a(t) - nn1 gives 2-1 Y 1+1 1 (-1)n1+1 1 0 n1! (nl + 1)2 sin r(a(t)- ) (l) nl=0 (5.20) rF(a(t)- 1ni + 1)) x3)] Repeating the steps for n and evaluating at the pole nl = -1, we finally obtain 16 = _ir25/(t) 1 1 2 Xl (2- x3)](t)+x (5.21) sin 7r(f(t)+ 1) r(f(t)+ 2) X1)X3x3) (5.21) That the first pole at f3(t) = -1 cancels the amplitude (5.18) can be seen by writing P(fo(t) + 2) = (f/(t) + 1)F(/3(t) + 1), and then canceling the pole coming from the sine function against the zero in r-1(/(t) + 1). The remaining poles are represented by the states (a_)-la on 0) and cancel unphysical poles generated by the daughters of the a(t) trajectory. 44 Repeating the six-point computation for a general critical point defined by z4 = x\ and z = x3 produces the (r 1)h- daughter of the /(t) sister. The complete set of states corresponding to /(t) and all its daughters is given by (a-1)-ma" 2 I 0), for m > 0 and n2 > 0. The set can only exist in totality, and results from a complete saturation of the propagator with the oscillator a-1. In the present context, we see that the 0/(t) sister does not appear in either the four- or five-point function, or on the zi and z3 propagators of the six-point function, because coupling the propagator to two on-shell states at any vertex prevents total saturation. This analysis suggests that to obtain the second sister trajectory, Y(t), we must first saturate the appropriate propagator with the oscillator a-i, permitting 03(t) to exist, and then with a-2. To verify this, we again consider the eight-point diagram of of Fig. 3. Inserting the identity operator on adjacent sides of the propagator z3, and using the four-dimensionality constraints, yields the form A8g dzldz2dz4dz5f(zl, z2, z4, z5) oo oo 1 1 (5.22) Z1 rnrnr! an (lj (t)(Hr r=1 nr=0 where t t- 3, and Hr = (z xl)(z x2)(z4 x4)(zr x5). (5.23) and where the x's are as before. Recall, we can obtain the leading 7(t) trajectory if we twist all the noncen- tral propagators and expand about the point z1 = xl, z2 = x2, z4 = VI--, and z5 = V/px. For (xi, x2) and (/4, v/i-) to represent double critical points will also require we continue both nl and n2 to negative values. Assuming this to be case, we have A8 f(x1,x2,' /,vV')0 0 E 1 (IHr)nr dzldz2dz4dz5 r=3 nr=0r r - x [gzlz2(v4 X4)(x5 x5)] 0 1 x 2(x _)( 2 )]n 1 n2=0 2n2 (5.24) Each pair of integrals is the same form as in (5.12). The first pair generates a double pole at nl = -1 and moves us onto a 3(t) trajectory. Subsequently, the second pair gives a double pole at n2 = -1 which now transfers us to a 7(t) trajectory. Integrating and analytically continuing to nl = -1 we immediately find Ag ~ 4i7r (22x4x5(x x)(2 X2)(1 V/f4)(1 V 1)) 00 00 1 x f(xl1,x2, vX, v ) Hr)n (5.25) r=3 nr=0 00 2+1 1 (-1)n2+1 1 Sn! (n+ 1)2 -1 + - n2=0 where the exact form of vo is not important. We can approach the final state (a_la_2)-1 10) by moving along either the leading /(t) or 7(t) trajectories. The nonsense poles obtained in two cases must cancel. For the first path, we set nr = 0 for r = 2,3,..., and perform a Sommerfeld-Watson transformation on n2, to get A8 47r2 22x445(x2- x1)(}x x2)(1 v/)(1 1))- 1 / (5.26) x f (xl, x 2 x4, V x- 3 t) t ' Similarly, for n3 : 0, sliding down the -(t) trajectory we obtain the result 4 2 x o \-1 A8 7r 2(2x4x5(x2 x)(x2 x2)(1 Vi)(1 V/)) 1 S 1 (5.27) x f(x1, x2, V 4, ) H)+ )()2 1 1 3) 3 sin r(7(t)t + 2) r(7(t) + 3) 46 where, for -7(t) = -2, this cancels the pole (5.26). Extending these results to the most general case, suggests that the rth sister trajectory forms when the propagator becomes successively saturated by the oscillators a-m, starting with m = 1, and eventually reaching m = r-1. In another words, to get to the trajectory am(t) we begin by moving down either a leading or daughter a(t) trajectory curve to either a leading or daughter 0(t) trajectory, which we reach by analytical continuation, etc. The resultant sister and its daughter trajectories are represented by open string states of the form (a_1)-ml ... (ar+1)-m (a-r)"r 10), for m1, -,mr-1 > 0,nr > 0, (5.28) where the leading trajectory is given by mr = ... mr1 = 1, nr = 0. By analogy we can immediately write down the corresponding closed string sister states by replacing the open string oscillator a_- with the closed string oscillators a_i&_i everywhere. An important point that needs to be stressed here for applying the proce- dure we have presented, is that it be possible to completely isolate the sister propagator. In the case of the double-loop four-point interaction, this crite- rion adds justification to our approach in Chap. 3 where we factorized the amplitude so that we could treat as individual objects the two loops and the connecting propagator. In the state analysis approach, projecting the physical states onto the connecting propagator gives 1 1 01 4 A(4;2) = H E rnrnr! ,nj a(t) d dzi r=1 n=0 i=1 x (0| exp[ + CL a(c n, ,(c n 0) (5.29) x (0 (a ) p- CI) ,o> 47 After simplifying, the calculation leading to the sister proceeds exactly as in the six-point case given earlier in this chapter, and reproduces the results of Chap. 3. Exposing the open string sister in the four-point closed string diagram of Fig. 8 presents a new difficultly, however, since we must look for a triple Regge pole that also is not a leading order term. The expression AG2 =( 2 ) 1d2zld2z2cl(0,p4 Vf(p3,z l,1 ) x o(O I T(At,a) i )c, x IAoI x c(0 T(A,at) | 0)o x V(p2,z2, 2) 0,Pi)ci. (5.30) reduces, after some algebra, to 00 00 AG2 12 nr 1 d2z1d2z2 47 rnnr! = jnj + 3 2a(t) ,1< r=1 nr= 0 I zlz2 I-2-(t) [-2p2 p3(z/2 + /2)( r/2 + r/2)ven (5.31) x [2P2 P3r2 Cr n z- )( z- 2 k,n=1 m=0 We used the fact here that the even and odd oscillator parts can be treated separately. To obtain the open string /3o(t) trajectory we set nr = 0 for r = 3, 4, ..., which allows the sums to carried out. In the high energy limit, we find 1 2 1 1 1 d2zld2z2 AG2 ) ( n1 n! 2n2n2 n1 + 2n2 + 3 2a(t) J< d1d x I z12 -2-a(t) [-s(z1 + )(z2 + z2)] n2 x [s ((1 zl) (1 )) ((1 z2) (1 -2))]n (5.32) Of course, using the integral representation for the propagator we can easily recover the corresponding expression of Chap. 4. Now set z = pei'. Expanding about the critical points at 0 = 0 and r gives Z 1 2n2 1 1 AG2 =2 + 2n + 3 (t) dpd2(P1P2)n+n2-1-at ni=O n2=0 sni+n2 E d01 d02(9) (1 P 2)( 2 ] + [(1 + pI)0(1 + p2)-1()n1n2), (5.33) where the first term in the last factor is for 01 and 02 being expanded about the same value, while the second term is the converse case. The 0 integrals are executed as before, giving (1 + (-)nl)262nl+2/(n1 + 1)2. Each p integration produces a factor B(n1 +n2 -at, 1- n1). Combining the terms then produces the factor (1 + (-)n1+n2). This leads to the result Ak=2 = 1 1 2n2 1 sni+n2 47r2 n2 n1 + 2n2 + 3 2a(t) n=0o n2=0 1 2 e2n"+2 x B(n + n2 (t), 1 l)2 + (-n2)(1 + (-)nl) E 12, 2 (n1 + 1)2' (5.34) where we have used (1 + (-)nl)2 = 2(1 + (-)nl). Utilizing the Sommerfeld-Watson transformation, the residue due to the double pole at n1 = -1 yields the result Ak=2 1r) 2n2 1 + (-)n2 B(- 1 + n2 a(),3/2)2-1+n2 472 =(-i2! 2 + 2n2 2(t) n2=0 (5.35) Writing the Beta function in terms of F functions, and pulling out the first three poles from one of the F's gives Ak=2 =-i 2n 1+(-)fn -i+n2 87 n 2!(1 + n2 a(t))3 n2=0 (5.36) x [(-1 r(2 + n2 a(t))r(3/2) ] 2 (-1 + n2 a(t))(n2 a(t))F( + n2 a(t))J 49 Thus, we have recovered the triple pole at n2 = a(t) 1. Computing the residue by taking the second derivative of the energy factor, gives the final result i9 1_ -_+__8_ _ Ak=2 = 9+(t 1 e- 1 Io(t) in2 s. (5.37) 7r sin 7r(,3o(t) + 1) LF(lo(t) + 2) To show that these poles cancel, we again start with (5.34) but now pick up the single pole at n1 = 2a(t) 3 2n2 to get k=2 1 0 2n2 F(-3 n2 + a(t))r(5/2 a(t) + n2) 2 4 n--2 2 r(-1/2) S2a(t)-3-n2( (-)n2)(1 (_)2a()-3-2n) 4a(t)-4-4n2 (2a(t)- 2- 2n2)2 (5.38) We must take the residue of the quadruple pole at n2 = a(t) 1, which will give a factor of 1 For this, we take two derivatives of the energy factor and, to get a nonvanishing result, one derivative of the factor (1 + ()2a(t)-3-2n2), which can be done 3 ways. This leads to Ak=2 31 1 1 + e-i(a()-) 1 r F(3/2) 2n2 s 3! 8 7r2 sin7r(a(t) -1) r(a(t)) [(-1/2)J (2s)a(-2(2)ln2s' (5.39) which reduces to the negative of (5.37). CHAPTER 6 CONCLUDING REMARKS In this thesis we have indicated that the Cerulus-Martin bound may not be violated in string theory if one includes higher order corrections to the tree diagram. To complete the proof requires that the entire perturbation series be summed to determine if the coefficients of the amplitudes have any effect upon the result. It is not clear, however, how to take the fixed angle limit in the high energy analysis employed above. Instead, the proper approach may be to adopt the techniques used by Gross and Mende.4 Basically, for closed strings, this means searching for saddle points on an N sheeted Riemann sur- face defined by an appropriate algebraic curve. To have the sisters produce the dominant behavior would require that we consider the limiting situation where the Riemann surface is divided in two, separated by a thread representing the sister propagator. The hope is that the uncontrollable higher order corrections which plagued the work of Ref. 4 would now be absent. Although this may remove one of the objections to locality, it should not imply that in any way have we shown that string theory is, or can be, a local theory. There still remains serious objections which may be more difficult, if not impossible, to overcome. For example, in string field theory, Eliezer and Woodard6 note that the cubic formulation of the field theory produces an infinite number of Abelian solutions.32 This causes a breakdown of the initial value problem since it requires an infinite amount of initial data. They show that attempts to restore this loss of predictability result in acausal behavior, 51 which, again, leads to nonlocality. Another problem in string field theory, is that the individual elements of the perturbative S-matrix still violate the CM bound. Possibly, with the help of the sisters, one can find a local field theory where strings are produced non-pertubartively and appear as bound states. This would probably resolve most of the locality problems in the field theory. We should point out, though, that it is not even clear if field theory should be the fundamental formulation for strings. Finally, let us comment on a physical interpretation for the sister states. Recall, the Regge slope a' is related to the string tension T, or energy per unit length, by a (6.1) 7rT This shows that, for instance, the first sister 03(t) has twice the tension of a(t). We visualize this occurring by bending over the string once to create a double strand, giving a 'folded' string. This picture is in accord with a reduction of the fundamental length scale I = vW1. The notion of folded strings originally dates back to the early 1970's where it was noted that pure states of the form anr 1 0) have Regge slope reduced by a factor of 1. Thus, the state analysis of Chap. 5 furthers the identification of the sister trajectories with folded strings. In conclusion, the motivation for studying sister trajectories is that they may eventually lead to a useful description of the short distance behavior of string theory. Presently, there is a growing belief that the current version of string theory is nonlocal at a fundamental level. The sister trajectories, and their interpretation as folded strings, may be the necessary ingredient for reformulating string theory to produce a local theory. APPENDIX WEIGHT DIAGRAMS AND LAX OPERATORS Recently, matrix models have received a great deal of attention as nonper- turbative descriptions of string theory.7,8,9 Since initial advancements, progress has proceeded in many different directions. In particular, Douglas33 has shown that the limited number of known matrix model solutions can be derived from the Lax pair formalism usually associated with the KdV equations. This iden- tification with integrable systems greatly increases the number of classifiable matrix models since it was shown a long time ago34 that Lax operators are associated with affine Lie algebras. For example, the models discussed by Douglas are related to the canonical representations of A(1). More recently, Di Francesco and Kutasov35'36 have discussed D(1) based matrix models which the standard matrix techniques37,38,39 have yet to solve. Thus, it may be worthwhile to focus on the integrable systems approach. Several approaches to constructing the Lax operators have been developed. The matrix procedure discussed by Drinfel'd and Sokolov34 defines first a ma- trix eigenvalue equation. The system incorporates knowledge of the Cartan subalgebra and root system of some embedding affine Lie algebra g. Starting with an affine Lie algebra facilitates the construction of an integrable system from the resulting Lax pair operators. To fix the gauge invariance in the matrix system, the gradation conventions of Drinfel'd and Sokolov require that one of the simple roots, say the mth must be removed from the affine system. The resulting system is denoted by (, cm,). This is equivalent to deleting the mth Dynkin vertex. For the most part, Drinfel'd and Sokolov choose the "canoni- 53 cal" gauge in which to express the coordinate dependent terms. In this gauge, Lax operators generate the regular KdV hierarchy equations. The modified KdV(mKdV) equations can be generated by expressing the coordinate term q(x) in the "diagonal" gauge. The canonical Lax operators can then be recovered using the well-known Miura transformations. The diagonal gauge is technically simpler than the canonical gauge. Furthermore, the final Lax operator is in a factorized form which has been used to quantize the theory.40 In this appendix our focus will be on the explicit construction of the Lax (pseudo)differential operators in the diagonal gauge using a simple diagram- matic technique.41 In most cases this technique arrives at these operators much quicker than a direct application of the scheme of Drinfel'd and Sokolov. Fur- thermore, the scheme also applies to higher representations of the embedding affine Lie algebra. In the first section we briefly review the construction of weight diagrams corresponding to representations of affine and non-affine Lie algebras. From there we review the matrix method of Drinfel'd and Sokolov for building Lax operators. Next is a presentation of our method, which re- places the matrix procedure with a scheme utilizing cyclic weight diagrams of representations of affine Lie algebras. We then present a proof that the diagrammatic algorithm produces the correct Lax operator. Finally, we dis- cusses the generalization to Lax operators based on supersymmetric affine Lie algebras. Review of Weight Diagrams As noted in the introduction, each Lax operator can be associated with a representation of some affine Lie algebra. Thus, in this section, we give a 54 brief review for constructing weight diagrams corresponding to these represen- tations. Recall, one can associate uniquely to every irreducible representation of a basic Lie algebra a highest weight vector.42 For each highest weight one can construct a weight diagram which encodes all relevant information concerning the particular representation studied, e.g., from it one can build explicit matrix representations of the generators of the Cartan subalgebra as well as the various raising and lowering operators. The level of a weight is the number of lowering operators applied to the highest weight which produces that weight. Finally, the height 7 of the weight diagram is the level of the lowest weight. Weight diagrams are generated by subtracting rows of the Cartan matrix initially from the highest weight vector written in the Dynkin basis. Rules of construction can be summed up as follows: 1. Subtract the iii row of the Cartan matrix n times from a weight vector whose iAh component has a positive value n. 2. When weight vectors have more than one positive component, subtract all possible permutations of the appropriate Cartan rows. A theorem due to Dynkin43 states that the final weight diagram is always "spindle shaped". In other words: i) the number of weight vectors at the level k is equal to the number at level -y7 k, ii) the number of weights at level k + 1 is greater than or equal to the number at level k for k < i. For an explicit example consider the algebra A2. Though this is almost a trivial case, the results will be useful for the next section. The Dynkin diagram is given by 1 0 0--- 55 where, recall, the single bar represents 1200. The Cartan matrix is then easily found to be A=(2 -1). (A.1) The highest weight vector of the fundamental representation is (10). 44 Since a positive one appears in the first place, we subtract the first row of the Cartan matrix one time. This gives the weight (-11). Now, due to the one in the second place, we subtract the second row of the Cartan matrix once to get (0 -1). This completes the process since no positive components remain. The result is the height two weight diagram (10), 11 (-11)2 (A.2) (0-1)3 where the subscripts on the weight vectors indicate a counting of the vectors. The ones adjacent to the arrows represent the normalization factors of the cor- responding negative simple roots. These values are fixed by the commutation relations of the Lie algebra. To simplify our diagrams, we will not display val- ues of unity. Later, we will see that the procedure for building weight diagrams is slightly modified in the supersymmetric case. For an affine Lie algebra,45 since there exists a linear combination among simple roots, weight diagrams of affine representations generally have infinite extent. However, some affine representations give cyclic weight diagrams of finite extent. In fact, it is these cyclic cases that are crucial to the scheme below. To produce a cyclic weight diagram, the affine component which is appended to the highest weight vector of the underlying non-affine Lie algebra, unlike the non-affine weight components, may have to be assigned a negative value. -1 2 -1 -1 A -1 -1 2 (a) (b) (-1 1 0 ) -, (0 -1 1 ) 4, I ( 1 0 -1)-- (c) Figure 10. The (-110) representation of A'). (a) Dynkin diagram; (b) Cartan matrix; (c) Cyclic weight diagram. The dashed arrow is the deleted root. As an explicit example, consider the non-twisted affine algebra A1). To generate the cyclic weight diagram corresponding to the canonical representa- tion, start with the weight vector (-110), where -1 corresponds to the affine root. Figure 10 gives the Dynkin diagram and subsequent Cartan matrix which then generates the displayed resultant weight diagram. This particular cyclic weight diagram can further be thought of as the affine extension of the highest weight diagram based on the fundamental representa- tion of the basic Lie algebra A2. This is easy to see by removing everywhere the component due to the affine root. However, this is not always the case. For example, Fig. 11 displays the cyclic weight diagram constructed with the weight vector (-211), where now the affine component is -2. Although (11) 57 (-2 1 1) /I \ (-1-1 2) (-1 2 -) \ L4- / I I I L4 ----- ------ (1 1 -2) (1 -2 1) (2 -1-1) Figure 11. Cyclic weight diagram of A(') from the weight (-211). The dashed arrows are the deleted root. (1 1) (-1 2 ) ( 2 -1) (0 ) (0 0o) 4, ,4, ( 1 -2) (-2 1 ) (-1-1) Figure 12. Highest weight diagram of the adjoint representation of A2, (11). generates the highest weight diagram of the adjoint representation of A2, we see by comparing with Fig. 12 that the affine extension contains an extra zero weight (000). 58 In general, the affine component in the affinely extended vector, associated with the highest weight vector of a basic Lie algebra, will always be negative. However, we will give an example below showing that some supersymmetric cases require positive affine components. Standard Construction of Lax Operators The Lax operator46 L(x, t) is defined to be linear and Hermitian. Further- more, it satisfies the characteristic equation L(x,t)o(x,t) = yo(x,t), (A.3) where the eigenvalue p is required to be constant under nonlinear evolution. In other words, the nonlinear behavior of the eigenfunctions O(x, t) are governed by some operator A(t), which may be nonlinear, via the equation t A(t)(x,t). (A.4) Futhermore, A(t) enters into the differential scalar Lax equation OL( t) = [A(t),L(x, t)], (A.5) which generates the integrable KdV equations. The matrix construction of Lax operators utilizing generators of some em- bedding affine Lie algebra ?(k), reviewed by Drinfel'd and Sokolov,34 begins with a matrix operator of the form L = I + A + q(x), (A.6) ax where I denotes the N x N dimensional unit matrix, and to simplify notation we have suppressed the argument t. The third term is discussed below. The second term is generated by the negative simple roots Ei of the embedding 59 affine Lie algebra. In the gradation conventions of Drinfel'd and Sokolov, we have the circulant matrix r A = ciEi (A.7) i=O The procedure is then to reduce the system of linear equations given by the kernel matrix equation L(x) = 0, (A.8) where O = (N',... N), to the linear differential eigenvalue equation (A.3), where the vacuum solution 4 is a function of the components of the eigen- function b. Drinfel'd and Sokolov show that such reduction is possible if one removes a simple root, say the mth from the affine root system. They denote this situation (O(k), cm), which is in the homogeneous or standard gradation.47 The coefficients in (A.7) are then assigned the values ciom = 1 and Cm = A, where A is a constant function of the spectral parameter z. Removing an element from the simple root system is equivalent to deleting the corresponding vertex from the Dynkin diagram. Thus, when an extremal vertex is deleted, the system (G(k), Cm) represents a single residual basic Lie algebra. Removing the affine vertex obviously gives G. On the other hand, deleting internal vertices splits the Dynkin diagram into two sections, corre- sponding to a pair of basic Lie algebras. For example, splitting A(2) at the mrh vertex gives Lax operators in the Bn-m and Cm series. Furthermore, the (pseudo)differential operator associated with the Dn series is derived using the embedding algebra D() In both situations, a pair of (pseudo)differential operators is found whose product gives the Lax operator L of Eq.(A.5). To determine the exact form of the vacuum solution 4 recall that negative simple roots are lowering operators on system eigenstates. Further, removal of a root in the affine system produces the simple root system of a non-affine 60 Lie algebra. Thus, due to the linear combination among the roots of the affine system, one root must be singled out to act as a conventional state raising operator. This role is given to the removed root. Thus, the vacuum eigenstate will be annihilated by a vacuum projection operator A- defined by r A = Eiom + AEm = A- + AEm. (A.9) i=O This requirement fixes the scalar vacuum solution 0 by setting it equal to a linear combination of the components of such that A-= 0, (A.10) is satisfied. A direct relation between the scalar operator L and the matrix operator will be given in the next section. For the kernel equation (A.8) to produce a unique solution, we require that the number of independent degrees of freedom equal the rank of the embedding affine Lie algebra O(k), or equivalently the residual system ( (k), Cm). The extra degrees of freedom generate gauge invariance. To fix the gauge invariance, one must find a matrix operator S(x) that enforces the gauge transformation 0 = ead S, (A.11) where ad denotes the adjoint mapping. The gauge freedom in Eq.(A.8) al- lows one freedom in determining the form of the coordinate dependent term qo(x),i. e., C O = I + A + qo(x). (A.12) Drinfel'd and Sokolov find the sufficient condition that S E CX(R rl), where rT is generated by the positive simple roots Fi, i /- m. Many authors, including Drinfel'd and Sokolov, work most frequently in the "canonical" gauge. However, in this paper we choose to work in their "diagonal" gauge which has the form /q 0 ... 0 0 0 q2 ... 0 0 qdiag(x) = : .. : (A.13) 0 0 ... qN-1 0 0 0 ... 0 qN This gauge leads to the convenient form r qdiag = E vi( q2",., qN)Hi, (A.14) i=0 which is in the canonical or principal gradation.47 Here, Hi are the generators of the Cartan subalgebra and the functions vi are linear combinations of the elements qi. In this gauge, the gauge term qdiag associated with (g(k), Cm) is the special case where the sum excludes i = m. The Lax operator Ldiag generates the mKdV equations, and is related to Lcan via the well-known Miura transformations. Diagrammatic Construction of Lax Operators To exploit gauge invariance of the Lax operators, one should choose a q(x) gauge most suited to ones needs. Here, we are interested in developing a diagrammatic scheme for constructing L. In this regard, the diagonal gauge proves more useful than the other choices. In this section, we will demonstrate how the diagonal gauge allows one to build Lax operators directly from cyclic weight diagrams of representations of affine Lie algebras. To motivate the algorithm, we first review the construction of L by solving the matrix system L = 0. For the present discussion, it will be sufficient to consider embedding algebras of the form (g(k), co) where the affine vertex is deleted. Thus, the diagonal gauge simply reduces to the form r qdiag(x) = i(x)Hi, (A.15) i=l1 62 where we have excluded H0 from the sum. Consider again the canonical representation of the embedding affine Lie algebra (A21, co) presented in Fig. 10. Matrix representations of the Cartan matrix can be read-off from the cyclic weight diagram. The matrix element (Hi)jj is extracted from the ith element of the j1th weight vector, while the off- diagonal elements are set to zero. The matrix entry of the negative simple root (Ei)jk is assigned its normalization factor if the kth weight vector branches into the jth weight vector as a result of subtracting the iA row of the Cartan matrix in the process. The other entries are by default zero. Thus, the matrix representations of the simple roots are easily found to give A = (1 0 (A.16) 0 1 0 where the effect of the affine root, indicated in Fig. 10 by the dashed arrow line, is assigned the value A. Plugging these values into the kernel equation (A.8) produces the system of equations [09+ vl]b1 = -A3, [0- v + v.'L, = -i01, (A.17) [0 V.] 3 = -2 Here, on the right-hand side we have placed the terms due to the matrix A. The vacuum condition (A.10) determines the scalar function to be 0 = ,'3. Thus, we must solve by starting with the last equation. First, we multiply this equation through by [9 v1 + v2], and then eliminate ('.? using the second equation. Then, multiplying through by [0 + vi] and using the top equation gives the scalar Lax eigenvalue equation L(A2 o) = [0 + [ -V1 + v21][ 2]3 (A.18) (A.18) = -A3, 63 where the spectral parameter is given by p = -A. Imposing the field redefini- tions qI = vi, q2 = v2 v1, (A.19) gives the standard form L(A(,co) = [9 + q1][9 + q2][( 1 q2]. (A.20) This example exhibits a common feature relevant for our scheme below. When the vacuum condition (A.10) requires the scalar eigenfunction to be given by a single component of the eigenfunction, say = ,, then the resulting characteristic equation satisfies Ly, = p ,. (A.21) Consequently, the system reduction must start with the ith equation in the matrix system, and proceed upward till the top equation is reached. If i < N the process continues with the bottom equation and moves upward until the ith equation is reached again. We shall refer to this case as trivial since the cor- responding cyclic affine weight diagram is linear, containing no branch points. A second feature brought out in this example, is that the number of factors in the resultant Lax operator (A.20) is equal to the number of weights in the weight diagram. Unfortunately, this is valid only for trivial cases. Nevertheless, this last observation is key to our scheme. To highlight one more property of the general procedure we turn to a non- trivial example. For this, we require a representation of an affine Lie algebra whose cyclic weight diagram has at least one branching point. Thus, consider the canonical representation of the affine algebra (D1), co). Fig. 13 presents the Dynkin diagram, Cartan matrix and corresponding cyclic weight diagram ( 2 0 -1 0 2 -1 -1 -1 2 0 0 -1 0 0 -1 0 0 o 0 0 -1 -1 i; 2 0 0 2 -----"- (-1 1 0 0 0) I 4, S(-1 -1 1 0 0) ------ S^4 (0 0 -1 1 1) (0 0 0 -1 1) (0 0 0 1 -1) (0 0 1 -1 -1) -----------(1 1-10 0) -1 0 0 --------- (1 -1 0 0 O)------- (c) Figure 13. The (-11000) representation of D4. (a) Dynkin diagram; (b) Cartan matrix; (c) Cyclic weight diagram. The dashed arrow is the deleted root. which has two branch points. Reading off from the weight diagram gives 000 000 100 010 0 0 0 001 000 000 0 0 0 (A.22) The branch points have manifested themselves by placing more than one non- zero entry in the second and sixth rows. Now, further reading off the elements 65 of the Cartan matrices gives the system of equations [8+ v+] 0 = -A07, [8 v1 + V2 ]2 = -01 A08, [8 V2 + V3 + v4]03 = -02, [8 v3 + v4]4 = 3, (A.23) [8 + v3 V4]s = -03, [8 + v2 6 = 05, [8 + v1 v2]07 = -06, [a Vl]0,8 = -b7. The vacuum condition (A.10) produces two distinct solutions, '8 and the linear combination 04 bs. Here, we consider the first case. Proceeding as before, we eliminate 46 and 07 in the last two equations to get [8 + v2 va v4][8 + vi v2][8 vi]08 = -04 05. (A.24) Now, we encounter a well-known technical problem not found in the trivial case. The components 04 and 05 can not both be simultaneously eliminated since the expressions [8 v3 + v4] and [8 + v3 v4] do not commute. This dilemma is directly linked to the fact the corresponding cyclic weight diagram has a branch point connecting the fourth and fifth weights to a single weight located below them. To overcome this obstacle the pseudo-differential operator p-1 must be introduced. Its operation on any function f(x) is given by the expansion 81lf(x) = (-1)if(i)(x)8l1-i. (A.25) i=0 Utilizing the pseudo-differential operator, we rewrite the fourth equation in (A.23) as 04 = -[0 -V3 + V41-1 (A.26) 66 Thus, the combined effect of the bottom five equations is 03 ={[9 v3 + v4-1 + [ + v3 v4]-1}[ + v2 v3 V41 (A.27) x [0 + v1 v21[Q v1]8- A helpful identity we use repeatedly is {A-1 + B-}-1= {A-[A + B]B-l}-1 = B[A + B]-'A. (A.28) When applied to Eq.(A.27), a cancellation occurs among the via's appearing in the curly brackets. This simplifies the expression to 1 03 = -[--v3 +v4]-1 [O +v3 v4] [ +2 v3 4][+VI v2][-v]s. (A.29) 2 Continuing, incorporating the next two equations in (A.23) requires a second application of the relation (A.28). Finally, the Lax operator based on (D1), co) with vacuum V)8 is L =-1[9 + V1 ][9 V1 + V2[19 V2 + V3 + V4][O V3 + V4] 4 (A.30) 4 -1[10 + V3 V4][9 + V2 V3 V41[+V V1 ][- V]. Using the field redefinitions qi = vi, q2 = V2 v, (A.31) q3 = -V2 + V3 4, q4 = -v3 + v4, we get L= 1 -l[+q ][9+q2 [O+q3[a+q4-l [9-q4][O-q3]['-q2][9-ql]. (A.32) which is proportional to the standard result. We have chosen these two examples because they introduce the techniques needed to generate Lax operators associated with even the most complicated algebraic systems. Furthermore, they show how closely the structure of cyclic affine weight diagrams is linked with the construction of general Lax operators. 67 As a result, we propose a set of four steps which allows one to construct Lax operators associated with cyclic representations of affine Lie algebras. First, we propose that to every weight vector of an affine cyclic weight diagram one can associate an operator as follows: Step 1: (aoala2...) --) [Ox + alvi(x) + a2v2(x) +-..]. (A.33) The coefficient ao does not appear on the right-hand side as it corresponds to the deleted vertex. Next, we introduce a step which is designed to facilitate the construction of Lax operators when branch points exist in the corresponding weight diagram. Essentially, this step reduces more complicated non-trivial cases to a sum of manageable trivial cases by reducing the branched weight diagram to a sum a linear subdiagrams. Step 2: Replace branching weight diagrams by the sum of linear subdiagrams, each representing a vertical route (A.34) beginning with, and ending on, the vacuum weightss. For example, Fig. 14 presents the four linear subgraphs associated with the canonical representation of D(1) In drawing cyclic weight diagrams, it is important that the arrows gener- ated by the deleted vertex are distinguished from the others. Our convention is to use dashed lines. Furthermore, the direction of the arrows must also be noted. The Lax operators associated with each subdiagram are then con- structed as follows: Step 3: Circulate around the loop beginning with the vacuum solution, such that the flow is opposite most of the arrows. If a weight vector is approached by an arrow's a) tail, append its weight factor to the operator's left side, b) head, append the weight factor's inverse to the operator's left side For weights at the tail end of both connecting arrows, do nothing. Multiply by the product of the corresponding normalization factors. (A.35) The loop is to be circulated in a direction opposite most of the arrows so that the leading term of the Lax operator L = (n + ... has positive exponent. i.e., n > 0. For trivial cases, this completes the computation of L. However, for non- trivial cases with branching weight diagrams we can not naively build the final Lax operator from a sum of its constituent linear subgraphs.Instead, as we shall prove in the next section, they are added together analogously to how one computes total resistance of resistors in parallel. Step 4: The Lax operator is given by the inverse of the sum of (A.36) the inverses, of the operators resulting from step three. For example, in the non-trivial case (D4 co). with 0 = 5 s as before, we build four operators corresponding to the linear subdiagrams in Fig. 14., L1 =[9- v 1[(9 + V1] [- V1 + V][(9 t"'. + V3 + V4][9 V3 + V41 (A.37) X [O + v2 v3 v4][O + v1 v2][O v1], L2 =[9- v1]-l[a + v1][O v1 + v2[][ v2 + v3 + v4][a + v3 v4] (A.38) x [Q + vi v3 v\][o + v1 V,][Q V1], L3 =[a v, + V_][o v2 + V3 + v4][o V3 + v4][9 + v2 V3 v31 (A.39) x [o + v, v]}[9 v,], r ------ (-1 1 0 0 0) (-1 -1 1 0 0) <------- (0 0 -1 1 1) (0 0 0 -1 1) (0 0 0 1 -1) (0 0 1 -1 -1 ) --------(1 1-1 0 0) (1 -1 0 0 0)-------- (-1 1 0 0 0) ,4, (-1 -1 1 0 0) <--------, (0 0 -1 1 1) 4, = (0 0 0 -1 1) 4, (0 0 1 -1 -1 ) (1 1 -1 0 0) (1 -1 0 0 0) <- (1 1 -1 0 0) (-1 1 0 0 0) (-1 -1 0 0) (-1 -1 1 0 0) <- (-1 -1 1 0 0)(-I (-1-1 1 0 0)<-- -4, ... (0 0-1 1 1) (0 0 -1 1 1) (0 0 -1 1 1) + (0 0 1 -1) + (0 0 0-1 1) + (0 0 0 1 -1) (0 0 1 -1 -1) (0 0 1 -1 -1) (0 0 1 -1 -1) (1 1 -1 0 ) (1 1 -1 0 0) (1 1 -1 0 0) (1 -1 0 0 0) -(1 1 -1 0 0) (1-1 0 0 0)- (1-1 0 0 0)- Figure 14. Subdiagram of the cyclic weight diagram of D 1). The dashed arrows are the deleted root. and L4 =[0 v1 + v2][( v2 + V3 + v4][O + v3 v4][O + v2 va v3] x [9 +v1 v2][a- Vi]. Factoring out common terms, we find 4 L-1 = (Li)-1 i=1 = [- v11-1[0 + V1 v21-] [a+ v- V3 V4]-1 X {[O9 v3 + v4]-1 + [ + v3 v41-1 }[ V2 + V3 + V]-1 x [O-v,1+v2]- [+-v1l]-l{1 +[-v -1['+vV,]}. (A.40) (A.41) 70 By taking the reciprocal, and simplifying, we reproduce the previous result Eq.(A.30). To end this section we consider the alternative vacuum choice 13 04. It should be obvious that since we are dealing with cyclic weight diagrams, Lax operators associated with other vacuum states can be achieved by cyclically permuting factors in the primary Lax operator. Thus, this second vacuum choice immediately gives the Lax operator L = 1[o-q3][O-q2][O-q]9-l[O+q1][O+q][9+q93][Oq4]-1[O-q4]. (A.42) 4 Proof of Diagrammatic Scheme. To prove the equivalence between the matrix system C0 = 0 and the diagrammatic algorithm, we begin by rewriting the former as 7D(x) = -A (x), (A.43) where, to simplify notation, we have defined D A = I + q(x). (A.44) ox The structure of the associated cyclic weight diagram is encoded entirely in the matrix A. Specifically, recall that the general matrix element Aij is pro- portional to A if the difference between the ith weight and the connecting jth weight equals the eliminated root of the embedding affine simple root system. All other connecting weights Ai are proportional to 1. Otherwise, the matrix element is assigned the value 0. In all cases the proportionality constant is the normalization factor of the connecting root. We construct the proof in stages. For the first stage we consider the trivial case, i.e., a single entry in each row and column of A. Removing a root produces 71 only a single vacuum state. This stage corresponds to weight diagrams with no branch points and only one arrow associated with the eliminated root. Clearly, we can rearrange the matrix equations in L = 0 such that the vacuum state equation appears last. Furthermore, it can be arranged such that A is lower triangular with ones located along a diagonal once removed from the main diagonal, except for the eliminated root whose coefficient A appears in the upper right-hand corner, i.e., A is a circulant matrix. Thus, the Lax eigenvalue equation becomes LV4N(x) = pON(X). (A.45) Since D9 is diagonal, the kth equation in (A.43) can be written N Dkk Akii, k = 2,3,... ,N. (A.46) i=1 Clearly, since A is a circulant matrix as specified above, the inequality i < k holds for k 0 1. Next, by repeatedly replacing the function 0j, appearing on the right-hand side, with the it- matrix equation we eventually reach the expression Ak1 1, i.e., N NN = N ANb,. i=l 1 NN = D, AN i A j (A.47) i=1 j=1 (A.47) N N = (_)7 ... AT1iAZ ... D k Akll i=1 k=l1 where 7 is the height of the cyclic weight diagram. Due to the successive applications of the state lowering operators A1y with i > j, this equation is interpreted as taking the highest state 71P and lowering it to the vacuum state ON- Replacing 01 through S0, = -ANo 4,,, (A.48) where we have used A1N = .A0 which excites the level of the state since 1 < N, gives N N DVNN = -(-)IA'A ".. ANMD A, A DA AklD-I 'N. (A.49) i=1 k=1 where No is the normalization factor of the affine root. Finally, moving terms to the left-hand side we recover (A.45) where N N L = {N0o ... D 1ANiD 1 ... Ak,) }-1, (A.50) i=1 k=1 and the spectral parameter is given by P = -(-)A. (A.51) Since each row and column of A contain only one entry the sum will generate a single term, i.e., L-1 = {NDV 1D_ D-.. }-1, (A.52) where N is the product of the normalization factors. Now, each D is a weight factor as defined in step 1. Thus, there is a direct mapping between the order of the weight factors and their location in the corresponding weight diagram. Now, suppose we permit multiple row entries in A, i.e., branch points in the weight diagram. First consider the case where such multiple entries occur above the Nth row. As before, there is a single vacuum state, and the constant A is located in the upper right-hand corner of A. Therefore, the constraint i < k, for k : 1, remains in effect for Eq.(A.46). Hence, the derivation leading to (A.50) follows through unchanged. Now, each new entry in A causes an additional final term in (A.50). Clearly, per step 4 of the diagrammatic algorithm, the 73 final Lax operator is obtained by taking the reciprocal of the sum of terms generated by (A.50). Next, suppose the multiple row entries in A, due to the branch point, occur in the NLh row. The vacuum condition (A.10) shows that this is equivalent to a degenerate vacuum state with, say, degeneracy d. Subsequently, this row will be associated with the eliminated root, and the d integers will be assigned the value A. In fact, A appears only in this row. Clearly, in the weight diagram the d weights share the same level. Let us first discuss the case where the coefficients A occur in the first row. Thus, Eq.(A.46) remains valid, keeping intact the constraint i < k, for k 4 1. Furthermore, the scalar eigenfunction O(x) is now a linear combination of the components ON, wN-1,... v-. ,, and the equation for 1 becomes N Di1 = E An, ,. (A.53) s=N-d+l Consequently, Eq.(A.49) is modified to N N N -E E -'A.,-' i=1 k=lj=N-d+l where f is the number of field replacements performed. The characteristic equation is obtained by multiplying both sides by Ai, and then summing over s, i.e., N N N N SA,. = -(-) ... A,D,-'Ai ... s=N-d+l s=1 k=l j=N-d+l (A.55) x )1Akl D1 AljPj. Note, the sum over s on the right-hand side has been extended to the entire range for convenience. 74 Each term in A1, contains the factor A, which can then be factored out. As a result, the scalar eigenfunction is found to be N A= A A, L,, (A.56) s=N-d+l and the Lax operator, N N L = A{5 ... Y A,, TA, 1 ...AklD1}-1. (A.57) s=1 k=1 Clearly, this has the same interpretation as the non-degenerate branching case. For the last stage of the proof, we relax the condition that multiple occur- rences of A must all be in the first row of A. In the weight diagram this means not all the arrows associated with the eliminated root point to the bottom level. Recall from the discussion surrounding Eq.(A.9), the eliminated root with coefficient A acts as a state raising operator. Thus, every occurrence of A will appear in the upper triangular portion of A, and the unit coefficients of the state lowering roots are in the lower triangular portion. For A in the kth row of A, k : 1, Eq.(A.46) is modified to N Dj 4' -= 1 A-o,'-,, (A.58) nAk where, since A corresponds to the state raising operator, k > j. Consider the case where j is the largest such index to satisfy this equation. Then, allowing degenerate vacuum states, we have N N : .. ED-(--)- "1 A,-i)l. Aj1 LA ' i=1 j=1 (A.59) N N N = (-) ... )7 DIAiEI -.AjEI[ ,.\(.'. + E A-j,(.], i=l j=l n k where, again, j is the number of field replacements performed. The effect of the factor in front of C" is to first, due to A, raise this state to ij and then 75 to lower it till the vacuum state 0i, is reached. Now, since the corresponding weight diagram is cyclic, there must exist some factor that will circulate ', back to gO. First, as was the case with (D(1), co), consider the situation where bk is an intermediate state in (A.59), i.e., N N = (_-) -... D E AsDi-1 Ajkk. (A.60) i=1 k=l1 This gives N N -. = (-)a{-j ... )1A -I 1 }-,,. (A.61) i=1 k=1 Thus, per step 3 of the diagrammatic algorithm, the factor Dk, associated with the weight vector at the tail end of both connecting arrows does not appear. Further, proceeding from higher weights to lower weights in the weight diagram contributes factors of )-1 in the operator defined in step 2 for the linear subdiagram. Finally, if ik does not appear as an intermediate state of the vacuum state , in (A.59), then it must occur as an intermediate state for one of the other vacuum states. Again, since the weight diagram is cyclic, there is some closed path going from ', to each of these other vacua. However, to write down a final expression is too unweilding. Nevertheless, it should be clear that the general rules of the diagrammatic algorithm are complete and provide an accurate mapping between weight diagrams and the scalar Lax equation. Supersymmetric Lax Operators In this section we generalize the diagrammatic scheme to supersymmetric affine Lie algebras. The classification of all possible supersymmetric exten- sions of the basic Lie algebras has been given by Kac.48 In addition to the 76 bosonic simple roots of the basic Lie algebra, the simple root system of the supersymmetric algebras contains two distinct kinds of fermionic roots. The Dynkin symbol of the first type is sometimes given by a shaded vertex repre- senting a non-zero norm. The second fermionic root type has zero norm whose Dynkin symbol is given correspondingly by a crossed out vertex. As always, the bosonic root is denoted by a white vertex. A new feature occurring in the supersymmetric Lie algebras is that they may have several non-equivalent simple root systems, corresponding to differ- ent Dynkin diagrams and Cartan matrices. In other words, the different root systems can not be transformed into each other through standard Weyl rota- tions. Instead, they are obtained by performing the "Weyl" transformation with respect to the nilpotent fermionic root. For more details, see Frappat et al. 49 which also presents a large collection of Dynkin diagrams associated with all of the classical contragradient supersymmetric cases, those of the affine and twisted affine supersymmetric algebras. Non-equivalent simple root systems which represent the same supersym- metric Lie algebra differ in the distribution of bosonic and fermionic roots. However, here we are interested in considering a natural extension of the Drinfel'd-Sokolov procedure to the supersymmetric case. This restricts the possible choices for the simple root system used for building the supers-ym- metric Lax operators.50 Recall, in the bosonic case the mKdV Lax operator constructed with the gradation choice of Drinfel'd and Sokolov generates Toda lattice models.34,51,52 For supersymmetric algebras it has been shown53 that Toda lattices are possible only for simple root systems composed purely of fermionic roots. Supersymmetric Lie algebras with purely fermionic root sys- 77 teams have been given by Leites et al. 54 SL(n + 1 I n), OSp(m I 2n) (m = 2n, 2n + 2, 2n 1), D(2 11; a). (A.62) Furthermore, the infinite-dimensional affine supersymmetric Lie algebras with purely fermionic simple root systems are SL(n I n)(1), OSp(2n + 2 | 2n)(1), D(2 | 1; a)(1), (A.63) while the infinite-dimensional twisted affine cases are SQ(2n + 1)(2), SL(n I n)(2), OSp(2n | 2n)(2). (A.64) The supersymmetric extension of the KdV equations was first discussed in Manin and Radul. 55 They suggested replacing the bosonic derivative 0x by its supersymmetric analog, i.e., x -- D = + x (A.65) Note that D2 = '. The system of matrix equations of Drinfel'd and Sokolov can then be generalized to50 Cb(x, ) =[D + Q(x, 0) + A]h(x, 0) = 0, (A.66) where A is generated by the purely negative fermionic roots, and Q(x, 0) is a Grassmann odd fermionic superfield which can be expanded as r Q(x,9) = H iJi(x, 0), (A.67) i=1 where now Hi are elements of the Cartan-Kac subalgebra. The vacuum con- dition is as before, A-O(X, 0) = 0. (.8 (A.68) 78 Since the second type of fermionic root is nilpotent, they deserve special treatment when constructing cyclic weight diagrams. To illustrate how this comes about, consider the fundamental representation of the supersymmetric algebra OSp(2 I 2). The Dynkin diagram of the purely fermionic root system is given by 1 1 where both fermionic roots are denoted as having zero norm, and where we have indicated the choice (11) for a highest weight vector. The Cartan matrix is then easily found to be A=(02 2). (A.69) To construct the highest weight diagram we proceed as before. Since a positive one appears both in the first and second places we have two permutations of subtraction to perform. In particular, we can start by subtracting the first row of the Cartan matrix giving (13), and then subtracting the second row resulting in (33). However, unlike the bosonic case, we may not subtract the first row of the Cartan another time from the weight (13). This is because here the fermionic weight vectors are nilpotent and subtracting any Cartan row twice gives a decoupled state. Similarly, we can start by subtracting the second Cartan row once(and only once) and then the first row giving (33). Thus, we find the weight diagram with height two: (11)i (13)2 (31)3 (A.70) (33)4 The decoupling which occurs when constructing a cyclic weight diagram for an affine supersymmetric algebra is almost as straightforward. For example, in Fig. 15 we display the partially decoupled weight diagram of (SL(2 I 2)(1), co) (1 -1 0 2) 2 --- b bb (0 -1 1 2) 4 --- 0 (1 0 0 0) <-I Jb3 (0 1 1 0)3 b (0 1 1 0)5 (1 1 0 0)6 - Figure 15. The (1001) representation of SL(2 I2)(1). (a) Dynkin diagram; (b) Cartan matrix; (c) Cyclic weight diagram. The dashed arrows are the deleted root. where states were decoupled as we went from top to bottom. There are several ways to decouple the remaining weights since the lowering operators b0, b2 and b3 still appear more than once. The only way for a cyclic weight diagram to emerge is by decoupling the weights outside the box. To see that this is also consistent, note that all paths leading from weight 5 to weights 2 or 4 require two applications of b0. To construct a super-Lax operator let us take the vacuum solution 4 = -4. We easily find the super-Lax operator to be (A.71) 80 As in the non-supersymmetric case, Lax operators corresponding to the other three vacua are obtained through cyclic permutations of the above operator. Discussion In this appendix we have shown how one can read off from cyclic weight diagrams, associated with representations of affine Lie algebras, Lax operators in the diagonal gauge. This method is most useful when tables of matrix representations are not at hand and must be generated by weight diagrams anyway. Furthermore, this procedure can easily be implemented on computer by virtue of the fact that computer generated algorithms currently exist for building highest weight diagrams.56 With minor modifications, these programs can be adapted for cyclic weight diagrams. It remains to be seen whether higher representations lead to any new physics. If so then a program of categorizing these results might be pursued to identify redundant solutions. This might be easier to answer for supersym- metric algebras since nilpotency projects out decoupled weight vectors. What is clear though, at least for the non-supersymmetric cases is that these higher representations lead to integrable systems. Recall, to prove the integrability of KdV systems Drinfel'd and Sokolov found the necessary infinite set of con- served currents to be given by the coefficients of the Laurent expansion of C in the affine parameter A. Our conclusion follows from the fact that every representation of a basic Lie algebra has an affine extension, and that defining properties of affine Lie algebras are representation independent. Finally, it would be interesting to see if our procedure could be modified to directly generate Lax operators in other gauges. Furthermore, in light of recent work47 on generalizations of the Drinfel'd and Sokolov scheme, one may 81 also consider different gradations of the affine Lie algebra from which to obtain the matrix A and the form of q(x). REFERENCES 1. D. J. Gross and P. F. Mende, Phys. Lett. 197B (1987) 129; Nucl. Phys. B303 (1988) 407 2. F. Cerulus and A. Martin, Phys. Lett. 8 (1964) 80 3. G. Veneziano, Nuovo Cim. 57A (1968) 190 4. P. F. Mende and H. Ooguri, Nucl. Phys. B339 (1990) 641 5. A. Martin, Nuovo Cimento 37 (1965) 671 6. D. A. Eliezer and R. P. Woodard, Nucl. Phys. B325 (1989) 389 7. M. Douglas and S. Shenker, Nucl. Phys. B335 (1990) 635 8. E. Brezin and V.A. Kazakov, Phys. Lett. B236 (1990) 144 9. D. Gross and A.A. Migdal, Nucl. Phys. B340 (1990) 333 10. P. Hoyer, N. A. T6rnqvist and B. R. Webber, Phys. Lett. 61B (1976) 191 11. Chan Hong-Mo, P. Hoyer and P. V. Ruuskanen, Nucl. Phys. B38 (1974) 125 12. V. Alessandrini, D. Amati, and B. Morel, Nuovo Cim. 7A (1972) 797 13. P. Hoyer, Phys. Lett. 63B (1976) 50 14. P. Hoyer, N. A. T6rnqvist and B. R. Webber, Nucl. Phys. B115 (1976) 429 15. W. J. Zakrzewski, Nucl. Phys. B130 (1977) 164 16. M. Sarbishaei and W. J. Zakrzewski, Nucl. Phys. B132 (1977) 268; Nucl. Phys. B132 (1977) 294 17. C. Barratt, Nucl. Phys. B120 (1977) 147 18. M. Sarbishaei, W. J. Zakrzewski and C. Barratt, Nucl. Phys. B132 (1978) 478 19. M. Quir6s, Ni-l. Phys. B155 (1979) 509 82 83 20. C. Barratt, Nucl. Phys. B126 (1977) 133 21. P. Hoyer and J. Kwiecinski, Nucl. Phys. 145 (1970) 409 22. M. Quir6s, Nucl. Phys. B160 (1979) 269 23. P. Di Vecchia, F. Pezzella, M. Frau, K. Hornfeck, A. Lerda and S. Scuito, Nucl. Phys. B322 (1989) 317 24. P. Di Vecchia, M. Frau, A. Lerda and S. Scuito, Nucl. Phys. B298 (1988) 526 25. J. A. Shapiro and C. B. Thorn, Phys. Rev. D 36 (1987) 432 26. M. B. Green, Phys. Lett. 266B (1991) 325 27. J. Goldstone, unpublished (1976). 28. W. Nahm, Nucl. Phys. B120 (1977) 125. 29. T. L. Curtright and C. B. Thorn, Nucl. Phys. B274 (1986) 520. 30. T. L. Curtright, C. B. Thorn and J. Goldstone, Phys. Lett. 175B (1986) 47. 31. T. L. Curtright, G. I. Ghandour and C. B. Thorn, Phys. Lett. 187B (1986) 45. 32. G. T. Horowitz, J. Morrow-Jones, S. P. Matrin and R. P. Woodard, Phys. Lett. 60B (1088) 261 33. M. Douglas, Phys. Lett. B238 (1990) 176 34. V. Drinfeld and V. Sokolov, J. Sov. Math. 30 (1984) 1975 35. P. Di Francesco and D. Kutasov, Nucl. Phys. B342 (1990) 589 36. P. Di Francesco and D. Kutasov, Princeton preprint PUPT-1206 (1990) 37. D. Bessis, C. Itzykson and J.-B. Zuber, Adv. Appl. Math. 1 (1980) 109 38. M. L. Mehta, Commun. Math. Phys. 79 (1981) 327 39. S. Chadha, G. Mahoux and M. L. Mehta, J. Phys A14 (1981) 579 40. V. A. Fateev and S. L. Luk'yanov, Int. J. Mod. Phys. A3 (1988) 507 41. S. L. Carbon and E. J. Piard, J. Math. Phys. 33 (1992) 2664 42. R. Slansky, Phys. Rep. 79 (1981) 1 43. E. B. Dynkin, Amer. Math. Soc. Trans. Ser. 2,6 (1957) 111 and 246 84 44. M. R. Bremner, R. V. Moody and J. Patera, Tables of Dominant Weight Mul- tiplicities for Representations of Simple Lie Algebras (Dekker, New York, 1985) p. 1 45. P. Goddard and D. Olive, Int. J. Mod. Phys. Al (1986) 303 46. A. Das, Integrable models (World Scientific,Singapore,1989) p. 1 47. M. F. De Groot, T. J. Hollowood and J. L. Miramontes, Princeton preprint IASSNS-HEP-91/19, PUPT-1251 (1991) 48. V. G. Kac, Adv. in Math. 26 (1977) 8 49. L. Frappat, A. Sciarrino and P. Sorba, Comm. Math. Phys. 121 (1989) 457 50. T. Inami and H. Kanno, Comm. Math. Phys. 136 (1991) 519 51. A. Bilal and J.-L. Gervais, Phys. Lett. B206 (1988) 412; Nucl. Phys. B314 (1989) 646 52. P. Forgacs, A. Wipf, J. Balog, L. Feher and L. O'Raifeartaigh, Phys. Lett. B227 (1989) 214; J. Balog, L. Feher, L. O'Raifeartaigh, P. Forgacs and A. Wipf, Ann. Phys. 203 (1990) 76 53. M. A. Olchanetsky, Comm. Math. Phys. 88 (1983) 63 54. D. A. Leites, M. V. Saveliev and V. V. Serganova, Serpukhov preprint, IHEP 85-81 (1985) 55. Yu. I. Manin and A. 0. Radul, Comm. Math. Phys. 98 (1985) 65 56. W. G. McKay, J. Patera, and D. W. Rand, SimpLie User's Manual and Software (CRM, Montreal, 1990) p. 1 BIOGRAPHICAL SKETCH Steve Carbon was born in Evanston, Illinois, on May 21, 1958. It was while was attending high school near Chicago that he was fascinated by his first physics course. Later, he graduated from the Miami Public School system and enrolled at the University of Central Florida in Orlando. He obtained B.S. degrees in both math and physics while at UCF before obtaining a job with RCA at the Kennedy Space Center as a software analyst. While with RCA, he earned an M.S. degree in math at UCF and started on an M.S. degree in physics. He then left his job with RCA to complete his physics masters. Af- terwards, he came to the University of Florida and subsequently began doing research under the supervision of Professor Charles Thorn. His research inter- ests have included Skyrmions, matrix models, covariant quantization of string string, and the high energy behavior of string theory. I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Charles B. Thorn, Chairman Professor of Physics I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate. in scope and quality, as a dissertation for the degree of Doctor of Philosophy. re Ramond Professor of Physics I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Pierre Sikivie Professor of Physics I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Richard D. Field Professor of Physics I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Christopher Stark Associate Professor of Mathematics This dissertation was submitted to the Graduate Faculty of the Department of Physics in the College of Liberal Arts and Sciences and to the Graduate School and was accepted as partial fulfillment of the requirements for the degree of Doctor of Philosophy. May 1993 Dean, Graduate school UNIVERSITY OF FLORIDA 3 I1262 08553 9244 3 1262 08553 9244 |

Full Text |

xml version 1.0 encoding UTF-8
REPORT xmlns http:www.fcla.edudlsmddaitss xmlns:xsi http:www.w3.org2001XMLSchema-instance xsi:schemaLocation http:www.fcla.edudlsmddaitssdaitssReport.xsd INGEST IEID EV3LPM15L_IZV0QY INGEST_TIME 2017-07-13T21:51:10Z PACKAGE AA00003652_00001 AGREEMENT_INFO ACCOUNT UF PROJECT UFDC FILES PAGE 2 6,67(5 75$-(&725,(6 ,1 675,1* 7+(25< %\ 67(9(1 / &$5%21 $ ',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 3 $&.12:/('*(0(176 RZH PDQ\ WKDQNV WR 3URIHVVRU &KDUOHV 7KRUQ P\ WKHVLV DGYLVRU IRU PDQ\ HQOLJKWHQLQJ GLVFXVVLRQV DQG IRU KLV HQFRXUDJHPHQW ZRXOG DOVR OLNH WR WKDQN WKH 8QLYHUVLW\ RI )ORULGD SDUWLFOH WKHRU\ JURXS IRU SUHVHQWLQJ D VWLPXODWLQJ DWPRVSKHUH LQ ZKLFK WR ZRUN OL PAGE 4 7$%/( 2) &217(176 3DJH $&.12:/('*(0(176 LL $%675$&7 LY ,1752'8&7,21 6,67(5 75$-(&725,(6 7UHH /HYHO 6L[3DUWLFOH 6FDWWHULQJ 7UHH /HYHO (LJKW3DUWLFOH 6FDWWHULQJ '28%/(/223 )2857$&+<21 6&$77(5,1* 23(1 675,1* 6,67(56 ,1 &/26(' 675,1* 6&$77(5,1* 26&,//$725 5(35(6(17$7,21 2) 6,67(5 75$-(&725,(6 &21&/8',1* 5(0$5.6 $33(1',; :(,*+7 ',$*5$06 $1' /$; 23(5$7256 5HYLHZ RI :HLJKW 'LDJUDPV 6WDQGDUG &RQVWUXFWLRQ RI /D[ 2SHUDWRUV 'LDJUDPPDWLF &RQVWUXFWLRQ RI /D[ 2SHUDWRUV 3URRI RI 'LDJUDPPDWLF 6FKHPH 6XSHUV\PPHWULF /D[ 2SHUDWRUV 'LVFXVVLRQ 5()(5(1&(6 %,2*5$3+,&$/ 6.(7&+ LLL PAGE 5 $EVWUDFW RI 'LVVHUWDWLRQ 3UHVHQWHG WR WKH *UDGXDWH 6FKRRO RI WKH 8QLYHUVLW\ RI )ORULGD LQ 3DUWLDO )XOILOOPHQW RI WKH 5HTXLUHPHQWV IRU WKH 'HJUHH RI 'RFWRU RI 3KLORVRSK\ 6,67(5 75$-(&725,(6 ,1 675,1* 7+(25< %\ 67(9(1 / &$5%21 0D\ &KDLUPDQ &KDUOHV % 7KRUQ 0DMRU 'HSDUWPHQW 3K\VLFV ,W LV VKRZQ WKDW E\ XVLQJ KLJKHURUGHU FRUUHFWLRQV WKDW LQFOXGH VLVWHU WUDMHFn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f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n LW\ EHFDXVH WKH LQWHUDFWLRQ RI VWULQJV LV ORFDO 7KH WUHH OHYHO REMHFWLRQ WR ORFDOLW\ LQ VWULQJ WKHRU\ LV WKDW LQ WKH OLPLW RI KLJK FHQWHU RI PDVV HQHUJ\ V fÂ§! RR IRU IL[HG VFDWWHULQJ DQJOH VLQff m fÂ§ M LW GRHV QRW EHKDYH DV GR WKHRULHV IRU QRQH[WHQGHG REMHFWV &HUXOXV DQG 0DUWLQ &0f IRXQG WKDW IRU JHQHUDO WKHRn ULHV XQGHU FHUWDLQ DVVXPSWLRQV ZKLFK LQFOXGH ORFDOLW\ WKH VFDWWHULQJ DPSOLWXGH REH\V WKH ULJRURXV ORZHU ERXQG $VWf _! Han?$OQVF 6WULQJ WKHRU\ RQ WKH RWKHU KDQG KDV WKH WUHH OHYHO IL[HGDQJOH EHKDYLRU $VWf _fÂ§! ZKLFK ZDV SRLQWHG RXW HYHQ LQ 9HQH]LDQRfV RULJLQDO SDSHUn *URVV DQG 0HQGH DQG ODWHU 0HQGH DQG 2RJXUL DWWHPSWHG WR GHWHUPLQH LI VWULQJ\ SHUWXUEDWLYH FRUn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n WLRQ RI XQLIRUP SRO\QRPLDO ERXQGHGQHVV VWDWHV WKDW IRU IL[HG I WKH DPSOLWXGH $VWf LV ERXQGHG E\ ZKHUH 1 GRHV QRW GHSHQG RQ V RU W 6WULQJ WKHRU\ GRHV QRW REH\ WKLV FRQGLWLRQ HLWKHU IRU W 1 VLQFH LWV IL[HG W EHKDYLRU JRHV DV VL 1RU GRHV WKLV EHKDYLRU REH\ HYHQ WKH ZHDNHU FRQGLWLRQ ZKHUH 1 r ZKLFK 0DUWLQ VKRZHG DOVR JLYHV WKH &0 ERXQG )XUWKHUPRUH LQ TXDQWXP ILHOG WKHRU\ SRO\QRPLDO ERXQGHGQHVV LV D FRQVHTXHQFH RI ORFDOLW\ $V D UHVXOW LW LV JHQHUDOO\ WKRXJKW WKLV SRZHU EHKDYLRU RI V OHDGV WR WKH &0 ERXQG YLRODWLRQ $OWKRXJK UHVWRULQJ WKH &0 ERXQG LV QHFHVVDU\ IRU D ORFDO VWULQJ WKHRU\ LW LV QRW VXIILFLHQW 1HYHUWKHOHVV UHVROYLQJ WKLV LVVXH PD\ EH LPSRUWDQW IRU IXWXUH GHYHORSPHQW RI VWULQJ WKHRU\ ,Q WKLV WKHVLV ZH ZLOO VKRZ KRZ VWULQJ\ FRUUHFWLRQV FDQ VORZ WKH H[SRQHQn WLDO IDOORII RI WKH VFDWWHULQJ DPSOLWXGH IRU W fÂ§! fÂ§RR +RZHYHU XQOLNH *URVV DQG 0HQGH ZKR H[DPLQHG WKH KLJK HQHUJ\ EHKDYLRU WKDW GRPLQDWHV RYHU WKH HQWLUH PRGXOL VSDFH ZH ZLOO IRFXV RQ SDUWLFXODU SURFHVVHV WKDW GRPLQDWH MXVW D VPDOO UHJLRQ 2XU DQDO\VLV ZLOO H[SRVH DQ RYHUDOO VXEGRPLQDQW EHKDYLRU WKDW LV FRQVLVWHQW ZLWK WKH ORZHU OLPLW RI WKH &0 ERXQG 7KLV VXJJHVWV WKDW WKH PRUH GRPLQDQW EHKDYLRU FRQVLGHUHG E\ *URVV DQG 0HQGH PD\ DFWXDOO\ H[FHHG WKH ORZHU ERXQG +RZHYHU ZH KDYH \HW WR VXP WKH SHUWXUEDWLYH VHULHV WR YHUn LI\ LI LQ IDFW WKH ERXQG LV REH\HG %\ FRQVLGHULQJ VXEGRPLQDQW EHKDYLRU ZH KRSH WKDW LQ SHUIRUPLQJ WKH SHUWXUEDWLYH VXP LW PD\ EH SRVVLEOH WR DYRLG WKH XQFRQWUROODEOH FRUUHFWLRQV ZKLFK REVWUXFWHG *URVV DQG 0HQGH 7KH SODQ LV DV IROORZV ,Q &KDS ZH UHYLHZ WKH HPHUJHQFH RI WKH OLQn HDU 5HJJH WUDMHFWRULHV FDOOHG VLVWHUV ZKLFK KDYH PRUH JUDGXDO VORSHV WKDQ WKH PAGE 8 VWDQGDUG DWf 5HJJH WUDMHFWRULHV DW ODUJH QHJDWLYH WUDQVIHU PRPHQWXP VTXDUHG W 5DWKHU WKDQ ZRUNLQJ GLUHFWO\ LQ WKH IL[HG DQJOH OLPLW ZH ILQG LW PRUH FRQYHn QLHQW WR WDNH V fÂ§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n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f DnW RT ZKHUH ZH FKRRVH WKH RSHQ VWULQJ VORSH D DQG LQWHUFHSW 42 7KLV OHDGV WR D WDFK\RQ PDVV RI PÂ fÂ§ fÂ§ ,Q WKH VDPH XQLWV IRU WKH FORVHG VWULQJ ZH KDYH DWf A 7KH WUDMHFWRU\ DWcf LV DVVRFLDWHG ZLWK WKH PRPHQWXP WUDQVIHU VTXDUHG DFURVV WKH SURSDJDWRU ]Â )LQDOO\ WKH WUDMHFWRU\ DVcMf LV GHILQHG ZLWK UHVSHFW WR WKH HQHUJ\ 6^M fÂ§SÂ SÂL Â‘ Â‘ f SMf PAGE 10 &+$37(5 6,67(5 75$-(&725,(6 $ FHQWUDO IHDWXUH RI WKH GXDO UHVRQDQFH WKHRU\nr ZDV WKDW LQ WKH KLJK HQHUJ\ OLPLW V fÂ§! RR WKH VFDWWHULQJ DPSOLWXGH VFDOHV DV $V Wf RF VDA? IRU IL[HG W 8QWLO WKH PLGfV LW ZDV WKRXJKW WKDW WKH 5HJJH WUDMHFWRULHV DWf ZHUH OLQHDU DQG SDUDOOHO LH DWf DnW44 GLIIHULQJ RQO\ LQ WKHLU LQWHUFHSW TT 7KHQ LQ +R\HU 7RUQTYLVW DQG :HEEHUr GLVFRYHUHG WKDW WKH WKHRU\ DOVR SUHGLFWHG D QHZ ffVLVWHUf WUDMHFWRU\ ZLWK D VORSH KDOI WKDW RI WKH OHDGLQJ 5HJJH WUDMHFWRU\ 7KH\ ZHUH OHG WR WKLV UHVXOW E\ D FDUHIXO H[DPLQDWLRQ RI WKH VL[ SDUWLFOH VFDWWHULQJ WUHH DPSOLWXGH RI )LJ +R\HU HW DO DUJXHG DV IROORZV ,Q WKH OLPLW V RR WKH VL[SRLQW DPSOLWXGH IDFWRUL]HV DV IROORZV $T a 'DDf9DD DEf'DEf9DE DFf'DFf f +HUH WKH SURSDJDWRUV '^Df KDYH ]HURV IRU D fÂ§ fÂ§ 2Q WKH RWKHU KDQG WKH YHUWLFHV 9DDFNEf KDYH XQSK\VLFDO SROHV IRU DDDE fÂ§fÂ§ ZKLFK KDYH WKH XQGHVLUDEOH SURSHUWLHV RI QHJDWLYH VSLQQRQVHQVHf DQG ZURQJ VLJQDWXUH 1RZ IRU DE WKHUH LV RQO\ RQH ]HUR FRPLQJ IURP WKH FHQWUDO SURSDJDn WRU ZKLOH WKHUH DUH WZR SROHV FRPLQJ IURP WKH DGMDFHQW YHUWLFHV 7KLV OHDYHV DQ XQSK\VLFDO SROH ZKLFK GRHV QRW DSSHDU LQ WKH H[DFW H[SUHVVLRQ IRU WKH VFDWWHUn LQJ DPSOLWXGH )RU WKH WKHRU\ WR EH FRQVLVWHQW WKHUH PXVW EH VRPH PHFKDQLVP WR FDQFHO WKLV XQZDQWHG SROH PAGE 11 6 )LJXUH 7KH ILUVW VLVWHU IOWf UHTXLUHV WZLVWV RQ ERWK ]? DQG $W WKDW WLPH PRVW SHRSOH ZRUNHG ZLWK IRXUGLPHQVLRQDO PRGHOV $OWKRXJK VWULQJ WKHRULHV DUH VLPSOHVW LQ FULWLFDO VSDFHWLPH GLPHQVLRQV RI DQG ZH FDQ VWLOO DGRSW WKLV UHVWULFWLRQ LQ WKH KLJK HQHUJ\ OLPLW E\ SHUPLWWLQJ PRPHQWXP WR JURZ ODUJH LQ RQO\ IRXU GLPHQVLRQV ,Q WKLV FDVH IRU VL[ SDUWLFOH VFDWWHULQJ WKHUH DUH HLJKW NLQHPDWLF GHJUHHV RI IUHHGRP \HW QLQH IUHH SDUDPHWHUV 7KXV RQH PXVW DSSO\ D IRXUGLPHQVLRQDOLW\ FRQVWUDLQW )RU WKH KLJKHU GLPHQVLRQ VWULQJ WKHRULHV RQH ZRXOG KDYH WR FRQVLGHU D KLJKHU 1SRLQW IXQFWLRQ ZLWK FRUUHVSRQGLQJ GLPHQVLRQDOLW\ FRQGLWLRQ 7KH IROORZLQJ GLVFXVVLRQ KROGV LQ DQ\ FDVH 7R H[DPLQH WKH KLJK HQHUJ\ EHKDYLRU RI WKH DPSOLWXGHV LQ D ZD\ WKDW PDNHV VHQVH UHTXLUHV WKDW RQH ILUVW DQDO\WLFDOO\ FRQWLQXH WKH HQHUJ\ LQWR WKH FRPSOH[ SODQH HJ V fÂ§r Â :KDW +RYHU HW DO REVHUYHG ZDV WKDW SUHYLRXV DQDO\VHV KDG LPSRVHG WKH GLPHQVLRQDOLW\ FRQGLWLRQ RQO\ DIWHU WKH HQHUJ\ KDG EHHQ DQDO\WLFDOO\ FRQWLQXHG EDFN WR WKH SK\VLFDO SODQH 7KH HIIHFW RI QRW PDLQn WDLQLQJ WKH FRQVWUDLQW WKURXJKRXW WKH FDOFXODWLRQ LV WKDW VRPH FULWLFDO SRLQW UHPDLQV KLGGHQ )L[LQJ WKLV RYHUVLJKW DQG LPSRVLQJ WKH FRQVWUDLQW EHIRUH WKH EDFN FRQWLQXDWLRQ DOORZV D IDFWRUL]DWLRQ WR RFFXU LQ WKH DPSOLWXGH ZKLFK WKHQ PAGE 12 H[SRVHV WKH FULWLFDO SRLQW ,QWHJUDWLRQ DERXW WKLV SRLQW WKHQ OHDGV WR WKH EHn KDYLRU $VWf _fÂ§! ZKHUH IWf ADWf fÂ§ LV WKH ILUVW VLVWHU WUDMHFWRU\ IRU Wf fÂ§ RU HTXLYDOHQWO\ DWf fÂ§ fÂ§ )XUWKHUPRUH DW DWf fÂ§ fÂ§ LW ZDV H[SOLFLWO\ VKRZQ WKDW WKH SROH GXH WR WKH VLVWHU WUDMHFWRU\ Wf SUHFLVHO\ FDQFHOV WKH UHPDLQLQJ XQSKYVLFDO SROH FRPLQJ IURP WKH DWf WUDn MHFWRU\ ,Q DGGLWLRQ WKH VLVWHU WUDMHFWRU\ KDV DVVRFLDWHG GDXJKWHUV ZKLFK FDQFHO WKH SROHV DW DWf fÂ§ fÂ§ 5HI DOVR QRWHG WKDW WKH VLVWHU WUDMHFWRU\ KDG EHHQ HOXVLYH LQ WKH SDVW EHFDXVH DW HDFK YHUWH[ LW FDQ QRW FRXSOH WR PRUH WKDQ RQH RQVKHOO VWDWH 7KLV GHFRXSOLQJ FDQ HDVLO\ EH XQGHUVWRRG E\ FRQVLGHULQJ WKH IDFWRUL]DWLRQ f ,Q WKH KLJK HQHUJ\ OLPLW WKH HQG YHUWLFHV WKDW FRXSOH DQ LQWHUPHGLDWH SURSDJDWRU WR WZR RQVKHOO VWDWHV DUH UHSUHVHQWHG E\ IDFWRUV RI XQLW\ ZKLFK REYLRXVO\ GR QRW KDYH SROHV 6KRUWO\ DIWHU WKH GLVFRYHU\ RI WKH ILUVW VLVWHU LQ WKH VL[SRLQW DPSOLWXGH ZRUNLQJ LQ WKH KHOLFLW\SROH OLPLW +R\HUn VKRZHG WKDW WKH HLJKWSRLQW WUHH DPSOLWXGH SUHGLFWV D VHFRQG VLVWHU WUDMHFWRU\ Wf MFWf fÂ§ 7KH SXUSRVH RI WKLV VHFRQG VLVWHU LV WR FDQFHO XQSK\VLFDO SROHV RFFXUULQJ RQ WKH #^Wf VLVWHU WUDMHFWRU\ IRU cWf fÂ§ fÂ§ +H WKHQ SURSRVHG WKH JHQHUDOL]DWLRQ DPWf a aP f f P ZKHUH WKH PfÂ§ VLVWHU ILUVW DSSHDUV DW WUHH OHYHO IRU P LQWHUDFWLQJ SDUWLFOHV +R\HU HW DO WKHQ VKRZHG WKDW WKH ILUVW VLVWHU Wf FRXOG EH REWDLQHG XVLQJ WKH PRUH JHQHUDO VLQJOH5HJJH OLPLW DV RSSRVHG WR WKH KHOLFLW\SROH OLPLW XVHG LQ 5HI 2WKHU ZRUN IROORZHG ZKLFK H[DPLQHG WKH VLVWHU WUDMHFWRULHV XQGHU WKH HYHQ OHVV JHQHUDO PXOWL5HJJH OLPLW 6LVWHUV ZHUH VXEVHTXHQWO\ IRXQG LQ WKH 1HYHX6FKZDU] VHFWRU RI WKH 165 VXSHUVWULQJn DQG UHODWHG SKHQRPHQRORJLFDO LPSOLFDWLRQV ZHUH GLVFXVVHGA PAGE 13 )LJXUH 3ORW VKRZLQJ WKH OHDGLQJ DWf 5HJJH WUDMHFWRU\ DQG WKH ILUVW WZR VLVWHUV Wf DQG Wf 4XLUV VKRZHG WKDW WKH VLVWHU Wf DSSHDUV LQ WKH VLQJOHORRS VL[SRLQW GLDJUDP DQG WKDW LW UHQRUPDOL]HV WKH FRUUHVSRQGLQJ WUHH OHYHO VLVWHUn )XUWKHU VHYHUDO SDSHUV DOVR FRQVLGHUHG WKH FORVHG ERVRQLF VWULQJ ZKLFK IRXQG WKDW DV LQ WKH RSHQ VWULQJ VLVWHUV DSSHDU DW WUHH OHYHO ZKHQ WKHUH DUH DW OHDVW VL[ LQWHUDFWLQJ SDUWLFOHV 7KH ILUVW WZR VLVWHU WUDMHFWRULHV DUH VKRZQ LQ )LJ DORQJ ZLWK WKH OHDGLQJ 5HJJH WUDMHFWRU\ 'XH WR VXFFHVVLYHO\ PRUH JUDGXDO VORSHV WKH QHW EHKDYLRU RI WKH VLVWHUV LV FOHDUO\ QRW OLQHDU DV W fÂ§ fÂ§ :H FDQ ILQG WKH DV\PSWRWLF EHKDYLRU E\ FRQVLGHULQJ WKH LQWHUVHFWLRQ RI WZR QHLJKERULQJ FXUYHV DP>Wf DQG DPLIf DQG WKHQ OHWWLQJ P fÂ§} (TXDWLQJ WKHVH XVLQJ f ZH HDVLO\ ILQG WKDW WR ORZHVW RUGHU DWf a fÂ§ AP &RPSDULQJ WKLV ZLWK f JLYHV DP^Wf a P \D^Wf m f PAGE 14 )RU IL[HG DQJOH VFDWWHULQJ ZH FRQFOXGH ZKLFK LV WKH &0 ERXQG 7KLV VKRUW FDOFXODWLRQ DOVR GHPRQVWUDWHV ZK\ ORFDOLW\ PD\ EH UHVSRQVLEOH IRU WKH &0 ERXQG YLRODWLRQ ,Q VWULQJ WKHRU\ WKH IXQGDPHQWDO OHQJWK VFDOH LV GHILQHG E\ VDr )XUWKHU WKH OLPLW P fÂ§! RF LV FRPSOHWHO\ HTXLYDOHQW WR D fÂ§! 7KXV WKH OHQJWK VFDOH O DVVRFLDWHG ZLWK HDFK VLVWHU DSSURDFKHV ]HUR DV WKH RUGHU RI WKH VLVWHU LQFUHDVH 7KLV PHDQV WKDW HDFK VXFFHVVLYH VLVWHU DSSHDUV PRUH ORFDO WKDQ WKH SUHYLRXV RQH DQG LQ WKH DV\PSWRWLF OLPLW ZH UHDFK SRLQWn OLNH EHKDYLRU 7KH SRVVLELOLW\ IRU UHVWRULQJ WKH &0 ERXQG LQ IRXUSRLQW VFDWWHULQJ H[LVWV LI ZH FDQ VKRZ WKDW WKH HQWLUH VHW RI VLVWHUV DPIf LV SUHVHQW 6LQFH DW HDFK YHUWH[ VLVWHUV GR QRW FRXSOH WR PRUH WKHQ RQH RQVKHOO VWDWH ZH PXVW FRQVLGHU KLJKHU RUGHU FRUUHFWLRQV :H PDNH DQ LQLWLDO VWHS LQ WKLV GLUHFWLRQ E\ VKRZLQJ WKDW WKH ILUVW VLVWHU Wf FRXSOHV DW WKH GRXEOHORRS OHYHO %HFDXVH W LV WR EH KHOG IL[HG DQG LQ RUGHU WR ZRUN XQGHU WKH PRVW JHQHUDO FRQGLWLRQV ZH DSSO\ WKH VLQJOH5HJJH OLPLW V fÂ§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f LV VHHQ LQ WKH KLJK HQHUJ\ OLPLW RI WKH VL[SRLQW IXQFWLRQ RQO\ LI WZLVWV DUH SODFHG RQ ERWK RI WKH DGMRLQLQJ SURSDJDWRUV DV VKRZQ PAGE 15 LQ )LJ :H EHJLQ KRZHYHU ZLWK WKH FRUUHVSRQGLQJ XQWZLVWHG DPSOLWXGH 7KLV LV HDVLO\ FDOFXODWHG LQ WKH )XELQL9HQH]LDQR IRUPDOLVP IURP 0 SL 9Sf$9^Sf$9Sf$9Sf Sf f ,Q JHQHUDO SDVVLQJ WKH YHUWH[ RSHUDWRUV WKURXJK HDFK RWKHU SURGXFHV IDFWRUV RI WKH IRUP H[S 3L Â‘ 3M 7 O ]IQQ /fÂ§n Q QfÂ§ f ZKHUH LV WKH SURGXFW RI FRRUGLQDWHV =^ ZKLFK DUH DVVRFLDWHG ZLWK WKH SURSDn JDWRUV FRQQHFWLQJ WKH YHUWLFHV )LQDO H[SUHVVLRQV IRU WKH DPSOLWXGH DUH XVXDOO\ ZULWWHQ LQ WHUPV RI WKH ULJKWKDQG IDFWRUV 7KH OHIWKDQG IRUP LV PRUH FRQYHn QLHQW KRZHYHU IRU ORFDWLQJ FULWLFDO SRLQWV LQ WKH KLJK HQHUJ\ OLPLW V fÂ§! LRR &RQVHTXHQWO\ ZH XVH WKH OHIWKDQG VLGH RI f LI RQH RI WKH FRQQHFWLQJ SURSDn JDWRUV VHHV WKH HQHUJ\ V DQG WKH ULJKWKDQG VLGH IRU QRQRYHUODSSLQJ TXDQWLWLHV ,Q WKH SDUWLFXODU FDVH RI )LJ WKH FRPSOHWH H[SRQHQWLDO IDFWRU LV WKHQ HDVLO\ IRXQG WR EH ]Q H[S 3f f 3 7 3]Vf f 6XEVWLWXWLQJ LQ WKH PRPHQWXP VFDODU SURGXFWV WKH IXOO DPSOLWXGH EHFRPHV $ a -nG]G]G]]n>OFLKf]0]ADWf [ LfBBT6f fTVmf f RR \Q ; H[S fÂ§ Â‘Vf V fÂ§ Â£ff Â‘ Q 8 ZKHUH ZH KDYH GHILQHG ,VJL f ,Q ZULWLQJ f ZH KDYH DOVR GURSSHG WHUPV LQ WKH H[SRQHQWLDO ZKLFK FDQ EH VDIHO\ QHJOHFWHG LQ WKH KLJK HQHUJ\ OLPLW PAGE 16 :H DUH QRZ LQ D SRVLWLRQ WR LPSRVH WKH IRXUGLPHQVLRQDOLW\ FRQVWUDLQW ZKLFK LQ WKH KLJK HQHUJ\ OLPLW UHGXFHV WR V V f $SSO\LQJ WKLV FRQVWUDLQW WR f DOORZV WKH DUJXPHQW RI WKH H[SRQHQWLDO WR EH IDFWRUL]HG JLYLQJ \ , fÂ§ fÂ§TLf fÂ§ fÂ§WL f fÂ§ fÂ§ Tf ? fÂ§ O fÂ§ T 6 f $ a G]LG]G]A]O n] n] OAOf n ; fBBDVfH[S>V [Lf B[ QfÂ§ f ZKHUH rL LAfB, Q LAfB 9 6 9 A LE f 7R GLVFXVV WKH KLJK HQHUJ\ OLPLW ZH PXVW OHW fÂ§! RRH ZKHUH WKH UHDO SDUW RI V LV KHOG IL[HG DQG LV VXFK WKDW WKH UHDO SDUW RI V LV LQ WKH VWULS RI FRQYHUJHQFH 7KH UHVXOW LV D )RXULHU LQWHJUDO ZKRVH DV\PSWRWLF EHKDYLRU LV GRPLQDWHG E\ LWV FULWLFDO SRLQWVA )RU [S[f WR EH D XVHIXO FULWLFDO SRLQW LW PXVW IDOO ZLWKLQ WKH LQWHJUDWLRQ UHJLRQ ]?]A &ULWLFDO SRLQWV WDNHQ DW WKH ERXQGDULHV GR QRW SURGXFH VLVWHUV 6LQFH WKH ERXQGDU\ RI WKH LQWHJUDWLRQ UHJLRQ LV QRW LQFOXGHG WKH IDFWRUV LQ f RWKHU WKDQ WKH H[SRQHQWLDO FDQ EH LJQRUHG GXULQJ LQWHJUDWLRQ 7R UHFRYHU WKH SURSHU OLPLW V fÂ§! RRHnff ZH REWDLQ D GRXEOH FULWLFDO SRLQW E\ FKRRVLQJ WKH SKDVHV Â£LÂ£ rrH LE f LE AA fÂ§f! rrH 7KLV LV FRPSOHWHO\ HTXLYDOHQW WR WZLVWLQJ WKH SURSDJDWRUV FRUUHVSRQGLQJ WR DQG VLQFH HQHUJLHV WKDW RYHUODS DQ RGG QXPEHU RI WZLVWHG SURSDJDWRUV FKDQJH VLJQ ,Q RWKHU ZRUGV WKH UROH RI WKH WZLVWV KHUH LV WR SODFH WKH FULWLFDO SRLQW LQVLGH WKH LQWHJUDWLRQ UHJLRQ PAGE 17 7R REWDLQ WKH OHDGLQJ VLVWHU WUDMHFWRU\ E\ HYDOXDWLQJ f DERXW WKH FULWLFDO SRLQWV ZH NHHS RQO\ WKH ORZHVW RUGHU WHUPV LQ WKH H[SRQHQWLDO DQG LQWHJUDWH DERXW m fÂ§ DUW f fÂ§ e H f IRU H VPDOO &KRRVLQJ KLJKHU SRZHUV RI ]? DQG ZRXOG OHDG WR GDXJKWHU WUDMHFWRULHV $IWHU VKLIWLQJ ]? DQG ZH REWDLQ $ a OaD^W?O [fDVfL B ;fTVf-" f ZKHUH DQG N! Â G] -R ODIRf V]?F H G]?G] H[S V]]?] F [L[[L Of[ f 6HWWLQJ \ fÂ§Â JLYHV Â G]] DAAHV]F I Â=O I G\ H[SÂHB -2 -fÂ§H -fÂ§XQ f f ,T L^HV< \]Lf f f ZKHUH MT fÂ§ LHÂ]V ,QWHJUDWLQJ RYHU ]? ZH HDVLO\ ILQG K Van 9UAfHrr UO\ -2 -fÂ§9Q H[SL\f H[SL\f f YR \ ZKHUH WKH \ LQWHJUDO LV V\PPHWULF ,I ZHUH QRW FULWLFDO WDNLQJ WKH OLPLW H fÂ§! QRZ ZRXOG JLYH ,T XVH G\ a Hf 7KLV GHPRQVWUDWHV WKH QHHG IRU D GRXEOH FULWLFDO SRLQW 1RZ GHILQH fÂ§V]AF ZKLFK JLYHV 6Ffff}!E U G]]U -R -R G\ H[S L\f H[SL\f f )RU XV WR FRQVLVWHQWO\ ZULWH \AH rFZH PXVW KDYH H L r WR UHDFK WKH ORZHU OLPLW fÂ§ IRU IL[HG \ &RQVTXHQWO\ LQ WKH KLJK HQHUJ\ OLPLW PAGE 18 V fÂ§ LRR \R ar DQG VR WKH LQWHJUDWLRQ RYHU \ JLYHV LQ 1H[W WKH LQWHJUDO JLYHV U fÂ§ cc fÂ§ Aff ZKLFK LV YDOLG RQO\ IRU Af fÂ§ 7KXV WKH FRPSOHWH DPSOLWXGH LV A LQ fÂ§VFf Uff ADÂffpL 4WLfDWf f [ [fBBDVfO [fBBTVf :H QRZ DQDO\WLFDOO\ FRQWLQXH WKH HQHUJ\ EDFN E\ PDNLQJ WKH UHSODFHPHQW fÂ§V fÂ§! HaLZV 'HILQLQJ Wf AAf fÂ§ ZKLFK FRUUHVSRQGV WR WKH ILUVW VLVWHU WUDMHFWRU\ DQG VLPSOLI\LQJ ZH ILQDOO\ DUULYH DW $T ALQHaLQA8faOKnfaOVAXf7OWf Of[I ÂfDOf["fBff f ; ,f:f46f B ;f"Wf46fB 6LQFH HDFK RI WKH HQHUJLHV FRPSULVLQJ L RYHUODSV ZLWK 6 WKH 5HJJH EHKDYLRU MGf VKRZV WKDW WKH FHQWUDO SURSDJDWRU LQ )LJ VHHV WKH VLVWHU 8VLQJ WKH IRXUGLPHQVLRQDOLW\ FRQVWUDLQW f ZH FDQ ZULWH Â 6 Âc Â£ Â£ VILL a VOfV a Â£Of! DQG HDVLO\ UHFRYHU (T % RI 5HI ([DPLQLQJ WKH 7 IXQFWLRQ LQ f ZH VHH WKDW WKH SROHV RI WKH VLVWHU WUDMHFWRU\ DUH IRU "f fÂ§ 2XU DSSURDFK PDNHV LW SDUWLFXODUO\ HDV\ WR GHWHUPLQH WKH VLJQDWXUH U RI WKHVH SROHV 7ZLVWLQJ WKH VLVWHU SURSDJDWRU FKDQJHV WKH VLJQ RI DOO RYHUODSSLQJ HQHUJLHV $OWKRXJK ERWK WKH QXPHUDWRU DQG GHQRPLQDWRU RI [M DQG [ FKDQJH VLJQ LQ (T f WKH VLJQV RI WKH HQHUJ\ UDWLRV UHPDLQ XQFKDQJHG 7KXV WKH WZLVWHG DQG XQWZLVWHG GLDJUDPV FDQ EH DGGHG WRJHWKHU JLYLQJ DQ RYHUDOO IDFWRU U 7KHUHIRUH WKH SROHV RI Âf KDYH SXUH SRVLWLYH VLJQDWXUH 6LQFH WKHVH SROHV FRUUHVSRQG WR RGG YDOXHV RI VSLQ LH mÂf fÂ§ WKH\ KDYH XQSK\VLFDO ZURQJVLJQDWXUH )RU WKH H[LVWHQFH RI WKH VLVWHU LW ZDV QHFHVVDU\ WKDW WKH DUJXPHQW RI WKH H[SRQHQWLDO IDFWRUL]H SURGXFLQJ D WXSOH FULWLFDO SRLQW ,QWHJUDWLQJ RYHU ERWK PAGE 19 )LJXUH $W WUHH OHYHO WKH VHFRQG VLVWHU Wf ILUVW DSSHDUV LQ HLJKWSRLQW VFDWWHULQJ &RQFXUUHQWO\ DQG ] VHH "eff FRRUGLQDWHV LQ HIIHFW UHPRYHG WKH OLQHDU SRZHU RI WKH SURSDJDWRU YDULDEOH IURP WKH H[SRQHQWLDO ,Q JHQHUDO LQWHJUDOV RI WKH IRUP I 4QH[SFPf f -R LQ WKH OLPLW F fÂ§! LQWHJUDWH WR a fÂ§U fFP P IRU T fÂ§ Q f P ? P PV 7KXV VLVWHUV GR QRW DSSHDU LQ RU SW VFDWWHULQJ VLQFH ERWK UHWDLQ WKH OLQHDU SRZHU RI )XUWKHUPRUH WR SURGXFH WKH VHFRQG VLVWHU Lf ERWK WKH OLQHDU DQG TXDGUDWLF SRZHUV RI PXVW EH LQWHJUDWHG DZD\ OHDYLQJ WKH FXELF SRZHU 7KLV RFFXUV ZKHQ WKH FULWLFDO SRLQW LV D WXSOH ZKLFK ILUVW DULVHV LQ WKH HLJKWSRLQW VFDWWHULQJ DPSOLWXGH 7UHH /HYHO (LJKW3DUWLFOH 6FDWWHULQJ ,Q WKLV VHFWLRQ ZH ZLOO H[SRVH WKH VHFRQG VLVWHU If LQ WKH RSHQ VWULQJ WUHH GLDJUDP RI )LJ ZKHUH WKH VLVWHU DSSHDUV DFURVV WKH SURSDJDWRU ZLWK ,Q WKH FRUUHVSRQGLQJ DPSOLWXGH ZH LVRODWH WKH UHOHYDQW WHUPV E\ LQFOXGLQJ LQ WKH H[SRQHQWLDO RQO\ TXDQWLWLHV ZKLFK RYHUODS WKH FHQWUDO SURSDJDWRU :H JDWKHU WKH RWKHU WHUPV LQWR D IXQFWLRQ I^]? f ZKRVH H[DFW IRUP FDQ PAGE 20 EH LJQRUHG VLQFH DV VKRZQ LQ WKH ODVW VHFWLRQ WKH VLVWHUV GHSHQG RQO\ RQ WKH H[SRQHQWLDO IDFWRU 7KH DGYDQWDJH RI XVLQJ WKH VLQJOH5HJJH OLPLW RYHU SDVW DSSURDFKHV EHFRPHV PRUH DSSDUHQW LQ WKLV H[DPSOH )URP )LJ ZLWKRXW WZLVWVf ZH LPPHGLDWHO\ ZULWH GRZQ A Â‘X Â L LI]L]]]f]V fÂ§ DWf RR BQ [ H[S > < fÂ§ L3=?= a 3= Sf f S S] S]J ]Jf f fÂ§ 6XEVWLWXWLQJ LQ WKH KLJK HQHUJ\ OLPLW YDOXHV RI WKH PRPHQWXP VFDODU SURGXFWV JLYHV WKH HLJKWWDFK\RQ DPSOLWXGH UO $ DWf L fQ ; H[S ?A< fÂ§]O]ÂVE a Vf ]O]]ÂV a V V a Vf Q O Q f a a A ]V a Vf ]V a VGf a ]JV a Âf ] V a VE a Â£f ]J ] ] V a V Â£ fÂ§ ff ZKHUH QRZ V 6 f V Vf f $SSO\LQJ WKH IRXUGLPHQVLRQDOLW\ FRQVWUDLQWV eOe ee B eOe B eOe B [ V n VV n f VEV IDFWRUL]HV WKH DUJXPHQW RI WKH H[SRQHQWLDO \LHOGLQJ f Q Â f UL Da Q G]LI]L=]=f] f fÂ§ fÂ§Df ; H[S _V < = [Of] [f] [f]J a [Ef fÂ§ f PAGE 21 ZKHUH [L [ 6 a a Â£ ; V A [ Â£ V Â£ a Â‘V f Â£ a m V )RU WKH FULWLFDO SRLQW [L;;;f WR EH LQVLGH WKH LQWHJUDWLRQ UHJLRQ ZH PXVW SODFH WZLVW RQ HDFK RI WKH DVVRFLDWHG SURSDJDWRUV DQG DSSO\ DQ DGGLWLRQDO IRXUGLPHQVLRQDOLW\ FRQVWUDLQW Â£Â£ &RQVHTXHQWO\ GXH WR WKH WZLVWV ZH KDYH WKH VLJQ FKDQJHV LG !V!! arrH Â‘ f cf 7R UHPRYH WKH ILUVW WZR SRZHUV RI LQ WKH H[SRQHQWLDO LQ f DQG WR REWDLQ D OHDGLQJ WUDMHFWRU\ ZH ZLOO LQWHJUDWH DURXQG ]? a UL H [ O + a 9A O f ?A O H &OHDUO\ WKLV LV MXVW RQH RI PDQ\ FULWLFDO SRLQWV WKDW ZH FRXOG KDYH FKRVHQ %\ ZULWLQJ A fÂ§ ; fÂ§ Y[Bf \7Tf a fÂ§ A)-f HWF DQG VKLIWLQJ WKH ]nV ZH ILQG YfÂ§fÂ§DWDf HV]bF $ aI[O[9[?[Ef Â ; G]L AH[S>A+AfYAf [ G]$ G] H[S>A[[[" [Lf[_ [f@ ZKHUH R F [M [Tf[ [f[_ [f[_ [f 7KH ODVW IRXU LQWHJUDOV LQ f FDQ EH GRQH LQ SDLUV UHVXOWLQJ LQ f a WW AV[[[ [Lf[ [f YA,; 9Aff ; IL[ 9A YAf n c;fBffnfmÂn-F M f PAGE 22 8VLQJ f WKHQ JLYHV $ f AI[L;A[A Y[cffULFf4AfUADWf f [ Vf;;[O [Lf[? ;fO \A; a \[AM Â‘ f $JDLQ ZH DQDO\WLFDOO\ FRQWLQXH EDFN E\ UHSODFLQJ fÂ§V ZLWK HaLQV 7KXV ZH ILQG WKH 5HJJH EHKDYLRU $J RF Â4EfBO ÂAfA ZKLFK FRUUHVSRQGV WR WKH VHFRQG VLVWHU WUDMHFWRU\ 7KH ILUVW SROH DW Af fÂ§ FDQFHOV WKH SROH RI WKH ILUVW VLVWHU WUDMHFWRU\ DW "Lf fÂ§ 7KH GDXJKWHUV RI Af FDQFHO WKH RWKHU SROHV DW "f fÂ§ fÂ§ :KHQ WKH FHQWUDO SURSDJDWRU LQ )LJ FDUULHV WKH VHFRQG VLVWHU If WKH DGMDFHQW SURSDJDWRUV DQG VHH WKH ILUVW VLVWHU SWf (DFK RI WKHVH VLVWHUV FDQ HDVLO\ EH FRPSXWHG E\ FRQVWUXFWLQJ WKH H[SRQHQWLDO WHUP LQ f IURP WKH DSSURSULDWH RYHUODS TXDQWLWLHV WDNLQJ WKH FRUUHVSRQGLQJ KLJK HQHUJ\ OLPLW DQG WKHQ LQWHJUDWLQJ RYHU D WXSOH )LQDOO\ LI ZH KDG FKRVHQ WR LQLWLDOO\ LQWHJUDWH RYHU D WXSOH FULWLFDO SRLQW IRU WKHQ ZH ZRXOG KDYH IRXQG WKH ILUVW VLVWHU WUDMHFWRU\ Wf PAGE 23 &+$37(5 '28%/(/223 )2857$&+<21 6&$77(5,1* :H QRZ DGRSW RXU SURFHGXUH WR KDQGOH ORRS FRUUHFWLRQV $V LQ WKH WUHH OHYHO FDVH ZH PXVW ILUVW LVRODWH WHUPV LQ WKH FRUUHVSRQGLQJ DPSOLWXGH ZKLFK RYHUODS WKH DSSURSULDWH SURSDJDWRU ,Q SDUWLFXODU WR VHDUFK IRU VLVWHUV LQ WKH GRXEOHn ORRS IRXUSRLQW DPSOLWXGH ZH FRQVLGHU WKH OLPLWLQJ VLWXDWLRQ ZKHUH WKH WZR ORRSV DUH VXIILFLHQWO\ VHSDUDWHG VXFK WKDW WKH\ DQG WKH FRQQHFWLQJ SURSDJDWRU FDQ EH WUHDWHG DV LQGLYLGXDO REMHFWV 7ZR VXFK WRSRORJLHV DUH VKRZQ LQ )LJ %RWK PD\ EH FRQVWUXFWHG E\ VHZLQJ WRJHWKHU WZR VLQJOHORRS GLDJUDPV )RU WKLV ZH XVH WKH IRUPDOLVP IURP DSSHQGL[ RI 'L 9HFFKLD HW DO ZKHUH WKH RSHQ VWULQJ 1SRLQW PXOWLORRS YHUWH[ KDV WKH IRUP 9 1Jf RF GASH[ S L QS Â‘ W Â‘ S S f % & f DQG ZKHUH U LV WKH SHULRG PDWUL[ &RPSOHWLQJ WKH VTXDUH DQG LQWHJUDWLQJ RYHU WKH ORRS PRPHQWXP S JLYHV 7 H[S LQ% f UB % & nf GHWI7UUfA 7KH IDFWRUL]HG IRXUWDFK\RQ GRXEOHORRS DPSOLWXGH LV WKHQ UO U U L %Â @ $ f -T G] PAGE 24 )LJXUH 7ZR GLVWLQFW WRSRORJLHV IRU SURGXFLQJ WKH Wf VLVWHU LQ GRXEOHn ORRS IRXUSRLQW VFDWWHULQJ KDV EHHQ UHGXFHG WR WKH VLQJOHORRS FDVH U QOQ IF ZKHUH N ZLOO EH GHILQHG EHORZ 7KH GHWDLOV RI WKH PHDVXUH GS ZKLFK LV D IXQFWLRQ RI N? DQG NfÂ PD\ EH VXSSUHVVHG LQ WKH DQDO\VLV EHORZ DV ORQJ DV ZH DYRLG WKH ERXQGDULHV RI WKH LQWHJUDWLRQ UHJLRQ ,Q WKH PXOWLORRS FDVH WKH FRHIILFLHQW %A LQ f LV JLYHQ E\ZLWK Dn f E 9Â Â JOID\ 0 A LQ A a ?f f 9 D GS 7DOL^]ff WY 7D]Rfn= ZKHUH ]TWLA DQG DUH IL[HG SRLQWV DQG D SURGXFW RI 6FKRWWN\ JURXS HOHPHQWV LV GHILQHG E\ US T7OL T7O T7OU OD F!LO U OVU QÂH=^` Sc s SLL f ZKHUH J LV WKH JHQXV QXPEHU $OVR : PHDQV WKDW WKH VXP LV RYHU DOO HOHPHQWV RI WKH 6FKRWWN\ JURXS H[FHSW WKDW WKH OHIWPRVW HOHPHQW LQ 7D FDQ QRW EH 6-@ ,Q WKH VLQJOHORRS FDVH 7D fÂ§ f DQG 6r1\f NQ\ ZKHUH N LV WKH PXOWLSOLHU DQG UHODWHG WR WKH UDGLXV RI WKH ORRS +HUH KRZHYHU WKH VXP UHVWULFWLRQ OHDYHV MXVW WKH LGHQWLW\ )LQDOO\ IRU RQH ORRS fÂ§! RR"L fÂ§} 7KXV GURSSLQJ WKH ORRS LQGH[ % RR Lf PfÂ§ Â 9L]f f PAGE 25 ZKHUH WKH SURMHFWLYH WUDQVIRUPDWLRQ LV H[SOLFLWO\ JLYHQ E\ =MO=M ]LLf] =L]L =LBLf B DLD 9L]f ; ÂÂOf ÂO =LBOf 7R UHGXFH f ZH ZLOO QHHG WKH FRPPXWDWRU f Y ; -fÂ§cHOQ9Zf f n 3DUWLDO GHULYDWLYHV RI WKH SURMHFWLYH WUDQVIRUPDWLRQ FDQ HDVLO\ EH WDNHQ JLYLQJ G]9L] f fÂ§ RfÂ a OfO a ]Lf rWO a ]Lf f RU PRUH JHQHUDOO\ G"9L] f P.fÂ§DfULL ODLD f fP =Lf f ]L LfP 7KH VLQJOHORRS WKUHHSRLQW GLDJUDP LV FRQVWUXFWHG E\ VHZLQJ WRJHWKHU WZR OHJV RI D ILYHSRLQW GLDJUDP DQG WKHQ IL[LQJ WKUHH RI WKH SURMHFWLYH FRRUGLQDWHV )RU WKDW FDVH IROORZLQJ 'L 9HFFKLD HW DO ZH VHZ WRJHWKHU OHJV DQG DQG WKHQ FKRRVH N] DQG ,Q WKH SUHVHQW FDVH ZH ZLOO DVVRFLDWH WKH FRRUGLQDWH ZLWK WKH FRQQHFWLQJ OHJ FRRUGLQDWH ]F 7KLV JLYHV G]9F] f fÂ§ DORQJ ZLWK Gr19F] f IRU P 7KXV WKH FRPPXWDWRU f EHFRPHV >%DtFn@ 9e P fr1O&FnLff! YX\ 22 9 9O A P f 1H[W WKH FRHIILFLHQW & LQ f LV JLYHQ E\ F ((4RffÂDULLQYLnfM frf ( ( AU fmf}f9:@3 & rf L (VU AQ A Q PO Mf f 4f Dr1 GfG"?QAL^\fn9M]ff LM QO PO L \ 9L\f 9M]f \ ] PAGE 26 ZKHUH WKH SULPH IRUP LV GHILQHG E\ Q A [7OO] 74Zf Z 7D]f (^]Zf ^]Zf ,, fÂ§ US 9 9 7D]f Z 7DZf f DQG WKH n LQGLFDWHV WKH WKH LGHQWLW\ LV QRW LQFOXGHG )RU D VLQJOH ORRS WKH SULPH IRUP UHGXFHV WR 7AL ] fÂ§ NQZ Z fÂ§ NQ] (]Zf ] Zf fÂ§ fÂ§Â‘ ss ] fÂ§ NQ] Z fÂ§ NQZ f %HORZ ZH ZLOO QHHG WKH FRPPXWDWRU _>&&f@_3Âf \ RR RR LM]F P RR UD 3L P fÂ§ f G7+YA]f@ ] P fÂ§ f V((AULfQ GOQ ,Q>]^ 9F]f?J F NQ9F]f 9F]f NQ]L L P P fÂ§ f $QÂ )F]f NQ9F]f ] f ZKHUH SF S 'XH WR PRPHQWXP FRQVHUYDWLRQ ZH FDQ QHJOHFW WKH VHFRQG GHQRPLQDWRU LQ WKH ODVW WHUP )XUWKHU LQ WKH KLJK HQHUJ\ OLPLW V fÂ§ RR ZH KDYH SL f 3 V SL f 3 V 3 Â‘ 3 fÂ§rf V DQG 3 Â‘ 3 rf V 7KHVH LPSO\ 3 f 3L 3L 3f f 3L f &RQVHTXHQWO\ VRPH RI WKH WHUPV LQ f GR QRW VXUYLYH WKH KLJK HQHUJ\ OLPLW LQ f 7KLV SHUPLWV XV WR GURS WKH HQWLUH ILUVW WHUP DQG WKH L F WHUP LQ WKH ODVW VXP 5HDUUDQJHPHQW WKHQ \LHOGV _>FTFf@_Scf RR 3L LAF P O P fÂ§ f RR [ GO ,Q Q O]L NQYF]f? -_>)&Âf NU=L@ Q U O ]fÂ§ f PAGE 27 7DNLQJ WKH GHULYDWLYHV JLYHV >&&f@ _S! AIWn>( LAF LFUQQ9Fn]ffP UL rL a NQ9F]ffP U?9Â=fU Q RR f IOmr! aNÂ‘rr! P ZKLFK VLPSOLILHV WR >&4c-OFf@ SLf 3L A Â 22 LEPQO =Of LA& P Q P ( Nf=Lfn f :H DOVR QHHG WKH VLQJOHORRS UHVXOW % H[S ,Q N & 33f LfXY3 33f f ZKHUH LM! DULVHV LQ SODQDU ORRS DPSOLWXGHV DQG FDQ EH H[SUHVVHG LQ WHUPV RI WKH -DFREL WKHWD IXQFWLRQ 6XEVWLWXWLQJ f DQG f LQWR f WKHQ JLYHV f rfa-4 G]]aaD^Wf --4 S3 [ H[S 22 A( P O =P^? fÂ§ OfPO fÂ§ f P ( 3L f 3,Q =M ,Q ]M ,Q N? ,Q Nn (( r t ,Q RR 3U AKD LQ r 22 fÂ§ \ fÂ§ O/ÂÂ•A]LfUD Â & rAfP 3LOQ (Q r" aEfP A L r"rÂ‘fr M L 8 WL rr rr L ( 3L Â‘ A ( r OfP ( B IFQ=IfP f r rr Y Y L [(WSKMA (BAfÂ‘}ffn U Y U Y Â‘rn f ZKHUH Â DQG M FRUUHVSRQG WR WKH GLIIHUHQW ORRSV DQG ZH KDYH GURSSHG D PRn PHQWXP LQGHSHQGHQW IDFWRU ZKLFK FDQ EH LJQRUHG LQ WKH KLJK HQHUJ\ OLPLW PAGE 28 5HSODFLQJ WKH PRPHQWXP VFDODU SURGXFWV E\ WKHLU KLJK HQHUJ\ OLPLWV DOORZV XV WR IDFWRUL]H WKH DUJXPHQW RI WKH H[SRQHQWLDO WR JHW $fa Â G]]aOaD^Wf I G?[ -A> ÂÂ9AfBB4Af -2 M [ H[S V RR ( ]\ P JP^]L=NLfJP]=Nf cf ZKHUH ZH KDYH GHILQHG JP[ \ Nf ,Q [ ,Q N \ fÂ§ MIHQ [fP 22 ( NQ[f P [ \ f 7KH IXQFWLRQ JLQ[\Nf LV IRU RULHQWDEOH SODQDU ORRSV DQG LV HVVHQWLDOO\ WKH PfÂ§ GHULYDWLYH RI ,Q 7KXV ZH FDQ LPPHGLDWHO\ ZULWH GRZQ WKH H[SUHVVLRQ LQ WKH QRQRULHQWDEOH FDVH QR P RR LQ D rf PQ RR fÂ§ 8NfQ[f P \fÂ§fÂ§ NfQ[fU ; \ f DQG IRU WKH QRQSODQDU FDVH XS ,Q L A fPN NQ[fP RR A A NQ[fP Umc! f 1RZ ZH VHDUFK IRU FULWLFDO SRLQWV ZKLFK GR QRW UHVLGH RQ WKH ERXQGDU\ RI WKH LQWHJUDWLRQ UHJLRQ 8QIRUWXQDWHO\ GXH WR LWV FRPSOLFDWHG IRUP RQH PXVW QXPHULFDOO\ VHDUFK IRU ]HURV LQ JP[\Nf ,W LV IRXQG WKDW JP[ \ Nf IRU DOO P GRHV LQGHHG SRVVHVV ]HURV WKDW DUH H[FOXVLYHO\ ZLWKLQ WKH LQWHJUDWLRQ UDQJH 7KHVH ]HURV JHQHUDWH WKH FULWLFDOSRLQW FXUYH [ 3\Nf IRU VRPH IXQFWLRQ 3\Nf ZKLFK VDWLVILHV JP3\Nf\Nf ,Q DGGLWLRQ QXPHULFDOO\ DQDO\VLV LQGLFDWHV WKDW ERWK QRQRULHQWDEOH DQG QRQSODQDU FDVHV DOVR SRVVHVV FULWLFDO SRLQW FXUYHV ,Q DOO WKHVH FDVHV WKH ]HURV GR QRW VHHP WR EH FRQILQHG WR DQ\ SDUWLFXODU UHJLRQ RI LQWHJUDWLRQ VSDFH PAGE 29 7KLV FDVH GLIIHUV IURP WKH WUHH FDOFXODWLRQ LQ WZR UHVSHFWV )LUVW WR IDFWRUL]H (T f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fÂ§! RRHA f EHFRPHV f $fa G]] D^Wf M GG AAf fÂ§ ?fÂ§DWfV]K f [ H[S ]fJL]L] Âf Nf ZKHUH A ROf OAf AffAf :H ZLOO HYDOXDWH DERXW WKH FULWLFDO FXUYH 3tLf H 3]Nf H f ([SDQGLQJ WKH J?nV DERXW WKLV FXUYH DQG WKHQ VKLIWLQJ ]? DQG JLYHV !aL G]]anaP 6 A / / AAAAfn Â‘fÂ§ DWfHV]K f ZKHUH G]?G]b H[S>VLLT@ KL 3] ÂLffO 3]s Nff f [ 3]L KLf N?fJ?]b 3 Nf $f DQG KLT DQG A DUH QRZ HYDOXDWHG RQ WKH FULWLFDO FXUYH 7KH LQWHJUDWLRQ RI DQG SURFHHGV DV EHIRUH JLYLQJ Of U MfÂ§fÂ§F -R $$R? aLQV G]] ] fAf GIM r ; GG9fL"fffBDfKfHVn L -2 f PAGE 30 6LPLODUO\ WKH LQWHJUDWLRQ LV DOVR HDVLO\ GRQH JLYLQJ f a QUHAVA0P f-GWO ; K f ZKLFK H[KLELWV WKH ILUVW VLVWHU WUDMHFWRU\ Wf 6LQFH WKH LQWHJUDQGV LQYROYH GHULYDWLYHV RI WKH -DFREL WKHWD IXQFWLRQV ZH DUH XQDEOH WR FRPSOHWH WKH FDOn FXODWLRQ VKRZLQJ H[SOLFLWO\ WKDW WKH VLVWHU GRHV QRW GHFRXSOH )RU WKH SODQDU GLDJUDP KRZHYHU LQ WKH VSHFLDO FDVH Wf fÂ§ LW FDQ HDVLO\ EH VKRZQ WKDW WKH VLJQV RI HDFK RI WKH LQWHJUDQG IDFWRUV DUH WKH VDPH RYHU WKH HQWLUH LQWHJUDn WLRQ UHJLRQ 2Q WKH RWKHU KDQG WR VKRZ WKDW GHFRXSOLQJ GRHV QRW RFFXU LQ WKH QRQRULHQWDEOH DQG QRQSODQDU FDVHV LV PRUH GLIILFXOW DOWKRXJK WKH UHVXOWV RI WKH QH[W FKDSWHU LQGLFDWH WKDW WKH VLVWHU VXUYLYHV WKH ODWWHU FDVH 7KH H[LVWHQFH RI WKH WKH VHFRQG VLVWHU UHTXLUHV WKDW WZR RI WKH JnV VKDUH WKH VDPH FULWLFDO SRLQW 8VLQJ O [fUfO DUfU UHr f LW IROORZV WKDW JU[ \ Nf JUL[\ Nf @7 >NQ[HaN8[ NQ\HaNQ\AM Q f 6LQFH WKH GLIIHUHQFH LV LQGHSHQGHQW RI WKH LQGH[ U IRU DQ\ JLYHQ FULWLFDO SRLQW HLWKHU RQH JU YDQLVKHV UHVXOWLQJ LQ D VLQJOH VLVWHU RU WKH\ DOO YDQLVK VLPXOWDQHn RXVO\ ,Q WKH ODWWHU FDVH f UHVXOWV LQ WKH IRUP $f I G]] Tf GIL > ÂAAAf 4f -R -R [ G]LG]AH[S (P = WP P f PAGE 31 )LJXUH *HQHUDO ""UORRS IRXUSRLQW GLDJ VLVWHU DPWf DPD OX ,QWHJUDWLQJ RYHU ]? DQG ZH REWDLQ f $ $ f 76 G]] GM U? B ; M G]A]99faa4AA A ]PAPf f 7KH ULJKW IDFWRU JLYHV D LQ OHDGLQJ RUGHU &RQVHTXHQWO\ WKH LQWHJUDO JHQHUDWHV D OHDGLQJ SROH DW DWf fÂ§ ZKHUHDV WKH VHFRQG VLVWHU UHTXLUHV DWf 3UHVXPDEO\ WKH Lf WUDMHFWRU\ LV SUHVHQW LI WKHUH DUH DW OHDVW WZR ORRSV RQ ERWK VLGHV RI WKH SURSDJDWRU :H VXVSHFW WKDW LQ WKLV FDVH WKHUH ZRXOG EH D IDFWRUL]DWLRQ RI WKH IRUP *P[\N Af P[\ KfP[\Nf f ZKHUH N? DQG $n FRUUHVSRQG WR VDPHVLGH ORRSV ,Q )LJ ZH GLVSOD\ WZR GLVWLQFW SRVVLEOH PXOWLORRS WRSRORJLHV IRU SURGXFLQJ WKH KLJKHU RUGHU VLVWHUV PAGE 32 W V Df ; ; W V Ef )LJXUH 7KH 5HJJH FXW EHKDYLRU LV DFURVV WKH GRWWHG OLQHV ,Q ERWK FDVHV WKH FHQWUDO SURSDJDWRU PD\ DOORZ XS WR WKH PfÂ§ VLVWHU LI WKHUH DUH DW OHDVW P ORRSV RQ HLWKHU VLGH +RZHYHU HYDOXDWLQJ )LJ D LV QRW SUDFWLFDO VLQFH WKH 6FKRWWN\ UHSUHVHQWDWLRQ RI WKH SULPH IRUP f LV PXFK WRR IRUPDO ZKHQ WZR RU PRUH XQIDFWRUL]HG ORRSV DUH SUHVHQW 2Q WKH RWKHU KDQG VLQFH )LJ E FRPSOHWHO\ IDFWRUL]HV WKH ORRSV LW UHTXLUHV QR PRUH WKDQ WKH WHFKQLTXHV SUHVHQWHG LQ WKLV FKDSWHU 7KH VLVWHU WUDMHFWRULHV PD\ DOVR DSSHDU DFURVV SURSDJDWRUV ZKLFK DUH HPn EHGGHG LQ DQ LUUHGXFLEOH GLDJUDP $Q H[DPSOH LV WKH GRXEOHORRS GLDJUDP GLVSOD\HG LQ )LJ 7KH VLVWHU KHUH PD\ EH DFURVV RQH RI WKH KRUL]RQWDO SURSDn JDWRUV 6XFK GLDJUDPV DUH KRZHYHU GRPLQDWHG E\ WKH EHKDYLRU RI 5HJJH FXWV ,Q WKH SUHVHQW FDVH WKH FXW LQ )LJ D JLYHV $ f OQVf3 IRU VRPH S DW IL[HG W 7KH FXW KDV WKH VDPH 5HJJH VORSH DV WKH ILUVW VLVWHU \HW LWV D>WfLQWHUFHSW LV KLJKHU ,Q JHQHUDO WKH QfÂ§ FXW RFFXUV DW WKH VDPH RUGHU DV WKDW RI WKH QfÂ§ VLVWHU EXW ZLWK D WUDMHFWRU\ O\LQJ DERYH WKH VLVWHU 7KLV LPSOLHV WKDW WKH FROOHFWLYH EHKDYLRU RI WKH FXWV ZRXOG DFWXDOO\ H[FHHG WKH PAGE 33 &0 ERXQG $ KLJK HQHUJ\ DQDO\VLV RI WKH HQWLUH PRGXOL VSDFH VXFK DV WKDW RI *URVV DQG 0HQGH ZRXOG EH GRPLQDWHG E\ WKH FXWV 7KLV LV VXSSRUWHG LQ SDUW E\ WKHLU SURSRVDO WKDW WKH IL[HG W EHKDYLRU KDYH WKH IRUP $ OQ V H} OQVfA f ZKHUH J LV WKH JHQXV QXPEHU 7KH VLQJOHORRS DPSOLWXGH FRPSXWHG ILUVW LQ WKH IL[HG DQJOH OLPLW ZDV VKRZQ H[SOLFLWO\ WR UHGXFH WR f IRU S LQ WKH IL[HG W OLPLW fÂ§! PAGE 34 &+$37(5 23(1 675,1* 6,67(56 ,1 &/26(' 675,1* 6&$77(5,1* $Q XQH[SHFWHG UHVXOW RI WKH ODVW FKDSWHU LV XQFRYHUHG E\ FRQVLGHULQJ WKH QRQSODQDU GLDJUDP LQ )LJ 7KH FHQWUDO SURSDJDWRU WKDW FDUULHV WKH VLVWHU IWf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n DWH RSHQ VWULQJ SURSDJDWRU WDNHV WKH IRUP D* fÂ§f L YZSLUf[ }r L L r!G YU -?=L? [ $ [ F 7$RIf fR [ 9S]]f ? SLfF f $PRQJ WKH PDQ\ H[SUHVVLRQV DSSHDULQJ LQ WKH OLWHUDWXUH IRU WKH WUDQVLWLRQ RSn HUDWRU 7 EHWZHHQ WKH RSHQ DQG FORVHG VWULQJ VWDWH ZH ZLOO XVH WKDW RI 6KDSLUR DQG 7KRUQ :H ZLOO LJQRUH KHUH WKH JKRVWV WHUPV JLYHQ LQ WKHLU H[SOLFLW H[SUHVn VLRQ IRU 7 7KHVH JLYH D QRQWULYLDO FRQWULEXWLRQ RQO\ LI ORRSV DUH SUHVHQW (YHQ WKHQ WKH JKRVWV FDQ EH LJQRUHG VLQFH WKH\ KDYH QR EHDULQJ RQ WKH FDOFXODWLRQ ZKLFK IRFXVHV RQ WKH H[SRQHQWLDO FRQWULEXWLRQV DZD\ IURP WKH LQWHJUDWLRQ PAGE 35 )LJXUH 1RQSODQDU GRXEOHORRS IRXUSRLQW GLDJUDP 7KH ORRSV FRQWDLQ FORVHG VWULQJ SROHV VWULQJ SURSDJDWRU ERXQGDU\ UHJLRQ 7KH WUDQVLWLRQ RSHUDWRU LV WKHQ JLYHQ E\ 22 7$DWf H[S fÂ§ < &QP$VQ f DBPO < &QPDQO n QP QP ZKHUH DQG a < Urf D DV / 67 D A &QP$Q n QP 9 PBL n DfÂ§P fÂ§ R QP fÂ§ Uf B nnQP fÂ§ B fQP U LQ f U LQ f P fÂ§ Q Q P B?QP LQ f U f Q P Q P Uf B fI A U U f f Q P f f f Q P f PAGE 36 7KH VLQH DQG FRVLQH RVFLOODWRUV RI WKH FORVHG VWULQJ DUH JLYHQ E\ $U fÂ§ fÂ§M $U $Uf $VU $U$ U f ZKHUH $U DQG $U FRUUHVSRQG WR WKH OHIW DQG ULJKW PRYHUV UHVSHFWLYHO\ ,Q WHUPV RI WKH VLQH DQG FRVLQH PRGHV WKH FORVHG VWULQJ YHUWH[ RSHUDWRU LV ZULWWHQ $& 930f SLfFL Hn3.f U}mf H[S S Â‘ e ]f / ]fÂ§n ? Q UL fÂ§ fÂ§ ]f 6Qff_ fFL Q f ZKHUH WKH ILUVW WZR IDFWRUV DUH WKH ]HUR PRGHV ,Q (T f ZH KDYH ZULWWHQ WKH FORVHG VWULQJ WUDMHFWRU\ DWf RFWf 7R HOLPLQDWH FRQIXVLQJ QRWDWLRQ ZH ZLOO ZULWH WKH RSHQ VWULQJ WUDMHFWRU\ DOVR LQ WHUPV RI DWf LH DWf DWf fÂ§ 7KHQ ZH KDYH ]/Ra fR ]Sa fR ]A fR ]D9f fR f 3XVKLQJ 7 WR WKH ULJKW WR WKH OHIW PRYLQJ WKH SURSDJDWRU WR WKH ULJKW DQG WKHQ XVLQJ PRPHQWXP FRQVHUYDWLRQ WR HOLPLQDWH SDUW RI WKH ]HUR PRGHV (T PAGE 37 f EHFRPHV DS Uf >n GBRf I Ur} 9"U -2 -?]L?O ; R H[S A DQrO =?f 9 f Q RR r &+Q4POU B rQf P RR ; H[S A &QPrQ f 4P QPfÂ§ RR If f B f UQPOf ; H[3 A A &fPDBQ f DfÂ§PfÂ§ QP U L [ H[S fÂ§S f A DQQA I f Y f Q O RR r A &QPDBPBOAP fÂ§ Aff fR P f :H FDQ HDVLO\ PRYH WKH HYHQ RVFLOODWRUV WKURXJK WR WKH YDFXXP VWDWHV VLQFH WKH\ RQO\ DSSHDU DW WKH IDU OHIW DQG IDU ULJKW 7KLV SURGXFHV WKH IDFWRU H[S B B ? n A -OQQ Q?c Q Q 3 f 3 ÂBÂ a] ]? ]? f] =f Q O ]]]f Orf S 3 f ZKLFK FDQ EH SXOOHG RXWVLGH WKH YDFXXP VWDWHV 1H[W SXVKLQJ WKH TXDGUDWLF WHUPV SDVW HDFK RWKHU SURGXFHV WKH IDFWRU H[SO 9 r Of&f&mfO f DPrW! f NQP DQG DQ RVFLOODWRU LQGHSHQGHQW H[SRQHQWLDO ZKLFK ZH FDQ QHJOHFW VLQFH LW ZLOO QRW VXUYLYH LQ WKH KLJK HQHUJ\ OLPLW 0RYLQJ WKH TXDGUDWLF WHUPV WR WKH YDFn XXP VWDWHV ZLOO SURGXFH QR RWKHU SHUPDQHQW HIIHFW DV WKH\ SDVV E\ WKH YHUWH[ PAGE 38 RSHUDWRUV 3XVKLQJ WKH IDFWRU f SDVW WKH ULJKW YHUWH[ WKHQ UHVXOWV LQ ? I $ fVf O G]]A GÂf]?GÂ= T] ; ]]L]f ]]?=f ][G S 3 DWf ; R H[S aL9S f &QQ2LPO]O a ]? Q O PfÂ§ [ H[S c$Q Â‘ < P+PA bff Q O P [ O P Of ÂIF OfRBMO&@IF>Amr}f@ _ff MIF f $JDLQ WKH TXDGUDWLF WHUP ZLOO QRW OHDYH DQ\ SHUPDQHQW LPSULQW DIWHU PRYLQJ WR WKH OHIWKDQG VLGH )LQDOO\ L ? UO G]], G]]aTÂf G][G] + OBB4Lf Af Af [ H[S A f 3 S3 e UQ OfPOFOFLP]L r^fr" Â"f IFQ O P f Â 7R SHUIRUP WKH VXPV LQ WKH VHFRQG H[SRQHQWLDO ZH JR WR WKH OLPLW V fÂ§ RRH DQG NHHS RQO\ WKH WHUP OLQHDU LQ ] 7KH VXPV FDQ QRZ EH GRQH E\ QRWLQJ f U U Q Q Q 7KXV RR Q RR rf Â‘fÂ§n fÂ§ Q Q O Q =f f PAGE 39 (T f WKHQ EHFRPHV $4 a /f I G]]!0Wf I $D mf 9"U \ \_]M_O ; f AOAf VfÂ§6 f [ H[S V]O f ff f ff $V LQ WKH GRXEOHORRS FDVH VLQFH WKHUH DUH RQO\ IRXU LQWHUDFWLQJ SDUWLFOHV WKH DUJXPHQW RI WKH H[SRQHQWLDO DSSHDUV LQ D IDFWRUL]HG IRUP ZLWKRXW UHVRUWLQJ WR D GLPHQVLRQDOLW\ FRQVWUDLQW ([DPLQLQJ WKH VHFRQG H[SRQHQWLDO WHUP LQ f ZH VHH WKDW WKHUH LV D FULWLFDO SRLQW ZKHQ ]? ]? RU :ULWLQJ SH LPSOLHV RU U 7R LQWHJUDWH f DERXW WKHVH SRLQWV ZH UHWXUQ WR WKH 7D\ORU VHULHV H[SDQVLRQ LQ f DQG f KH $RaL L]]aDPO UOD9!O [ H[S [ H[S V] 33 FRV ? FRV V Â &N4&Q23O3 VLQNGLf VPQ NQ :H H[SDQG E\ VHWWLQJ VLQQf t Q IRU DQG VLQQf UV fÂ§ fQQ IRU U 7KH S VXPV IRU FDQ HDVLO\ EH FDUULHG RXW DV IROORZV ? A Q Af ? ^S\ G A BS?Q n Q f aASO Sf Q O Q O / :LWK D VLPLODU H[SUHVVLRQ IRU U HT f EHFRPHV $r a -R G]]aD^f / GSLGSASLSn!aO OL^nfM W ? B [ H[S V]?3?SfL^? SLf 3f VSS BL ? H[S _fÂ§6!33 3Lf +a 3f fÂ§ ASLS ZKHUH WKH ILUVW H[SRQHQWLDO LV IRU ? DQG H[SDQGHG DURXQG WKH VDPH YDOXH DQG WKH VHFRQG IRU WKH FRQYHUVH FDVH f PAGE 40 ,QWHJUDWLRQ RYHU \LHOGV UHr\ B HaL\ 9 "LfO 3f H[S VSLS=A\ f ZKHUH WKH H[DFW H[SUHVVLRQ IRU MT LV QrW QHHGHG 7KH LQWHJUDWLRQ RYHU \ JLYHV ÂW 8QOLNH WKH SUHYLRXV H[DPSOHV WKH VLVWHU LV QRW QHFHVVDULO\ WKH GRPLQDQW EHKDYLRU 7KLV UHTXLUHV WKDW ZH H[WHQG RXU FRQVLGHUDWLRQV WR KLJKHU RUGHUV %\ 7D\ORU H[SDQGLQJ fÂ§ Sf DQG Sf ZH REWDLQ PDQ\ WHUPV ZKLFK PD\ LQGLFDWH WKH SUHVHQFH RI WKH RSHQ VWULQJ VLVWHU IÂ‘ 7R PDNH D ILUP GHWHUPLQDn WLRQ UHTXLUHV VRPH FDUH VLQFH WKH > WUDMHFWRU\ LV GHJHQHUDWH ZLWK WKH GLODWRQ WUDMHFWRULHV WKDW PD\ DSSHDU DFURVV WKH DGMDFHQW FORVHG VWULQJ SURSDJDWRUV 7KHUH LV QR GRXEW KRZHYHU ZKHQ f JHQHUDWHV D WULSOH SROH 7KH IRUP RI WKH UHTXLUHG VROXWLRQ LV VXJJHVWHG E\ WKH SDUWLDO ZDYH DQDO\VLV WHUP / & DWf WfDFWf WfDFWf Wf V VD^Wf?QV f IRU WKH FDVH DWf FLFLf (T f \LHOGV WKLV UHVXOW LI ZH VHOHFW WKH S] H[SDQVLRQ WHUPV IRU ERWK S? DQG S 7KLV JLYHV f /HW Z SLS 7KHQ f PAGE 41 7KH S LQWHJUDO HDVLO\ JLYHV fÂ§ OQ LH 1H[W GHILQLQJ Z ] \ ZH LQWHJUDWH RYHU WR REWDLQ $U OQHr} Hrf r Ln G\\rP AVQ GDWfA -T 9 Ap67 L HV\ HV\\ f )LQDOO\ WKH HQG UHVXOW LV $J a Â7 A Hf!IWLfffIW:UO Lffr:OQV f ZKHUH WKH RSHQ VWULQJ VLVWHU Wf DWf fÂ§ 2QH SRVVLEOH FRQFHUQ WKDW PD\ DULVH LQ WKH DERYH FDOFXODWLRQ LV WKDW LQ ZULWLQJ (T f ZH KDYH GLVFDUGHG WKH WHUP ]]L=fO ]]?=f^ ]]?=fO ]]?=f R A f :KHQ WDFK\RQV DUH SUHVHQW WKLV PD\ GLYHUJH DW WKH FULWLFDO SRLQWV LQ WKH QHLJKn ERUKRRG ] s L s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n MHFWRULHV 7KH EDVLF QDWXUH RI VLVWHU VWDWHV ZLOO GLIIHU IURP WKH VWDWHV DVVRFLDWHG ZLWK WKH VWDQGDUG 5HJJH WUDMHFWRU\ DWf VLQFH WKH FRUUHVSRQGLQJ VLVWHU SROHV DUH QRW SK\VLFDO ,Q WKH VSDFHOLNH W UHJLRQ WKH SROHV KDYH WKH PDQLIHVW XQSK\VLFDO FKDUDFWHULVWLF RI QRQVHQVH LH QHJDWLYH VSLQ 7KH WLPHOLNH UHVRQDQFHV DUH QRW SK\VLFDO HLWKHU ,Q H[DFW H[SUHVVLRQV WKH UHVLGXHV DVVRFLDWHG ZLWK SK\VLn FDO UHVRQDQFHV FDQ DOZD\V EH ZULWWHQ DV SRO\QRPLDOV LQ WKH HQHUJLHVA $W WUHH OHYHO ZH FDQ VHH IURP WKH ILQDO H[SUHVVLRQV f DQG f WKDW WKH HQHUJLHV RYHUODSSLQJ WKH VLVWHU SURSDJDWRU DUH QRW LQ WKLV IRUP +RZHYHU WKLV LV QRW WKH FDVH LQ WKH GRXEOHORRS H[SUHVVLRQ f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fÂ§ f ZKHUHDV GXH WR JDXJH LQYDULDQFH VWULQJ VWDWHV ILOO PXOWLSOHWV RI WKH WUDQVYHUVH JURXS fÂ§ f )RU PAGE 43 WKH FDVH RI IRXU VSDFHWLPH GLPHQVLRQV WKH FRXQWLQJ SUREOHP ZDV VROYHG LQ ZKHQ *ROGVWRQH SUHVHQWHG WKH JHQHUDWLQJ IXQFWLRQ RR RR [[Mf >AL[U@(-UnUnfffnnfLUf Lf Q O U O 7KLV KDV VLQFH EHHQ JHQHUDOL]HG WR KLJKHU GLPHQVLRQV DQG IRU WKH 6XSHUVWULQJ DQG +HWHURWLF VWULQJ ([SDQGLQJ RXW f WKH FRHIILFLHQW RI [ FRXQWV WKH QXPEHU RI f UHSUHVHQWDWLRQV RI VSLQ ZKLOH WKH H[SRQHQW LV WKH FRUUHn VSRQGLQJ PDVV OHYHO 7KH FRQQHFWLRQ WR VLVWHUV FDQ EH PDGH LI LQ f RQH VHWV WKH [ H[SRQHQW LQ WKH VHFRQG IDFWRU HTXDO WR LH -U0f KU f f U 6LQFH SROHV LQ WKH DWf SODQH DUH ODEHOHG E\ 0A -f ZH FDQ LGHQWLI\ f ZLWK f 7KH VWDWH DQDO\VLV RI *ROGVWRQH LV LQ WKH WLPHOLNH W UHJLRQ ZKLOH WKH KLJK HQHUJ\ DQDO\VHV H[SRVHV GRPLQDQW EHKDYLRU LQ WKH VSDFHOLNH UHJLRQ 'HWHUPLQn LQJ WKH VWDWH UHSUHVHQWDWLRQ RI WKH VLVWHU WUDMHFWRULHV ZLOO SURYLGH D PRUH GLUHFW OLQN EHWZHHQ WKHVH WZR DSSURDFKHV 7KH XQLI\LQJ IHDWXUH RI SROH FDQFHOODWLRQ FDQ EH VHHQ LQ )LJ ZKLFK IRU DWf GLVSOD\V WKH ORZHVW PDVV OHYHOV REWDLQHG IURP *ROGVWRQHfV IRUPXOD f 7KH ILJXUH VKRZV KRZ WKH YDULRXV WUDMHFWRULHV FRQVSLUH WR IRUP WKH SK\VLFDO VWDWHVfVROLG GRWVf DQG UHPRYH VRPH RI WKH SXUH JDXJH VWDWHVfFURVVHVf DQG WKDW WKH Wf WUDMHFWRU\ LQ ERWK UHJLPHV HQWHUV ZLWK WKH RSSRVLWH VLJQ WR WKH DWf DQG Wf WUDMHFWRULHV 7KH VWDWH UHSUHVHQWDWLRQV RI WKH SK\VLFDO VWDWHV GHILQHG DW WKH SROHV RI WKH VWDQGDUG 5HJJH DWf WUDMHFWRULHV DQG LWV GDXJKWHUV DUH ZHOO NQRZQ 7KH ILUVW WKUHH VWDWHV RI WKH OHDGLQJ WUDMHFWRU\ GLVSOD\HG LQ )LJ DUH JLYHQ E\ WKH WDFK\RQ f WKH fSKRWRQf DOBA f DQG WKH PDVVLYH VSLQ WZR V\PPHWULF VWDWH f ZKHUH WKH WUDQVYHUVH LQGH[ L ef fÂ§ 6XSSUHVVLQJ WKH VSDFHWLPH LQGH[ WKH JHQHUDO OHDGLQJ VWDWH LV JLYHQ E\ Df M f Q ,W LV PAGE 44 RWf )LJXUH /RZHU PDVV VWDWHV LQ WKH RSHQ ERVRQLF VWULQJ 'RWV GHQRWH SK\Vn LFDO VWDWHV DQG FURVVHV GHQRWH SXUH JDXJH VWDWHV 7KH FRHIILFLHQWV LQGLFDWH FRQn WULEXWLRQV IURP WKH YDULRXV OHDGLQJ DQG GDXJKWHU WUDMHFWRULHV 1RQVHQVH SROHV DUH IURP KLJK HQHUJ\ VFDWWHULQJ DQDO\VLV DQG VHQVH SROHV DUH IURP *ROGVWRQHnV IRUPXOD LPSRUWDQW WR QRWH WKDW WKH PRGH QXPEHU RI WKH VWDWHV DORQJ WKH DWf WUDMHFWRU\ GLIIHU E\ RQH 7KLV LPSOLHV WKDW E\ YDU\LQJ QU LQ WKH JHQHUDO RSHQ VWULQJ VWDWH f f f DU f f f rf! f ZH PRYH DORQJ D SDWK LQ )LJ WKDW SDUDOOHOV WKH VLVWHU WUDMHFWRU\ $On WKRXJK WKH SROHV RI WKH VLVWHU WUDMHFWRU\ GR GLIIHU E\ PRGH QXPEHU U WKH FRUUHVSRQGLQJ VLVWHU VWDWHV FDQ QRW EH UHSUHVHQWHG E\ WKH SK\VLFDO VWDWHV f ,QVWHDG E\ DQDORJ\ ZLWK WKH KLJK HQHUJ\ DQDO\VLV ZH PXVW DQDO\WLFDOO\ FRQWLQXH DZD\ IURP WKH VWDWHV GHILQHG E\ f 7R SURFHHG ZH ZLOO ZRUN GLUHFWO\ ZLWK WKH IDFWRUL]HG VFDWWHULQJ DPSOLWXGH 7KLV LVRODWHV WKH DSSURSULDWH SURSDJDWRU ZKLFK DOORZV XV SURMHFW RQWR LW DOO SRVVLEOH FODVVHV RI SK\VLFDO VWDWHV :H DUH WKHQ IUHH WR VHOHFW WKH VWDWHV ZKLFK OHDG WR WKH VLVWHUV )RU WKH VL[SRLQW PAGE 45 GLDJUDP RI )LJ DJDLQ LJQRULQJ WKH WZLVWV ZH SURMHFW WKH SK\VLFDO VWDWHV RQWR WKH FHQWUDO SURSDJDWRU 7R SUHVHUYH XQLWDULW\ ZH LQVHUW WKH FRUUHVSRQGLQJ LGHQWLW\ RSHUDWRU RQ DGMDFHQW VLGHV RI WKH SURSDJDWRU LH L $T GaOGaG^SL 9S]O Of9SOf,UAA,9SOf9S]Lf Sf ]O] /T f 22 ZKHUH RR ( A ( Â7 f rf r ff Â‘ f m m 7KH QRUPDOL]DWLRQV LQ f DUH IL[HG E\ WKH SURMHFWRU FRQGLWLRQ Â DQG WKH FRPPXWDWLRQ UHODWLRQV >DUDLDQ@ fÂ§ Â‘ f :KHQ ZH SURMHFW RQWR WKH FHQWUDO SURSDJDWRU ZH HDVLO\ REWDLQ L [ } L LfÂ§ 77 \ fÂ§fÂ§UD Rf f R D"fÂ§fÂ§ fÂ§ /R L7aRU QU 7M L-QM a DIf f f ZKHUH W WM 6XEVWLWXWLQJ WKLV LQWR f XVLQJ WKH IRXUGLPHQVLRQDOLW\ FRQn VWUDLQW f DQG WKHQ WDNLQJ WKH KLJK HQHUJ\ OLPLW 6 fÂ§! \LHOGV $T a IO G]LG]]]OaDKf ]=OaDW?O ]fnBf6fO faBT6f -2 rr rr ??7O ;,, ( UQfU ]Y M B Df .! r f@ UfÂ§ QU 8 ZKHUH [?[b DQG V DUH DV EHIRUH 1RWH E\ XVLQJ WKH LQWHJUDO UHSUHVHQWDWLRQ eeO -R ZH FDQ UHSODFH WKH VXPV LQ f E\ H[SRQHQWLDO IXQFWLRQV WR JHW $T a Ln G=OG]]G]]AOaDWOf]AOaDW?O LfTVf U4Vf U [ a4fH[S>LA \r [LfÂ -f U f f PAGE 46 7KXV ZH KDYH FRPSOHWHO\ UHFRYHUHG (T f ,Q IDFW DW YLUWXDOO\ HDFK VWHS RI RXU FRPSXWDWLRQV EHORZ WKHUH LV D SDUDOOHO VWHS XVLQJ WKH H[SRQHQWLDWHG IRUP 7KLV SURYLGHV D XVHIXO FKHFN RQ RXU UHVXOWV DQG DOORZV XV WR EH EULHI LQ PXFK RI WKH GHULYDWLRQ $V LQ &KDS WR REWDLQ WKH OHDGLQJ Wf WUDMHFWRU\ ZH UHTXLUH WKDW DT ef EH D FULWLFDO SRLQW ,Q WKLV FDVH QRW RQO\ GR ZH WZLVW WKH SURSDJDWRUV ]? DQG EXW WKH LQGH[ Q? PXVW EH DQDO\WLFDOO\ FRQWLQXHG WR D QHJDWLYH YDOXH 7KH VWDQGDUG SURFHGXUH LV WR UHSODFH WKH LQILQLWH VXP E\ D 6RPPHUIHOG:DWVRQ FRQWRXU LQWHJUDO DQG WKHQ SXVK EDFN WKH FRQWRXU H[SRVLQJ WKH SROHV RQ WKH QHJDWLYH UHDO D[LV 7R JHQHUDWH WKH QHFHVVDU\ SROH LQ Q? ZH ILUVW HYDOXDWH f DW WKH FULWLFDO SRLQW $ERXW ef WKH DPSOLWXGH f LV DSSUR[LPDWHO\ $T a eB4Ofe B4Âf eLffTVf ;fmAf f ZKHUH DIWHU VKLIWLQJ ]? DQG WKH LQWHJUDO EHFRPHV rr rr U f Lm Q H U r U fÂ§ -H G]LG]AV]L]Vf 7KH GRXEOH LQWHJUDO LV HDVLO\ SHUIRUPHG JLYLQJ Q H >r [ Af LLfQLL U QUfÂ§ m [ < QLA QOO (7 O-QMRWf P f f ZKHUH \J HL $V ZH ZLOO VHH VKRUWO\ LW LV FUXFLDO IRU WKH VLVWHU WKDW U]M DSSHDU DV D GRXEOH SROH $V D SUHOXGH WR UHSODFLQJ WKH GLVFUHWH YDULDEOH Q? E\ D FRQWLQXRXV RQH ZH PXVW UHSODFH Q?? E\ LWV JDPPD IXQFWLRQ UHSUHVHQWDWLRQ 7QL f 7R DQDO\Wn LFDOO\ FRQWLQXH WR WKH SROH DW Q? fÂ§ ZH PXVW EH FDUHIXO VLQFH (T f PAGE 47 YDQLVKHV IRU RGG EHFDXVH RI WKH QXPHUDWRU LQ WKH ODVW IDFWRU 8VLQJ WKH 6RPPHUIHOG:DWVRQ WUDQVIRUPDWLRQ WR FRQYHUW WKH VXP RYHU Q LQWR D FRQWRXU LQWHJUDO ZH JHW RR RR rRrVrRU UfÂ§ QUfÂ§ [\ G[ OOf f OF VLQ Q[ 7[ f [ B D L[ Of ZKHUH ZH KDYH VHSDUDWHG RXW DQG GLVSOD\HG WKH RGG Q FRQWULEXWLRQV 7R FRQWLQXH EDFN WR WKH SROH ZH PXVW ILUVW VLJQDWXUL]H WKH ODVW IDFWRU E\ VHWWLQJ fÂ§ HOU 3XVKLQJ EDFN WKH FRQWRXU WKHQ H[SRVHV WKH GRXEOH SROH ZLWK UHVLGXH 22 RR B HLQ[Of U QU G G[ \ R f /VLQ Q[ 7[ f [ MQM fÂ§ FWIfe 6LQFH WKLV H[SUHVVLRQ YDQLVKHV ZKHQ ZH VHW [ fÂ§ LQ WKH ODVW IDFWRU ZH RQO\ QHHG WR GLIIHUHQWLDWH WKLV WHUP 7KH UHVXOW LV fÂ§LQ 7KH UHDVRQ ZK\ ZH UHTXLUH D GRXEOH FULWLFDO SRLQW VKRXOG QRZ EH FOHDU ,I RQO\ ]? RU ZHUH FULWLFDO D VLQJOH Q? SROH ZRXOG UHVXOW ZKRVH UHVLGXH YDQLVKHV 1H[W LQ WKH OLPLW D fÂ§!Â‘ fÂ§ VLQ U[7D f fÂ§! f 7KXV Q rr1 ,, ( SrfÂ§ >Â r ; Af@ n U QU f [ n, QM a D ,Q WKH VSHFLDO FDVH QU IRU U ZH KDYH WKH QRQVHQVH SROH DW DWf fÂ§ LH ,T fÂ§Â7Â O4If f PAGE 48 6LQFH WKH DQDO\WLFDO FRQWLQXDWLRQ ZDV DORQJ WKH FXUYH GHVFULEHG E\ f LW LV FOHDUV WKDW WKLV SROH LV JHQHUDWHG E\ WKH OHDGLQJ 5HJJH WUDMHFWRU\ DWf 7KH DQDO\VLV DERYH VKRZV WKDW WKLV SROH LV JLYHQ E\ f f 7KH LQYHUVH RVFLOODWRU LQGLFDWHV WKH XQSK\VLFDO QDWXUH RI WKLV VWDWH 1RZ ZH PXVW H[SOLFLWO\ VKRZ WKDW WKH SROH f LV FDQFHOHG E\ D FRUn UHVSRQGLQJ SROH RQ WKH OHDGLQJ ILUVW VLVWHU WUDMHFWRU\ If 7R DQDO\WLFDOO\ FRQWLQXH WR WKLV SROH DORQJ If ZH PXVW FRQYHUW WKH 8 VXP LQ f WR D FRQWRXU LQWHJUDO ZKHUH ZH KDYH VHW QU IRU U 3LFNLQJ XS WKH SROH DW [ ADLf fÂ§ JLYHV f 5HSHDWLQJ WKH VWHSV IRU Q? DQG HYDOXDWLQJ DW WKH SROH Q? fÂ§ ZH ILQDOO\ REWDLQ "rf [M fÂ§ [Lf DUMM fÂ§ ef f ÂWW VAa VLQ"UIf f 7If f 7KDW WKH ILUVW SROH DW If fÂ§ FDQFHOV WKH DPSOLWXGH f FDQ EH VHHQ E\ ZULWLQJ 7If f If Of7Wf f DQG WKHQ FDQFHOLQJ WKH SROH FRPLQJ IURP WKH VLQH IXQFWLRQ DJDLQVW WKH ]HUR LQ 7 A^^Wf f 7KH UHPDLQLQJ SROHV DUH UHSUHVHQWHG E\ WKH VWDWHV DBLfBDfU f DQG FDQFHO XQSK\VLFDO SROHV JHQHUDWHG E\ WKH GDXJKWHUV RI WKH DWf WUDMHFWRU\ PAGE 49 5HSHDWLQJ WKH VL[SRLQW FRPSXWDWLRQ IRU D JHQHUDO FULWLFDO SRLQW GHILQHG E\ ]> fÂ§ [L DQG ; SURGXFHV WKH U fÂ§ ffÂ§ GDXJKWHU RI WKH If VLVWHU 7KH FRPSOHWH VHW RI VWDWHV FRUUHVSRQGLQJ WR If DQG DOO LWV GDXJKWHUV LV JLYHQ E\ DBLfaPDQA f" IrU P DQG Q 7KH VHW FDQ RQO\ H[LVW LQ WRWDOLW\ DQG UHVXOWV IURP D FRPSOHWH VDWXUDWLRQ RI WKH SURSDJDWRU ZLWK WKH RVFLOODWRU TBM WKH SUHVHQW FRQWH[W ZH VHH WKDW WKH If VLVWHU GRHV QRW DSSHDU LQ HLWKHU WKH IRXU RU ILYHSRLQW IXQFWLRQ RU RQ WKH ]? DQG SURSDJDWRUV RI WKH VL[SRLQW IXQFWLRQ EHFDXVH FRXSOLQJ WKH SURSDJDWRU WR WZR RQVKHOO VWDWHV DW DQ\ YHUWH[ SUHYHQWV WRWDO VDWXUDWLRQ 7KLV DQDO\VLV VXJJHVWV WKDW WR REWDLQ WKH VHFRQG VLVWHU WUDMHFWRU\ If ZH PXVW ILUVW VDWXUDWH WKH DSSURSULDWH SURSDJDWRU ZLWK WKH RVFLOODWRU DBL SHUPLWWLQJ If WR H[LVW DQG WKHQ ZLWK DB 7R YHULI\ WKLV ZH DJDLQ FRQVLGHU WKH HLJKWSRLQW GLDJUDP RI RI )LJ ,QVHUWLQJ WKH LGHQWLW\ RSHUDWRU RQ DGMDFHQW VLGHV RI WKH SURSDJDWRU DQG XVLQJ WKH IRXUGLPHQVLRQDOLW\ FRQVWUDLQWV \LHOGV WKH IRUP a G]?G]G]AG]AI^]? ]f 22 22 9 f A Â‘ A UQUQU?e -Q B DIf UaO QU f ZKHUH I I DQG +U ]? ;Lf] ;; a AfA rf f DQG ZKHUH WKH [nV DUH DV EHIRUH 5HFDOO ZH FDQ REWDLQ WKH OHDGLQJ If WUDMHFWRU\ LI ZH WZLVW DOO WKH QRQFHQn WUDO SURSDJDWRUV DQG H[SDQG DERXW WKH SRLQW ]? DQG \[i )RU [L[f DQG ) \[Af WR UHSUHVHQW GRXEOH FULWLFDO SRLQWV ZLOO DOVR UHTXLUH ZH FRQWLQXH ERWK Q? DQG Q WR QHJDWLYH YDOXHV $VVXPLQJ WKLV WR PAGE 50 EH FDVH ZH KDYH RR RR f $b aI^[?;\[O \[(f fÂ§V+UfQU G]?G]AG]?G]A f R fÂ§Q A D 9 f -fÂ§ ( QL ( Q! QLO UfÂ§ QUfÂ§ 6=O=\; [fY[cn ef m QQ R R rrV[D[A;L f[ ;f (MAL LQM 4Âff f (DFK SDLU RI LQWHJUDOV LV WKH VDPH IRUP DV LQ f 7KH ILUVW SDLU JHQHUDWHV D GRXEOH SROH DW Q? fÂ§ DQG PRYHV XV RQWR D IWf WUDMHFWRU\ 6XEVHTXHQWO\ WKH VHFRQG SDLU JLYHV D GRXEOH SROH DW Q fÂ§ ZKLFK QRZ WUDQVIHUV XV WR D Wf WUDMHFWRU\ ,QWHJUDWLQJ DQG DQDO\WLFDOO\ FRQWLQXLQJ WR QM fÂ§ ZH LPPHGLDWHO\ ILQG R ? A $J a "7 Â ;;[O [?f[ a [fO ?I[a?f^O \[f f ; IL[,[\; \[(f Q( aQ7 UfÂ§ QU \ OI Â2 A aO AÂ -Q" a frf n ZKHUH WKH H[DFW IRUP RI XT LV QRW LPSRUWDQW :H FDQ DSSURDFK WKH ILQDO VWDWH DBLDBf f E\ PRYLQJ DORQJ HLWKHU WKH OHDGLQJ Wf RU Wf WUDMHFWRULHV 7KH QRQVHQVH SROHV REWDLQHG LQ WZR FDVHV PXVW FDQFHO )RU WKH ILUVW SDWK ZH VHW QU IRU U DQG SHUIRUP D 6RPPHUIHOG:DWVRQ WUDQVIRUPDWLRQ RQ Q WR JHW $V a 7 Â6[;[ [?f^[? [fO \[Of^O \[( ; I[,[\; \[(f *f fÂ§ DWf 6LPLODUO\ IRU Qb A VOLGLQJ GRZQ WKH Wf WUDMHFWRU\ ZH REWDLQ WKH UHVXOW ? A A Â [[[ ;Of[ ;fO 9AfO \[(ff f ; IL[ [ \ID \IL(fAf+c^; VLQ UWf f 7Lf fn PAGE 51 ZKHUH IRU Wf fÂ§ WKLV FDQFHOV WKH SROH f ([WHQGLQJ WKHVH UHVXOWV WR WKH PRVW JHQHUDO FDVH VXJJHVWV WKDW WKH UfÂ§ VLVWHU WUDMHFWRU\ IRUPV ZKHQ WKH SURSDJDWRU EHFRPHV VXFFHVVLYHO\ VDWXUDWHG E\ WKH RVFLOODWRUV 4P VWDUWLQJ ZLWK P DQG HYHQWXDOO\ UHDFKLQJ P U fÂ§ ,Q DQRWKHU ZRUGV WR JHW WR WKH WUDMHFWRU\ DPWf ZH EHJLQ E\ PRYLQJ GRZQ HLWKHU D OHDGLQJ RU GDXJKWHU DWf WUDMHFWRU\ FXUYH WR HLWKHU D OHDGLQJ RU GDXJKWHU IWf WUDMHFWRU\ ZKLFK ZH UHDFK E\ DQDO\WLFDO FRQWLQXDWLRQ HWF 7KH UHVXOWDQW VLVWHU DQG LWV GDXJKWHU WUDMHFWRULHV DUH UHSUHVHQWHG E\ RSHQ VWULQJ VWDWHV RI WKH IRUP RBOf PL f f f DBULf UDDBUfQn f IRU P? f Â‘ fPUBL QU f ZKHUH WKH OHDGLQJ WUDMHFWRU\ LV JLYHQ E\ P? PUB? OQU %\ DQDORJ\ ZH FDQ LPPHGLDWHO\ ZULWH GRZQ WKH FRUUHVSRQGLQJ FORVHG VWULQJ VLVWHU VWDWHV E\ UHSODFLQJ WKH RSHQ VWULQJ RVFLOODWRU DBÂ ZLWK WKH FORVHG VWULQJ RVFLOODWRUV Â HYHU\ZKHUH $Q LPSRUWDQW SRLQW WKDW QHHGV WR EH VWUHVVHG KHUH IRU DSSO\LQJ WKH SURFHn GXUH ZH KDYH SUHVHQWHG LV WKDW LW EH SRVVLEOH WR FRPSOHWHO\ LVRODWH WKH VLVWHU SURSDJDWRU ,Q WKH FDVH RI WKH GRXEOHORRS IRXUSRLQW LQWHUDFWLRQ WKLV FULWHn ULRQ DGGV MXVWLILFDWLRQ WR RXU DSSURDFK LQ &KDS ZKHUH ZH IDFWRUL]HG WKH DPSOLWXGH VR WKDW ZH FRXOG WUHDW DV LQGLYLGXDO REMHFWV WKH WZR ORRSV DQG WKH FRQQHFWLQJ SURSDJDWRU ,Q WKH VWDWH DQDO\VLV DSSURDFK SURMHFWLQJ WKH SK\VLFDO VWDWHV RQWR WKH FRQQHFWLQJ SURSDJDWRU JLYHV f $ LI f a+f"R UnU Q! DWf Â‘! nKf K Gnf $ &f?QU Ff?}O I ; Df M fff ? D? H[SA FW ,Q NR /U f PAGE 52 $IWHU VLPSOLI\LQJ WKH FDOFXODWLRQ OHDGLQJ WR WKH VLVWHU SURFHHGV H[DFWO\ DV LQ WKH VL[SRLQW FDVH JLYHQ HDUOLHU LQ WKLV FKDSWHU DQG UHSURGXFHV WKH UHVXOWV RI &KDS ([SRVLQJ WKH RSHQ VWULQJ VLVWHU LQ WKH IRXUSRLQW FORVHG VWULQJ GLDJUDP RI )LJ SUHVHQWV D QHZ GLIILFXOWO\ KRZHYHU VLQFH ZH PXVW ORRN IRU D WULSOH 5HJJH SROH WKDW DOVR LV QRW D OHDGLQJ RUGHU WHUP 7KH H[SUHVVLRQ $* *ff G]LG=F3$ Y?SV]LO]LOf ; 7$WDf fF [ ,$, [ F 7$DAf fR [ 9^S=]f SLfF f UHGXFHV DIWHU VRPH DOJHEUD WR A ÂfQ] RR RF U$ $ Q R U7OUQUL -QL f D: fn0L / ]LGA] ; ][] LfÂ§fÂ§DLf U UY U U[UHYHQ L QU 33Âr U rfA n rn fm ; S Â‘ 3UÂ A Of fAPO f IFQ O P :H XVHG WKH IDFW KHUH WKDW WKH HYHQ DQG RGG RVFLOODWRU SDUWV FDQ EH WUHDWHG VHSDUDWHO\ 7R REWDLQ WKH RSHQ VWULQJ IWf WUDMHFWRU\ ZH VHW QU IRU U ZKLFK DOORZV WKH VXPV WR FDUULHG RXW ,Q WKH KLJK HQHUJ\ OLPLW ZH ILQG RR RR O* W]f 6 ; 9"Un Q QL UG;R QQQO Q DWf -?=L?O O?] Ga ] M G ] [ ]L] aDWf aV]L ]Lf] ]f Q ; VO =Lf LfAO =f =ff f 2I FRXUVH XVLQJ WKH LQWHJUDO UHSUHVHQWDWLRQ IRU WKH SURSDJDWRU ZH FDQ HDVLO\ UHFRYHU WKH FRUUHVSRQGLQJ H[SUHVVLRQ RI &KDS PAGE 53 1RZ VHW SHrp ([SDQGLQJ DERXW WKH FULWLFDO SRLQWV DW DQG Q JLYHV L f $J f e e P R Q UH GOG QLc rfÂ§rQ Q? Q? Q n DWf -R GSLGSSLSf A A fÂ§ fÂ§ 2I ; 6 OO7O eeer}r OSLf??SfO Q L > 3Of + 3< Q L f Q L I Q fÂ‘ > ZKHUH WKH ILUVW WHUP LQ WKH ODVW IDFWRU LV IRU ? DQG EHLQJ H[SDQGHG DERXW WKH VDPH YDOXH ZKLOH WKH VHFRQG WHUP LV WKH FRQYHUVH FDVH 7KH LQWHJUDOV DUH H[HFXWHG DV EHIRUH JLYLQJ fÂ§fQL fHQLQL Of (DFK S LQWHJUDWLRQ SURGXFHV D IDFWRU %Q ? Q fÂ§ FWÂ fÂ§ ccQLf &RPELQLQJ WKH WHUPV WKHQ SURGXFHV WKH IDFWRU fÂ§fQLULf 7KLV OHDGV WR WKH UHVXOW RQ $N B/ \ O\ eB 7 QM Q? Q? ULLQ P R Q Q fÂ§ DWf LL [ %QL Q DWf QLfÂO ff ff 77R QL OU f ZKHUH ZH KDYH XVHG fÂ§ fQLf fÂ§ fQLf 8WLOL]LQJ WKH 6RPPHUIHOG:DWVRQ WUDQVIRUPDWLRQ WKH UHVLGXH GXH WR WKH GRXEOH SROH DW Q? fÂ§ \LHOGV WKH UHVXOW U,\ afr %L QDWff9} WW QR QR DWf 9 Zf n f Q f $NB fÂ§ÂWWf :ULWLQJ WKH %HWD IXQFWLRQ LQ WHUPV RI 7 IXQFWLRQV DQG SXOOLQJ RXW WKH ILUVW WKUHH SROHV IURP RQH RI WKH 7fV JLYHV $N =r 9 A !: Q 6Q Q? Q DI ff [ > U QR4fUf n > Q DWffQ DLff7 Q DWff f PAGE 54 7KXV ZH KDYH UHFRYHUHG WKH WULSOH SROH DW Q DWf fÂ§ &RPSXWLQJ WKH UHVLGXH E\ WDNLQJ WKH VHFRQG GHULYDWLYH RI WKH HQHUJ\ IDFWRU JLYHV WKH ILQDO UHVXOW $N Br$!Lf a !QRWf V"r!OQV f 7 VLQ[RWf f 7IWf f 7R VKRZ WKDW WKHVH SROHV FDQFHO ZH DJDLQ VWDUW ZLWK f EXW QRZ SLFN XS WKH VLQJOH SROH DW Q\ DWf fÂ§ fÂ§ U" WR JHW $ N Q B\ \ B 7 n m 7fÂ§ Q DWff7 DWf mf UfÂ§Lf ; DfrfQ fQfO BfDQM HDffÂ§fÂ§Q DIf fÂ§ fÂ§ Qf n f :H PXVW WDNH WKH UHVLGXH RI WKH TXDGUXSOH SROH DW FWWf fÂ§ ZnKLFK ZLOO JLYH D IDFWRU RI M\ )RU WKLV ZH WDNH WZR GHULYDWLYHV RI WKH HQHUJ\ IDFWRU DQG WR JHW D QRQYDQLVKLQJ UHVXOW RQH GHULYDWLYH RI WKH IDFWRU fÂ§fDLffÂ§QA ZKLFK FDQ EH GRQH ZD\V 7KLV OHDGV WR KAff L _a J7f L $N sAeOB\LBOBLB VI:LUUfOQV U VLQWTf fÂ§ f 7DIff >7 fÂ§OfB f ZKLFK UHGXFHV WR WKH QHJDWLYH RI f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n IDFH GHILQHG E\ DQ DSSURSULDWH DOJHEUDLF FXUYH 7R KDYH WKH VLVWHUV SURGXFH WKH GRPLQDQW EHKDYLRU ZRXOG UHTXLUH WKDW ZH FRQVLGHU WKH OLPLWLQJ VLWXDWLRQ ZKHUH WKH 5LHPDQQ VXUIDFH LV GLYLGHG LQ WZR VHSDUDWHG E\ D WKUHDG UHSUHVHQWLQJ WKH VLVWHU SURSDJDWRU 7KH KRSH LV WKDW WKH XQFRQWUROODEOH KLJKHU RUGHU FRUUHFWLRQV ZKLFK SODJXHG WKH ZRUN RI 5HI ZRXOG QRZ EH DEVHQW $OWKRXJK WKLV PD\ UHPRYH RQH RI WKH REMHFWLRQV WR ORFDOLW\ LW VKRXOG QRW LPSO\ WKDW LQ DQ\ ZD\ KDYH ZH VKRZQ WKDW VWULQJ WKHRU\ LV RU FDQ EH D ORFDO WKHRU\ 7KHUH VWLOO UHPDLQV VHULRXV REMHFWLRQV ZKLFK PD\ EH PRUH GLIILFXOW LI QRW LPSRVVLEOH WR RYHUFRPH )RU H[DPSOH LQ VWULQJ ILHOG WKHRU\ (OLH]HU DQG :RRGDUGp QRWH WKDW WKH FXELF IRUPXODWLRQ RI WKH ILHOG WKHRU\ SURGXFHV DQ LQILQLWH QXPEHU RI $EHOLDQ VROXWLRQVnr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f KDV WZLFH WKH WHQVLRQ RI DWf :H YLVXDOL]H WKLV RFFXUULQJ E\ EHQGLQJ RYHU WKH VWULQJ RQFH WR FUHDWH D GRXEOH VWUDQG JLYLQJ D nIROGHGf VWULQJ 7KLV SLFWXUH LV LQ DFFRUG ZLWK D UHGXFWLRQ RI WKH IXQGDPHQWDO OHQJWK VFDOH 9FG 7KH QRWLRQ RI IROGHG VWULQJV RULJLQDOO\ GDWHV EDFN WR WKH HDUO\ nV ZKHUH LW ZDV QRWHG WKDW SXUH VWDWHV RI WKH IRUP DnUU f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frfA 6LQFH LQLWLDO DGYDQFHPHQWV SURJUHVV KDV SURFHHGHG LQ PDQ\ GLIIHUHQW GLUHFWLRQV ,Q SDUWLFXODU 'RXJODVnn KDV VKRZQ WKDW WKH OLPLWHG QXPEHU RI NQRZQ PDWUL[ PRGHO VROXWLRQV FDQ EH GHULYHG IURP WKH /D[ SDLU IRUPDOLVP XVXDOO\ DVVRFLDWHG ZLWK WKH .G9 HTXDWLRQV 7KLV LGHQn WLILFDWLRQ ZLWK LQWHJUDEOH V\VWHPV JUHDWO\ LQFUHDVHV WKH QXPEHU RI FODVVLILDEOH PDWUL[ PRGHOV VLQFH LW ZDV VKRZQ D ORQJ WLPH DJRn WKDW /D[ RSHUDWRUV DUH DVVRFLDWHG ZLWK DIILQH /LH DOJHEUDV )RU H[DPSOH WKH PRGHOV GLVFXVVHG E\ 'RXJODV DUH UHODWHG WR WKH FDQRQLFDO UHSUHVHQWDWLRQV RI 0RUH UHFHQWO\ 'L )UDQFHVFR DQG .XWDVRYAfA KDYH GLVFXVVHG G?A EDVHG PDWUL[ PRGHOV ZKLFK WKH VWDQGDUG PDWUL[ WHFKQLTXHVnfnrff KDYH \HW WR VROYH 7KXV LW PD\ EH ZRUWKZKLOH WR IRFXV RQ WKH LQWHJUDEOH V\VWHPV DSSURDFK 6HYHUDO DSSURDFKHV WR FRQVWUXFWLQJ WKH /D[ RSHUDWRUV KDYH EHHQ GHYHORSHG 7KH PDWUL[ SURFHGXUH GLVFXVVHG E\ 'ULQIHOfG DQG 6RNRORYn GHILQHV ILUVW D PDn WUL[ HLJHQYDOXH HTXDWLRQ 7KH V\VWHP LQFRUSRUDWHV NQRZOHGJH RI WKH &DUWDQ VXEDOJHEUD DQG URRW V\VWHP RI VRPH HPEHGGLQJ DIILQH /LH DOJHEUD 4 6WDUWLQJ ZLWK DQ DIILQH /LH DOJHEUD IDFLOLWDWHV WKH FRQVWUXFWLRQ RI DQ LQWHJUDEOH V\VWHP IURP WKH UHVXOWLQJ /D[ SDLU RSHUDWRUV 7R IL[ WKH JDXJH LQYDULDQFH LQ WKH PDWUL[ V\VWHP WKH JUDGDWLRQ FRQYHQWLRQV RI 'ULQIHOfG DQG 6RNRORY UHTXLUH WKDW RQH RI WKH VLPSOH URRWV VD\ WKH PfÂ§ PXVW EH UHPRYHG IURP WKH DIILQH V\VWHP 7KH UHVXOWLQJ V\VWHP LV GHQRWHG E\ 4FPf 7KLV LV HTXLYDOHQW WR GHOHWLQJ WKH PfÂ§ '\QNLQ YHUWH[ )RU WKH PRVW SDUW 'ULQIHOfG DQG 6RNRORY FKRRVH WKH fFDQRQL PAGE 58 FDOf JDXJH LQ ZKLFK WR H[SUHVV WKH FRRUGLQDWH GHSHQGHQW WHUPV ,Q WKLV JDXJH /D[ RSHUDWRUV JHQHUDWH WKH UHJXODU .G9 KLHUDUFK\ HTXDWLRQV 7KH PRGLILHG .G9P.G9f HTXDWLRQV FDQ EH JHQHUDWHG E\ H[SUHVVLQJ WKH FRRUGLQDWH WHUP T[f LQ WKH fGLDJRQDOf JDXJH 7KH FDQRQLFDO /D[ RSHUDWRUV FDQ WKHQ EH UHFRYHUHG XVLQJ WKH ZHOONQRZQ 0LXUD WUDQVIRUPDWLRQV 7KH GLDJRQDO JDXJH LV WHFKQLFDOO\ VLPSOHU WKDQ WKH FDQRQLFDO JDXJH )XUWKHUPRUH WKH ILQDO /D[ RSHUDWRU LV LQ D IDFWRUL]HG IRUP ZKLFK KDV EHHQ XVHG WR TXDQWL]H WKH WKHRU\p ,Q WKLV DSSHQGL[ RXU IRFXV ZLOO EH RQ WKH H[SOLFLW FRQVWUXFWLRQ RI WKH /D[ SVHXGR fGLIIHUHQWLDO RSHUDWRUV LQ WKH GLDJRQDO JDXJH XVLQJ D VLPSOH GLDJUDPn PDWLF WHFKQLTXH ,Q PRVW FDVHV WKLV WHFKQLTXH DUULYHV DW WKHVH RSHUDWRUV PXFK TXLFNHU WKDQ D GLUHFW DSSOLFDWLRQ RI WKH VFKHPH RI 'ULQIHOnG DQG 6RNRORY )XUn WKHUPRUH WKH VFKHPH DOVR DSSOLHV WR KLJKHU UHSUHVHQWDWLRQV RI WKH HPEHGGLQJ DIILQH /LH DOJHEUD ,Q WKH ILUVW VHFWLRQ ZH EULHIO\ UHYLHZ WKH FRQVWUXFWLRQ RI ZHLJKW GLDJUDPV FRUUHVSRQGLQJ WR UHSUHVHQWDWLRQV RI DIILQH DQG QRQDIILQH /LH DOJHEUDV )URP WKHUH ZH UHYLHZ WKH PDWUL[ PHWKRG RI 'ULQIHOnG DQG 6RNRORY IRU EXLOGLQJ /D[ RSHUDWRUV 1H[W LV D SUHVHQWDWLRQ RI RXU PHWKRG ZKLFK UHn SODFHV WKH PDWUL[ SURFHGXUH ZLWK D VFKHPH XWLOL]LQJ F\FOLF ZHLJKW GLDJUDPV RI UHSUHVHQWDWLRQV RI DIILQH /LH DOJHEUDV :H WKHQ SUHVHQW D SURRI WKDW WKH GLDJUDPPDWLF DOJRULWKP SURGXFHV WKH FRUUHFW /D[ RSHUDWRU )LQDOO\ ZH GLVn FXVVHV WKH JHQHUDOL]DWLRQ WR /D[ RSHUDWRUV EDVHG RQ VXSHUV\PPHWULF DIILQH /LH DOJHEUDV 5HYLHZ RI :HLJKW 'LDJUDPV $V QRWHG LQ WKH LQWURGXFWLRQ HDFK /D[ RSHUDWRU FDQ EH DVVRFLDWHG ZLWK D UHSUHVHQWDWLRQ RI VRPH DIILQH /LH DOJHEUD 7KXV LQ WKLV VHFWLRQ ZH JLYH D PAGE 59 EULHI UHYLHZ IRU FRQVWUXFWLQJ ZHLJKW GLDJUDPV FRUUHVSRQGLQJ WR WKHVH UHSUHVHQn WDWLRQV 5HFDOO RQH FDQ DVVRFLDWH XQLTXHO\ WR HYHU\ LUUHGXFLEOH UHSUHVHQWDWLRQ RI D EDVLF /LH DOJHEUD D KLJKHVW ZHLJKW YHFWRUf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fÂ§ URZ RI WKH &DUWDQ PDWUL[ Q WLPHV IURP D ZHLJKW YHFWRU ZKRVH LfÂ§ FRPSRQHQW KDV D SRVLWLYH YDOXH Q :KHQ ZHLJKW YHFWRUV KDYH PRUH WKDQ RQH SRVLWLYH FRPSRQHQW VXEWUDFW DOO SRVVLEOH SHUPXWDWLRQV RI WKH DSSURSULDWH &DUWDQ URZV $ WKHRUHP GXH WR '\QNLQnr VWDWHV WKDW WKH ILQDO ZHLJKW GLDJUDP LV DOZD\V fVSLQGOH VKDSHGf ,Q RWKHU ZRUGV Lf WKH QXPEHU RI ZHLJKW YHFWRUV DW WKH OHYHO N LV HTXDO WR WKH QXPEHU DW OHYHO fÂ§ N LLf WKH QXPEHU RI ZHLJKWV DW OHYHO N LV JUHDWHU WKDQ RU HTXDO WR WKH QXPEHU DW OHYHO N IRU N )RU DQ H[SOLFLW H[DPSOH FRQVLGHU WKH DOJHEUD 7KRXJK WKLV LV DOPRVW D WULYLDO FDVH WKH UHVXOWV ZLOO EH XVHIXO IRU WKH QH[W VHFWLRQ 7KH '\QNLQ GLDJUDP LV JLYHQ E\ R R PAGE 60 ZKHUH UHFDOO WKH VLQJOH EDU UHSUHVHQWV r 7KH &DUWDQ PDWUL[ LV WKHQ HDVLO\ IRXQG WR EH 7KH KLJKHVW ZHLJKW YHFWRU RI WKH IXQGDPHQWDO UHSUHVHQWDWLRQ LV f 6LQFH D SRVLWLYH RQH DSSHDUV LQ WKH ILUVW SODFH ZH VXEWUDFW WKH ILUVW URZ RI WKH &DUWDQ PDWUL[ RQH WLPH 7KLV JLYHV WKH ZHLJKW fÂ§ f 1RZ GXH WR WKH RQH LQ WKH VHFRQG SODFH ZH VXEWUDFW WKH VHFRQG URZ RI WKH &DUWDQ PDWUL[ RQFH WR JHW fÂ§f 7KLV FRPSOHWHV WKH SURFHVV VLQFH QR SRVLWLYH FRPSRQHQWV UHPDLQ 7KH UHVXOW LV WKH KHLJKW WZR ZHLJKW GLDJUDP f f $f Of ZKHUH WKH VXEVFULSWV RQ WKH ZHLJKW YHFWRUV LQGLFDWH D FRXQWLQJ RI WKH YHFWRUV 7KH RQHV DGMDFHQW WR WKH DUURZV UHSUHVHQW WKH QRUPDOL]DWLRQ IDFWRUV RI WKH FRUn UHVSRQGLQJ QHJDWLYH VLPSOH URRWV 7KHVH YDOXHV DUH IL[HG E\ WKH FRPPXWDWLRQ UHODWLRQV RI WKH /LH DOJHEUD 7R VLPSOLI\ RXU GLDJUDPV ZH ZLOO QRW GLVSOD\ YDOn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f f ?K Of Ff )LJXUH 7KH fÂ§f UHSUHVHQWDWLRQ RI $M Df '\QNLQ GLDJUDP Ef &DUWDQ PDWUL[ Ff &\FOLF ZHLJKW GLDJUDP 7KH GDVKHG DUURZ LV WKH GHOHWHG URRW $V DQ H[SOLFLW H[DPSOH FRQVLGHU WKH QRQWZLVWHG DIILQH DOJHEUD A 7R JHQHUDWH WKH F\FOLF ZHLJKW GLDJUDP FRUUHVSRQGLQJ WR WKH FDQRQLFDO UHSUHVHQWDn WLRQ VWDUW ZLWK WKH ZHLJKW YHFWRU fÂ§ f ZKHUH fÂ§ FRUUHVSRQGV WR WKH DIILQH URRW )LJXUH JLYHV WKH '\QNLQ GLDJUDP DQG VXEVHTXHQW &DUWDQ PDWUL[ ZKLFK WKHQ JHQHUDWHV WKH GLVSOD\HG UHVXOWDQW ZHLJKW GLDJUDP 7KLV SDUWLFXODU F\FOLF ZHLJKW GLDJUDP FDQ IXUWKHU EH WKRXJKW RI DV WKH DIILQH H[WHQVLRQ RI WKH KLJKHVW ZHLJKW GLDJUDP EDVHG RQ WKH IXQGDPHQWDO UHSUHVHQWDn WLRQ RI WKH EDVLF /LH DOJHEUD $Â 7KLV LV HDV\ WR VHH E\ UHPRYLQJ HYHU\ZKHUH WKH FRPSRQHQW GXH WR WKH DIILQH URRW +RZHYHU WKLV LV QRW DOZD\V WKH FDVH )RU H[DPSOH )LJ GLVSOD\V WKH F\FOLF ZHLJKW GLDJUDP FRQVWUXFWHG ZLWK WKH ZHLJKW YHFWRU fÂ§f ZKHUH QRZ WKH DIILQH FRPSRQHQW LV fÂ§ $OWKRXJK f PAGE 62 f )LJXUH &\FOLF ZHLJKW GLDJUDP RI IURP WKH ZHLJKW fÂ§f 7KH GDVKHG DUURZV DUH WKH GHOHWHG URRW f ,Â f f YK VK f f V3 f f ,f )LJXUH +LJKHVW ZHLJKW GLDJUDP RI WKH DGMRLQW UHSUHVHQWDWLRQ RI f JHQHUDWHV WKH KLJKHVW ZHLJKW GLDJUDP RI WKH DGMRLQW UHSUHVHQWDWLRQ RI ZH VHH E\ FRPSDULQJ ZLWK )LJ WKDW WKH DIILQH H[WHQVLRQ FRQWDLQV DQ H[WUD ]HUR ZHLJKW f PAGE 63 ,Q JHQHUDO WKH DIILQH FRPSRQHQW LQ WKH DIILQHO\ H[WHQGHG YHFWRU DVVRFLDWHG ZLWK WKH KLJKHVW ZHLJKW YHFWRU RI D EDVLF /LH DOJHEUD ZLOO DOZD\V EH QHJDWLYH +RZHYHU ZH ZLOO JLYH DQ H[DPSOH EHORZ VKRZLQJ WKDW VRPH VXSHUV\PPHWULF FDVHV UHTXLUH SRVLWLYH DIILQH FRPSRQHQWV 6WDQGDUG &RQVWUXFWLRQ RI /D[ 2SHUDWRUV 7KH /D[ RSHUDWRUA /[Wf LV GHILQHG WR EH OLQHDU DQG +HUPLWLDQ )XUWKHUn PRUH LW VDWLVILHV WKH FKDUDFWHULVWLF HTXDWLRQ /[ WfI![ Wf QI!^[ Wf $f ZKHUH WKH HLJHQYDOXH S LV UHTXLUHG WR EH FRQVWDQW XQGHU QRQOLQHDU HYROXWLRQ ,Q RWKHU ZRUGV WKH QRQOLQHDU EHKDYLRU RI WKH HLJHQIXQFWLRQV ![Wf DUH JRYHUQHG E\ VRPH RSHUDWRU If ZKLFK PD\ EH QRQOLQHDU YLD WKH HTXDWLRQ fÂ§ $WfM![Wf $f RW )XWKHUPRUH $>Wf HQWHUV LQWR WKH GLIIHUHQWLDO VFDODU /D[ HTXDWLRQ >$^Wf/[Wf@ $f ZKLFK JHQHUDWHV WKH LQWHJUDEOH .G9 HTXDWLRQV 7KH PDWUL[ FRQVWUXFWLRQ RI /D[ RSHUDWRUV XWLOL]LQJ JHQHUDWRUV RI VRPH HPn EHGGLQJ DIILQH /LH DOJHEUD UHYLHZHG E\ 'ULQIHOnG DQG 6RNRORY EHJLQV ZLWK D PDWUL[ RSHUDWRU RI WKH IRUP & ,fÂ§ $ T[f $f R[ ZKHUH GHQRWHV WKH 1 [ 1 GLPHQVLRQDO XQLW PDWUL[ DQG WR VLPSOLI\ QRWDWLRQ ZH KDYH VXSSUHVVHG WKH DUJXPHQW W 7KH WKLUG WHUP LV GLVFXVVHG EHORZ 7KH VHFRQG WHUP LV JHQHUDWHG E\ WKH QHJDWLYH VLPSOH URRWV (c RI WKH HPEHGGLQJ PAGE 64 DIILQH /LH DOJHEUD ,Q WKH JUDGDWLRQ FRQYHQWLRQV RI 'ULQIHOnG DQG 6RNRORY ZH KDYH WKH FLUFXODQW PDWUL[ U $ PAGE 65 /LH DOJHEUD 7KXV GXH WR WKH OLQHDU FRPELQDWLRQ DPRQJ WKH URRWV RI WKH DIILQH V\VWHP RQH URRW PXVW EH VLQJOHG RXW WR DFW DV D FRQYHQWLRQDO VWDWH UDLVLQJ RSHUDWRU 7KLV UROH LV JLYHQ WR WKH UHPRYHG URRW 7KXV WKH YDFXXP HLJHQVWDWH ZLOO EH DQQLKLODWHG E\ D YDFXXP SURMHFWLRQ RSHUDWRU $ GHILQHG E\ U $ n\ A $ (P $ $(9UM $f Â 7KLV UHTXLUHPHQW IL[HV WKH VFDODU YDFXXP VROXWLRQ ]! E\ VHWWLQJ LW HTXDO WR D OLQHDU FRPELQDWLRQ RI WKH FRPSRQHQWV RI VXFK WKDW $fA $f LV VDWLVILHG $ GLUHFW UHODWLRQ EHWZHHQ WKH VFDODU RSHUDWRU / DQG WKH PDWUL[ RSHUDWRU & ZLOO EH JLYHQ LQ WKH QH[W VHFWLRQ )RU WKH NHUQHO HTXDWLRQ $f WR SURGXFH D XQLTXH VROXWLRQ ZH UHTXLUH WKDW WKH QXPEHU RI LQGHSHQGHQW GHJUHHV RI IUHHGRP HTXDO WKH UDQN RI WKH HPEHGGLQJ DIILQH /LH DOJHEUD 4A? RU HTXLYDOHQWO\ WKH UHVLGXDO V\VWHP FPf 7KH H[WUD GHJUHHV RI IUHHGRP JHQHUDWH JDXJH LQYDULDQFH 7R IL[ WKH JDXJH LQYDULDQFH RQH PXVW ILQG D PDWUL[ RSHUDWRU 6DUf WKDW HQIRUFHV WKH JDXJH WUDQVIRUPDWLRQ &T HDG6& $f ZKHUH DG GHQRWHV WKH DGMRLQW PDSSLQJ 7KH JDXJH IUHHGRP LQ (T$6f DOn ORZV RQH IUHHGRP LQ GHWHUPLQLQJ WKH IRUP RI WKH FRRUGLQDWH GHSHQGHQW WHUP e ,fÂ§$ T[f $f 'ULQIHOfG DQG 6RNRORY ILQG WKH VXIILFLHQW FRQGLWLRQ WKDW 6 ( &AI5 Uf ZKHUH 7@ LV JHQHUDWHG E\ WKH SRVLWLYH VLPSOH URRWV )c L A P 0DQ\ DXWKRUV LQFOXGLQJ 'ULQIHOnG DQG 6RNRORY ZRUN PRVW IUHTXHQWO\ LQ WKH fFDQRQLFDOf JDXJH +RZHYHU LQ WKLV SDSHU ZH FKRRVH WR ZRUN LQ WKHLU PAGE 66 fGLDJRQDOf JDXJH ZKLFK KDV WKH IRUP TGLD[f 7KLV JDXJH OHDGV WR WKH FRQYHQLHQW IRUP U GLDJ T[ 9 TL SYO 9R 4Q L R $f $f ZKLFK LV LQ WKH FDQRQLFDO RU SULQFLSDO JUDGDWLRQAn +HUH +M DUH WKH JHQHUDWRUV RI WKH &DUWDQ VXEDOJHEUD DQG WKH IXQFWLRQV DUH OLQHDU FRPELQDWLRQV RI WKH HOHPHQWV T ,Q WKLV JDXJH WKH JDXJH WHUP TGLD DVVRFLDWHG ZLWK A?FPf LV WKH VSHFLDO FDVH ZKHUH WKH VXP H[FOXGHV L P 7KH /D[ RSHUDWRU /GLD JHQHUDWHV WKH P.G9 HTXDWLRQV DQG LV UHODWHG WR /FDQ YLD WKH ZHOONQRZQ 0LXUD WUDQVIRUPDWLRQV 'LDJUDPPDWLF &RQVWUXFWLRQ RI /D[ 2SHUDWRUV 7R H[SORLW JDXJH LQYDULDQFH RI WKH /D[ RSHUDWRUV RQH VKRXOG FKRRVH D T[f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f ZKHUH WKH DIILQH YHUWH[ LV GHOHWHG 7KXV WKH GLDJRQDO JDXJH VLPSO\ UHGXFHV WR WKH IRUP U TGWD[f YL[f+LL L $f PAGE 67 ZKHUH ZH KDYH H[FOXGHG +T IURP WKH VXP &RQVLGHU DJDLQ WKH FDQRQLFDO UHSUHVHQWDWLRQ RI WKH HPEHGGLQJ DIILQH /LH DOJHEUD A "FRf SUHVHQWHG LQ )LJ 0DWUL[ UHSUHVHQWDWLRQV RI WKH &DUWDQ PDWUL[ FDQ EH UHDGRII IURP WKH F\FOLF ZHLJKW GLDJUDP 7KH PDWUL[ HOHPHQW +L fMM LV H[WUDFWHG IURP WKH HOHPHQW RI WKH MfÂ§ ZHLJKW YHFWRU ZKLOH WKH RII GLDJRQDO HOHPHQWV DUH VHW WR ]HUR 7KH PDWUL[ HQWU\ RI WKH QHJDWLYH VLPSOH URRW LV DVVLJQHG LWV QRUPDOL]DWLRQ IDFWRU LI WKH NfÂ§ ZHLJKW YHFWRU EUDQFKHV LQWR WKH MfÂ§ ZHLJKW YHFWRU DV D UHVXOW RI VXEWUDFWLQJ WKH URZ RI WKH &DUWDQ PDWUL[ LQ WKH SURFHVV 7KH RWKHU HQWULHV DUH E\ GHIDXOW ]HUR 7KXV WKH PDWUL[ UHSUHVHQWDWLRQV RI WKH VLPSOH URRWV DUH HDVLO\ IRXQG WR JLYH $ ? $ $f 92 ZKHUH WKH HIIHFW RI WKH DIILQH URRW LQGLFDWHG LQ )LJ E\ WKH GDVKHG DUURZ OLQH LV DVVLJQHG WKH YDOXH $ 3OXJJLQJ WKHVH YDOXHV LQWR WKH NHUQHO HTXDWLRQ $f SURGXFHV WKH V\VWHP RI HTXDWLRQV >G YO@WSO $9! >G LT Y`[c! $f >G Y@L! [S +HUH RQ WKH ULJKWKDQG VLGH ZH KDYH SODFHG WKH WHUPV GXH WR WKH PDWUL[ $ 7KH YDFXXP FRQGLWLRQ $ f GHWHUPLQHV WKH VFDODU IXQFWLRQ WR EH Lcf 7KXV ZH PXVW VROYH E\ VWDUWLQJ ZLWK WKH ODVW HTXDWLRQ )LUVW ZH PXOWLSO\ WKLV HTXDWLRQ WKURXJK E\ >G fÂ§ UT A@ DQG WKHQ HOLPLQDWH [cf XVLQJ WKH VHFRQG HTXDWLRQ 7KHQ PXOWLSO\LQJ WKURXJK E\ >G LT@ DQG XVLQJ WKH WRS HTXDWLRQ JLYHV WKH VFDODU /D[ HLJHQYDOXH HTXDWLRQ /$ 9FrfA >G LT@>G 9L Yf>G Y@US $9! $f PAGE 68 ZKHUH WKH VSHFWUDO SDUDPHWHU LV JLYHQ E\ S fÂ§$ ,PSRVLQJ WKH ILHOG UHGHILQLn WLRQV L mL Y ]T $f JLYHV WKH VWDQGDUG IRUP /$Frf >D L@>Â! T@>G D@ $f 7KLV H[DPSOH H[KLELWV D FRPPRQ IHDWXUH UHOHYDQW IRU RXU VFKHPH EHORZ :KHQ WKH YDFXXP FRQGLWLRQ $ f UHTXLUHV WKH VFDODU HLJHQIXQFWLRQ WR EH JLYHQ E\ D VLQJOH FRPSRQHQW RI WKH HLJHQIXQFWLRQ VD\ WKHQ WKH UHVXOWLQJ FKDUDFWHULVWLF HTXDWLRQ VDWLVILHV /[SL ILLSL $f &RQVHTXHQWO\ WKH V\VWHP UHGXFWLRQ PXVW VWDUW ZLWK WKH ]fÂ§ HTXDWLRQ LQ WKH PDWUL[ V\VWHP DQG SURFHHG XSZDUG WLOO WKH WRS HTXDWLRQ LV UHDFKHG ,I ] 1 WKH SURFHVV FRQWLQXHV ZLWK WKH ERWWRP HTXDWLRQ DQG PRYHV XSZDUG XQWLO WKH ]fÂ§ HTXDWLRQ LV UHDFKHG DJDLQ :H VKDOO UHIHU WR WKLV FDVH DV WULYLDO VLQFH WKH FRUn UHVSRQGLQJ F\FOLF DIILQH ZHLJKW GLDJUDP LV OLQHDU FRQWDLQLQJ QR EUDQFK SRLQWV $ VHFRQG IHDWXUH EURXJKW RXW LQ WKLV H[DPSOH LV WKDW WKH QXPEHU RI IDFWRUV LQ WKH UHVXOWDQW /D[ RSHUDWRU $f LV HTXDO WR WKH QXPEHU RI ZHLJKWV LQ WKH ZHLJKW GLDJUDP 8QIRUWXQDWHO\ WKLV LV YDOLG RQO\ IRU WULYLDO FDVHV 1HYHUWKHOHVV WKLV ODVW REVHUYDWLRQ LV NH\ WR RXU VFKHPH 7R KLJKOLJKW RQH PRUH SURSHUW\ RI WKH JHQHUDO SURFHGXUH ZH WXUQ WR D QRQn WULYLDO H[DPSOH )RU WKLV ZH UHTXLUH D UHSUHVHQWDWLRQ RI DQ DIILQH /LH DOJHEUD ZKRVH F\FOLF ZHLJKW GLDJUDP KDV DW OHDVW RQH EUDQFKLQJ SRLQW 7KXV FRQVLGHU WKH FDQRQLFDO UHSUHVHQWDWLRQ RI WKH DIILQH DOJHEUD 'A?FTf )LJ SUHVHQWV WKH '\QNLQ GLDJUDP &DUWDQ PDWUL[ DQG FRUUHVSRQGLQJ F\FOLF ZHLJKW GLDJUDP PAGE 69 2 2 2 Ef } L L R 2 2 OÂ f 2 A OÂ 2 2 f VK L f 2 fa f f n f f Ff )LJXUH 7KH fÂ§ f UHSUHVHQWDWLRQ RI Df '\QNLQ GLDJUDP Ef &DUWDQ PDWUL[ Ff &\FOLF ZHLJKW GLDJUDP 7KH GDVKHG DUURZ LV WKH GHOHWHG URRW ZKLFK KDV WZR EUDQFK SRLQWV 5HDGLQJ D B $ f ? 7KH EUDQFK SRLQWV KDYH PDQLIHVWHG WKHL IURP WKH ZHL JKW GLDJUDP JLYHV $ ? $ Â $ f A UHV E\ SODFLQJ PRUH WKDQ RQH QRQn ]HUR HQWU\ LQ WKH VHFRQG DQG VL[WK URZV 1RZ IXUWKHU UHDGLQJ RII WKH HOHPHQWV PAGE 70 RI WKH &DUWDQ PDWULFHV JLYHV WKH V\VWHP RI HTXDWLRQV > 9L@LSL fÂ§ $ > fÂ§ 9L Y@LS L $ > Y Y LT@ >G 9 I@ $f > 8 9@ f > 8 m 9@ f > WT O!@ > LT@ 7KH YDFXXP FRQGLWLRQ $ f SURGXFHV WZR GLVWLQFW VROXWLRQV DQG WKH OLQHDU FRPELQDWLRQ fÂ§ +HUH ZH FRQVLGHU WKH ILUVW FDVH 3URFHHGLQJ DV EHIRUH ZH HOLPLQDWH DQG LQ WKH ODVW WZR HTXDWLRQV WR JHW > Y X X@> LT X@> WT@ $f 1RZ ZH HQFRXQWHU D ZHOONQRZQ WHFKQLFDO SUREOHP QRW IRXQG LQ WKH WULYLDO FDVH 7KH FRPSRQHQWV DQG FDQ QRW ERWK EH VLPXOWDQHRXVO\ HOLPLQDWHG VLQFH WKH H[SUHVVLRQV > fÂ§ X X@ DQG > Y fÂ§ X@ GR QRW FRPPXWH 7KLV GLOHPPD LV GLUHFWO\ OLQNHG WR WKH IDFW WKH FRUUHVSRQGLQJ F\FOLF ZHLJKW GLDJUDP KDV D EUDQFK SRLQW FRQQHFWLQJ WKH IRXUWK DQG ILIWK ZHLJKWV WR D VLQJOH ZHLJKW ORFDWHG EHORZ WKHP 7R RYHUFRPH WKLV REVWDFOH WKH SVHXGRGLIIHUHQWLDO RSHUDWRU Âf PXVW EH LQWURGXFHG ,WV RSHUDWLRQ RQ DQ\ IXQFWLRQ I[f LV JLYHQ E\ WKH H[SDQVLRQ RR GAI[f PAGE 71 7KXV WKH FRPELQHG HIIHFW RI WKH ERWWRP ILYH HTXDWLRQV LV ^ > fÂ§ O! 9? A ?G 9 fÂ§ 8@ A A ?G 8 fÂ§ O! fÂ§ 8@ [ >G Y Y@>G X[@ $ KHOSIXO LGHQWLW\ ZH XVH UHSHDWHGO\ LV $f ^$ %aO`a ^$aO>$ %@%aO`a %>$ %@aO$ $f :KHQ DSSOLHG WR (T$f D FDQFHOODWLRQ RFFXUV DPRQJ WKH XÂnV DSSHDULQJ LQ WKH FXUO\ EUDFNHWV 7KLV VLPSOLILHV WKH H[SUHVVLRQ WR Â>GY Y$@G >Â! 8f@> XAA@> 8LX@>"@ $f &RQWLQXLQJ LQFRUSRUDWLQJ WKH QH[W WZR HTXDWLRQV LQ $f UHTXLUHV D VHFRQG DSSOLFDWLRQ RI WKH UHODWLRQ $f )LQDOO\ WKH /D[ RSHUDWRU EDVHG RQ FTf ZLWK YDFXXP LV A O>G XL@>G 9L Y@>G Y KY X@>G Y X@ [ GaO>G Y Y@>G Y Y Y@>G LT Y@>G 8L@ 8VLQJ WKH ILHOG UHGHILQLWLRQV $f T 98 T Y YX T Y Y Y T Y X $f ZH JHW / fÂ§G A^G?TL?>G T?>G T@>G TAG A^G fÂ§ Tr?>G fÂ§ T?>G fÂ§ TAOG fÂ§ T?? $f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f DY[f Â‘ff@ $f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f EHJLQQLQJ ZLWK DQG HQGLQJ RQ WKH YDFXXP ZHLJKWVf )RU H[DPSOH )LJ SUHVHQWV WKH IRXU OLQHDU VXEJUDSKV DVVRFLDWHG ZLWK WKH FDQRQLFDO UHSUHVHQWDWLRQ RI 'A? ,Q GUDZLQJ F\FOLF ZHLJKW GLDJUDPV LW LV LPSRUWDQW WKDW WKH DUURZV JHQHUn DWHG E\ WKH GHOHWHG YHUWH[ DUH GLVWLQJXLVKHG IURP WKH RWKHUV 2XU FRQYHQWLRQ LV WR XVH GDVKHG OLQHV )XUWKHUPRUH WKH GLUHFWLRQ RI WKH DUURZV PXVW DOVR EH QRWHG 7KH /D[ RSHUDWRUV DVVRFLDWHG ZLWK HDFK VXEGLDJUDP DUH WKHQ FRQ PAGE 73 VWUXFWHG DV IROORZV 6WHS &LUFXODWH DURXQG WKH ORRS EHJLQQLQJ ZLWK WKH YDFXXP VROXWLRQ VXFK WKDW WKH IORZ LV RSSRVLWH PRVW RI WKH DUURZV ,I D ZHLJKW YHFWRU LV DSSURDFKHG E\ DQ DUURZfV Df WDLO DSSHQG LWV ZHLJKW IDFWRU WR WKH RSHUDWRUfV OHIW VLGH Ef KHDG DSSHQG WKH ZHLJKW IDFWRUnV LQYHUVH WR WKH RSHUDWRUfV OHIW VLGH )RU ZHLJKWV DW WKH WDLO HQG RI ERWK FRQQHFWLQJ DUURZV GR QRWKLQJ 0XOWLSO\ E\ WKH SURGXFW RI WKH FRUUHVSRQGLQJ QRUPDOL]DWLRQ IDFWRUV $f 7KH ORRS LV WR EH FLUFXODWHG LQ D GLUHFWLRQ RSSRVLWH PRVW RI WKH DUURZV VR WKDW WKH OHDGLQJ WHUP RI WKH /D[ RSHUDWRU / GQ KDV SRVLWLYH H[SRQHQW LH Q )RU WULYLDO FDVHV WKLV FRPSOHWHV WKH FRPSXWDWLRQ RI / +RZHYHU IRU QRQn WULYLDO FDVHV ZLWK EUDQFKLQJ ZHLJKW GLDJUDPV ZH FDQ QRW QDLYHO\ EXLOG WKH ILQDO /D[ RSHUDWRU IURP D VXP RI LWV FRQVWLWXHQW OLQHDU VXEJUDSKV,QVWHDG DV ZH VKDOO SURYH LQ WKH QH[W VHFWLRQ WKH\ DUH DGGHG WRJHWKHU DQDORJRXVO\ WR KRZ RQH FRPSXWHV WRWDO UHVLVWDQFH RI UHVLVWRUV LQ SDUDOOHO 6WHS I 7KH /D[ RSHUDWRU LV JLYHQ E\ WKH LQYHUVH RI WKH VXP RI $f WKH LQYHUVHV RI WKH RSHUDWRUV UHVXOWLQJ IURP VWHS WKUHH )RU H[DPSOH LQ WKH QRQWULYLDO FDVH 'A?FTf ZLWK DV EHIRUH ZH EXLOG IRXU RSHUDWRUV FRUUHVSRQGLQJ WR WKH OLQHDU VXEGLDJUDPV LQ )LJ /? >G XL@B> L!L@>G XM Yf>G Y Y Y@>G Y Y@ $f [ >G Y fÂ§ Y fÂ§ Y@>G 9L fÂ§ Y@>G fÂ§ 8L@ Â >G AL@>A XL@>G 9L Y?>G Y Y X@>G Y Y? $f [ >G Y Y Y@>G Y fÂ§ Y@>G L!M@ / ?G fÂ§ 9? Y@>F" fÂ§ Y Y X@> fÂ§ Y X@> Y fÂ§ Y fÂ§ X@ [ >G L!L X@> 9L@ $f PAGE 74 ! f 7 f f D R R f fc L,= f f A f f 7 f 7 f f f 7 f 7 f f f V8 f Â‘ 7 f f f VK f A f f f f U V8 f f VK f f VK 7 f f f f 7 VK f f)LJXUH 6XEGLDJUDP RI WKH F\FOLF ZHLJKW GLDJUDP RI DUURZV DUH WKH GHOHWHG URRW f f 7KH GDVKHG DQG Â >Â" a 9@> fÂ§ 9 9 I@ >F" 9 fÂ§ 9@ > 9 9 8@ [ >G Y[ fÂ§ Yf>G fÂ§ XL@ )DFWRULQJ RXW FRPPRQ WHUPV ZH ILQG OfÂ§ ?G fÂ§ L!L@ A ?G 8L fÂ§ Y@ ?G 8 fÂ§ Y fÂ§ A@ $f [ ^?G fÂ§ 8 8@ A ?G 9 fÂ§ @ A` ?G fÂ§ Y 7 9 f@ [ >GYL Y@ AG XL@ ;^ >G XL@ > XL@` $f PAGE 75 %\ WDNLQJ WKH UHFLSURFDO DQG VLPSOLI\LQJ ZH UHSURGXFH WKH SUHYLRXV UHVXOW (T$f 7R HQG WKLV VHFWLRQ ZH FRQVLGHU WKH DOWHUQDWLYH YDFXXP FKRLFH A fÂ§ 9K ,W VKRXOG EH REYLRXV WKDW VLQFH ZH DUH GHDOLQJ ZLWK F\FOLF ZHLJKW GLDJUDPV /D[ RSHUDWRUV DVVRFLDWHG ZLWK RWKHU YDFXXP VWDWHV FDQ EH DFKLHYHG E\ F\FOLFDOO\ SHUPXWLQJ IDFWRUV LQ WKH SULPDU\ /D[ RSHUDWRU 7KXV WKLV VHFRQG YDFXXP FKRLFH LPPHGLDWHO\ JLYHV WKH /D[ RSHUDWRU / ?G fÂ§ T@>G fÂ§ TL@>G fÂ§ T[?G A >G TL@>G T@>GT?>G T?G A>G fÂ§ T@ $f 3URRI RI 'LDJUDPPDWLF 6FKHPH 7R SURYH WKH HTXLYDOHQFH EHWZHHQ WKH PDWUL[ V\VWHP &LS DQG WKH GLDJUDPPDWLF DOJRULWKP ZH EHJLQ E\ UHZULWLQJ WKH IRUPHU DV 9LIW[f fÂ§$ [f $f ZKHUH WR VLPSOLI\ QRWDWLRQ ZH KDYH GHILQHG 9 &$ ,fÂ§T[f $f R[ 7KH VWUXFWXUH RI WKH DVVRFLDWHG F\FOLF ZHLJKW GLDJUDP LV HQFRGHG HQWLUHO\ LQ WKH PDWUL[ $ 6SHFLILFDOO\ UHFDOO WKDW WKH JHQHUDO PDWUL[ HOHPHQW $cM LV SURn SRUWLRQDO WR $ LI WKH GLIIHUHQFH EHWZHHQ WKH rfÂ§ ZHLJKW DQG WKH FRQQHFWLQJ MfÂ§ ZHLJKW HTXDOV WKH HOLPLQDWHG URRW RI WKH HPEHGGLQJ DIILQH VLPSOH URRW V\VWHP $OO RWKHU FRQQHFWLQJ ZHLJKWV $ c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e VXFK WKDW WKH YDFXXP VWDWH HTXDWLRQ DSSHDUV ODVW )XUWKHUPRUH LW FDQ EH DUUDQJHG VXFK WKDW $ LV ORZHU WULDQJXODU ZLWK RQHV ORFDWHG DORQJ D GLDJRQDO RQFH UHPRYHG IURP WKH PDLQ GLDJRQDO H[FHSW IRU WKH HOLPLQDWHG URRW ZKRVH FRHIILFLHQW $ DSSHDUV LQ WKH XSSHU ULJKWKDQG FRUQHU LH $ LV D FLUFXODQW PDWUL[ 7KXV WKH /D[ HLJHQYDOXH HTXDWLRQ EHFRPHV /L-MQ[f QA1^[f $f 6LQFH 9 LV GLDJRQDO WKH NfÂ§ HTXDWLRQ LQ $f FDQ EH ZULWWHQ 1 9NLcfN A$KR N 1 $f Â &OHDUO\ VLQFH $ LV D FLUFXODQW PDWUL[ DV VSHFLILHG DERYH WKH LQHTXDOLW\ L N KROGV IRU N A 1H[W E\ UHSHDWHGO\ UHSODFLQJ WKH IXQFWLRQ LS DSSHDULQJ RQ WKH ULJKWKDQG VLGH ZLWK WKH LfÂ§ PDWUL[ HTXDWLRQ ZH HYHQWXDOO\ UHDFK WKH H[SUHVVLRQ $NLLSN LH 7$Y Y 1 fÂ§$1LLSL L 1 1 @ D1Mn3M $LMLSM L OM O $f 1 1 fÂ§\ A $1L9L $\ Â‘ f f 7!N $IFLLT Â N ZKHUH LV WKH KHLJKW RI WKH F\FOLF ZHLJKW GLDJUDP 'XH WR WKH VXFFHVVLYH DSSOLFDWLRQV RI WKH VWDWH ORZHULQJ RSHUDWRUV $A ZLWK L M WKLV HTXDWLRQ LV LQWHUSUHWHG DV WDNLQJ WKH KLJKHVW VWDWH L DQG ORZHULQJ LW WR WKH YDFXXP VWDWH [SZ PAGE 77 5HSODFLQJ LSL WKURXJK 7K9K fÂ§ fÂ§$R ;LS1 $f ZKHUH ZH KDYH XVHG $-9 $nR$ ZKLFK H[FLWHV WKH OHYHO RI WKH VWDWH VLQFH 1 JLYHV 1 1 91A1 \[? < Â‘ f f < $}n9aO$nL Â‘ Â‘ f 9aONNO9aO[O!1 $f Â N O ZKHUH $J LV WKH QRUPDOL]DWLRQ IDFWRU RI WKH DIILQH URRW )LQDOO\ PRYLQJ WHUPV WR WKH OHIWKDQG VLGH ZH UHFRYHU $f ZKHUH 1 1 9aO?1L9aO f f Â‘ ?N[9aO`a? $f Â N O DQG WKH VSHFWUDO SDUDPHWHU LV JLYHQ E\ NL \ $ $f 6LQFH HDFK URZ DQG FROXPQ RI $ FRQWDLQ RQO\ RQH HQWU\ WKH VXP ZLOO JHQHUDWH D VLQJOH WHUP KH / $f ZKHUH $f LV WKH SURGXFW RI WKH QRUPDOL]DWLRQ IDFWRUV 1RZ HDFK 7f LV D ZHLJKW IDFWRU DV GHILQHG LQ VWHS 7KXV WKHUH LV D GLUHFW PDSSLQJ EHWZHHQ WKH RUGHU RI WKH ZHLJKW IDFWRUV DQG WKHLU ORFDWLRQ LQ WKH FRUUHVSRQGLQJ ZHLJKW GLDJUDP 1RZ VXSSRVH ZH SHUPLW PXOWLSOH URZ HQWULHV LQ $ KH EUDQFK SRLQWV LQ WKH ZHLJKW GLDJUDP )LUVW FRQVLGHU WKH FDVH ZKHUH VXFK PXOWLSOH HQWULHV RFFXU DERYH WKH URZ $V EHIRUH WKHUH LV D VLQJOH YDFXXP VWDWH DQG WKH FRQVWDQW $ LV ORFDWHG LQ WKH XSSHU ULJKWKDQG FRUQHU RI $ 7KHUHIRUH WKH FRQVWUDLQW L N IRU N UHPDLQV LQ HIIHFW IRU (T$f +HQFH WKH GHULYDWLRQ OHDGLQJ WR $f IROORZV WKURXJK XQFKDQJHG 1RZ HDFK QHZ HQWU\ LQ $ FDXVHV DQ DGGLWLRQDO ILQDO WHUP LQ $f &OHDUO\ SHU VWHS RI WKH GLDJUDPPDWLF DOJRULWKP WKH PAGE 78 ILQDO /D[ RSHUDWRU LV REWDLQHG E\ WDNLQJ WKH UHFLSURFDO RI WKH VXP RI WHUPV JHQHUDWHG E\ $f 1H[W VXSSRVH WKH PXOWLSOH URZ HQWULHV LQ $ GXH WR WKH EUDQFK SRLQW RFFXU LQ WKH 1fÂ§ URZ 7KH YDFXXP FRQGLWLRQ $ f VKRZV WKDW WKLV LV HTXLYDOHQW WR D GHJHQHUDWH YDFXXP VWDWH ZLWK VD\ GHJHQHUDF\ G 6XEVHTXHQWO\ WKLV URZ ZLOO EH DVVRFLDWHG ZLWK WKH HOLPLQDWHG URRW DQG WKH G LQWHJHUV ZLOO EH DVVLJQHG WKH YDOXH $ ,Q IDFW $ DSSHDUV RQO\ LQ WKLV URZ &OHDUO\ LQ WKH ZHLJKW GLDJUDP WKH G ZHLJKWV VKDUH WKH VDPH OHYHO /HW XV ILUVW GLVFXVV WKH FDVH ZKHUH WKH FRHIILFLHQWV $ RFFXU LQ WKH ILUVW URZ 7KXV (T$f UHPDLQV YDOLG NHHSLQJ LQWDFW WKH FRQVWUDLQW L N IRU N A )XUWKHUPRUH WKH VFDODU HLJHQIXQFWLRQ FI![f LV QRZ D OLQHDU FRPELQDWLRQ RI WKH FRPSRQHQWV LS1LLS1K f f frS1GL DQG WKH HTXDWLRQ IRU WT EHFRPHV 1 $9fL A $f V 1fÂ§GO &RQVHTXHQWO\ (T$f LV PRGLILHG WR Y \U 1 1 1 f ( ( pafDWY L NfÂ§O M 1fÂ§GO $f [ 7!N A$IFL=A r$LMWSM ZKHUH LV WKH QXPEHU RI ILHOG UHSODFHPHQWV SHUIRUPHG 7KH FKDUDFWHULVWLF HTXDWLRQ LV REWDLQHG E\ PXOWLSO\LQJ ERWK VLGHV E\ $OV DQG WKHQ VXPPLQJ RYHU V LH 1 ( $r fr ( V 1fÂ§GO 1 1 1 ( ( V O Â M 1fÂ§GO /$ $f [ ;$M7 ?NO9aO 1RWH WKH VXP RYHU V RQ WKH ULJKWKDQG VLGH KDV EHHQ H[WHQGHG WR WKH HQWLUH UDQJH IRU FRQYHQLHQFH PAGE 79 (DFK WHUP LQ $OV FRQWDLQV WKH IDFWRU $ ZKLFK FDQ WKHQ EH IDFWRUHG RXW $V D UHVXOW WKH VFDODU HLJHQIXQFWLRQ LV IRXQG WR EH 1 W! A $OV[SW $f V 1fÂ§GO DQG WKH /D[ RSHUDWRU 1 1 / f f f ( $L},fU$} f f f $QAU U $f V N &OHDUO\ WKLV KDV WKH VDPH LQWHUSUHWDWLRQ DV WKH QRQGHJHQHUDWH EUDQGOLQJ FDVH )RU WKH ODVW VWDJH RI WKH SURRI ZH UHOD[ WKH FRQGLWLRQ WKDW PXOWLSOH RFFXUn UHQFHV RI $ PXVW DOO EH LQ WKH ILUVW URZ RI $ ,Q WKH ZHLJKW GLDJUDP WKLV PHDQV QRW DOO WKH DUURZV DVVRFLDWHG ZLWK WKH HOLPLQDWHG URRW SRLQW WR WKH ERWWRP OHYHO 5HFDOO IURP WKH GLVFXVVLRQ VXUURXQGLQJ (T$f WKH HOLPLQDWHG URRW ZLWK FRHIILFLHQW $ DFWV DV D VWDWH UDLVLQJ RSHUDWRU 7KXV HYHU\ RFFXUUHQFH RI $ ZLOO DSSHDU LQ WKH XSSHU WULDQJXODU SRUWLRQ RI $ DQG WKH XQLW FRHIILFLHQWV RI WKH VWDWH ORZHULQJ URRWV DUH LQ WKH ORZHU WULDQJXODU SRUWLRQ )RU $ LQ WKH NfÂ§ URZ RI $ N A (T$f LV PRGLILHG WR 1 7!MUM!M $nR$9!W A $f QAN ZKHUH VLQFH $ FRUUHVSRQGV WR WKH VWDWH UDLVLQJ RSHUDWRU N M &RQVLGHU WKH FDVH ZKHUH M LV WKH ODUJHVW VXFK LQGH[ WR VDWLVI\ WKLV HTXDWLRQ 7KHQ DOORZLQJ GHJHQHUDWH YDFXXP VWDWHV ZH KDYH 1 1 L Y M Â‘ f f $ MLI!M 1 1 1 f$ ,M9O>$U4?!N<$aMQn3Q Â M L QAN $f ZKHUH DJDLQ LV WKH QXPEHU RI ILHOG UHSODFHPHQWV SHUIRUPHG 7KH HIIHFW RI WKH IDFWRU LQ IURQW RI "!r LV WR ILUVW GXH WR $ UDLVH WKLV VWDWH WR 9f DQG WKHQ PAGE 80 WR ORZHU LW WLOO WKH YDFXXP VWDWH LS LV UHDFKHG 1RZ VLQFH WKH FRUUHVSRQGLQJ ZHLJKW GLDJUDP LV F\FOLF WKHUH PXVW H[LVW VRPH IDFWRU WKDW ZLOO FLUFXODWH LS EDFN WR LSN )LUVW DV ZDV WKH FDVH ZLWK ^'A? FTf FRQVLGHU WKH VLWXDWLRQ ZKHUH LSN LV DQ LQWHUPHGLDWH VWDWH LQ $f LH r! 79 !f( Â 79 ( N 9L$L9 $ cNLSr $f 7KLV JLYHV r3N Â‘ 79 f+( 1 ( N 9O$f'UO f$MN` OLS DL 7KXV SHU VWHS RI WKH GLDJUDPPDWLF DOJRULWKP WKH IDFWRU n'N DVVRFLDWHG ZLWK WKH ZHLJKW YHFWRU DW WKH WDLO HQG RI ERWK FRQQHFWLQJ DUURZV GRHV QRW DSSHDU )XUWKHU SURFHHGLQJ IURP KLJKHU ZHLJKWV WR ORZHU ZHLJKWV LQ WKH ZHLJKW GLDJUDP FRQWULEXWHV IDFWRUV RI LQ WKH RSHUDWRU GHILQHG LQ VWHS IRU WKH OLQHDU VXEGLDJUDP )LQDOO\ LI LSN GRHV QRW DSSHDU DV DQ LQWHUPHGLDWH VWDWH RI WKH YDFXXP VWDWH LS LQ $f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n VLRQV RI WKH EDVLF /LH DOJHEUDV KDV EHHQ JLYHQ E\ .DFL ,Q DGGLWLRQ WR WKH PAGE 81 ERVRQLF VLPSOH URRWV RI WKH EDVLF /LH DOJHEUD WKH VLPSOH URRW V\VWHP RI WKH VXSHUV\PPHWULF DOJHEUDV FRQWDLQV WZR GLVWLQFW NLQGV RI IHUPLRQLF URRWV 7KH '\QNLQ V\PERO RI WKH ILUVW W\SH LV VRPHWLPHV JLYHQ E\ D VKDGHG YHUWH[ UHSUHn VHQWLQJ D QRQ]HUR QRUP 7KH VHFRQG IHUPLRQLF URRW W\SH KDV ]HUR QRUP ZKRVH '\QNLQ V\PERO LV JLYHQ FRUUHVSRQGLQJO\ E\ D FURVVHG RXW YHUWH[ $V DOZD\V WKH ERVRQLF URRW LV GHQRWHG E\ D ZKLWH YHUWH[ $ QHZ IHDWXUH RFFXUULQJ LQ WKH VXSHUV\PPHWULF /LH DOJHEUDV LV WKDW WKH\ PD\ KDYH VHYHUDO QRQHTXLYDOHQW VLPSOH URRW V\VWHPV FRUUHVSRQGLQJ WR GLIIHUn HQW '\QNLQ GLDJUDPV DQG &DUWDQ PDWULFHV ,Q RWKHU ZRUGV WKH GLIIHUHQW URRW V\VWHPV FDQ QRW EH WUDQVIRUPHG LQWR HDFK RWKHU WKURXJK VWDQGDUG :H\O URWDn WLRQV ,QVWHDG WKH\ DUH REWDLQHG E\ SHUIRUPLQJ WKH f:H\Of WUDQVIRUPDWLRQ ZLWK UHVSHFW WR WKH QLOSRWHQW IHUPLRQLF URRW )RU PRUH GHWDLOV VHH )UDSSDW HW DO ZKLFK DOVR SUHVHQWV D ODUJH FROOHFWLRQ RI '\QNLQ GLDJUDPV DVVRFLDWHG ZLWK DOO RI WKH FODVVLFDO FRQWUDJUDGLHQW VXSHUV\PPHWULF FDVHV WKRVH RI WKH DIILQH DQG WZLVWHG DIILQH VXSHUV\PPHWULF DOJHEUDV 1RQHTXLYDOHQW VLPSOH URRW V\VWHPV ZKLFK UHSUHVHQW WKH VDPH VXSHUV\Pn PHWULF /LH DOJHEUD GLIIHU LQ WKH GLVWULEXWLRQ RI ERVRQLF DQG IHUPLRQLF URRWV +RZHYHU KHUH ZH DUH LQWHUHVWHG LQ FRQVLGHULQJ D QDWXUDO H[WHQVLRQ RI WKH 'ULQIHOnG6RNRORY SURFHGXUH WR WKH VXSHUV\PPHWULF FDVH 7KLV UHVWULFWV WKH SRVVLEOH FKRLFHV IRU WKH VLPSOH URRW V\VWHP XVHG IRU EXLOGLQJ WKH VXSHUV\Pn PHWULF /D[ RSHUDWRUVnA 5HFDOO LQ WKH ERVRQLF FDVH WKH P.G9 /D[ RSHUDWRU FRQVWUXFWHG ZLWK WKH JUDGDWLRQ FKRLFH RI 'ULQIHOfG DQG 6RNRORY JHQHUDWHV 7RGD ODWWLFH PRGHOV SRU 68SHU6\PPHWULF DOJHEUDV LW KDV EHHQ VKRZQWKDW 7RGD ODWWLFHV DUH SRVVLEOH RQO\ IRU VLPSOH URRW V\VWHPV FRPSRVHG SXUHO\ RI IHUPLRQLF URRWV 6XSHUV\PPHWULF /LH DOJHEUDV ZLWK SXUHO\ IHUPLRQLF URRW V\V PAGE 82 < WHUQV KDYH EHHQ JLYHQ E\ /HLWHV HW DO r 6/Q Qf 26SP Qf P Q Q Q s f '> Df $f )XUWKHUPRUH WKH LQILQLWHGLPHQVLRQDO DIILQH VXSHUV\PPHWULF /LH DOJHEUDV ZLWK SXUHO\ IHUPLRQLF VLPSOH URRW V\VWHPV DUH 6/Q QfAf 26SQ QfA?' Dfrf $f ZKLOH WKH LQILQLWHGLPHQVLRQDO WZLVWHG DIILQH FDVHV DUH 64Q fA ? 6 /Q ? 26SQ ? QfA. $f 7KH VXSHUV\PPHWULF H[WHQVLRQ RI WKH .G9 HTXDWLRQV ZDV ILUVW GLVFXVVHG LQ 0DQLQ DQG 5DGXO UUn 7KH\ VXJJHVWHG UHSODFLQJ WKH ERVRQLF GHULYDWLYH G[ E\ LWV VXSHUV\PPHWULF DQDORJ LH U? 4 G[A' fÂ§ fÂ§ $f RG R[ 1RWH WKDW 'a 7KH V\VWHP RI PDWUL[ HTXDWLRQV RI 'ULQIHOnG DQG 6RNRORY FDQ WKHQ EH JHQHUDOL]HG WRfrp &UI[f >' 4[f $@[I[f $f ZKHUH $ LV JHQHUDWHG E\ WKH SXUHO\ QHJDWLYH IHUPLRQLF URRWV DQG 4[f LV D *UDVVPDQQ RGG IHUPLRQLF VXSHUILHOG ZKLFK FDQ EH H[SDQGHG DV U 4[Gf nA+LnL!L[f $f Â ZKHUH QRZ +c DUH HOHPHQWV RI WKH &DUWDQ.DF VXEDOJHEUD 7KH YDFXXP FRQn GLWLRQ LV DV EHIRUH $ aLcI^[f $f PAGE 83 6LQFH WKH VHFRQG W\SH RI IHUPLRQLF URRW LV QLOSRWHQW WKH\ GHVHUYH VSHFLDO WUHDWPHQW ZKHQ FRQVWUXFWLQJ F\FOLF ZHLJKW GLDJUDPV 7R LOOXVWUDWH KRZ WKLV FRPHV DERXW FRQVLGHU WKH IXQGDPHQWDO UHSUHVHQWDWLRQ RI WKH VXSHUV\PPHWULF DOJHEUD 26S f 7KH 'YQNLQ GLDJUDP RI WKH SXUHO\ IHUPLRQLF URRW V\VWHP LV JLYHQ E\ ZKHUH ERWK IHUPLRQLF URRWV DUH GHQRWHG DV KDYLQJ ]HUR QRUP DQG ZKHUH ZH KDYH LQGLFDWHG WKH FKRLFH f IRU D KLJKHVW ZHLJKW YHFWRU 7KH &DUWDQ PDWUL[ LV WKHQ HDVLO\ IRXQG WR EH $ $f 7R FRQVWUXFW WKH KLJKHVW ZHLJKW GLDJUDP ZH SURFHHG DV EHIRUH 6LQFH D SRVLWLYH RQH DSSHDUV ERWK LQ WKH ILUVW DQG VHFRQG SODFHV ZH KDYH WZR SHUPXWDWLRQV RI VXEWUDFWLRQ WR SHUIRUP ,Q SDUWLFXODU ZH FDQ VWDUW E\ VXEWUDFWLQJ WKH ILUVW URZ RI WKH &DUWDQ PDWUL[ JLYLQJ f DQG WKHQ VXEWUDFWLQJ WKH VHFRQG URZ UHVXOWLQJ LQ f +RZHYHU XQOLNH WKH ERVRQLF FDVH ZH PD\ QRW VXEWUDFW WKH ILUVW URZ RI WKH &DUWDQ DQRWKHU WLPH IURP WKH ZHLJKW f 7KLV LV EHFDXVH KHUH WKH IHUPLRQLF ZHLJKW YHFWRUV DUH QLOSRWHQW DQG VXEWUDFWLQJ DQ\ &DUWDQ URZ WZLFH JLYHV D GHFRXSOHG VWDWH 6LPLODUO\ ZH FDQ VWDUW E\ VXEWUDFWLQJ WKH VHFRQG &DUWDQ URZ RQFHDQG RQO\ RQFHf DQG WKHQ WKH ILUVW URZ JLYLQJ f 7KXV ZH ILQG WKH ZHLJKW GLDJUDP ZLWK KHLJKW WZR f ? f f ? f $f 7KH GHFRXSOLQJ ZKLFK RFFXUV ZKHQ FRQVWUXFWLQJ D F\FOLF ZHLJKW GLDJUDP IRU DQ DIILQH VXSHUV\PPHWULF DOJHEUD LV DOPRVW DV VWUDLJKWIRUZDUG )RU H[DPSOH LQ )LJ ZH GLVSOD\ WKH SDUWLDOO\ GHFRXSOHG ZHLJKW GLDJUDP RI ^6/^ fAFTf PAGE 84 Ff )LJXUH 7KH f UHSUHVHQWDWLRQ RI 6/ fA Df '\QNLQ GLDJUDP Ef &DUWDQ PDWUL[ Ff &\FOLF ZHLJKW GLDJUDP 7KH GDVKHG DUURZV DUH WKH GHOHWHG URRW ZKHUH VWDWHV ZHUH GHFRXSOHG DV ZH ZHQW IURP WRS WR ERWWRP 7KHUH DUH VHYHUDO ZD\V WR GHFRXSOH WKH UHPDLQLQJ ZHLJKWV VLQFH WKH ORZHULQJ RSHUDWRUV EfÂ DQG VWLOO DSSHDU PRUH WKDQ RQFH 7KH RQO\ ZD\ IRU D F\FOLF ZHLJKW GLDJUDP WR HPHUJH LV E\ GHFRXSOLQJ WKH ZHLJKWV RXWVLGH WKH ER[ 7R VHH WKDW WKLV LV DOVR FRQVLVWHQW QRWH WKDW DOO SDWKV OHDGLQJ IURP ZHLJKW WR ZHLJKWV RU UHTXLUH WZR DSSOLFDWLRQV RI T 7R FRQVWUXFW D VXSHU/D[ RSHUDWRU OHW XV WDNH WKH YDFXXP VROXWLRQ n) :H HDVLO\ ILQG WKH VXSHU/D[ RSHUDWRU WR EH / fÂ§ >' rD@>' A n-@>' A n,O+' n/O@ $f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n :LWK PLQRU PRGLILFDWLRQV WKHVH SURJUDPV FDQ EH DGDSWHG IRU F\FOLF ZHLJKW GLDJUDPV ,W UHPDLQV WR EH VHHQ ZKHWKHU KLJKHU UHSUHVHQWDWLRQV OHDG WR DQ\ QHZ SK\VLFV ,I VR WKHQ D SURJUDP RI FDWHJRUL]LQJ WKHVH UHVXOWV PLJKW EH SXUVXHG WR LGHQWLI\ UHGXQGDQW VROXWLRQV 7KLV PLJKW EH HDVLHU WR DQVZHU IRU VXSHUV\Pn PHWULF DOJHEUDV VLQFH QLOSRWHQF\ SURMHFWV RXW GHFRXSOHG ZHLJKW YHFWRUV :KDW LV FOHDU WKRXJK DW OHDVW IRU WKH QRQVXSHUV\PPHWULF FDVHV LV WKDW WKHVH KLJKHU UHSUHVHQWDWLRQV OHDG WR LQWHJUDEOH V\VWHPV 5HFDOO WR SURYH WKH LQWHJUDELOLW\ RI .G9 V\VWHPV 'ULQIHOfG DQG 6RNRORY IRXQG WKH QHFHVVDU\ LQILQLWH VHW RI FRQn VHUYHG FXUUHQWV WR EH JLYHQ E\ WKH FRHIILFLHQWV RI WKH /DXUHQW H[SDQVLRQ RI & LQ WKH DIILQH SDUDPHWHU $ 2XU FRQFOXVLRQ IROORZV IURP WKH IDFW WKDW HYHU\ UHSUHVHQWDWLRQ RI D EDVLF /LH DOJHEUD KDV DQ DIILQH H[WHQVLRQ DQG WKDW GHILQLQJ SURSHUWLHV RI DIILQH /LH DOJHEUDV DUH UHSUHVHQWDWLRQ LQGHSHQGHQW )LQDOO\ LW ZRXOG EH LQWHUHVWLQJ WR VHH LI RXU SURFHGXUH FRXOG EH PRGLILHG WR GLUHFWO\ JHQHUDWH /D[ RSHUDWRUV LQ RWKHU JDXJHV )XUWKHUPRUH LQ OLJKW RI UHFHQW ZRUNAn RQ JHQHUDOL]DWLRQV RI WKH 'ULQIH)G DQG 6RNRORY VFKHPH RQH PD\ PAGE 86 DOVR FRQVLGHU GLIIHUHQW JUDGDWLRQV RI WKH DIILQH /LH DOJHEUD IURP ZKLFK WR REWDLQ WKH PDWUL[ $ DQG WKH IRUP RI T^[f PAGE 87 5()(5(1&(6 *URVV DQG 3 ) 0HQGH 3K\V /HWW % f 1XF 3K\V % f ) &HUXOXV DQG $ 0DUWLQ 3K\V /HWW f 9HQH]LDQR 1XRYR &LP $ f 3 ) 0HQGH DQG + 2RJXUL 1XF 3K\V % f $ 0DUWLQ 1XRYR &LPHQWR f $ (OLH]HU DQG 5 3 :RRGDUG 1XF 3K\V % f 0 'RXJODV DQG 6 6KHQNHU 1XF 3K\V % f ( %UH]LQ DQG 9$ .D]DNRY 3K\V /HWW % f *URVV DQG $$ 0LJGDO 1XF 3K\V % f 3 +R\HU 1 $ 7RUQTYLVW DQG % 5 :HEEHU 3K\V /HWW % f &KDQ +RQJ0R 3 +R\HU DQG 3 9 5XXVNDQHQ 1XF 3K\V % f 9 $OHVVDQGULQL $PDWL DQG % 0RUHO 1XRYR &LP $ f 3 +R\HU 3K\V /HWW % f 3 +R\HU 1 $ 7RUQTYLVW DQG % 5 :HEEHU 1XF 3K\V % f : =DNU]HZVNL 1XF 3K\V % f 0 6DUELVKDHL DQG : =DNU]HZVNL 1XF 3K\V % f 1XFO 3K\V % f & %DUUDWW 1XF 3K\V % f 0 6DUELVKDHL : =DNU]HZVNL DQG & %DUUDWW 1XF 3K\V % f 0 4XLUV 1XFO 3K\V % f PAGE 88 & %DUUDWW 1XF 3K\V % f 3 +R\HU DQG .ZLHFLQVNL 1XFO 3K\V f 0 4XLUV 1XFO 3K\V % f 3 'L 9HFFKLD ) 3H]]HOOD 0 )UDX +RUQIHFN $ /HUGD DQG 6 6FXLWR 1XFO 3K\V % f 3 'L 9HFFKLD 0 )UDX $ /HUGD DQG 6 6FXLWR 1XFO 3K\V % f $ 6KDSLUR DQG & % 7KRUQ 3K\V 5HY f 0 % *UHHQ 3K\V /HWW % f *ROGVWRQH XQSXEOLVKHG f : 1DKP 1XFO 3K\V % f 7 / &XUWULJKW DQG & % 7KRUQ 1XFO 3K\V % f 7 / &XUWULJKW & % 7KRUQ DQG *ROGVWRQH 3K\V /HWW % f 7 / &XUWULJKW *KDQGRXU DQG & % 7KRUQ 3K\V /HWW % f 7 +RURZLW] 0RUURZ-RQHV 6 3 0DWULQ DQG 5 3 :RRGDUG 3K\V /HWW % f 0 'RXJODV 3K\V /HWW % f 9 'ULQIHOG DQG 9 6RNRORY 6RY 0DWK f 3 'L )UDQFHVFR DQG .XWDVRY 1XFO 3K\V % f 3 'L )UDQFHVFR DQG .XWDVRY 3ULQFHWRQ SUHSULQW 3837 f %HVVLV & ,W]\NVRQ DQG -% =XEHU $GY $SSO 0DWK f 0 / 0HKWD &RPPXQ 0DWK 3K\V f 6 &KDGKD 0DKRX[ DQG 0 / 0HKWD 3K\V $ f 9 $ )DWHHY DQG 6 / /XNf\DQRY ,QW 0RG 3K\V $ f 6 / &DUERQ DQG ( 3LDUG 0DWK 3K\V f 5 6ODQVN\ 3K\V 5HS f ( % '\QNLQ $PHU 0DWK 6RF 7UDQV 6HU f DQG PAGE 89 0 5 %UHPQHU 5 9 0RRG\ DQG 3DWHUD 7DEOHV RI 'RPLQDQW :HLJKW 0XOn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n WHUZDUGV KH FDPH WR WKH 8QLYHUVLW\ RI )ORULGD DQG VXEVHTXHQWO\ EHJDQ GRLQJ UHVHDUFK XQGHU WKH VXSHUYLVLRQ RI 3URIHVVRU &KDUOHV 7KRUQ +LV UHVHDUFK LQWHUn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Â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|