> >@=%` >bjbj .h̟̟6l4/0d///////$O0h2>/ >/S/--- /- /---$0GoQx&>-/i/0/-3(X3-3-8Z@-.4bl>/>/,/ Breakdown of Special Relativity at the Length Scale below Subhadronic Region
E. J. Jeong
Department of Physics, The University of Michigan, Ann Arbor,
Michigan 48109
Abstract
According to the Einsteins mass energy relations, EMBED Equation.3 , EMBED Equation.3 is a fixed constant with no known parametric dependences. However, the recent study of the scale dependent mass relations suggests that the rest mass of the elementary particles may not be, in general, constants. Although special relativity was developed as a theory with no interactions present, the notion that the rest mass is an indisputable constant and there is only the corresponding interaction potential may not be justified. In the conventional field theoretical consideration, the interaction potential energy is not directly translated into the rest mass of the particles. However, in the recent result of the study of the quark confinement problem which can easily be explained by the consideration of the scale dependent rest mass, it is suggested that the interaction potential energy may represent the major part of the rest mass of the elementary particles.
1. Introduction
There have been many suggestions and suspicions about the validity of special relativity in the small length scale (3). However, the experimental results have been in good agreement with the theoretical predictions that such considerations have largely been ignored. In this connection, we have discussed and explained the quark confinement problem using the scale dependent mass relations (2) without any specific contradictions (4). Since there is no such known phenomenon as the scale dependent variation of the rest mass of a particle in special relativity, we suspect that special relativity may be an approximation of a theory that incorporates the scale dependencies in it.
2. Einsteins Mass Energy Relation
Einstein has published the theory of special relativity (1) in 1905. One of the results of his theory was the mass energy relation given by EMBED Equation.3 . [1]
The validity of this relation has been verified with great accuracy (5). The above equation can also be written
EMBED Equation.3 , [2]
where EMBED Equation.3 is tacitly assumed to be a constant. Since the mass energy relation was initially developed for the macroscopic objects, there was no question regarding the constancy of the rest mass [6]. Even in the strong gravitational field, the rest mass would be a constant and according to general relativity, it is the warping of the space time that causes the motion of the stellar objects. However, there are two unnatural factors about the equation [2].
1. There is no room for the varying potential energy.
2. There is no concept of the interaction between the particles which is thought, in general, to be related to the energy of the particles under investigation.
In fact, 1 and 2 are closely related to each other and make us feel that EMBED Equation.3 may well not be a constant, so that if special relativity has to be broken down, it will start with the redefinition of EMBED Equation.3 . In most of the cases of the non interacting particles, there is no issue of the contradiction, however, since we have observed this clue in our previous notes (2) (4), we claim that EMBED Equation.3 is not a constant contrary to the tacit assumption of special relativity as it was originally formulated.
3. Definition of the Scale
We define the scale as an arbitrary quantity with the dimension of mass. We say the scale is large if the interacting charges are close together and small if they are far apart. Therefore, the scale is, in general, dependent on the relative distance between the two interacting charges. And also the scale depends on the relative momentum of the charged particles due to the uncertainty relation
EMBED Equation.3 [3]
In essence, our definition of the scale is the same as the one commonly named momentum scale (7). It is emphasized that when we mention the scale, we have in mind the scale dependent running coupling constant which has been derived using the renormalization group equations(8).
4. Scale Dependent Rest Mass Formula
We have shown in the previous notes(2) that the rest mass of a particle can be written for an electron,
EMBED Equation.3 , [4]
and for a quark
EMBED Equation.3 , [5]
where the constancy of EMBED Equation.3 has been demonstrated in our discussion on the quark confinement problem(4). The definitions [4] and [5] were derived using the first loop self energy diagram(2) in the limit EMBED Equation.3 , EMBED Equation.3 , EMBED Equation.3 . Therefore, we effectively singled out the pure self energy contribution which does not include the kinetic energy. We call this mass the potential mass to distinguish it from the conventional rest mass. In the macroscopic scale, the potential mass is approximately the same as the rest mass.
Since the coupling constants in quantum field theory depend, in general, (7) on the arbitrary scale parameter EMBED Equation.3 as shown from the study of the renormalization group equation(8), the relations [4], [5] indicate that the potential masses are not in general constants.
We have shown in the previous notes (4), how this scale dependent rest mass concept can be used to explain the problem of the quark confinement. Since this potential mass excludes the kinetic portion of the energy involved, we claim that it should be in fact the conventional rest mass in special relativity. Therefore, we are confronted with a contradiction between special relativity and the running coupling constant in quantum field theory. In fact, we have two choices either to discard the concept of the running coupling constant as false or to revise special relativity on the ground that the constant rest mass term is unnatural. We take the view that special relativity needs modification despite all the beauty and successes of its predictions. The reasons for this choice are;
1. The concept of the running coupling constant explained the asymptotic freedom (9).
2. There exists experimental evidence that the coupling constants do depend on the scale (10).
3. The quark confinement problem can be explained easily by using the concept of the scale dependent rest mass (4).
One may say that the concept of the running coupling constant is all right but the relation [4[, [5] are wrong. However, it must be noted that the conventionally known potential energy can not be translated into a mass in the classical treatment of the problem, the concept of which was essential for the explanation of the quark confinement.
We propose here a set of the general rules that the concept of the scale dependent rest mass brings into the new physics.
5. Postulates
(1) The potential mass of a charged particle depends on the coupling constants the particle is subjected to.
EMBED Equation.3
(2) All the coupling constants of the renormalizable gauge theories depend on the scale.
(3) The scale is an unknown function of the relative 4 coordinates or the 4 momentum between the given interacting charges.
(4) The total energy of the system in consideration is independent of the scale.
One can rephrase (1), (2), (3) simply, The potential mass of a charged particle, in general, depends on the relative 4 coordinates or the momentum between the interacting charged particles.
6. Modification to Einsteins Mass-Energy Relation
Suppose that a particle m is subjected to a coupling constant g, then, we can write the total energy of the system, according to our postulate
EMBED Equation.3 [6]
where EMBED Equation.3 is the potential mass and EMBED Equation.3 is the spatial momentum. The EMBED Equation.3 dependence of EMBED Equation.3 is a necessary consequence of our postulates. For a macroscopic body, the coupling constants involved are usually more than one. Even for an electron, we may not know how many coupling constants are involved depending on the composite nature of its substructures. The only possible guiding principle is the constancy of EMBED Equation.3 (4).
Therefore, we can write the energy equation for an electron
EMBED Equation.3 [7]
and for a quark
EMBED Equation.3 [8]
If we suggest that the scale dependent running coupling constant is valid beyond the perturbation calculation limit (7), we may write for an electron
EMBED Equation.3 [9]
and for a quark, if the electric contribution can be ignored,
EMBED Equation.3 [10]
Where EMBED Equation.3 and EMBED Equation.3 are the coupling constants at EMBED Equation.3 respectively, EMBED Equation.3 and EMBED Equation.3 , EMBED Equation.3 and EMBED Equation.3 are the group structure dependent constants (7).
Now, we have two nontrivial energy equations, which have additional parameter EMBED Equation.3 dependence. Equation [9] shows that in the electron-electron collision experiment, the potential mass of each of the electrons increase when they come close together, which will be subsequently reduce the kinetic mass( defined in [13]) of the electron at the same time. Equation [10] says instead that in the quark-quark interaction case, the potential mass of each quark decrease when they come close together, which is in the asymptotic region, however, if they are about to be separated wide, the potential mass increase drastically (4), which is the cause of the quark confinement. According to the equation [9], the increase of potential mass is observable only when the two charges are close together, which may well have escaped the experimental discrimination. Therefore, rhw success of QED does not violate the equation [9]. On the other hand, the asymptotic freedom and the quark confinement are the natural consequences of the equation [10]. To see the scale variation dependence of EMBED Equation.3 , we take a partial derivative over EMBED Equation.3 on the equation [6].
EMBED Equation.3 [11]
which shows that as the coupling constant increases, the corresponding momentum decreases along the variation of the scale. Leaving behind the many potential applications of this result on the cosmological problems, we define for convenience,
EMBED Equation.3 = potential mass [12]
EMBED Equation.3 = kinetic mass [13]
In terms of these definitions, the equation [6] can be written,
EMBED Equation.3 [14]
EMBED Equation.3 , [15]
where M is the usual relativistic mass, which doesnt depend on the scale. As we have suggested in the beginning, the unnaturalness of the constancy of the rest mass and that there is no room for the potential energy variation are remedied by this prescription. We may be able to state a law at this point A particle moves in space toward the region where the potential mass is the minimum.
7. Conclusions
Since the equation [6] is valid for a particle charged only with one type of interaction and the corresponding coupling constant, a generalization is necessary. For a particular case of the proton and the neutron, one can express the total energy by the sums of the three quarks potential mass and the kinetic mass contributed from both the color and electric charges respectively. However, we doubt the necessity of describing a macroscopic body in terms of quark coupling constant. At the level of the gravitation, the only charge that matters would be the mass of the macroscopic body. In this case we can write,
EMBED Equation.3 EMBED Equation.3 [16]
where EMBED Equation.3 is the rest mass and EMBED Equation.3 is the potential mass contribution due to the external gravitational interaction and EMBED Equation.3 the momentum due to the external interaction of the gravitation. EMBED Equation.3 include the internal gravitational potential mass and also the kinetic mass coming from the internal motion of the constituents. An easy place to test the theory would be in the subhadronic behavior of the quarks using the equation [10] and [11].
There remain several questions regarding on how the Lorentz transformation has to be reconciled with this variation and how this result will affect quantum field theory especially in the renormalization program. And also finally how general relativity will be affected by this result? We leave these questions for the future research.
References
(1) A. Einstein, Annln. Phys. 17 (1905a) 891. A. Einstein, Annln. Phys. 18 (1905b) 639.
(2) E. J. Jeong, A method of radiative correction 1984
(3) R. Hofastadter, International Symposium on Lepton and Photon Interactions at High Energy (Stanford, Calif., 1975) 869; IL-T. Cheon, Lett, Nuovo Cim. 2b (1979) 604.
(4) E. J. Jeong, Quark Confinement 1984
(5) N. M. Smith, Jr., Phys. Rev. 56 (1939) 548.
(6) A. Einstein, The meaning of relativity, Princeton University Press (1946)
(7) P. Ramond, Field Theory: A Modern Primer, The Benjamin/Cummings Publishing Company (1981).
(8) Gell-Mann and Low, Phys. Rev. 95 (1954) 1300; C. Callan, Phys. Rev. D5 (1972) 3202; K. Symanzik, Comm. Math. Phys. 23 (1971) 49; t Hooft, Nucl. Phys. B61 (1973) 455; S. Weinberg, Phys. Rev. D8 (1973) 3497; S. Coleman, Lecture Erice summer school (1974) 75; L. Wilson, Revs. Modern Phys. 47 (1975) 773; E.C.G. Stueckelberg and A. Peterman, Helv. Phys. Acta 26 (1953) 499; K. Wilson, Phys. Rev. D3 (1971) 1818.
(9) D. Gross and P. Wilczek, Phys. Rev. Lett. 26 (1973) 1343; H. D. Politzer, Phys. Rev. Lett. 26 (1973) 1346.
(10) W. K. H. Panofsky, in Proc. 14th Int. Conf. on High-Energy Physics, Vienna, Austria, September 1968; P. Carruthers, Phys. Reports 1C (1971) 30, references.
5=>OP P Q i
bqr9<h:9hTlh^yhH}hbh?_hU2hhuLjh8ihhuLEHUjܾ0L
hhuLCJUVaJhjh8ihhuLEHUj0L
hhuLCJUVaJjh8iUhzth8i:MNOP\DEhigd8igdi+>DEhi
,39^9쵨왌ĈĈĀqdjh h EHUj0L
h CJUVaJjh UhNj h;hqgEHUjW0L
hqgCJUVaJjh;h;EHUj0L
h;CJUVaJhqgjh8ih;EHUj0L
h;CJUVaJjh;Uh;h h:9h8ihTl'`a67\]vw,<bgdi+9:MNOP 5678KLMN\]+<=P{njh ajch~Wh
EHUj.L
h
CJUVaJjh
Uh
h8ij
hU2hU2EHUj0L
hU2CJUVaJjhU2UhU2h;hqgjh h EHUj
h h EHUj0L
h CJUVaJh jh U'PQRSUayz@ATUVWXYZmnoprsǸǧuhYjlw0L
haQCJUVaJjhKghaQEHUjgm0L
haQUVhi+haQjhKghaQEHUjTm0L
haQUVjhaQUhaQjh1chaQEHUj0L
haQCJUVaJh1cjh1cUh ah
jh
Ujh~Wh
EHUjw0L
h
CJUVaJ"#6789A B !!!r"""""""J#K#j#k#~#######W$X$Y$$$$$% %k%l%¾¾⾺h8ihU2h9j>$hhEHUjs1L
hCJUVaJhjhUh ah[hhNjI"haQhaQEHUjc0L
haQCJUVaJh
haQjhaQUj hi+haQEHU2B !!q"r""""j####X$Y$$$m%n%o%%%2&3&T&U&[(\((gdi+l%m%o%%%%3&4&G&H&I&J&[&\&o&p&q&r&&&&&&&&&&&&&&&&&&&-'{'Ÿqd`hrj.hhEHUj1L
hCJUVaJj,hhEHUj*1L
hCJUVaJjt*hhEHUj1L
hCJUVaJjQ(h}DhEHUjԶ1L
hUVj%hhEHUjƶ1L
hCJUVaJjhUhh ahzt%{''(((?(@(S(T(U(V(Z([(o((((((((((((((((((j)))))))̵̤̤̄̄̕|mj2L
h0+CJUVaJjh0+Uh0+j76hyChyCEHUj~2L
hyCCJUVaJhztj&3hyChyCEHUj}2L
hyCCJUVaJjhyCUhyChj0hpfhuEHUj2L
huCJUVaJjhpfUhrhpf%(((((((()))))))*
**#+$+////00!1R1S11gdi+))))))********(*)***+*/*0*C*D*E*F*d*e*x*y*z*{*****淪梞sf[j#2L
huUVj6EhuhuEHUjՁ2L
huCJUVaJj/Ch0+huEHUj2L
huCJUVaJhujhuUj(Ah0+huEHUj2L
huCJUVaJj<h0+h0+EHUjU2L
h0+CJUVaJhzth0+jh0+UjV9h0+h0+EHU!*********************"+$+r+s+++++/k/l//滮柒掊sfbZbjhUhjOhCyhCyEHUj3L
hCyCJUVaJjhCyUhCyhztjMhuhuEHUjh2L
huCJUVaJjKhuhuEHUjP2L
huCJUVaJjIhuhuEHUj02L
huCJUVaJhujhuUjnGh2huEHU //////////////////0011111 1!1"151617181R1Ȼר{s{dWs{jD[h)hv EHUjg3L
hv CJUVaJjh)Uh)jXhh)EHUj3L
h)CJUVaJjVhhEHUj3L
hCJUVaJhztjThhEHUj03L
hCJUVaJhjhUjQhhEHUj3L
hCJUVaJ R111111111111111112g3h3i3y3z3555555566666 6!6&6ЬШРqd`hNqjehR~rhR~rEHUj5L
hR~rCJUVaJj5chR~rhR~rEHUj5L
hR~rCJUVaJhR~rjhR~rUh7bhhM;Qj`h?hI2EHUj3L
hI2CJUVaJhztj]hI2hI2EHUj3L
hI2CJUVaJhI2jhI2Uhv %11111h3i3j3y3z3556 6&8'8v9w9x9y9z9{9|99999::gd.gdi+&6'6:6;6<6=6R6S6f6g6h6i666666677*7+7,7-7.75777%8&8'809L9u9v9|9999Ȼwsssoskgcohhhzth.h}rSjmhR~rhNqEHUje5L
hNqCJUVaJjkhNqhNqEHUj5L
hNqCJUVaJhNqjhNqUjxihR~rhNqEHUj5L
hNqCJUVaJjqghR~rhR~rEHUje5L
hR~rCJUVaJhR~rjhR~rU&99999::::z::::::::!;";r;s;;;;;.</<<<<<<<=0=~==>>>>hV4hV4H*hV4h.h4C(:::::!;";r;s;;;====>>,1h/ =!"#$%Dd
0b
c$A??3"`?20vʝomeD`!0vʝomeHxڥS=KPﾘ4RD`A覈T"*
K*/PGqҭC']
k).
wKU?<8{}Ih61BXA Q?k
kvEhћ
jeJbgm&4aO#L/n.[.-$Dh>[{&fmUjX#Sg$:To1꽻fPQm̦{yE'P·L#<7̸g.-nYp!3\J
[K|"鰯#8ڠ+-ٟ
d1G҉|aK0'<2w
y) >oC{EC=8o7d?|q
pKN!٥obv=8NؕN]4|K;z
Dd
@hb
c$A??3"`?2T8ݶo u0`!(8ݶo u@ |xcdd``> @c112BYL%bpuobAi,@u@@ڈc ؽc$;vv0o8L+KRsA2u(2tA4Ag!!v120e8KDd
0b
>
c$A-??3"`?20vʝome`!0vʝomeHxڥS=KPﾘ4RD`A覈T"*
K*/PGqҭC']
k).
wKU?<8{}Ih61BXA Q?k
kvEhћ
jeJbgm&4aO#L/n.[.-$Dh>[{&fmUjX#Sg$:To1꽻fPQm̦{yE'P·L#<7̸g.-nYp!3\J
[K|"鰯#8ڠ+-ٟ
d1G҉|aK0'<2w
y) >oC{EC=8o7d?|q
pKN!٥obv=8NؕN]4|K;zmDd
|b
?
c$A.??3"`?2:,JZzHWˍ`!:,JZzHWˍ `8>0Yxcdd``dd``baV d,FYzP1n:&n! KA?H1ā깡jx|K2B*RvfRv,L!
~
Ay+V}.
_']aF\_ҹ@Z*a|qNT>ooaAc!is#IadZ8J߄/-YQ[A\8ߍwD'F .Ff_Pԃ
Ma`x5}G!p3\p4C^NSLLJ%W@0u(2tA4T}b@3XYDd
@hb
@
c$A/??3"`?2QԴ Kl}RP-5
`!%Դ Kl}RP@ |xcdd``> @c112BYL%bpuobM|i,@u@@ڈc Xg~%ļ?m`b[u=؇`!v 0y{aĤ\Y\ 2Cb>1,M JDd
@hb
<
c$A+??3"`?2P0$9uIbٸX>F,<`!$0$9uIbٸX>F@ |xcdd``> @c112BYL%bpuob;ҼXt9@Jy
3Dh2!z.C``ÄI)$5d.P"CDHg!!v120eL`Dd
@hb
c$A ??3"`?2P0$9uIbٸX>F,B`!$0$9uIbٸX>F@ |xcdd``> @c112BYL%bpuob;ҼXt9@Jy
3Dh2!z.C``ÄI)$5d.P"CDHg!!v120e
!"#$%&'()*+,-./012346789:;<?BlCDFEGIHJLKMNOQPRSTVUWYXZ\[]_^`abecdfghjikmnopqsrutvxwy{z|}~Root Entry FroQAGData
5oWordDocument.hObjectPool^?@oQroQ_1278263045'F?@oQ?@oQOle
CompObjfObjInfo "%(+./2569>CHMRUVY\_dinqruxyz}
FMicrosoft Equation 3.0DS EquationEquation.39q>uL|
E=mo
c2
/
1"v2
c2Equation Native _1278263004F?@oQ?@oQOle
CompObj
f
FMicrosoft Equation 3.0DS EquationEquation.39q>u
m0
FMicrosoft Equation 3.0DS EquationEquation.39qObjInfo
Equation Native 6_1278265329F?@oQ?@oQOle
CompObj
fObjInfoEquation Native _1278265431F?@oQAoQ>o
E2
=mo2
c4
+p2
c2
FMicrosoft Equation 3.0DS EquationEquation.39q>ď
moOle
CompObjfObjInfoEquation Native 6_1278266120FAoQAoQOle
CompObjfObjInfo
FMicrosoft Equation 3.0DS EquationEquation.39q>"
mo
FMicrosoft Equation 3.0DS EquationEquation.39qEquation Native 6_1278266881OFAoQAoQOle
CompObj fObjInfo! Equation Native !E_1278129921$FAoQAoQOle
#>)uA
xpe"!/2
FMicrosoft Equation 3.0DS EquationEquation.39qaJ
Me
=CompObj#%$fObjInfo&&Equation Native '}_12782448391 )FAoQAoQe2
82
Ce
FMicrosoft Equation 3.0DS EquationEquation.39qs̽
Mq
=C3
g2
82
CqOle
)CompObj(**fObjInfo+,Equation Native -_1278267853.FAoQAoQOle
0CompObj-/1fObjInfo03
FMicrosoft Equation 3.0DS EquationEquation.39q>%
Ce
=Cq
=2.22971013
cm"1Equation Native 4_1278242132"63FAoQPLCoQOle
7CompObj248f
FMicrosoft Equation 3.0DS EquationEquation.39q#xiLX
m0
!0
FMicrosoft Equation 3.0DS EquationEquation.39qObjInfo5:Equation Native ;?_1278242151;8FPLCoQPLCoQOle
<CompObj79=fObjInfo:?Equation Native @1_1278244716=FPLCoQPLCoQԃ
n!4
FMicrosoft Equation 3.0DS EquationEquation.39qLA
2p!0
FMicrosoft Equation 3.0DS EqOle
ACompObj<>BfObjInfo?DEquation Native E5_1278268259,JBFPLCoQPLCoQOle
FCompObjACGfObjInfoDIuationEquation.39q>
QC
FMicrosoft Equation 3.0DS EquationEquation.39qxflt
Equation Native J)_1278326643GFPLCoQPLCoQOle
KCompObjFHLfObjInfoINEquation Native O#_1278326470LFPLCoQPLCoQOle
P
FMicrosoft Equation 3.0DS EquationEquation.39qHwA
E2
=Fg()()2
c4
+p2
()c2CompObjKMQfObjInfoNSEquation Native T_1278326484@YQFPLCoQPLCoQOle
WCompObjPRXfObjInfoSZEquation Native [H
FMicrosoft Equation 3.0DS EquationEquation.39q,PN<
Fg()()
FMicrosoft Equation 3.0DS EquationEquation.39q_1278326517VFPLCoQPLCoQOle
]CompObjUW^fObjInfoX`yo
p()
FMicrosoft Equation 3.0DS EquationEquation.39q
GA
Equation Native a5_1278326570TE[FPLCoQPLCoQOle
bCompObjZ\cfObjInfo]eEquation Native f)_1278326655c`FPLCoQPLCoQOle
gCompObj_ahfObjInfobjEquation Native k)_1278378513reFPLCoQPLCoQ
FMicrosoft Equation 3.0DS EquationEquation.39q
H L
p
FMicrosoft Equation 3.0DS EquationEquation.39qOle
lCompObjdfmfObjInfogoEquation Native p h,Y
Ce
=Cq
=2.22971013
cm"1
FMicrosoft Equation 3.0DS EquationEquation.39q
E2
=_1278377457jFPLCoQPLCoQOle
sCompObjiktfObjInfolvEquation Native w_1278377648hwoFPLCoQPLCoQOle
{CompObjnp|fe2
()82
Ce
()2
c4
+pe2
()c2
FMicrosoft Equation 3.0DS EquationEquation.39q|\
E2
=ObjInfoq~Equation Native _1278378368mtFPLCoQPLCoQOle
C3
g2
()82
Cq
()2
c4
+pq2
()c2
FMicrosoft Equation 3.0DS EquationEquation.39qCompObjsufObjInfovEquation Native g_1278378325yFPLCoQPLCoQKPD
E2
=eo2
1"eo2
62
lno
Ce
82
()()2
c4
+pe2
()c2
FMicrosoft Equation 3.0DS EqOle
CompObjxzfObjInfo{Equation Native uationEquation.39qiA
E2
=C3
go2
1+go2
82
113Cad
"43Cf
()lno
Cq
82
()()2
c4
+pq2
()c2
FMicrosoft Equation 3.0DS EquationEquation.39qPz
eo_1278378388~FDoQDoQOle
CompObj}fObjInfoEquation Native 6_1278378419|FDoQDoQOle
CompObjf
FMicrosoft Equation 3.0DS EquationEquation.39qd
go
FMicrosoft Equation 3.0DS EquationEquation.39qObjInfoEquation Native 6_1278378453FDoQDoQOle
CompObjfObjInfoEquation Native >_1278378531FDoQDoQ"P
=o
FMicrosoft Equation 3.0DS EquationEquation.39q]
Ce
=Cq
=2.22971013
cm"1Ole
CompObjfObjInfoEquation Native _1278378544FDoQDoQOle
CompObjfObjInfo
FMicrosoft Equation 3.0DS EquationEquation.39q(X
C,ad
FMicrosoft Equation 3.0DS EqEquation Native D_1278378576FDoQDoQOle
CompObjfuationEquation.39qԉ
Cf
FMicrosoft Equation 3.0DS EquationEquation.39qPT
C3ObjInfoEquation Native 6_1278378600FDoQDoQOle
CompObjfObjInfoEquation Native 6_1278463112FDoQDoQOle
CompObjfObjInfoEquation Native )
FMicrosoft Equation 3.0DS EquationEquation.39q
tA
FMicrosoft Equation 3.0DS EquationEquation.39q_1278463733FDoQDoQOle
CompObjfObjInfohsB
p
FMicrosoft Equation 3.0DS EquationEquation.39q
h
Equation Native 6_1278463792FDoQDoQOle
CompObjfObjInfoEquation Native )_1278463946FYFoQYFoQOle
CompObjfObjInfoEquation Native _1278464257FYFoQYFoQ
FMicrosoft Equation 3.0DS EquationEquation.39qvPgA
P"P"="C2
F"F"g"g"
FMicrosoft Equation 3.0DS EqOle
CompObjfObjInfoEquation Native juationEquation.39qN
Fg()()=Mo
()
FMicrosoft Equation 3.0DS EquationEquation.39q_1278464359FYFoQYFoQOle
CompObjfObjInfoEquation Native i_1278464721FYFoQYFoQOle
CompObjfM`%
P()c=Mo'
()
FMicrosoft Equation 3.0DS EquationEquation.39qp
E2
=M2
c4
=Mo2
(ObjInfoEquation Native _1278464699FYFoQYFoQOle
)c4
+M'
o2
()c2
FMicrosoft Equation 3.0DS EquationEquation.39qw$ą
M2
=Mo2
()+M'
oCompObjfObjInfoEquation Native _1278574578FYFoQYFoQ2
()
FMicrosoft Equation 3.0DS EquationEquation.39q`r
E2
=Mo2
c4
+Mog2
()c4
+pg2
(Ole
CompObjfObjInfoEquation Native )c2
FMicrosoft Equation 3.0DS EquationEquation.39qvA
FMicrosoft Equation 3.0DS Eq_1278574535FYFoQYFoQOle
CompObjfObjInfoEquation Native #_1278574693F0GoQ0GoQOle
CompObjfuationEquation.39qP~
Mo
FMicrosoft Equation 3.0DS EquationEquation.39q+>
Mog
(ObjInfoEquation Native 6_1278574759FYFoQYFoQOle
CompObjfObjInfoEquation Native G_1278574834FYFoQ0GoQ)
FMicrosoft Equation 3.0DS EquationEquation.39q'A
Pg
()Oh+'0Ole
CompObj
fObjInfoEquation Native
CL`Dd
@hb
!
c$A!??3"`?2P0$9uIbٸX>F,H`!$0$9uIbٸX>F@ |xcdd``> @c112BYL%bpuob;ҼXt9@Jy
3Dh2!z.C``ÄI)$5d.P"CDHg!!v120eL`YDd
@b
"
c$A"??3"`?2œ5(3"dN`!wœ5(3"d Excdd``vdd``baV d,FYzP1n:&B@?b u
ㆪaM,,He` @201d++&1X|#+|-×fTTgUXA|J,3
Qp()Jd:@${inpenR~>ge7<3ȈpoHRxag3Ȅ=`+*Qp@JNLLJ%v {: @>1,F~+Dd
(b
3
c$A"??3"`? 2/[f:D`!/[f:D^@ drxڥR=KA\.g-D,"lX4V<H \'Xiec&,X' &h.ͼ772K@+%wG3a$
s73ˢgN^{!`I~#QL2dTmVº1^Pgв^cZ{}$ڳWy*r@4vkAՕxj9A_k2Ut_)S#ekӷ飪aM
ƧÄ7Bɾ}i_4Hh|8[5|?W~Ibsڻ!`vqczz?^;!IlDWMz^9H+SId2\:z @c112BYL%bpucQu\e.[`m|
&^Fc?@!{2Lt |pd(F"X@&`Ss Z=\Ĥ\Y\ːCb>1,Zĩ~Dd
h0
6
#A%2~j
lZ^`!Rj
lB@8| xcdd``Vdd``baV d,FYzP1n:&&V! KA?H1ZX
ㆪaM,,He`H @201d++&1X|#+|-LʠT1TBSOy 梸 dnpenR~C'ЖB8 1^#&0eȽ
V r+p0@ZCF&&\@]
@p2}qDd
0
7
#A&
2n*F[3RKJ``!B*F[3RK dHxcdd`` @c112BYL%bpu7sSDd
T6
2
3A!?2WHa$
b$`!WHa$
b
XJkxcdd``d!0
ĜL0##0KQ*
Wä2AA?H1Zc@øjx|K2B*R\``0I3Dd
b
c$A??3"`?2
˃#x1Y4%`!˃#x1Y4*``X)$xxcdd``eb``baV d,FYzP1n:A! KA?H1
@P5<%!`31{vL,LI9x2-
1X|#+|-̛TT1.fiը$dm^cdU 5W?\!
~
Ay8 1j<F0"/*70,@c!jG{џ_/$\a\lp#Es2p{x@8ߙU^O'0np͝7!+dn.ܞYl>xAO9
' ķ*.pG?4O 0y{yI)$5dP"CXP!ĀGf~)>#Dd
pT0
#A2ːh#OGԑ{(`!sːh#OGԑ XJAxcdd``ed``baV d,FYzP1n:&V! KA?H1Z@eǀqC0&dT201
@201W&00<|@aŪXk34YB]j mU vF%!^r16D@2.M
2
M-VK-WMNPs}ݏL[@f̥&UrAccv0o8021)W2ȂaPdk17-Dd
0@b
c$A??3"`?2w+O>$$x?$S*`!K+O>$$x?$k xcdd`` @c112BYL%bpu
1X|#+|-ɛTT@ͭv #T`+KRs@1u(2t5t,L07Dd
b
c$A??3"`?2>ӗ<צeRjL.`!ӗ<צeRjL:Rxcdd`` @c112BYL%bpuobox6si#.,J.-\`0M!I)$5dP"CXHB;3X?58Dd
b
c$A??3"`?25SguX4@W0`!5SguX4@W@hxcdd``> @c112BYL%bpuXQ=ε
>2b -Hb̡g(dsa|/fp(S$Ȅ\lbj.4r48@S;+KRs4A0u(2tA4T}b@33XlvDd
b
c$A??3"`? 2[\X}np|ǈ@ϴ7j3`!/\X}np|ǈ@ϴ 8xcdd``Ng2
ĜL0##0KQ*
W ,d3H1)fY[Pr+PT
obIFHeA*p
t!f0-`ek wfRQ9A $37X/\!(?71aŪXkyWZ'LF]F\!LEB F0W>T#nP
,<
(Ɗ?yeAFhV3A|
(_<o3#g
Q.F.&ݕw8<|qnTy6@Sn k%z|3R~ ̷I`ȞܤyL\
x)cE~&O6w(*bW0Q],~<Ù@|J.hl;\021)W2tԡ"b>193X?=[^Dd
b
c$A??3"`?
2i?,',sE{6`!=?,',sp 8xڥKA]U7BXLA3
?H!R`-IH'"R&Z?Pp3ofnؽ(ĸ0ss~|0HX%4=5bb!cbZ#_P/Ⅾjmk8؊/43g&[&>+xϼ$ா\y{}HHږ
C|犅ʴOO`Qд}[2(Dd
@b
c$A??3"`?2Ҝ"if0T9`!Ҝ"if0T` M(xڥKhAgfgw'yѶx"ZDX+ŀrV#Mi#a=yDT As ZXLq^;
>Ͽ6D0N`Ǥ' A(%6*?]Yq$(רtZ/-DELCjl3EnDAI]TzA%'A|B2~wvp,P-@ba.sPɌxh̡/W嚨xC]|MKTa$WK_[e;U @c112BYL%bpuobXt9 g#7lnj%\`2M% #RpeqIj.C\E. ,s;.FfMyDd
,hb
c$A??3"`?2QBXj-h<-sC`!%BXj-h<@|xcdd``> @c112BYL%bpu䂆8$C``ÞI)$5Ad.P"CXYO`cQjDd
0
#A2IkT_
༥G`!IkT_
༥Yhxcdd``> @c112BYL%bpuob:Yt9?VJ?ȉg>#gc'P~#;H9L";=#?O@. Ma`8ղNB\~FX|b6v]j.[7em|q&^Fc?@!3Lt |pd(FY@&F`Ss Z=\Ĥ\Y\ C.<b@33X6ɮDd
|b
c$A??3"`?2_'F$s9\>;J`!3'F$s9\>`0xcdd`` @c112BYL%bpu`55Qds%4T!6b;LLJ%A2u(2t5=;aDd
T|b
c$A??3"`?2Q09wJvz-1L`!%09wJvz `XJ0xcdd``> @c112BYL%bpu @c112BYL%bpu @c112BYL%bpudKF&&\/ s: @>1,MDd
b
c$A??3"`?2:Uubp!r1TT`!Uubp!r1:Rx=NOq}3R=I8|_"EioNr}7_Mf~{00/!e1Q0TT|yYӚ\AdG泠He"Nr#/^5ƕ{`wcu*55ƅ]rZz`zؚAJ@s/#0 26*0AoSy8HDd
b
c$A
??3"`?2D51uWg>DV`!D51uWg>v
&dxڕR;KA-D,.v R
Slp> g BJG%X+&>a
fٙo,
@#9'LC!!."8@2~
I&` {0O
qctX-vxPf2K=eC9ubzwu>76UD+e[N|&߸bf,~5ZNJ&,m+8߁e{| %CHS(fP.q֩-m-68ķY|}xcdd`` @c112BYL%bpu8Y@Ӵ;W+KRsnePdk1cDd
lb
c$A??3"`?2|]3ӈfQ[`!|]3ӈfQ~ (xcdd``$d@9`,&FF(`Ts A?dmbEәzjx|K2B*Rj8 :@u!f0109Y@V%3ȝATN`gbM-VK-WMcXsF V0Z~3Mb&#.# 0;3>gBgaCWOL`}c262
Ma`Q 2\0BgZ~" |fnOP%=+'pCOCf\ДNhf%F&&\E@]`
"HRDd
b
c$A??3"`?2:V-'ϳy$hg^`!:V-'ϳy$hg2!@CxڥJALl6&T,V/E``$QW&@.F-F,bE!>l'F0?30o`)q'y®h
y""OML$`\qM]VKlA&_5copY pLM!nllXUch]%͘{wZ~7CCO&RthDDˌ'D)Nr~) "3YbpR<%8:/ׁDt-ӈQ~d4$뗊}Nퟎ9={>Y,4gbX=G|'}t))/
^ǜUuk轭8=70:eC1u>fu^)Dd
b
c$A??3"`?24_ꑹn``!4_ꑹn͢`z@CxڥS=KP=3i%Ct(YbDXɂ8 \qw(I0ABݓ{; Bē`1e4G#eYur
ad9茯RXLɢ\omV]IO%X"ΐnN^qjí˞h/7(y{G>.d1/$U^ buh?#(:UZxRD8J
G >km0N߭ ݏ~
GҿZlOk:5J(<ϯkϥ!n(k?xy
0~cQZ\1uL)'^M0y$16Q- 8Tg!9&3߹TDd
H
b
c$A??3"`?2iH4boWOhyc`!iH4boWOh2@H hxڥ;KPϽ}т!*">(B'E:"nRVB
%Nu:梋/Z{nV9Ͻ矓s D|`wz%čmGd}y1ZKIE^Ae|Ǣ$@GݜdݜU.ywbQSPs{FWU}dy٪^[";u?˛aaL)pԏKSmU8"fPIvKOwp)'o"|\sۢ0_Y0rFp3 '>Y_eY<~#\*z]En2J8\Yqk=>:e>:a-EyK'{TqgP:J:bҠ}\Dd
Tb
c$A??3"`?24yS 6f`!4yS ȽXJkxcdd``d!0
ĜL0##0KQ*
Wä2AA?H1Zc@øjx|K2B*R\``0I3Dd
|hb
c$A??3"`?2Q)ӛRѯ>oS{-g`!%)ӛRѯ>oS{`@0|xcdd``> @c112BYL%bpu +ssL9:|a>#dO3\
%s>`
t,%\L:0LM炆>8
{v0o8N+KRsA2u(2tA4Ag!t?0eְrT=Dd
|b
c$A??3"`?2?c +ck`![?c +R`@20)xcdd``dd``baV d,FYzP1n:&! KA?H1Z깡jx|K2B*R
vfjv,L!
~
Ay+V}.
_˙9ӖUsi#FofVJ<2?27)?gG!32H0ͅ\
1jF KucP
}.p`p221)W2,ԡRY`31sDd
|hb
c$A??3"`?2Q)ӛRѯ>oS{-9n`!%)ӛRѯ>oS{`@0|xcdd``> @c112BYL%bpu #%&')*,-./12(1:>!$(+03>"
9MO
7KM<PRy@TVYmor"68j~3GI[oq? S U !!!!"""("*"/"C"E"d"x"z"""""""""""""r###k'''''''''())!)5)7)))))))----..&.:.<.R.f.h..../*/,/6::::::::::::::::::::::::::::::::::::::::::::::::1e:2eLQ3eQ4eܗW5eW6eWw669*urn:schemas-microsoft-com:office:smarttagsplace9*urn:schemas-microsoft-com:office:smarttagsState8*urn:schemas-microsoft-com:office:smarttagsCity=*urn:schemas-microsoft-com:office:smarttags PlaceType=*urn:schemas-microsoft-com:office:smarttags PlaceName=h:EV[7NX[]_y"9:<=Bj3JPSUZ[rsuvy? Y \ e !!!!!!!""""""."D"I"J"M"d""""""""""""r####&&i''''''''''''''''''()))))!)9):)A)B)F)))))))-... .%.&.?.@.C.D.H.R.k.l.o.p.y.......//11111112#2.22222222222z334484@4e4j4l4p4445%585<5D5H555555555]6g66svSaR7[,/bj3SU- Z !!!!!"""#$##C''''''( )!)Q)))))))-. ././5/111122222 333,434`4d4445*5v5}5555563333333333333333333333nq 7Ny"9j3J[r? V !!!""+"/"F"d"{"""""""""r##k''''''()!)8)))))-.&.=.R.i...111111222222222222!3"3r3s33333.4/4444444&5'50505x5y55556667Ny"9j3J[r? V !!!""+"/"F"d"{"""""""""r##k''''''()!)8)))))-.&.=.R.i...6HGH}R8
>:o%I2aQKNrn[F!'i+U2:9Lg<4CyC