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IJERA (International journal of Engineering Research and Applications) is International online, ... peer reviewed journal. For more detail or submit your article, please visit www.ijera.com

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- 1. A. Lakshmi, S. N. Hasan / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 2, Issue 5, September- October 2012, pp.578-584 Spinors as a tool to compute orbital ephemeris A. Lakshmi1 and S. N. Hasan2 1 Department of Mathematics, Osmania University, Hyderabad -500007 2 Deparment of Astronomy, Osmania University, Hyderabad – 500007.ABSTRACT In this paper, the Spinors of the physical 3-D Let i1 , i2 ,...., im be the elements of S in the ascendingEuclidean spaces of Geometric Algebras over a finite order. We definedimensional vector space are defined. Action of Spinors eS ei1 ei2 ........ eim and e 1L .on Euclidean spaces is discussed. The technique for thecalculation of ephemeris using Spinors is illustrated. In We shall identify e{i} with ei .this process sequences of Spinors in their half angle form Note that if is a permutation of {1, 2, ….., m}, thenas well as their matrix form are used and compared. It isshown that when large number of rotations is involved, ei (1) ei ( 2 ) ........e i = (-1) m ei1 ei2 ........eim (m)Spinor matrix methods are advantageous over theSpinor half angle method. m stands for order of permutation. 1L 2 ei … and ei e j e j ei if i jKeywords - Ephemeris, Euclidean space, Euler angles,Rotations, Spinors. C (E ) = ⊕ A k where Ak = { ∑ ses } a sI. INTRODUCTION s =k Hestenes [1] defined Spinors as a product of twovectors of i-plane and also proved that they can be treated dim A k = nC k and dim C (E ) = 2 n .as rotation operators on i-plane. Spinors are also defined as The operation ‘Geometric Product’ of vectors denoted byelements of a minimal left ideal [2], [3]. Rotation operators play a key role in the determination of orbits of celestial ab , is defined asbodies such as Satellites and Spacecrafts. Geometric Algebra develops a unique and coordinate free method in ab = a . b + a ∧ b ………….. (1)this regard. There are various techniques using Quaternions, Matrices and Euler angles. Geometric Algebra unifies all b a = b . a + b ∧ a = a . b - a ∧b ……….. (2)these systems into a unique system called Spinors. Geometric Algebra not only integrates the above mentioned Here a.b = ab 0 is the scalar part and a ∧b = ab is 2systems but also establishes the relation among these the bivector part. Elements of Geometric Algebra are calledsystems and thus providing a clear passage to move from multivectors as they are in the form A = A 0 + A 1 + ….one system to the other. In this paper we study different parameterizations used for + A n . A multivector is said to be even (odd) ifrepresenting Spinors and rotations. We apply the Euler A r = 0 whenever r is odd (even) [5].angle parameterization of Spinors to solve the problem offinding the orbital ephemeris of celestial bodies [4]. A k ∈ A k , denotes the k - vector part of the multivector A.II. GEOMETRIC ALGEBRA There exists a unique linear map † : C(E) → C(E) that Let E be an n - dimensional vector space over R,the field of real numbers and g be a symmetric, positive takes A to A † satisfyingdefinite, bilinear form g : E × E → R denoted by (i) eS ei1 ei2 ........ eim then g ( x, y) = x. y x, y E .There exists a unique Clifford m (m-1)Algebra (C (E ), ) which is a universal algebra in which E eS† = ei m ei m-i ........ei 2 ei1 = (-1) 2 eSis embedded. Clifford algebra is also called Geometric (ii) (AB)† = B†A †Algebra as all elements and operation used in it can be ‘ † ’ is called the reversion operator.interpreted geometrically. We shall identify E with (E) .We choose and fix an orthonormal basis 2.1 Euclidean nature of Geometric Algebra Bn = {e1, e2 ,...., en } for E . 2.1.1 Definition Norm of a multivector To everyLet N ,2,...., n, n = dim E and S ⊆ N 1 multivector, A C (E) the magnitude or modulus of A is 1 † defined as A A A 2 . With this definition of norm, 0 578 | P a g e
- 2. A. Lakshmi, S. N. Hasan / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 2, Issue 5, September- October 2012, pp.578-584C(E) becomes a Euclidean algebra. The inverse of a non- 2.3.2 Definition Spinor i - space The Spinorzero element of A of C (E) , is also a multivector, defined i - space S 3 ’ is defined as † A by A 1 2 . { S3 = R / R = x y, x , y ∈ i - space} A2.1.2 Definition k sp a c e Every k- vector Ak + + i S 3 can also be denoted by C3 (E ) or C3 () . ‘ R ’ is a multivector, has a scalar part ‘ ’ and a bivector partdetermines a k space . ‘ i1 + i 2 + i 3 ’. Spinors do not satisfy commutative2.2 Euclidean space E3 property with respect to the operation ‘Geometric product’Denote the Clifford Algebra constructed over a three in view of the above definition (2.2.1).dimensional vector space E with C3 (E ) . III. ACTION OF SPINORS ON EUCLIDEANFor a trivector A3 , designate a unit trivector ‘ i ’ proportional SPACE - ROTATIONSto A3 . That is A3 A3 i . Spinors of Euclidean space also can be treated as rotation operators on i - space of vectors. Rotation‘ i ’ represents the direction of the space represented by A3 . operators can be constructed by considering the groupThe set of all vectors x which satisfy the equation action of Spinors by conjugation (Hestenes 1986). x i 0 , is called the Euclidean 3- dimensional vector Consider a Spinor R = v u , where u and v are unitspace corresponding to ‘ i ’ and is denoted by ‘E3’. E3 is vectors of E. Define a rotation operator R on E asalso be called an i space , the trivector i is called the † R x R xR u v x v u . pseudoscalar of the space as every other pseudoscalar is ascalar multiple of it. Unlike rotations in two dimensions, rotations in threeFactorize i = 123 where 1, 2 and 3 are dimensions are more complex as (i) the operation to be considered is the group action by conjugation, givingorthonormal vectors. They represent a coordinate frame in similarity transformations and (ii) the axis about which thethe i space . x x1 1 x2 2 x3 3 is a rotation takes place is also to be specified. The resultingparametric equation of the i space . x1 , x2 and x3 are vector changes as the axis of rotation changes. This can be shown in the following examples. called the rectangular components of vector x with Rotation of the vector x about the axis 1 , the axisrespect to the basis 1 , 2 , 3 . i space of vectors is perpendicular to the plane 2 3 is represented by thea 3 – dimensional vector space with the above basis.2.2.1 Definition Dual of vector i1 = 1i = 23 ; bivector i1 1i 2 3 . i 2 2i 3 1 ; i3 3i 1 2 Let x i space of vectors and x x1 1 x2 2 x3 3 .i1 i 2 = - i 3 = - i1 i 2 , i 2 i 3 = - i1 = - i 3 i 2 and † i 3 i1 = - i 2 = - i1 i3 i1 x i1 3 2 x 2 3 3 2 x11 x2 2 x3 3 2 3 x11 x2 2 x3 3 iThe set of bivectors in C3 (E ) (also denoted by C3 () )is a 3-dimensional vector space with basis i1, i2 , i3 . Every Rotation of the vector x about the axis 2, the axisbivector B is a dual of a vector b that is B = b i = i b . perpendicular to the plane 3 1 is represented by the bivector i2 2i 3 1 . 2.3 Spinors of Euclidean space C3 () i † 2.3.1 Definition Spinor The Geometric product of two i2 x i2 1 31x 31 1 3 x11 x2 2 x3 3 31 x2 2 x11 x3 3 vectors in the i - space is called a Spinor and is denoted by‘R’. Thus 3.1 Composition of Rotations R = x y = x . y + x ∧ y∀ x , y ∈ E . Two rotations R x R1† xR1 and 1 † ‘ R ’ is a multivector, has a scalar part ‘ = x . y ’ and R x R2 xR2 can be combined to give a new rotation 2 †a bivector part ‘ i1 + i 2 + i 3 = x ∧ y ’. R x R R x R2 R1† xR1R2 R3 xR3 † 3 2 1 Note that the order of rotations R R is opposite to that 2 1 of the corresponding Spinor R1 R2 . 579 | P a g e
- 3. A. Lakshmi, S. N. Hasan / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 2, Issue 5, September- October 2012, pp.578-5843.2 Proposition Product of two rotations is also a (iv) Spinors can be converted into other forms easily as androtation. when required.Proof: As Geometric product of Spinors is a binaryoperation, R3 is also a Spinor. 3.3 Representing a rotation using Euler angles Rotations are orthogonal transformations. They transformR3 R3 R2 R1† R1R2 1 as R1† R1 1 R2 R2 † † † one coordinate frame XYZ into another coordinate frame xyz preserving the angle between them. Euler stated thatHence R3 is also a unitary Spinor and it can be proved that every rotation can be expressed as a product of two or threedeterminant of R is one (Hestenes 1986). As a rotations about fixed axes of a standard basis in such a way 3 that no two successive rotations have the same axis ofconsequence of this R is a rotation. rotation. This theorem is known as ‘Euler’s theorem’. Thus 3 every Spinor can be divided further into a product of two orProduct of two rotations in three dimensional spaces is not three Spinors that represent rotations about base vectors.commutative in general. Euler angles are widely used to represent rotations.The rotations determined by R R and R R are To represent the rotation using Euler angles, we select an 1 2 2 1 axis for the first rotation among the three axes in theconsidered as two different rotations as their sequential sequence. Then according to the rule that no two successiveorder is not the same. rotations have the same axis of rotation, we will have two axes to choose for the second rotation , and for the third,3.2.1 Spinors of the i - space in half angle form again two options are there to choose a different one fromAs a unit vector is treated as a representation of the the previous. Thus we get totally 3 x 2 x 2 = 12 sets ofdirection of a vector, a unit bivector can be treated as a Euler angles. Hence, one can represent the same rotationrepresentation of an angle, which is a relation between two using different sets of Euler angles. If the axes chosen fordirections. the first rotation is same as that of third, such a sequence is called a Symmetric set or a Classical set of Euler angles. i ˆ ˆHence for x , y ∈ C3 () , let x, y be their directions whichare elements of the i - space . 3.4 Finding the Matrix form of the Euler angle sequenceFrom the definition of a Spinor of the i - space , the Spinor of Spinors We consider the 3-2-1 symmetric sequence of rotations; the ˆ ˆ ˆ ˆ ˆ ˆR = x y = x . y + x∧ y . Spinor that represents the required rotation is given as a sequence of three rotations about the base vectors ˆ = cos (1 / 2) A + A sin (1 / 2) A is the half angle form of the 1, 2 , 3 is defined bySpinor R . R = R Q R ,S = C + (E ) if dim E ≤3 . S ’ can be related to complexnumbers for dim E = 2 and it can be related to Quaternion Where R = e 1/ 2 i 3 = cos + 1 2 sin ,Algebra for dim E = 3. ( ) ˆ 2 2R = cos (1 / 2) A + A sin (1 / 2) A can also be written as e1 / 2 A . Q = e(1/ 2) i 2 = cos + 31 sin , 2 2 ˆ x yHere A = ˆ ∧ ˆ , the bivector representing the plane of R = e(1/ 2) i 1 = cos + 2 3 sinrotation and A gives the magnitude of the angle through 2 2which the rotation takes place. The new set of axes after rotation are given by3.2.2 Matrix form of a Spinor There are different matrices to represent rotations. Instead, ek R k R † k Rthe use of Spinors to represent a rotation gives the matrix † † †elements directly by the formula = R Q R k R Q R This can be converted into the matrix form by calculating e jk = j . ek = j . ( k ) . R the elements of the matrix [e jk ] given asThere are also other forms such as Exponential form,Quaternion form etc. The advantages in using Spinor e jk = j .ek = R kAlgebra as a substitute for all the above algebras † † †(i) The coordinate free nature of Spinors. e1 = R1 = R Q R 1R Q R(ii) The existence of Spinors in every dimension facilitatingto perform rotations in higher dimensional spaces. † † †(iii) Representation of rotations using Spinors enables us to = R Q ( R 1R )Q Rfind the magnitude of the angle of rotation as well as the † †orientation of the rotation which is not the case with the = R Q (1(cos + 12 sin ))Q Rmatrix representation. 580 | P a g e
- 4. A. Lakshmi, S. N. Hasan / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 2, Issue 5, September- October 2012, pp.578-584 † † † † = R Q (1 cos + 2 sin )Q R = R (Q 3Q ) R † † † = R [(Q 1Q ) cos + (Q 2Q ) sin ]R † = R [3 (cos + 31 sin )]R † = R [1(cos + 31 sin ) cos + 2 sin ]R † = R (3 cos + 1 sin ) R † = R (1 cos cos - 3 sin cos + 2 sin ) R † † = ( R 3 R ) cos + ( R 1R ) sin † † † = ( R 1R ) cos cos - ( R 3R ) sin cos + ( R 2 R ) sin = 3 (cos + 23 sin ) cos + 1 sin = 3 cos cos - 2 sin cos + 1 sin = 1 cos cos - 3 (cos + 23 sin ) sin cos + 2 (cos + 23 sin ) sin = 1 sin - 2 sin cos + 3 cos cos = 1 cos cos - 3 cos sin cos + 2 sin sin cos + 2 cos sin + 3 sin sin The matrix obtained is as follows cos cos - cos sin sin = 1(cos cos ) + 2 (sin sin cos + cos sin ) + 3 (sin sin - cos sin cos ) sin sin cos + cos sin cos cos - sin sin sin - sin cos † † † sin sin - cos sin cos sin cos + cos sin sin cos cos e2 = R2 = R Q R 2 RQ R † † † This sequence is used in aerospace applications and to find = R Q ( R2 R )Q R orbital ephemeris. † † = R Q [2 (cos + 12 sin )]Q R IV EULER ANGLES AND EQUIVALENT ROTATIONS † † The set of Euler angles that represent a particular = R Q (2 cos - 1 sin )Q R rotation are not unique. Representing the same rotation by † † † two different Euler angle sequences gives Equivalent = R (Q 2Q ) cos - (Q 1Q ) sin ) R rotations. In this paper we use sequences of Spinors in their matrix form and also in their half angle form in place of † rotation sequences to obtain the relationship between the = R (2 cos - 1(cos + 31 sin ) sin ) R two sets of angles (i) orbital ephemeris set ( L , and † = R (2 cos - 1 cos sin + 3 sin sin ) R ) and (ii) orbital parameters ( , and ) set. Thus established the equivalence between Spinor methods and † † † Quaternion methods. = ( R 2 R ) cos - ( R 1R ) cos sin + ( R 3R ) sin sin = 2 (cos + 23 sin ) cos - 1 cos sin + 3 (cos + 23 sin ) sin sin = 2 cos cos + 3 sin cos - 1 cos sin + 3 cos sin sin - 2 sin sin sin = 1(- cos sin ) + 2 (cos cos - sin sin sin ) + 3 (sin cos + cos sin sin ) † † † e3 = R3 = R Q R3RQ R † † † = R Q ( R 3R )Q R † † = R Q (3 )Q R 581 | P a g e
- 5. A. Lakshmi, S. N. Hasan / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 2, Issue 5, September- October 2012, pp.578-584 Figure 1: ephemeris sequence frame of reference (XYZ, NED frame of reference) throughL = Earth-Latitude of orbiting body the orbital parameters ( , and ). This can be done by 0 using 3-1-3 sequence of Euler angles. = Ephemeris path direction angle Earth-Longitude of orbiting body0 Greenwich with respect to X axis = Angle to the orbit ascending node, also called ‘RightAscension of the ascending node’. It is the angle subtendedat the center of the Earth from the Vernal Equinox (positiveX axis) to the ascending node ‘N’. The angle of inclination. It is the angle between orbitalplane and equatorial plane. The argument of the latitude to orbiting bodyThe position vector of the object is given by the vector OR . ‘P’ is the point on the surface of the Earth in theradial direction of the object. The reference frame XYZ isthe Equatorial frame of reference that is the plane thatcontains the Earth’s equator. X, Y axes are contained in the Figure 2: Orbital sequenceequatorial plane of the Earth. The Z axis is normal to thisXY plane such that XYZ frame forms a right handed frame 4.1.2 Orbital parameter sequenceof reference. ‘NOR’ is the orbital plane. The trajectory of In this application, the aircraft’s body frame xyz is relatedthe object is as indicated in the figure 1. ON is the line to the NED reference coordinate frame XYZ (refer fig 3)segment in which the orbit trajectory and the reference defined as XY plane is the Tangent plane to the Earthframe intersect each other. pointing towards North and East directions respectively. Z axis points towards the centre of the Earth (NED frame of4.1 Matrix method: reference). The positive x axis of the body frame is directed along the longitudinal axis. The positive y axis is directed4.1.1 Orbit ephemeris sequence along its right wing and the positive z axis is perpendicular A tabulation of data of the Earth longitude and Latitude of to the xy plane such that xyz forms a right handed system.a body, as a function of time is called the orbit ephemeris of These two frames are related by the heading and attitudethe body. sequence of rotations followed by a third and final rotation In this sequence the body frame (xyz) of the object will be about the newest x axis which is the position vector of therelated to an inertial frame of reference (XYZ) through the spacecraft through an angle as depicted by the figure 2.orbital Ephemeris ( L , 0 , ). This requires This requires the 3-1-3 symmetric sequence of rotations; thethe 3-2-1 symmetric sequence of rotations; the Spinor that Spinor that represents the required rotation is given as arepresents the required rotation is given as a sequence of sequence of three rotations about the base vectors is definedthree rotations about the base vectors is defined by by † † † †R R Q LR R Q R k R Q L R , † † R R Q R R Q R k RQ R , (1/ 2) i 3 Where R = e(1 / 2) i 3 = cos + 12 sin ,Where R = e = cos + 1 2 sin , 2 2 2 2 (1 / 2) i 2 (- L) Q = e(1 / 2) i 1 = cos + 23 sin ,Q(- L) = e 2 2 R e1 / 2 i 3 cos 1 2 sin , (- L) (- L) L L= cos + 31 sin = cos - 31 sin 2 2 2 2 2 2 The matrix obtained is (1/ 2) i 1 R = e = cos + 2 3 sin cos cos - sin cos sin - sin cos cos - sin cos sin sin 2 2The matrix is obtained is as follows cos sin + cos cos sin cos cos cos - sin sin - cos sin sin sin sin cos cos (cos L cos ) (- cos L sin ) - sin L As the orbital sequence and ephemeris sequence represent= (- sin sin L cos + cos sin ) (cos cos + sin sin L sin ) - sin cos L the same body frame of reference, they can be equated to (sin sin + cos sin L cos ) sin cos - cos sin L sin cos cos L get the relations between the orbital ephemeris in terms ofIn this application we relate the body frame (xyz) of a the orbital elements. Thus we obtainspacecraft or a near earth orbiting satellite to an inertial 582 | P a g e
- 6. A. Lakshmi, S. N. Hasan / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 2, Issue 5, September- October 2012, pp.578-584 sin L sin sin ⇒ cos cos = cos cos cos - sin cos sin = cos cos( + )....(1) cos sin - sin sin = cos cos sin + sin cos cos = cos sin( + )....( 2) tan cos tan cos sin cos = cos sin cos + sin sin sin = sin cos( - )...(3) cos L sin (tan + tan cos) tan = = cos L cos 1 - tan cos tan sin cos = cos sin sin - sin sin cos = sin sin( - )...( 4) 4.2 Spinor half angle method Equating the orbit sequence with Ephemeris sequence Dividing (3) by (1) and (2) by (4) gives R Q L R R Q R cos( - ) , tan = tan cos( + ) R e1 / 2 i 3 cos 1 2 sin ; tan cos( - ) 2 2 ⇒ = ....(5) L L tan cos( + ) Q L e1 / 2 i 2 L cos 31 sin 2 2 1 sin( + ) - tan = tan sin( - ) R e1 / 2 i 1 cos 2 3 sin ; 2 2 sin( + ) sin( + ) 1 / 2 i 1 ⇒ - tan tan = = ...(6) R e cos 1 2 sin ; sin( - ) sin( - ) 2 2 Multiplying (5) and (6) gives Q = e (1 / 2) i 2 = cos + 1 2 sin and 2 2 cos( - ) sin( + ) tan( + ) - tan 2 = = .....( 7) R = e (1 / 2) i 3 = cos + 1 2 sin sin( - ) cos( + ) tan( - ) 2 2 dividing (6) by (5) gives RQ L R = RQ R ⇒ sin( + ) cos( + ) sin 2( + )RQ- L = RQ RR -1 = RQ RR- = RQ R- = RQ R - tan 2 = = sin( - ) cos( - ) sin 2( - ) ⇒ R Q-L = R Q R 1 - tan 2 but cos 2 = To avoid half angles 1 + tan 2 = 2, L = 2, = 2, = 2, = = 2, = 2 sin 2( - ) + sin 2( + ) tan 2 = = - R Q- L = R Q R ⇒ R2Q-2 = R2 Q2 R2 sin 2( - ) - sin 2( + ) tan 2 Gives tan 2 = - cos 2 tan 2 (cos + 2 3 sin )(cos + 31 sin ) substituting 2 , 2 , 2 = (cos + 1 2 sin )(cos + 2 3 sin )(cos + 1 2 sin ) gives ⇒ cos cos + 12 sin sin + 23 sin cos - 31 sin cos tan = - cos tan tan - tan [ = (cos + 1 2 sin ) cos cos + 1 2 cos sin + 23 sin cos + 31 sin sin ] - cos tan = tan = tan( - ) = 1 + tan tan After simplification we get cos tan + tan = cos cos cos + 1 2 cos cos sin + 2 3 cos sin cos + 31 cos sin sin + tan = 1 - tan cos tan 1 2 sin cos cos - sin cos sin + 1 3 sin sin cos + 2 3 sin sin sin Similarly we get the other relations. = cos cos cos - sin cos sin + 12 (cos cos sin + sin cos cos ) 4.3 Singularities in Euler sequences + 23 (cos sin cos + sin sin sin ) + 31(cos sin sin - sin sin cos ) A singularity occurs in every sequence of Euler angles. For example in the 3-1-3 sequence, 583 | P a g e
- 7. A. Lakshmi, S. N. Hasan / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 2, Issue 5, September- October 2012, pp.578-584 e sin cos sin tan = 12 = = . Singularity occurs at e11 cos cos cos = . 2But Euler angles work well for small angles or Infinitesimalrotations. Hence in the problems like finding the orientationof the spacecraft and tracking an aero plane employment ofEuler angles is advantageous.V. DISCUSSIONS Even though there are many representations forrotations in 3-D namely Quaternions, Spinors, Euler axis-angle and sequences of Euler angles, each one can betransformed into the other. Each system has its own meritsor demerits over the other systems. Singularities occur forevery set of Euler angles but they work well in infinitesimalrotations. Comparatively, the other methods of usingQuaternions and Euler axis and angle for representingrotations provide a better procedure as they need fourparameters to be defined and they can be converted into anyother convenient form depending upon the informationavailable. In the case of Spinors, whose parametric form is given as s i , the number of parameters reduce further as the parameter is not independent of .VI. CONCLUSIONS The technique of using Spinors can replace theconventional methods and also provide a richer formalism.If large number of rotations is involved Spinor matrixmethods are advantageous over the Spinor half anglemethod.ACKNOWLEDGMENTS I would like to thank Professor S. Uma Devi, deptof Engineering Mathematics, Andhra University for hervaluable and encouraging suggestions while writing thepaper.REFERENCESBooks: [1] Hestenes D., New Foundations for Classical Mechanics, D. Reidel Publishing, 1986. [2] Hestenes D., Space-Time Algebra, Gordon and Breach (1966). [3] Lounesto P., Clifford Algebras and Spinors, 2nd Edition, Cambridge University Press (2001). [4] Kuipers J.B., Quaternions and Rotation Sequences, Princeton University Press, Princeton, New Jersey (1999). Thesis: Hasan S. N., Use of Clifford Algebra in Physics: A Mathematical Model, M.Phil Thesis, University of Hyderabad, India (1987). 584 | P a g e

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