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The Parameterization of Orthogonal Matrices: ME. tCOA Review Mainly for Statisticians. .OU0) 6. PERFORMING ORG. REPORT NUMBER

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Department of StatisticsNuclear Sciences Center, University of FloridaGainesville, FL 32611

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I. SUPPLEMENTARY NOTES

19. KEY WOMS (Continue on reverse side If nocessary and Identify by block number.,

• Cayley"ks transformation; Diagonalization, simultaneous; Eulerian angles;

Haar measure; Orthogonal group; Rotation groups.,-

20. ABSTRACT (Continue on reee,. old@. It Oeo ,m,,ad Identify by block numb it)

This paper reviews methods for the parameterization of orthogonal matrices.Examples of where such parameterization can be used are presented.

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THE PARAMETERIZATION OF ORTHOGONAL MATRICES:

A REVIEW MAINLY FOR STATISTICIANS

By

I. J. Good Andre I. Kliuri

Department of Statistics Department of Statistics

Virginia Polytechnic Institute University of Florida

and State University Gainesville, FL 32611

Blacksburg, VA 24060

" -.

Technical Report Number 293 .-

Department of Statistics --

University of Florida . 'v.,% az I;or

Gainesville, FL 32611

November 1987

" - - -- C- -- -- - -- _ _; - L -- ' - r: C-. -- - - - . - --- -- -=-, - " ~ - = - ---- . -

THE pAqAMETEU= OTZoN OF ORTHOGONAL MATRICES:

A REYVEW MA.INLY FOR STAT STZICANS'

Byx. j. Good and Andre I. Xlurl

virginia Polytechnic nuatute & stake University and Universty of Florlda

Abstract

Tecluiques and appialtns for the parameterizatici of orthogonal maktrice

am guvey d, mainly for the beefMt of statisticlaz.

By an orthogonal matrix 2 we shall mean a real square matrix of order n x n

for which CC' is the identity matix, where CO denots the transpose of C.

orthogonal matrices are used frequently in Statistics, especially in linea models

and MuilivarLake analysis (M, for emmple, Graybil. 1961. Chap. 11; Searle,

1971, Cnap. 2; Anderson, 1965; and Zames, 1954.).

Orthgonmal matrice of dete*rminan 1 represent elements of the rotation group

In n dImmandns. Gel'tand et al. (1963) and Murnaghan (1936) give exeive,

discussi of the repreeton of the n-dimensional rotation group. Hoffman et

al. (1972) and Raffenetti and lRuedenberg (1970) represented the cls of

n-dimensional orthogonal matrices in terms of generalized Eulerian angles. The

'Thds work was supported In part by a grant #GM18870 from the National1 ikukes of Health and one from the Office of Naval ResearchW00014-6-X-0059.

AKS 1960 subject lassdffltions. Primary 65P30; Secondary 62399, 5AS7.Key Uards awd pfheses: Cayley's transformation; Diagonalization,

u -- ; Kulerian angles, generalized; Haar measmre; Representation of theorthogonal groupl Rotation groups.

1

representation of three-densional or l D w fte* ME-

been used in the study of the motion of a rigid body (Euler, 1776, Whittaker,

1927, PP. 9-10). Per further ideas and a three-dimensional application, see

moran (1975).

The nz elements of an orthogonal matrix are subject $o %n(n+) constraints.

it is therefore not surrirsing that they can be represented by only nz - Ijn(n+1)

In(n-1) independent pLameters. A representation is convenient if the whole of a

matrix can be quloty computed from the Isn(n-1) parameters. Such a represetation

facilitates the search for an orthogonal matrix that sati fies some optimality

criterion. The Independent parameters can be used also in integrating a *acion

over the n-dimmionil orthogonal group (Nurnaghan, 1939, pp. 230-242), wtdch in

defined as the set of all n x n orthogonal mattices with the operatio of matrix

uu:liplication.

As far as we know, the various methods of parameterixing orthogonal matrices

are not available in the statistical literature. Our aim is to bring seeral of them

methods to the attention of the statistical profseeion, and we do not claim

mathematical originality, although probably a few of ou cow nt have some

novelty.

Firft we mention some applications. The cited references may be consuted

for moe details. The rotation group of n x n orthogonal matrices win be denoted

by o(n).

2. some Applications In Statistics of Parameterization of the Orthogonal rou.

(a) A parameterization of O(n) can be used to define an invariant (SUr)

measure, ;L, on 0(n). This meaire is invariant in the seme that If a set G of

orthogonal n by n matrices has measure jL(G) then, for every orthogonal n by n

2

Sm

matrix C;, L(G) = i&(G) =A(EG) and tip (lAM aJ dn6beid 51

obtained by multiplying all the matrices in G on the righ (left] by C. Haar

meas re is unique up to a positive multiplicative constant. In accordance, for

example, With James (195., p. 53), the Haar Meaare (G), if it exsts, Is given

by the %n(n-1)-fold integral

If

A(G) an fa ja C. C (M)U (Cj), GCO(n)

where and cj are the it and j cum vetrs of an ortogo.a i C(i

ds9 denotes a scalar Product, and te product of differential form is to be

interpreted as an exterior product. (Exterior products are explained, for

example, by James, 1954, p. 46.) For example, if n = 2, the differential form,

n(cj • d) , for- i.

. Cox sin 1C=I 0 (e<2W,

-sin 9 Co e

i s> ~a dO.

The Haar measure ham been used to derive the distribution of the canonical

correlation coefficients (James, 195.). Andermn (1965) used the represetsabkt

given by formula (3.2) of Section 3 below to obtain the joint distribuUon of the

elgenvalues of the sample variance -covariance ma rix. See also, for example,

WVIsan (1957), Chattopadhyay et al. (1976).

(b) When carrying out simulation experiments in rerosson heo,ry t

condition umber of XOZ Is of interest, where X is the matrix associated with the

regression model. This condition number is defined (Bartree, 195, p. 153) as

the ratio of the largest to the smallest of the eigenvalues of X'X. George Terrell

3

pouted ut (private commucajoi) that by appyitW aaml tUagMW

transformations to X, we can generate several different X matrices for which the

condition numbers are all equal. This would be convenient in the simulation

experiments. The generation of suc.h random orthogonal tranuforaloaw can be

facilitated by utilzing the independent parameters obtalnd through paat i

of the orthogonal matrices representing Uese tra a .

(c) Twe sulLaneam dliagonalzatlon of several matrices Is of Is'derable

iterest In statistics, especially in the analysis of random - or mid -efects

models. Let_£, .6, ... , ,Ag be A (A 2) positive definite symmetrc matrices of

o der p- x p. suppose that Uhs matrices are not altneusly dia' __ -Uxblo

(because they do not comnite), and we wish to find an orthogonal matrix, B, which

makes them sm uuly was diagonal as pomsiblew. Firy and Gaaatsct (299k, p.

170) itroducd a meamure of simltaneous deviation of U matrices, .. .

... , BAI, from diagonality. This measure is given by the function

X(D , A,, *-- ,,.a n., n., .. ng)

U (Idiag(j3)l/, )lnA I)n , (2.1.)

where dag(,8ka) is a diagonal matrix having the same diagonal elements as--MB

(I - 1, 2, .... 1), I1 denotes the determinant of a matrix, and n1 , nz, .

ng are positive weights. n is known that X ; 1 with equality occurring If and only

f3 diaonalizes the Ajs. n is therefore of interest to find the minimum of X over

the rotation group O(p).

The determinaUon of the optimal orthogonal matrix, 3, that minimizes X in

(2.1) can be reduced to an optimization problem in p(p-j) dimensions since this

I

is "the number of independent parameters that represent the elements of O(P).

Flury and Gautschi (1986) did not follow this procedure; instead, they introduced

an iterative algorithm whereby a converging sequence of orthogonal matrices, A(o),a() , was derived such that X(!(9+)) G xIM)), j - 0, 1, 2, All

pairs of columns of the orthogonal matrix D() obtained in the J- terat=i (3 = 0,

1, 2, ... ) are subjected to rotations by a specific 2 x 2 orthogonal matrix

parametfried by a single parameter. Tis yields the matrix N( + J) and the

process is repate until some convergme criterion is met.

(d) The need for paramete arises in response surface analysis.

Consider a linear model of the form

T - d . (2.2)

where T is the vector of observations, 0 Is a vector of unnown parameters, and X

is a fnown matrix of order n x p and rank p. The elements of x are functions of

the settings of m Input variables denoted by x, x , ... , xm . The n x m matrix,

D = (xui), where xui is the setting of the 1th variable at the uth experimental run

(u =1, 2,... n; j = 1, 2,... ,m), is called the design matrx. The

elemnts of the error vector d in (2.2) are asmed to be independently. andam

identically distributed random variables with zero means and variances equal to oz.

Tests of signiflcance concerning the parameter vector A in (2.2) depend on

the assumption that * is normally distributed. The effect of nonnormality of the

error distribution on these tests has been studied by several ators (see, for

example, Pearson, 1931; Geary, 1947; Gayen, 1950; and David and Jolawson,

1951a, 1951b). Box and Watson (1962) pointed out tha the sensitivity to

nonnormality depends very much on the settings of the input variables specified in

Ute design matrix. This was also demonstrated by Vuchkov and Solakov (1980).

5

Thus, a properly chosen design maltik dm la l ies. Lhe i rf sm , a-

robust, to failure of the normality assumption.

Box and Watson (1962) introduced a design robustness criterion for a model

of the form (2.2) of degree one (a first-order model), which can be rewritten an

= + Dr + a, (2.3)

Where 1. is a column vector of one of order n x 1, D Is the design matrix such

that X -- _3,, and (0o, TO)* 9. Box and Watson's criterion is for testng the

-0 a

using the men squam ratio

f. F~~(D) - [y DD'D)D'y/Y.,,-,. .. , .- " ,,[XK)1 X', ,,y )T / (n m- )] "

They showed that in nonnormal zlbmations, 7(D) Is distributed approximately under

Ho as an IP-distritutIon with modified degrees of freedom given by vs - im and va

= r(n-m-1). Here the corrective factor 7) is equal to unity if g(D) = 0 regardless

of the yes or their distribution, where

g(D) - d - m(m+2)(n-1)/[n(n+1)] , (2.5)

and d =£ dj vith dji being the ith diagonal element of D(DD)-LD'.

Since 71 is equal to unity when the error distribution is normal, a design

matriz satisfying

Z(D) = 0 (2.6)

will result in an approximate F statistic with degrees of freedom identi-

6

cal to those obtained under normalitY; Cdsequenijy, the deegtS mkix2

can determine whether the distribution of FCP) is insensitive (robust)

to nonnormality. The matrix D(DDEa)-D'. being idemptent, can be ex-

pressed in the form

P diag(I.,0)

where P is an orthogonal matrix. In this way, the parameterization of

D( D -1Dcan be reduced to that of an orthogonal matrix. The quantity,

d, in (2.5) can be regarded as a function, h(P), of P and (2.5) becomes

g(D) - h(P) - i(m.2)(n-1)/(n(n4.1)] .(2.7)

For a fixed n, one method L~o implement the Box-Watson criterion is to find P go

that g()has an abeolut minimum over *the rotation group 0(n). if this absolute

minimum is zero, then the Box-Watson criterion is satisfied. Otherwise, the optimal

design that minimizes gz(D) is the design thak comes -closest- to satisfying this

criterion. For more details concerning this matter see Khu.ri and Myer. (1961).

The minimization process can be carried out by expressing h( F) in (2.7) In

term of the parameters of the orthogonal matrix P.

We ow describe various methods of parameterization.

3. Now to Parameterize an Orthogonal Marx

The four methods we shall discuss are (i) expressing an orthogonal matrix in

the form el where T is skew-symmetric, (Ui) using cayley's transformation which

also given an expression in terms of a skew -symmetric matrix, (iii) using Zulerian

angles for the three-dimensional case, and (iv) using generalized Eulerlan angles

for the n-dimensional case.

(1) If 9 is an n x n orthogonal matrix with determinant 1, then it can be

7

vriwUen in the form

(3.1)

vwire T is a skew-symmetziC maLZ (3e, for example, Ganblacher, 1959, p.

28). (The expoeia is of counse definable, as an Infnite series.) The

element ofZ, above its main diagonal, can be ued to pam.1rz Q.

now, if a is given, then to find T we can fir find Ue eftenvalues of Q.

Thas are necessarily of the form e • :j , ... , e *:L q , 1 , wwer the

eige value I is of multiplicity (n - 2q), and none of the eal nmbers j

*jis a multiple of 2V (j - 1, 2, ... , q). (Those that are odd multiples

of r give an even mber of eigenvalues equal to -1.) If we donate the

matrix~- a by [a + bi], a notation that is reasonable because the

f 2 x 2 matrices form a representation of colex numbers, then Q can be

written (GzrMaeber, 1959, p. 28), as the product of three real matrices,

Q2= dia(Ee' ] ... ,e 1eI], 1 ... , 1)QL . (3.2)

orme is an ozhogonal matrix of the form

,.= (Z 1 , Z., =Z,,, -, ... , 'Yq -=-q.. --*,*

such that~j + iZ is an eigenvector with eigenvalue eiJ (j = 1, ... q),

and is an eigenvector with eigenvalue I (k = 2q + 1, ... , n).

We now define the sksv-symmtric matrix T by the equation

_T , X d i ag ( [ i @ , ., , [ i eq ) , 0 , . . , o )Q j . ( 3 .3 )

since • [ i =j ] = (eioJ3 we have Q = e! as required.

9

-I -x

we note that

q-i

_9= 11 !ja=o

where

_1j - dia&gXaj. ., j - 0...., q .i

and wherejaj and .3 -,j-2 are Ue identity matrices of dimensions 23 and n-2J-2,

repectively. If Ij+l is a plane parallel to both th vectors !I*, and , then

Rj represents a rotation of the n-space leaving fixed every point in the

(n-2)-space ortUwg l to 11+ 1 (VItali, 1928). Tius Q repront th product of q

such simple rotations. When n - 3 and q - 1, this Is a familiar fact in dynamic,

known to Euler. To obtain a power Qc we could multiply all the angles of rotation

.. by a.

We note also that

T(zj + yij ) = i0j(.xj + iyj) , j - 1, 2 ... , q ,(3. )

ThA = 0 , k = 2q+1, ... , n

Equations (3.4) state that + iyj is an eigenvector of T with eigenvalue i4t, and

that is an eigenvector of T with zero eigenvalue. Since the eigenvaLues of T

are purely imaginary or zero (at least one of them is zero if n is odd), and the

Imaginary ones pair off in conjugate pairs ±Ati, ... * *1%q, it follows that (3.3)

gives a representation of a real skew-symmetric matrix in terms of its eigenvalues

and eigenvectors (Gantmacher, 1959, p. 285). This observation enables us to

calculate .I if T is given and if (3.1) is known to be true. We first put T in the

form (3.3), then we calculate Q by (3.2). For convenience in computing the

9

t I l 1

eigenvalues and eigenvectors of T, we noe& Lhak z# azad j 1, .,. , q) are

eigenvectors of Ue symmetric matrix & with eigenvahue .Ojz. ti's can be easily

seen from (3.3), which when both sides are squared gives

= q, 2 (z j * +:i

Examle. n - 3. Xn three dimensions any orthogonal matrix Q, with

determimt 1, has the form

= eW -c 0 ab -a 0

The eigenvalues of

0 c -b

I-c 0 a

b -a o

are ip, -ip, and o, where p = (az + b& + cz)S. The normalized eigenvector

with eigenvalue 0 is v = p-'Ca, b, C]' and the other normalized eigen-

vectors can be chosen as u t iv where

u = (b z + cz)'S[O, c, -b]'

and

v- p-'(bz + cZ)'5(bz + c z , -ab, -ac]'

The vectors u and v are orthonormal eigenvectors of Ta with the eigenvalue

10

-p.

P (3.2), we have

Q = [u, Y, w]diag([e'i t ],X)[u, v, v]'

The matrix Q represents a rotation of the 3-space about a line whose di-

rection cosines are (a/p, b/p. c/p).

(I.) if q Is an orthogonal matrix that does not have the eigenvalue

-1, then it my be written in Cayley's form (Gant~mcher, 1959, p. 239).

Q - (I - S)(Z + S) "L (3.5)

where Z is a sicew-symmetric matrix. [I + 3 is never singular If 3 is

skew-Symmetric. A pro of this fact is given by Ferrar (1950, p. 163) who

attrlbte the fact to Cayley. A simpler pLoof Is that all the eigenvalims of a

skew-symmetric matrix S are purely imaginary or zero (because It is Hermitian) and

therefore -1 cannot be an eiewalhe of 3.] The form (3.5) has the advantage of

defining 2 as a one-to-one 6fckion of 3, but it has the disadvantage of being

resricted to those orthogonal matrices that do not have -the eigenvab -1. If Us

conUon is not amied, 9 can still be written in the form (Ferrar, 190, p.

16)

= J( - S)(X + s-. (3.6)

where 3 is a diagonal matrix in which each element on the diagonal is either I or

-1 (and where S is skew-symmetric). This representation can be made unique by

Inisting that all the plus l's precede all the minus I's along the diagonal of J.

When this repesentation is used we must supplement the n(n-l) parameters in S

with an aaumption for the number, r, of plus l's in 3.

11

(iii) In three dimensions orthogonal tfansiofrations with dbtermbAh 1 are of

course represented by the rotation of a rigid body free to turn about a point 0.

The motion of this body is determined by three independent angles known as Euleran

angles (Euler, 1776; Whittaker, 1927, p. 9; Condon, 1959, p. 6). These angles

can be introduced in various ways. The following one is used by Condonx

Let OXYZ be a right-handed system of rectangular axes fixed In spae. rAt

Oxyz be rectangular axes fixed relatively to the body and moving with it, such that

before the displacement the two sets of axes OXYZ and 072 are colncident in

powition. rot vZ and z be unit vectors in the positive dlrectix:n of tm Z and x

axes, respectively, and lot .. ,Z x 5 be their vector product. Deno t the

angles ZOz, XOK, KOz by 0, *, *, respectvely. These are the thrme Wlerian

angles.

By a remark made in Section 3(i) the total movement of Ute body is equivalent

to a rotation. This rotation is represented by the prodct of three mat ices,

Q = diag(Cei*], I)diag(1, [ei])diag([ei, 1)

Coso cos* - sinO Cose sin* si.* co* + con case sin* sine Bii

- , -coo* sli- - sino cose cos* -sin* sn* + cos. cose coft sine co"

8si * sine -Cos* sine case

(3.7)

This then provides a parameterization in three dimensions. The parameterization is

unique if, for example, 0 4 0 < 2w, 0 ( 0 < 2w1, and 0 4 * 4 a.

(iv) A generalization of the concept of the three-dimensional Eul.rian angles

to n dimensios was given by Raffenetti and Ruedenberg (1970). Formulae and a

computer program were derived that expressed an arbitrary orthogonal matrix of

12

order n x n in terms of '!n(n-l) angulAt v&tlablE, A general h-dismniOmal

orthogonal matrix, , can be constructed by the sequence of recurrence relations

[ ) = ±1

.q(2) = dag(Q0 -), 1) , i = 2, 3, ... , n

A() = ai..,iaj_z,j al i , 1 = 2, 3, , n

QM±) =A( 1 ),q(i) , = 2, 3, ... , n ,

where (aij) (i - 1, 2, ., n; j = 1, 2, ... , n; i < J) is an n x n matrix

whoe diagonal elements are 1 except for the diagonal elements in the it and j

columns., which are cosyjn; all off-diagonal elements are 0 except for the one

corresponding o tha intersection of the jth row and the P co , which is uiny-,

and that on the irsection of the P row and the ih columns which is -sinVU.

The choice Q(± I yields an orthogonal matrix Q - (Qjj) with determinant 1, and

the choice Q(2.) = -1 yields a matrix Q with determinant -1.

ConveTsely, if an orthogonal matrix, 9, = (Uij), is given, then the

corresponding angular variables are found by minimizing the function f(y)

= :Car - Qjj(V)]z, and where = (V.z, V , - - (±, j * 1,

• t,j2, ... , n , and the elements of Q are found by a prescribed recurrence

scheme. Hoffman et al (1972), however, give algebraic inversion formula*, and a

compuer program, for the angular variables in terms of the elements of the

orthogonal matrix.

/. A Problem Concerning Haar Measure.

It would be of value for numerical integration over the orthogonal group if

Haar measure could be generated by "sn(n-1) statistically independent random angles

13

each uniformly distributed. For we could then approximate the- integration over te-

orthogonal group by a discrete sum over points uniformly spaced on circles. in

three dimensions the Eulerian angles do not serve this purpose . For suppose that

0, 0, and * are each uniformly distributed. Then coes cannot have the sam

distribution as the other elements on the diagonal of the matrix in (3.7), so that

the -tranormatLon is not symmetrical with respect to the aces. Thus the distribution

cannot be invariant under all rotations.

One can arrive at Har meaure for the three -dimensional rotation group by

choosing an axis of rotation with uniform polar angles in (0, w) and (0, 2w) and

then choosing an angle wi of rotation with uniform distribution In (0, 2w). The

matrix giving this orthogonal transformation, expressed In terms of w and the

direction cosines, cosh, coup, and cosi, of the axis of rotation, is

-222ba 2c~ch 3 2c,.csbz + CRS

2Lcczbh + C3s I - 2zha 2czcahz - CLS , (4.1)

II I ±chZ -C22 2c~cbh + CJLs 1 - 22sba

where s " siunw, h sinWa, SL = sinA, 22 = sinw, s 3 = snv, c, cash, CZ =

oCne/, C3 = conwv * Ttis remal can be readily derived from Whittaker (1927, p. 0 ).

Foruula (4.1) provides a matrix representation of the rotation group and it ix

equipped with Haar measure if A, ;, w and w have uniform distributions in (0, 2ff),

(0, 2w), (0, 2w), and (0, V) respectively. It seems to be difficult to generalize

this remlt to n dimensions.

1/.

References.

[1] Anderson, G. A. (1965). An asmptotic expansion for the distribution

of the latent roots of the estimated covariance matrix. Ann. matw.

Statist. 36, 1153-1173.

[2] Box, G. E. P., and Watson, G. A. (1962). Robustness to non-normality

of regresion tests. Blometrfla 49, 93-106.

[3] Chattopadhyay, A. K., PllUal, K. C. 3., and U, S. C. (1976). Kaximi-

zation of an integral of a matrix funn and asymptotic expansions

of dis~lbutios of latent roots of two matrices. Ann. Statist. 4,

796-606.

(4] Condon, E. U. (1956). Kinematcs, Chapter 2 of Rabndook of Physcs (ed.

E. U. Condon A H. Odishaw). New York: McGraw-Hill.

(5] David, P. N., and Johnson, N. L. (1951a). A method of investigating

the effect of non-normality and heterogeneity of variance on tests of

the general linear hypothesis. Ann. Math. Statist. 22, 332-392.

[6] David, F. N., and Johmson, N. L. (1951b). The effect of non-normality

on the power function of the P test in the analysis of variance. Bio

mefrfta 38, 43-57.

(7] EuIer, L. (1776). Novi Comment. Petrop 20, p. 206 ff. (Reference from

Whttakker, 1927, p. 8.)

[] rerrar, W. L. (1950). Algebra. Oxfordt Clarendon Pro.

E9] Flury, B. N., and Gaut chi, W. (1936). An algorithm for simultaneous

orthogonal transformation of sveral positive definite symmetric matrices

to neawly diagonal form. SIAM J. S€. Comput. 7, 169-184.

10] Ganitmacher, P. R. (1959). Tre Theory of Matrices Z. New York:

15

Chelsea.

([.] Gayen, A. K. (1950). The diOribution Of the variance raio in random

samples of any size drawn from nonc-normal unverses. Biouetrikh 37,

236-255.

[12] Geary, R. C. (194.7). Tesing for normality. Biometrfka 34, 209-V42.

(13] Gel'iand, Z. M., NMilos, R. A., and Shapiro, Z. Ya. (1963). Represen-

tations of the Rotation and Lorentz Groups and their Applications.

Naw York: Macmfllan; Oxdord: Pergamon.

[14] Graybil, F. A. (1961). An Zatroduction to Liahmr Statistical Models,

Vol. I. New York: McGraw-Hll.

[15] Hartre, D. R. (1955). Numerical Analysis. Oidord Clarendon resm.

[16] Hoffman, D. K., Raffenetl, R. C., and Ruedenberg, K. (1972). Goner-

ailsn m of Euler angles to N-dUmminal othogonal matrices. J.

Ma. Phys. 13, 528-533.

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