q(l;f) = ^jtf(xn)

mathematics of computationvolume 58, number 197january 1992, pages 305-314

THE EXISTENCE OF EFFICIENT LATTICE RULESFOR MULTIDIMENSIONAL NUMERICAL INTEGRATION

HARALD NIEDERREITER

Dedicated to Professor Edmund Hlawka on the occasion of his 75th birthday

Abstract. In this contribution to the theory of lattice rules for multidimen-

sional numerical integration, we first establish bounds for various efficiency

measures which lead to the conclusion that in the search for efficient lattice

rules one should concentrate on lattice rules with large first invariant. Then we

prove an existence theorem for efficient lattice rules of rank 2 with prescribed

invariants, which extends an earlier result of the author for lattice rules of

rank 1.

1. Introduction

For s > 2 an s-dimensional lattice is the set of all linear combinations

with integer coefficients of s linearly independent vectors in R5. We only

consider lattices which contain Zs as a sublattice. If L is such a lattice, then

L n [0, X)s is a finite set consisting, say, of the distinct points Xt,..., x#. The

5-dimensional lattice rule corresponding to L (or, by a slight abuse of language,

the lattice rule L) approximates the integral /(/) of a function / over [0, 1]J

by

Q(L;f) = ̂ JTf(xn).n=l

We write X(L) = L n [0, 1)J = {xi, ..., xjv} for the set of nodes in the latticerule L. If we want to emphasize that the number of nodes in a lattice rule is

N, then we speak of an N-point lattice rule. To avoid a trivial case, we always

assume that N > 2.Lattice rules were originally designed for the numerical integration of periodic

functions having [0, X]s as their period interval, and they were introduced by

Sloan [15] and Sloan and Kachoyan [16]. Later, the applicability of latticerules was extended to nonperiodic integrands by Niederreiter and Sloan [14].

A special class of lattice rules has been known for a long time as the method

of good lattice points, which goes back to Korobov [5] and Hlawka [3]. Werefer to Lyness [9] for a recent survey of lattice rules and to Hua and Wang [4]

and Niederreiter [11, 13] for expository accounts of the method of good lattice

points.

Received April 12, 1990; revised January 17, 1991.1991 Mathematics Subject Classification. Primary 65D32; Secondary 11K45.

© 1992 American Mathematical Society0025-5718/92 $1.00 + $.25 per page

305

License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use

306 HARALD NIEDERREITER

An important classification of lattice rules was established by Sloan and Ly-

ness [18]. They showed that for any s-dimensional lattice rule L there exist a

uniquely determined integer r (called the rank) with X < r < s and positive

integers «i,..., nr (called the invariants) with «,+i|«, for i = 1, ... , r - 1

and «r > 1 such that the node set X(L) consists exactly of all fractional parts

(1) ^¿"Z'f with 1 <£,<«, for 1 </<r,

and with suitable zi,... , xr G Zs. Here the fractional part {t} of t =

(tx, ... , ts) e Rs is defined by

{t} = ({tx},...,{ts})e[0,xy,

where {t} = t - [t\ for t e R. The points listed in (1) are all distinct, and so

the number N of nodes satisfies N - nx ■ • ■ nr. The lattice rules that are used

in the method of good lattice points are precisely the lattice rules of rank 1.

In the present paper we will, first of all, present evidence that in the search

for efficient lattice rules one should concentrate on lattice rules with large first

invariant «i (see §2). Then we will prove an existence theorem for lattice rules

of rank 2 which shows what kind of efficiency one can expect if the invariants

«i and «2 are prescribed (see §3). This theorem can be viewed as an extension

of the existence theorem for efficient lattice rules of rank 1 in Niederreiter [12].

To assess the efficiency of lattice rules, we use a standard procedure in nu-

merical integration, namely to consider the order of magnitude of error bounds

(for suitable classes of integrands) in terms of the number 7Y of nodes. To de-

scribe the most important error bounds, we introduce the following definitions

and notations.

Definition 1. The dual lattice L1- of a lattice L is defined by

Lx = {h e R* : h • x e Z for all x e L},

where h • x denotes the standard inner product of h and x.

For a lattice rule L we have ¿DZ1, and so it follows that L± ç Zs. For

« e Z we put r(h) = max(l, |«|), and for h = («i, ... , «s) e Zs we put

Kh) = n<=i'-(A.).Definition 2. For any lattice rule L and for any real a > X define

he// v ;M>

Now suppose that the integrand / is periodic with period interval [0, 1]*and that / is represented by its absolutely convergent Fourier series with Fourier

coefficients /(h) satisfying /(h) = 0(r(h)~Q) for some a > X. Then Sloan

and Kachoyan [ 17] have shown the error bound

(2) Q(L;f)-I(f) = 0(Ra(L)),

where the implied constant depends only on /. Thus, an efficient lattice rule

should have a small value of Ra(L). To get a criterion independent of a, we


LATTICE RULES FOR MULTIDIMENSIONAL NUMERICAL INTEGRATION 307

introduce the quantity RX(L). Put

C(N) = {« e Z: - N/2 < « < N/2}, C*(N) = C(N)\{0},Q(N) = {(«i ,...,hs)eZs: hi e C(N) for 1 < i < s},

Q(N) = Q(N)\{Q}.

Definition 3. For any /V-point lattice rule L let

he£(L) v ;

where E(L) = Q(N) ni1.

Note that E(L) is nonempty by [14, Proposition 3]. We remark that in

the definition of RX(L) we cannot use the same range of summation as in the

definition of Ra(L), a > X, since the resulting infinite series would diverge.

The advantage of RX(L) is that all quantities Ra(L), a > X, can be bounded

in terms of Rx (L). In fact, in Theorem 1 we will show that Ra(L) = 0(RX (L)a)

for all a > X. Thus, a small value of RX(L) guarantees small values of Ra(L)

for all a > X.

Definition 4. The figure of merit p(L) of a lattice rule L is defined by

p(L) = min r(h).heLxM)

The quantities Ra(L), a > X, can be bounded in terms of the figure of

merit p(L). For a = X we note first that we also have p(L) = minh6£(L) r(h)

with E(L) as in Definition 3, according to [14, Proposition 1]. Hence, for any

A/-point lattice rule L we have

(3) -L-<Rl(L) = 0^l^sp(L) - ,v ' V P{L)

where the upper bound was shown in the proof of [14, Theorem 2]. For a > X

we have

W -4-</?a(L) = 0((1+1°^^"'Vp(L)° aK ' V P(L)a J

where the upper bound was shown in the proof of [17, Theorem 4]. The im-

plied constants in (3) and (4) depend only on a and 5. An error bound for

nonperiodic integrands is based on the following notion.

Definitions. The discrepancy D(L) of the node set X(L) of an TV-point lattice

rule L is defined by

| card(*(L) n /)D(L) = sup

jVol(7)

N

where the supremum is extended over all half-open subintervals / of [0, 1 )s

of the form J = n¡=i[M¡ > vi) and where Vol(/) denotes the volume of /.

Now let the integrand / be of bounded variation on [0, 1]* in the sense of

Hardy and Krause. Then we have

(5) Q(L;f)-I(f) = 0(D(L))



by the Koksma-Hlawka inequality (see [6, Chapter 2]), where the implied con-

stant depends only on /. By combining Theorem 1 in [14] and inequality (4)

in [14], we obtain

(6) D(L)<s/N+±Rx(L)

for any A/-point lattice rule L. Therefore, a small value of RX(L) guarantees a

small discrepancy D(L) and thus a small error bound in (5). The discrepancy

D(L) can also be bounded in terms of the figure of merit p(L). According to

results in [14] we have

c, ^ „,T^c's(XoëNy< D(L) <

p(L) - * ' - p(L)

for any TV-point lattice rule L, where the positive constants cs and c's depend

only on 5.

In view of the results above, we see that an efficient lattice rule L can be

characterized as having a small value of RX(L) or a large value of p(L), and

that these two characterizations are basically equivalent because of (3). For

a detailed discussion of various ways of assessing the efficiency of integration

rules such as lattice rules we refer to Lyness [8].

2. Some simple bounds

We show first that the quantities Ra(L), a > X, can be bounded in terms of

RX(L). Let £(a) = ]C/¡ti h~a > a > 1, be the Riemann zeta-function.

Theorem 1. For any TV-point lattice rule L and any a > X we have

Ra(L) <(X + 2C(a)N~ay - X + (X + 2Ç(a)yRx(L)°.

Proof. By [17, Lemma 1] we have Nx e Zs for all x e L, hence L1- contains

TVZ*. We write

(7) Ra(L)= £ r(h)-"+ Y, Kb)-a=:E,+E2-

Now

h£NZs h€Z,J-\.iVZs

E, = £ r(h)~a - 1 = Y, r(Nhx)-a---r(Nhs)-a-X

h€iVZs h,.hs£Z

= Í£r(TV«H -l = íl+2¿(/V«H -1

= (l + 2Ç(a)A/-a)i-l.

To bound Y,2 > we use that every h ^ NZS can be uniquely represented in the

form h = k + TVm with k G C*(N) and meZ1. We have heL1 if and onlyif k e L1-. Thus, the h e L-L\ArZî are exactly given by all points k + TVm withk 6 E(L) and msZ1, where E(L) is as in Definition 3. Therefore,

^2=E E r(k + Nm)-a.m€Zs YeE(L)

We claim that

(8) r(k + TVm) > r(k)r(m) for k e CS(N), m e Zs.



It suffices to show

r(k + Nm) > r(k)r(m) for k e C(N), m e Z.

This inequality is trivial whenever km = 0. If km ^0, then

TVr(k + Nm) = \k + Nm\ > N\m\ - \k\ > N\m\ - —

TV= y(2|w| - 1) > |A:| |ra| = r(k)r(m).

Thus (8) is proved. Using (8), we get

£2 < ¿2 ¿2 r(m)-ar(k)-°meZ* ke£(L)

= ( y r(m)~a) ( E r(krQ) =(i+2C(a))s E r(k)"Q

\m€ZJ / \ke£(L) / V€E{L)

<(X+2C(a)y\ > r(k)-M =(X + 2C(a)yRx(Lr.

In view of (7), this establishes the result. D

We note that (1 + 2Ç(a)7V_a)i - 1 < c(s, a)N~a with a constant c(s, a)

depending only on s and a. Furthermore, we have RX(L) > p(L)~l > TV-1,

where the first inequality is obtained from (3) and the second inequality follows

from the bound p(L) < TV shown in [14, Proposition 2]. Hence, Theorem 1

yields Ra(L) = 0(Rx(L)a) with an implied constant depending only on s and

a.

We consider now lattice rules of arbitrary rank r and with invariants «1, ... ,

nr as described in § 1. Note that «1 is the largest invariant. The following result

is an improvement on the bound p(L) < N = nx ■ • • nr mentioned above.

Proposition 1. We always have p(L) < nx.

Proof. Since the invariants n2, ... ,nr are divisors of «1, it follows from thedescription of the node set X(L) in (1) that the coordinates of all points of Lare rationals with denominator «1 . Therefore Lx contains «iZs. In particu-

lar, we have h0 = («1, 0, ... , 0) e LL , hence p(L) < r(hn) = «1. D

The argument in the proof of Proposition 1 also yields general lower bounds

for the quantities Ra(L), a > X. Here and later on, we use the expression

(9) S(m) = Y \n\~l for integers m > X,heC'(rn)

where for m = 1 we use the standard convention that an empty sum has the

value 0. The following result is obtained from [12, Lemmas 1 and 2].

Lemma 1. For any m > X we have

S(m) = 2 log m + C + e(m) with \e(m)\ < 4/m2,

where C = 2y - log4 = -0.23 • ■ • with y being the Euler-Mascheroni constant.



Proposition 2. For any N-point lattice rule we have

Ra(L)>(X+2Ç(a)nx-ay-X fora>X,

Proof. We have shown in the proof of Proposition 1 that LL D nxZs. Thus

for a > X,

Ra(L)> Y r(h)~a= Y r(h)-a-X = (X+2i;(a)nx-ay-X,

hen,Zs heniZ/

where the last identity is shown like the formula for £ i in the proof of Theorem

1. For a = X we note that E(L) D C*(N) n (nxZs) and that the elements of

the latter set are exactly all points «ih with he C*(TV/«i). Therefore,

Rx(L)> Y rCnih)-1 = E r(nxhTl-lh6C;(V/«,) h€Cs(N/ni)

Y r(nxh)-1) -l=(l + l Y \h\~l) -1^6C(JV/n,) / \ ' h€C'(N/m) )

-(•♦MS)'—It follows from Proposition 2 and Lemma 1 that if factors depending only on

5 and a are suppressed, then Ra(L) is at least of the order of magnitude nxa

for a > X and at least of the order of magnitude nxl Xog(N/nx) for a = 1.

For the discrepancy D(L) we have the following general lower bound.

Proposition 3. We always have D(L) > X/nx .

Proof. By the proof of Proposition 1, the coordinates of all points of X(L)

are rationals with denominator «i . For 0 < e < «j"1 let Je be the interval

[e, n\~x) x [0, l)i_1. Then Jc contains no point of X(L), and so

D(L)> card(X(L)nJA_Yol{jE)TV

Letting e —> 0+ we get the desired result. D

= Vol(/e) = -j- - e."i

The bounds established in the propositions above basically carry the same

information, but the consequences are displayed most clearly by Proposition 1.

To assess the efficiency of a lattice rule, one has to relate the figure of meritp(L) to the number TV = «i • • • «r of nodes. For lattice rules of rank r > 2,

Proposition 1 says that the "relative" figure of merit p(L)/N satisfies

00) m<_ 'TV «2 • • • «r

We want p(L)/N to be as large as possible for an efficient lattice rule, but (10)shows that this becomes more unlikely the larger the invariants n2, ... , nr.

Indeed, (10) suggests that if we want to look for efficient lattice rules of rank

> 2, then our best bet is to consider lattice rules with large first invariant «i. In



particular, we could consider lattice rules of rank 2 with small second invariant

«2 . This is also supported by the results of the explicit search for efficient latticerules carried out by Sloan and Walsh [20] which yielded lattice rules of preciselythis type.

We will now concentrate on lattice rules of rank 2. In §3 we establish results

which show the existence of lattice rules L of rank 2 for which the quanti-

ties Ra(L), a > 1, are small, and these results are the better the smaller theinvariant «2.

3. Existence theorems for lattice rules of rank 2

We consider lattice rules which have a useful additional property, namely

that of projection regularity. If L is an s-dimensional lattice rule with node

set X(L), then for 1 < d < s we define Xd(L) to be the subset of [0, X)dobtained by retaining only the first d coordinates of each point of X(L). Then

L is called projection regular if card(Xd(L)) = n\ • < • n¿ for 1 < d < r, where

r is the rank and «1, ... , nr are the invariants of L. A characterization ofprojection-regular lattice rules was given by Sloan and Lyness [19].

For lattice rules of rank 1 a general existence theorem for efficient lattice rules

was established in Niederreiter [12]. It was shown that for every dimension

s > 2 and every integer TV > 2 there exists a projection-regular TV-point latticerule L of rank 1 with

(11) Rx(L) = 0(N-l(XogNy),

where the implied constant depends only on 5. This result is in fact best

possible since it was proved by Larcher [7] that for any TV-point lattice rule L

of rank 1, Rx(L) is at least of the order of magnitude TV~'(logTV)*.

We now establish an analogous existence theorem for lattice rules of rank 2.

We recall that for such lattice rules we have two invariants «1 > 1 and «2 > 1

with «2|«i, and the number TV of nodes is given by TV = «i«2. For a detailed

discussion of lattice rules of rank 2, see Lyness and Sloan [10]. We now fix the

dimension s >2 and the invariants «1 and «2, and we put

Z, = {z G Z: 0 < z < «, and gcd(z, «,) =1} for / = 1, 2.

Let Sf = ^(s; «1, «2) be the family of all ¿-dimensional lattice rules L ofrank 2 with prescribed invariants «1 and «2 for which the node set X(L)consists exactly of all fractional parts

< —Z] + —z2 \ with 1 < kx < «1, 1 < k2 < «2l"i "2 J

as in (1), where zi and Z2 have the special form

(12) z, = (z[X),...,z\s)), z2 = (0,zf,...,zf)

with z, e Zi, X < j < s, and z2;) G Z2, 2 < j < s. It follows immediatelyfrom [19, Theorems 2.1 and 3.3] that every lattice rule L G Sf is projectionregular. For each LgÍ' the corresponding lattice in Rs consists exactly of

all linear combinations with integer coefficients of the vectors

b, = — z¡ for i = X, 2, b, = e, for 3 < i < s,«i



where e, is the unit vector with 1 in the z'th coordinate and 0 elsewhere. Let

v ; Les?

be the average value of T?i(L) as L runs through Sf. Note that card(J?) =

<f>(nx)s4>(n2y~l, where <j> is Euler's totient function.

Theorem 2. For every dimension s > 2 and any prescribed invariants nx and

«2 we have

M^)<c,(S^r + ̂ L)

with a constant cs depending only on s. In particular, for every s >2 and any

«i and «2 there exists a projection-regular s-dimensional lattice rule L of rank

2 with invariants nx and n2 such that

Corollary 1. For every s > 2 and any prescribed invariants nx and n2 there

exists a projection-regular s-dimensional lattice rule L of rank 2 with invariants

«i and «2 such that

Ra(L)<c(s,a)^fl + ^y foralla>X,

where the constant c(s, a) depends only on s and a.

Proof. This follows from Theorems 1 and 2. D

Corollary 2. For every s > 2 and any prescribed invariants nx and n2 there

exists a projection-regular s-dimensional lattice rule L of rank 2 with invariants«i and «2 such that the discrepancy of the node set satisfies

with a constant cs depending only on s.

Proof. This follows from (6) and Theorem 2. G

These results guarantee the existence of efficient lattice rules provided that

the invariant «i is sufficiently large, or equivalently, that the invariant «2 issufficiently small (if lattice rules with the same number N = nxn2 of nodes are

compared). This is in accordance with a conclusion that was reached in §2 by

different arguments, namely that among lattice rules of rank 2 the most likely

candidates for efficient lattice rules are those with small second invariant «2.In view of (2) and (5), the bounds in Corollaries 1 and 2 yield information on

the error bounds that can be achieved for suitable lattice rules L g Sf .

If we consider the order of magnitude of the bound for RX(L) in Theorem 2,

then we observe that the first term TV-•(logTV)5 is the same as the best possible

order of magnitude of RX(L) for lattice rules of rank 1 (compare with (11) and

the remarks following it). The second term «j-1 log TV is nearly best possible

since it follows from the remarks after Proposition 2 that Rx(L) is at least

of the order of magnitude «¡"1log«2. If «1 « «2 (i.e., if «1 and «2 are of



the same order of magnitude), then the term «j-1 log TV is in fact best possible,

since we then have log TV = log(«i«2) « log«j « log «2 . If powers of log TV are

ignored, then the bounds for Ra(L), a > X, and D(L) in Corollaries 1 and 2,

respectively, are best possible, since Ra(L) and D(L) are at least of the order

of magnitude «¡~a and «f1, respectively, by results in §2.

We emphasize that Theorem 2 provides an upper bound for the average value

of RX(L) as L runs through the family Sf. This means that the bound for

Rx(L) in Theorem 2 is met by "random" choices of L G S?. This has the

following practical implication when searching for efficient lattice rules of rank

2 with prescribed invariants: choose lattice rules Le^ "at random," thenthere is a good chance that after a reasonably small number of trials a latticerule can be found for which the bounds in Theorem 2 and Corollaries 1 and 2

hold. For lattice rules of rank 1 this "randomized" search procedure was already

suggested by Haber [1].The rest of the paper, which can be found in the Supplement section of this

issue, is devoted to the proof of Theorem 2. In §4 we establish some auxiliary

results that are needed for the proof, and in §5 we complete the proof. The

basic ideas of the proof would also work for lattice rules of rank > 2, but the

details become exceedingly more complicated.

Bibliography

1. S. Haber, Parameters for integrating periodic functions of several variables, Math. Comp. 41

(1983), 115-129.

2. G. H. Hardy and E. M. Wright, An introduction to the theory of numbers, 4th ed., ClarendonPress, Oxford, 1960.

3. E. Hlawka, Zur angenäherten Berechnung mehrfacher Integrale, Monatsh. Math. 66 (1962),

140-151.

4. L. K. Hua and Y. Wang, Applications of number theory to numerical analysis, Springer,Berlin, 1981.

5. N. M. Korobov, The approximate compulation of multiple integrals, Dokl. Akad. Nauk

SSSR 124 (1959), 1207-1210. (Russian)

6. L. Kuipers and H. Niederreiter, Uniform distribution of sequences, Wiley, New York, 1974.

7. G. Larcher, A best lower bound for good lattice points, Monatsh. Math. 104 ( 1987), 45-51.

8. J. N. Lyness, Some comments on quadrature rule construction criteria, Numerical Integration

III (H. Braß and G. Hämmerlin, eds.), ISNM85, Birkhäuser, Basel, 1988, pp. 117-129.

9. _, An introduction to lattice rules and their generator matrices, IMA J. Numer. Anal. 9

(1989), 405-419.

10. J. N. Lyness and I. H. Sloan, Some properties of rank-2 lattice rules, Math. Comp. 53

(1989), 627-637.

11. H. Niederreiter, Quasi-Monte Carlo methods and pseudo-random numbers, Bull. Amer.

Math. Soc. 84 (1978), 957-1041.

12. -, Existence of good lattice points in the sense of Hlawka, Monatsh. Math. 86 (1978),203-219.

13. _, Quasi-Monte Carlo methods for multidimensional numerical integration, Numerical

Integration III (H. Braß and G. Hämmerlin, eds.), ISNM85, Birkhäuser, Basel, 1988, pp.157-171.

14. H. Niederreiter and I. H. Sloan, Lattice rules for multiple integration and discrepancy, Math.Comp. 54(1990), 303-312.

15. I. H. Sloan, Lattice methods for multiple integration, J. Comput. Appl. Math. 12/13 (1985),131-143.



16. I. H. Sloan and P. J. Kachoyan, Lattices for multiple integration, Proc. Centre Math. Anal.

Austral. Nat. Univ., vol. 6, Austral. Nat. Univ., Canberra, 1984, pp. 147-165.

17._, Lattice methods for multiple integration: theory, error analysis and examples, SIAM

J. Numer. Anal. 24 (1987), 116-128.

18. I. H. Sloan and J. N. Lyness, The representation of lattice quadrature rules as multiple sums,

Math. Comp. 52 (1989), 81-94.

19. _, Lattice rules: projection regularity and unique representations, Math. Comp. 54 ( 1990),

649-660.

20. I. H. Sloan and L. Walsh, A computer search of rank-2 lattice rules for multidimensional

quadrature, Math. Comp. 54 (1990), 281-302.

Institute for Information Processing, Austrian Academy of Sciences, Sonnenfels-

gasse 19, A-1010 Vienna, AustriaE-mail address : [email protected]


mathematics of computationvolume 58, number 197january 1992, pages s7-s16

Supplement to

THE EXISTENCE OF EFFICIENT LATTICE RULESFOR MULTIDIMENSIONAL NUMERICAL INTEGRATION

HARALD NIEDERREITER

This supplement contains the proof of Theorem 2 in the main part of the paper. In §4

we establish some auxiliary number-theoretic results that are needed for the proof, and in

§5 we complete the proof. The numbering of lemmas and equations is continued from the

main part and the references axe to the bibliography in the main part.

4. AUXILIARY RESULTS

We need various concepts and results from number theory (see [2] as a general refer-

ence). Recall that a function F on the set N of positive integers is called multiplicative if

F(mn) = F(m)F(n) whenever gcd(rrt,n) = 1. We write J2d\n ^or a sum over ^ positive

divisors d of n € N. If F and G are multiplicative functions, then the summatory function

Y^i\n F(d) °f F anc^ ̂ e Dhichlet convolution J2d\n F(n/d)G(d) of F and G are multiplica-

tive functions of n £ N. Let p. be the Möbius function and note that p. is multiplicative.

From now on we abbreviate gcd(m, n) by (m, n). The symbol p will always denote a prime

number. For n € N let tf(n) be the largest exponent such that pc'^\n (if p\n, then

ep(n) = 0).

Lemma 2. For k,m,n € N we have

fl<»-.»>-£^)«*)(».^H0«"> »Sa

Proof. For dgNwe have

(13) WN-«*)(^^)-(M)(^)

since d/(d, k) and k/(d, k) are coprime. Therefore,

B(k,m,n) = ^/¿(-)(d,fcm).'d>

din

© 1992 American Mathematical Society

0025-5718/92 $ 1.00 + $.25 per page

S7


SS SUPPLEMENT

-3

tí

o ■

a»

■a*

1•g

vi S

o **

t. °-E i

M2C

o

*, hio,

P §IIy5"!

il

cl

gh

° I

H .S

33

a fe,

s -S -s -S« -3

_ aE J3

Ü'sST

íTpw

h

i?

5"

fe

ST

ii'S"-1

3

Z

e

Z

S»,

E

g * g01 -2 -a

» e §■r S m"p . ïS * .so fl

O S *.a «

Il ~ S

a."3 £

^ Êg-tE ¿Il

JL il!"° «4-1 2 w

A * S -

B,

6Je'

5"

uï

Zw

ES

?" « eI J3 ^** 0

K e

■Sas.

Il g

U X ? S

QfQf

Il "*.

*kï

eII

a

•S

•3 ä

g jes ?

= o.

'Rrg

g M S?

» -il5 *- £ *

■g o | ~= a-

■- % " ~^

» i,^cy 11

H h aîl3,

s«5c> ¡ g"Ö _; m " S• g o .S „ ¿3• -3 11 V M *>-3 II «, ^ q

^jh S O ¡

t-8l8ïTau a^ _

^ 9 | ¿ »

Sa d Ö -5Xi .5 o as -3

II

E

%

II

e"

E

Ê"1Il "

e"'Z, vi

•Si

• n CD

Jîo >v v

=3?


SUPPLEMENT S9

II

as& SJ|g oc.

■o■3"

•* .2

~? -oA S

A-o

te

M 1)

00

sir*~ ** 9■o ^;'fl■B*¡X¡ t> m

S? " «

a c v

"a O

3 -=.; »^•S S a« 'G -S

a *i, td ft,

A s'l

I■ËSu

e lis

II

e"

E

íl

a Ja

E¿

E

o?

W4

Eto *

2 5"S. ?

►3a

II

ir6

z

E_

Sí

— a. i

5 o.-

VI

í^* oT .a

E

II

's"E

« a -«

t. Z 2

^ ^ M•3 í .3

-2=3 §

•fl ^ 00

9 a 3

E^

111

Èá S

gÜ11

elTJ

ztil

C

.3

K

I18ST•o■5"

»3. ¡

3.

tu

£2

3o

Sz

A a

II "3

b, a

¿P"■*

I*-"ï7 "o."a IIII í~~

„ a.

* jf

3 s■o Sai.

ll V

3-11

¿*

\/l <-> «VI u— 8 -§S3 ̂ 5 ÏÏ

w-

6Ä ,


SIO SUPPLEMENT

% ■*

i S

s i;E c

I ¡?e <;

I 1« *..9 >.

II

¡aÔio

¡¡5

w<•a3

1

Of

■p

¡3

i* 1

Cl

o.

c4

e?

o,

c4

- si-

o; vr

o JB

■J3

„ I II i.+

B,

+ ^S B.M

+

Il B.+

"»If if"-B. »■

W'

B. |

II "a.

"o. -»

Er

bObO h

E

Sí

;. » H H

o. S - o

<l? > & a

*„ 02a c a oJ-í H -a

2 ■« E ■—u Íj a «

S « S <3£ u ä .a

»o. -a a na -t>

í A 1a

1

.2 Í3 »•f8 - "3 zco; ■ «¡CU«

v|SSD"ü — _ ■

!~í "3 **- í¡■as.i §

¡si» I•-■a ¿ u

ï! c ■

il

6

I

cS

-s.

S

II

c

o

or


SUPPLEMENT Sil

^2

■S1

¿3

^T- 3!S -a2- H

OCN

m «"^. e ft-2 " g.

»I «* ¿ l g 8 I •M a "* b ü tí*3

Sî s ^"9 u-*s .3 «-;g s *v¡ as•TJ 2 - " Ja ,

_ a X -g s H!i C ™ u u Í3

S S S!t3-a l"^J.f, 'E -a s-, croc*':■S a. _:£* m g

8 §•?? a<

.3 g

lA

o "* J-is .a a

<ScS ̂CÎ ;

»V

B, 03-~ .a

o.rcS 9

s C"

e

c? e,*Ow ^

Í3 «

5e "OO V

; C?

í B.Í

a,

25

r»Il * ..

iif g c5•° "s

xi , Mh

i^°

K|sr? gi

2@1•Il

-^ tu a>

.~ -° j

s § *.a O „

1 * +

S <a \"

°1.2 .o

-a 2 S■» g aüs 82

Qr

il

¡2

+

t. ==2 II

22(a. B.

U U

»1,

«•N

a,

c4

ri «1.9»

|£»i w "H-

: VI-5

*. c '

B.

2 -i»-

W€ Jl^ 2

W,

"a. o.C- oj

2 -aB.

- l-s .9

Cl

S 1 l 13 j¿

1


S12 SUPPLEMENT

.S

1

Or

0 K|

1 Sq.

il

¡s?

faß I -ï2 i °-la a;

W"

o.

or

S.2U cö

ja i«

Ca-ll

c?

c?

O

i9

ii

00

3_

o,| r,a Its

5? I •O

W|W|

II

fe -pn =9

3

Sa.

Wi

4!

zHIc

a.

W4 5

-B.US

D*i2■E1g8

+al

-a'S

9 g

.2 Z

li

3H

c

2

VI

"i?ST

-tvs

ii ¿" a

2 Q>Si

- a" jí so ,2f . a

II

lis

1 £ H|t»Äi § £

-lt. °

Cf

2 O

Qf

■a

*■32

M o■2 ||SJ pj

I*5 -o £ x

WÍ1

ero cl-o

wl wí

1 h, tu d

Ja j3

' -w v2*

i" °-£ï

e l-o

w*

J3

g g"

n •§

B -a

If■b|.

S.'SS "*=-d _i

113,2"^ K

■s sf ■*18■^l^=9g a6 810 v"â «

1 5

II

W4WI cle e

C? Ó

KÍ «Til» i1<5

c

6

c|<S


SUPPLEMENT SI3

Il AC C

¿"I e

SwiJi"|-o

ÎW!

wi w^

■«1 e

ÏWÏII

N

W5

[Wî

il

£

B«¿3 3

5Ö

.2 -a

S1OJ >■S «

a^3S ~a

1 "IS OJ■S Ja

f

M

ÍBa.

Wl

w?

y.

ow

H

O

OOKP-

■sz

Si1 gI-2 S

=9 J

JäJS

il

WsWS

Wí

JJ§

!I3

■3

wl

ja

eB

1 a

si

wl

II

N

J=

IE

H H>-) M

OJ TUL

M J3

S3 S3

-H O

W*x

H*

X

-a

WsWS

■a

H «5

« 25 9^ s

n °

n| m« I C

+

H

¡rlê•* I e•WI*w»

."WÍ îW]

.-WS .-W]

WE

& 2


S14 SUPPLEMENT

O"e

= S

22o +>

I sII1-8B, ï

i.»

Ala

!=3

e

CM

I O

S3

c?

e

Sh

tS

a M

w?i

+a;

e

o-

^3c? i

■S 43

•3.9

2 i*

.-i o,

S Al

¡s.

I

+

fei

- 00

* CMC ers

•* .5

S23 ja

5=3

2 ¡3=3

2 *9

A Scj et;

«2

a ja

S i?S 92 -ailII

ja *

», 6

rh>

3

1

■>

fCï

-lï3^ .o

Wï

"B S

*ï H

Jl1H•* -o -¿■- I «

1,5 i?2

g 9 i

9.1^

•Jt-a-s-o ■<= <»i-J

jf$ h

F-T J -^

': ¡s

fe

I

+fe.

WS I

•e

53 WE5

i

e \S

a.

a- 1^

CT^ vie l-a ^j

W^ i

wiwifW; VI


SUPPLEMENT S15

-a

o c2 -*

S -o"g S

+Ci

j-a-S

J) T) O^ tU 'Tí

¡alÍ 5 '

, ta

S*

¿s jaH 9

M O -Ü -g

S « o o

-1 S 2

:» S.

11

o>

W^

2 B.§

22-9•a 2 «oa o ^

.0 js "«-2 a ¿j

O ^ ci

^ a «> J3

J S g g

-•*••* u ■

9 9 oc?TW II II

c c<5<5

2 fëCJ I

+

» c fe;|

2 = -g

M

§■

J

J3

U

i

si w

2.3i S0 03

s la1"

2 .S

Wï

w§

rie" s

ï2

•° 2■a 9

2 g¡E 03

i» V

w «

2 O

¡sà

3 "S


S16 SUPPLEMENT

E

fe

I%

ja

í

fe;a

[WI

W,'

•6-

W^

WlE

fe?

¿Wl I+ ?

W4

H

Z

feja

+

fe.

+ £

V

Wl

4 W^V

er

u «

i- 2

'-S Sl- S

CD V

>9~

OJ T3

Ja J3S -G

¡•s g

V

Ci

t


q(l;f) = ^jtf(xn)

Documents