Tilburg University

A new triangulation of the unit simplex for computing economic equilibria

Dang, C.; Talman, A.J.J.

Publication date:1989

Link to publication in Tilburg University Research Portal

Citation for published version (APA):Dang, C., & Talman, A. J. J. (1989). A new triangulation of the unit simplex for computing economic equilibria.(CentER Discussion Paper; Vol. 1989-35). Unknown Publisher.

General rightsCopyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright ownersand it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights.

• Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain • You may freely distribute the URL identifying the publication in the public portal

Take down policyIf you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediatelyand investigate your claim.

Download date: 27. Apr. 2021

~~IIZ Discussionfor a erEconomic Research

35

imi iiiii i~ii i iu iiuui ~niuuiNU m iN i

No. 8935A NEW TRIANGULATION OF THE UNIT SIMPLEX

FOR COI~UTING ECONOMIC EQUILIBRIA~. ..-.

by Chuangyin Dangand Dolf Talman ~ ~~ ~ ( ~ ~

July, 1989

1

A NBii TRIANGULATION OF TF~ UNIT SIl~LER

FOR COl~UTING ECONOMLIC EQUILIBRIA

Chuangyin DANG and Dolf TALMAN

Tilburg, The Netherlands

ABSTRACT - In order to compute economic equilibrie in an exchange

economy by means of a simplicial algorithm, we propose a new

simpliciel subdivision of the underlying price space, being the unit

simplex. We call this subdivísion the Ti-triangulation. We show that

the Ti-triangulation is superior to the well lrnown Ki- and Ji-

triangulations according to measures of efficiency of

triangulations.

1. INTRODUCTION

Since van der Laan and Talman [8] proposed the first simplicial

variable dimenaion algorithm without an extra dimension, a lot of

contributions to this field have appeared, e.g., see [3], [4], [6],

[8], and [11]. These algorithms can be used to find en economic

equilibrium in a pure exchange economy. They subdivide the price

space, being the n-dimensional unit simplex Sn, into simplices and

search for a simplex that yields an approximate equilibrium. So far,

we might say that the triengulations which underly all these

elgorithms are adaptions or generalizations to Sn of the well lmown

K1- and J1-triangulations of Rn. In [2], we proposed a new

2

simplicial subdivision of Rn, called the D1-triangulation, which ia

superior to the K1- end J1-triangulations. In order to improve the

algorithms on Sn, we want to construct a simpliciel subdivision of

Sn by using the D1-triangulation, which is auperior to the other

triangulations of Sn.

Section 2 introduces the new triangulation of Sn, called the

Ti-triangulation. We give the pivot rules of the Ti-triangulation in

section 3. It is compared with the Ki- and Ji-triangulations in

section 4.

2. A NEW TRIANGULATION OF THE UNIT SIMPLEX

Let m be a positive 3nteger. Let Cn(m) -{x E Rn ~ m 2 xl 2 x2

2... 2 xn 2 0}. Clearly, Cn(m) is an n-dimensional simplex in Rn

We first triangulate Cn(m) into n-dimensional simplices and then

trensform Cn(m) to the n-dimensional unit simplex Sn ~{x E Rnrl I

~iix~ - 1, and x~ 2 0 for j- 1, ..., n41}. For h- 1, ..., n, let

uh denote the h-th unit vector in Rn. Let T~c -{y E Cn(m) ~ all

components of y are odd}. Let y0 - m and yn{1 - 0. Let y E TOc Let

I- I(Y) -{i ~ yirl ~ Yi ~ Yi-1}.

I} - I4(Y) -{i ~ Yi~l - Yi ~ yi-1}.

I- ' I-ÍY) -{i ~ Yiil ~ Yi L Yi-1}.and J- J(Y) -{i I Yi.l - yi s yi-1}.Let N-{1, 2, .. , n}. It is obvious that for each y E TOc~

N- I(Y) u J(Y) ~ I~ÍY) v I-(Y)-

3

Let s L(sl' s2' " ' sn)T be a sign vector such that ai E{-1, tl}

fori z 1, 2, .. , n. LetT~T(y, s) -{i ~ iEI, oriEI`andsi3 1, or i E I- and si ~ -1}. Let rt be a permutation of the elementaof N. Let q- q(n) be a nonnegative integer such that rt(i) E T for

i- n-q.l, ... , n and n(n-q) ~ T. Let 0 5 p 5 q-1 be an integer.Assume n 2 2.

If q- n and p - 0, let y~ a y,

yk s y i sn(k)urt(k) k- 1, 2. ..., n.IfqLnandpx 1, lety0ayta,

Yk - Yk-1- sR(k)urt(k) k- 1. 2. ... . P-1,

Yk - Y ~ sR(k)urt(k). k- P, .... n.

When25q( n, lety~syt s,

Yk - Yk-1 - srt(k)urt(k) k- 1, 2, .. , n-q-1,

and if p - 0, let yn-q ~ y,

yk s y~ srt(k)urt(k), k~ n-q~l, .. , n,

and if p 2 1, letYk L yk-1 - srt(k)urt(k) k- n-q. . ... n-9~P-1,

Yk - Y t s~(k)urt(k) k- n-q'P. ..., n.

Whenq C2, lety~-yts,

Yk - Yk-1 - sR(k)un(k) k z 1. 2, .... n.~t y0 yl~ yn ye p~uced in the above manner, then it ís

obvious that they are affinely independent. Let o:(y0 yl~ .~

yn] be the convex hull of y0 yl yn Thus cs is an n-dimensio-nal simplex, which will be denoted by T1(y, n, s, p, q). Let T1(m)

be the set of all such simplices T1(y, R, s, p, q) in Cn(m).

4

Lemma 2.1. ~ c- Cn(m),vET1(m)

Proof. Let x E Cn(m). Let~xi~ .l, if ~xi~ is even end ~xi~ ( m,

yi - ~xi~-1, if ~xi~ is even and lxi~ - m,~xi~ , otherwise,

and let

(-1, if ~xi~ is even and ~xiJ ( m or ~xi~ i s odd and ~xi~ - m,si I

1, otherwise,for i- 1, 2, .. , n. Obviously, y-(y1, y2, . . , yn)T E T~c. Let rtbe a permutation of N such that 0 5 sn(1)(xn(1)

- yn(1)) Ssrt(2)(xn(2) - yn(2)) 5... s sn(n)(xn(n) - Yn(n)) 5 1, where in case

sn(i)(xrt(i) - Yn(i)) - sn(~)(xn(~) - Yrt(~)) and i~ j . then n(i) (

rt(j) if sn(i) --1. and n(i) ~ n(j) if sn(i) - 1. Let 4- 4(rt).When q( 2, let y0 - Y t s. Yk - Yk-1 - sn(k)urt(k) k: 1 2~

.... n, and let q~ - sn(1)(xn(1) - Yn(1)). 91 - sn(2)(xrt(2) - yn(2))- sn(1)(xrt(1) - yn(1)). ... . qn-1 ' sn(n)(xn(n) - yn(n)) -

srt(n-1)(xrt(n-1) - Yn(n-1)). ~ - 1 - ~z~qk ~ 1 - sn(n)(xn(n) -

~.n kyrt(n)). Obviously, ~:Cqk - 1, qk z 0 for all k, and x~ Lk-Oqky 'So x E C' [YC. Y1. .. . Yn] ~ Cn(m).

When q - n, the proof is similar to that of Lemma 2.2 in [2].Suppose 2 s q( n. If

Lk-n-9sn(k)(xn(k) - Yn(k)) - qsrt(n-q)(xn(n-q) - Yn(n-q))5 1,

5

let q~ - sn(1)(xrt(1) - yn(1)), 91 - Sn(2)(xrt(2) - Yn(2)) -

sn(1)(xrt(1) - Yn(1))' " ' `~n-q-1 - art(n-q)(xrt(n-q) - Yrt(n-q)) -

srt(n-q-1)(xn(n-q-1) - Yn(n-q-1)). ~-9 - 1 - (Lk:n-9sn(k)(xn(k) -yR(k)) - qsrt(n-qI(xn(n-q) - yrt(n-q))). ~-9~1 - srt(n-941)(xn(n-q~l)- Yrt(n-qfl)) - srt(n-q)(xrt(n-q) - yrt(n-q))'

... , qn a srt(n)(xrt(n) -yR(n)) - sR(n-q)(xrt(n-q) - Yrt(n-q)). P- 0. YO ' Y~ s. Yk - Yk-1 -

srt(k)un(k) k- 1. 2. .... n-q-1. yn-9 : Y. Yk ' Y} sn(k)un(k)~ k 3

n-qal, .. , n, then qk 2 0 for all k, ~~Oqk - 1, and x~~~Cqkyk

Thus x E v- IYO. Y1. -... Yk~ C Cn(m). If

Lk-n-9srt(k)(xrt(k) - Yrt(k)) - 9sn(n-q)(xrt(n-q) - Yn(n-q)) ) 1,

we show that there exists an integer 1 5 p 5 q-1 such that the

following system has a nonnegative solution,

q0 - srt(1)(xR(1) ' YR(1)).

q1 - aR(2)(xR(2) - yR(2)) - sR(lI(xrt(1) ' Yn(1)).

qn-q'P-2 - sn(n-q'P-1)(xrt(n-9fP-1) - YR(n-9tP-1)) - sn(n-9tP-2)

(xn(n-q'P-2) - YR(n-9rP-2)).

qn-9~P-1 - -srt(n-9~P-1)(xrt(n-9tP-1) - Yn(n-q~p-1)) }

(~-n-9}Psrr(k)(xrt(k) - YR(k)) - 1)I(q-P),

qk - gn(k)(xrt(k) - yR(k)) t( 1 - Lken-q~Pgrt(k)(xn(k) -Yrt(k)))I(q-P). k - n-q'P. .... n.

If qn-q~p-1 2 0 for p- q-1, it is clear that qk 2 0 for all k.

If not, then since ~k-n-qgn(k)(xn(k) - YR(k)) - qsn(n-q)(xn(n-q) -

yR(n-q)) ) 1, there exists an integer 1 S p~ 5 q-2 such that

6

-sR(n-9'P~-1)(xrt(n-9tP~-1) - yn(n-4'PD 1)) 4 ( Lk-n-9tP~srt(k)(xrt(k) -

yR(k)) - 1)I(9-PD) 2 0.

-sn(n-9~Pp)(xn(n-94Pp) - yrt(n-9~PC)) ; (Lk-n-4tpC.lsrt(k)(xrt(k) -

yn(k)) - 1)I(9-P~1) ( 0.

Hence. srt(n-9~P~)(xrt(n-9tP~) - yn(n-9tP~)) ' (1 - Lk-n-9tP~srt(k)

(xrt(k) - YR(k)))I(4-PQ) 2 sn(n-9'P~)(xn(n-9fP0) - yrt(n-9{P~)) ' ( -

(9-PO-1) an(n-94PC)(xrt(n-9tPC) - Yrt(n-9tP0)) - srt(n-9tP0)(xn(n-9;PC)

- Yn(n-q~p )))I(9-PC) - 0. Thus when P- PQ. 4k 2 0 for ell k.0

ObviouslY. Lk-09k ~ 1. Let YD - Y' s. Yk ' Yk-1 - sR(k)un(kI~ k-

1. 2. .... n-9iP-1. and yk - y. srt(k)urt(k). k- n-4tP. .... n. We

eesily obtain that x -~:Oqkyk. Therefore, x E o-(y0 yl~ .

Yn~ C Cn(m).

From the above conclusions, the lemma follows immediately.

Theorem 2.2. T1(m) is a triangulation of Cn(m).

Proof. From Lemma 2.1 and the definition, the theorem follows

ímmediately.

We call the triangulation T1(m) the T1-triangulation of Cn(m).

Let P be the (n41)xn matrix defined by

-11 -1

P z 1 ' .-11

and let ui -(1, 0, .. , O)T E Rnii. Clearly, Sn - m iPCn(m) t{ui}.

Let Tí :- m-iPTi .{ui} -{m iPa t{ui} I a E T1}.

Then T1 is s triangulation of Sn. It is called the Ti-triangulation

with grid aize m-1. For n- 2 and m S 4 and for n- 3 and m s 2 the

Ti-triangulation of Sn with grid size m-1 ís illustrated in the

figures 1 end 2, respectively.

3. THE PIVOT RULES OF THE Ti-TRIANGULATION

For convenience, we consider the pivot rules of the T1-

triangulation of Cn(m).

Let a z (Y0. Y1. .... Yn~ ' T1(Y. n. s, P. q) be given. We want

to obtain the parameters of ~-~y0 yi yn] - T1(Y n B pr

q) such that all vertices of a are its vertices except yi, in case

the facet of a opposite to vertex yi does not lie in the boundary of

Cn(m). In Table (3-1), we show how y, n, s, p, and q are determined

from y, R, s, p, q, and i. From this table, it ia easy to obtain

each vertex yk of c~, k- 0, 1, .. , n, and in particular its new

vertex.

8

4. THE COMIPARISON OF THE Ki-, Ji-, AND Ti-TRIANOULATIONS OF Sn

In [2], we have introduced the D1-triangulation of Rn and

demonstrated Lhat it is superior to the well known K1- and J1-

triangulations (see [1], [5], [10]) according to the number of

simplices in the unit cube, the diameter, the average direction

density, and the surface density. From the definition of the T1-

tríangulation of Cn(m), it can be seen that we use the D1-

triangulation to triangulate the interior of Cn(m) and the J1-

triangulation to triangulate the boundary of Cn(m), i.e., we obtain

the T1-triangulation of Cn(m) by combining the D1-triangulation and

the J1-triangulation. Hence, the T1-triangulation ia superiar to the

triangulations of Cn(m), which are obtained Prom the K1- and J1-

triangulations. Therefore, the Ti-triangulation is auperior to the

well Imown Ki- and Ji-triangulations (see [9] and [10]) of Sn

according to the same measures, where the K1- and Ji-triangulations

have been obtained from the K1- and J1-triangulationa in the same

way as the Ti- from the T1-triangulation.

9

Figure 1. Ti- triangulation of S2 with m'1 - U4.

ul

Figure 2. Ti- triangulation of S3 with m'i - 1~2

u4

u3

u2 u3

Table (3-1) The Pivot Rules of the T1-Triangulation

i P 9 n Y s n P 9

0 y s rt pal q

1 Y s n P-1 qn

25p5q-1 n(1)EI;(Y)uI-(Y) Y s-2s un(1)rt(1) n P-1 9-1

rt(1)~I~(Y)uI-(Y) Y s-2s un(1)n(1) rt P q0

n(1)EJ(Y) Y s-2s un(1)n(1) rt P 90

rt(1)EI`(Y)uI-(Y) Y s-2s un(1)rt(1) n P 9a11n-

rt(1)EJ(Y) Y s-2s un(1)rt(1) n P 91p2

n(1)EI'(Y)uI-(Y) Y s-2s un(1)n(1) rt p.l q.l

i P 9 n Y s rt P 9

0 q~n-1 y s-2s un(1)rt(1) rt P 9

n(1)EI}(Y)~I-ÍY) Y s-2s urt(1)n(1) rt P 4-10

rt(1)f~I}(Y)~I-(Y) Y s-2s un(1)rt(1) n P 9

i(p-1 Y s (n(1).....rt(itl). P 9rt(i).....n(n))

15isni-p-1 n y g n P-1 q

i)p-1 Y S ín(1).....n(P-1), p~1 qlsp(n-1 rtíi).rtÍP)....,

n(i-1).níi{1)....,rt(n))

n-1 y.2sn(n)un(n) s-2sn(n)unín) n 9(n)-1 9(n)n-1

n y.2s un(n-1)n(n-1) s-2s urt(n-1)

n(n-1) (ní1).....n(n). 9(rt)-1 q(n)n(n-1))

i P 9 rt Y s n P 4

isi5 Y s ~rt~l).....rt~iil). P 9n-q-2 rt~i).....rtin))

~ Y s Írt~l).....rtii.l). P 9~1n(i),...,rt(n))

1 Tnín-q- )E iY.s)n-q-1 pil Y s Írtil).....n~ifl). Ptl qtil

n(i).....rt(n))25q~n

rt~n-q-1)iLT~Y.s) Y s ~nÍl).....rtii41). P 9n~i).....rt~n))

0 y s rt ptl q

1 y s rt p-1 q

n-qp22 y s (nll).....rt~ial). P-1 9-1

rt(i).....rt(n))

1 P 9 n Y s rt P 9

n-q~i Y s írt(1).....rtÍitl). P 9Cn-9 n(i).....n(n))tp-1

n-9 P22 Y s n P-1 q ItP-1

n-q'P isp~9-1 y s n Pfl q-1(i

25q~nn-1 y.2srt(n)urt(n) s-2sn(n)unÍn) rt 9Ín)-1 9(n)

q-1n y.2s un(n-1)

rt(n-1) s-2s urt(n-1)n(n-1) (n(1).....n(n). 9(rt)-1 9(n)

.n(n-1))

n-9 0 Y s (n(1).....nín-q-1). P 9-1(i n(i).n(n-q)....,sn n(i-1).n(itl),...,

n(n))

i p q n y s rt P 9

15i Y s (rt(1).....n(ifl), P 9~n-2 rt(i).....n(n))

n Yt2srt(n}urt(n) s-2srt(n)~rt(n) n 9(rt)-1 9(rt)qC2

rt(i) E T(Y.s) Y g (rt(1).....rt(i.l}. P q41n(i)....,rt(n))

n-2rt(i) a T(Y.s) Y s (n(1).....n(i'1). P 9

rt(i)....,rt(n))

15

REFERENCES

[1] E.L. Allgower and K. Georg, "Simplicial and continuation

methods for approximating fixed points and solutions to systems

of equations", SIAM Review 22, 1980, pp. 28 - 85.

[2] C. Dang, "The D1-triangulation of Rn for simplicial slgorithms

[3]

for computing solutions of nonlinear equations", Discussion

Paper 8928, CentER for Economic Research, Tilburg University,

The Netherlends, 1989

T.M. Doup and A.J.J. Talman, "The 2-ray algorithm for solving

equilibrium problems on the unit simplex", Methods of

Operations Research 5~, 1987. PP. 269 - 285.

[4] T.M. Doup, G. van der Laan and A.J.J. Talman, "The (2n}1-2)-ray

algorithm: a new simplicial algorithm to compute economic

equilibria", Mathematical Programming 39, 1987, pp. 241 - 252.

[5] B.C. Eaves, A Course in Triangulations for Solving Equations

with Deformations, Springer Verlag, Berlin, 1984.

[6] M. Kojima and Y. Yamamoto, "A unified approach to the

[Ï]

implementation of several restart fixed point algorithms and a

new variable dimension algorithm", Mathematical Programming 28,

1984, PP. 288 - 328.

H.W. Kuhn, "Simplicial approximation of fixed points",

Proceedings National Academy of Sciences U.S.A. 61, 1968, pp.

1238 - 124z.

[8] G. van der Laan and A.J.J. Talman, "A restart algorithm for

computing fixed points without an extra dimension",Mathematical Programming 17, 19~9, pp. 74 - 84.

16

[9] H. Scarf and T. Hansen, The Computation of Economic Equilibria,

Yale University Press, New Haven, 1973.

[10] M.J. Todd, The Computation of Fixed Points and Applications,

Springer Verlag, Berlin, 1976.

[11] A.H. Wright, "The octahedral algorithm, a new simplicial fixed

point algorithm", Mathematical Progremming 21, 1981, pp. 4~ -

69.

Discussion Paper Series, CentER, Tilburg University. The Netherlands:

No. Author(s)

8801 Th. van de Klundertand F. van der Plceg

Title

Fiscal Policy and Finite Lives in Interde-pendent Economies with Real and Nominal WageRigidity

8802 J.R. Magnus andH. Pesaran

8803 A.A. Weber

8804 F. van der Ploeg andA.J. de Zeeuw

8805 M.F.J. Steel

8806 Th. Ten Raa endE.N. Wolff

8807 F. van der Ploeg

8901 Th. Ten Ras andP. Kop Jansen

8902 Th. Nijman and F. Palm

8903 A. van Soest,I. Woittiez, A. Kapteyn

8904 F. van der Plceg

8905 Th. van de Klundert andA. van Schaik

8906 A.J. Markink andF. van der Plceg

8907 J. Osiewalski

8908 M.F.J. Steel

The Bias of Forecasts from a First-orderAutoregression

The Credibility of Monetary Policies, Policy-makers' Reputation and the EMS-Hypothesis:Empirical Evidence from 13 Countries

Perfect Equilibrium in a Model of CompetitiveArms Accumulation

Seemingly Unrelated Regression EquationSystems under Diffuse Stochastic PriorInformation: A Recursive Analytical Approach

Secondary Products and the Measurement ofProductivity Growth

Monetary and Fiscal Policy in InterdependentEconomies with Capital Accumulation, Deathand Population Growth

The Choice of Model in the Construction ofInput-Output Ccefficients MatricesGeneralized Least Squares Estimation ofLinear Models Containing Rational FutureExpectations

Labour Supply, Income Taxes and HoursRestrictions in The Netherlands

Capital Accumulation, Inflation and Long-Run Conflict in International Objectives

Unemployment Persistence and Loss ofProductive Capacity: A Keynesian Approach

Dynamic Policy Simulation of Linear Modelswith Rational Expectations of Future Events:A Computer Package

Posterior Densities for Nonlinear Regressionwith Equicorrelated Errors

A Bayesian Analysis of Simultaneous EquationModels by Combining Recursive Analytical andNumerical Approaches

No. Author(s)

8909 F. van der Ploeg

8910 R. Gradus andA. de Zeeuw

8911 A.P. Barten

8912 K. Kamiye andA.J.J. Talman

8913 G. van der Laan andA.J.J. Talman

8914 J. Osiewalski andM.F.J. Steel

8915 R.P. Gilles, P.H. Ruysend J. Shou

8916 A. Kapteyn, P. Kooremanand A. van Soest

8917 F. canova8918 F. van der Ploeg

8919 W. Bossert andF. Stehling

8920 F. van der Ploeg

8921 D. Canning

8922 C. Fershtman andA. Fishman

8923 M.B. Canzoneri andC.A. Rogers

8924 F. Groot, C. Withagenand A. de Zeeuw

8925 O.P. Attanasio andG. Weber

Title

Two Essays on Political Economy(i) The Political Economy of Overvaluation(ii) Election Outcomes and the StockmarketCorporate Tax Rate Policy and Publicand Private Employment

Allais Characterisation of PreferenceStructures and the Structure of Demand

Simplicial Algorithm to Find Zero Pointsof a Function with Special Structure on aSimplotope

Price Rigidities and Rationing

A Bayesian Analysis of Exogeneity in ModelsPooling Time-Series and Cross-Section Data

On the Existence of Networks in RelationalModels

Quantity Rationing and Concavity in aFlexible Household Labor Supply Model

Seasonalities in Foreign Exchange Markets

Monetary Disinflation, Fiscal Expansion andthe Current Account in an InterdependentWorld

On the Uniqueness of Cardinally InterpretedUtility Functiona

Monetary Interdependence under AlternativeExchange-Rate Regimes

Bottlenecks and Persistent Unemployment:Why Do Booms End?

Price Cycles end Booms: Dynamic SearchEquilibrium

Is the European Community an Optimal CurrencyArea? Optimal Tax Smoothing veraus the Costof Multiple Currenciea

Theory of Natural Exhaustible Resources:The Cartel-Versus-Fringe Model Reconsidered

Consumption, Productivity Growth and theInterest Rate

No. Author(s)

8926 N. Rankin

8927 Th. van de Klundert

8928 c. Dang

8929 M.F.J. Steel andJ.F. Richard

893o F. van der Ploeg

8931 H.A. Keuzenkamp

8932 E, van Damme, R. Seltenand E. Winter

8933 H. Carlsson andE. van Damme

8934 H. Huizinga

8935 c. Dang anaD. Talman

Title

Monetary and Fiscal Policy in a'Hartian'Model of Imperfect CompetitionReducing External Debt in a World withImperfect Asset and Imperfect CommoditySubstitution

The D1-Triangulation of Rn for SimplicialAlgorithms for Computing Solutions ofNonlinear Equations

Bayesian Multivariate Exogeneity Analysis:An Application to a UK Money Demand Equation

Fiscal Aspects of Monetary Integration inEurope

The Prehistory of Rational Expectations

Alternating Bid Bargaining with a SmallestMoney Unit

Global Payoff Uncertainty and Risk Dominance

National Tax Policies towards Product-Innovating Multinational Enterprises

A New Triangulation of the Unit Simplex forComputing Economic Equilibria

pn Rnx an~~~ ~nnn ~ F T~~ R~ ~Rr TI-IF NFTI-IFRLANBibliotheek K. U. Brabantui ~ u nwiw~~ M mw u~i~u