Osterreichisch-Ungarisch-Slowakisches Kolloquium iiber Zahlentheorie

G r a z - M a r i a t r o s t , J u n i 1992

Franz Ha l t e r -Korh \md Robert T i c l i y

He ra u sgeb e r

Bericht Nr . 318(1993)

Grazer Mathematische Berichte

}>ormah

B e r i c h t e der ina then ia t i s ch - s t a t i s t i s chen Sek t i on

in der Forschungsgesel lschaft J o a n n e u m G r a z

Herausgeber:

R. B u r k a r d , U . D i e t e r , R. D o m i a t y , P. F l o r , H . F l o r i a i i

F . H a l t e r - K o d i , L . R e i c h , R. T i c h y , H . Vog l e r

O s t e r r . - U n g . - S l o w . K o U . u b e r Z a h l e n t h e o r i e Q r a i - M a r i a t r o s t , 1 9 9 2 G r a z e r M a t h . B e r . 3 1 8 ( 1 9 9 2 )

T R A N S F O R M A T I O N S T H A T P R E S E R V E

U N I F O R M D I S T R I B U T I O N , i r

O t o S t r a u c h Ste fan P o n i h s k y

1. I n t r o d u c t i o n

In [1]^ we in t roduced t l i e not ion of a un i form distr i l )Ut ion preserving t r ans f o rmat i on and

established some their fundamenta l properties.^ Here we say t h a t a map T : [0, 1] [0, 1]

is a uniform distribution preserving t rans fo rmat ion^ i f {T{x„))'^^^ is a un i f o rm ly d i s t r i bu t ed

sequence'* in [0,1] for every u .d . sequence ^ = ix„)^-^ C [0, 1].

The paper is devoted to the fo l lowing prob lem:

Given a Riernann integrable function T : [0,1] [0,1] and a Jordan measurable set

Y C [ 0 ,1 ] , f ind condi t ions under w l i i rh the contract ion T\Y can be extended to a u .d .p .

t r ans f o rma t i on 7' of t l ie whole unit interval [ 0 , ! ] . In other words, f ind condi t ions for the

existence of a t rans fo rmat ion T : [0. 1] — [0, 1] for w^liich

( i ) TJY = f\r

( i i ) r is a u .d .p . t rans fo rmat i on .

This question is mot iva ted by an idea of an integrat ion method whict i arose sa a byp roduc t

of the next theorem. We proved in [4, l l i eo rem 1] a special case {k = 1) of the fo l lowing

general theorem:

T h e o r e m 1 . Functions l\ [0, 1] — [0. 1] : [0,1] ^ [0, 1] are u.d.p. transformations if and only if for every fiiemann inlegrab/e function f{xi,..., J t ) defined on the k dimensional unit cube, the composition f {l\{x^),... ,Tt^{xit]) is also Riernann integrable and

I / ( T , ( j , ) ll{X),})(\xi...dxk= f ... f f{xu...,Xk)dx^...dxk. (1) Jo Jo Jti

Theorem 1 allows us to remove one or more of the functions T, f rom the l e f t -hand side

of (1 ) . Th i s lechni ip ie is not completely new and already (a l though not exp l i c i t l y ) loomed in

in t eg ra t i on of integrals of the type

/ f{{qx})dx = / f[x)dx, Jo Jo

' R e s e a r c h s u p p o r t e d hy t l i t S l o v a k A t a d e n i y o f S c i e n c e s ( J r a i i t

' p a r t s o f [4] w e r e d e l i v e r e d a t t h e T " * C z e t h o s i u v a k N t l i n b e r T l i e o r y C o n f e r e n c e h e l d i n R a c k o v a d o l i n a ,

S e p t e m b e r » 1 4 , 1 9 8 5 , C z e c h o s l o v a k i a

Mparts o f [ 4 ] o v e r l a p witli r e s u l t s o f [ 1 ]

• ' a b b r e v i a t e d a u . d . p . t r a n s f o r m a t i o n

* a b b r e v i a t e d a u . d . s e q u e n c e

- 173 -

where / ; [0,1] — [0,1] is a l i i emai i r i integrable funct ion and {i/,f) t l ie f ract ional par t oS qx and q is a posit ive integer.

Previous idea can be generalized in the fo l lowing way: given two Hieniann integrable funct ions / : [0,1] — R and T : [0.1] — [0,1] and a Jordan measurable set V C [0,1] such t h a t 7'lv is extendable to a u.d .p . t rans fo rmat i on then we can compute the integra l /„' HT{x))i\x as foNows:

f J{T(x))Ax= [' f(x)dx+ f (f(r{x))- f{f{x)])dx. (2) Jo Jo .'|i).il\ ^ '

A n o t h e r aspect of the formula (2) is that it can also be used to est imate the difference

f f(r(x))i\x- f !(x)Ax < / Jo Jo J\oM\y

T h e fo l lowing no ta t i on w i l l be used in the paper:

j(T(x)) - f(r(x)) Ax. (3 )

• i f T ; [0, 1] [0, 1] is a Lebesgue measurable funct ion then gr : [0,1] [0,1] is defined

by ff-,(.r) = | 7 - ' ( [ 0 , i ) ) | for all J- 6 [0, 1].

Here |V'| is the Lebesgue measure of Y. Funct ion gj is non-decreasing and i t can be

viewed as a d i s t r i b u t i o n funct ion o f 7', i f T is considered as a r andom variable.

• i f V C [0, 1] then

fl7M(-r) = r ^ " ' ( [ 0 , x ) ) n r| for all J £ [0, 1].

2 . M a i n r e s u l t s

T h e extension prob lem of T\y can be answered in a f l i rmat ivo i f the funct ion gi\y can be comfo r t ab l y l iand led , as the next residt shows.

T h e o r e m 2. Le i 7' : [0,1] [0,1] and Y C [0.1] be Jordan measurable. Then the contrac­tion T\y can be extended to a T : [0,1] — [0,1] satisfying 0) 71K = f\y (ii) T is a u.d.p. transformation

if and onfy if the function x - gr.Yi^) 's nonderrea.sing for x £ [0 ,1 ] .

f'roof. Theorem 1 o f [4] says tha t a map

7- : [0,1] ^ [ 0 , 1 ]

is a u .d .p . t r ans f o rmat i on i f and only i f T is measurable in the Jordan sense and s imultaneously

gf(x) = X. ^

Now, i f 7' is an extension of T\y then

ffj=(jr) = 9 r , y ( - r ) + - g f , [ o , i ] \ r ( ^ ' -

- m -

' l l i i i s i f 7'|y is extendable to a u.d.p. t rans fo rmat ion f. then

^ - 9 7 M ( J - ) = 9f|o,,,\,(-^)

is nondecreasing. in the opposite d irect ion suppose that x — gxxi^) is nondecreasing. I f i i ' ; [0 .1] — [0.1] is the functioiL which graph is symmetr ic to the g raph of the funct ion •i - 57,1 ( j ' ) w i l l i respect to the line x = y then one of the possible extensions of r|i- is

T 'Cx) = tM|[0 . . r ]n ( [0 .1 ]\>- )| ) , for x € [0, 1] \.

Note t h a t the nondecreasing proper ty of x -gr.vix) implies tha t the d i s t r i b u t i o n funct ion gr.vlx) is cont inuous. Moreover, the fact that gr.Y(x) is cont i imous is equivalent w i t h the proper ty that T is nonconstant on each subinterval of 5'.

The next example shows an apphcat ion of Theorem 2:

E x a m p l e 1: Let T{x) = v/|sin(2fc7rj-)|

and

r = 7 - ' ( [ 0 . ! / ] ) for y e [0 ,1 ] .

.Again a rout ine c omputa t i on gives

2 grix) = - arcsin x^

n and

arcsin a- , for 0 < x < y

^ arcsin t/', for y < x < I.

Theorem 2 impl ies tha t the map 7'|v is extendable to a u .d .p . t rans fo rmat i on i f and only i f

^ - ffr,)(^) is nondecreasing, tha t is if and only i f

- arcsin r ^ j < 1 for 16(0,1/).

Thus the m a x i m a l ( w i t h respect to the inclusion) set V for which T|>' is s t i l l extendable t o a u .d .p . t r ans f o rmat i on on [0,1] is given by

. v/162 + .|jr-« - 16

Then

i n = . 7 , l d ) = A r c s i n ( ^ ^ ^ - " ' 1 . 0 . 3 1 6 . . . ir

and the re lat ion (.1) gives that for every nonnegative continuous funct ion / : [0,1] ^ R we have

^ ' / (v'|sin(2A-jr,r)|) <]x - j ' f { x ) d , < max / ( x ) - ( l - |V|) x6[0,l]

< 0 . 6 8 4 , , . max f ( x ) . r6[0,l|

O n t l i e o ther haru i , the direct c omputa t i on in this case gives

Ju Ju

= 0.718. /(Od(.

In most cases i t is not an easy task to find the funct ion gT.vix) for given T. Therefore we shall he in teres t ing in f inding such cr i te r ia whicl i express the nondecreasing p rope r t y of ^ -gr,\{x) in te rms of integrals f rom funct ion T|i . The next result contains an in f in i t y of such cond i t i ons :

T h e o r e m 3. Let 7" : [0, 1] — [0,1] a m i Y C [0, 1] be measurable in Jordan sense and assume that T is nonconstant on each subinterval o f V . f u t

-I- x)Ax.

I'hen T\Y is extendable to a u.d.p. transformation if and only if

1 < '7 - !)('•+;;;+')

(4)

for every i f i , fc = 0 , 1 , 2 ,

Proo/. .\<cording to l lausdor f f ' s theorem [2] the moment problem

Sk = f x''dg{i). k = 0 , 1 , 2 , . . . ( 5 ) Jo

is un ique ly solvable in g i f and only i f

^ ( - 1 ) ' ( . > 0

for every m , A: = 0, 1 ,2 , . . . . Replacing g{x) by x — gT.yi-r) in (5) we find tha t = j - ^ — s^. Since

the mono t on i c i t y of x — gr.Y ecjuivalent to (4 ) . .\on negat i v i ty o f x — gT.y[x) follows f rom the c on t i nu i t y o{ gfy{x) at 0 and the proo f is finished.

The next c r i t e r i on contains only a finite number o f condi t ions. For i ts f o r m u l a t i o n we shall need the fo l lowing adajited def init ions and no ta t i on from \r,\.

Let

J'(9)={i g{-r)dx. f xg{x)dxj g\x)dx \Jo Ju Jo /

- w -

for a R io i i i a i i i i - i i i t eg rab le f\iiiction g : [0. 1] — (0, I j . Then define

!! = {/•((;): <j ; [0, 1] — [0. \].g is nondecreasing}

and for a given nondec : [0, 1] [0. 1] t l ie fo l lowing de format ion of U

f i jo = ( " ' " -^ 9(-f ) f fo( j ) ( l j-^ : s : [ 0 . l ] - [ 0 . l ] ,<7is nondecreasing

Let i/ (u i , e i , U2, t ' i ) denote the d i s t r i bu t i on funct ion in x defined by

( 0, for 0 < X < ui

g(u,.v^.U2.V2)= I 7 ^ J - + « i - I ' l ^ , for u, < x <

I 1, for f2 < I < 1.

Define for ( A ' l , .V^. .V3) 6 [O. l j^ the fo l lowing auxi l iary funct ions

ffi

92

93

94

95

9fi

97

g'j

= 9 (^0, (1 - A , ) - ^fm~Xt - .\l~2.\2). 1 , ( 1 - A , ) + y 3 ( 2 A i - A 7 - 2 A 2 ) ^ ,

ff(A, - 3 ( 2 A 2 - A , ) . 0. A i + 3 ( 2 A 2 - A i ) , 1 ) ,

/ 2( 1 - A , ^ , 3 ( 1 - 2 A 2 ) \

/ 3.V2 - 2 A , 2A,^ \

A , ' 3 ( A , - A 2 ) ' J '

/ (1 - A|)^ ^ , (1 - A , \ - 2 X 2 \

.V?

= 9 0 ,

A? 2 A , - A i

^ V 2 ( A , -.V2)' .V, • 2 ( A , - A 2 ) 3 - 8A

, 1

= g 1

-. 2 A 2 - 1, 1

- 1 ( 1 - 2 . V 2 ) ' • • 4 ( 1 - 2 A 2 ,

Analog ica l ly define the fo l lowing regions in [0 ,1 ] ' '

V3 I I I = = | ( A , . A 2 , A 3 ) ; w i t h .V3 = A , - ^ / ^ A , - A? - 2A2 ,

A , - ; A,' - min ( ^( 1 - A , ) ^ ^ A ? ) < A2 < A , - ^ A/ , l_

2'

where 0 < .V, < i

II2 =

II3 =

|( A , , A2 , A3 ) ; w i t h A3 = A ? + 3(2A2 - A , ) ^ IA, < A j < ^ A ,

+ i m i n ( A , , ( l - A ' l ) ) , where 0 < A , < l l , o

| ( A , , A 2 , A 3 ) ; w i t h A-3 = 2 A , - 1 + ^ •

- 177 -

r. + ^ V , < .v., < : - ^( 1 - A• , )^ wl.ere ~ < A , < 1 (> , i 2 . i 2

l i s =

(A,,A2. A3); w i l h A3 = l A f

9 ( A , - A 2 ) ' 2 ,.

3 = A, < A2 < A, - ^A^ where 0 < A, < ^ |

(A , ,A2 ,A3) ; w i t h A3 = 2 A , - 1 + ( I - A i f

1 - 2 A 2 '

^A, < A 2 < A, - JA'', where 0 < A, < ^

(A,,A2,A3); w i t h A3 = A f

2(A, - A2)'

^ A ' l < A2 < A ' l - ^ A ? , where i < A , < 1

Let i l l w h a t follows P r ( I I , ) denote pro jec t ion o f I I , i n to the plane A| X A'2.

T h e o r e m 4 . Let T : [0,1] — [0,1] and 5' C [0.1] f>e Jo/dan measurable and

Xl'' = ^--\y\ I T(x)ii

A 3 " ' = i - | i T ^ + | V | ^ T ( x ) d x + i | ^ ^ J T ( x ) - 2 ' ( j / ) | d x d t f ,

Then a necessary condition for extendabiUty ofT\Y to a u.d.p. transformation on the whole [0, 1] is the existence of an i v.ith 1 < i < 1, such that

{xr'.xV)ePHn,i

o r equivalently the existence o f a j with ^ < j <~ such that

[xl'\xV) 6 i ' n i i , ) .

If such an i or j exists tlyen a sufficient condition for extendahility ofTly is A J ' 6 (.-l,,.4j] wdere

= J ' ^ (\y\ l>1 ak[x)i\x - ^ T{x)dr - j ' ^ |,r - T(y)\dgk{x)dy^ Ak =

for k = i,j (In case j = 7 He taAe l i ie maxima/ . I j for g : and g j ) .

Proof. Let go : [0,1] — [0,1] be a fixed given d i s t r i bu t i on func t ion . Func t i on i - ga{x) is nonnegat ive and nondecreasing i f and only i f there exits a d i s t r i b u t i o n func t i on g : [0,1] — [0, 1] such tha t

g ( j ) + go ( j ) _ X

2 ~ 2 ' for e [ o , i ] .

- m-

It follows f rom [5] t l i a t this equivalent to the existence of such g for which T ^2l£i±ffii£l^ =

( j , |, which in t u r n is equivalent to ( j . | ) -/ " (po ) £ ^^go- Thus the problem is reduced to

the descr ipt ion of the body Qg^. According to [5] w i t h respect to the plane A'] X A ' j the upper

surface of the body Q is U j . j f l , and the lower surface eq\ials U^_| I I , . Moreover, the surfaces f l ,

are bu i l t f rom the images of /"(g,). for i = 1 , . . . , 7. It is proved here that il is convex in direc­

t i on of the axes .V^ and A '3. Th is and the con t inu i t y o f the map J-{g)+ ^0, 0, g{x)gQ{x)(ij:^

i m p l y t h a t if T{g,) and J^(gj) coincide in the first coordinate then contains whole seg­

ment w i t h endpoints J'lg,) + (o.Ojg g,{x)g„lx)dx^ and J^ig^) + [o,Q, £ g j ( i ) g o ( x ) d x ) . Let

I - S o ( ^ ) be nondecreasing and nonnegative. Since Pr(flj(|) = P r ( f l ) , there exist indices i , j w i t h 1 < i < 4 ,5 < j < 7 such that T(g,),J^(g,) and ( ] , i , i ) - /'(go) have ident i ca l f irst

two coordinates. Consequently, i f

1 f ' - - S ^ ( x ) d x 6 [ . 4 , ( g o ) , ^ j ( S o ) l ,

where ^ ^

Aklgo) = / gl'\x + / g t ( x ) g „ ( x )dx , Jo Jo

then ( 1 . 1 , l ) - J F ( g , ) 6 n , „ .

I t remains to show that i f go = gry then ( ^ , 5 . 5 ) - -/(ffo) = (.V,^'', .Vj^ '^, . V j ' ' ' ) and

[ .4 , , / l j l = [ . l , (So) ,/l j (9o) ] . To do this let 7'o : [0, 1] — [0. 1] be a Jordan measurable random variable which d i s t r i b u t i o n

funct ion is So(x) . Then the above relations conta in ing integrals o f go(:r) can be r ewr i t t en in to re lat ions conta in ing integrals of 7'o(x) using the fo l lowing equalit ies:

/ go(.r)dx = 1 - / 7o (x )dx Jo Jo

/' 1 1 f '

X ! ;o ( i )dx = 2 " 2 i , / / |x - y|ds(x)d(y„(x) = I g{x)dx+ f ga{x)<\x-2 f g{x]gulx)i]x

Jo Jo Jo Jo Jo

I I |x - i/|d9(x)dyu(x) =11 |x -To ( !/ )|dg (x )dg . Jo Jo Jo Jo

These equalit ies cannot be d irect ly applied to gjy for the de f in i t ion o f ^^.^(x) depends od T\y which however is not defined on the whole interval [0 ,1 ] . From this reason in t roduce the funct ion

r, = r ( x )x i ' ( x ) ,

where as usual \ i - is the indicator of V . Then 57',(x) = 97-,i ' ( i ) -(- (1 - |V|) and then use g3 , ( x ) - (1 - |)'|) instead of go(x) . F inal ly the relations

/ / |7 ' (x )x i (x ) -7 ' ( j/ )\ i ( !/ )|dxdi/ = / / | 7 - ( x ) - T ( y ) | d x d ! / Jo Jo J Jvx)

+-2(l-\Y\)J^T(x)<ix

- 179 -

+ (1 - \ Y \ ] ( ^ l - ^ g(xyU

end the proo f o f t l i e Theorem.

No te t h a t for an a r b i t r a r y nondecreasing go ; [0.1] — [0,1] we have

AAgu) < Aj(go)- ( 6 )

T h i s can be proved using the fo l lowing two step:

1. I f (fi) holds for two nondecreasing go = g i and gu = g j then ( 6 ) also holds for go =

o g i + /^g2 w i t h a r b i t r a r y nonnegat ive Q,/3 satisfying a -j- >i = 1.

2. Inequa l i t y ( 6 ) holds for each nondecreasing go : [0,1] [0, 1] hav ing a stej) equal to 1.

These t w o facts i m p l y t h a t (C) holds for nondecreasing step funct ions go. Then due to

con t inuous i t y o f in tegra l the re lat ion ( 6 ) is t rue for all go.

F ina l l y note t h a t in the proo f o f 1. i t is not necessary to consider al l the possible

combina t i ons bu t (using the mapp ing $ f rom [5]) only fo l lowing three cases =

( l , 5 ) , ( 2 , 5 ) , ( l , . 5 )

T h e next c r i t e r i a concern more special t rans format ions . As the first case take the case

when r|v is a piecewise l inear funct ion and Y is a union of a f in i te number of intervals . Then

as in [4, p.497] the interva l [0,1] can be decomposed in to a f ini te number of open intervals

Jj = ( ! / j - i i y j ) . j = 1 . 2 , . . . , u

such t h a t

0 = go < !/ i < !/2 < • • • < !/n = 1 w i t h the p rope r t y t h a t for every j = 1,2 n the set 7 '|y ' ( J j ) can be w r i t t e n in the f o r m *

T\Y\JJ) = [J1J..CY, 1=1

where 7j^,,t = l , 2 . . . . , n j are disjoint open intervals w i t h

I ' l l ( f j . i ) = ' ^ j and T l / j , is l inear for every i = 1 ,2 , . . . , H J .

A l o n g the s imi lar lines as in [4] the fo l lowing analogon of Propos i t ion 7 of [4] can be proved.

T h e o r e m 5, A piecewise linear contraction T|i- where V is a finite nnion of intervals in [0.1]

can be extended to a u.d.p. transformation on the whole [0, 1] if and only if

M ; l > | / „ l l + |fj,2l + . . . + | / . , „ , |

for every j = 1. 2 . . . , n .

^ e i n p l y m i i o n s a r e a l s o a l l o w e d

- 180 -

Siu i i iar ly , PropDai l iou ti of [4] can be generalized in the fo lk iwing way.

T h e o r e m 6. Let ) be a finite union of open intervals in [0,1] and T|j- be p i e r eH i . s e d/fferen-tiable. Then T\y can tie extended to a u.d.p. trausforniation on the whole [0, 1] if and only

for all but a finite number of pcjints y € 7"|i-(i ).

T h e o r e m 7. Let ( I n ) , ^ , be a sequence of .Jordan measurable subsets in [0,1] such that

Then a transformation T : [0,1] — [0.1] is a u.d.p. transformation on [0, I j i Tand onfy ' { T \ Y „ is extenilable to a u.d.p. transformation [0,1] for every n.

Proof. T h e necessary c o n d i t i o i L is t r i v i a l and thus take a (bounded ) Riernann integrable funct ion / : [0,1) — R. It follows f rom (2) tha t

Th is means tha t (7) implies that for every Riemann integrable funct ion / ; [0,1] —* R we have

and the above ment ioned Theorem 1 of [4] (or Theorem 1) finishes the proof . The previous theorem provides an intermediary step between the un i f o rm convergence

o f sequences of u .d .p . t rans format ions and pointwise convergence of such sequences. In Propos i t ion 5 ofj-t] it is proved tha t the uni form convergence preserves un i f o rm d i s t r i b u t i o n , However i t is proved in Theorem 3 of [6] tha t the pointwise h m i t does not necessarily preserve this proper ty .

T h e o r e m 8. Le( ( T , ; ) ^ j be a sequence of u.d.p. transformations which pointwise converges to T in such a way that Egorov's theorem with respect to the Jordan measure is true, i.e. to every £ > 0 there exists a Jordan measurab/e i , C [0,1] such that 1 - |)',| < r and 7'„|y, uniformly converges to 7'|i-,. Then T is a u.d.p. transformation.

Proof. Due t t ) our hypothesis T , i are Jordan measurable and Tily^ are extendable to u.d.p. t rans fo rmat i ons and thus x — gr„,Y, are nondecreasing for every n. The un i f o rm convergence o f T„|i', to 7'|i', impl ies the Jordan measurabi l i ty o f 7"|v, un i f o rm convergence x - gr„,Y, to ^ - 9T,V,• f 'onseqi ient ly x - gjy^ are nondecreasing for every £ > 0, and then Theorems 2 and 7 finish the p r o o f

if

l i t " I r , . I = 1. (7)

-181 -

R e f e r e n c e s

[1] 1)0SC'II,VV.; F i i i i r t i o i i s t l i d t preserve u i i i f o rm ( l i s t r i l )u t ion . Trans, .\nier. M a t h . Soc. 307

(1) (1988) , 113 iry2

[2] l l A t I . S I ) O R F F , F . ; Moinentenproblenie f i i r ein enilliches In te rva l . Ma th .Ze i t s chr i f t 16 (1923) , 220-248

[3] P O R l ' l t S K Y . S . : Functions tha t preserve un i f o rm d i s t r i b u t i o n l p r e l i m i n a r y announce­m e n t ) . A c t a M a t h . Un iv . Comenianae , i0-51 (1987/88), 223-22.5

[4] P O R U B S K Y , S . SA I , . \T ,T . - . STRAUCH,0 . : Trans format ions t h a t preserve u n i f o r m dis­

t r i b u t i o n . A c t a A r i t h m . 49 (1988) , 4.59-479

[5] .STR. '\UCH,0.: A new moment problem of d i s t r i bu t i on funct ions in the u n i t in te rva l . M a t h . Slovaca ( submi t t ed )

[6] T 1 ( T 1 Y , R . F . - W 1 N K L E R , R . : Un i f o rm d i s t r i bu t i on preserving mappings . A c t a A r i t h m . 60 ( 2 ) ( 1 9 9 1 ) , 177-189

M a t h e m a t i c a l I n s t i t u t e Slovak Academy of Sciences Stefanikova u l . 19 814 73 l i r a t i s l ava Czechoslovakia

Depar t inen t of .Mathematics Faculty of Electr ica l Engineer ing

Slovak Technical Un ivers i ty l lkov icova 3

812 19 l i ra t i s l ava Czechoslovakia

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