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(1)

REVUE D'ANALYSE NUMÉRIQUE

ET DE LA TIIÉORIE Dn rÁppRoxIMATIoN,

Tomc

3,

No

1,

1974, Pp,

ll-zl

ON THE GENERALIZED SUM OF INTERVALS

by

STEFAN N (cluj)

l.

Complex sum and generalized sum

of

intervals

In the

set

I of real

closed intervals

A : lar, ør], B : lbt br),

.

..

the

complex

sum (or simply:

sum)

of intervals [2] is

defined as

A + B : {u + alu e A &a t B} : lø, *

br,

ør*

brf.

11

ø, e R

(ø,

is a real

number)

then it is

customary

to

denot'e a',

- - lør,

a1], hence

R C I'

The

pair (H,

@) is ca11ed. a generøl,ized, sum

of intervals, if the

follo-

wing

cond.itions

are

satisfied :

SJ H is

a set of some ord.ered pairs of intervals'

Sr)

@

is a rule

which associates

with

each ord.ered

pait (A, B) = tI

an unique interval,

denoted bY

A

@ B.

Sa)

If (A, B) = H n

R'z

(i'e' A

anð'

B

are numbers anð'

(A' B) e H)'

then

A@ B: A +

B

We

can give

the following

examples.

1) If H:

I2 and the rule

is (A, B): fu ! alu e A &a e B),

then

we

have

the

complex

sum of

intervals.

(2)

THE GENERALIZED SUM OF INTERVALS 13

72 STEFAN N. BERTr

2) I,el à be the width of ,4

(i.e.

Ã: a,z- a').

11

hz ll2

an'd

H: {(A, B)

1.4

<

B}

then by the

rule

ø,B)-\W I

rve

have tl¡e

h-q,uas,isuøt,

of intervals [1]. The

Þ-quasisum

is

denoted

by A@hB,

3) If / : (Ír, Ír, Í", /r) is a

s¡zstem

of 4

real

'umbers, then for

H: {(A, B)lÃ(f, - /,) + Bu, - /,) <

0}

the

rule

(A, B) * lh I

b,

I ÃÍ, * Bfr,

o,

-f ó' * Ãf"'l

Bfo)

clefines

a

generalized st1m, clenotecl

hy A@¡B'

4) If u

and

u

are functions

of 4 real

arguments,

then

for

n : {(A, B)

| u(ar, Q.2, b1, br)

- u(øt,

a.t,

bt,

b')

a(ar,

ar, br,

br)

-

a(ar,

øt, bt,

bt))

the

rule

(A, B) - let l-

å,

* u(at, ør, br,

br)

-

w(ør,

ar, br,

br),

h I

br

l

a(ar, &2, by, br)

-

u(ør,

at, bt,

bt)l

clefines

a

generalized sum denotecl

b)' A

@,,,, B.

5)

For the generalized. sum we call give the

following

representation:

1et

a and p be

functions

of 4 reaT

arguments such

that rot

a.t, Ór

=

R

it

follows :

a(ø1,

ør, br,

br)

:

þ(at, a'1,

bt,

br)

: ørl bti

then

for

H: {(A, B)l

a(a.r,

ar, br,

br)

4

þ(øt, ør,

b',

br)}

we have the generalized sum given by the mapping

(A, B) ,-,

la(ø1,

6r, bt,

br), þ(at,

ør, br,

br))'

In the

present papef we shal1

give

sorne results about

the

sums 2),

3)

and. 4).

2. On thc

quasisums

of

intervals

In

connection

with

some interval- qamsioþerations:

if HcI3, f:ll+1, 2 rrn

Rs

the real

equation

f(A,.B, X X :'A f B, then the interval

solution

Í(A, B, X) : g(A, B, X) (A, B, X, = l)

is

called. quasisum.

It

is immediate

the

getetaTization

for a

quasioperation generated

by a

real

binary or

n-aÍy operation.

In the þaper [1] is

given the proof of

the

following theorem:

tnponirw-l . iti tu":1'-o I (å: l, 2, 3,

.

")

the-

møþs

d'etined' by

sr(A,

B) - A - B,

so(A,

B): A - (A - sr-'(/, B))(å:2,3,4' "')

ahereA

- B - {r -rlu e A &u e B\.

Th-e'interua'1,

equøtionto(\'4):

: B

høs a.n,inl,eiuø1,-solrttion

iJ

and,

onty'

i,f

A < B.

The sol'ul'ion

is

d,eno-

ted, by

A

@h

B

ønd,

is

equø|,

with

tke interuø|' 2 3

(ø, I ø, -l- ót * br)h - a'- b,

2h-t

A e)þB

-.--.

-l

a,

I \ +

bz)k

-

aÀ-

2h-l

This is q.n

interuø\, øl,so

for

rea.l,

h

genu.ølized'

qw,øsisim

A

@o

B (h,-q + B)

then

tke h-quøs,isum

is

tke c

we define for å

sums

A@rpB: (-oo, +oo)

anó'

AO- B : "fib'-!L'

where A @rpB is the

improper

interval

(the

set of real

numbers) and ,4

@* B is a real

number.

Èor the wid.th of quasisum we have the formula

B--A

2h-l

(3)

4 5 THE GENERALIZED Sulvl OF ]NTERVALS 15

74 $TEFAN N. BERTI

In

the case of complex operations the inclusions

ACC

and

BÇD

(denoted

(A,

B)

C(C,D)) imply that

AoBÇCoD,

i.e. the interval arithmetic is

inclusion neonotonic

l2].

For

the

quasioperations the

interval

arithmetic

is not

inclusion mono-

tonic. First

we give

the

following theorem'

THEoREM 2,.

çe, Ð Cç, D)

ønd'

à < E, e <D, then ue

can

giae tke Jol,touing þarørnetricøl' forn+s

for

tkese interuøl's :

A:

lar,

ql al, B: lbr,lbri

ø

+ bl,

C

: løt-(b-+cld')sr.2r*o.t +

+

c

I

d')(r

-

s)rl.

D:lbr-c,btlø*b+d),

where

&1,

bre

R, ø, b,

c,

d

e R+

(i.e. are positive numtrers) an!' 0

< s, I <

1.

eroo¡.

The

above parameirical

forms follow lot A, B

and'

C,

hence from

A < B

and

B QD.

From -4 CCitfollows:C:lør- x,atløly]

where

x, ye R+.

From

e < D

we have

the

inequations

x<b*c*d, Y<blcld-x,

which has the

solution

,:

(b

{cld)s, y:

(b

-l¿* d')(t-s)t

and the

theo

For the q ypot

ha¡re

THEoREM and'

raal,s

A @uB,

C @nD

exist, then

for "Í A, B,

C,

D

giuen

in

theorern

2, ue

høae

A

@e

" :lo i

øt

*

b,

+ #,

a'

+ at I h +#rl'

-sAfs)

This form of

quasisums

follows immediately from the

Theorem

I

THEoRÞM 4.

For

tke quasisurns giaen

in

theorem

3 ue

ha'ae:

A@nB:A+Beh:hr:!tb. - 2øIb

ønd'

B :ølb

Ã+E 2ølb

where O

< s'< I is

ørbitrøry.

The proof

is

immediately.

D alblc*d'

" c+n 2(ø¡b+a+d)-st(btald') alb*cld'

C@nD:C+D+k'-hz:

2(a

t

b

-l

a 1- d)

-

st(b

I

a

{

d)

We denote s

: I - s

a'nd'

t :

7

-

t.

Proof.

We have (for the

parametrical

forms given in

theorem 2)

Ã:ø, B-:ø+b,

e

:a'ibtcld'-st(b *c-l d)' D:alb!c{d',

therefore (theorem 1)

ftl-t-_

The theorem

is

proved.

TrrEoREM 5.

For the

quas'isums giaen

in

tkeorem

3 ae

ka'ue

(A

@u

B : A+ B

&

c @nD:c +

D)

' +ä : " b(alb*a+d)+a(otd') !\ølb +

a+d)

&h: atb 2ølb

The theorem

is

an immed.iately consequence of theorem 4.

THEOR_EM

6. The

quøsisums

A @nB ønd c @nD

øre comþtrex sums

for

ø common aa,lue

of h, if

and' onl'y

if

:î b(atb*cl-d) b(alb+a+d)+ø(c+d)

i,e.

for -ald)-b-d+

þ

+c+

d\(

(l -

s)rÞ

b(ø*btrc*.d\, b(a+b+c+d)+a(ald)s'

" - t(o+b+c+d,l

+

øþ¡

d.)s"

' b(ølb-l

a-ld)

|a(ctd,)

C@nD:lonø'I\*

h(b-ctd')

2k-l

-l

c )- (b

I

c

t

d')((t

-

s) th -sÞ

-

(1

-

s)l)

ø*hIb'l

2h-l

(4)

16 $TEFAN N' BERTI 6

For the following

theorem

we recall the

d.efinitions

of

some bìnaty

,"latîJÅ -ã-i

ti,".";"ú;"1.-;T I2;

see [1])

: for the intervars A and

B we have

A <

B

+

ø2

1br, A )

B

+

øv <bL

<

ø,

1br'

A

C

B

+

b'

1

ø'

1

ø2

1bz' A> B+ B 1A, Al B+ B )A, Al Bo B CA'

IHDoREM6.Ifp={(,},l,l,C'))then'inttrt'elcyþotkesisof

lheorerns

2

and'

3 it

fol'l'ous

(A

@oB)p(C @nD)

o l-b, o)

plkþ

-b -

d

+ qs'

i

a

-

4(1

- s)ll'

uhere

þ :

d

-

c

|

(b

+ t+

d)((1

-

s)'

- s)

ønd

I :

b

*

c

*

d'

Proof.

By

ad'dition

to the

endpoints

of the

intervals

[-ó, 0)

and' lhþ

-

b

-

d'

+

qs,

+

c

- q(l-

s)']

oftheconstantnumberk,b,next,bydivisionwiththepositivenumber

2k

- 1 and next ¡î-ä¿ãiti"" of ihe

constant number

I *-9t { ót

we

ããt"i" th"

""ãpãi"is'otJrtã

quasisums

A @nB

anð' C

@nD'The

theorem

is

proved.

IHEoRÐM

7.

With'

the

notøtions

of

tkeorern

6

we haae the fol'loaing equiualences :

A

@u

B;' c

@r D

+

<q(l -

s)t

-

b

-

c'

A@*Bl-C@nD+

q(t-s)t,-c iÎ

(1

--s)lf s.#*

oq(l-s)r-b-c<kþ< if (1-s)rfsrf-i.,

T 'rHE cENERALIzED suM oF INTERVALS t7

A@nB4C@eDo

A

@n

B I

C @nD

e

h'þ

>

b

+

d

-

qt'

We give only

th

ences relateð't'o

A 9., F l--9

9u

D' Th"

pioãls -

ot itre can

be made

similarly. We

have r i.,-r,rih"otem 6) :

,4 t-ó,01 l-

lhþ

-

b

-

d'

I

qs'

hþl

+*;l;(i''iít* --b <hþ l' - c(l -

s)ú

{

0

+

o q(l -

s),

-

b

-

c

t

<min

(e(t

- s)t-c, d-q.t) "

(q(1

-

s)l

-

ó

- -;'< < q(r -

s)

I -

c &

q(r -

s)ú

-

c

<

d

-

qs) V (g(1

-

s)t

-

b

- -

c

<

< d-

4s & d

-

qs

< q(l -

s)l

-

c)'

Since

q(l -

s)t

-

c

<

d

-qs e

(1

-

s),

+ t <ry.:L q btald'

and

q(r- s)r- c>d,- qsë(1 -s)rf s> #h

the

equivalence fo11orvs.

fn the

end

of ifrir

putograph we

give an

exarnple.

We

consider the iutervals

A : lør, ør* 31, B : lh *6, a, * 101,

c

: lør-2' øtl 4l' D:lh*5, qtlltu)

where

¿r € R

and. u

e

R+.

these

intervals satisfy the cond.itions of theorems

2

and.3

with the

Parameters

s

I

aL br a. b C d s t

a,r

at*6 l+u

u

o 1

I l*u 3lu l-Fu 3+u r+

d-qs In this way

we have

Ae)þB:

and

+

19)A

-

2ø'

-

l0

2h-r

(4ørf19)h-2a1-9

2h-l

(4at

A@nBcc @hD.*(1-s)rf

s

. #heq$ -

s)t

-

c

<

<

d

- q"

A

@nB)C @hD.'(1 -s)rl-s, # *&'d -

qs

t

< q(l -

s)t

-

c' (4øt I 18J_w)k-2ør- I-tt' (4o, 2h-l

l-18+u)h-2øt-

c@þB: 2h-l

2 - Rcvue d'analyse numérique et de la théorie de l'aPproximation' torne 3' no 1' 1974

(5)

I I

THE GENERALIZED SUM OF INTERVALS 19

1B STEFAN N. BERTI

For

the

the

following

comparison

of the

quasisums

A@hB

anð' C @o

D we

have

equivalences :

hence f.or

u :2.

'We have

the

complex sum

for the point

P

with u:2 A@þB{

c@o D

+u> 1 & h> h+

(u,h)

e D'

A@hBl

C@o D

+u> 1 & I <k' a-.+ (u'k) e D'

A@hBCC@o D+w> 1 & îtu (1e (u'k) eD"

4@uB]

C@e D

+u< 1 &

h

<7 + (u'k) e

Dn

A@þ Bl-

c@o D

eu< 1 & | <k a¿+(ø'h)eDu A@hBlc@¡ D+u<1& hr¿+@'h) =Dc' In the figure 1

are represented

the

domains

D' - D*

anð,

k: !

7

(Fig.

1.).

The

monotonic 1aw

valid for P (interval

arithmetic sum)

is

valid. also

in a

neighbourhood

of P, namely in

the

i.e.

complex

domain Dr.

3.

Some eonsiderations on

the

generalized surn ,4 @o B($ 1

;

3)

I1

Í : (lr, Ír, f",

Ín)

is

a system

of 4

real numbers,

then

for

H: {(/, B) lA(f, -,f,) + BU,- /,) <

0}

k

we define the generalized. sum

A

B

as

the

interval lør

l

b,

* ÃÍ,I BÍ",

ú,

i

b,

-l ÃÍ, -f Bf^l

D6 Dl TrrEoREM B.

For

the generølized, sutn

A

@r

B ue

høae one

of

the follo-

uing

3 formes:

+\\

.,{r

,

D5 D,

l) Í: (Ír, fr*8, f'+Í, Íù, B <4!;

c

2)

Í : U, * f,

.fr, .fr, .f,

* g), B > 4!;

3).f : (f', Í,, h+l, fr] g),

ö tl

7

D3

--¿P___

!

2

2 u uhere

fr,,f, = n

ønd

f, g e R+

øre

arbitrar

real, nunt'bers, The quasisuno

A

B

ha.s the

form 2),

nømely witlt'

Fig'l

Only

the

points of

the domain

u

> 0 * O' + defi

@u

if";1;; B and Ã

C

oJ 9, ã

O.

""a

C @È

D the

comPlex sum

for h,

iL:.

ft:.fz:

2h-1h-l ønd

f : s:;-,

1 (B

<Ã)

therefore

| +u

u

6+u A@hB:A@t h Þ-t h-t h \B

I 2h

-1'

2h=j' 2k

-r'

2n-tl

l+1t

4(3*u)

Ð"

3+u

(6)

20 $TEFAN

N. BERTI 10

of

11 THE GENERALIZED SUM OF ]NTERVALS 2L

the set of

elements

AB

where

A

is

unicuelv

d.etermincd,

but is

not

ichte'Präuss. Akad.

\Miss' Berlin ected

with inversability of

complex ervals)

is the

scope

of the

investi- sums and generally of the quasiop- Ptoof

. From the

conditi on

.4'(f1- /t) + B(f, - /t) < 0 holds

one

the

folloiving Possibilities :

I

Í"_ f,>o & ro-r,<o & B.n(!"-!',)'

Í,-Ín

Í"-l',<0 & Ín-f,>0 & B '++'

or r"-Ír>o & fn-r'>o

and

the

parametrical forms given

in

theorem

hold' An

immediately proof sives

thal

the quasisum has the form Z'

rHÊoREM

9. Ti,;';r;;;.t;;;'t-i** Àø, B

giaes ø qør,øsisutn

if

a'nd, only

if

fr- l":1,*fn:l

ønd'

Ã(f"-f) + BUo-lr) >

0'

tine þroof

is

verY simPle'

RDFERENCDS

[1] B et!i, S. N., Aritmeti'ca çi analizø interoalel'or' Rev' de anal' nu[r' 9i teot' apr' l' 1'

2t-39

(1e72).

[2]

Moor,

E.

RaÍl

on, Interual' analysis' Prentice-Hall series

in

automatic computa-

tion, 1966.

Receivetl 28. II. 19?3.

lttsl,itutul d,e ealcul ddn Clu'i aI Aca' dcmiei Reþubl'icii Socialisle Romd'nxa

4. An

example related

to the

gencralizeil sum

/

@¡,e

B (s1,

4)

¡I\he above sum

is not in the

present papef in general'

We give

only

the following

examPle,

If

u(ø7, ø2, br, br)

:

o't6z

* brb,

and

a(øt'

ø2' b1' b')

:

(øt

t

bt)(ø'

*

b')'

then

A

@rop B

is the

interval

lør* brl øú, *

brb,

- ø!-

b?'

ø'+

b''

* (ø'ib')(ø'lb') - (ø'+b')'l'

d.efined- for

Ãb,.+Bar<0.

5. A

historical remarh

Thecomplexsulnshavebeenconsidered(in'1895),bvl'"}.ROEBENIUS

(1g4g_1917) remarking that. for "ottit"".ìr

(subsêti.

of a

group) we

òan define Úr"

group'opeìation

; "u-"'ly if

Q[ and

!$ are

subsets ot

Referințe

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•  A TS groups a sequence of events of one character or a stable group of characters over a period of. story Kme, which is uninterruptedly told in a span

However, the sphere is topologically different from the donut, and from the flat (Euclidean) space.. Classification of two

The purpose of the regulation on volunteering actions is to structure a coherent approach in order to recognise the period of professional experience as well as the

The number of vacancies for the doctoral field of Medicine, Dental Medicine and Pharmacy for the academic year 2022/2023, financed from the state budget, are distributed to

Adrian Iftene, Faculty of Computer Science, “Alexandru Ioan Cuza” University of Iași Elena Irimia, Research Institute for Artificial Intelligence “Mihai Drăgănescu”, Romanian