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(1)

REVUE D'ANALYSE NUMÉRIQUE

Tomc

No

1974, Pp,

by

STEFAN N (cluj)

### l.

Complex sum and generalized sum

intervals

set

closed intervals

.

complex

sum)

defined as

br,

brf.

11

(ø,

number)

customary

denot'e a',

a1], hence

The

### pair (H,

@) is ca11ed. a generøl,ized, sum

follo-

cond.itions

satisfied :

### SJ H is

a set of some ord.ered pairs of intervals'

@

which associates

each ord.ered

denoted bY

@ B.

Sa)

R'z

anð'

are numbers anð'

then

B

can give

examples.

I2 and the rule

then

have

complex

### sum of

intervals.

(2)

THE GENERALIZED SUM OF INTERVALS 13

72 STEFAN N. BERTr

(i.e.

11

an'd

1.4

B}

rule

## ø,B)-\W I

rve

h-q,uas,isuøt,

Þ-quasisum

denoted

s¡zstem

real

0}

rule

b,

o,

Bfo)

clefines

### a

generalized st1m, clenotecl

and

are functions

arguments,

for

### n : {(A, B)

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

a.t,

b')

a(ar,

br)

a(ar,

bt))

rule

å,

br)

w(ør,

br),

br

### l

a(ar, &2, by, br)

u(ør,

bt)l

clefines

### a

generalized sum denotecl

@,,,, B.

### 5)

For the generalized. sum we call give the

representation:

1et

functions

arguments such

a.t, Ór

R

follows :

a(ø1,

br)

þ(at, a'1,

br)

then

a(a.r,

br)

þ(øt, ør,

### b',

br)}

we have the generalized sum given by the mapping

la(ø1,

br), þ(at,

br))'

### In the

present papef we shal1

sums 2),

and. 4).

quasisums

intervals

connection

### with

some interval- qamsioþerations:

Rs

equation

solution

### is

called. quasisum.

is immediate

getetaTization

### for a

quasioperation generated

real

n-aÍy operation.

### In the þaper [1] is

given the proof of

### the

following theorem:

.

the-

d'etined' by

sr(A,

so(A,

ahereA

Th-e'interua'1,

### : B

høs a.n,inl,eiuø1,-solrttion

and,

i,f

The sol'ul'ion

d,eno-

ted, by

@h

ønd,

equø|,

### with

tke interuø|' 2 3

(ø, I ø, -l- ót * br)h - a'- b,

-.--.

a,

bz)k

aÀ-

interuø\, øl,so

rea.l,

genu.ølized'

qw,øsisim

@o

then

tke h-quøs,isum

tke c

sums

anó'

improper

(the

numbers) and ,4

### @* B is a real

number.

Èor the wid.th of quasisum we have the formula

### 2h-l

(3)

4 5 THE GENERALIZED Sulvl OF ]NTERVALS 15

74 \$TEFAN N. BERTI

### In

the case of complex operations the inclusions

and

(denoted

B)

### i.e. the interval arithmetic is

inclusion neonotonic

For

### the

quasioperations the

arithmetic

inclusion mono-

we give

### the

following theorem'

THEoREM 2,.

ønd'

### Ã < E, e <D, then ue

can

giae tke Jol,touing þarørnetricøl' forn+s

### for

tkese interuøl's :

lar,

ø

C

c

d')(r

s)rl.

where

&1,

R, ø, b,

d

### e R+

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

1.

eroo¡.

### The

above parameirical

and'

hence from

and

where

From

we have

inequations

solution

(b

(b

theo

ha¡re

raal,s

C @nD

exist, then

C,

giuen

theorern

høae

@e

øt

b,

a'

quasisums

Theorem

THEoRÞM 4.

### For

tke quasisurns giaen

theorem

ha'ae:

ønd'

where O

ørbitrøry.

The proof

immediately.

2(a

b

a 1- d)

st(b

a

d)

We denote s

a'nd'

7

t.

Proof.

parametrical

theorem 2)

e

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

therefore (theorem 1)

ftl-t-_

The theorem

proved.

TrrEoREM 5.

quas'isums giaen

tkeorem

ka'ue

@u

&

D)

a+d)

The theorem

### is

an immed.iately consequence of theorem 4.

THEOR_EM

quøsisums

### A @nB ønd c @nD

øre comþtrex sums

ø common aa,lue

and' onl'y

i,e.

þ

d\(

s)rÞ

+

d.)s"

a-ld)

c )- (b

c

d')((t

s) th -sÞ

(1

s)l)

### ø*hIb'l

2h-l

(4)

16 \$TEFAN N' BERTI 6

theorem

d.efinitions

some bìnaty

see [1])

B we have

B

ø2

B

øv <bL

ø,

A

B

b'

ø'

ø2

lheorerns

and'

fol'l'ous

@oB)p(C @nD)

plkþ

d

a

4(1

uhere

d

c

(b

d)((1

s)'

ønd

b

c

d'

Proof.

endpoints

intervals

and' lhþ

b

d'

qs,

c

s)']

2k

constant number

we

quasisums

anð' C

theorem

proved.

IHEoRÐM

With'

notøtions

tkeorern

### 6

we haae the fol'loaing equiualences :

@u

@r D

s)t

b

c'

(1

### oq(l-s)r-b-c<kþ< if (1-s)rfsrf-i.,

T 'rHE cENERALIzED suM oF INTERVALS t7

@n

C @nD

h'þ

b

d

qt'

We give only

### th

ences relateð't'o

9u

pioãls -

### similarly. We

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

lhþ

b

d'

qs'

s)ú

0

s),

b

c

(e(t

(q(1

s)l

ó

s)

c &

s)ú

c

d

qs) V (g(1

s)t

b

c

4s & d

qs

s)l

c)'

Since

s)t

c

d

(1

s),

and

### the

equivalence fo11orvs.

end

putograph we

exarnple.

### We

consider the iutervals

c

where

and. u

R+.

### these

intervals satisfy the cond.itions of theorems

and.3

Parameters

s

### I

aL br a. b C d s t

a,r

u

o 1

we have

and

19)A

2ø'

l0

2h-l

(4at

s

s)t

c

d

A

qs

s)t

### -

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

### 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

the

following

comparison

quasisums

anð' C @o

have

equivalences :

hence f.or

'We have

complex sum

P

c@o D

(u,h)

C@o D

C@e D

h

Dn

c@o D

are represented

domains

anð,

7

1.).

monotonic 1aw

arithmetic sum)

valid. also

neighbourhood

the

complex

domain Dr.

### 3.

Some eonsiderations on

### the

generalized surn ,4 @o B(\$ 1

3)

I1

Ín)

a system

real numbers,

for

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

0}

k

we define the generalized. sum

as

interval lør

b,

ú,

b,

### -l ÃÍ, -f Bf^l

D6 Dl TrrEoREM B.

### For

the generølized, sutn

@r

høae one

the follo-

3 formes:

+\\

.,{r

D5 D,

c

2)

.fr, .fr, .f,

ö tl

7

D3

--¿P___

2

2 u uhere

ønd

øre

### arbitrar

real, nunt'bers, The quasisuno

A

ha.s the

nømely witlt'

Only

points of

u

@u

C

O.

C @È

comPlex sum

iL:.

2h-1h-l ønd

1 (B

therefore

u

I 2h

2h=j' 2k

2n-tl

4(3*u)

Ð"

### 3+u

(6)

20 \$TEFAN

N. BERTI 10

of

11 THE GENERALIZED SUM OF ]NTERVALS 2L

elements

where

is

d.etermincd,

### but is

not

\Miss' Berlin ected

complex ervals)

scope

### of the

investi- sums and generally of the quasiop- Ptoof

conditi on

one

### the

folloiving Possibilities :

I

Í,-Ín

and

### the

parametrical forms given

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

a'nd, only

ønd'

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'

(1e72).

[2]

E.

### RaÍl

on, Interual' analysis' Prentice-Hall series

### in

automatic computa-

tion, 1966.

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

example related

gencralizeil sum

@¡,e

4)

¡I\he above sum

### is not in the

present papef in general'

only

examPle,

### If

u(ø7, ø2, br, br)

o't6z

and

ø2' b1' b')

(øt

bt)(ø'

b')'

then

@rop B

interval

brb,

b?'

b''

d.efined- for

### 5. A

historical remarh

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

(subsêti.

group) we

group'opeìation

Q[ and

### !\$ are

subsets ot

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