K/79MINTERNATIONAL CENTRE FOR

THEORETICAL PHYSICS

A S Y M P T O T I C N U M B E R S - I I :

ORDER AND INTERVAL TOPOLOGY

Todor D. Todorov

INTERNATIONALATOMIC ENERGY

AGENCY

UNITED NATIONSEDUCATIONAL,

SCIENTIFICAND CULTURALORGANIZATION 1979 MIRAMARE-TRIESTE

IC/79M

International Atomic Energy Agency

and

United Nations Educational Scientific and Cultural Organization

INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS

A S Y M P T O T I C N U M B E R S - I I :

OHDER AND IHTEHVAL TOPOLOGY •

Todor D. Todorov**

Internat iona l Centre for Theoret ica l Phys ics , T r i e s t e , I t a l y .

MIRAMARE - TRIESTE

July 1979

Submitted for p u b l i c a t i o n .Some of the r e s u l t s of t h i s paper were reported at the Conference"Operatoren - Dis tr ibut ionen und Verwandte [Jon-Standard Metboden",Oberwolfach, Federal Republic of Germany, 2-8 July 1978.

On leave of absence from'. I n s t i t u t e of Nuclear Research and Nuclear Energy,Boul. Lenin 72 , 1113 S o f i a , Bulgaria.

ABSTRACT

In f&J i t was shown that the set of asymptotic numbers A is a sy»t™

of generalized numbers including isomorphically the set of real nuabert

R as veil as the set (field) of formal pover (asymptotic) aeries.

In the present paper, which is .1 continuation of /8 / , a relation of order on

the set A is introduced due to A turning out to be a totally-ordered *e$.

the consistency between the order and the algebraic operations In A i s

investigated and in particular, i t is shown that the inequalities in i.

can be added and multiplied as in the set of the real nunbers. The notions

of infinitesimals, finite and infinitely large numbers are introduo«d|

A turns out to be a non—archimedean set. The usage of infinitesimals

as well as infinitely l<irg« numbers along with the rani numbers is tbe

renson vhy the terms and the notions introduced in this paper

are very much alike those of the non—standard analysis + + + / T / »

In connection with the order relation, an interval topology of A is intro-

duced e,oA sonio of i ts properties are established. The theory of the asympto-

t ic functions as well as the application.! to the quantum theory are "lilt off

for a future paper.

The notions of the usyantotic numbers /2 ,4 ,8 / as well aa at th,e aeyppit©tie

functions /3,§7 :lre introduced as a subsidiary device fop tbe inr»etiga.ti»»

of some problems in quantum theory. For further details about th« aotlratiaa

of this work we advise the reader to refer to £z,3,4,5,8/. But the knawlefga

of fkj is iiuite sufficient for the understanding of the preaoat pap«r.

- 1 -

1 . IS THL ai T THfc ASm'TO'l'IC

1 (Order) Lot a, b g A . ( i ) ><e shal l say thiit a i s not larger

than b and we write t h i s as a ^ b , i f for every choice of U. & a there

ex i s t s ,3 S . b such that J,(u) ^ ^ ( s ) for a l l su f f i c i en t ly small s£ : (0 , l7

(then- e x i s t s £ e ( 0 , l ) such tliat -t- (;-)^=/f (s) for a l l s e ( U , £ ) ) |

( i i ) .Vc shal l say that a ia smaller than b or that b i s larger than a

and then we -write o ji. to, i f a ^ b and i ^ k b ,

itB-AUk j We shall denote by K, No, Z, It, H+, and C the se t s of naturals ,

naturals and zero ( i . e. {<i, 1 , 2 , . . . 4 ) , in tegers , real , pos i t i ve real and

complex numbers respect ive ly . »'a sh.ill denote by A the set of a l l realasymptotic numbers /8, Definit ion 5 / ,•fe shall introduce some subsidiary notions with the help of which the above

d e f i n i t i o n could he formulated in a more convenient form:

DEFINITION 2 (F i l t er &) ( i ) 1V0 denute by $ the set of a l l subsets of ( 0 , f /

which contain an interval of the Lype ( 0 , £ ) , £ £ ( 0 , l ) ( £ i s d i f ferent

( in general) for the dif ferent elements o f £ ) .

It i s e a s i l y ver i f ied that 0 is a f i l t e r on (O,lJ , i . e . & possesses the

following ( f i l t e r ) propert ies:

( i ) <*> 1 * £ •

(a)

S,T & £ imp l i e S f) T e £ ,

S e £ " » l S E T S (0,1) imply

the help of Cf , Definition 1 could be periphrased as follows (ve areto formulate i t as a

LJV.JVU Let a , b e A . Then a -^ b i f and only if for every choice 4. ^ athere exists Q & b such that

i'tiOPF 1 The equivalence between Llefinition 1 and LemvUjL follows from the

fji.et that the assert ion " ^si <C(s)^ift ( s j j & & " i s eriuivalent to the

asser t ion " *L (s'jii'/Z (s) holds fur -til su f f i c i en t ly small s £ : ( 0 , l l "

Further, iff? arc going to use LewmAl ..ugetlier with ( l ) , (^j and (:i) f i r s t at al

iL.'-Aifly 1 Let us note that " a ^.h" is not equivalent LM "for every choice

of <C a there e x i s t s Ii & b such that

(3) ^si J. ( *! ^J3i*>iEi£».

In order to ciiiiviiic:! i m r . i ^ u , >.> '•' u. v \,-.\ --uyntio ued remark, i t would "be

sufficient t consider the case a = b.

- 2 -

P.jj:. '|JV_- !Jf

LJiiMAJtlf a,b@A and a - b ^ ^ , thc«s (i) a / b if and only i f for every

choice of J. & a ani ^ e b (5) holdsrfiSIa -i b i f and only i f tbera exist-s

4. g a and B e b such that (5) holds.

PHOOFj The validity of (&) for al l J. £j a and al l S E b implies, of cour«et

a ^- b as veil as a-^ b implies the existence of -i g a and fleb suchthat (5) holds, bearing in mind that { s i <i(a) - y S ( s ) j e ^ i s not possiblein the case a - b ^-& • (i) Let a 4. b and let (5) hold for ^ £ i and^ B b ( j si -C (s) - /3 ( sJ^E.^ should mean a - b e & ) • Let us set(for the sake of convenience) a - b = c. The ntf £$implies that every repre-sentative y e e c a n b e represented in the form )"{s) •> j ^ o ^ * + 'iS(s),where Jf S Z, ^ o e ^ i J e ^ O an<1 ^ i s t h e s o m e f o i r a 1 1 jTe c an<*l i g y i { a y y = 0 . If y-^-J3 » then we obtain y o - ^ 0 , i .e . jat i r t f l J ^ j

for all JTEc (but not only for Y ~ ** ~fi )• £ u t t h e latter means that (5)

holds for all d E a and all yB e bj (ii) Let there exist <^s a and 8 e b

such that (5) holds. Consequently, there exists }f 6 o / that £st W») •**-OjgE £

which implies again yo - - 0, i . e . £si y ( s ) -^ 0^ t5 S for al l ^ £ c,

i . e . (s) holds for al l <4 & a and all ^Hgb, i . e . a b. On the otherhand, since a - b 0 6 ? , a » b i s impossible, i . e . a «<£ b. Tha

proof is finished.

A and a - b & & , then a ^ b i f and only i f i g l ,a - b).^ b implieSj obviously, a ^r b» to believe that

we can aet J -fi £ a » Let a ^ b and let us as same a 3 b. This, means,in particular, that for the accuracies ^o. of a and ijp of b, Va^-ifr holdi(and consequently, 1>^ JLV ) and every <J. e u can be represented in the form<L (s) = P(a) + JA(B), where p(a) is the main part of a, lim A(11)/«**•. 0

and every representative J3 e b can he represented in th*- form fl(s) = P(a) ++ + iAC5 where ytf0 is a reul number and . 4

J?!»^(s)/s - 0.

LOU!A3 If a,( t g b means a c b orPltOCFt a ^ b and even

If we choose A(s ) - a"fT , i . o . *C(

exists no ^ J e b such that \s\ 4.(s)

dicts a ^t b, Thv leontis proved.

- p(a) *

$

, we see that thereT ho la t ter '"contra-

Let us note that the re la t ion a Cl b makes sense for the complex

asymptotic numbers too and not only for tlie real ones. /S, StC-Jt/

Til£0RIii 1 The relation ia a relation &f non-striotly order in A,i . e . i t i s reflexive, anti-symmetric und transitive.

( i ) a^-t, (reflexive) folloxs from Len.rod / by a = b

-3 -

1

(ii) in order to prove that * £ b

,*l let

and ,^-it. implies u = b (unti-symmetria)

icJ,j /jg_b be chosen arbitrarily.

in. to Leiarwt^the soU S = {.si X (s) ^Ji{-}\ ^

\ * i s : * ( s ) ^ / j ( s ) l belong to £ , i .o . S,T 5 ^ i and corresponding

^ [ . i s n T (=• £ . on the other hand, (5) implies S I) I - 0 , which contra-

diets (1), i . e . tlu- .. ucption a - b P is iwt true. Let a - b e & .

Corresponding to Leama. J, ft b and b a are reduced -.0 a £ b and b & a ,

i . e . a - bf ( i i i ) Let a -£ b and b ^ c . *e must prove a ^ c ( t rans i t ive) .

Let * t e * be chosen a rb i t ra r i ly . Corresponding to Lanital, there exist ^HEb

and V& c s u c l" U m t t I i e s e t s a = }&l "M^^/SMi a n d T - is»/j(»)^J^8)

are elements of £ . Hence, with the help of (ii), we obtain a fl T & <S .

Let us set 0 = s . «i (a) ^ ^(s)} . Obviou.sly, Sf| T C Ii C (0,1 ) , which ,

corresponding to ( j ) , leads to U e £ , i . e . a ^ c. lhe proof is. completed.

TilEDitQI 2 The relation " ^ - " is a relation of s t r ic t order in A, i . e .

it is» iiianti-reflexive, i.e. T^ZTZ for all a £ A (" " means "the

logical contradiction")? (*_*) anti-symiiietric, i . e . i ^ b implies b - * a |

( i i i ) t rans i t ive , i . e . a . b and b c imply a^io. furthermore, "a ^ , b"

if aid only if "a-^-b or n = h".

FHOOFi The bheorem is a (standard) consequence of Definition 1 and Theoraj 1,

T.iEljilnU 3 Every two asymptotic numbers a and b are orderable, i . e . for

every i , b e A one (and only one) of the assertions j a «<£ b, a • b and

a > b holds. (In other words, tlie order in A is linear .or, which ia

the same, the order possesses the property of .trichotomy).

FBOOF) Let a,b S A ami let us set a - b - c. If c (? ( i . e . a fl b - 0 ) ,

then there exists Yt> ^5 ''> Yo^j*"®* such that every representative J"&o

can be represented in the fona J^(s) - Ycs + ,(") wLore /** &: Z is the

power of e and li^BA(s)/y" - 0. If ya ^ 0 s iben Jst «C(s)^ yfl(s) \%~£

tor iill J E a and /J e b, i . e . a <t M hold:, (see Lemma 2, (i)) • If J^o > 0,

then i s i ^ , ( s ) ^ / l ( s ) j E s for a l l W £ n and all /3 £ bj i . e . a » b.

-fit c e . y . Then, correspouding to / S , Tlioor m 3/ , a 0 b - 0 and

one (and only one) of the relations u c b, a = b and a ^ b holds' these

relations are equivalent to a < b , a = "b and a > b, respectively. The theorem

is proved.

3 (Positive and negative numbers) Lt-t a e A- "o say that a is

positive if a > 0 and we say that a is negative if m i f l , The set of

.til positive asymptotic numbers is denoted 1>y A+.

- I t -

6j of .1.a a L U i k i C o r r e p i ; / « , j

Tlie use of the notation 0 instead of 0*" LB based on /$, Definition la / .

IlIiPaEM 4 (i) If a E .V, a <j£(5> , then <i £ 0 if and only if -^» - ^ 0

(respectively) where A* i» the po-«tr of a and . is the correHponding

(for k = M ) coefficient in the normal additive form {&, (50)j of a((f y(ii) The inequal LUoat 0 a 0 * * / . . . ^ O ^1 0 . . . , ">> fe Z, hold.

VHOOFi The theorem could be ptoved juat lilie the theorems exposed so far. life

notice that tlie inerjuulities a ^ 0 have sen«e in A and ^» ^ 0 - in ft,

1 If a,b £. A and a - b ^ , then a -1 b if unl only if a, -

a - b ^ . 0 .

COitOLLilHY 2^jjivery asymptotic zero different from 0 i s positive, i . e . 0 j> 0

for a l l V e Z | ( i i ) If a Z 0, then a <5? .

THUQRQl 5 If a,lu g A jnd a - b & ^ , then a ^ b if and only if "^ ^ *Jr

(respectively) r where i^ and are the accuracies of a and b respectively.

Consequence of the preceding theorem (or of £j, Theorem 3/)-

Lij-lftY: If 1 and 1 > J 2 , - ^ £ ' V . are two asymptotic units, . i .e. 1 , 1 f=.I,

then 1* sfe 1 if and only if J4 > J .

If a£A, tnen a <0 implies -a > 0 / § , Definition 9/(

and a.$fc-tr , then a > 0 implies -a JCO. (rt'e shall recall

that -u" - 011 ^ , Theorem llj).

If

PACK)Ft (_O_ Let a 0. Corresponding to Corollary 2 of Theorem 4, aconsequently, ^ * .*£ 0 which follova to ~"^* > 0 » i > e * ~a > ° J correspondingto Tlieoran A\ ( i i ) ia proved analogously to ( i ) .

! T Let i £ A and " 1. The reciprocal number a"1 /8, Definition %}

is positive (negative) if and only if a is positive (negative).

FROOfi The theorem follows directly from 'fheoruiu 4, bearing in mini, a

a, fiaoEa AND AL

NOT we are going to discuss the consistency between tlie order and the alge-

braic operations in A. rt'e should like to point out, in advance, that the essen-

t ia l points of this section are Theorem IS, Theorem 17 and Theorem lfl. The

reader who is interested only in the final results could pay attention to

these three theorems only (as well *.a Theorem 21, perhaps).

B If ± , b , c f e A, them ( 0 a . b implies a + o ^ b + a» ( i i ) a . ^ 0

and b>;0 imply

- 5 -

,-X(a)

arid J ' E c 1)0 chosen a r b i t r a r i l y nnd

£ holi lc Obviously, -is) <4 (s) •&/?[&11 "

+• ¥{*) ^ «vsj * f^s/i> which implio* a + c 2= b + c I ( i i ) Let

it ^ 0 and b > 0 . If (at iea.it) one of the numbers a and b is asymptotic.

zero, then a . b ^ O is obvious. Let a, b ^ - 6> . Corresponding to . LeAiiuV .2, ( i ) ,

lor every choice of 4. e a, 3 & b, C&(«)e <J" aiui 2 ( s ) s °°"

{si i ( s ) > A ( s ) l e ^ » m i {si / 3 ( S ) & r2u)3 S £ hold. Let ^

on (l.Oj (for example, 2S(s) - £{*> = txi>(-l/si) or just 4(s) = /its) - 0

on (1,0) ) . tie have that

holdsjfrom which, -bearing in mind (2) and (3) , we obtain |s« 4 ( s ) . /3(s) >

^ ( s ) . ^ ( s ) 4 E . £ The l a t t e r means ,i,b > 0 , 'bearing i u mind Lean" i , ( i i ) and

/ J , Theorem 18?. Ti.e proof is completed.

CUKOLLAKY» a 2 i 0 for a l l a E. A and a 0 if and only if a • 0 .

TiiBp.iEtl 9 If u , b , c , d e A , then a ^ b and C d implies a + c ^ b + d.

FROOFi The proof i s standard ( i . e . the asser t ion of the theorem follows fro»

Theorem 8 as in any r ing or f ield) > a. - ^ b implies a + c ^ b + ot. c^,d

implies c + b ^ d + b from which ;i + c ^ b 4 d follows.

DEFINITION 4 Let A i A X A be the set ol a l l t r i p l e s (a,b,c) where a , b , e (

'iSfe shal l separate the following subsets of A X A X A:

(0)

(a)

DJJ = i ( a , b, c) ! a ci b, c < 0 ^ ,

D - Dj U D<> F

t /*L, and M, are the po«ers of a and b respectively*

TliJJOIlEH 10 If a , b , c E A and (a, b, c) ^ 0, then a ^ b implies a . c ^ b .c

by c ^ 0 and a ^ b implies a. c > b. c by c -^ 0,

FltOOt't (ij_ Let t i c - b.c g^ ^ . B e a r i n g i;t mind the generalized d i s t r i bu t i ve

law ft, (2l)Jt (b - a ) . c + 0 =• b. c - a. c ( V i'a the accuracy of b.c - a . a ) ,

we get (b - a ) . c £.£) which, as 0 is a. simple ideal of A f&y Theorem 18/ t

follows to b - a, c afe Q . By Corollary 1 of Theorem 4, a A. b implies

b ^ a ^ 0 which together n i th c ^ 0 follows to (b - a ) . c > 0. We must agaia

use the generalized d i s t r i b u t i v e law and we obtain b.c — a . c ^ O which,

by Corollary 1 of Theorem 4, implies a.c ^ i b . o . I f c ^ 0 , corresponding to

Theorenj 6, -c > 0 and coilsequently, - c . (b - a) ^ 0 , from which wo get

a.c - b. c 5« ° J i - 0 - a .c S t . c | ( i i ) Let a. ? - b.c (p . If we assuiue also

that e g ( 9 ^ i ° i ( t h i s ' " pa r t i cu la r , means c > 0 ) , then we have-ft* ^-A1* o n < l

a.C ^ b.c wil l reduce to j A ^ C j A " ' ' (because of ^ ^Mf ) . Let osfm& .

If c > 0 , then u -6 b implies a . e r f b.c - the cu.se a - b^ffvc already

considered tibove and for a r b e (? , i . e . for a C b , a . c ^ b . C is obvious.

If c ^ . 0 , then a - £ b (together aitl i (7)) implies e i ther a - \>&& which

t«c considered already, or a = b which implies a.c = b . c . The theorem i s proved.

TIUJHtEH II If u.,b,c,de=A and ( a , b , c ) g H i , then a ^ b implies a.c > b . c

(in sp i te of c > 0 ) .

FHOOF; In fact , in t h i s case a.c > b . c is reduced (because of A

£ to fi/*»+^3yVf"1^f , wliere l i i s the accuracy of c, i . e

TttEOREM 12 If ( a , b , c ) e D ; j > t n e » a ^ i b implies a.c ^ b.c (in sp i te of c<£0).

i>ilOOFs In t h i s case, corresponding to (7) , a ^-b i s reduced, to a CZ. b and

corresponding to Theorem 4, c <£. 0 means, in fac t , J£ ^-0 . where J i and K, axe

the power and the corresponding coefficient in the normal addit ive form / j )

of c, respectively. Hence, we get a . c c t i . c , i . e . $ . c ^ . b . c .

13 If a,b,c,d^A, then 0,

iin.l O

:b and 0 -c d imply a.c

c rd imply (a,b,c),(c,d,b) 0.PflCOP; First of a l l , 0 -4c

Indeed, corresponding to Theorem 4 (/a*6b and ^ / ^ i /*t are possible only in

the case i ^ ^.0 4n M^ ( 'L^ and A^ are the first coefficients in the normal

additive forms of a and b respectively), which contradicts ( 1 / , S | i .e .

(a,b,c) S D j . On the other hand, 0 Ac shows that (a tb,c)fit£» , i . e .

(a,b,c) qfeD, In the same way we obtain (c,d,b)^, D. Further on the proof is

standard ' (as in any ordered ring or field)i We multiply the first inequality

hy c and the second by b and we obtain a . c ^ b . c and c . b ^ d . b , i .e.

a.c ^ b . d . The proof is completed.

COriumit*. If a, b e A, then 0 - £ a - i b implies a" ^ b" for every n-1,2

The str ict inequality » 4- » in A h*s,with respect to the algebraic operations,

properties quite analogous to those of the non-strict inequality " •At " .

Because of the strange al«abraic properties of A, however, we can obtain an

equality after adding u.n asymptotic number to a strict inequality (to

both sides of a strict inequality); this Broperty "does not have'ati analogue in

the set of real numbers iL The following lines are devoted Just to this upe-

cial feature of the strict inequalities in A .

-6- - 7 -

DiUMTlON & Me shu.ll denote by £+ the following subset of AlAlA

, B - 1) <£ Q ,J**-/> A \ U i ( a , b , c ) >

where />a / i s the power ol1 a - b arid T^t i s the accuracy of o .

THimUSM 14 If u,t>,c ^ a. i then a -<- b irsplies u + c .**_ b + e in tlie case

( i i , b , c ) ^ E + iind a+ c b + L in the cutf (a,b, c) g£ JS+ «

PftfJOF; iVe showed already that i ^ b implies a + i > b + Q , Consequently,

we must only specify the cases (»|b,c)^fc.E+ *"d (a>t>ic) s . Cf • To th i s end

i t i s convenient to put down a, b and c in the normal additive form [8,

Definition 13]:

*,! > ^i" and >£ i i r e t t t e Polfers of a, b and c respectively, , * •

= minj/fc y»y ' > n "* ° » ^ * n •^/* ( ' a a n d /*„ " 0>y"4= n ^ j - * ' a n d v» i *>

and >i are the accuracies of a , b and c respectively. We obtain 4 mJ3 ,

/ * 4i n i A . / tind in the case a / li, a - b e j ^ f i . e , » c b ) ^ - ^ J ,

y * ^ n - 6 Tit hold also. I t i s suff ic ient to use / 8 , Xheorem ii6/ only. The

proof i s completed.

OiiFINITION C \ie shall denote by EQ the following^ subset of A X A X A I

(10) E a JU, !>,<>) i a . b e A ^ U i(a ,b,c) . a.b, c

heriy i 4 , ^ ^ u.Tulytt0^ are the powers of a, b ind a - b , respectively, and

•lie i s the re lat ive accuracy of c .

15 If i i , b , c £ , L , then u . h implies a .c^Lb.c in tk« case (a,b,c)

f£ Eu and a.c - b.c in the cas« (u,b,c) gIL, .

The iroof i s f:uitc analogous to tjiut of Theorem 14. It i s convenient

to put down tiie numbers a, b and c in normal multipl icative fora

Jt, Definition 14/ and to use /&, TliooreB 2 tj/.

In >, i tc of Die results oi iheorefci 1-J and Theorem 15 most of the

j.rop-rLieH of the str ict inequ&litites in ii arc vulid in A too, e .g . t

TiUJOItat 16 If a , b & A , then a > I) and b > 0 implies a .b->0 .

J J i l 0 0 F ' Corresponding to Vhcor™ B, wo luve i i . b i . 0 . Futhermore, A does not h«ve

div isor , of zero fg, Theorcn l], i.e. u .b^=0, which iapl'iea a . b > 0 .

lVUDiiLa: 17 If a, b,c,d e s A , tfcen a ^ b ,vnd c *L d iaply a + c ^ b + d .

P a 0 0 F > L e t a ^ h : ! i d c ^ " * " e " ' " " -;»°* «t fir^t that, (at l e l i s l ) o n e o f

-8 -

the triples (a,b,c) and (c,d,b) does not belong to I^.Let us assume the •

i .e . thiit (a,b,c) , (c,d,b) g K , . (jO Let <u,t least) one of the differences

a - b and c - d not be an asymptotic zero - l e t us assume, for instance,

that & - b £ 4 p . ( 9 ; implies. $r i . ruin. (^ »s) = ^ i/fc_y > ij, *•

*" H- , / * . / * i w ^ V> * • * • 3 ^ ^ ^ which is a contradiction; ( i i ) Let a -

and c - d e.@ . rfe obtain "^ i* - -^ and t& >% , corresponding to TheoCea 5

and Jr^_f^)>a tlI"i

m//i-(i ^zfy > corresponding to (.9). On the other hand,

•} = mirt-fli, i>e) s %^ fc/^yi 4 and i^ = k»*/»^^ m.J = %^ ^ / t V > J*,

i . e . l j ^ V ^ , whicii contradictid I t > 1^/ . The sutsidiajry assertion i s proved.

Without loss of generality we can assuate that (a,b,c) ^ Ef . We obtain

a + c •£- b + c and c + b ^ d + b , corresponding to Theorem 14 and Theo-

rem S respect ively , i . e . a + c ^ - b + d . The proof i s finished.

THieOiiElLi la If ti,l), c,d E A , then 0 - t a s. b and 0 -<. c -e- d imply a. c «£ b.d .

PHOOFi Let 6 A H Z b and 0 c «cd . ( i ) »'e siiall show that (at leusti} one

of the t r ip l e s (a,b,c) and (c,d,b) does not belong to t,, . Let us assume

that ( a , b , c ) , ( c , d , b ) e ^ • ^e obtainy*^ = /^ . s ^ ' •/'»-/—/*''> -^ t

t corresponding to (lOj , iVe have also I

^ ^ ^ j y t . / ^ ^ ^ W<rta,^^l// > > .e. >^ ,which i ( a contradiction. Futher, l e t us assiuce that (a,b,c) js. E , . On the

other hand, as we already showed by proving Theorem 13, (u ,b ,c) , (c,d,b) jfe D .

Corresponding to Theorem 10 and Theorem 15, c.b ^rd .b atid a.e*£-b,c, i . e .

a.c ^ b.d . The theorem i s proved.

OiaOLUHit If a , b © A and n£»N (U = 1 , - ' , . . . ) , then 0 ^ .a b implies

a" ^ b a .

TlIH);tiai 19 Each , (oi,itive usyuptotic tMji ber cannot possess more than one

sp,uure root.

Kftflflfc Let ; i > 0 und a = b"" = c aud 0 *- b . c . Froin tlie above Corollary

we obtain b" -^ c^ , which contradicts b" -.. c2 .

-•fliUOma. vQ If l t,b e. A, then a - i b implies a -d (a + b) /s ^ b and

(a + b)/2 - h only in the case it - h £ # ( _ £ . ,£** ft, Definition 12/) .

HOOf1: ( j ) « i b iiaolios d s (b - n)/_' > 0 . Corresponding to Theoren 10,

•A & a. + A - (a *- b)/i . ftr thy uo»-er>4l of 4 , >* r ) ^ holds.

Coni(t-,1Uently, thq assimilution of A Iron a /J , Theorem 137 is impossible

because O ^ u + A , i . e . " 0 ^ (a + b ) / ^ . .<« add b/2 to both

- 9 -

of a/2 <C b/2

then ( i /2 ,b /2 ,b /a ) ^

and we obtain {u. bf ( i i ) If a - b

E+ and consequently, corresponding to Tleorem ]5,

tt/j^b/2 implies (ft+b)/i? .^b . I f ;i - b g ; ^ , then ( a / i , b / 2 , b / 2 ) e . E*. and

a/a *i- b/i implies (a + b ) / - « b . The theorem i s proved.

THBDlUAi 21 For ever/ , b E . > there exists c ^ A such Ihiit a ji. c ^ b .

(in otlier words, A i s a lensu set with respect to the order.)

PitOOFs ( i ) If a - h^.P, then, corresponding to tile above theorem, a ^L c/L b

for c =(a + b)/2 f ( i i ) Let a J£- b und a - b £ ^ ( i . ( . a C b ) . Corre-

sponding to Theorem 5, ^ >• > where if, and Jr are the accuracies of u. and

b respectively. Let £ be an arbitrary positive asymptotic number different

from asymptotic zero, i . e . f e f c p ' , uritn power /<£ •= *{», • Then- c = a + g

nossesses the property we are looking for) a < c < b . Indeed, the posit ivity of g.implies • .Aa ^r c ind A a ^ implies a j£c, i . e . . L / C , Besides, corresponding to

Lemma 2, a C. b, which (because u f A = i* ) leads to c C- *> > i - e . c •£ b 4 The

jiroof i s completed.

tlEkiAU&t The condition £ ^ & was required in the above proof in order to com-

prise the cases a - i ) g ^ and Va = ^ +. 1. If %!.>% + 1 the choice £<&&

i s possible also.

The theorems exposed ao fur show that the jjioperties of the order in A with

resfuct to the algebraic operations in A are distinguished from the properties

of u:iy order ring or f ie ld , (see, for instance, / l / ) . Aa ire sav;hovevert the

pecul iar i t i es of the order in A (in respect to R, for instance) refer to

rather "narrow" subsets of A i A X A - the sets 0, E,. and Eo • Tb.lt

gives us a poss ib i l i ty to work in most enses with the inequalit ies in A

by the usual (as in d] rules. In th i s respect the non^strict inequality

" ^» " i s more convenii'iit because tlie p e c u l i a r i t i e s aentionrid above refer

only to the set D .

We shall define some nations direct ly conucted with the order of the asymptgtio

numbers.

PJaiMTIOM 7 (liagnitude) By the term magnitude | a | of a given asympto-

t i c number a s A we shall understand the extension of the function 1*1, i e S ,

on A at the point x = a. / | , Definition «7«

R£ii.VJtK) Corresponding to [S, Definition «/ , in order to obtain jaf , a. e JL

ve nust fora the set juj* . j | ^ j i ^ e u] , The smallest asymptotic number

with respect to the inclusion " £ " which covers | a | * i i . e . / a j ' c a,

-10-

i» the aagnitude |a( of a . We shalL recall, that »e called point a "perfect"

if l&J*" 1*1 and*imperfect

THHDRQJ 24

fa I ( s t r ic t ) .i f | a f

(i) |aj exists and | a j e . A for every choice qf a £ A . Th«

numbers a ^ ^ a r e perfect and the numbers a Q w e imperfect points of

the magnitude i ( i i ) Moreover, a = max(-a.a) holds for every a£A.

PBOOFi (i) Let a £ A \ p and let M and <* be the pover anB tb« correspon-

ding coefficient in the normal additive form $,(.58}] of a {a.(^& implies

JtgZ and ^»^>0). Every representative U 6 . *- c a n D e represented in the

form «t(s) " ^* jf + ,?A(«) where lim >/i(s)^* - 0. Hence, we draw the con-

clusion that the point s=0 i s not a non-trivial adherent point of the set at

al l jeros of •*, ( i . e . a l l points s at which *i(s) - 0 ) . Consequently,,

for every -i. ei a there exists E g S such that <£(g) does not Changs i t s

sigh . on E . Besides, the sign of «t(s) on E does not depend on the choice

of •£&*., namely, <t(s) > 0 , a&iu i f - £ - > 0 and ^ ( s ) ^ 0 , •&£ if

^H ^ 0 . That means ]«<-(s)| - U(E) on E if a >0 and l«i(«)| - - ^(s)

on E i f t z O . The theorem is proved in the considere'd case \ ( i i ) L«ti

a - 0 V e . ^ . We have (obviously) JO'l^C- 01* ( s tr i c t ) , from which fol io*

th« existence of |0wl as well ,'as the imp erf ectnass of | a| at a » 0 .

Futhermore, ve obtain I 0vl »lOVJ , i . e . Jov/- mojc (0^, -0^) ho Ida too (baoaux

0^ i 0 • and -V - 0 )• The pro«f is completed.

THEDBOI 23 If t , b , c g i , then*

|M ; t 0 and |a) - 0 i f and only if a - 0 ,

| |« | - |b| 1^ . i» ± b| *x | a | + I b | i

| a . b | - | a | . |bj ,

Ja/b| - la|/ |b| , b £.& •

The proof i s analogous to that of Theorem 22 - we shall not give i t .

HQUllKi Definition T, Theorem 24, ( i ) and Theorem 23 are valid not only for

real asymptotic numbers but also for the complex ones'. [6! Sec.2] .

3 . ORDER IN SOME SUBSETS OF A

la / 8 , Definition 10/ we defined the sets of asymptotic number* R% It"*" and

jf &ud we proved / 3 , Theorem 20/ that E° and A*" are isoaorphic (with res-

pect to the algebraic operations) to the f ie ld of real numbers .8 and H

maps homonorphicully on K . The correspondences realizing the iao*Brphima

and homoaiorphlsm mentioned above were given byi

- 1 1 -

. - . - . , . - • . . ! * • • _ •;„: »-

(11)

(12)

a* - x + 0

= 1-1-0

respectively where the numbers from fl , It"" and R are written in normal addi-tive form. Now we ahull consider the properties of the sets H-,K"* and B aawell as the correspondences (.11) and (.12) with respect to the order inA and H respectively.

TiiHOUEM 2V R. and if" are order fields. The correspondence (l l) preserve*the order (strict and non-strict) in A and ft respectively, i . e . if a,bfi &* = 0,«* and i , y g H are their images by (ll) respectively, then a <^bif and only if x .*. y .

PHPOFi The first part of the theorem - that it" and H~* are order fields -follows directly from Theorem 8 . Further, let us take note that if a tb6l)VV • 0,«* , then a-ftt b and a - hfeff are equivalent to each other. That provesthe theorem.T.HBP&BU 2JS- The mapping (lg) of ft on R preserves the non-strict order, i .e .if i , y g R are the images of a,b£g R by (12) respectively, then a^tbimplies x * y .

The proof is analogous to that of Theorem 23. We notice th«t if a , b e ftand a - b g 7 i then o ^ b implies x « y (but not x ^ y), i .e . (12) doesnot preserve (in general) tlia strict inequality.

In /J, Theorem 23/ we showed that the set A*"" of all asymptotic number* with. infinite accuracies is a field which is isomorphic to the field of the for-

mal power (asymptotic) series (with real coefficients in the case of realasymptotic numbers). With respect to the order,the following theorem holds.

THEOREM 26 A** is an order field. Furthermore, if a g A*" , then. a^O if-and only if •£» » 0 respectively,where^^*** is the power and U* is the firstlcoefficient of the main part of a respectively.

PftDOFi The theorem is an Immediate consequence of Theorem 8.

In /8,Sec.£7we introduced an additive [8, Definition 13]:

(13) a - q + 0* , a gA, q e f i 0*6^ ,

and a multiplicative {%, Definition 147;

(!•*) a - r. I* , t g l S ^ , rgi*" , r*ff I ,

fora of a . Further, we developed an algebraic techique [&, See. £] with the

-12-

•ymbols (ll) and (12), which allowed ua to express the algebraic operations andalgebraic properties of A by the algebraic operations and algebraic proper-ties of A"* t & and I • T n e sense of this approach is in the fact that JC~\B a f ie ld fb% Theorem 23/ and, as we know from Theorea Z$, A** i s an order

field, and & and I huve comparatively simple algebraic properties, including'comparative^ a imp le order (see Theorem 4, and Corollary of Theorem 0). Thefollowing theorems allow ua to express the order in A by the. order of if" ,.9 and I .

THEOREM £? If a - q + tf* and a* = IJ'+ 0* , then, a X. »' if and only if 1

' in the ease a - Q » (ii) 0w' in the case

THEOREM 28 If a - r . l J 1 then •<£ •»'in the a/*,'e I .

and a*- r'.l"1* , where a ,a 'g . k\6

i f and only i f i (ij_r^.r/.' in the case a / a ' ^ I , ( i i ) !"*-£ l

TiitJORQl g?. Let a . p + 0v a ,nda«p.l J be the normal additive / S , DefiBition

13/ and multiplicative /g, Definition 14/ forms of a respectively. Them *

i s negative, i . e . a ^ 0, i f and only i f p i s negative, i . e . Tp-^ 0 .

Theorem 2 ? , Theorem 2} and.Theorem £9 are an ismiediate periphrasis of matt-

rial exhibited so far, therefore we do not rewrite their proofs,.

4. INflMITESIMALS, FINITE AMP INflWITELlf LARGE ASMOT0TIC SUMBEB5

DEFINITION 8 ( inf inites inals , finite and 1 nfinitely large lumbers), ( i ) Ifa c i , then a wil l be called inf inite ly small or an iofiniteaimal i f |a|for a l l i g t , 1 ^bO. The set of a l l infinitesimals wil l be denoted by ^ >( i i ) The number a e . A will be called f in i te i f there exists I S K suchthat |a|^-(xt . The set of a l l f in i te number* will b« denoted by JX, t( i i i ) The asymptotic number a will be called inf ini te ly large i f | x | ^ | a jfor a l l x 5= ft. The set of a l l inf inite ly large numbers will be denoted byflOiABK L As We know from fit, Theorem 20J, A i s isonorphic to K° and B*"^ j , Definition 10/ which, on their part, are sutiseta of A. So, as we vrite

JiJ

*l> we have in mind either x EH or x c= If*.

~The asymptotic numbers a » s + 0 , b « 2+ s + 0 and c > l/» +«give us examples of an infinitesimal, finite and infinitely large number,respectively. In other words, A possesses infinitesimals (different from sent)1,finite and infinitely large numbers. The latter could be formulated in the follow-ing wayi Is a non-archimedean order set. (A totally-ordered set F whichcent a ins N 4» called arcMmedean if for everj a ^ F there existssuch that |a|-=- \n' see , for instance, [*])* It is clear also, that

REH.VliK 3 Hie above definition makes sense also for the complex asymptotic members.

The following theorem establishes a connection between the notions jast introduced

and the properties of the representatives of tho asymptotic numbers.

THEOREM 30 (i) The asv^ptotic number a is an infinitesimal, i . e . a S .£ B- iif and only if there exists d S a such that lim-*.(«) - 0. Besides, *cm*t£.£lt

implies lim^(s) « 0| ( i i ) The asymptotic number a i s finite, i . t . a e i L Jif and only if for every i g a there exists E e ^ such that •£ i s bounded on E..,Besides, i f ac^l.then there exists I E S such that llnni(a) = x for &11«E <X ; ,

»-»0(.iii) The asymptotic number a i s infinitely large , . i . e . a ejt^, t i f and onlyif ther exists •* G. a such that <C i s unbounded on every £ s £ . Moreover,

if ngJU.'1"1 »f^ j 0 " " * n& Ni ' t h e n ^^in the case i / 0 ()

PROOFt The theorem follows immediately from /fi, Theorem St/ and Theorttoor Theorem 29.

The following theorem characterizes the sets i l» , ,/L an

of the asymptotic numbers.

THflORHI 31 Uy* is the power of a e A, them (i)_ a & J L , if and only if

y* i-0 or » - 0° i (ij) ae . JXT i f a n d 0 D l y i*J** °> ( i 1 1 ) » S J L . i*

and only i f f*^- 0 •

pHOQFt The theorem follows directly from / 8 , Theorem Vl].

THEORat 32 The asymptotic zeros are either infinitesimals or infinitely larg*

and lim<£.(s) - «— for all ci S a in the case a .> 0 .

through the power*

numbers. In particular, 0 &.Shj tor V =• o, ! , . • . , » - and Ci^^Si^^ for"* — 1 , - 2 , . . . . Ihe asymptotic units 1 , - = 0 , 1 , . . . , o- are finite numbers and,more strictly, I C-fl. \ , f l . ( l i s the set of all aeraptotio .units.).

PflOOFt Triviitl.

COROUHRYt

iii) \ a- S At j a | ^ 0 $ ,

^a e l l |al -i. O"1! ,

TllEOKBi 33 If a. e.A and a ffe (P, then a is an infinitesimal or infinitely

large if jind o«ly if a" is infinitely large or an infinitesimal, respectively,

and a ^ Sl\Slm it and only it a"1 & /iA.Jlv

PrtOOF: The theorem follows directly from Definition 8.

THEOREM 34 The set of finite numbers SL i s closed with respect to the ope-

implies

, implies

rations addition, subtraction and multiplication and a , b £ j \ , a n d

a /b e. £i if and only if b i s not an infinitesimal, i . e .With respect to the order, J l is an archimedean se,t and 0 ^ a Ac.

a e >iV for every choice of a and b .

PROOF; Let a,b e J^- • Corresponding to Definition 8, there exist i , y £ Rsuch that jaf^ |xj and |bU|yj . We obtain U ± bl |a | + l b | ^ \ x | + \j\,

i . e . a i b £ j \ , and |a.b| - |aj . |bj : x.y , i . e . a.b E j J . . The a*sertioaof the theorem about the division follows from Theorem 33. The property that : :Jl is an archimedean set follows directly from Definition 8. Let 0«£ a * b S j / 1 .

The latter means that |af = a ^ b = 1 b I -i |x | for any x S H, i .« . > g J l . Th«theorem is proved.

THEORai 3&, The set of all infinitesimals JLo i* a simple ide»l in the set •! ^te.finite numbers St > Furthermore, .Q, posseases the property] For every choice »fa , b E A , O i s / i b e j i , implies a e XU •

f^ppft (i) Let a . b e i V . , i . e . |a.l^|xj itud tb |^ |x/ for every choice of the iwtreal number x + 0 . We have (a + b | ^ |a| + [ b| ^ 2x, i . e . a + b e J J « |fii^Obvloutlv. aejlpi™plies -a&>A1, I i l i i l Let » e < f i a n d bG W »hieV-means thu.i there exists a real number x0 such that l a ) ^ ) x o | . We have)a.b| - | a | . | b | Z xo |x | for all real number x^icO, i . e . a . b g j l , . We

proved that jto is an1 ideal in «f\. . (iv) We Must prove that &„ is a simpleideal in Q , i . e . that a . b £ j X « implies either a e / U o r b £JU. Let uaassume the opposite, i . e . that a , b ^ J7O . We have [a |^ |x 0 ] and \b\

for some real numbers xQ •*. 0 and yo 0, i .«; ^|io.yo|, i.e.

which i s a contradiction! (v) Let 0 4, a b S J l . , i . e . [a.| |b| )x| for »U

real 1 y» 0 . We obtain \&\&.J\.O 1 i>e- > e - J l « • The theorem i* proved.

DEFIMITIOM 9 ^Infinitesimal relation). We shall gay that a , b e A ore infinite-

ly close and ve write this i s a ^ b, if a - b i s an infinitesimal, i . e .

if t - b ei2« •

THEOREM 36 (i) The relation " - ^ " i s an equivalence relation io the B«t of .

a l l f ini te number* J7. | ( i j ) If a,b, c,d e *fl , then a r b and can A

implies a * ^ a - ' b + d, a.c-'s? b.d and. if, furthemore, a/c £ J^. (or b/de^jfj

then a/c AS b/d f ( i i i ) Let a,beJ"l.and a b. Then i ^ b implies

-llt-

for all c , d g J l such that a ss c and b » d | (iv) The factor-setis isomorphic to the f ie l* of real numbers R (or to the field of tn» complex

numbers C in the case of the complex asymptotic numbers).

PBOOFi Consequence of Theorem 30, ( i ) .

HillAltK. I As A (and J"), too) is not a ring ££, Theorem fQ f • the propertiesof the ideals, we know from the theory of rings, do not bold (in general) i v

A (or in SI ) • In particular, i f k i s an ideal of A (or If i s an ideal in

J ^ ) , then the relation " a — b, i f a - b & W" i s not always an equivalencerelation in A ( i n *Q.)ji-for example, & does not define such a relation /'./j£, Theorem 18/, Consequently, the assertion of the above theorem ig n»t a standarcorollary of Theorem 35. On the other hand, SXo i s not a DOXIBAL ideal, ia Sfc.

(in spite of JT/J7, being a f ield) , p.g. SUvU*(\£l),} - 0 , 1 , . • . , * - , ar . al«e>ideal* in SL There XA were defined in ft, Definition llj.

itaiARK 2. Elements of •&•/&> (the equivalence classes) we shall denote by tbeirImages in B , i . e . as real numbers.

.DEFINITION 10 (Real image • f ) • Th* cononicn.1 homomorpbism of Si. onto ft

(the homomorphism of Q onto R with kernel Slo ) wil l be denoted by f ,

If a i s a f ini te number, i . e . a e j l , then i t s image in B Ti l l be den*t«4

by t{o.), i . e . T (&)eR for every & ^ 47- • ( I n t h * c a s e o f coaplex aavapt*-*t i c nvsbers T ( a ) e C for a l l f inite complex asymptotic numbers a . ) T<*)wiIVbe called real (or complex, reap.) image of o. .Corresponding to Theorem 30, i f 4 6 X V and * S H, tyien f(a) - I i f and omly

i f there exist* a representative <*•&.& such that l m «C (s) - x. In fact,l ig>t(s) • x for al l - i s a .

Cll I f <* G. Q-^fL , then f(u) - <4> vbere Uo is the correspo«dlng(the first, i .e. for k-o) coefficient in the normal additive form [

f (a) - 0 for all a e -iU i i.«- for all infinitesimal*.nf

fitOOFi Trivial .

3% The homomorphism <f possesses the following properties) If a,b &them ( i ) a ^ b implies rX») ^ f(b) ( (ij,) a ^i b implies r'(a) .i f and only i f » * b | ( i i ,p f ( l a | ) - l f ( a ) | f (iv) f (») £: 0 i f antif ml ~& % .

1'flOOFi ft'e shall prove only the property ( i )^ the remaining ones are proved

analogously. Let a 4, b, i . e . (by definition) there exist. <Cea and j t t k

such that j s i -i-(a) ^ / ? ( s ) J gs .The latter means i.e.

-16-

aaiAHKs We proved that a unique real (or ctunplei) number x « Y{») corresponds

to every f inite asymptotic number a . On the other hand, Corresponding to

If, Theorem 20j, H is isomorphic to two eubspaces H and R"*of A {Sy Defi-nition IO/, i .e . R B° and fl^ it5*". Now,let us rcatvlet ourselves to only onejfthese isomorphisinsi either HSg B° or H ^ R"* . Then the following assertion

holdsi Every f i n i t e real asymptotic number a ia in f in i te ly close to a unique

real number, namely, f ( a ) , i . e . a as ^{a) for a l l a e JJ. •

5. INTERVAL TOPOLOG* OF A

DEFINITION 11 (Interval topology), ( i ) A non-empty subset A of A will be

called interval of A i f a,b&iS , x e A and a, * X ^ h implies x e.£rt

( i i ) The subset CE of A wi l l be called an open set of A i f for every x e ^

there ex i s t s an interval A Q Of such that I E ^ . (The empty set 0 as well

aa A are open se ts of A) | ( i i i ) The family of a l l open sets of A wil l be

called the interval topology of A f (iv) . By the term a neighbourhood of

a e A we shall understand every subset k of A for which there ex i s t s an

open set C of A so that H E if i 'S: H I (v) Let t ^ i and i D g l , nEs-N .

Then we shall say that the sequence | t tn 1 n e k ] i s convergent (with respect

to the interval topology) and a i s i t s l imit and we write th i s as LimaB - a

(in contrast to "lim" which wi l l be preserved for the ordinary topdlgy of ft)

i f for every neighbourhood U of a there exists no e N- such that aB e. M

for a l l n > n,, j (vi) Let » n e A» n E N. The sequence j an 1 n e N i w i l l becalled fundumenttil if for every £ ^ A+ ( A+ is the set of a l l positive asympto-t i c numbers) there exists n0 3 N such that 1 ftp - aq |-*1 ( for a l l P i q g Nsuch that p >• no and q >• nQ . (see, far instance, fij) <

We ought to find a. more convenient und simpler, i f passible, base of neigh-bourhood a of every asymptotic number. In the case Lim an » a JE; A*~ / 5 , Defi-nition l l / the fumily of a l l intervals of . type (a -t , a +£ ) "JisAifx-awhere f & A+ turns out to be such a base. The following theorem i« validi

THEOREM 38(The case Lim an e A~ ) . Let i,, e A , n e S and a e.ST . Then

Lim a • a i f and only i f for every. / ^ A. there exists no e ^ such that

U B - t for al l g

yaOOF: Je must prove that the family of intervals (a -r,a +f), / e A+ i s a

base of neighbourhoods of a . Indeed, a.^ a." GL A, u. £ A*" and a ' ^ a ^ a."

implies a ' - a, a - a " ^ 0 • Corresponding to Theorem 21, a ' ^ a - / '*

a +g"JZ a " for &'=* (a - a ' ) /2 ^ud / = ( a " - a) /2 .Let us put £ -

/e obtain a' 1 ii - f i n _ a. + £ ^. ,a'*« Viic tuoorma ia prove'

-1?-

The following theorem gives a t e s t for convergence in the case Lim__an

THBQiLEti 38(A teat for convergence in the Case £4

n e N and a ts A~* be chosen a r b i t r a r i l y and l e t

A"" ) . Let

be the i r normal additive forma {&, Definition 13] where^** and >*« are the

are the power and accuracy of » ,

if and only if the following three conditions

1*™ »n =•»- f t i i i J for every k £ i ,

powers and accuracies of a and A1 and

respectively. Then Lim an » a

a r e v a l i d i ( i ) Ji™/1 n =/* I

there exists N such that for all n

and lim 1)^- mean /* und V are the unique

a m ' ^ n * " G " J respectively.

IU&AH3-.) As usual, lim/tt -A*

adherent points of the sequences

In our case^* -J */**} ^ S z U-i°*}and consequently, M and V are

either trivial adherent points of the above sequences respectively (the sequen-

ces are trivial) or these sequences increase unboundedly.

Iff >JjPROOF: Let us assume that ( i ) , ( i i ) ani ( i i i ) hold and let £ &

are the powers of ^ n « [a^ - a | ,a g l l , then l^Mi-^ » i-" and consequently,

A^A^ >A*l holds for every suff ic iently large n where yfj. i s the power of t.

Uencc, re obtain |a n - a,|^. £ for ul l suff iciently large n, i . e . Lin aD - a •

L t Li a hold und let us assume that ffzl V f»"*t i s an adherent point

If we choose £ & A+ such tlmt A y « ' then

for infinitely many n , vhich contradict*

s mi j,dliercnt uoint of | v I It E M j &nd '

of

we

n&Nj[

obtain

Lira an = a

|a n - a| > €

If »£i I) i«-J

then ior £ 0 we obi . in ^ contradiction aiso. Let us assume that there

exists k €= Z such that k fc S* ..nd at. -*LIS !n ^^

latter means ^/*^*- — k f o r i"f initely many

choose £ & A such thnt A*t -* k wei n f i n i t e l y many n • The proof i s completed.

COHOLLAiti 1 I f V n E. Z U \ c \ , n e N and Iii

^. for infinitely many a. The

n and,consequently, if we

obtain \&n - a| > £ for

V . a « . then

f o r any M &L Z and any «iL £

COKOLLiHY 2 In the case " V,

c ) , k

. —" for a l l nH (iQ) reduces to

= L ,• " I 0 ) ,

COItOLLARY 3.

(2*1 Lim

holds for arb i t ra r i ly chosen

- 0

( o r C), n « N .

4 Liia 6"n>

0 holds.

,/e notice that, corresponding to ^'J, Definition 1^/ and £^, TheorifceSS/, w» treat

evory formal power series us a number of A*" ; as we did in (2^) and (i.%) where

the elements of the sequences on the left-hand sides'of these formulas are numbers

from A»- . '

40. Let A» n e N . Then the sequence i s fundamental

i f and only i f i t i s convergent and Lim * n A"1" .

PBOOFi The proof i s quite analogous to that of Theorem jfl. We shall not give i t .

DEFIKITIOK 1^ (Series in A>. (i) Let J* s Z and ftk e A, k ^ ^ + l , , , , be an

arbitrary asymptotic numbers. The sequence {Sjjj n - / f , y * + 1 , . . . ^ f There

3n - fl ak ,will be called an inf inite series,uiid for this sequence the notation

will be used. The elements of the sequence {$nt n - y tM + 1 , . . \ are called the

partial suns of the inf inite series} ( i i ) An inf inite series will be called

convergent i f the sequence of i t s partial sums i s convergent and the limit of

this sequence wil l be called turn of the series , 'ito shall not use different

Dotations for convergent infinite aeries and their sum;

'i'J-EOit£h 41 For any

the aeries

(23)

and any (o . *, * +1 f

I <,i s convergent o n J^p f i . e , for e v e ry infinitesimal h, and (2J) i s the additive

form ^ i , Definition Vi] of i ts sum.

In fact, the uliove theorem i s a tr iv ia l generalization of Corollary 2 of

Theorem $t (vo could obtain the case described in this corollary by h . E

which i s an infinitesimal. Jo should s t i l l notice that for infinitesimals h

with f in i te tccumcy U ( i . e . for h e A \ / | - ) the sum (21) coincides with

one of the partial <}um*t of the series (i.3), n;unely, with i,, .

(23) defines it napkin- of .ft, i , ,to „ , i . e . (ki) l s : lI1 a 3 v l l l o U t i c

function defined on

-18- -19-

tfe considered the cuse Lhn^atl - a e A** only. The case IJ^fn - a A*" i»

more complicated and not too important for us. That i s wily we shall briefly

expose the results in this case without giving the proofs-

DEFINITION 13 (The set E(a)). If a EA, then we shall put

und i.- the power of £ .{it V» • - , i . e .where D is the accuracy of a

• e A"* , then £(») - A+.)

TUEORili 42 (The Case Lin Bn - a fife. AT*)* If the accuracy V of the asymptali*,'number a i s f in i t e , i . e . V g ^ i then the family of intervals (a*, a +g)

obtained by al l n 'gj l such that a V a and a'- ae.fi? (*•«. l ' £ » ) andal l E £ E(a) i s a base (which i s simple and convenient, in some ways) ofthe neighbourhoods of a .

THB3KEM 43 A i» Hausdorff's set (see Theorem 31). i

THBDREU 44 A i s not Dedekind complete. (An ordered s«t i s called Dsdekiftdcomplete i f every non-empty subset which i s bounded above has a least upperbound.) In other words. A i s di(connected. ~*

VSOOFi Indeed, the sot A^*) of a l l asymptotic number* with the same p©v*rj* ^

i s bounded above, namely, every positive asymptotic number a with pawr J^ /*

i s an upper bound of A('u). But A_p*) has not a least upper tOTn4-

A* ve knov fb, Theorem SO?, the set of the real numbers R i s isffnorphi<f<*p

the sets a and tC [%, Definition 10/. Consequently, the topology of A^

induces two topologies on R: B - topology (by means of the lsoa»rpbl*m,R V R°) and HT-topolagr (by means of tbe isoooVphian H S 8 " ) . The ^

theorems refer exactly to these tvo cases.

THEOREM 45 (R°-topology). Let an N, i.e. an - xn + 0* where

n e N. Their the sequence lant ne.K| is convergent (vita respect to the

of A) if and only if one of the following tvo conditio'n*. is vallfT i (i)(i) j x n t i s convergent with respect to the topology of K andfor all sufficiently large n g N where x - lim xn . In this ca«e him a, • a,

° ( ° °where a + 0° i .e .In this case Lio an

m(x_ + 0°) - lim *„ » 0° I ( i i ) l i s x . • « •o*" n a—Mai " ' * n*^b* •

"(tn + 0°) . Q'1 .

THEOHEH 40 (IT*-topology). Let a n c R°~, n Nt i .e . »,, - xn + 0"* where

D € N . Then the sequence {&ni i s ) l ] i a coarergent (with respect t« ik«

gy of A)if and only i f one of the following two conditions is validt

-20-

( i ) i s t r iv ia l , i . e . there exists such that an

for n i l su f f i c i en t ly large n & N . In th i s case Lin an = a, of course)

( i i ) lim x n = — . I n th i s case Li u » n - 0~ .

TiiEUHJSIt 47 Let ^ani n e ^ and {bni n g J i j be two convergent sequence* of

asymptotic numbers and l e t Lim aQ - a and Lin bn - b , Then the convergence

of the sequences \ a^ * bn i n& Nl , | a n . bn > n e M J or ^Hn/bn » n S S J

implies ^>jjan * ba) - a ± b, Liu(an, bn) - a. b or i^i^o/^o) • * /b respac-

txvely ( in the case of d iv i s ion ve ought to require s t i l l ba,bjs,& , n e S ) ,

We shal l stop invest igat ing the properties of the set of asymptotio nuabersv&s

A . (We have done i.t perhaps in more detail than/necessary for our purpose* i o •

phys ics ) . Now, ve are ready to define the asymptotic functions and, in par t i -

cular, Dirac'a delta-function as v e i l as i t s square, e t c . , in accordance aithlour

promise a^de at " the beginning of / 8 / . We shall put tttJ-S off for • "

a future paper.. • •

ACKSOWLEDGMENTS

T h i s paper has been prepared under t h e g u i d a n c e o f Acad. Cbr, Ya.

Christov. The author would l ike to thank him for the help. The author alsowishes to thank Professor Abdus Salam, the International Atomic Energy Agencyand UNESCO for hospitality at the International Centre for Theoretical Physics,

• Trieste.

-21-

CURRENT ICTP PHEPlilWTS AND

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J . LOHENO, J . PRZYSTAWA and A.P. CHACKNELL:subduction c r i t e r ion .

A comment on the chain

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* Internal Reports 1 Limited distributionTHESE PREPRINTS AR3 AVAILABLE FROM THE PUBLICATIONS OFFICE, ICTP, P.O. BOX 586,

I-341OO TRIESTE, ITALY.

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T.N, SHERHT: Comment on t!;e question of pauge h ie ra rch ies .

IC/79/51 A.O. BARUTs Mafjnetic resonances between massiva and classless apin-Jparticles with mapnistic mojuents.

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M. TOSSOUF: A possible real izat ion of Einstein 's causal theory under-lying quantum mechanics.

H.F. KOTKATA, E..A, HAHMOUD and M,K. EI-H0U3LY: Kinetics of crystalgrowth in amorphous solid and supercooled liquid TeS-gQ using OTA andd .c . conductivity measurements.

FAHG LI ZHI and ft. iUTFFINIi On the douplara S3433,

I.H. EL-SHIRAFYJof spiral p la te .

First and second fundamental boundary value problems

On electrodynamics with internalA.O. BAKUT, I . HABUPTO and 0. VITISLLOifermionic exci tat ions.