Một số vấn đề xung quanh chuẩn tam giác Acsimet. pot

Bai baa de c p mot so van de xung quanh ham sinh doi voi cac dang chuan tam giac Acsimet.. Chung diroc sd-dung ro g rai trong cac mo hlnh heuristic dira tren lap luan khong chac chan voi

T; pchi Tin hoc vaDieu khien hQC, T.2 , S.4 (2004), 373~384

MQT 50 VAN DE XUNG QUANH CHUAN TAM GIAC AC51MET

Vi~n Gong ngh~ thOng tin

Abstract This article deals with some problems relating to decrea ing and increasing generators (addit ve generators) of Archimedean Trian ular n rms.

Tom t~t Bai baa de c p mot so van de xung quanh ham sinh doi voi cac dang chuan tam giac Acsimet.

Chuan tam giac, goi th la T-chwln va T - dc)ichuan, la lap cac ham 2 bien mo ro g cua hai phep toan logic va va ho~c Chung diroc sd-dung ro g rai trong cac mo hlnh heuristic dira tren lap luan khong chac chan voigia tri l<\l,pluan nKm tro g doan [0,1] Kho g d n gian nhir hai p ep toan va va hoiic, ca cap T-ch an, T - doi chuan la mot loat cac tu ch n khac nhau ma tro g qua trlnh lap luan, h tho g co the lira chon tuy thuoc VaGcac yeu to chi phoi nhir trlnh dQ chuyen gia, ng o thu thap tin Trong [1] chung toi dii trlnh bay nhirng kien thirc ca ban ve Tvchuan, T - doi chuan cling nhir mot so danh gia toan hoc xu g quanh phep phu dinh va lap luan khong chac chan Trong bai bao nay chung toi trlnh bay nhimg n hien ciru tep tuc x ng quanh cac ham sinh cua Tvchuan, T - doi chuan dang Acsimet

Cau true bai bao nlnr sau: Muc 2 danh cho viec gioi thieu ham sinh cua T-chuan, cling mot so ket qua chimg minh toan h9C xung quanh c c ham sinh cua T-chuan Acsimet T-d i chuan Acsimet va ham sinh tu n irng dirc trlnh bay tro g Muc 3 Muc 4 trlnh bay mdi quan h~ giira phep phu dinh manh va cac ham sinh Phan cu i bai bao la quan h~ giira cac ham sinh va mot so cap Tvchua , T - doi chuan teu bieu

J , ,

2 T-CHUAN VA HAM SINH Nhimg van de ca ban ve Tvchuan va T-doi chuan d diroc trlnh bay trong [1],de tien theo d i, chung toi nhac lai dinh nghia cua chung

Djnh nghia 2.1 Tvchuan la ham so T :[0,1] x [a, 1]-+ [0,1] sao cho voi moi x, y, z, t E [0,1] luon co:

(i) T(x, 1) = x (dieu kien bien p ai);

(ii) T(x, y) 2 T(z, t), neu x 2 z va y 2 t (tfnh dondieu);

(iii) T(x, y) =T(y, x) (tfnh giao hoan):

(iv) T(x, T(y, z) = T(T(x, y), z) (tfnh ket h p)

Tir c c dieu kien (i), (ii) va (iii) de dan su ra tinh chat sau cua Tschuan:

(v) T(x,a) =T(O,x) =° (ton tai phan td-0)

374 L HAl K H l , D~NG XU AN H ONG , NGUYEN L UONG D ONG

T<chuan OlIQ'C g i la A c simet neu va chi neu n6 thoa man them 2 dieu kien sau:

(vi) T la lien tuc;

(vii) T(x,x) <x, '\I x E (0,1)

T -chuan Acsimet OlIQ'C goi la chif , t (strict) neu va chi neu n6 thoa man them oieu kien:

(viii) T la tang chat tron (0,1) x (0,1), tire la neu Xl < X2 va Y1 < Y2 thi T(X1' yd <

T (X2 ' Y 2 )

D!nh ly 2.2 (xem, chan han, [4])Ham ss r [0,1] x[0,1] -+ [0,1]la mot T- c huan A csi m e t

n e u v a c h i n e u ton ttii m¢t ham s o f l ien iu c va gidm c h if, t t i [0,1] s ang [0,00], uo i f ( 1 ) =0,

sao c ho :

T(x , y) = f [- l J (J(x) +f(y)) , ' \ I x , y E [0,1]'

trong d6 f [ l J c ho bdi c ong t u i c

(2.1)

f[-l J (Z) = {f-1(Z), neu Z E [0, (O)],

(2.2)

Ham f neu tren OlIQ'C goi la ham sinh gidm (decreasing generator cua Tvchuan, con f [ -l J

OlIQ'C goi la ham gid·nguqc cua f.

Nhan xet 2.3 Do tinh chat giam cua ham f nen ta c6: ° ~ f(x) + f(y) ~ 2f(0) ~ +00, '\ Ix, y E [0,1] VI the mien xac dinh [0,00] cua ham gia n iroc f[-lJ neu trong Dinh ly

2.2 co the lam chinh xac hem (cu the la o01;1-n[0,2f(0)] nhir sau:

- Neu f(O) < +00 thi

f [- lJ( z = {f - 1(Z), neu z E [0 ,f(O) ,

0, neu z E (f(0), 2f(0)].

- Neu f(O) = + 0 thl

f [ lJ(Z) = f-1(z), ' \Iz E [0, +00]

- T-chuan Acsimet la chif,t neu va chi neu n6 OlIQ'C sinh boi mot ham sinh giam f nhir

tren va voi f(O) = 00 Khi 06 ham f OlIQ'C goi la ham sinh gidm chif,t. Tro g tnrong h p

khong chat Tvchuan Acsimet OlIQ'C goi la Tvchuan nilpotent voi f(O) = 1va ham sinh f khi

0 OlIQ'C goi la ham sinh g i dm c h suin

- MQi ham sinh giam va ham gia ngircc cua n6 aeu thoa man h~ thirc: f [- lJ (J(x)) =x ,

' \Ix E [0,1]' va

f(J[-lJ(X)) = {X' neu X E [0 ,f(O)],

f(O), neu X E (f(0), 00 ].

Vi du 2.4 f(x) = 1 - x P, p> 0 Day la mot ham sinh giam chuan (do f(O) = 1) Khi 06

Tvchuan OlIQ'C xay dung tir ham f(x) tren nhir sau:

Tir f(x) ta xay dung h m gia n iroc theo cong thirc

(2.3)

f [ -l J (X) = {f - 1(X) = (1- x)~, neu x E [0,1]'

MOT s6V A N ElE XUN G QUANH C H AN TAM G IAC AC S I M ET 375 Khi do T -chuan se la:

T( x, y) =f [- l ] (U( x ) +f(y)) = f [- 1 ] (2 - x P - y P)

-= {(XP +yP - 1)i , neu 2 - x P - yP E [0,1]

1

Nhtr v~ chung ta thay r~ g, veri bat ki mot ham f lien tuc va giam chat nao tir [0, 1]

san [0 ,00 ] ' voi f(l) = °luon co the t9-0 ra mot ham Tvchuan Acsimet th ng qua cong tlnrc

(2.1) Diroi day chung ta xet mot so ham sa cap voi dfeu kien giam chat trong doan [0,1] (ham sinh ra T -chuan Acsimet chat)

• Xet la cac ham phan thirc hiru ti bac nhat - ham hypecb l v o g goc, voi x E [0,1]:

ax +b

f( x )= d ' ei -0, a d- bei - 0

ex+

Nlnr di biet, ham nay lien tuc va luon dong bien hoac nghich bien tren tung khoan xac

dinh, & day cluing ta din tim ra cac dieu kien de ham la n hich bien trong doan [0,1]

Xet dieu kien f(l) =0:

a+b

=°¢:}a+b =0

e+d

VI ham sinh giarn chat f nhan true tung lam tern can dung, do do:

d

= °¢ :} d = 0 e

Ng ai ra de f( x ) n hich bien trong (0,1] thl phai co f'( x ) ~ 0, Vx E (0,1] va dau bang xay

ra chi tai c c diem rei rac V~y la

-be

f'(x) = ( ex )2 ~ 0, Vx E (0,1]

Do ei - 0, va bi - °(vI dau b~ g xay ra chi tai cac diem roi rac), nen b , e phai cung d u, Han

nira, VIa +b =°nen a i - 0 Khi do co the viet 1 -if( x ) nhir sau:

f (x) = ax - a = 1 ~x.

ex -a x

£)~t -% =A Do b , e cung da , nen a, c phai trai dau VI the, A > 0, chung ta diroc

I-x

f( x ) = A , A> 0

NgU'C!c1 -igia S11 co (2.4), chiing ta se chimg minh r~ g f(x) & dang (2.4) la mot ham

giam chat cua mot T -chuan nao d That v~y, tir (2.4) ch ng ta co l'(x) = ~ Nhu vay,

1'(x) <0, Vxi - 0, VA> 0, nen f(x) la mot ham giam chat tren (0,1]

376 LE H A l KHOl , D , 6 , N G XUAN H N , NGUYEN LUONG DO N

luo sinh ra diroc mot T-chucfn Acsimet

Chung ta co ket qua sau

Dinh ly 2.5 Ham p lu in t l uic h ii u t b ~c r du i t J( x ) = ~; :~ la m9 t ham sinh gid m c hi f, t cua

I- x

Nhan xet 2.6 Vci viec bieu dien T-chucfn qua ham sinh nhir (2.1) thl hing so A ( A > 0)

kho g lam thay d i dang Tvchuan do no tao ra Noi each khac, viec nhan ham sin giarn ch~t

v i mot so dirong cling se ch mot ham sinh giam chat moi va kho g lam thay d i T-chuan

tao ra Thirc ra, dieu nay khong chi dung cho triro g hop ham sin thoa man tinh chat giam

chat, ma con dung cho ca trirong hop ham sinh chuan Chung ta co dinh l y sau

Chun g m i nh Xet ham so g( x ) = a.J( x ) a > 0, v a Tvchuan

d 9 sinh ra Khi do se co

Th~t v~y,

* Neu z E [0,J ( O) ] (khi do a E [0, aJ( O )] = [0,g( O )]) , thl

* Neu z E [J(0), 2J (0)] , thi oz E [ a J(O) , a2J(0)] = [g(0),2g(0)] , suy ra J[ - I](z) = 0 =

Nhir vay, g [- I (a z ) = J [- I (z) , \ ;j E [0,2 J ( 0 )]. T do su ra

a

Dinh ly duoc chirng minh

Nhan xet 2 8 Tir ket qua tren suy ra co the viet lai diroc (2.5) diroi dang chin tiic:

•

MQT s6 VAN D E XUNG Q A H CHUAN TAM GIAC ACSIMET 377 Vci ham sinh giam chat nay, de dang tim ra T -chuan Acsimet thoa man dieu kien giam chat

x

Tvchuan duy nhat

x E [0,1]:

-x - d ' a , r:0, tir va m a khong c6 nghiern chung

x+e

Di'eu kien f(l) =0ch h thirc

a+b +c

d =° {: }a+b+c=° (d +e: 0).

+e

VI ham sinh giarn chat f nhan true tung lam ti~m can dirng, do 06 - ~ = ° { :}e = 0 VI

the, ccling phai khac 0 oe ham f(x) kho g suy bien

Xet 0 -0h m

/ a x2 - d e

f ( x ) = (dx)2 , 'Ix E (0,1]

De ham so n hich bien thl phai c6 f'( x ) ~ °trong (0,1]' dau ban xay ra tai cac diem n'1i

rac Di'eu nay c6 n hia la g( x ) = ad x 2 - cd ~ °tro g (0,1], dau ban xay ra tai cac diern roi

- Tlnr nhat ad <0 Khi 06 yeu cau bai toan tirorig duorig vci

maxg( x ) =g(O) = + cd ~ 0,

[ 0 , 1 ]

ma c d : f.0, nen ta diroc cd > 0

- Thir hai ad > 0 Khi 06 yeu cau bai toan tuorig dirong voi

m a xg(x) =g(l) =ad - c d ~ 0 [ 0 , 1]

Chung ta c6 ket qua sau

Diuh ly 2.9. Ham pluin tlui c hiiu ti d! ; mg f(x) = a x 2d: ! x e+ c , x E [0,1], u uuit ham sinh

gi d m cU a T 'c hnu i r: Acsi m e t c hi j, t n e u v a ch i n e u n c6 d! ; m g s a :

f( x ) =- a x2 + dxb X+c,b a + +c=O va, ' h ovc {ad < °

c d >0

{ad > °

ho ij,c

"". """ •• , .

3 T-DOI CHUAN VA HAM SINH

Dinh n g hi a 3.1 T-doi chuan la ham so S :[0,1] x [0,1] -+ [0,1] sao cho vci moi x, y, z,t E

[0, Ij luon co:

(i)' S( O ,x) = x (dieu kien bien trai):

(ii)' S( x , y) :2S(z , t) , neu x :2z va y :2t (Hnh don die };

(iii)' S(x , y) = S(y , x ) (tfnh giao h an);

(iv)' S( x, S(y , z )) = S(S(x , y) ,z ) (tfnh ket hop)

Theo dieu kien (i)', (ii)' va (iii)' ta de dang su ra tinh chat sau cua T-d i chuan:

(v)' S(x , 1) = S(1 , x) = 1 (ton tai p an tu 1)

T -doi chuan duoc goi la Acsime t neu va chi neu no thoa man them 2 di'eu kien sau: (vi)' S la lien tuc:

(vii S( x, x ) >x, Vx E (0,1)

T- doi chuan Acsimet duoc goi la c h ~ t neu va chi neu no thoa man them dieu kien: (viii)' S la tang chat tron (0,1) x (0,1), tire la neu Xl < X2 va Y1 < Y2 t hi S( X1' Y1) <

S(X2 , Y2)

Dinh l y 3 2. (xem, chang han, [4])H am so S: [0,1] x [0,1] -+ [0,1] la m9t T - aoi ct u in

A c sime t n e u va c h i n e u ton t ai mot h am so 9 l ien tuc v a t ang c h ~ t t re n [0,1]' v ai g(O) =0,

sao c h o:

S( x, y) = g [ -l ] (g(x) +g(y)), Vx, Y E [0,1]' (3.1)

trong a h am g[- l] ixic ajn h tren [0,+00] diio c c h bd i ciitu; iluic

g[- l] (Z) = { g- l (Z), neu z E[ O , g ( l )] ,

Ham 9 nhir tren diroc goi la ham sin h t a ng (increasing generator) cua T-doi chuan S , va

g [-l] d oc g i la ham gid ngu q c cua g

Cling nhir doi voiTvchuan, co the lam chinh xac han mien xac dinh cua g[-l ] , cu the la dean [0,2g( 1 )] , nhir sau:

- Neu g( l ) <+00 thl

g [ - l ] (z) = { g- l (Z),

1,

- Ne g( l ) = +00 thl

neu z E [ 0 ,g ( l )] ,

neu z E ( g ( l ), 2g( 1 )].

g[ - l ](Z ) =g -l( z ) , Vz E [0,+00]

Cling nhir trtrcng hop Tvchuan, co the thay r~ng (xem, chang han, [3]):

- T-doi chuan Acsimet la c h ~ t neu va chi neu no diroc sinh boi mot ham sinh tang 9 nhir

tren va voig( l ) = 00 Khi do g ' du qc goi la ham sin h t a g c h ~ t Tro g triro g hop kho g

chat, ta goi T-doi ch an Acsimet d la T- doi chuan ni l potent voi g( 1 ) = 1va ham sinh tang

9 khi do diroc goi la ham sinh t ang chsuin.

- M9i ham sinh tang va ham gi ngiroc cua no deu thoa man: g [ - l] (g(x)) =x , Vx E [0,1]'

M QT s6 VA N B E XU N G QUA N CHUA N TAM G I A C AC S I MET 379

g(g[- l J (X)) = {X' neu X E [O,g ( l ) ]

d T - doi chuan diroc xay dung tir ham g ( x ) n ir sau:

Tir g (x ) ta xay dun ham gia n uoc theo cong thirc

g l J (X) = { g - l( X) = 1- (1- x )~ , neu X E [0,1]'

Khi do T -doi chuan se la:

S( x, y) =g lJ ((g( x ) +g(y)) =g l J (2 - (1- x)P - (1- y ) P )

= {I - ((1- x ) P +(1 - y) P - 1) ~, neu (1- x)P +(1- y )P - 1> °

1

Ket hop voiVi du 2.4, cluin ta dUClC c~p T-chuan, doi ch an nilpotent sau:

{ I

Day chinh la cap Tvch an, doi ch an do Schweizer va Sklar tirn ra nam 19 3

Nhan xet 3 4. Tien han cac lap luan va chirn min tirong tv n ir tro g P an 2 cling se

qua nay clng co the co dUCl C tu mdi q an h~ gira cac ham sin f va 9 tro g cac phan trin

bay tep theo

Tro g [1],mot so ket qua xung quanh phep p u dinh da diroc trinh bay, p an nay tiep

phu dinh

Dinh nghia 4.1 Phep phu dinh la ham so N : [0,1] * [0,1] sao ch voimoi x, Y E [0,1]

luo co:

(i) N( l ) =°va N( O ) = 1 (dieu kien bien);

(ii) I c / x, y E [0,1]' neu x : : ;; y thi N(x) 2' : N(y) (tfnh dan dieu)

Tren thirc te, n irci ta thuorig quan tam c c ham phu dinh manh, tire la ham phu dinh

thoa man them 2 dieu kien sau:

3 0 L H l KH O I , B A G X U N H N , N G Y EN L U N B O G

(iii) N la mot ham lien tIC;

(iv) N(N ( x ) =x

Dinh l y 4.2 C l io N l a mo t h am so t i t [0,1] -+ [0,1] K h i fl 6 N l a mo t h am ph u f l ~n h ma r d : neu v c i neu t o'n i ai mo t h am so f l ie n t uc t i t [0,1] -+ [0,00], sao c h f l a g idm c i u i

f ) =0, N(x) =f 1( 0 ) - f (x)) , VxE [0,1]' f 1 l a h m nguQ ' c c a f

Chung min h

o c« ki ¢ n di n: Giii sir N (x ) la mot ham phu dinh manh X a y dirng ham f(x) nlur sau:

- Ch f O ) = co st >0 bat ki

- f (x) = ~ f( O ) [1 - x +N( x ) ] , Vx E (0,1]

lim f( x ) = -21f(O) [1 - 0 +N (O) ] = ~f(0) 2 = f(O),

Mat khac, do cac ham so 1- x va N(x) a u la giarn chat tren [0,1] va f(O) > 0, nen f (x)

ciing la giam ch t tren [ 0, 1 ]

Cluing ta lai co

f(x) + f (N(x)) = " 2f O ) [1- x + N (x)] + "2f O ) [1- N( x) + N(N(x)) ],

1

= " 2f O ) [1- x + N(x) + 1- N( x ) + N (N(x)) ]

Thoo dinh nghia e LW ham phu dinh 1I1 -nhthl N (N(x)) = x , VI the ta co

f (x ) +f N( x )) = ~f ( 0 ) 2 = f O) ¢ } f (N(x ) = f O ) - f x ) ¢} N(x ) = f 1 (1(0) - f x ))

V?-yv ri1I1 iham phu dinh man N ( x ) luon ton tai ham so f (x ) : [0,1] -+ [0,+00) lien

tICsao ch f ) = 0, f giam chat va N( x ) = f 1(1(0) - f( x )) , Vx E [0,1]

D i e u k i ¢ f u: Ta co N (O) = f 1(f(0) - f(O)) = f 1(0) = 1 do r: la ham ngiroc cua f va

f( l =O

N(I) = f 1( f 0 ) - f )) = f -1( f(0)) = O

f O) - f (x d < f(O) - f X2 ).

[0,00] -+ [0,1] S y ra f 1 ( 0 ) - f( x d) > f 1 ( 0 ) - f( X2 )), hay N(xd > N( X2 ) '

Xet N(N(x )) = N( f 1( f 0 ) - f (x))) = f 1( f 0 ) - f -1 ( 0 ) - f (x)))) = f 1( f 0 )-( 0) - f(x ))) = f 1( f x )) = x

V?-yN(x) = f 1 ( 0 ) - f( x )) la mot ham phu dinh man

Nhan xet 4.3 Verip ep ph dinh chuan ta co: f x ) = 1- x, con veri ho phu dinh Yager ta

co: fw (x) = 1 - xw, w > o

MO T s6VA NDEXUNG QUAN H C H UAN TAM G I AC ACS I MET 381

Tucng ttr nhir Dinh ly 4.2, c ung ta co k ~tqua sau

Dinh If 4 4. Clio N la m9t ham ti [0,1] - t [0,1] Khi a N la m9t ham ph ' li ajnh manl

n eu va c h i neu ta'n tr ;r im9t ham 9 lien iu c tit [0,1] - t [0,00], sao cho g(O) = 0, 9 to , ta n c hij , t

va N(x) =g-l (g(1) - g(x) , Vx E [0,1]' g-l la ham nqu o»: c7 la g

Nhan xet 4.5 Voi phep phu dinh ch an ta co: g(x) =x. Vci ho phu dinh Sugeno ta co:

( ) = log (1+AX) \ -1 \ -/ °

g> - X A ' / > , / r

Voiho phu dinh Yager ta co: gw ( x ) =x W ,w >0

~ ,, " ;/ ' - ' J

, ,

T- DOl CHUAN TIEU BIEU

Muc nay trinh bay hai phuang p -ip xay dimg ham sin dira tren cac ham sinh dii ca

sinh gidm c hiu i n c ho bo i cang tlui c:

J I(x) = 1 - f(1 - x)

ciiru; s f la m9t h m sinh gidm chsuin.

Tuang tu nhir vay, ch ng ta ciing co the xay dung dirc mot ham sinh tan chuan gl (x)

mci dira tren ham sinh tang chuan biet t.nroc g (x) bang cong thirc:

91( x ) = 1 - 9 (1 - x).

Menh de 5.2 V6i mo i lu i tt i si n h g i dm f( x ), c l n uu; ta c o thl ! x ay dung m o t h am sinh !Jilim m6 i thOng qu a cang th o c

J I (x) =f(g(x)),

v 6i g(x) to ,m9t ham sinh tang cluuit».

moidira tren ham sinh tang biet truce g(x) bang cong tnrc:

gl(X) =g(f(x)) ,

trong do f(x) la mot ham sinh giam chuan

Phan diroi day chung ta se xem xet mot so cap T-chuan, T -d i chuan teu biP1l

Vi du 5.3 f(x) = (1- x )p , » >0, Do f(O) = 1 nen day la mot ham sinh giam chuan Xay dung ham gia n iroc:

f [ l ] X) = {f-1(~) =1 - x~, neu x E [0,1]

3 2 L H l KHOl ElA G XUAN H N , NGUYEN LUON ElO G

X ay dung Tvchuan:

T ( x, y) = f [- I](( f X ) +f(y ) = f [ - ] ( - X)p + (1- y )P )

= {I - ((1 - x )P + (1 - Y )P)*, neu (1 - x)P + (1 - y) P <1

0, neu (1- x ) P + (1 - y )P > 1

1

= 1 - (min ( 1,(1 - x ) P +(1 - y)P) ) p

D e dang tirn ra T -d i ch an tuang irng voi Tschuan tren la:

S(x, y) = min (1, {/x p +y p ) ,

tuang ling voi ham sinh tang g( x ) =f(l - x) =x",

Day cling la c~p T-chuan nilpotent do Yager tirn ra n m 1980

Vi du 5 4. Trong vi du nay cluing ta xet mot lap T-chuan/d i chuan dirc tham so h a, cu

the:

p - 1

f (x) = logp -, » > 0, p # - l

p x - 1

RiSran day la ham sinh giam chat co tap xac dinh [0,1]' voi f(l) =0 va f(O) = 00.

f [- I ] (X) =f l(x) =logP (P p-1x + 1).

Xay du g Tvchuan:

( p - 1 P-1)

T ( x, y ) = r : ' ( ( x ) +f(y ) = i= logp p x _ 1+logppY _ 1

P (p x - 1)(pY - 1) P P - 1

day chinh la h Tvchuan Frank

HQ T - doi chuan Frank tuan ling la

S (x , y ) = 1 _ logp (1 + (p l -x - l )(p l -Y - 1)

p - 1

v ' i ham sinh

p-1

g ( x ) =logp I x "

-VI g (x) = f - x), S (x, y ) = 1- T ( x , 1- y).

Duoi day xet triro g hop khi p -+ 00 lieu ham gioihan co con la mot ham sin giam chat

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f (x) =logp - =logp +logp- +logp-

p X -1 P p X p X - 1