The 7 th Balkan Conference on Operational Research BACOR 05 Constanta, May 2005, Romania

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1 The 7 th alan onerence on Oeratonal Reearch AOR 5 ontanta, May 5, Roana THE ESTIMATIO OF THE GRAPH OX DIMESIO OF A LASS OF FRATALS ALIA ÃRULESU Ovdu Unverty, ontanta, Roana Abtract Fractal denon are the ot ortant attrbute o ractal and the o countn denon wdely ued. Uually t not o eay to deterne denon. In oe aer we have condered a cla o uncton and we have tuded the ntude o the Haudor h-eaure o the rah, Ã, o an eleent o th cla. In th aer we deterne the o denon o Ã. Keyword: o denon, Haudor h-eaure, Hadaard condton. ITRODUTIO The ortance o the ractal et n cence ncreae n the lat year. The ot ortant attrbute o ractal are the denon. For the ecovtch uncton, ven by t co t, where, and l, the ractal denon have been etated, n oe cae [5], but ther eact ractal denon unnown. Denton Let R n be the Eucldean n - denonal ace. I r a ven nuber, then, a contnuou uncton hr, dened on [, r, nondecrean and uch that l hr called a eaure uncton. r I, E a nonety and bounded ubet o R n and h a eaure uncton then, the Haudor h - eaure o E dened by: Hh E l n h,

2 n ben condered over all cover o E wth a countable nuber o here o rad. Partcularly, when r r, eaure and denoted by. h the obtaned eaure called the -denonal Haudor H ate the Hadaard condton there Denton It ad that the euence * * et uch that, or every. It nown that the rah o a uncton : D R the et,, D. In our aer [] - [4], the uncton co t ro wa relaced, 3,, 3 and the ollown uncton wa ntroduced,,,,, 3 where ven n and * R a euence that ate the Hadaard condton. Theore [], [3] I h a eaure uncton uch that h t ~ t,, the uncton dened n 3,, and * R a euence that ate Hadaard condton, then H. The reult rean true and. In what ollow we hall deterne the o denon o the rah o the uncton ven n 3. There are any euvalent denton [6] o the o denon, but we hall ue the ollown one. Denton 3 Let be a otve nuber and let E be a nonety and bounded ubet o R. onder the - eh o R,,,,, I the nuber o the - eh uare that nterect E, then the uer and lower E o denon o E are dened by lo E lo E d E l ; d E l. lo lo I thee lt are eual, the coon value called the o denon o E and denoted by d E. For any ven uncton :, R and t, t,, we denote by R t,t the ocllaton o on the nterval t, that R t t u t. For brely, any,,, 5 derent value.,t, u t t, ut n th aer ndcate a otve contant that ay have

3 . RESULTS In th art o the aer we hall ue the ollown reult: Lea [6]. Let R, : be a contnuou uncton, and be the leat nteer reater than or eual to. I the nuber o the uare o the - eh that nterect, then,, R R. Lea Hölder neualty. I *, b a, R,, and,, then b a b a. Theore I the uncton ven n 3, wth,, then. d Proo.. We rove that. d The rt art o the roo ollow that o the theore. Let u conder, all enouh and * uch that: 4 Then. Un the Hadaard condton t can be deduced that,, where and are contant that don t deend on and. Thu,

4 . 5 Fro 4 and 5 we obtan. 6 Fro lea and 6, we deduce that R,. Snce, and,, then and ro the revou relaton t reult that, where. Thereore, lo lo lo l lo lo l d o,. d. We rove that. d I we conder, and * uch that. 7 and *, then. 8 We hall etate the odulu ro the orula 7, un lea. Let, be any nuber and. Then. 9 Un the Hadaard condton and,, we obtan

5 A ont called an ecetonal ont or a uncton the dervatve ' For the ecetonal ont,, For the non-ecetonal ont, or every, Then I ', I ', Thereore, then h h '., when h. h h h ' h h, h. ' h h h h. then h h h h h. h h h, h. Partcularly, or the non-ecetonal ont, or all enouh. So, or, Snce, and un 7, t can be een that, 4. doen't et.

6 4 5 Thu, the relaton 8 - ve: ow, ro lea,.,., R thu, d. lo lo l l lo lo Fro the art and, t reult that d. orollary I the uncton ven n,, and then d.,,, orollary I any erodc z - za uncton,, and then d.,, ILIOGRAPHY [] ãrbulecu, A., On the h-eaure o a et, Revue Rouane de Mathéatue ure et aluée, toe XLVII, o 5-6, ; [] ãrbulecu, A. 3, ew reult about the h-eaure o a et, Analy and Otzaton and Derental Syte, Kluwer Acadec Publher, ; [3] ãrbulecu, A., Soe reult on the h - eaure o a et, ubtted; [4] ãrbulecu, A., On a cla o uncton wth the rah o denon, ubtted; [5] ecovtch, A.S., Urell, H.D., 937, Set o ractonal denon V: On denonal nuber o oe contnuou curve, London Math. Soc.J.,,.8-5; [6] Falconer, K.J., 99, Fractal eoetry: Matheatcal oundaton and alcaton, J.Wley &Son Ltd.