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## ANNALES. UNIVERSITATIS SCIENTIARUM BUDA PESTINENSIS DE ROLANDO Eorvos NOMINATAE SECTIO MATHEMATICA TOMUS XXVIII RE DIGIT A.

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1 ANNALES UNIVERSITATIS SCIENTIARUM BUDA PESTINENSIS DE ROLANDO Eorvos NOMINATAE SECTIO MATHEMATICA TOMUS XXVIII RE DIGIT A. CSASZAR ADI UV ANTIBU S M.ARATO, M. BOGNAR, K.B0R0CZKY, E. FRIED A. HAJNAL, J. HORVATH, F. KARTESZI, I. KATA !, A. KOSA, L. LOVASZ, J. MOGYORODI, J. MOLNAR, P. REVESZ, F. SCHIPP, T. SCHMIDT, Z. SEBESTYEN, M. SIMONOVITS, GY. SOOS, V. T. SOS, J. SURANYI, L. VARGA, 1. VINCE 1986

2 ANNALES UNMRSITATIS SCIENTIARUM BUDAPESTINENSIS DE ROLANDO Eorvos NOMINATAE sectio BIOLOGICA inccpit annotated MCMLVII sectio CHIMICA incepit annotated MCMLIX sectio CLASSICA incepit annotated MCMXXIV sectio COMPUTATORICA inccpit annotated MCMLXXVIII sectio GEOGRAPHICA incepit annotated MCMLXVI sectio GEOLOGICA incepit annotated MCMLVII sectio HISTORICA incepit annotated MCMLVII sectio Iuridica incepit annotated MCMLIX sectio LINGUISTICA incepit anno MCMLXX SECT IO MATHEMATICA incepit anno MCMLVIII SECTIO PAEDAGOGICA ET PSYCHOLOGICA incepit anno MCMLXX SECTIO PHILOLOGICA incepit anno MCMLVII SECTIO PHILOLOGICA HUNGARICA incepit anno MCMLXX SECTIO PHILOLOGICA MODERIL an

3 SPLINE FUNCTIONS AND CAUCHY PROBLEMS, VI. APPROXIMATE SOLUTION OF THE DIFFERENTIAL EQUATION y (n) = f (x, y) WITH SPLINE FUNCTIONS By THARWAT FAWZY Suez Canal University, Ismailia, Egypt (Received September 13, 1979) 1. Introduction and description of the method In the recent papers [ 1] - [7] the approximate solution by spline functions of differential equations with given initial value conditions has been studied. In this paper a method to approximate the solution of the initial value problem y

4 4 THARWAT FAWZY We assume that f: R xr-r is defined and continuous with its first, second, ..., r-th derivatives in D: Jx-x 0 J (xk + 1 Yk + 1), q = 0, 1, ..., r, and the corresponding approximate values are defined as (7) ji ~ 1; _q) = J

5 SPLINE FUNCTIONS AND CAUCHY PROBLEMS, VI. 5 wherei = 0, I, ..., (n-1) and n + r y (j) (9) y ~ (t) = 2; ~ - (t-xk) j, xk: s: .f:; s;:, xk + i j = o j. Here, it is convenient to write down the Taylor polynomial of the exact solution for xk:; s;:, f:; s;:, xk + i as (I 0) n + r-1 y * y (.; k) y (t) = 2; ~ I (f-xk) i + (t-xk) n + r, i = d J. (n + r)! Xk <.; k *

*6 6 THARWAT FAWZY PROOF. By using (11), (5), (8), the Lipschitz condition (4), the expansion (9) and (IO) it is easy to get ni-1 e (i + j) n + re *

*7 SPLINE FUNCTIONS AND CAUCHY PROBLEMS, VI. 7 THEOREM 2. Let ycn + q)) (xk + 1), given by (6), be the higher derivatives of tlze exactsolutionof (l) - (2) / orq = 0, I, ..., r. lfthecorrespondingapproximate values} ii ~ iq) are given by (7), then the error is bounded by the inequality e (n + q) , yiq>, ..., y} 3>, q = 0, I, ..., n + r there is a unique spline function S Ll (x) = S L1 (y; x) interpolated on L1 to the set y and satisfying the following conditions: (18) (19) S (.Y; x) = SL! (x ) ECn + r [O, bj, Skq) (xk) = _Hql, S} .. 1 ~ 1 (xN) = y) sl where q = 0, I, ..., n + rand k = 0, 1 , ..., (N - 1), Also for (20) PROOF. From the continuity condition (18) using (19) and (20) it is easy to get (21)*

*8 8 THARWAT FAWZY where (22) n + rt y- (p = 1, 2 ,. .., n + r + 1) are the unknowns to be determined. The system of linear equations (21) in the unknowns a ~ k) has a unique solution since its determinant Dr, cO Here, and it is easy to prove that Dr = ld 11 1, i & j = 1, 2, ..., n + r + 1, d 11 = (n: r + j) (il)! hj n + r Dr = hl / 2 (n + r) (n + r + l) JI ii 1 = 0 and this does not equal to zero since h, c. 0. If we replace the p-th column in Dr by the column (F ~ k) 1 p ~ k>, ... , F}. ~,) T and denote the resulting determinant by D ~, then the solution of the system (21) will be (23) ap tk) - -, Df p - 1, 2, ..., n + r + 1 D, and after factorizing Df in terms of F ~ k>, Fik>, ..., F}. ~ ,, the solution (23) will take the form (24) and this completes the proof. l n + ra lk) = "" 'cp *

*9 SPLINE FUNCTIONS AND CAUCHY PROBLEMS, VJ. g PROOF. Using (24) we get 1 n + r (25) yes j. p hp-1 LJ pt I 1 = 0 Now, using the Taylor expansion of y (l> (x), ie: n + rt-1 yli + t) y *

(x) -S ~> (x) Is Kw, (h) hn + rq jorallxe [o, b] andallq = 0, I, ..., n + r. PROOF. Using (26) and (20), it is easy to get n + rq eti + q> Jy(1:) -y (n + r> I jy (x) -s ~> (x) I s ~ _k _.- hi + <> kt k hn + rq + i = o JI (n + rq)! + n + ft ql [n + r + p) la ~ k> jhp + n + rq. p = lq Taking the h elp of Theorem I, Theorem 2, the definition of the modulus of continuity of y(x) and the lemma 2, the above inequality becomes jy (x) -s ~> (x) Is cj * w, (h) hn + rq

*10 10 THARWAT FAWZY where cef * (q = 0, I, ..., n + r) are constants independent of h. Taking K = = max c; * (q = 0, l, ..., n + r), we get Jy (x) I :: o; JS ~ n) (x) -y <"> (x) J + Jy *

*11 A NOTE ABOUT CERTAIN ZERO SETS OF BROKEN CHAINS by G.RAMHARTER Institute for Analysis, Technische Universitat, Vienna (Received on) If a is any natural number, then, as is well known, for almost all real numbers xe (0, I) is the mean frequency, with the a occurs as a partial denominator in the regular continued fraction expansion x = =: [av a 2,] a1 + a2 +, present and equal to 2 log (l + l / (a 2 + 2a)). In contrast to this, in the semi-circular development with partial denominators of ~ 2, the al - a2 - frequency of occurrence of every number a2 = 3 is almost everywhere equal to zero. Since the proof of the first mentioned theorem, already assumed by GAuss, by LEVY and (independently) KuzMJN, numerous other mean value properties of the denominators have been found. However, the exception zero sets that occur have also been examined more closely, with the Hausdorff dimension being particularly suitable as a finer differentiating feature (cf. for example [I] - [4], [7], [8]). I. J. Goo D in particular has systematically studied the relationship between the growth properties of the denominators and the dimensional numbers. P. ERDOS suggested to investigate the set E of the chain breaks with different partial denominators in pairs. We will see that, contrary to expectations, their dimension is coarse, even if one excludes any (finite) number of values for the partial denominators or allows only increasing partial denominator successes beyond this. It has been shown that the semi-regular version of the problem can be applied to the asymmetrical Lagrange spectrum investigated in [5], [6]. The sets Gq n E considered in the following contain precisely the real numbers that are the worst one-sidedly approximable. For a fixed q E'N F q denotes the set of numbers en x = [a 11 a 2,], whose partial denominators are all ~ q, and Gq (c Fq) denotes the set of numbers x for which, moreover, ( qs) a 1 sa 2 s ... applies.*

*12 12 G.RAMHARTER SET. It applies as well as d. (F E) I (loglogq) 2 logq 1m qn = - + o --- (q- 00). According to ([2], Th. 2), dim Fq has the same asymptotic behavior as dim (FqnE). The additional condition an ~ as (n ~ s) surprisingly has no influence on the dimension in either case. Identical statements apply in the semi-regular version if q ?: 3 is assumed. BEWEJS. For every at most reducible system S of intervals Ii with lengths I / ii we set Lsff5) = .EII; ls. For a set Hc [O, I] and an e> o denote As ,, (H) = inf LsCJ), where the infimum is to extend over all systems of intervals which cover f-1 and whose lengths are all bounded by e are. Then there exists hs (h) = Jim As ,, (H), the e-0 so-called s-dimensional Hausdorff measure of H with respect to the measure function ts. Furthermore, there is (as can be shown) a uniquely determined number d E [O, I] (the so-called Hausdorff dimension dim H of H) such that lzs (h) = = for S d (In the case of d = 0 or d = I, the first and second of these statements do not apply). We call a closed interval / Cn) = / (av ..., an) with endpoints [av ..., an], [av ..., an + 1] (n, av ..., anen) fundamental ( with respect to H) of the n-th order exactly if there is an element x EH with x = [av ... a, j) ...]. Obviously, H for every fixed n is covered by the system SCn) of all fundamental intervals of order n with respect to H; we denote 3cn) as the fundamental nth order cover system of H. From the definition of the dimension it follows immediately that dim H = 5 s if Jim inf Ls (SCn)) *

*30 30 M. j. SCHOEMAN ourselves to those endomorphisms *

*34 34 M. J. SCHOEMAN LEMMA 3.1. Let E denote the extension 0 - B --AL ... A / B --0 and let *

* (a + b) = Is injective we have aeb, proving thatbnim 1p = 0. For any aea we have jip (a + B) = *

*42 42! VAN CHAJDA Hence there exist c, d of 2! such that (i) (c, d) ET (x, y) (ii) (1): Let 2! E *

*52 52 M.O.DESHPANDE-0.0.HAMEDANI (v) the condition (T 5) if and only if! _ ** o), then (x-ax) (y-ay) 2 '= 0 and hence condition (T 4) can be expressed as a 2 (x- y) 2 s a2 {(x- ax) 2 + (y- ay) 2}. (I - a) 2 II Thus (T 4) also holds for -sa <, where Tis taken on some sub- 3 I + Y2 paces of R. (iii) and (v) are essentially similar, so we will only prove (iii). Case 1 (O *

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