Is RD Sharma enough for IIT JEE

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 - O> '- 0 We can define the total q-th derivative ofjw.rt x, expressed as a function of x and y only, as (3) dq -f (x, y) = j (x, y) & where da -f (x, y (x)) = yes (x, y (x)). dxa 1 *

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 (x, Y2) I ~ L IY1 -Y2 I, q = 0, I ,, r. Let y: [O, b] -R be the unique solution of (I) - (2). Our purpose is to construct a polynomial spline function of degree m $ 2n + 2r + I approximating the solution y on the interval [O, b]. This spline function will be denoted by s, i (x) where LI is the mesh points LI: 0 = X 0 y (fn-;)) dfn-i ... df1. Xk (ni) times xk The relation (5) gives the values ​​of the exact solution and its i-th derivatives at x = xk + i where i = 0, 1, ..., (n-1). If i = n, n + 1, ..., n + r then the exact values of the higher derivatives, using the definition (3), will be (6) Yk "+" \ ql = 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 p hp-1, l ,,, j. pt t 1 = 0 4. Convergence of the spline function to the solution Before we prove the convergence of the constructed s pline function to the solution of the differential equation (1) - (2), we first prove the following lemma: LEMMA 2. The following inequalities are true 1a 1 :: s; AP w (h) p hp r where AP are constants independent of hand p = 1, 2, .., n + r + I. '

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 kt (x-xk) n + r-1 J = O J! (n + r-t)! wherexk (t: kt) Yn1 = L _k _.- hj + "'hn + r-1 i = o JI (n + rt)! and if we used this result with (22) we can get n + r-1 eu + 1> ((27) 1F?> 1 shl-nr-1 e ~ i1 + 2: ~ hi +. j = o j. + I 1y (x), then (27) becomes (28) IF tk> 1 * w, (h) to 1 1 sc 1 -, = ,, ..., n + rh where cf are constants independent of h. Using the help of this result (28) in (25), the proof of this lemma will be complete. THEOREM 4. Let y (x) be the solution of (1) - (2) and Let fecr ([o, b ] xr, where re J +. If S ,, (x) is the spline junction constructed in Theorem 3, then there exists a constant K, independent of h, such that Jy (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 (x) -S ~ n) (x) J = = l / (x, SJ (x)) - f (x, y (x)) I + i / n> (x) -s ~ n \ x) J. Taking the help of the Lipschitz condition and Theorem 4, this becomes JS ~ n) (x) -S ~ n \ x) J :: 5 LKw, (h) hn + r + Kw, (h) hr :: 5 M * wr (h) hr. Hence the proposition. REMARK. We have proved that the method is stable in [8]. References [ 1] OH. MIC ULA: Numerical Integration of Diferential Equation y (n) = f (x, y) by Spline Functions, Rev. Roum. Math. Pures et Appl., 17 (1972), [2] THARWAT FAwzv: Spline Functions and Cauchy Problems, I. Approximate solution of y "= f (x, y, y ') with spline functions, Anna / es Univ. Sci., Budapest, Sectio Comp., 1 (1978), [3] THARWAT FAwzv: Spline Functions and Cauchy Problems, II. Approximate solution of y "= f (x, y, y ') with spline functions, Acta Math. Acad. Sci. Hungar., 29 (1977), [4] THARWAT FAwzv: Spline Functions and Cauchy Problems, III. Approximate solution of y '= f (x, y) with spline functions, Anna / es Univ. Sci. Budapest , Sectio Comp., 1 (1978), [5] THARWAT FAwzv: Spline Functions and Cauchy Problems, IV. On the stability of the method, Acta Matil. Acad. Sci. Hungar., 30 (1977), [6] THARWAT FAwzv, K6HEGVI janos and FEKETE ISTVAN: Spline Functions and Cauchy Problems, V. Applications programs to the method, Anna / es Univ. Sci. Budapest, Sectio Comp. 1 (1978), [7] THARWAT FAwzv: Spline Functions and Cauchy Problems , Ph.D. Thesis. The Hungarian Academy of Science. Institute of Math. Researches. Budapest / T. [8] THARWAT F AWZY: Spline functions and Cauchy Problems, VII, Anna / es Univ. Sci. Budapest, Sectio Mathematica, 24 (1981),

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)) O does not generally imply dim H?: S. This makes access to the lower estimates, with which we want to start here, difficult. It is clear that because of the monotony of the dimension it is sufficient to show (1) The idea is now to consider suitable subsets of G / IE with not too small (and at most polynomial) growth of the subscribers, for which one Conclusion as the above mentioned is still just admissible, and thus to approximate the dimension from below.Let Gq, p (c Gq n E), q, pen, the set of numbers x = [a 1, a 2,], for which q = 5a 1

13 ABOVE CERTAIN ZERO QUANTITIES OF L o, I> S> S-t> O, q, PEN, and a cover system U of Gq, P of the type described with fineness L (m :!) holds. 5 The argumentation is essentially the same as for Gooo ((2], S), except for the observation that the requirement g (n) = O (IogP n), which is sufficient for the purposes there, is given by the weaker reg (n) = O (nP ) (Note in particular the relation (4.7)). We further claim: Let qen be given arbitrarily and p, 2P- I> Q, and s = _! _! chosen 2 p. Then we have for every finite cover system m:! of Gq, P of arbitrary fineness Ls (m:!)> I. It is clear that from (2) and (3) taken together dim Gq, P- ~ + (p- =) for every fixed q follows and from this (1): If qn = q (a 11, an) is the denominator of the reduced fraction Pnfqn = = [a 1 ,, an], then one results for the length Interval! (Av ..., an) using known rules IJ (n) I = l [av ..., an] - [av., An-v an + l] i = l / (qn (qn + qn-1)), together with the trivial estimates qn-l II (a 1,., Anw> 2-s ( (a 1 + I) ... (an + l)) - 2 s. The fact that finite systems consisting of fundamental intervals are sufficient to cover Gq P is immediately clear from the definition of these sets. For the proof of (3) one can assume without restriction of generality that there are no two intervals of such an upper cover system m :! have inner points in common. If an interval / (av ..., an) occurs, remove all sub-intervals / (av ..., an, k 11 k 2,) that may occur. In doing so, L 5 (m :!) is certainly not enlarged. If a cover system of the type under consideration now contains an intervalJ f (gv ..., ilm-v ilm, ..., il,), SO it contains for each / c = llm-1 f J, ..., g (m ) an interval of the form I (a 11, am-v k, bv ..., bv (k)) with order v (k) + m generally dependent on k. All of them are taken into account

14 14 G.RAMHARTER these facts and relation (4b), one obtains, if n denotes the largest occurring order of intervals in m5, then Ls (m5) = L; I J (m) Is> J (ai, ..., am) efil \, men g (l) g (n-1) g (n)> 2-s L; (a1 + 1) -2s ... L; (an-1 + l) -2s L: (an + 1) -2s. Now an-1 = an-2 + I an = an-1 + I and because of p "2 = 2 this is" 2 = 2 for n EN, as one can easily see. Continuing in this way, one confirms the correctness of (3). We turn to the above estimates. For m = 0, 1, 2, ... let Hm denote the set of numbers with an expansion of the form [(1) m, a 11 a 2,], with a sequence of m ones at the beginning and the remaining part denominators 2 ~ a 11 ~ a 2 ~ should apply. Obviously = 01 = U HmU {[(lL]}. M = o According to [2, Lemma 1), dim H 111 = dim H 0 and therefore (dim oq ~) dim 0 1 = = dim H 0 According to a Remark from above, it is sufficient to show the following statements about the fundamental cover systems sen) of H 0 and F qn E): (5) Let 2> a = 2s = 1 + t> 1 be given arbitrarily. Then Ls (S (n) (Ho)) <1 is sufficiently large for all (those n EN. (6) / if, moreover, q> (l / t) 1 1 1, then for all n Ls (SCn> (Fq + inE)) <1. (The proof of (6) would of course be dispensable, since the upper estimate of dim (FqnE) because of FqnEcFq also follows from the mentioned result of Gooo on dim Fq. Here we give a simple proof independent of this) . Because of (4a) we have - "" "" IJ (n) ls <2 al'1 2 a2a ... 2 a ;; a ~ -a -a J -ad (-1 1) -I (1 1) -I L .. Gn

15 ABOVE CERTAIN ZERO QUANTITIES OF BROKEN CHAINS 15 and so on until} 1) (1 1) 21 an-i -a _, ~ an - a -0 against 0 for n- 00 From this the correctness follows of 5). Again because of (4a), starting from Ls (S

16

17 NOTES ON LACUNARY INTERPOLATION BY SPLINES, I. (0,3) INTERPOLATION By THARWAT FAWZY Suez Canal University, lsmailia, Egypt (Received August 11, 1982) 1. Introduction. P. Tu RAN and J. BALAZS [I] in 1957 have initiated the study of "Lacunary Interpolation". Recently, A. MEIR and A. SHARMA [2], B. K. SwARTZ and R. S. VARGA [3], S. DEMKO [4], A. K. VARMA [5] and J. PRASAD and A. K. VARMA [6] considered special Iacunary interpolation problems. In this series of papers titled Notes on Lacunary Interpolation by Splines, we present new methods for the cases (0,3), (0,2), (0,4), (0, I, 3), (0, I , 4) and (0,2,4) interpolation and we get results of the same order as that of best approximation for the functions and their all possible derivatives. Moreover, the stability of interpolation in each case is proved. In this paper we begin with the case (0,3) interpolation and it is convenient to state the results of J. PRASAD and A. K. VARMA [6] in the following two theorems. THEO REM A. Given arbitrary numbers f (x;), i = 0, I, ..., n; j

(z1), i = 0, I, ..., n-1, p = 0, 3; 2z; = x 1 + xi + 1; f '(x 0), f' (xn); there exists a unique Sn ES ~~> 5 such that Sn (x 1) = f (x;), i = 0, I, ..., n, (1.1) s ~) (z;) = j

(z;), i = 0, I, ..., n-1; p = 0, 3, S ~ (xn) = f '(xn), S ~ (x 0) = f' (x 0). THEOREM B. Let fe Cr [O, I]. Then for the unique quintic spline Sn (x) associated with f and satisfying (I. I), we have (1.2) / S ~ t (x) -jCi> (x) / 2 PJ, r6r-Jwr (6), j = O, I, 2 and r = 3, 4, 5, (1.3) IS ~ J> (x) - j (x) I: = f3 1, r6 6-1 max IJ <0> ( x) I, j = 0, I, 2 and r = 6, O "'x"' I 2 ANNALES-Sectio Mathematica-Tomus XXVIII.

18 18 THARWAT FAWZY where wr () denotes the modulus of continuity of J (z"), k = 0, 1, ..., n- 1. PROOF. For xe [x ", xk + ij and k = 0, I, ..., n-1, set (2.4) Sn (x) = f (x1 <) + a1 <(xx ") + (l / 2) b 1 c (x-xk) 2 + (1/3!) C1 <(x-x1 <) 3 Then the values ​​(2.5) ak = (I / hk) [4f (zk) - 3f (xk) - f (xk + 1) + (hv 12) j <3 l (z1 <)], (2.6) bk = [4f (Xu 1) + 4f (xk) - 8f (zk) - (lzv2) j <3 l (zk)] / hi and (2.7) prove the theorem. THEOREM 2.2. Let fe C 3 [O, 1]. Thell for the unique cubic spline Sn associated witlz f and given in Theorem 2.1. we have for all x E [O, l] IS ~ l (x) - j (x) I: c 3, i1z 3 -iw 3 (/ z) i = 0, 1, 2, 3, where w 3 () denotes the modulus of continuity ofj <3 l, lz = max hk, k = 0, I, ..., n- I and the values ​​of c 3, 1 are C3, o = 2/3, C3,1 = 4/3, C3,2 = 5/3, C3,3 = J.

19 NOTES ON LACUNARY INTERPOLATION BY SPLINES, I. 19 (2.8) PROOF. We have for XE [xk, xk + ij, Xk <~ k k = I, 2, ..., n - l S1r (x) = f (xk) + ak (x-xk) + (l / 2) bix-xk) 2 + (1/31) ck (xx 1 J 3 + (3.6) + (1 / 4l) dk (x-xk) 4, (3.7) bk = (l / h 2) [j (xk + i) -2f (xk) + / ( xk_ 1) - (h 3 / l 2) (! <3) (zk) -j <3) (zk_ 1))] 2 *

20 20 THARWAT FAWZY and 3.8) S 0 (x} = f (x 1) + a 1 (xx 1) + (I / 2) b 1 (xx 1) 2 + (I / 3!) C 1 (XX 1 ) (I / 4!) D1 (X - X1) 4 Then, fork = I, 2, ..., n- I, the values ​​dk = - (- 8 -) 4! [/ (Xk + 1) -f (xk) - (h 2/4) bk-2f (zk) + 2f (xk)] + (24 / 5h) / <3) (zk), 5h 4 (3.9) ck = (!) 4! [/ (xk + il - f (xk) - (h 2/4) bk - 2f (zk) + 2f (xk)] - (7/5) / <3) (zk) 5h 3 (3.10) and (3.11) ak = (I / h) [/ (xk + 1) - f (xk) - (h 2/2) bk- (h 3/3!) ck - (h 4/4!) dk] prove the theory. THEOREM 3.2. Let fe C 4 [0, 1]. Then for t! Ze unique spline S Ll (x) associated with f and given in Theoretn 3. I, we have for all xe [x, <'xk + ij, k = 1, 2, ..., n-1 (3. J 2) IS ~) (x) -J , C4, o = 23/30, C4,1 = 49/30, C 4,2 = 179/60, C4,3 = 23/5, C4 , 4 = 17/5, ct 0 = 97/120, c!, 1 = 18/10, ct 2 = 209/60, c !, 3 = 28/5, c!, 4 = 22/5. Before proving this theorem, we state and prove sotne letntnas which will help us in arriving at the proof. LEMMA 3.1. For bk given in (3.7), the estimation lbk-f "(xk) i: 2 (h 2/12) w 4 (h) holds true for all k = I, 2, ..., PROOF. We have for all k = 1, 2, ..., n- l, (3.14) 3 f (xk + i) = L; W) / j!) j

21 NOTES ON LACUNARY INTERPOLATION BY SPLINES, I. 21 and (3.17) pa> (zk-1) = pa> (xk) - (h / 2) / (4) (riik-1>), zk-1 (x) i ~ (17 / S) wih) holdstrueforallxe (xk, xk + l] andk = 1,2, ..., n -1. PROOF. We have for all k = I, 2, ..., n-1, (3.19) 3 f (zk) = L (hj / 2jj!) J (xk) + (h 4 / 2 4 4!) J <4> (17 ~ kl), xk <'Y / ~ k) (xk) i ~ (6/5) hw 4 (h) holds true for all k = I, 2, ..., n- I. PROOF . Using (3.14), (3.16), (3.19) and Lemma 3.1 in (3.10), we get the required results. LEMMA 3.4. For ak given in (3.11), the inequality I ak-f '(xk) i ~ (23/60) h ​​3 w 4 (h) holds true for all k = 1, 2, ..., n-1. PROOF. Using (3.14), Lemma 3.1 and Lemma 3.3 in (3.11), Lemma 3.4 easily follows. PROOF of THEOREM 3.2. We have for all XE (xk, Xk + il and all k = 1, 2, ....., n-1 3 j (xk) / (j - i)! ] (x-xk) (;} k>) (x-xk) <4-1> j.:i (3.20) where xk <~} k> (x 1) / (j - i)!] (X -x 1)

22 22 THARWAT FAWZY If i = 4, then IS (x) - J <4> (x) I = I di - J <4> (x) I: I d1 - J <4> (x1) I + IJ <4> (x1) - J <4> (x) I; : (17/5) w 4 (h) + w 4 (h) = (22/5) w 4 (h) and this completes the proof of Theorem Case C. In this case f EC 5 [O, I] and we consider the partition where, xk + 1 -xk = h and k = 0, 1, ..., n-1. THEOREM 4.1. Given arbitrary numbers f (xk), k = 2, 3, ..., n-1, j

(zk), k = 0, I, ..., n-1; p = 0.3; 2zk = xk + xk + 1; then there exists a unique spline Sil such that (4.1) SilE: rr: 5 on each [xk, xk + d, k = 0, I, ..., n-1 (4.2) SLJ (xk) = f (xk ), k = 2, 3, ..., n-1, (4.3) s ~> (zk) = j

(z,), k = 0, I, ... 'n-1; p = 0, 3, (4.4) Sil E c (zk + 1) -2j <3> (zk) + J <3> (zk_ 1)] / h 2 and k = I, 2, ..., n-2. (4.6) k = 1, 2, ..., n -2, where, Sri (x) = f (z 0) + S ~ (Z0) (x - Z 0) + (I / 2) S'r / (Z 0) (x - Z 0) 2 + (1/31) j <3 l (z 0) {XZ 0) 3 + (4.8) + (I / 4!) S ~ 4 l (z0) ( x -Z 0) 4 + (I / 5!) S ~ 5) (z0) (x - z 0) 5, (4.9) 5 S 0 (x) = 2: [Sii> (z 1) / j! ] (x - z 1 F, j = o

23 NOTES ON LACUNARY INTERPOLATION BY SPLINES, I. 23 Sk (x) = f (zk) ak (x-zk) + (1/2!) Bk (x-zk) 2 + (1/3!) Ck (x -zk) 3 + (4.10) + (1/4!) dk (xz 1) 4 + (l / 5!) e "(x-zk) 5, Sn-1 (X) = f (zn-1) + S ~ -2 (Zn-1) (X-Zn-1) + (1/2) S; (_ 2 (Zn-1) (X-Zn-1) (1/3!) / <3> ( zn_1) (x- Zn_ 1) 3 + (1/4!) S ~ (Zn-i) (x- Zn_ 1) 4 + (4.11) + (l / 5!) 5 ~ (Zn-i) (X -Zn-i) 5 and e "in (4.10) is given by (4.6). Thus, fork = I, 2, ..., n- 2, the values ​​(4.12) dk = [J <3> (zk + 1) - j <3> (z 1 J - (/ z 2/2) e "] / h, bk = (4 / h 2) (/ (zk + 1) + f (zk) - (3h 3/4 !) j <3> (z 1 J- 2f (xk + 1) - (4.13) - (8 7 ~; 1) d1c - (/ :. ~!) e 1,] and (4.14) a1c = [/ ( z1c + i) - f (zk) - (fz 3/3!) j <3> (z1c) - (h 2/2) b1c- (h 4/4!) dk- (h 5/5!) e1c ] / h complete the proof of Theorem 4.1. THEOREM 4.2. Let fe C ~ [O, I]. Tlzerz for tlze unique spline 5.1 given in Theorem 4.1, we have for all xe [z1c, Z1c + il and k = 1, 2, ..., n- 2 (4.15) (4.16) (4.17) (4.18) I 5 ~ 2. 1 (x) - J

24 24 THARWAT FAWZY IX4 = 5/2, IX5 = 3/2, {J 4 = 13/4, {J 5 = 5/2, y4 = 5, Ys = 7/2, «54 = 15/4,« 55 = 5/2. Before proving this theorem, we prove some lemmas which will help us in arriving at the proof of Theorem 4.2. LEMMA 4.1. For ek given in (4.6) we have le "- J <5) (x) i ~ (3/2) w 5 (h) which holds true for all XE [z", zk + i] and all k = I , 2, ..., n-2. PROOF. From (4.6) we have for all XE [z ", zk + 1] and all k = I, 2, ..., n-2 I ek - j (g ~" l) - e "I, z" <; ~ ") (; & ">) - e") + + (24 ~ 5!) Hs (e "- psiua))] -f" (z ") I where z" <~~ "l

25 NOTES ON LACUNARY INTERPOLATION BY SPLINES, I.25 Using Lemma 4.2, it is easy to prove Lemma 4.3. LEMMA 4.4. For ak given in (4.14), we have for all k = I, 2, ..., n- 2 I ak-f '(zk) I :: (I / 8) h 4 w 5 (1z) PROOF. We have from (4.14) I ak- J '(zk) I = (h / 2) I bk - f "(zk) I + (h 3/4!) I dk - j <4> (zk) I + + (h4 / 5!) jek- ps> (; ~ '')) j where Zk <~ ak> (zk) i + + (h 5/5!) I ek- J <5> (rio) I wherezk <'yjo k = 0, I ,. .., n, (5.2) Sn (zk) = fjk, 0, k = 0, I, ..., n-1, (5.3) Sh 3> (zk) = fjk, 3l k = 0, I, ..., n - I where we suppose that there exists a function F (f, n) such that (5.4) w 3 (h) h 3 F (f, n): maxlf (xk) -r: xk 0 1 , k '(5.5) w 3 (h) h 3 F (f, n): maxjf (zk) - {jk ol k' and (5.6)

26 26 THARWAT FAWZY Then there are constants K ;, independent of j, F and n, such that F (S, n) kih 3 -iw 3 (h) ~ I IDi (f-Sn) 11 _, i = 0, I, 2, 3 where 11 II- = II III- [o, 11 and K0 = 109/6, K 1 = 317/12, K 2 = 115/6 K3 = 2. PROOF. Analogous to (2.4), then for xe [x ", xk + i] and k = 0, I, ..., n- I (5. 7) Sn (x) = IXk, 0 + a ,, (x -xk) + (I / 2) bk (x- X1c) 2 + (I / 3!) ck (x - xk) 3 where (5.8) Qk = (l // zk) [4, b ,,, 0 -3cxk, o-cxl <+1.0 + (/ z 3/12), 8k, 3], (5.9) bk = (II hz) [4ixk + 1, O + 4cxk, 0-8, Bk, 0 - (M / 2), Bk, a] and (5.10) Using (2.5) with (5.8), (2.6) with (5.9) and (2.7) with (5.10), with the help of (5.4), (5.5 ) and (5.6), the following estimations could be easily obtained for all k = 0, I, ..., n- k (5. II) (5.12) and (5.13) I ak- ak I "2 (79 / 12) h 2 w 3 (h) F (f, n), lbk-bki "2 (33/2) hw 3 (h) f (f, n) Thus, the above three inequalitites (5.11) - (5.13) with (5.7) and (2.4) give for all XE [x ", xk + il and k = 0, 1, ..., 11- I, (5.14) (5.15) (5.16) and I Sn (x) - Sn (x)! "2 (35/2) h 3 w 3 (h) F (f, n), IS ~ (x) - S ~ (x) l" 2 (301fl2) hw 3 (h) f (f, n), IS;, '(x) -S:.' (X) I "2 (35/2) hw 3 (h) F (f, n) (5.17) IS ~ 3) (x) -S ~ 3) ( x) I -; w 3 (h) ff, n). Using (5.14) - (5.17) with the help of Theorem 2.2, we easily get I Sn (x) - '/ (x) i "2! Sn ( x) -Sn (x) I + ISn (x) - / (x)! "2 (I 09/6) h 3 w 3 (/ z) f (f, n), (5.18) IS ~ (x) - f '(x) I" 2 IS ~ (x) - S ~ (x ) I + IS ~ (x) - j '(x) I "2 (317 / I 2) / z 2 w 3 (h) f (f, 11), (5.l 9)

27 NOTES ON LACUNARY INTERPOLATION BY SPLINES, I. 27! S ~ (x) - / "(x) I;! S ~ (x) - S ~ (x) I + IS ~ (x) - /" (x) I; (115/6) hw 3 (h) F (f, n) (5.20) and (5.21)! S ~ 3 l (x) - j <3 l (x) I;! S ~ 3 l (x) - s ~ 3 l (x) i + ISA 3 l (x) - j <3 l (x) I; 2w 3 (11) F (f, n) and the proof of Theorem 5.1 is now complete. THEOREM 5.2. Let fe C 4 [0, l] and let S ~ be the unique spline constructed in the same manner as tlzat of Theorem 3.1 and satisfying tlzefollowing data: (5.22) k = l, 2, ..., n, (5.23) (5.24) s - :( zk) = fl't, o, s * k = I, 2, ..., n - I, k = I, 2, ..., n -1, (5.25) S ~ "(xk) = (l / h 2) [x1 + 1, 0-2xt, 0 + + afc_ 1, 0 - (lz 3/12) (fl !, 0 -fll 1, 0)], k = I, 2, ..., n-1, where we suppose that there exists a function F '~ (f, n) suclz tlzat (5.26) w4 (h) lz 4 F (f, n) ~ maxjf (xk) -at 0 1, k '(5.27) and (5.28) Then there are constants Kt, independent off, F * and n, such that Kfh 4 -iw 4 (h ) f (f, n) ~ I IDi (fS *) 11 _, i = 0, I, 2, 3, 4 where II II- = II lll_ro, 11 THEOREM 5.3. Let fe C 5 [0, I] and let S ~ * be the unique spline constructed in the same manner as that of Theorem 4. I and satisfying the following data: (5.29) -S ** (L1 Xk) - 17.k, ** U> k = 2, 3, ..., n - I, (5.30) (5.31) -S ** (d zk) = fl ** k, u, k = 0, I, ..., n - I, k = 0 , I, ..., n - 1,

28 28 THARWAT FAWZY where we wppose that there exists a / unction F ** (f, n) such that (5.33) w 5 (h) h 5 F ** (f, n) ~ maxj / (xk) -attj, k '(5.34) w 5 (h) h 5 F ** (f, n) ~ maxj / (zk) -pttl, k' (5.35) Then there exist constants Kt *, independent off, F ** and n, such that K1 * h 5 -iw 5 (h) ~ lldi (f -S ~ *) ll =. i = 0, l, 3, 4, 5 where II II == 11 lll = ro, tj As we have mentioned before, the proofs of the last two theorems are similar to that of Theorem 5.1. Finally, to illustrate our method, a numerical example is given. The method described in Case B is applied to the function / (x) = l + xex and the following results are obtained for x = 0.86 and h = 0, I. Numerical value Exact value The error /(0.86) (I0 ) - 6 f '(0.86) (10) - 4 /"(0.86) (10) - 3 j <3) (0.86) (10) - 2 j <4) (0.86) Note that for h = 0.1, W4 (/ z) = References [1] BALAZS, j. and TuRAN, P .: Notes on interpolation II, III, IV, Acta Math. Acad. Sci. Hungar., 8 (1957),, 9 (1958),, 9 (1958), [2] MEIR, A. and SHARMA, A .: Lacunary interpolation by splines, SIAM.]. Num. Anal. 10 (1973),, [3] Sw ARTZ, B, K. and VARGA, RS: A note on lacunary interpolation by splines, SI AM. J. Num. Anal., 10 (1973), [4] DEMKO, S .: Lacaunary polynomial spline interpolation, SI AM.]. Num. Anal. 13 (1976 ),, [5] VARMA, AK: Lacunary interpolation by splines I. II, Acta Math. Sci. Hungar., 31 (1978),, [61 PRASAD, J. and VARMA, AK: Lacunary interpolation by quintic splines, SI AM.]. Num. Anal., 16 (1979),

29 GENERALIZED DIRECT SUMMANDS OF ABELIAN GROUPS By MJ SCHOEMAN University of Pretoria (Received December 8, 1982) Dedicated to F. Loonstra, a great teacher of mathematics 1. Introduction The investigation into the existence of an automorphism rx of a group A such that rx acts as the identity on the torsion subgroup T of A and as - I on the factor group A / T, has attracted attention in various papers. From [4] we know conditions under which the existence of such an rx implies that A is splitting. It is still an open question whether A splits if there exists to each pair (fl, y) of automorphisms of T and A / T respectively, an automorphism inducing fl and y. MADER [4] considered the following more general situation. Let B be an arbitrary subgroup of A and rx an automorphism of A which acts as the identity on Band as - I on A / B. We then have the following ([4] Proposition 2.2): (*) Let such an rx exist and let A / B [2] = 0. If either 2B = B or 2 (A / B) = = A / B, then B is a direct summand of A. In [3] it was noted that if rx is an automorphism of A which acts as the identity on T and as - I on A / T, then 2A :: = :; T EB C for some subgroup C of A, and hence A is quasi-splitting (in the sense of WALKER [5]). It is the purpose of this paper to elaborate on results in [3] by considering generalized direct summands of a group A; defined as subgroups B of A such that r A :: = :; B EB C for some subgroup C of A and some non-zero integer r. In section 2 we shall consider endomorphisms of A which act as multiplication by (different) integers n and m on the subgroup Band the factor group A / B respectively. It will be seen (Proposition 2.4) that if either B [n-m] = 0 or (n-m) a EB, a EA, implies a EB; then such an endomorphism exists if and only if (11- m) a :: = :; B EB C for some subgroup C of A. Conditions under which generalized direct summands are direct summands (in the ordinary sense) are also given, showing, inter alia, how MADER's result mentioned above, fits into a much more general theory. In section 3 we go a step further by considering subroups B of A for which there exists a subgroup C of A such that A :: = :; B EB C, for some endomorphism Of A. For reasons which will become clear, it will be necessary to restrict

30 30 M. j. SCHOEMAN ourselves to those endomorphisms

31 GENERALIZED DIRECT SUMMANDS OF = ABELIAN GROUPS 31 (2.2) If r As: B EB Cs A, 0 ~ re Z, then there exists a commutative triangle Fig. 2. with exact row. Conversely, if such a triangle exists and if r- 1 B = B then ra ,, ;; bffi Im {J. PROOF. If aea and ra = b + c, beb, cec, then fj defined by {J (a + B) = c is a homomorphism making the triangle commutative. Conversely, if such a fj exists then {J (a + B) + B = ra + b for all aea, and hence ra ,, ;; B + Im {J. Furthermore, if for some aea we have {J (a + B) EB then {J (a + B) + B = ra + b = Band so r- 1 B = B implies that aeb, that is {J (a + B) = 0. Hence B 11 Im fj = 0 and thus ra: $ b EB Im fj s: A. B It is well-known that there is a one-to-one correspondence between direct summands of A and endomorphisms n of A satisfying n 2 = n. Using the same techniques as in (2. I) and (2.2) it is not difficult to show the following analogy for generalized direct summands. (2.3) If r A: $ B EB C ,, ;; A, 0,, ​​o r E Z, then there exists an endomorphism of A such that 2 = r . Conversely, if B satisfies condition S (r) and if : A-Bis an epimorphism with 2 = r , then ras: b EB (r-rp) A. Let B be a subgroup of A and suppose there exists an endomorphism a of A such that ax = nx, xeb, and aa-maeb, aea, where n and mare integers. Note that if rz = m then always such an endomorphism exists, namely a = Ti.A. This trivial case will henceforth be excluded. If B is fully invariant in A then we can describe this situation by saying that ae E (A) induces the pair (n, m) ee (b) XE (A / B). The first proposition gives a correspondence between generalized direct summands of A and endomorphisms of A acting in the way mentioned above. PROPOSITION 2.4. Let B be a subgroup of A and a an endomorphism of A such that ax = nx, x EB and aa-maeb, aea, where m and n are integers, m ~ 11. If B satisfies condition S (n- m) then (nm) a ,, ;; B EB C for some C == :: A. Conversely, if r A ::, ;; B EB Cs: A for some integer r ~ 0, then for any pair 11, m of integers witlz 11-m = r, there exists an aee (a) such that ax = nx, xeb, and aa-maeb, aea. PROOF. If aea then the hypothesis on a implies that aa -maeb. Define {J: A-B by {Yes = aa-ma, and y: A / B-A by 11 (a + B) = na-aa, aea. We have commutative triangles

32 32 M. J. SCHOEMAN with exact rows and so the second parts of (2.1) and (2.2) complete the first part of the theorem. a ~., B --A / (nm) I, / B and Fig. 3. A - '~', A / a ~ ot (nm) 'A / B Conversely, if ra ,, ;;; bffic, , ;;; a, O :: ZrEZ, then (2.1) and (2.2) imply that we have commutative triangles and Fig. 4. with exact rows. Let r = n-m and define rx: A- .. A by rxa = A + a + ip (a + B) EB. We now give two conditions under which generalized direct summands are direct summands. LEM.MA 2.5 [3]. Let B be a direct summand of r A for some 0 :: Z re Z. If rb = B then B is a direct summand of A. COROLLARY 2.6. If ra ,, ;;; bef) c ,, ;;; a, O :: ZrEZ, and rb = B then Bis a direct summand of A. PROOF. If ra ,, ;;; bffic then ra + b = Bffi (Cn (rA + B)). Since rb = B we have ra = Bffi (Cn (rA + B)), and so Lemma2.5 completes the proof. LEMM.A 2.7. If ra ,, ;;; bef) c ,, ;;; a, O :: ZrEZ, and r (a / b) = A / B, then B is a direct summand of A. PROOF. Since any aea is of the form a = rx + b, xea, beb, the result follows immediately. We are now ready to show how MADE R's result (mentioned in the introduction) fits into the more general theory described above. It also shows how the requirement A / B [2] = 0 can be replaced by the weaker condition that either A / B [2] = 0 or B [2] = 0. (The reader should bear in mind that an endomorphism of A which acts as the identity on B and as - 1 on A / B, is in fact an automorphism of A).

33 GENERALIZED DIRECT SUMMANDS OF ABELIAN GROUPS 33 PROPOSITION 2.8. Let B be a subgroup of A satisfying condition S (r) and n, mezwithn-m = r.supposefurtherthateitherrb = Borr (A / B) = A / B. Then B is a direct summand of A if and only if there exists an o: ee (a) such that rxx = nx, xeb and rxa-maeb, aea. PROOF. If Bis a direct summand of A, say A = B $ C, and aea with a = b + c, beb, cec; then rx defined by rxa = nb + mc is an endomorphism of A having the required properties. The converse follows from Proposition 2.4, Corollary 2.6 and Lemma 2.7. The following places Proposition 2.4 of [4] in perspective. COROLLARY 2.9. Let A be a torsion group and B a subgroup of A such that either A [p] = 0 of A / B [p] = 0, pa prime number. Then Bis a direct summand of A if and only if there exists an ocee (a) such that rxx = nx, xeb, and rxa-maeb, a EA, where m and n are integers with n-m = p. PROOF. A torsion group which does not contain elements of prime order p, is divisible by p. 3. A more general situation In this section we go a step further by considering a subgroup B of A and a pair (o: ', f) ee (b) x E (A / B) where rx' can be extended to an endomorphism oc of Band "" $ can be lifted to an endomorphism f3 of A, oc r '- {j. We wish to find conditions under which (cc- {j) a :: s; B EB C for some subgroup C of A, thus extending the results of the previous section to, what we believe, the limit. Let Ann B = {JEE (A) IJB = O} and EB (A) = {<1> E (A) i <1> BsB}. Every <1> EEB (A) thus induces (by restriction) an endomorphism <1> 'of B and an endomorphism <1> of E (A / B), the latter being defined by <1> (a + B) = 'and <1> induce endomorphisms <1> ~ and if> * of Ext (A / B, B). These maps are given by <1> ~: E - <1> 'E and *: E - Eif> where <1>' E and E "ii> are defined in [1] (Section 50). Since the above diagram commutes we have <1> 'E = E ~ = Ker

EEB (A) , let - 1 B = {aea [<1> a EB}. Then <1> is injective if and only if - 1 B = B, while <1> 'is injective if and only if B n Ker <1> = 0. 3 ANNALES-Sectio Mathetnatica- Tomus XXVI II.

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

~ = Ker

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

35 GENERALIZED DIRECT SUMMANDS OF ABELIAN GROUPS 35 domorphism {J of A, rx-cp {j. Supposefurther that a ee 8 (A) exists such that ee 8 (A) exists such that < Px = cxx, xeb and cxa- (jaeb, aea. PROOF. To prove the first part of the theorem, we notice that the hypothesis on rx 'and fimplies the commutativity of the diagrams E: o ... B ... A ... A / B - + - 0 E: o ... B ... AA / B ... o I a 'I a I ~ and I 1r I ~ I 7i t + t E: 0-B- + -A - + - A / B - + - OE: O- ~ B ... AA / B - + - 0 We therefore have rx'e = e ~ and {J'E = Elf. On the other hand, we also have a commutative diagram E: 0 - + - B - + - A - + - A / B ... O la 'la> l7i + + t E: o ... BAA / B - 0 which implies that rx'e = ef. Thus rx'e = {j'e and so EE Ker (rx '- / j'k Since rt ..- {3ee 8 (A) we may apply Lemma 3.1 to complete the first part of the theorem. Conversely, if ex, / JEE8 (A) are such that (rx- (j) a: sbef) c: sa, then the first part of Lemma 3.1 implies that rx'e = fj'e and since {J'E = Elf we have rx'e = ep. Hence there e xists a commutative diagram (cf [i]) E: o ... B ... AA / B-0 I a 'J, II tt cx'e: O - + - BA 1 -AJB-0 II I y II - t E (J: o ... B ... A 2 -A / B - + - O II 16 I ii t + E: OB ... A ... A / B --- 0 It is clear that = oys is an endomorphism of A having the desired properties. LEMMA 3.3. Let A: SBEf) C: SA for some ee 8 (A). If either 'Is an automorphism of B or Is surjective, then Bis a direct summand of A. 3 *

36 36 M.J. SCHOEMAN PROOF. First assume that 'is an automorphism of B. We have A + + B = B ffi (Cn ( A + B)) and hence a = b ffi (Cn ( A + B)). It is not difficult to see that A = BffiC 'where C' = {xeal xec}. On the other hand, if ~ (A / B) = A / B then for all XEA we have x = a + b, aea, beb, and hence xebffic. COROLLARY 3.4. Let B be a subgroup of A and (ex. ', F) ee (b) xe (a / b) where (1. 1 can be extended to an ixee (a) and 7J be lifted to a / 3EE (A) , (/.~~3. Suppose further that either (i) ((/..-/3) 'is an automorphism of B and f (cx .- {3) = (a- {3) f for all feann B or (ii) (cx .- {3) is an automorphism of A / B. Then Bis a direct summand of A if and only if there exists a ee 8 (A) such that x = cx .x, xeb and a- {3aeb, aea. References [1) L. FuCHs: Infinite abelian groups, Volume 1. Academic Press (1 970). [2] L. FucHs: Infinite abelian groups, Volume 2 Academic Press (1973). [3] F. LOONSTRA and MJ SCHOEMAN: On a paper by Mader (Submitted). [4] A. MADER; Ori the automorphism group and endomorphism ring of an abclian group Annales Univ. Sci. Budapest , Sectio Mathematica, 8 (1965), [5] CP WALKER: Properties of Ext and quasi-splitting of abelian groups. Acta Mat / 1. Acad. Sci. Hungar., 15 (1964),

37 RELATIVES OF 3-PERMUTABILITY AND PRINCIPAL TOLERANCE TRIVIAL VARIETIES By IVAN CHAJDA Pi'erov, Czechoslovakia (Received September 30, 1982) JT BALDWIN and J. BERMAN [I] described a close connection between definability of principal congruences and relatives of congruence permutability and 3-permutability. Some of these results and problems can be translated into the terminology of tolerance relations, [3], [5] and can be useful for characterizing so called principal tolerance trivial varieties. These varieties form a very large class of varieties with "nice" properties which can be used in applications. 1. Permutability and its relative By a tolerance on an algebra 21 = (A, F) is meant a reflexive and symmetric binary relation on 21 having the Substitution Property with respect to F, i.e. it is a symmetric and diagonal subalgebra of 21 x21. It is easy to show that the set LT (21) of all tolerances on 21 forms an algebraic lattice with respect to set inclusion, [2], [3], [5]. If a, b are elements of 21, denote by T (a, b) or T A (a, b) the least tolerance on 21 containing the pair (a, b). By Con (21) we denote the congruence lattice of 21 and by e (a, b) ore A (a, b) the principal congruence on m containing the pair (a, b). An algebra m is tolerance trivial if every tolerance on m is a congruence; m is principal tolerance trivial if T (a, b) b) for each two elements a, b of m. T (a, b) is called a principal tolerance. A variety

38 38 IVAN CHAJDA This result motivated our effort to characterize principal tolerance trivial varieties by relatives of permutability and use such characterizations for creating polynomial conditions. PROPOSITION 2. Let x, y, a, b be element of an algebra m. Then (x, y) et (a, b) if and only if there exist a (2 + n) -ary polynomial p and elements c 11 ..., en of m such that X = p (a, b, C 11 .., Cn) y = p (b, a, C 11, Cn) For the proof, see eg Lemma 1 in [2]. Thus, in the terminology of [I], principal tolerance trivial algebras are exactly algebras with I-step principal congruences. We can adopt another concept on [I]: \ J {has 3-permutable principal congruences if E> (a, b). E> (c, d). E> (a, b) = E> (c, d) E> (a, b) E> (c, d) for each elements a, b, c, d of Ill, or equivalently, if E> (a , b) VE> (c, d) = E> (a, b) E> (c, d) E> (a, b) in Con (Ill). Now we can translate Theorem 3.7 in [I] for n = I in our terminology: PROPOSITION 3. Let ECon (2l) and every elements a, b of m, e v E> (a, b) = e. E> (a, b). e in Con (\ JI), i.e. principal tolerance trivial varieties have 3-permutable principal congruences. It is worthy to say that principal tolerance trivial varieties constitute a very large class of varieties containing among others these "nice" varieties: (i) all permutable varieties (ie all varieties of groups, quasigroups, rings, modules, etc.) as follows by Proposition I; (ii) a variety of distributive lattices, varieties of distributive p-algebras, a variety of Heyting algebras etc., see e.g. [6]. It follows that some theorems on permutable varieties remain true also for e.g. the variety of distributive lattices provided those proofs use only principal congruences. One such case will be given in the third part. It is evident that the identity from Proposition 3 is equivalent to the inclusion e.e (a, b) E> 2E> (a, b) EJ E> (a, b). However, it is not characterizable by a Mal'cev condition. Such characterization is possible for the converse inclusion:

39 RELATIVES OF 3-PERMUTABILITY 39 THEOREM I. For a variety ({), the following conditions are equivalent: (I) For each 2! Em, each e E Con ​​(2!) And each element a, b of 2 !, e.e (a, b) Bc ;; G (a, b) G G (a, b); (2) there exist 6-ary polynomials Pv ..., Pn such that x = p 1 (x, y, z, z, x, y, z, z) y = Pn (z, z, x, y, x, y, z, z) p / z, v, x, y, x, y, z, v) = Pi + 1 (x, y, z, v, x, y, z, v) for i = I , ..., n-1. PROOF. (1) => (2): Let 2! = F, 1 (x, y, z, v) be a free algebra of ({) with four free generators x, y, z, v. Put e = G (x, z) V G (v, y). Then (x, y) E G E> (z, v) 0, thus, by (I), (x, y) E G (z, v) e E> (z, v). i.e. there exist elements c, d of 2! search that (x, c) ee> (z, v), (c, d) eg, (d, y) eg (z, v). Hence, there exist 6-ary polynomials Pv ..., Pn such that c = p 1 (x, y, z, v, x, y, z, v) P; (z, v, x, y, x, y, z, v) = P; + 1 (x, y, z, v, x, y, z, v) for i = I, ..., n- ld = Pn (z, v, X, y , x, y, z, v) as follows from (c, d) EG (x, z) VG (v, y) in F 4 (x, y, z, v). EE> (z, v) and (d, y) EG (z, v) give immediately x = p 1 (x, y, z, z, x, y, z, z) y = Pn (z, z, x, y, x, y, z, z). (2) => (1): Let 2XE E> (a, b) E>. Then there exist c, d of m such that (x, c) ee>, (c, d) ee> (a, b), (d, y) ee>. Put r = p 1 (x, y, c, d, x, y, c, d), s = Pn (c, d, x, y, x, y, c, d). Thus (r, s) EE> (x, c) VE> (d, y) c ;; e> and, by the identities of (2), However (x, c) E (x, r) = (P1 (X. y, c, c, x, y, c, c), P1 (X, y, c, d, x, y, c, d)) EE> (a, b) (s, y) = (Pn (c, d, x, y, x, y, c, d), Pn (c, c, x, y, x, y, c, c)) EQ (a, b), ie (x, y) ee> (a, b) B E> (a, b) proving (1). I!

40 40 IVAN CHAJDA REMARK l. The identity (I) of Theorem I is probably the best approximation of 3-permutability of principal congruences which can be characterized by a Mal'cev condition, since there exist varieties whose free algebras have 3-permutable principal congruences but the whole ({) has not this property, see eg [I]. REMARK 2. If we replace (I) of Theorem I by an analogous identity for tolerances, namely: (1) For each 2lE (2): Let \ Jf.E ({), a, b, c, d be elements of SJ (. And (x, y) E ET (a, b) T (c, d) T (a, b). Then, by (1), (x, y) EG (a, b) G (c, d). G (a, b).

41 RELATIVES OF 3-PERMUTABILJTY 41 By Proposition 2, m has I-step principal congruences and, by the remark after Theorem 3.5 in [I], h (b (a, b)) = B (h (a), h ( b)) for any homomorphism h of m. Let h be a canonical homomorphism of m onto m / e (c, d), thus ie proving (1): Let m T (a, b) T (c, d) T (a, b) ~ T (c, d). T (a, b). T (c, d), E (2): Let m = F 2 + n (x, y, Z 1,., Zn) be a free algbera over

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

43 RELATIVES OF 3-PERMUTABILITY Some applications A variety ({) has directly decomposable corzgrounce if for all m, l8 of ({) and each E> ECon (filx58) there exist 0 1 ECon (fil) and 0 2 ECon (lb) such that 0 = 0 1 X0 2 GA FRASER and A. HORN [8] gave a Mal'cev conditions characterizing such varieties. This condition is, however, rather long and complicated. Here we show how it can be simplified if ({) is assumed to be principal tolerance trivial: THEOREM 4. Let ({) be a principal tolerance trivial variety, then the following conditions are equivalent: (I) ({) has directly decomposable congruences; (2) there exist a (2 + n) -ary polynomial p, binary polynomials qv ..., qn and ternary polynomials rv ..., rn such that X = p (x, y, q 1 (x, y) , .., Pn (x, Y)) y = p (y, x, q 1 (x, y), ..., qn (x, y)) z = p (x, y, r, (x , y, z), ..., rn (x, y, z)) = p (y, x, r 1 (x, y, z), ..., rn (x, y, z)). PROOF. (1) => (2): Let m = F 2 (x, y), F = F 3 (x, y, z) be free algebras over ({) and let ({) have directly decomposable congruences. By Theorem 4 in [8], we have ([x, z], [y, z]) ee> ([x, x], [y, y]). By (l) it means ([x, z], [y, z]) et ([x, x], [y, y]) and, by Proposition 2, there exist a (2 + n) -ary polynomial p and elements C 1, ..., Cn of fil X58 such that [x, z] = p ([x, x], [y, y], c 1, ..., en) [y, z] = p ([y, y], [x, x], c 1, ..., en) Since Ci Em xm, we have C; = [q; (x, y), r; (x, y, z)] for some binary or ternary polynomials q; or r ;, respectively. If we write it componentwise, we obtain (2). (2) => (1): Let fil, 58 E ({) and av a 2 be elements of fil and bv b 2, b be elements of) 8. Put Ci = [qi (av a 2,), r; (b 11 b 2, b)]. By (2), ([av b], [a 2, b]) = ([p (a 1, a 2, q (a 11 a 2))), p (bv b 2, r (b 11 b 2 , b))], [p (a 2, a 11 q (a 11 a 2)), p (b 2, bv r (b 11 b 2, b))]) = = (p ([av bi] , [a 2, b 2], Cv ..., en), p ([a 2, b 2], [av b 1], C 11, en)) EET ([av b 1], [a 2 , b 2]). Thus ([av b], [a 2, b]) c0 ([a 11 b 1], (a 2, b 2]) and by Theorem 4 in [8], (I) is proved.

44 44 IVAN CHAJDA The easy way of using the condition (2) of Theorem 4 can be illustrated by these examples: EXAMPLE I. Let CO be a variety of rings with unit element. Since

45 RELATIVES OF 3-PERMUTABILITY 45 The inclusion <: ;; :: is evidently true in any case, hence CEP is equivalent to the converse inclusion only. Since ({) is principal tolerance trivial, CEP is equivalent to (I) = (2): Let 2i = F 2 + n (x, y, Z 1 ,, zn) be a free algebra of ({) with free generators x, y, Zv ..., zn and let p be a (2+ n) -ary polynomial over ({). Denote by c = p (x, y, z), d = p (y, x, z)., Let Q3 be an algebra of ({) generated by the generators x, y, c, d. Then clearly ~ is a subalgebra of Ill and, by Proposition 2, (c, d) ET A (x, y) n (\ Bx \ B). By (* *) it implies (c, d) ET B (x, y), thus, by Proposition 2, there exists a 6-ary polynomial q such that c = q (x, y, x, y, c, d) d = q (y, x, x, y, c, d) proving (2). (2) = (I): Suppose W, Q3 E ({), Q3 is a subalgebra of W, c, d, x, y are elements of Q3 and (c, d) ET A (x, y) n ( \ Bx \ B). By Proposition 2, there exist a (2+ n) -ary polynomial p and elements a 11., an of m such that c = p (x, y, a1 '..., an) d = p (y, x , al> ... 'an) By (2), there exists a 6-ary polynomial q with c = q (x, y, x, y, c, d) d = q (y, x, x, y ,CD). Since x, y, c, dare elements of \ B, it yields (c, d) ET 8 (x, y). 4. Connections with modularity Now, we can study connections of varieties investigated in the first part and congruence modularity. A variety ({) is modular if Con (W) is modular for each WE ({). It is well-known that the 3-permutability of congruences implies the modularity. The first theorem shows what can be said in this sense on principal tolerance trivial varieties and the second one is a strengthening of the result on 3-pennutability. THEOREM 6. Let

46 46 IVAN CHAJDA Let PROOF. Suppose me (a, b) -e> r: ;; e> (a, b) -8 E> (a, b). Then in Con (2!). (A) Firstly suppose R, TECon (~ l) and 0 (c, d) r: ;; r. In a routine way, analogous to that of the proof of Theorem 6, we can easily obtain R / \ (E> (c, d) VT) = 6 (c, d) V (R / \ T). (B) Now suppose the general case R, S, TECon (2), Sr: ;; R and proceed to prove the modular identity by an induction, using (A) as an induction hypothesis. Let (x, y) e R / \ (S VT). Clearly S = V {0 (ca, da); rxe /} in Con (ill), thus the previous formula implies (x, y) E V {E> (ca, da); ci: E /} VT.

47 RELATIVES OF 3-PERMUTABILITY 47 By the Mal'cev lemma, there exists a finite subset, say {1, ..., n} of I such that n (x, y) E (V 0 (c ;, d; )) VT. i = i Clearly 0 (c ;, d;) t ;; s for i = I, ..., n and i.e. Applying (A), we have n (x, y) ER / \ [(V 0 (c;, d;)) VT], i = l 11 (x, y) e R / \ [0 (cv d 1 ) V (V 0 (c; id;) VT)]. i = 2 11 (x, y) E 0 (cl 'd 1) V (R / \ [V 0 (c ;, d;) VT]) i = 2 and after n steps of this procedure we conclude proving the modularity of

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49 ON SOME FIXED POINT THEOREMS AND THEIR COMPARISONS By M. 0. DESHPANDE and G. 0. HAMEDANI Department of Mathematics, Statistics, and Computer Science Marquette University Milwaukee (Received July 28, 1983) 0. Introduction While considering conditions under which a mapping T of a complete metric space X into itself has a unique fixed point, KANNAN [3] and subsequently FISHER [I], (2] have considered the first three of the six conditions listed below as (T 1) - (T 6) . For any x, yex: (T 1) d (tx, Ty) sc {d (x, Tx) + d (y, Ty)}, l where Osc <-. 2 I (T 2) d (tx, Ty) sc {d (x, Ty) + d (y, Tx)}, where Osc <(T 3) {d (tx, Ty)} 2 :: s ;; c {d (x, Tx) d ( x, Ty) + d (y, Ty) d (y, Tx)}, whereosc <(T,) {d (tx, Ty)} 2 sc {{d (x, Tx)} 2 + {d (y , Ty)} 2}, where Osc <(T 5) {d (tx, Ty)} 2sc {d (x, Tx) d (y, Tx) + d (x, Ty) d (y, Ty)} , where Osc <2.1 (T 6) {d (tx, Ty)} 2 sc {d (x, Tx) d (y, Ty) + d (x, Ty) d (y, Tx)}, where Osc <2. KANNAN has proved that (T 1) implies a unique fixed point for T which is the limit of the sequence {Px} for arbitrary xex. Fisher later proved that (T 2} and (T 3) likewise imply a unique fixed point for T. The object of this note is to show (in part I below) that each of the conditions (T 4), (T 5) , (T 8) imply the same result; and then to study (in part 2) the relation between conditions (T 1) - (T 8). Specifically, we show that (T 1) =? (T,), which implies 4 ANNALES-Sectlo Mathematlca-Tomus XXVlll.

50 50 M. 0. DESHPANDE HAMEDANI that our Theorem 1 is stronger than that of KANNAN; that the theorem is strictly stronger is shown by means of an example. Various examples are then given to show that (T 1), (T 2) and (T 3) are all uncomparable and also that (T 6) does not imply any of the other five conditions. In the special case of a linear mapping Tx = ax on (R, d), where R is the set of all real numbers and d is the usual metric, exact intervals have been obtained in which the coefficient a must be in order that each of the conditions (T 1) - (T 6) may hold. All of our counterexamples such as (T 1) no => (t 2), (T 1) no => (t 2), etc., come from this linear mapping, in some cases restricted to a;:; mailer domain . While there are some implications for which we neither have proofs nor counterexamples, it has been demostrated that the linear mappings on R cannot be employed for counterxamples in these cases, and some mappings on other complete metric spaces might well be useful. 1. Fixed Point Theorems THEOREM I. If Tis a mapping of the complete metric space X into itself, satisfying (T 4), then T has a unique fixed point. PROOF. Let x be an arbitrary point in X. Then which implies that {d (px, p + ix)} 2 = 5 c {{d (rn-1x, Px)} 2 + {d (px, p + ix)} 2}, d (tnx, Tn + 1 x): (-c-) 112 d (tn- 1 x, Tnx) 1-c. for n = 1, 2, ... Since 0: 5 c <_! _, {Px} is a Cauchy sequence in X and hence 2 .. has a limit z in X. We now have which implies that {d ( px, Tz)} 2: 5 c {{d (p- 1 x, Px)} 2 + {d (z, Tz)} 2}, {d (z, Tz)} 2 sc {d (z, Tz )} 2 Since Osc <+, it follows that d (z, Tz) = 0, and hence T; = z, so that z is aa fixed point of T. If z 'is a second fixed point of T, then {d (z, z')} 2 = {d (tz, Tz ')} sc {{d ( z, Tz)} 2+ {d (z ', Tz')} 2} = 0, so z = z '. THEOREM 2. If T is a mapping of the complete metric space X into itself satisfying (T 5), then T has a unique fixed point.

51 ON SOME FIXED POINT THEOR.EMS 51 PROOF. Let x be an arbitrary point in X. If for some positive integer n, d (tnx, yn + lx) = 0, then we have a fixed point; if not, then from (T 5) we obtain d (px, yn + 1 x): s: (l ~ c) d (tn-ix, Px) for n = 1, 2, ... Now the rest of the proof is similar to that of Theorem I. TREOREM 3. If T is a mapping of the complete metric space X into itself, satisfying (T 6), then T has a unique fixed point. PROOF. Let x be an arbitrary point in X. If for some positive integer n, d (tnx, yn + ix) = 0, then we have a fixed point; if not, then from (T 6) we obtain d (tnx, yn + lx): s: cd (tn-ix, Tnx) for n = 1, 2, ... Since Osc

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

53