Showing papers on "Quintic function published in 1979"

Lacunary Interpolation by Quintic Splines

Abstract: Here we study the quintic splines satisfying certain conditions and show that the convergence is of the same order as that of best approximation by quintic $C^2 $ splines.

The Euclidean condition in pure cubic and complex quartic fields

Abstract: In this paper we prove that a field Q( \\Jd) is euclidean with respect to the ordinary norm if and only if d = 2, 3 or 10. We also prove that certain fields of the 4 /form Q(\\J — d),d > 0, are or are not euclidean. The purpose of this research is to determine which pure cubic fields are euclidean with respect to the ordinary norm, and partially to determine the same for fields QA$J—d), d>0. More precisely, a field is said to be euclidean for the ordinary norm (just euclidean, for short) if its ringof integers/? has the following property: Va, b GR 3p, r E R s.t. a = pb + r, \\N(r)\\ < W(b)\\. We prove the following: Theorem A. Q($Jd) is euclidean if and only if d = 2,3 or 10. Theorem B. If d = 2,3 or 1, then Q(\\\\fird) \" euclidean. If d i= 12, 44, 67 or the preceding values, and if neither d nor 2d is a perfect square, then Q(\\V~ d) is not euclidean, d > 0. Pure Cubic Fields By a pure cubic field we mean a cubic field of the form Q(\\ß), d £ Z. Any such field has one real embedding and a pair of conjugate complex embeddings and, hence, has one fundamental unit and negative discriminant. The three fields proven to be euclidean are the pure cubics of smallest discriminant (in absolute value). Cassels [1] proved that a cubic field of negative discriminant D cannot be euclidean if D > 4202 = 176,400. This result reduces our problem to a finite number of cases, a number which is reduced much further by the necessity of unique factorization. Notation. We consider fields Q(\\ß): d will always be used in this context. R: the ring of integers of Q(\\Vd), e: the fundamental unit of Q(\\ß), D: the discriminant of Q(\\fd), 8: Iß, (b): the ideal bR, for b G Q(\\ß), N(b) (resp. N(p)): the norm of the element b (resp. of the ideal £), b(c): the residue class of b mod c, for b, c G R. Received June 14, 1977; revised December 2, 1977. AMS (MOS) subject classifications (1970). Primary 12A30. © 1979 American Mathematical Society 0025-5718/79/0000-0029/$03.50 389 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 390 VINCENT G. CIOFFARI Latin letters refer to field elements and German letters to ideals. Note: in a field Q(\\ß), N(a + b3\\/d + c3s/d2) = a3 + b3d + c3d2 3abcd. I. Preliminary Results. (a) Class Number. We determine criteria for a field to have class number one, a necessary condition for the euclidean property. The following lemma will be used again later to prove that certain fields are not euclidean. Lemma. Let Kbe a field of odd prime degree q. Let p be a prime totally ramified in K, p p 1 iq), and let ip) = pq denote the prime ideal factorization of ip). Let u £ R, the ring of integers of K. Then u = b mod p, where b is the unique integer in the set {0, 1, . . . , p 1} such that bq = Niu) mod p. Proof. In this diagram, ax and a2 are the canonical maps; N is the norm map; ^ is the map which associates to each class in R/p the unique integer mod p which belongs to that class; \\p is the map which sends each element to its »7th power. All the maps are multiplicative homomorphisms, and y? and ,-) = p¡, and bq = Pie\ ■ ■ • ekrir, i=l,...,r + 2,kif £Z. License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use THE EUCLIDEAN CONDITION IN PURE CUBIC FIELDS 391 It can be deduced that some »7th power-free rational integer s, divisible by some p¡s but not by pr+2, is a »7th power in K. This implies that K = Q(^/s). We have a contradiction; hence, h ¥= 1. It is easily deduced that q\\h. Q.E.D. There are forty-two fields satisfying Cassels' bound which are not excluded by these propositions. Consulting class number lists [2], we find that the following thirty-one actually have class number one: 2, 3, 5, 6, 10, 12, 17, 23, 29, 33, 41, 44, 45, 46, 53, 55, 59, 69, 71, 82, 99, 107, 116, 145, 179, 188, 197, 226, 332, 404 and 575. (b) Standard Form for d. In view of Proposition 2, we will always assume d to be of one of the following forms: (i) d = p, a prime, (ii) d = pxp2, the product of two primes, (iii) d = pxp\\, px > p2. Subject to these conditions, no two values of d generate the same field. (c) Basis and Discriminant. We state without proof the following well-known results:

Solution of 3-dimensional time-dependent viscous flows. Part 1: Investigation of candidate algoriths

Abstract: There is considerable interest in developing a numerical scheme for solving the time dependent viscous compressible three dimensional flow to aid in the design of helicopter rotors. Numerical algorithms are examined to determine their overall suitability for the efficient and routine solution of an appropriate system of partial differential equations. It is concluded that a consistently split time linearized block implicit scheme using either quintic B spline collocation or the generalized operator compact implicit approach to generate a fourth order accurate algorithm is particularly well suited for use on the present problem. High cell Reynolds number behavior leads to favoring the generalized operator compact implicit approach over the quintic B spline collocation method.