Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí toán học quốc tế đề tài: The well-rounded linear function. | The well-rounded linear function Roger B. Eggleton Mathematics Department Illinois State University Normal IL 61790-4520 USA roger@ To Aviezri Fraenkel friend and mentor of many years. Abstract The generic linear function ax b of a real variable with a b x OR is usually evaluated as a scale function product followed by a translation sum . Our main result shows that when such a function is variously combined with rounding functions floor and ceiling exactly 67 inequivalent rounded generic linear functions result of which 38 are integer-valued and 29 are not. Several related results are also established with elucidation of the relevant equivalence class structures. Submitted June 27 2000 Accepted August 15 2000. Mathematical Reviews Subject Classification 2000 05A15 11A99 26A09. 1 Introduction The floor function 9J and ceiling function are familiar integer-valued functions of a real variable which may be collectively referred to as rounding functions. They are defined for all x 2 R by _xJ n precisely when n x n 1 and n eZ x n precisely when n 1 x n and n eZ. The half-bracket notation and the names floor function and ceiling function were introduced by Iverson 8 in a book on programming published in 1962. The floor function is identical with the integer part function a terminology and notation still widely encountered in number theory. I am not sure when or by whom this notation was introduced. Dickson for example used it when describing Legendre s result for the power of a prime in a factorial see 2 page 263 but Legendre see 11 page 10 actually used the notation E taking the initial E from the French entier . The ceiling function is THE ELECTRONIC JOURNAL OF COMBINATORICS 8 no. 2 2001 R6 1 identical with the so-called post office function which rounds intermediate weights up to the next scale point for postal charges. Many notations have denoted it but none was widely adopted prior to Iverson s half bracket notation. A charming discourse on the rounding
