Carter Bays

A comparison of next-fit, first-fit, and best-fit

“Next-fit” allocation differs from first-fit in that a first-fit allocator commences its search for free space at a fixed end of memory, whereas a next-fit allocator commences its search wherever it previously stopped searching. This strategy is called “modified first-fit” by Shore [2] and is significantly faster than the first-fit allocator. To evaluate the relative efficiency of next-fit (as well as to confirm Shores results) a simulation was written in Basic Plus on the PDP-11, using doubly linked lists to emulate the memory structure of the simulated computer. The simulation was designed to perform essentially in the manner described in [2]. The results of the simulation of the three methods show that the efficiency of next-fit is decidedly inferior to first-fit and best-fit when the mean size of the block requested is less than about 1/16 the total memory available. Beyond this point all three allocation schemes have similar efficiencies.

Improving a Poor Random Number Generator

The BASIC interactive systems of both the IBM 360 under ITF and the PDP 11/45 under RSTS contain built-in random number functions whmh have the short period of 8192. Moreover, subsequences of every fourth and eighth number behave very nonrandomly. To improve these generators, we employ an algorithm which resembles the MacLaren-Marsaglia method, but which reqmres only one random number generator and less time. Not only is the period lengthened, but local nonrandomness subsides. The results indicate that in the absence of knowledge about some particular generator, our method can only be beneficial. Empirical and analytm results are given.

A new bound for the smallest x with p(x) li (x)

Let π(x) denote the number of primes li(x). This improves earlier bounds of Skewes, Lehman, and te Riele. We also plot more than 10000 values of π(x)-li(x) in four different regions, including the regions discovered by Lehman, te Riele, and the authors of this paper, and a more distant region in the vicinity of 1.617 x 10 9608 , where π(x) appears to exceed li(x) by more than.18x 1/2 /log x. The plots strongly suggest, although upper bounds derived to date for li(x) - π(x) are not sufficient for a proof, that π(x) exceeds li(x) for at least 10 311 integers in the vicinity of 1.398 x 10 316 . If it is possible to improve our bound for π(x)-li(x) by finding a sign change before 10 316 , our first plot clearly delineates the potential candidates. Finally, we compute the logarithmic density of li(x) - π(x) and find that as x departs from the region in the vicinity of 1.62 x 10 9608 , the density is 1 - 2.7 x 10 -7 =.99999973, and that it varies from this by no more than 9 x 10 -8 over the next 10 30000 integers. This should be compared to Rubinstein and Sarnak.

The segmented sieve of eratosthenes and primes in arithmetic progressions to 1012

The sieve of Eratosthenes, a well known tool for finding primes, is presented in several algorithmic forms. The algorithms are analyzed, with theoretical and actual computation times given. The authors use the sieve in a refined form (the “dual sieve”) to find the distribution of primes in twenty arithmetic progressions to 1012. Tables of values are included.

The reallocation of hash-coded tables

When the space allocation for a hash-coded table is altered, the table entries must be rescattered over the new space. A technique for accomplishing this rescattering is presented. The technique is independent of both the length of the table and the hashing function used, and can be utilized in conjunction with a linear reallocation of the table being rescattered. Moreover, it can be used to eliminate previously flagged deletions from any hash-coded table, or to change from one hashing method to another. The efficiency of the technique is discussed and theoretical statistics are given.

On the fluctuations of Littlewood for primes of the form 4̸=1

Let rb,c(x) denote the number of primes 6 x which are c (mod b). Among the first 950,000,000 integers there are only a few thousand integers n with r4,3(n) < or4jl(n). The authors find three new widely spaced regions containing hundreds of millions of such integers; the density of these integers and the spacing of the regions is of some importance because of their intimate connection with the truth or falsity of the analogue of the Riemann hypothesis for L(s). The discovery that the majority of all integers n less than 2 X 1010 with r4,3(n) < ir4,1(n) are the 410,000,000 (consecutive) integers lying between 18,540,000,000 and 18,950,000,000 is a major surprise; results are carefully corroborated and some of the implications are discussed.

Zeroes of Dirichlet L -functions and irregularities in the distribution of primes

Seven widely spaced regions of integers with π4,3(x) < π4,1(x) have been discovered using conventional prime sieves. Assuming the generalized Riemann hypothesis, we modify a result of Davenport in a way suggested by the recent work of Rubinstein and Sarnak to prove a theorem which makes it possible to compute the entire distribution of π4,3(x) - π4,1(x) including the sign change (axis crossing) regions, in time linear in x, using zeroes of L(s,x),x the nonprincipal character modulo 4, generously provided to us by Robert Rumely. The accuracy with which the zeroes duplicate the distribution (Figure 1) is very satisfying. The program discovers all known axis crossing regions and finds probable regions up to 10 1000 Our result is applicable to a wide variety of problems in comparative prime number theory. For example, our theorem makes it possible in a few minutes of computer time to compute and plot a characteristic sample of the difference li(x)-π(x) with fine resolution out to and beyond the region in the vicinity of 6.658 x 10 370 discovered by te Riele. This region will be analyzed elsewhere in conjunction with a proof that there is an earlier sign change in the vicinity of 1.39822 × 10 316 .

A review of portable random number generators

Abstract Portable random number generators for personal computers can be evaluated quickly and effectively by plotting their two- and three-dimensional serial patterns. An examination of about 40 published generators yielded only a few that had acceptable serial patterns.

Details of the first region of integers with _{3,2}()<_{3,1}()

Since the time of Chebyshev [4] there has been interest in the magnitude of the smallest integer x with ir3,2(x) < ir3 1(x), where rb, C(x) denotes the number of positive primes < x and -c (mod b). The authors have recently reached this threshold with the discovery that ir3,2(608981813029)73,1(608981813029) =1. This paper includes a detailed numerical and graphical description of values of 1r3,2(x) ir3 1(x) in the vicinity of this long sought number.

Introduction to Cellular Automata and Conway’s Game of Life

Although cellular automata has origins dating from the 1950s, widespread popular interest did not develop until John Conway’s “Game of Life” cellular automaton was initially revealed to the public in a 1970 Scientific American article (Gardner in Sci. Am. 223:120–123, 1970). The feature of his “game” that probably evoked this intensive interest was the discovery of “oscillators” (periodic forms) and “gliders” (translating oscillators). This introductory chapter is for those who are either unfamiliar with the game, or feel that a brief “refresher course” would be appropriate.