8,005 research outputs found
On a Deterministic Property of the Category of -almost Primes: A Deterministic Structure Based on a Linear Function for Redefining the -almost Primes (, ) in Certain Intervals
In this paper based on a sort of linear function, a deterministic and simple
algorithm with an algebraic structure is presented for calculating all (and
only) -almost primes ( , ) in
certain interval. A theorem has been proven showing a new deterministic
property of the category of -almost primes. Through a linear function that
we obtain, an equivalent redefinition of the -almost primes with an
algebraic characteristic is identified. Moreover, as an outcome of our
function's property some relations which contain new information about the
-almost primes (including primes) are presented.Comment: 10 pages. Accepted and presented article in the 11th ANTS, Korea,
2014. The 11th ANTS is one of international satellite conferences of ICM
2014: The 27th International Congress of Mathematicians, Korea. (Expanded
version
Two Compact Incremental Prime Sieves
A prime sieve is an algorithm that finds the primes up to a bound . We say
that a prime sieve is incremental, if it can quickly determine if is
prime after having found all primes up to . We say a sieve is compact if it
uses roughly space or less. In this paper we present two new
results:
(1) We describe the rolling sieve, a practical, incremental prime sieve that
takes time and bits of space, and
(2) We show how to modify the sieve of Atkin and Bernstein (2004) to obtain a
sieve that is simultaneously sublinear, compact, and incremental.
The second result solves an open problem given by Paul Pritchard in 1994
Setting-up early computer programs: D. H. Lehmer's ENIAC computation
A complete reconstruction of Lehmer's ENIAC set-up for computing the exponents of p modulo two is given. This program served as an early test program for the ENIAC (1946). The reconstruction illustrates the difficulties of early programmers to find a way between a man operated and a machine operated computation. These difficulties concern both the content level (the algorithm) and the formal level (the logic of sequencing operations)
Shortest Path in a Polygon using Sublinear Space
\renewcommand{\Re}{{\rm I\!\hspace{-0.025em} R}}
\newcommand{\SetX}{\mathsf{X}} \newcommand{\VorX}[1]{\mathcal{V} \pth{#1}}
\newcommand{\Polygon}{\mathsf{P}} \newcommand{\Space}{\overline{\mathsf{m}}}
\newcommand{\pth}[2][\!]{#1\left({#2}\right)} We resolve an open problem due
to Tetsuo Asano, showing how to compute the shortest path in a polygon, given
in a read only memory, using sublinear space and subquadratic time.
Specifically, given a simple polygon \Polygon with vertices in a read
only memory, and additional working memory of size \Space, the new algorithm
computes the shortest path (in \Polygon) in O( n^2 /\, \Space ) expected
time. This requires several new tools, which we believe to be of independent
interest
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