We describe a new exact-arithmetic approach to linear programming when the number of variables n is much larger than the number of constraints m (or vice versa). The algorithm is an implementation of the simplex method which combines exact (multiple precision) arithmetic with inexact (floating point) arithmetic, where the number of exact arithmetic operations is small and usually bounded by a f...

—Interval arithmetic has become a popular tool for general optimization problems such as robust root finding and global maximum/minimum finding. This has been effectively applied in many computer fields, such as computer graphics. However, interval arithmetic suffers from two significant weaknesses. First, its evaluations are often very conservative, making it useless in many practical situatio...

This paper compares the performance and efficiency of different function range interval methods for plotting f (x, y)= 0 on a rectangular region based on a subdivision scheme, where f (x, y) is a polynomial. The solution of this problem has many applications in CAGD. The methods considered are interval arithmetic methods (using the power basis, Bernstein basis, Horner form and centred form), an...

In this short paper, we study sums of the shape $$\sum _{n\leqslant x}{f([x/n])}/{[x/n]},$$ where f is Euler totient function $$\varphi $$ , Dedekind $$\Psi sum-of-divisors $$\sigma or alternating $$\beta .$$ We improve previous results when $$f=\varphi and derive new estimates $$f=\Psi f=\sigma $$f=\beta

The objective of this paper is to investigate palindromes in three Smarandache-type arithmetic functions,namely, the Smarandache function S(n), pseudo Z(n), and Sandor-Smarandache SS(n).

A minimal r-complete partition of an integer m is a partition of m with as few parts as possible, such that all the numbers 1, . . . , rm can be written as a sum of parts taken from the partition, each part being used at most r times. This is a generalization of M-partitions (minimal 1-complete partitions). The number of M-partitions of m was recently connected to the binary partition function ...