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We prove that it is NP-hard to dissect one simple orthogonal polygon into another using a given number of pieces, as is approximating the fewest pieces to within a factor of $1+1/1080-\varepsilon$.

We prove various theorems on approximation using polynomials with integer coefficients in the Bernstein basis of any given order. In the extreme, we draw the coefficients from $\{ \pm 1\}$ only. A basic case of our results states that for…

Let $f(x)$ be a polynomial of degree $n \ge 1$ with real coefficients and let $X \ge 2$ and $\delta \ge 0$ be real numbers. Let $\|\cdot\|$ be the distance to the nearest integer. We obtain upper bounds for the number of solutions to the…

This paper is a continuation of a previous paper. Here, as there, we examine the problem of finding the maximum number of terms in a partial sequence of distinct unit fractions larger than 1/100 that sums to 1. In the previous paper, we…

We improve on Gonek-Montgomery's quantitative version of Kronecker's approximation theorem.

A numerical scheme is presented to solve the one source near field refractor problem to arbitrary precision and it is proved that the scheme terminates in a finite number of iterations. The convergence of the algorithm depends upon proving…

We reveal a relationship between the prime counting function and an operation performed on a unique subsequence of the primes.

We present a complete algorithm for finding an exact minimal polynomial from its approximate value by using an improved parameterized integer relation construction method. Our result is superior to the existence of error controlling on…

We consider the problem of approximation of a continuous function $f$ defined on a compact metric space $X$ by elements from a sum of two algebras. We prove a de la Vall\'{e}e Poussin type theorem, which estimates the approximation error…

For a given irrational number, we consider the properties of best rational approximations of given parities. There are three different kinds of rational numbers according to the parity of the numerator and denominator, say odd/odd, even/odd…

If we cannot obtain all terms of a series, or if we cannot sum up a series, we have to turn to the partial sum approximation which approximate a function by the first several terms of the series. However, the partial sum approximation often…

We construct explicit easily implementable polynomial approximations of sufficiently high accuracy for locally constant functions on the union of disjoint segments. This problem has important applications in several areas of numerical…

We study approximation by arbitrary linear combinations of $n$ translates of a single function of periodic functions. We construct some methods of this approximation for functions in a class induced by the convolution with a given function,…

In this paper we find the formula that gives the n-th smallest term in a given finite sequence of real numbers.

We introduce a $2$-approximation algorithm for the minimum total covering number problem.

In this paper we construct approximations for the Caputo derivative of order $1-\alpha,2-\alpha,2$ and $3-\alpha$. The approximations have weights $0.5\left((k+1)^{-\alpha}-(k-1)^{-\alpha}\right)/\Gamma(1-\alpha)$ and…

An exact upper bound on the sum of squared nearest-neighbor distances between points in a rectangle is given.

In this note, we illustrate the computation of the approximation of the supply curves using a one-step basis. We derive the expression for the L2 approximation and propose a procedure for the selection of nodes of the approximation. We…

Let $n,d$, and $k$ be positive integers where $n$ and $d$ are coprime. Our two main results are Theorem 1. There is a partition of the infinite interval $[kd,\infty)$ of positive integers into a family of finite sets $X$ for which the sum…

We consider the problem of computing the nearest matrix polynomial with a non-trivial Smith Normal Form. We show that computing the Smith form of a matrix polynomial is amenable to numeric computation as an optimization problem.…