On the maximum values of the additive representation functions

Let $A$ and $B$ be sets of nonnegative integers. For a positive integer $n$ let $R_{A}(n)$ denote the number of representations of $n$ as the sum of two terms from $A$. Let $\displaystyle s_{A}(x) = \max_{n \le x}R_{A}(n)$ and \displaystyle d_{A,B}(x) = \max_{\hbox{t: a_{t} \le x$or$b_{t} \le x}}|a_{t} - b_{t}|. In this paper we study the connection between $s_{A}(x)$, $s_{B}(x)$ and $d_{A,B}(x)$. We improve a result of Haddad and Helou about the Erd\H{o}s - Tur\'an conjecture

On minimal additive complements of integers

Let $C,W\subseteq \mathbb{Z}$. If $C+W=\mathbb{Z}$, then the set $C$ is called an additive complement to $W$ in $\mathbb{Z}$. If no proper subset of $C$ is an additive complement to $W$, then $C$ is called a minimal additive complement. Let $X\subseteq \mathbb{N}$. If there exists a positive integer $T$ such that $x+T\in X$ for all sufficiently large integers $x\in X$, then we call $X$ eventually periodic. In this paper, we study the existence of a minimal complement to $W$ when $W$ is eventually periodic or not. This partially answers a problem of Nathanson.Comment: 13 page

Solution of the Least Squares Method problem of pairwise comparison matrices

The aim of the paper is to present a new global optimization method for determining all the optima of the Least Squares Method (LSM) problem of pairwise comparison matrices. Such matrices are used, e.g., in the Analytic Hierarchy Process (AHP). Unlike some other distance minimizing methods, LSM is usually hard to solve because of the corresponding nonlinear and non-convex objective function. It is found that the optimization problem can be reduced to solve a system of polynomial equations. Homotopy method is applied which is an efficient technique for solving nonlinear systems. The paper ends by two numerical example having multiple global and local minima

Three-term idempotent counterexamples in the Hardy-Littlewood majorant problem

The Hardy-Littlewood majorant problem was raised in the 30's and it can be formulated as the question whether $\int |f|^p\ge \int|g|^p$ whenever $\hat{f}\ge|\hat g|$. It has a positive answer only for exponents $p$ which are even integers. Montgomery conjectured that even among the idempotent polynomials there must exist some counterexamples, i.e. there exists some finite set of exponentials and some $\pm$ signs with which the signed exponential sum has larger $p^{\rm th}$ norm than the idempotent obtained with all the signs chosen + in the exponential sum. That conjecture was proved recently by Mockenhaupt and Schlag. \comment{Their construction was used by Bonami and R\'ev\'esz to find analogous examples among bivariate idempotents, which were in turn used to show integral concentration properties of univariate idempotents.}However, a natural question is if even the classical $1+e^{2\pi i x} \pm e^{2\pi i (k+2)x}$ three-term exponential sums, used for $p=3$ and $k=1$ already by Hardy and Littlewood, should work in this respect. That remained unproved, as the construction of Mockenhaupt and Schlag works with four-term idempotents. We investigate the sharpened question and show that at least in certain cases there indeed exist three-term idempotent counterexamples in the Hardy-Littlewood majorant problem; that is we have for 0 $\int_0^{\frac12}|1+e^{2\pi ix}-e^{2\pi i([\frac p2]+2)x}|^p > \int_0^{\frac12}|1+e^{2\pi ix}+e^{2\pi i([\frac p2]+2)x}|^p$. The proof combines delicate calculus with numerical integration and precise error estimates.Comment: 19 pages,1 figur

Solving the Least Squares Method problem in the AHP for 3 X 3 and 4 X 4 matrices

The Analytic Hierarchy Process (AHP) is one of the most popular methods used in Multi-Attribute Decision Making. The Eigenvector Method (EM) and some distance minimizing methods such as the Least Squares Method (LSM) are of the possible tools for computing the priorities of the alternatives. A method for generating all the solutions of the LSM problem for 3 × 3 and 4 × 4 matrices is discussed in the paper. Our algorithms are based on the theory of resultants

Conclusions

Publication within the project “The V4 towards migration challenges in Europe. An analysis and recommendations” is financed by Visegrad Fund

An inconsistency control system based on incomplete pairwise comparison matrices

Incomplete pairwise comparison matrix was introduced by Harker in 1987 for the case in which the decision maker does not fill in the whole matrix completely due to, e.g., time limitations. However, incomplete matrices occur in a natural way even if the decision maker provides a completely filled in matrix in the end. In each step of the total n(n–1)/2, an incomplete pairwise comparison is given, except for the last one where the matrix turns into complete. Recent results on incomplete matrices make it possible to estimate inconsistency indices CR and CM by the computation of tight lower bounds in each step of the filling in process. Additional information on ordinal inconsistency is also provided. Results can be applied in any decision support system based on pairwise comparison matrices. The decision maker gets an immediate feedback in case of mistypes, possibly causing a high level of inconsistency

On the variances of a spatial unit root model

The asymptotic properties of the variances of the spatial autoregressive model $X_{k,\ell}=\alpha X_{k-1,\ell}+\beta X_{k,\ell-1}+\gamma X_{k-1,\ell-1}+\epsilon_{k,\ell}$ are investigated in the unit root case, that is when the parameters are on the boundary of domain of stability that forms a tetrahedron in $[-1,1]^3$. The limit of the variance of $n^{-\varrho}X_{[ns],[nt]}$ is determined, where on the interior of the faces of the domain of stability $\varrho=1/4$, on the edges $\varrho =1/2$, while on the vertices $\varrho =1$