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

Partitions of the set of nonnegative integers with the same representation functions

For a set of nonnegative integers S let RS(n) denote the number of unordered representations of the integer n as the sum of two different terms from S. In this paper we focus on partitions of the natural numbers into two sets affording identical representation functions. We solve a recent problem of Chen and Lev. © 2017 Elsevier B.V

Tendances évolutives du latin tardif dans la Britannia romaine (d’après les inscriptions).

Summary: This study presents some of the most important phonological and grammatical phenomena which show the evolution of Late Latin in the Roman province of Britannia. The investigation is based on a corpus of inscriptions on stone (established by Collingwood and Wright). The Vulgar Latin of Britannia seems not very different from that of other provinces, but the progression of certain changes is slower. The author insists on the different origins of soldiers and colonists who took part in the romanization of the island

Effect of age and magnesium supply on the free radical and anti-oxidant content of plants

The scientific literature signs that the shortage of magnesium may increase the free radical content in the animal and human tissues, but nothing can be kown about a similar effect in the plants. This is why we decided to research the plant tissues to know how the radical content and the concentration of antioxidants which may eliminate the free radicals. We have researched the wheat (Triticum aestivum var. GK Pinka), maize (Zea mays var. Furio) and table beet (Beta vulgaris var. Rubra). We measured the concentrations in the coleoptyle of wheat and maize and in the roots of table beet. On the basis of our measurements we may conclude that the increase of magnesium concentration in the nutrient solution does decrease the radical content (HO•, LPO) and the activity of antioxidants. It is known, that the production of free radicals does increase as the uncoupling of oxidative phosphorilation is increasing according to its measure. Nevertheless the reduction of concentration of magnesium increases the measure of uncoupling of oxidative phosphorilation and the production of free radicals. On the basis of these effects one can understand why the radical content is lower after the addition magnesium ions into the nutrient solution. The decrease of activity of activity of GSH, catalase and FRAP-value may be understood also: the real cause is the fact that the promoter free radicals needed to the “de novo synthesis” have decreased. The last conclusion may be the next: the magnesium does influence the of free radicals and antioxidants in the plant tissues