Abstract. Random allocations of balls into boxes are considered. Properties of the number of boxes containing a fixed number of balls are studied. A moment inequality is obtained. A merge theorem with Poissonian accompanying laws is proved. It implies an almost sure limit theorem with a mixture of Poissonian laws as limiting distribution. Almost sure versions of the central limit theorem are obtained when the parameters are in the central domain.

In a former paper, motivated by a recent theory of relators (families of relations), the first author has investigated increasingly regular and normal functions of one preordered set into another instead of Galois connections and residuated mappings of partially ordered sets. A function f of one preordered set X into another Y has been called (1) increasingly g-normal, for some function g of Y into X if for any x 2 X and y 2 Y we have f .x/ Ä y if and only if x Ä g.y/; (2) increasingly '-regular, for some function ' of X into itself if for any x 1 ; x 2 2 X we have x 1 Ä '.x 2 / if and only if f .x 1 / Ä f .x 2 /. In the present paper, for instance, we shall show that if ' is an increasinglyregular function of X into itself, then ' Ä if and only if ' ı Ä , and if f i is an increasingly g i-normal function of X into Y for each i D 1; 2, then f 1 Ä f 2 if and only if g 2 Ä g 1. Moreover, for instance, we shall show that if f is an increasingly ' i-regular function of X into Y for each i D 1; 2, then f is increasingly ' 1 ı ' 2-regular, and if f is an increasingly g-normal function of X into Y and h is an increasingly k-normal function of Y into Z, then h ı f is g ı k-normal.

Given a graph G = (V, E), a dominating set is a subset D ⊆ V such that every vertex in V \ D is adjacent with at least one vertex in D. The domination number of G, denoted by γ(G), is the minimum cardinality of a dominating set in G. Assuming that G is connected, a subset D ⊆ V is said to be a connected dominating set if it is a dominating set and the subgraph G[D] induced by D is connected. The minimum cardinality of a connected dominating set is termed the * corresponding author 1 connected domination number, denoted by γ c (G). Connected dominating sets serve as important tools for eciently designing backbone networks in ad hoc wireless networks.Comparing γ(G) and γ c (G) for a random graph with constant edge probability p, we obtain that the two parameters are asymptotically equal with probability tending to 1 as the number of vertices gets large. We also consider nonconstant edge probability p n tending to zero (where n is the number of vertices). Among other results, we extend an asymptotic formula of Gilbert on the probability of connectivity.

In this paper the Decomposition Theorem for functional equations is shown. As an application of this Theorem the two times continuously differentiable solution of the functional equation G 1 (x(x + y)) + F 1 (y) = G 2 (y(x + y)) + F 2 (y) can be given with unknown functions G i , F i : R + → R (i = 1, 2) where the Equation is fulfilled for all x, y ∈ R + (where R + := {x ∈ R | x > 0}).

In this work, we present some proposals for the transfer and storage of information with the help of dominating sets in graph. In traffic science, the right flow of information is very important: the right person must receive the information at the right time, with the help of which the necessary decision can be made. It is important to note that many disasters could have been prevented or avoided with the appropriate flow of information. If we also add to this that the necessary theoretical results are available and the flow of information can be ensured with minimal financial expenditure, then it becomes clear why publications of this nature are so important. Of course, there are several methods and approaches for theoretical descriptions, but all of them can be used and none of them can be said to be the best: they all have their advantages. Thus we select the method, which our needs and we use the appropriate IT background.